plane trigonometry. - The University of Chicago Library

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TIlEATilSE-

R

ON

SURVEYING, CONTAINING

THE THEORY AND PRACTICE: TO WHICH IS PREFIXED

A PERSPICUOUS SYSTEM OF

PLANE TRIGONOMETRY. THE WHOLE CLEARLY DEMONSTRATED AND ILLUSTRATED BY A LARGE NUMBER OF APPROPRIATE EXAMPLES,-

PARTICULARLY ADAPTED TO THE USE OF SCHOOLS. BY JOHN GUMMERE, A.M., FELLOW OF THE AMERICAN PHILOSOPHICAL S OCTET T, A NR'CORRESPONDING MEMBER OF THE ACADEMY OP NATURAL SCIENCES, PHILADELPHIA.

FIFTEENTH EDITION, CAREFULLY REVISED, AND ENLARGED BY THE ADDITION OF ARTICLES ON THE

THEODOLITE, LEVELLING, AND TOPOGRAPHY. ALSO,

HINTS TO YOUNG SURVEYORS, AND RULES FOR SURVEYING THE PUBLIC LANDS OF THE UNITED STATES.

BY GEORGE II. HOLLIDAY, M. A.

PHILADELPHIA:

URIAPI HUNT & SON. CINCINNATI: APPLEGATE & CO.

1855.

ENTERED,

according to Act of Congress, in the year 1853, by URIAH HUNT,

in the Clerk’s Office of the District Court of the United States in and for the Eastern District of Pennsylvania.

CONTENTS,

PAGE

Logarithms

...

.

.

.

.

9

Geometrical Definitions ......

25

Geometrical Problems .

29

Plane Trigonometry .

.

.

.

.

37

Application of Plane Trigonometry to the Mensuration of Distances and Heights.

.

.

.

.

.

.

Practical Questions .

70 80

Dimensions of a Survey ....

83

Supplying Omissions in the Dimensions of a Survey

109

Problems for finding the Content of Land

120

Laying out and Dividing Land . Variation of the Compass

.

Miscellaneous Questions .

.

Hints to Young Surveyors

.

.

.

Theodolite . . . Levelling . . . Topography .

.

'

.

. .

.

.

165 .

.

. .

205 215

.

219 239

.

. ...

257 282

PREFACE. THE following compilation originated in the belief that our schools are in want of a Treatise on Surveying, adapted to the methods practised in this country, and freed from the defects of the systems now in use. Notwithstanding the importance of the science, and the large number that make it an object of study, it is believed we are not in possession of a treatise on this subject, suited to the wants of the student. The works of Gibson and Jess are the only ones at present in general use; the former, though much the better of the two, is deficient in many respects. It may be sufficient here, merely to advert to its want of exam¬ ples, which renders it entirely unsuitable for a school book. From the latter, the student would in vain expect to become acquaint¬ ed with the principles of the science, or the rationale of any of the rules, necessary in performing the various calculations.* In order to understand the principles of surveying, a previous knowledge of Geometry is absolutely necessary; and this know¬ ledge will be best acquired from a regular treatise on the subject. In the demonstrations, therefore, throughout this work, the stu¬ dent is supposed to be acquainted with the elements of that science. The references are adapted to Playfair’s Geometry but they will in general apply equally well to Simson’s transla¬ tion of Euclid’s Elements. As there are many who wish to obtain a practical knowledge of Surveying, whose leisure may be too limited to admit of their

• Each of these works has lately gone through a new edition, in which consi¬ derable additions are stated to have been made. On examination, however, it does not appear, that those additions are such as to supply the deficiencies. The additions made to Gibson, consist principally of some nautical problems quite foreign to a treatise on Surveying. Those made to Jess, consist of a few extracts from Gibson, in one of which the Pennsylvania method of calculation is introduced, as being quite different from that given by Jess; whereas it is well known to he the method given by that author, and used, as well in the preceding, as in the subsequent part of his work. 5

vi

PREFACE.

going through a course of Geometry, the author has adapted his work to this class, by introducing the necessary geometrical defi¬ nitions and problems, and by giving plain and concise rules, en¬ tirely detached from the demonstrations; the latter being placed in the form of notes at the bottom of the page. Each rule is ex¬ emplified by one wrought example; and the most of them by several unwrought examples, with the answers annexed. In the laying out and dividing of land, which forms the most difficult part of surveying, a variety of problems is introduced, adapted to the cases most likely to occur in practice. This part of the subject, however, presents such a great variety of cases, that we should in vain attempt to give rules that would apply to all of them. It cannot therefore be too strongly recommended to every one, who has the opportunity, to make himself well ac¬ quainted with Geometry, and also with Algebra, previous to en¬ tering on the study of Surveying. Furnished with these useful auxiliaries, and acquainted with the principles of the science, the practitioner will be able to perform, with ease, any thing likely to occur in his practice. The compiler thinks proper to acknowledge, that in the ar¬ rangement of the work, he availed himself of the advice of his learned preceptor and friend, E. Lewis of New-Garden; and that several of the demonstrations were furnished by him.

ADVERTISEMENT TO THE FOURTH EDITION

In preparing this edition for the press, several alter¬ ations have been made, which, it is believed, will be found to be real improvements. A number of new Pro¬ blems has been introduced, and a more methodical arrangement of the whole has been adopted. Instead of three different rules for calculating the content of a Survey, one general rule, including these, is now given. It may be further added, that the rules for solving seve¬ ral of the problems in Division of Land, have been considerably simplified. The Mathematical Tables have been stereotyped, after carefully revising them and comparing them with the most correct European Edi¬ tions. THE AUTHOR.

ADVERTISEMENT TO THE FOURTEENTH EDITION.

To meet the wants of the Student of Civil Engi¬ neering, this edition has been enlarged by the addition of several chapters, in which the Theodolite and Le¬ velling Instrument are described, the methods of ad¬ justing and using them are given, and the principles and practice of Levelling and Topography are ex¬ plained and illustrated.—The whole has been carefully revised and the few typographical errors existing in former editions have been corrected. J. G. 7

CONTENTS.

PAOE

1

Logarithms, 9 Geometrical Definitions, - -- -- -- -- -- -- --25 Geometrical Problems, ------^------.---29 Plane Trigonometry, - -- -- -- -- -- -- -- -35 Application of Plane Trigonometry to the Mensuration of Distances and Heights, - -- -- -- -- -- -.-----68 Practical Questions, - -- -- -- -- -- -- -- --78 Dimensions of a Survey, - -- -- -- -- -- -- --81 Supplying Omissions in the Dimensions of a Survey, - -- -- -107 Problems for finding the Content of Land, - -- -- -- --118 Laying out and Dividing Land, - - - - - - -- -- -- - 163 Variation of the Compass, - -- -- -- -- - - -- -- 202 Miscellaneous Questions, - -- -- -- -- -- -- -- 213 Theodolite, ------------------- - 217 Levelling, ------------ - 237 Topography, - -- -- -- -- -- -- -- -- -- 254

EXPLANATION OF THE

CHARACTERS USED IN THIS WORK.

4- signifies plus, or addition. — minus, or subtraction. X multiplication.

-7- division. : :: : proportion. = equality. y/ square root.

GO difference between two quantities when it is not known winch is the greater.

A

TREATISE ON SURVEYING.

OF LOGAKITHMS. LOGARITHMS are a series of numbers so contrived, that by them the work of multiplication is performed by addition, and that of division by subtraction.

If a series of numbers in arithmetical progression be placed as indices, or exponants, to a series of numbers in geometrical progression, the sum of difference of any two of the former, will answer to the product or quotient of the two corresponding terms of the latter. Thus, 0. 1. 2. 3. 4. 5. 6. 7. &c., arith. series, or indices. 1. 2. 4. 8. 16. 32. 64. 128. &c., geom. series. Now 2+3=5. also 7—3=4. And 4X8=32. and 128+8=16. Therefore the arithmetical series, or indices, have the same properties as logarithms; and these properties hold true, whatever may be the ratio of the geometrical series. There may, therefore, be as many different systems of logarithms, as there can be taken different geometrical series having unity for the first term. But the most B

9

10

OF LOGARITHMS.

convenient system is that in which the ratio of the geome¬ trical series is 10; and this is the one in common use. Thus, 0. 1. 2. 3. 4. 5. &c. indices or logar. 1. 10. 100. 1000. 10000. 100000. &c. natural numbers In this system the log, of 1. is 0,* the log. of 10 is 1. the log. of 100, is 2, &c. Hence it is plain that the log. of any number between 1 and 10, will be expressed by a decimal, the log. of any number between 10, and 100, by 1 and a decimal, the log. of any number between 100 and 1000, by 2 and a decimal, &c. The numbers, 0, 1, 2, 3, &c. that stand before the deci¬ mal part of logarithms, are called indices and are always less by unity, than the number of figures in the integral part of the corresponding natural number. The index of t.e logarithm of a number, consisting in whole, or in part of integers, is affirmative; but if the number be a decimal, the index is negative, and is mark¬ ed by a negative sign (—) placed either before or above it. If the first significant figure of the decimal be ad¬ jacent to the decimal point, the index is,—1, or 1; if there be one cipher between them, the index is —2, or 2; if there be two ciphers between them, the index is —3 or 3, &c. The decimal parts of the logarithms of numbers, con¬ sisting of the same figures and in the same order, are the same, whether the number be integral, fractional, or mixed. This is illustrated as follows : * In every system the logarithm of 1 is 0.

OF LOGARITHMS.

Number 18960 1896 189.6 18.96 1.896 .1896 .01896 .001896 .0001896

II

Logarithm 4.27784 3.27784 2.27784 1.27784 0.27784 —1.27784 —2.27781 —3.27784 —4.27784

The method of finding logarithms in the tables, and of multiplying, dividing, &c. by them, is contained in the following problems. PROBLEM I. To find the Logarithm of a given number. If the given number consists of one or two figures only, find it in the column marked No. in the first page of the table, and against it, in the next column, marked log. is the logarithm. Thus the log. of 7 will be found 0.84510, and the log. of 85 will be found 1.92942. But if the given number be either wholly or in part decimal, the index must be changed accordingly. Ob¬ serving that the index must always be one less than the number of figures in the integral part of the given num¬ ber ; also, when the given number is wholly a decimal, the index is negative, and "must be one more than the number of the ciphers between the decimal point and first significant figure on the left hand. Thus the log. of .7 is—1.84510, and the log. of .0085 is—3.92942. If the given number consists of three figures, find it in one of the other pages of the table, in the column marked No. and against it, in the next column, is the decimal

12

OF LOGARITHMS.

part of the logarithm. The index must be placed before it agreeably to the above observation. Thus the log. of 421 is 2.62428, the log. of 4.21 is 0.62428, and the log. of .0421 is —2.62428. If the given number consists of four figures, find the three left hand figures in the column marked No. as be¬ fore, and the remaining, or right hand figure at the top of the table; in the column under this figure, and against the other three, is the decimal part of the logarithm. Thus the log. of 5163 is 3.71290, and the log. of .6387 is—1.80530. If the given number consist of five or six figures, find the logarithm of the four left hand figures as before ; then take the difference between this logarithm and the next greater in the table. Multiply this difference by the remaining figure or figures of the given number, and cut off one, or two figures to the right hand of the pro¬ duct, according as the multiplier consists of one, or two figures; then add the remaining figure or figures of the product to the logarithm first taken out of the table, and the sum will be the logarithm required. Thus, let it be required to find the logarithm of 59686; then, Logarithm of 5968 is The next greater log. is - -

77583 77590

Difference - -• - - 7 Remaining figure 6 Product ----- 4,2 To 77583 Add 4 Decimal part of the log.

77587

13

LOGARITHMS.

The natural number consisting of five integers, the in dex must be 4; therefore the log .of 59686 is4.77587. \

Again, let it be required to find the log. of .0131755; then, Logarithm of 1317 is The next greater log. is Difference Remaining figures

-

-

32 - - 55 -

Product To Add - Decimal part of the log.

11959 11991

1760

- - 11959 18* -

11977

As the given number is a decimal, and has one cipher between the decimal point and first significant figure, the index must be—2; therefore the log. of .0131755 is —2.11977. EXAMPLES.

1. 2. 3. 4. 5. 6. 7.

Required the log. of 4.3 Required the log. of 7986 Required the log. of .3754 Required the log. of 596.87 Required the log. of 785925 Required the log. of 6543900 Required the log. of .0027863 * Because 17.0 is nearer 18 than 17.

2

Ans. 0.63347 Ans. 3.90233 Ans. -1.57449 Ans. 2.77588 Ans. 5.89538 Ans. 6.81583 Ans. -3.44503

14

LOGARITHMS.

PROBLEM II. Zb Jind the natural number corresponding to a given logarithm. If four figures only be required in the answer, look in the table for the decimal part of the given logarithm, and if it cannot be found exactly, take the one nearest to it, whether greater or less; then the three figures in the first column, marked No. which are in a line with the logarithm found, together with the figure at the top of the table directly above it, will form the number re¬ quired. Observing, that when the index of the given logarithm is affirmative, the integers in the number found, must be one more than the number expressed by the index; but when the index of the given logarithm is negative, the number found will be wholly a decimal, and must have one cypher less, placed between the decimal point and first significant figure on the left hand, than the number expressed by the index. Thus the natural number corresponding to the logarithm 2.90233 is 798.6, the natural number corresponding to the logarithm 3.77055 is 5896, and the natural number corresponding to the logarithm —3.36361 is .00231. If the exact logarithm be found in the table, and the figures in the number corresponding do not exceed the index by one, annex ciphers to the right hand till they do. Thus the natural number corresponding to the lo¬ garithm 6.64068 is 4372000. If Jive or six figures be required in the answer, find, in the table, the logarithm next less than the given one, and take out the four figures answering to it as before. Subtract this logarithm from the next greater in the table, and also from the given logarithm; to the latter difference, annex one or two ciphers, according as five

LOGARITHMS.

15

or six figures are required, and divide the number thus produced, by the former difference; annex the quotient to the right hand of the four figures already found, and it will give the natural number required. Thus let it be required to find the natural number corresponding to the logarithm 2.53899 true to Jive figures;then, Given logarithm - .53899 Next less - - .53895 the natural number corresponding is 3459 Diff. with one cipher annexed 40 Next less log. - - .53895 Next greater - - .53908 Difference 13 Divide 40 by 13 and the quotient will be 3, which, an¬ nexed to the right hand of 3459, the four figures already found, makes 34593 ; therefore as the index is 2, the re¬ quired natural number is 345.93. Again let it be required to find the natural number corresponding to the logarithm 4.59859, true to six figures; then, Given logarithm - - .59859 Next less - - - - .59857, the natural number an■ swering to it is 3968. Diff. with two ciphers annexed 200 Next less log. - - - 59857 Next greater - - - 59868 Difference - - - - 11 Divide 200 by 11, and the quotient will be 18, which annexed to the right hand of 3968 the four figures al-

16

LOGARITHMS.

ready found, makes 396818; therefore as the index is 4, the required natural number is 39681.8. EXAMPLES.

1. Required the natural number answering to garithm 1.88030. Ans. 75.91. 2. Required the natural number answering to garithm 5.37081. Ans. 234861. 3. Required the natural number answering to garithm 3.11977. Ans. 1317.56. 4. Required the natural number answering to garithm—2.97435. Ans. .094265.

the lo¬ the lo¬ the lo¬ the lo¬

PROBLEM III. To multiply numbers by means of logarithms. Case 1.—When all the factors are whole or mixed numbers. RULE.

Add together the logarithms of the factors, and the sum will be the logarithm of the product. EXAMPLES.

1. Required the product of 84 by 56. Logarithm of 84 is 1.92428 Do. of 56 is 1.74819 Product 4704 Sum 3.67247 2. Required the continued product of 17.3, 1.907 and 34. Logarithm of 17.3 is 1.23805 Do. 1.907 is 0.28035 Do. 34. is 1.53148 Product 1121.71

Sum 3.04988

17

I.OGARITHM9.

3. Find by logarithms the product of 76.5 by 5.5 Ans. 420.75. 4. Find by logarithms the continued product of 42.35, 1.7364, and 1.76. Ans. 129.424. CASE 2.—When some

or all of the factors are decimal

numDers. BULB.

Add the decimal parts of the logarithms as before, and if there be any to carry from the decimal part, add it to the affirmative index or indices, or else subtract it from the negative. Then add the indices together, when they are all of the same kind; that is, all affirmative or all negative; but when they are of different kinds, take the difference be¬ tween the sums of the affirmative and negative ones, and prefix the sign of the greater. Note.—When the index is affirmative, it is not neces¬ sary to place any sign before it; but when it is negative, the sign must not be omitted. EXAMPLES.

1. Required the continued product of 349.17, 25.43, 93521 and .00576. Logarithm of 349.17 Do. 25.43 Do. .93521 Do. .00576 Product 47.83 2*

% . IS

is is - is Sum

C

2.54303 1.40535 -1.97090 -3.76042 1.67970

18

LOGARITHMS.

In this example there is 2 to carry from the decimal part of the logarithms, which added to 3, the sum ot the affirmative indices, makes 5; from this taking 4, the sum of the negative indices, the remainder is 1, which is the index of the sum of the logarithms, and is affirmative, because the sum of the affirmative indices, together with the number carried, exceed the sum of the negative indices. 2. Required the continued product of and .003179. Logarithm of .0839 is -2.92376 Do. .7536 is -1.87714 Do. .003179 is -3.50229 Product .000201

Sum -4.30319

In this example there is 2 to carry from the decimal part of the logarithms, which subtracted from 6, the sum of the negative indices, leaves 4, which is the index of the sum of the logarithms, and is negative, because the sum of the negative indices is the greater. 3. Required the continued product of 13.19, .3765, and .00415. Ans. .02061. 4. Required the continued product of 343, 1.794, 5.41, and .019. Ans. 63.25. PROBLEM IY. To divide numbers by means of Logarithms. CASE 1.—When the dividend and divisor are both whole or mixed numbers.

LOGARITHMS.

19

RULE.

From the logarithm of the dividend, subtract the lo¬ garithm of the divisor, the remainder will be the loga¬ rithm of the quotient. Note.—When the divisor exceeds the dividend, the question must be wrought, by the rule given in the next case. EXAMPLES.

1.

Required the quotient of 3450 divided

Logarithm of 3450 Do. 23 Quotient 150

is 3.53782 is 1.36173

Remainder 2.17609

2. Required the quotient of 420.75 divided by 76.5. Ans. 5.5. 3. Required the quotient of 37.1542 divided by 1.73958. Ans. 21.3585. CASE 2.—When the dividend or divisor, or both of them, are decimal numbers.

RULE.

Subtract the decimal parts of the logarithms as before, and if 1 be borrowed in the left hand place of the deci¬ mal part, add it to the index of the divisor when that in¬ dex is affirmative, but subtract it when negative. Then conceive the sign of the index of the divisor changed from affirmative to negative, or from negative to affirmative ; and if, when changed, it be of the same name with that of the dividend, add the indices together

20

LOGARITHMS.

but if it be of a different name, take the difference of the indices, and prefix the sign of the greater. EXAMPLES.

1. Required the quotient of .7591 divided by 32.147 Logarithm of .7591 is -1.88030 Do. 32.147 is 1.50714 Quotient .02361

Remain. -2.37316

In this example, the index of the divisor, with its sign changed, is —1, which added to —1, the index of the dividend, makes —2, for the index of the quotient. 2. Required the quotient of .63153 divided by .00917. Logarithm of .63153 is -1.80039 Do. .00917 is -3.96237 Quotient 68.8683

Remain. 1.83802.

In this example there is 1 to carry from the decimal part of the logarithm, which subtracted from —3, the index of the divisor, leaves —2; this, with its sign changed, is-j-2; from which subtracting 1, the index of the dividend, the remainder is 1, and is affirmative, be¬ cause the affirmative index is the greater. 3.

Required the quotient of 13.921 divided

Logarithm of 13.921 Do. 7965.13 Quotient .001748

is is

1.14367 3.90125

Remain. -3.24242

LOGARITHMS.

21

In this example there is 1 to carry from the decimal part of the logarithm, which added to 3, the index of the divisor, makes 4; this, with its sign changed, is, —4 j from which subtracting 1, the index of the dividend, the remainder is —3. 4. Required the quotient of 79.35 divided by .05178. Ans. 1532.46. 5. Required the quotient of .5903 divided by .931. Ans. .63404. PROBLEM V. To involve a number to any power, that is, to find the square, cube, Sfc. of a number, logarithmically. RULE.

Multiply the logarithm of the given number by the index of the power, viz. by 2 for the square, by 3 for the cube, dzc. and the product will be the logarithm of the power. Note.—When the index of the logarithm is negative, if there be any to carry from the decimal part, instead of adding it to the product of the index and multiplier, subtract it, and the remainder will be the index of the logarithm of the power, and will always be negative. EXAMPLES.

1. Required the square of 317. Logarithm of 317 is 2.50106 2 Square 100489

5.00212

22

LOGARITHMS.

2. Required the 5th power of 1.735. Logarithm of 1.735

is

0.23930 5

5th power 15.7218 1.19650 3.

Required the cube of .08761.

Logarithm of .08761

is -2.94255 3

Cube .0006724 -4.82765 4. Required the cube of 7.503. Ans. 422.37. 5. Required the 7th power of .32513. Ans. .0003841, PROBLEM VI. To extract any root of a number logarithmically. RULE.

Divide the logarithm of the given number by the index of the root, that is, by 2 for the square root, by 3 for the cube root, &c. and the quotient will be the logarithm of the required root.' Note.—When the index of the logarithm is negative, and does not exactly contain the divisor, increase the index by a number just sufficient to make it exactly divi¬ sible by it, and carry the units borrowed, as so many tens, to the left hand figure of the decimal part; then proceed with the division in the usual manner.

23

LOGARITHMS. EXAMPLES.

1.

Required the cube root of 391.27.

Logarithm of 391.27 is

3)

2.59248

Cube root 7.314 0.86416 2.

Required the square root of .08593.

Logarithm of .08593 is

2)

—2.93414

Square root .29314 —146707 3.

Required the cube root of .7596.

3) . Logarithm of .7596 is —1.88058 Cube root .9124 —1.96019 4.

Reauired the cube root of .0000613. 3) Logarithm of .0000613 is —5.78746 Cube root .03943 —2.59582

5. Required the square root of 365. Ans. 19.105. 6. Required the 5th root of .9563. Ans. .9911. 7. Required the 4th root of .00079. Ans. .16765 Of the Arithmetical Complements of Logarithms. When it is required to subtract several logarithms from others, it will be more convenient to convert the subtraction into an addition, by writing down, instead of the logarithms to be subtracted, what each of them wants of 10.00000, which may readily be done, by writ-

24

LOGARITHMS.

ing down what the first figure, on the right hand, wanta of 10, and what every other figure wants of 9; this re¬ mainder is called the Arithmetical Complement. Thus, if the logarithm be 2.53061, its arithmetical complement will be 7.46939. If one or more figures to the right hand be ciphers, write ciphers in their place, and take the first significant figure from 10, and the remaining figures from 9. Thus, if the logarithm be 4.61300, its arithmetical complement will be 5.38700. In any operation, where the arithmetical complements of logarithms are added to other logarithms, there must be as many tens subtracted from the sum, as there are arithmetical complements used. As an example, let it be required to divide the product of 76.4 and 35.84, by the product of 473.9 and 4.76. 473.9 Ar. Co. 7.32431 4.76 Ar. Co. 9.32239 35.84 log. 1.55437 76.4 log. 1.88309 Quotient 1.214

0.08416

GEOMETRY. DEFINITIONS.

is that science wherein the properties of magnitude are considered. 1. GEOMETRY

2. A point is that which has position, but not magni¬ tude.

3. A line has length but not breadth. 4. A straight, or right line, is the shortest line that can be drawn between any two points. 5. A superficies or surface is that which has length and breadth, but not thickness. 6. A plane superficies is that in which any two points being taken, the straight line which joins them lies wholly in that superficies. Rig. l.

7. A plane rectilineal angle is the incli¬ nation of two straight fines to one ano¬ ther, which meet together, but are not in the same straight fine, as A, Fig. 1. A

Note.—When several angles are o formed about the same point, as at B, Fig. 2, each particular angle is expressed by three letters, whereof the middle letter shows the angu¬ lar point, and the other two the A fines that form the angle; thus, CBD or DBC signifies the angle formed by the fines CB and DB. D

25

26

GEOMETRY.

8. depends on the inclination which the lines that form it have to each other, and not on the length of those lines. Thus the angle DBE is greater than the angle ABC, Fig. 3. 9.

T

When a straight line stands on ano line so as to incline to neither side, but C E / makes the angles on each side equal, / then each of those angles is called a / right angle, and the line which stands / on the other is said to be perpendicular A D B to it. Thus ADC and BDC are right angles, and the line CD is perpendicular to AB, Fig. 4. Fig. 4.

10. An acute angle is that wrhich is less than a right angle, as BDE, Fig. 4. 11. An obtuse angle is that which is greater than a right angle, as ADE, Fig. 4. Fi 5 s- 12. Parallel straight lines are those D c which are in the same plane, and A_ B which, being produced ever so far both ways, do not meet, as AB, CD, Fig. 5.

13. A figure is a space bounded by one or more lines. Fig. 6. c

14. A plane triangle is a figure bounded by three straight lines, as ABC, Fig. 6. B

Fi

s- 8-

15. An equilateral triangle has its three sides equal to each other, a^ A, Fig. 7.

16. An isosceles triangle has only two of its sides equal, as B, Fig. 8.

27

GEOMETBY.

17. A scalene triangle has three uneq ABC, Fig. 6. Fig. 9. o

18. A right angled triangle has one ri angle, as ABC, Fig. 9; in which the side AC, opposite to the right angle, is called the hypothenuse. 19. An obtuse angled triangle has one obtuse angle, as C, Fig. 10.

Fig. 10.

20. An acute angled triangle has all its angles acute, as ABC, Fig. 6.

21. Acute and obtuse angled triangles oblique angled triangles.

22. Any plane figure bounded by four r called a quadrilateral. Fig.whose 11. 23. Any quadrilateral, oppo¬ site sides are parallel, is called & paral¬ lelogram, as D, Fig. 11.

Fig. 12. 24. A parallelogram, whose angles are all right angles, is called a rectangle, as E, Fig. 12.

25. A parallelogram whose sides are all equal, and angles right, is called a square, as F, Fig. 13.

Fig. 13.

p

ft

26. A rhomboides is a parallelogram, whose opposite sides are equal, and angles oblique, as D, Fig. 11. 27. A rhombus is a parallelogram, whose sides are all equal and angles oblique, as G, Fig. 14.

Fig. 14.

28

GEOMETRY.

28. Any quadrilateral figure that is not a parallelo¬ gram, is called a trapezium. «r

29. A trapezium that has two parallel sides is called a trapezoid. 30. A right line joining any two opposite angles of a quadrilateral figure, is called a diagonal. 31. That side upon which any parallelogram, or triangle is supposed to stand, is called the lose; and the perpendicular falling thereon from the opposite angle is called the altitude of the c Fig. 15. parallelogram, or triangle. Thus AD is the base of the parallelogram ABEC, or triangle ABC, and CD is 1 A D the altitude, Fig. 15.

/

32. All plane figures contained unde four sides, are called polygons; of which those having five sides, are called pentagons ; those having six sides, hexagons, and so on.

33. A regular polygon is one whose a as sides, are all equal. Fig. 16.

34. A by one curve line called the circumfer¬ ence or periphery, every part of which ^ is equally distant from a certain point within the circle; and this point is called the centre, Fig. 16.

35. The radius of a circle is a straigh from the centre to the circumference, as CB, Fig. 17.

36. The diameter of a circle is a straigh through the centre, and terminated both ways by the

29

GEOMETRY.

circumference, as AE, Fig. 17. It di¬ vides the circle into two equal parts, called semicircles.

Fig 17. B

37. A quadrant is one quarter of a circle, as ACB, Fig. 17. Note.—The fourth part of the cir¬ cumference of a circle is also called a quadrant. 38. A segment of a circle is the figure contained by a right line, and the part of the circumference it cuts off: thus AEBA and AEDA are segments of the circle ABED, Fig. 16. 39. An arc of a circle is any part of the circumfer ence; as AD or DE, Fig. 17. 40. Ratio is a mutual relation between two quantities of the same kind with respect to magnitude. Note.—A ratio is generally expressed, either by two numbers or by two right lines. 41. When two quantities have the same ratio as two other quantities, the four quantities taken in order are called proportionals; and the last is said to he a fourth proportional to the other three. a

42. When three quantities of the same kind are such that the first has to the second the same ratio which the second has to the third, the third is called a third proportioned to the first and second, and the second is called a mean proportional between the first and third. 8*

30

GEOMETRICAL PROBLEMS.

GEOMETRICAL PROBLEMS. PROBLEM I. To bisect a right line, AB, Fig. 18. Fig. 18.

\c/

\

I

A

/D\

Open the dividers to any dis¬ tance more than half the line AB, and with one foot in A, describe the arc CFD; with the same opening, and one foot in B, describe the arc CGD, meet¬ ing the first arc in C and D; from C to D draw the right line CD, cutting AB in E, which will be equally distant from A and B. PROBLEM II.

At a given point A, in a right line EF, to erect a per¬ pendicular, Fig. 19. Fig. 19. B

From the point A, lay off on each side, the equal distances AC, \ AD; from C and D, as centres, with any radius greater than AC or AD, describe two arcs intersect¬ ing each other in B; from A to B, draw the line AB, which will be the perpendicular re¬ quired.

31

GEOMETRICAL PROBLEMS.

PROBLEM III. To raise a, ’perpendicular on the end B of a right line AB, Pig. 20. Fig. 20.

Take any point D not in the line AB, and with the distance from D to B, describe a circle cutting AB in E; from E through D draw the right line EDO, cutting the periphery in A— C, and join CB, which will be perpendicular to AB.

j>y

PROBLEM IV. To let fall a perpendicular upon a given line BC, from a given point A, without it, Fig. 21. Fig. 21.

In the line BC take any point D, and with it as a cen¬ tre and distance DA describe an arc AGE, cutting BC in G; with G as centre, and distance B GA, describe an arc cutting AGE in E, and from A to E draw the line AFE; then AF will be perpendicular to AB.

,G

PROBLEM V. Through a given point A to drau) f Fis- 22a right line AB, parallel to a given / right line CD, Fig. 22. —; From the point A to any point F, in the line CD, draw the right

0

jr-

<3

32

GEOMETRICAL PROBLEMS.

line AF; with F as a centre and distance FA, describe the arc AE, and with the same distance and centre A describe the arc FG; make FB equal to AE, and through A and B draw the line AB, and it will be parallel to CD. PROBLEM VI.

'

At a given point B, in a given right line LG, to make an angle equal to a given angle A, Fig. 23. p With the centre A and any distance AE, describe the arc DE, and with the same distance and centre B describe the arc FG; make HG equal to DE, and through B and H draw the line BH; then will the angle HBG be equal to the angle A. Fig- 23.

PROBLEM VII. To bisect any right lined angle BAC, Fig. 24. F;

g- 24- o In the lines AB and AC, from the point A, set off equal distances, AD and AE; with the centres D and E and any distance more than half B DE, describe two arcs cutting each other in F; from A through F draw the line AG, and it will bisect the angle BAC.

33

GEOMETRICAL PROBLEMS.

PROBLEM

yin.

To describe a triangle that shall have its sides respect¬ ively equal to three right lines, D, E, and F, of which any two must be together greater than the third, Fig. 25. F! 25 Make AB equal to D; D swith the centre A and distance equal to E, de- F. describe an arc, and with the centre B and distance equal to F describe another arc, cutting the former in C; draw AC and BC, and ABC is the triangle required.

PROBLEM IX. Upon a given line AB to describe a square, Fig. 26. At the end B of the line AB, by Problem III. erect the perpendicular BC, and make it equal to AB; with A and C as centres, and distance AB or BC, describe two arcs cutting each other in D; draw AD, and CD, then will ABCD be the square required.

Fig. 26.

PROBLEM X. To describe a circle that shall pass through the angular points A, B, and C, of a triangle ABC, Fig. 27.. By Problem I. bisect any two of the sides, as AC, BC, by the perpen¬ diculars DE and FG; the point II where they intersect each other will be the centre of the circle: with this centre, and the distance from E

34

GEOMETRICAL PROBLEMS.

. it to either of the points A, B, or C, describe the circle. PROBLEM XI. To divide a given right line AB into any number of equal parts, Pig. 28. Draw the indefinite right line AP, making an angle with AB, also draw BQ, parallel to AP, in each of which, take as many equal parts AM, MN, &c. Bo, on, &c., as the line AB is to be divided into; then draw Mm, Nn, &c., intersecting AB in E, F, &c., which will divide the line as required. PROBLEM XII. To malee a plane diagonal scale, Fig. 29. DraAv eleven lines parallel to, and equidistant from each other; cut them at right angles by the equidistant lines BC; EF; 1, 9; 2, 7; &c. then will BC, &c. be divided into ten equal parts ; divide the lines EB, and FC, each into ten equal parts; and from the points of division on the line EB, draw diagonals to the points of division on the line FC: thus join E and the first division on FC, the first division on EB, and the second on FC, &c. Fig. 29. ' H

5

4 1

1)

1

3

3 I

I

:

i>

2

1 1

^

7

9

E 2 4 6 8

71 L i __LL_Lo i l * \! 1 -4 __L . . _L ±_.-6 -LL i j_L_ -8 i i All 1C E LULL

1'

Note.—Diagonal scales serve to take off dimensions or numbers of three figures. If the first large divisions be units, the second set of divisions, along EB, will be

35

GEOMETRICAL PROBLEMS.

10th parts, and the divisions in the altitude, along BC, will be 100th parts. If HE be tens, EB, will be units, and BC will be tenth parts. If HE be hundreds, BE will be tens, and BC units. And so on, each set of divisions being tenth parts of the former ones. For example, suppose it were required to take off 242 from the scale. Extend the dividers from E to 2 towards II; and with one leg fixed in the point 2, extend the other till it reaches 4 in the line EB; move one leg of the di¬ viders along the line 2, 7, and the other along the line 4, till they come to the line marked 2, in the line BC, and that will give the extent required. PROBLEM XIII. 7o find a third proportional to two given right lines, A and B. Draw two right lines, CD, CE B containing any angle; make CF A equal A, and CG, CH, each equal B; join FG and draw HL paralIel to it: then will CL be the third x proportional required. C

— G

L

E

PROBLEM XIV. 7b find a fourth proportional to three given right lines, A, B and C. A Draw two right lines, DE, DF containing any angle; make DG equal A, DH equal B, and DL equal C; join GH and draw LM parallel to it: DM will be a fourth proportional to A, B, and C.

B c

36

GEOMETRICAL PROBLEMS.

PROBLEM XV. To find a mean proportional between two given right lines A and B. Draw any right line CE and in it take CD equal A, and DE equal A B; bisect CE in F, and with the E centre F and radius FC or FE describe the semicircle CGE; draw DG perpendicular to CE: / then DG will be a mean propor- (_ tional between A and B. C «

PROBLEM XVI.

To divide a given right line AB into two parts that shall have the same ratio to each other as two given lines C and D. Draw AE making an angle with AB; in AE take AF equal p C and FE equal D: join EB and draw FG parallel to it; then AG will have to GB the same ratio & that C has to D. PROBLEM XVII. To divide a given right line AB in two parts in the point D, so that AD may be to DB in the rat io of two given num' bers m and n. For example, let m=3, and n=4. Draw AC making any angle with AB; take the number m from any convenient scale of equal parts, and lay it on AC, from A to E; and take the number n from the same scale, ^ and lay it from E to C; join CB and draw ED ' / parallel to it; then AB will be divided as re- /_ A ] quired.

PLANE TRIGONOMETRY. DEFINITIONS. 1. PLANE TRIGONOMETRY is the art by which, when any three parte of a plane triangle, except the three angles, are given, the others are determined.

2. The periphery of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds, &c. 3. The measure of an angle is the arc of a circle, contained between the two lines that form the angle, the angular point being the centre; thus the angle ABC, Fig. 30, is measured by the arc DE, and contains the same number of degrees that the arc does. The measure of a right angle is therefore 90 degrees; for DH, Fig. 31, which measures the right angle DCIl, is one-fourth part of the circumference, or 90 degrees. Note.—The degrees, minutes, seconds, &c., contained in any arc, or angle, are written in this manner, 50° 18' 35"; which signifies that the given arc or angle contains 50 degrees, 18 minutes, and 35 seconds. 4. The complement of an arc, or of an angle, is what it wants of 90°; and the supplement of an arc, or of an angle, is what it wants of 18,0°. * 5. The chord of an arc, is a right line drawn from one extremity of the arc to the other: thus the line BE is the chord of the arc BAE or BDE, Fig. 31. 37

38

PLANE TRIGONOMETRY.

Fjg 31

‘ ‘

- 6. The sine of an arc, is a right line drawn from one extremity of the arc perpendicular to the diar meter which passes through the other extremity: thus BF is the sine of the arc AB or BD, Fig. 31.

7. T part of the diameter which is in¬ tercepted between the sine and the centre : thus CF is the cosine of the arc AB, and is equal to BI, the sine of its complement HB, Fig. 31

8. The versed sine of an arc, is that part of which is intercepted between the sine and the arc: thus AF is the versed sine of AB; and DF of DB, Fig. 31. 9. The tangent of an arc, is a right line circle in one end of the arc, being perpendicular to the diameter which passes through that end, and is termi¬ nated by a right line drawn from the centre through the other end: thus AG is the tangent of the arc AB, Fig. 31. 10. The secant of an arc, is the right line drawn from the centre and terminating the tangent; thus CG is the secant of AB, Fig. 31. 11. The cotangent of an arc, is the tangent of the complement of that arc; thus HK is the cotangent of AB, Fig. 31. • 12. The cosecant of an arc, is the secant of the com¬ plement of that arc; thus CK is the cosecant of AB, Fig. 31. 13. The sine, cosine, &c., of an angle is the same as the sine, cosine, &c., of the arc that measures the angle. *

39

PLANE TRIGONOMETRY.

PROBLEM I.

Fig 82

-

r

'

To construct the lines of cords, sines, tangents, and secants, to any radius. Fig. 32. Describe a semicircle with any convenient radius CB; from the centre C draw CD perpendicular to AB, and produce it to F; draw BE parallel to CF and join AD. Divide the arc AD into nine «o equal parts, as A 10; 10, 20, &c., and with one foot of the dividers so in A, transfer the distances A, 10; ^ A, 20, &e., to the right line AD; m then will AD be a line of chords con¬ / / S WSx \ V structed to every V ■/, ten degrees.

/

vyim

Divide BD into nine equal parts, and from the points 0 of division, 10, 20, 30, &c., draw lines parallel to CB,* and meeting CD in 10, 20, 30, &c., and CD will he a line of sines. From the centre C, through the divisions of the arc BD, draw lines meeting BE, in 10, 20, 30, &c., and BE will be a line of tangents. With one foot of the dividers in C transfer the dis¬ tances from C to 10, 20, &c., in the line BE to the line CF, which will then be a line of secants. * To avoid confusion, these lines are not drawn in the figure.

40

PLANE TRIGONOMETRY.

By dividing the arcs AD and BD each into 90 equal parts, and proceeding as above, the lines of chords, sines, &c., may be construed to every degree of the quadrant. PROBLEM II. At a given point A. in a given right line AB, to maize an angle of any proposed number of degrees, suppose 88 degrees. Fig. 33. Fig. 33.

With the centre A, and a radius

equal to 60 degrees, taken from a scale of chords, describe an arc, cutting AB in m; from the same scale of chords, fake 38 degrees and apply it to the arc from m to n, and from A through n draw the line AC; then will the angle A contain 88 degrees. Note.—Angles of more than 90 degrees are taken off at twice. PROBLEM III. To measure a given angle A. Fig. 34.

Fig. 34.

Describe the arc mn with the chord of 60 degrees, as in the last problem. Take the arc mn between the dividers, and that extent applied to the scale of chords will show the degrees in the given angle.

Note.—When the distance mn exceeds 90 degrees, it must be taken off at twice, as before.

PLANE TRIGONOMETRY.

41

OF TIIE TABLE OF

LOGARITHMIC OR ARTIFICIAL SINES, TANGENTS, &c.

THIS table contains the logarithms of the sine, tan¬ gent, &c. to every degree and minute of the quadrant, the radius being 10000000000, and consequently its lo¬ garithm 10.

Let the radius CB, Fig. 32, be supposed to consist of 10000000000 equal parts as above, and let the qua¬ drant DB be divided into 5400 equal arcs, each of these will therefore contain 1'; and if from the several points of division in the quadrant, right lines be drawn perpen¬ dicular to CB, the sine of every minute of the quadrant to the radius CB will be exhibited. The lengths of these lines being computed and arranged in a table, constitute what is usually termed a table of natural sines. The logarithms of those numbers taken from a table of loga¬ rithms, and properly arranged, form the table of loga¬ rithmic or artificial sines. In like manner the logarithmic tangents and secants are to be understood. The method by which the sines are computed is too abstruse to be explained in this work, but a familiar ex¬ position of this subject, as well as of the construction of logarithms may be seen in Simpson's trigonometry.

To find, by the table, the sine, tangent, Sfc. of an arc or angle. If the degrees in the given angle be less than 45, look for them at the top of the table, and for the minutes, in

4*1?

42

PLANE TRIGONOMETRY.

the left hand column; then in the column marked at the top of the table, sine, tangent, &c. and against the mi¬ nutes, is the sine, tangent, &c. required. If the degrees are more than 45, look for them at the bottom of the table, and for the minutes, in the right hand column; then in the column marked at the bottom of the table, sine, tangent, &c. and against the minutes, is the sine, tangent, &c. required. Note.—The sine of an angle and of its supplement being the same, if the given number of degrees be above 90, subtract them from 180°, and find the sine of the re¬ mainder. EXAMPLES.

1. Required the sine of 2. Required the tangent of 3. What is the secant of 4. What is the sine of

32° 27' 57° 39' 89° 31' 157° 43'

Ans. Ans. Ans. Ans.

9.72962. 10.19832. 12.07388. 9.57885,

7o find the degrees and minutes corresponding to a given sine, tangent, &c. Find, in the table, the nearest logarithm to the given one, and the degrees answering to it will be found at the top of the table, if the name be there, and the mi¬ nutes on the left hand; but if the name be at the bottom of the table, the degrees must be found at the bottom, and the minutes at the right hand. EXAMPLES.

1. Required the degrees and minutes in the angle whose sine is 9.G4390. Ans. 26° 8'.

PLANE TRIGONOMETRY.

43

2. Required the degrees and minutes in the angle whose tangent is 10.47464. Ans. 71° 28'.

ON GUNTER’S SCALE. scale is an instrument by which, with a pair of dividers, the different cases in trigonometry, and many other problems, may be approximately solved. GUNTER’S

It has on one side, a diagonal scale, and also the lines of chords, sines, tangents, and secants, with several others. On the other side there are several logarithmic lines as follow: The line of numbers marked Num., is numbered from the left hand of the scale towards the right, with 1, 2,3, 4, 5, 6, 7, 8, 9, 1, which stands in the middle of the scale; the numbers then go on 2, 3, 4, 5, 6, 7, 8, 9, 10, which stands at the right hand end of the scale. These two equal parts of the scale are similarly divided, the distances between the first 1, and the numbers 2, 3, 4, Ac. being equal to the distances between the middle 1, and the numbers 2, 3, 4, Ac. which follow it. The sub¬ divisions of the two parts of this line are likewise simi¬ lar, each primary division being divided into ten parts, distinguished by lines of about half the length of the pri¬ mary divisions. The primary divisions on the second part of the scale, are estimated according to the value set upon the unit on the left hand of the scale. If the first 1 be considered as a unit, then the first 1, 2, 3, Ac. stand for 1,2, 3, Ac the middle 1 is 10, and the 2, 3, 4, Ac. following stand

44

PLANE TRIGONOMETRY.

for 20, 30, 40, &c. and the ten at the right hand for 100. If the first 1 stand for 10, the first 2, 3, 4, &c. must be counted 20, 30, 40, &c. the middle 1 will be 100, the second 2,3, 4, <$zc. will stand for 200, 300, 400, &c. and the 10 at the right hand for 1000. If the first 1 be considered as TV of a unit, the 2, 3, 4, &c. following will be A, TO , TV , &c. and the middle 1, and the 2,3, 4, &c. following, will stand for 1, 2, 3,4, &c. The intermediate small divisions must be estimated according to the value set upon the primary divisions. Sines.—The line of sines, marked Sin., is numbered from the left hand of the scale towards the right, 1, 2, 3, 4, &c. to 10, then 20, 30, 40, &c. to 90, where it termi¬ nates just opposite 10 on the line of numbers. Tangents.—The line of tangents, marked Tan., begins at the left hand, and is numbered 1, 2, 3, &c. to 10, then 20, 30, 40, 45, where there is a brass pin, just under 90 in the line of sines; because the sine of 90° is equal to the tangent of 45°. From 45 it is numbered towards the left hand 50, 60,70,80, &c. The tangent arcs of above 45° are therefore counted backward on the line, and are found at the same points of the line as the tangents of their complements. There are several other lines on this side of the scale, as Sine Rhumbs, Tangent Rhumbs, Versed Sines, &c.; but those described are sufficient for solving all the pro¬ blems in plane trigonometry. Remarks on Angles, Triangles, Sfc. 1. If from a point D in a right line AB, one or more right fines be drawn on the same side of it, the angles thus formed at the point D will be together equal to two

45

PLANK TRIGONOMETRY.

right angles, or 180°; thus ADE +EDB=two right angles, or 180°: also ADC + CDE + EDB = two right angles, or 180°. Fig. 35. 2. Since the angles thus formed A at the point D, on the other side of AB, would also be equal to two right angles, the sum of all the angles formed about a point is equal to four right angles, or 360°. 3. If two right lines cut one another, the opposite angles will be equal: thus AEC=BED, and AED = CEB, Fig. 36.

c

K

s-

36

-

4. The sum of the three angles of a plane triangle is equal to two rightangles, or 180°.

5. If the sum of two angles of a tria tracted from 180°, the remainder will be the third angle.

6. If one angle of a triangle be subtracte the remainder will be the sum of the other two angles. 7. In right-angled triangles, if one of the acute angles be subtracted from 90°, the remainder will be the other acute angle. 8. The angles at the base of an isosceles triangle are equal to one another. Fi 9. If one side of a triangle be proc s;.§7 duced, the external angle will be equal to the sum of the two exter/ \ nal and opposite angles: thus the j; » external angle CBD, of the triangle ABC, is equal to the sum of the internal and opposite angles A and C, Fig. 37.

46

PLANE TRIGONOMETRY.

Fig. 88. A

10. T circle is double of the angle at the circumference, upon the same base, that is upon the same part of the circumference: thus the angle BEC is double of the angle BAC, Fig. 38.

11. T ment of a circle are equal to one another: thus the angle BAD is equal to the angle BED; also the angle BCD is equal to the angle BFD, Fig. 39.

12. T right angle: thus the angle ECF, Fig. 45, is a right angle.

13. This mark ' placed on the sides or i of a triangle, indicates that they are given; and this mark 0 placed in the same way, indicates that they are required. PRACTICAL RULES FOR SOLVING ALL THE CASES OF PLANE TRIGONOMETRY. CASE 1.

The angles and one side of any plane tr iangle being given, to find the other sides. „

RULE.

As the sine of the angle opposite the given' side, Is to the sine of the angle opposite the required side, So is the given side, To the required side.* * DEMONSTRATION. Let ABC, Fig 40, be any plane triangle ; take BF= AC, and upon AB let fall the perpendiculars CD and FE, which will bo

PLANE TRIGONOMETRY.

47

Note 1.—The proportions in trigonometry are worked by logarithms: thus, from the sum of the logarithm of the second and third terms, subtract the logarithms of the first term, and the remainder will be the logarithm of the fourth term. 2. The logarithmic sine of a right angle or 90° is 10.00000, being the same as the logarithm of the radius. EXAMPLES. 9

1. In the triangle ABC, there are given the angle A = 32° 15', the angle B=114° 24', and consequently the angle C = 33° 21', and the sides AB=98 ;* required the sides AC and BC. By Construction, Fig. 41. Fig. 41. Make AB equal to 98 by a scale of equal parts, and draw AC, making the angle A =32° 15'; also make the angle B=114° 24', and produce BC, AC, till they meet in C, then is ABC the triangle required; and AC measured by the same scale of equal parts, is 162, and BC is 95. the sines of the angles A and B to the equal radii AC and BF. Now the triangles BDC and BEF being similar, we have CD : FE : : BC : BF : or AC; that is sin. A : sin. B : : BC : AC. In like manner it is proved, that jsin. A : sin. C : : BC : AB. When one of the angles is obtuse, the demonstration is the same. Hence it appears, that in any plane triangle, the sides are to one another as the sines of their opposite angles. * This 98 may express so many feet, or yards, &e., and the other sides will be of the same denomination as the given.

48

PLANE TRIGONOMETRY.

By Calculation.

As sine of the angle C 33° 21'

-

- 9.74017

Is to sine of the angle B 14° 24' So is AB 98

- 9.95937 1.99123

i

11.95060 9.74017 To AC 162.3 As sine of C 33° 21'

2.21043 -

-

-

- 9.74017

Is to sine of A 32° 15’- - So is AB 98

-

9.72723 1.99123 11.71846 9.74017

To BC 95.12

1.97829 By Gunter's Scale.

Extend the compasses, on the line of sines, from 33° 21' to 65° 36' the supplement of the angle B; that extent will reach, on the line of numbers, from 98 to 162, the side AC. Extend the compasses from 33’ 21' to 32° 15' on the line of sines; that extent will reach, on the line of num bers, from 98 to 95, the side BC. 2. In the right-angled triangle ABC, are given the hypothenuse AC=480, and the angle A=53° 8'. To find the base AB and perpendicular BC.

49

PLANE TRIGON03IETET.

From 90° subtract the angle A=53° 8'; the remain¬ der 36° 52' will be the angle C. The angle B, being a right angle, is 90°. By Construction, Fig. 42. This may be constructed as in the preced- Fls- 42ing example, or otherwise thus, Draw the line AB of any length, and draw AC, making the angle A— 53° 8'; make AC = 480 by a scale of equal parts, and from C i draw CB perpendicular to AB, then ABC is A the triangle required. AB measured by the same scale of equal parts, will be 288, and BC will be 384. By Calculation. As sine ofB 90 - - - - - - 10.00000 Is to sine of A 53° 8' - - - - 9.90311 So is AC 480 - 2.68124 12.58435 To BC 384

2.58435

As sine ofB90°

- 10.00000

Is to sine of C 36° 52' - - - - 9.77812 So is AC 480 2.6812412.45936 To AB 288

2.45936

0

-

50

PLANE TRIGONOMETRY.

By Gunter's Scale. Extend the compasses, on the line of sines, from 90° to 53° 8', that extent will reach, on the line of numbers, from 480 to 384, the perpendicular BC. Extend the compasses, on the line of sines, from 90° to 36° 52', the complement of the angle A; that extent will reach, on the line of numbers, from 480 to 288, the base AB. 3. In the triangle ABC, are given the angle A=79° 23', the angle B=54° 22', and the side BC=125; re¬ quired AC and AB. Ans. AC=103.4, and AB=91.87. 4. In a right-angled triangle, there are given the angle A=56° 48', and the base AB=53.G6, to find the perpendicular BC and hypothenuse AC. Ans. BC=82 and AC=98. 5. In the right-angled triangle ABC, are given the angle A=39° 10', and the perpendicular BC=407.37, to find the base AB and hypothenuse AC. Ans. AB=500.1, and AC=645. CASE

2.

Two sides and an angle opposite one of them, being given, to find the other angles and side. RULE.

As the side opposite the given angle, Is to the other given side, So is the sine of the angle opposite the former, To the sine of the angle opposite the latter.* * This is evident from the demonstration of the rule in the preceding case.

PLANE TRIGONOMETRY.

51

Add the angle thus found to the given angle, and subtract their sum from 180°, the remainder will he the third angle. After finding the angles, the other side may he found by Case 1. Note.—The angle found by this rule is sometimes ambiguous; for the operation only gives the sine of the angle, not the angle itself; and the sine of every angle is also the sine of its supplement. When the side opposite the given angle is equal to, or greater than the other given side, then the angle opposite that other given side is always acute; but when this is not the case, that angle may^be either acute or obtuse, and is consequently ambiguous. EXAMPLES.

1. In the triangle ABC, are given the angle C = 33° 21', the side AB = .98 and the side BC = .7912; re¬ quired the angles A and B, and the side BC. By Construction, Fig. 43. Make BC = .7912 by a scale Fig. 43. of equal parts, and draw CA, '"X making the angle C =33° 21'; / with the side AB = .98, in the \ \ compasses, taken from the same • \ \ 1 scale of equal parts, and B as bV j a centre, describe the arc ab, cutting AC in the point A, and join BA; then is ABC the triangle required: the side AC, measured by the scale of equal parts, will be 1.54, and the angles A and B, mea¬ sured by a scale of chords, will be 26° 21' and 120° 18'. Here the arc ab cuts AC in one point only, because AB is greater than BC; therefore the angle A is acute, and not ambiguous.

52

PLANE TRIGONOMETRY.

By Calculation. As AB, .98 - - - - —1.99123 Is to BQ .7912 - - - - - —1.89829 So is sine of C, 33° 21' - 9.74017 •

To sine of A, 26° 21'

9.63846 9.64723

To the angle C=33° 21' add the angle A=26° 21' and the sum is 59° 42' which subtracted from 180° leaves the angle B=120° 18' As sine of C, 33° 21' - - -

9.74017

Is to sine of B, 120° 18*. 9.93621 So is AB, .98 ... - —1.99123 9.92744 To AC, 1.539 - - - - 0.18727 By Gunter's Scale. Extend the compasses from .98 to .79 on the line of numbers, that extent will reach from 33° 21' to 26° 21', the angle A, on the line of sines. Add the angle A=26° 21' to the angle C=33° 21', and the sum will be 59° 42'j then extend the compasses from 33° 21' to 59° 42', on the line of sines, that extent will reach from .98 to 1.54, the side AC, on the line of numbers. 2. In the triangle ABC, are given the angle C=33° 21', the side BC=95.12 and the side AB=60, to find the angles A and B, and the side AC.

53

PLANE TRIGONOMETRY.

By Construction, Fig. 44. This is constructed in the same Fis- 44manner as the preceding exam¬ ple ; only, AB being shorter than BC, the arc ab cuts AC in two points on the same side of BC; hence the angle A may be either acute or obtuse. The side required has also two values, as AC and AC. By Calculation. AsAB, 60

1.77815

Is to BC, 95.12 So is sine C, 33° 21'

1.97827 9.74017 11.71844

60° 38' acute 1 { 119 22 obtuse } 9'94029 The sum of the angles C and A subtracted from 180° leaves the angle B=86° 1' if A be acute, or 27° 17' if A be obtuse. To sine of A

To find the side AC answering to the acute value of the angle A. As sine of C, 33° 21' - - - - 9.74017 Is to sine of B< 86° 1' - - - - 9.99895 So is AB, 60 1.77815 11.77710 To AC, 108.9

2.03693

To find the side AC, answering to the obtuse value of the angle A.

54

PLANE TRIGONOMETRY.

As sine of C, 33° 21'

- -

-

9.74017

Is to sine of B, 27° 17 - - - 9.66124 So is AB, 60 - - - - 1.77815 j 1.43939 To AC, 50.03

.... - 1.69922

3. In a triangle ABC, the side AB is 274, AC 306. and the angle B 78° 13'; required the angles A and C. and the side BC. Ans. A=40° 33', C=61° 14', and BC=203.2. 4. In a right-angled triangle, there are given the hypothenuse AC=272, and the base AB=232; to find the angles A and C, and the perpendicular BC. Ans. A=31° 28' C=58° 32' and BC=142. 5. In a right-angled triangle ABC, the hypothenuse AC is 150, and one side BC 69; required the angles and other side. Ans. 0=62° 37', A=27° 23', and AB 133.2. CASE

3.

Two sides and the included angle being given, to find the other angles and side. RULE.

Subtract the given angle from 180°, and the remainder will be the sum of the two unknown angles. Then, As the sum of the two given sides, Is to their difference;

55

PLANE TRIGONOMETRY.

So is the tangent of half the sum of the two un¬ known angles, , To the tangent of half their difference.* This half difference of the two unknown angles, added to their half sum, will give the angle opposite the greater of the two given sides, and being subtracted * DEMONSTRATION. Let ABC, Fig. 45, be the proposed triangle, having the two given sides AB, AC, including the given angle A. About A as a centre, with AC the greater of the given i ^ sides for a radius, describe a circle meeting AB jproduced in E and F, and BC in D; join DA, EC, and FC, and draw FG parallel to BC, meet¬ ing EC produced in G.

Fig. 45.

G

The angl# EAC (32.1.) is equal to the sum of the unknown angles ABC, ACB ; and the angle EFC at the circumference, is equal to the half of EAC at the centre (20.3,;) therefore EFC is half the sum of the unknown angles; but (32.1.) the angle ABC is equal to the sum of the angles BAD and ADB, or BAD and ACB ; therefore FAD is the difference of the unknown angles ABC, ACB; and FCD, at the cir¬ cumference is the half of that difference; but because of the parallels DC, FG, the angles GFC, FCD are equal; therefore GFC is equal to half the difference of the unknown angles ABC, ACB; but since the angle ECF in a semicircle, is a right angle, EG is perpendicular to CF, and therefore CF being radius, EC, CG are the tangents of the angles EFC, CFG; it is also evident that EB is the sum of the sides BA, AC, and that BF is the difference; therefore since BC, FG are parallel, EB ; BF : : EC : CG (2.6.:) that is, the sum of the sides AC, AB, is to their difference, as the tangent of half the sum of the angles ABC, ACB, is to the tangent of half their, difference. To demonstrate the latter part of the Fig. 46. rule, let AC and AB, Fig. 46, represent C i ft any two magnitudes whatever; in AB pro- * duced, take BD equal to AC the less, and bisect AD in E.

D

Then because AE is equal to ED, and AC to BD, CE is equal to EB; therefore AE or ED is half the sum of the given magnitudes AB, AC, and CE, or EB is half their difference; but AB the greater is equal to AE, EB, that is to half the sum added to half the difference; and AC the less, is equal to the excess of AE, half the sum, above CE, half the dif¬ ference.

56

PLANE TRIGONOMETRY.

from the half sum will give the angle opposite the less given side. After finding the angles, the other side may be found by Case 1. EXAMPLES.

1. In the triangle ABC, there are given AB=128, AC = 90, and the angle A=48° 12', to find the angles B and C, and the side BC. By Construction, Fig. 47. Fig. 47.

c

Draw AB = 128, and make the angle A=48° 12'; draw AC=90, and join BC. The angle B will measure 44° 37', the angle C 87° 11', and the side BC 95.5. By Calculation.

AB AC

128 90

Angle A

180° 0' - - - - - 48 12

Sum of the angles B and C 131 48 218 Sum 05 54 Half sum do. Difference 38 2.31 1846 As the sum of the sides, AB, AC, 218 - Is to their difference, 38 1.57978 So is the tangent of half the sum of the 1 34938 angles B and C, 65° 54' J 11.92916 To tang, of half their difference, 21° 17' - - 9.59070 Half sum of the angles B and C - - 65° 54' Add and subtract half their difference - 21 17 Angle C--Angle B

87

11

44 37

57

PLANE TRIGONOMETRY.

To find the side BC. As sine of B, 44° 37' [s to sine of A, 48 12 So is AC, 90 -

-

9.84656 9.87243 1.95424 11.82667 1.98011

To BC, 95.52 By Gunter’s Scale.

Extend the compasses from 218, the sum of the sides, to 38, their difference, on the line of numbers, and apply this extent to the line of tangents from 45° to the left hand; then keeping the left leg of the compasses fixed, move the other leg to 65° 54' the half sum of the angles; that distance will reach from 45° on the same line, to 21° 17', the half difference of the required angles. Whence the angles are obtained as before. To extend the second proportion, proceed as directed in Case 1st. 2. In a triangle ABC, are given AB=109, BC=76, and the contained angle B=101° 30', to find the other angles and side. ( Ans. The angle A=30° 57', C=47° 33', and the \ side AC= 144.8. 3. Given, in a right-angled triangle, the base AB=890 and the perpendicular BC=787, to find the angles and hypothenuse. ( Ans. The angle A=41° 29', C=48° 31', and ( the hypothenuse AC=1188. II

58

PLANE TRIGONOMETRY.

CASE

4.

Given the three sides, to find the angles. RULE 1.

Consider the longest side of the triangle as the base, and on it let fall a perpendicular from the opposite angle. This perpendicular will divide the base into two parts, called segments, and the whole triangle into two right-angled triangles. Then, As the base, or sum of the segments, Is to the sum of the other two sides; So is the difference of those sides, To the difference of the segments of the base.* Fig. 48. * DEMONSTRATION. Let ABC, Fig. 48, be a triangle, and CD be perpendicular upon AB. / yJ About C as a centre, with the less side BC for / \ a radius, describe a circle, meeting AC proI I ^uce^> G and E, and AB in F. Then it is \a/ \. / evident that AE is equal to the sum of the \/ sides AC, BC, and that AG is equal to their difference; also because CD bisects FB (3.3,) it is plain that AF is the difference of the seg¬ ments of the base ; but AxBAF = AEx AG (36.3. cor.;) therefore AB : AE : : AG : AF )16.6;) that is, the base, is to the sum of the sides, as the difference of the sides, is to the difference of the segments of the base. Cor. If AF be considered the base of the triangle AFC, then will CD be a perpendicular on the base produced; AE will be equal to the sum of the sides AC, FC, and AG will be equal to their difference; also AB will be equal to the sum of the segments AD, FD. But bj the preceding demonstration, and (16.5,) AF : AE : : AG : AB; hence, when the per¬ pendicular falls without the triangle, the base is to the sum of the sides, as the difference of the sides is to the sum of the segments of the base. A rule might, therefore, be given, making either side of a triangle the base; and such a rule would be rather more convenient, in some cases, than the one above: but then, on account of the perpendicular, sometimes falling within and sometimes without the triangle, it would require twc cases, and consequently would be less simple.

59

PLANE TRIGONOMETRY.

To half the base, add half the difference of the seg¬ ments, and the sum -will be the greater segment; also from half the base, subtract half the difference of the segments, and the remainder will be the less segment. Then, in each of the two right-angled triangles, there will be known two sides, and an angle opposite to one of them; consequently the other angles may be found by Case 2. 1. In the triangle ABC, are given AB = 426, AC=365, and BC = 230; required the angles. By Construction, Fig. 49. Draw AB=426; with AC = 365 in the dividers, and one foot in A, de¬ scribe an arc, and with BC = 230, and one foot in B describe another arc, cutting the former in C ; join AC, BC, and ABC will be the triangle required. The angles measured by a scale of chords, will be A =32° 39', B = 58° 56', and C = 88° 25'. By Calculation. AC BC Sum

365 ------- 230 - -

595

Difference -----

135

PLANK TRIGONOMETRY.

60

As the base AB ------ 426 2.62941 Is to the sum of the sides AC, BC - 595 2.77452 So is the diff. of the sides AC, BC - 135 2.13033 4.90485 2.27544

To the diff. of the segments AD, DB 188.6 Half diff. of the segments Half base Segment AD Segment BD

-

-

- 94.3213.

------- 307.3 ------- 118.7 .

As AC - 365 2.56229 Is to AD 307.3 2.48756 So is sine of ADC - - 90° 10.00000 12.48756 To sine of ACD

Angle A

-

-

-

-----

As BC Is to BD So is sine of BDC

-

-

57° 21' 90 00

9.92527

- 32 39

- - -

230 2.36173 118.7 2.07445 90° 10.00000 12.07445

To sine of BCD 31° 4' 90 0 Angle B

----- -

9.71272

58 56

From 180° subtract the sum of the angles A, and B, 91° 35', and the remainder 88° 25' is the angle C.

PLANE TRIGONOMETRY.

61

By Gunter's Scale. Extend the compasses from 426 to 595 on the line of numbers, that extent will reach on the same line from 135 to 188.6 the difference of the segments of the base. Whence the segments of the base are found as before. To extend the other proportions, proceed as directed in Case 2. 2. In a triangle ABC, there are given AB=64, AC— 47. and BC=34; required the angles. Ans. Angle A —31a 9', B=45° 38’, and C=103° 13'. ,3. In a triangle ABC, are given AC=88, AB=108, and BC=54, to find the angles. Ans. Angle A—29° 49’, B=54° 7, and C=96° 4'. RULE

2.

Add together the arithmetical complements of the lo¬ garithms of the two sides containing the required angle, the logarithm of the half sum of the three sides, and the logarithm of the difference between the half sum and the side opposite the required angle. Then half the sum of these four logarithms will be the logarithmic cosine of half the required angle.*

* DEMONSTRATION. Let ABC, be a triangle of which the side AB, is greater than AC: make AD=AC, join DC, bisect it in E, and join AE; draw EH parallel and equal to CB; join IIB and produce it to meet AE produced in G. •

Now in the triangles AED, AEC, all the sides of the one are equal to the sides of the other, each to each; therefore (8.1) the angle EAD *=EAC, and AED= AEC; consequent¬ ly AED is a right angle.

62

PLANE TRIGONOMETRY* EXAMPLES.

1. Ill the triangle ABC, are given AB=426, AC=365 and BC=230; required the angle A. By Calculation.

BC 230 AC 365 AB 426 2)1021 Half sum 510.5 Difference

280.5

Ar, Co. Ar. Co.

7.43771 7.37059

log.

2.70800

log.

2.44793 2)19.96423

Cos. iA 16° 20' 2

9.98211

5

Angle A 32 40 Because EH is equal and parallel to BC, BH is also equal and parallel to EC (33.1;) now in the triangles EDF and HBF, the angle EFD=BFH, the angle FED=FHB (29.1) and ED=EC=BH; therefore (26.1) EF=FH, and FD=FB. Again, the angle HGE=DEA=a right angle; if therefore with the centre F, and radius FE=FH, a circle be described, it will pass through the point G (31.3.) Now 2 AF=2 AD + 2 DF= AD + AD + DB= AD -f AB= AC-f AB; there¬ fore AF= ^AC+JAB; also FK=^IK— ^EH=^BC; hence, by adding equals to equals, AF-f FK= £ AC-f JAB+^BC, or AK=|(AC+AB-f BC;) again/ AI= AK—IK= |(AC-fAB-f BC)—BC. But (Dem. to rule, case 1st.) AD : AE:: sin. AED: sin. ADE:: rad.: cos. EAD (cos. £BAC.) Also, AB : AG : : sin. AGB : sin. ABG : : rad.: cos. BAG (cos. £BAC.) Hence (c. 6) AB x AD: AG x AE:: rad.2: (cos. JBAC)2. But AB x AD= AB x AC, and (cor. 36.3) AG X AE= AK X AL=£(AC-f AB-f BC) X [KAC-f AB-bBC)—BC;] therefore ABxAC: £(AC+AB+BC)x[KAC-r AB+BC)—BC] :: rad.2 : (cos. ^BAC)2. Now it is evident, that in working this proportion by logarithms

PLANE TRIGONOMETRY.

63

If the other angles are required, they may he found by Case 1. 2. In a triangle ABC, are given AB = 64, AC=4 7, and BC = 34, to find the angle B. Ans. Angle B = 45° 38'. 3. In a triangle ABC, are given AC= 88, AB = 108, and BC =54, to find the angle C. Ans. C = 96° 4'. The preceding rules solve all the cases of plane trian¬ gles, both righfiangled and oblique. There are, how¬ ever, other rules, suited to right-angled triangles, which are sometimes more convenient than the general ones. Previous to giving these rules, it will be necessary to make the following Eemarlcs on right-angled triangles. 1. ABC, Pig. 50, being a right-angled Fig. 50. triangle, make one leg AB radius, that is, with the centre A, and distance AB, de¬ scribe an arc BF. Then it is evident that the other leg BC represents the tangent of the arc BF, or of the angle A, and the hypothenuse AC the secant of it. 2. In like manner, if the leg BC, Fig. 51, Jje made radius; then the other leg AB will represent the tangent of the arc BG, or angle C, and the hypothenuse AC the secant of it.

Fig. 51. C

Tang. 0

and taking tlie arithmetical complements of the logarithms of the first term, viz. of the two sides, including the required angle, if we omit the logarithm of the square of radius, which is 20, it is just equivalent to re¬ jecting 20 from the sum of the logarithms, which would otherwise have to be done.

64

PLANE TRIGONOMETRY.

Fig. 62. c

3. But if the hypothenuse be made radius, then each leg will represent the sine of its opposite angle; namely, the leg AB, Fig. 52, the sine of the arc AE or angle C, and the leg BC the sine of the arc CD, or angle A.

E

The angles and one side of a right-angled triangle being given, to find the other sides. RULE.

Call any one of the sides radius, and write upon it the word radius; observe whether the other sides be¬ come sines, tangents, or secants, and write these words on them accordingly. Call the word written upon each side the name of that side. Then, As the name of the side given, Is to the name of the side required; So is the side given, To the side required.* Two sides of a right-angled triangle being given, to find the angles and other side. RULE.

Call either of the given sides radius, and write on them as before. Then, Fig. 53.

_ * DEMONSTRATION. Let ABC, Fig. 53, be a rightangled triangle; then it is evident that BC is the tan¬ gent, and AC the secant of the angle A, to the radius AB. Let AD represent the radius of the tables, and draw DE perpendicular to AD, meeting AC produced in E; then DE is the tangent, and AE the secant of the angle A, to the radius AD. But because of the similar triangles ADE, ABC, AD : DE : : AB : BC; that is, the tabular radius : tabular tangent: : AB : BC. Also AD : AE : : AB : AC; that is, the tabular radius : tabular secant *. : AB : AC. These proportions correspond with the rule. When either of the other sides is made radius, the demonstrar tion will be similar. *

/!E

65

PLANE TRIGONOMETRY.

As the side made radius, Is to the other given side; So is radius, To the name of that other side.* After finding the angle, the other side is found as in O O the preceding rule. 7

EXAMPLES.

1. In a right-angled triangle ABC, are given the base AB=208, and the angle A ==35° 16', to find the hypothenuse AC and perpendicular BC.

By Calculation.

The hypothenuse AC being radius. As the sine of C, 54° 44' - - - 9.91194 Is to radius - - - - - -10.00000 So is AB 208 2.31806 12.31806 To AC 254.8

2.40612

As the sine of C 54° 44' - Is to the sine of A, 35 16 - : So is AB 208

-

9.91194 9.76146 2.31806 12.07952

To BC, 147.1

2.16758

This is the converse of the preceding' rule. 6*

I

66

TLANE TRIGONOMETRY.

The base AB being radius. As radius - - - 10.00000 Is to the secant of A, 35° 16' - - 10.08806 So is AB 208 2.31806 12.40612 To AC 254.8

• 2.40612 t

As radius ------- 10.00000 Is to tangent of A, 35° 16' - - 9.84952 So is AB, 208 ------ 2.31806 12.16758 To BC 147.1

2.16758

The perpendicular BC being radius. As tangent of C 54° 44' - - - - 10.15048 Is to secant of C, 54 44 - - - - 10.23854 So is AB, 208 ------ 2.31806 12.55660 To AC 254.8

2.40612

As tangent of C, 54° 44' - - - - 10.15048. Is to radius, ------- 10.00000 So is AB 208 2.31806 ♦

12.31806 To BC, 147.1 - - -

- - 2.16758

2. In a right-angled triangle ABC, there are given the hypothenuse AC=272, and the base AB=232 ; required the angles A and C, and the perpendicular BC.

67

PLANE TRIGONOMETRY.

By Calculation,

The hypothenuse AC being radius. As AC, 272 2.43457 Is tp AB, 232 ------- 2.36549 So is radius ------- 10.00000 ' 12.36549 To sine ofC, 58° 32' - - - - 9.93092 As radius 10.00000 Is to sine of A, 31° 28’ - - - - 9.71767 So is AC, 272 2.43457 12.15224 ToBC, 142

2.15224

The base AB being radius. As AB, 232 2.36549 Is to AC, 272 - - - 2.43457 So is radius ------- 10.00000 12.43457 To secant of A, 31° 28' -

-

-

- 10.06908

As radius ------- 10.00000 Is to tangent of A, 31° 28' - - - 9.78675 So is AB, 232 ------ 2.36549 12.15224 ToBC, 142

2.15224

68

PLANE TRIGONOMETRY.

3. In a right-angled triangle, are given the hypothenuse AC=36.57, and the angle A=27° 46', to find the base AB, and perpendicular BC. Ans. Base AB=32.36, and perpendicular BC= 17.04. 4. In a right-angled triangle, there are given, the perpendicular=193.6, and the angle opposite the base 47° 51'; required the hypothenuse and base. Ans. Hypothenuse=288.5, and base=2l3.9. 5. Required the angles and hypothenuse of a rightangled triangle, the base of which is 46.72, and perpen¬ dicular 57.9. ^ns ^ Angle opposite the base 38° 54', angle opposite ( the perpendicular 51° 6', and hypothenuse 74.4. When twro sides of a right-angled triangle are given, the other side may be found by the foliowring rules, with¬ out first finding the angles. I. When the hypothenuse and one leg are given, to find the other leg. RULE.

Subtract the square of the given leg from the square of the hypothenuse; the square root of the remainder will be the leg required.* Or by logarithms thus, To the logarithm of the sum of the hypothenuse and given side, add the logarithm of their difference ; half this sum wall be the logarithm of the leg required. * DEMONSTRATION. The square of the hypothenuse of a right-angled tri¬ angle is equal to the sum of the squares of the sides (47.1.) Therefore the truth of the first part of each of the rules is evident.

PLANE TRIGONOMETRY.

69

2. When the two legs are given to find the hypothenuse. .

RULE.

Add together the squares of the two given legs; the square root of the sum will he the hypothenuse.* Or by logarithms thus, From twice the logarithm of the perpendicular, sub¬ tract the logarithm of the base, and add the correspond¬ ing natural number to the base; then, half the sum of the logarithms of this sum, and of the base, will be the logarithm of the hypothenuse. EXAMPLES.

1. The hypothenuse of a right-angled triangle is 272, and the base 232 ; required the perpendicular. Calculation by logarithms. Hypothenuse - - 272 Base - - - - 232 Sum - - - - 504 log. 2.70243 Difference - - - 40 1.60206 2)4.30449 Perpendicular - - 142 * Put h=the hypothenuse, 6=the

2.15224 base, and p=the perpendicular,

then (47.1) p2=h2—7>2=(5.2 cor.) h-\-bxh—b, or p—*/hJrbxh—b; whence, from the nature of logarithms, tlie latter part of the first rule is evident Also (47.1) A2=i2+/=5x6+^_ or (6x6+y ) which, solved by logarithms, will correspond with the latter part of the second rule.

70

APPLICATION OF

2. Given the base 186, and the perpen to find the hypothenuse. Calculation by Logarithms. Perpendicular

Base

-

-

152 log. 2.18184 • 2 -

186

4.36368 2.26951

124.2

2.09417

310.2

log. 2.49164 2)4.76115 log. 2.38057

240.2

Hypothenuse

2.26951

3. The hypothenuse being given equal 403, and one leg 321; required the other leg. Am. 243.7. 4. What is the hypothenuse of a right-angled triangle, the base of which is 31.04 and perpendicular 27.2 ? Am. 41.27. The following examples, in which trigonometry is applied to the mensuration of inaccessible distances and heights, will serve to render the student expert in solv¬ ing the different cases, and also to elucidate its use. Fig. 54.

THE

APPLICATION OF PLANE

TRIGONO¬

METRY TO THE MENSURATION OF DIS¬ TANCES AND HEIGHTS. EXAMPLE

1, Fig. 54.

Being on one side of a river and want¬ ing to know the distance to a house on the other side,

71

PLANE TRIGONOMETRY.

I measured 500 yards along the side of the river in a right line AB, and found the two angles* between this line and the object to be CAB = 74° 14' and CBA=49° 23'. ‘ Required the distance between each station and the object. Calculation. The sum of the angles CAB and CBA is 123° 37', which subtracted from 180° leaves the angle ACB = 56° 23'. Then by Case 1; s. ACB : s. CBA : : AB : AC 56° 23' 49° 23' 500 455.8 and s. ACB : s. CAB : : BA : BC 56° 23' 74° 14' 500 577.8 EXAMPLE

2, Fig. 55.

Suppose I want to know the dis¬ tance between two places, A and B, accessible at both ends of the line AB, and that I measured AC = 735 yards, and BC= 840; also the angle ACB = 55° 40'. What is the distance be¬ tween A and B ?

Fig. 55.

Calculation. The angle ACB = 55° 40', being subtracted from 180°, leaves 124° 20'; the half of which i» 62° 10'. Then by Case 3, * The angles may be taken with a common surveyor’s compass; or more accurately with an instrument called a theodolite.

72

APPLICATION OP

CAB—CBA 2 1575 105 62° 1CK 7° 12/ CAB+CBA CAB—CBA To and from ' 2 =62° 1CK add and sub. 2 12', and we shall have CAB = 69° 22', and CBA=54° 58'. Then, BC + AC : BC—AC

: :

s. ABC : 54°58/

tangent

CAB+CBA 2

tangent

s. ACB : : AC : AB 55° 40/ * 735 741.2

3, Fig. 56. Wanting to know the distance between two inaccessible objects A and B, I measured a base line CD = 300 yards: at C the angle BCD was 58° 20' and ACD 95° 20'; at D the angle CDA was 53° 30' and CDB 98° 45'. Re¬ quired the distance AB.

EXAMPLE Fig. 56.

Calculation. 1. In the triangle ACD, are given the an 95° 20', ADC = 53° 30', and the side CD =300, to find AC=465.98. 2. In the triangle BCD, are given the angle BCD= 58° 20', BDC = 98° 45', and side CD=300, to find BC = 761.47. 3. In the triangle ACB we have now given the angle ACB=ACD—BCD = 37°, the side AC = 465.98 and BC = 761.47, to find AB = 479.8 yards, the distance required. EXAMPLE

4, Fig. 57.

Being on one side of a river and observing three objects, A, B and C stand on the other side, whose distances apart I knew to be, AB = 3 miles, AC

73

PLANE TRIGONOMETRY.

= 2, and BC = 1.S, I took a station D, in a straight line with the objects A and C, being nearer the former, and found the angle ADB =17° 47'. Required mj distance from each of the objects. Construction.

With the three given distances, describe the triangle ABC; from B, draw BE parallel to CA, and draw BD making the angle EBD=17° 47' (the given angle ADB) and meeting CA produced, in D : then AD, CD and BD will be the distances required* Calculation.

1. In the triangle ABC we have all the sides given, to find the angle C = 104° 8'. 2. Subtract the sum of the angles D and C from 180°, the remainder 58° 5' will be the angle DBC; theft in the triangle BCD we know all the angles and the side BC to find DC = 5.002 and BD=5.715; therefore DA=DC—AC=3.002. EXAMPLE

5, Fig. 58.

From a station at D, I perceived three objects, A, B and C, whose distances from each other I knew to be as follows: AB= 12 miles, BC = 7.2 miles, and AC = 8 miles; at D, I took the angle CDB=25° and ADC = 19°. Hence it is required to find my distance from each of the objects.

Fig. ^58.

* DEMONSTRATION. By construction, the distances AB, BO and AC are iqual to the given distances; also the angle (29.1) BDC = the angle DBE =tlic given angle.

K

74

APPLICATION OF

Cmstructioii.

With the given distances describe the triangle ABC; at B, make the angle EBA=19°=the given angle ADC, and at A, make the angle EAB=25°—the given angle BDC; draw AE, and BE meeting in E, and (by prob. 10,) describe a circle that shall pass through the points A, E and B: join CE and produce it to meet the circle in D, and join AD, BD, then will AD, CD, and BD be the distances required.* Calculation.

1. In the triangle ABC, all the sides are given, to find the angle BAC=35° 35'. 2. In the triangle AEB, are given all the angles, viz. EAB=25°, EBA=19°, and AEB=136°, and the side AB=12, to find AE= 5.624. * DEMONSTRATION. The angle ADC standing on the same arc with the angle ABE is equal to it (21.3.) For the same reason the angle BDC is equal to the angle BAE; but by construction the angles ABE and BAE are equal to the given angles ; therefore the angles ADC and BDC are equal to the given angles.

Note.—When the given angles ADC, BDC are respectively equal to the angles ABC, BAG, the point E will fall on the point C, the circle will pass through the points A, C, and B, and the point D may be any where in the arc ADB; consequently, in this case, the situation of the point D, or its distance from each of the objects A, B, C, cannot be determined from the data given. It may not be improper also to observe, that even when the angle ADR, which is the sum of the given angles, is equal to the sum of the angles ABC, BAG, 01 which is the same thing, is the supplement of the angle ACB, the circle passes through the points A, C, B; but then the angles ADC, BDC, unless they have been erroneously taken, will be respectively equal to the angles ABC, BAG.

75

PLANE TRIGONOMETRY.

3. In tlie triangle CAE we have given the side AC = 8, AE= 5.624, and the angle CAE=BAC—EAI3 = 10° 35', to find the angle ACE=22° 41/. 4. In the triangle DAC, all the angles are given, viz. ADC=19°, ACD=22° 41' and CAD=180°—the sum of the angles ADC and ACD=138° 19', and the side AC = 8, to find AD=9.47 miles, and CD = 16.34. In the triangle ABD, we have the angle ADB=ADC +BDC=44°, the angle BAD=CAD—BAG=102° 44', and the side AB=12, to find BD=16.85 miles. EXAMPLE

6, Eig. 59.

A person having a triangular field, the sides of which measure AB=50 perches, AC=46 perches, and BC = 40 perches, wishes to have a well dug in it, that shall he equally dis¬ tant from the corners A, B and C. What must be its distance from each corner, and by what angle from the corner A, may its place be found ?

Pig. 59.

Construction.

With the given sides construct the triangle ABC, and (by Prob. 10.) describe a circle that shall pass through the points A, B, and C; then the centre E of this circle is the required place of the well.* Calculation.

1. In the triangle ABC, all the sides are given, to find the angle ABC=60° 16'. 2. Join CE and produce it to meet the circumference in D; also join AE and AD; then the angles ADC, ABC * The demonstration of this is plain (1.3 cor.)

76

APPLICATION' OP

being angles in the same segment, are equal; also the angle DAC being an angle in a semicircle, is a righhangle: therefore in the right-angled triangle DAC, we have the angle ADC=ABC=60° 16', and the side AC, to find CD=52.98 perches. The half of CD is=26.49 perches = CE=the distance of the well from each corner. 3. The angle ACD=90°—ADC=29° 44; but be¬ cause AEC is an isosceles triangle, the angle CAE= ACE=29° 44' the angle required. 7, Fig. 61. Wishing to know the height of a steeple situated on a horizontal plane, I measured 100 feet in a right line from its base, and then took the angle of elevation* of the top, which I found to. be 47° 30', the centre of the quadrant being 5 feet above the ground: required the height of the steeple..

EXAMPLE Fife. 61.

Fig. 60. * Angles of elevation, or of depression, A are usually taken with an instrument called a quadrant, the arc of which is divided into 90 equal parts or degrees, and those, when the instrument is sufficiently large, may he subdivided into halves, quarters, &c. From the centre a plummet is suspended by a fine silk thread. Fig. 60 is a representation of this instrument. To take an angle of elevation, hold the quadrant in a vertical position, and the degrees being numbered from B towards C, with the eye at C, look along the side CA, moving the quadrant till the top of the object is seen in a range with this side ; then the angle BAD made by the plummet with the side BA, will be the angle of elevation required. Angles of depression are taken in the same manner, except that then the eye is applied to the centre of the quadrant. Note.—In finding the height of an object, it is best to take such a posi¬ tion that the observed angle of altitude may bo about 45°; for when the observed angle is 45°, a small error committed in taking it, makes the least error in the computed height of the object.

PLANE TRIGONOMETRY,

77

Calculation. In tlie right-angled triangle DEC, we have the angle CDE=47° 30', and the base DE = AB = 100 feet, to find CE = 109.13 feet; to CE add EB = DA = 5 feet, the height of the quadrant, and it will give BC = 114.13 feet, the required height of the steeple. 8, Fig. 62. Fig. 62. Wishing to know the height of a tree situated in a bog, at a sta¬ tion D, which appeared to be on a level with the bottom of the tree, I took the angle of eleva¬ tion BDC = 51° 30'; I then measured DA = 75 feet in a direct line from the tree, and at A, took the angle of elevation BAC = 26° 30'. Required the height of the tree. Calculation. 1. Because the exterior angle of a triangle is equal to the sum of the two interior and opposite ones, the angle BDC = DAC + ACD; therefore ACD = BDC— DAC = 25° : now in the triangle ADC we have DAC = 26° 30', ACD=25°, and AD=75, to find DC = 79.18. 2. In the right-angled triangle DBC are given DC = 79.18, and the angle BDC=51° 30'to find BC = 61.97 feet, the required height of the tree. EXAMPLE

EXA.MPLE

Wanting to know the height of a tower EC, which stood upon a hill, at A, I took the angle of elevation CAB=44° ; I then measured AD 134 yards, on level ground, in a straight line towards the

9, Fig. 63.

78

APPLICATION OP

tower; at D the angle CDB was 67° 50' and EDB 51°. Required the height of the tower and also of the hill. Calculation. 1. In the triangle ADC we have the angle DAC = 44°, the angle ACD = BDC—DAC=23° 50', and the side AD, to find DC = 230.4. 2. In the triangle DEC all the angles are given, viz. CDE=BDC—BDE=16° 50', DCE=90°—BDC=22° 10', DEC = 180° = tlie sum of the angles CDE and DCE = 141°, and CD=230.4, to find CE = 106 yards, the height of the tower. 3. In the right-angled triangle DBC, we have the angle BDC = 67° 50', and the side DC=230.4, to find BC=213.4; then BE=BC—CE=213.4—106=107.4 yards, the height of the hill. 10, Fig. 64. An obelisk AD standing on the top of a declivity, I measured from its bottom a distance AB=40 feet, and then took the angle ABD=41°; going on in the same direction 60 feet farther to C, I took the an¬ gle ACD=23° 45': what was the height of the obelisk ? EXAMPLE

Calculation. 1. In the triangle BCD, we have given the angle BCD=23° 45', the angle BDC=ABD—BCD=17° 15', and side BC = 60, to find BD=81.49. 2. In the triangle ABD are given the side AB=40, BD=81.49, and the angle ABD=41°, to find AD= 57.64 feet, the height of the obelisk.

PLANE TRIGONOMETRY.

EXAMPLE

79

11, Fig. 65.

Wanting to know the height of an object on the other side of a river, but which appeared to be on a level with the place where I stood, close by the side of the river; and not having F room to go backward on the same plane, on account of the immediate rise of the bank, I placed a mark where I stood, and measured in a direct line from the object, up the hill, whose ascent was so regular that I might account it a right line, to the distance of 132 yards, ■where I perceived that I was above the level of the top of the object; I there took the angle of depression of the mark by the river’s side equal 42°, of the bottom of the object equal 27°, and of its top equal 19° : re¬ quired the height of the object. Calculation. 1. In the triangle ACD, are given the angle CAD= EDA=27°, ACD=180°—CDE (FCD)=138°, and the side CD=132, to find AD=194.55 yards. 2. In the triangle ABD, we have given ADB=ADE— BDE = 8°, ABD=BED+BDE=109 ° and AD = 194.55, to find AB = 28.64 yards, the required height of the object. Fig. 66. EXAMPLE 12, Fig. 66. c A May-pole whose height was 100 feet stand¬ ing on a horizontal plane, was broken by a blast of wind, and the extremity of the top part struck the ground at the distance of 34 feet from the bottom of the pole : required the length of each part.

APPLICATION OF

Construction. Draw AB=34, and perpendicular to it, make BC— 100; join AC and bisect it in D, and draw DE perpendi¬ cular to AC, meeting BC in E; then AE=^CE=. the part broken off.* Calculation. 1. In the right-angled triangle ABC, we have AB=34 and BC=100, to find the angle C=18° 47'. 2. In the right-angled triangle ABE, we have AEB== ACE4-CAE=2ACE=37° 34', and AB=34, to find AE =55.77 feet, one of the parts; and 100—55.77=44.23 feet the other part. PRACTICAL QUESTIONS. 1. At 85 feet distance from the bottom of a tower, the angle of its elevation was found to be 52° 30': required the altitude of the tower. Ans. 110.8 feet. 2. To find the distance of an inaccessible object, I measured a line of 73 yards, and at each end of it took * DEMONSTRATION. In the triangles AED, DEC, the angle ADE=CDE, the side AD=CD, and DE is common to the two triangles, therefore (4.1) AE=CE.

Note.—This question may be neatly solved in the following man¬ ner without finding either of the angles. Thus, draw DF perpendi¬ cular to BC, then (31.3 and cor. 8.6) FC : DC :: DC : CE; conseDC2 AC2 AB2+BC2 quently CE=pQ j but DC2=—j-= ^ , and FC—-|BC ; thereAB2+BC2 2BC same as before nearly, fore CE=

342+10(T 1156+10000 11156■ =55.79, the 200 200 200

PLANE TRIGONOMETRY.

81

the angle of position of the object and the other end, and found the one to be 90°, and the other 61° 45': required the distance of the object from each station. Ans. 135.9 yards from one, and 154.2 from the other. 3. Wishing to knoAv the distance between two trees C and D, standing in a bog, I measured a base line AB= 339 feet; at A the angle BAD was 100° and BAC 36° 30'; at B the angle ABC was 121° and ABD49°: required the distance between the trees. • • Ans. 697i feet. 4. Observing three steeples, A, B and C, in a town at a distance, whose distances asunder are known to be as follows, viz. AB=213, AC=404, and BC=262 yards, I took their angles of position from the place D where I stood, which was nearest the steeple B, and found the angle ADB=13° 30'; and the angle BDC=29° 50'. Re¬ quired my distance from each of the three steeples. Ans. AD=571 yards, BD=389 yards, and CD=514 yards. 5. A May-pole, whose top was broken off by a blast of wind, struck the ground at 15 feet distance from the foot of the pole : what was the height of the whole Maypole, supposing the length of the broken piece to be 39 feet ? Ans. 75 feet. 6. At a certain place the angle of elevation of an in¬ accessible tower was 26° 30'; but measuring 75 feet in a direct line towards it, the angle was then found to be 51° 30': required the height of the tower and its distance from the last station. Ans. Height 62 feet, distance 49.

7. From the top of a tower by the sea side high, I observed that the angle of depression of a ship’s L

82

APPLICATION, &C.

bottom, then at anchor, was 35°; what was its distance from the bottom of the wall ? Ans. 204.2 feet. 8. There are two columns left standing upright in the ruins of Persepolis; the one is 64 feet above the plane, and the other 50; in a right line between these stands an ancient statue, the head of which is 97 feet from the summit of the higher, and 86 from that of the lower column; and the distance between the lower column and the centre of the statue’s base is 76 feet: required the distance between the tops of the columns. Ans. 157 feet.

SURVEYING.

CHAPTER I. ON THE DIMENSIONS OF A SURVEY. 1. SURVEYING

is the art of measuring, laying out and

dividing land.

2. A Four-Pole Chain is an instrument used for mea¬ suring the boundaries of a survey. It is, as its name im¬ ports, 4 poles or 66 feet in length, and is divided into 100 equal parts or links. The length of a link is therefore 7.92 inches. Note.—A Four-pole Chain is frequently called simply a chain. 3. A Two-pole Chain is 2 poles or 33 feet in length, and is usually divided into 50 equal parts or links. When it is thus divided, the links are of the same length as in a four-pole chain; and the measures taken with it are reduced to four-pole chains previous to using them in calculation. Sometimes the two-pole chain is divided into 40 links; in which case, each two links is the one-tenth of a perch. Measures taken with a two-pole chain, thus divided, are usually expressed in perches and tenths.

84

DIMENSIONS OF A SURVEY. [CHAP. I.

4. The Distance of a line in surveying, is its length, estimated in a horizontal direction. It is generally ex¬ pressed either in chains and links, or in perches and tenths. 5. A Meridian or Meridian Line is any line that runs due north or south. Note.—All the meridians passing through any survey of moderate extent may be considered as straight lines, parallel to one another.*

6. The Bearing or Course of a line, i which it makes, with a meridian passing through . one end | and it is reckoned from the North or South Points of the horizon towards the East or West Points. Fig. 77.

%

Thus, supposing the line NS, Fig. 77, to be a meridian, and the angle SAB to be 50°; then the hearing of AB from the point A, is 50° to the east of south; which is usually ex¬ pressed thus: S. 50° E, and read, south, fifty degrees east.

7. T • the bearing taken from the other end of the line.

Note.—The bearing and the reverse bearing of a line, are angles of the same magnitude,f but lying between * The meridians are, in reality, curve lines which meet in the north and south poles of the earth. No two of them are therefore exactly parallel; but in usual surveys their deviation from parallelism is so very small, that there is no sensible error in considering them so. f As the meridians are not exactly parallel, this is not strictly true, except in a few cases ; but the difference is too small to be observed in practice. In

CHAP. 1.]

DIMENSIONS OF A SURVEY.

directly opposite points. Thus, if the bearing of AB, from the end A, is S. 50° E., the bearing of the same line from the end B, is N. 50° W. 8. A Circumferentor or Surveyor’s Compass, is an in¬ strument used to take the bearings of lines. The circumference of its face is divided into degrees, and in some of the larger ones into half degrees, in such manner that two opposite points may be exactly in the direction of the sights with which the instrument is fur¬ nished. These points are the north and south points of the instrument. Midway between them, on the circum¬ ference, are the east and west points. The degrees are numbered from 0° to 90°, each way from the north and south points to the east and west ones. In the centre of the face is a pin, finely pointed, which supports a Magnetic Needle, moving freely within the instrument. The instrument, when used, is placed on a staff, having a pointed iron at the bottom, and a ball and socket at the top. The Chain and Compass are the instruments with which the dimensions of surveys in this country are generally taken. It is important to have them accurately made. In the selection of a compass, particular atten¬ tion should be, directed to the settling of the needle. If, when the needle has been moved out of its natural posi¬ tion, it settles very soon, it is defective; either its mag¬ netic virtue is weak, or it does not move with sufficient freedom on the pin. 9. The Difference of Latitude, or the Northing or the latitude of Philadelphia the greatest difference between the bearing and reverse bearing of a line, a mile in length, is only 44". In higher latitudes the difference is greater.

8

86

DIMENSIONS OF A SURVEY. [CHAP. I.

Southmg of a line, is the distance that one end is further north or south than the other end ; or it is the distance which is intercepted on a meridian passing through one end, between this end and a perpendicular to the meri¬ dian, from the other end. Thus, if NS, Fig. 77, be a meridian passing through the end A, of the line AB, and B6 be perpendicular to NS, then is Ab the difference of latitude or southing of AB. 10. The Departure or the Easting or Westing of a line is the distance that one end is further east or west than the other end; or it is the distance from one end, perpen¬ dicular to a meridian passing through the other end. Thus B5, Fig. 77, is the departure or easting of the line AB. But if ns be a meridian, and AC perpendicular to it, and if the bearing of the line be taken from B to A, then is BC the difference of latitude or northing, and AC the departure or westing, of the line AB. Note.—It is evident from the definitions, that the Dis¬ tance, Difference of Latitude, and the Departure form the sides of a right-angled triangle ; in which, considering the departure as the base, the perpendicular is the differ¬ ence of latitude, the hypothenuse is the distance, and the angle at the perpendicular is the bearing. 11. The Meridian Distance of any station, is its dis¬ tance from a meridian passing through the first station of the survey, or any other assumed point. 12. The 'Traverse Table, is a table containing the dif-

CHAP. I.] DIMENSIONS OF A SURVEY.

87

Terences of latitude and the departures, computed to dif¬ ferent courses and distances. 13. The Area or Content of a tract of land is the hori¬ zontal surface included within its boundaries, expressed in known measures, as Acres, Roods, and Perches. 14. In Ogoing round a tract of lando and returning to o the place of beginning, it is evident that the whole northing which has been made, must be equal to the southing, and the easting to the westing; or in other words, that the sum of all the northings must be equal to that of the southings, and the sum of the eastings, to that of the* westings. This principle enables us to judge of the accuracy of a survey, when the bearings and distances of all the sides have been taken. If the sums of the computed north¬ ings and southings are equal, and also those of the east¬ ings and westings ; or, if, though not exactly equal, they are very nearly so, we may conclude that the survey has been correctly made; as very small differences in these sums may be imputed to little, unavoidable errors in taking the bearings and measuring the distances. But when the sum of the northings differs considerably from that of the southings, or that of the eastings from that of the westings, we must infer that an error has been made, too great to be admitted. In this case a re-survey should be taken. It is a rule with some of our best practical surveyors, that when the difference between the sums of the north¬ ings and southings, called the error in latitude, or that between the sums of the eastings and westings, called the, error in departure, exceeds one link for every five chains in the sum of the distances, a re-survey ought to be made.

DIMENSIONS OF A SURVEY. [cHAP. I.

When the errors in latitude and departure fall within the limits just mentioned, they should be properly appor¬ tioned among the several latitudes* and departures; we shall thus obtain what are called the corrected latitudes and departures. The method of doing this will be given in one of the following problems. PROBLEM I. To reduce two-pole chains and links to four-pole chains and links. RULE.

1. If the number of chains is even, divide it by 2, and to the quotient annex the given number of links. 2. If the number of chains is odd, divide by 2 as be¬ fore, for the chains; and for the 1 that is off, add 50 to the given number of links. EXAMPLES.

1. In 16 two-pole chains and 37 links, how many fourpole chains and links ? Ans. 8 ch. 37 links, or 8.37 ch. 2. How many four-pole chains and links are there in 17 two-pole chains and 42 links ? Ans. 8.92 ch. 3. IIow many four-pole chains and links are there in 22 two-pole chains and 7 links? Ans. 11.07 ch.

* In order to conciseness of expression, difference of latitude is frequently called simply, latitude.

CHAP.

l.J

DIMENSIONS OF A SURVEY.

89

PROBLEM II. To reduce two-pole chains and links to perches and hundredths of a perch. RULE.

Multiply the links by 4, for the hundredths, and the chains by 2, for the perches. If the hundredths exceed 100, set down the excess, and add 1 to the perches. Note.—This rule supposes the two-pole chain to be di¬ vided into 50 links. EXAMPLES.

1. Reduce 17 two-pole chains and 21 links to perches o.nd hundredths. Ans. 34.84 per. 2. Reduce 15 two-pole chains and 38 links to perches and hundredths. Ans. 31.52 per. 3. Reduce 57 two-pole chains and 49 links to perches and hundredths. Ans. 115.96 per. PROBLEM III. To reduce square four-pole chains to acres. RULE.

Divide by 10, and the quotient will be the acres. If there is a decimal in the quotient, multiply it,by 4, for the roods; and the decimal of these py 40, for the perches. 8*

M

90

DIMENSIONS OP A SURVEY. [cHAJ». I.

EXAMPLES.

1.

Reduce 523.2791 square chains to ac 10)523.2791 52.32791 4 1.31164 40 12.46560 Ans. 52 ac. 1 r. 12 p,

2. Reduce 41.9682 square chains to acres. Ans. 4 ac. 0 r. 31 p. 3. Reduce 132.925 square chains to acres. Ans. 13 ac. 1 r. 6.8 p. PROBLEM IV. To reduce acres, roods and perches to square chains. RULE.

Divide the perches by 40 and prefix the roods; divide the result by 4 and prefix the acres; then this latter re¬ sult, multiplied by 10, will give the square chains. Or reduce the given quantity to perches and divide 16. EXAMPLES.

1. Reduce 13 ac. 1 r. 10jo. to square chains. 40)10 4)1.25 13.3125 ^4=133.125 sq. ch.

CHAP. I.] DIMENSIONS OF A SURVEY. 91

2. Reduce 127 etc. 3r. 23 p. to square chains. Ans. 1278.9375 sq. ch. 3. Reduce 35 ac. 0 r. 20 p. to square chains. Ans. 351.25 sq. ch. PROBLEM V. To find the hearing of a line. 1. Let a stake of six or eight feet in length be set up perpendicularly, at the far end of the line. Set up the compass staff perpendicularly, at the beginning of the line, and placing the compass on the staffs adjust it to a horizontal position; the ball and socket admitting a motion for that purpose. ,This position can be deter¬ mined with sufficient accuracy, by observing whether, when the compass is turned round, the ends of the needle remain at the same height above the face of the instrument. 2. Turn the compass round so as to bring the south end of it towards the stake at the far end of the line. Then applying the eye to the sight at the north *end, move the compass gently round till the stake can be seen through the fine slits in both sights, and let it re¬ main in this position. 3. When the needle has settled, observe the number of degrees and parts of a degree, that are intercepted between the south end of the needle and the north or south point of the compass, to whichever it is nearest; which will be the bearing, reckoning it from that point, towards the east if the south end of the needle is to the right hand, but towards the west if it is to the left hand, Note 1.—The bearing thus obtained may be, and should be, verified by going, to the far end of the line,

92

DIMENSIONS OF A SURVEY. [CHAP. I.

and from thence taking the bearing to the first end; which, if both bearings are correct will be the reverse of the former. Note 2.—When there is a fence on the side, or other obstacle in the way, preventing the stake at one end from being seen through the compass sights at the other end, the bearing may be obtained by setting up the compass and stake at small equal distances to the right or left, so that the line joining them may be parallel to the side. Note 3.—The method of obtaining the bearing between two stations when there are obstacles in the way, which also prevent a parallel bearing being readily taken, or when the stations are too distant to be seen from each other, will be noticed in the next chapter. PROBLEM YI. To measure the distance of a line. For convenience in marking the termination of the chain in measuring, ten iron pins should be provided, about a foot in length, and terminated at top by a small ring, to which a piece of red flannel or other conspicuous substance should be tied, in order that the pins may be readily found, when set up among high grass or in other situations where they would not otherwise be easily discovered. Let the person who is to go foremost in carrying the chain, take nine of the pins in his left hand, and one end of the chain and the other pin in his right hand; then he moving on in the direction of the line, let another person take the other end of the chain and hold it at the begin¬ ning of the line. When the leader has moved on till the chain is stretched tight, he must set down the pin, per-

CHAP. I.] DIMENSIONS OF A SURVEY.

93

pendicularly, exactly at the end of the chain, the hinder chain-man taking care that the cjiain is in the direction of the line; which is readily determined by observing whether it is in a range with a stake previously set up at the far end of the line. When the leader has not his end of the chain in the direction of the line, the hinder chain-man can direct him which way to move, by a mo¬ tion of his left hand. When the distance of one chain or half chain* has been thus determined, the carriers, taking hold of the two’ends of the chain, move on till the hinder one comes to the pin which was set up by the other; then the chain being stretched, the person at the fore end of it sets up another pin as before ; the hinder chain-man then taking up the pin at his end, they proceed to a third distance of the chain; and so on. When the person at the fore end of the chain has set up all his pins, he still moves on another length of the chain, and then setting his foot on 4t to keep it in place, he cries “ out.'' .The hinder chain man then comes forward, and counts to him the ten pins; and he setting up one of them at the end of the chain, again moves on, drag¬ ging the chain after him, till he is checked by the hinder chain-man, who, getting the hind end of the chain, applies it as before to the pin set up. The number of outs should be carefully noticed ; each out being ten chains, when a four-pole chain is used, but only five, when the measuring is done, with a two-pole chain. When arri¬ ved at the end of the line, the number of pins, which the one at the fore end of the chain has set up since the last out, and the number of links from the last pin to the end of the line, must be carefully noted. From these, and

* When a two-polc chain is used, one length of it may properly bo called a half chain.

94

DIMENSIONS OF A SURVEY.

[CHAP.

I.

the number of outs, the distance measured is readily determined. ■4

All slant or inclined surfaces, as the sides of a hill, should be measured horizontally, and not on the plane or surface of the hill. To effect this, the hind end of the chain, in ascending a hill, should be raised from the ground till it is on a level with the fore end, and, by means of a plummet and line, or when the hill is not very steep, by estimation, should be liejd perpendicularly above the termination of the preceding chain. In descending a hill, the fore end of the chain should be raised in the same manner, and the plummet being suspended from it will show the commencement of the succeeding chain. PKOBLEM VII. To protract a Survey, having the hearings and distances of the sides given. The method of doing this will be best understood by an example. Thus, Suppose the following field notes to be given, it is required to protract the survey. Ch. 1. N. 50° E. 9.60 2. S. 32° E. 16.38 3. S. 41° W. 6.30 4. West 8.43 5. N. 79° W. 10.92 6. N. 5° E. 11.25 7. S. 83° E.' 6.48 Method 1st. Draw NS, Fig. 75, to represent a meridian line; then N stand¬ ing for the north and

CHAP. I.] DIMENSIONS OF A SURVEY.

95

S for the south, the east will be to the right hand, and the west to the left. In NS take any convenient point as A for the place of beginning, and apply the straight edge of the protractor to the line, with the centre to the point A, and the arch turned towards the east, be¬ cause the first bearing is easterly; then holding the protractor in this position, prick off 50° the first bear¬ ing, from the north end, because the bearing is from the north; through this point and the point A, draw the line AB on which lay 9.60 chains, the first distance from A to B. Now apply the centre of the protractor to the point B, with the arch turned toward the east, because the second bearing is easterly, and move it till the line AB produced, cuts the first bearing 50°; the straight edge of the protractor will then be parallel to the meridian NS; hold it in this position, and from the south eiid prick off the second bearing 32° ; draw BC and on it lay the second distance 16.38 chains. Pro¬ ceed in the same manner at each station, observing always, previous to pricking off the succeeding bearing, to have the arch of the protractor turned easterly or westerly, according to that bearing, and to have its straight edge parallel to the meridian; this last may always be done by applying the centre to the station point, and making the preceding distance line produced if necessary, cut the degrees of the preceding bearing. It may also be done by drawing a straight line through each station, parallel to the first meridian. When the survey is correct, and the protraction accurately performed, the end of the last distance will fall on the place of beginning. Method 2d. With the chord of 60° describe the circle NESW,

96

DIMENSIONS OF A SURVEY. [CHAP. I.

Fig. 76, and draw the diameter NS. Take the several bearings from the line of chords, and lay them off on the circumference from N or S according as the bearing is northerly or southerly, and towards E or W accord¬ ing as it is easterly or westerly, and number them 1,2, 3, 4, &c., as in the figure. From A, the centre of the circle, to 1 draw A 1, on which lay the first distance from A to B; parallel to A 2 draw BC, on which lay the second distance from B to C ; parallel to A 3 draw CD, on which lay the third distance from C to D; pro¬ ceed in the same manner with the other bearings and distances. Fig. 76.

2. The following field ‘notes are given, to protract the survey. Ch. 1. N. 15° 00' E. 20 2. N. 37° 30' E. 10 3. East 7.50 4. S. 11° 00' E. 12.50 5. South 13.50 6. West 10. 7. S. 36° 30' W. 10. 8. N. 38° 15'W. 8.50 EXAMPLE

PROBLEM VIII. The bearing of two lines from the same station being given, to find the angle contained between them. RULE.

When they run from the same point of the compass, towards the same point, sxibtract the less from the greater.

CHAP. I.] DIMENSIONS OF A SURVEY.

97

When they run from the same point, towards different points, add them together. When they run from different points, towards the same point, add them together, and take the supple¬ ment of the sum. When they run from different points, towards different points, subtract the less from the greater, and take the supplement of the remainder. Note.—When the bearing of one of the lines is given toicards the station, instead of from it, take the reverse bearing of such line; the angle may then be found by the above rule. EXAMPLES.

Fi

s-

67



1. Given the bearing of the line AB, Fig. 67, N. 34° E., and AD, N. 58° E.; required the angle A. AD, N. 58° E. AB, N. 34° E. Angle A = 24° 2. Given the bearing of BA, Fig. 57, S. 34° W., and BC, S. 35° E.; required the angle B. Am. B = 69°. 3. Given the bearing of BC, Fig. 67, S. 35° E., and CD, S. 87° W.; required the angle C. Ans. 58°. 4. Given the bearing of DC, Fig. 67, N. 87° E., and DA, S. 58° W.; required the angle D. Ans. 151°. 9

N

98

DIMENSIONS OF A SURVEY.

[CHAP.

I.

PROBLEM IX. To change the hearings of the sides of a survey in a cor¬ responding manner, so that any particular one of them may become a Meridian. RULE.

Subtract the bearing of the side that is to be made a meridian, from those bearings that are between the same points that it is, and also from those that are between points directly opposite to them. If it is greater than either of the bearings from which it is to be subtracted, take the difference, and change E. to W., or W. to E. Add the bearing of the side which is to be made a meridian, to those bearings which are neither between the same points that it is, nor between the points that are directly opposite to them. If either of the sums ex¬ ceeds 90°, take the supplement and change N. to S., or S. to N * Note.—When the bearings of some, or all, of the sides of a survey have been thus changed, and by calculation the changed bearing of another side or line has been © © * The changing of the hearings so as to make a given side become a meridian, may be illustrated by means of a protracted survey. If a pro¬ tracted survey or plot is held horizontally, with the meridian in a north and south direction, the north end being towards the north, the bearings of the sides of the plot will then correspond with the bearings of the sides of the survey. If then, keeping the paper horizontal, it be turned round till any particular side of the plot has a north and south direction, or be¬ comes a meridian, the bearings of all the other sides of the plot will have been changed by a like quantity. But it is evident, that neither the relation of the different parts of the plot to one another, the area nor the lengths of the sides will have been altered by this change. It may be here observed, that some calculations in surveying are considerably shortened by changing the bearings so as to make a certain side become a meridian. The method was communicated to me by Robert Patterson, late Professor of Mathematics and Natural Philosophy in the University of Pennsylvania.

CHAP. I.] DIMENSIONS OF A SURVEY.

99

found, its true bearing will be obtained by applying to the changed bearing, the bearing of the side which was made a meridian, in a contrary manner to what is directed in the rule; that is, by adding in the case in which the rule directs to ‘subtract, and by subtract¬ ing in the case in which it directs to add. EXAMPLES.

1. Given the bearings of the sides of a survey as fol¬ low; 1st. S. 454° W.; 2d. N. 50° W. j 3d. North; 4th. N. 85° E.; 6th. S. 47° E.; 6th. S. 204° W.; and 7th. N. 51 i° W. Required the changed bearings, so that the 5th side may be a meridian. 1st. S. 454° W. 47 924 180 chang. bear. N. 874° W. 2d. N. 50° W. 47 chang. bear. N. 3° W. 3d. N. 0° E. 47 chang. bear. N. 47° E. 4th. N. 85° E. 47 132 180 chang. bear. S. 48° E.

100

DIMENSIONS OF A SURVEY.

[CHAP. 1.

5th. side, changed bearing, South. 6th.

S. 201° W. 47

chang. bear. S. 671 7th.

N. 51° W. 47

chang bear. N. 41 W. 2. Given the following bearings of the sides of a sur¬ vey ; 1st. S. 401° E.; 2d. N. 54° E.; 3d. N. 291° E.; 4th. N. 281° E.; 5th. N. 57° W.; and 6th. S. 47° W.; to find the changed bearings so that the 2d. side may be a me¬ ridian. Ans. 1st. N. 851° E.; 2d. North; 3d. N. 241° W.; 4th. N. 251° W.; 5th. S. 69° W.; 6th. S. 7° E. 3. Given the bearings as in the 1st. example ; viz. 1st. S.451°W.; 2d.N.50°W.; 3d.North; 4th.N.85°E.; 5th. S. 47° E.; 6th. S. 201° W; 7th. N. 511° W.; to find the changed bearings so that the 6th side may be a me¬ ridian. Ans. 1st. S. 25° W.; 2d. N. 701° W.; 3d. N. 201° W.; 4th N. 641° E.; 5th. S. 671° E.; 6th. South ; 7th. N. 713° W. PROBLEM X. Of the bearing, Distance, Difference of Latitude and De¬ parture, any two being given, to find the other two. RULE.

When the bearing and distance are given. As Rad.: cos. of bearing : : distance r dif. of latitude. Rad. : sin. of bearing : : distance; departure.

CHAP. I.] DIMENSIONS OF A SURVEY.

101

When the hearing and difference of latitude are given. As Rad.: sec. of bearing :: diff. lat.: distance. Rad.: tang, of bearing :: diff. lat.: departure. When the hearing and departure are given. As Rad.: cosec. of bearing :: departure : distance, • Rad.: cotang, of bearing :: departure : diff. lat. When the difference of latitude and the departure are given. As diff. lat.: departure :: rad. : tang, of bearing. Rad. : sec. of bearing :: diff. lat. : distance. When the distance and difference of latitude are given. *1

As Diff. lat.: distance : : rad.: sec. of bearing. Rad.: tang, of bearing : : diff. lat.: departure. When the distance and departure are given. As Distance : departure :: rad.: sin. of bearing. Rad.: cos. of bearing :: distance : diff. lat. Note.—It is evident the above proportions are the solu¬ tions of a right-angled triangle, having for its sides the distance, difference of latitude, and departure. EXAMPLES.

1 Given the bearing of a line, N. 53° 20* E., distance 13.25 ch.; to find the difference of latitude and the de¬ parture. Ans. Diff. lat. 7.91 N.: dcp. 10.63 E. 9

*

102

DIMENSIONS OF A SURVEY.

[CHAP. 1.

2. Given the bearing of a line, S. 32° 30 departure 10.96 ch. to find the distance and difference of latitude. Ans. Dist. 20.40 ch.; diff. lat. 17.20 S. 3. Given the distance of a line, running between the north and east, 44 ch. and its difference of latitude 34.43 ch.; to find the bearing and departure. Ans. Bearing, N. 38° 30' E.; dep.‘ 27.39 ch. E. 4. The bearing of a line S. 32° 30' E., and the differ¬ ence of latitude 17.21 ch. being given, to find the dis¬ tance and departure. Ans. Dist. 20.41 ch.; dep. 10.96 E. 5. Given the difference of latitude of a line 27.92 N., and the departure 5.32 E.; to find the bearing and dis¬ tance. Ans. Bearing, N. 10° 47' E.; dist. 28.42. 6. The distance of a line, running between the north and west, is 35.35 ch., and its departure 15.08 ch., re¬ quired the bearing and difference of latitude. Ans. Bearing N. 25° 15' W.; diff. lat. 31.97 N. PROBLEM XI. To find the difference of latitude and departure correspond¬ ing to any given hearing and distance, by means of the Traverse Table. RULE.

When the distance is any number of whole chains or perches, not exceeding 100. Find the given bearing at the top or bottom of the table, according as it is less or more than 45°. Then against

CHAP.

I.J

DIMENSIONS OF A SURVEY.

103

the given distance, found in the column of distances at the side of the table, and under or over the given bearing, is the difference of latitude and departure ; which must be taken as marked at the top of the table, when the bearing is at the top; but as marked at the bottom, when the bearing is at the bottom. When the distance is a number of whole chains or perches, exceeding 100. Separate the distance into parts that shall not exceed 100 each; and find, as before, the difference of latitude and departure, corresponding to the given bearing and to each of those parts; the sums of these will be the dif ference of latitude and departure required. When the distance is expressed by chains or perches and

Find, as at corresponding to the givdn bearing and to the whole chain or perches. Then considering the decimal part as a whole number, find the difference of latitude and de¬ parture corresponding to it, and remove the decimal point in each of them, two figures to the left hand if there are two decimal figures in the distance, or one figure to the left if there is but one; then these added to the former will give the difference of latitude and departure required. Note. When the number of whole chains or perches is less than 10, and the second decimal figure is a cipher, the difference of latitude and departure may Ire taken out at once, by considering the mixed number, rejecting

104

DIMENSIONS OP A SURVEY.

[CHAP. I.

the cipher, as a whole number. The difference of lati¬ tude and departure thus found, must have the decimal point in each, removed one figure to the left hand. EXAMPLES.

1. Given the bearing of a line S. 351° E., dist. 79 ch.; required the difference of latitude’ and departure by the traverse table. Ans. Diff. lat. 64.51 S.; dep. 45.59 E. 2. A line bears N. 201° E., 117 ch.; required the dif¬ ference of latitude and the departure. Dist. 100, gives diff. lat. 93.67 and dep. 35.02 17 15.92 5.95 Whole dist. 117

diff lat. 109.59 N.

dep. 40.97 E.

3. Required the difference of latitude parture of a line which bears, S. 41 J° W., 57.36 ch. Dist. 57.00 gives diff. lat. 42.53 and dep. 37.96 36 .27 .24 Whole dist. 57.36

diff. lat. 42.80 S.

dep. 38.20 W.

4. Required the difference of latitude an of a line which bears, N. 72° W., 124.37 ch. Dist. 100.0 gives diff. lat. 30.90 and dep. 95.11 24.00 7.42 22.83 .37 .11 .35 Whole dist. 124.37

diff. lat. 38.43 N. dep. 118.29 W.

•CHAP. I.] DIMENSIONS OF A SURVEY.

105

5. Given the bearing and distance of a line, N. 391° W. 15.20 ch., to find its difference of latitude and de¬ parture. Ans. Diff. lat. 11.72 N., and dep. 9.67 \V. 6. The bearing and distance of a line are N. 46° E., 27.25 ch.; required its difference of latitude and de¬ parture. Ans. Diff. lat. 18.93 N. and dep. 19.60 E. 7. The bearing and distance of a line are S. 37£° W., 137.50 ch.; required its difference of latitude and de¬ parture. Ans. Diff. lat. 109.45 S., and dep. 83.23 W. 8. Required ^he difference of latitude and departure of a line, whose bearing and distance are S. 6i° E., 5.60 ch. Ans. Diff. lat. 5.56 S., and dep. 0.63 E. PROBLEM XII. Given the heatings and distances of all the sides of a tract of land to obtain the corrected latitudes and departures. RULE.

1. Rule a table as in the annexed example, in the first vertical column of which, place the letters designating the sides, or the numbers denoting the stations at the be¬ ginning of each side ; in the second column, place the bearings ; and, in the third, the distances. 2. 'Find, by the last problem, the difference of latitude and the departure, corresponding to each side, and place them in the next four columns, under their proper heads of N. or S., E. or W. Add up the northings and south¬ ings ; and if the sums are not equal, find their difference 5 O

106

DIMENSIONS OP A SURVEY. [CHAP.

f.

which will be the error of the survey in difference of latitude; which call by the same name as the least sum. Proceed in the same manner with the eastings and west¬ ings, and find the error in departure. Also add up the column of distances. Then it will be, As the sum of the distances, Is to any particular distance, So is the error in lati ude or departure To the correction of latitude or departure, correspond¬ ing to that distance. 3. Find, by the above proportion, the corrections of latitude and departure corresponding 'to all the sides calculating them to the nearest two decimal figures, and place them in the next two columns, heading them with the same names as the errors in latitude and departure. If the sums of these corrections, are not respectively equal to the errors in latitude and departure, which, in consequence of the fractions neglected, will sometimes be the case, alter some of them by a unit in the second decimal figure, to make them so. 4. Apply these corrections to their corresponding dif¬ ferences of latitude and departures, by adding when of the same name, but by subtracting when of different names, and the corrected differences of latitude and de¬ partures will be obtained; wdiich may be placed in the four succeeding columns. In these the sums of the northings and southings will be equal, and also those of the eastings and westings.* * The directions given in the rule, for correcting the errors in difference oflatitude and departure, are deduced from the rule given and demonstrated in No. 4, of a periodical work, called the Analyst, by Nathaniel Bowdilch, A. M., and also by the editor, Professor Adrain. The demonstration is too long, and not of a nature for insertion here.

CHAP. I.] DIMENSIONS OF A SURVEY.

101

Note 1.—In the proportion for finding the correction of the latitude or departure, the decimal parts of the sum of the distances and of the particular distance may be omitted, taking, in each case, the nearest number of whole chains. 2. The corrections may be frequently estimated with sufficient accuracy without the trouble of working out the proportions. 3. When one or two of the sides are hilly, or when there are other difficulties in the way of obtaining their bearing or distances with accuracy, it is better to allow a considerable part of the errors, on the latitudes and de¬ partures corresponding to them, and afterwards to ap¬ portion the remaining part among the others. EXAMPLES.

1. Given the bearings and distances of the sides of a tract of land as follow : 1st. S. 404° E. 31.80 ch.; 2nd. N. 54° E. 2.08 ch.; 3rd. N. 29?° E. 2.21 ch.; 4th. N. 28^E. 35.35 ch.; 5th. N. 57° W. 21.10 ch.; 6th. S. 47° W. 31.30 ch. Required the corrected differences of latitude and departures. Sta. Courses.

Cor. Cor. Dist. Ch. N. L. S. L. E. D. W.D. S. E. N. L. s. L. |E. D. W.D.

1 S. 40£ E. 31.80

24.18 20.65

24.21 20.70

.03 .05

t

2.08 1. 23

1.68

.00 .00 1.23

1.68

3 N. 29JE. 2.21 1.92

1.08

.00

1.92

1.08

4 N. 28£ E. 35.35 31.00

17.00

.04 .05 30.96

2 N. 54 E.

<=> 1 ° i

5 N. 57 W. 21.10 11.49 6 8.47. W. 31.30

17.05 1 1

17.69 .02 .03 11.47 21.34

22.89 .03 .04

21.37

17.66 22.85

123.84 45.64 45.52 40.41 40.58 .12 .17 45.58 45.58j40.51 40.51 45.52 40.41 12 Er. s.

.17 Er. E.

108 As 124 124 24 124 124

DIMENSIONS OF A SURVEY. 32 : : : 2 :: : 35 :;: : 21 : : : 31 :::

.12 .12 .12 .12 .12

:: : : :

.03 As 124 : 32 .00 124 :: 2 .03 or .04 124 ;: 35 .02 124 :: 21 .03 124 :: 31

: : : : :

[CHAP* I.

.17 .17 .17 .17 .17

: : : : :

.04 .00 .05 .03 .04

2. Given the bearings and distances of the sides of a tract of land as follow: 1st. N. 75° E. 13.70 ch.; 2d. N. 204 E. 10.30 ch.; 3d. East 16.20 ch.; 4th. S. 334° W. 35.30 ch.; 5th. S. 76 W. 16 ch.; 6th. North 9 ch.; 7th. S. 84° W. 11.60 ch.; 8th. N. 534° W. 11.60 ch.; 9th. N. 361° E. 19.36 ch.; 10th. N. 224° E. 14 ch.; 11th. S. 764° E. 12 ch.; 12th. S. 15° W. 10.85 ch.; 13th. S. 18° W. 10.62 ch.; to the place of beginning. Required the corrected latitudes and departures. Ans. 1st. 3.56 N. 13.26 E.; 2d. 9.66 N. 3.62 E.; 3d. 0.02 N. 16.22 E.; 4th. 29.39 S. 19.44 W.; 5th. 3.85 S. 15.50 W.; 6th. 9.01 N. 0.01 E.; 7th. 1.19 S. 11.52 W.; 8th. 6.96 N. 9.27 W.; 9th. 15.54 N. 11.61 E.; 10th. 12.95 N. 5.38 E.; 11th. 2.73 S. 11.70 E.; 12th. 10.46 S. 2.80 W.; 13th, 10.08 S. 3.27 W.

* When, as in this case, the correction is found to be nearly midway be¬ tween two numbers, it is best to note them both. Then, if in using the one that is nearest to the true value, the sum of the corrections does not equal the whole error, the other should be taken.

CHAP.

II.J

109

SUPPLYING OMISSIONS.

CHAPTER II. On supplying omissions in the dimensions of a survey. When the bearings and distances of all the sides of a survey are known, except one bearing and one distance, or two bearings, or two distances, these can be obtained rby calculation, provided those that are known can be depended on, as sufficiently accurate. This may some¬ times be necessary when there are obstacles in the way of obtaining one or two of the bearings or distances: or when, after they have all been taken on the ground, the notes of one or two of them happen to be obliterated. As, however a bearing, or distance thus obtained, must be affected by any error or errors that may have been made in taking the others, it is better, when practicable, to have the bearings and distances of all the sides, as taken on the ground. PROBLEM I. The hearings and distances of all the sides of a tract of land, except the bearing and distance of one side, being given, to find these. RULE.

Find by prob. 11, of the preceding chapter, the differ¬ ences of latitude and the departures for the sides whose bearings and distances are given, and place them in their proper columns in a table ruled for the purpose: Add up the northings and southings, and taking the dif10



°

110

SUPPLYING OMISSIONS. [CHAP. II.

ference of their sums, place it opposite the unknown side, in the column whose sum is the least. The sums of the two columns will then be equal. This is called balancing the latitudes. Do the same with the eastings and westings. The two numbers inserted to make the latitudes and the departures balance, will be the differ¬ ence of latitude and the departure of the unknown side: with which its bearing and distance may be found, by prob. 10, of the preceding chapter. • Note 1.—By the application of this rule, the bearing and distance of a line joining two corners or stations, may be found, when there are obstacles in the way which prevent our going directly from one corner to the other, or when one cannot be seen from the other. To do this, let one or two, or more stations, if necessary, be taken out of the line, and take the bearing and distance from the first corner to the first assumed station ; from this station to the second; and so on, to the second corner. Then considering these bearings and distances, as the bearings and distances of the sides of a survey, the re¬ quired bearing and distance of the line may be found by the above rule. The bearing thus found must be revers¬ ed, in order to have the bearing from the first corner to the second. 2. In the same way the bearing and distance of a straight road to run between two given places, may be found, by taking the several bearings and distances of the old road if there is one; or of lines joining assumed stations and extending from one of the places to the other. EXAMPLES.

1. The bearings and distances of the side of a tract of land, except the bearing and distance of one side which

CHAP. II.J

111

SUPPLYING OMISSIONS.

are not known, arg as in the following field-notes; re¬ quired the unknown bearing and distance. Chains.

Chains.

1. S. 45°* W. 15.16 2. N. 50° \V. 22.10 3. North 18.83 4. N. 85° E. 35.65 Sta.

Bearings.

5. 6. S. 201 W°. 23.80 7 N. 51* W°. 26.47

Dist.

S.

N.

l S. 451 W. 15.16

E.

10.62

10.81

2 N. 50 W. 22.10 14.20 3 4

North

w.

16.93

18.83 18.83

N. 85 E. 35.65 3.11

35.52

V

5

(19.79) (21.20)

6 S. 201 W. 23.80

22.29

8.33

7 N. 5li W. 26.47 16.56

20.65 0

52.70 52.70

56.72 56.72

As diff. oflat. * 19.79 S. Ar. Co. 8.70355 : dep. 21.20 E. - - - 1.32674 :: rad. - - 10.00000 : tang. bear. S. 47°

E.

-

-

- 10.03029

As rad. 10.00000 : sec. bearing 47° 10.16622 :: diff. lat. 19.79 - - - - 1.29645 : dist. -

- 29.02

-

-

-

-

Ans. S. 47° E. 29.02 ch.

1.46267

112

SUPPLYING OMISSIONS.

[CHAP. II.

2. Given the bearings and distances of the sides of a tract of land, as follow: 1st. N. 155 W 9.40 ch.; 2dN. 631 E. 10.43 ch.; 3d. S. 49° E. 8.12 ch.; 4th. S. 13* E. 8.45 ch.; 5th. S. 165 E. 6.44 ch.; 6th. Unknown ; 7th. N. 60° W. 9.72 ch.; 8th. N. 17i E. 7.65 ch.; required the bearing and distance of the 6th. side. Ans. S. 60° 8' W. 12.27 ch. 3. One side of a tract of land of which a survey is to be taken, passes through a pond. Two stations are therefore taken on one side of the pond as represented in Fig. 80. The bearings and distances from the first end of the side to the first station, from that to the second, and thence to the other end of the side are; 1st. S. 52° W. 10.70 ch.; 2d. S. 7*° W. 13.92 ch.; and 3d. S. 34j E. 9 ch. Required the bearing and distance of the side. Ans. S. 10° 33' W. 28.31 ch. 4. Given the bearings and distances of an old road, running between two places, as follow; 1st. S. 10° E. 92.20 ch.; 2d.* S. 15° W. 120.50 ch.; 3d.. S. 18* VV. 205. ch.; 4th. S. 71* E. 68 ch. Required the bearing and distance of a straight road, that shall connect the two places. Ans. S. 2° 8' W. 423.47 ch. PROBLEM II. Given all the hearings and distances of the sides of a sur¬ vey, except the distances of two sides, to find these. RULE.

, By prob. 9, of the preceding chapter, change all the given bearings, in a corresponding manner, so that one of the sides whose bearings only are given, may become a meridian. With the changed bearings and given dis¬ tances find the corresponding differences of latitude, and the departures. Add up the eastings and westings, and take the difference of their sums, which will be the

CHAP. II.]

113

SUPPLYING OMISSIONS*

departure of that unknown side, which is not made a me¬ ridian. With this departure and the changed bearing, find by prob. 10, of the preceding chapter, the distance and difference of latitude of this side, which place in their proper columns. Now add up the northings and southings, and take their difference, which will be the distance of the side made a meridian.* EXAMPLES.

Given the following bearings and distances of the sides of a survey; 1st. S. 451 W. 15.16 ch.; 2d. N. 50° W. 22.10 ch. j 3d. North 18.83 ch. ; 4th. N. 85° E. 35.65 ch.; 5th. S. 47° E. dist. unknown; 6th. S. 20J W. disk unknown; 7th. N. 511 W. 26.47 ch. to the place of be¬ ginning. Required the unknown distances. Changed bearings.

Dist.

N.

S. 45i° W.

N. 87^° W.

15.16

0.66

2

N. 50 W.

N. 3 W.

22.10 22.07

3

North

N. 47 E.

18.83

4

N. 85 E.

S. 48 E.

35.65

5

S. 47 E.

South

(29.02)

(29.02)

6

s. 2(q w.

S. 67-1

! (23.80)

(9.11)

7

N. 51J W.

N. 4J W.

Sta.

Bearings.

1

w.

E.

W. 15.15 I» 1.16

12.85

13.77 23.85 26.49

26.47 26.40

I

S.

(21.99) 1.96

61.98 61.98 40.26 40.26

* The reason of the rule is obvious. For as the side made a meridian has no departure, the difference of the sums of the departures, must be the de¬ parture of the other unknown side. And when the difference of latitude of this side has been found and placed in its proper situation, the difference of the sums of the latitudes must evidently be the difference of latitude of the side made a meridian ; or which, in this case, is the same thing*, its distance.

10*

P

114

SUPPLYING OMISSIONS.

[CHAP. II.

As rad. 10.00000 s cosec. chang. bearing 674° - - 10.03438 :: dep. 2l’.99 L.34223 :

Dist. 6th side -

- 23.80 - 1.37661

As rad. - - - -- -- -- 10.00000 : cotang, chang. bearing 674 - - 9.61722 :: dep. ----- 21.99 - 1.34223 :

diff. lat. 6th. side - - 9.11 - 0.95945 Ans. 5th. side 29.02 ch. and 6th. side 23.80 ch.

2. Given the bearings and distances of a tract of land as follow: 1st. S. 404 E. 31.80 ch.; 2d. N. 54° E. dist. unknown; 3d. N. 294 E. 2.21 ch.; 4th. N. 284 E. 35.35 ch.; 5th. N. 57° W. dist. unknown; 6th. S. 47° W. 31.30 ch.; to the place of beginning. Required the distances of the 2d. and 5th. sides. Ans. 2d. side. 2.08 ch. and 5th. side 20.90 ch. PROBLEM III. Given the hearings and distances of all the sides of a sur¬ vey except two ; one of which has only its hearing given, and the other, the distance and the points of the compass between which it runs ; to find the unknown hearing and distance. RULE.

As in the last problem, change all the given bearings, so that the side whose bearing only is given, may be¬ come a meridian. Find the differences of latitude and the departures, corresponding to the changed bearings and the given distances. Take the difference of the sums of the eastings and westings, which will be the de-

CHAP. II.]

SUPPLYING OMISSIONS.

115

parture of the side whose bearing is not given With the given distance and this departure, find by chap. 1. prob. 10. the changed bearing and difference of latitude, and place them in their proper columns. From the changed bearing, the true bearing may be readily found by note to prob. 9, chap. 1. Lastly, take the difference of the sums of the northings and southings, and it will be the distance of the side, changed to a meridian. Note.—The changed bearing as found by the rule, must be reckoned from the north, or the south point of the compass, according as the one, or the other, will ren¬ der the true bearing when found from it, conformable to the given points. The point from which the changed bearing must be reckoned determines also the column in which the difference of latitude must be placed. Some¬ times the changed bearing when reckoned from either north or south, will render the true bearing conformable to the given points. In such cases, there are two differ¬ ent bearings and distances that will answer the condi¬ tions of the problem ; and we can only know which of them is the right one by previously knowing the required bearing nearly. EXAMPLES.

1. Given the bearings and distances of a survey as fol¬ low : 1st. S. unknown W. 15.16 ch.; 2d. N. 50° W. 22.10 ch.; 3d. N. 18.83 ch.; 4th. N. 85° E. 35.65 ch.; 5th. S. 47° E. 29.02 ch.; 6th. S. 20$ W. dist. unknown; 7th. N. 51 i° W. 26.47 ch. Required the unknown bear¬ ing and distance.

116

SUPPLYING OMISSIONS.

Sta.

Changed bearings.

N.

Dist.

rt*

j

8.

i

13.73

S. (45° 36') W. (S. 25° 6' W.) 15.16

CD,

CO /—\ 1

Bearings.

[CHAP. II.

2

N. 50 W.„

N. 70^ W.

22.10

3

North

N. 20\ W.

18.83 17.64

4

N. 85 E.

N. 64} E.

35.65 15.35

5

S. 47 E.

S. 674 E-

29.02,

6

S. 20J W.

South

(23.81)

7

N. 51} W.

N. 71} W.

26.47

E.

7.37 |

w.

!

20.83

j

j

6.59 | 32.18

1 1

11.11 26.81 (23.81) 8.29

25.14

48.65 48.65 58.99 58.99

As dist. 1st, side 15.16 Ar. Co. 8.81930 : dep. do. 6.43 0.80821 :: rad. - -- -- -- -- 10.00000 : sin. chan. bear. 25° 6'

- - - 9.62751

As rad. - -- -- -- -- 10.00000 : cos. chang. bearing 25° 6' - - 9.95692 :: dist. 15.16 - - 1.18070 :

diff. lat. - - - - 13.73 - - 1.13762 Ans. 1st. S. 45° 36' W.; 6th, 23.81 ch.

2. Given the following bearings and distances of a survey; 1st. S. 40i° E. 31.80 ch.; 2d. N. 54^ E. dist. unhnoion 3rd. N. 291° E. 2.21 ch.; 4th. N. unknown E. 35.35 ch.; 5th. N..57° W. 20.90 ch.; 6th. S. 47° W. 31.30 ch.; to place of beginning. Required the bearing of the 4th. side and distance of the 2d. side. Ans. Bearing of 4th. side N. 281° E.; dist. of 2d. side, 2.09 ch.

CHAP. II.]

SUPPLYING OMISSIONS.

117

PROBLEM IV. Given all the hearings and distances of the sides of a tract of land, except the bearings of two sides, to find these hearings. RULE.

1. Find the difference of latitude and the departure of each side, whose bearing and distance are both given. Take the difference of the sums of the northings and southings of these sides, and alsos the difference of the sums of the eastings and westings. These differences will be the difference of latitude, and the departure of a line, which, with those sides, would form a closing survey; and which may therefore be called a closing line. 2. With the difference of latitude and departure of the closing line, find, by prob. 10. chap. 1, its bearing and distance. Take the closing line and the two sides whose bearings are not given, for the three sides of a triangle, and calculate the angles. 3. To the bearing of thevclosing line, apply, by addi¬ tion or subtraction, as the case may require, the angle contained between it, and the side which is the one coming first in the order of the survey; and it will give the bearing of that side. Then to the reverse bearing of that side, apply in a proper manner, the angle con¬ tained between the two sides which are sides of the sur¬ vey, and it will give the bearing of the second of those sides.* * It is easy to see the reason of the rule, by considering that the two sides whose beatings arc not given, being made to form with the closing line, the

118

SUPPLYING OMISSIONS.

[CHAP.

II.

EXAMPLES.

1. Given the bearings and distances of the sides of a tract of land as follow : 1st. S. unknown W. 15.16 ch.; 2d. N. 50° W. 22.10ch.; 3d. North 18.83 ch.; 4th. N. 85° E. 35.65 ch.; 5th. S. unknown E. 29.02 ch.; 6th S. 205° W. 23.80 ch.; and 7th. N. 511 W. 26.47. ch. Re¬ quired the unknown bearings. Sta*

l 2

3

Bearings.

S.

Disk

E.

w.

W. 15.16

N. 50° W. 22.10 14.20 North

16.93

18.83 18.83

4

N. 85 E. 35.65 3.11

5

S.

6

s.

N.

j

35.52

E. 29.02 22.29

S. 20i W. 23.80

7 N. 511 W. 26.47 16.56

8.33 20.65

52.70 22.29 35.52 45.91 22.29 35.52 30.41 S.

10.39 E.

sides of a triangle, the sum or difference of their differences of latitude, will necessarily be equal to the difference of latitude of the closing line; and that, therefore, their differences of latitude will he such as to make the sums of the northings and southings of the whole survey equal; and the same for the departures.

CHAP. II.] SUPPLYING OMISSIONS.

119

As diff.'lat. - - 30.41 S. Ar. Co. 8.51698 : dep. - 10.39 E. 1.01662 : ; rad. - -- -- -- -- -- 10.00000 : tang, of bear, of clos. line, S. 18° 52' 9.53360 As rad. - - - 1Q.00000 : diff. lat. - 30.41 - - 1.48302 : : sec. of bear, of clos. line - 18° 52' - 10.02398 dist. of clos. line

-

-

- 32.14 1.50700

Let DE, Fig. 78, represent the closing line, DF, the 1st side of the survey, and FE, the 5th side. Then, DE 32.14 Ar. Co. 8.81930 DF 15.16 FE 29.02 8.53730 2)76.32 Half sum 38.16 Eem.

log. 1.58161

6.02

— 0.77960

Cos. i F 43° 44'

2)19.71781

F 87 28 9.85890 As DE 32.14 : FE 29.02 : : sin. F 87° 28' sin. D 64° 26' DE, S. 18° 52' E. Angle D 64 26 1st side, S. 45 34 W.

Ar. Co. 8.49295 1.46270 9.99958 9.95523 FD, N. 45° 34' E. Angle F 87 28 133 2 180 00 5th side, S. 46 58 E.

120

CONTENT OF LAND.

[CHAP. III.

2. Given the bearings and distances of the sides of a tract of land as follow: 1st. S. unhwivn E. 31.80 ch.; 2d. N. 54° E. 2.08 ch.; 3d. N. 291 E. 2.21 ch.; 4th. N. 281° E. 35.35 ch.; 5th. N. 57° W. 20.90 ch.; and 6th. S unknown W. 31.30 ch. to the place of beginning. Re¬ quired the unknown bearings. Am. 1st. S. 40° 29' E.; and 6th. S. 47° W.

CHAPTER III. Problems for finding the Content of Land. When the sides of a survey are right lines, and all the bearings and distances are given, the area may be found by a problem that will be given in this chapter. If one or two of the bearings or distances are not known, they may be found by the problems in the last chapter. Al¬ though the problem alluded to, is general, and may be applied whatever number of sides there may be, yet there are some particular rules for finding the areas of triangles and quadrilaterals, which are often useful. These, and also rules for finding the areas of circles and ellipses, are given in the first part of the chapter. When a part of the boundary of a tract of land, is irregular, as is frequently the case, if one or more of the sides are bounded by water, it is sometimes very trouble¬ some to take all the bearings and distances requisite to obtain the area with accuracy. In these cases, it is usual to run one or more straight lines, called stationary lines, near to such boundary, and so as to connect the straight sides of the survey. In measuring these sta¬ tionary lines, perpendicular distances are measured from them, to each bend in the irregular boundary. These perpendicular distances are called off-sets. The lengths

CHAP.

III.J

CONTENT OF LAND.

121

of the off-sets, and the distance of the foot of each, from the commencement of the stationary line, should be care¬ fully noted in the field book; observing also that such a number of off-sets^should be taken, that the part of the irregular boundary intercepted between each adjacent two, may, without material error, be considered a straight line. From these notes, the area or areas of the land contained between the stationary line or lines, and the irregular boundary, may readily be calculated. This area added to the area enclosed by the stationary lines, and straight sides of the survey, when they are on the outside of the stationary lines, or subtracted from it, when on the inside, will give the area of the survey. In those cases in which water is a boundary of a tract of land, if that water is a brook or rivulet, it is usual to consider a fine running through its middle as the true boundary; and the off-sets mast be measured accordingly. When tide water is the boundary, the land is considered as extending to the line of low water mark. t If the bearings of all the corners of a tract of land from two stations, taken either within or out of the tract are given, and also the bearing and distance of these stations from each other, the area may be calculated. It is however necessary, that the two stations should be so taken that they shall not be in a straight line, or very nearly in a straight line, with either of the corners of the land. This method of obtaining the area, though not practically so accurate as where the bearings and dis¬ tances of the sides are correctly given, may sometimes be found useful. Some surveyors, in order to calculate the area of a survey, first protract it; then dividing the plot into tri¬ ll Q

122

CONTENT OF LAND. [CHAP. III.

angles and trapeziums by lines joining opposite corners, they measure with the scale and dividers the lengths of such lines and perpendiculars as are requisite for calcu¬ lating the areas of these. The sum of the areas thus obtained, is the area of the survey. When the survey is carefully protracted, and proper attention is given to take the measures with the utmost precision, this method serves to give a near value of the content; but is by no means to be depended on as equally accurate with the general problems mentioned above. The area of a field or small tract of land, the corners of which can be seen from one another, may readily be found by means of the chain only. To do this, the lengths of the sides must be measured, and also the length of diagonals joining opposite corners, so as to divide the field into triangles. Or instead of the diagonals, the dis¬ tances from some assumed point within the field, to the several corners, may be used. Having then all the sides of the several triangles, the area of each may be found; and the sum of these areas will be the area of the tract. PROBLEM I. To find the area of a Parallelogram, whether it be a Square, a Rectangle, a Rhombus, or a Rhomboides. RULE.

Multiply the length by the height or perpendicular breadth, and the product will be the area.* Fig. 68. D

__ A1

I 11

I

11 1

* DEMONSTRATION. Let ABCD (Fig. 68) be C a rectangle; and let its length AB and CD, and its breadth AD and BC, be each divided into as many equal parts, as are expressed by the number of times they contain the lip lineal measuring unit; and let all the oppo¬ site points of division be connected by right

CHAP. I1I.J

CONTENT OF LAND.

±40

Note.—Because the length of a square is equal to its height, its area will be found by multiplying the side by itself. EXAMPLES.

1. Required the area of a square field, a side of which measures 7.29 four-pole chains. 7.29 Ch. 7.29 6561 1458 5103 10)53.1441 Area 5 A. 1 R. 10 P. 5.31441 4 1.25764 40 10.30560 2. Required the area of a rectangular field whose length is 13.75 chains, and breadth 9.5 chains. lines. Then, it is evident that these lines divide the rectangle into a num¬ ber of squares, each equal to the superficial measuring unit; and that the number of these squares is equal to the number of lineal measuring units in the length, as often repeated as there are lineal measuring units in the breadth, or height; that is, equal to the length multiplied by the breadth. But the area is equal to the number of squares or superficial measuring units; and therefore the area of a rectangle is equal to the product of the length and breadth. Again, a rectangle is equal to any oblique parallelogram of an equal length and perpendicular height (36.1;) therefore the area of every parallelogram is equal to the product of its length and height

124

CONTENT OF LAND.

[CHAP. UI.

13.75 Ch. 9.5 6875 12375 10)130.625 Area 13 A. OR. 10P 13.0625 4 .2500 40

10.0000

3. Required the area of a field, in the for boides, whose length AB is 42.5 perches, and perpen¬ dicular breadth CD is 32 perches. Fig. 15. 42.5 P. 32 850 1275 410)13610.0 *4)34 8 A. 2 R. 4. What is the area of a square tract of land whose side measures 176.4 perches? Ans. 194 A. 1 R. 36.96 P. 5. What is the area of a rectangular plantation whose length is 52.25 chains, and breadth 38.24 chains ? Ans. 199 A. 3 R. 8.6 P. 6. The length of a field, in the form of a rhombus, measures 16.54 chains, and the perpendicular breadth 12.37 chains: required the area. Ans. 20 A. 1 R. 33.6 P.

CHAP. III.]

CONTENT OP LAND.

125

7. Required the area of a field in the form of a rhomboides, whose length is 21.16 chains, and perpendicular breadth 11.32 chains. A?is. 23 A. 3R. 32.5 P. PROBLEM II. To find the area of a triangle when the base and perpen¬ dicular height are given. RULE.

Multiply the base by the perpendicular height, and half the product will be the area.# EXAMPLES.

1. The base AB of a triangular piece of ground, measures 12.38 chains, and the perpendicular CD 6.78 chains: required the area. Fig. 49. 12.38 Ch. 6.78 9904 8666 7428 2)83.9364 10)41.9682 Area, 4 A. OR. 31P. 4.19682 4 .78728 40 31.49120 * DEMONSTRATION. A triangle is half a parallelogram of the same base and altitude, (41.1) and therefore the truth of the rule is evident

11*

126

CONTENT OP LAND. [CHAP. 111.

2. Required the area of a triangular field, one side of which measures 18.37 chains, and the distance from this side to the opposite angle, 13.44 chains. Ans. 12 A. 1 R. 15 P. 3. What is the area of a triangle whose base is 49 perches and height 34 perches ? Ans. 5 A. OR. 33 P. PROBLEM III. To find the area of a triangle when two sides and their included angle are given. RULE.

As radius, Is to the sine of the included angle; So is the rectangle of the given sides, To double the area.* EXAMPLES.

1. In a triangular lot of ground ABC, the side AB measures 64 perches, the side AC 40.5 perches, and their contained angle CAB 30°: required the area. Fig. 49. * DEMONSTRATION. In the triangle ABC, Fig. 49, let AB and AC be the given sides, including the given angle A, and let CD be perpendicular on AB. Then by trig. rad.: sin. A :: AC : CD;but (cor. 1.6) AC : CD :: ACx AB: CDxAB; therefore (11.5) rad. : sin. A :: ACx AB : CDx AB; but CDX AB is equal to twice the area of the triangle : hence the truth of the rule is evident

CHAP. III.]

127

CONTENT OP LAND.

As radius - Is to sin. A, 30°

10.00000

-

9.69897 1.80618 1.60746

So is AB, AC

13.11261 To double the area 1296 perches

3.11261

410)641 4)16 8 4 A. OR. 8P. 2. What is the area of a triangle, two sides of which measure 15.36 chains and 11.46 chains respectively, and their included angle 47° 30' ? Ans. 6 4. 1 R. 38 P. 3. One side of a triangular field bears N. 12° E. dis tance 18.23 chains, and at the same station the other ad¬ jacent side bears N. 78° 30' E. distance 13.84 chains: required the area. Ans. 11 A. 2R. HP. 4. Required the area of a triangular piece of ground, one side of which bears N. 82° 30' W. dist. 19.74 chains, and at the same station the other adjacent side S. 24° 15 E. dist. 17.34 chains. Ans. 14 A. 2 R. 8 P. PROBLEM IV. To find the area of a triangle when one side and the two adjacent angles are given. RULE.

Subtract the sum of the two given angles from 180°; the remainder will be the angle opposite the given side. Then,

128

CONTENT OF LAND.

[CHAP. III.

As the rectangle of radius and the sine of the angle opposite the given side, Is to the rectangle of the sines of the other angles, So is the square of the given side, To double the area.* EXAMPLES.

1. In a triangular field ABC, the side AB measures 76 perches, the angle A 60°, and the angle B 50°: re¬ quired the area. Fig. 47. The angle ACB=180° — the sum of the angles A and B,=70°. rad. sin. C. 70° sin. A. 60° sin. B. 50° < AB 76 £ AB 76

As rad.xsin. C, : sin. A x sin. B,

:: AB3=ABx AB, : double area in perches

Ar. Co. 0.00000 Ar. Co. 0.02701 9.93753 9.88425 1.88081 1.88081

4078

3.61041

40)2039 4)50 39 12 A. 2R. 39 P. * DEMONSTRATION. Let AB, Fig. 49, be the given side of the triangle ABC, and A and B the given angles; also let CD be perpendicular on AB : Then by trig. sin. ACB : sin. B. : : AB : AC rad. : sin. A : : AC : CD. Therefore (23.6) rad. X sin. ACB : sin Ax sin. B :: ABxAC : CDx AC :: (Cor. 1.6) AB : CD : : AB-: ABxCD; but ABxCD is equal to double the area of the triangle ABC ; therefore (11.5) rad. X sin. ACB : sin. A X sin. B :: AB'^; double the area of the triangle ABC.

CONTENT OF LAND. CHAP. III.] CONTENT

«

129

2. One side of a triangle measures 24.32 chains, and the adjacent angles are 63° and 74° : required the area. Ans. 37 A. 0 R. 22 P. 3. What is the area of a triangular field, one side of which is 17.36 chains, and the adjacent angles 37° 30', and'48° 15'? Ans. 6 A. 3 R. 18 P. PROBLEM V. To find the area of a triangle when the three sides are given. RULE. %

From half the sum of the three sides subtract each side severally; multiply the half sum and the three re¬ mainders continually together, and the square root of the last product will be the area.* * DEMONSTRATION. Let ABC, Fig. 69, Fig. 69. be the triangle. Bisect any two of the A angles, BAC, ABC, by the straight lines AG, BG, meeting in G; let fall on the three sides of the triangle, the perpen¬ diculars GD, GF, GE, and join GC; also produce AB, AC, and bisect one of the exterior angles, IIBC, by the line BK, meeting AG* produced in K, join KC, and let fall the perpendiculars KH, KM, and KL. Then (26.1) AD is equal to AE and DG to GE; also BD is equal to BF, and DG to GF; hence GF and GE are equal, and consequently (47.1) CF is equal to CE. In like manner it may be proved that AII is equal to AL, BII to BM, and CM to CL; as likewise that KH, KM, and KL are equal to each other. Now since BII is equal to BM and CL to CM, it is manifest that AH and AL together, are equal to the sum :f the three sides AB, AC, and BC; hence AII or AL is equal to the semiperimeterof the triangle ABC. But since twice AD, twice BD, and twice CF are, together equal to the sum of the sides of the triangle, or twice AII, it is ob-

♦ The nngle BAC is less than the angle IIBC (16.1;) consequently BAG is less than IIBK, and BAG, KBA, are together less than IIBK, KBA; but HBK, KBA, aro together equal to two right angles; hence BAG, KBA, are less than two right angles j therefore (cor. 29.1) the line BK will meet the line AG produced.

R

130

CONTENT OF LAND.

[CHAP. HI

EXAMPLES.

1. Required the area of a triangular tract of land whose three sides are 49.00, 50.25 and 25.G9 chains. vious that AD, BD and CF together, are equal to AH; consequently CF is equal to BH or BM; hence CM or CL is equal to BF or BD* and therefore DII and BC are equal. Hence, if from the semiperimeter AH, the three sides AB, AC and BC be severally taken, the remainders will be BH, CL, (or BD) and AD re¬ spectively. Again, since the angles DBF and DGF are together equal to two right angles, as likewise DBF and FBH together equal to two right angles, it is manifest that the angle DGF is equal to the angle HBF; and the angle DGB to the angle HBK; the triangles DBG and HKB are therefore similar. Hence BD : DG :: KH : HB; also in the similar triangles ADG, AHK, AD : DG : : AH : HK; therefore (23.6) AD x BD : DG2: : AH : HB : : AH2: AH XHB. If therefore we take between AD and BD, and between AH and HB, the mean proportionals M and N respectively, the foregoing analogy will become 2 2 2 M2: DG :: AH : N ; hence (22.6) M : DG :: AH : N; consequently the rectangle M X N is equal to the rectangle AH x DG; therefore ABC=ABG-f

BCG+ACG=:AHxDG=MxN=>/(ADxBD)x v' (AHxHB)=V (AD X BD x HB X AH.)

131

CONTENT OF LAND.

CHAP. 111.]

49.00 50.25 25.69 124.94

Sum

log.

62.47 C 13.47 { 12.22 ( 36.78

Half sum Remainders

1.79567 1.12937 1 08707 1.56561 2)5.57772 2.78886

615 chains 61.5 Acres=61 A. 2R.

2. What is the area of a triangular field whose sides measure 10.64,12.28, and 9.00 chains ? Am. 4 A. 2 R. 26 P. 3. What quantity of land is contained in a triangle, the sides of which are 20, 30 and 40 chains ? Am. 29 A. OR. 7P PROBLEM VL To find the area of a trapezium, when one of the diagonals and the two perpendiculars, let fall on it from the op¬ posite angles, are given. RULE.

Multiply the sum of the perpendiculars by the diago¬ nal, and half the product will be the area.*

* DEMONSTRATION.

The area of the triangle ABC (Fig. 70) = ACXBF ; 2

and the area of the triangle ADC

ACXDE 2

therefore the sum of these

132

CONTENT OF LAND.

[CHAP. 1IT.

Note.—When all the sides and one of the diagonals are given, the trapezium will be divided into two trian¬ gles, the area of each of which may be found by the last problem. The sum of these areas will be the area of the trapezium. EXAMPLES. Fig. 70. B

1. I form of a trapezium, the diago¬ nal AC measures 20.64 chains, the perpendicular BF 6.96 chains, and DE 5.92 chains; required the area. Fig. 70. Ch. 6.96 5.92 12:88 20.64 5152 7728 2576 2)265.8432

132.9216 Ch.=13 A. 1R. 6 P. 2. Required the area of a trapezium whose diagonal measures 16.10 ch. and the perpendioulars 6.80 ch. and 3.40 ch. Aus. 8 A. OR. 331P. 3. The diagonal of a trapezium is 24 ch. and the perpendiculars are 8.27 ch., and 12.43 ch.; what is the area ? Am. 24 A. 3 R. 14 P. ACXBF areas, or the area of the trapezium ABCD=

x AC.

ACXDE

BF+DE

CHAP. III.]

CONTEXT OF LAND.

133

PROBLEM VII. To find the area of a trapezium, when all the angles and two opposite sides are given. Note.—When three of the angles are given, the fourth may he found, by subtracting their sum from 360°. RULE.

Consider one of the given sides and its adjacent an¬ gles, or their supplements when their sum exceeds 180°, .as the side and adjacent angles of a triangle, and find its double area by prob. 4. Proceed in the same man¬ ner with the other given side and its adjacent angles : Half the difference of the areas thus found will be the area of the trapezium.* EXAMPLES. Fig. 71. 1. In a four-sided field ABCD, there are given the following bearings and distances, viz. AB, N. 24° E. dist. 6.90 ch.; BC, N. 64° '40' E.; CD, S. 35° 20' E. dist. 11.50 ch.; and DA, S. 88° W.: required the area. Fig. 71.

From the given bearings, the angles may be found as follows: AD, N. 88° E. AB, N. 24 E. BAD =64°

CB, S. 64° 40' W. CD, S. 35 20 E. BCD = 100° 00

Let AB, CD, Fig. 71, be the given sides of the tra¬ Produce DA, CB, to meet in E ; then 2ABCD = 2EDC * 2 EDC—2 EAB 2EAB or ABCD= — , Hence the truth of the rule is evident. * DEMONSTRATION.

pezium ABCD.

134

CONTENT OF LAND.

BC, N. 64° 40' E. BA, S. 24 00 W.

[CHAP. III.

DC, N. 35° 20' W. DA, S. 88 00 W.

40 40 123 20 180 00 180 00 ABC =139 20

ABC=56 40 Construction.

Make AB=6.90, and draw DA, CB, making the angle DAB=64°, and ABC=139° 20'; produce DA and make the angle EAF=56° 40'=the given angle ADC; lay off AF=11.50 = the given side BD, and parallel to AD draw FC, meeting BC in C; lastly draw CD parallel to AF, meeting AD in D; then will ABCD be the trapezium.* Calculation.

The angle E=180°—the sum of the angles BCD, ADC=23° 20'. f rad. Ar. Co. 0.00000 1 sin. E. 23° 20' Ar. Co. 0.40222 J sin.EAB116°00' 9.95366 : sin. EABXsin. EBA, 1 sin. EBA 40° 40' 9.81402 JAB 6.90 0.83885 2 :: AB , \ AB 6.90 0.83885

As rad.Xsin. E,

: 2 EAB 70.405

1.84760

* DEMONSTRATION. By construction FC is parallel to AD and CD to AF; therefore (34.1) CD=AF and (29.1) the angle ADC=EAF; hence it is evident that the sides AB, CD, and the angles of the trapezium ABCD are respectively equal to the given sides and angles.

!

CHAP. III.] CONTENT OF LAND.

As radxsin. E, sin.ECDxsin. EDC, ::CD*,

135

rad. Ar. Co. 0.00000 sin. E. 23° 20' Ar. Co. 0.40222 sin. ECD 100° 00* 9.99335 sin. EDC 56 40 9.92194 CD 11.50 1.06070 CD 11.50 1.06070

: 2 EDC 274.731 2.43891 2 EAB 70.405 2ABCD 204.326 ABCD= 102.163 Ch.=10 A. OR. 34.6P. 2. In a trapezium ABCD, the angles are, A=65°, B —81°, C=120°, and consequently Drr:94°; also the side AB=20 ch. and CD=11 ch.: required the area. Ans. 22 A. 2R. 27 P. 3. Required the area of a four-sided piece of land, bounded as follows: 1. N. 12° 30' E. 2. N. 81 00 E. dist. 23.20 ch. : 3. S. 36 00 W. 4. N. 89 00 W. dist. 12.90 ch. Ans. 27 A. 2R. 24 P. PROBLEM VIII. To find the area of a trapezium when three sides and the two included angles are given. RULE.

As radius, Is to the sine of one of the given angles; So is the rectangle of the sides including this angle. To a certain quantity.

136

CONTENT OF LAND.

[CHAP.

in.

As radius, Is to the sine of the other given angle; So is the rectangle of the sides including this other angle, To a second quantity. Take the difference between the sum of the given angles and 180°; Then, As radius, Is to the sine of this difference; So is the rectangle of the opposite given sides, To a third quantity. • If the sum of the angles be less than 180°, subtract the third quantity from the sum of the other two, and half the difference will he the area of the trapezium. But if the sum of the given angles exceed 180°, add all the three quantities together, and half the sum will be the area.* Fig. 72. _ * DEMONSTRATION. Let ABCD (Fig. 72 or 73) be the trapezium, having the given sides, AD, AB, BC, and given angles DAB, ABC. Complete the paral¬ lelograms ABCE, ABFD, and join ED, CF; then because EC, DF, are each parallel and equal to AB, they are (30.1) parallel and equal to each other, and (33.1) ECFD is a parallelogram; there¬ fore ABFD=ABHG+GHFD=(35.1) ABCE+ECFD = (34.1) ABCE+ 2ECD; to the first and last of these equals add ABCE, then ABFD+ ABCE = 2 ABCE+2ECD=2 ABCDE. But Fig. 72, when the sum of the given angles DAB, ABC, is less than 180°, 2ABCDE = 2ABCD+2EAD; therefore in this case ABFD+ABCE = 2ABCD + 2EAD; or ABFD + ABCE—2EAD = Fig. 73. 2ABCD. D F And, Fig. 73, when the sum of the given angles DAB, ABC, exceeds 180°, 2ABCDE = 2ABCD— 2EAD ; therefore ABFD+ABCE = 2ABCD— 2EAD; or, ABFD+ABCE+2EAD = 2ABCD. But by prob. 3, one of the first two proportions gives 2B AD (= ABFD,) and the other gives 2 ABC (=ABCE;) also because the angle EAD is the

CHAP. III.] CONTENT OF LAND.

137

EXAMPLES.

1 In a trapezium ABCD, there are given AD=23.32 ch., AB=25.70 ch., and BC= 15.84 ch., the angle DAB= 6-4°, and ABC=82°: required the area. As rad. -----: sin. DAB, 64° - - - i n / A n JS AD 23.32 : : ADxAB, AB 25 7Q

Ar. Co. 0.00000 9.95366 1.36773 -_ -_ 1.40993

: first quantity 538.66 -

-

-

2.73132

As rad. -----: sin. ABC, 82° - - - -

Ar. Co. 0.00000 9.99575 1.40993 £V AB BC 25.70 15.84 1.19975

.. ABxBC Auxi^, ' •• f

: second quantity 403.12 DAB ABC

2.60543

64° 82 146 180

Difference

34°

As rad. -----: sin. difference 34° - -

-

Ar. Co. 0.00000 9.74756 1.36773 ..ATWT*r •ADXBC, jS AD BC 23.32 15.84-- -1.19975 : third quantity 206.55 -

-

-

2.31504

difference between the sum of the given angles and 180°, and the side EA=BC, the third proportion gives 2EAD: hence the truth of the rule is manifest.

12*

S

138

CONTENT OF LAND.

[CHAP.

in.

1st quantity 538.66 2d “ 403.12 941.78 “ 206.55

3d

2)735.23 367.615 ch.=36 A. 3 E. 2 P. 2. What is the area of a four-sided lot of ground, three sides of which, taken in order, measure 6.15, 8.46, and 7.00 chains, respectively; the angle contained by the first and second sides 56°, and that contained by the second and third sides 98° 30'? Am. 4 A. 0 E. 25 P. 3. One side of a quadrilateral piece of land hears S. 7} E. dist. 17.53 ch., the second, N. 87 E. dist. 10.80 ch. and the third, N. 251 E. dist. 12.92 ch.: what is the area ? Am. 21 A. 3 E. 2 P. PEOBLEM IX. To find the area of a trapezoid. RULE.

Multiply the sum of the parallel sides by their per¬ pendicular distance, and half the product will be the area.* EXAMPLES.

1. Eequired the area of a trapezoid ABCD, of which the parallel sides AD, BC measure 6.14 and 9.48 chains, Fig. 74. C

* DEMONSTRATION. The trapezoid ABCD, Fig. 74, = the triangle ABD + BDC = (by prob. 2,) ApxBF+BCXDU = (becauseBF=I)E>) ADXBF

, BCXBF—AD+BCXBF *

2

2

CHAP,

in.]

CONTENT OF LAND.

139

respectively, and their perpendicular distance BF or DE, 7.80 chains. Ch. 6.14 9.48 15.62 7.80 124960 10934 2)121.8360 60.9180 Ch.= 6 A. 0 R. 15 P. 2. The parallel sides of a trapezoid are 12.41 and 8.22 chains, and their perpendicular distance 5.15 chains: required the area. Ans. 5 A. 1 R. 10 P. 3. Required the area of a trapezoid whose parallel sides are 11.34 and 18.46 chains, and their perpendicular distance 13.25 chains. Ans. 19 A. 2 R. 39 P. PROBLEM X. To find the area of a circle, or of an ellipsis* RULE.

Multiply the square of the circle’s diameter, or the * If two pins bo set upright in a plane, and a thread, the length of which is greater than twice the distance between the pins, haying the ends tied together, be put about the pins; and if the point of a pin or pencil applied to the thread, and held so as to keep it uniformly tense, be moved round, till it return to the place from which the motion began; then the point of the pin or pencil will have described on the plane, a curved line called an Ellipsis.

140

CONTENT OF LAND.

[CHAP.

Ill

product of the two diameters of the ellipsis, by .7854, for the area.* Note 1.—If the diameter of a circle he multiplied by 3.1416, the product will he the circumference; also if the circumference be divided by 3.1416, the quotient will be the diameter. 2. If the area of a circle he divided by .7854, the square root of the quotient will he the diameter. EXAMPLES.

1. How many acres are in a circle a mile in diameter?. 1 mile=80 ch. 80 6400 .7854 3141600 47124 5026.5600 Sq. Ch.=502 A. 2 R. 25 P. nearly. Or by Logarithms. Square of 80 | gjj .7854 5026.56 Sq. Ch.

log. 1.90309 1.90309 —1.89509 3.70127

2. Required the area of an ellipsis, the longer diameter of which measures 5.36 ch. and the shorter 3.28 ch. * The demonstration of this rule is too abstruse to admit of a place in tills work. The student who wishes to see a demonstration is referred to a treatise on Mensuration or Fluxions.

CHAP. III.]

CONTENT OF LAND.

141

Ch. 5.36 3.28 4288 1072 1608 17.5808 .7854 703232 879040 1406464 1230656 13.80796032 Sq. Ch.=l A. 1R. 20.9 P. PROBLEM XI. The hearings and distances of the sides of a tract of land being given, to calculate thd area. RULE.

1. Rule a table and head it as in the annexed example; observing that the letters E. D. D. and W. D. D., stand for East Double Departure and West Double Departure. 2. Find by prob. 12. chap. 1., the corrected differ¬ ences of latitude and the departures, corresponding to the several sides, placing them in their proper places in the table. 3. When the departures corresponding to the first and last sides are of the same name, add them together, and place the sum opposite the first side, in the column of double departures, which is of that name; but when they are of different names, take their difference, and place it

142

CONTENT OF LAND.

[CHAP. III.

in the column of double departures, which is of the same name with the greater departure. Proceed in the same manner with the departures corresponding to the first and second sides, placing the result opposite the second side ; with those corresponding to the second and third sides, placing the result opposite the third side; and so on to the last. 4. Commencing with any side of the survey at plea¬ sure, assume any number whatever for a multiplier cor¬ responding to that side, and place it in the column of multipliers, opposite to the side, marking it w ith the let¬ ter E, fop east. If this multiplier and the double depar¬ ture, corresponding to the next side, are of the same name, take their sum for the next multiplier, marking it with that name ; but if they are of different names, take their difference, marking it with the name of the greater. Proceed in the same way with this multiplier and the next double departure; and so on till multipliers have been found corresponding to all the sides. 5. Multiply each of the corrected differences of lati¬ tude by its corresponding multiplier; and when the mul¬ tiplier is east, place the product in that column of areas, which is of the same name with the difference of latitude; but when it is west, place the product in the column of areas, which is of a different name from that of the dif¬ ference of latitude. 6. Add up the numbers in the columns of areas, and taking the difference of their sums, divide it by 2; the result will be the area of the survey.*

* DEMONSTRATION.

Let ABCDEFG, Fig. 79, be a plot of a survey; and

let the east and west line AL, represent the assumed multiplier. From the points 13 and L draw BM parallel, and LM perpendicular to AL, meeting in

CHAP. 111.] CONTENT OF LAND.

140

Note 1.—If the double departures have been correctly found, the sums of the numbers in the two columns, will be equal. Also, if the multipliers have been correctly ob¬ tained, the sum or difference of the multiplier last found Fig. 79. N

H

s

M; and bisect BM by the meridian NS. Draw the other east and west lines, Cc, mdy ne, rf and ug; and also the meridians hk, Bp, Cn, Z>$, ter, Fu>, and Gx. Then it is evident that the differences of latitude and the de¬ partures corresponding to the several sides will be as in the following table. Also according to the rule, Bh-{- Av=Bx, is the double departure correspond¬ ing to the first side; Bh-\- Cl—Ck, is that for the second side; Cl—Dm—Bp, is that for the third; and so on to the last. E. W. E.D. D. j W.D.D Multipliers.j JN. Areas.

Dist. IN. AB Ah

S. Areas.

Bh

Bx

Aa-f 2>6, E 2Aab B

BC

BI Cl

Ck

Bb+Cc, E

2 BbcC

CD

Cm

Dp

Cc+Dd, E

2 CcdD

DE

~D<, Eq

En

Dd-\~ Ec, E

2 DdeE

EF

Er

Fr

Fs

Ee+Ff, E

2 EefF

Gt

Gu

Ff+Off,E 2 F/g a

Dm

FG Ft GA Gv

Av

AID

Gg-\-Aa,E 2 Gga A

144

CONTENT OF LAND.

[CHAP. III.

and the next double departure, according as they are of the same or different names, will be equal to the assumed multiplier. 2. It is best in general, to assume 0, for the first mul¬ tiplier ; as by so doing there is one multiplication less to be performed, and the other multipliers are mostly smaller numbers, than they would otherwise be. fl

By construction, the assumed multiplier AL=Aa-f- !L— Aa-\-bM—Aa +B6. By proceeding with this multiplier and the double departures, as directed in the rule, we shall evidently have the other multipliers as re¬ presented in the table. It is also plain that the products of the differences of latitude hv these multipliers, will be as represented in the columns of north and south areas. The sum Qf the north areas is 2 BAGF/iB ; and the sum of the south areas is 2 BCDEFfbB. The difference of these is 2 ABCDEFGA; the half of which is the area of the survey. The preceding demonstration may easily be extended to the case in which the assumed multiplier is so small as to make the meridian NS pass through the survey. Thus, suppose Aic-fHB to be the assumed multiplier, the meridian in this case coinciding with FII. Then the multipliers will be equal to the differences between the above multipliers and the quantity, iva-\-bll or its equal 26H. We may therefore represent them, and the products, as in the following table ; in which the multipliers are marked, and the products placed, in conformity with the rule. f S.

Dist.

N.

1 AB 1

Ah j

Multipliers.

N. Areas.

S. Areas.

(Aa+Bb)—2 6//, E. 2 AaBb—2wabH

BC

\Ul

(Bb+Cc)—2bIl E.

2 BbcC—2HbcK

CD

Cm (Cc+Dd)—2 bH E.

2 CcdD—2KcdP

DE

\Dq (ZM+JEe)—'2 bll E.

2 DdeE—2PdcQ

EF

j Er (Ec+ Ff)—2 bll, E.

2 EefF—2 QffF

Gx)

FG

Ft J

VV.

2 Ffyl—2 FfgO

GA

Gv 1 2bll—(Ghr-f-Art) W.

2t.gaw—2 GgaAs

2 bll—(Ff+

CHAP. III.]

CONTENT OF LAND.

145

3. Instead of assuming the first multiplier east, it might with equal propriety be assumed west. Also instead of finding the multipliers from the departures, they might be found in a similar manner, from the differ¬ ences of latitude; using, in that case, the departures for multiplicands. 4. When one or two bearings or distances are omitted, they may be found by the problems in the last chapter; and in these cases the differences of latitude and depar¬ tures are to be used as first obtained from the tables, there being no means of correcting them. EXAMPLES.

1. Given the bearings and distances of the sides of a tract of land as follows: 1st. 40i° E. 31.80 ch.; 2d. N. 54° E. 2.08 ch.; 3d. N. 291° E. 2.21 ch.; 4th. N. 281° E. 35.35 ch.; 5th. N. 57° W. 21.10 ch.; and 6th S. 47° W. 31.30 ch.; to the place of beginning. Required the area of the tract. The sum of the north areas is 2 AabB—2 Hbaw; and the sum o^the south areas is 2 BCDEF/6B+2 wafF — 2 H6/F-—2 AGFfa=2 BCDETy7>B+ 2 wafF—2 waf F—2 Hbaw — 2 AGF fa—2 BCDEF/6B~2H6tzw—2AGF/o =2BCDEiy 6B — 2 AGFfa—2 Hbaw. If now the sum of the north areas be subtracted from that of the south areas, the remainder will be 2 BCDEF/’bB —2 AQF/a—2 AabB=2 BCDEF/6B—2 BAGF/b—2 ABCDEFGA.

13

T

146

CTJ

1.08

1.68

42.66 E.

2.15 E.

0.61

40.51

489.3102

1904.7456| 45.9455

45.9455 -

21.37 q CO

00 dd

q

17.66 N. 57 W.|

d o* q d qq dd TJW

31.30

21.10 11.49

W W H-+ nh 05 GO (M C* fc £ CO

a>

N

00

a

00

CM o

CO

'« • CO'

H >4 fa i>

05 C5 <35 U

<

m

W «

00 ct

CO

S. 47 W.

1.92 2.21

35.35 31.00

CD

1.23 31.80 W Hw o

o o

00

to d

00 q d ryi t> i—t C* r-4

CO

2.08

Dist. Bearings.

&

2*

o o

c* CO

q d ^ oo q d

O q

N. 54 E.

CD

24.18 20.65

w

17.69 .02 loi 11.47

q

Ill

c*

22.89

48.2688 25.14 E.

43.27 E. 1339.6392

2.76

18.13

o q o q

.05 30.96;

1.92

o ©m q CO o q ©

w.

s

-

SH

1.23

CD

0 00 01 ot

17.0,5

00 q

[CHAP.

0* CO

24.21

1.68

|

W. o ©

Cor.l

j

00 CO d

j

2.15

122.38 E. 1 27.5274

0.00 E.

Multipliers.

|

$ E.D.D.

1

P P

| WWSf

N. Areas. S. Areas.

CONTENT OP LAND.


2. Required the area of a tract of land, bounded as follows: 1st. N. 75° E. 13.70ch.; 2d. N. 201° E. 10.30 ch.; 3d. East, 1G.20 ch.; 4th. S. 331° W. 35.30 ch.; 5th. S 76° W. 16 ch.; 6th. North, 9ch.; 7th. S. 84°

147

CHAP III.] CONTENT OF LAND. 0

04*

04 04 -'~N

U07.8684

379.8838 1 2595.6207 379.8838

6.07 9.99 W ! 100.6992

04 00

CO CO b

S'*

M .

92.02i

£

3.92 W

29.90 W

00 04

46.89 W

49.23 W

£

20.79

11.51 16.93 W

00 CT5


41.0032

387.2050

728.6706

342.6408 33.8436

984.5650 5.5440 15.49 1.44 W

34.94 33.50 E

16.88 E 163.0608

3.22 36.72 E

N. Areas. pliers.

0.00 E

Multi¬

W.D.D.

CO 1-

CO

05

Q O* GO

ft CO ft

8.90

04 05

3.27

2.80

15.50

CO t"

115.52

o jo

1 CO 1©

q. ~oT

o

q

04

q

o

q

04

05 04 05

q

y*

CO

5

a

co

£

CO C5

W

W

10.48

••ciht CO

£

£

GO

00

05

H

04

H«*

03 u0

GO

O

o o

q 04

w

w o 04

£

iO t'" 00 t-* iO iG>

q

q o

IH

00

.•oht CO 04 CO .04

£

00

o

GO 04

q

04 O

o 04

co

S. 15 W.| 10.85

CO

CO*

£ £

1 rn 04

li.eoj

rH|d CO CO

o o

North.

04

16.20

H(M

35.30

W

o iO

6.94

o o

9.65 10.30

13.70 W

rH

oq CO*

b

East.

Sta. ] Bearing.

i—

q

i> HO 04 : CO q d d 05 iH

00*6

Dist.

3.54

TJ

05

2.75] 11.68

CO

11.60

CO

19.36! 15.51

CO

1.21

04 CO

5.36

1

q

04 <=?

c

05

3.28 .02

o

i> {>

61.93| 61.68

1 -i

iC

o

N

2.8 ij

1 [

04

.02

o

.02 .02

04

00

q d

.02

9.01

6.96

04



29.44

W

*©" 04

©

U <

00

61.80 1

11.70

5,38

11.61 05

10.46

iO 00 CO

.02 15.59

19.44

16.22 19.49 .05

.01

£

**

o CO

.02 I iO

o

•■4

C3

i>

|

9.66 j

IN

o Jq Io .02

.02 .02 j 3.56

cor. E.

04 q

q 29.39

3.62

13.26

.

0Q fc#

q 04 C3

1

w.

w

17.08

w 1

16.991

®

2.341

ft

19.84

(3

16.88

I

co

ft

04

£ 00 f—1 GQ co iH

q

00

04

j •«3iy,8 1

152.5393

W. 11.60 ch.; 8th. N. 534 W. 11.60 eh.; 9th. N. 36i E. 19.36 eh.; 10th. N. 224° E. 14 eh.; 11th. S. 764° E. 12 ch.; 12th. S. 15° W. 10.85 ch.; 13th. S. 18° W. 10.62 ch.; to the place of beginning.

148

CONTENT OP LAND.

[CHAP. III.

3. Given the boundaries of a tract of land viz. 1st. S. 35i° W. 11.20 ch.; 2d. N. 45° W. 24.36 ch.; 3d. N. 15i° E. 10.80 ch.; 4th. S. 77° E. 16. ch.; 5th. N. 87i° E. 21.50 ch.; 6th. S. 60° E. 14.80 ch.; South 10.91 ch.; 8th. N. 85° W. 29.28 ch.; to the place of beginning: required the area. Ans. 85 A. 3 R. 17 P.

4. Given the boundaries of a tract of land as follow, viz. 1st. N. 19° E. 27 ch.; 2d. S. 77° E. 22.75 ch.; 3d. S. 27° E. 28.75 ch.; 4th. S. 52° W. 14.50 ch.; 5th. E. 19 ch.; 6th. West, 17.72 ch.; 7th. N. 36° W. 11.75 ch.; 8th. North, 16.07 ch.; 9th. N. 62° W. 14.88 ch.; to the place of beginning: required the area. Ans. 152 A. 2 R. 6 P. 5. Required the area of a tract of land bounded as follows: 1st. S. 62° W. 7.57 ch.; 2d. N. 43s° W. 5.89 ch.; 3d North, 5.82 ch.; 4th. N. 33i° W. 8.83 ch.; 5th. N. 48° E. 4.81 ch.; 6th. N. 12° E. 4.66 ch.; 7th. N. 62^° E. 5.27 ch.; 8th. S. 6£°E. 5.60 ch.; 9th. S. 40£°E. 5.87 ch.; 10th. East, 6.54 ch.; 11th. North, 5.52 ch.; 12th. N. 68£° E. 3.10 ch.; 13th. S. 30° E. 7.90 ch.; 14th. S. 23° W. 8.80 ch.; 15th. S. 3l£° E. 6.42 ch.; 16th. S. 50° W. 8.40 ch.; 17th. N. 44° W. 6.85 ch. to the place of beginning. Ans. 44 A. 2 R. 22 P. 6. Given the following field-notes to find the area of the survey; also the bearings and distance of the 3d side, which were omitted to be taken on account of obstacles in the way. Ch. 1. S. 85i° E. 23.30 2. S. 19 E. 31.12 3. 4. N.64 W. 29.72

CHAP. III.]

CONTENT OF LAND.

149

Ch. 5. N. 15i° W. 22.46 6. N. 58 E. 25.94 7. S. 271 E. 6.60 Ans. Area 182 A. 0 E. 21.7 P. and the bearing and distance of the 3d side, S. 66° 23' W. 28.06 ch. 7. Being furnished with the field-notes of a tract of land, and requested to calculate the area, I found on examining them, that the figures expressing the angles of bearing of the 4th and 5th sides were so defaced as to be illegible : but as the remaining data are sufficient, the area is required. The field-notes are as follows : Ch. l.S. 601° W. 10.34 2. N. 27i W. 17.88 3. N. 51 E. 15.85 4. N. — E. 9.61 5. S. — E. 19.18 6. S. 161 E. 22.21 7. S. 71* W. 16.66 8. N. 71* W. 5.76 Am. 81 A. 2 E. 23 P. 8. In a survey, represented Fig^si. Fig. 81, the corner at A was inaccessible, occasioned by the overflowing of water; but being a tree, it can be seen from the adjacent corners B and L. I therefore set my instrument at B and took the bearing to A, which I reversed, and set in my field-book as the first bear- D ing. I then proceeded to take the bearings and distances of the several sides to L; and at L, I took the bearing of the side LA. The field-notes being as follows, the length' of the sides AB and LA, and the area are required. 13

*

150

CONTENT OF LAND.

[CHAP. III.

Fig. 80.

AB,N.51i°W. Ch. BC, S. 451 W. 15.16 CD, N. 50 W. 22.10 DE, North, 18.83 EF, N. 48 E. 22.60 FG, N. 251 W. 20.17 GH, East, 26.57 III, S. 301 E. 22.86 IK, S. 44 W. 15.04 KL, S. 47 E. 28.55 LA, S. 201 W. Ans. AB, 26.47 ch.; LA, 23.81 ch.; and the area 244 A. 3 It. 13 P. N

9. In taking a survey of a tract of land bounded by six straight sides, Fig. 80,1 was prevented going directly from the 3d to the 4th comer by a pond of water. I therefore set up two stakes near the edge of the pond, and took the bearing and distance from the 3d corner to the first stake, from the first stake to the second, and from the second to the 4th corner, and noted them in my field-book as all belonging to the 3d station of the survey. The field-notes being as follows, the bearing and distance of the 3d side, and the area of the survey are required. 1. North, 7.81 Ch. 2. S. 761° E. 18.15 fS. 52 W. 10.701 MS. 71 W. 13.92 \ IS. 33i E. 9.00J 4. N. 841 W. 27.12 5. N. 41 W. 22.00 6. East, 16.58 Ans. 3d side, S. 10° 47' W. 28.42 ch.; and area 80 A. 0 R. 25 P.

CHAP. III.]

CONTENT OF LAND.

151

PROBLEM XII. To find the area, when ofi-sets are taken. RULE.

1. Find by the last problem, the area enclosed by the stationary lines and straight sides of the survey. 2. Subtract the stationary distance of each off-set, from that of the one immediately following; the remain¬ ders will be the distances, intercepted on the stationary line, between each two adjacent off-sets. Place these under one another in a column as in the annexed exam¬ ples. Also take the sums of each two adjacent off-sets, and place them in the next column, so as to correspond with the intercepted distances. 3. Multiply the sum of each two adjacent off-sets by their intercepted distance on the stationary line; then, half the sum of the products will be the area of the off¬ sets on that line. 4. If there are off-sets on more than one stationary line, proceed in the same manner with the others. 5. When the stationary lines are within the boundary of the survey, add the areas of the off-sets to the area enclosed by the stationary lines and straight sides; but when the stationary lines are without the boundaiy, sub¬ tract the areas of the off-sets.# * DEMONSTRATION. Considering the boundary as straight between the ends of each two adjacent off-sets, it is plain that the area contained between the stationary line and boundary will be divided by the off-sets into trap©zoidcs and triangles. Hence the truth of the rule is evident.

152

CONTENT OF LAND. EXAMPLE

[CHAP. HI.

1, Fig. 82.

Enquired the area of a piece of meadow, bounded on one side by a brook; the field-notes being as follows : Left-hand off-sets on the stat line. Stat. Dist. Off-sets.

1. N. 161°E. 14.85Ch. No. 1. 2. East, 7.82 2. 3. S. 3i W. 14.45 Stat. line. 3. 4. N.86* W. 11.07 4.

0.00Ch. 0.30Ch. 0.95 0.84 2.03 0.86 3.28 0.50

CHAP. III.] CONTENT OP LAND.

To find the area ofi the off-sets. No.

Sta. Dist. l Off-sets. Ch. Ch.

Intercep. Sums of Off-sets. Products. Dist.

I

0.00

0.30

2

0.95

0.84

0.95

1.14

1.0830

3

2.03

0.86

1.08

1.70

1.8360

4

3.28

0.50

1.25

1.36

1.7000

5

5.20

1.80

1.92

2.30

4.4160

6

7.43

2.35

2.23

4.15

9.2545

7

8.98

1.45

1.55

3.80

5.8900

8

10.46

1.08

1.48

2.53

3.7444

9

11.71

1.85

1.25

2.93

3.6625

10

14.45

0.35

2.74

2.20

6.0280

o\ AA 2)37.6144

18.8072 Ch. =1 A. 3 R. 21P. A. R. P. Area of ABCD 13 1 11 Do. of off-sets 1 3 21 Whole area EXAMPLE

15 0 32 2. Fig. 83. O

Required the area of a survey from the following field notes. U

154

1. 2. 3. 4. 5. 6.

N. N. N. S. S. S.

CONTENT OP LAND.

36|° 56J 26J 71J 26J 45

W. E. E. E. E. W.

Ch. 30.00 21.60 stat line. 13.44 Do. 18.96 Do. 13.46 Do. 42.41

[CHAP.

III.

Left hand off-sets. 1st Stationary Line. 3d Stat Line. Sta. Dist Off-sets. Sta. Dist. Off-sets. No. Ch. Ch. No. Ch. Ch. 1. 0.00 0.50 1. 0.00 0.55 2. 6.10 3.40 2. 4.20 2.50 3. 10.15 3.10 3. 8.05 3.20 4. 14.08 3.96 4. 15.15 2.45 5. 19.20 2.70 5. 18.96 0.50 6. 21.60 0.55 4th Stat. Line. 1. 0.00 0.50 2d., Stat. Line. 1. 0.00 0.55 2. 5.12 2.75 2. 13.44 0.55 3. 10.00 1.90 4. 13.46 0.70

The area within the stationary lines and straight sides, found by the last problem, is 1152.5381 square chains. To find the area of the off-sets. 1st. Stationary Line. No.

Sta. Dist. Off-sets. Intercep. Sums of Off-sets. Products. Ch. Ch. Dist.

I

0.00 0.50

2

6.10 3.40

6.10

3.90 23.7900

3

10.15 3.10

4.05

6.50 26.3250

4

14.08 3.96

3.93

7.06 27.7458

5

19.20 2.70

5.12

6.66 34.0992

6

21.60 0.55

2.40

3.25

7.8000

2d. Stationary Line. No.

i 2

Sta. Dist Off-sets. Intercep. Sums of Dist. Off-sets. Products. Ch. Ch. 0.00 0.55 13.44 0.55 13.44

1.10 14.7840

CHAP. III.J CONTENT OP LAND.

155

3d. Stationary Line.

No.

Sta. Disk Off-seta Ch. Ch.

Intercep. Sums of Disk Off-seta Producta

l

0.00

0.55

2

4.20

2.50

4.20

3.05

12.8100

3

8.05

3.20

3.85

5.70

21.9450

4

15.15

2.45

7.10

5.65

40.1150

5

18.96

0.50

3.81

2.95

11.2395

4th. Stationary line.

1

0.00

0.50

2

5.12

2.75

5.12

3.25

16.6400

3 10.00

1.90

4.88

4.65

22.6920

4 13.46

0.70

3.46

2.60

8.9960

2)268.9815 Area of the off-sets ------ 134.49075 Ch. Area within the stationary lines

- 1152.5381 1287.02885 Ch. 128.702885 Acr 4 2.811540 40 32.46160

Area of the survey, 128 A. 2 R. 32 P.

J 56

CONTENT OF LAND.

[CHAP. IH.

3. Required the area of a meadow from the following field-notes. EXAMPLE

Left-hand off-sets on the stat. line. Sta. Off-sets. Dist.

l.N.41i°E. 14.35 Ch. No. 1. 0.00 Ch. 0.38 Ch. 2. S. 42i E. 14.71 Sta. line. 2. 2.65 2.35 3. S. 54 W. 16.32 3. 3.80 1.70 4. N. 32 i W. 11.50 4. 6.00 2.75 5. 7.50 1.40 6. 9.60 3.20 2.72 7. 12.38 0.42 8. 14.71 Am. Area 22 A. 3 R. 27 P. EXAMPLE

4.

The following field notes are given, to find the area of the survey. Left-hand off-sets. On the 1st stat. line. On the 2d stat. line Sta. Dist. Off-sets. Sta. List. Off-sets

1. S. 69 2. S. 28 3. S. 321 4. N.48 5. N.26I

Ch. No. Ch. Ch. No. Ch. E. 16.14 sta. line 1. 0.00 0.44 1. 0.00 E. 9.38 Do. 2. 3.80 2.00 2. 2.67 W. 21.20 3. 7.04 3.79 3. 6.20 W. 22.47 4. 9.87 2.34 4. 9.38 5.13.24 3.00 E. 19.00 6.16.14 0.31 Am. 56 A. 2 R. 18

Ch. 0.31 2.94 2.62 0.39

P.

PROBLEM XIII. Given the hearing and distance of two statiom from each other and the hearings of all the corners of a tract of land from these stations, to find the area of the tract. The method of doing this will be best explained by an example.

ClfAP. III.]

157

CONTENT OF LAND. EXAMPLE 1.

Let ABCDEFGA, Fig. 84, represent a field, all the angles of which can be seen from two stations, H and I, without it. The bearing and distance of the stations, and the hearings of all the angles of the field, from each station, being as follow, it is required to find the area. Fig. 84. D

The station H bears from the station I, North, dist. 28. Ch. Bearings.

Bearings.

IIA S. 811° E. IA N. 28i°E. HB S. 85S E. IB N. 421 E. IIC S. 68

E. IC N. 511 E.

IID S. 581 E. ID N. 71

E.

IIE S. 351 E. IE S. 82 i E. HF S. 28i E. IF N.731 E. IIG S. 40

E. IG N.60 E. Construction. Draw III according to the given bearing and distance; and from the points II and I, draw IIA, IIB, IIC, &c.„

158

CONTENT OF LAND. [CHAP 1/1,

and IA, IB, IC, Ac. according to the given bearings; then will the intersections A, B, C, Ac. of the corres¬ ponding bearings HA and IA, HB and IB, HC and IC, Ac. be the angular points of the field. Calculation. In each of the triangles IHA, IIIB, IHC, Ac. we have the side IH ; and from the bearings of the sides, we have all the angles, to find the sides IA, IB, IC, Ac. Then in each of the triangles, IAB, IBC, ICD, Ac. we have two sides, and the included angle; whence the areas may be found by prob. III. From the sum of the areas of the triangles IAB, IBC, ICD, and IDE, which is equal to the area IABCDEI, subtract the sum of the areas of the triangles I AG, IGF and IFE, which is equal to the area IAGFEI; the re¬ mainder will be the area of the field ABCDEFGA. Note.—-In working the proportions for finding the sides IA, IB, Ac. it will be unnecessary, when the area only is required, to take out the natural numbers correspond¬ ing to the logarithms of those sides ; because in the pro¬ portions for finding the areas it will be sufficient to know the logarithms of the sides, without knowing their real lengths. To find the log. of IA. As sin. IIAI, 70° 00' : sin. AHI, 81 30 : : IH, 28

9.97299 9.99520 1.44716 11.44236

:

IA,

log.

1.46937

CHAP. III.J CONTENT OP LAND. 159 To find the log. of IB. As sin. HBI 52° 00' : sin. BHI85 45 - - - - : : IH 28 -

9.89653 9.99880 1.44716 11.44596

: IB ------ log. 1.54943 To find the log. of IC. As sin. HC1 60° 30' 9.93970 : sin. CHI 68 00 9.96717 : : III28 - 1.41716 11.41433 : IC - - - - - - log. 1.47463 To find the log. of ID. As sin. HDI50° 45' - - - - - 9.88896 : sin. DHI58 15 ----- 9.92960 : : IH 28 ------- 1.44716 11.37676 * ID

log. 1.48780 To find the log. of IE.

As sin. HEI47° 00' - - - - - 9.86413 : sin. EHI35 30 ----- 9.76395 : : IH 28 - - 1.44716 11.21111 5 IE

log. 1.34698

160

CONTENT OF LAND. [CHAP.

To find the log. of IF. As sin. HFI 78° 00' : sin. FHI28 30 : : III 28 -

9.99040 9.67866 1.44716 11.12582

: IF

log.

1.13542

To find the log. of IG. As sin. HGI 80° 00' ------ 9.99335 : sin. GUI 40 00 9.80807 : : IH 28 -------- 1.44716 11.25523 i

IG

Jog. 1.26188 To find the double area of the triangle IAB.

As rad. - - sin. AIB 13° 45' IA : IA X IB IB :

-

10.00000

9.37600 log.' 1.46937 1.54943 2.39480

2IAB 248.2

To find the double area of the triangle IBC. As rad. - : sin. BIG 9° 15' IB : : IB XIC, IC :

2IBC 169.9

10.00000 9.20613 log. 1.54943 1.47463

-

2.20319

CONTENT OF LAND.

CHAP. III.]

To find the double area of the triangle ICD. As rad. - - : sin. CID 19° 30'

-

10.00000

9.52350 log. 1.47463 1.4S780

. :ICxID,

2.48593

: 2ICD, 306.15

To find the double area of the triangle IDE. 10.00000 As rad. 9.64953 sin. DIE 26° 30' (ID----- log. 1.48780 T„

::IDXIE,JJE

1.34698

: 2IDE 305.007 -

2.48431

To find the double area of the triangle IEF. 10.00000 9.60931 log. 1.34698 1.13542

As rad. - - sin. EIF 24° 00' IE : IEXIF. IF

-

•I

: 2IEF 123.511

2.09171

To find the double area of the triangle IFG. As rad - - - : sin. FIG 13° 30' IFXIG,{£

;

;

2IFG 58^274 - - . 14* X

- 10.00000 - - - 9.36818 - log. 1.13542 • . 1.26188

162

CONTENT OF LAND.

[CHAP. III.

To find the double area of the triangle IAG. As rad. : sin. AIG 31° 30' : : IAXIGJ

:

IIG

- - - - 10.00000 - - - - 9.71809 - - - log. 1.46937 - . . _ 1.26188

2IAG 281.412 -

-

-

-

- 2.44934

Ch.

2IAB - - 2IBC - - 2ICD 2IDE ---

-

CE.

248.2 169.9 306.15 305.007

2IEF 123.511 2IFG 58.274 2 IAG 281.412 2IAGFEI 463.197

2IABCDEI 2IAGFEI

1029.257 463.197

2ABCDEFGA

566.060

ABCDEFGA

283.03 Ch.=21 A. 1 E. 8 P.

The bearings and distances of the sides, if required, might readily be obtained. For, having found the distances IA, IB, we have in the triangle IAB, two sides, and an included angle; whence the angle IAB and side AB may be found. The angle IAB applied to the bearing of IA, will give the bearing of AB. In the same manner the bearings and distances of the other sides may be found. EXAMPLE 2.

Being required to calculate the a^ca of a field, the owner of which refuses permission to go on it, I choose two stations, F and G, in the adjacent land, from whence

CHAP. III.] CONTENT OF LAND.

163

all the angles of the- field are visible. The bearing and distance of the stations, and the bearings of the angles, from each station, are as follow. What is the area of the field ? The station G bears from the station F, N. 43° W. 20 ch. Bearings.

.Bearings.

FA N 25J° E. GA S. 66° E. FB N. 19 W. GB N. 23 E. FC N. 5 W. GC

N. 38iE.

FD N. 16 E. GD N. 60JE. FE N. 6O5 E. ]GE S. 84 E. Aiis. 33 A. 1R. 7 P. PROBLEM XIV. To find the area of a survey by protracting it, and dividing the plot into triangles and trapeziums. The method of doing this will be easily understood from the following example. EXAMPLE 1.

Given the bearings and distances of the sides of a tract of land as follow: 1st. N. 50° E. 9.60 ch.; 2d. S. 32° E. 16.38 ch.; 3d. S. 41° W. 6.30ch.; 4th. West, 8.43 eh.; 5th. N. 79° W. 10.92 ch.; 6th. N. 5° E. 11.25 ch.; 7th. S. 83° E. 6.48 ch.; to the place of beginning. Required the area. Fig. 75, is a plot of this survey: and by drawing the lines as in the plot, it is divided into two trapeziums AGFE, AEDF, and a triangle BDC. Measure the

164

CONTENT OP LAND*

[CHAP. III.

several bases and perpendiculars, on the same scale that was used in the protraction, and find the double areas of the triangle and trapeziums by probs. 2 and 6; the sum of these will be the double area of the survey. Bases. Perpens.

EG 16.68 x | ^ | =203.6628=2 AGFE EB 19.17 x | BD 19.23 X

IQ

Ce 5.16

j =267.4215=2 AEDB = 99.2268=2 BDC 2)570.3111 ch.=2 ABCDEFG 285.15555 ch.=28A. 2R. 2P. = the area required. EXAMPLE 2.

The following field-notes are given to protract the survey and find the area. Ch. 1. N. 15° 00* E. 20 2. N. 37° 30' E. 10 3. East 7.50 4. S. 11° 00' E. 12.50 5. South 13.50 6. West 10. 7. S. 36° 30' W. 10. 8. N. 38° 15'W. 8.50

An$. 46 A. 2 R. 9 P

CHAPTER IV.

LAYING OUT AND DIVIDING LAND. PROBLEM I.

To lay oat a given quantity of land in a square form. ♦

RULE.

Reduce the given quantity to chains or perches, and extract the square root, which will be the length of a side, of the same denomination to which the given quan¬ tity is reduced. EXAMPLES.

1. Required the side of a square that shall contain 9 A. 3R. 28 P. 40)28 Per. 4)3.7 R. 9.925 A.=99.25 ch. Ch. 99.25(9.96 ch. the length of a side. 81 189)1825 1701 1986)12400 11916 484 165

166

LAYING OUT • AND DIVIDING LAND.

[CHAP. IV.

2. Required the side of a square tract of land that shall contain 325 acres. Ans. 57 chains.

PROBLEM II. To lay out a given quantity of land in a rectangular form, having one side given. RULE.

Divide the given content by the length of the given side, the quotient will be the length of the required side. EXAMPLES.

1. It is required to lay out 120 acres in a rectangumr form, the length of one side being given, equal 100 perches. Acres. 120 4 480 40 1,00)192,00 192 Per. the length of the other side, 2. The length of a rectangular piece of land is 8 chains; what must be its breadth, that the content may be 5 acres ? Ans. 6.25 chains.

CHAP. IV."1

167

LAYING OUT AND DIVIDING LAND.

PROBLEM III. To lay out a given quantity of land in a rectangular form, having the length to the breadth in a given ratio. RULE.

As the less number of the given ratio, Is to the greater; So is the given area, To a fourth term.* The square root of this fourth term will be the length required. Having the length, the breadth maj be found by the preceding problem. Or it may be found in the same manner as the length. Thus, As the greater number of the given ratio, Is to the less; So is the given area, To a fourth term. The square root of this fourth term will be the breadth required. '

EXAMPLES.

1. It is required to lay out 864 acres in a rectangular form, having the length to the breadth in the ratio of 5 to 3. * DEMONSTRATION. Let ABCD, Fig. 85, be a rectangle, and ABFE and AIIGD be squares on the greater and less sides respectively : then (1.6) AD : AE (AB) : : the rectangle AG : square AF. Also AB : All (AD) : : the rectangle AC : square AG. Hence the truth of the rule is evident.

Fig. 85. 35

F

J)

G

A]

B

-

168

LAYING OFT AND DIVIDING LAND.

[CHAP. IV.

864 A.=138240 P. Sq. P. Sq. P. As 3 : 5 : : 138240 : 230400 \Z23040Q=480 Perches, the length required. Sq. P. Sq. P. As 5 : 3 : : 138240 : 82944 V82944 =288 Perches, the breadth required. 2. It is required to lay out 27 A. 3 R. 20 P.in a rect¬ angular form, having the length to the breadth in the ratio of 9 to 7. Ans. Length 75.725 P. Breadth 58.897 P. PROBLEM IV. To lay ov,t a given quantity of land in a rectangular form, having the length to exceed the breadth by a given difference. RULE.

To the given area, add the square of half the given difference of the sides, and extract the square root of the sum; to this root, add half the given difference for the greater side, and subtract it therefrom for the less.* Fig. 86. * DEMONSTRATION. Let ABCD, Fig. 86, be a rectangle ; in DC let DE be taken equal DA or j\ BC, and let EC be bisected in F ; then (6.2) DF2 | \ —DCXDE+FC3=DCXAD+FC2=the rectangle j \ AC+the square of half the difference of the sides DC, DA; also DF+FC=DC, the greater side, and DF—FC = DE or DA, the less side. This problem may be neatly constructed thus: take EC equal the given difference of the sides and bisect it in F; make EG perpendicular to EC and equal to the square root of the given area, and with the centre F and radius FG describe the arc DG meeting CE produced in D : make DA perpendicular to DC and equal to DE, and complete the rectangle ABCD, which will be the one required. Since (47.1) FG2=EG2 + EF2=the given area + the square of half the given difference of the sides, the truth of the construction is plain, from the preceding demonstration.

CHAP. IV.]

LAYING OUT AND DIVIDING LAND.

169

EXABIPLES.

1. It is required to lay out 47 A. 2 R. 16 P. in a rect¬ angle, of which the length is to exceed the breadth by 80 perches. 47 A. 2 R. 16P.=7616 Per. 2)80 P. — 1600 40 ^9216=96 40 — half diff. add and subtract 40 — 1600 length 136 breadth 56 2. It is required to lay out 114 A. 2R. 33.4 P. in a rect¬ angular form, having the length to exceed the breadth by 15.10 ch. Ans. Length 42.25 ch. Breadth 27.15 ch. PROBLEM V. To lay out a given quantity of land in the form of a tri¬ angle or parallelogram, one side and an adjacent angle being given. RULE.

For a triangle. As the rectangle of the given side and sine of the given angle, Is to twice the given area; So is radius, To the other side, adjacent to the given angle. Then having two sides and the included angle given, the other angles and side, if required, may be found by trig, case 3. Y 15

170

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

For a parallelogram. As the rectangle of the given side and sine of the given angle, • Is to the given area; So is radius, To the other side, adjacent to the given angle.* EXAMPLES. Fig. 87 C

1. Let AB, BC, Fig. 87, be two sides of a tract of land; the bear¬ ing of AB is S. 874° W. dist. 16.25 ch. and the bearing of BC, N. 274° E.; it is required to lay off 10 acres by a straight line AD, running from the point A, to the side BC. Bearing of BA, N. 874° E. BC, N. 274 E.

Angle B, 60° AB 16.25 ch. - - Ar. Co. 8.78915 As ABXsin. B sin. B, 60° - - - 0.06247 : twice the given area 200 sq. ch. - - - 2.30103 : : rad. - -- -- -- -- -- - 10.00000 : BD 14.21 ch.

1.15265

* DEMONSTRATION. It is demonstrated, prob. 3, chap. 3, Content of Land, that rad. : sin. B : : ABXBD : 2ABD (see Fig. 87); therefore (1.6 cor.) rad.XAB : sin. BXAB : : ABXBD : 2ABD, or (16.5) sin. BXAB : 2ABD : : rad.XAB : ABXBD : : rad. : BD. Since ABDF is equal to 2ABD, the truth of the rule for the parallelogram is evident. This problem may be constructed as follows; take AB equal the given side and draw BC making the angle B equal to the given angle; make BE perpendicular to AB, and equal twice the given area of the triangle divided by the given side, or equal the given area of the parallelogram divided by the. given side; and parallel to AB, draw EF cutting BC in D, and join DA; then will ABD be the triangle required; or complete the parallelo¬ gram ABDF, for the one required. The reason of the construction is plain.

CHAP. IV.]

LAYING OUT AND DIVIDING LAND.

171

2. Given the side AB, Fig. 15, of a p equal 20 eh. and the angle A 63° 30'; required the side AC, that the content may be 21 £ acres. A »r> . • » AB A 20 63 ch.o As ABxsin. A |$ gin>

3(),

Ar. Co.

: the given area 215 sq. ch. :: rad. -----: AC 12.01 ch.

8.69897 0.04821 2.33244

10.00000 1.07962

3. Given one side of a triangle, equal 30 perches, an angle adjacent to this side 71° 15', and the area 2 acres ; required the other side adjacent to the given angle. Ans. 22.53 perches. 4. Given one side of a parallelogram, equal to 32.26 ch., an angle adjacent to this side 83° 30', and the area 74 acres; required the other side adjacent to the given angle. Ans. 23.09 ch.

PROBLEM VI. The area and base of a triangle being given, to cut off a given part of the area by a line running from the angle opposite the base. RULE.

As the given area of the triangle, Is to the area of the part to be cut off; So is the given base, To the base corresponding to that area.* * The truth of this rule is manifest from 1.0.

172

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

EXAMPLES. Fig. 88.

c

A

F

1. Given tlie area of the triangle ABC, Fig. 88, equal 650 square perches, and the length of the base AB, 40 perches; it is required to cut off 290 perches to\ wards the angle A, by a line running from the angle C to the base. ABC. ADC. AB. AD. As 650 : 290 : : 40 : 17.85 per.

2. In a triangle ABC, there are given the area 27 A. 1R. 16 P. and the base AB 85.20 ch., to cut off 10 acres towards the angle B, by a line CD running from the angle C to the base: the part BD of the base is required. Ans. 12.87 ch. PROBLEM VII. The area and two sides of a triangle being given, to cut off a triangle containing a given area, by a line running from a given point in one of the given sides, and falling on the other. RULE.

As the given area of the triangle, Is to the area of the part to be cut off; So is the rectangle of the given sides, To a fourth term. Divide this fourth term by the distance of the given point from the angular point of the two given sides; the quotient will be the distance of the required point from the same angle.* * DEMONSTRATION. From the demonstration to prob. 3, chap. 3, we have, Fig. 89, rad. : sin. A : : ABx AC : 2ABC, and rad. : sin. A : : APxAG: 2APG; therefore (11 & 16.5) 2ABC : 2APG : : ABx AC : APxAG, or (15.5) ABC : APG : : ABx AC : APx AG; hence the truth of the rule is manifest.

CHAT. IV.]

LAYING OUT AND DIVIDING LAND.

173

EXAMPLES.

1. Given the area of the triangle ABC, Fig. 89, 5 acres; the side AB 50 perches, the side AC 40 perches, and the distance of a point P from the angle A, 36 jerches; it is required to find a point G to which, if a ine be drawn from the point P, it shall cut off a triangle APG containing 3 A. 0 R. 20 P. As the triangle ABC 800 sq. p. : the triangle APG 500 - •:ABXAC^B 50 - :

APx AG - AP 36 - AG 34.72 per.

Ar. -

Co. -

7.09691 2.69897 1.69897 1.60206

3.09691 log. 1.55630 -

1.54061

2. Given the area of a triangle ABC, 12 A. 1 R. 23 P. the side AB 20 ch., the side AC 16.25 ch., and the dis¬ tance of a point P in the side AB, from the angle A 8.50 ch.; it is required to find the distance AG of a point G in the line AC, so that a line drawn from P to G may cut off a triangle APG containing 3 acres. Ans. 9.25 ch. PROBLEM VIII. T

Fhe area and base of a triangle being given, to cut off a triangle containing a given area, by a line running pa¬ rallel to one of the sides. RULE.

As the given area of the triangle, Is to the area of the triangle to be cut off; So is the square of the given base, To the square of the required base.

174

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

The square root of the result will be the base of the required triangle.* EXAMPLES. Fig. 90. c

1. Given the area of the triangle ABC, Fig. 90, 500 square perches, and the base AB 40 perches; it is required to cut off 120 sq. per. towards the angle A, by a line DG running parallel to the side BC.

As the triangle ABC 500 - - - Ar. Co. 7.30103 2.07918 : the triangle ADG120 1.60206 JAB 40 : : AB2 (AB 40 ------ 1.60206 : AD2 AD 19.6 per.

2)2.58433 -

1.29216

2. Given the area of a triangle ABC, 10 acres, and the base AB 25 eh., to find BD a part of the base, so that a line DG running from the point D, parallel to the side AC, may cut off a triangle BDG containing Ah acres. Am. BD=16.77 ch.f * The truth of this rule is manifest from 19.6. This problem may be neatly constructed as follows: Let ABC, Fig. 90, be the given triable, and AB the given base; on AB describe the semi¬ circle AEB, and take AF to AB in the ratio of the part to be cut off, to the whole triangle; draw FE perpendicular to AB, meeting the semicircle in E, join AE, and make AD equal to AE ; from D draw DG parallel to BC, and the thing is done. For, join EB, and we have, by similar triangles, AB : AE : : AE : AF; therefore (20.6 cor. 2) AB : AF :: AB2: AE2 (AD2) : : [19.6] ABC : ADG. f If it be required to produce two sides of a given triangle so far that the triangle formed by these sides produced, and a line drawn betweon them parallel to the third side, may contain a given area, it may be done by the above rule. Thus, Fig. 90, ADG : ABC : : AD2: AB2.

' CHAP.

IV.]

LAYING

OUT

AND DIVIDING LAND.

175

PROBLEM IX. The hearings of two adjacent sides AD, AE, Fig. 91, of a tract of land being given, to cut off a triangle ABC containing a given area by a line BC running a given course. RULE. A

B

D

From the given bearings of the lines, find the angles A, B, and C; then, As the rectangle of the sines of the angles A and B, Is to the rectangle of radius and sine of the angle C; So is twice the given area,’ To the square of the side AB.* In like manner the other sides may be found; or having found one side, the others may be found by trig, case 1. EXAMPLES.

1. Let the bearing of AD, Fig. 91, be N. 87° 30' E. and of AE, N. 27° 30' E.; it is required to cut off 10 acres by a line BC running N. 38° W. • * The truth of this rule is evident from the demonstration to prob. 4, chap. 3. Construction. Draw AD, AE, (Fig, 92,) according to the given bearings, and in AD take AF equal the square root of the given area, and on it describe the square AFGII; make IE = AI, and draw ED, according to the reverse bearing of the division line BC, meeting AD in D; on AD describe a semicircle, and produce GF to meet it in K, join AK and make AB equal to it; draw BC parallel to DE, and ABC will be the triangle required. For join IF, EF and KD; then (31.3, and cor. 8.G) AD : AK (AB) : : AK (AB) : AF; or (cor. 19.6) AD : AF :: ADE : ABC; but (1.6) AD : AF : : ADE : AFE; therefore (11.5) ADE : ABC : : ADE : AFE, and consequently (9.5) ABC = AFE; but because AI = IE, AFE = 2AFI — (41.1) AFGII; therefore ABC = AFGII —the given area of the triangle.

176

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

AD, N. 87° 30' E. BA, S.:87° 30' W. CA, S. 27° 30'W. AE, N. 27 30 E. BC, N. 38 00 W. CB, S. 38 00 E. Angle A, 60 00

125 30 180 00

Angle C, 65 30

Angle B, 54 30 A sin. • AS/ As A Xsm. TJ

v : rad. X sm. n C,

fA 60° 00' ^ 54 30 -

Ar. Co.

J C 65 30 - - - < ra(j

: : twice the given area, 200 sq. ch.

0.06247 0.08931 9.95902 10.00000 2.30103

: AB2 - - - - 2)2.41183 AB 16.07 1.20591 2. Given the bearing of one side of a tract of land, S. 53° 15' E., and the bearing of an adjacent side taken at the same angle, N. 55° 00' E., to cut off 4 acres bj a line running N. 4° 00' W.; required the distance on the first side. Am. 9.76 ch. PROBLEM X. The bearings of three adjacent sides, EA, AB, BF, Fig. 93 or 94, of a tract of land, and the length of the middle side AB, being given, to cut off a trapezoid ABCD, containing a given area, by a line DC, parallel to AB. RULE.

From the given bearings find the angles A and B; add these together, and take the difference between their sum and 180°, and call it P. Then, As the product of the sines of A and B, Is to the product of radius and sine of P:

CHAP. IV.]

LAYING OUT AND DIVIDING LAND.

177

So is twice the area to be cut off, To a fourth term. When the sum of the angles A and B is greater than 1S0°, add this fourth term to AB2; but when the sum of these angles is less than 180°, take the difference between this fourth term and AB2. The square root of the re¬ sult will be DC. Then, As the sine of P, Is to the sine of B, So is the difference between DC and AB, To AD.* * DEMONSTRATION. Produce EA and FB, Fig. 95, to meet in P. Then (19.6) PDC : PAB : ; CD2: AB2, or (17.5) ABCD : PAB : : CD2—AB2 : AB2, or (A.5) PAB : ABCD : : AB2: CD2—AB2, or (15.16.5) 2PAB : AB2: : 2ABCD : CD2—ABs. But by the demonstration to prob. 4, chap. 3, 2PAB : AB2: : sin. AXsin. B : rad.Xsin. P. Consequently, (11.5)

sin. AXsin. B : rad,Xsin. P : : 2ABCD : CD2—AB2.

The latter part of the rule does not require demonstration. Construction. Draw AI, Fig. 93, perpendicular to AB, and make it equal to the quotient of twice the given area divided by AB. From I, draw III parallel to AB meeting AE and BF, in G and II, and on GII, describe the semicircle GMII. From A, draw AL parallel to BF ; and make LM perpen¬ dicular to GII. With the distance IIM and centre II, describe the arc MN; and from N, draw ND parallel to AL. Lastly, draw DC parallel to AB, and it will be the division line required. For join BD, BG, and CG, Fig. 95. Then by similar triangles PG : PD : : GII : DC : : GII : IIM : : IIM : IIL : : DC : AB : : PC : PB. Hence (3.6) CG is parallel to BD; and consequently the triangle BDC is equal to BGD. To each of these, add ABD. Then wo have ABCD = ABC. But it is plain from the construction that ABG is equal to the given area. Hence ABCD is equal to the given area. When the sum of the angles A and B is less than 180°, as in Fig. 94,

178

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

EXAMPLES.

1. Given the bearing of EA, Fig. 93. West, AB, N. 10° E. dist. 15 ch.; and BF, N. 58° 30' E. to cut off 10 acres by a line CD, running parallel to AB. Required the length of the division line and the distance AD. AE, N. 90° E. BF, N. 58° 30' E. A, 80° O' AB, N. 10 E. BA, S. 10 0 W. B, 131 30 A =80° 48° 30 211 30 180 0 180 0 B=131 30 As sin. A x sin. B,

A, 80° 00' B, 131 30

: Rad. x sin. P, |

"30"

;: twice the given area 200 sq. ch. : fourth term 141.68 AB2=225.

Ar. Co. 0.00665 0.12554 10.00000 - - 9.71809 - - 2.30103 -

-

P=31°30

- -

-



-

- 2.15131

DC= / 366.68=19.15 ch. As sin. P, 31° 30' - - Ar. Co. 0.28191 : sin. B, 131 30 9.87446 : s DC—AB, 4.15 0.61805 :

AD

5.95 - -

-

-

-

- 0.77442

the semicircle must be described on AB; the point L must be deter¬ mined by drawing* GL parallel to FB: and the arc MN must be described with the radius BM and centre B. The other parts of the construction are the same as before.

CHAP. IV.]

LAYING OCT AND DIVIDING LAND.

179

2. Given the bearings of three adjacent sides of a tract of land and the length of the middle one as follows: 1st. N. 20° W.; 2d. N. 60° 30' E. dist, 6 ch.; 3d. S. 61° 30' E.; to cut off a lot containing 2i acres, by a line parallel to the 2d side. Required the length of the division line and the distance on the 1st side. Ans. Division line 8.70 ch.; distance on 1st side 3.45 ch. 3. Given as follows: 1st side N. 31° 15' W.; 2d N. 58° 45' E. dist. 13.50 ch.; 3d S. 14° 45' E.; to cut off 8 acres by a line parallel to the 2d side. The length of the division line and the distance on the 1st side are required. Ans. Division line 11.61 ch.; distance on the 1st side 6.38 ch. PROBLEM XI. The hearings of three adjacent sides EA, AB, BF, Fig. 96, of a tract of land, and the length of the middle side AB being given, to cut off a trapezium, ABCD, con¬ taining a given area, by a line CD, running a given course. RULE.

Draw AS parallel to BF, meeting CD or CD pro¬ duced, in S. From the given bearings, find the interior angles A, B, C, and D; add A and B together, and take the difference between their sum and 180°, and call it P. Then, As the product of the sines of C and D, Is to the product of radius and sine of P; So is twice the area to be cut off, To a fourth term.

180

LAYING OUT AND DIVIDING LAND.

[CHAP. IY.

Also, as the product of the sines of C and D, Is to the product of the sines of A and B; SoisAB2, To a fourth term. When the sum of the angles A and B, is greater than 180°, add these two fourth terms together; but when the sum of these angles is less than 180°, take the difference of the fourth terms. The square root of the result will he CD. Then, As sin. C : sin. B : : AB : CS. The difference between CD and CS, gives DS. As sin. P : sin. C : : DS : AD.* Fig. 97.

(17.6) ARX BU=YW2.

Then,

* DEMONSTRATION. Produce EA and FB, Fig. 97, to meet in P; draw Alt and BU, each pa¬ rallel to CD, and let YW, also parallel to CD, make the trian¬ gle PYW equal to PAB. Then (15.6) PA : PY : : PW : PB; hut (4.6) PA : PY : : AR YW, and PW : PB : ; YW BU. Therefore (11.5) AR YW : : YW : BU, and hence But by trigonometry,

As sin. ARB (sin. C) : sin. B : : AB : AR, sin. AUB (sin. D) : sin. A : : AB : BU. Hence (23:6) As sin. CXsin. D : sin. AX sin. B : : AB2 : ARXBU; Or, sin. CXsin. D : sin. AX sin. B : : AB2 : YW2. And by the demonstration to the rule in the last problem, we have sin. CXsin. D : rad.Xsin. P : : 2YWCD (2ABCD) : CD2—YW2. The sum of these fourth terms gives CD2. The demonstration is similar, when the sum of the angles A and B is less than 180°. Construction. From B, Fig. 96, draw BU according to the reverse bear¬ ing of the division line DC, meeting EA or EA produced in U. Make AI perpendicular to AB, and equal to the quotient of twice the given area, divi¬ ded by AB. Draw IG parallel to AB ; GII parallel to BU ; and UL parallel to BF On GII, describe the semicircle GMII, and make LM perpendicular to GII; with the radius IIM and centre II, describe the arc MN; and from N, draw ND parallel to UL. From D, draw the division line DO parallel to BU.

CHAP. IV.]

18]

LAYING OUT AND DIVIDING LAND.

EXAMPLES.

1. Let the bearings of EA, Fig. 96, be N. 80° 30' W.; AB, North, dist. 12 ch.: and BF, N. 58° K; it is re¬ quired to cut off 10 acres by a line DC, running N. 14° 30' W. AE, S. 80° 30' E. AB, N. 0 00 E.

BF, N. 58° E, BA, S. 0 E.

80 30 180 0

58 180

A= 99° 30/

B=122

CB, S. 58° 00' W. CD, S. 14 30 E. C=72° 30'

A = 99° 30* B=122 00

DA, N. 80° 30'W. DC, N. 14 30 W. D—66° 00'

221 30 180 00 P=41 30

, • • n
_

- Ar. Co.

- - :: twice the given area 200 sq. ch. - : rad. X sin. P,

|

30

.

.

0.02058 0.03927 - 10.00000 - 9.82126 - 2.30103

. fourth term 152.1 ------- 2.18214

When the sum of the angles A and B is less than 180°, the semicircle must be described on BU; the point L must be determined by drawing GL parallel to BF, and the arc MN must be described with the radius BM and centre B. The demonstration is exactly the same as that for the construction of the last problem.

1G

182

LAYING OU

AND DIVIDING LAND.


As sin. Cxsin. D,

: fourth term fourth term

[CHAP. IV.

Ar. Co. -

138.24 152.10

-

-

0.02058 0.03927 9.99400 9.92842 1.07918 1.07918

- 2.14063

DC= y/290.34= 17.04 As sin. C 72° 30' Ar. Co. : sin. B 122 00 : : AB 12 - - -

0.02058 9.92842 1.07298

: CS 10.67 - DC 17.04

1.02818

DS

-

6.37

As sin. P 41° 30' ------ Ar. Co. 0.17874 : sin. C 72 30 _- - 9.97942 : : DS 6.37 - -- -- -- -- -- 0.80414 : AD 9.17

------

0.96230

2. Given the bearings of three adjacent sides of a tract of land and the length of the middle one as follow: 1st., N. 31° 15' W.; 2d. N. 58° 45' E. dist. 13.50 ch.; 3d. S. 14° 45' E.; to cut off 8 acres by a line from the 1st. side to the 3d. running S. 87° 30' E.; required the length of the division line and the distance on the 1st side. Ans. Division line 12.76 ch.; distance 1st. side 2.69 ch. 3. Given as follow; 1st side, N. 74° 45' W.; 2d. N. 37° E. dist. 17.24 ch.; 3d. N. 84° E.; to cut off a field containing 20 acres, by a line from the 1st. side to the 3d-

CHAP. IT.]

183

LAYING OUT AND DIVIDING LAND.

running N. 20° E. Tlie length of the division line and the distance on the 1st side are required. Ans. Division line 19.G8 ch.; distance on 1st side 14.01 ch. PROBLEM XII. The bearings of several adjacent sides, EA, AV, VW, WX, XB, BF, Fig. 98, of a tract of land, and the distance of each, except the first and last, being given, to cut off a given area, by a line DC, running a given course. RULE.

Join AB and calculate the area of AVWXBA, and the bearing and distance of AB. Subtract the area of AVWXBA from the area to be cut off, the remainder will be the area ABCD. Then with the bearings of EA, AB, BF, DC, the distance AB, and the area of ABCD, proceed as in the last problem. X

EXAMPLES.

Fig. 98.

1. Let the bearing F of EA, Fig. 98, be N. 48° 80' W.; AV, S. 78° W. dist. 8 ch.; VW, N. 26° 30' W. dist. 11.08 ch.;WX, N. 38° 30' E. dist. 12.82 ch.; XB, S. 64° E. dist. 10.86 E ch.; and BF, S. 86° E. It is required to cut off 30 acres by a line DC, running N. 32° 15' E. Stat. Bearing. Dist. AV

N.

S. 78° W 8.00

VW N.26.JW 11.08

S.

12.78

12.78 W

126.6498

9.75 W

97.7925

17.74

7.99 E

38.0324

(4.96)

4.80

12.79 E

172.9208

19.94 17.74 17.74

25.57

4.95 7.98

(13.52) 19.94

Multi¬ pliers. N.Ar. S. Areas. 0.00 E

476

BA

E. D.D. W.D.D.

7.83

9.91

S. 64 E. 10.86

w.

12.79

1.66

WX N 38£ E 12.82 10.03 XB

E.

3.03 |

9.76

25.57

435.3955

Sq. ch. 217.697751

184

LAYING OUT AND DIVIDING LAND. Sq.

[CHAP. IV.

ch.

Area to be cut off 300. Area of AVWXBA 217.69775 Area of ABCD

82.30225

As difld of lat. of BA, 13.52 S. Ar. Co. 8.86902 : dep. do. - - 4.96 W. ------ 0.69548 :: rad. - -- -- -- -- -- - 10.00000 : tang of bearing of BA, S. 20° 9' W. -

-

-

9.56450

As rad. - -- -- -- -- Ar. Co. 0.00000 : sec. of bearing 20° 9' ------- 10.02743 :: difF. of lat. 13.52 1.13098 : BA 14.40

1.15841

The angles, found from the bearings, are A=lll° 21', 1° 45', D=80° 45' and P=37<3 30'. .$C,61°45'- - - Ar. Co. 0.05508 As sin. Cxsin. D 0.00568 ( D, 80 45 - - 10.00000 < rad. -----: rad. X sin. P, 9.78445 1 P, 37 30 2.21643 ABCD 164.6 sq. ch. - ..25 : fourth term 115.25

2.06164

$C, 61° 45' - Ar. Co. ’ ) D, 80 45 S A, 111 21 - - - : sin. A x sin. B, £B, 106 9 - - - X AB, 14.40 - - - : : AB2 \ AB, 14.40 - - - -

0.05508 0.00568 9.96912 9.98251 1.15836 1.15836

do. DC-

:.36 165.25

yj 328.61 = 18.13

2.32911

CHAP. IV.]

LAYING OUT AND DIVIDING LAND.

As sin. C, 61° 45' ; sin. B, 106 9 - : : AB, 14.40 - - -

Ar. Co. 0.05508 9.98251 1.15836

: CS, 15.70 As sin. P, 37° 30' : sin. C. 61 45 - : : DS, 2.43 - - - -

1.19595 Ar. Co. 0.21555 9.94492 0.38561

185

: AD, 3.52 .... 0.54608 2. Given as follow; 1st side, N. 62° 15' W.; 2d N. 19° E. dist. 18 ch.; 3d S. 77° E. dist, 15.25 ch.; 4th S. 27° E.; to cut off 35 acres by a line, from the first side to the last, running N. 82° 30' E. Required the length of the division line, and the distance on the first side. Am. Division line 22.98 ch.; distance on 1st side 5.14 ch. PROBLEM XIII.

ings of ;ill the given sides, so as to make the side, on which the division line is to fall, a meridian. 1(5* 2 A

186

LAYING OUT AND DIVIDING LAND.

[CHAP. IV

With the given distances and changed bearings, find the corresponding differences of latitude and departures; add together the numbers in each departure column, and take the difference of their sums, which will be the de¬ parture of the division line, and must be placed in the proper column, opposite said line. Then having all the departures, find the double departures, as in Prob. 11, of the last chapter. Find also the multipliers, beginning with the one to correspond with the division line, and assuming it 0; multiply the known differences of latitude by their corresponding multipliers, and place the pro¬ ducts in the proper columns of north and south areas. Add together the products in each of the columns of areas, and subtract the less sum from the greater; take the difference between the remainder and double the area to be cut off, and divide it by the multiplier correspond¬ ing to the side on which the division line is to fall; the quotient will be the difference of latitude of this side, which place against it, in the column of north or south latitude, according as its changed bearing is north or south. Then add together the numbers in each latitude column, and take the difference of their sums, which will be the difference of latitude of the division line, of the same name with the less sum. With the difference of latitude and the departure of the division line, find, by Prob. 10, chap. 1, its changed bearing and its distance. Then find the true bearing by note to the rule in Prob. 9, chap. 1.* EXAMPLES.

s

1. Let the bearing of AB be N. 62i° W. 14.75 ch.; BC, N. 19° E. 27 ch.; CD, S. 77° E. 22.75 ch.; and DE,S. 27° E.; it is required to cut off 70 acres by a line All, run¬ ning from the angle A and falling on the side DE. * The reason of this rule is sufficiently obvious without a demonstration.

HAP. IV.}

187

LAYING OUT AND DIVIDING LAND. “ X >

o

x CD

bO W

ol

a

> a

F

cc 1

? , .

25

X o

CO

■H*-

<» <* H

w

©

to o

GO a?

cr

CD

e:i c

p

era «

ao

©

©

©

m

H

bo bO

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© © •H-

o

CD

CC

©

CO

N.

© 00 ©

CO © bo ©

© © *— © ©

Ol

bo bO s

© CD ©

>4-

pr

^ P c* 3 -H
©J

o

<1 © to © 0*

CO

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bO

/



CO

1—< -a 0*

E.

03 bo

© 03

bo

b5

© bS

bO 00 CO

00 ©

CO

3 H

© pi

© ©

b5 © pi i—i

bo rf*

©

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CD i—<

b

00

bS

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CO

GO

k

b tf* b

© bO

© 00 © CO © w w w

h_

b5 ©

CD

CO CO

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w © bo tr* 1 © • 25 >’

© CD CO 00

O cO QO )fk

©0* <1*-* b» bo

CO CO -vj ©

CO V—X bO CO

00

t-*

o

CO CO ^ CO CD

►4*

As diff. lat. of IIA, 5.22 S. dcp. do. 28.33 W. rad.

£ © © © © © ©

© 00

CD CD

© ^ ©

© CD © © bO

00

© © 00

U1

© © © © ©

> CD

g

Ar. Co. 9.28233 - - - 1.45225 - - 10.00000

: tang, changed hear, of IIA, S. 79° 34' W. 1073458 Subtract 27 00 True bearing of IIA, S. 52 34 W,

188

LAYING OUT AND DIVIDING LAND.

[CHAP. TV.

As rad. Ar. Co. 0.00000 : sec. changed bearing of HA, 79° 34' - - 10.74210 : : diff. lat. 5.22 0.71767 : dist. AH, 28.83 - - 1.45977 Hence AH, bears N. 52° 34' E. dist. 28.83 cb. 2. Given as follow: 1st side S. 78° W. 8 ch.; 2d N. 261° W. 11.08 cb.; 3d N. 381° E. 12.82 ch.; 4th S. 64° E. 10.86 ch.; 5th S. 231° E.; to cut off 25 acres by a line running from the place of beginning, and falling on the 5th side; required its bearing and distance. Am. N. 45° 1' E. dist. 10.67 ch.

Fig. m

PROBLEM XIY. T7ie ddes AB, BC, CA, Fig. 100, of a triangular piece of ground being given, to divide it into two parts having a given ratio, by a line FE, running parallel to one of the sides as BC. RULE.

As the sum of the numbers expressing the ratio of the parts, Is to that number of the ratio which corresponds to the part to be adjacent to A; So is the square BC, To the square of FE. Then, As BC : AB : : FE : AF.* * DEMONSTRATION. Let m to n be the ratio of the part AFE to the part FECB; then (18.5) m + ABC : ADE : (19.G) BC2: FE2.

Construction, On AB describe the semicircle AMB, and by Prob. IT, Page 3G, divide AB in K, so that AK may be to KB in the given ratio of the part AFE to the part FECB ; draw KM perpendicular to AB, meeting the

CHAP. IV.]

189

LAYING OUT AND DIVIDING LAND.

EXAMPLES.

1. Let AB be 21.26 ch.; BC, 12.76 ch.; and AC, 19.30 ch.; it is required to divide the triangle by the line FE, parallel to BC, so that the part AFE may be to the part FECB as 2 to 3. As 5 : 2 : : 12.762 : FE2= 65.12704. FE=V65-12704 = 8-07As 12.76 : 21.26 :: 8.07 : AF=13.45. 2. The three sides of a triangular piece of land, taken in order, measure 15,10, and 13 chains respectively; it is required to divide it into two equal parts by a line parallel to the second side. What will be the length of the division line and its distance from the place of beginning, measured on the first side ? Ans. Division line 7.07 ch.; dist. on 1st side 10.61 ch. PROBLEM XV. The hearings and distances of the sides AB, BC, CA, Fig. 101, of a triangular piece of ground being given, to divide it into two parts having a given ratio, by a line FE, running a given course.

Fi 101

s-

-

* RULE.

As the product of the sines of F and E, Is to the product of the sines of B and C; semicircle in M, and with the radius AM and centre A, describe the arc MF. From F, draw the division line FE parallel to BC. Since, (35.3, and cor. 8.G) AB : AM (AF) : : AM (AF): AK, we have (20.6 cor. 2) AB : AK :: AB2: AF2:: ABC : AFE. Hence the truth of the construction is evident.

190

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

So is the square of BC, To a fourth term. Multiply this fourth term by that number of the ratio which corresponds to the part to be adjacent to the angle A, and divide the product by the sum of the numbers expressing the ratio. The square root of the result will he FE. Then, As sin. A : sin. E : : FE : AF.*

EXAMPLES.

1. Let the bearing of AB, be S. 821° E. dist. 14.17 ch.; BC, N. 181° W. 8.51 ch.; and CA, S. 61i° W. dist. 12.87 ch.; it is required to divide the triangle by the line FE, running N. 141° E. so that the part AFE may be to the part FECB in the ratio of 2 : 3. Eig. 102.

R

* DEMONSTRATION. Draw CG and BR, Fig. 102, parallel to EF; and let YW, also parallel to EF, make the triangle AYW equal to ABC. Then (15.6) AB : AY : : AW : AC; but (4.6) AB : AY : : BR : YW, and AW : AC : : YW: GC. Therefore BR : YW : : YW : GC ; and hence (17.6) BR X GC = VW2. But by trigo¬ nometry, As sin. R (sin. E) : sin. C : : BC : BR, sin. G (sin. F) : sin. B : : BC : GC.

Hence (23.6) sin. EX sin. F : sin. BXsin. C : : BC2: BRXGC : : BC2: YW2. Also, m + ABC : AFE : : AYW : AFE : : (19.6) YW2: FE2. Construction. From C, Fig. 101, draw CG according to the reverse bear¬ ing of FE, and on AG describe the semicircle AMG. By prob. 17, page 36, divide Ap in K, so that AK may be to KB in the given ratio of the part AFE to the part FECB. Draw KM perpendicular to AB, and with the radius AM and centre A, describe the arc MF. From F, and parallel to GC, draw FE the required division line. For join KC, Fig. 102; tfien (1.6) KC divides the triangle in the given ratio. Now AC : AE : : AG : AF (AM) : : (cor. 8.6) AM (AF) : AK; therefore (11.5) AC : AE : : AF : AK, and hence (15.6) the triangle AFE is equal to AKC. Consequently FE divides the triangle in the given ratio.

CHAP. IV.]

191

LAYING OUT AND DIVIDING LAND.

Angle A=364°, B=631°. 0=804°, E=461° and 1=83° “ 46° 45' AR. Co. 0.13765 fE As sin. E X sin. E, s -^3 gg QQ 0.00325 ... - 9.95179 ' B’ 63 30 : sin. B X sin. C, 1 C, 80 15 9.99368 j BC, 8.51 0.92993 2 : : BC , - - - - 0.92993 ]BC, 8.51

J B,

: fourth term

88.35

------- 1.94623

2 5)176.70 FE=\/ 35.34=5.95 As sin. A, 36° 15' ------ Ar. Co. 0.22819 : sin. E, 46 45 - -- -- -- -- 9.86235 : : FE, 5.95 0.77379 : AF,

7.32

0.86433

2. The bearings and distances of a triangular piece of land ABC are, AB, S. 69° E. 21.40 ch.; BC, N. 311 E. 18.66 ch.; and CA, S. 741 W. 30.85 ch.; and it is re¬ quired to divide it by a line FE, running due north, so that the part AEF, may be to the part FECB, as 4 to 5. What will be the length of the division line FE, and the distance AF ? Am. FE 10.74, and AF 17.40. PROBLEM XVI. The hearings and distances of the sides AB, BC, CD, DA, Fig. 103, of a trapezoidal tract of land being given, to divide it into two parts having a given ratio, by a line FE, running parallel to the parallei sides AB, CD.

Fi 103

s-

-

RULE.

Multiply the square of AB, by that number of the

192

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

ratio which, corresponds to the part, to be adjacent to CD, and the square of CD, by the other number of the ratio. Add the two products together, and divide the sum by the sum of the numbers expressing the ratio. The square root of the quotient will give FE. Then, As DC—AB : FE—AB : : AD : AF.* EXAMPLES.

1. Let the bearing of AB be N. 14° E. dist. 10 ch. BC, N. 55i° E. dist. 18.67 ch.; CD, S. 14° W. dist. 20.98 ch.: and DA, W. 12.70 ch.; it is required to divide the trape¬ zoid into two parts by a line FE, parallel to AB or DC, Fig. 104.

c

* DEMONSTRATION. Produce DA and CB, Eig. 104, to meet in P, Then (19.6) PDC : PAB : : CD2: AB2, or (17.5) ABCD : : PAB : : CD2—AB2: AB2, or (16.5) ABCD : CD2—AB2: : PAB : AB2. In like manner ABEF: FE2—AB2: : PAB : AB2. Hence (11.5) ABCD : CD2 —AB2: : ABEF : FE2—AB2, or (16.5) ABCD : ABEF : : CD2—AB*: rE2—AB2. Therefore m + n : m : :

CD2—AB2: FE2—AB2, or multiplying extremes and means, m + n FE2— m + n. AB2=m. CD2—m. AB2. But (m + n.) AB2 = m. AB2 + n. AB2. There¬ fore, adding equals to equals, we have m + n. FE2=m. CD2 + n. AB2. Hence the truth of the rule is evident. Construction. Join CA, Fig. 103, and parallel to it, draw BII, meeting DA produced in IF. Divide (prob. 17, page 36) I1D in K so that IIK may be to KD in the given ratio of ABEF to FECD, and draw KL parallel to BC. On CD, describe the semicircle CMD, and draw LM perpendicular to CD. With the radius CM and centre C, describe the arc MN, and from N draw NF parallel to KL. From F, draw the division line FE parallel to AB or CD. For join IvC, Fig. 104, and draw KU parallel to AB. Then since BII is parallel to AC, the triangle AIIC is equal to ABC; and adding ADC to each, we have CUD = ABCD. Now (4.6) PC : PE : : CD : FE (CM):: (cor. 8.6) CM (EF) : CL (UK) : : PF; PK. Therefore (11.5) PC : PE : : PF : PK; and hence (15.6) the triangle PEF is equal to PCK, Consequently CKD = FECD. But it has been proved that CIID = ABCD; hence taking equals from equals we have 1ICK=ABEF. But (1.6) IICK : CKD : : IIK : KD : : m : n. Therefore ABEF : FECD ::m: n.

CHAP. IV.]

193

LAYING OUT AND DIVIDING LAND.

so that the part ABEF may he to the part FECD as 3 to 2. 2. AB3=200 3. CD2=1320.4812 5)1520.4812 FE= V304.0962=17.44 As 10.98 : 7.44 : : 12.70 : AF = 8.61 2. The boundaries of a trapezoidal field ABCD are given as follow; viz. AB, N. 80° W. 60 per.; BC, N. 39 i° W. 45.5 per.; CD, S. 80° E. 89.4 per.; and DA, South, 30 per.; and it is required to divide it into two equal parts by a line FE parallel to AB or CD. What will be the length of the division line FE, and the distance AF? Ans. FE 76.13 per., and AF 16.46 per.

PROBLEM XVII. The hearings and distances of the sides AB, BC, CD, DA, Fig. 105, of any quadrilateral tract of land being given, to divide it into two parts having a given ratio, by a line H' FE, running parallel to one of the sides as AB or CD.

Fig. 105.

A

K

F

o

a

D

RULE.

Call the side to which the division line is to be paral¬ lel, the parallel side; and the one opposite to this, the opposite side. From the bearings, find the angles. Take the difference between the sum of the angles adjacent to the parallel side and 180°, and call it P. Then, 17

2B

194

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

As the product of the sines of the angles adjacent to the parallel side, Is to the product of the sines of the angles adjacent to the opposite side: So is the square of the opposite side, To a fourth term. Multiply this fourth term by that number of the ratio which corresponds to the part to be adjacent to the parallel side, and to the product add the product of the square of the parallel side by the other number of the ratio; and divide the sum by the sum of the numbers expressing the ratio. The square root of the quotient will be the length of the division line FE. Then, As the sine of P, Is to the sine of E; So is the difference between FE and the parallel side, To the distance of F from the adjacent end of the parallel side.* Fig. 106. R

* DEMONSTRATION. Produce DA and CB, Fig. 106, to meet in P. Draw DR and CG each parallel to AB ; and let VW, also parallel to AB, make the triangle PVW equal to PCD. Then (15.6) PD : PY : PW : PC. But (4.6) PD : PY : DR : VW, and PW : PC : : YW CG. Therefore (11.5) DR : YW : : YW : CG; and hence (17.6) DRXCG =VW2. But by trigonometry, As sin. CRD (sin. B) : sin. C : : CD : DR. sin. CGD (sin. A) : sin. D : : CD : CG. Hence (23.6) sin. AX sin. B : sin. CXsin. D : : CD2: DRXCG. Or, sin. AX sin. B : sin. CXsin. D : : DC2: YW2. But by the demonstration to the rule in the last problem, we have m + n. FE2=m. VW2+n. AB2. Hence the truth of the rule is evident. Construction. From C, Fig. 105, draw CG parallel to AB, and on it de scribe the semicircle CMG.

CHAP. IV.J LAVING OUT AND DIVIDING LAND.

195

EXAMPLES.

1 Let the bearing of AB be North, 12 ch.; BC, N 56£° E. 20.78 ch.; CD, S. 33i° E. 22.21 ch.; and DA, S. 80^° W. 30 ch.; it is required to divide the tract into two parts by a line FE, parallel to AB, so that the part ABEF may be to the part FECD as 3 to 5. Angle A=80J°, B=123i°, C=90°, D=66°, and P=24°. Ar. Co. 0.00600 (A, 80°30' As sin. A x sin. B, JB, 123 30 0.07889 10.00000 C, 90 00 : sin. Cxsin. D, - - - 9.96073 }D, 66.00 - - - 1.34655 C CD, 22.21 2 ::CD - - - 1.34655 CD, 22.21

\

I

: fourth term

547.92 3

------ 2.73872

1643.76 5AB2=720.00 8)2363.76 FE= y/ 295.47=17.19 As sin. P, 24° 00' Ar. Co. 0.39069 : sin. E, 123 30 ------ 9.92111 :: FE-AB, 5.19 0.71517 : AF, 10.64

1.02697

Join CA, and draw BH parallel to it meeting DA produced in H. Divide HD in K, so that I IK may be to KD in the given ratio of the part ABEF to FECD. Draw KL parallel to BC and LM perpendicular to CG. With the radius CM and centre C, describe the arc MN. Draw NF parallel to KL and FE parallel to AB. Then will FE be the division line. When the di¬ vision line is to be parallel to CD, the semicircle must be described on CD; and the line CG need not be drawn. The demonstration of the construction is the same as for the last problem.

196

LAYING OUT AND DIVIDING LAND.

[CHAP. TV.

2. The boundaries of a field ABCD are given as fol¬ low : viz. AB, S. 101° W. 7.20 ch.; BC, S. 67° W. 12.47 ch.; CD, N. 23° W. 13.33 ch.; and DA, S. 89° E. 18 ch.; and it is to be divided into two parts by a line FE, paral¬ lel to the side AB, so that the part ABEF may be to the part FECD, as 3 to 4. Required the length of the divi¬ sion line FE, and the distance AF. Am. FE 10.69 ch.; and AF 7.15 ch. 3. Given the boundaries of a field the same as in the preceding example, to divide it into two parts by a line FE, parallel to the side CD, so that the part ABEF may be to the part FECD, as 3 to 4. Required FE and AF Am. FE 10.14, and AF 10.16. PROBLEM XVIII. J’/)e bearings and distances of tike

Fig. 107.^

sides AB, BC, CD, DA, Fig. '107, of any quadrilateral tract of land, being given, to divide it into two parts having a given ratio by a lim FE, running a given course from some point in AD to another in BC. RULE.

From A, draw AS parallel to BC, meeting FE in S. From the bearings find the angles A, B, C, D, E, and F* Take the difference between the sum of the angles A and B, and 180°, and call it P. Then, As the product of the sines of the angles E and F, : the product of the sines of the angles A and B; : : the square of AB, : a fourth term. Also, As the product of the sines of the angles E and F, : the product of the sines of the angles C and D; * It is immaterial whether it is the angle BEF or CEF, that is found, also whether, AFE or DFE.

CHAP. IV.]

LAYING OUT AND DIVIDING LAND.

197

:: the square of CD, : to a fourth term. Multiply this latter fourth term by that number of the ratio which corresponds to the part to be adjacent to AB, and the other fourth term by the other number of the ratio; add the two products together, and divide the sum by the sum of the numbers expressing the ratio. The square root of the quotient will give the length of the di¬ vision line FE. Then, As sin. E. : sin. B : : AB : ES. The difference between FE and ES, gives FS. Then, As sin. P.: sin. E ; : FS : AF.# EXAMPLES.

1. Let the bearing of AB be North, dist. 12 ch.; BC N. 56|° E. 20.78 ch.; CD, S. 33£0 E. 22.21 ch.; and DA, S. 80£° W. 30 ch.; it is required to divide the tract into two parts by a line FE, running N. 20° W. so that the part ABEF may be to the part FECD as 3 to 5. f Angle A=80h°, B-123i°, C=90°, D=66°, E-76|°, F=79i° and P=24°. E, 76° 30' Ar. Co. 0.01217 As sin. E x sin. F, F, 79 30 0.00733 A, 80 30 9.9940C : sin. A X sin. B, B, 123 30 9.92111 AB, 12 1.07918 2 :: AB AB, 12 1.07918

i

: fourth term

123.87 5

2.09297

619.35 ♦ The truth of this rule is evident from the demonstration to the rule in the last problem. Construction. Draw CG according to the reverse bearing of FE, and then proceed with the construction exactly as in the last problem.

17 *

198

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

76° 30' - Ar. Co. 0.01217 79 30 0.00733 90 00 - - - - 10.00000 : sin. Cxsin. D, 5 Ci 66 00 - - - - 9.96073 - - -, - 1.34655 22.21 2 SCB, *****8 CD - - - - 1.34655 (CD, 22.21

As sin. Ex sin. F, SE,

*F,

* fourth term 471.33 ------- - 2.67333 3 1413.99 619.35 8)2033.34 FE= y/ 254.17=15.94 As sin. E, 76° 30' - - - - - - Ar. Co. 0.01217 : sin. B, 123 30 - - 9.92111 1.07918 : : AB, 12 - - - - : ES, 10.29 - - - -

1.01246

As sin. P, 24° 00' - - - - - - Ar. Co. 0.39069 9.98783 : sin. E, 76 30 - - 0.75205 :: FS, 5.65 - - - : AF, 13.51

1.13057

2. The boundaries of a field ABCD are given as fol¬ low : viz. AB, S. 10J° W. 7.20 ch.; BC, S. 67° W. 12.47 ch.: CD, N. 23° W. 13.33 ch.; and DA, S. 89° E. 18 ch.; and it is to be divided into two parts ABEF and FECD, in the ratio of 3 to 4, by a line FE, running due South. Required the length of the division line FE and the distance AF. Ans. FE 10.10 and AF 8.12.

CHAP. IV.]

LAYING OUT AND DIVIDING LAND.

199

PROBLEM XIX. The boundaries of a tract of land ABCDEFGHIA, Fig. 108, being given, to divide it into two equal parts by a line IN running from the corner I, and falling on the opposite side CD.

Fig. 108.

RULE.

Suppose lines drawn from I, to C and D, and calculate the area of the whole tract. Take the corrected latitudes and departures* of IA, AB, and BC, and by balancing find the latitude and departure of Cl; also calculate the area of the part IABCI; from half the area of the whole tract, subtract the area of the part IABCI, the remainder will he the area of the triangle ICNI. Take the latitudes and departures of IC and CD, and by balancing find the latitude and departure of DI, and calculate the area of the triangle ICDI. Then, As the area of the triangle ICDI, Is to the area of the triangle ICNI; So is the latitude of CD, To the latitude of CN. Also, As tli
200

LASING OUT AND DIVIDING LAND.

[CHAP. IV.

Now take the latitudes and departures of IC and CN, and by balancing find the latitude and departure of the division line NI; with which, find its bearing and distance.* EXAMPLES.

1. Let the bearing of AB be N. 19° E. dist. 27 ch.; BC, S. 77° E. 22.75 ch.; CD, S. 27° E. 28.75 ch.; DE, S. 52° W. 14.50 ch.; EF, S. 15£° E. 19 ch.; FG, West, 17.72 ch.; GH, N. 36° W. 11.75ch.; HI, North, 16.07ch., and IA, N. 62° W. 14.88 ch.; it is required to divide the tract into two equal parts by a line IN running from th(^ corner I, and falling on the opposite side CD. First calculate the whole area, thus: * This rule needs no demonstration.

CHAP. IV.j

201

LAYING OUT AND DIVIDING LAND, H-i >

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202

LAYING OUT AND DIVIDING LAND.

[CHAP. IV.

To find the latitude and departure ofi Cl, and area of the part IABCI. Sta. IA

1

N

*

S.

E.

6.97

w.

E.D.D. W.D.D. | Mult. N. A. S. Areas.

13.13

AB 2551

8.80

30.98

0.00 E

4.133

4.33 W i

110.4583

i

BC

5.13 22.18

Cl

(27.35)

30.98

26.65 E

136.7145

4.33

30.98 E

847.3030

(17.85)

32.48 32.48 30,98 30.98

35 31

35.31

1094.4758

Area of IABCI (sq. ch.) 547.2379 Half area of ABCDEFGHIA 762.6990 Area of ICNI (sq. ch.)

215.4611

To find the area ofi ICDL Sta. N.

S.

IC 27.35 CD DI

E.

W.

E.D. D. W.D. D. Mult. N. A. S. Areas. i

13.07

17.85 25.64 13.07 (1.71)

30.92 E

30.92 17.85

(30.92)

27.35 27.35 30.92 30.92

30.92

0.00 E

13.07 E

30.92 Area ofICDI (Sq. ch.)

As area of ICDI, 407.57 Ar. Co. : area of ICNI, 215.46 :: latitude of CD, 25.64 S. : latitude of CN, 13.55 S.

792.7888 22.3497 815.1385 407.5692

7.38980 2.33337 1.40892

------ 1.13209

As area of ICDI, 407.57 Ar. Co. 7.38980 : area of ICNI, 215.46 2.33337 : : departure of CD, 13.07 E. - ----- 1.11628 : denarture of CN, 6.91 E.

0.83945

CHAP. 1V.J

203

LAYING OUT AND DIVIDING

To find, the latitude and departure, (f: NX Sta. N.

S.

E.

W.

17.85

IC 27.35 (

13.55 6.91

CN

(13.80)

NI 27.35

(24.76)

27.35 24.76 24.76

To find the hearing and distance o/’NI. As diff. of lat. of NI, 13.80 S. Ar. Co. 8.86012 : dep. do. 24.76 W. 1.39375 :: rad. - - - - - - - - - - - - 10.00000 : tang, bearing of NI, S. 60° 52' W.

- - - 10.25387

As rad. - -- -- -- -- Ar. Co. 0.00000 : sec. bearing of NI, 60° 52'- - - - -10.31261 : : diff. lat. do. 13.80 - - - - - - 1.13988 : dist. NI, 28.35 ch. - -- -- -- - 1.45249 Hence IN bears, N. 60° 52' E. dist. 28.35 ch. 2. Given the boundaries of a tract of land as follow; VIz. 1st. S. 35i° W. 11.20 ch.; 2d. N. 45° W. 24.36 ch. ; 3d. N. 15£° E. 10.80 ch.; 4th. S. 77° E. 16 ch.; 5th. N. 87F E. 21.50 ch.; 6th. S. 60° E. 14.80 ch.; 7th. South, 10.91 ch.; 8th. N. 85° W. 29.28 ch. to the place of be¬ ginning ; to divide the tract into two equal parts by a line running from the first station and falling on one of the opposite sides; the bearing and distance of the di¬ vision line are required. Ans. N. 7° 18' E. 15.28 ch.

CHAPTER V.

VARIATION OF THE COMPASS.

A MERIDIAN indicated by the magnetic needle is not, in general, a true one; for the needle does not point truly to the north point of the horizon, but varies from it, in some places to the eastward, and in others to the westward. The angle contained between the true meridian and that indicated by the needle, is called the variation of the compass. The variation is named east or west, according as the north end of the needle points to the eastward or west¬ ward of the true north. As the variation is different in different places, so also, in the same place, it does not remain the same, but differs sensibly in the course of a few years. Hence, in running a line that was run a number of years pre¬ viously, the bearing will be found different from what it was at that time; this, together with some difference in compasses, causes many difficulties, and frequently in¬ accuracies, in tracing old lines.

CHAP. V.] VARIATION OF THE COMPASS.

205

The easiest way to guard against those difficulties and inaccuracies would be to make and return the surveys according to the true, and not the magnetic bearings. In order to do this, it will be necessary to know the va¬ riation of the compass for the place in which the survey is made; and this may readily be found by first tracing a meridian line in the following manner. To draio a true meridian line by means of the greatest elongation of the pole star.

The pole star is situated about 1J° from the true pole, and therefore apparently revolves round it, in a small cir¬ cle, once in about 23 h. 56 m. When at its greatest distance east or west from the true pole, it is said to be at its greatest east or west elongation. It is therefore evident that in the course of one apparent revolution it must be twice at its greatest elongation, once to the east and once to the west. The following tables exhibit the times, nearly, of the greatest eastern elongations of the pole star for six months of the year, and of the greatest western elonga¬ tions for the other six months. The other greatest elon¬ gations take place in the day time, and are therefore in¬ visible. Some of those inserted in the tables are also invisible; because they occur, either before daylight is gone, in the evening, or after it has returned, in the morning. The most of those in the 3d, 4th, 9th, and 10th months are in this situation. The time in the tables is reckoned from noon; and therefore when it is less than 12 hours, the greatest elon¬ gation takes place in the evening of the same day; but when it exceeds 12 hours, if 12 hours be subtracted from it, the remainder will be the time of greatest elongation in the morning of the following day.

206

VARIATION OF THE COMPASS.

[CHAP. V.

Eastern Elongation. i Days. 4 mo. (Ap.) 5 mo. (Ma.) 6 mo. (Ju.)^7 mo. (July) 8mo.(Aug.) 9 mo. (Sep.)

1 7 13 19 25

H.

M.

H.

M.

H.

M.

H.

M.

18 17 17 17 16

18 56 34 12 49

16 16 15 15 14

26 3 40 17 53

14 14 13 13 12

24 0 35 10 45

12 11 11 11 10

20 10 55 9 31 9 7 9 43 8

H.

M.

H.

16 8 53 7 30 7 8 7 45 6

M.

20 58 36 15 53

Western Elongation. Days. 10 mo. (Oc.) 11 ma(No.) 12 mo.(De.)|l mo. (Ja.) 2 mo. (Feb.) 3 mo.(Mar.) 1 7 13 19 25

H. M.

H. M.

H. M.

H.

M.

H. M.

H.

M.

18 17 17 17 16

16 15 15 15 14

14 19 13 53 13 27 13 00 12 34

12 11 11 10 10

2 36 10 44 18

9 50 9 26 9 2 8 39 8 16

8 7 7 6 6

1 38 16 54 33

18 56 34 12 49

22 59 35 10 45

In order to determine a true meridian, by the method here used, it is necessary to know the bearing of the pole star, called its azimuth, at the time of its greatest elon¬ gation. This depends on the latitude of the place, and the distance of the star from the pole. This distance is called the polar distance of the star. It is subject to a small annual diminution, which is called its annual preces¬ sion. The polar distance of the star on the 1st. of the 1st. month (January) 1830, was 1°35' 51"; and its annual precession is 19.3". The polar distance may be found for any subsequent time by multiplying 19.3", by the interval between the 1st. of the year 1830, and the given time, and subtract ing the product from 1° 35' 51”. Thus, suppose the polar

CHAP. V.]

VARIATION OF THE COMPASS.

207

distance of the pole star was required for the 1st. of the 7th. month, (July) 1845. The interval is 15.5 years, and 19.3" X 15.5 gives 299.15"=4' 59". From 1° 35' 51", take 4' 59 , and we have 1° 30' 52" for the polar distance of the star, at the time proposed.* When the polar distance of the pole star is known, its azimuth, at the time of greatest elongation may be found by the following proportion. As radius, Is to the secant of the latitude of the place, So is the sine of the polar distance, To the azimuth. This azimuth will be cast or west, according as the elongation is east or west; and consequently its name will be known from the preceding tables. As an example, let the azimuth of the pole star, at Philadelphia, latitude 39° 57', be required, for the 1st. of the 7th. month (July) 1845. The polar distance found above is 1° 30' 52"; and it may be taken 1° 31' without material error. As radius ------- Ar. Co. 0.00000 : secant of lat. 39° 57' 10.11543 :: sin. of pol. dist. 1 31 ------ - 8.42272 : sin. of azimuth 1° 59'E.

------ 8.53815

* The polar distance obtained as above, is called the mean polar distance ; and it is sufficiently accurate for our present purpose. To obtain the true polar distance, two small corrections called aberration and nutation, would have to be applied.

208

VARIATION OF THE COMPASS.

[CHAP. V.

In order to observe the greatest elongation of the pole star, it will he necessary to prepare the following simple apparatus. Place two posts firmly in the ground, about three feet apart, and nearly east and west from each other; the heights of the posts, which should be the same, may be about two or three feet; on those posts, place a thick board or plank, five or six inches wide, and nail it fast to each of them, taking care that it be level or nearly so : take a piece of board, a foot or eighteen inches long and four or five wide, and near the middle of it fasten a com¬ pass-sight perpendicularly; this board is to slide on the horizontal one already mentioned. Take a stiff pole 18 or 20 feet in length, and fix it in an inclined position, in such a manner that a plumb line suspended from the upper end, may be nearly north, from the middle of the horizontal board, and about ten feet distant from it; the elevation of the pole must be such that the pole star, when viewed through the com¬ pass sight placed on the horizontal board, may appear a few inches below its upper end ; when in this position the lower end should be fastened in the ground, and the. pole should be supported by a couple of crotchets placed near the middle. The plumb should weigh a pound or more, and should swing in a vessel of water, in order to prevent the line being agitated by the motion of the air. The apparatus being prepared, proceed, about 15 or 20 minutes previous to the time of greatest elongation as indicated by the table, to make the observation as fol¬ lows : Let an assistant hold a lighted candle near the

CHAP.

V.J

VARIATION OF THE COMPASS. ~0d

plumb line, so as to illuminate it and render it distinctly visible; place the small board with the compass-sight attached to it, on the horizontal one, and move it east or west as the case may require, till the pole star, plumb line, and aperture in the compass-sight are all m a direct range. If the star should be deviating to the east, it will leave the plumb-line to the west; and the contrary, if deviating to the west; keep therefore shifting the sight, till the star appears stationary behind the plumb-line ■, which it will do for several minutes at the time of its greatest elongation, and will then recede from the line on the contrary side from which it did before it became stationary. The compass-sight must not be moved after the star has Attained its greatest elongation ; but the ap¬ erture in it being then in a direct range with the plumb line and star, the board to which the sight is fixed, must be fastened to the one on which'it slides, by a small tack passing through each end. This being done let an assistant take a straight stake, with a piece of lighted candle stuck on it, and go north to the distance of 30 or 40 perches; then looking through the compass-sight, direct him to set up the stake perpendicularly, and m such a situation that the candle stuck on the top may appear exactly behind the plumb-line; w hen thus placed, let it be firmly fixed in the ground. Next, let another straight stake be set up in the same manner near the plumb-line; the remaining part of the work may then be left till morning. O %

Measure accurately the distance between the two stakes. Then, As radius, Is to the tangent of the azimuth; So is the distance between the stakes, in feet, To a fourth term, in feet. 2D

210

VARIATION OF THE COMPASS. [CHAP. V.

Lay off the distance contained in this fourth term from the northerly stake, and perpendicular to a line join¬ ing the two stakes; it must be laid off towards the west if the azimuth is east, but towards the east if the azimuth is west. Next remove the northerly stake, and set it up at the other extremity of the distance thus laid off; then a straight line joining the two stakes will be a true meri¬ dian line. To obtain the variation, set up a compass in the place of the southerly stake, and direct the sights truly to the northerly one; the needle will then point out the varia tion, which will be east or west, according as the north end of the needle points to the east or west of the north point of the compass. The whole process is so simple, that an example is deemed unnecessary. It has already been observed, that the greatest elon¬ gations of the pole star are invisible during the greater part of the 3d and 4th months, and also of the 9th and 10th; consequently a meridian line cannot be obtained by the preceding method, during those periods. But as the surveyor may generally choose his time for tracing a meridian line, and as, when this is*done, he can at any time obtain the variation, it is thought unnecessary to introduce other methods. Those, however, who would wish to be acquainted with simple and accurate methods of tracing a meridian line at any season of the year, may consult a pamphlet on the subject by Andrew Elicott, A.M. from which the substance of the preceding method is extracted; and which contains others suited to those times of the year in which this cannot be applied. It may not be improper also to observe, that the second volume of the American Philosophical Transactions con-

CHAP. V.]

VARIATION OF THE COMPASS.

211

tains an essay by Robert Patterson, Professor of Mathe¬ matics in the University of Pennsylvania, in which is given a method for obtaining the variation to a sufficient degree of accuracy for any purpose in surveying, and which has this advantage, that the observation may be made at any season of the year, and at any time in the evening. There are also other methods besides those alluded to above, by which a meridian line may be traced, or the variation of the compass determined; but as the most of them require expensive instruments for making the observations, it is thought unnecessary to notice them in this work. To obtain the true bearings of a survey, from the magnetic ones, the variation being given. If the variation be east, add it to the north-easterly and south-westerly bearings, and subtract it from those that are north-westerly or south-easterly; but if the variation be west, add it to the north-westerly and south¬ easterly bearings, and subtract it from those that are north-easterly or south-westerly; this being done, the true bearings are obtained. To find the difference between the present variation, and that at a time when a tract of land was formerly surveyed, in order to trace or run out the original lines. Go to any papt of the premises, where any two adja¬ cent corners are known; and if one can be seen from the other, take its bearing; which compared with that of the same line in the former survey, shows their differ¬ ence. But if one comer cannot be seen from the other, run the line according to the given bearing, and measure

212

VARIATION OF THE COMPASS. [CHAP. V.

the nearest distance between the line so run, and the corner; then, As the length of the given line, Is to the said distance ; So is 57.3 degrees.* To the difference of variation required. EXAMPLE,

Suppose it be req'uired to run a line, which some years ago bore N. 45° E. dist. 20 ch. and in running this line by the given bearing, the corner is found 20 links to the left hand; what allowance must be made on each bear ing to trace the old lines ; and what is the present bear¬ ing, by the compass, of this particular line ? L, L. Deg. As 2000 : 20 : : 57.3 20

2000)1146.0(0° 34' Consequently 34 minutes, or a little more than half a degree, is the allowance required; and the line in ques¬ tion bears N. 44° 26' E. Note. The above rule is simple and sufficiently ac¬ curate when the distance between the sought corner and

* 57.3 is the radius (nearly) of a circle in such parts as the circumference contains 360.

CHAP. V.] VARIATION OP THE COMPASS

213

random line, is small. But when this distance is con¬ siderable, it will be better to find the angle by trigono¬ metry.

ON LOCAL ATTRACTION. is well known that iron or any ferruginous substance attracts the magnetic needle,and consequently when near, will draw it aside from the position in which it would otherwise settle. And as the earth in many places con¬ tains, near its surface, substances of this kind, the needle will not unfrequently be attracted from its true direction. The surveyor ought therefore, at each station, to take a back sight to the preceding one; and if he arrive at one at which the compass does not reverse truly, he may conclude, provided no error was committed in taking the bearing at the last station, that at the present one, the needle is affected by some local attraction. In such a case, he should first determine whether any error was committed at the last station, and if none is found, take the difference between the bearing from the last station and the reverse bearing, which will be the local variation of the needle at the present station. This variation must be applied, according to its name, to the bearing of the following station. IT

If at the first and second station of a survey the com¬ pass is found not to reverse truly, the surveyor will be at loss to know which of them is affected by.attraction. But by taking another station, either within or without lhe survey, and taking its bearing from each of those

214

VARIATION OF THE COMPASS. [CHAV. V.

stations, and the bearing of each of those from it, he may, in general, determine at which of them the attraction exists. Note.—The area of the survey is not affected by the general variation, because it is the same at each station. But where local attraction exists and causes a variation in the position of the needle, as this variation will be dif¬ ferent at different stations, it will, unless ascertained, and allowed on the corresponding bearings, materially affect the truth of the survey.

CHAPTER VL

MISCELLANEOUS QUESTIONS.

1. A circular fish-pond is to be dug in a garden, that shall take up just half an acre: what must be the length of the cord that strikes the circle ? Ans. 27.75 yards. 2. Two sides of a triangle are 20 and 40 perches re¬ spectively : required the third side, so that the content may be just an acre. Ans. Either 23.099 or 58.876 perches. 3. In 110 acres of statute measure, in which the pole is 5.5 yards, how many Cheshire acres, where the cus¬ tomary pole is 6 yards; and how many of Ireland, where the pole in use is 7 yards ? Ans. 92 A. 1 R. 29 P. Cheshire ; 67 A. 3 R. 25 P. Irish. 4. The ellipse in Grosvenor square, London, measures 840 links the longer way, and 612 tiie shorter, within the rails ; now the wall being 14 inches thick, it is requi¬ red to find what quantity of ground it encloses, and how much it stands upon. Ans. It encloses 4 A. OR. 6P. and stands on 1760^ square feet 5. Required the dimensions of an elliptical acre, with the greater and less diameters in the ratio of 3 to 2. Ans. 17.481 by 11.654 perches.

216

MISCELLANEOUS

QUESTIONS.

[CHAP. VI.

6. The three sides of a triangular field, containing 6 A. 1 R. 12 P. are in the ratio of the three numbers, 9, 8, 6, respectively ; required the sides. Ans. 59.029, 52.47, and 39.353 perches. 7. In a pentangular field, beginning with the south side and measuring round towards the east, the first ox south side is 27.35 ch., the second 31.15 ch., the third 23.70 ch., the fourth 29.25 ch., and the fifth 22.20 ch.; also the diagonal from the first angle to the third is 38 ch., and that from the thirl to the 5th. 40.10 ch.; required the area of the field. Ans. 117 A. 2R. 39 P. 8. Required the dimensions of an oblong garden, con¬ taining three acres, and bounded by 104 perches of pale fence.* Ans. 40 P. by 12. 9. How many acres are contained in a square mea¬ dow, the diagonal of which is 20 perches longer than either of its sides ? Ans. 14 A. 2 II. IIP. 10. A gentleman has a garden 100 feet long and 80 broad, and a gravel walk is to be made of equal width half round it; what must be the width of the walk, so that it may take up just one fourth of the ground. Ans. 11.8975 feet. . II. A person has a circular yard that is 150 feet in diameter, and wishes a walk of equal width made round it within the fence: required the width of the walk so that it may occupy a fifth part of the ground. Ans. 7.918 feet. * This question maybencatly constructed by 28, 6 Playfair's Geometry, It may not be improper also to observe, that the 2d question, and all those following the 8th, admit of neat geometrical constructions.

CHAP. VI.] MISCELLANEOUS

QUESTIONS.

217

•12. From a point within a triangular field the sides of which were equal, I measured the distances to the three angles, and found them 12.5,10, and 7.5 chains, re¬ spectively ; required the area. Ans. 12 A. 1 R 23 P. 13. On examining the field-notes of a lot of ground of which I wished to know the content, I found them as fol¬ low : 1st. S. 72° W. 24 per., 2d. North, 38 per., 3d. N. 82J E. 41 per., 4th. , 20 per., 5th. S. 80° E. 11.5 per., 6th. S. 26° W. 22 per., and 7th. , 37 per., to the place of beginning. The bearings of the 4th and 7th boundary lines were illegible ; but the data remaining being suf¬ ficient, the area is required. Aiis. 12 A. 3 R. 2 P. 14. It is required to lay out 4A acres of land in a tri¬ angular form, so that the length of one side may be 15 chains, and the lengths of the other sides in the ratio of 2 to 3 ; what hiust bedhe lengths of those sides ? Ans. 7.7914 and 11.6871 chains; or 29.58536 and 44.37804 chains. 15. It is required to lay out five acres of ground in a triangular form to be bounded by 135 perches of fence; the length of one side is to be 50 perches ; what must be the lengths of the other sides ? Ans. 33.3785 and 51.6215 perches. 16. The area of a rectangular field is acres and the length of the diagonal 50 perches: required the sides. Ans. 30 and 40 perches. 17. In a rectangular tract of land, containing 58 A. 3R. 8 P. the difference of the lengths of the sides is just equal to the difference between the lengths of the longer side and the diagonal; hence the sides are required. Ans. 21 and 28 chains. .

1
2E

218

MISCELLANEOUS

QUESTIONS. [cHAP. VI.

18. The boundaries of a tract of land are as follow : 1st. N. 14° W. 15.20 ch.; 2d. N. 70i° E. 20.43 ch.; 3d. S. 6° E. 22.79 ch.; 4th. N. 86|° W. 18 to the place of beginning ; within the tract there is a spring, the bearing ► and distance of which, from the 2d corner, is S. 75° E. 7.90 ch. It is required to cut off 10 acres from the west side of this tract by a straight line running through the spring; what must be the distance of the division line from the 1st corner, measured on the fourth side. Ans. 4.6357 chains. 19. The boundaries of a quadrilateral tract of land are as follow : 1st. N. 35i° E. 23 ch.; 2d. N. 75|-° E. 30.50 ch.; 3d. S. 3i° E. 46.49 ch. and 4th. N. 66^° W. 49.64ch., to the place of beginning. This tract is to be divided into four equal parts by two straight lines, one of which is to run parallel to the 3d side ; required the distance of the parallel division line from the first corner, measur¬ ed on the 4th side ; also the bearing of the other division line, and its distance from the same corner, measured on the first side. Ans. Distance of the parallel division from the first corner 32.50 chains; the bearing of the other, S. 88° 22 E. and its distance from the same corner 5.99 chains.

CHAPTER VII.

OF THE THEODOLITE.

A VERTICAL ANGLE is any angle in a plane perpen¬ dicular to the horizon. Consequently angles of eleva¬ tion and depression are vertical angles. A Theodolite is an instrument used to measure both horizontal and vertical angles. By the aid of a theodolite, purveys may be much more accurately made than with the compass, espe¬ cially when the tract is large or the ground hilly, or where there is local attraction. Before describing the theodolite, it will be best .to notice separately one or two of its appendages. Of the Spirit Level. The Spirit Level consists principally of a cylindrical glass tube, nearly, but not entirely, filled with alcohol, or some other fluid. The inner part of that side of the tube which, when in use, is to be uppermost, is ground from end to end into a regular curve, having its convexity upwards. In consequence of this curva¬ ture in the upper part of the cavity, the air bubble, or part not occupied by the fluid, must be at the middle of the tube w hen the axis of the level, that is, the straight line passing lengthwise through its middle, is in a horizontal position. And conversely the axis must be horizontal when the instrument is so placed that the bubble stands at the middle of the tube.

220

OF THE THF.ODOLITE.

[CHAP. VII

Of the Vernier. A Vernier is a graduated index which serves to sub¬ divide the smallest divisions of a graduated arc or straight line. Verniers are somewhat different according to the value of the divisions which they are to subdivide and the degree of precision they are designed to give; but the principle of construction is the same in all. The following description applies to the vernier attached to the common theodolite, the limb of which is divided into degrees and half degrees, numbered from 0° to 360°; and the vernien subdivides the half degrees to minutes. In Fig. 115, AB represents a part of the graduated limb, and CD the vernier; the whole being drawn on an enlarged scale. The extent of the vernier is ex¬ actly equal to 29 half degrees, or 870 minutes, on the limb, and is divided into 30 equal parts, numbered from 0 to 30. Each division of the vernier contains there¬ fore 29 minutes. Hence it is evident that when the 0, or zero line of the vernier exactly coincides, as in the figure, with a division line on the limb, the first divi¬ sion line of the vernier must be one minute behind the line on the limb first following that with which the zero of the vernier coincides; the second line of the ver¬ nier must be two minutes behind the second on the limb; the third line on the vernier must be three minutes behind the third on the limb; and so on. If then, the vernier were to be moved forward till its first division line coincided with the first on the limb, the zero of the vernier would be one minute past the line oif the limb with which it before coincided; if moved till the second lines coincided, the zero of the vernier

CHAV. VII.J

OF THE THEODOLITE.

221

would be two minutes past the same line; and in like manner for other coincidences. Jt therefore follows that for any position of the vernier, the number of the division line on it which coincides with one on the limb, must express the number of minutes that the zero of the vernier is past the division line on the limb, next preceding it. . The arc indicated by the vernier index in any posi¬ tion is the arc intercepted in the direction in which the degrees are numbered between the zero line of the limb and the zero line of the vernier. Hence to find the arc, or as it is technically expressed, to read off the arc, we must add to the arc expressed by the number of whole degrees intercepted and the odd half degree, if there is one, the number of minutes indicated by the number of that division line on the vernier which coin¬ cides, or is the nearest to coincidence, with one on the limb. Thus, as the zero of the vernier in Fig. 115 exactly coincides with the division line on the limb, numbered 50, the arc indicated is 50°. In Fig. 116, the zero of the vernier, is past the line on the limb de¬ noting 52° 30'*and the 21st division line of the ver¬ nier coincides with a line on the limb. The arc indi¬ cated is therefore 52° 51'. Sometimes it is requisite that the vernier should serve to read off an arc on either side of the zero line In this case the zero of the vernier is placed at its mid¬ dle, and^the division lines have two sets of numbers, as in Fig. 117. The upper numbers are used when the zero of the veTnier is to the right of the zero of the arc, and the lower ones when it is to the left. The arc indicated by the vernier in the position that it has in the figure, is 2° 12' to the right. To determine the reading of a vernier, applied to a 19*

222

OF THE THEODOLITE. [CHAP. VII.

graduated arc in which the value of the least division is more or less than half a degree, we must divide this value by the number of divisions on the vernier, and the quotient will indicate the reading that the vernier is intended to give. In the nautical instrument called a sextant, the value of the least division on the limb is usually 10 minutes, and the vernier is divided into 60 equal parts. Hence 10 minutes, or 600 seconds, di¬ vided by 60, gives 10 seconds for the reading of the vernier. '

Description of the Theodolite.

The Theodolite is represented in Fig. 109. When in use, the lower circular plate AB is screwed fast to a three-legged stand or tripod, the upper part of which is represented in the figure. The circular plate CD has a hollow axis passing through it at right angles at its centre, and firmly attached to it. The lower part of the axis terminates in a ball, to which it is fastened by a screw. This ball is partly enclosed in a socket projecting from the upper side of the jilate AB; the opening in the upper part of the socket being larger than the axis, so as to allow the latter some motion in every direction. In the lower plate AB, four screws called levelling screws, are inserted, standing opposite each other in pairs, the'tops of which press against the under side of the plate CD. These screws are turned by milled projections or heads, two qf which are shown in the figure. By turning them in opposite directions, the position of the plate CD may be changed so that it may be made level, even when the lower plate AB has considerable inclination. The part GII of the instrument consists of twro cir¬ cular plates in close contact with each other. The

CHAP. VII.J

OF THE THEODOLITE.

223

lower one is called the graduated plate, and its cham¬ fered edge, which is usually made of silver, is called the limb of the instrument. To the central part of this plate, a hollow axis is attached, the cavity of which fits to the outside of the axis of the plate CD, so that the former may move easily and steadily about the latter. The lower part of the axis is embraced by a clamping-pi»ce that may be tightened or loosened by the screw E. When the clamp is loose, the graduated plate may be turned round by the hand ; but when it is made fast, the axis of the graduated plate, and conse¬ quently the plate itself, becomes firmly connected with the plate CD, and also, when the levelling screws are tight, with the stand of the instrument. The gradu¬ ated plate may however, when thus connected, be moved a small distance either way by turning the screw F, called a tangent-screw, which gives it a slow motion. Around the axis of the graduated plate is another hol¬ low axis, with which a telescope, called the lower telei scope, is connected. When the axis of the graduated plate is clanjped fast, this axis and the telescope may be made to revolve by turning the milled head I, and ma*y be secured in any position by the screw K, which clamps the axis to the graduated plate. The plate immediately above the graduated plate is called the vernier plate. It has, at its centre, a solid axis which fits into the cavities of the hollow axes of both the graduated plate and plate CD. When the screw L, which serves to clamp the vernier plate to the graduated plate, is loosened, tjie former may be turned round by hand; and when this screw is made tight, a slow motion may be given to the vernier plate by the tangent-screw M. The vernier plate has two, or sometimes three, verniers at different parts of its

224

OF THE THEODOLITE. [CHAP. VII

edge, which are chamfered for the purpose. When there are only two, they are placed directly opposite to each other. When there are three, they are placed at equal distances around the edge. The microscope S is used to enable the eye more certainly to distin¬ guish the line on the vernier that coincides with one on the limb. Two levels are placed on the vernier plate, at right angles to each other; ope of which is shown in the figure. The frame which supports the vertical semicircle and upper telescope with its attached level, is attached to the vernier plate by three screws at equal distances from one another. The immediate supports of this telescope are called wyes, or Y’s, from their resem¬ blance to the letter Y. The telescope is held in its place by two curved pieces, moveable on joints, which pass over it and are fastened by the pins T and U. When these pins are taken out, the pieces may be turned back and the telescope taken from its place and reversed, that is changed end for end. In the tube of the telescope a flat ring(is placed at right angles to its axis, and connected with the tube by four screws, opposite to each other in pairs; the heads of three of which, 6, c, and d, are shown. Two spi¬ der’s lines or very fine wires, at right angles to each other, are attached to the ring, in the directions re¬ spectively of each pair of screws, and intersectipg each other in the centre of the ring. At each end of the principal tube, and fitting it on the inside, a short moveable tube is inserted. The one at the end to which the eye is applied, and which usually contains several glass lenses, may be moved out or in, by a small pin attached to its lower side, and may thus be so adjusted as to render the spider’s lines distinctly

CHAP. VII.] OF THE THEODOLITE.

220

visible. The tube at the other end, which contains one lens, may be moved out or in, by turning the milled head V, and may be thus so placed as to render an object to which the telescope is directed, as distinct as its distance permits. A straight line in the direction in which the inter¬ section of the spider’s lines or wires is seen by an eye placed at the eye end of the telescope, is the line of sight of the telescope, and is technically called the line of collimation. The telescope is said to be directed to any point in an object, when it is so placed that the line of collimation passes through the point; or which is the same, when the point is directly behind the in¬ tersection of the spider’s lines. The semicircle PQ is called the vertical limb of the instrument. That face of it which is not seen in the figure is divided into degrees and half degrees, num¬ bered each way from a line taken as a zero or 0 line; and the arc is read by a vernier numbered as in Fig. 117. The other face lias two sets of unequal divi¬ sions on it, numbered each way from a zero line. One of these denotes the difference between the real dis¬ tance of an object to which the upper telescope is di¬ rected and its horizontal distance, expressed in 100th parts of the distance. The other denotes the vertical distance of the object, above or below a horizontal line, passing through the instrument, expressed in 100th parts of the horizontal distance. The vertical limb is moved so as to give to the telescope different inclina¬ tions to the horizon, by means of the milled head N. At the bottom of the frame which supports the ver¬ tical limb and upper telescope, and directly over the centre of the vernier plate, there is a compass box, containing a magnetic needle. This part of the

226

OF THE THEODOLITE. [CHAP. VII

instrument may be used to take the bearing of an object in the same manner as a common compass. ADJUSTMENTS OF THE THEODOLITE.

In order that the theodolite may be in a good state for use, the line of collimation of the upper telescope should exactly coincide with the axis of the telescope : the axis of the attached level should be parallel to this line; the axes of the levels on the vernier plate should be parallel to this plate; the line of collimation of the telescope should, when the milled head N is turned, move in a plane perpendicular to the vernier plate: and when this line is brought parallel to the vernier plate, the zero of the vertical limb should coincide with the zero of its vernier. Previous therefore to using the instrument, the different parts should be examined, and adjusted if necessary. This may be done by the following methods. FIRST ADJUSTMENT.

To make the line of collimation coincide with the axis of the telescope. The instrument being firmly screwed on the tripod and the legs of the latter being sufficiently extended to ensure its remaining fixed in its position, loosen the clamp screw L, and turn the vernier plate till the tele¬ scope points to some distant object having on it a small well-defined point. Then, having fastened the screw L, move the telescope by the tangent-screw M, and milled head N, till the line of collimation is di¬ rected exactly to the point. Revolve now the telescope in its Y’s, half round, that is, till the level, from being directly below the tele-

CHAP. VII.] OF THE THEODOLITE.

227

scope is directly above it. If the horizontal spider’s line still coincides with the point, it requires no adjust¬ ment; but if it does not, diminish the distance onehalf, by loosening one of the screws c and d and tight¬ ening the other; and then bring the line to coincide with the point by means of the milled head N. Re¬ volve the telescope round to its first position, and if the horizontal line and point do not then coincide, repeat the operation till the coincidence has place in both positions. In a similar manner the vertical line may be adjusted. When both adjustments are com¬ plete, the line of collimation should coincide with the same point during a complete revolution of the tele¬ scope in its Y’s. SECOND ADJUSTMENT.

To malce the axis of the level attached to the upper tel&scope parallel to the line of collimation. Turn the vernier plate till the telescope comes di¬ rectly over two of the levelling screws; and if the telescope is not nearly in a horizontal position, make it so by turning the milled head N. Then turn the levelling screws over which the telescope stands, in opposite directions, till the bubble of the level stands exactly at the middle of the tube; observing to keep the screws firm against the plate CD. When this is done, reverse the telescope in its Y’s, and if the bubble does not stand at the middle of the tube, correct half of the deviation by one of the screws m and n, which serve to raise or depress the ends of the level, and then, by the levelling screws, bring the bubble to the middle. Again reverse the telescope and repeat the correction if necessary.

228

OF THE THEODOLITE [CHAP. VII.

Revolve now the telescope in its Y’s so as to bring the level a considerable distance from its proper or lowest position, and if the bubble deviates from the middle of the tube, make the requisite correction by means of two screws, p and q, which move.the end of the level laterally, one of which is shown at p. When this part of the adjustment has been so made that the bubble will remain in the middle of the tube while the telescope is revolved either way, the first part of the adjustment should, by again reversing the telescope, be examined, and repeated if necessary. THIRD ADJUSTMENT.

To make the axes of the levels on the vernier plate paral¬ lel to that plate. Turn the vernier plate till the upper telescope stands over two of the levelling screws; then one of the levels on this plate will be parallel to these two screws, and the other will be parallel to the other two. By means of these levelling screws bring the bubbles of both levels to stand in the middles of their respective tubes. Then move the vernier plate 180°, and if the bubble in either of the levels deviates from the middle, correct one-half of the deviation by one of the screws at its ends, and the other half by the levelling screws, parallel to it; and if the bubble of the other level also deviates from the middle, proceed in the same way to correct the deviation. Repeat the operations till the bubbles of both levels will remain at the middles of their respective tubes during a complete revolution of the vernier plate.

CHAP. VII.]

OF THE THEODOLITE.

229

FOURTH ADJUSTMENT

To make the axis about which the vertical limb revolves, parallel to the vernier plate.

Turn the vernier plate till the upper telescope is directed towards a well-defined, elevated point, on a house or other object, not very remote, and having clamped the plate, direct the telescope by means of the tangent-screw M and milled head N, exactly to the point. Then turn the milled head N till the line of collimation coincides with some well-defined point near the ground; or, if none such is found, let an assistant make a suitable mark in the direction of the line of collimation when thus brought nearly horizon¬ tal. This being done, reverse the telescope in its Y’s; and, proceeding as before, direct it to the elevated point. Then, if by turning the milled head N, the line of collimation is brought to coincide with the lower point or mark, the axis of the vertical limb is parallel to the vernier plate; but if this is not the case, the adjustment must be made by the screws which attach the upper frame to this plate. FIFTH ADJUSTMENT.

To adjust the vernier of the vertical limb, or determine the correction which should be made, to allow for its deviation from correct adjustment.

Direct the upper telescope to some elevated point and note the angle of elevation indicated by the ver¬ nier. Reverse the telescope in its Y’s, and again direct it to the same point, and note the angle of elevation. If the angles are the same, the vernier is properly 20

OF THE THEODOLITE. [CHAP. Til.

adjusted. If they differ, the position of the vernier requires adjustment; and this may be effected by the screw v, which is one of those that fasten the upper frame to the vernier plate. Instead of changing the position of the vernier, we may take half the difference of the two angles of ele¬ vation as a correction, to be added to all those angles of elevation and subtracted from those of depression, that are taken with the telescope in that position in its Y’s, which gave the least of the two angles; but to be applied in a reverse manner when the position of the telescope is that which gave the greater angle SIXTH ADJUSTMENT.

To make the axis about which the lower telescope revolves vertically, perpendicular to the axis of the instrument. Turn the vernier plate till the levels on it are re¬ spectively parallel to a pair of opposite levelling screws, and by means of these screws bring the bubbles to stand in the middles of the tubes; the instrument is then levelled, and its axis is perpendicular to the hori¬ zon. Suspend a plumb-line of considerable length at a short distance from the instrument, and loosening; the clamping screw K, turn the milled head I, till the telescope is directed to the line, and then fasten the clamping-screw. Now, applying the hand to the eyeend of the telescope, move it vertically, and observe whether it continues directed to the line, throughout its whole extent. If it does, the axis is properly ad¬ justed ; if it does not, the adjustment may be made by two small screws which move the remote end of the axis vertically. The lower telescope being used only as a guard to

CHAP. VII.]

OF THE THEODOLITE.

231

ascertain whether or not the horizontal limb of the instrument remains fixed while the vernier plate is made to revolve, in order to direct the upper telescope to an object, requires no other adjustment than the above, except those for distinct vision of the spider’s lines and object.* In extensive practical operations, the above adjust¬ ments should be examined, and corrected if necessary, not only before the commencement, but every day or two during their continuance. Of the arc of the horizontal limb, corresponding 10 a given position of the vernier plate. When there is but one vernier, the arc indicated by it is regarded as the corresponding arc. When there are two or three, it is usual to distinguish them by calling one A, another B, and the third, if there is a third, C, and to consider the arc indicated by the ver* When the foregoing adjustments have been completed for the first time, there is another examination which it is well to make. Direct the upper telescope to some well-defined point near the horizon, and note the arcs on the horizontal limb, indicated by the different verniers. Then having di¬ rected the lower telescope to the same point, fasten its clamping-screw K; examining first, however, to see that the upper telescope has not changed its position, or if it has, bringing it back by the tangent-screw F. Reverse now the upper telescope in its Y’s, and again direct it to the point; exa¬ mining at the same time to see that the lower telescope has not changed its position, or, if it has, bringing it back by the tangent-screw F. Note the arcs indicated by each of the verniers; and subtract the arc indicated by each verniei in the former position from that in the latter, increasing that of the latter by 360°, if necessary. Add together the remainders, and divide the sum by the number of verniers. If the quotient is 180°, we infer that the axis about which the upper telescope revolves is, as it ought to be, at right angles to the line of eollimation of the telescope. If the quotient dif¬ fers materially from 180°, the instrument is imperfect, except one of the Y’s is laterally adjustable. Where this is not the case, the imperfection can only be well remedied by an instrument-maker. If the remainders men¬ tioned above differ much from each other, we infer that the instrument has not been well centered, or not well divided.

232

OF THE THEODOLITE [CHAP. VII.

nier A, as expressing the position of the vernier plate, at least very nearly ; the readings of the other ver¬ niers being used merely as a test, or means of cor¬ recting the former. We therefore note the arc indi¬ cated by the vernier A. Then note the minutes of the arc indicated by the vernier B; but instead of the number of degrees indicated by it, we take either the number indicated by the vernier A, or this number in¬ creased or diminished by a unit, so that the arc set down for the vernier B, may be very nearly the same as that for the vernier A. If there is a third vernier C, we proceed in the same manner. Then the sum of the arcs obtained for each vernier, divided by the num¬ ber of verniers,* will give the value of the arc corre¬ sponding to the given position of the vernier plate, or, as it is sometimes expressed, the arc indicated by the verniers. Thus, if we suppose the instrument to have three verniers, and that for a given position of the plate, the reading of the vernier A is 142° 2', of B, 261° 58', and of C, 22° 3'; then, instead of these quan¬ tities for B and C, we write for B, 141° 58', and for C, 142° 3'. Adding together 142° 2', 141° 58', and 142° 3', and dividing the sum by 3, we obtain 142° 1', for the arc indicated by the verniers. PROBLEM I. To measure with the Theodolite the horizontal angular distance of two objects, as seen from a given station. By means of a plumb-line suspended from the centre of the plate which forms the top of the tripod, set the * When, as is commonly the case, the number of the degrees of the arc for each vernier is the same, we need only divide the sum of the minutes of the arcs, by the number of verniers, and annex the quotient to that num¬ ber of degrees.

CIIA1’. VII.]

OF THE THEODOLITE.

233

centre of the instrument directly over the stationmark, and then level it, as directed in the sixth adjust¬ ment. Direct the upper telescope to the object which stands to the left, when the face is turned towards the angle to be measured, and note the readings of the verniers. Then, having directed the lower telescope to the same object and fastened its clamping-screw K, direct the upper telescope to the other object, and note the readings of the verniers; observing first however, whether there has been any change in the position of the lower telescope ; and if there has, bringing it back to the object at the left by means of the tangent-screw F, and again by the tangent-screw M, adjusting the direction of the upper telescope to the object at the right. Subtract the arc which expresses the first posi¬ tion of the vernier plate, found from the readings of the verniers, from that which expresses the second po¬ sition, found in like manner, increasing the latter by 360°, if necessary, and the remainder will be the angle required.*. * Sometimes, in order to give greater precision to the result, the operation is repeated two or three or more times. When this is done, the verniers need only be read at the first and last positions of the vernier plate. The method of proceeding is as follows : After the upper telescope has been directed to the object at the right, loosen the clamping-screw E, and turn the instrument round till the upper telescope points towards the object at the eft, then fasten the screw, and by means of the tangent-screw F, direct the telescope accurately to the object. Then, having directed the lower tele¬ scope to this object, and fastened its clamping-screw, direct the upper tele¬ scope again to the object at the right, in the usual way. The vernier plate must then have moved, from its first position, a distance equal to twice the measure of the angular distance of the objects. If the operation be again repeated, the distance moved by the plate must be equal to three times the measure of the angle; and so on. Hence, if the arc which expresses the first position of the vernier plate be subtracted from that which expresses its last position, and the remainder, or, when the plate has made more than a complete revolution, the remainder increased by 3G0°, be divided by the number which denotes the number of times the operation has been per¬ formed, the quotient will be the armle required.

20*

°2G

234

OF THE THEODOLITE.

[CHAP. VII

PROBLEM II. To measure a vertical angle with the Theodolite. The instrument being placed and levelled, direct the upper telescope to the point, of which the angle of ele¬ vation or depression is to be taken. Then, if the ver¬ nier of the vertical limb is accurately adjusted, the angle indicated by it will be the angle required. When this is not known to be the case, after having noted the arc indicated by the vernier, move the vertical limb till the telescope is horizontal, as indicated by the bubble of its level standing in the middle of the tube, and note the arc then indicated by the vernier. If in both positions the zero of the limb is on the same side as the zero of the vernier, subtract one arc from the other; but if on different sides, add the two arcs toge¬ ther. The result will be the required angle. PROBLEM III. To measure with the Theodolite, the angles of a tract of land, as ABCDEFGHA, Fig. 110; and having these and the measures of the sides, to find the content. . Set up the instrument at one of the angles as A, and also set up sight poles at II and B, or at some distant points in the sides AH and AB, and then measure the angle as directed in Prob. I. Proceed in the same way to measure the other angles of the tract, observing that the angle II, which is called a re-ent^aiu angle, is to be regarded as measured by the arc aoc, and is therefore greater than 180°. When there are fences or other obstructions along the sides, the instrument must be set up at some dis¬ tance from the corner, and the poles must be placed

CHAP. VII.]

OF THE THEODOLITE.

285

respectively as far from each side as the instrument is from the range of that side, so that straight lines from the instrument to the poles, may be parallel to the sides. With the theodolite, or, which is better, with a com¬ pass, take the bearing of one of the sides, as AH. With this bearing and the angle A, find the bearing of AB. Then with the bearing of BA, which is the reverse of the bearing of AB, and the angle B, find the bearing of BC; and proceed thus, to find the bearings of the other sides. 1 Having measured the horizontal lengths of the sides, and obtained the bearings from the measured angles, the content may be found by prob. 12, chap. III.; ob¬ serving that as the differences of latitude and the departures are given in the traverse table, only for every quarter of a degree, they should be computed for each side by prob. 10, chap. I.; or, if they are taken from the traverse table, proportional parts should be applied for the odd minutes, above full quarters of a degree. Note 1. When the angles of the tract have all been measured, the accuracy of the measures may be tested as follows. Subtract 2 from the number of sides of the tract, and multiply 180° by the remainder; the product will express the sum of all the interior angles (32. 3. cor. 1.) Hence if the sum of the measured angles is equal, or very nearly equal, to this sum, they may be regarded as having been correctly taken; but should the sums differ considerably, an error in one or more of the angles must have been committed, and the operations should be repeated. 2. In measuring the lengths of the sides, if any of them have a considerable ascent or descent, and if

236

OF THE THEODOLITE. [cHAP. VII.

this ascent or descent is tolerably regular, it will be more convenient to measure their real lengths along the ground, and then by the theodolite determine for each of them the quantity that must be subtracted in order to obtain the horizontal lengths. To do this, let a mark be made on the pole at the farther end of the line, at the same height from the ground as the axis of the vertical limb of the theodolite, and diiect the upper telescope to this mark. The index to that arc of the vertical limb, which is graduated for the purpose, will then indicate the number of links which must be subtracted from each chain in the measured length, to obtain the horizontal length. 3. Instead of measuring all the sides and angles, we may take two or three or more stations within the tract, and apply the method given in the 13th problem of chap. III. The stations should be so chosen that none of the angles contained between the lines which connect any two of the stations with a corner of the tract, should be either very acute or very obtuse. When the stations are judiciously chosen, their hori¬ zontal distances accurately measured, and the requi¬ site angles, contained between the lines joining the stations with one another and with the corners of the tract, are carefully taken with a good theodolite, the content may in this way be very correctly determined. We may also, if it is required, compute the sides and angles of the tract; and thence, if the bearing of one of the sides or of one of the lines joining a station with a corner, has been taken, the bearings of all the sides may be obtained.

CHAP. VII.]

237

OF THE THEODOLITE.

PROBLEM IV. To run a line with the Theodolite from one end B, of a given line AB, Fig. 118, that shall make a given angle with that line. Let the theodolite be placed at B and levelled. Then, a sight-pole being set up at A, direct the tele¬ scopes to it and note the arc indicated by the verniers. Add the given angle to this arc, and turn the vernier plate till the arc indicated by its verniers is equal to the sum, and clamp it in that position. Look through the lower telescope to ascertain whether the graduated plate has changed its position, and if it has, bring it back by the tangent-screw F. If the upper telescope is not nearly parallel with the ground in the direction in which it points, make it so, by turning the milled head IN’, and then let a sight-pole be set up at some distant point exactly in the direction of the line of collimation. Suppose C to be such a point. Then will BC be the required line. This line may be, if desired, marked by driving a number of stakes into the ground at convenient distances along it, using the telescope to have them accurately placed. If it is required to extend the line beyond the point C, let a sight-pole be set up at B, and the theodolite bo removed to C. Then proceed to set up a pole at an¬ other point D, exactly as directed above, except that the given angle must now be considered as being 180°, in order that CD may be in the same direction with BC. Or, after having directed the telescope to the i

238

OF THE THEODOLITE. [CHAP. VII.

pole at B, we may, instead of turning the vernier plate 180°, let it remain unmoved, and reverse the telescope in its Y’s, which will amount to the same. Proceeding thus from point to point, we may extend the line to any required distance.

CHAPTER VIII.

LEVELLING.

The surface of an expanse of tranquil water or any similar surface concentric with it, is called a Level Sur¬ face. Any points situated in a level surface are said to be on the same level; and any line traced in such a surface is called a line of level. If through any place, a level surface be conceived to' pass, the distance which another place is from this sur¬ face, either above or below, measured on a line per¬ pendicular to it, is called the difference of level of the two places. Levelling is the art of determining the difference or differences of level of two or more placesi In consequence of the globular figure of the earth, a level surface is not, as it appears to be, a plane surface. It is nearly, though not exactly, spherical. In the ope¬ rations of levelling we may, without sensible error, regard a level surface at any place, as being strictly a spherical surface, with a radius equal to 3956 miles, the mean radius of the earth, or, which is more exact, with a radius equal to 3968 miles the centre being * The earth, if we disregard the inequalities in its surface, is an oblate spheroid. Tts polar diameter is 7899 miles, and its eouatorial diameter 7925 miles. A level surface is therefore a spheroidal surface. The radius of the spherical surface which most nearly coincides with any small portion of this spheroidal surface, changes slightly with the latitude of the placeFor any place in the United States, the greatest error which can occur from 239

240

LEVELLING.

CHAP. VIII.

in a straight line conceived to be drawn downwards from the place, perpendicular to the level surface. And for places not very remote from each other, we may regard the spherical surfaces of level passing through •hem, as having a common centre. Let A and B, Fig. 120, be two places, not very remote from each other, and C, the common centre of the spherical surfaces of level passing through them. With the centre C and radius CA, describe the arc Aa. Then will Aa be a line of leyel of the place A, and Ba will be the difference of level of the two places A and B. •

The line Ad, drawn perpendicular to CA, is called a line of apparent level of the place A. It is the line of level that would be indicated by an accurately adjusted levelling instrument placed at A. The distance ad is called the correction of the apparent level. This correc¬ tion must be subtracted from the height Bd of the ap¬ parent level to obtain the height Bo, of the true level. The correction varies as the square of the distance from the place.* The following table contains the considering the level surface passing through it, as a spherical surface with a radius equal to the mean radius of the earth, is about i of an inch for a distance of two miles. For a spherical surface with a radius 5 or 6 miles greater than the equatorial radius of the earth, the greatest error is about of an inch, at the same distance. It may be further observed, that the greatest error or greatest deviation of the spherical from the spheroidal sur¬ face, varies as the square of the distance from the place. * We have (36.3), (2Ca -f ad), ad == Ad*. But as ad, for any distance to which a single sight in levelling is ever extended, is extremely small in comparison with 2Ca, we may, without sensible error, take 2Ca instead of (2C« -f- ad).

Ad3

We shall thus have 2Ca, ad = Ad3; or ad = .

Conse¬

quently as Ca is constant, ad varies as the square of the distance Ad varies. Let the value of ad be required for a distance of 100 chains or 6600 feet. Then taking Ca = 3968 miles, we have 2Ca = 7936 miles = 7936 x 5289

241

LEVELLING,

CHAP. VIII.J

value of the correction in decimal parts of a foot, for each chain of distance, from 1 to 120. TABLE, Giving the differences between the true and apparent level, for distances from 1 to 120 chains. i

Chains.

Feet.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

.000 .000 .001 .002 .003 .004 .005 .007 .008 .010 .013 .015 .018 .020 .023 .027 .030 .034 .038 .042 .046 .050 .055 .060

> '

feet.

Chains.

Feet.

Chains.

25 .065 49 26 .070 50 27 .076 51 28 .082 52 29 .087 53 30 .094 54 31 .100 55 32 .106 56 33 .113 57 34 .120 58 35 .127 59 36 .135 60 37 .142 61 38 .150 62 39 .158 63 40 .166 64 41 .175 65 42 .183 66 43 .192 67 44 .201 68 45 .211 69 46 .220 70 47 .230 71 48 .240 72

Hence ad —

66002

Feet.

Chains.

Feet.

Chains.

Feet.

.250 .260 .270 .281 .292 .303 .314 .326 .338 .350 .362 .374 .387 .400 .413 .426 .439 .453 .467 .481 .495 .509 .524 .539

73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

.554 .569 .585 .600 .616 .632 .649 .665 .682 .699 .716 .734 .751 .769 .787 .805 .823 .842 .861 .880 .899 .919 .938 .958

97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

.978 .998 1.019 1.040 1.060 1.082 1.103 1.124 1.146 1.168 1.190 1.213 1.235 1.258 1.281 1.304 1.327 1.351 1.375 1.399 1.423 1.447 1.472 1.497



1003 X 663 1003 X 66

1.0396 ft. 7936 X 5280 7936 X 5280 “7936 X 80 Having the correction for 100 chains, we easily obtain it for any other

distance by the following proportion: As 1002 : square of given distance in chains :: 1.0396 : the required correction, in feet.

21

2H

242



LEVELLING. [CHAP. VIII.

When a ray of light passes obliquely through por¬ tions of air of different densities, it becomes bent from a straight line, and enters the eye so as to make the point from which it proceeds appear in a direction slightly different from its true direction. This effect is called Refraction; and when the point, or body from which the light proceeds, is on or near the earth, it is called Terrestrial Refraction. Terrestrial Refraction generally makes the point appear to be more elevated than it really is. Thus it is not actually the point d of the line CE, Fig. 120, that, to an eye at A, appears to be at d, but another point c, a little below d. If therefore we wish to notice the effect of refraction, we must take ac instead of ad, for the correction of apparent level. In temperate climates, cd is, in the usual state of the atmosphere, about i of ad. Consequently we may ob¬ tain the correction of apparent level with allowance for refraction, by diminishing the correction given in the preceding table by a i part. As the effect of refraction in the common operations of levelling is always small, and is subject to considera¬ ble variations depending on the state of the air between the object and place of observation, it is commonly dis¬ regarded. It would, however, in general be better to allow for it as above, except when the object is but little distant from the place of observation, or when, by the method noticed in the next article, the desired result is obtained independently of the correction of apparent level. Let CD, Fig. 121, be the line of apparent level indiFig. 121.

cated by a levelling instrument L, placed midway be¬ tween the places A and B; also let the arc be be the

CHAP. Till.] LEVELLING.

243

line of true level passing through the instrument, the arc A a, concentric with the former, the line of true level of the place A, and AC and BD, lines perpen¬ dicular to A«. Then in consequence of the equality of the distances LC and LD, we have bC equal to cD; and therefore as Ab is equal to ac, we have AC equal to aD. Consequently Ba, the difference of level of the places A and B, is obtained by subtracting AC, the height of the apparent level of the instrument, at A, from BD, its height at B. We thus obtain the dif¬ ference of level of the two places, independently of the correction of apparent level. It is also indepen¬ dent of refraction, as the effect of refraction would be sensibly the same for the two points C and D, and in the same direction. It may be observed that it is not necessary the instrument should be directly between the places. It may be placed in any convenient po¬ sition on either side of the line joining them, pro¬ vided its distances from the places are equal, or very nearly so. OF THE LEVELLING INSTRUMENT.

The Levelling Instrument, or Level as it is frequently called, is an instrument used to denote the line of ap¬ parent level. It is supported, like the theodolite, on a tripod. The instrument, without the tripod, is repre¬ sented in Fig. 119. Its lower part is nearly similar to the corresponding part of the theodolite. The plate IIII, which screws on the tripod, has four levellingscrews inserted in it, by means of which the plate GG may be placed in a horizontal position, even when the former is considerably inclined. S is a clamp-screw,

244

LEVELLING. [CHAP. Till.

and O a tangent-screw. The bar F,E is firmly at¬ tached at right angles to an axis, which terminates in a ball, enclosed in the socket R. In some instruments this bar is enlarged at the middle so as to form a com¬ pass box, in which a magnetic needle is placed. The telescope AB is supported on two wyes, Yl and Y2, one of which, Yl, is firmly connected with the bar EE, and the other, Y2, is moveable a small distance up or down by the screw N, or in some instruments by two screws, M and P. The eye-tube L is moveable in or out by hand, so as to render the spider’s lines distinctly visible, and the tube which contains the object-glass is moved by turning the milled head d, and may thus be so adjusted as to give distinct vision of an object to which the telescope is directed. One end, C, of the spirit-level CD is moveable up or down by the screw m, and the other end is moveable laterally by two opposite screws p and q, of which only p appears in the figure. The following adjustments of the level should be examined, and corrected if necessary, before using it in practical operations. FIRST ADJUSTMENT.

To make the line of collimation coincide with the axis of the telescope. Having screwed the level to the tripod and set up the instrument so as to stand firmly, loosen the clampscrew, turn the telescope towards some distant welldefined object, and fasten the screw. By means of the tangent-screw move the telescope slowly till the

LEVELLING.

CHAP. VIII.]

245

line of collimation is directed exactly to some distinct point in the object, and then proceed according to the instructions in the second paragraph of the first ad¬ justment of the theodolite. SECOND ADJUSTMENT.

To make the axis of the level CD parallel to the line of collimation. By turning the screw N, bring the bubble of the level to stand at the middle of the tube. Reverse the telescope in its wyes, and, if the bubble does not then stand in the middle, correct one half of the deviation by the screw m, and the other half by the screw N. Again reverse the telescope in its wyes, and repeat the correction if necessary. Now by revolving the telescope in its wyes, bring the level to some distance on one side of its lowest or proper position, and if the bubble then deviates from the middle, the deviation must be corrected by means of the screws p and y, which move one end of the level laterally, the correction being continued till the bubble will remain at the middle while the telescope is revolved so as to bring the level to a considerable distance on either side of its lowest position. When this has been done, the first part of the adjustment should again be examined and corrected if neces¬ sary.* * A method by which the second adjustment may be made for an instru¬ ment in which the telescope is not reversible, will be found in the first of the following problems.

21*

246

LEVELLING.

[CHAP. VIU.

THIRD ADJUSTMENT.

To make the line of collimation parallel to the bar EE, or which is the same, at right angles to its axis. Turn the telescope till it stands directly over two of the levelling-screws, and by means of them bring the bubble to stand at the middle of the tube. Then turn the telescope half round, that is, till it stands over the same screws, but pointing in the opposite direction, and if the bubble does not remain in the middle, correct one half of the deviation by the levelling-screws and the other half by the screw N. Now place the telescope over the other levelling-screws, and proceed in a similar manner. Continue the correc¬ tions till the bubble will remain in the middle of the tube during an entire revolution. These adjustments having been carefully made, the instrument is ready for use. When on the ground it must, in each new position in which it is placed, be levelled. This is done by placing the telescope over two of the levelling-screws and by their means bring¬ ing the bubble of the level to the middle of the tube, then doing the same with the telescope over the other two, and again over the first two. Then, if the third adjustment has been accurately made, the bubble will stand in the middle of the tube in any position of the telescope. OF THE LEVELLING STAFF.

A Levelling Staff consists of a square or rectangu¬ lar staff and a small circular or rectangular board,

CHAP. VIII.] LEVELLING.

247

called a Vane, which is so attached to the staff as to be moveable along it from end to end. It is used for measuring the height of the line of apparent level passing through the telescope of the levelling instru¬ ment, above the place where the staff is placed. The face of the vane, represented in Fig. Ill, is divided into four equal parts by two straight lines intersecting each other at right angles; one line being horizontal, and consequently the other vertical. Two opposite parts of the face are painted white and the other two black; thus the lines and their intersection are easily distinguished even at a considerable dis¬ tance. A screw, the head of which is shown at m, serves to clamp the vane to the staff in any required position. The staff is composed of two rectangular bars of wood, between five and six feet long, placed side by side, and forming together a square staff, the breadth of each side of which is about an inch and a quarter. The bars are so connected that one, which is two or three inches the shorter of the two, may be made to slide along the other or principal bar, and thus, when necessary, increase the length of the staff. In order to this, the front or sliding bar has throughout its length, on the side next the other, a projection which is terminated by a brass plate a little wider than the projection and firmly attached to it; and the principal bar has a groove in it to receive the projection and plate of the former. The forms of the projection and groove are exhibited in Fig. 112, which repre¬ sents a section of the bars, at right angles to their length.

248

LEVELLING. [CHAP. VIII.

The staff is represented in Fig. 113, in which beefa is the principal bar, which is capped at the bottom with a brass plate; bd is the sliding bar, A is the vane, seen edgewise, and n is a screw which serves to clamp the bars together in any given position. One side of each bar is divided, from bg upwards, into feet and hundredths of a foot; the feet being numbered as in the figure. The subdivisions of the feet are omitted as they could not well be all exhibited. On the cham¬ fered edge of a brass plate connected with the vane, a fine line a is drawn, directly opposite to the horizontal line of the vane. The line a serves therefore to de¬ note on the side of the principal bar, the height that the vane is above the line bg, when that height does not exceed 5 feet. The height of the vane above the line bg is usually called the height of the vane, although it is less than its true height above the ground or place on which the staff stands, by the length of the part be. This however produces no error in the use of the staff in levelling, as the difference in level of two places is found from the difference in the heights at the places, of a line of apparent level passing through the level¬ ling instrument, and this difference will evidently be the same whether both heights are measured from the ground, or both from the line bg. When it is required to raise the vane to a greater height than 5 feet, it must be slid up to a pin at p, which checks it at that height, and be fastened there by the screw m, Fig. 111. Then loosening the screw n, Fig..114, the front bar, which carries the vane, may be slid upwards till the vane acquires the required

CHAP. VIII.]

LEVELLING.

249

height. When this is done, that point on the graduated side of the sliding bar, which corresponds to the line marked 5, on the principal bar, will evidently indicate the height of the vane. SCHOLIUM.

The theodolite, levelling-instrument, and levellingstaff, are made of various forms and of different de¬ grees of perfection. Those which have been described are, when well made, good and convenient instruments; and from the descriptions which have been given of them, the student will find it easy to understand the manner of adjusting and using others which may differ from them in form. PROBLEM I. To test the adjustment of the level.

Select a place where the ground is tolerably level for a distance of 15 or 20 chains, and at each extremity of the distance chosen, as at A and. B, Fig. 122, drive Fig. 122.

a short stake. Set up the level by the stake at A, placing it so that the eye-end of the telescope may be over or nearly over the stake, and level the instru¬ ment. Place the levelling-staff on the stake, raising or lowering the vane till its horizontal line is at ex¬ actly the same height as the centre of the eye-end of the telescope, and note the height. Now let an as¬ sistant take the staff and set it up vertically on the stake at B. Direct the telescope to the middle of the breadth of the staff, and then, by raising or lowering the hand as a signal, direct the staff-bearer to raise or 21

250

LEVELLING. [cHAK VIII.

lowei the vane, repeating the signal till its centre ap¬ pears precisely in the direction of the line of collimation, or at least exactly of the same height. When this is the case, by a circular motion of the hand, direct the vane to be clamped, and again sight to it to ascertain that in clamping, its height has not been changed. Note the height of the vane, and subtract from it the correction of apparent level corresponding to the distance between the stakes, taken from the preceding table, or rather this correction diminished by a j part, to allow for refraction. Then, if the instrument is accurately adjusted and the observations have been carefully made, the difference between the height of the vane at A, and its corrected height at B, will be the true difference of level of the tops of the two stakes; the higher being that at which the height of the vane is the less. Placing now the in¬ strument at B, proceed in the same manner to find again the difference of level of the stakes. If this difference is the same as the former, the adjustment of the instrument is correct. But if there is any mate¬ rial difference in the two results, we infer that the axis of the level is not parallel to the line of collimation. To make it so, take half the difference of the results obtained, and let the vane, taken in its last po¬ sition, be elevated or depressed by that quantity, accord¬ ing as the result obtained at the more elevated of the two stakes, is less or greater than that obtained at the other.* Then, by the screw N, bring the line of colli* Let ab be the line of apparent level through the instrument when placed at A, and ae, the line of apparent level indicated by the instrument. Then ed being the line of apparent level through the instrument when placed at B, the line cf making the angle fhd equal to bhe, and consequently dj equal to be, will be the line of apparent level indicated by the instrument.

CHAP. VIII.]

LEVELLING.

2-51

mation to point exactly to the horizontal line of the vane, and with the telescope in that position, bring the bubble again to the middle of the tube by the screw m.

Note.

In this manner the axis of the level may be made parallel to the line of collimation when the tele¬ scope is not reversible, as is the case in some instru¬ ments. EXAMPLE.

Let the height A a, be 5.295ft. ;* Be, 2.063ft.; Be, 5.527ft.; and A/, 8.935ft.; and the horizontal distance from A to B, 20 chains. The distance being 20 chains, the correction for ap¬ parent level, taken from the table, is 0.042ft. From this deducting ± of 0.042, we have 0.035ft. for the cor¬ rection to be subtracted from the heights Be and Af. This gives, for the corrected heights, Be = 2.028ft., and Af= 8.900ft. Hence, taking 2.028 from 5.295, we have 3.227ft. for the true difference of level obtained with the instrument at A; the place B being higher than A. And taking 5.527 from 8.900, we have 3,373ft. for the difference of level obtained with the instrument at B. Half the difference of these results gives 0.053ft. for the value of df, the error due to the error in the It is therefore evident that the difference of level of the two places obtained with the instrument at A, is less than the true difference by the quantity be, and the difference obtained with the instrument at B, is greater than the true difference, by the equal quantity df. Hence the difference of the two results must be twice df the error in height produced by the error in the adjustment of the instrument. * Although the station staff is only divided to hundredths of a foot, we may with tolerable precision estimate the thousandths ; and when great accuracy is desired, it is better to do so.

252

LEVELLING.

[CHAP. VIII.

adjustment of the instrument. Hence, as the difference of level given by the instrument at the more elevated place is greater than the other, the vane must be lowered 0.053 ft. from its last height. The adjustment may then be made as directed. PROBLEM II. To determine the difference of level of two places, A and B, Fig. 123, when they are visible from, each other and do not differ in level more than 8 or 10 feet. Place the level in some position C, about equally distant from the places, either in the line joining them or on either side as may be most convenient, and level the instrument. Let the staff-bearer set up the level¬ ling-staff at A, and having sighted to it and obtained the height of the vane at that station, let the staff be removed to B, and do the same. Then the difference of the two heights, without any corrections, will be the difference of level of the places; that place being the higher, at which the height of the vane is the less. If the surface of the ground between the places is such that when the level is placed at equal or nearly equal distances from them, the line of apparent level of the instrument would pass below one place or too high above the other, as in Fig. 124, it may be placed in any position C, that will permit the sights to be taken to both places. Then, having measured the horizontal distances from A to C, and from C to B, we proceed as above, except that the observed heights Aa and B5 must be corrected by the differences between the true and apparent level, as taken from the table:

CHAP. YHI.] LEVELLING.

253

or, which is generally better, by the tabular quantities diminished by a & part. It may, however, be observed that when neither of the distances AC nor BC, exceeds five or six chains, the corrections are so small that they may be generally omitted. When a valley intervenes between the two places so that there is no suitable intermediate situation for the instrument, it may be placed at one of them; and then the difference of level may be determined as in the pre¬ ceding problem. EXAMPLES.

1. Let the observed height Aa, Fig. 123, be 7.343 ft. and B5, 3.635 ft.; then the difference of these, 3.708 ft. is the difference of level of A and B, the place B being higher than A. 2. Let the observed height Aa, Fig. 124, be 8.457 ft.; Fig. 124.

B6, 1.525 ft.; the distance from A to C, 24.1 ch.; and the distance from C to B, 8.2 ch. The correction for the distance 24.1 ch., taken from the table and diminished by a & part, is 0.050 ft., and for the distance 8.2 ch., it is 0.006 ft. The corrected heights are therefore Aa 8.407 ft., and B6 1.519 ft. Hence the difference of level is 6.888 ft. 22

254

LEVELLING.

[CHAP.

ym.

PROBLEM III. To find the difference of level of two places not visible from each other, or if visible, dif¬ fering considerably in level. Let A and F, Fig. 125, be the two places. Place the level in some position P, that will permit a sight to be taken to A, and also to some other place towards F, at about the same distance from the in¬ strument. Having levelled the instrument, let the staffbearer set up the staff at A, and when the sight has been taken, let him note the height of the vane as a first back¬ sight. Then let the staff be taken to some station B, about as far from the instrument as that is from the station A, and when the sight has been taken, let the height of the vane be noted as the first fore-sight. Next, the staffbearer remaining at B, take the level to some suitable place Q, beyond B, and pro¬ ceed to take a back-sight to B, and then a fore-sight to a new station C. In the same manner the operation must be continued from C to D, from D to E, and from E to F; the number of intermediate stations necessary to be taken, depending on the irregularities in the

CHAP. Tin.] LEVELLING.

255

ground and on the difference of level of the given places. Take the difference between the sum of the back¬ sights and the sum of the fore-sights, and it will be the difference of level of E and F; the place F being higher or lower than A, according as the sum of the back-sights is greater or less than that of the fore¬ sights.* Note,—It is not necessary that either the interme¬ diate stations or the places of the instrument should be in the direct line between the given places; and frequently it will be found convenient to deviate con¬ siderably from that line. It may further be observed that although it is generally best to take each pair of the sights at equal or nearly equal distances, as the correction for apparent level is thus avoided, and also a slight deviation of the axis of the level from paral¬ lelism with the line of collimation will not then sensibly affect the accuracy of the result, yet sometimes in order to diminish the number of stations the sights are taken at unequal distances. When this is done, the distances must be measured and the sights be corrected as directed in such case, in the last problem. * Let GF be a line of level through the place F; then AG is the dif¬ ference of level of the two places A and F. Now we have, Sum of back-sights, = A a + Be + Ce + Dg + E£=Aa + be + B6 + Ce + D g + EZ. Also, Sum of fore-sights, = B6 + Cd + Df+ E& + Dm — Db + Ce + ed + Dg + gf+Dl + lh + D — Fm + lh + gf+ ed +B6 + Ce + Dg + EZ= qd + B6 + Ce + Dg + E l =pc + B6 + Ce + Dg + E£ = AG+A a + 6c + B6 + Ce + Dg + EZ. The difference of these sums is AG, the difference of level of the places A and F.

256

LEVELLING.

[CHAP. VIII.

EXAMPLE.

Let the back-sights and fore-sights, Fig. 125, be as following table. Back-sights.

i. 2. 3. 4. 5.

7.103 ft. 9.227 1.236 1.610 2.125

Sum 21.301

Fore-sights.

1.566 ft. 3.178 9.415 6.367 9.910 Sum 30.436 21.301 Diff. 9.135

Hence the difference of level of A and F is 9.135 ft., and as the sum of the back-sights is less than that of the fore-sights, the place F is lower than A.

Fig. 119. a

Fig. 109.

CHAPTER IX.

TOPOGRAPHY.

is a branch of surveying,* the object of which is to determine and designate on a map, the various undulations and inequalities in the surface of a particular place, tract of land or district of country. A map in which these inequalities, the courses of streams, and sometimes other circumstances, as the positions and extents of forests, marshes, &c., are de¬ signated, is called a topographical map. TOPOGRAPHY

In addition to the boundaries and content of a tract of land, it is frequently required that the various slopes and irregularities of the surface should be determined and designated, in order to give a more complete view of the ground, and to afford the means for an appro¬ priate location of buildings or works of any kind that may be designed to be erected on it. If we assume the surface of a tract of land to be intersected by a number of level surfaces or horizontal planest at equal distances from one another, and trans¬ fer all the lines of level in which these planes meet the surface of the ground to an assumed horizontal * The term surveying is here used in a more extended sense than as de¬ fined in the first chapter. f In tracts of any moderate extent, the surfaces of level may, for the purpose for which they are here introduced, he regarded as horizontal planes.

22*

2K

258

TOPOGRAPHY. [CHAP. IX.

plane passing through the lowest point, making them occupy positions on that plane, corresponding with their positions on their respective planes, the varia¬ tions in the distances of the lines from one another, when thus transferred, will indicate the variations in the inclination of the ground. For as the difference of level from line to line is the same, it is evident that the horizontal distances of the lines, taken in any di¬ rection, will diminish as the inclination in that direction increases. Thus in ABCD, Fig. 129, which represents a small tract, of which the length AB is 1000 feet and the breadth AD, 800 feet, the lines 10, 20, 30, &c., represent lines of level in which horizontal planes at the distance of 10 feet from one another, intersect the surface of the ground. The lowest level passes through the point F, at which the stream EF leaves the tract. From F the ground rises more rapidly to the left than to the right, as is indicated by the lines of level being nearer to one another on that side than on the other. In passing from F towards the corner B of the tract, we may observe that the. acclivity, which is gentle, increases till we come to the 30 feet level, then diminishes to the 40 feet level; it then again increases, and more rapidly, to the 60 feet level, and lastly slightly diminishes to the 70 feet, or highest level. In descending, the declivity continually dimin¬ ishes to B. From the point l in the side AD, the ground descends moderately towards the corner A, the declivity diminishing till the surface becomes nearly level; and from the stream, towards A, the acclivity, which is slight, diminishes, so that between the two *>ranches of the 30 feet line of level and the corner A, the ground is nearly level.

CHAP. IX.] TOPOGRAPHY.

259

The distance which should be taken for the distance of the assumed planes from one another, must depend on the extent of the survey, the inequalities in the sur¬ face, and the degree of minuteness with which it is required that they should be designated. It may vary from 3 or 4 to 20 or 30 feet, according to circum¬ stances. The levelling required for a topographical survey may be performed either with a level or theodolite. Where there is considerable ascent or descent in the ground, the latter is the most convenient instrument; and although the results obtained with it are not in general as accurate as those that may be obtained with a good level, they are, when due care is taken, sufficiently so for the object in view. When the theo¬ dolite is used, the sight should be taken to a point at the same height above the ground at the station, as the axis about which the upper telescope revolves is above the ground at the place of the instrument. To do this, let an assistant place and clamp the vane of a levelling-staff at that height, or make a mark at the same height on a pole; and when he has taken the staff or pole to the station and set it up vertically, sight to the vane or mark. When this is done, the difference of level between the station and place of the instrument, expressed in 100th parts of the horizontal distance of the two, will be indicated on the vertical limb. It is however better, except when the horizon¬ tal distance is quite small, to obtain the difference of level from that distance and the angle of elevation or depression.

260

TOPOGRAPHY.

[CHAP. IX.

PROBLEM I. Having given the back-sights and fore-sights taken to a number of consecutive stations, or to two or more con¬ nected series of stations, to determine the heights of the stations above a line of level or surface of level through the lowest, 1. When there is but one series of stations. Assume for the height of the first station above some assumed line of level, any quantity taken at pleasure, observing however to make it sufficiently great for the assumed line of level to be lower than the lowest station, or at least as low. To the assumed height of the first sta¬ tion add the first back-sight, and from the sum subtract the first fore-sight, and the remainder will be the height of the second station above the assumed line of level. With this height and the second back-sight and second fore-sight proceed in like manner to find the height of the third station; and thus on to the last. Now subtract the least of the heights obtained, which must be that of the lowest station, from each of the others, and the remainders will be the heights of the other stations above the line of level passing through the lowest. 2. When there are two series of stations connected by intervening sights between the first station of the first series and the first of the second series. Assume for the height of the first station of the first series, a quantity sufficiently great for the assumed surface of .'evel to be below all the stations, or at least as low as

CIIAP. IX.] TOPOGRAPHY.

261

the lowest, and proceed as directed above, to obtain the heights of the other stations of that series above the assumed surface of level. Then commencing again with the assumed height of the first station, proceed in like manner with the sights connecting that with the first station of the second series, and with the sights to the stations in this series, to find the heights of all these stations. When this is done, subtract the least of all the heights obtained from each of the others, and the remainders will be the required heights. 3. In like manner, whatever be the number of the series of stations, the heights of all the stations, above a surface of level passing through the lowest, may be obtained. Note. If it is required to find the heights of the stations above a line or surface of level at a given dis¬ tance below the lowest or any other given station, it is easily performed by applying to the heights of the stations above the assumed level, the difference be¬ tween the height of that station above the assumed level and its height above the given level. EXAMPLES.

1. Taking to the nearest length of a foot, the back¬ sights and fore-sights given in the example to the last problem of the preceding chapter, it is required to find the heights of the stations A, B, C, &ic. Fig. 125, above a line of level passing through the lowest. Assume the height AL, of the station A, above an assumed line of level LM to be 15 feet. Then we have

262

TOPOGRAPHY.

[CHAP. IX.

Feet.

15, height of A, above LM, iC 15 +7.1—1.6=20.5 “ B, « 20.5 + 9.2—3.2=26.5 “ C, a 26.5 + 1.2—9.4 = 16.3 « D, a 16.3 + 1.6—6.4=11.5 “ E, u 11.5 + 2.1—9.9= 5.7 “ F, Subtracting 5.7 from each of the above heights, we have the heights above GF, the line of level through F. These are, for A, 9.3ft.; B, 14.8 ft.; C. 20.8 ft.; D, 10.6 ft.; E, 5.8 ft.; and F, 0 ft. 2. Let the hack-sights and fore-sights, taken from A to B, Fig. 126, from A to C, and from C to D, be as Fig. 126. Cl

given below; to find the heights of the stations along AB, AC, and CD, above a surface of level through the lowest station. AB B-sts.

1 2 3 4 5 6

8.7 ft. 1.1 0.7 2.2 1.9 7.4

CD

AC B-sts.

F-sts.

3.8 ft. 9.9 9.3 8.0 7.1 4.2

1 2

F-sts.

8.5 ft. ,5.2 ft. 9.1 0.6

1 2 3 4 5 6 7

B-sts.

F-sts.

3.1ft. 2.6 1.5 4.2 3.8 9.3 2.8

8.6 ft. 7.7 9.3 7.9 7.3 2.2 7.4

By proceeding as directed in the rule, we obtain the following heights of the stations above the surface of level passing through the lowest.

263

TOPOGRAPHY.

CHAP. IX.]

CD AC 25.6 ft. 30.8 ft. 1 1 34.1 2 20.1 2 25.6 3 15.0 3 7.2 4 4 3.5 5 5 0.0 6 6 7.1 7 7. 2.5 8 PROBLEM II. To determine the inequalities in the surface of the ground along a line running in a given direction, and to draw an irregular or curved line to represent them. AB 30.8 ft. 1 35.7 2 26.9 3 18.3 12.5 7.3 10.5

Let short stakes be driven at the beginning and end of the line, and at each point along it where there is any material change in the inclination of the ground; and let the horizontal distance of each stake from the beginning of the line, or the distances from stake to stake, be measured. Then level from stake to stake, using intermediate stations whenever the difference of level between any two is too great to permit sights to be taken to both from a single position of the instru¬ ment. Find, by the last problem, the heights of the stations, where the stakes are placed, above a line of level passing through the lowest, or above any assumed or given line of level. Draw a straight line ef Fig. 127, to represent the Fig. 127. ¥

c

a

j

os

1

i—i

;—i_

L..-i—J

I

;

cdghlmnpqf

line of level to which the heights of the stations are referred, and on it make ea, eh, es, ec, &c., equal to the distances of the stations from the beginning of the line. From the points e, a, h, s, &c., draw lines per-

264

TOPOGRAPHY. [CHAP. IX.

pendicular to ef, and make them equal to the heights of the respective stations. Through the tops of these perpendiculars, draw the curved line EF, which will be the line required to be draw’ll. The line EF is called a Profile of the ground in the direction of the given line. Note. The heights of the perpendiculars are fre¬ quently taken from a scale three or four times as great as that used in laying ofF the horizontal distances. When this is done, the curved line, or profile, as it is still called, indicates more distinctly the lesser changes in the inclination of the ground.

EXAMPLE.

The distances of the stations along a given line, measured from the beginning of the line, and their heights above a given line of level, determined by the last problem, from sights taken on the ground, being as below, the profile of the ground in the direction of the line, obtained by taking the heights from a scale three times as great as that from which the distances are taken, will be that of Fig. 127. ' St a.

1 2 3 i 4 5 ! 6 . 7

Diet.

Oft. 90 230 336str. 445 505 555

Ht.

41.1ft. 38.3 22.0 15.3 21.1 32.5 36.5

Sta.

8 9 10 11 12 13 14

Dist.

610ft. 655 725 770 850 915 1000

Ht.

41.9ft. 55.0 69.2 72.0 69.1 56.5 50.1

CHAP. IX.]

TOPOGRAPHY.

265

At the fourth station a stream of water crosses the line, and is noted by the letters sir. placed by the side of the distance. PROBLEM III. To determine those points along a line running in a given direction, that are at given heights above a given line or surface of level. Proceed, as directed in the last problem, to.find the heights above the given line or surface of level, of those points in the line where there is any material change in the inclination of the ground, and also their distances from the commencement of the line. Observe between which two of the heights obtained, any one of those given falls, and take their difference. Also take the difference between the given height and that one of the two which appertains to the point nearest the beginning of the line. Then, as the first difference : the second : : the horizontal distance between the points : to a fourth term, which added to the distance of the point nearest the beginning of the line, will give the distance of the required point. Proceed in the same manner with the other given heights. Or we may draw, by the last problem, a profile of the ground, as EF, Fig. 128, in reference to the given Fig. 128.

line of level
2L

266

TOPOGRAPHY.

[CHAP. IX,

in the figure; and from the points in which these meet the profile EF, draw lines parallel to ea. The dis¬ tances from e, at which these last lines meet ef, will be the horizontal distances of the required points, from the beginning of the line. EXAMPLE.

Let the data found on the ground be the same as in the example to the last problem, and let the given heights of the required points be 10, 20, 30, 40, &c. feet. The distances and heights of the stations being the same as for the profile in Fig. 127, we shall obtain a similar profile EF, Fig. 128. Drawing the lines paral¬ lel to ef at distances from it, equal to 10, 20, 30, &c. feet, we find that the first line above ef does not meet the profile EF. Consequently there is no point in the latter so low as 10 feet above the given level. Each of the other parallels meets the line EF in two points. There are therefore two points in the line at the height of 20 feet above the given level; two at the height of 30 feet; and so on to 70 feet. The distances e, 40; e, 30; &c., on ithe line ef, are the distances of the required points from the beginning of the line. In illustration of the first method of finding the dis¬ tances of the required points from the beginning of the line, as the given heights of the stations evidently show that there is no point in the line so low as 10 feet above the given level, let it be required to find a point at the height of 20 feet. On examining the given heights of the stations, wre perceive that there must be two such points; one between the 3d and

CHAP. IX.]

TOPOGRAPHY.

267

1th stations, and one between the 4th and 5th. For the first of these we take the difference between 15.3 and 22.0, which is 6.7; the difference between 20.0 and 22.0, which is 2.0; and the difference between the distances 230 and 336, which is 106. Hence, as 6.7 : 2.0 : : 106 : 32. Adding therefore 32 to 230, we have 262 feet for the distance of this point. PEOBLEM IV. To determine the undulations and inequalities of the sur¬ face in a tract of land ABCD, Fig. 129, and to draw a topographical map designating them. Fig. 129. E

"With a compass or theodolite, run a number of lines, ab, cd, ef &c., across the tract, parallel to one of the sides, as AB; making them nearer together or farther apart, according to the inequalities in the ground and the degree of minuteness with which it is intended to designate them. Drive stakes at the beginnings and ends of the lines AB, ah, cd, &c., and at all the points along them where there is any material change in the inclination of the ground, and proceed to level from

268

TOPOGRAPHY.

[CHAP. IX.

stake to stake, along tliese lines and the line AC; also measure the distances of the stakes from the com¬ mencement of the lines, or from one another. Then, by problem I., find the heights of all the stations where the stakes are driven, above a surface of level passing through the lowest station. Now, having drawn the lines db. cd, efy &c., in their proper positions on the map, determine, by the last problem, the points in these lines and the lines AB and DC, that correspond to heights in the lines on the ground, of 10, 20, 30, 40, &c., feet, above the surface of level passing through the lowest station, or to any other heights, increasing by equal differences, that may be deemed expedient. Through each set of points appertaining to the same height, draw a curve line. The curve lines thus drawn will represent lines of level of 10 feet, 20 feet, &c., or of the number of feet of the heights used in obtaining them, wrhatever that may be. These lines serve to indicate the changes in the inclination of the ground. But it is usual, instead of drawing them distinctly with ink, to draw them with a pencil only, or faintly with Indian-ink, and then to shade the map by short straight lines, drawn perpen¬ dicularly from each curve of higher level, to that of the next lower; the lines being drawn closer together and rather heavier as the distance between the lines of level diminishes. For those parts of the ground that are level, or very nearly so, the shading is omitted. The greater or less darkness of the shading on the dif¬ ferent parts of the map, therefore, indicates a greater or less inclination in the ground in those parts; and the omission of the shading in any pails, indicates that

CHAP. IX.] TOPOGRAPHY.

269

in those places the surface is level, or very nearly so In Fig. 130, we have a map of the tract, shaded as described above. Fig. 130. E

130

* Note.—It is not necessary that the lines on which the levels are taken should be run parallel to one another. They may be run making given angles with any given lines of the survey. Sometimes it is desired that the surface of a particular part of the tract should be designated with more minuteness than is important for other remote parts. In this case it will be found convenient to determine the position of some point near the middle of the part which it is desired par¬ ticularly to designate, and then to run lines from this point, in directions making angles of 20° or 30° with one another. It may be further remarked, that instead of taking the surface of level through the lowest point, as the plane of reference, we may, if preferred, take that through the highest. The former is however the one generally taken. 23*

270

TOPOGRAPHY.

[CHAP. IX.

The operations to be performed in this problem, so far as observations on the ground and numbers are concerned, are merely repetitions of those which have been exemplified in the preceding problems. An ex¬ ample is not therefore necessary.

APPENDIX.

CONTAINING INSTRUCTIONS FOR SURVEYING THE PUBLIC LANDS OF THE UNITED STATES. BY GEOKGE H. HOLLIDAY, A.M.

THE following statement of the mode of making the surveys of public lands of the United States, and the proper method of subdividing, is prepared to meet the wants of landowners and surveyors in those surveys. In most of the leading principles I take pleasure in acknowledging my indebtedness to a pamphlet of instructions prepared, I believe, by the late excellent Sur¬ veyor-General for the district of Illinois and Missouri, D. A. Spaulding, Esq., and furnished to applicants at that office, for information. 1. The surveys of the public lands of the United States are based on a line running east and west, called the base line. 2. From certain noted points, as the mouths of principal rivers, are lines run due north, called principal meridians. 3. Townships are bodies of land, six miles square, subdivided into thirty-six parts called sections. The following represents a township with the mode of numbering sections:

(271)

272

INSTRUCTIONS FOR SURVEYING NORTH.

6

5

4

3

7

8

9

18

17

19

2

1

10

11

12

16

15

14

13

20

21

22

23

24

30

29

28

27

26

25

31

32

33

34

35

36

*

4. Those lines of townships running north and south are called range lines. Rows or tiers of townships running north and south are called ranges. Tiers of townships east and west are called townships, and the lines dividing them are called township lines. Townships are numbered from the base line and the principal meridians, affording an easy method of de¬ scribing lands. Thus, if the township above be considered township three north of the base line, in range four west of the third principal meridian, the one joining it on the north will be township four range four, the one to the west township three range five, which cannot be mistaken for any other. Sections and parts of sections are described with the same ease and certainty. 5. In running out townships as described hereafter, many errors arise which it would be inconvenient to carry to any con¬ siderable distance; to obviate these, generally at intervals of thirty miles, or five townships, a line is run due east and west from the principal meridians, called a standard or correction line. This line also serves to correct the difficulty otherwise arising from the convergence of the meridians. 6. A chain is a measure of four rods. It is more convenient in practice to use a half-chain of two rods, divided into fifty links; each link thus is the one-hundreth part of a chain, and is so expressed and used in taking field-notes and making cal¬ culations.

THE PUBLIC LANDS.

079

A {o

7. In running standard lines, corners are established every 40 chains, for quarter section and section corners; at the end of every sixth mile a township corner is made. These corners are starting points for surveys to the north of this line, and do not apply to any surveys to the south. 8. The mode of laying off a township is as follows:—The south line is a standard line or the north line of a township already surveyed. The east line is a principal meridian or the west line of the township to the east; it only remains to establish the west and north lines. The surveyor therefore proceeds to the south-west corner of the township already established, and runs due north, establishing, as he proceeds, corners at every 40 chains for quarter-section and section corners. These cor¬ ners were formerly made in timber, by taking two trees as wit¬ nesses, by boxing into them, marking with the number of the section, and accurately taking their course and distance; now the regulations require four such witness-trees at section and township corners, and two at quarter-section corners. Formerly in prairie a small mound was thrown up, now one of consider¬ able size is required, in which must be buried stones of size and shape to be described in the field-notes, or a certain quantity of charcoal. At the end of six miles the surveyor establishes his township corner, and the west line is finished. All the corners now established on this line belong to the township lying west, and not to the one now to be run out. The corners for this township are to be made as described hereafter. The surveyor next proceeds to the north-east corner of the township already established and runs west, setting at every 40 chains a tempo¬ rary post, proceeding until he strikes the west line of the town¬ ship, and notes down the distance he falls to the north or south of the township corner. If there is any excess or deficiency of six miles in this line, it is all thrown into the last 40 chains, or the west half of the north line of section 6; then if the line run is not a straight line from township corner to corner, he returns, and by offsets from the temporary posts already set he esta¬ blishes all the corners in a straight line. All the corners on this line are intended for the township north, and not for the one now laid off. In case the township closes on a standard 2M

274

INSTRUCTIONS FOR SURVEYING

line, the north-west corner is made at the intersection of the west line with the standard line, whether it be more or less than six miles, and without regard to the township corner already established for the townships north, the surveyor giving in his notes the length of the lines and the distance between his corner and the one on the standard line. 9. The township lines being established, the surveyor proceeds to the subdivision. He commences at the corner to sections 35 and 36 on the south line of the township and runs north, makes a corner at 40 chains, being the quarter-section corner between 35 and 36, and one at 80 chains, corner to 25, 26, 35, 36. Here he turns east, and at 40 chains sets a temporary post, and proceeds in the same line until he comes to the corner to 25 and 36 on range line; he notes the distance and the falling to the north or south of this corner, then returns, and by offsets from the line run establishes the quarter-section corner between sections 25 and 36, at an equal distance from each section cor¬ ner, and in a straight line with them; then, taking a new start from the corner to 25, 26, 35, 36, proceeds in the same way to run out sections 25, 24, 13, and 12. Then from the corner to sections 1, 2, 11, 12, he runs north 40 chains, establishes the quarter-section corner between 1 and 2, then runs north until he strikes the north line of the township; here he establishes his corner, giving in his notes the length of the line and the distance from the corner already established for the township north. He then returns to the south line of the township, and commencing at corner to sections 34 and 35, runs out another tier of sections in the same manner as the first, and so on until he has completed four tiers of sections. He then returns to cor¬ ner to sections 31 and 32, and proceeds in the same way with the tier of sections on the east; but at every section corner he also runs wTest, makes a quarter-section corner at 40 chains, and proceeds in the same line until he strikes the range line, where he make3 his corner, giving the falling to the north or south of the corner already established for the township west. 10. It will be perceived that all the section corners are esta¬ blished on lines running north, and these, as well as the quartersection corners on these lines, are established permanently in

275

THE PUBLIC LANDS.

the first instance, and all the quarter-section corners on east and west lines, are corrected to an average distance and in a straight line with the section corners, except those in the western tier of sections, which are established permanently in the first instance. It will also be seen that all excess or deficiency of measure¬ ment is thus thrown into the north half of the sections on the north and the west half of the sections on the west of township; these sections are hence called fractional sections. A complete township with fractions would occupy too much space. Below are three fractional sections, one in the north, one in the north¬ west, and one in the west of townships; they are taken from the government surveys in township 11 north, range 8 west of third principal meridian:— SEC. 3.

SEC. 6.

The numbers on the lines represent the length in chains and

276

INSTRUCTIONS FOR SURVEYING

hundredths; the numbers at offsets on range and township lines represent the difference in links with corners in other townships ; the number in the south-east corner of sec. 3 shows the falling north in running the south line. The general principles of making the public surveys are now before the reader; a knowledge of them is not only important but indispensable to every surveyor. * * 11. No corners are established in the interior of sections, and as the public lands are sold in tracts of 40, 80,160, 320, or 640 acres, it devolves upon other surveyors to make the divisions. This at first view would seem a very simple affair; but the diffi¬ culties continually arising, and the frequent conflicting surveys by different surveyors, and even clashing surveys by the same, prove the necessity of some well-grounded system being adopted and universally acted upon to insure justice to all parties inte¬ rested. 12. As the land was originally owned and surveyed by the United States, and conveyed by the corners thus established, it is of the first importance to find these corners before any divi¬ sions are attempted. This is frequently a difficult matter. Many witness-trees were killed by boxing; many others die, and even the roots disappear before the land is occupied. In prairie there are many hills that may be mistaken for mounds, the rooting of hogs destroys many more, and there is reason to believe that in many cases, particularly in prairies, corners never were established. In case of the loss of these corners they must be renewed in the same manner in which they should first have been established. 13. In case of loss of corners on a standard line or principal meridian, the line must be followed and measured from one known correct corner to another on this line. This is necessary in order to be assured that both the course and measurement is the same used by the first surveyors; many causes of disagree¬ ment in both are frequent, the causes of which it is not here necessary to explain. The missing corner should be placed in a straight line between the original corners, and at an equal or proportional distance. 14. Should any discrepancy, however, appear in either the

THE PUBLIC LANDS.

277

course or length of this line, it will be necessary to examine the section lines intersecting it. The corners on the standard line should be at the points of intersection of the sectional lines originating or terminating in it. The finding of these points as called for by the field-notes, frequently relieves the surveyor of a great deal of difficulty, and throws unexpected light on some of the most complicated cases of difficulty. This remark applies also to all other lines of townships. 15. In running a township corner, the better plan is to com¬ mence at the nearest known corner on the range line, and then run north or south past the township corner, until another cor¬ ner is found, then place the missing corner at an equal or pro¬ portional distance as required by the field-notes, and in a straight line; should this, from any circumstances, appear not to have been the point where the original corner was established, take the range of the corners on the township lines to the east and west, and should these unite in one point on the north and south line, the presumption is that that is the true place for the corner, and it should be there established. 16. Section and quarter-section corners on the north and south lines are to be renewed in the same manner, by starting at some corner on these lines, and running past the missing corners until some reliable corner is found, then establishing the missing corner in a straight line and at its proper distance. If the surveyor, in the extension of these lines, strikes a township line, he is to proceed no further on the same course, but the corner here, if missing, must be renewed, as in sec. 13, before any further steps are taken. It is important to re-establish all section corners on north and south lines, because they were originally established in this mode, on the same course and at equal distances, while the lines of sections running east and west are of very unequal lengths and in almost all cases make an angle at section corners. Should any discrepancy, however, arise, these lines should be examined as in establishment of town¬ ship corners. 17. When all the corners of the original survey are found or renewed, the surveyor is ready to make the subdivisions, and 24

278 not before. pursued:

INSTRUCTIONS FOR SURVEYING

The following plan will represent the course to be NORTH.

A

G

B

i

i 0 F

E

Suppose, for instance, a quarter-section is to be surveyed; say the north-west quarter. The corners at A, B, and C, are already established; it only remains to establish the corner at 0. Experience shows that the quarter-section comers on the north and south of sections, required to be made on corrected lines, are much more frequently out of place than those on the east and west, made on the first line run; consequently, as it is desirable to carry errors in the establishment of original corners no farther than possible, sections should be divided by a straight line between the quarter-section corners on the east and west. These errors thus never pass the centre of the section. The surveyor should therefore commence at C and run east, set a random stake at 40 chains on his line, continue the same line until he intersects the section line on the east at D, and divides the distance and establishes his corner at 0 accordingly, with¬ out any regard to the corners at B and E. 18. Should a smaller division be necessary, as, for instance, the north-east quarter of the north-west quarter. On the line from C to D a temporary stake must be set at F, as well as 0. These, corrected, will be the southern corners of the east half of the quarter-section. A line must be run from A to B, and at an equal distance the corner G established, then a line run

THE PUBLIC LANDS.

279

from Gr to F, and also from 0 to B, at equal distances from these points, corners are to be established, and the tract is sur¬ veyed. If in timber, it only remains to mark the south line, a measurement is unnecessary. 19. The peculiar value of this system consists in never run¬ ning a line witnout a corner at the starting point and one at the termination. By this means both errors in course and dis¬ tance are corrected, and if corners are not established in pre¬ cisely the same place they originally occupied, justice, as far as practicable, is done to all interested parties. 20. Errors in the original surveys often cause a great deal of trouble; those most frequent deserve mention. The measure¬ ment was kept by outs, each out being the distance one set of pins carries the chain-carriers when the pins are changed to the fore chain-man. Eight outs make the 40 chains, or half-mile, the distance between the original corners. Sometimes the chain-carriers only made seven outs, by mistake, between cor¬ ners, and sometimes nine. In all cases, however, where the original corners are found they are immovable, however much they may be out of the way. Sometimes a pin was lost, and, consequently, 50 links in every out were lost until the fact was discovered. When chain-carriers are fresh, they carry the chain much tighter than when they are tired; thus a great deal of want of uniformity may be accounted for. 21. Crooked lines are sometimes found. These are frequent in section lines running east and west; the quarter-section cor¬ ner being made sometimes on a random line without correcting; sometimes the surveyor made his offset in the wrong direction; and, generally in the older surveys, great carelessness was manifested. Sometimes considerable crooks are found in town¬ ship lines, so as to lead to the opinion that all the corners were made at random. In dealing with these great care must be exercised, and no opinions formed without the most decisive evidence. 22. In the taking of witness-trees there are often considerable discrepancies in the notes. In distance I have often found 10 links, sometimes a chain, more or less than the notes. Some¬ times there is reason to believe the distance to one of the trees

280

INSTRUCTIONS FOR SURVEYING

was lost, and filled up at random in copying. In bearing there are fewer errors, but more important. In two instances I have found 30° mistaken for 50°. A very frequent error is made by letting the eye rest upon some leading division upon the face of the compass, and counting degrees in the wrong direction. In all cases where there are two trees these are easily rectified; there are four data for determining the corner—the distance and bearing to each tree—when three of these agree, the pre¬ sumption is that it is correct. 23. The labours of the surveyor, to result in any lasting good, should be spread on record; this is the duty of the county sur¬ veyor. But not only does the office fall into the hands of men incompetent to perform the work, but in no instance with which I am acquainted are the records kept in a manner likely ever to settle disputes, or render satisfaction to any intelligent person consulting them. Below is the plan adopted by me, and which should be followed in all records of surveys. It is simple and satisfactory. It will be seen that in all sectional lines and their parallels I make the variations entirely by the scale, noting by that the degree I run on. FEBRUARY 23, 1854. For David Holmes, S. E. J, N. W. £• Sec. 21, T. 10, R. 9. Magnetic var. E. 8° 00'. Chain-carriers, Riley Hicks and A. B. Carr, sworn. Chains & Courses.

Began at J section corner, between secs. 20 and 21, Stake in original mound; thence Index 8° 00'. North. 13 00 Entered brush and young timber, bears E. and W. 43 Branch 6 links wide, runs E. 27 32 47 Left brush, entered Hubbard’s field. 40 00 Mound ploughed up, set stake ; thence 20 27 Left field, entered hazel. 30 00 Entered edge of prairie, as in field-notes. 40 00 Mound ploughed up between secs. 16 and 17. 80 00 Mound to 8, 9, 16, 17, lost; thence 00 Enter barrens, N. E. and S. W. as in field-notes. 26

THE PUBLIC LANDS.

,281

Chains & Courses.

40 00 47 00 50 00

Mound between 8 and 9 lost, land broken. Joe’s Creek, 10 links wide, bears S. W. Former bed of creek, 5.00 chains farther north than called for by field-notes. 79 85 14 links east of corner to 4, 5, 8, 9, set by me in survey of Jayne’s land. Then corrected stake corner to sections 16, 17, 20, 21, thence East. Index 8° 00'. 20 00 Set stake. 40 44 6 links south of J sec. corner between sections 16 and 21, S. W. witness standing, N. E. cut down, renewed by marking elm 9 inches diameter, N. 63° E. 49 links. Then from this corner South. Index 8° 00'. 20 00 Set stake. 41 06 57 links east of corner centre of section, set by me in survey of Carr’s land. Corrected this line back, and set stake above at average distance, and took witness black oak, 32 inches diameter. N. 57° E. 9 links. Then from corrected stakes, N. W. corner of E. f, N. W. J this section. South. Index 7° 40'. 20 00 Set stake. 40 48 8 links west of corner set by me in survey of Carr’s land corrected corner on above line and took witness hickory, 8 inches diameter. S. 5° W. 93 links. 24*

2N

HINTS TO YOUNG SURVEYORS. BY A PRACTICAL SURVEYOR.

YOUNG surveyors who have not had the advantage of instruction from careful practical operators often meet with difficulties, and are much puzzled for want of the readiest means of overcoming them. In such cases, it is all im¬ portant to keep themselves cool and collected; otherwise they may expect to be pestered with officious suggestions or mortify¬ ing remarks from the lookers-on, thus adding to their confusion, and rendering the chance of overcoming the difficulty more uncertain. It is presumed that young men undertaking the business have made themselves familiar with all the necessary calculations: in which event a little practice will enable them to adapt the proper means to the circumstances of the case. A few examples are here inserted, embracing some of the obstructions which most frequently occur in the practice, with a view to familiarize the student with them, in order that he may at the outset be the better able to encounter them.

EXAMPLE

1.

Fig. 1.

A. has agreed to convey to B. a part of his tract of land, situate in the township of 0 , county of M , state of D , at a certain price per acre. The parties having fixed upon the several corners A, B, C, D, E, F, G, and II, call upon Q. to survey it, and make out a description to be recited in the deed. After examining the ground, the corner at A is fixed upon

(282)

HINTS TO YOUNG SURVEYORS.

283

as the best place to commence. Placing the instrument at A, direct the telescope to a staff placed perpendicular at B, a Fig. 1.

stone for a corner; ascertain the bearing of AB, N. 48° 82' W. The line AB runs through a pond of water, and cannot be measured with a chain. The distance must therefore be ascertained by trigonometry. From the point A measure any angle, say 21°. West of the line AB measure any distance on this line, say 80 P. to a. Place the transit at A, and measure the angle AaB, which proves to be 139° 38', the angle AB a is therefore 19° 22'. Now we have the three angles and one side of the triangle ABa, to find the side AB, which is found to be 156.25 P. The corner C is a spike driven near the middle of a white oak stump. It cannot be seen from B, nor can the points B

284

HINTS TO YOUNG SURVEYORS.

and C be seen from any intermediate place, on account of hills intervening. Place the instrument at B, and run a random line as near the direction of C as you can judge, say S. 51° 35' W.; measure this line. At 40 P. it crosses a small run of water. Thence 62.8 P. to a point opposite C (whole dis¬ tance 102.8). The line is 8 links East of the point C, the difference therefore of the bearing of the true line is 10' West of the random line, or S. 51° 45' W., 102.8 P. The corner at D is a hole in a high rock, and cannot be measured to with the chain. You must therefore resort to trigo¬ nometry. From C, the corner at D bears N. 41° 51' W. With this line measure any angle, say 40°, to the East; measure any distance on this line, say 92 P. to a point where you can see objects at C, D, and E (marked 6). From b measure the angle C6D which proves to be a right angle, and DJE, which is 112° 34'. Measure the line 6E, 84.62 P. to a stake in the middle of a public road. You have now one side and two angles of the triangle C5D, to find the side CD, which is 120.1 P., and the side 6D, which is found to be 77.2 P. And two sides and enclosed angle of the triangle D5E, to find the side DE and its bearing, N. 52° 41' E. 134.8 P. From E you proceed along the middle of the road S. 71° 42' E. 14.5 P. to a small run of water, and 76.62 P. to another stake in the middle of the road. Thence S. 84° 16' E. 52.48 P. to a stone at G. The next line, GH, is inaccessible in consequence of thick bushes and trees, extending nearly its whole length. Measure back any distance on your last line that will give you a clear sight, say 8 P. to c, and from II on the same course, measure 8 P. to d, and place a staff there. From C you ascer¬ tain the course cd, which is parallel wTith GH, to be S. 51° 42' E., measure the distance from c to d, which is 144.2 P. If the staff has been placed in the proper direction from II, the dis¬ tance of GH will be equal to cd. Test this by placing the instrument at cZ, and take the bearing of FG, S. 84° li|&E., this strikes 5 links short of the corner at II. I must there¬ fore add 0.2 P. to the length of the line cd, for the length of GII. GII is therefore S. 51° 42' E., 144.4 P. The last line IIA cannot be measured, or its bearing taken from either end. It is, however, ascertained by traversing the survey to be S. 42° 42' W. 94.72 P.

HINTS TO YOUNG SURVEYORS.

285

The accuracy of the work may be tested by calculating the bearing and distance of the corner H or A from either of the other opposite corners. Thus: In the triangle ABH, you have the two sides and their bearings, to find the bearing and distance of the line BIT. This having been found by calculation to be S. 79° 13' E., 185.6 P., the instrument placed upon B and directed to II will test the accuracy of the whole survey. In making out the description for the conveyancer, it is best to commence at some well-defined corner of the survey. This, however, is a mere matter of convenience. The description of the above may be as follows. Beginning at a stake set for a corner in the middle of a public road leading from R to C , at the distance of 82.6 P. south-eastward from a stone, a corner of John Dunns land. Thence along the middle of said road S. 71° 42' E., 91.12 P. to a stake, and S. 84° 16' E., 52.48 P. to a stone for a corner. Thence by land of Peter Binder, S. 51° 42' E., 144.4 P., to a white oak sapling* marked. Thence by land of John White, S. 42° 42' W.? 94.72 P., to a stone set for a corner. Thence by other ground of the said A., N. 48° 32' W., 156.25 P., crossing a pond of water to a stone for a corner. Thence still by said A.’s land, N. 51° 45' W., 102.8 P., to a spike in a white oak stump. Thence by same land, N. 41° 51' W., 120.1 P. to a hole in a rock, in a line of Peter Dunn’s land. Thence by said land, N. 52° 41' E., 134.8 P., to the place of begnining. Containing 222 A. 3 qr. 23 P. of land.

EXAMPLE 2.

It is required to survey a piece of ground described as follows: Beginning at a corner in the middle of a road leading from H to B . Thence by ground of W. B. C., N. 50° 30' W., 7^24 P. to a black oak tree for a corner. Thence by land of A. G., S. 40° W., 204 P. to a stone. Thence by the same land, S. 84° 30' W., 48.4 P. to a white oak tree, standing by the bank of P—— creek, and 1 P. to the middle of the creek. Thence up the said creek, the several courses, 42.5 P. Thence by ground of J. A., N. 40° E., 1 P., to a pin oak on the

286

HINTS TO YOUNG SURVEYORS.

bank of said creek, and 32.8 P. to a corner. Thence by ground of J. F., S. 50° 30' E., 18.8 P., to a stone, and N. 38° 30/ 38.4 P. to a stone. Thence by land of J. N., S. 50° 30' E., 93.2 P. to a corner in the middle of said road. Thence along the said road, S. 1° 45' E., 51.4 P. to the place of beginning. Containing 45 acres of land more or less.

Upon examining the ground, I do not find any of the land¬ marks that may be depended upon, except the stones at the two ends of the line GH [See plan). These are firm in the ground, and have each a small hole near the centre. This line is recited N. 38° 30' E., 38.4 P. I find the bearing at the present time to be N. 37° 42' E., consequently the variation of the needle since the former survey, is 48' W. I measure the line GH, and find it to correspond with the former measure, 38.4 P. It rarely if ever happens that the bearings of recent surveys correspond with those of the same lines taken years b J?k, and very generally there is a difference between the old and recent measures of lines. It is best to make the corrections, as well of the bearings as of the distances of the whole survey before proceeding further, in order to have as little calculating as possible during the survey. Some surveyors rule a table with

287

HINTS TO YOUNG SURVEYORS.

four columns like the following, placing the recited bearings in the first, corrected bearings in the second, recited distances in the third, and corrected distances in the fourth, thus: Recited Distances.

Corrected Distances.

N. 50° 30' W. N. 51° 18' W.

74.24

74.62

S. 40° W. S. 39° 12' W.

20.4

S. 84° 30' W. S. 83° 42' W.

48.4

Recited Bearings.

| Corrected Bearings.

Creek. N. 40° E. N. 39° 12'E.

32.8

S. 50° 30' E. S. 51° 18' E.

18.8

N. 38° 30' E. ;N. 37° 42' E. , 38.4 S. 50° 30' E. S. 51° 18' E.

93.2

S. 1° 45'

51.4

S. 2°

33' E.

38.6

The variation being found to be 48' W., the bearings are corrected by adding 48' to NW and SE, and deducting 48' from the SW and NE courses. If the measure of the line between the two stones do not correspond with the recited measure, make the correction of all the lines in that proportion, and place the result in the 4th column. Thus, suppose GH should measure 38.6 P., place that in the 4th column, opposite the line GII. Then, as 38.4 : 38.6 :: 74.24 : 74.62, &c. The old and new measures agree in the present case, there¬ fore the 4th column is unnecessary. Having ascertained the bearing and distance of the line GH, I proceed from II to I. I take my course S. 51° 18' E., and measure upon that course 73.2 P. which is 20 perches short of the distance required, where I meet with a thicket through which I cannot see or measure. From this point, marked d, and with the line «, I measure an angle of 60°, and measure 20 P. to b. From be and with the line bd I measure an angle of 60° and measure 20 P., which must bring me to I, where

288

HINTS TO YOUNG SURVEYORS.

I put a stake, the old land-mark being lost. From I to A there is no obstruction. I therefore place the instrument at I, and direct the telescope S. 2° 33' E., and measure 51.4 P. At this point we find a stone corresponding with the last course and distance; this proves the correctness of the stake at I. The line AB is inaccessible ; I therefrom run a line parallel with BC (S. 39° 12'), 20.4 P., marked c. From C I run a line parallel with AB (N. 51° 18' W.) 74.25 P. to C, a stone. From C run a line N. 39° 12' E. 20.4 P. to B; this was for¬ merly a black oak tree. The tree is gone, but the stump remains. From C I run a line S. 83° 42' W. 42.98 P., which brings me to a white oak tree on the bank of the creek, and T P. to the middle of the creek. I now return to the stone at G, and run a line from G, N. 51° 30' W., 18.8 P. to F. The line EF is also inaccessible. I find by calculation that the point E should bear S. 68° 54' W. from G., 37.95 P. Run the line GE accordingly, and strike the pin oak tree recited in the deed, on the bank of the creek, and 1 P. from the middle thereof. From E I find D bears S. 19° 38' E., measure 42.98 P. At 10 perches from E make an off-set to the creek; at the further distance of 20 P. make another off-set. The first off-set measures 3.2 P. to the middle of the creek, and the second 9.42 P. I find this tract to contain 45 A. 3 R. 27 P., including half the creek and half the road. It is best to place but little reliance upon the needle, and it may not be too much to say that it is impossible to complete a survey of any considerable tract of land with perfect accu¬ racy by the use of the needle alone. It is of great use, how¬ ever, and not to be altogether rejected. It is well to take the bearing of some one well defined line of the land to be surveyed, and by sighting back upon the lines as you advance, ascertain the other bearings by measuring the angles.

Establishing Disputed Lines. This is the most delicate business that a young surveyor can undertake, and in which he should exercise the greatest care. A mistake in the early period of his practice may affect his

HINTS TO YOUNG SURVEYORS.

289

reputation through life. In this branch, old surveyors, whc are well acquainted with the neighbouring land-marks, have a decided advantage. But young surveyors need not therefore be discouraged. A little patience, and a little more trouble in finding the land-marks of the neighbouring tracts, which have a relation to the line in question, will enable him to come to as correct a conclusion as those who may presume upon their longer experience, and better knowledge, and take less care in their operations. It is not to be supposed that a sur¬ veyor by mere intuition, or some mysterious art, unknown to the uninitiated, can find out property lines or direct his com¬ pass with unerring certainty to a contested corner. He must be governed by the best evidence that the case will admit of, and should exercise due diligence to make himself acquainted with all the facts of the case. The best evidence of boundary¬ lines is monuments, such as stones firmly fixed in the ground, or trees known and acknowledged as land-marks. The next best is fences that have been for a long time received and acknowledged as lines, or ditch-banks thrown up to mark lines. Where no visible evidence is to be found on the ground, or where these are disputed, other means are to be resorted to. Before commencing to adjust a disputed line, it is advisable to examine the deeds as well of the lands which the line divides, as also the deeds of the lands adjoining these, and plot the whole upon a paper of convenient size to carry into the field, noting the recited land-marks upon the plan. If no definite marks are to be found upon the disputed line, it may be that it is part of a longer line. If satisfactory marks are to be found on the extended line, trace them to the line in dispute. If nothing of this kind is to be found, measure the lines which terminate in the disputed line, if definite starting points can be found. If not, these points must be found by tracing the other lines connected with them. Where care has not been taken to preserve ancient land-marks, it often becomes neces¬ sary to trace the lines of several adjacent tracts before any satisfactory conclusion can be arrived at. Roads. In tracing a road from the record, the same rules should be observers in other surveys. It frequently happens that roads

290

HINTS TO YOUNG SURVEYORS.

have been laid out upon the lines dividing properties, and in many cases, the stones marking the lines of the road have been lost. In such cases their proper position must be found either by tracing the bearings and distances from the record, making allowance for the variation of the needle, or by measuring the property lines which terminate in the road.

MATHEMATICAL TABLES DIFFERENCE

LATITUDE AND DEPARTURE;

LOGARITHMS, FROM

I

TO

10,000;

ARTIFICIAL SINES, TANGENTS, AND SECANTS.

STEREOTYPE EDITION, CAREFULLY REVISED AND CORRECTED.

2

TRAVERSE TABLE, g u>*

s o
4

5 6 7 8 9 10

i

.

Deg.

1 Deg.

i a f* ? oo Dep. ?

1 Deg.

Lat.

Dep.

Lat.

Dep.

Lat.

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

0.00

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

0.01 0.02 0.03 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

0.01 0.03 0.04 0.05 0.07 0.08 0.09 0.10 0.12 0.13

9 10

11.00

11.00

11.00

0.01 0.01 0.02 0.02 0.03 0.03 0.03 0.04 0.04

1 2 3 4 5 6 7

8

11 12 13 14 15 16 17 18 19 20

12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00

0.05 0.05 0.06 0.06 0.07 0.07 0.07 0.08 0.08 0.09

12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00

0.10 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.17

12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00

0.14 0.16 0.17 0.18 0.20 0.21 0.22 0.24 0.2£ 0.26

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00 30.00

0.09 0.10 0.10 0.10 0.11 0.11 0.12 0.12 0.13 0.13

21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00 30.00

0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.24 0.25 0.26

21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00 30.00

0.27 0.29 0.30 0.31 0.33 0.34 0.35 0.37 0.38 0.39

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

31.00 32.00 33.00 34.00 35.00 36.00 37.00 38.00 39.00 40.00

0.14 0.14 0.14 0.15 0.15 0.16 0.16 0.17 0.17 0.17

31.00 32.00 33.00 34.00 35.00 36.00 37.00 38.00 39.00 40.00

0.27 0.28 0.29 0.30 0.31 0.31 0.32 0.33 0.34 0.35

31.00 32.00 33.00 34.00 35.00 36.00 37.00 38.00 39.00 40.00

0.41 0.42 0.43 0.45 0.46 0.47 0.48 0.50 0.51 0.52

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

41.00 42.00 43.00 44.00 45.00 46.00 47.00 48.00 49.00 50.00

0.18 0.18 0.19 0.19 0.20 0.20 0.21 0.21 0.21 0.22

41.00 42.00 43.00 44.00 45.00 46.00 47.00 48.00 49.00 50.00

0.36 0.37 0.38 0.38 0.39 0.40 0.41 0.42 0.43 0.44

41.00 42.00 43.00 44.00 45.00 46.00 47.00 48.00 49.00 50.00

0.54 0.55 0.56 0.58 0.59 0.60 0.62 0.63 0.64 0.65

41 42 43 44 45 46 47 48 49 50

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o

is9)

£

89| Deg.

89J Deg.

89J Deg.

© o p £5 in

s

TRAVERSE TABLE. d

i Deg.

4 Deg.

3 1 Deg.

d

s*

Cfl* p o ®

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

51.00 52.00 53.00 54.00 55.00 56.00 57.00 58.00 59.00 60.00

0.22 0.23 0.23 0.24 0.24 0.24 0.25 0.25 0.26 0.26

51.00 52.00 53.00 54.00 55.00 56.00 57.00 58.00 59.00 60.00

0.45 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.51 0.52

51.00 52.00 53.00 54.00 • 55.00 56.00 57.00 57.99 58.99 59.99

0.67 0.68 0.69 0.71 0.72 0.73 0.75 0.76 0.77 0.79

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

61.00 62.00 63.00 |64.00 65.00 66.00 67.00 68.00 69.00 70.00

0.27 0.27 0.27 0.28 0.28 0.29 0.29 0.30 0.30 0.31

61.00 62.00 63.00 64.00 65.00 66.00 67.00 68.00 69.00 70.00

0.53 0.54 0.55 0.56 0.57 0.58 0.58 0.59 0.60 0.61

60.99 61.99 62.99 63.99 64.99 65.99 66.99 67.99 68.99 69.99

0.80 0.81 0.82 0.84 0.85 0.86 0.88 0.89 0.90 0.92

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

71.00 72.00 73.00 74.00 75.00 76.00 77.00 78.00 79.00 80.00

0.31 0.31 0.32 0.32 0.33 0.33 0.34 0.34 0.34 0.35

71.00 72.00 73.00 74.00 75.00 76.00 77.00 78.00 79.00 80.00

0.62 0.63 0.64 0.65 0.65 0.66 0.67 0.68 0.69 0.70

70.99 71.99 72.99 73.99 74.99 75.99 76.99 77.99 78.99 79.99

0.93 0.94 0.96 0.97 0.98 0.99 1.01 1.02 1.03 1.05

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

81.00 82.00 83.00 84.00 85.00 86.00 87.00 88.00 89.00 90.00

0.35 0.36 0.36 0.37 0.37 0.38 0.38 0.38 0.39 0.39

81.00 82.00 83.00 84.00 85.00 86.00 87.00 88.00 89.00 90.00

0.71 0.72 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79

80.99 81.99 82.99 83.99 84.99 85.99 86.99 87.99 88.99 89.99

1.06 1.07 1.09 1.10 1.11 1.13 1.14 1.15 1.16 1.18

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

91.00 92.00 93.00 94.00 95.00 96.00 97.00 98.00 99.00 100.00

0.40 0.40 0.41 0.41 0.41 0.42 0.42 0.43 0.43 0.44

91.00 92.00 93.00 94.00 95.00 96.00 97.00 98.00 99.00 100.00

0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.86 0.87

90.99 91.99 92.99 93.99 94.99 95.99 96.99 97.99 98.99 99.99

1.19 1.20 1.22 1.23 1.24 1.26 1.27 1.28 1.30 1,31

91 92 93 94 95 96 97 98 . 99 100

a>

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

a

1

5

89£ Deg.

p o a

o6 a

a

89£ Deg,

89$ Deg.

C/3 ’ —i

Q

4

TRAVERSE TABLE.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

1 2 3 4 5 6 7 8 9 10

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 "

0.02 0.03 0.05 0.07 0.09 0.10 0.12 0.14 0.16 0.17

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

0.02 0.04 0.07 0.09 0.11 0.13 0.15 0.17 0.20 0.22

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

0.03 0.05 0.08 0.10 0.13 0.16 0.18 0.21 0.24 0.26

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.25 0.28 0.31

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00

0.19 0.21 0.23 0.24 0.26 0.28 0.30 0.31 0.33 0.35

11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00

0.24 0.26 0.28 0.31 0.33 0.35 0.37 0.39 0.41 0.44

11.00 12.00 13.00 14.00 14.99 15.99 16.99 17.99 18.99 19.99

0.28 0.31 0.34 0.37 0.39 0.42 0.45 0.47 0.50 0.52

10.99 11.99 12.99 13.99 14.99 15.99 16.99 17.99 18.99 19.99

0.34 0.37 0.40 0.43 0.46 0.49 0.52 0.55 0.58 0.61

11 12 13 14 J5 16 17 18 19 20

2*1 22 23 24 25 26 27 29 30

21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00 30.00

0.37 0.38 0.40 0.42 0.44 0.45 0.47 0.49 0.51 0.52

21.00 21.99 22.99 23.99 24.99 25.99 26.99 27.99 28.99 29.99

0.46 0.48 0.50 0.52 0.55 0.57 0.59 0.61 0.63 0.65

20.99 21.99 22.99 23.99 24.99 25.99 26.99 27.99 28.99 29.99

0.55 0.58 0.60 0.63 0.65 0.68 0.71 0.73 0.76 0.79

20.99 21.99 22.99 23.99 24.99 25.99 26.99 27.99 28.99 29.99

0.64 21 0.67 22 0.70 - 23 0.73 24 0.76 25 0.79 26 0.83 27 0.86 28 0.89 29 0.92 30

31 32 33 34 35 36 37 38 39 40

31.00 32.00 32.99 33.99 34.99 35.99 36.99 37.99 38.99 39.99

0.54 0.56 0.58 0.59 0.61 0.63 0.65 0.66 0.68 0.70

30.99 31.99 32.99 33.99 34.99 35.99 36.99 37.99 38.99 39.99

0.68 0.70 0.72 0.74 0.76 0.79 0.81 0.83 0.85 0.87

30.99 31.99 32.99 33.99 34.99 35.99 36.99 37.99 38.99 39.99

0.81 0.84 0.86 0.89 0.92 0.94 0.97 0.99 1.02 1.05

30.99 31.99 32.98 33.98 34.98 35.98 36.98 37.98 38.98 39.98

0.95 0.98 1.01 1.04 1.07 1.10 1.13 1.16 1.19 1.22

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 43 49 50

40.99 41.99 42.99 43.99 44.99 45.99 46.99 47.99 48.99 49.99

0.72 0.73 0.75 0.77 0.79 0.80 0.82 0.84 0.86 0.87

40.99 41.99 42.99 43.99 44.99 45.99 46.99 47.99 48.99 49.99

0.89 0.92 0.94 0.96 0.98 1.00 1.03 1.05 1.07 1.09

40.99 41.99 42.99 43.99 44.99 45.99 46.99 47.98 48.98 49.98

1.07 1.10 1.13 1.15 1.18 1.20 1.23 1.26 1.28 1.31

40.98 41.98 42.98 43.98 44.98 45.98 46.98 47.98 48.98 49.98

1.25 1.28 1.31 1.34 1.37 1.40 1.44 1.47 1.50 1.53

41 42 43 44 45 46 47 48 49 50

V

Dep.

Lat.

Dep.

Lat.

Dep. 1 Lat.

Distance.

Lat.

is

o

1 Deg.

If Deg.

« in

3

89 Deg.

88} Deg.

If

U Deg.

j)

80i Deg-.

||f

l(

Deg.

Dep. | Lat.

6

881 Deg.

1

o

5

TRAVERSE TABLE, Distance.

5

14 Deg.

1 Deg;.

1J Deg.

14 Deg.

6

P

8

Lat.

Dep.

51 52 63 54 55 56 57 53 59 60

50.99 51.99 52.99 53.99 54.99 55.99 56.99 57.99 58.99 59.99

0.89 0.91 0.92 0.94 0.96 0.98 0.99 1.01 1.03 1.05

61 62 63 64 65 66 67 68 69 70

60.99 61.99 62.99 63.99 64.99 65.99 66.99 67.99 68.99 69.99

1.06 : 60.99 1.08 s 61.99 1.10 ; 62.99 1.12 | 63.98 1.13 : 64.98 1.15 65.98 1.17 66.98 1.19 67.98 1.20 68.98 1.22 69.98

71 72 . 73 74 75 76 77 78 79 80

70.99 71.99 72.99 73.99 74.99 75.99 76.99 77.99 78.99 79.99

1.24 1.26 1.27 1.29 1.31 1.33 1.34 1.36 1.38 1.40

81 82 83 84 85 86 87 88 89 90

80.99 81.99 82.99 83.99 84.99 85.99 86.99 87.99 88.99 89.99

1.41 1.43 1.45 1.47 1.48 1.50 1.52 1.54 1.55 1.57

'1 i ,1 | ! j

91 92 93 94 95 96 ■ 97 98 99 100

1 90.99 i 91.99 ! 92.99 1 ! 93.99 ij 94.99 95.99 9€ .99 97.99 98.98 99.98

0)o

Dep.

rta

3

Dep.

Lat.

Dep.

Lat.

50.98 51.98 52.98 53.98 54.98 55.98 56.98 57.98 58.98 59.98

1.34 1.36 1.39 1.41 1.44 1.47 1.49 1.52 1.54 1.57

50.98 51.98 52.98 53.97 54.97 55.97 56.97 57.97 58.97 59.97

1.56 1.59 1.62 1.65 1.68 1.71 1.74 1.77 1.80 1.83

51 52 53 54 55 56 57 58 59 60

1.33 1.35 1.37 1.40 1.42 1.44 1.46 1.48 1.51 1.53

60.98 61.98 62.98 63.98 64.98 65.98 66.98 67.98 68.98 69.98

1.60 1.62 1.65 1.68 1.70 1.73 1.75 1.78 1.81 1.83

60.97 61.97 62.97 63.97 64.97 65.97 66.97 67.97 68.97 69.97

1.86 1.89 1.92 1.95 1.99 2.02 2.05 2.08 2.11 2.14

61 62 63 64 65 66 67 68 69 70

70.98 71.98 72.98 73.98 74.98 75.98 76.98 77.98 78.98 79.98

1.55 1.57 1.59 1.61 1.64 1.66 1.68 1.70 1.72 1.75

70.98 71.98 72.97 73.97 74.97 75.97 76.97 77.97 78.97 79.97

1.86 1.88 1.91 1.94 1.96 1.99 2.02 2.04 2.07 2.09

70.97 71.97 72.97 73.97 74.97 75.96 76.96 77.96 78.96 79.96

2.17 2.20 2.23 2.26 2.29 2.32 2.35 2.38 2.41 2.44

71 72 73 74 75 76 77 78 79 80

80.98 81.98 82.98 83.98 84.98 85.98 86.98 87.98 88.98 89.98

1.77 1.79 1.81 1.83 1.85 1.88 1.90 1.92 1.94 1.96

80.97 81.97 82.97 83.97 84.97 85.97 86.97 87.97 88.97 89.97

2.12 2.15 2.17 2.20 2.23 2.25 2.28 2.30 2.33 2.36

80.96 81.96 82.96 83.96 84.96 85.96 86.96 87.96 88.96 89.96

2.47 2.50 2.53 2.57 2.60 2.63 2.66 2.69 2.72 2.75

81 82 83 84 85 86 87 88 89 90

1.59 90.98 1.61 91.98 1.62 92.98 1.64 ^ 93.98 1.66 94.98 1.68 95.98 1.69 96.98 1.71 ; 97.98 1.73 98.98 1.75 99.98

1.99 2.01 2.03 2.05 2.07 2.09 2.12 2.14 2.16 2.18

90.97 91.97 92.97 93.97 94.97 95.97 96.97 97.97 98.97 99.97

2.38 2.41 2.43 2.46 2.49 2.51 2.54 2.57 2.59 2.62

90.96 91.96 92.96 93.96 94.96 95.96 96.95 97.95 98.95 99.95

2.78 91 2.81 92 2.84 93 2.87 94 2.90 95 2.94 96 2.96 97 2.99 98 3.02 99 3.05 100

Lat.

Lat.

Dep.

Lat.

Dep. •

Lat.

89 Deg.

Lat.

1 :

Dep.

50.99 1.11 51.99 1.13 52.99 1.16 53.99 - 1.18 54.99 1.20 55.99 1.22 56.99 1.24 1.27 57.99 58.99 1.29 1.31 59.99

Dep.

88J Deg.

884 Deg.

ij

881 Deg.

a)o

rta

3

TRAVERSE TABLE,

t>

i g ri* P

2 Deg.

2i Deg.

2* Deg.

2| Deg.

g oo’ P 3 O

fto

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

1 2 3 4 5 6 7 8 9

1.00 2.00 3.00 4.00 5.00 6.00 7.00 7.99 8.99 9.99

0.03 0.07 0.10 0.14 0.17 0.21 0.24 0.28 0.31 0.35

1.00 2.00 3.00 4.00 5.00 6.00 6.99 7.99 8.99 9.99

0.04 0.08 0.12 0.16 0.20 0.24 0.27 0.31 0.35 0.39

1.00 2.00 3.00 4.00% 5.00 5.99 6.99 7.99 8.99 9.99

0.04 0.09 0.13 0.17 0.22 0.26 0.31 0.35 0.39 0.44

1.00 2.00 3.00 4.00 4.99 5.99 6.99 7.99 8.99 9.99

0.05 0.10 0.14 0.19 0.24 0.29 0.34 0.38 0.43 0.48

1 2 3 4 5 6 7 8 9 10

10.99 11.99 12.99 13.99 14.99 15.99 16.99 17.99 18.99 19.99

0.38 0.42 0.45 0.49 0.52 0.56 0.59 0.63 0.66 0.70

10.99 11.99 12.99 13.99 14.99 15.99 16.99 17.99 18.99 19.98

0.43 0.47 0.51 0.55 0.59 0.63 0.67 0.71 0.75 0.79

10.99 11.99 12.99 13.99 14.99 15.99 16.98 17.98 18.98 19.98

0.48 0.52 0.57 0.61 0.65 0.70 0.74 0.79 0.83 0.87

10.99 11.99 12.99 13.98 14.98 15.98 16.98 17.98 18.98 19.98

0.53 0.58 0.62 0.67 0.72 0.77 0.82 0.86 0.91 0.96

11 12 13 14 15 16 17 18 19 20

23 24 25 26 27 28 29 30

20.99 21.99 22.99 23.99 24.98 25.98 26.98 27.98 28.98 29.98

0.73 0.77 0.80 0.84 0.87 0.91 0.94 0.98 1.01 1.05

20.98 21.98 22.98 23.98 24.98 25.98 26.98 27.98 28.98 29.98

0.82 0.86 0.90 0.94 0.98 1.02 1.06 1.10 1.14 1.18

20.98 21.98 22.98 23.98 24.98 25.98 26.97 27.97 28.97 29.97

0.92 0.96 1.00 1.05 1.09 1.13 1.18 1.22 1.26 1.31

20.98 21.97 22.97 23.97 24.97 25.97 26.97 27.97 28.97 29.97

1.01 1.06 1.10 1.15 1.20 1.25 1.30 1.34 1.39 1.44

23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 ; 40

30.98 31.98 32.98 33.98 34.98 35.98 36.98 37.98 38.98 39.98

1.08 1.12 1.15 1.19 1.22 1.26 1.29 1.33 1.36 1.40

30.98 31.98 32.97 33.97 34.97 35.97 36.97 37.97 38.97 39.97

1.22 1.26 1.30 1.33 1.37 1.41 1.45 1.49 1.53 1.57

30.97 31.97 32.97 33.97 34.97 35.97 36.96 37.96 38.96 39.96

1.35 1.40 1.44 1.48 1.53 1.57 1.61 1.66 1.70 1.75

30.96 31.96 32.96 33.96 34.96 35.96 36.96 37.96 38.96 39.95

1.49 1.54 1.58 1.63 1.68 1.73 1.78 1.82 1.87 1.92

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

40.98 41.97 42.97 43.97 44.97 45.97 46.97 47.97 48.97 49.97

1.43 1.47 1.50 1.54 1.57 1.61 1.64 1.68 1.71 1.74

40.97 41.97 42.97 43.97 44.97 45.96 46.96 47.96 48.96 49.96

1.61 1.65 1.69 1.73 1.77 1.81 1.85 1.88 1.92 1.96

40.96 41.96 42.96 43.96 44.96 45.96 46.96 47.95 48.95 49.95

1.77 1.83 1.88 1.92 1.96 2.01 2.05 2.09 2.14 2.18

40.95 41.95 42.95 43.95 44.95 45.95 46.95 47.95 48.94 49.94

1.97 2.02 2.06 2.11 2.16 2.21 2.25 2.30 2.35 2.40

41 42 43 44 45 46 47 48 49 50

fto ft

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

ft0

10

11 12

13 14 15 16 17 18 19 20

21 22

Q

88 Deg.

871 Deg.

> I

87$ Deg.

87* Deg.

ft

21 22

1 on 3

7

TRAVERSE TABLE,

Lat.

Dep

2i Deg.

2| Deg.

2£ Deg.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

2 Deg.

51 52 53 54 55 56 57 58 59 60

50.97 51.97 52.97 53.97 54.97 55.97 56.97 57.96 58.96 59.96

1.78 1.81 1.85 1.88 1.92 1.95 1.99 2.02 2.06 2.09

50.96 51.96 52.96 53.96 54.96 55.96 56.96 57.96 58.95 59.95

2.00 2.04 2.08 2.12 2.16 2.20 2.24 2.28 2.32 2.36

50.95 51.95 52.95 53.95 54.95 55.95 56.95 57.94 58.94 59.94

2.22 2.27 2.31 2.36 2.40 2.44 2.49 2.53 2.57 2.62

50.94 51.94 52.94 53.94 54.94 55.94 56.93 57.93 58.93 59.93

2.45 2.50 2.54 2.59 2.64 2.69 2.73 2.78 2.83 2.88

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

60.96 61.96 62.96 63.96 64.96 65.96 66.96 67.96 68.96 69.96

2.13 2.16 2.20 2.23 2.27 2.30 2.34 2.37 2.41 2.44

60.95 61.95 62.95 63.95 64.95 65.95 66.95 67.95 68.95 69.95

2.39 2.43 2.47 2.51 2.55 2.59 2.63 2.67 2.71 2.75

60.94 61.94 62.94 63.94 64.94 65.94 66.94 67.94 68.93 69.93

2.66 2.70 2.75 2.79 2.84 2.88 2.92 2.97 3.01 3.05

60.93 61.93 62.93 63.93 64.93 65.92 66.92 67.92 68.92 69.92

2.93 2.97 3.02 3.07 3.12 3.17 3.21 3.26 3.31 3.36

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

70.96 71.96 72.96 73.95 74.95 75.95 76.95 77.95 78.95 79.95

2.48 2.51 2.55 2.58 2.62 2.65 2.69 2.72 2.76 2.79

70.95 71.94 72.94 73.94 74.94 75.94 76.94 77.94 78.94 79.94

2.79 2.83 2.87 2.91 2.94 2.98 3.02 3.06 3.10 3.14

70.93 71.93 12.93 73.93 74.93 75.93 76.93 77.93 78.92 79.92

3.10 3.14 3.18 3.23 3.27 3.31 3.36 3.40 3.45 3.49

70.92 71.92 72.92 73.91 74.91 75.91 76.91 77.91 78.91 79.91

3.41 3.45 3.50 3.55 3.60 3.65 3.70 3.74 3.79 3.84

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

80.95 81.95 82.95 83.95 84.95 85.95 86.95 87.95 88.95 89.95

2.83 2.86 2.90 2.93 2.97 3.00 3.04 3.07 3.11 3.14

80.94 81.94 82.94 83.94 84.93 85.93 86.93 87.93 88.93 89.93

3.18 3.22 3.26 3.30 3.34 3.38 3.42 3.45 3.49 3.53

80.92 81.92 82.92 83.92 84.92 85.92 86.92 87.92 88.92 89.91

3.53 3.58 3.62 3.66 3.71 3.75 3.79 3.84 3.88 3.93

80.91 81.91 82.90 83.90 84.90 85.90 86.90 87.90 88.90 89.90

3.89 3.93 3.98 4.03 4.08 4.13 4.17 4.22 4.27 4.32

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

90.95 91.94 92.94 93.94 94.94 95.94 96.94 97.94 98.94 99.94

3.18 3.21 3.25 3.28 3.32 3.35 3.39 3.42 3.46 3.49

90.93 91.93 92.93 93.93 94.93 95.93 96.93 97.92 98.92 99.92

3.57 3.61 3.65 3.69 3.73 3.77 3.81 3.85 3.89 3.93

90.91 91.91 92.91 93.91 94.91 95.91 96.91 97.91 98.91 99.91

3.97 4.01 4.06 4.10 4.14 4.19 4.23 4.27 4.32 4.36

90.90 91.89 92.89 93.89 94.89 95.89 96.89 97.89 98.89 99.88

4.37 91 4.41 92 4.46 93 4.51 94 4.56 95 4.61 96 4.65 97 4.70 98 4.75 99 4.80 100

©

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

ca

n

S i

:

© O

S3

88 Deg.

871 Deg.

87} Deg.

|

87} Deg.

fS «

s

8

TRAVERSE TABLE 2 OB* ST

3 Deg.

3* Deg.

3§ Deg.

5 P

a o

Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

1.00 2.00 3.00 3.99 4.99 5.99 6.99 7.99 8.99 9.99

0.05 0.10 0.16 0.21 0.26 0.31 0.37 0.42 0.47 0.52

1.00 2.00 3.00 3.99 4.99 5.99 6.99 7.99 8.99 9.98

0.06 0.11 0.17 0.23 0.28 0.34 0.40 0.45 0.51 0.57

1.00 2.00 2.99 3.99 4.99 5.99 6.99 7.99 8.98 9.98

11 12 13 14 15 16 17 18 19 20

10.98 11.98 12.98 13.98 14.98 15.98 16.98 17.98 18.98 19.97

0.58 0.63 0.68 0.73 0.79 0.84 0.89 0.94 0.99 1.05

10.98 11.98 12.98 13.98 14.98 15.97 16.97 17.97 18.97 19.97

0.62 0.68 0.73 0.79 0.85 0.91 0.96 1.02 1.08 1.13

20.97 21.97 23. 22.97 24 23.97 25 24.97 26 25.96 27 26.96 28 27.96 29 28.96 30 29.96

1.10 1.15 1.20 1.26 1.31 1.36 1.41 1.47 1.52 1.57

20.97 21.96 22.96 23.96 24.96 25.96 26.96 27.95 28.95 29.95

31 32 33 34 35 36 37 38 39 40

30.96 31.96 32.95 33.95 34.95 35.95 36.95 37.95 38.95 39.95

1.62 1.67 1.73 1.78 1.83 1.88 1.94 1.99 2.04 2.09

41 42 43 44 45 46 47 48 49 50

40.94 41.94 42.94 43.94 44.94 45.94 46.94 47.93 48.93 49.93



Dep.

21

22

1 3

3f Deg.

Dep.

3 O CD

Lat.

Dep.

0.06 0.12 0.18 0.24 0.31 0.37 0.43 0.49 0.55 0.61

1.00 2.00 2.99 3.99 4.99 5.99 6.99 7.98 8.98 9.98

0.06 0.13 0.20 0.26 0.33 0.39 0.46 0.52 0.59 0.65

1 2 3 4 6 6 7 8 9 10

10.98 11.98 12.98 13.97 14.97 15.97 16.97 17.97 18.96 19.96

0.67 0.73 0.79 0.85 0.92 0.98 1.04 1.10 1.16 1.22

10.98 11.97 12.97 13.97 14.97 15.97 16.96 17.96 18.96 19.96

0.72 0.78 0.85 0.92 0.98 1.05 1.11 1.18 1.24 1.31

11

1.19 1.25 1.3* 1.36 1.42 1.47 1.53 1.59 1.64 1.70

20.96 21.96 22.96 23.96 24.95 25.95 26.95 27.95 28.95 29.94

1.28 1.34 1.40 1.47 1.53 1.59 1.65 1.71 1.77 1.83

20.96 21.95 22.95 23.95 24.95 25.94 26.94 27.94 28.94 29.94

1.37 1.44 1.50 1.57 1.64 1.70 1.77 1.83 1.90 1.96

23 24 25 26 27 28 29 30

30.95 31.95 32.95 33.95 34.94 35.94 36.94 37.94 38.94 39.94

1.76 1.81 1.87 1.93 1.98 2.04 2.10 2.15 2.21 2.27

30.94 31.94 32.94 33.94 34.93 35.93 36.93 37.93 38.93 39.93

1.89 1.95 2.01 2.08 2.14 2.20 2.26 2.32 2.38 2.44

30.93 31.93 32.93 33.93 34.92 35.92 36.92 37.92 38.92 39.91

2.03 2.09 2.16 2.22 2.29 2.35 2.42 2.49 2.55 2.62

31 32 33 34 35 36 37 38 39 40

2.15 2.20 2.25 2.30 2.36 2.41 2.46 2.51 2.56 2.62

40.93 41.93 42.93 43.93 44.93 45.93 46.92 47.92 48.92 49.92

2.32 • 40.92 2.38 41.92 2.44 42.92 2.49 43.92 2.55 44.92 2.61 45.91 2.66 46.91 2.72 47.91 2.78 48.91 2.83 49.91

2.50 2.56 2.63 2.69 2.75 2.81 2.87 2.93 2.99 3.05

40.91 41.91 42.91 43.91 44.90 45.90 46.90 47.90 48.90 49.89

2.68 2.75 2.81 2.88 2.94 3.01 3.07 3.14 3.20 3.27

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Lat.

Dep.

Lat.

87 Deg.

861 Deg.

Dep. j

86* Deg.

12

13 14 15 16 17 18 19 20 21 22

oJ

V

a

86| Deg.

3

,

9

TRAVERSE TABLE 1 Distance.

50.89 51.89 52.89 53.88 54.88 55.88 56.88 57.88 58.87 59.87

3.34 3.40 3.47 3.53 3.60 3.66 3.73 3.79 3.86 3.92

51 52 53 54 55 56 57 58 59 60

3.72 3.79 3.85 3.91 3.97 4.03 4.09 4.15 4.21 4.27

60.87 61.87 62.87 63.86 64.86 65.86 66.86 67.85 68.85 69.85

3.99 4.05 4.12 4.19 4.25 4.32 4.38 4.45 4.51 4.58

61 62 63 64 65 66 67 68 69 70

70.87 71.87 72.86 73.86 74.86 75.86 76.86 77.85 78.85 79.85

4.33 4.40 4.46 4.52 4.58 4.64 4.70 4.76 4.82 4.88

70.85 71.85 72.84 73.84 74.84 75.84 76.84 77.83 7-8.83 79.83

4.64 4.71 4.77 4.84 4.91 4.97 5.04 5.10 5.17 5.23

71 72 73 74 75 76 77 78 79 80

4.59 4.65 4.71 4.76 4.82 4.88 4.93 4.99 5.05 5.10

80.85 81.85 82.85 83.84 84.84 85.84 86.84 87.84 88.83 89.83

4.94 5.01 5.07 5.13 5.19 5.25 5.31 5.37 5.43 5.49

80.83 81.82 82.82 83.82 84.82 85.82 86.81 87.31 88.81 89.81

5.30 5.36 5.43 5.49 5.56 5.62 5.69 5.76 5.82 5.89

81 82 83 84 85 86 87 88 89 90

90.85 91.85 92.85 93.85 94.85 95.85 96.84 97.84 98.84 99.84

5.16 5.22 5.27 5.33 5.39 5.44 5.50 5.56 5.61 5.67

90.83 91.83 92.83 93.82 94.82 95.82 96.82 97.82 98.82 99.81

5.56 5.62 5.68 5.74 5.80 5.86 5.92 5.98 6.04 6.10

90.81 91.80 92.80 93.80 94.80 95.79 96.79 97.79 98.79 99.79

5.95 91 6.02 92 6.08 93 6.15 94 6.21 95 6.28 96 6.34 97 6.41 98 6.47 99 6.54 100

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Lat. | Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

50.93 51.93 52.93 53.93 54.92 55.92 56.92 57.92 58.92 59.92«

2.67 2.72 2.77 2.83 2.88 2.93 2.98 3.04 3.09 3.14

50.92 51.92 52.91 53.91 54.91 55.91 56.91 57.91 58.91 59.90

2.89 2.95 3.00 3.06 3.12 3.17 3.23 3.29 3.34 3.40

50.90 51.90 52.90 53.90 54.90 55.90 56.89 57.89 58.89 59.89

3.11 3.17 3.24 3.30 3.36 3.42 3.48 3.54 3.60 3.66

61 62 63 64 65 66 67 68 69 70

60.92 61.92 62.91 63.91 64.91 65.91 66.91 67 91 68.91 69.90

3.19 3.24 3.30 3.35 3.40 3.45 3.51 3.56 3.61 3.66

60.90 61.90 62.90 63.90 64.90 65.89 66.89 67.89 68.89 69.89

3.46 3.51 3.57 3.63 3.69 3.74 3.80 3.86 3.91 3.97

60.89 61.88 62.88 63.88 64.88 65.88 66.88 67.87 68.87 69.87

71 72 73 74 75 76 77 78 79 80

70.90 71.90 72.90 73.90 74.90 75.90 76.89 77.89 78.89 79.89

3.72 3.77 3.82 3.87 3.93 3.98 4.03 4.08 4.13 4.19

70.89 71.88 72.88 73.88 74.88 75.88 76.88 77.87 78.87 79.87

4.03 4.08 4.14 4.20 4.25 4.31 4.37 4.42 4.48 4.54

81 82 83 84 85 86 87 88 89 90

80.89 81.89 82.89 83.88 84.88 85.88 86.88 87.88 88.88 89 $88

4.24 4.29 4.34 4.40 4.45 4.50 4.55 4.61 4.66 4.71

80.87 81.87 82.87 83.86 84.86 85.86 86.86 87.86 88.86 89.86

91 92 93 94 95 96 97 98 99 100

90.88 91.87 92.87 93.87 94.87 95.87 96.87 97.87 98.86 99.86

4.76 4.81 4.87 4.92 4.97 5.02 5.08 5.13 5.18 5.23

« o

Dep.

Lat.

3 Deg.

3£ Deg.

3i Deg.

3| Deg.

Lat.

Cl

.a Q

87 Deg.

25

36} Deg.

864 Deg.

20

86i Deg.

6 o

1w

s

-

TRAVERSE TABLE

10 o 5' ST <6a

Lat.

Dep.

Lat.

1 2. 3 : 4 5 6 7 8 9 10

1.00 2.00 2.99 3.99 4.99 5.99 6.98 7.98 8.98 9.98

0.07 0.14 0.21 0.23 0.35 0.42 0.49 0.56 0.63 0.70

1.00 1.99 2.99 3.99 4.99 5.98 6.98 7.98 8.98 9.97

0.07 0.15 0.22 0.30 0.37 0.44 0.52 0.59 0.67 0.74

1.00 1.99 2.99 3.99 4.98 5.98 6.98 7.98 8.97 9.97

0.08 0.16 0.24 0.3i 0.39 0.47 0.55 0.63 0.71 0.78

11 12 13 14 15 16 17 18 19 20

10.97 11.97 12.97 13.97 14.96 15.96 16.96 17.96 18.95 19.95

0.77 0.84 091 0.98 1.05 1.12 1.19 1.26 1.33 1.40

10.97 11.97 12.96 13.96 14.96 15.96 16.95 17.95 18.95 19.95

0.82 0.89 0.96 1.04 1.11 1.19 1.26 1.33 1.40 1.48

10.97 11.96 12.96 13.96 14.95 15.95 16.95 17.94 18.94 19.94

0.86 0.94 1.02 1.10 1.18 1.26 1.33 1.41 1.49 1.57

21 22 23 24 25 26 27 28 29 30

20.95 21.95 22.94 23.94 24.94 25.94 26.93 27.93 28.93 29.93

1.46 1.53 1.60 1.67 1.74 1.81 1.88 1.95 2.02 2.09

20.94 21.94 22.94 23.93 24.93 25.93 26.93 27.92 28.92 29.92

1.56 1.63 1.70 1.78 1.85 1.93 2.00 2.08 2.15 2.22

20.94 21.93 22.93 23.93 24.92 25.92 26.92 27.91 28.91 29.91

31 32 33 34 35 36 37 38 39 40

30.92 31.92 32.92 33.92 34.91 35.91 36.91 37.91 38.90 39.90

2.16 2.23 2.30 2.37 2.44 2.51 2.58 2.65 2.72 2.79

30.91 31.91 32.91 33.91 34.90 35.90 36.90 37.90 38.89 39.89

2.30 2.37 2.45 2.52 2.59 2.67 2.74 2.82 2.89 2.96

41 42 43 44 45 46 47 48 49 50

40.90 41.90 42.90 43.89 44.89 45.89 46.89 47.88 48.88 49.88

2.86 2.93 3.00 3.07 3.14 3.21 3.28 3.35 3.42 3.49

40.89 41.88 42.88 43.88 44.88 45.87 46.87 47.87 48.87 49.86

<5a

Dep.

Lat.

Dep.

4 Deg.

4i Deg. Dep.

Deg.

41

Lat.

Dep.

4| Deg. Lat. 1.00 1.99 2.99 3.98 4.98 5.98 6.97 7.97 8.97 9.97 ,

Dep.

Q

86 Deg.

a n

f&

0.08 0.17 0.25 0.33 0.41 0.50 0.53 0.66 0.75 0.83

8 9 10

10.96 11.96 12.96 13.95 14.95 15.95 16.94 17.94 10.93 19.93

0.91 0.99 1.08 1.16 1.24 1.32 1.41 1.49 1.57 1.66

11 12 13 14 15 16 17 18 19 20

1.65 1.73 1.80 1.88 1.96 2.04 2.12 2.20 2.28 2.35

20.93 21.92 22.92 23.92 24.91 25.91 26.91 27.90 28.90 29.90

1.74 1.82 1.90 1.99 2.07 2.15 2.24 2.32 2.40 2.48

21 22 23 24 25 26 27 28 29 30

30.90 31.90 32.90 33.90 34.89 35.89 36.89 37.88 38.88 39.88

2.43 2.51 2.59 2.67 2.75 2.82 2.90 2.98 3.06 3.14

30.89 31.89 32.89 33.88 34.88 35.88 36.87 37.87 38.87 39.86

2.57 2.65 2.73 2.82 2.90 2.98 3.06 3.15 3.23 3.31

31 32 33 34 35 36 37 38 39 40

3.04 3.11 3.19 3.26 3.33 3.41 3.48 3.56 3.63 3.71

40.87 41.87 42.87 43.86 44.86 45.86 46.86 47.85 48.85 49.85

3.22 40.86 3.30 41.86 3.37 42.85 3.45 43.85 3.53 44.85 3.61 45.84 3.69 j 46.84 3.77 47.84 3.84 48.83 3.92 49.83

3.40 3.48 3.56 3.64 3.73 3.81 3.89 3.97 4.06 4.14

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat. 1 Dep.

Lat.

opV 3 cn

a

•4-1

g 5T

851 Deg.

85$ Deg.

85$ Deg.

1 2 3 4 5 6

s

TRAVERSE TABLE.

11

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

50.88 51.87 52.87 53.87 54.87 55.86 56.86 57.86 58.86 59.85

3.56 3.63 3.70 3.77 3.84 3.91 3.98 4.05 4.12 4.19

50.86 51.86 52.85 53.85 54.85 55.85 56.84 57.84 58.84 59.84

3.78 3.85 3.93 4.00 4.08 4.15 4.22 4.30 4.37 4.45

50.84 51.84 52.84 53.83 54.83 55.83 56.82 57.82 58.82 59.82

4.00 4.08 4.16 4.24 4.32 4.39 4.47 4.55 4.63 4.71

50.82 51.82 52.82 53.81 54.81 55.81 56.80 57.80 58.130 59.79

4.22 4.31 4.39 4.47 4.55 4.64 4.72 4.80 4.89 4.97

51 52 > 53 54 55 ; 56 57: 58 59 60

61 62 63 64 65 66 ' 67 68 69 70

60.85 61.85 62.85 63.84 64.84 65.84 66.84 67.83 68.83 69.83

4.26 4.32 4.39 4.46 4.53 4.60 4.67 4.74 4.81 4.88

60.83 61.83 62.83 63.82 64.82 65.82 66.82 67.81 68.81 69.81

4.52 . 4.59 4.67 4.74 4.82 4.89 4.97 5.04 5.11 5.19

60.81 61.81 62.81 63.80 64.80 65.80 66.79 67.79 68.79 69.78

4.79 4.86 4.94 5.02 5.10 5.18 5.26 5.34 5.41 5.49

60.79 61.79 62.78 63.78 64.78 65.77 66.77 67.77 68.76 69.76

5.05 5.13 5.22 5.30 5.38 5.47 5.55 5.63 5.71 5.80

61 62 63 64 65 66 67 ■ 68 69 70

71 72 73 74 75 76 77 78 79 80

70.83 71.82 72.82 73.82 74.82 75.81 76.81 77.81 78.81 79.81

4.95 5.02 5.09 5.16 5.23 5.30 5.37 5.44 5.51 5.58

70.80 71.80 72.80 73.80 74.79 75.79 76.79 77.79 78.78 79.78

5.26 5.34 5.41 5.48 5.56 5.63 5.71 5.78 5.85 5.93

70.73 71.78 72.77 73.77 74.77 75.77 76.76 77.76 78.76 79.75

5.57 5.65 5.73 5.81 5.88 5.96 6.04 6.12 6.20 6.28

70.76 71.75 72.75 73.75 74.74 75.74 76.74 77.73 78.73 79.73

5.88 5.96 6.04 6.13 6.21 6.29 6.38 6.46 6.54 6.62

71 : 72 73 74 75 ' 76 77 78 i 79 ; 80

81 82 83 84 85 86 87 88 89 90

80.80 81.80 82.80 83.80 84.79 85.79 86.79 1! 87.79 88.78 89.78

5.65 5.72 5.79 5.86 5.93 6.00 6.07 6.14 6.21 6.28

80.78 81.78 82.77 83.77 84.77 85.76 86.76 87.76 88.76 89.75

6.00 6.08 6.15 6.23 6.30 6.37 6.45 6.52 6.60 6.67

80.75 81.75 82.74 83.74 84.74 85.73 86.73 87.73 88.73 89.72

6.36 6.43 6.51 6.59 6.67 6.75 6.83 6.90 6.98 7.06

80.72 81.72 82.71 83.71 84.71 85.70 86.70 87.70 88.70 89.69

6.71 6.79 6.87 6.96 7.04 7.12 7.20 7.29 7.37 7.45

81 82 83 84 85 86 87 88 89 : 90

91 92 93 94 95 96 97 98 99 100

90.78 91.78 92.77 93.77 94.77 95.77 96.76 97.76 98.76 99.76

6.35 6.42 6.49 6.56 6.63 6.70 6.77 6.84 6.91 6.98

90.75 91.75 92.74 93.74 94.74 95.74 96.73 97.73 98.73 99.73

6.74 6.82 6.89 6.97 7.04 7.11 7.19 7.26 7.34 7.41

90.72 91.72 92.71 93.71 94.71 95.70 96.70 97.70 98.69 99.69

7.14 7.22 7.30 7.38 7.45 7.53 7.61 7.69 7.77 7.85

90.69 91.68 92.68 93.68 94.67 95.67 96.67 97.66 98.66 99.66

7.54 91 7.62 92 7.70 93 7.78 94 7.87 95 7.95 96 8.03 97 8.12 98 8.20 99 8.28 100

«

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o

a tn

A.i

4 Deg.

4| Deg.

41 Deg.

4| Deg.

0)

o

d 86 Deg.

85} Deg.

rt 85!

851 Deg*

n

s

12 g ‘ Ml*

TRAVERSE TABLE. S Deg.

5} Deg.

54 Deg. !

5} Deg.

p

o ?

Lat.

Dep.

Lat.

Dep.

Lat*

Dep.

1.00 1.99 2.99 3.98 4.98 5.98 6.97 7.97 8.97 9.96

0.09 0.17 0.26 0.35 0.44 0.52 0.61 0.70 0.78 0.87

1.00 1.99 2.99 3.98 4.98 5.97 6.97 7.97 8.96 9.96

0.09 0.18 0.27 0.37 0.46 0.55 0.64 0.73 0.82 0.92

1.00 1.99 2.99 3.98 4.98 5.97 6.97 7.96 8.96 9.95

11 . 12 13 14 15 16 • 17 , 18 ' 19 20

10.96 0.96 11.95 1.05 12.95 1.13 13.95 1.22 14.94 . 1-31 15.94 1.39 16.94 1.48 1.57 17.93 18.93 1.66 19.92 1.74

10.95 11.95 12.95 13.94 14.94 15.93 16.93 17.92 18.92 19.92

21 22 23 , 24 - 25 26 ' 27 28 29 30

20.92 21.92 22.91 23.91 24.90 25.90 26.90 27.89 28.89 29.89

1.83 1.92 2.00 2.09 2.18 2.27 2.35 2.44 2.53 2.61

31 ■ 32 33 34 35 36 37 38 39 40

30.88 31.88 32.87 33.87 34.87 35.86 36.86 37.86 38.85 39.85

‘ 41 ■ 42 43 44 45 • 46 ’ 47 48 49 50

1 2 3 . 4 5 6 : *7 8 0 10

oi

' .

o a rt m

s

g o»* P 3 O

Lat.

Dep.

0.10 0.19 0.29 0.38 0.48 0.58 0.67 0.76 0.86 0.96

0.99 1.99 2.98 3.98 4.97 5.97 6.96 7.96 8.95 9.95

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

1.01 10.95 1.10 11.94 1.19 12.94 1.28 13.94 1.37 ' 14.93 1.46 15.93 1.56 16.92 1.65 17.92 1.74 18.91 1.83 19.91

1.05 1.15 1.25 1.34 1.44 1.53 1.63 1.73 1.82 1.92

10.94 11.94 12.93 13.93 14.92 15.92 16.91 17.91 18.90 19.90

1.10 1.20 1*30 1.40 1.50 1.60 1.70 1.80 1.90 2.00

20.91 21.91 22.90 23.90 24.90 25.89 26.89 27.88 28.88 29.87

1.92 2.01 2.10 2.20 2.29 2.38 2.47 2.56 2.65 2.75

20.90 21.90 22.89 23.89 24.88 25.88 26.88 27.87 28.87 29.86

2.01 20.89 2.11 21.89 2.20 22.88 2.30 23.88 2.40 24.87 2.49 25.87 2.59 26.86 2.68 27.86 2.78 28.85 2.88 ‘29.85

2.10 2.20 2.30 2.40 2.50 2.60 2.71 2.81 2.91 3.01

23 24 25 26 27 28 29 30

2.70 2.79 2.88 2.96 3.05 3.14 3.22 3.31 3.40 3.49

30.87 31.87 32.86 33.86 34.85 35.85 36.84 37.84 38.84 39.83

2.84 2.93 3.02 3.11 3.20 3.29 3.39 3.48 3.57 3.66

30.86 31.85 32.85 33.84 34.84 35.83 36.83 37.83 38.82 39.82

2.97 3.07 3.16 3.26 3.35 3.45 3.55 3.64 3.74 3.83

30.84 31.84 32.83 33.83 34.82 35.82 36.81 37.81 38.80 39.80

3.11 3.21 3.31 3.41 3.51 3.61 3.71 3.81 3.91 4.01

31 32 33 34 35 36 37 38 39 40

40.84 41.84 42.84 43.83 44.83 45.82 46.82 47.82 48.81 49.81

3.57 3.66 3.75 3.83 3.92 4.01 4.10 4.18 4.27 4.36

40.83 41.82 42.82 43.82 44.81 45.81 46.80 47.80 48.79 49.79

3.75 3.84 3.93 4.03 4.12 4.21 4.30 4.39 4.48 4.58

40.81 41.81 42.80 43.80 44.79 45.79 46.78 47.78 48.77 49.77

3.93 4.03 4.12 4.22 4.31 4.41 4.50 4.60 4.70 4.79

40.79 41.79 42.78 43.78 44.77 45.77 46.76 47.76 48.75 49.75

4.11 4.21 4.31 4.41 4.51 4.61 4.71 4.81 4.91 5.01

41 42 43 44 45 46 47 48 49 50

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

85 Deg.

84} Deg.

1

84* Deg.

|

84} Deg.

ct>

1 2 3 4 5 a

7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22

V o s «

s

13

TRAVERSE TABLE, Distance.

g

5 Deg.

5| Deg.

5| Deg.

54 Deg.

u>‘ r*P D O
Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

50.81 51.80 52.80 53.79 54.79 55.79 56.78 57.78 58.78 59.77

4.44 4.53 4.62 4.71 4.79 4.88 4.97 5.06 5.14 5.23

50.79 51.78 52.78 53.77 54.77 55.77 56.76 57.76 58.75 59.75

4.67 4.76 4.85 4.94 5.03 5.12 5.22 5.31 5.40 5.49

50.77 51.76 52.76 53.75 54.75 55.74 56.74 57.73 58.73 59.72

4.89 4.98 5.08 5.18 5.27 5.37 5.46 5.56 5.65 5.75

50.74 51.74 52.73 53.73 54.72 55.72 56.71 57.71 58.70 59.70

5.11 5.21 5.31 5.41 5.51 5.61 5.71 5.81 5.91 6.01

51 52 53 54 55 i 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

60.77 61.76 62.76 63.76 64.75 65.75 66.75 67.74 68.74 69.73

5.32 5.40 5.49 5.58 5.67 5.75 5.84 5.93 6.01 6.10

60.74 61.74 62.74 63.73 64.73 65.72 66.72 67.71 68.71 69.71

5.38 5.67 5.76 5.86 5.95 6.04 6.13 6.22 6.31 6.41

60.72 61.71 62.71 63.71 64.70 65.70 66.69 67.69 68.68 69.68

5.85 5.94 6.04 6.13 6.23 6.33 6.42 6.52 6.61 6.71

60.69 61.69 62.68 63.68 64.67 65.67 66.66 67.66 68.65 6 9.65

6.11 6.21 6.31 6.41 6.51 6.61 6.71 6.81 6.91 7.01

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

70.73 71.73 72.72 73.72 74.71 75.71 76.71 77.70 78.70 79.70

6.19 6.28 6.36 6.45 6.54 6.62 6.71 6.80 6.89 6.97

70.70 71.70 72.69 73.69 74.69 75.68 76.68 77.67 78.67 79.66

6.50 6.59 6.68 6.77 6.86 6.95 7.05 7.14 7.23 7.32

70.67 71.67 72.66 73.66 74.65 75.65 76.65 77.64 78.64 79.63

6.81 6.90 7.00 7.09 7.19 7.28 7.38 7.48 7.57 7.67

70.64 71.64 72.63 73.63 74.62 75.62 76.61 77.61 78.60 79.60

7.11 7.21 7.31 7.41 7.51 7.61 7.71 7.81 7.91 8.02

71 72 73 74 75 76 77 78 79 80

81 82 , 83 84 85 86 87 88 89 90

80.69 81.69 82.68 83.68 84.68 85.67 86.67 87.67 88.66 89.66

7.06 7.15 7.23 7.32 7.41 7.50 7.58 7.67 7.76 7.84

80.66 81.66 82.65 83.65 84.64 85.64 86.64 87.63 88.63 89.62

7.41 7.50 7.59 7.69 7.78 7.87 7.96 8.05 8.14 8.24

80.63 81.62 82.62 83.61 84.61 85.60 86.60 87.59 88.59 89.59

7.76 7.86 7.96 8.05 8.15 8.24 8.34 8.43 8.53 8.63

80.59 81.59 82.58 83.58 84.57 85.57 86.56 87.56 88.55 89.55

8.12 8.22 8.32 8.42 8.52 8.62 8.72 8.82 8.92 9.02

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

90.65 91.65 92.65 93.64 94.64 95.63 96.63 97.63 98.62 99.62

7.93 8.02 8.11 8.19 8.28 8.37 8.45 8.54 8.63 8.72

90.62 91.61 92.61 93.61 94.60 95.60 96.59 97.59 98.59 99.58

8.33 8.42 8.51 8.60 8.69 8.78 8.88 8.97 9.06 9.15

90.58 91.58 92.57 93.57 94.56 95.56 96.55 97.55 98.54 99.54

8.72 8.82 8.91 9.01 9.11 9.20 9.30 9.39 9.49 9.58

90.54 91.54 92.53 93.53 94.52 95.52 96.51 97.51 98.50 99.50

9.12 91 9.22 92 9.32 93 9.42 94 9.52 95 9.62 96 9.72 97 9.82 98 9.92 99 10.02 100

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.



«

o

a a

85 Deg. •

&

84J Deg.

84} Deg.

k

V CJ

s 84i Deg.

s



14 o u»*

TRAVERSE TABLE. 6 Deg.

6i Deg.

6$ Deg.

6| Deg. Dep.

£s o
0.99 1.99 2.98 3.97 4.97 5.96 6.95 7.94 8.94 9.93

0.12 0.24 0.35 0.47 0.59 0.71 0.82 0.94 1.06 1.18

1 2 3 4 5 6 7 8 9 10

1.25 1.36 1.47 1.59 1.70 1.81 1.92 2.04 2.15 2.26

10.92 11.92 12.91 13.90 14.90 15.89 16.88 17.88 18.87 19.86

1.29 1.41 1.53 1.65 1.76 1.88 2.00 2.12 2.23 2.35

11 12 13 14 15 16 17 , 18 19 20

20.87 21.86 22.85 23.85 24.84 25.83 26.83 27.82 28.81 29.81

2.38 2.49 2.60 2.72 2.83 2.94 3.06 3.17 3.28 3.40

20.85 21.85 22.84 23.83 24.83 25.82 26.81 27.81 28.80 29.79

2.47 2.59 2.70 2.82 2.94 3.06 3.17 3.29 3.41 3.53

21 22 23 24 25 26 27 28 29 30

3.37 3.48 3.59 3.70 3.81 3.92 4.03 4.14 4.25 4.35

30.80 31.79 32.79 33.78 34.78 35.77 36.76 37.76 38.75 39.74

3.51 3.62 3.74 3.85 3.96 4.08 4.19 4.30 4.41 4.53

30.79 31.78 32.77 33.76 34.76 35.75 36.75 37.74 38.73 39.72

3.64 3.76 3.88 4.00 4.11 4.23 4.35 4.47 4.58 4.70

31 32 33 34 35 36 37 38 39 40

40.76 41.75 42.74 43.74 44.73 45.73 46.72 47.71 48.71 49.70

4.46 4.57 4.68 4.79 4.90 5.01 5.12 5.23 5.34 5.44

40.74 41.73 42.72 43.72 44.71 45.70 46.70 47.69 48.69 49.68

4.64 4.76 4.87 4.98 5.09 5.21 5.32 5.43 5.55 5.66

40.72 41.71 42.70 43.70 44.69 45.68 46.67 47.67 48.66 49.65

4.82 4.94 5.05 5.17 5.29 5.41 5.52 5.64 5.76 5.88

41 42 43 44 45 46 47 48 49 50

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

6 «>o

L&ti

Dep.

Lat.

Dep.

Lat. J Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.99 1.99 2.98 3.98 4.97 5.97 6.96 7.96 8.95 9.95

0.10 0.21 0.31 0.41 0.52 0.63 0.73 0.84 0.94 1.05

0.99 1.99 2.98 3.98 4.97 5.96 6.96 7.95 8.95 9.94

0.11 0.22 0.33 0.44 0.54 0.65 0.76 0.87 0.98 1.09

0.99 1.99 2.98 3.97 4.97 5.96 6.96 7.95 8.94 9.94

0.11 0.23 0.34 0.45 0.57 0.68 0.79 0.91 1.02 1.13

11 12 13 14 15 16 17 18 19 20

10.94 11.93 12.93 13.92 14.92 15.91 16.91 17.90 18.90 19.89

1.15 1.25 1.36 1.46 1.57 1.67 1.78 1.88 1.99 2.09

10.93 11.93 12.92 13.92 14.91 15.90 16.90 17.89 18.89 19.88

1.20 1.31 1.42 1.52 1.63 1.74 1.85 1.96 2.07 2.18

10.93 11.92 12.92 13.91 14.90 15.90 16.89 17.88 18.88 19.87

21 22 23 24 25 26 27 28 29 30

20.88 21.88 22.87 23.87 24.86 25.86 26.85 27.85 28.84 29.84

2.20 2.30 2.40 2.51 2.61 2.72 2.82 2.93 3.03 3.14

20.88 21.87 22.86 23.86 24.85 25.85 26.84 27.83 28.83 29.82

2.29 2.40 2.50 2.61 2.72 2.83 2.94 3.05 3.16 3.27

31 32 33 34 35 36 37 38 39 40

30.83 31.82 32.82 33.81 34.81 35.80 36.80 37.79 38.79 39.78

3.24 3.34 3.45 3.55 3.66 3.76 3.87 3.97 4.08 4.18

30.82 31.81 32.80 33.80 34.79 35.79 36.78 37.77 38.77 39.76

41 42 43 44 45 46 47 48 49 50

40.78 4.29 41.77 4.39 42.76 . 4.49 43.76 4.60 44.75 4.70 45.75 4.81 46.74 4.91 47.74 5.02 48.73 5.12 49.73 5.23 Dep.

<£ o

c« f/i

s

Lat.

g .

o0)

rt 84 Deg. j| 83f Deg.

, 834 Deg.

831 Deg.

S

1

15

TRAVERSE TABLE.

Dep.

Lat.

Dep.

Lat. | Dep.

6f Deg. Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

50.72 51.72 52.71 53.70 54.70 55.69 56 69 57.68 58.68 59.67

5.33 5.44 5.54 5.64 5.75 5.85 5.96' 6.06 6.17 6.27

50.70 51.69 52.68 53.68 54.67 55.67 56.66 57.66 58.65 59.64

5.55 5.66 5.77 5.88 5.99 6.10 6.21 6.31 6.42 6.53

50.67 51.67 52.66 53.65 54.65 55.64 56.63 57.63 58.62 59.61

5.77 5.89 6.00 6.11 6.23 6.34 6.45 6.57 6.68 6.79

50.65 51.64 52.63 53.63 54.62 55.61 56.60 57.60 58.59 59.58

5.99 6.11 6.23 6.35 6.46 6.58 6.70 6.82 6.93 7.05

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

60.67 61.66 62.65 63.65 64.64 65.64 66.63 67.63 68.62 69.62

6.38 €.48 6.59 6.69 6.79 6.90 7.00 7.11 7.21 7.32

60.64 61.63 62.63 63.62 64.61 65.61 66.60 67.60 68.59 69.58

6.64 6.75 6.86 6.97 7.08 7.19 7.29 7.40 7.51 7.62

60.61 61.60 62.60 63.59 64.58 65.58 66.57 67.56 68.56 69.55

6.91 7.02 7.13 7.25 7.36 7.47 7.58 7.70 7.81 7.92

60.58 61.57 62.56 63.56 64.55 65.54 66.54 67.53 68.52 69.51

7.17 7.29 7.40 7.52 7.64 7.76 7.88 7.99 8.11 8.23

61 62 63 64 65 66 67 68 69 70 .

71 72 73 74 75 76 77 78 79 80

70.61 71.61 72.60 73.59 74.59 75.58 76.58 77.57 78.57 79.56

7.42 7.53 7.63 7.74 7.84 7.94 8.05 8.15 8.26 8.36

70.58 71.57 72.57 73.56 74.55 75.55 76.54 77.54 78.53 79.53

7.73 7.84 7.95 8.06 8.17 8.27 8.38 8.49 8.60 8.71

70.54 71.54 72.53 73.52 74.52 75.51 76.51 77.50 78.49 79.49

8.04 8.15 8.26 8.38 8.49 8.60 8.72 8.83 8.94 9.06

70.51 71.50 72.49 73.49 74.48 75.47 76.47 77.46 78.45 79.45

8.35 8.46 8.58 8.70 8.82 8.93 9.05 9.17 9.29 9.40

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

80.56 81.55 82.55 83.54 84.53 85.53 86.52 87.52 88.51 89.51

8.47 8.57 8.68 8.78 8.88 8.99 9.09 9.20 9.30 9.41

80.52 81.51 82.51 83.50 84.50 85.49 86.48 87.48 88.47 89.47

8.82 8.93 9.04 9.14 9.25 9.36 9.47 9.58 9.69 9.80

80.48 81.47 82.47 83.46 84.45 85.45 86.44 87.43 88.43 89.42

9.17 9.28 9.40 9.51 9.62 9.74 9.85 9.96 10.08 10.19

80.44 81.43 82.42 83.42 84.41 85.40 86.40 87.39 88.38 89.38

9.52 9.64 9.76 9.87 9.99 10.11 10.23 10.34 10.46 10.58

91 92 93 94 95 96 97 98 99 100

90.50 91.50 92.49 93.49 94.48 95.47 96.47 97.46 98.46 99.45

9.51 9.62 9.72 9.83 9.93 10.03 10.14 10.24 10.35 10.45

90.46 91.45 92.45 93.44 94.44 95.43 96.42 97.42 98.41 99.41

9.91 10.02 10.12 10.23 10.34 10.45 10.56 10.67 10.78 10.89

90.42 91.41 92.40 93.40 94.39 95.38 96.38 97.37 98.36 99.36

10.30 10.41 10.53 10.64 10.75 10.87 10.98 11.09 11.21 11.32

90.37 91.36 92.36 93.35 94.34 95.33 96.33 97.32 98.31 99.31

10.70 91 10.81 92 10.93 93 11.05 94 11.17 95 11.28 96 , 11.40 97 11.52 98 11.64 99 11.75 100

Distance.

Lat.

6* Deg. | i

6$ Deg.

Distance.

Distance.

6 Deg.

Dep.

Lat.

Dep. | Lat.

Dep.

Lat.

Dep.

Lat.

83} Deg.



83* Deg.

|

83$ Deg.

. .

*

81 82 83 84 85 86 87 88 89 ' 90

Distance.

84 Deg.

.

16

TRAVERSE TABLE U Deg.

UJ*

i o ?

0.13 0.26 0.39 0.52 0.65 0.78 0.91 1.04 1.17 1.31

0.99 1.98 2.97 3.96 4.95 5.95 6.94 7.93 8.92 9.91

0.13 0.27 0.40 0.54 0.67 0.81 0.94 1.08 1.21 1.35

1 2 3 4 5 6 7 8 9 10

10.91 11.90 12.89 13.88 14.87 15.86 16.85 17.85 18.84 19.83

1.44 1.57 1.70 1.83 1.96 2.09 2.22 2.35 2.48 2.61

10.90 11.89 12.88 13.87 14.86 15.85 16.84 17.84 18.83 19.82

1.48 1.62 1.75 1.89 ‘2.02 2.16 2.29 2.43 2.56 2.70

11 12 13 14 15 16 17 18 19 20

2.65 2.78 2.90 3.03 3.15 3.28 3.41 3.53 3.66 3.79

20.82 21.81 22.80 23.79 24.79 25.78 26.77 27.76 28.75 29.74

2.74 2.87 3.00 3.13 3.26 3.39 3.52 3.65 3.79 3.92

20.81 21.80 22.79 23.78 24.77 25.76 26.75 27.74 28.74 29.73

2.83 2.97 3.10 3.24 3.37 3.51 3.64 3.78 3.91 4.05

21 22 23 24 25 26 27 28 29 30

30.7S 31.74 32.74 33.73 34.72 35.71 36.70 37.70 38.69 39.68

3.91 4.04 4.16 4.29 4.42 4.54 4.67 4.80 4.92 5.05

30.73 31.73 32.72 33.71 34.70 35.69 36.68 37.67 38.67 39.66

4.05 4.18 4.31 4.44 4.57 4.70 4.83 4.96 5.09 5.22

30.72 31.71 32.70 33.69 34.68 35.67 36.66 37.65 38.64 39.63

4.18 1 31 4.32 32 4.45 33 ’ 4.58 ! 34 4.72 35 4.85 36 4.99 37 5.12 38 5.26 39 5.39 40

40.70 41.69 42.68 43.67 44.67 45.66 46.65 47.64 48.63 49.63

5.00 40.67 5.12 41.66 5.24 42.66 5.36 43.65 5.48 44.64 5.6! 45.63 5.73 ; 46.62 5.85 47.62 5.97 48.61 6.09 49.60

5.17 5.30 5.43 5.55 5.68 5.81 5.93 6.06 6.18 6.31

40.65 41.64 42.63 43.62 44.62 45.61 46.60 47.59 48.58 49.57

5.35 5.48 5.61 5.74 5.87 6.00 6.13 6.27 6.40 6.53

I 40.63 : 41.62 j 42.61 43.60 | 44.59 45.58 46.57 1 47.56 1 48.55 49.54

5.53 5.66 5.80 5.93 6.07 6.20 6.34 6.47 6.61 6.74

Dep.

Lat.

Lat.

Dep. | Lat.

' Dep.

Lat.

Lat.

1 2 3 4 5 6 7 8 9 10

0.99 1.99 2.98 3.97 4.96 5.96 6.95 7.94 8.93 9.93

0.12 0.24 0.37 0.49 0.61 0.73 0.85 0.97 1.10 1.22

0.99 1.98 2.98 3.97 4.96 5.95 6.94 7.94 8.93 9.92

11 12 13 14 15 16 17 18 19 20

10.92 11.91 12.90 13.90 14.89 15.88 16.87 17.87 18.86 19.85

1.34 1.46 1.58 1.71 1.83 1.95 2.07 2.19 2.32 2.44

10.91 11.90 12.90 13.89 14.88 15.87 16.86 17.86 18.85 19.84

21 22 23 24 25 26 27 28 29 30

20.84 21.84 22.83 23.82 24.81 25.81 26.80 27.79 28.78 29.78

2.56 2.68 2.80 2.92 3.05 3.17 J3.29 3.41 3.53 3.66

31 32 33 34 35 36 37 38 39 40

30.77 31.76 32.75 33.75 34.74 35.73 36.72 37.72 38.71 39.70

3.78 3.90 4.02 4.14 4.27 4.39 4.51 4.63 4.75 4.87

41 42 43 44 45 46 47 48 49 50

£to

£

Dep.

Dep.

5

7| Deg. Lat.

Lat.

o 0

71 Deg.

Distance.

7 Deg.

83 Deg

Dep.

Lat.

Dep.

0.99 1.98 2.97 3.97 4.96 5.95 6.94 7.93 8.92 9.91

1.39 1.51 1.64 1.77 1.89 2.02 2.15 2.27 2.40 2.52

20.83 21.82 22.82 23 81 24.80 25.79 26.78 27.78 28.77 29.76

Dep

0.13 0.25 0.38 0.50 0.63 0.76 0.88 1 1.01 1.14 1.26

Deg.

821 Deg.

824 Deg.

41 42 43 44 45 46 47 48 49 50 6

o

| t w

s

17

TRAVERSE TABLE Distance.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

51 52 53 54 55 56 57 58 59 60

50.62 51.61 52.60 53.60 54.59 55.58 56.58 57.57 58.56 59.55

6.22 6.34 6.46 6.58 6.70 6.82 6.95 7.07 7.19 7.31

50.59 51.58 52.58 53.57 54.56 55.55 56.54 57.54 58.53 59.52

6.44 6.56 6.69 6.81 6.94 7.07 7.19 7.32 7.45 7.57

50.56 51.56 52.55 53.54 54.53 55.52 56.51 57.50 58.50 59.49

6.66 6.79 6.92 7.05 7.18 7.31 7.44 7.57 7.70 7.83

50.53 51.53 52.52 53.51 54.50 55.49 56.48 57.47 58.46 59.45

6.88 7.01 7.15 7.28 7.42 7.55 7.69 7.82 7.96 8.09

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

60.55 61.54 62.53 63.52 64.52 65.51 66.50 67.49 68.49 69.48

7.43 7.56 7.68 7.80 7.92 8.04 8.17 8.29 8.41 8.53

60.51 61.50 62.50 63.49 64.48 65.47 66.46 67.46 68.45 69.44

7.70 7.82 7.95 8.08 8.20 8.33 8.46 8.58 8.71 8.83

60.48 61.47 62.46 63.45 64.44 65.44 66.43 67.42 68.41 (?9.40

7.96 8.09 8.22 8.35 8.48 8.61 8.75 8.88 9.01 9.14

60.44 61.43 62.42 63.42 64.41 65.40 66.39 67.38 68.37 69.36

8.23 8.36 8.50 8.63 8.77 8.90 9.04 9.17 9.30 9.44

61 62 63 64 65 66 67 68 69 70

71 , 72 73 74 75 76 77 78 79 80

70.47 71.46 72.46 73.45 74.44 75.43 76.43 77.42 78.41 79.40

8.65 8.77 8.90 9.02 9.14 9.26 9.38 9.51 9.63 9.75

70.43 71.42 72.42 73.41 74.40 75.39 76.38 77.38 78.37 79.36

8.96 9.09 9.21 9.34 9.46 9.59 9.72 9.84 9.97 10.10

70.39 71.38 72.38 73.37 74.36 75.35 76.34 77.33 78.32 79.32

9.27 9.40 9.53 9.66 9.79 9.92 10.05 10.18 10.31 10.44

70.35 71.34 72.33 73.32 74.31 75.31 76.30 77.29 78.28 79.27

9.57 9.71 9.84 9.98 10.11 10.25 10.38 10.52 10.65 10.79

71 72 73 74 75 76 77 78 79 80

81 82 83 ; 84 85 86 87 88 89 . 90

80.40 81.39 82.38 83.37 84.37 85.36 86.35 87.34 88.34 89.33

9.87 9.99 10.12 10.24 10.36 10.48 10.60 10.72 10.85 10.97

80.35 81.34 82.34 83.33 84.32 85.31 86.30 87.30 88.29 89.28

10.22 10.35 10.47 10.60 10.73 10.85 10.98 11.23 11.36

80.31 81.30 82.29 83.28 84.27 85.26 86.26 87.25 88.24 89.23

10.57 10.70 10.83 10.96 11.09 11.23 11.36 11.49 11.62 11.75

80.26 81.25 82.24 83.23 84.22 85.21 86.21 87.20 88.19 89.18

10.92 11.06 11.19 11.33 11.46 11.60 11.73 11.87 12.00 12.14

81 82 83 , 84 85 86 87 88 89 90

91 92 93 94 95 96 97 , 98 99 100

90.32 91.31 92.31 93.30 94.29 95.28 96.28 97.27 98.26 99.25

11.09 11.21 11.33 11.46 11.58 11.70 11.82 11.94 12.07 12.19

90.27 91.26 92.26 93.25 94.24 95.23 96.22 97.22 98.21 99.20

11.48 11.61 11.74 11.86 11.99 12.12 12.24 12.37 12.49 12.62

90.22 91.21 92.20 93.20 94.19 95.18 96.17 97.16 98.15 99.14

11.88 12.01 12.14 12.27 12.40 12.53 12.66 12.79 12.92 13.05

90.17 91.16 92.15 93.14 94.13 95.12 96.11 97.10 98.10 99.09

12.27 91 12.41 92 12.54 93 12.68 94 12.81 95 12.95 96 13.08 97 , 13.22 98 13.35 99 13.49 100

!

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.


o art

3

7 Deg.

83 Deg.

n

7} Deg.

7J Deg.

Deg.

11.11

82| Deg.

824 Deg.

2P

|

821 Deg.

o<0 ' £3

a tn

3

(8

TRAVERSE TABLE

2 £S* p" •3

8 Deg.

8i Deg.

8£ Deg.

O a

Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.99 1.98 2.97 3.96 4 < 95 5.94 6.93 7.92 8.91 9.90

0.14 0.28 0.42 0.56 0.70 0.84 0.97 1.11 1.25 1.39

0.99 1.98 2.97 3.96 4.95 5.94 6.93 7.92 8.91 9.90

0.14 0.29 0.43 0.57 0.72 0.86 1.00 1.15 1.29 1.43

0.99 1.98 2.97 3.96 4.95 5.93 6.92 7.91 8.90 9.89

11 13 14 15 16 17 18 19 20

10.89 11.88 12.87 13.86 14.85 15.84 16.83 17.82 18.82 19.81

1.53 1.67 1.81 1.95 2.09 2.23 2.37 2.51 2.64 2.78

10.89 11.88 12.87 13.86 14.85 15.84 16.83 17.81 18.80 19.79

1.58 1.72 1.87 2.01 2.15 2.30 2.44 2.58 2.73 2.87

21 22 23 24 25 26 27 28 29 30

20.80 21.79 22.78 23.77 24.76 25.75 26.74 27.73 28.72 29.71

2.92 3.06 3.20 3.34 3.48 3.62 3.76 3.90 4.04 4.18

20.78 21.77 22.76 23.75 24.74 25.73 26.72 27.71 28.70 29.69

31 32 33 34 35 36 37 38 39 40

30.70 31.69 32.68 33.67 34.66 35.65 36.64 37.63 38.62 39.61

4.31 4.45 4 59 4.73 4.87 5.01 5.15 5.29 5.43 5.57

41 42 43

40.60 41.59 42.53 43.57 44.56 45.55 46.54 47.53 48.52 49.51 Dep.

12

44

45 46 47 ? 48 49 50 • 6o A

Dep.

O pS C3

Lat.

Dep.

0.15 0.30 0.44 0.59 0.74 0.89 1.03 1.18 1.33 1.48

0.99 1.98 2.97 3.95 4.94 5.93 . 6.92 7.91 8.90 9.88

0.15 0.30 0.46 0.61 0.76 0.91 1.06 1.22 1.37 1.52

1 2 3 4 5 6 7 8 9 10

10.88 11.87 12.86 13.85 14.84 15.82 16.81 17.80 18.79 19.78

1.63 1.77 1.92 2.07 2.22 2.36 2.51 2.66 2.81 2.96

10.87 11.86 12.85 13.84 14.83 15.81 16.80 17.79 18.78 19.77

1.67 1.83 1.98 2.13 2.28 2.43 2.59 2.74 2.89 3.04

11 12 13 14 15 16 17 18 19 20

3.01 3.16 3.30 3.44 3.59 3.73 3.87 4.02 4.16 4.30

20.77 21.76 22.75 23.74 24.73 25.71 26.70 27.69 28.68 29.67

3.10 3.25 3.40 3.55 3.70 3.84 3.99 4.14 4.29 4.43

20.76 21.74 22.73 23.72 24.71 25.70 26.69 27.67 28.66 29.65

3.19 3.35 3.50 3.65 3.80 3.96 4.11 4.26 4.41 4.56

21

30.68 31.67 32.66 33.65 34.64 35.63 36.62 37.61 38.60 39.59

4.45 4.59 4.74 4.88 5.02 5.17 5.31 5.45 5.60 5.74

30.66 31.65 32.64 33.63 34.62 35.60 36.59 37.58 38.57 39.56

4.58 4.73 4.88 5.03 5.17 5.32 5.47 5.62 5.76 5.91

30.64 31.63 32.62 33.60 34.59 35.58 36.57 37.56 38.55 39.53

4.72 4.87 5.02 5.17 5.32 5.48 5.63 5.78 5.93 6.08

31 32 33 34 35 36 37 38 39 40

5.71 5.85 5.98 6.12 6.26 6.40 6.54 6.68 6.82 6.96

40.58 41.57 42.56 43.54 44.53 45.52 46.51 47.50 48.49 49.48

5.88 6.03 6.17 6.31 6.46 6.60 6.74 6.89 7.03 7.17

40.55 41.154 42.53 43.52 44.51 45.49 46.48 47.47 48.46 49.45

6.06 6.21 6.36 6.50 6.65 6.80 6.95 7.09 7.24 7.39

40.52 41.51 42.50 43.49 44.48 45.46 46.45 47.44 48.43 49.42

6.24 6.39 6.54 6.69 6.85 7.00 7.15 7.30 7.45 7.61

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

?

22

23 24 25 26 27 28 29 30

oV

rt

a 4-> m

3 |1

8| Deg.

82 Deg. ’

81} Deg.

, i'

811 Deg.

1

81i Deg.

tn

s

19

TRAVERSE TABLE.

g w

8 Deg.

8\ Deg.

8£ Deg.

p 3

8| Deg. Dep.

2 : B

O A

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

51 52

59 60

50.50 51.49 52.48 53.47 54.46 55.46 56.45 57.44 58.43 59.42

7.10 7.24 7.38 7.52 7.65 7.79 7.93 8.07 8.21 8.35

50.47 51.46 52.45 53.44 54.43 55.42 56.41 57.40 58.39 59.38

7.32 7.46 7.61 7.75 7.89 8.04 8.18 8.32 8.47 8.61

50.44 51.43 52.42 53.41 54.40 55.38 56.37 57.36 58.35 59.34

7.54 7.69 7.83 7.98 8.13 8.28 8.43 8.57 8.72 8.87

50.41 51.39 52.38 53.37 54.36 55.35 56.34 57.32 58.31 59.30

7.76 7.91 8.06 8.21 8.37 8.52 8.67 8.82 8.98 9.13

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

60.41 61.40 62.39 63.38 64.37 65.36 66.35 67.34 68.33 69.32

8.49 8.63 8.77 8.91 9.05 9.19 9.32 9.46 9.60 9.74

60.37 61.36 62.35 63.34 64.33 65.32 66.31 67.30 68.29 69.28

8.75 8.90 9.04 9.18 9.33 9.47 9.61 9.76 9.90 10.04

60.33 61.32 62.31 63.30 64.29 65.28 66.26 67.25 68.24 69.23

9.02 9.16 9.31 9.46 9.61 9.76 9.90 10.05 10.20 10.35

60.29 61.28 62.27 63.26 64.24 65.23 66.22 67.21 68.20 69.19

9.28 9.43 9.58 9.74 9.89 10.04 10.19 10.34 10.50 10.65

61 62 63 64 65 66 67 68 69 70

71 70.31 72 71.30 73 72.29 74 73.28 75 74.27 76 75.26 77 76.25 78 77.24 79 |j 78.23 80 jI 79.22

9.88 10.02 10.16 10.30 10.44 10.58 10.72 10.86 10.99 11.13

70.27 71.25 72.24 73.23 74.22 75.21 76.20 77.19 78.18 79.17

10.19 10.33 10.47 10.62 10.76 10.91 11.05 11.19 11.34 11.48

70.22 71.21 72.20 73.19 74.18 75.17 76.15 77.14 78.13 79.12

10.49 10.64 10.79 10.94 11.09 11.23 11.38 11.53 11.68 11.82

70.17 71.16 72.15 73.14 74.13 75.12 76.10 77.09 78.08 79.07

10.80 10.95 11.10 11.26 11.41 11.56 11.71 11.87 12.02 ,12.17

71 72 73 74 75 76 77 78 . 79 80

81 82 83 84 85 86 87 88 89 90

80.21 81.20 82.19 83.18 84.17 85.16 86.15 87.14 88.13 89.12

11.27 11.41 11.55 11.69 11.83 11.97 12.11 12.25 12.39 12.53

80.16 81.15 82.14 83.13 84.12 85.11 86.10 87.09 88.08 89.07

11.62 11.77 11.91 12.05 12.20 12.34 12.48 12.63 12.77 12.91

80.11 81.10 82.09 83.08 84.07 85.06 86.04 87.03 88.02 89.01

11.97 12.12 12.27 12.42 12.56 12.71 12.86 13.01 13.16 13.30

80.06 81.05 82.03 83.02 84.01 85.00 85.99 86.98 87.96 88.95

12.32 12.47 12.63 12.78 12.93 13.08 13.23 13.39 13.54 13.69

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

90.11 91.10 92.09 93.09 94.08 95.07 96.06 97.05 98.04 99.03

12.66 12.80 12.94 13.08 13.22 13.36 13.50 13.64 13.78 13.92

90.06 91.05 92.04 93.03 94.02 95.01 96.00 96.99 97.98 98.97

13.06 13.20 13.34 13.49 13.63 13.78 13.92 14.06 14.21 14.35

90.00 90.99 91.98 92.97 93.96 94.95 95.93 96.92 97.91 98.90

13.45 13.60 13.75 13.89 14.04 14.19 14.34 14.49 14.63 14.78

89.94 90.93 91.92 92.91 93.89 94.88 95.87 96.86 97.85 98.84

13.84 91 14.00 92 14.15 93 14.30 94 14.45 95 14.60 96 14.76 97 14.91 98 15.06 99 15.21 100

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

53 54 55 56 57

58


1

s

82 Deg.

81J Deg.

?

o

rtE3

81J Deg.

8l4^Deg. 'f

s m

20 d U3*

TrtAVERSE TABLE. 9 Deg.

9} Deg. |

9$ Deg.

d £ 3a

°l Deg.

P



Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.99 1.98 2.96 3.95 4.94 5.93 6.91 7.90 8.89 9.88

0.16 0.31 0.47 0.63 0.78 0.94 1.10 1.25 1.41 1.56

0.99 1.97 2.96 3.95 4.93 5.92 6.91 7.90 8.88 9.87

0.16 032 0.<4O 0.64 0.80 0.96 1.13 1.29 1.45 1.61

0.99 1.97 2.96 3.95 4.93 5.92 6.90 7.89 8.88 9.86

0.17 0.33 0.50 0.66 0.83 0.99 1.16 1.32 1.49 1.65

0.99 1.97 2.96 3.94 4.93 5.91 6.90 7.88 8.87 9.86

0.17 0.34 0.51 0.68 0.85 1.02 1.19 1.35 1.52 1.69

11 12 13 14 15 16 17 18 19 20

10.86 11.85 12.84 13.83 14.82 15.80 16.79 17.78 18.77 19.75

1.72 1.88 2.03 2.19 2.35 2.50 2.66 2.82 2.97 3.13

10.86 11.84 12.83 13.82 14.80 15.79 16.78 17.77 18.75 19.74

1.77 1.93 2.09 2.25 2.41 2.57 2.73 2.89 3.05 3.21

10.85 11.84 12.82 13.81 14.79 15.78 16.77 17.75 18.74 19.73

1.82 1.98 2.15 2.31 2.48 2.64 2.81 2.97 3.14 3.30

10.84 11.83 12.81 13.80 14.78 15.77 16.75 17.74 18.73 19.71

1.86 2.03 2.20 2.37 2.54 2.71 2.88 3.05 3.22 3.39

11 12 13 14 15 16 17 18 19 20

20.74 21.73 22.72 23.70 24.69 25.68 26.67 27.66 28.64 29.63 , ,

3.29 3.44 3.60 3.75 3.91 4.07 4.22 4.38 4.54 4.69

20.73 21.71 22.70 23.69 24.67 25.66 26.65 27.64 28.62 29.61

3.38 3.54 3.70 3.86 4.02 4.18 4.34 4.50 4.66 4.82

20.71 21.70 22.68 23.67 24.66 25.64 26.63 27.62 28.60 29.59

3.47 3.63 3.80 3.96 4.13 4.29 4.46 4.62 4.79 4.95

20.70 21.68 22.67 23.65 24.64 25.62 26.61 27.60 28.58 29.57

3.56 3.73 3.90 4.06 4.23 4.40 4.57 4.74 4.91 5.08

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

30.62 31.61 32.59 33.58 34.57 35.56 36.54 37.53 38.52 39.51

4.85 5.01 5.16 5.32 5.48 5.63 5.79 5.94 6.10 6.26

30.60 31.58 32.57 33.56 34.54 35.53 36.52 37.51 38.49 39.48

4.98 5.14 5.30 5.47 5.63 5.79 5.95 6.11 6.27 6.43

30.57 31.56 32.55 33.53 34.52 35.51 36.49 37.48 38.47 39.45

5.12 5.28 5.45 5.61 5.78 5.94 6.11 6.27 6.44 6.60

30.55 31.54 32.52 33.51 34.49 35.48 36.47 37.45 38.44 39.42

5.25 5.42 5.59 5.76 5.93 6.10 6.27 6.44 6.60 6.77

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

40.50 41.48 42.47 43.46 44.45 45.43 46.42 47.41 48.40 49.38

6.41 6.57 6.73 6.88 7.04 7.20 7.35 7.51 7.67 7.82

40.47 41.45 42.44 43.43 44.41 45.40 46.39 47.38 48.36 49.35

6.59 6.75 6.91 7.07 7.23 7.39 7.55 7.72 7.88 8.04

40.44 41.42 42.41 43.40 44.38 45.37 46.36 47.34 48.33 49.32

6.77 6.92 7.10 7.26 7.43 7.59 7.76 7.92 8.09 8.25

40.41 41.39 42.38 43.36 44.35 45.34 46.32 47.31 48.29 49.28

41 42 43 44 45 46 47 48 49 50;

ogoi

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

6.94 7.11 7.28 7.45 7.62 7.79 7.96 8.13 8.30 8.47 j ! Lat.

21 22 23 ‘ 24 25 26 27 28 29 30

Dep.

Lat.

Dep.

ca

s

81 Deg.

80| Deg.

80£ Deg.

80| Deg.

CD

i

1 2 3 4 5 6 7 8 9 10

1

6O +J

1

3

|

21

TRAVERSE TABLE,

g

g OB* P oo

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

u>‘ p £3 O a>

51 52 53 54 55 56 57 58 59 60

50.37 51.36 52.35 53.34 54.32 55.31 56.30 57.29 58.27 59.26

7.98 8.13 8.29 8.45 8.60 8.76 8.92 9.07 9.23 9.39

50.34 51.32 52.31 53.30 54.28 55.27 56.26 57.25 58.23 59.22

8.20 8.36 8.52 8.68 8.84 9.00 9.16 9.32 9.48 9.64

50.30 51.29 52.27 53.26 54.25 55.23 56.22 57.20 58.19 59.18

8.42 8.58 8.75 8.91 9.08 9.24 9.41 9.57 9.74 9.90

50.26 51.25 52.23 53.22 54.21 55.19 56.18 57.16 58.15 59.13

8.64 8.81 8.98 9.14 9.31 9.48 9.65 9.82 9.99 10.16

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

60.25 61.24 62.22 63.21 64.20 65.19 66.18 67.16 68.15 69.14

9.54 9.70 9.86 10.01 10.17 10.32 10.48 10.64 10.79 10.95

60.21 61.19 62.18 63.17 64.15 65.14 66.13 67.12 68.10 69.09

9.81 9.97 10.13 10.29 10.45 10.61 10.77 10.93 11.09 11.25

60.16 61.15 62.14 63.12 64.11 65.09 66.08 67.07 68.05 69.04

10.07 10.23 10.40 10.56 10.73 10.89 11.06 11.22 11.39 11.55

60.12 61.10 62.09 63.08 64.06 65.05 66.03 67.02 68.00 68.99

10.33 10.50 10.67 10.84 11.01 11.18 11.35 11.52 11.69 11.85

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

70.13 71.11 72.10 73.09 74.08 75.06 76.05 77.04 78.03 79.02

11.26 11.42 11.58 11.73 11.89 12.05 12.20 12.36 12.51

70.08 71.06 72.05 73.04 74.02 75.01 76.00 76.99 77.97 78.96

11.41 11.57 11.73 11.89 12.06 12.22 12.38 12.54 12.70 12.86

70.03 71.01 72.00 72.99 73.97 74.96 75.94 76.93 77.92 78.90

11.72 11.88 12.05 12.21 12.38 12.54 12.71 12.87 13.04 13.20

69.97 70.96 71.95 72.93 73.92 74.90 75.89 76.87 77.86 78.84

12.02 12.19 12.36 12.53 12.70 12.87 13.04 13.21 13.38 13.55

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

80.00 80.99 81.98 82.97 83.95 84.94 85.93 86.92 87.90 88.89

12.67 12.83 12.98 13.14 13.30 13.45 13.61 13.77 13.92 14.08

79.95 80.93 81.92 82.91 83.89 84.88 85.87 86.86 87.84 88.83

13.02 13.18 13.34 13.50 13.66 13.82 13.98 14.15 14.31 14.47

79.89 80.88 81.86 82.85 83.83 84.82 85.81 86.79 87.78 88.77

13.37 13.53 13.70 13.86 14.03 14.19 14.36 14.52 14.69 14.85

79.83 80.82 81.80 82.79 83.77 84.76 85.74 86.73 87.71 88.70

13.72 13.89 14.06 14.23 14.39 14.56 14.73 14.90 15.07 15.24

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 '100

89.88 90.87 91.86 92.84 93.83 94.82 95.81 96.79 97.78 98.77

14.24 14.39 14.55 14.70 14.86 15.02 15.17 15.33 15.49 15.64

89.82 90.80 91.79 92.78 93.76 94.75 95.74 96.73 97.71 98.70

14.63 14.79 14.95 15.11 15.27 15.43 15.59 15.75 15.91 16.07

89.75 90.74 91.72 92.71 93.70 94.68 95.67 96.66 97.64 98.63

15.02 15.18 15.35 15.51 15.68 15.84 16.01 16.17 16.34 16.50

89.69 90.67 91.66 92.64 93.63 94.61 95.60 96.58 97.57 98.56

15.41 91 15.58 92 15.75 93 15.92 94 16.09 95 16.26 96 16.43 97 16.60 98 16.77 99 16.93 100

o
Dep.

Lat.

Dep.

Lat,

Dep.

Lat.

Dep.

Lat.

s

9 Deg.

11.11

81 Deg.

26

H Deg.

80J Deg.

9£ Deg.

80£ Deg.

9| Deg.

80i Deg.


o

1 s .

22 s 09*

TRAVERSE TABLE. 10} Deg.

10 Deg.

i 101 Deg.

10} Deg.

P

o

g ST a o

re

Lai.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.98 1.97 2.95 3.94 4.92 5.91 6.89 7.88 8.86 9.85

0.17 0.35 0.52 0.69 0.87 1.04 1.22 1.39 1.56 1.74

0.98 1.97 2.95 3.94 4.92 5.90 6.89 7.87 8.86 9.84

0.18 0.36 0.53 0.71 0.89 1.07 1.25 1.42 1.60 1.78

0.98 1.97 2.95 3.93 4.92 5.90 6.88 7.87 P.85 9.83

0.18 0.36 0.55 0.73 0.91 1.09 1.28 1.46 1.64 1.82

0.98 1.96 2.95 3.93 4.91 5.89 6.88 7.86 8.84 9.82

0.19 0.37 0.56 0.75 0.93 1.12 1.31 1.49 1.68 1.87

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10.83 11.82 12.80 13.79 14.77 15.76 16.74 17.73 18.71 19.70

1.91 2.08 2.26 2.43 2.60 2.78 2.95 3.13 3.30 3.47

10.82 11.81 12.79 13.78 14.76 15.74 16.73 17.71 18.70 19.68

1.96 2.14 2.31 2.49 2.67 2.85 3.03 3.20 3.38 3.56

10.82 11.80 12.78 13.77 14.75 15.73 16.72 17.70 18.68 19.67

2.00 2.19 2.37 2.55 2.73 2.92 3.10 3.28 3.46 3.64

10.81 11.79 12.77 13.75 14.74 15.72 16.70 17.68 18.67 19.65

2.05 2.24 2.42 2.61 2.80 2.98 3.17 3.36 3.54 3.73

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

20.68 21.67 22.65 23.64 24.62 25.61 26.59 27.57 28.56 29.54

3.65 3.82 3.99 4.17 4.34 4.51 4.69 4.86 5.04 5.21

20.66 21.65 22.63 23.62 24.60 25.59 26.57 27.55 28.54 29.52

3.74 3.91 4.09 4.27 4.45 4.63 4.80 4.98 5.16 5.34

20.65 21.63 22.61 23.60 24.58 25.56 26.55 27.53 28.51 29.50

3.83 4.01 4.19 4.37 4.56 4.74 4.92 5.10 5.28 5.47

20.63 21.61 22.60 23.58 24.56 25.54 26.53 27.51 28.49 29.47

3.92 4.10 4.29 4.48 4.66 4.85 5.04 5.22 5.41 5.60

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

30.53 31.51 32.50 33.48 34.47 35.45 36.44 37.42 38.41 39.39

5.38 5.56 5.73 5.90 6.08 6.25 6.42 6.60 6.77 6.95

30.51 31.49 32.47 33.46 34.44 , 35.43 36.41 37.39 38.38 39.36

5.52 5.69 5.87 6.05 6.23 6.41 6.58 6.76 6.94 7.12

30.48 31.46 32.45 33.43 34.41 35.40 36.38 37.36 38.35 39.33

5.65 5.83 6.01 6.20 6.38 6.56 6.74 6.92 7.11 7.29

30.46 31.44 32.42 33.40 34.39 35.37 36.35 37.33 38.32 39.30

5.78 5.97 6.16 6.34 6.53 6.71 6.90 7.09 7.27 7.46

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

40.38 41.36 42.35 43.33 44.32 45.30 46.29 47.27 48.26 49.24

7.12 7.29 7.47 7.64 7.81 7.99 8.16 8.34 8.51 8*68

40.35 41.33 42.31 43.30 44.28 45.27 46.25 47.23 48.22 49.20

7.30 7.47 7.65 7.83 8.01 8.19 8.36 8.54 8.72 8.90

40.31 41.30 42.28 43.26 44.25 45.23 46.21 47.20 48.18 49.16

7.47 40.28 7.65 41.26 7.84 42.25 8.02 43.23 8.20 44.21 8.38 45.19 8.57 46.18 8.75 47.16 8.93 1 48.14 9.11 , 49.12

7.65 7.83 8.02 8.21 8.39 8.58 8.77 8.95 9 14 9.33

41 42 43 44 45 46 47 48 49 50

6 o

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Dep.

Lat.

Jja
Q

re

V o

Jj 80 Deg.

°

79| Deg.

f

79*. Deg.

79* Deg.

s

23

TRAVERSE TABLE,

s| 10i Deg.

10£ Deg.

lOf Deg.

Distance.

Distance.

! 10 D eg.

Lat.

Dep.

9.29 9.48 9.66 9.84 10.02 10.21 10.39 10.57 10.75 10.93

50.10 51.09 52.07 53.05 54.03 55.02 56.00 56.98 57.9658.95

9.51 9.70 9.89 10.07 10.26 10.45 10.63 10.82 11.19

51 52 53 54 55 56 57 58 59 60

59.98 60.96 61.95 62.93 63.91 64.89 65.88 66.86» 67.84 68.83

11.12 11.30 11.48 11.66 11.85 12.03 12.21 12.39 12.57 12.76

59.93 60.91 61.89 62.88 63.86 64.84 65.82 66.81 67.79 68.77

11.38 11.56 11.75 11.94 12.12 12.31 12.50 12.68 12.87 13.06

61 62 63 64 65 66 67 68 69 70

12.63 12.81 12.99 13.17 13.35 13.52 13.70 13.88 14.06 14.24

69.81 70.79 71.78 72.76 73.74 74.73 75.71 76.69 77.68 78.66

12.94 13.12 13.30 13.49 13.67 13.85 14.03 14.21 14.40 14.58

69.75 70.74 71.72 72.70 73.68 74.67 75.65 76.63 77.61 78.60

13.24 13.43 13.62 13.80 13.99 14.18 14.36 14.55 14.74 14.92

71 72 73 74 , 75 : 76 77 78 79 so:

79.71 80.69 81.68 82.66 83.64 84.63 85.61 86.60 87.58 88.56

14.41 14.59 14.77 14.95 15.13 15.30 15.48 15.66 15.84 16.01

79.64 80.63 81.61 82.59 83.58 84.56 85.54 86.53 87.51 88.49

14.76 14.94 15.13 15.31 15.49 15.67 15.85 16.04 16.22 16.40

79.58 80.56 81.54 82.53 83.51 84.49 85.47 86.46 87.44 88.42

15.11 15.29 15.48 15.67 15.85 16.04 16.23 16.41 16.60 16.79

81 82 83 . 84 85 86 87 88 89 90

15.80 15.98 16.15 16.32 16.50 16.67 16.84 17.02 17.19 17.36

89.55 90.53 91.52 92.50 93.48 94.47 95.45 96.44 97.42 98.40

16.19 16.37 16.55 16.73 16.90 17.08 17.26 17.44 17.62 17.79

89.48 90.46 91.44 92.43 93.41 94.39 95.38 96.36 97.34 98.33

16.58 16.77 16.95 17.13 17.31 17.49 17.68 17.86 13.04 18.22

89.40 90.39 91.37 92.35 93.33 94.32 95.30 96.28 97.26 98.25

16.97 91 17.16 92 17.35 93 17.53 94 17.72 95 17.91 96 18.09 97 18.28 98 18.47 99 18.65 100

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

51 52 53 54 55 56 57 58 59 60

50.23 51.21 52.19 53.18 54.16 55.15 56.13 57.12 58.10 59.09

8.86 9.03 9.20 9.38 9.55 9.72 9.90 10.07 10.25 10.42

50*19 51.17 52.15 53.14 54.12 55.11 56.09 57.07 58.06 59.04

9.08 9.25 9.43 9.61 9.79 9.96 10.14 10.32 10.50 10.68

50.15 51.13 52.11 53.10 54.08 55.06 56.05 57.03 58.01 59.00

61 62

10.59 10.77 10.94

64 65 66 67 . 68 69 70

60.07 61.06 62.04 63.03 64.01 65.00 65.98 66.97 67.95 68.94

11.29 11.46 11.63 11.81 11.98 12.16

60.03 61.01 61.99 62.98 63.96 64.95 65.93 66.91 67.90 68.88

10.85 11.03 11.21 11.39 11.57 11.74 11.92 12.10 12.28 12.46

71 72 73 74 75 76 77 78 79 80

69.92 70.91 71.89 72.88 73.86 74.85 75.83 76.82 77.80 78.78

12.33 12.50 12.68 12.85 13.02 13.20 13.37 13.54 13.72 13.89

69.87 70.85 71.83 72.82 73.80 74.79 75.77 76.76 77.74 78.72

81 82 83 84 85 86 87 88 89 90

79.77 80.75 81.74 82.72 83.71 84.69 85.68 86.66 87.65 88.63

14.07 14.24 14.41 14.59 14.76 14.93 15.11 15.28 15.45 15.63

91 92 93 94 95 96 97 98 99 100

89.62 90.60 91.59 92.57 93.56 94.54 95.53 96.51 97.50 98.48


Dep.

Dep. |

r

Lat.

63

o

1

Q

11.11

11.00


80 Deg.

79| Deg.

79£ Deg.

79* Deg.

5

TRAVERSE TABLE,

24 1 Distance.

g

11 Deg.

1H Deg.

p o

©

1 2 3 4 5 6 ' 7 8 9 10

Lat. 0.98 1.96 2.94 3.93 4.91 5.89 6.87 7.85 8.83. 9.82

Dep.

114 Deg.

11* Deg.

Lat.

Dep.

Lat.

0.19 0.38 0.57 0.76 0.95 1.14 1.34 1.53 1.72 1.91

0.98 1.96 2.94 3.92 4.90 5.88 6.87 7.85 8.83 9.81

0.20 0.39 0.59 0.78 0.98 1.17 1.37 1.56 1.76 1.95

0.98 1.96 2.94 3.92 4.90 5.88 6.86 7.84 8.82 9.80

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.59 1.79 1.99

0.98 1.96 2.94 3.92 4.90 5.87 6.85 7.83 8.81 9.79

0.20 0.41 0.61 0.82 1.02 1.22 1.43 1.63 1.83 2.04

1 2 3 4 5 6 7 8 : 9 10

Dep.

Lat.

Dep.

11 12 13 14 15 16 17 18 19 20

10.80 11.78 12.76 13.74 14.72 15.71 16.69 17.67 18.65 19.63

2.10 2.29 2.48 2.67 2.86 3.05 3.24 3.43 3.63 3.82

10.79 11.77 12.75 13.73 14.71 15.69 16.67 17.65 18.63 19.62

2.15 2.34 2.54 2.73 2.93 3.12 3.32 3.51 3.71 3.90

10.78 11.76 12.74 13.72 14.70 15.68 16.66 17.64 18.62 19.60

2.19 2.39 2.59 2.79 2.99 3.19 3.39 3.59 3.79 3.99

10.77 11.75 12.73 13.71 14.69 15.66 16.64 17.62 18.60 19.58

2.24 2.44 2.65 2.85 3.06 3.26 3.46 3.66 3.87 4 07

11 12 13 14 15 16 17 18 19 20 *

21 22 23 24 25 26 27 28 29 30

4.01 20.61 4.20 21.60 4.39 22.58 23.56 • 4.58 4.77 24.54 4.96 25.52 26.50 5.15 5.34 27.49 28.47 5.53 5.72 29.45

20.60 21.58 22.56 23.54 24.52 25.50 26.48 27.46 28.44 29.42

4.10 4.29 4.49 4.68 4.88 5.07 5.27 5.46 5.66 5.85

20.58 21.56 22.54 23.52 24.50 25.48 26.46 27.44 28.42 29.40

4.19 20.56 4.39 21.54 4.59 ■ 22.52 4.78 23.50 4.98 24.48 5.18 25.46 5.38 26.43 5.58 27.41 5.78 23.39 29.37 5.98

4.28 4.48 4.68 4.89 5.09 5.30 5.50 5.70 5.91 6.11

21 1 22 23 24 25 26 27 . 28 29 30 31 32 33 34 35 36 37 38 39 40

: 31 32 33 34 35 36 37 ' 38 39 40

30.43 31.41 32.39 33.38 34.36 35.34 36.32 37.30 38.28 39.27

5.92 6.11 6.30 6.49 6.68 6.87 7.06 7.25 7.44 7.63

30.40 31.39 32.37 33.35 34.33 35.31 36.29 37.27 38.25 39.23

6.05 6.24 6.44 6.63 6.83 7.02 7.22 7.41 7.61 7.80

30.38 31.36 32.34 33.3J2 34.30 35.28 36.26 37.24 38.22 39.20

6.18 6.38 6.58 6.78 6.98 7.18 7.33 7.58 7.78 7.97

30.35 31.33 32.31 33.29 34.27 35.25 36.22 37.20 38.18 39.16

6.31 6.52 6.72 6.92 7.13 7.33 7.53 7.74 7.94 8.15

41 42 43 44 45 46 47 I 48 i 49 so

40.25 41.23 42.21 43.19 44.17 45.15 46.14 47.12 48.10 49.08

7.82 8.01 8.20 8.40 8.59 8.78 8.97 9.16 9.35 9.54

40.21 41.19 42.17 43.15 44.14 45.12 46.10 47.08 48.06 49.04

8.00 8.19 8.39 8.58 8.78 8.97 9.17 9.36 9.56 9.75

40.18 41.16 42.14 43.12 44.10 45.08 46.06 47.04 48.02 49.00

8.17 8.37 8.57 8.77 8.97 9.17 9.37 9.57 9.77 9.97

40.14 41.12 42.10 43.08 44.06 45.04 46.02 46.99 47.97 48.95

8.35 8.55 8.76 8.96 9.16 9.37 9.57 9.78 9.98 10.18

O

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

;

O

03

s

79 Deg.

1 78| Deg.

78* Deg.

j

78* Deg.

41 ■ 42 43 44 45 46 47 48 49 50 o

J 5

25

TRAVERSE TABLE,

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Lep.

Distance.

51 52 53 54 55 56 57 53 59 60

50.06 51.04 52.03 53.01 53.99 54.97 55.95 56.93 57.92 58.90

9.73 9.92 10.11 10.30 10.49 10.69 10.88 11.07 11.26 11.45

50.02 51.00 51.98 52.96 53.94 54.92 55.90 56.89 57.87 58.85

9.95 10.14 10.34 10.53 10.73 10.93 11.12 11.32 11.51 11.71

49.98 50.96 51.94 52.92 53.90 54.88 55.86 56.84 57.82 58.80

10.17 10.37 10.57 10.77 10.97 11.16 11.36 11.56 11.76 11.96

49.93 50.91 51.89 52.87 53.85 54.83 55.81 56.78 57.76 58.74

10.39 10.59 10.79 11.00 11.20 11.40 11.61 11.81 12.01 12.22

51 52 53 54 55 56 57 58 59 60

i 61 62 63 64 65 66 67 68 69 70

5S 88 60.86 61.84 62.82 63.81 64.79 65.77 66.75 67.73 68.71

11.64 11.83 12.02 12.21 12.40 12.59 12.78 12.98 13.17 13.36

59.83 60.81 61.79 62.77 63.75 64.73 65.71 66.69 67.67 68.66

11.90 12.10 12.29 12.49 12.68 12.88 13.07 13.27 13.46 13.66

59.78 60.76 61.74 62.72 63.70 64.68 65.66 66.63 67.61 68.59

12.16 12.36 12.56 12.76 12.96 13.16 13.36 13.56 13.76 13.96

59.72 60.70 61.68 62.66 63.64 64.62 65.60 66.58 67.55 68.53

12.42 12.63 12.83 3.03 13.24 13.44 13.64 13.85 14.05 14.25

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

69.70 70.68 71.66 72.64 73.62 74.60 75.59 76.57 77.55 78.53

13.55 13.74 13.93 14.12 14.31 14.50 14.69 14.88 15.07 15.26

69.64 70.62 71.60 72.58 73.56 74.54 75.52 76.50 77.48 78.46

13.85 14.05 14.24 14.44 14.63 14.83 15.02 15.22 15.41 15.61

69.57 70.55 71.53 72.51 73.49 74.47 75.45 76.43 77.41 78.39

14.16 14.35 14.55 14.75 14.95 15.15 15.35 15.55 15.75 15.95

69.51 70.49 71.47 72.45 73.43 74.41 75.39 76.37 77.34 78.32

14.46 14.66 14.87 15.07 15.27 15.48 15.68 15.88 16.09 16.29

71 72 73 74 75 76 77 78 79

81 82 83 84 85 86 87 88 89 90

79.51 80.49 81.48 82.46 83.44 84.42 85.40 86.38 87.36 88.35

15.46 15.65 15.84 16.03 16.22 16.41 16.60 16.79 1698 17. .7

79.44 80.42 81.41 82.39 83.37 84.35 85.33 86.31 87.29 88.27

15.80 16.00 16.19 16.39 16.58 16.78 16.97 17.17 17.36 17.56

79.37 80.35 81.33 82.31 83.29 84.27 85.25 86.23 87.21 88.19

16.15 16.35 16.55 16.75 16.95 17.15 17.35 17.54 17.74 17.94

79.30 80.28 81.26 82.24 83.22 84.20 85.18 86.16 87.14 88.11

16.49 16.70 16.90 17.11 17.31 17.51 17.72 17.92 18.12 18.33

81 82 83 84 85 86 87 88 89

91 92 93 94 95 96 97 98 99 100

89.33 90.31 91.29 92.27 93.25 94.24 95.22 96.20 97.18 98.16

17.36 17.55 17.75 17.94 18.13 18.32 18.51 18.70 18.89 19.08

89.25 90.23 91.21 92.19 93.17 94.16 95.14 96.12 97.10 98.08

17.75 17.95 18.14 18.34 18.53 18.73 18.92 19.12 19.31 19.51

89.17 90.15 91.13 92.11 93.09 94.07 95.05 96.03 97.01 97.99

18.14 18.34 18.54 18.74 18.94 19.14 19.34 19.54 19.74 19.94

89.09 90.07 91.05 92.03 93.01 93.99 94.97 95.95 96.93 97.90

18.53 91 18.74 92 18.94 93 19.14 94 19.35 95 19.55 96 19.75 97 19.96 98 20.16 99 20.36 100

6

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o

3

11} Deg.

79 Deg;.

i 20 *

78} Deg.

11} Deg.

Ill Deg.

78$ Deg.

78} Deg. i

ty

Id., -nr.

Distance. 1

Lat.

11 Deg.

80

90

,

« o

1 s ;

TRAVERSE TABLE

26

gw*

12 Deg.

124 Deg.

12£ Deg.

Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.98 1.96 2.93 3.91 4.89 5.87 6.85 7.83 8.80 9.78

0.21 0.42 0.62 0.83 1.04 1.25 1.46 1.66 1.87 2.08

0.98 1.95 2.93 3.91 4.89 5.86 6.84 7.82 8.80 9.77

0.21 0.42 0.64 0.85 1.06 1.27 1.49 1.70 1.91 2.12

0.98 1.95 2.93 3.91 4.88 5.86 6.83 7.81 8.79 9.76

11 12 13 14 15 16 17 18 19 20

10.76 11.74 12.72 13.69 14.67 15.65 16.63 17.61 18.58 19.56

2.29 2.49 2.70 2.91 3.12 3.33 3.53 3.74 3.95 4.16

10.75 11.73 12.70 13.68 14.66 15.64 16.61 17.59 18.57 19.54

2.33 2.55 2.76 2.97 3.18 3.39 3.61 3.82 4.03 4.24

21 22 23 24 25 26 27 28 29 30

20.54 21.52 22.50 23.48 24.45 25.43 26.41 27.39 28.37 29.34

4.37 4.57 4.78 4.99 5.20 5.41 5.61 5.82 6.03 6.24

20.52 21.50. 22.48 23.45 24.43 25.41 26.39 27.36 28.34 29.32

31 32 33 34 35 36 37 38 39 40

30.32 31.30 32.28 33.26 34.24 35.21 36.19 37.17 38.15 39.13

6.45 6.65 6.86 7.07 7.28 7.48 7.69 7.90 8.11 8.32

41 42 43 44 45 46 47 48 49 50
Dep.

Lat.

Dep.

tn

a o 9

0.22 0.98 0.43 . 1.95 0.65 2.93 0.87 3.90 1.08 4.88 1.30 5.85 1.52 6.83 1.73 7.80 1.95 8.78 2.16 9.75

0.22 0.44 0.66 0.88 1.10 1.32 1.54 1.77 1.99 2.21

1 2 3 4 5 6 7 8 9 10

10.74 11.72 12.69 13.67 14.64 15.62 16.60 17.57 18.55 19.53

2.38 2.60 2.81 3.03 3.25 3.46 3.68 3.90 4.11 4.33

10.73 11.70 12.68 13.65 14.63 15.61 16.58 17.56 18.53 19.51

2.43 2.65 2.87 3.09 3.31 3.53 3.75 3.97 4.19 4.41

11 12 13 14 15 16 17 18 19 20

4.46 4.67 4.88 5.09 5.30 5.52 5.73 5.94 6.15 6.37

20.50 21.48 22.45 23.43 24.41 25.38 26.36 27.34 28.31 29.29

4.55 4.76 4.98 5.19 5.41 5.63 5.84 6.06 6.28 6.49

20.48 21.46 22.43 23.41 24.38 25.36 26.33 27.31 28.28 29.26

4.63 4.86 5.08 5.30 5.52 5.74 5.96 6.18 6.40 6.62

21 22 23 24 25 26 27 28 29 30

30.29 31.27 32.25 33.23 34.20 35.18 36.13 37.13 38.11 39.09

6.58 6.79 7.00 7.21 7.43 .7.64 7.85 8.06 8.27 8.49

30.27 31.24 32.22 33.19 34.17 35.15 36.12 37.10 38.08 39.05

6.71 6.93 7.14 7.36 7.58 7.79 8.01 8.22 8.44 8.66

30.24 31.21 32.19 33.16 34.14 35.11 36.09 37.06 38.04 39.01

6.84 7.06 7.28 7.50 7.72 7.95 8.17 8.39 8.61 8.83

31 32 33 34 35 36 37 38 39 40

40.10 41.08 42.06 43.04 44.02 44.99 45.97 46.95 47.93 43.91

8.52 40.07 8.73 41.04 8.94 42.02 9.15 43.00 9.36 43.98 9.56 44.95 9.77 45.93 9.98 ‘ 46.91 10.19 47.88 10.40 48.86

8.70 8.91 9.12 9.34 9.55 9.76 9.97 10.18 10.40 10.61

40.03 41.00 41.98 42.96 43.93 44.91 45.89 46.86 47.84 48.81

8.87 9.09 9.31 9.52 9.74 9.96 10.17 10.39 10.61 10.82

39.99 40.96 41.94 42.92 43.89 44.87 45.84 46.82 47.79 48.77

9.05 9.27 9.49 9.71 9.93 10.15 10.37 10.59 10.81 11.03

41 42 43 44 45 46 47 48 49 50

Dep.

Lat.

Lat.

Dep. 1 Lat.

Dep.

Lat.

Dep.

A 03

s

g ?



o

12} Deg.

78 Deg.

77| Deg.

77£ Deg.

77* Deg.

(U a

1

3

27

TRAVERSE TABLE.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

49.89 50.86 51.84 52.82 53.80 54.78 55.75 56.73 57.71 58.69

10.60 10.81 11.02 11.23 11.44 11.64 11*. 85 12.06 12.27 12 .'47

49.84 50.82 51.79 52.77 53.75 54.72 55.70 56.68 57.66 58.63

10.82 11.03 11.25 11.46 11.67 11.88 12.09 12.31 12.52 12.73

49.79 50.77 51.74 52.72 53.70 54.67 55.65 56.63 57.60 58.58

11.04 11.25 11.47 11.69 11.90 12.12 12.34 12.55 12.77 12.99

49.74 50.72 51.69 52.67 53.64 54.62 55.59 56.57 57.55 58.52

11.26 11.48 11.70 11.92 12.14 12.36 12.58 12.80 13.02 13.24

51 52 53 54 55 56 ' 57 58 59 . 60

61 62 63 64 65 66 67 68 69 70

59.67 60.65 61.62 62.60 63.58 64.56 65.54 66.51 67.49 68.47

12.68 12.89 13.10 13.31 13.51 13.72 13.93 14.14 14.35 14.55

59.61 60.59 61.57 62.54 63.52 64.50 65.47 66.45 67.43 68.41

12.94 13.16 13.37 13.58 13.79 14.00 14.22 14.43 14.64 14.85

59.55 60.53 61.51 62.48 63.46 64.44 65.41 66.39 67,. 36 68.34

13.20 13.42 13.64 13.85 14.07 14.29 14.50 14.72 14.93 15.15

59.50 60.47 61.45 62.42 63.40 64.37 65.35 66.32 67.30 68.27

13.46 13.68 13.90 14.12 14.35 14.57 14.79 15.01 15.23 15.45

61 62 63 64 65 ‘ 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

69.45 70.43 71.40 72.38 73.36 74.34 75.32 76.30 77.27 78.25

14.76 14.97 15.18 15.39 15.59 15.80 16.01 16.22 16.43 16.63

69.38 70.36 71.34 72.32 73.29 74.27 75.25 76.22 77.20 78.18

15.06 15.28 15.49 15.70 15.91 16.13 16.34 16.55 16.76 16.97

69.32 70.29 71.27 72.25 73.22 74.20 75.17 76.15 77.13 78.10

15.37 15.58 15.80 16.02 16.23 16.45 16.67 16.88 17.10 17.32

69.25 70.22 71.20 72.18 73.15 74.13 75.10 76.08 77.05 78.03

15.67 15.89 16.11 16.33 16.55 16.77 16.99 17.21 17.44 17.66

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

79.23 80.21 81.19 82.16 83.14 84.12 85.10 86.08 87.06 88.03

16.84 17.05 17.26 17.46 17.67 17.88 18.09 18.30 18.50 18.71

79.16 80.13 81.11 82.09 83.06 84.04 85.02 86.00 86.97 87.95

17.19 17.40 17.61 17.82 18.04 18.25 18.46 18.67 18.88 19.10

79.08 80.06 81.03 82.01 82.99 83.96 84.94 85.91 86.89 87.87

17.53 17.75 17.96 18.18 18.40 18.61 18.83 19.05 19.26 19.48

79.00 79.98 80.95 81.93 82.90 83.88 84.85 85.83 86.81 87.78

17.88 18.10 18.32 18.54 18.76 18.98 19.20 19.42 19.64 19.86

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

89.01 89.99 90.97 91.95 92.92 93.90 94.88 95.86 96.84 97.81

18.92 19.13 19.34 19.54 19.75 19.96 20.17 20.38 20.58 20.79

88.93 89.91 90.88 91.86 92.84 93.81 94.79 95.77 96.75 97.72

19.31 19.52 19.73 19.94 20.16 20.37 20.58 20.79 21.01 21.22

88.84 89.82 90.80 91.77 92.75 93.72 94.70 95.68 96.65 97.63

19.70 19.91 20.13 20.35 20.56 20.78 20.99 21.21 21.43 21.64

88.76 89.73 90.71 91.68 92.66 93.63 94.61 95.58 96.56 97.53

20.08 91 20.30 92 20.52 93 : 20.75 94 20.97 95 21.19 96 21.41 97 21.63 98 21.85 99 22.07 100

6 a &

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

o

.3 o

12 Deg.

78 Deg.

121 Deg.

12i Deg.

77} Deg.

Tti Deg.

1

12} Deg.

Lat.

77} Deg.

<0

o

1 S :

TRAVERSE TABLE.

28 Distance.

g w’ sr

13 Deg.

13* Deg.

13J Deg

13* Deg.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

2 3 4 5 6 7 8 9 iO

0.97 1.95 2.92 3.90 4.87 5.85 6.82 7.80 8.77 9.74

0.23 0.45 0.67 0.90 1.12 1.35 1.57 1.80 2.02 2.25

0.97 1.95 2.92 3.89 4.87 5.84 6.81 7.79 8.76 9.73

0.23 0.46 0.69 0.92 1.15 1.38 1.60 1.83 2.06 2.29

0.97 1.95 2.92 3.89 4.86 5.83 6.81 7.78 8.75 9.72

0.23 0.47 0.70 0.93 1.17 1.40 1.63 1.87 2.10 2.33

0.97 1.94 2.91 3.89 4.86 5.83 6.80 7.77 8.74 9.71

0.24 0.48 0.71 0.95 1.19 1.43 1.66 1.90 2.14 2.38

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10.72 11.69 12.67 13.64 14.62 15.59 16.57 17.54 18.51 19.49

2.47 2.70 2.92 3.15 3.37 3.60 3.82 4.05 4.27 4.50

10.71 11.68 12.65 13.63 14.60 15.57 16.55 17.52 18.49 19.47

2.52 2.75 2.98 3.21 3.44 3.67 3.90 4.13 4.35 4.58

10.70 11.67 12.64 13.61 14.59 15.56 16.53 17.50 18.48 19.45

2.57 2.80 3.03 3.27 3.50 3.74 3.97 4.20 4.44 4.67

10.68 11.66 12.63 13.60 14.57 15.54 16.51 17.48 18.46 19.43

2.61 2.85 3.09 3.33 3.57 3.80 4.04 4.28 4.52 4.75

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

20.46 21.44 22.41 23.38 24.36 25.33 26.31 27.28 28.26 29.23

4.72 4.95 5.17 5.40 5.62 5.85 6.07 6.30 6.52 6.75

20.44 21.41 22.39 23.36 24.33 25.31 26.28 27.25 28.23 29.20

4.81 5.04 5.27 5,50 5.73 5.96 6.19 6.42 6.65 6.88

20.42 21.39 22.36 23.34 24.31 25.28 26.25 27.23 28.20 29.17

4.90 5.14 5.37 5.60 5.84 6.07 6.30 6.54 6.77 7.00

20.40 21.37 22.34 23.31 24.28 25.25 26.23 27.20 28.17 29.14

4.99 5.23 5.47 5.70 5.94 6.18 6.42 6.66 6.89 7.13

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

30.21 31.18 32.15 33.13 34.10 35.08 36.05 37.03 38.00 38.97

6.97 7.20 7.42 7.65 7.87 8.10 8.32 8.55 8.77 9.00

30.17 31.15 32.12 33.09 34.07 35.04 36.02 36.99 37.96 38.94

7.11 7.33 7.56 7.79 8.02 8.25 8.48 8.71 *6.94 9.17

30.14 31.12 32.09 33.06 34.03 35.01 35.98 36.95 37.92 38.89

7.24 7.47 7.70 7.94 8.17 8.40 8.64 8.87 9.10 9.34

30.11 31.08 32.05 33.03 34.00 34.97 35.94 36.91 37.88 38.85

7.37 7.61 7.84 8.08 8.32 8.56 8.79 9.03 9.27 9.51

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

39.95 40.92 41.90 42.87 43.85 44.82 45.80 46.77 47.74 48.72

9.22 9.45 9.67 9.90 10.12 10.35 10.57 10.80 11.02 11.25

39.91 40.88 41.86 42.83 43.80 44.78 45.75 46.72 47.70 48.67

9.40 9.63 9.86 10.08 10.31 10.54 10.77 11.00 11.23 11.46

39.87 40.84 41.81 42.78 43.76 44.73 45.70 46.67 47.65 48.62

9.57 9.80 10.04 10.27 10.51 10.74 10.97 11.21 11.44 11.67

39.63 40.80 41.77 42.74 43.71 44.68 45.65 46.62 47.60 48.57

9.75 9.98 10.22 10.46 10.70 10.93 11.17 11.41 11.65 11.88

41 42 43 44 45 46 47 48 49 50

OJ

Dep.

Lat. j

Dep.

Lat.

Dep. | Lat.

Dep.

Lat.

o

p

1

o C)

ri

s

77 Deg.

76J Deg.

76h Deg.

76* Deg.

o

1to 3

TRAVERSE TABLE, 13i Deg.

13J Deg.

13J Deg.

Distance.

Distance. '

13 Deg.

29

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

49.69 50.67 51.64

49.64 50.62 51.59 52.56 53.54 54.51 55.48 56.46 57.43 58.40

11.69 11.92 12.15 12.38 12.61 12.84 13.06 13.29 13.52 13.75

49.59 50.56 51.54 52.51 53.48 54.45 55.43 56.40 57.37 58.34

11.91 12.14 12.37 12.61 12.84 13.07 13.31 13.54 13.77 14.01

49.54 50.51 51.48 52.45 53.42 54.40 55.37 56.34 57.31 58.28

12.12 12.36 12.60 12.84 13.07 13.31 13.55 13.79 14.02 14.26

51 52 53 54 55 56 57 58 59 60

51 52 53 54 55 56 57 53 59 60

53.59 54.56 55*54 56.51 57.49 58.46

11.47 11.70 11.92 12.15 12.37 12.60 12.82 13.05 13.27 13.50

61 62 63 64 65 66 67 68 69 70

59.44 60.41 61.39 62.36 63.33 64.31 65.28 66.26 67.23 68.21

13.72 13.95 14.17 14.40 14.62 14.85 15.07 15.30 15.52 15.75

59.38 60.35 61.32 62.30 63.27 64.24 65.22 66.19 67.16 68.14

13.98 14.21 14.44 14.67 14.90 15.13 15.36 15.59 15.81 16.04

59.31 60.29 61.26 62.23 63.20 64.18 65.15 66.12 67.09 68.07

14.24 14.47 14.71 14.94 15.17 15.41 15.64 15.87 16.11 16.34

59.25 60.22 61.19 62.17 63.14 64.11 65.08 66.05 67.02 67.99

14.50 14.74 14.97 15.21 15.45 15.69 15.93 16.16 16.40 16.64

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

69.18 70.15 71.13 72.10 73.08 74.05 75.03 76.00 76.98 77.95

15.97 16.20 16.42 16.65 16.87 17.10 17.32 17.55 17.77 18.00

69.11 70.08 71.06 72.03 73.00 73.98 74.95 75.92 76.90 77.87

16.27 16.50 16.73 16.96 17.19 17.42 17.65 17.88 18.11 18.34

69.04 70.01 70.98 71.96 72.93 73.90 74.87 75.84 70.82 77.79

16.57 16.81 17.04 17.28 17.50 17.74 17.98 18.21 18.44 18.68

68.97 69.94 70.91 71.88 72.85 73.82 74.79 75.76 76.74 77.71

16.88 17.11 17.35 17.59 17.83 18.06 18.30 18.54 18.78 19.01

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

73.92 79.90 80.87 81.85 82.82 83.80 84.77 85.74 86.72 87.69

18.22 18.45 18.67 18.90 19.12 19.35 19.57 19.80 20.02 20.25

78.84 79.82 80.79 81.76 82.74 83.71 84.68 85.66 86.63 87.60

18.57 18.79 19.02 19.25 19.48 19.71 19.94 20.17 20.40 20.63

78.76 79.73 80.71 81.68 82.65 83.62 84.60 85.57 86.54 87.51

18.91 19.14 19.38 19.61 19.84 20.08 20.31 20.54 20.78 21.01

78.68 79.65 80.62 81.59 82.56 83.54 84.51 85.48 86.45 87.42

19.25 19.49 19.73 19.97 20.20 20.44 20.68 20.92 21.15 21.39

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

88.67 89.64 90.62 91.59 92.57 93.54 94.51 95.49 96.46 97.44

20.47 20.70 20.92 21.15 21.37 21.60 21.82 22.05 22.27 22.50

88.58 89.55 90.52 91.50 92.47 93.44 94.42 95.39 96.36 97.34

20.86 21.09 21.32 21.54 21.77 22.00 22.23 22.46 22.69 22.92

88.49 89.46 90.43 91.40 92.38 93.35 94.32 95.29 96.26 97.24

21.24 21.48 21.71 21.94 22.18 22.41 22.64 22.88 23.11 23.34

88.39 89.36 90.33 91.31 92.28 93.25 94.22 95.19 96.16 97.13

21.63 91 21.87 92 22.10 93 22.34 94 22.58 95 22.82 96 23.06 97 23.29 98 23.53 99 23.77 100

«5o iS

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

3

52.62

77 Deg.

76J Deg.

76J Deg.

76* Deg.

oo

c £

Q

TRAVERSE TABLE

30 o o a

14* Deg.

14 Deg.

14* Deg.

14| Deg.

g ST 3

o ?

Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.97 1.94 2.91 3.88 4.85 5.82 6.79 7.76 8.73 9.70

0.24 0.48 0.73 0.97 1.21 1.45 1.69 1.94 2.18 2.42

0.97 1.94 2.91 3.88 4.85 5.82 6.78 7.75 8.72 9.69

0.25 0.49 0.74 0.98 1.23 1.48 1.72 1.97 2.22 2.46

0.97 1.94 2.90 3.87 4.84 5.81 6.78 7.75 8.71 9.68

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

0.97 1.93 2.90 3.87 4.84 5.80 6.77 7.74 8.70 9.67

0.25 0.51 0.76 1.02 1.27 1.53 1.78 2.04 2.29 2.55

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10.67 11.64 12.61 13.58 14.55 15.52 16.50 17.47 18.44 19.41

2.66 2.90 3.15 3.39 3.63 3.87 4.11 4.35 4.60 4.84

10.66 11.63 12.60 13.57 14.54 15.51 16.48 17.45 18.42 19.38

2.71 2.95 3.20 3.45 3.69 3.94 4.18 4.43 4.68 4.92

10.65 11.62 12.59 13.55 14.52 15.49 16.46 17.43 18.39 19.36

2.75 3.00 3.25 3.51 3.76 4.01 4.26 4.51 4.76 5.01

10.64 11.60 12.57 13.54 14.51 15.47 16.44 17.41 18.37 19.34

2.80 3.06 3.31 3.56 3.82 4.07 4.33 4.58 4.84 5.09

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

20.38 21.35 22.32 23.29 24.26 25.23 26.20 27.17 28.14 29.11

5.08 5.32 5.56 5.81 6.05 6.29 6.53 6.77 7.02 7.26

20.35 21.32 22.29 23.26 24.23 25.20 26.17 27.14 28.11 29.08

5.17 5.42 5.66 5.91 6.15 6.40 6.65 6.89 7.14 7.38

20.33 21.30 22.27 23.24 24.20 25.17 26.14 27.11 28.08 29.04

5.26 5.51 5.76 6.01 6.26 6.51 6.76 7.01 7.26 7.51

20.31 21.28 22.24 23.21 24.18 25.14 26.11 27.08 28.04 29.01

5.35 5.60 5.86 6.11 6.37 6.62 6.87 7.13 7.38 7.64

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

30.08 31.05 32.02 32.99 33.96 34.93 35.90 36.87 37.84 38.81

7.50 7.74 7.98 8.23 8.47 8.71 8.95 9.19 9.44 9.68

30.05 31.02 31.98 32.95 33.92 34.89 35.86 36.83 37.80 38.77

7.63 7.88 8.12 8.37 8.62 8.86 9.11 9.35 9.60 9.85

30.01 30.98 31.95 32.92 33.89 34.85 35.82 36.79 37.76 38.73

7.76 8.01 8.26 8.51 8.76 9.01 9.26 9.51 9.76 10.02

29.98 30.95 31.91 32.88 33.85 34.81 35.78 36.75 37.71 38.68

7.89 8.15 8.40 8.66 8.91 9.17 9.42 9.67 9.93 10.18

31 32 33 34 35 36 37 38 39 . 40

41 42 43 44 45 46 47 48 49 50

39.78 40.75 41.72 42.69 43.66 44.63 45.60 46.57 47.54 48.51

9.92 10.16 10.40 10.64 10.89 11.13 11.37 11.61 11.85 12.10

39.74 40.71 41.68 42.65 43.62 44.58 45.55 46.52 47.49 48.46

10.09 10.34 10.58 10.83 11.08 11.32 11.57 11.82 12.06 12.31

39.69 40.66 41.63 42.60 43.57 44.53 45.50 46.47 47.44 48.41

10.27 10.52 10.77 11.02 11.27 11.52 11.77 12.02 12.27 12.52

39.65 40.62 41.58 42.55 43.52 44.48 45.45 46.42 47.39 48.35

10.44 10.69 10.95 11.20 11.46 11.71 11.97 12.22 12.48 12.73

41 42 43 44 45 46 47 48 49 50

V

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o

S

76 Dtg. 1

J

Dep.

Lat.

Dep.

1

| | I ; ;

V

o 0 ti

75| Deg.

75* Deg.

75* Deg.

s■s

TRAVERSE TABLE

31

Distance.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

51 52 53 54 55 56 57 58 59 60

49.49 50.46 51.43 52.40 53.37 54.34 55.31 56.28 57.25 58.22

12.34 12.53 12.82 13.06 13.31 13.55 13.79 14.03 14.27 14.52

49.43 50.40 51.37 52.34 53.31 54.28 55.25 56.22 57.18 58.15

12.55 12.80 13.05 13.29 13.54 13.78 14.03 14.28 14.52 14.77

49.38 50.34 51.31 52.28 53.25 54.22 55.18 56.15 57.12 58.09

12.77 13.02 13.27 13.52 13.77 14.02 14.27 14.52 14.77 15.02

49.32 50.29 51.25 52.22 53.19 54.15 55.12 56.09 57.06 58.02

12.98 13.24 13.49 13.75 14.00 14.26 14.51 14.77 15.02 15.28

51 52 53 54 55 56 57 58 , 59 60

61 62 63 64 65 66 67 68 69 70

59.19 60.16 61.13 62.10 63.07 64.04 65.01 65.98 66.95 67.92

14.76 15.00 15.24 15.48 15.72 15.97 16.21 16.45 16.69 16.93

59.12 60.09 61.06 62.03 63.00 63.97 64.94 65.91 66.88 67.85

15.02 15.26 15.51 15.75 16.00 16.25 16.49 16.74 16.98 17.23

59.06 60.03 60.99 61.96 62.93 63.90 64.87 65.83 66.80 67.77

15.27 15.52 15.77 16.02 16.27 16.53 16.78 17.03 17.28 17.53

58.99 59.96 60.92 61.89 62.86 63.83 64.79 65.76 66.73 67.69

15.53 15.79 16.04 16.29 16.55 16.80 17.06 17.31 17.57 17.82

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 : 80

68.89 69.86 70.83 71.80 72.77 73.74 74.71 75.68 76.65 77.62

17.18 17.42 17.66 17.90 18.14 18.39 18.63 18.87 19.11 19.35

68.82 69.78 70.75 71.72 72.69 73.66 74.63 75.60 76.57 77.54

17.48 17.72 17.97 18.22 18.46 18.71 18.95 19.20 19.45 19.69

68.74 69.71 70.67 71.64 72.61 73.58 74.55 75.52 76.48 77.45

17.78 18.03 18.28 18.53 18.78 19.03 19.28 19.53 19.78 20.03

68.66 69.63 70.59 71.56 72.53 73.50 74.46 75.43 76.40 77.36

18.08 18.33 18.59 18.84 19.10 19.35 19.60 19.86 20.11 20.37

71 72 73 74 75 76 77 ’ 78 79 80

81 82 83 84 85 86 87 88 89 90

78.59 79.56 80.53 81.50 82.48 83.45 84.42 85.39 86.36 87.33

19.60 19.84 20.08 20.32 20.56 20.81 21.05 21.29 21.53 21.77

78.51 79.48 80.45 81.42 82.38 83.35 84.32 85.29 86.26 87.23

19.94 20.18 20.43 20.68 20.92 21.17 21.42 21.66 21.91 22.15

78.42 79.39 80.36 81.32 82.29 83.26 84.23 85.20 86.17 87.13

20.28 20.53 20.78 21.03 21.28 21.53 21.78 22.03 22.28 22.53

78.33 79.30 80.26 81.23 82.20 83.17 84.13 85.10 86.07 87.03

20.62 20.88 21.13 21.39 21.64 21.90 22.15 22.41 22.66 22.91

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

88.30 89.27 90.24 91.21 92.18 93.15 94.12 95.09 96.06 97.03

22.01 22.26 22.50 22.74 22.98 23.22 23.47 23.71 23.95 24.19

88.20 89.17 90.14 91.11 92.08 93.05 94.02 94.98 95.95 96.92

22.40 22.65 22.89 23.14 23.38 23.63 23.88 24.12 24.37 24.62

88.10 89.07 90.04 91.01 91.97 92.94 93.91 94.88 95.85 96.81

22.78 23.04 23.29 23.54 23.79 24.04 24.29 24.54 24.79 25.04

88.00 88.97 89.94 90.90 91.87 92.84 93.80 94.77 95.74 96.70

23.17 91 23.42 92 23.68 93 ’ 23.93 94 24.19 95 24.44 96 24.70 97 ' 24.95 98 25.21 99 25.46 100

<0 o a «s

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Q

14 Deg.

14$ Deg.

14$ Deg.

14j Deg.

/ Lat.

oo CJ

76 Deg.

751 Deg.

a

75J Deg.

75$ Deg.

•4-1

Q

32

TRAVERSE TABLE.

g

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

g S' p co

10

0.97 1.93 2.90 3.86 4.83 5.80 6.76 7.73 8.69 9.66

0.26 0.52 0.78 1.04 1.29 1.55 1.81 2.07 2.33 2.59

0.96 1.93 2.89 3.86 4.82 5.79 6.75 7.72 8.68 9.65

0.26 0.53 0.79 1.05 1.32 1.58 1.84 2.10 2.37 2.63

0.96 1.93 2.89 3.85 4.82 5.78 6.75 7.71 8.67 9.64

0.27 0.53 0.80 1.07 1.34 1.60 1.87 2.14 2.41 2.67

0.96 1.92 2.89 3.85 4.81 5.77 6.74 7.70 8.66 9.62

0.27 0.54 0.81 1.09 1.36 1.63 1.90 2.17 2.44 2.71

2 3 4 5 6 7 8 9 10

11 : 12 13 14 : 15 16 17 18 19 20

10.63 11.59 12.56 13.52 14.49 15.45 16.42 17.39 18.35 19.32

2.85 3.11 3.36 3.62 3.88 4.14 4.40 4.66 4.92 5.18

10.61 11.58 12.54 13.51 14.47 15.44 16.40 17.37 18.33 19.30

2.89 3.16 3.42 3.68 3.95 4.21 4.47 4.73 5.00 5.26

10.60 11.56 12.53 13.49 14.45 15.42 16.38 17.35 18.31 19.27

2.94 3.21 3.47 3.74 4.01 4.28 4.54 4.81 5.08 5.34

10.59 11.55 12.51 13.47 14.44 15.40 16.36 17.32 18.29 19.25

2.99 3.26 3.53 3.80 4.07 4.34 4.61 4.89 5.16 5.43

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

20.28 21.25 22.22 23.18 24.15 25.11 26.08 27.05 28.01 28.98

5.44 5.69 5.95 6.21 6.47 6.73 6.99 7.25 7.51 7.76

20.26 21.23 22 19 23.15 24.12 25.08 26.05 27.01 27.98 28.94

5.52 5.79 6.05 6.31 6.58 6.84 7.10 7.36 7.63 7.89

20.24 21.20 22.16 23.13 24.09 25.05 26.02 26.98 27.95 28.91

5.61 5.88 6.15 6.41 6.68 6.95 7.22 7.48 7.75 8.02

20.21 21.17 22.14 23.10 24.06 25.02 25.99 26.95 27.91 28.87

5.70 5.97 6.24 ' 6.51 6.79 7.06 7.33 7.60 7.87 8.14

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

29.94 30.91 31.88 32.84 33.81 34.77 35.74 36.71 37.67 38.64

8.02 8.28 8.54 8.80 9.06 9.32 9.58 9.84 10.09 10.35

29.91 30.87 31.84 32.80 33.77 34.73 35.70 36.66 37.63 38.59

8.15 8.42 8.68 8.94 9.21 9.47 9.73 10.00 10.26 10.52

29.87 30.84 31.80 32.76 33.73 34.69 35.65 36.62 37.58 38.55

8.28 8.55 8.82 9.09 9.35 9.62 9.89 10.16 10.42 10.69

29.84 30.80 31.76 32.72 33.69 34.65 35.61 36.57 37.54 38.50

8.41 8.69 8.96 9.23 9.50 9.77 10.04 10.31 10.59 10.86

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

39.60 40.57 41.53 42.50 43.47 44.43 45.40 46.36 47.33 48.30

10.61 10.87 11.13 11.39 11.65 11.91 12.16 12.42 12.68 12.94

39.56 40.52 41.49 42.45 43.42 44.38 45.35 46.31 47.27 48.24

10.78 11.05 11.31 11.57 11.84 12.10 12.36 12.63 12.89 13.15

39.51 40.47 41.44 42.40 43.36 44.33 45.29 46.25 47.22 48.18

10.96 11.22 11.49 11.76 12.03 12.29 12.56 12.83 13.09 13.36

39.46 40.42 41.39 42.35 43.31 44.27 45.24 46.20 47.16 48.12

11.13 11.40 11.67 11.94 12.21 12.49 12.76 13.03 13.30 13.57

41 42 43 44 45 46 47 48 49 50

V

Dep.

Lat.

Dep.

Lat. .

Dep.

Lat.

Dep.

Lat.

o

CD

1

.

2 3 4 5 6 7 8 9

o

15 Deg.

15* Deg.

15* Deg.

15f Deg.

»

1



j3 OJ

'S Q

75 Deg.

74| Deg.

74* Deg.

74i Deg.

s

TRAVERSE TABLE. 154 I>eg-

15i Deg.

j

15J Deg.

g

Distance.

15 Deg.

33

ST P

Dep.

o

49.09 50.05 51.01 51.97 52.94 53.90 54.86 55.82 56.78 57.75 i

13.84 14.11 14.39 14.66 14.93 15.20 15.47 15.74 16.01 16.29

51 52 53 54 55 56 57 58 59 60

16.30 16.57 16.84 17.10 17.37 17.64 17.90 18.17 18.44 18.71

58.71 59.67 60.63 61.60 62.56 63.52 64.48 65.45 66.41 67.37

16.56 16.83 17.10 17.37 17.64 17.92 18.19 18.46 18.73 19.00

61 62 63 64 65 66 67 68 69 70

68.42 69.38 70.35 71.31 72.27 73.24 74.20 75.16 76.13 77.09

18.97 19.24 19.51 19.78 20.04 20.31 20.58 20.84 21.11 21.38

68.33 69.30 70.26 71.22 72.18 73.15 74.11 75.07 76.03 77.00

19.27 19.54 19.82 20.09 20.36 20.63 20.90 21.17 21.44 21.72

71 72 73 74 75 76 77 78 79 80

21.31 21.57 21.83 22.09 22.36 22.62 22.88 23.15 23.41 23.67

78.05 79.02 79.98 80.94 81.91 82.87 83.84 84.80 85.76 86.73

21.65 21.91 22.18 22.45 22.72 22.98 23.25 23.52 23.78 24.05

77.96 78.92 79.88 80.85 81.81 82.77 83.73 84.70 85.66 86.62

21.99 22.26 22.53 22.80 23.07 23.34 23.62 23.89 24.16 24.43

81 82 83 84 85 86 87 88 89 90

87.80 88.76 89.73 90.69 91.65 92.62 93.58 94.55 95.51 96.48

23.94 24.20 24.46 24.72 24.99 25.25 25.^1 25.78 26.04 26.30

87.69 88.65 89.62 90.58 91.54 92.51 93.47 94.44 95.40 96.36

24.32 24.59 24.85 25.12 25.39 25.65 25.92 26.19 26.46 26.72

87.58 88.55 89.51 90.47 91.43 92.40 93.36 94.32 95.28 96.25

24.70 91 24.97 92 25.24 93 25.52 94 25.79 95 26.06 96 26.33 97 26.60 98 26.87 99 27.14 100

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

49.26 50.23 51.19 52.16 53.13 54.09 55.06 56.02 56.99 57.96

13.20 13.46 13.72 13.98 14.24 14.49 14.75 15.01 15.27 15.53

49.20 50.17 51.13 52.10 53.06 54.03 54.99 55.96 56.92 57.89

13.41 13.68 13.94 14.20 14.47 14.73 14.99 15.26 15.52 15.78

49.15 50.11 51.07 52.04 53.00 53.96 54.93 55.89 56.85 57.82

13.63 13.90 14.16 14.43 14.70 14.97 15.23 15.50 15.77 16.03

61 62 63 64 65 66 67 68 69 70

58.92 59.89 60.85 61.82 62.79 63.75 64.72 65.68 66.65 67.61

15.79 16.05 16.31 16.56 16.82 17.08 17.34 17.60 17.86 18.12

58.85 59.82 60.78 61.75 62.71 63.68 64.64 65.61 66.57 67.54

16.04 16.31 16.57 16.83 17.10 17.36 17.62 17.89 18.15 18.41

58.78 59.75 60.71 61.67 62.64 63.60 64.56 65.53 66.49 67.45

71 72 73 74 75 76 77 78 79 80

68.58 69.55 70.51 71.48 72.44 73.41 74.38 75.34 76.31 77.27

18.38 18.63 18.89 19.15 19.41 19.67 19.93 20.19 20.45 20.71

68.50 69.46 70.43 71.39 72.36 73.32 74.29 75.25 76.22 77.18

18.68 18.94 19.20 19.46 19.73 19.99 20.25 20.52 20.78 21.04

81 82 83 84 85 86 87 88 89 90

78.24 79.21 80.17 81.14 82.10 83.07 84.04 85.00 85.97 86.93

20.96 21.22 21.48 21.74 22.00 22.26 22.52 22.78 23.03 23.29

78.15 79.11 80.08 81.04 82.01 82.97 83.94 84.90 85.87 86.83

91 92 93 94 95 96 97 98 99 100

87.90 88.87 89.83 90.80 91.76 92.73 93.69 94.66 95.63 96.59

23.55 23.81 24.07 24.33 24.59 24.85 25.11 25.36 25.62 25.88

6 o 0

Dep.

Lat.

iS 1

75 -Deg.

Lat.

?

©

o

rt 74J Deg.

587

2 li

74$ Deg.

74i Deg.


5

34

TRAVERSE TABLE,

o o

16 Deg.

16* Deg.

16* Deg.

16f Deg.

O ST s o

(D

Lat.

Dep.

Lat.

1 2 3 4 6 7 8 9 10

0.96 1.92 2.88 3.85 4.81 5.77 6.73 7.69 8.65 9.61

0.28 0.55 0.83 1.10 1.38 1.65 1.93 2.21 2.48 2.76

0.96 1.92 2.88 3.84 4.80 5.76 6.72 7.68 8.64 9.60

0.28 0.56 0.84 1.12 1.40 1.68 1.96 2.24 2.52 2.80

0.96 1.92 2.88 3.84 4.79 5.75 6.71 7.67 8.63 9.59

0.28 0.57 0.85 1.14 1.42 1.70 1.99 2.27 2.56 2.84

0.96 1.92 2.87 3.83 4.79 5.75 6.70 7.66 8.62 9.58

0.29 0.58 0.86 1.15 1.44 1.73 2.02 2.31 2.59 2.88

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10.57 11.54 12.50 13.46 14.42 15.38 16.34 17.30 18.26 19.23

3.03 3.31 3.58 3.86 4.13 4.41 4.69 4.96 5.24 5.51

10.56 11.52 12.48 13.44 14.40 15.36 16.32 17.28 18.24 19.20

3.08 3.36 3.64 3.92 4.20 4.48 4.76 5.04 5.32 5.60

10.55 11.51 12.46 13.42 14.38 15.34 16.30 17.26 18.22 19.18

3.12 3.41 3.69 3.98 4.26 4.54 4.83 5.11 5.40 5.68

10.53 11.49 12.45 13.41 14.36 15.32 16.28 17.24 18.19 19.15

3.17 3.46 3.75 4.03 4.32 4.61 4.90 5.19 5.48 5.76

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

20.19 21.15 22.11 23.07 24.03 24.99 25.95 26.92 27.88 28.84

5.79 6.06 6.34 6.62 6.89 7.17 7.44 7.72 7.99 8.27

20.16 21.12 22.08 23.04 24.00 24.96 25.92 26.88 27.84 28.80

5.88 6.16 6.44 6.72 7.00 7.28 7.56 7.84 8.11 8.39

20.14 21.09 22.05 23.01 23.97 24.93 25.89 26.85 27.81 28.76

5.96 6.25 6.53 6.82 7.10 7.38 7.67 7.95 8.24 8.52

20.11 21.07 22.02 22.98 23.94 24.90 25.85 26.81 27.77 28.73

6.05 6.34 6.63 6.92 7.20 7.49 7.78 8.07 8.36 8.65

21

31 29.80 32 30.76 33 31.72 34 32.68 35 33.64 36 34.61 37 35.57 38 jj 36.53 39 | 37.49 40 ;i 38.45

8.54 8.82 9.10 9.37 9.65 9.92 10.20 10.47 10.75 11.03

29.76 30.72 31.68 32.64 33.60 34.56 ‘35.52 36.48 37.44 38.40

8.67 8.95 9.23 9.51 9.79 10.07 10.35 10.63 10.91 11.19

29.72 30.68 31.64 32.60 33.56 34.52 35.48 36.44 37.39 38.35

8.80 9.09 9.37 9.66 9.94 10.22 10.51 10.79 11.08 11.36

29.68 30.64 31.60 32.56 33.51 34.47 35.43 36.39 37.35 38.30

8.93 9.22 9.51 9.80 10.09 10.38 10.66 10.95 11.24 11.53

31 32 33 34 35 36 37 38 39 ' 40

41 42 43 44 45 46 47 48 49 50

39.41 40.37 41.33 42.30 43.26 44.22 45.18 4C. 14 47.10 48.06

11.30 11.58 11.85 12.13 12.40 12.68 12.95 13.23 13.51 13.78

39.36 40.32 41.28 42.24 43.20 44.16 45.12 46.08 47.04 48.00

11.47 11.75 12.03 12.31 12.59 12.87 13.15 13.43 13.71 13.99

39.31 40.27 41.23 42.19 43.15 44.11 45.06 46.02 46.98 47.94

11.64 11.93 12.21 12.50 12.78 13.06 13.35 13.63 13.92 14.20

39.26 40.22 41.18 42.13 43.09 44.05 45.01 45.96 46.92 47.88

11.82 12.10 12.39 12.68 12.97 13.26 13.55 13.83 14.12 14.41

41 42 43 44 45 46 47 48 49 50

«

Dep.

Lst.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

og• 3Cfl

5

o

0 3n

s

74 Deg.

Dep.

73$ Deg.

Lat.

j

Dep.

73* Deg.

Lat.

Dep.

73* Deg.

a

22

23 24 25 26 27 28 29 30

1

s

35

TRAVERSE TABLE, g CJO* p a

16 Deg.

16$ Deg.

161 Deg.

16|

2 w' p o

©

Lat.

Dep,

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

49.02 49.99 50.95 51.91 52-87 53.83 54.79 55.75 56.71 57.68

14.06 14.33 14.61 14.88 15.16 15.44 15.71 15.99 16.26 16.54

48.96 49.92 50.88 51.84 52.80 53.76 54.72 55.68 56.64 57.60

14.27 14.55 14.83 15.11 15.39 15.67 15.95 16.23 16.51 16.79

48.90 49.86 50.82 51.78 52.74 53.69 54.65 55.61 56.57 57.53

14.48 14.77 15.05 15.34 15.62 15.90 16.19 16.47 16.76 17.04

48.84 49.79 50.75 51.71 52.67 53.62 54.58 55.54 56.50 57.45

14.70 14.99 15.27 15.56 15.85 16.14 16.43 16.72 17.00 17.29

61 62 63 64 65 ' 66 67 68 69 70

58.64 59.60 60.56 61.52 62.48 63.44 64.40 65.37 66.33 67.29

16.81 17.09 17.37 17.64 17.92 18.19 18.47 18.74 19.02 19.29

58.56 59.52 60.48 61.44 62.40 63.36 64.32 65.28 66.24 67.20

17.07 17.35 17.63 17.91 18.19 18.47 18.75 19.03 19.31 19.59

58.49 59.45 60.41 61.36 62.32 63.28 64.24 65.20 66.16 67.12

17.32 17.61 17.89 18.18 18.46 18.74 19.03 19.31 19.60 19.88

58.41 59.37 60 33 61.28 62.24 63.20 64.16 65.11 66.07 67.03

17.58 61 17.87 62 18.16 63 18.44 64 18.73 65 19.02 66 19.31 i 67 19.60 68 19.89 69 20.17 70

71 72 73 ■ 74 75 76 77 78 79 80

68.25 69.21 70.17 71.13 72.09 73.06 74.02 74.98 75.94 76.90

19.57 19.85 20.12 20.40 20.67 20.95 21.22 21.50 21.78 22.05

68.16 69.12 70.08 71.04 72.00 72.96 73.92 74.88 75.84 76.80

19.87 20.15 20.43 20.71 20.99 21.27 21.55 21.83 22.11 22.39

68.08 69.03 69.99 70.95 71.91 72.87 73.83 74.79 75.75 76.71

20.17 20.45 20.73 21.02 21.30 21.59 21.87 22.15 22.44 22.72

67.99 68.95 69.90 70.86 71.82 72.78 73.73 74.69 75.65 76.61

20.46 20.75 21.04 21.33 21.61 21.90 22.19 22.48 22.77. 23.06

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 ‘ 88 89 90

77.86 78.82 79.78 80.75 81.71 82.67 83.63 84.59 85.55 86.51

22.33 77.76 22.60 78.72 22.88 79.68 23.15 80.64 23.43 , 81.60 23.70 ) 82.56 23.98 83.52 24.26 84.48 24.53 85.44 24.81 86.40

22.67 22.95 23.23 23.51 23.79 24.07 24.35 24.62 24.90 25.18

77.66 78.62 79.58 80.54 81.50 82.46 83.42 84.38 85.33 86.29

23.01 23.29 23.57 23.86 24.14 24.43 24.71 24.99 25.28 25.56

77.56 78.52 79.48 80.44 81.39 82.35 83.31 84.27 85.22 86.18

23.34 23.63 23.92 24.21 24.50 24.78 25.07 25.36 25.65 25.94

81 82 83 84 85 86 87 88 89 90

91 , 92 93 94 95 96 97 98 99 100

87.47 88.44 89.40 90.36 91.32 92.28 93.24 94.20 95.16 96.13

25.08 25.36 25.63 25.91 26.19 26.46 26.74 27.01 27.29 27.56

87.36 88.32 89.28 90.24 91.20 92.16 93.12 94.08 95.04 96.00

25.46 25.74 26.02 26.30 26.58 26.86 27.14 27.42 27.70 27.98

87.25 88.21 89.17 90.13 91.09 92.05 93.01 93.96 94.92 95.88

25.85 26.13 26.41 26.70 26.98 27.27 27.55 27.83 28.12 28.40

87.14 88.10 89.05 90.01 90.97 91.93 92.88 93.84 94.80 95.76

26.23 91 26.51 92 26.80 93 i 27.09 94 27.38 95 27.67 96 27.95 97 28.24 98 28.53 99 28.82 100

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

6

o

a

.2

Q

74 Deg.

fl>

51 52 53 ■ 54 55 56 57 58 59 60

o o

a as

731 Deg.

|

73$ Deg.

73$ Deg. Q

TRAVERSE TABLE.

3d

gca*

17 Deg.

17* Deg.

1

17 h Deg.

. 17} Deg.

V

' oCD

gtn" 5>s o

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep. »

1 2 3 4 5 6 7 8 9 10

0.96 1.91 2.87 3.83 4.78 5.74 6.69 7.65 8.61 9,56

0.29 0.58 0.88 1.17 1.46 1.75 2.05 2.34 2.63 2.92

0.95 1.91 2.87 3.82 4.78 5.73 6.69 7.64 8.60 9.55

0.30 0.59 0.89 1.19 1.48 1.78 2.08 2.37 2.67 2.97

0.95 1.91 2.86 3.81 4.77 5.72 6.68 7.63 8.58 9.54

0.30 0.60 0.90 1.20 1.50 1.80 2.10 2.41 2.71 3.01

0.95 1.90 2.86 3.81 4.76 5.71 6.67 7.62 8.57 9.52

0.30 0.61 0.91 1.22 1.52 1.83 2.13 2.44 2.74 3.05

1 2 3 4 5 6 7 8 9 10

11 13 14 15 16 17 18 19 20

10.52 11.48 12.43 13.39 14.34 15.30 16.26 17.21 18.17 19.13

3.22 3.51 3.80 4.09 4.39 4.68 4.97 5.26 5.56 5.85

10.51 11.46 12.42 13.37 14.33 15.28 16.24 17.19 18.15 19.10

3.26 3.56 3.85 4.15 4.45 4.74 5.04 5.34 5.63 5.93

1Q.49 11.44 12.40 13.35 14.31 15.26 16.21 17.17 18.12 19.07

3.31 3.61 3.91 4.21 4.51 4.81 5.11 5.41 5.71 6.01

10.48 11.43 12.38 13.33 14.29 15.24 16.19 18.10 19.05

3.35 3.66 3.96 4.27 4.57 4.88 5.18 5.49 5.79 6.10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

20.08 21.04 21.99 22.95 23.91 24.86 25.82 26.78 27.73 28.69

6.14 6.43 6.72 7.02 7.31 7.60 7.89 8.19 8.48 8.77

20.06 21.01 21.97 22.92 23.88 24.83 25.79 26.74 27.70 28.65

6.23 6.52 6.82 7.12 7.41 7.71 8.01 8.30 8.60 8.90

20.03 20.98 21.94 22.89 23.84 24.80 25.75 26.70 27.66 28.61

6.31 6.62 6.92 7.22 7.52 7.82 8.12 8.42 8.72 9.02

20.00 20.95 21.91 22.86 23.81 24.76 25.71 26.67 27.62 28.57

6.40 6.71 7.01 7.32 7.62 7.93 8.23 8.54 8.84 9.15

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

29.65 30.60 31.56 32.51 33.47 34.43 35.38 36.34 37.30 38.25

9.06 9.36 9.65 9.94 10.23 10.53 10.82 11.11 11.40 11.69

29.61 30.56 31.52 32.47 33.43 34.38 35.34 36.29 37.25 38.20

9.19 9.49 9.79 10.08 10.38 10.68 10.97 11.27 11.57 11.86

29.57 30.52 31.47 32.43 33.38 34.33 35.29 36.24 37.19 38.15

9.32 9.62 9.92 10.22 10.52 10.83 11.13 11.43 11.73 12.03

29.52 30.48 31.43 32.38 33.33 34.29 35.24 36.19 37.14 38.10

9.45 9.76 10.06 10.37 10.67 10.98 11.28 11.58 11.89 12.19

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47, 48 49 50

39.21 11.99 40.16 12.28 41.12 12.57 42.08 12.86 43.03 13.16 43.99 13.45 44.95 13.74 45 90 ; 14.03 46.86 14.33 47.82 14.62

39.16 40.11 41.07 42.02 42.98 43.93 44.89 45.84 46.80 47.75

12.16 12.45 12.75 13.05 13.34 13.64 13.94 14.23 14.53 14.83

39.10 40.06 41.01 41.96 42.92 43.87 44.82 45.78 46.73 47.69

12.33 12.63 12.93 13.23 13.53 13.83 14.13 14.43 14.73 15.04

39.05 40.00 40.95 41.91 42.86 43.81 44.76 45.71 46.67 47.62

12.50 12.80 13.11 13.41 13.72 14.02 14.33 14.63 14.94 15.24

41 42 43 44 45 46 47 48 49 50

Dep.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

orjo>

12

V o a m

s

Lat.

73 Deg.

72} Deg.

|

72* Deg.

1~.\A

72* Deg.

CD

s

TRAVERSE TABLE ;

g 5’ p S3 O

17 Deg.

m Beg.

17* Deg. Lat.

37 17} Deg.

g w'

p S3 O

Dep.

Lat.

Dep.

15.34 15.64 15.94 16.24 16.54 16.84 17.14 17.44 17.74 18.04

48.57 49.52 50.48 51.43 52.38 53.33 54.29 55.24 56.19 57.14

15.55 15.85 16.16 16.46 16.77 17.07 17.38 17.68 17.99 18.29

51 52 53 54 55 56 57 58 59 60

Lat.

Dep.

Lat.

Dep.

51 52* 53 54 55 56 57 58 59 60

48.77 49.73 50.68 51.64 52.60 53.55 54.51 55.47 56.42 57.38

14.91 15.20 15.50 15.79 16.08 16.37 16.67 16.96 17.25 17.54

48.71 49.66 50.62 51.57 52.53 53.48 54.44 55.39 56.35 57.30

15.12 15.42 15.72 16.01 16.31 16.61 16.90 17.20 17.50 17.79

48.64 49.59 50.55 51.50 52.45 53.41 54.36 55.32 56.27 57.22

61 62 63 64 65 66 67 68 69 70

58.33 59.29 60.25 61.20 62.16 63.12 64.07 65.03 65.99 66.94

17.83 18.13 18.42 18.71 19.00 19.30 19.59 19.88 20.17 20.47

58.26 59.21 60.17 61.12 62.08 63.03 63.99 64.94 65.90 66.85

18.09 18.39 18.68 18.98 19.28 19.57 19.87 20.16 20.46 20.76

58.18 59.13 60.08 61.04 61.99 62.95 63.90 64.85 65.81 66.76

18.34 18.64 18.94 19.25 19.55 19.85 20.15 20.45 20.75 21.05

58.10 59.05 60.00 60.95 61.91 62.86 63.81 64.76 65.72 66.67

18.60 18.90 19.21 19.51 19.82 20.12 20.43 20.73 21.04 21.34

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

67.90 68.85 69.81 70.77 71.72 72.68 73.64 74.59 75.55 76.50

20.76 21.05 21.34 21.64 21.93 22.22 22.51 22.80 23.10 23.39

67.81 68.76 69.72 70.67 71.63 72.58 73.54 74.49 75.45 76.40

21.05 21.35 21.65 21.94 22.24 22.54 22.83 23.13 23.43 23.72

67.71 68.67 69.62 70.58 71.53 72.48 73.44 74.39 75.34 76.30

21.35 21.65 21.95 22.25 22.55 22.85 23.15 23.46 23.76 24.06

67.62 68.57 69.52 70.48 71.43 72.38 73.33 74.29 75.24 76.19

21.65 21.95 22.26 22.56 22.86 23.17 23.47 23.78 24.08 24.39

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

77.46 78.42 79.37 80.33 81.29 82.24 83.20 84.15 85.11 86.07

23.68 23.97 24.27 24.56 24.85 25.14 25.44 25.73 26.02 26.31

77.36 78.31 79.27 80.22 81.18 82.13 83.09 84.04 85.00 85.95

24.02 24.32 24.61 24.91 25.21 25.50 25.80 26.10 26.39 26.69

77.25 78.20 79.16 80.11 81.07 82.02 82.97 83.93 84.88 85.83

24.36 24.66 25.96 25.26 25.56 25.86 26.16 26.46 26.76 27.06

77.14 78.10 79.05 80.00 80.95 81.91 82.86 83.81 84.76 85.72

24.69 81 25.00 82 25.30 83 25.61 84 25.91 85 26.22 86 26.52 ‘ " 87 26.83 88 27.13 89 27.44 no

91 92 93 94 95 96 97 98 99 100

87.02 87.98 88.94 89.89 90.85 91.81 92.76 93.72 94.67 95.63

26.61 26.90 27.19 27.48 27.78 28.07 28.36 28.65 28.94 29.24

86.91 87.86 88.82 89.77 90.73 91.68 92.64 93.59 94.55 95.50

26.99 27.28 27.58 27.87 28.17 28.47 28.76 29.06 29.36 29.65

86.79 87.74 88.70 89.65 90.60 91.56 92.51 93.46 94.42 95.37

27.36 27.66 27.97 28.27 28.57 28.87 29.17 29.47 29.77 30.07

86.67 87.62 88.57 89.53 90.48 91.43 92.38 93.33 94.29 95.24

27.74 91 28.05 92 28.35 93 28.66 94 28.96 95 29.27 96 29.57 97 29.88 98 30.18 99 30.49 100

flj

Dep.

Lat.

Dep

Lat.

Dep.

Lat.

Dep.

Lat.

CD

O

4

rj

CD

p

o

rtG

«3 72J Deg.

73 Deg. 1 1

27*

72J Deg.

72* Deg.

Q

38 g

TRAVERSE TABLE, 18 Deg.

18$ Deg.

18| Deg.

18i Deg.

g W* ST P o

S' g o ?

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.95 1.90 2.85 3.80 4.76 5.71 6.66 7.61 8.56 9.51

0.31 0.62 0.93 1.24 1.55 1.85 2.16 2.47 2.78 3.09

0.95 1.90 2.85 3.80 4.75 5.70 6.65 7.60 8.55 9.50

0.31 0.63 0.94 1.25 1.57 1.88 2.19 2.51 2.82 3.13

0.95 1.90 2.84 3.79 4.74 5.69 6.64 7.59 8.53 9.48

0.32 0.63 0.95 1.27 1.59 1.90 2.22 2.54 2.86 3.17

0.95 1.89 2.84 3.79 4.73 5.68 6.63 7.58 8.52 9.47

0.32 0.64 0.96 1.29 1.61 1.93 2.25 2.57 2.89 3.21

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10.46 11.41 12.36 13.31 14.27 15.22 16.17 17.12 18.07 19.02

3.40 3.71 4.02 4.33 4.64 4.94 5.25 5.56 5.87 6.18

10.45 11.40 12.35 13.30 14.25 15.20 16.14 17.09 18.04 18.99

3.44 3.76 4.07 4.38 4.70 5.01 5.32 5.64 5.95 6.26

10.43 11.38 12.33 13.28 14.22 15.17 16.12 17.07 18.02 18.97

3.49 3.81 4.12 4.44 4.76 5.08 5.39 5.71 6.03 6.35

10.42 11.36 12.31 13.26 14.20 15.15 16.10 17.04 17.99 18.94

3.54 3.86 4.18 4.50 4.82 5.14 5.46 5.79 6.11 6.43

11 12 13 14 15 16 17 18 19 20

22 23 24 25 26 27 28 29 30

19.97 20.92 21.87 22.83 23.78 24.73 25.68 26.63 27.58 28.53

6.49 6.80 7.11 7.42 7.73 8.03 8.34 8.65 8.96 9.27

19.94 20.89 21.84 22.79 23.74 24.69 25.64 26.59 27.54 28.49

6.58 6.89 7.20 7.52 7.83 8.14 8.46 8.77 9.08 9.39

19.91 20.86 21.81 22.76 23.71 24.66 25.60 26.55 27.50 28.45

6.66 6.98 7.30 7.62 7.93 8.25 8.57 8.88 9.20 9.52

19.89 20.83 21.78 22.73 23.67 24.62 25.57 26.51 27.46 28.41

6.75 7.07 7.39 7.71 8.04 8.36 8.68 9.00 9.32 9.64

21 22 23 24 25 26 27 28 ! 29 I 30 J

31 32 33 34 35 36 37 38 39 40

29.48 30.43 31.38 32.34 33.29 34.24 35.19 36.14 37.09 38.04

9.58 9.89 10.20 10.51 10.82 11.12 11.43 11.74 12.05 12.36

29.44 30.39 31.34 32.29 33.24 34.19 35.14 36.09 37.04 37.99

9.71 10.02 10.33 10.65 10.96 11.27 11.59 11.90 12.21 12.53

29.40 30.35 31.29 32.24 33.19 34.14 35.09 36.04 36.98 37.93

9.84 10.15 10.47 10.79 11.42 11.74 12.06 12.37 12.69

29.35 30.30 31.25 32.20 33.14 34.09 35.04 35.98 36.93 37.88

9.96 10.29 10.61 10.93 11.25 11.57 11.89 12.21 12.54 12.86

31 fj 32 1 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

38.99 39.94 40.90 41.85 42.80 43.75 44.70 45.65 46.60 47.55

12.67 12.98 13.29 13.60 13.91 14.21 14.52 14.83 15.14 15.45

38.94 39.89 40.84 41.79 42.74 43.69 44.64 45.59 46.54 47.48

12.84 13.15 13.47 13.78 14.09 14.41 14.72 15.03 15.35 15.66

38.88 39.83 40.78 41.73 42.67 43.62 44.57 45.52 46.47 47.42

13.01 13.33 13.64 13.96 14.28 14.60 14.91 15.23 15.55 15.87

38.82 39.77 40.72 41.66 42.61 43.56 44.51 45.45 46.40 47.35

13.18 13.50 13.82 14.14 14.46 14.79 15.11 15.43 15.75 16.07

41 42 43 44 45 46 47 48 49 50


Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

21

o

Dep.

Lat.

Dep.

Lat.

11.11

Dep.

rt

s

72 Deg.

711

Deg.

714 Deg.

|

m Deg.

CD

o o

J

3

1

39

TRAVERSE TABLE.

Distance.

1 1 Distance.

Dep.

16.18 i 48.29 49.24 16.50 50.19 16.82 51.13 17.13 17.45 52.08 53.03 17.77 53.98 18.09 18.40 54.92 55.87 18.72 19.04 56.82

16.39 16.71 17.04 17.36 17.68 18.00 18.32 18.64 18.96 19.29

51 52 53 54 55 56 57 58 59 60

57.85 58.80 59.74 60.69 61.64 62.59 63.54 64.49 65.43 66.38

19.36 19.67 19.99 20.31 20.62 20.94 21.26 21.58 21.89 22.21

57.76 58.71 59.66 60.60 61.55 62.50 63.44 64.39 65.34 66.29

19.61 19.93 20.25 20.57 20.89 21.22 21.54 21.86 22.18 22.50

61 62 63 64 65 66 67 68 69 70

22.23 22.55 22.86 23.17 23.49 23.80 24.11 24.43 24.74 25.05

67.33 68.28 69.23 70.18 71.12 72.07 73.02 73.97 74.92 75.87

22.53 22.85 23.16 23.48 23.80 24.12 24.43 24.75 25.07 25.38

67.23 68.18 69.13 70.07 71.02 71.97 72.91 73.86 74.81 75.75

22.82 23.14 23.47 23.79 24.11 24.43 24.75 2-5.07 25.39 25.72

71 72 73 74 75 76 77 78 79 80

76.93 77.88 78.83 79.77 80.72 81.67 82.62 83.57 84.52 85.47

25.37 25.68 25.99 26.31 26.62 26.93 27.25 27.56 27.87 28.18

76.81 77.76 78.71 79.66 80.61 81.56 82.50 83.45 84.40 85.35

25.70 26.02 26.34 26.65 26.97' 27.29 27.61 27.92 28.24 28.56

76.70 77.65 78.60 79.54 80.49 81.44 82.38 83.33 84.28 85.22

26.04 26.36 26.68 27.00 27.32 27.64 27.97 28.29 28.61 28.93

81 82 83 84 85 86 87 88 89 90

28.12 28.43 28.74 29.05 29.36 29.67 29.97 30.28 30.59 30.90

86.42 87.37 88.32 89.27 90.22 91.17 92.12 93.07 94.02 94.97

28.50 28.81 29.12 29.44 29.75 30.06 30.38 30.69 31.00 31.32

86.30 87.25 88.19 89.14 90.09 91.04 91.99 92.94 93.88 94.83

28.87 29.19 29.51 29.83 30.14 30.46 30.78 31.10 31.41 31.73

86-17 87.12 88.06 89.01 89.96 90.91 91.85 92.80 93.75 94.69

29.25 91 29.57 92 29.89 93 30.22 94 30.54 95 30.86 96 31.18 97 31.50 98 31.82 99 32.14 100

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

18 Deg.

18} Deg.

18i Deg.

Lat.

Dep.

Lat.

Dep.

Lat.

51 52 53 54 55 56 57 58 59 60

48.50 49.45 50.41 51.36 52.31 53.26 54.21 55.16 56.11 57.06

15.76 16.07 16.38 16.69 17.00 17.30 17.61 17.92 18.23 18.54

48.43 49.38 50.33 51.28 52.23 53.18 54.13 55.08 56.03 56.98

15.97 16.28 16.60 16.91 17.22 17.54 17.85 18.16 18.48 18.79

48.36 49.31 50.26 51.21 52.16 53.11 54.05 55.00 55.95 56.90

61 62 63 64 65 66 67 68 69 70

58.01 58.97 59.92 60.87 61.82 62.77 63.72 64.67 65.62 66.57

18.85 19.16 19.47 19.78 20.09 20.40 20.70 21.01 21.32 21.63

57.93 58.88 59.83 60.78 61.73 62.68 63.63 64.58 65.53 66.48

19.10 19.42 19.73 20.04 20.36 20.67 20.98 21.30 21.61 21.92

71 72 73 74 75 76 77 78 79 80

67.53 68.48 69.43 70.38 71.33 72.28 73.23 74.18 75.13 76.08

21.94 22.25 22.56 22.87 23.18 23.49 23.79 24.10 24.41 24.72

67.43 68.38 69.33 70.28 71.23 72.18 73.13 74.08 75.03 75.98

81 82 83 84 85 86 87 88 89 90

77.04 77.99 78.94 79.89 80.84 81.79 82.74 83.69 84.64 85.60

25.03 25.34 25.65 25.96 26.27 26.58 26.83 27.19 27.50 27.81

91 92 93 94 95 96 97 98 99 100

86.55 87.50 88.45 89.40 90.35 91.30 92.25 93.20 94.15 95.11

6 o0

Dep.

Dep.

18} Deg. Lat.

rt CD

s

72 Deg.

711 Deg.

71 i Deg.

7H Deg.

V

o Q

1

TRAVERSE TABLE

40

g uo ’

19 Deg.

19* Deg.

19* Deg.

19| Deg. |

ST

o

re

Lat.

Dep.

Lat.

1 2 3 4 6 7 8 9 10

0.95 1.89 2.84 3.78 4.73 5.67 6.62 7.56 8.51 9.46

0.33 0.65 0.98 1.30 1.63 1.95 2.28 2.60 2.93 3.26

0.94 1.89 2.83 3.78 4.72 5.66 6.61 7.55 8.50 9.44

0.33 0.66 0.99 1.32 1.65 1.98 2.31 2.64 2.97 3.30

0.94 1.89 2.83 3.77 4.71 5.66 6.60 7.54 8.48 9.43

11 12 13 14 15 16 17 18 19 20

10.40 11.35 12.29 13.24 14.18 15.13 16.07 17.02 17.96 18.91

3.58 3.91 4.23 4.56 4.88 5.21 5.53 5.86 6.19 6.51

10.38 11.33 12.27 13.22 14.16 15.11 16.05 16.99 17.94 18.88

3.63 3.96 4.29 4.62 4.95 5.28 5.60 5.93 6.26 6.59

21 22 23 24 25 26 27 28 29 30

19.86 20.80 21.75 22.69 23.64 24.58 25.53 26.47 27.42 28.37

6.84 7.16 7.49 7.81 8.14 8.46 8.79 9.12 9.44 9.77

19.83 20.77 21.71 22.66 23.60 24.55 25.49 26.43 27.38 28.32

31 32 33 34 35 36 37 38 39 40

29.31 30.26 31.20 32.15 33.09 34.04 34.98 35.93 36.88 37.82

10.09 10.42 10.74 11.07 11.39 11.72 12.05 12.37 12.70 13.02

41 42 43 44 45 46 47 48 49 50

38.77 39.71 40.66 41.60 42.55 43 49 44.44 45.38 46.33 47.28

6 o

Dep.

5

js

Q

Dep.

Lat.

Dep.

Lat.

Dep.

g cm'

P 3 O O

0.33 0.67 1.00 1.34 1.67 2.00 2.34 2.67 3.00 3.34

0.94 1.88 2.82 3.76 4.71 5.65 6.59 7.53 8.47 9.41

0.34 0.68 1.01 1.35 1.69 2.03 2.37 2.70 3.04 3.38

9 10

10.37 11.31 12.25 13.20 14.14 15.08 16.02 16.97 17.91 18.85

3.67 4.01 4.34 4.67 5.01 5.34 5.67 6.01 6.34 6.68

10.35 11.29 12.24 13.18 14.12 15.06 16.00 16.94 17.88 18.82

3.72 4.06 4.39 4.73 5.07 5.41 5.74 6.08 6.42 6.76

11 12 13 14 15 16 17 18 19 20

6.92 7.25 7.58 7.91 8.24 8.57 8.90 9.23 9.56 9.89

19.80 20.74 21.68 22.62 23.57 24.51 25.45 26.39 27.34 28.28

7.01 7.34 7.68 8.01 8.35 8.68 9.01 9.35 9.68 10.01

19.76 20.71 21.65 22.59 23.53 24.47 25.41 26.35 27.29 28.24

7.10 7.43 7.77 8.11 8.45 8.79 9.12 9.46 9.80 10.14

21 22 23 24 25 26 27 28 29 30

29.27 30.21 31.15 32.10 33.04 33.99 34.93 35.88 36.82 37.76

10.22 10.55 10.88 11.21 11.54 11.87 12.20 12.53 12.86 13.19

29.22 30.16 31.11 32.05 32.99 33.94 34.88 35.82 36.76 37.71

10.35 10.68 11.02 11.35 11.68 12.02 12.35 12.68 13.02 13.35

29.18 30.12 31.06 32.00 32.94 33.88 34.82 35.76 36.71 37.65

10.48 10.81 11.15 11.49 11.83 12.17 12.50 12.84 13.18 13.52

31 32 33 34 35 36 37 38 39 40

13.35 13.67 14.00 14.32 14.65 14.98 15.30 15.63 15.95 16.28

38.71 39.65 40.60 41.54 42.48 43.43 44.37 45.32 46.26 47.20

13.52 13.85 14.18 14.51 14.84 15.17 15.50 15.83 16.15 16.48

38.65 39.59 40.53 41.48 42.42 43.36 44.30 45.25 46.19 47.13

13.69 14.02 14.35 14.69 15.02 15.36 15.69 16.02 16.36 16.69

38.59 39.53 40.47 41.41 42.35 43.29 44.24 45.18 46.12 47.06

13.85 14.19 14.53 14.87 15.21 15.54 15.88 16.22 16.56 16.90

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o u e:

-

1

71 Deg.

70} Deg.

70* Deg.

i|

70* Deg.

1 2 3 4 5 6 7 8

5

41

TRAVERSE TABLE, 1

g p «o

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

48.22 49.17 50.11 51.06 52.00 52.95 53.89 54.84 55.79 56.73

16.60 16.93 17.20 17.58 17.91 18.23 18.56 18.88 19.21 19.53

43.15 49.09 50.04 50.98 51.92 52.87 53.81 54.76 55.70 56.65

16.81 17.14 17.47 17.80 18.13 18.46 18.79 19.12 19.45 19.78

48.07 49.02 49.96 50.90 51.85 52.79 53.73 54.67 55.62 56.56

61 62 63 64 65

69 70

57.68 58.62 59.57 60.51 61.46 62.40 63.35 64.30 6524 66 19

19.86 20.19 20.51 20.84 21.16 21.49 21.81 22.14 22.46 22.79

57.59 58.53 59.48 60.42 61.37 62.31 63.25 64.20 65.14 66.09

20.11 20.44 20.77 21.10 21.43 21.76 22.09 22.42 22.75 23.08

71 72 73 74 75 76 , 77 78 79 80

67.13 68.08 69.02 69.97 70.91 71.86 72.80 73.75 74.70 75.64

23.12 23.44 23.77 24.09 24.42 24.74 25.07 25.39 25.72 26.05

67.03 67.97 68.92 69.86 70.81 71.75 72.69 73.64 74.58 75.53

81 82 ' 83 84 85

76.59 77.53 78.48 79.42 80.37 81.31 82.26 83.21 84.15 85.10

26.37 26.70 27.02 27.35 27.67 28.00 28.32 28.65 28.98 29.30

91 92 93 94 95 96 97 98 99 .100

86.04 86.99 87.93 89.82 90.77 91.72 92.66 93.61 94.55

29.63 29.95 30.28 30.60 30.93 31.25 31.58 31.91 32.23 32.56

d

Dep.

Lat.

66

67 68

86

' 87 88

89 90

o

19 Deg.

i9*:Deg.

19* Deg.

j

191 Deg.

P

88.88

Lat.

Dep.

a n

17.02 17.36 17.69 18.03 18.36 18.69 19.03 19.36 19.69 20.03

48.00 48.94 49.88 50.82 51.76 52.71 53.65 54.59 55.53 56.47

17.23 17.57 17.91 18.95 18.6b 18.92 19.26 19.60 19.94 20.27

51 52 53 54 55 56 57 58 59 , 60 ,

57.50 58.44 59.39 60.33 61.27 62.21 63.16 64.10 65.04 65.98

20.36 20.70 21.03 21.36 21.70 22.03 22.37 22.70 23.03 23.37

57.41 58.35 59.29 60.24 61.18 62.12 63.06 64.00 64.94 65.88

20.61 20.95 21.29 21.63 21.96 22.30 22.64 22.98 23.32 23.65

61 ’ 62 63 64 65

23.41 23.74 24.07 24.40 24.73 25.06 25.39 25.72 26.05 26.38

66.93 67.87 68.81 69.76 70.70 71.64 72.58 73.53 74.47 75.41

23.70 24.03 24.37 24.70 25.04 25.37 25.70 26.04 26.37 26.70

66.82 67.76 68.71 69.65 70.59 71.53 72.47 73.41 74.35 75.29

23.99 24.33 24.67 25.01 25.34 25.68 26.02 26.36 26.70 27.03

71 72 73 ' 74 75 : 76 77 78 . 79

76.47 77.42 78.36 79.30 80.25 81.19 82.14 83.08 84.02 84.97

26.70 27.03 27.36 27.69 28.02 28.35 28.68 29.01 29.34 29.67

76.35 77.30 78.24 79.18 80.12 81.07 82.01 82.95 83.90 84.84

27.04 27.37 27.71 28.04 28.37 28.71 29.04 29.37 29.71 30.04

76.24 77.18 78.12 79.06 80.00 80.94 81.88 82.82 83.76 84.71

27.37 27.71 28.05 28.39 28.72 29.06 29.40 29.74 30.07 30.41

81 82 83 84 85

85.91 87.80 88.74 89.60 90.63 91.58 92.52 93.46 94.41

30.00 30.33 30.66 30.99 31.32 31.65 31.98 32.31 32.64 32.97

85.78 86.72 87.67 88.61 89.55 90.49 91.44 92.38 93.32 94.26

30.38 30.71 31.04 31.38 31.71 32.05 32.38 32.71 33.05 33.38

85.65 86.59 87.53 88.47 89.41 90.35 91.29 92.24 93.18 94.12

30.75 91 31.09 92 31.43 93 31.76 94 32.10 95 32.44 96 32.78 97 33.12 98 33.45 99 33.79 100

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

86.86

j

CJ

71 Deg.

66

67 68

69 70 ,

so

86

87 88

89 90 '

cd

rtC2

al

5

g

70f Deg.

70* Deg.

2S

70* Deg.

n

a

TRAVERSE TABLE

42

g

i 20 Deg;.

20} Deg.

|

20* Deg.

g

20} Deg.

s» a ?

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.94 1.88 2.82 3.76 4.70 5.64 6.58 7.52 8.46 9.40

0.34 0.68 1.03 1.37 1.71 2.05 2.39 2.74 3.08 3.42

0.94 1.88 2.81 3.75 4.69 5.63 6.57 7.51 8.44 9.38

0.35 0.69 1.04 1.38 1.73 2.08 2.42 2.77 3.12 3.46

0.94 1.87 2.81 3.75 4.68 5.62 6.56 7.49 8.43 9.37

0.35 0.70 1.05 1.40 1.75 2.10 2.45 2.80 3.15 3.50

0.94 1.87 2.81 3.74 4.68 5.61 6.55 7.48 8.42 9.35

0.35 0.71 1.06 1.42 1.77 2.13 2.48 2.83 3.19 3.54

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

10.34 11.28 12.22 13.16 14.10 15.04 15.97 16.91 17.85 18.79

3.76 4.10 4.45 4.79 5.13 5.47 5.81 6.16 6.50 6.84

10.32 11.26 12.20 13.13 14.07 15.01 15.95 16.89 17.83 18.76

3.81 4.15 4.50 4.85 5.19 5.54 5.88 6.23 6.58 6.92

10.30 11.24 12.18 13.11 14.05 14.99 15.92 16.86 17.80 18.73

3.85 4.20 4.55 4.90 5.25 5.60 5.95 6.30 6.65 7.00

10.29 11.22 12.16 13.09 14.03 14 % 15.90 16.83 17.77 18.70

3.90 4.25 4.61 4.96 5.31 5.67 6.02 6.38 6.73 7.09

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

19.73 20.67 21.61 22.55 23.49 24.43 25.37 26.31 27.25 28.19

7.18 7.52 7.87 8.21 8.55 8.89 9.23 9.58 9.92 10.26

19.70 20.64 21.58 22.52 23.45 24.39 25.33 26.27 27.21 28.15

7.27 7.61 7.96 8.31 8.65 9.00 9.35 9.69 10.04 10.38

19.67 20.61 21.54 22.48 23.42 24.35 25.29 26.23 27.16 28.10

7.35 7.70 8.05 8.40 8.76 9.11 9.46 9.81 10.16 10.51

19.64 20.57 21.51 22.44 23.38 24.31 25.25 26.18 27.12 28.05

7.44 7.79 8.15 8.50 8.86 9.21 9.57 9.92 10.27 10.63

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

29.13 30.07 31.01 31.95 32.89 33.83 34.77 35.71 36.65 37.59

10.60 10.94 11.29 11.63 11.97 12.31 12.65 13.00 13.34 13.68

29.08 30.02 30.96 31.90 32.84 33.77 34.71 35.65 36.59 37.53

10.73 11.08 11.42 11.77 12.11 12.46 12.81 13.15 13.50 13.84

29.04 29.97 30.91 31.85 32.78 33.72 34.66 35.59 36.53 37.47

10.86 11.21 11.56 11.91 12.26 12.61 12.96 13.31 13.66 14.01

28.99 29.92 30.86 31.79 32.73 33.66 34.60 35.54 36.47 37.41

10.98 11.34 11.69 12.05 12.40 12.75 13.11 13.46 13.82 14.17

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

38.53 39.47 40.41 41.35 42.29 43.23 44.17 45.11 46.04 46.98

14.02 14.36 14.71 15.05 15.39 15.73 16.07 16.42 16.76 17.10

38.47 39.40 40.34 41.28 42.22 43.16 44.09 45.03 45.97 46.91

14.19 14.54 14.88 15.23 15.58 15.92 16.27 16.61 16.96 17.31

38.40 39.34 40.28 41.21 42.15 43.09 44.02 44.96 45.90 46.83

14.36 14.71 15.06 15.41 15.76 16.11 16.46 16.81 17.16 17.51

38.34 39.28 40.21 41.15 42.08 43.02 43.95 44.89 45.82 46.76 |

14.53 14.88 15.23 15.59 15.94 16.30 16.65 17.01 17.36 17.71

41 42 43 44 45 46 47 48 49 50

6

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep. 1 Lat.

o

is S

70 Deg;.

Dep. 1

69} Deg.

Lat.

Dep.

69} Deg.

Lat.

J

ST o a

Dep.

69} Deg.


rtc

j

s

TRAVERSE TABLE, 20| Deg.

20$ Deg.

20f Deg.

Distance.

Distance.

20 Deg.

43

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 99 60

47.92 48.86 49.80 50.74 51.68 52.62 53.56 54.50 55.44 56.38

17.44 17.79 18.13 18.47 18.81 19.15 19.50 19.84 20.18 20.52

47.85 48.79 49.72 50.66 51.60 52.54 53.43 54.42 55.35 56.29

17.65 18.00 18.34 18.69 19.04 19.38 19.73 20.07 20.42 20.77

47.77 48.71 49.64 50.58 51.52 52.45 53.39 54.33 55.26 56.20

17.86 18.21 18.56 18.91 19.26 19.61 19.96 20.31 20.66 21.01

47.69 48.63 49.56 50.50 51.43 52.37 53.30 54.24 55.17 56.11

18.07 18.42 18.78 19.13 19.49 19.84 20.19 20.55 20.90 21.26

51

61 62 63 64 65 66 67 68 69 70

57.32 58.26 59.20 60.14 61.03 62.02 62.96 63.90 64.84 65.78

20.86 21.21 21.55 21.89 22.23 22.57 22.92 23.26 23.60 23.94

57.23 58.17 59.11 60.04 60.98 61.92 62.86 63.80 64.74 65.67

21.11 21.46 21.81 22.15 22.50 22.84 23.19 23.54 23.88 24.23

57.14 58.07 59.01 59.95 60.88 61.82 62.76 63.69 64.63 65.57

21.36 21.71 22.06 22.41 22.76 23.11 23.46 23.81 24.16 24.51

57.04 57.98 58.91 59.85 60.78 61.72 62.65 63.59 64.52 65.46

21.61 21.97 22.32 22.67 23.03 23.38 23.74 24.09 24.45 24.80

61 ’ 62 63 64 65 66 67 68 ; 69 70

71 72 73 74 75 76 77 78 79 80

66.72 67.66 68.60 69.54 70.48 71.42 72.36 73.30 74.24 75.18

24.23 24.63 24.97 25.31 25.65 25.99 26.34 26.68 27.02 27.36

66.61 67.55 68.49 69.43 70.36 71.30 72.24 73.18 74.12 75.06

24.57 24.92 25.27 25.61 25.96 26.30 26.65 27.00 27.34 27.69

66.50 67.44 68.38 69.31 70.25 71.19 72.12 73.06 74.00 74.93

24.86 25.21 25.57 25.92 26.27 26.62 26.97 27.32 27.67 28.02

66.39 67.33 68.26 69.20 70.14 71.07 72.01 72.94 73.88 74.81

25.15 25.51 25.86 26.22 26.57 26.93 27.28 27.63 27.99 28.34

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

76.12 77.05 77.99 78.93 79.87 80.81 81.75 82.69 83.63 84.57

27.70 28.05 28.39 28.73 29.07 29.41 29.76 30.10 30.44 30.78

75.99 76.93 77.87 73.81 79.75 80.68 81.62 82.56 83.50 84.44

23.04 28.38 28.73 29.07 29.42 29.77 30.11 30.46 30.80 31.15

75.87 76.81 77.74 78.68 79.62 80.55 81.49 82.43 83.36 84.30

28.37 28.72 29.07 29.42 29.77 30.12 30.47 30.82 31.17 31.52

75.75 28.70 76.68 29.05 77.62 . 29.41 78.55 29.76 79.49 30.11 80.42 30.47 81.36 30.82 82.29 31.18 83.23 31.53 84.16 31.89

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

85.51 86.45 87.39 88.33 89.27 90.21 91.15 92.09 93.03 93.97

31.12 31.47 31.81 32.15 32.49 32.83 33.18 33.52 33.86 34.20

85.38 86.31 87.25 88.19 89.13 90.07 91.00 91.94 92.88 93.82

31.50 31.84 32.19 32.54 32.88 33.23 33.57 33.92 34.27 34.61

85.24 86.17 87.11 88.05 88.98 89.92 90.86 91.79 92.73 93.67

31.87 32.22 32.57 32.92 33.27 33.62 33.97 34.32 34.67 35.02

85.10 86.03 86.97 87.90 88.84 89.77 90.71 91.64 92.58 93.51

32.24 91 32.59 92 32.95 93 33.30 94 33.66 95 34.01 96 34.37 97 34.72 98 35.07 99 35.43 100

o

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

rta «

s

70 Deg. »! !i1

69J Deg.

69i Deg.

691 D eg.

52:

53 54 55 56 57 : 58 59 60 ,

0 0

a «n

3

:

44

21i Deg. Lat.

2U Deg. .

Dep.

1 2 3 4 5 6 7 8 9 10

0.93 1.87 2.80 3.73 4.67 5.60 6.54 7.47 8.40 9.34

0.36 0.72 1.08 1.43 1.79 2.15 2.51 2.87 3.23 3.58

0.93 1.86 2.80 3.73 4.66 5.59 6.52 7.46 8.39 9.32

0.36 0.72 1.09 1.45 1.81 2.17 2.54 2.90 3.26 3.62

0.93 1.86 2.79 3.72 4.65 5.58 6.51 7.44 8.37 9.30

0.37 0.73 1.10 1.47 1.83 2.20 2.57 2.93 3.30 3.67

0.93 1.86 2.79 3.72 4.64 5.57 6.50 7.43 8.36 9.29

0.37 0.74 1.11 1.48 1.85 2.22 2.59 2.96 3.34 3.71

11 12 13 14 15 16 17 18 19 20

10.27 11.20 12.14 13.07 14.00 14.94 15.87 16.80 17.74 18.67

3.94 4.30 4.66 5.02 5.38 5.73 6.09 6.45 6.81 7.17

10.25 11.18 12.12 13.05 13.98 14.91 15.84 16.78 17.71 18.64

3.99 4.35 4.71 5.07 5.44 5.80 6.16 6.52 6.89 7.25

10.23 11.17 12.10 13.03 13.96 14.89 15.82 16.75 17.68 18.61

4.03 4.40 4.76 5.13 5.50 5.86 6.23 6.60 6.96 7.33

10.22 11.15 12.07 13.00 13.93 14.86 15.79 16.72 17.65 18.58

4.08 4 45 4.82 5.19 5.56 5.93 ,6.30 6.67 7.04 7.41

11 12 13 14 15 16 17 18 19 , 20

21 22 23 24 25 26 27 28 29 30

19.61 20.54 21.47 22.41 23.34 24.27 25.21 26.14 27.07 28.01

7.53 7.88 8.24 8.60 8.96 9.32 9.68 10.03 10.39 10.75

19.57 20.50 21.44 22.37 23.30 24.23 25.16 26.10 27.03 27.96

7.61 7.97 8.34 8.70 9.06 9.42 9.79 10.15 10.51 10.87

19.54 20.47 21.40 22.33 23.26 24.19 25.12 26.05 26.98 27.91

7.70 8.06 8.43 8.80 9.16 9.53 9.90 10.26 10.63 11.00

19.50 20.43 21.36 22.29 23.22 24.15 25.08 26.01 26.94 27.86

7.78 8.15 8.52 8.89 9.26 9.63 10.01 10.38 10.75 11.12

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

28.94 29.87 30.81 31.74 32.68 33.61 34.54 35.48 36.41 37.34

11.11 11.47 11.83 12.18 12.54 12.90 13.26 13.62 13.98 14.33

28.89 29.82 30.76 31.69 32.62 33.55 34.48 35.42 36.35 37.28

11.24 11.60 11.96 12.32 12.69 13.05 13.41 13.77 14.14 14.50

28.84 29.77 30.70 31.63 32.56 33.50 34.43 35.36 36.29 37.22

11.36 11.73 12.09 12.46 12.83 13.19 13.56 13.93 14.29 14.66

28.79 29.72 30.65 31.58 32.51 33.44 34.37 35.29 36.22 37.15

11.49 11.86 12.23 12.60 12.97 13.34 13.71 14.08 14.45 14.82

31 32 33 34 35 36 37 38 39 , 40

41 42 43 44 45 46 47 48 49 50

38.28 39.21 40.14 41.08 42.01 42.94 43.88 44.81 45.75 46.68

14.69 15.05 15.41 15.77 16.13 16.48 16.84 17.20 17.56 17.92

38.21 39.14 40.08 41.01 41.94 42.87 43.80 44.74 45.67 46.60'

14.86 15.22 15.58 15.95 16.31 16.67 17.03 17.40 17.76 18.12

38.15 39.08 40.01 40.94 41.87 42.80 43.73 44.66 45.59 46.52

15.03 15.39 15.76 16.13 16.49 16.86 17.23 17.59 17.96 18.33

38.08 39.01 39.94 40.87 41.80 42.73 43.65 44.58 45.51 46.44

15.19 15.56 15.93 16.30 16.68 17.05 17 42 17.79 18.16 18.53

41 42 43 44 45 46 47 48 49 50

6

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o

■ ,

2

69 Deg.

Dep.

21 i Deg.

Distance.

Distance.

21 Deg. Lat.

cs m

j.

TRAVERSE TABLE.

68| Deg.

Lat.

Dep.

68£ Deg.

Lat.

Dep.

681 Deg.

1 2 3 4 5 6 7 8 9 10

©

O rt -g

s

45

TRAVERSE TABLE. g c*’

21 Deg.

21i Deg.

21i Deg.

21f :Deg.

g Cfl* p 3 O CD

p 0 o »

Lat,

Dep.

Lat.

Dep.

Lat.

51 52 53 54 55 56 57 58 59 60

47.61 48.55 49.48 50.41 51.35 52.28 53.21 54.15 55.08 56.01

18.28 18.64 18.99 19.35 19.71 20.07 20.43 20.79 21.14 21.50

47.53 48.46 49.40 50.33 51.26 52.19 53.12 54.06 54.99 55.92

18.48 18.85 19.21 19.57 19.93 20.30 20.66 21.02 21.38 21.75

47.45 48.38 49.31 50.24 51.17 52.10 53.03 53.96 54.89 55.83

18.69 ! 47.37 48.30 19.06 49.23 19.42 50.16 19.79 51.08 20.16 52.01 20.52 52.94 20.89 53.87 21.26 54.80 21.62 55.73 21.99

18.90 19.27 19.64 20.01 20.38 20.75 21.12 21.49 21.86 22.23

51 1 52 53 54 55 56 * 57 58 59 60 ‘

61 62 63 64 65 66 67 68 69 70

56.95 57.88 58.82 59.75 60.68 61.62 62.55 63.48 64.42 65.35

21.86 22.22 22.58 22.94 23.29 23.65 24.01 24.37 24.73 25.09

56.85 57.78 58.72 59.65 60.58 61.51 62.44 63.38 64.31 65.24

22.11 22.47 22.83 23.20 23.56 23.92 24.28 24.65 25.01 25.37

56.76 57.69 58.62 59.55 60.48 61.41 62.34 63.27 64.20 65.13

22.36 22.72 23.09 23.46 23.82 24.19 24.56 24.92 25.29 25.66

56.66 57.59 58.52 59.44 60.37 61.30 62.23 63.16 64.09 65.02

22.60 22.97 23.35 23.72 24.09 24.46 24.83 25.20 25.57 25.94

61 62 63 64 65 66 ' 67 68 69 70 ;

71 72 73 74 75 76 77 78 79 80

66.28 67.22 68.15 69.08 70.02 70.95 71.89 72.82 73.75 74.69

25.44 25.80 26.16 26.52 26.88 27.24 27.59 27.95 28.31 28.67

66.17 67.10 68.04 68 97 69.90 70.83 71.76 72.70 73.63 74.56

25.73 26.10 26.46 26.82 27.18 27.55 27.91 28.27 28.63 29.00

66.06 66.99 67.92 68.85 69.78 70.71 71.64 72.57 73.50 74.43

26.02 26.39 26.75 27.12 27.49 27.85 28.22 28.59 28.95 29.32

65.95 66.87 67.80 68.73 69.66 70.59 71.52 72.45 73.38 74.30

26.31 26.68 27.05 27.42 27.79 28.16 28.53 28.90 29.27 29.64

*71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

75.62 76.55 77.49 78.42 79.35 80.29 81.22 82.16 83.09 84.02

29.03 29.39 29.74 30.10 30.46 30.82 31.18 31.54 31.89 32.25

75.49 76.42 77.36 78.29 79.22 80.15 81.08 82.02 82.95 83.88

29.36 29.72 30.08 30.44 30.81 31.17 31.53 31.89 32.26 32.62

75.36 76.29 77.22 78.16 79.09 80.02 80.95 81.88 82.81 83.74

29.69 30.05 30.42 30.79 31.15 31.52 31.89 32.25 32.62 32.99

75.23 76.16 77.09 78.02 78.95 79.88 80.81 81.74 82.66 83.59

30.02 30.39 30.76 31.13 31.50 31.87 32.24 32.61 32.98 33.35

81 82 83 84 85 86 87 88 89 90

91 , 92 , 93 94 95 96 97 98 99 100

84.96 85.89 86.82 87.76 88.69 89.62 90.56 91.49 92.42 93.36

32.61 32.97 33.33 33.69 34.04 34.40 34.76 35.12 35.48 35.84

84.81 85.74 86.68 87.61 88.54 89.47 90.40 91.34 92.27 93.20

32.98 33.34 33.71 34.07 34.43 34.79 35.16 35.52 35.88 36.24

84.67 85.60 86.53 87.46 88.39 89.32 90.25 91.18 92.11 93.04

33.35 33.72 34.08 34.45 34.82 35.18 35.55 35.92 36.28 36.65

84.52 85.45 86.38 87.31 88.24 89.17 90.09 91.02 91.95 92.88

33.72 91 34.09 92 34.46 93 34.83 94 35.20 95 : 35.57 96 35.94 97 36.31 98 36.69 99 ! 37.06 100

Dep.

Lat.

Dep.

Lat.

1

<6 o d ci

s

Dep.

Dep. | Lat.

Lat.

Dep.

Dep. 1 Lat.

28

68} Deg.

68J Deg.

68^ Deg.

:

d o

fl


69 Deg.

,

2

,

46 o ST o

TRAVERSE TABLE, 22 Deg.

22} Deg.

22} De?.

5 «*’

22| Deg.

ST

Lat.

Dep.

Lat.

1 2 3 ; 4 5 6 7 8 9 10

0.93 1.85 2.7 ^ 3.71 4.64 5.56 6.49 7.42 8.34 9.27

0.37 0.75 1.12 1.50 1.87 2.25 2.62 3.00 3.37 3.75

0.93 1.85 2.78 3.70 4.63 5.55 6.48 7.40 8.33 9.26

0.38 0.76 1.14 1.51 1.89 2.27 2.65 3.03 3.41 3.79

0.92 1.85 2.77 3.70 4.62 5.54 6.47 7.39 8.31 9.24

0.38 0.77 1.15 1.53 1.91 2.30 2.68 3.06 3.44 3.83

0.92 1.84 2.77 3.69 4.61 5.53 6.46 7.38 8.30 9.22

0.39 0.77 1.16 1.55 1.93 2.32 2.71 3.09 3.48 3.87

1 2 3 4 , 5 6 7 8 9 10

11

13 14 15 16 17 18 19 20

10.20 11.13 12.05 12.98 13.91 14.83 15.76 16.69 17.62 18.54

4.12 4.50 4.87 5.24 5.62 5.99 6.37 6.74 7.12 7.49

10.18 11.11 12.03 12.96 13.88 14.81 15.73 16.66 17.59 18.51

4.17 4.54 4.92 5.30 5.68 6.06 6.44 6.82 7.19 7.57

10.16 11.09 12.01 12.93 13.86 14.78 15.71 16.63 17.55 18.48

4.21 4.59 4.97 5.36 5.74 6.12 6.51 6.89 7.27 7.65

10.14 11.07 11.99 12.91 13.83 14.76 15.68 16.60 17.52 18.44

4.25 4.64 5.03 5.41 5.80 6.19 6.57 6.96 7.35 7.73

11 12 13 , 14 15 16 17 18 19 20

21 22 23 24 25 26 : 27 28 29 30

19.47 20.40 21.33 22.25 23.18 24.11 #5.03 25.96 26.89 27.82

7.87 8.24 8.62 8.99 9.37 9.74 10.11 10.49 •10.86 11.24

19.44 20.36 21.29 22.21 23.14 24.06 24.99 25.92 26.84 27.77

7.95 8.33 8.71 9.09 9.47 9.84 10.22 10.60 10.98 11.36

19.40 20.33 21.25 22.17 23.10 24.02 24.94 25.87 26.7£ 27.72

8.04 8.42 8.80 9.18 9.57 9.95 10.33 10.72 11.10 11.48

19.37 20.29 21.21 22.13 23.05 23.98 24.90 25.82 26.74 27.67

8.12 8.51 8.89 9.28 9.67 10.05 10.44 10.83 11.21 11.60

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

28.74 29.67 30.60 31.52 32.45 33.38 34.31 35.23 36.16 37.09

11.61 11.99 12.36 12.74 13.11 13.49 13.86 14.24 14.61 14.98

28.69 11.74 29.62 12.12 30.54 12.50 31.47 ! 12.87 32.39 13.25 33.32 13.63 34.24 14.01 35.17 14.39 36.10 14.77 37.02 15.15

28.64 29.56 30.49 31.41 32.34 33.26 34.18 35.11 36.03 36.96

11.86 12.25 12.63 13.01 13.39 13.78 14.16 14.54 14.92 15.31

28.59 29.51 30.43 31.35 32.28 33.20 34.12 35.04 35.97 36.89

11.99 12.37 12.76 13.15 13.53 13.92 14.31 14.70 15.08 15.47

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

38.01 38.94 39.87 40.80 41.72 42.65 43.58 44.50 45.43 46.36

15.36 15.73 16.11 16.48 16.86 17.23 17.61 17.98 18.36 18.73

37.95 38.87 39.80 40.72 41.65 42.57 43.50 44.43 45.35 46.28

15.52 15.90 16.28 16.66 17.04 17.42 17.80 18.18 18.55 18.93

37.88 38.80 39.73 40.65 41.57 42.50 43.42 44.35 45.27 46.19

15.69 16.07 16.46 16.84 17.22 17.60 17.99 18.37 18.75 19.13

37.81 38.73 39.65 40.58 41.50 42.42 43.34 44.27 45.19 46.11

15.86 16.24 16.63 17.02 17.40 17.79 18.18 18.56 18.95 19.34

41 42 43 44 45 46 47 48 49 50

«

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

»

12 ,

o a

s

Dep.

1

E3 O A

Lat.

Dep.

Lat.

Dep.

u

o

G .

68 Deg.

67f Deg.

67} Deg.

67} Deg.

rt

5

47

TRAVERSE TABLE.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

| Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

47.29 48.21 49.14 50.07 51.00 51.92 52.85 53.78 54.70 55.63

19.10 19.43 19.85 20.23 20.60 20.98 21.35 21.73 22.10 22.48

47.20 48.13 49.05 49.98 50.90 51.83 52.76 53.68 54.61 55.53

19.31 19.69 20.07 20.45 20.83 21.20 21.58 21.96 22.34 22.72

47.12 48.04 48.97 49.89 50.81 51.74 52.66 53.59 54.51 55.43

19.52 19.90 20.28 20.66 21.05 21.43 21.81 22.20 22.58 22.96

47.03 47.95 48.88 49.80 50.72 51.64 52.57 53.49 54.41 55.33

19.72 20.11 20.50 20.88 21.27 21.66 22.04 22.43 22.82 23.20

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

56.56 57.49 58.41 59.34 60.27 61.19 62.12 63.05 63.98 64.90

22.85 23.23 23.60 23.97 24.35 24.72 25.10 25.47 25.85 26.22

56.47 57.38 58.31 59.23 60.16 61.09 62.01 62.94 63.86 64.79

23.10 23.48 23.85 24.23 24.61 24.99 25.37 25.75 26.13 26.51

56.36 57.28 58.20 59.13 60.05 60.98 61.90 62.82 63.75 64.67

23.34 23.73 24.11 24.49 24.87 25.26 25.64 26.02 26.41 26.79

56.25 57.18 58.10 59.02 59.94 60.87 61.79 62.71 63.63 64.55

23.59 23.98 24.36 24.75 25.14 25.52 25.91 26.30 26.68 27.07

61 62 63 64 65 66 67 68 . 69 70

71 72 73 74 75 76 77 78 79 80

65.83 66.76 67.68 68.61 69 54 70.47 71.39 72.32 73.25 74.17

26.60 26.97 27.35 27.72 28.10 28.47 28.84 29.22 29.59 29.97

65.71 66.64 67.56 68.49 69.42 70.34 71.27 72.19 73.12 74.04

26.88 27.26 27.64 28.02 28.40 28.78 29.16 29.53 29.91 30.29

65.60 66.52 67.44 68.37 69.29 70.21 71.14 72.06 72.99 73.91

27.17 27.55 27.94 28.32 28.70 29.08 29.47 29.85 30.23 30.61

65.48 66.40 67.32 68.24 69.17 70.09 71.01 71.93 72.85 73.78

27.46 27.84 28.23 28.62 29.00 29.39 29.78 30.16 30.55 30.94

71 72 73 74 75 76 77 78 79 80

, 81 82 . 83 84 * 85 86 87 88 89 90

75.10 76.03 76.96 77.88 78.81 79.74 80.66 81.59 82.52 83.45

30.34 30.72 31.09 31.47 31.84 32.22 32.59 32.97 33.34 33.71

74.97 75.89 76.82 77.75 78.67 79.60 80.52 81.45 82.37 83.30

30.67 31.05* 31.43 31.81 32.19 32.56 32.94 33.32 33.70 34.08

74.83 75.76 76.68 77.61 78.53 79.45 80.38 81.30 82.23 83.15

31.00 31.38 31.76 32.15 32.53 32.91 33.29 33.68 34.06 34.44

74.70 75.62 76.54 77.46 78.39 79.31 80.23 81.15 82.08 83.00

31.32 31.71 32.10 32.48 32.87 33.26 33.64 34.03 34.42 34.80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

84.37 85.30 86.23 87.16 88.08 89.01 89.94 90.86 91.79 92.72

34.09 34.46 34.84 35.21 35.59 35.96 36.34 36.71 37.09 37.46

84.22 85.15 86.08 87.00 87.93 88.85 89.78 90.70 91.63 92.55

34.46 34.84 35.21 35.59 35.97 36.35 36.73 37.11 37.49 37.86

84.07 85.00 85.92 86.84 87,77 88.69 89.62 90.54 91.46 92.39

34.82 35.21 35.59 35.97 36.35 36.74 37.12 37.50 37.89 38.27

83.92 84.84 85.76 86.69 87.61 88.53 89.45 90.38 91.30 92.22

35.19 91 35.58 92 35.96 93 36.35 94 36.74 95 37.12 96 ; 37.51 97 37.90 98 38.28 99 38.67 100


Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

22 Deg.

22* Deg.

£a

' Q

68 Deg.

67f Deg.

22^ Deg.

67$ Deg.

22f Deg.

67$ Deg.

«o |

s

1

TRAVERSE TABLE.

48 o w*

23 Deg.

23} Deg.

23J Deg.

23J Deg.

g

p

o ©

*3

Lat.

Dep.

Lat.

Dep.

1 2 3 4 5 6 7 8 9 10

0.92 1.84 2.76 3.68 4.60 5.52 6.44 7.36 8.28 9.20

0.39 0.78 1.17 1.56 1.95 2.34 2.74 3.13 3.52 3.91

11 12 13 14 15 16 17 18 19 20

10.13 11.05 11.97 12.89 13.81 14.73 15.65 16.57 17.49 18.41

4.30 4.69 5.08 5.47 5.86 6.25 6.64 7.03 7.42 7.81

10.11 11.03 11.94 12.86 13.78 14.70 15.62 16.54 17.46 18.38

4.34 4.74 5.13 5.53 5.92 6.32 6.71 7.11 7.50 7.89

21 22 23 24 25 26 27 28 29 30

19.33 20.25 21.17 22.09 23.01 23.93 24.85 25.77 26.69 27.62

8.21 8.60 8.99 9.38 9.77 10.16 10.55 10.94 11.33 11.72

19.29 20.21 21.13 22.05 22.97 23.89 24.81 25.73 26.64 27.56

31 32 33 34 35 36 37 38 39 40

28.54 29.46 30.38 31.30 32.22 33.14 34.06 34.98 35.90 36.82

12.11 12.50 12.89 13.28 13.68 14.07 14.46 14.85 15.24 15.63

41 42 43 44 ' 45 46 47 48 49 50

37.74 38.66 39.58 40.50 41.42 42.34 43.26 44.18 45.10 46.03 Dep.

a

o

J s n

0.39 0.79 1.18 1.58 1.97 2.37 2.76 3.16 3.55 3.95

Dep.

Lat.

Dep.

o CD

0.92 1.83 2.75 3.67 4.59 5.50 6.42 7.34 8.25 9.17

0.40 0.80 1.20 1.59 1.99 2.39 t. 79 3.19 3.59 3.99

0.92 1.83 2.75 3.66 4.58 5.49 6.41 7.32 8.24 9.15

0.40 0.81 : .21 : .61 2.01 2.42 2.82 3.22 3.62 4.03

2 3 4 5 6 7 8 9 10

10.09 11.92 12.84 13.76 14.67 15.59 16.51 17.42 18*. 34

4.39 4.78 5.18 5.58 5.98 6.38 6.78 7.18 7.58 7.97

10.07 10.98 11.90 12.81 13.73 14.64 15.56 16.48 17.39 18.31

4.43 4.83 5.24 5.64 6.04 6.44 6.85 7.25 7.65 8.05

11 12 13 14 15 16 17 18 19 20

8.29 8.68 9.08 9.47 9.87 10.26 10.66 11.05 11.45 11.84

19.26 20.18 21.09 22.01 22.93 23.84 24.76 25.68 26.59 27.51

8.37 8.77 9.17 9.57 9.97 10.37 10.77 11.16 11.56 11.96

19.22 20.14 21.05 21.97 22.88 23.80 24.71 25.63 26.54 27.46

8.46 8.86 9.26 9.67 10.07 10.47 10.87 11.28 11.68 12.08

21 22 23 24 25 26 27 28 29 30

28.48 29.40 30.32 31.24 32.16 33.08 34.00 34.91 35.83 36.75

12.24 12.63 13.03 13.42 13.82 14.21 14.61 15.00 15.39 15.79

28.43 29.35 30.26 31.18 32.10 33.01 33.93 34.85 35.77 36.68

12.36 12.76 13.16 13.56 13.96 14.35 14.75 15.15 15.55 15.95

28.37 29.29 30.21 31.12 32.04 32.95 33.87 34.78 35.70 36.61

12.49 12.89 13.29 13.69 14.10 14.50 14.90 15.30 15.71 16.11

31 32 33 34 35* 36 37 38 39 40

16.02 16.41 16.80 17.19 17.58 17.97 18.36 18.76 19.15 19.54

37.67 38.59 39.51 40.43 41.35 42.26 43.18 44.10 45.02 45.94

16.18 16.58 16.97 17.37 17.76 18.16 18.55 18-95 19.34 19.74

37.60 38.52 39.43 40.35 41.27 42.18 43.10 44.02 44.94 45.85

16.35 16.75 17.15 17.54 17.94 18.34 18.74 19.14 19.54 19.94

37.53 38.44 39.36 40.27 41.19 42.10 43.02 43.93 44.85 45.77

16.51 16.92 17.32 17.72 18.12 18.53 18.93 19.33 19.73 20.14

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

67 Deg".

0.92 1.84 2.76 3.68 4.59 5.51 6.43 7.35 8.27 9.19

Lat.

66} Deg.

11.00

66£ Deg.

66| Deg.

1

0? o a £

Q

49

TRAVERSE TABLE,

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

j Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

46.95 47.87 48.79 49.71 50.63 51.55 52.47 53.39 54.31 55.23

19.93 20.32 20.71 21.10 21.49 21.88 22.27 22.66 23.05 23.44

46.86 47.78 48.70 49.61 50.53 51.45 52.37 53.29 54.21 55.13

20.13 20.53 20.92 21.32 21.71 22.11 22.50 22.90 23.29 23.68

46.77 47.69 48.60 49.52 50.44 51.36 52.27 53.19 54.11 55.02

20.34 20.73 21.13 21.53 21.93 22.33 22.73 23.13 23.53 23.92

46.68 47.60 48.51 49.43 50.34 51.26 52.17 53.09 54.00 54.92

20.54 20.94 21.35 21.75 22.15 22.55 22.96 23.36 23.76 24.16

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

56.15 57.07 57.99 58.91 59.83 60.75 61.67 62.59 63.51 64.44

23.83 24.23 24.62 25.01 25.40 25.79 26.18 26.57 26.96 27.35

56.05 56.97 57.88 58.80 59.72 60.64 61.56 62.48 63.40 64.32

24.08 24.47 24.87 25.26 25.66 26.05 26.45 26.84 27.24 27.63

55.94 56.86 57.77 58.69 59.61 60.53 61.44 62.36 63.28 64.19

24.32 24.72 25.12 25.52 25.92 26.32 26.72 27.11 27.51 27.91

55.83 56.75 57.66 58.58 59.50 60.41 61.33 62.24 63.16 64.07

24.57 24.97 25.37 25.78 26.18 26.58 26.98 27.39 27.79 28.19

61 62 63 64 65 • 66 67 68 69 1 70

71 72 73 74 75 76 77 78 79 80

65.36 66.28 67.20 68.12 69.04 69.96 70.88 71.80 72.72 73.64

27.74 28.13 28.52 28.91 29.30 29.70 30.09 30.48 30.87 31.26

65.23 66.15 67.07 67.99 68.91 69.83 70.75 71.67 72.58 73.50

28.03 28.42 28.82 29.21 29.61 30.00 30.40 30.79 31.18 31.58

65.11 66.03 66.95 67.86 68.78 69.70 70.61 71.53 72.45 73.36

28.31 28.71 29.11 29.51 29.91 30.30 30.70 31.10 31.50 31.90

64.99 65.90 66.82 67.73 68.65 69.56 70.48 71.39 72.31 73.22

28.59 29.00 29.40 29.80 30.21 30.61 31.01 31.41 31.82 32.22

71 72 73 74 75 76 77 78 79 80

61 82 83 84 85 86 87 88 89 90

74.56 75.48 76.40 77.32 78.24 79.16 80.08 81.00 81.92 82.85

31.65 32.04 32.43 32.82 33.21 33.60 33.99 34.38 34.78 35.17

74.42 75.34 76.26 77.18 78.10 79.02 79.93 80.85 81.77 82.69

31.97 32.37 32.76 33.16 33.55 33.95 34.34 34.74 35.13 35.53

74.28 75.20 76.12 77.03 77.95 78.87 79.78 80.70 81.62 82.54

32.30 32.70 33.10 33.49 33.89 34.29 34.69 35.09 35.49 35.89

74.14 75.06 75.97 76.89 77.80 78.72 79.63 80.55 81.46 82.38

32.62 33.03 33.43 33.83 34.23 34.64 35.04 35.44 35.84 36.25

81 82 83 84 85 86 87 88 , 89 90

91 92 93 94 95 96 97 98 99 100

83.77 84.69 85.61 86.53 87.45 88.37 89.29 90.21 91.13 92.05

35.56 35.95 36.34 36.73 37.12 37.51 37.90 38.29 38.68 39.07

83.61 84.53 85.45 86.37 87.29 88.20 89.12 90.04 90.96 91.88

35.92 36.32 36.71 37.11 37.50 37.90 38.29 38.68 39.08 39.47

83.45 84.37 85.29 86.20 87 12 88.04 88.95 89.87 90.79 91.71

36.29 36.68 37.08 37.48 37.88 38.28 38.68 39.08 39.48 39.87

83.29 84.21 85.12 86.04 86.95 87.87 88.79 89.70 90.62 91.53

36.65 91 37.05 92 37.46 93 37.86 94 38.26 95 38.66 96 39.07 97 39.47 98 39.87 99 40.27 100

o c

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

in

3

23 Deg.

23i Deg.

231 Deg.

23} Deg.

do rt s S3

67 Deg.

66} Deg.

28 *

66* Deg.

2T

66} Deg.

on

1

i

TRAVERSE TABLE.

50 Distance.

g UJ"

24 Deg.

24$ Deg.

24$ Deg.

24} Deg.

V

3 o »

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

1 2 3 4 5 6 7 8 9 10

0.91 1.83 2.74 3.65 4.57 5.48 6.39 7.31 8.22 9.14

0.41 0.81 1.22 1.63 2.03 2.44 2.85 3.25 3.66 4.07

0.91 1.82 2.74 3.65 4.56 5.47 6.38 7.29 8.21 9.12

0.41 0.82 1.23 1.64 2.05 2.46 2.87 3.29 3.70 4.11

0.91 1.82 2.73 3.64 4.55 5.46 6.37 7.28 8.19 9.10

0.41 0.83 1.24 1.66 2.07 2.49 2.90 3.32 3.73 4.15

0.91 1.82 2.72 3.63 4.54 5.45 6.36 7.27 8.17 9.08

0.42 0.84 1.26 1.67 2.09 2.51 2.93 3.35 3.77 4.19

10

11 12 13 14 15 16 17 18 19 20

10.05 10.96 11.88 12.79 13.70 14.62 15.53 16.44 17.36 13.27

4.47 4.88 5.29 5.69 6.10 6.51 6.92 7.32 7.73 8.13

10.03 10.94 11.85 12.76 13.68 14.59 15.50 16.41 17.32 18.24

4.52 4.93 5.34 5.75 6.16 6.57 6.98 7.39 7.80 8.21

10.01 10.92 11.83 12.74 13.65 14.56 15.47 16.38 17.29 18.20

4.56 4.98 5.39 5.81 6.22 6.64 7.05 7.46 7.88 8.29

9.99 10.90 11.81 12.71 13.62 14.53 15.44 16.35 17.25 18.16

4.61 5.02 5.44 5.86 6.28 6.70 7.12 7.54 7.95 8.37

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 ' 27 23 29 30

19.18 20.10 21.01 21.93 22.84 23.75 24.67 25.58 26.49 27.41

8.54 8.95 9.35 9.76 10.17 10.58 10.98 11.39 11.80 12.20

19.15 20.06 20.97 21.88 22.79 23.71 24.62 25.53 26.44 27.35

8.63 9.04 9.45 9.86 10.27 10.68 11.09 11.50 11.91 12.32

19.11 20.02 20.93 21.84 22.75 23.66 24.57 25.48 26.39 27.30

8.71 9.12 9.54 9.95 10.37 10.78 11.20 11.61 12.03 12.44

19.07 19.98 20.89 21.80 22.70 23.61 24.52 25.43 26.34 27.24

8.79 9.21 9.63 10.05 10.47 10.89 11.30 11.72 12.14 12.56

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

28.32 29.23 30.15 31.06 31.97 32.89 33.80 34.71 35.63 36.54

12.61 13.02 13.42 13.83 14.24 14.64 15.05 15.46 15.86 16.27

28.26 29.18 30.09 31.00 31.91 32.82 33.74 34.65 35.56 36.47

12.73 13.14 13.55 13.96 14.38 14.79 15.20 15.61 16.02 16.43

28.21 29.12 30.03 30.94 31.85 32.76 33.67 34.58 35.49 36.40

12.86 13.27 13.68 14.10 14.51 14.93 15.34 15.76 16.17 16.59

28.15 29.06 29.97 30.88 31.78 32.69 33.60 34.51 35.42 36.33

12.98 13.40 13.82 14.23 14.65 15.07 15.49 15.91 16.33 16.75

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

37.46 38.37 39.28 40.20 41.11 42.02 42.94 43.85 44.76 45.68

16.68 17.08 17.49 17.90 18.30 18.71 19.12 19.52 19.93 20.34

37.38 38.29 39.21 40.12 41.03 41.94 42.85 43.76 44.68 45.59

16.84 17.25 17.66 18.07 18.48 18.89 19.30 19.71 20.13 20.54

37.31 38.22 39.13 40.04 40.95 41.86 42.77 43.68 44.59 45.50

17.00 17.42 17.83 18.25 18.66 19.08 19.49 19.91 20.32 20.73

37.23 17.16 38.14 : 17.58 39.05 18.00 39.96 18.42 40.87 18.84 41.77 19.26 42.68 19.68 43.59 20.10 44.50 20.51 45.41 20.93

41 42 43 44 45 46 47 48 49 50

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o

Dep.

1 2 3 4 5 6 7 8 9

Lat.

11

□ te

66 Deg.

1

651 Deg.

65 $ Deg.

65$ Deg.

1

•2

Q

,

I

1 i t

51

TRAVERSE TABLE, 24} Deg.

24£ Deg.

24f Deg.

g U1" p

Distance.

24 Deg.

S3 o

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

46.59 47.50 48.42 49.33 50.24 51.16 52.07 52.99 53.90 54.81

20.74 21.15 21.56 21.96 22.37 22.78 23.18 23.59 24.00 24.40

46.50 47.41 48.32 49.24 50.15 51.06 51.97 52.88 53.79 54.71

20.95 21.36 21.77 22.18 22.59 23.00 23.41 23.82 24.23 24.64

46.41 47.32 48.23 49.14 50 05 50.96 51.87 52.78 53.69 54.60

21.15 21.56 21.98 22.39 22.81 23.22 23.64 24.05 24.47 24.88

46.32 47.22 48.13 49.04 49.95 50.86 51.76 52.67 53.58 54.49

21.35 21.77 22.19 22.61 23.03 23.44 23.86 24.28 24.70 25.12

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

55.73 56.64 57.55 58.47 59.38 60.29 61.21 62.12 63.03 63.95

24.81 25.22 25.62 26.03 26.44 26.84 27.25 27.66 28.06 28.47

55.62 56.53 57.44 58.35 59.26 60.18 61.09 62.00 62.91 63.82

25.05 25.46 25.88 26.29 26.70 27.11 27.52 27.93 28.34 28.75

55.51 56.42 57.33 58.24 59.15 60.06 60.97 61.88 62.79 63.70

25.30 25.71 26.13 26.54 26.96 27.37 27.78 28.20 28.61 29.03

55.40 56.30 57.21 58.12 59.03 59.94 60.85 61.75 62.66 63.57

25.54 25.96 26.38 26.79 27.21 27.63 28.05 28.47 28.89 29.31

61 62 63 64 65 66 67 68 69 70

71 : 72 73 74 75 76 77 78 79 80

64.86 65.78 66.69 67.60 68.52 69.43 70.34 71.26 72.17 73.08

28.88 29.28 29.69 30.10 30.51 30.91 31.32 31.73 32.13 32.54

64.74 65.65 66.56 67.47 68.38 69.29 70.21 71.12 72.03 72.94

29.16 29.57 29.98 30.39 30.80 31.21 31.63 32.04 32.45 32.86

64.61 65.52 66.43 67.34 68.25 69.16 70.07 70.98 71.89 72.80

29.44 29.86 30.27 30.69 31.10 31.52 31.93 32.35 32.76 33.18

64.48 65.39 66.29 67.20 68.11 69.02 69.93 70.84 71.74 72.65

29.72 30.14 30.56 30.98 31.40 31.82 32.24 32.66 33.07 33.49

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

74.00 74.91 75.82 76.74 77.65 78.56 79.48 80.39 81.31 82.22

32.95 33.35 33.76 34.17 34.57 34.98 35.39 35.79 36.20 36.61

73.85 74.76 75.68 76.59 77.50 78.41 79.32 80.24 81.15 82.06

33.27 33.68 34.09 34.50 34.91 35.32 35.73 36.14 36.55 36.96

73.71 74.62 75.53 76.44 77.35 78.26 79.17 80.08 80.99 81.90

33.59 34.00 34.42 34.83 35.25 35.66 36.08 36.49 36.91 37.32

73.56 74.47 75.38 76.28 77.19 78.10 79.01 79.92 80.82 81.73

33.91 34.33 34.75 35.17 35.59 36.00 36.42 36.84 37.26 37.68

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

83.13 84.05 84.96 85.87 86.79 87.70 88.61 89.53 90.44 91.35

37.01 37.42 37.83 38.23 38.64 39.05 39.45 39.86 40.27 40.67

82.97 83.88 84.79 85.71 86.62 87.53 88.44 89.35 90.26 91.18

37.38 37.79 38.20 38.61 39.02 39.43 39.84 40.25 40.66 41.07

82.81 83.72 84.63 85.54 86.45 87.36 88.27 89.18 90.09 91.00

37.74 38.15 38.57 38.98 39.40 39.81 40.23 40.64 41.05 41.47

82.64 83.55 84.46 85.37 86.27 87.18 88.09 89.00 89.91 90.81

38.10 91 38.52 92 38.94 93 39.35 94 39.77 95 40.19 96 40.61 97 41.03 98 41.45 99 41.87 100

6

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

o



Lat.

rt

3

66 Deg.

65j Deg.

65} Deg.

65} Deg.

a

o Ct m

s

!

52

TRAVERSE TABLE

g S’

25 Deg.

2o\ Deg.

25£ Deg.

251 Deg.

p

o

Dep.

ST s o o

0.43 0.86 1.29 1.72 2.15 2.58 3.01 3.44 3.87 4.31

0.90 1.80 2.70 3.60 4.50 5.40 6.30 7.21 8.11 9.01

0.43 0.8’'' 1.30 1.74 2.17 2.61 3.04 3.48 3.91 4.34

1 2 3 4 5 6 7 8 9 10

9.93 10.83 11.73 12.64 13.54 14.44 15.34 16.25 17.15 18.05

4.74 5.17 5.60 6.03 6.46 6.89 7.32 7.75 8.18 8.61

9.91 10.81 11.71 12.61 13.51 14.41 15.31 16.21 17.11 18.01

4.78 5.21 5.65 6.08 6.52 6.95 7.39 7.82 8.25 8.69

11 12 13 14 15 16 17 18 19 20

8.96 9.38 9.81 10.24 10.66 11.09 11.52 11.94 12.37 12.80

18.95 19.86 20.76 21.66 22.56 23.47 24.37 25.27 26.17 27.08

9.04 9.47 9.90 10.33 10.76 11.19 11.62 12.05 12.48 12.92

18.91 19.82 20.72 21.62 22.52 23.42 24.32 25.22 26.12 27.02

9.12 9.56 9.99 10.43 10.86 11.30 11.73 12.16 12.60 13.03

21 22 23 24 25 26 27 28 29 30

28.04 28.94 29.85 30.75 31.66 32.56 33.46 34.37 35.27 36.18

13.22 13.65 14.08 14.50 14.93 15.36 15.78 16.21 16.64 17.06

27.98 28.88 29.79 30.69 31.59 32.49 33.40 34.30 35.20 36.10

13.35 13.78 14.21 14.64 15.07 15.50 15.93 16.36 16.79 17.22

27.92 28.82 29.72 30.62 31.52 32.43 33.33 34.23 35.13 36.03

13.47 13.90 14.34 14.77 15.21 15.64 16.07 16.51 16.94 17.38

31 32 33 34 35 36 37 38 39 40

17.33 17.75 18.17 18.60 19.02 19.44 19.86 20.29 20.71 21.13

37.08 37.99 38.89 39.80 40.70 41.60 42.51 43.41 44.32 45.22

17.49 17.92 18.34 18.77 19.20 19.62 20.05 20.48 20.90 21.33

37.01 37.91 38.81 39.71 40.62 41.52 42.42 43.32 44.23 45.13

17.65 18.08 18.51 18.94 19.37 19.80 20.23 20.66 21.10 21.53

36.93 37.83 38.73 39.63 40.53 41.43 42.33 43.23 44.13 45.03

17.81 18.25 18.68 19.12 19.55 19.98 20.42 20.85 21.29 21.72

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Dep.

Lat.

Lat.

Dep.

Lat.

1 2 3 4 5 7 8 9 10

0.91 1.81 2.72 3.63 4.53 5.44 6.34 7.25 8.16 9.06

0.42 0.85 1.27 1.69 2.11 2.54 2.96 3.38 3.80 4.23

0.90 1.81 2.71 3.62 4.52 5.43 6.33 7.24 8.14 9.04

0.43 0.85 1.28 1.71 2.13 2.56 2.99 3.41 3.84 4.27

0.90 1.81 2.71 3.61 4.51 5.42 6.32 7.22 8.12 9.03

11 12 13 14 15 16 17 18 19 20

9.97 10.88 11.78 12.69 13.59 14.50 15.41 16.31 17.22 18.13

4.65 5.07 5.49 5.92 6.34 6.76 7.18 7.61 8.03 8.45

9.95 10.85 11.76 12.66 13.57 14.47 15.38 16.28 17.18 18.09

4.69 5.12 5.55 5.97 6.40 6.83 7.25 7.68 8.10 8.53

21 22 : 23 24 ' 25 26 27 28 29 ; 30

19.03 19.94 20.85 21.75 22.66 23.56 24.47 25.38 26.28 27.19

8.87 9.30 9.72 10.14 10.57 10.99 11.41 11.83 12.26 12.68

18.99 19.90 20.80 21.71 22.61 23.52 24.42 25.32 26.23 27.13

31 32 33 34 35 36 37 38 39 40

28.10 29.00 29.91 30.81 31.72 32.63 33.53 34.44 35.35 36.25

13.10 13.52 13.95 14.37 14.79 15.21 15.64 16.06 16.48 16.90

41 42 43 44 45 46 47 48 49 50

37.16 38.06 38.97 39.88 40.78 41.69 42.60 43.50 44.41 45.32

« o

Dep.

! Q

55"

Lat.

ft>

6

g

65 Deg.

Dep.

64| Deg.

Lat.

Dep.

64| Deg.

Dep. 1 Lat. 64i Deg.

o Cti 11>

s

TRAVERSE TABLE. g

25 Deg.

25i Deg.

25£ Deg.

53 25f Deg.

ST

g w'

»

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

g o »

51 52 53 54 55 56 57 58 59 60

46.22 47.13 48.03 48.94 49.85 50.75 51.66 52.57 53.47 54.38

21.55 21.98 22.40 22.82 23.24 23.67 24.09 24.51 24.93 25.36

46.13 47.03 47.94 48.84 49.74 50.65 51.55 52.46 53.36 54.27

21.75 22.18 22.61 23.03 23.46 23.89 24.31 24.74 25.17 25.59

46.03 46.93 47.84 48.74 49.64 50.54 51.45 52.35 53.25 54.16

21.96 22.39 22.82 23.25 23.68 24.11 24.54 24.97 25.40 25.83

45.94 46.84 47.74 48.64 49.54 50.44 51.34 52.24 53.14 54.04

22.16 22.59 23.03 23.46 23.89 24.33 24.76 25.20 25.63 26.07

51 52 53 54 55 56 57 ‘ 58 59 60

61 62 , 63 64 65 66 67 68 69 70

55.28 56.19 57.10 58.00 58.91 59.82 60.72 61.63 62.54 63.44

25.78 26.20 26.62 27.05 27.47 27.89 28.32 28.74 29.16 29.58

55.17 56.08 56.98 57.89 58.79 59.69 60.60 61.50 62.41 63.31

26.02 26.45 26.87 27.30 27.73 28.15 28.58 29.01 29.43 29.86

55.06 55.96 56.86 57.77 58.67 59.57 60.47 61.38 62.28 63.18

26.26 26.69 27.12 27.55 27.98 28.41 28.84 29.27 29.71 30.14

54.94 55.84 56.74 57.64 58.55 59.45 60.35 61.25 62.15 63.05

26.50 26.94 27.37 27.80 28.24 28.67 29.11 29.54 29.98 30.41

61 62 63 64 65 66 67 ' 68 69 70

71 73 74 75 76 77 78 79 80

64.35 65.25 66.16 67.07 67.97 68.88 69.79 70.69 71.60 72.50

30.01 30.43 30.85 31.27 31.70 32.12 32.54 32.96 33.39 33.81

64.22 65.12 66.03 66.93 67.83 68.74 69.64 70.55 71.45 72.36

30.29 30.71 31.14 31.57 31.99 32.42 32.85 33.27 33.70 34.13

64.08 64.99 65.89 66.79 67.69 68.60 69.50 70.40 71.30 72.21

30.57 31.00 31.43 31.86 32.29 32.72 33.15 33.58 34.01 34.44

63.95 64.85 65.75 66.65 67.55 68.45 69.35 70.25 71.16 72.06

30.85 31.28 31.71 32.15 32.58 33.02 33.45 33.89 34.32 34.76

71 , 72 73 74 75 76 77 : 78 79 80

81 82 83 - 84 85 86 87 88 89 90

73.41 74.32 75.22 76.13 77.04 77.94 78.85 79.76 80.66 81.57

34.23 34.65 35.08 35.50 35.92 36.35 36.77 37.19 37.61 38.04

73.26 74.17 75.07 75.97 76.88 77.78 78.69 79.59 80.50 81.40

34.55 34.98 35.41 35.83 36.26 36.68 37.11 37.54 37.96 38.39

73.11 74.01 74.91 75.82 76.72 77.62 78.52 79.43 80.33 81.23

34.87 35.30 35.73 36.16 36.59 37.02 37.45 37.88 38.32 38.75

72.96 73.86 74.76 75.66 76.56 77.46 78.36 79.26 80.16 81.06

35.19 35.62 36.06 36.49 36.93 37.36 37.80 38.23 38.67 39.10

81 82 83 . 84 85 86 87 88 89 90 -

91 92 93 94 95 ' 96 97 ; 98 99 100

82.47 83.38 84.29 85.19 86.10 87.01 87.91 88.82 89.72 90.63

38.46 38-88 39.30 39.73 40.15 40.57 40.99 41.42 41.84 42.26

82.31 83.21 84.11 85.02 85.92 86.83 87.73 88.64 89.54 90.45

38.82 39.24 39.67 40.10 40.52 40.95 41.38 41.80 42.23 42.66

82.14 83.04 83.94 84.84 85.75 86.65 87.55 88.45 89.36 90.26

39.18 39.61 40.04 40.47 40.90 41.33 41.76 42.19 42.62 43.05

81.96 39.53 91 82.86 39.97 92 83.76 40.40 93 84.67 40.84 94 85.57 i» 41.27 95 86.47 ! 41.71 96 87.37 42.14 97 88.27 42.58 98 89.17 43.01 99 90.07 43.44 100

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

£3 O



72

u

o □

• s ; .a Q

Dep.

Lat.

o a C!

65 Deg.

64| Deg.

64J Deg.

64^ Deg.

rt 5

TRAVERSE TABLE.

54 5 sr o

26 Deg.

261 Deg.

26 ^ Deg.

g 5' ST Da A

26| Deg.

Lat.

Dep.

Lat.

0.90 1.80 2.70 3.60 4.49 5.39 6.29 7.19 8.09 8.99

0.44 0.88 1.32 1.75 2.19 2.63 3.07 3.51 3.95 4.38

0.90 1.79 2.69 3.59 4.48 5.38 6.28 7.17 8.07 8.97

0.44 0.88 1.33 1.77 2.21 2.65 3.10 3.54 3.98 4.42

0.89 1.79 2.68 3.58 4.47 5.37 6.26 7.16 8.05 8.95

0.45 0.89 1.34 1.78 2.23 2.68 3.12 3.57 4.02 4.46

0.89 1.79 2.68 3.57 4.46 5.36 6.25 7.14 8.04 8.93

0.45 0.90 1.35 1.80 2.25 2.70 3.15 3.60 4.05 4.50

9 10

12 13 14 15 16 17 18 19 20

9.89 10.79 11.68 12.58 13.48 14.38 *15.28 16.18 17.08 17.98

4.82 5.26 5.70 6.14 6.58 7.01 7.45 7.89 8.33 8.77

9.87 10.76 11.66 12.56 13.45 14.35 15.25 16.14 17.04 17.94

4.87 5.31 5.75 6.19 6.63 7.08 7.52 7.96 8.40 8.85

9.84 10.74 11.63 12.53 13.42 14.32 15.21 16.11 17.00 17.90

4.91 5.35 5.80 6.25 6.69 7.14 7.59 8.03 8.48 8.92

9.82 10.72 11.61 12.50 13.39 14.29 15.18 16.07 16.97 17.86

4.95 5.40 5.85 6.30 6.75 7.20 7.65 8.10 8.55 9.00

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

18.87 19.77 20.67 21.57 22.47 23.37 24.27 25.17 26.06 26.96

9.21 9.64 10.08 10.52 10.96 11.40 11.84 12.27 12.71 13.15

18.83 19.73 20.63 21.52 22.42 23.32 24.22 25.11 26.01 26.91

9.29 9.73 10.17 10.61 11.06 11.50 11.94 12.38 12.83 13.27

18.79 19.69 20.58 21.48 22.37 23.27 24.16 25.06 25.95 26.85

9.37 9.82 10.26 10.71 11.15 11.60 12.05 12.49 12.94 13.39

18.75 19.65 20.54 21.43 22.32 23.22 24.11 25.00 25.90 26.79

9.45 9.90 10.35 10.80 11.25 11.70 12.15 12.60 13.05 13.50

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 . 38 39 40

27.86 28.76 29.66 30.56 31.46 32.36 33.26 34.15 35.05 35.95

13.59 14.03 14.47 14.90 15.34 15.78 16.22 16.66 17.10 17.53

27.80 28.70 29.60 30.49 31.39 32.29 33.18 34.08 34.98 35.87

13.71 14.15 14.60 15.04 15.48 15.92 16.36 16.81 17.25 17.69

27.74 28.64 29.53 30.43 31.32 32.22 33.11 34.01 34.90 35.80

13.83 14.28 14.72 15.17 15.62 16.06 16.51 16.96 17.40 17.85

27.68 28.58 29.47 30.36 31.25 32.15 33.04 33.93 34.83 35.72

13.95 14.40 14.85 15.30 15.75 16.20 16.65 17.10 17.55 18.00

31 32 33 34 35 36 37 38 39 40

41 42 ' 43 44 45 46 47 48 49 50

36.85 17.97 18.41 37.75 38.65 ’ 18.85 39.55 19.29 40.45 19.73 41.34* 20.17 42,24 20.60 43.14 21.04 44.04 21.48 44.94 21.92

36.77 37.67 38.57 39.46 40.36 41.26 42.15 43.05 43.95 44.84

18.13 18.58 19.02 19.46 19.90 20.35 20.79 21.23 21.67 22.11

36.69 37.59 38.48 39.38 40.27 41.17 42.06 42.96 43.85 44.75

18.29 18.74 19.19 19.63 20.08 20.53 20.97 21.42 21.86 22.31

36.61 37.51 38.40 39.29 40.18 41.08 41.97 42.86 43.76 44.65

18.45 18.90 19.35 19.80 20.25 20.70 21.15 21.60 22.05 22.50

41 42 43 44 45 46 47 48 49 50 .

Dep.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

?

1 2 3 4 5 6 7 8 9 10

n

<6

o

rt Q

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

1 2 3 4 5

6 7

8

o6

S3 Of

S3

64 Deg.

631 Deg.

63£ Deg.

C3J Deg.

|

s

55

TRAVEKSE TABLE,

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

45.84 46.74 47.64 48.53 49.43 50.33 51.23 52.13 53.03 53.93

22.36 22.80 23.23 23.67 24.11 24.55 24.99 25.43 25.86 26.30

45.74 46.64 47.53 48.43 49.33 50.22 51.12 52.02 52.92 53.81

22.56 23.00 23.44 23.88 24.33 24.77 25.21 25.65 26.09 26.54

45.64 46.54 47.43 48.33 49.22 50.12 61.01 51.91 52.80 53.70

22.76 23.20 23.65 24.09 24.54 24.99 25.43 25.88 26.33 26.77

45.54 46.43 47.33 48.22 49.11 50.01 50.90 51.79 52.69 53.58

22.96 23.41 23.86 24.31 24.76 25.21 25.66 26.11 26.56 27.01

51 52 53 . 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

54.83 55.73 56.62 57.52 58.42 59.32 60.22 61.12 62.02 62.92

26.74 27.18 27.62 28.06 28.49 28.93 29.37 29.81 30.25 30.69

54.71 55.61 56.50 57.40 58.30 59.19 60.09 60.99 61.88 62.78

26.98 27.42 27.86 28.31 28.75 29.19 29.63 30.08 30.52 30.96

54.59 55.49 56.38 57.28 58.17 59.07 59.96 60.86 61.75 62.65

27.22 27.66 28.11 28.56 29.00 29.45 29.90 30.34 30.79 31.23

54.47 55.36 56.26 57.15 58.04 58.94 59.83 60.72 61.62 62.51

27.46 27.91 28.36 28.81 29.26 29.71 30.16 30.61 31.06 31.51

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

63.81 64.71 65.61 66.51 67.41 68.31 69.21 70.11 71.00 71.90

31.12 31.56 32.00 32.44 32.88 33.32 33.75 34.19 34.63 35.07

63.68 64.57 65.47 66.37 67.27 68.16 69.06 69.96 70.85 71.75

31.40 31.84 32.29 32.73 33.17 33.61 34.06 34.50 34.94 35.38

63.54 64.44 65.33 66.23 67.12 68.01 68.91 69.80 70.70 71.59

31.68 32.13 32.57 33.02 33.46 33.91 34.36 34.80 35.25 35.70

63.40 64.29 65.19 66.08 66.97 67.87 68.76 69.65 70.55 71.44

31.96 32.41 32.86 33.31 33.76 34.21 34.66 35.11 35.56 36.01

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

72.80 73.70 74.60 75.50 76.40 77.30 78.20 79.09 79.99 80.89

35.51 35.95 36.38 36.82 37.26 37.70 38.14 38.58 39.01 39.45

72.65 73.54 74.44 75.34 76.23 77.13 78.03 78.92 79.82 80.72

35.83 36.27 36.71 37.15 37.59 38.04 38.48 38.92 39.36 39.81

72.49 73.38 74.28 75.17 76.07 76.96 77.86 78.75 79.65 80.54

36.14 36.59 37.03 37.48 37.93 38.37 38.82 39.27 39.71 40.16

72.33 73.22 74.12 75.01 75.90 76.80 77.69 78.58 79.48 80.37

36.46 36.91 37.36 37.81 38.26 38.71 39.16 39.61 40.06 40.51

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

81.79 82.69 8'5.59 84.49 85.39 86.28 87.18 88.08 88.98 89.88

39.89 40.33 40.77 41.21 41.65 42.08 42.52 42.96 43.40 43.84

81.62 82.51 83.41 84.31 85.20 86.10 87.00 87.89 88.79 89.69

40.25 40.69 41.13 41.58 42.02 42.46 42.90 43.34 43.79 44.23

81.44 82.33 83.23 84.12 85.02 85.91 86.81 87.70 88.60 89.49

40.60 41,05 41.50 41.94 42.39 42.83 43.28 43.73 44.17 44.62

81.26 82.15 83.05 83.94 84.83 85.73 86.62 87.51 88.40 89.30

40.96 91 , 41.41 92 41.86 93 42.31 94 42.76 95 43.21 96 43.66 97 44.11 98 44.56 i 99 45.01 100

(6 c

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

i? en

s

26 Deg.

64 Deg.

26J Deg.

63f Deg.

26$ Deg.

63$ Deg.

26| Deg.

63J Deg.

«

o G

.a Q

.

56 '

d CJD

TRAVERSE TABLE.

n Deg.

274 Deg.

27$ Deg.

27} Deg.

d

P o a

Lat.

1 2 3 4 5 6 7 8 9 10

0.89 1.78 2.67 3.56 4.45 5.35 6.24 7.13 8.02 8.91

0.45 0.91 1.36 1.82 2.27 2.72 3.18 3.63 4.09 4.54

0.89 i. 78 2.67 3.56 4.45 5.33 5.22 7.11 8.00 8.89

0.46 0.92 1.37 1.83 2.29 2.75 * 3.21 3.66 4.12 4.58

0.89 1.77 2.66 3.55 4.44 5.32 6.21 7.10 7.98 8.87

0.46 0.92 1.39 1.85 2.31 2.77 3.23 3.69 4.16 4.62

0.88 1.77 2.65 3.54 4.42 5.31 6.19 7.08 7.96 8.85

0.47 0.93 1.40 1.86 2.33 2.79 3.26 3.72 4.19 4.66

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

9.80 10.69 11.58 12.47 13.37 14.26 15.15 16.04 16.93 17.82

4.99 5.45 5.90 6.36 6.81 7.26 7.72 8.17 8.63 9.08

9.78 10.67 11.56 12.45 13.34 14.22 15.11 16.00 16.89 17.78

5.04 5.49 5.95 6.41 6.87 7.33 7.78 8.24 8.70 9.16

9.76 10.64 11.53 12.42 13.31 14.19 15.08 15.97 16.85 17.74

5.08 5.54 6.00 6.46 6.93 7.39 7.85 8.31 8.77 9.23

9.73 10.62 11.50 12.39 13.27 14.16 15.04 15.93 16.81 17.70

5.12 5.59 6.05 6.52 6.98 7.45 7.92 8.38 8.85 9.31

11 12 13 | 14 15 16 17 , 18 19 20

21 22 23 24 25 26 27 28 29 30

13.71 19.60 20.49 21.38 22.28 23.17 24.06 24.95 25.84 26.73 .

9.53 9.99 10.44 10.90 11.35 11.80 12.26 12.71 13.17 13.62

18.67 19.56 20.45 21.34 22.23 23.11 24.00 24.89 25.78 26.67

9.62 10.07 10.53 10.99 11.45 11.90 12.36 12.82 13.28 13.74

18.63 19.51 20.40 21.29 22.18 23.06 23.95 24.84 25.72 26.61

9.70 10.16 10.62 11.08 11.54 12.01 12.47 12.93 13.39 13.85

18.58 19.47 20.35 21.24 22.12 23.01 23.89 24.78 25.66 26.55

9.78 10.24 10.71 11.17 11.64 12.11 12.57 13.04 13.50 13.97

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

27.62 28.51 29.40 30.29 31.19 32.08 32.97 33.86 34.75 35.64

14.07 14.53 14.98 15.44 15.89 16.34 16.80 17.25 17.71 18.16

27.56 28.45 29.34 30.23 31.12 32.00 32.89 33.78 34.67 35.56

14.19 14.65 15.11 15.57 16.03 16.48 16.94 17.40 17.86 18.31

27.50 28.38 29.27 30.16 31.05 31.93 32.82 33.71 34.59 35.48

14.31 14.78 15.24 15.70 16.16 16.62 17.08 17.55 18.01 18.47

27.43 28.32 29.20 30.09 30.97 31.86 32.74 33.63 34.51 35.40

14.43 14.90 15.37 15.83 16.30 16.76 17.23 17.69 18.16 18.62

31 32 33 34 35 36 37 ' 38 39 40

41 42 43 44 45 46 ' 47 48 49 50

36.53 37.42 38.31 39.20 40.10 40.99 41.83 42.77 43.66 44.55

18.61 19.07 19.52 19.98 20.43 20.88 21.34 21.79 22.25 22.70

36.45 37.34 33.23 39.12 40.01 40.89 41.78 42.67 43.56 44.45

18.77 19.23 19.69 20.15 20.60 21.06 21.52 21.98 22.44 22.89

36.37 37.25 38.14 39.03 39.92 40.80 41.69 42.58 43.46 44.35

18.93 19.39 19.86 20.32 20.78 21.24 21.70 22.16 22.63 23.09

36.28 37.17 33.05 38.94 39.82 40.71 41.59 42.43 43.36 44.25

19.09 19.56 20.02 20.49 20.95 21.42 21.88 22.35 22.82 23.28

41 42 43 44 45 ’ 46 47 48 49 50

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

aJ o co -M

3

P D

Dep.

63 Deg.

Lat.

Dep.

62| Deg.

Lat.

Dep.

62£ Deg.

Lat.

Dep.

62* Deg.

O

9

i

o c &

s

57

TRAVERSE TABLE. Distance.

a w*

27£ Deg.

27} Deg.

27 Deg.

27| Deg.

p

3

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

; 51 l 52 | 53 54 55 56 57 58 59 60

45.44 46.33 47.22 48.11 49.01 49.90 50.79 51.68 52.57 53.46

23.15 23.61 24.06 24.52 24.97 25.42 25.88 26.33 26.79 27.24

45.34 46.23 47.12 48.01 48.90 49.78 50.67 51.56 52.45 53.34

23.35 23.81 24.27 24.73 25.18 25.64 26.10 26.56 27.01 27.47

45.24 46.12 47.01 47.90 48.79 49.67 50.56 51.45 52.33 53.22

23.55 24.01 24.47 24.93 25.40 25.86 26.32 26.78 27.24 27.70

45.13 46.02 46.90 47.79 48.67 49.56 50.44 51.33 52.21 53.10

23.75 24.21 24.68 25.14 25.61 26.07 26.54 27.01 27.47 27.94

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

54.35 55.24 56.13 57.02 57.92 58.81 59.70 60.59 61.48 62.37

27.69 28.15 28.60 29.06 29.51 29.96 30.42 30.87 31.33 31.78

54.23 55.12 56.01 56.90 57.79 58.68 59.56 60.45 61.34 62.23

27.93 28.39 28.85 29.30 29.76 30.22 30.68 31.14 31.59 32.05

54.11 54.99 55.88 56.77 57.66 58.54 59.43 60.32 61.20 62.09

23.17 28.63 29.09 29.55 30.01 30.48 30.94 31.40 31.86 32.32

53.98 54.87 55.75 56.64 57.52 58.41 59.29 60.18 61.06 61.95

28.40 23.87 29.33 29.80 30.2630.73 31.20 31.66 32.13 32.59

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 7G 77 78 79 80

63.26 64.15 65.04 65.93 66 83 67.72 68.61 69.50 70.39 71.28

32.23 32.69 33.14 33.60 34.05 34.50 34.96 35.41 35.87 36.32

63.12 64.01 64.90 65.79 66.68 67.57 68.45 69.34 70.23 71.12

32.51 32.97 33.42 33.88 34.34 34.80 35.26 35.71 36.17 36.63

62.98 63.86 64.75 65.64 66.53 67.41 68.30 69.19 70.07 70.96

32.78 33.25 33.71 34.17 34.63 35.09 35.55 36.02 36.48 36.94

62.83 63.72 64.60 65.49 66.37 67.26 68.14 69.03 69.91 70.80

33.06 71 33.-52 72 33.99 ’73 34.46 74 34.92 75 35.39 76 35.85 77 36.32 78 36.78 79 37.25 80

81 82 83 84 85 86 87 88 89 90

72.17 73.06 73.95 74.84 75.74 76.63 77.52 78.41 79.30 80.19

36.77 37.23 37.68 38.14 38.59 39.04 39.50 39.95 40.41 40.86

72.01 72.90 73.79 74.68 75.57 76.46 77.34 78.23 79.12 80.01

37.09 37.55 38.00 38.46 38.92 39.38 39.83 40.29 40.75 41.21

71.85 72.73 73.62 74.51 75.40 76.28 77.17 78.06 78.94 79.83

37.40 37.86 38.33 38.79 39.25 39.71 40.17 40.63 41.10 41.56

71.68 72.57 73.45 74.34 75.22 76.11 76.99 77.88 78.76 79.65

37.71 38.18 38.65 39.11 39.58 40.04 40.51 40.97 41.44 41.91

91 92 93 94 95 96 97 98 99 100

81.08 81.97 82.86 83.75 84.65 85.54 86.43 87.32 88.21 89.10

41.31 41.77 42.22 42.68 43.13 43.58 44.04 44.49 44.95 45.40

80.90 81.79 82.68 83.57 84.46 85.35 86.23 87.12 88.01 88.90

41.67 42.12 42.58 43.04 43.59 43.96 44.41 44.87 45.33 45.79

80.72 81.60 82.49 83.38 84.27 85.15 86.04 86.93 87.81 88.70

42.02 42.48 42.94 43.40 43.87 44.33 44.79 45.25 45.71 46.17

80.53 81.42 82.30 83.19 84.07 84.96 85.84 86.73 87.61 88.50

42.37 91 42.84 92 43.30 93 43.77 94 . 44.23 95 44.70 96 45.16 97 i 45.63 98 & 46.10 99 46.56 100

o

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

O

a>

rtd

3

Lat.

81 82 83 84 85 86 87 88 89 90

V CJ

Sj

63 Deg.

w”

62} Deg.

" 2 cr

62} Deg.

62} Deg.

s



58

TRAVERSE TABLE. Distance. |

Distance.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.83 1.77 2.65 3.53 4.41 5.30 6.18 7.06 7.95 8.83

0.47 0.94 1.41 1.88 2.35 2.82 3.29 3.76 4.23 4.69

0.88 1.76 2.64 3.52 4.40 5.29 6.17 7.05 7.93 8.81

0.47 0.95 1.42 1.89 2.37 2.84 3.31 3.79 4.26 4.73

0.88 1.76 2.64 3.52 4.39 5.27 6.15 7.03 7.91 8.79

0.48 0.95 1.43 1.91 2.39 2.86 3.34 3.82 4.29 4.77

0.88 1.75 2.63 3.51 4.38 5.26 6.14 7.01 7.89 8.77

0.48 0.96 1.44 1.92 2.40 2.89 3.37 3.85 4.33 4.81

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

9.71 10.60 11.43 12.36 13.24 14.13 15.01 15.89 16.73 17.66

5.16 5.63 6.10 6.57 7.04 7.51 7.98 8.45 8.92 9.39

.9.69 10.57 11.45 12.33 13.21 14.09 14.98 15.86 16.74 17.62

5.21 5.68 6.15 6.63 7.10 7.57 8.05 8.52 8.99 9.47

9.67 10.55 11.42 12.30 13.18 14.06 14.94 15.82 16.70 17.58

5.25 5.73 6.20 6.68 7.16 7.63 8.11 8.59 9.07 9.54

9.64 10.52 11.40 12.27 13.15 14.03 14.90 15.78 16.66 17.53

5.29 5.77 6.25 6.73 7.21 7.70 8.18 8.66 9.14 9.62

11 12 13 14 15 16 17 18 19 20

21 22* 23 24 25 26 27 28 29 30

18.54 19.42 20.31 21.19 22.07 22.96 23.84 24.72 25.61 26.49

9.86 10.33 10.80 11.27 11.74 12.21 12.68 13.15 13.61 14.08

18.50 19.38 20.26 21.14 22.02 22.90 23.78 24.66 25.55 26.43

9.94 10.41 10.89 11.36 11.83 12.31 12.78 13.25 13.73 14.20

18.46 19.33 20.21 21.09 21.97 22.85 23.73 24.61 25.49 26.36

10.02 10.50 10.97 11.45 11.93 12.41 12.88 13.36 13.84 14.31

18.41 19.29 20.16 21.04 21.92 22.79 23.67 24.55 25.43 26.30

10.10 10.58 11.06 11.54 12.02 12.51 12.99 13.47 13.95 14.43

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

27.37 28.25 29.14 30.02 30.90 31.79 32.67 33.55 34.43 35.32

14.55 15.02 15.49 15.96 16.43 16.90 17.37 17.84 18.31 18.78

27.31 28.19 29.07 29.95 30.83 31.71 32.59 33.47 34.35 35.24

14.67 15.15 15.62 16.09 16.57 17.04 17.51 17.99 18.46 18.93

27.24 28.12 29.00 29.88 30.76 31.64 32.52 33.39 34.27 35.15

14.79 15.27 15.75 16.22 16.70 17.18 17.65 18.13 18.61 19.09

27.18 28.06 28.93 29.81 30.69 31.56 32.44 33.32 34.19 35.07

14.91 15.39 15.87 16.35 16.83 17.32 17.80 18.28 18.76 19.24

31 32 33 34 35 36 37 : 38 39 40 ;

41 42 43 44 45 46 47 48 49 50

36.20 37.08 37.97 38.85 39.73 40.62 41.50 42.38 43.26 44.15

19.25 19.72 20.19 20.66 21.13 21.60 22.07 22.53 23.00 23.47

36.12 37.00 37.88 38.76 39.64 40.52 41.40 42.28 43.16 44.04

19.41 19.88 20.35 20.83 21.30 21.77 22.25 22.72 23.19 23.67

36.03 36.91 37.79 38.67 39.5* 40.43 41.30 42.18 43.06 43.94

19.56 20.04 20.52 20.99 21.47 21.95 22.43 22.90 23.38 23.86

35.95 36.82 37.70 38.58 39.45 40.33 41.21 42.08 42.96 43.84

19.72 20.20 20.68 21.16 21.64 22.13 22.61 23.09 23.57 24.05

41 42 43 44 45 46 47 48 49 50

V o

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

28 Deg.

28* Deg. Dep.

28* Deg. Lat.

Dep.

28| Deg. Lat.

Dep.

Dep. 1 Lat.

3

62 Deg.

61J Deg.


ri£3

rt 61* Deg.

6H Deg.

5

1

TRAVERSE TABLE,

59

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

Lat.

51 52 53 54 , 55 56 57 58 59 60

45.03 45.91 46.80 47.68 48.56 49.45 50.33 51.21 52.09 52.98

23.94 24.41 24.88 25.35 25.82 26.29 26.76 27.23 27.70 28.17

44.93 45.81 46.69 47.57 48.45 49.33 50.21 51.09 51.97 52.85

24.14 24.61 25.09 25.56 26.03 26.51 26.98 27.45 27.93 28.40

44.82 45.70 46.58 47.46 48.33 49 21 50.09 50.97 51.85 52.73

24.34 24.81 25.29 25.77 26.24 26.72 27.20 27.68 28.15 28.63

44.71 45.59 46.47 47.34 48.22 49.10 49.97 50.85 51.73 52.60

24.53 25.01 25.49 25.97 26.45 26.94 27.42 27.90 28.38 28.86

51 52 , 53 54 55 . 56 57 ■ 58 59 60

61 62 63 64 65 66 67 68 69 70

53.86 54.74 55.63 56.51 57.39 58.27 59.16 60.04 60.92 61.81

28.64 29.11 29.58 30.05 30.52 30.99 31.45 31.92 32.39 32.86

53.73 28.87 54.62 29.35 55.50 29.82 56.38 30.29 57.26 30.77 58.14 31.24 59.02 31.71 59.90 32.19 60.78 . 32.66 61.66 33.13

53.61 54.49 55.37 56.24 57.12 58.00 58.88 59.76 60.64 61.52

29.11 29.58 30.06 30.54 31.02 31.49 31.97 32.45 32.92 33.40

53.48 54.36 55.23 56.11 56.99 57.86 58.74 59.62 60.49 61.37

29.34 29.82 30.30 30.78 31.26 31.75 32.23 32.71 33.19 33.67

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

62.69 63.57 64.46 65.34 66.22 67.10 67.99 68.87 69.75 70.64

33.33 33.80 34.27 34.74 35.21 35.68 36.15 36.62 37.09 37.56

62.54 63.42 64.30 65.19 66.07 66.95 67.83 68.71 69.59 70.47

33.61 34.03 34.55 35.03 35.50 35.97 36.45 36.92 37.39 37.87

62.40 63.27 64.15 65.03 65.91 66.79 67.67 68.55 69.43 70.31

33.88 34.36 34.83 35.31 35.79 36.26 36.74 37.22 37.70 38.17

62.25 63.12 64.00 64.88 65.75 66.63 67.51 68.38 69.26 70.14

34.15 34.63 35.11 35.59 36.07 36.56 37.04 37.52 38.00 38.48

71 ' 72 . 73 74 75 76 ■ 77 78 79 80

81 82 83 84 85 86 87 88 89 90

71.52 72.40 73.28 74.17 75.05 75.93 76.82 77.70 78.58 79.47

38.03 38.50 38.97 39.44 39.91 40.37 40.84 41.31 41.78 42.25

71.35 72.23 73.11 73.99 74.88 75.76 76.64 77.52 78.40 79.28

38.34 38.81 39.29 39.76 40.23 40.71 41.18 41.65 42.13 42.60

71.18 72.06 72.94 73.82 74.70 75.58 76.46 77.34 78.21 79.09

38.65 39.13 39.60 40.08 40.56 41.04 41.51 41.99 42.47 42.94

71.01 71.89 72.77 73.64 74.52 75.40 76.28 77.15 78.03 78.91

38.96 39.44 39.92 40.40 40.88 41.36 41.85 42.33 42.81 43.29

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

80.35 81.23 82.11 83.00 83.88 84.76 85.65 86.53 87.41 88.29

42.72 43.19 43.66 44.13 44.60 45.07 45.54 46.01 46.48 46.95

80.16 81.04 81.92 82.80 83.68 84.57 85.45 86.33 87.21 88.09

43.07 43.55 44.02 44.49 44.97 45.44 45.91 46.39 46.86 47.33

79.97 80.85 81.73 82.61 83.49 84.37 85.25 86.12 87.00 87.88

43.42 43.90 44.38 44.85 45.33 45.81 46.28 46.76 47.24 47.72

79.78 80.66 81.54 82.41 83.29 84.17 85.04 85.92 86.80 87.67

43.77 91 44.25 92 44.73 93 45.21 94 45.69 95 46.17 96 46.66 97 47.14 98 47.62 99 48.10 100

«

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

28 i Deg.

28 Deg.

28£ Deg.

281 Deg.

%

!

o CJ Si tn

•5

62 Deg.

61J Deg.

61 i Deg.

6H Deg.

oof

£3 tn

s

TRAVERSE TABLE,

60 g CO*

29 Deg.

29} Deg.

29$ Deg.

F o»

Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.87 1.75 2.62 3.50 4.37 5.25 6.12 7.00 7.87 8.75

0.48 0.97 1.45 1.94 2.42 2.91 3.39 3.88 4.36 4.85

0.87 1.74 2.62 3.49 4.36 5.23 6.11 6.98 7.85 8.72

0.49 0.98 1.47 1.95 2.44 2.93 3.42 3.91 4.40 4.89

0.87 1.74 2.61 3.48 4.35 5.22 6.09 6.96 7.83 8.70

11 12 13 14 15 16 17 18 19 20

9.62 10.50 11.37 12.24 13.12 13.99 14.87 15.74 16.62 17.49

5.33 5.82 6.30 6.79 7.27 7.76 8.24 8.73 9.21 9.70

9.60 10.47 11.34 12.21 13.09 13.96 14.83 15.70 16.58 17.45

5.37 5.86 6.35 6.84 7.33 7.82 8.31 8.80 9.28 9.77

21 22 23 ‘24 25 26 27 28 29 30

18.37 19.24 20.12 20.99 21.87 22.74 23.61 24.49 25.36 26.24

10.18 10.67 11.15 11.64 12.12 12.60 13.09 13.57 14.06 14.54

18.32 19.19 20.07 20.94 21.81 22.68 23.56 24.43 25.30 26.17

31 32 33 34 . 35 36 37 38 39 40

27.11 27.99 28.86 29.74 30.61 31.49 32.36 33.24 34.11 34.98

15.03 15.51 16.00 16.48 16.97 17.45 17.94 18.42 18.91 19.39

41 42 43 44 45 46 47 48 49 50

35.86 36.73 37.61 38.48 39.36 40.23 41.11 41.98 42.86 43.73

6

Dep.

o

fl

Dep.

29} Deg.

(/>

3 O CD

Lat.

Dep.

0.49 0.98 1.48 1.97 2.46 2.95 3.45 3.94 4.43 4.92

0.87 1.74 2.60 3.47 4.34 5.21 6.08 6.95 7.81 8.68

0.50 0.99 1.49 1.98 2.48 2.98 3.47 3.97 4.47 4.96

1 2 3 4 5 6 7 8 9 10

9.57 10.44 11.31 12.18 13.06 13.93 14.80 15.67 16.54 17.41

5.42 5.91 6.40 6.89 7.39 7.88 8.37 8.86 9.36 9.85

9.55 10.42 11.29 12.15 13.02 13.89 14.76 15.63 16.50 17.36

5.46 5.95 6.45 6.95 7.44 7.94 8.44 8.93 9.43 9.92

11 12 13 14 15 : 16 17 18 19 20

10.26 10.75 11.24 11.73 12.22 12.70 13.19 13.68 14.17 14.66

18.28 19.15 20.02 20.89 21.76 22.63 23.50 24.37 25.24 26.11

10.34 10.83 11.33 11.82 12.31 12.80 13.30 13.79 14.28 14.77

18.23 19.10 19.97 20.84 21.70 22.57 23.44 24.31 25.18 26.05

10.42 10.92 11.41 11.91 12.41 12.90 13.40 13.89 14.39 14.89

21 22 23 24 25 26 27 28 29 30

27.05 27.92 28.79 29.66 30.54 31.41 32.28 33.15 34.03 34.90

15.15 15.64 16.12 16.61 17.10 17.59 18.08 18.57 19.66 19.54

26.98 27.85 28.72 29.59 30.46 31.33 32.20 33.07 33.94 34.81

15.27 15.76 16.25 16.74 17.23 17.73 18.22 18.71 19.20 19.70

26.91 27.78 28.65 29.52 30.39 31.26 32.12 32.99 33.86 34.73

15.38 15.88 16.38 16.87 17.37 17.86 18.36 18.86 19.35 19.85

31 32 33 34 35 36 37 38 39 40

19.88 20.36 20.85 21.33 21.82 22.30 22.79 23.27 23.76 24.24

35.77 36.64 37.52 38.39 39.26 40.13 41.01 41.88 42.75 43.62

20.03 20.52 21.01 21.50 21.99 22.48 22.97 23.45 23.94 24.43

35.68 36.55 37.43 38.30 39.17 40.04 40.91 41.78 42.65 43.52

20.19 20.68 21.17 21.67 22.16 22.65 23.14 23.63 24.13 24.62

35.60 36.46 37.33 38.20 39.07 39.94 40.81 41.67 42.54 43.41

20.34 20.84 21.34 21.83 22.33 22.83 23.32 23.82 24.31 24.81

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Dep.

Lat.

Dep. | Lat.

61 Deg.

601 Deg.

a

a

a

£

2 ST

60£ Deg.

60J Deg.

£

5



;

61

TRAVERSE TABLE Distance.

g

29 Deg.

29} Deg.

29| Deg.

29i Deg.

P

P O O

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

44.61 45.48 46.35 47.23 48.10 48.98 49.85 50.73 51.60 52.48

24.73 25.21 25.69 26.18 26.66 27.15 27.63 28.12 28.60 29.09

44.50 45.37 46.24 47.11 47.99 48.86 49.73 50.60 51.48 52.35

24.92 25.41 25.90 26.39 26.87 27.36 27.85 28.34 28.83 29.32

44.39 45.26 46.13 47.00 47.87 48.74 49.61 50.48 51.35 52.22

25.11 25.61 26.10 26.59 27.08 27.58 28.07 28.56 29.05 29.55

44.28 45.15 46.01 46.88. 47.75 48.62 49.49 50.36 51.22 52.09

25.31 25.80 26.30 26.80 27.29 27.79 28.28 28.78 29.28 29.77

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

53.35 54.23 55.10 55.98 56.85 57.72 58.60 59.47 60.35 61.22

29.57 30.06 30.54 31.03 31.51 32.00 32.48 32.97 33.45 33.94

53.22 54.09 54.97 55.84 56.71 57.58 58.46 59.33 60.20 61.07

29.81 30.29 30.78 31.27 31.76 32.25 32.74 33.23 33.71 34.20

53.09 53.96 54.83 55.70 56.57 57.44 58.31 59.18 60.05 60.92

30.04 30.53 31.02 31.52 32.01 32.50 32.99 33.48 33.98 34.47

52.96 53.83 54.70 55.56 56.43 57.30 58.17 59.04 59.91 60.77

30.27 30.77 31.26 31.76 32.25 32.75 33.25 33.74 34.24 34.74

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

62.10 62.97 63.85 64.72 65.60 66.47 67.35 68.22 69.09 69.97

34.42 34.91 35.39 35.88 36.36 36.85 37.33 37.82 38.30 38.78

61.95 62.82 63.69 64.56 65.44 66.31 67.18 68.05 68.93 69.80

34.69 35.18 35.67 36.16 36.65 37.14 37.62 38.11 38.60 39.09

61.80 62.67 63.54 64.41 65.28 66.15 67.02 67.89 68.76 69.63

34.96 35.45 35.95 36.44 36.93 37.42 37.92 38.41 38.90 39.39

61.64 62.51 63.38 64.25 65.11 65.98 66.85 67.72 68.59 69.46

35.23 35.73 36.22 36.72 37.22 37.71 38.21 38.70 39.20 39.70

71 72 73 74 75 : 76 77 78 79 80 .

81 82 83 84 85 86 87 88 89 90

70.84 39.27 71.72 39.75 72.59 40.24 73.47 40.72 74.34 41.21 75.22 : 41.69 76.09 42.18 76.97 42.66 77.84 43.15 78.72 43.63

70.67 71.54 72.42 73.29 74.16 75.03 75.91 76.78 77.65 78.52

39.58 40.07 40.56 41.04 41.53 42.02 42.51 43.00 43.49 43.98

70.50 71.37 72.24 73.11 73.98 74.85 75.72 76.59 77.46 78.33

39.89 40.38 40.87 41.36 41.86 42.35 42.84 43.33 43.83 44.32

70.32 71.19 72.06 72.93 73.80 74.67 75.53 76.40 77.27 78.14

40.19 40.69 41.19 41.68 42.18 42.67 43.17 43.67 44.16 44.66

81 82 ; 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

79.59 80.46 81.34 82.21 83.09 83.96 84.84 85.71 86.59 87.46

44.12 44.60 45.09 45.57 46.06 46.54 47.03 47.51 48.00 48.48

79.40 80.27 81.14 82.01 82.89 83.76 84.63 85.50 86.38 87.25

44.46 44.95 45.44 45.93 46.42 46.91 47.40 47.88 48.37 48.86

79.20 80.07 80.94 81.81 82.68 83.55 84.42 85.29 86.17 87.04

44.81 45.30 45.80 46.29 46.78 47.27 47.77 48.26 48.75 49.24

79.01 79.87 80.74 81.61 82.48 83.35 84.22 85.08 85.95 86.82

45.16 91 45.65 92 46.15 93 46.64 94 47.14 95 47.64 96 48.13 97 48.63 98 49.13 99 49.62 100

6

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o £3

3

61 Deg.

29*

1

•o

rtc 60} Deg.

604 Deg.

60} Deg.

m

3

62

TRAVERSE TABLE. 30i Deg. Dep.

30* Deg.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.87 1.73 2.60 3.46 4.33 5.20 6.06 6.93 7.79 8.66

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

0.86 1.73 2.59 3.46 4.32 5.18 6.05 6.91 7.77 8.64

0.50 1.01 1.51 2.02 2.52 3.02 3.53 4.03 4.53 5.04

0.86 1.72 2.58 3.45 4.31 5.17 6.03 6.89 7.75 8.62

0.51 1.02 1.52 2.03 2.54 3.05 3.55 4.06 4.57 5.08

0.86 1.72 2.58 3.44 4.30 5.16 6.02 6.88 7.73 8.59

0.51 1.02 1.53 2.05 2.56 3.07 3.58 4.09 4.60 5.11

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

9,53 10.39 11.26 12.12 12.99 13.86 14.72 15.59 16.45 17.32

5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00

9.50 10.37 11.23 12.09 12.96 13.82 14.69 15.55 16.41 17.28

5.54 6.05 6.55 7.05 7.56 8.06 8.56 9.07 9.57 10.08

9.48 10.34 11.20 12.06 12.92 13.79 14.65 15.51 16.37 17.23

5.58 6.09 6.60 7.11 7.61 8.12 8.63 9.14 9.64 10.15

9.45 10.31 11.17 12.03 12.89 13.75 14.61 15.47 16.33 17.19

5.62 6.14 6.65 7.16 7.67 8.18 8.69 9.20 9.71 10.23

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

18.19 19.05 19.92 20.78 21.65 22.52 23.38 24.25 25.11 25.98

10.50 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00

18.14 19.00 19.87 20.73 21.60 22.46 23.32 24.19 25.05 25.92

10.58 11.08 11.59 12.09 12.59 13.10 13.60 14.11 14.61 15.11

18.09 18.96 19.82 20.68 21.54 22.40 23.26 24.13 24.99 25.85

10.66 11.17 11.67 12.18 12.69 13.20 13.70 14.21 14.72 15.23

18.05 18.91 19.77 20.63 21.49 22.34 23.20 24.06 24.92 25.78

10.74 11.25 11.76 12.27 12.78 13.29 13.80 14.32 14.83 15.34

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

26.85 27.71 28.58 29.44 30.31 31.18 32.04 32.91 33.77 34.64

15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00

26.78 27.64 28.51 29.37 30.23 31.10 31.96 32.83 33.69 34.55

15.62 16.12 16.62 17.13 17.63 18.14 18.64 19.14 19.65 20.15

26.71 27.57 28.43 29.30 30.16 31.02 31.88 32.74 33.60 34.47

15.73 16.24 16.75 17.26 17.76 18.27 18.78 19.29 19.79 20.30

26.64 15.85 27.50 16.36 28.36 16.87 29.22 17.38 30.08 17.90 30.94 18.41 31.80 18.92 32.66 ! 19.43 33.52 19.94 34.38 20.45

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

35.51 36.37 37.24 38.11 38.97 39.84 40.70 41.57 42.44 43.30

20.50 21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00

35.42 36.28 37.14 38.01 38.87 39.74 40.60 41.46 42.33 43.19

20.65 21.16 21.66 22.17 22.67 23.17 23.68 24.18 24.68 25.19

35.33 36.19 37.05 37.91 38.77 39.63 40.50 41.36 42.22 43.08

20.81 21.32 21.82 22.33 22.84 23.35 23.85 24.36 24.87 25.38

35.24 36.10 36.95 37.81 38.67 39.53 40.39 41.25 42.11 42.97

20.96 21.47 21.99 22.50 23.01 23.52 24.03 24.54 25.05 25.56

41 42 43 44 45 46 47 48 49 50

6

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

u rtfl

o

11.00

Lat.

30} Deg.

Distance.

Distance.

1 30 Deg.

Dep.

Dep. | Lat.

Lat.

Dep.

rt

3

60 Deg.

59| Deg.

59* Deg.

59i Deg.

«

s

G3

TRAVERSE TABLE. Distance.

g

30 Deg.

30* Deg.

30J Deg.

30} Deg.

PT

3 O

©

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 62 53 54 55 56 57 58 59 60

44.17 45.03 45.90 46.77 47.63 43.50 49.36 50.23 51.10 51.96

25.50 26.00 26.50 27.00 27.50 28.00 28.50 29.00 29.50 30.00

44.06 44.92 45.78 46.65 47.51 48.37 49.24 50.10 50.97 51.83

25.69 26.20 26.70 27.20 27.71 28.21 28.72 29.22 29.72 30.23

43.94 44.80 45.67 46.53 47.39 48.25 49.11 49.97 50.84 51.70

25.88 26.39 26.90 27.41 27.91 28.42 28.93 29.44 29.94 30.45

43.83 44.69 45.55 46.41 47.27 48.13 48.99 49.85 50.70 51.56

26.08 26.59 27.10 27.61 28.12 28.63 29.14 29.65 30.17 30.68

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

52.83 53.69 54.56 55.43 56.29 57.16 58.02 58.89 59.76 60.62

30.50 31.00 31.50 32.00 32.50 33.00 33.50 34.00 34.50 35.00

52.69 53.56 54.42 55.29 56.15 57.01 57.88 58.74 59.60 60.47

30.73 31.23 31.74 32.24 32.75 33.25 33.75 34.26 34.76 35.26

52.56 53.42 54.28 55.14 56.01 56.87 57.73 58.59 59.45 60.31

30.96 31.47 31.97 32.48 32.99 33.50 34.01 34.51 35.02 35.53

52.42 53.28 54.14 55.00 55.86 56.72 57.58 58.44 59.30 60.16

31.19 31.70 32.21 32.72 33.23 33.75 34.26 34.77 35.28 35.79

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 73 79 80

61.49 62.35 63.22 64.09 64.95 65.82 66.68 67.55 63.42 69.28

35.50 36.00 36.50 37.00 37.50 33.00 38.50 39.00 39.50 40.00

61.33 62.20 63.06 63.92 64.79 65.65 66.52 67.38 68.24 69.11

35.77 36.27 36.78 37.28 37.78 38.29 38.79 39.29 39.80 40.30

61.18 62.04 62.90 63.76 64.62 65.48 66.35 67.21 68.07 68.93

36.04 36.54 37.05 37.56 38.07 38.57 39.08 39.59 40.10 40.60

61.02 61.88 62.74 63.60 64.46 65.31 66.17 67.03 67.89 68.75

36.30 36.81 37.32 37.84 38.35 38.86 39.37 39.88 40.39 40.90

71 72 73 74 75 76 77 78 79 . 80

81 82 83 84 85 86 87 88 89 90

70.15 71.01 71.88 72.75 73.61 74.48 75.34 76.21 77.08 77.94

40.50 41.00 41.50 42.00 42.50 43.00 43.50 44.00 44.50 45.00

69.97 70.83 71.70 72.56 73.43 74.29 75.15 76.02 76.88 77.75

40.81 41.31 41.81 42.32 42.82 43.32 43.83 44.33 44.84 45.34

69.79 70.65 71.52 72.38 73.24 74.10 74.96 75.82 76.68 77.55

41.11 41.62 42.13 42.63 43.14 43.65 44.16 44.66 45.17 45.68

69.61 70.47 71.33 72.19 73.05 73.91 74.77 75.63 76.49 77.35

41.41 41.93 42.44 42.95 43.46 43.97 44.48 44.99 45.51 46.02

81 82 83 84 85 86 87 83 89 90

91 78.81 92 79.67 93 ; 80.54 94 81.41 95 82.27 ■ 96 83.14 97 84.00 98 84.87 99 85.74 100 86.60

45.50 46.00 46.50 47.00 47.50 48.00 48.5C 49.00 49.50 50.00

78.61 79.47 80.34 81.20 82.06 82.93 83.79 84.66 85.52 86.38

45.84 46.35 46.85 47.35 47.86 48.36 48.87 49.37 49.87 50.38

78.41 79.27 80.13 80.99 81.85 82.72 83.58 84.44 85.30 86.16

46.19 46.69 47.20 47.71 48.22 48.72 49.23 49.74 50.25 50.75

78.21 79.07 79.92 80.78 81.64 82.50 83.36 84.22 85.08 85.94

46.53 91 47.04 92 47.55 93 48.06 94 48.57 95 49.08 96 49.60 97 93 ; 50.11 50.62 99 51.13 100

Lat.

Dep.

Lat.

Dep.

Lat.

« c

fl

Dep.

Dep. |

Lat.

fS in

3

60 Deg.

59} Deg.

59* Deg.

59* Deg.

o p a Q

04 s ST o A

TRAVERSE TABLE, 31 Deg.

31* Deg.

31* Deg.

311 Deg.

g P

S3

Lai.

Dep.

Lat.

Dep.

Lat,

Dep.

Lat.

Dep.

a a

1 2 3 4 5 6 7 8 9 10

0.86 1.71 2.57 3.43 4.29 5.14 6.00 6.86 7.71 8.57

0.51 1.03 1.55 2.06 2.58 3.09 3.61 4.12 4.64 5.15

0.85 1.71 2.56 3.42 4.27 5.13 5.98 6.84 7.69 8.55

0.52 1.04 1.56 2.08 2.59 3.11 3.63 4.15 4.67 5.19

0.85 1.71 2.56 3.41 4.26 5.12 5.97 6.82 7.67 8.53

0.52 1.04 1.57 2.09 2.61 3.13 3.66 4.18 4.70 5.22

0.85 1.70 2.55 3.40 4.25 5.10 5.95 6.80 7.65 8.50

0.53 1.05 1.58 2.10 2.63 3.16 3.68 4.21 4.74 5.26

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

9.43 10.29 11.14 12.00 12.86 13.71 14.57 15.43 16.29 17.14

5.67 6.18 6.70 7.21 7.73 8.24 8.76 9.27 9.79 10.30

9.40 10.26 11.97 12.82 13.68 14.53 15.39 16.24 17.10

5.71 6.23 6.74 7.26 7.78 8.30 8.82 9.34 9.86 10.38

9.38 10.23 11.08 11.94 12.79 13.64 14.49 15.35 16.20 17.05

5.75 6.27 6.79 7.31 7.84 8.36 8.88 9.40 9.93 10.45

9.35 10.20 11.05 11.90 12.76 13.61 14.46 15.31 16.16 17.01

5.79 6.31 6.84 7.37 7.89 8.42 8.95 9.47 10.00 10.52

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

18.00 18.86 19.71 20.57 21.43 22.29 23.14 24.00 24.86 25.71

10.82 11.33 11.85 12.36 12.88 13.39 13.91 14.42 14.94 15.45

17.95 18.81 19.66 20.52 21.37 22.23 23.08 23.94 24.79 25.65

10.89 11.41 11.93 12.45 12.97 13.49 14.01 14.53 15.04 15.56

17.91 18.76 19.61 20.46 21.32 22.17 23.02 23.87 24.73 25.58

10.97 11.49 12.02 12.54 13.06 13.58 14.11 14.63 15.15 15.67

17.86 18.71 19.56 20.41 21.26 22.11 22.96 23.81 24.66 25.51

11.05 11.58 12.10 12.63 13.16 13.68 14.21 14.73 15.26 15.79

21 22 23 24 25 26 , 27 28 29 30

31 32 33 34 35 36 37 38 39 40

26.57 27.43 28.29 29.14 30.00 30.86 31.72 32.57 33.43 34.29

15.97 16.48 17.00 17.51 18.03 18.54 19.06 19.57 20.09 20.60

26.50 27.36 28.21 29.07 29.92 30.78 31.63 32.49 33.34 34.20

16.08 16.60 17.12 17.64 18.16 18.68 19.19 19.71 20.23 20.75

26.43 27.28 28.14 28.99 29.84 30.70 31.55 32.40 33.25 34.11

16.20 16.72 17.24 17.76 18.29 18.81 19.33 19.85 20.38 20.90

26.36 27.21 28.06 28.91 29.76 30.61 31.46 32.31 33.16 34.01

16.31 16.84 17.37 17.89 18.42 18.94 19.47 20.00 20.52 21.05

31 32 33 34 35 36 37 38 39 40

« 41 42 43 44 45 46 - 47 48 49 50

35.14 36.00 36.86 37.72 38.57 39.43 40.29 41.14 42.00 42.86

21.12 21.63 22.15 22.66 23.18 23.69 24.21 24.72 25.24 25.75

35.05 35.91 36.76 37.62 38.47 39.33 40.18 41.04 41.89 42.75

21.27 21.79 22.31 22.83 23.34 23.86 24.38 24.90 25.42 25.94

34.96 35.81 36.66 37.52 38.37 39.22 40.07 40.93 41.78 42.63

21.42 21.94 22.47 22.99 23.51 24.03 24.56 25.08 25.60 26.12

34.86 35.71 36.57 37.42 38.27 39.12 39.97 40.82 41.67 42.52

21*57 22.10 22.63 23.15 23.68 24.21 24.73 25.26 25.78 26.31

41 42 ' 43 44 45 46 47 48 49 50

« o S

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

i

' Q

59 Deg.

11.11

58} Deg.

58* Deg.

V

o

rta 58} Deg.

so

s

:

1

TRAVERSE TABLE. 31$ Deg.

31} Deg.

31} Deg.

Distance.

Distance.

31 Deg. Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 ' 52 53 54 55 56 57 58 59 60

43.72 44.57 45.43 46.29 47.14 48.00 48.86 49.72 50.57 51.43

26.27 26.78 27.30 27.81 28.33 28.84 29.36 29.87 30.39 30.90

43.60 44.46 45.31 46.17 47.02 47.88 48.73 49.58 50.44 51.29

26.46 26.98 27.49 28.01 28.53 29.05 29.57 30.09 30.61 31.13

43.48 44.34 45.19 46.04 46.90 47.75 48.60 49.45 50.31 51.16

26.65 27.17 27.69 28.21 28.74 29.26 29.78 30.30 30.83 31.35

43.37 44.22 45.07 45.92 46.77 47.62 48.47 49.32 50.17 51.02

26.84 27.36 27.89 28.42 28.94 29.47 29.99 30.52 31.05 31.57

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

52.29 53.14 54.00 54.86 55.72 56.57 57.43 58.29 59.14 60.00

31.42 31.93 32.45 32.96 33.48 33.99 34.51 35.02 35.54 36.05

52.15 53.00 53.86 54.71 55.57 56.42 57.28 58.13 58.99 59.84

31.65 32.16 32.68 33.20 33.72 34.24 34.76 35.28 35.80 36.31

52.01 52.86 53.72 54.57 55.42 56.27 57.13 57.98 58.83 59.68

31.87 32.39 32.92 33.44 33.96 34.48 35.01 35.53 36.05 36.57

51.87 52.72 53.57 54.42 55.27 56.12 56.98 57.82 58.67 59.52

32.10 32.63 33.15 33.68 34.20 34.73 35.26 35.78 36.31 36.83

61 62 63 64 65 66 67 68 69 70

71 ' 72 73 74 75 76 77 78 79 80

60.86 61.72 62.57 63.43 64.29 65.14 66.00 66.86 67.72 68.57

36.57 37.08 37.60 38.11 38.63 39.14 39.66 40.17 40.69 41.20

60.70 61.55 62.41 63.26 64.12 64.97 65.83 66.68 67.54 68.39

36.83 37.35 37.87 38.39 38.91 39.43 39.95 40.46 40.98 41.50

60.54 61.39 62.24 63.10 63.95 64.80 65.65 66.51 67.36 68.21

37.10 37.62 38.14 38.66 39.19 39.71 40.23 40.75 41.28 41.80

60.37 61.23 62.08 62.93 63.78 64.63 65.48 66.33 67.18 68.03

37.36 37.89 38.41 38.94 39.47 39.99 40.52 41.04 41.57 42.10

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

69.43 70.29 71.14 72.00 72.86 73.72 74.57 75.43 76.29 77.15

41.72 42.23 42.75 43.26 43.78 44.29 44.81 45.32 45.84 46.35

69.25 70.10 70.96 71.81 72.67 73.52 74.38 75.23 76.09 76.94

42.02 42.54 43.06 43.58 44.10 44.61 45.13 45.65 46.17 46.69

69.06 69.92 70.77 71.62 72.47 73.33 74.18 75.03 75.88 76.74

42.32 42.84 43.37 43.89 44.41 44.93 45.46 45.98 46.50 47.02

68.88 69.73 70.58 71.43 72.28 73.13 73.98 74.83 75.68 76.53

42.62 43.15 43.68 44.20 44.73 45.25 45.78 46.31 46.83 47.36

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

78.00 78.86 79.72 80.57 81.43 82.29 83.15 84.00 84.86 85.72

46.87 47.38 47.90 48.41 48.93 49.44 49.96 50.47 50.99 51.50

77.80 78.65 79.51 80.36 81.22 82.07 82.93 83.78 84.64 85.49

47.21 47.73 48.25 48.76 49.28 49.80 * 50.32 50.84 51.36 51.88

77.59 78.44 79.30 80.15 81.00 81.85 82.71 83.56 84.41 85.26

47.55 48.07 48.59 49.11 49.64 50.16 50.68 51.20 51.73 52.25

77.38 78.23 79.08 79.93 80.78 81.63 82.48 83.33 84.18 85.04

47.89 91 48.41 92 48.94 93 49.47 94 49.99 95 50.52 96 51.04 97 51.57 f 98 52.10 99 52. G2 100

Dep.

Lat.

Dep.

Dep.

Lat.

Dep.

Lat.

V

o a ct

s m

59 Deg.

Lat.

V

o

jj 58} Deg.

| Hi

YX

58$ t)eg.

58$ Deg. Q

06 o 5*

TKAVERSE TABLE. 321 Deg.

32 Deg.

32* Deg.

O

32} Deg.

? a ?

Lat.

Dep.

Lat.

i 2 3 4 5 6 7 8 9 10

0.85 1.70 2.54 3.39 4.24 5.09 5.94 6.78 7.63 8.48

0.53 1.06 1.59 2.12 2.65 3.18 3.71 4.24 4.77 5.30

0.85 1.69 2.54 3.38 4.23 5.07 5.92 6.77 7.61 8.46

0.53 1.07 1.60 2.13 2.67 3.20 3.74 4.27 4.80 5.34

0.84 1.69 2.53 3.37 4.22 5.06 5.90 6.75 7.59 8.43

0.54 1.07 1.61 2.15 2.69 3.22 3.76 4.30 4.84 5.37

0.84 1.68 2.52 3.36 4.21 5.05 5.89 6.73 7.57 8.41

0.54 1.08 1.62 2.16 2.70 3.25 3 79 4.33 4.87 5.41

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

9.33 10.18 11.02 11.87 12.72 13.57 14.42 15.26 16.11 16.96

5.83 6.36 6.89 7.42 7.95 8.48 9.01 9.54 10.07 10.60

9.30 10.15 10.99 11.84 12.69 13.53 14.38 15.22 16.07 16.91

5.87 6.40 6.94 7.47 8.00 8.54 9.07 9.61 10.14 10.67

9.28 10.12 10.96 11.81 12.65 13.49 14.34 15.18 16.02 16.87

5.91 6.45 6.98 7.52 8.06 8.60 9.13 9.67 10.21 10.75

9.25 10.09 10.93 11.77 12.62 13.46 14.30 15.14 15.98 16.82

5.95 6.49 7.03 7.57 8.11 8.66 9.20 9.74 10.28 10.82

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 . 30

17.81 18.66 19.51 20.35 21.20 22.05 22.90 23.75 24.59 25.44

11.13 11.66 12.19 12.72 13.25 13.78 14.31 14.84 15.37 15.90

17.76 18.61 19.45 20.30 21.14 21.99 22.83 23.68 24.53 25.37

11.21 11.74 12.27 12.81 13.34 13.87 14.41 14.94 15.47 16.01

17.71 18.55 19.40 20.24 21.08 21.93 22.77 23.61 24.46 25.30

11.28 11.82 12.36 12.90 13.43 13.97 14.51 15.04 15.58 16.12

17.66 18.50 19.34 20.18 21.03 21.87 22.71 23.55 24.39 25.23

11.36 11.90 12.44 12.98 13.52 14.07 14.61 15.15 15.69 16.23

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

26.29 27.14 27.99 28.83 29.68 30.53 31.38 32.23 33.07 33.92

16.43 16.96 17.49 18.02 18.55 19.08 19.61 20.14 20.67 21.20

26.22 27.06 27.91 28.75 29.60 30.45 31.29 32.14 32.98 33.83

16.54 17.08 17.61 18.14 18.68 19.21 19.74 20.28 20.81 21.34

26.15 26.99 27.83 28.68 29.52 30.36 31.21 32.05 32.89 33.74

16.66 17.19 17.73 18.27 18.81 19.34 19.88 20.42 20.95 21.49

26.07 26.91 27.75 28.60 29.44 30.28 31.12 31.96 32.80 33.64

16.77 17.31 17.85 18.39 18.93 19.48 20.02 20.56 21.10 21.64

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 17 48 49 50

34.77 35.62 36.47 37.31 38.16 39.01 39.86 40.71 41.55 42.40

21.73 22.26 22.79 23.32 23.85 24.38 24.91 25.44 25.97 26.50

34.67 35.52 36.37 37.21 38.06 38.90 39.75 40.59 41.44 42.29

21.88 22.41 22.95 23.48 24.01 24.55 25.08 25.61 26.15 26.68

34.58 35.42 36.27 37.11 37.95 38.80 39.64 40.48 41.33 42.17

22.03 22.57 23.10 23.64 24.18 24.72 25.25 25.79 26.33 26.86

34.48 35.32 36.16 37.01 37.85 38.69 39.53 40.37 41.21 42.05

22.18 22.72 23.26 23.80 24.34 24.88 25.43 25.97 26.51 27.05

41 42 43 44 45 46 47 48 49 ' 50

V o

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

s s in

58 Deg.

Dep.

57| Deg. i

Lat.

Dep.

57* Deg.

Lat.

w n

Dep.

57} Deg.

V o c ec |

S

67

TRAVERSE TABLE.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

43.25 44.10 44.95 45.79 46.64 47.49 48.34 49.19 50.03 50.88

27.03 27.56 28.09 28.62 29.15 29.68 30.21 30.74 31.27 31.80

43.13 43.98 44.82 45.67 46.51 47.36 48.21 49.05 49.90 50.74

27.21 27.75 28.28 23.82 29.35 29.88 30.42 30.95 31.48 32.02

43.01 43.86 44*. 70 45.54 46.39 47.23 48.07 48.92 49.76 50.60

27.40 27.94 28.48 29.01 29.55 30.09 30.63 31.16 31.70 32.24

42.89 43.73 44.58 45.42 46.26 47.10 47.94 48.78 49.62 50.46

27.59 28.13 28.67 29.21 29.75 30.29 30.84 31.38 31.92 32.46

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

51.73 52.58 53.43 54.28 55.12 55.97 56.82 57.67 58.52 59.36

32.33 32.85 33.38 33.91 34.44 34.97 35.50 36.03 36.56 37.09

51.59 52.44 53.28 54.13 54.97 55.82 56.66 57.51 58.36 59.20

32.55 33.08 33.62 34.15 34.68 35.22 35.75 36.29 36.82 37.35

51.45 52.29 53.13 53.98 54.82 55.66 56.51 57.35 58.19 59.04

32.78 33.31 33.85 34.39 34.92 35.46 36.00 36.54 37.07 37.61

51.30 52.14 52.99 53.83 54.67 55.51 56.35 57.19 58.03 58.87

33.00 33.54 34.08 34.62 35.16 35.70 36.25 36.79 37.33 37.87

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

60.21 61.06 61.91 62.76 63.60 64.45 65.30 66.15 67.00 67.84

37.62 38.15 38.68 39.21 39.74 40.27 40.80 41.33 41.86 42.39

60.05 60.89 61.74 62.58 63.43 64.28 65.12 65.97 66.81 67.66

37.89 38.42 38.95 39.49 40.02 40.55 41.09 41.62 42.16 42.69

59.88 60.72 61.57 62.41 63.25 64.10 64.94 65.78 66.63 67.47

38.15 38.69 39.22 39.76 40.30 40.83 41.37 41.91 42.45 42.98

59.71 60.55 61.40 62.24 63.08 63.92 64.76 65.60 66.44 67.28

38.41 38.95 39.49 40.03 40.57 41.11 41.65 42.20 42.74 43.28

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

68.69 69.54 70.39 71.24 72.08 72.93 73.78 74.63 75.48 76.32

42.92 43.45 43.93 44.51 45.04 45.57 46.10 46.63 47.16 47.69

68.50 69.35 70.20 71.04 71.89 72.73 73.58 74.42 75.27 76.12

43.22 43.76 44.29 44.82 45.36 45.89 46.42 46.96 47.49 48.03

68.31 69.16 70.00 70.84 71.69 72.53 73.38 74.22 75.06 75.91

43.52 44.06 44.60 45.13 45.67 46.21 46.75 47.28 47.82 48.36

68.12 68.97 69.81 70.65 71.49 72.33 73.17 74.01 74.85 75.69

43.82 44.36 44.90 45.44 45.98 46.52 47.06 47.61 48.15 48.69

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

77.17 78.02 78.87 79.72 80.56 81.41 82.26 83.11 83.96 84.80

48.22 48.75 49.28 49.81 50.34 50.87 51.40 51.93 52.46 52.99

76.96 77.81 78.65 79.50 30.34 81.19 82.04 82.88 83.73 84.57

48.56 49.09 49.63 50.16 50.69 51.23 51.76 52.29 52.83 53.36

76.75 77.59 78.44 79.28 80.12 80.97 81.81 82.65 83.50 84.34

48.89 49.43 49.97 50.51 51.04 51.58 52.12 52.66 53.19 53.73

76.53 77.38 78.22 79.06 79.90 80.74 81.58 82.42 83.26 84.10

49.23 91 49.77 92 50.31 93 50.85 94 51.39 95 51.93 96 52.47 97 53.02 98 53.56 99 54.10 100

ao

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

32 Deg.

32i Deg.

|

32$ Deg.

32} Deg.

Lat.

rt

3

58 Deg.

57| Deg.

57J Deg.

57} Deg.

6 o S3 fit
s

TRAVERSE TABLE.

68 g 5’ P 3 O o

33 Deg.

33* Deg.

33J Deg.

Dep.

Lat.

Dep.

Lat.

2 3 4 5 6 ' 7 8 9 10

0.84 1.68 2.52 3.35 4.19 5.03 5.87 6.71 7.55 8.39

0.54 1.09 1.63 2.18 2.72 3.27 3.81 4.36 4.90 5.45

0.84 1.67 2.51 3.35 4.18 5.02 5.85 6.69 7.53 8.36

0.55 1.10 1.64 2.19 2.74 3.29 3.84 4.39 4.93 5.48

0.83 1.67 2.50 3.34 4.17 5.00 5.84 6.67 7.50 8.34

0.55 1.10 1.66

11 12 13 14 15 16 17 18 19 20

9.23 10.06 10.90 11.74 12.58 13.42 14.26 15.10 15.93 16.77

5.99 6.54 7.08 7.62 8.17 8.71 9.26 9.80 10.35 10.89

9.20 10.04 10.87 11.71 12.54 13.38 14.22 15.05 15.89 16.73

6.03 6.58 7.13 7.68 8.22 8.77 9.32 9.87 10.42 10.97

21 22 23 24 25 26 27 28 29 30

17.61 18.45 19.29 20.13 20.97 21.81 22.64 23.48 24.32 25.16

11.44 11.98 12.53 13.07 13.62 14.16 14.71 15.25 15.79 16.34

17.56 18.40 19.23 20.07 20.91 21.74 22.58 23.42 24.25 25.09

31 32 33 34 35 36 37 38 39 40

26.00 26.84 27.68 28.51 29.35 30.19 31.03 31.87 32.71 33.55

16.88 17.43 17.97 18.52 19.06 19.61 20.15 20.70 21.24 21.79

41 42 43 44 45 46 47 48 49 50

34.39 35.22 36.06 36.90 37.74 38.58 39.42 40.26 41.09 41.93

4

Dep.

) O

Dep.

Lat.

Dep.

s O to

0.56 1.11 1.67

2.76 3 31 3.86 4.42 4.97 5.52

0.83 1.66 2.49 3.33 4.16 4.99 5.82 6.65 7.48 8.31

9.17 10.01 10.84 11.67 12.51 13.34 14.18 15.01 15.84 16.68

6.07 6.62 7.18 7.73 8.28 8.83 9.38 9.93 10.49 11.04

9.15 9.98 10.81 11.64 12.47 13.30 14.13 14.97 15.80 16.63

6.11 6.67 7.22 7.78 8.33 8.89 9.44 10.00 10.56 11.11

11 12 13 14 15 16 17 18 19 20

11.51 12.06 12.61 13.16 13.71 14.26 14.80 15.35 15. Q0 16.45

17.51 18.35 19.18 20.01 20.85 .21.64 22.51 23.35 24.18 25.02

11.59 12.14 12.69 13.25 13.80 14.35 14.90 15.45 16.01 16.56

17.46 18.29 19.12 19.96 20.79 2*1.62 22.45 23.28 24.11 24.94

11.67 12.78 13.33 13.89 14.44 15.00 15.56 16.11 16.67

21 22 23 24 25 26 27 28 29 30

25.92 26.76 27.60 28.43 29.27 30.11 30.94 31.78 32.62 33.45

17.00 17.55 18.09 18.64 19.19 19.74 20.29 20.84 21.38 21.93

25.85 26.68 27.52 28.35 29.19 30.02 30.85 31.69 32.52 33.36

17.11 17.66 18.21 18.77 19.32 19.87 20.42 20.97 21.53 22.08

25.78 26.61 27.44 28.27 29.10 29.93 30.76 31.60 32.43 33.26

17.22 17.78 18.33 18.89 19.44 20.00 20.56 21.11 21.67 22.22

31 32 33 34 35 36 37 38 39 40

22.33 22.87 23.42 23.96 24.51 25.05 25.60 26.14 26.69 27.23

34.29 35.12 35.96 36.80 37.63 38.47 39.31 40.14 40.98 41.81

22.48 23.03 23.58 24.12 24.67 25.22 25.77 26.32 26.87 27.41

34.19 35.02 35.86 36.69 37.52 38.36 39.19 40.03 40.86 41.69

22.63 23.18 23.73 24.29 24.84 25.39 25.94 26.49 27.04 27.60

34.09 34.92 35.75 36.58 37.42 38.25 39.08 39.91 40.74 41.57

22.78 41 23.33 42 23.89 43 24.45 44 25.00 45 25.56 1 46 26.11 47 26.67 48 27.22 49 27.78 50

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

2.21

1 2 3 2.22 4 2.78 5 3.33 6 3.89 1 7 4.44 8 5.00 9 5.56 10

12.22

Lat.

rt 57 Deg,

5C| Deg.

56*:Deg.

o6

rtd

O

s

g . 5* V

Lat.

1

33} Deg.

56* :Deg.

Q |

69

TRAVERSE TABLE.

g00 p o

33 Deg.

33} Deg.

33* Deg.

33} Deg.

g as'

P j3

O

»

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

9

51 52 53 54 55 56 57 58 59 60

42.77 43.61 44.45 45.29 46.13 46.97 47.80 48.64 49.48 50.32

27.73 28.32 28.87 29.41 29.96 30.50 31.04 31.59 32.13 32.68

42.65 43.49 44.32 45.16 46.00 46.83 47.67 48.50 49.34 50.18

27.96 28.51 29.06 29.61 30.16 30.70 31.25 31.80 32.35 32.90

42.53 43.36 44.20 45.03 45.86 46.70 47.53 48.37 49.20 50.03

28.15 28.70 29.25 29.80 30.36 30.91 31.46 32.01 32.56 33.12

42.40 43.24 44.07 44.90 45.73 46.56 47.39 48.23 49.06 49.89

28.33 28.89 29.45 30.00 30.56 31.11 31.67 32.22 32.78 33.33

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

51.16 52.00 52.84 53.67 54.51 55.35 56.19 57.03 57.87 58.71

33.22 33.77 34.31 34.86 35.40 35.95 36.49 37.04 37.58 38.12

51.01 51.85 52.69 53.52 54.36 55.19 56.03 56.87 57.70 58.54

33.45 33.99 34.54 35.09 35.64 36.19 36.74 37.28 37.83 38.38

50.87 51.70 52.53 53.37 54.20 55.04 55.87 56.70 57.54 58.37

33.67 34.22 34.77 35.32 35.88 36.43 36.98 37.53 38.08 38.64

50.72 51.55 52.38 53.21 54.05 54.88 55.71 56.54 57.37 58.20

33.89 34.45 35.00 35.56 36.11 36.67 37.22 37.78 38.33 38.89

61 62 63 64 65 66 67 68 69 ' 70

71 72 73 74 75 76 77 78 79 80

59.55 60.38 61.22 62.06 62.90 63.74 64.58 65.42 66.25 67.09

38.67 39.21 39.76 40.30 40.85 41.39 41.94 42.48 43.03 43.57

59.38 60.21 61.05 61.89 62.72 63.56 64.39 65.23 66.07 66.90

38.93 39.48 40.03 40.57 41.12 41.67 42.22 42.77 43.32 43.86

59.21 60.04 60.87 61.71 62.54 63.38 64.21 65.04 65.88 66.71

39.19 39.74 40.29 40.84 41.40 41.95 42.50 43.05 43.60 44.15

59.03 59.87 60.70 61.53 62.36 63.19 64.02 64.85 65.69 66.52

39.45 40.00 40.56 41.11 41.67 42.22 42.78 43.33 43.89 44.45

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

67.93 68.77 69.61 70.45 71.29 72.13 72.96 73.80 74.64 75.48

44.12 44.66 45.20 45.75 46.29 46.84 47.38 47.93 48.47 49.02

67.74 68.58 69.41 70.25 71.08 71.92 72.76 73.59 74.43 75.27

44.41 44.96 45.51 46.06 46.60 47.15 47.70 48.25 48.80 49.35

67.54 68.38 69.21 70.05 70.88 71.71 72.55 73.38 74.22 75.05

44.71 45.26 45.81 46.36 46.91 47.47 48.02 48.57 49.12 49.67

67.35 68.18 69.01 69.84 70.67 71.51 72.34 73.17 74.00 74.83

45.00 45.56 46.11 46.67 47.22 47.78 48.33 48.89 49.45 50.00

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

76.32 77.16 78.00 78.83 79.67 80.51 81.35 82.19 83.03 83.87

49.56 50.11 50.65 51.20 51.74 52.29 52.83 53.37 53.92 54.46

76.10 76.94 77.77 78.61 79.45 80.28 81.12 81.96 82.79 83.63

49.89 50.44 50.99 51.54 52.09 52.64 53.18 53.73 54.28 54.83

75.88 76.72 77.55 78.39 79.22 80.05 80.89 81.72 82.55 83.39

50.23 50.78 51.33 51.88 52.43 52.99 53.54 54.09 54.64 55.19

75.66 76.50 77.33 78.16 78.99 79.82 80.65 81.48 82.32 83.15

50.56 91 92 51.11 51.67 93 52.22 94 52.78 95 53.33 96 53.89 97 54.45 98 55.00 99 55.56 100

Dep.

Lat.

Dep.

Lat.

Lat.

Dep.


o

|I

Dep, 1

Lat.

6 o

ri 57 Deg.

561 Deg.

56* Deg.

56^ Deg.

3

TRAVERSE TABLE.

70 g

34 Deg.

34* Deg.

34* Deg.

34* De5.

ST

g ST 3

g

Lat,

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.83 1.66 2.49 3.32 4.15 4.97 5.80 6.63 7.46 8.29

0.56 1.12 1.68 2.24 2.80 3.36 3.91 4.47 5.03 5.59

0.83 1.65 2.48 3.31 4.13 4.96 5.79 6.61 7.44 8.27

0.56 1.13 1.69 2.25 2.81 3.38 3.94 4.50 5.07 5.63

0.82 1.65 2.47 3.30 4.12 4.94 5.77 6.59 7.42 8.24

11 12 13 14 15 16 17 18 19 20

9.12 9.95 10.78 11.61 12.44 13.26 14.09 14.92 15.75 16.58

6.15 6.71 7.27 7.83 8.39 8.95 9.51 10.07 10.62 11.18

9.09 9.92 10.75 11.57 12.40 13.23 14.05 14.88 15.71 16.63

6.19 6.75 7.32 7.8& 8.44 9.00 9.57 10.13 10.69 11.26

21 22 23 24 25 26 27 28 29 30

17.41 18.24 19.07 19.90 20.73 21.55 22.38 23.21 24.04 24.87

11.74 12.30 12.86 13.42 13.98 14.54 15.10 15.66 16.22 16.78

31 32 33 34 35 36 37 38 39 40

25.70 26.53 27.36 28.19 29.02 29.85 30.67 31.50 32.33 33.16

17.33 17.89 18.45 19.01 19.57 20.13 20.69 21.25 21.81 22.37

41 42 43 44 45 46 47 48 49 50

33.99 34.82 35.65 36.48 37.31 38.14 38.96 39.79 40.6.2 41.45 Dep.

O

Dep.

Lat.

Dep.

Dep.

0.57 1.13 1.70 2.27 2.83 3.40 3.96 4.53 5.10 5.66

0.82 1.64 2.46 3.29 4.11 4.93 5.75 6.57 7.39 8.22

0.57 1.14 1.71 2.28 2.85 3.42 3.99 4.56 5.13 5.70

1 2 3 4 5 6 7 8 9 10

9.07 9.89 10.71 11.54 12.36 13.19 14.01 14.83 15.66 16.48

6.23 6.80 7.36 7.93 8.50 9.06 9.63 10.20 10.76 11.33

9.04 9.86 10.68 11.50 12.32 13 15 13.97 14.79 15.61 16.43

6.27 6.84 7.41 7.98 8.55 9.12 9.69 10.26 10.83 11.40

11 12 13 14 15 16 17 18 19 20

17.36 11.82 12.38 18.18 19.01 12.94 19,84 13.51 20.66 , 14.07 21.49 14.63 22.32 15.20 23.14 15.76 23.97 16.32 24.80 16.88

17.31 18.13 18.95 19.78 20.60 21.43 22.25 23.08 23.90 24.72

11.89 12.46 13.03 13.59 14.16 14.73 15.29 15.86 16.43 16.99

17.25 18.08 18.90 19.72 20.54 21.36 22.18 23.01 23.83 24.65

11.97 12.54 13.11 13.68 14.25 14.82 15.39 i5.96 16.53 17.10

21 22 23 24 25 26 27 28 29 30

25.62 26.45 27.28 28.10 28.93 29.76 30.58 31.41 32.24 33.06

17.45 18.01 18.57 19.14 19.70 20.26 20.82 21.39 21.95 22.51

25.55 26.37 27.20 28.02 28.84 29.67 30.49 31.32 32.14 32.97

17.56 18.12 18.69 19.26 19.82 20.39 20.96 21.52 22.09 22.66

25.47 26.29 27.11 27.94 28.76 29.58 30.40 31.22 32.04 32.87

17.67 18.24 18.81 19.38 19.95 20.52 21.09 21.66 22.23 22.80

31 32 33 34 35 36 37 38 39 40

22.93 23.49 24.05 24.60 25.16 25.72 26.28 26.84 27.40 27.96

33.89 34.72 35.54 36.37 37.20 38.02 38.85 39.68 40.50 41.33

23.07 23.64 24.20 24.76 25.33 25.89 26.45 27.01 27.58 28.14

33.79 34.61 35.44 36.26 37.09 37.91 38.73 39.56 40.38 41.21

23.22 23.79 24.36 24.92 25.49 26.05 26.62 27.19 27.75 28.32

33.69 34.51 35.33 36.15 36.97 37.80 38.62 39.44 40.26 41.08

23.37 23.94 24.51 25.08 25.65 26.22 26.79 27.36 27.93 28.50

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Dep.

Lat.

Dep. 1 Lat.

56 Deg.

55| Deg.

55* Deg.

0) o

a

1

ei o

a a

Lat.

«

55* Deg.

Q

71

TRAVERSE TABLE g

34 Deg.

34\ Deg.

34J Deg.

34} Deg.

g CO

9

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

p C O CO

51 52 53 54 55 56 57 58 59 60

42.28 43.11 43.94 44.77 45.60 46.43 47.26 48.08 48.91 49.74

28.52 29.03 29.64 30.20 30.76 31.31 31.87 32.43 32.99 33.55

42.16 42.98 43.81 44.64 45.46 46.29 47.12 47.94 48.77 49.60

28.70 29.27 29.83 30.39 30.95 31.52 32.08 32.64 33.21 33.77

42.03 42.85 43.68 44.50 45.33 46.15 46.98 47.80 48.62 49.45

28.89 29.45 30.02 30.59 31.15 31.72 32.29 32.85 33.42 33.98

41.90 42.73 43.55 44.37 45.19 46.01 46.83 47.66 48.48 49.30

29.07 29.64 30.21 30.78 31.35 31.92 32.49 33.06 33.63 34.20

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

50.57 51.40 52.23 53.06 53.89 54.72 55.55 56.37 57.20 58.03

34.11 34.67 35.23 35.79 36.35 36.91 37.46 38.03 38.58 39.14

50.42 51.25 52.08 52.90 53.73 54.55 55.38 56.21 57.03 57.86

34.33 34.89 35.46 36.02 36.58 37.15 37.71 38.27 38.83 39.40

50.27 51.10 51.92 52.74 53.57 54.39 55.22 56.04 56.86 57.69

34.55 35.12 35.68 36.25 36.82 37.38 37.95 38.52 39.08 39.65

50.12 50.94 51.76 52.59 53.41 54.23 55.05 55.87 56.69 57.52

34.77 35.34 35.91 36.48 37.05 37.62 38.19 38.76 39.33 39.90

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

58.86 59.69 60.52 61.35 62.18 63.01 63.84 64.66 65.49 66.32

39.70 40.26 40.82 41.38 41.94 42.50 43.06 43.62 44.18 44.74

58.69 59.51 60.34 61.17 61.99 62.82 63.65 64.47 65.30 66.13

39.96 40.52 41.08 41.65 42.21 42.77 43.34 43.90 44.46 45.02

58.51 59.34 60.16 60.99 61.81 62.63 63.46 64.28 65.11 65.93

40.21 40.78 41.35 41.91 42.48 43.05 43.61 44.18 44.75 45.31

58.34 59.16 59.98 60.80 61.62 62.45 63.27 64.09 64.91 65.73

40.47 41.04 41.61 42.18 42.75 43.32 43.89 44.46 45.03 45.60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

67.15 67.98 68.81 69.64 70.47 71.30 72.13 72.96 73.78 74.61

45.29 45.85 46.41 46.97 47.53 48.09 48.65 49.21 49 .’77 50.33

66.95 67.78 68.61 69.43 70.26 71.09 71.91 72.74 73.57 74.39

45.59 46.15 46.71 47.28 47.84 48.40 48.96 49.53 50.09 50.65

66.75 67.58 68.40 69.23 70.05 70.87 71.70 72.52 73.35 74.17

45.88 46.45 47.01 47.58 48.14 48.71 49.28 49.84 50.41 50.98

66.55 67.37 68.20 69.02 69.84 70.66 71.48 72.30 73.13 73.95

46.17 16.74 47.31 47.88 48.45 49.02 49.59 50.16 50.73 51.30

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

75.44 76.27 77.10 77.93 78.76 79.59 80.42 81.25 82.07 82.90

50.89 51.45 52.00 52.56 53.12 53.68 54.24 54.80 55.36 55.92

75.22 76.05 76.87 77.70 78.53 79.35 80.18 81.01 81.83 82.66

51.22 51.78 52.34 52.90 53.47 54.03 54.59 55.15 55.72 56.28

75.00 75.82 76.64 77.47 78.29 79.12 79.94 80.76 81.59 82.41

51.54 52.11 52.68 53.24 53.81 54.37 54.94 55.51 56.07 56.64

74.77 75.59 76.41 77.23 78.06 78.88 79.70 80.52 81.34 82.16

51.87 91 52.44 92 53.01 93 53.58 94 54.15 95 54.72 96 55.29 97 55.86 98 56.43 99 57.00 100

ci

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

p 3 O

o J3

rt

s w

56 Deg.

55} Deg.

55J Deg.

55| Deg

V o § Q

72

TRAVERSE TABLE,

Dep.

Distance.

Distance.

35 Deg

0.81 1.62 2.43 3.25 4.06 4.87 5.68 6.49 7.30 8.12

0.58 1.17 1.75 2.34 2.92 3.51 4.09 4.67 5.26 5.84

1 2 3 4 5 6 7 8 9 10

6.39 6.97 7.55 8.13 8.71 9.29 9.87 10.45 11.03 11.61

8.93 9.74 10.55 11.36 12.17 12.99 13.80 14.61 15.42 16.23

6.43 7.01 7.60 8.18 8.76 9.35 9.93 10.52 11.10 11.68

11 12 13 14 15 16 17 18 19 20

17.10 17.91 18.72 19.54 20.35 21.17 21.98 22.80 23.61 24.42

12.19 12.78 13.36 13.94 14.52 15.10 15.68 16.26 16.84 17.42

17.04 17.85 18.67 19.48 20.29 21.10 21.91 22.72 23.54 24.35

12.27 12.85 13.44 14.02 14.61 15.19 15.77 16.36 16.94 17.53

21 22 23 24 25 26 27 28 29 30

25.24 26.05 26.87 27.68 28.49 29.31 30.12 30.94 31.75 32.56

18.00 18.58 19.16 19.74 20.32 20.91 21.49 22.07 22.65 23.23

25.16 25.97 26.78 27.59 28.41 29.22 30.03 30.84 31.65 32.46

18.11 31 18.70 32 19.28 33 19.86 34 20.45 35 21.03 36 21.62 37 22.20 38 22.79 i 39 23.37 40

23.66 j 33.38 24.24 34.19 24.82 35.01 25.39 35.82 25.97 36.64 26.55 37.45 27.13 38.26 27.70 39.08 28.28 39.89 28.86 40.71

23.81 24.39 24.97 25.55 26.13 26.71 27.29 27.87 28.45 29.04

33.27 34.09 34.90 35.71 36.52 37.33 38.14 38.96 39.77 40.58

23.95 24.54 25.12 25.71 26.29 26.88 27.46 28.04 28.63 29.21

Lat.

Dep.

Lat.

351 Deg.

35^ Deg.

Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.82 1.64 2.46 3.28 4.10 4.91 5.73 6.55 7.37 8.19

0.57 1.15 1.72 2.29 2.87 3.44 4.01 4.59 5.16 5.74

0.82 1.63 2.45 3.27 4.08 4.90 5.72 6.53 7.35 8.17

0.58 1.15 1.73 2.31 2.89 3.46 4.04 4.62 5.19 ; 5.77

0.81 1.63 2.44 3.26 4.07 4.88 5.70 6.51 7.35 8.14

0.58 1.16 1.74 2.32 2.90 3.48 4.06 4.65 5.23 5.81

11 12 13 14 15 16 17 18 19 20

9.01 9.83 10.65 11.47 12.29 13.11 13.93 14.74 15.56 16.38

6.31 6.88 7.46 8.03 8.60 9.18 9.75 10.32 10.90 11.47

8.98 9.80 10.62 11.43 12.25 13.07 13.88 14.70 15.52 16.33

6.35 | 8.96 6.93 | 9.77 7.50 i 10.58 8.08 11.40 8.66 12.21 9.23 13.03 9.81 13.84 10.39 14.65 10.97 15.47 11.54 16.28

21 22 23 24 25 26 27 28 29 30

17.20 18.02 18.84 19.66 20.48 21.30 22.12 22.94 23.76 24.57

12.05 12.62 13.19 13.77 14.34 14.91 15.49 16.06 16.63 17.21

17.15 17.97 18.78 19.60 20.42 21.23 22.05 22.87 23.68 24.50

12.12 12.70 13.27 13.85 14.43 15.01 15.58 16.16 16.74 17.31

31 32 33 34 35 36 37 38 39 40

25.39 26.21 27.03 27.85 28.67 29.49 30.31 31.13 31.95 32.77

17.78 18.35 18.93 19.50 20.08 20.65 21.22 21.80 22.37 22.94

25.32 26.13 26.95 27.77 28.58 29.40 30.22 31.03 31.85 32.67

17.89 18.47 19.05 19.62 20.20 20.78 21.35 21.93 22.51 23.09

41 42 43 44 45 46 47 48 49 50

33.59 34.40 35.22 36.04 36.86 37.68 38.50 39.32 40.14 40.96

23.52 24.09 24.66 25.24 25.81 26.38 26.96 27.53 28.11 28.68

33.48 34.30 35.12 35.93 36.75 37.57 38.38 39.20 40.02 40.83

v

Dep.

Lat.

Dep.

o

Lat.

i j 1 ! i | 1 ! | I

Dep.

Dep.

35f Deg. Lat.

41 42 43 44 45 46 47 48 49 50 <0

a a

5

c ct

55 Deg.

54f Deg.

54£ Deg.

544 Deg.

|

i

31

73

TRAVERSE TABLE. 3oh Deg.

354 Deg.

35f Deg.

Distance.

j Distance.

35 Deg.

Lat.

Dep.

Lat.

Dep.

29.43 30.01 30.59 31.17 31.74 32.32 32.90 33.47 34.05 34.63

41.52 42.33 43.15 43.96 44.78 45.59 46.40 47.22 48.03 48.85

29.62 30.20 30.78 31.36 31.94 32.52 33.10 33.68 34.26 34.84

41.39 42.20 43.01 43.82 44.64 45.45 46.26 47.07 47.88 48.69

29.80 30.38 30.97 31.55 32.13 32.72 33.30 33.89 34.47 35.05

51 52 53 54 55 56 57 58 59 60

49.82 50.63 51.45 52.27 53.f)8 53.90 54.71 55.53 56.35 57.16

35.21 35.78 36.36 36.94 37.51 38.09 38.67 39.25 39.82 40. 40

49.66 50.48 51.29 52.10 52.92 53.73 54.55 55.36 56.17 56.99

35.42 36.00 36.58 37.16 37.75 38.33 38.91 39.49 40.07 40.65

49.51 50.32 51.13 51.94 52.75 53.56 54.38 55.19 56.00 56.81

35.64 36.22 36.81 37.39 37.98 38.56 39.14 39.73 40.31 40.90

61 62 63 64 65 66 67 68 69 70

40.72 41.30 41.87 42.44 43.02 43.59 44.17 44.74 45.31 45.89

57.98 58.80 59.61 60.43 61.25 62.06 62.88 63.70 64.51 65.33

40.98 41.55 42.13 42.71 43.29 43.86 44.44 45.02 45.59 46.17

57.80 58.62 59.43 60.24 61.06 61.87 62.69 63.50 64.32 65.13

41.23 41.81 42.39 42.97 43.55 44.13 44.71 45.29 45.88 46.46

57.62 58.43 59.24 60.06 60.87 61.68 62.49 63.30 64.11 64.93

41.48 42.07 42.65 43.23 43.82 44.40 44.99 45.57 46.16 46.74

71 72 73 74 75 76 77 78 79 80

66.35 67.17 67.99 68.81 69.63 70.45 71.27 72.09 72.90 73.72

46.46 47.03 47.61 48.18 48.75 49.33 49.90 50.47 51.05 51.62

66.15 66.96 67.78 68.60 69.41 70.23 71.05 71.86 72.68 73.50

46.75 47.33 47.90 48.48 49.06 49.63 50.21 50.79 51.37 51.94

65.94 66.76 67.57 68.39 69.20 70.01 70.83 71.64 72.46 73.27

47.04 47.62 48.20 48.78 49.36 49.94 50.52 51.1*1 51.68 52.26

65.74 66.55 67.36 68.17 68.98 69.80 70.61 71.42 72.23 73.04

47.32 47.91 48.49 49.08 49.66 50.25 50.83 51.41 52.00 52.58

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

74.54 75.36 76.18 77.00 77.82 78.64 79.46 80.28 81.10 81.92

52.20 52.77 53.34 53.92 54.49 55.06 55.64 56.21 56.78 57.36

74.31 75.13 75.95 76.76 77.58 78.40 79.21 80.03 80.85 81.66

52.52 53.10 53.67 54.25 54.83 55.41 55.98 56.56 57.14 57.71

74.08 74.90 75.71 76.53 77.34 78.16 78.97 79.78 80.60 81.41

52.84 53.42 54.01 54.59 55.17 55.75 56.33 56.91 57.49 58.07

73.85 74.66 75.48 76.29 77.10 77.91 78.72 79.53 80.35 81.16

53.17 91 53.75 92 54.34 93 54.92 94 55.50 95 56.09 96 56.67 97 57.26 98 57.84 99 58.42 100

tj o

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 59 60

41.78 42.60 43.42 44.23 45.05 45.87 46.69 47.51 48.33 49.15

29.25 29.83 30.40 30.97 31.55 32.12 32.69 33.27 33.84 34.41

41.65 42.47 43.28 44.10 44.92 45.73 46.55 47.37 48.18 49.00

61 62 63 64 65 65 67 68 69 70

49.97 50.79 51.61 52.43 53.24 54.06 54.88 55.70 56,52 57.34

34.99 35.56 36.14 36.71 37.28 37.86 38.43 39.00 39.58 40.15

71 72 73 74 75 76 77 78 79 80

58.16 58.98 59.80 60.62 61.44 62.26 63.07 63.89 64.71 65.53

81 82 83 84 85 86 87 88 89 90

i

iS

5

55 Deg.

30*

54£ Deg.

54f Deg.

2 V

Lat.

54* :Deg.

cJ o a

3

74

TRAVERSE TABLE.

» Distance.

Distance.

Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 4 5 6 7 8 9 10

0.81 1.62 2.43 3.24 4.05 4.85 5.66 6.47 7.28 8.09

0.59 1.18 1.76 2.35 2.94 3.53 4.11 4.70 5.29 5.88

0.81 1.61 2.42 3.23 4.03 4.84 5.65 6 45 7.26 8.06

0.59 1.18 1.77 2.37 2.96 3.55 4.14 4.73 5.32 5.91

0.80 1.61 2.41 3.22 4.02 4.82 5.63 6.43 7.23 8.04

0.59 1.19 1.78 2.38 2.97 3.57 4.16 4.76 5.35 5.95

0.80 1.60 2.40 3.20 4.01 4.81 5.61 6.41 7.21 8.01

0.60 1.20 1.79 2.39 2.99 3.59 4.19 4.79 5.38 5.98

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 ' 19 20

8.90 9.71 10.52 11.33 12.14 12.94 13.75 14.56 15.37 16.18

6.47 7.05 7.64 8.23 8.82 9.40 9.99 10.58 11.17 11.76

8.87 9.68 10.48 11.29 12.10 12.90 13.71 14.52 15.32 16.13

6.50 7.10 7.69 8.28 8.87 9.46 10.05 10.64 11.23 11.83

8.84 9.65 10.45 11.25 12.06 12.86 13.67 14.47 15.27 16.03

6.54 7.14 7.73 8.33 8.92 9.52 10.11 10.71 11.30 11.90

8.81 9.61 10.42 11.22 12.02 12.82 13.62 14.42 15.22 16.03

6.58 7.18 7.78 8.38 8.97 9.57 10.17 10.77 11.37 11.97

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

16.99 17.80 18.61 19.42 20.23 21.03 21.84 22.65 23.46 24.27

12.34 12.93 13.52 14.11 14.69 15.28 15.87 16.46 17.05 17.63

16.94 17.74 18.55 19.35 20.16 20.97 21.77 22.58 23.39 24.19

12.42 13.01 13.60 14.19 14.78 15.37 15.97 16.56 17.15 17.74

16.88 17.68 18.49 19.29 20.10 20.90 21.70 22.51 23.31 24.12

12.49 13.09 13.68 14.28 14.87 15.47 16.06 16.65 17.25 17.84

16.83 17.63 13.43 19.23 20.03 20.83 21.63 22.44 23.24 24.04

12.56 13.16 13.76 14.36 14.96 15.56 16.15 16.75 17.35 17.95

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

25.08 25.89 26.70 27.51 28.32 29.12 29.93 30.74 31.55 32.36

18.22 18.81 19.40 19.98 20.57 21.16 21.75 22.34 22.92 23.51

25.00 25.81 26.61 27.42 28.23 29.03 29.84 30.64 31.45 32.26

18.33 18.92 19.51 20.10 20.70 21.29 21.88 22.47 23.06 23.65

24.92 25.72 26.53 27.33 28.13 28.94 29.74 30.55 31.35 32.15

18.44 19.03 19.63 20.22 20.82 21.41 22.01 22.60 23.20 23.79

24.84* 25.64 26.44 27.24 28.04 28.85 29.65 30.45 31.25 32.05

18.55 19.15 19.74 20.34 20.94 21.54 22.14 22.74 23.33 23.93

31 32 33 34 35 . 36 37 38 39 40 j

41 42 43

33.17 33.98 34.79 35.60 36.41 37.21 38.02 38.83 39.64 40.45

33.06 24.10 33.87 24.69 34.68 25.27 35.48 25.86 26.45 | 36.29 27.04 i 37.10 37.90 27.63 38.71 28.21 39.52 28.80 40.32 29.39

24.24 24.83 25.43 26.02 26.61 27.20 27.79 28.38 28.97 29.57

32.96 33.76 34.57 35.37 36.17 36.98 37.78 38.59 39.39 40.19

24.39 24.98 25.58 26.17 26.77 27.36 27.96 28.55 29.15 29.74

32.85 33.65 34.45 35.26 36.06 36.86 37.66 38.46 39.26 40.06

24.53 25.13 25.73 26.33 26.92 27.52 28.12 28.72 29.32 29.92

41 42 43 44 45 46 47 48 49 50 |

Lat.

Dep.

Lat.

Dep.

Lat.

u 1

44

45 46 47 48 49

50 a> o

2

36 Deg.

Dep.

Lat.

54 Deg.

36* Deg.

Dep.

53| Deg.

36i Deg.

36| Deg.

Dep.

53J Deg.

Lat.

j

Dep.

53* Deg.

0J

j

a &

s |

75

TRAVERSE TABLE. g So '

ST i^i o A

36 Deg.

36* Deg.

36J Deg.

36j Deg.

g to r*

V P

O

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

51 52 53 54 55 56 57 58 ' 59 60

41.26 42.07 42.83 43.69 44.50 45.30 46.11 46.92 47.73 48.54

29.98 30.56 31.15 31.74 32.33 32.92 33.50 34.09 34.68 35.27

41.13 41.94 42.74 43.55 44.35 45.16 45.97 46.77 47.58 48.39

30.16 30.75 31.34 31.93 32.52 33.11 33.70 34.30 34.89 35.48

41.00 41.80 42.60 43.41 44.21 45.02 45.82 46.62 47.43 48.23

30.34 30.93 31.53 32.12 32.72 33.31 33.90 34.50 35.09 35.69

40.86 41.67 42.47 43.27 44.07 44.87 45.67 46.47 47.27 48.08

30.51 31.11 31.71 32.31 32.91 33.51 34.10 34.70 35.30 35.90

51 52 53 54 55 56 57 ! 58 59 60 '

61 62 63 64 65 66 67 68 69 70

49.35 50.16 50.97 51.78 52.59 53.40 54.20 55.01 55.82 56.63

35.85 36.44 37.03 37.62 38.21 38.79 39.38 39.97 40.56 41.14

49.19 50.00 50.81 51.61 52.42 53.23 54.03 54.84 55.64 56.45

36.07 36.66 37.25 37.84 38.44 39.03 39.62 40.21 40.80 41.39

49.04 49.84 50.64 51.45 52.25 53.05 53.86 54.66 55.47 56.27

36.28 36.88 37.47 38.07 38.66 39.26 39.85 40.45 41.04 41.64

48.88 49.68 50.48 51.28 52.08 52.88 53.68 54.49 55.29 56.09

36.50 37.10 37.69 38.29 38.89 39.49 40.09 40.69 41.28 41.88

61 62 63 64 65 66 67 68 69 70

71 72 /3 74 75 76 77 78 79 80

57.44 58.25 59.06 59.87 60.68 61.49 62.29 63.10 63.91 64.72

41.73 42.32 42.91 43.50 44.08 44.67 45.26 45.85 46.43 47.02

57.26 58.06 58.87 59.68 60.48 61.29 62.10 62.90 63.71 64.52

41.98 42.57 43.17 43.76 44.35 44.94 45.53 46.12 46.71 47.30

57.07 57.88 58.68 59.49 60.29 61.09 61.90 62.70 63.50 64.31

42.23 42.83 43.42 44.02 44.61 45.21 45.80 46.40 46.99 47.59

56.89 57.69 58.49 59.29 60.09 60.90 61.70 62.50 63.30 64.10

42.48 43.08 43.68 44.28 44.87 45.47 46.07 46.67 47.27 47.87

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

65.53 66.34 67.15 67.96 68.77 69.58 70.38 71.19 72.00 72.81

47.61 48.20 48.79 49.37 49.96 50.55 51.14 51.73 52.31 52.90

65.32 66.13 66.93 67.74 68.55 69.35 70.16 70.97 71.77 72.58

47.90 48.49 49.08 49.67 50.26 50.85 51.44 52.04 52.63 53.22

65.11 65.92 66.72 67.52 68.33 69.13 69.94 70.74 71.54 72.35

48.18 48.78 49.37 49.97 50.56 51.15 51.75 52.34 52.94 53.53

64.90 65.70 66.50 67.31 68.11 68.91 69.71 70.51 71.31 72.11

48.46 49.06 49.66 50.26 50.86 51.46 52.05 52.65 53.25 53.85

81 82 83 , 84 85 86 87 88 89 90

91 92 93 94 95 96 , 97 98 99 100

73.62 74.43 75.24 76.05 76.86 77.67 78.47 79.28 80.09 80.90

53.49 54.08 54.66 55.25 55.84 56.43 57.02 57.60 58.19 58.78

73.39 74.19 75.00 75.81 76.61 77.42 78.23 79.03 79.84 80.64

53.81 54.40 54.99 55.58 56.17 56.77 57.36 57.95 58.54 59.13

73.15 73.95 74.76 75.56 76.37 77.17 77.97 78.78 79.58 80.39

54.13 54.72 55.32 55.91 56.51 57.10 57.70 58.29 58.89 59.48

72.91 73.72 74.52 75.32 76.12 76.92 77.72 78.52 79.32 80.13

54.45 91 55.05 92 55.64 93 56.24 94 56.84 95 57.44 96 58.04 97 58.64 98 59.23 99 59.83 100

« o p es

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

m

5

54 Deg.

53| Deg.

53* Deg.

Lat.

53* Deg.

V

o <3

s

76

TRAVERSE TABLE

g35* ST S3 O

©

1

i

37 Deg.

37* Deg.

37* Deg.

; 37} Deg

gu> ’ p

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

a a p

3 4 5 6 7 8 9 10

0.80 1.60 2.40 3.19 3.99 4.79 5.59 6.39 7.19 7.99

0.60 1.20 1.81 2.41 3.01 3.61 4.21 4.81 5.42 6.02

0.80 1.59 2.39 3.18 3.98 4.78 5.57 6.37 7.16 7.96

0.61 1.21 1.82 2.42 3.03 3.63 4.24 4.84 5.45 6.05

0.79 1.59 2.38 3.17 3.97 4.76 5.55 6.35 7.14 7.93

0.61 1.22 1.83 2.43 3.04 3.65 4.26 4.87 5.48 6.09

0.79 1.58 2.37 3.16 3.95 4.74 5.53 6.33 7.12 7.91

0.61 1.22 1.84 2.45 3.06 3.67 4.29 4.90 5.51 6.12

7 8 9 10

11 12 13 14 15 16 17 18 19 20

8.78 9.58 10.38 11.18 11.98 12.78 13.58 14.38 15.17 15.97

6.62 7.22 7.82 8.43 9.03 9.63 10.23 10.83 11.43 12.04

8.76 9.55 10.35 11.14 11.94 12.74 13.53 14.33 15.12 15.92

6.66 7.26 7.87 8.47 9.08 9.68 10.29 10.90 11.50 12.11

8.73 9.52 10.31 11.11 11.90 12.69 13.49 14.28 15.07 15.87

6.70 7.31 7.91 8.52 4.13 9.74 10.35 10.96 11.57 12.18

8.70 9.49 10.28 11.07 11.86 12.65 13.44 14.23 15.02 15.81

6.73 7.35 7.96 8.57 9.18 9.80 10.41 11.02 11.63 12.24

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

16.77 17.57 18.37 19.17 19 97 20.76 21.56 22.36 23.16 23.96

12.64 13.24 13.84 14.44 15.05 15.65 16.25 16.85 17.45 18.05

16.72 17.51 18.31 19.10 19.90 20.70 21.49 22.29 23.08 23.88

12.71 13.32 13.92 14.53 15.13 15.74 16.34 16.95 17.55 18.16

16.66 17.45 18.25 19.04 19.83 20.63 21.42 22.21 23.01 23.80

12.78 13.39 14.00 14.61 15.22 15.83 16.44 17.05 17.65 18.26

16.60 17.40 18.19 18.98 19.77 20.56 21.35 22.14 22.93 23.72

12.86 13.47 14.08 14.69 15.31 15.92 16.53 17.14 17.75 18.37

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

24.76 25.56 26.35 27.15 27.95 28.75 29.55 30.35 31.15 31.95

18.66 19.26 19.86 20.46 21.06 21.67 22.27 22.87 23.47 24.07

24.68 25.47 26.27 27.06 27.86 28.66 29.45 30.25 31.04 31.84

18.76 19.37 19.97 20.58 21.19 21.79 22.40 23.00 23.61 24.21

24.59 25.39 26.18 26.97 27.77 28.56 29.35 30.15 30.94 31.73

18.87 19.48 20.09 20.70 21.31 21.92 22.52 23.13 23.74 24.35

24.51 25.30 26.09 26.88 27.67 28.46 29.26 30.05 30.84 31.63

18.98 19.59 20.20 20.82 21.43 22.04 22.65 23.26 23.88 24.49

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

32.74 33.54 34.34 35.14 35.94 36.74 37.54 38.33 39.13 39.93

24.67 25.23 25.88 26.48 27.08 27.68 28.29 28.89 29.49 30.09

32.64 33.43 34.23 35.02 35.82 36.62 37.41 38.21 39.00 39.80

24.82 ! ! 32.53 25.42 i ' 33.32 26.03 ! 34.11 26.63 ! 34.91 27.24 35.70 27.84 36.49 28.45 37.29 29.05 38.08 29.66 38.87 30.26 39.67

24.96 25.57 26.18 26.79 27.39 28.00 28.61 29.22 29.83 30.44

32.42 33.21 34.00 34.79 35.58 36.37 37.16 37.95 38.74 39.53

25.10 25.71 26.33 26.94 27.55 28.16 28.77 29.39 30.00 30.61

41 42 43 ,44 45 46 47 48 49 50

6 o

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

© o

2

!j i!

!1 j. 1 |

Lat.

Dep.

a

CS

s

53 Deg.

52| Deg.

52* Deg.

52* Deg.

1 2 3

4 5 6

J

s

7?

TRAVERSE TABLE.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

40.73 41.53 42.33 43.13 43.92 44.72 45.52 46.32 47.12 47.92

30.69 31.29 31.90 32.50 33.10 33.70 34.30 34.91 35.51 36.11

40.60 41.39 42.19 42.98 43.78 44.58 45.37 46.17 46.96 47.76

30.87 31.48 32.08 32.69 33.29 33.90 34.50 35.11 35.71 36.32

40.46 41.25 42.05 42.84 43.63 44.43 45.22 46.01 46.81 47.60

31.05 31.66 32.26 32.87 33.48 34.09 34.70 35.31 35.92 36.53

40.33 41.12 41.91 42.70 43.49 44.28 45.07 45.86 46.65 47.44

31.22 31.84 32.45 33.06 33.67 34.28 34.90 35.51 36.12 36.73

51 52 53 54 55 56 57 58 59 60

61 63 64 65 66 67 68 69 70

48.72 49.52 50.31 51.11 51.91 52.71 53.51 54.31 55.11 55.90

36.71 37.31, 37.91 38.52 39.12 39.72 40.32 40.92 41.53 42.13

48.56 49.35 50.15 50.94 51.74 52.54 53.33 54.13 54.92 55.72

36.92 37.53 38.13 38.74 39.34 39.95 40.55 41.16 41.77 42.37

48.39 49.19 49.98 50.77 51.57 52.36 53.15 53.95 54.74 55.53

37.13 37.74 38.35 38.96 39.57 40.18 40.79 41.40 42.00 42.61

48.23 49.02 49.81 50.60 51.39 52.19 52.98 53.77 54.56 55.35

37.35 37.96 38.57 39.18 39.79 40.41 41.02 41.63 42.24 42.86

61 62 63 64 65 ■ 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

56.70 57.50 58.30 59.10 59.90 60.70 61.49 62.29 63.09 63.89

42.73 43.33 43.93 44.53 45.14 45.74 46.34 46.94 47.54 48.15

56.52 57.31 58.11 58.90 59.70 60.50 61.29 62.09 62.88 63.68

42.98 43.58 44.19 44.79 45.40 46.00 46.61 47.21 47.82 48.42

56.33 57.12 57.91 58.71 59.50 60.29 61.09 61.88 62.67 63.47

43.22 43.83 44.44 45.05 45.66 46.27 46.87 47.48 48.09 48.70

56.14 56.93 57.72 58.51 59.30 60.09 60.88 61.67 62.46 63.26

43.47 44.08 44.69 45.30 45.92 46.53 47.14 47.75 48.37 48.98

71 72 73 74 75 76 77 78 79 80

: 81 82 83 84 85 86 87 88 89 90

64.69 65.49 66.29 67.09 67.88 68.68 69.48 70.28 71.08 71.88

48.75 49.35 49.95 50.55 51.15 51.76 52.36 52.96 53.56 54.16

64.48 65.27 66.07 66.86 67.66 68.46 69.25 70.05 70.84 71.64

49.03 49.63 50.24 50.84 51.45 52.06 52.66 53.27 53.87 54.48

64.26 65.05 65.85 66.64 67.43 68.23 69.02 69.82 70.61 71.40

49.31 49.92 50.53 51.14 51.74 52.35 52.96 53.57 54.18 54.79

64.05 64.84 65.63 66.42 67.21 68.00 68.79 69.58 70.37 71.16

49.59 50.20 50.81 51.43 52.04 52.65 53.26 53.88 54.49 55.10

81 82 83 84 85 86 87 88 89 90

91 92 , 93 94 95 96 97 98 99 100

72.68 73.47 74.27 75.07 75.87 76 67 77.47 78.27 79.06 79.86

54.77 55.37 55.97 56.57 57.17 57.77 58.38 58.98 59.58 60.18

72.44 73.23 74.03 74.82 75.62 76.42 77.21 78.01 78.80 79.60

55.08 55.69 56.29 56.90 57.50 58.11 58.71 59.32 59.92 6p.53

72.20 72.99 73.78 74.58 75.37 76.16 76.96 77.75 78.54 79.34

55.40 56.01 56.61 57.22 57.83 58.44 59.05 59.66 60.27 60.88

71.95 72.74 73.53 74.32 75.12 75.91 76.70 77.49 78.28 79.07

55.71 91 56.32 92 56.94 93 57.55 94 58.16 95 58.77 96 59.39 97 60.00 98 60.61 99 61.22 100

o p

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

62

s on

37 Deg.

53 Deg.

37* Deg.

52| Deg.

37| Deg.

37J Deg.

521 Deg.

i

524 Deg.

<0

o

J s

78 g

TRAVERSE TABLE, 38 Deg.

38i Deg.

38* Deg.

ST o

Lat.

1 2 3 4 5 6 7 8 9 10

0.79 1.58 2.36 3.15 3.94 4.73 5.52 6.30 7.09 7.88

0.62 1.23 1.85 2.46 3.08 3.69 4.31 4.93 5.54 6.16

0.79 1.57 2.36 3.14 3.93 4.71 6.50 6.28 7.07 7.85

0.62 1.24 1.86 2.48 3.10 3.71 4.33 4.95 5.57 6.19

0.78 1.57 2.35 3.13 3.91 4.70 5.48 6.26 7.04 7.83

11 12 13 14 15 16 17 18 19 20

8.67 9.46 10.24 11.03 11.82 12.61 13.40 14.18 14.97 15.76

6.77 7.39 8.00 8.62 9.23 9.85 10.47 11.08 11.70 12.31

8.64 9.42 10.21 10.99 11.78 12.57 13.35 14.14 14.92 15.71

6.81 7.43 8.05 8.67 9.29 9.91 10.52 11.14 11.76 12.38

8.61 9.39 10.17 10.96 11.74 12.52 13.30 14.09 14.87 15.65

21 22 23 24 25 26 ■ 27 28 29 30

16.55 17.34 18.12 18.91 19.70 20.49 21.28 22.06 22.85 23.64

12.93 13.54 14.16 14.78 15.39 16.01 16.62 17.24 17.85 18.47

16.49 17.28 18.06 18.85 19.63 20.42 21.20 21.99 22.77 23.56

13.00 13.62 14.24 14.86 15.48 16.10 16.72 17.33 17.95 18.57

31 32 33 34 35 36 37 38 39 40

24.43 25.22 26.00 26.79 27.58 28.37 29.16 29.94 30.73 31.52

19.09 19.70 20.32 20.93 21.55 22.16 22.78 23.40 24.01 24.63

24.34 25.13 25.92 26.70 27.49 28.27 29.06 29.84 30.63 31.41

41 42 43 44 45 46 47 48 49 50

32.31 33.10 33.88 34.67 35.46 36.25 37.04 37.82 38.61 39.40

25.24 25.86 26.47 27.09 27.70 28.32 28.94 29.55 30.17 30.78

a> o

Dep.

Lat.

s

M S3

Dep.

Dep.

Lat.

Dep. 0.62 | 1.24 1 1.87 2.49 ] 3.11 3.74 4.36 4.98 5.60 6.23

Lat.

Dep.

52 Deg.

a CD

0.78 1.56 2.34 3.12 3.90 4.68 5.46 6.24 7.02 7.80

0.63 1.25 1.88 2.50 3.13 3.76 4.38 5.01 5.63 6.26

1 2 3 4 5 6 7 8 9 10

6.85 7.47 8.09 8.72 9.34 9.96 10.58 11.21 11.83 12.45

8.58 9.36 10.14 10.92 11.70 12.48 13.26 14.04 14.82 15.60

6.89 7.51 8.14 8.76 9.39 10.01 10.64 11.27 11.89 12.52

11 12 13 14 15 16 17 18 19 20

16.43 17.22 18.00 18.78 19.57 20.35 21.13 21.91 22.70 23.48

13.07 13.70 14.32 14.94 15.56 16.19 16.81 17.43 18.05 18.68

16.38 17.16 17.94 18.72 19.50 20.28 21.06 21.84 22.62 23.40

13.14 13.77 14.40 15.02 15.65 16.27 16.90 17.53 18.15 18.78

21 22 23 24 25 26 27 28 29 30

19.19 19.81 20.43 21.05 21.67 22.29 22.91 23.53 24.14 24.76

24.26 25.04 25.83 26.61 27.39 28.17 28.96 29.74 30.52 31.30

19.30 19.92 20.54 21.17 21.79 22.41 23.03 23.66 24.28 24.90

24.18 24.96 25.74 26.52 27.30 28.08 28.86 29.64 30.42 31.20

19.40 20.03 20.66 21.28 21.91 22.53 23.16 23.79 24.41 25.04

31 ■ 32 j 33 34 35 36 37 38 39 40

32.20 32.98 33.77 34.55 35.34 36.12 36.91 37.70 38.48 39.27

25.38 26.00 26.62 27.24 27.86 28.48 29.10 29.72 30.34 30.95

32.09 32.87 33.65 34.43 35.22 36.00 36.78 37.57 38.35 39.13

25.52 26.15 26.77 27.39 28.01 28.64 29.26 29.88 30.50 31.13

31.98 32.76 33.53 34.31 35.09 35.87 36 65 37.43 38.21 38.99

25.66 26.29 26.91 27.54 28.17 28.79 29.42 30.04 30.67 31.30

41 42 43 44 45 46 47 48 49 50

Dep.

Lat.

Lat.

Dep;

Lat.

Dep. |

<5 :

g 5?

Lat.

CD

38} Deg.

511 Deg.

51* Deg.

51* Deg.

1

{

U 1, C !

$V)

I

3

!

79

TRAVERSE TABLE, g ST a a>

38 Deg.

38} Deg.

38} Deg.

38} Deg.

C 5’ r*

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

E5 O ft)

51 52 53 54 55 56 57 58 , 59 60

40.19 40.98 41.76 42.55 43.34 44.13 44.92 45.70 46.49 47.28

31.40 32.01 32.63 33.25 33.86 34.48 35.09 35.71 36.32 36.94

40.05 40.84 41.62 42.41 43.19 43.98 44.76 45.55 46.33 47.12

31.57 32.19 32.81 33.43 34.05 34.67 35.29 35.91 36.53 37.15

39.91 40.70 41.48 42.26 43.04 43.83 44.61 45.39 46.17 46.96

31.75 32.37 32.99 33.62 34.24 34.86 35.48 36.11 36.73 37.35

39.77 40.55 41.33 42.11 42.89 43.67 44.45 45.23 46.01 46.79

31.92 32.55 33.17 33.80 34.43 35.05 35.68 36.30 36.93 37.56

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

48.07 48.86 49.64 50.43 51.22 52.01 52.80 53.58 54.37 55.16

37.56 38.17 38.79 39.40 40.02 40.63 41.25 41.86 42.48 43.10

47.90 48.69 49.47 50.26 51.05 51.83 52.62 53.40 54.19 54.97

37.76 38.38 39.00 39.62 40.24 40.86 41.48 42.10 42.72 43.34

47.74 48.52 49.30 50.09 50.87 51.65 52.43 53.22 54.00 54.78

37.97 38.60 39.22 39.84 40.46 41.09 41.71 42.33 42.95 43.58

47.57 48.35 49.13 49.91 50.69 51.47 52.25 53.03 53.81 54.59

38.18 38.81 39.43 40.06 40.68 41.31 41.94 42.56 43.19 43.81

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 . 79 80

55.95 56.74 57.52 58.31 59.10 59.89 60.68 61.46 62.25 63.04

43.71 44.33 44.94 45.56 46.17 46.79 47.41 48.02 48.64 49.25

55.76 56.54 57.33 58.11 58.90 59.68 60.47 61.25 62.04 62.83

43.96 44 57 45.19 45.81 46.43 47.05 47.67 48.29 48.91 49.53

55.57 56.35 57.13 57.91 58.70 59.48 60.26 61.04 61.83 62.61

44.20 44.82 45.44 46.07 46.69 47.31 47.93 48.56 49.18 49.80

55.37 56.15 56.93 57.71 58.49 59.27 60.05 60.83 61.61 62.39

44.44 71 45.07 ■ 72 45.69 73 46.32 74 46.94 75 47.57 76 48.20 77 48.82 78 49.45 79 50.07 80

81 82 83 , 84 85 86 87 88 89 90 1

63.83 64.62 65.40 66.19 66.98 67.77 68.56 69.34 70.13 70.92

49.87 50.48 51.10 51.72 52.33 52.95 53.56 54.18 54.79 55.41

63.61 64.40 65.18 65.97 66.75 67.54 68.32 69.11 69.89 70.68

50.15 50.77 51.38 52.00 52.62 53.24 53.86 54.48 55.10 55.72

63.39 64.17 64.96 65.74 66.52 67.30 68.09 68.87 69.65 70.43

50.42 51.05 51.67 52.29 52.91 53.54 54.16 54.78 55.40 56.03

63.17 63.95 64.73 65.51 66.29 67.07 67.85 68.63 69.41 70.19

50.70 51.33 51.95 52.58 53.20 53.83 54.46 55.08 55.71 56.33

91 92 93 94 95 96 97 98 99 100

71.71 72.50 73.28 74.07 74.86 75.65 76.44 77.22 78.01 78.80

56 05 ' *1.46 56.64 72.25 57.26 73.03 57.87 73.82 58.49 74.61 59.10 75.39 59.72 76.18 60.33 76.96 60.95 77.75 61.57 78.53

56.34 56.96 57.58 58.19 58.81 59.43 60.05 60.67 61.29 61.91

71.22 72.00 72.78 73.57 74.35 75.13 75.91 76.70 77.48 78.26

56.65 57.27 57.89 58.52 59.14 59.76 60.38 61.01 61.63 62.25

70.97 71.75 72.53 73.31 74.09 74.87 75.65 76.43 77.21 77.99

56.96 91 57.58 92 58.21 93 58.84 94 59.46 95 60.09 96 60.71 97 61.34 98 61.97 99 62.59 100

oi

Dep.

Dep.

Lat.

Dep.

o

fl

JS

3

Lat.

52 Deg.

Dep.

Lat.

51J Deg.

i J

5IJ Deg.

Lat.

511 Deg.

81 82 83 84 85 86 87 88 89 90

€> O c ce tn

s

TRAVERSE TABLE.

00 3 CD *

39 Deg.

39J Deg.

39} Deg.

39* leg.

V

o c*

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

g ST ■ S3 O CD

1 2 3 4 5 6 ; ' 7 8 9 10

0.78 1.55 2.33 3.11 3.89 4.66 5.44 6.22 6.99 7.77

0.63 1.26 1.89 2.52 3.15 3.78 4.41 5.03 5.66 6.29

0.77 1.55 2.32 3.10 3.87 4.65 5.42 6.20 6.97 7.74

0.63 1.27 1.90 2.53 3.16 3.80 4.43 5.06 5.69 6.33

0.77 1.54 2.31 3.09 3.86 4.63 5.40 6.17 6.94 7.72

0.64 1.27 1.91 2.54 3.18 3.82 4.45 5.09 5.72 6.36

0.77 1.54 2.31 3.08 3.84 4.61 5.38 6.15 6.92 7.69

0.64 1.28 1.92 2.56 3.20 3.84 4.48 5.12 5.75 6.39

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 . 16 17 18 19 20

8.55 9.33 10.10 10.88 11.66 12.43 13.21 13.99 14.77 15.54

6.92 7.55 8.18 8.81 9.44 10.07 10.70 11.33 11.96 12.59

8.52 9.29 10.07 10.84 11.62 12.39 13.16 13.94 14.71 15.49

6.96 7.59 8.23 8.86 9.49 10.12 10.76 11.39 12.02 12.65

8.49 9.26 10.03 10.80 11.57 12.35 13.12 13.89 14.66 15.43

7.00 7.63 8.27 8.91 9.54 10.18 10.81 11.45 12.09 12.72

8.46 9.23 9.99 10.76 11.53 12.30 13.07 13.84 14.61 15.38

7.03 7.67 8.31 8.95 9.59 10.23 10.87 11.51 12.15 12.79

11 12 13 14 15 16 17 18 19 20

21 22 23 ; 24 25 26 27 . 28 29 30

16.32 17.10 17.87 18.65 19.43 20.21 20.98 21.76 22.54 23.31

13.22 13.84 14.47 15.10 15.73 16.36 16.99 17.62 18.25 18.88

16.26 17.04 17.81 18.59 19.36 20.13 20.91 21.68 22.46 23.23

13.29 13.92 14.55 15.18 15.82 16.45 17.08 17.72 18.35 18.98

16.20 16.98 17.75 18.52 19.29 20.06 20.83 21.61 22.38 23.15

13.36 13.99 14.63 15.27 15.90 16.54 17.17 17.81 18.45 19.08

16.15 16.91 17.68 18.45 19.22 19.99 20.76 21.53 22.30 23.07

13.43 14.07 14.71 15.35 15.99 16.63 17.26 17.90 18.54 19.18

21 22 23 24 25 26 27 28 ' 29 30

24.09 24.87 25.65 26.42 27.20 27.98 28.75 29.53 30.31 31.09 .

19.51 20.14 20.77 21.40 22.03 22.66 23.28 23.91 24.54 25.17

24.01 24.78 25.55 26.33 27.10 27.88 28.65 29.43 30.20 30.98

19.61 20.25 20.88 21.51 22.14 22.78 23.41 24.04 24.68 25.31

23.92 24.69 25.46 26.24 27.01 27.78 28.55 29.32 30.09 30.86

19.72 20.35 20.99 21.63 22.26 22.90 23.53 24.17 24.81 25.44

23.83 24.60 25.37 26.14 26.91 27.68 28.45 29.22 29.98 30.75

19.82 20.46 21.10 21.74 22.38 23.02 23.66 24.30 24.94 25.58

31 32 33 34 35 36 37 38 39 40

25.80 26.43 27.06 27.69 28.32 28.95 29.58 30.21 30.84 31.47

31.75 32.52 33.30 34.07 34.85 35.62 36.40 37.17 37.95 38.72

25.94 26.57 27.21 27.84 28.47 29.10 29.74 30.37 31.00 31.64

31.64 32.41 33.18 33.95 34.72 35.49 36.27 37.04 37.81 38.58

26.08 26.72 27.35 27.99 28.62 29.26 29.90 30.53 31.17 31.80

31.52 32.29 33.06 33.83 34.60 35.37 36.14 36.90 37.67 38.44

26.22 26.86 27.50 28.14 28.77 29.41 30.05 30.69 31.33 31.97

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

o C5 *

,

31 32 33 34 35 36 37 38 39 40

41 31.86 - 42 32.64 43 33.42 44 * 34.19 45 34.97 . 46 35.75 47 36.53 48 37.30 49 38.08 50 38.86 6 o

Dep.

rt Q

51 Deg.

50} Deg.

50} Deg.

50} Deg.

Q

81

TRAVERSE TABLE.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

39.63 40.41 41.19 41.97 42.74 43.52 44.30 45.07 45.85 46.63

32.10 32.72 33.35 33.98 34.61 35.24 35.87 36.50 37.13 37.76

39.49 40.27 41.04 41.82 42.59 43.37 44.14 44.91 45.69 46.46

32.27 32.90 33.53 34.17 34.80 35.43 36.06 36.70 37.33 37.96

39.35 40.12 40.90 41.67 42.44 43.21 43.98 44.75 45.53 40.30

32.44 33.08 33.71 34.35 34.98 35.62 36.26 36.89 37.53 38.16

39.21 39.98 40.75 41.52 42.29 43.06 43.82 44.59 45.36 46.13

32.61 33.25 33.89 34.53 35.17 35.81 36.45 37.09 37.73 38.37

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

47.41 48.18 48.96 49.74 50.51 51.29 52.07 52.85 53.52 54.40

38.39 39.02 39.65 40.28 40.91 41.54 42.16 42.79 43.42 44.05

47.24 48.01 48.79 49.56 50.34 51.11 51.88 52.66 53.43 54.21

38.60 39.23 39.86 40.49 41.13 41.76 42.39 43.02 43.66 44.29

47.07 47.84 48.61 49.38 50.16 50.93 51.70 52.47 53.24 54.01

38.80 39.44 40.07 40.71 41.35 41.98 42.62 43.25 43.89 44.53

46.90 47.67 48.44 49.21 49.97 50.74 51.51 52.28 53.05 53.82

39.01 39.65 40.28 40.92 41.56 42.20 42.84 43.48 44.12 44.76

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

55.18 55.95 56.73 57.51 58.29 59.06 59.84 60.62 61.39 62.17

44.68 45.31 45.94 46.57 47.20 47.83 48.46 49.09 49.72 50.35

54.98 55.76 56.53 57.31 58.08 58.85 59.63 60.40 61.18 61.95

44.92 45.55 46.19 46.82 47.45 48.09 48.72 49.35 49.98 50.62

54.79 55.56 56.33 57.10 57.87 58.64 59.42 60.19 60.96 61.73

45.16 45.80 46.43 47.07 47.71 48.34 48.98 49.61 50.25 50.89

54.59 55.36 56.13 56.89 57.66 58.43 59.20 59.97 60.74 61.51

45.40 46.04 46.68 47.32 47.96 48.60 49.24 49.88 50.52 51.16

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

62.95 63.73 64.50 65.28 66.06 66.83 67.61 68.39 69.17 69.94

50.97 51.60 52.23 52.86 53.49 54.12 54.75 55.38 56.01 56.64

62.73 63.50 64.27 65.05 65.82 66.60 67.37 68.15 68.92 69.70

51.25 51.88 52.51 53.15 53.78 54.41 55.05 55.68 56.32 56.94

62.50 63.27 64.04 64.82 65.59 66.36 67.13 67.90 68.67 69.45

51.52 52.16 52.79 53.43 54.07 54.70 55.34 55.97 56.61 57.25

62.28 63.04 63.81 64.58 65.35 66.12 66.89 67.66 68.43 69.20

51.79 52.43 53.07 53.71 54.35 54.99 55.63 56.27 56.91 57.55

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

70.72 71.50 72.27 73.05 73.83 74.61 75.38 76.16 76.94 77.71

57.27 57.90 58.53 59.16 59.79 60.41 61.04 61.67 62.30 62.93

70.47 71.24 72.02 72.79 73.57 74.34 75.12 75.89 76.66 77.44

57.58 58.21 58.84 59.47 60.11 60.74 61.37 62.01 62.64 63.27

70.22 70.99 71.76 72.53 73.30 74.08 74.85 75.62 76.39 77.16

57.88 58.52 59.16 59.79 60.43 61.06 61.70 62.34 62.97 63.61

69.96 70.73 71.50 72.27 73.04 73.81 74.58 75.35 76.12 76.88

58.19 91 58.83 92 59.47 93 60.11 94 60.75 95 61.39 96 62.03 97 62.66 98 63.30 99 63.94 100

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep. | Lat.

<0o

39 Deg.

i1

|! i! 1|

39i Deg.

! 39} Deg.

39* Deg.

(3 CTJ

an

3

51 Deg.

SI

50} Deg.

2Z

50* Deg.

1

50} Deg.

«o

rt0 an

s ,

82 g

TRAVERSE TABLE 40 Deg.

40} Deg.

40} Deg.

40} Deg.

P

a ?

Lat.

Dep.

Lat.

Dep.

Lat.

1 2 3 « 4 1 5 6 7 8 9 10

0.77 1.53 2.30 3.06 3.83 4.60 5.36 6.13 6.89 7.66

0.64 1.29 1.93 2.57 3.21 3.86 4.50 5.14 5.79 6.43

0.76 1.53 2.29 3.05 3.82 4.58 5.34 6.11 6.87 7.63

0.65 1.29 1.94 2.58 3.23 3.88 4.52 5.17 5.82 6.46

0.76 1.52 2.28 3.04 3.80 4.56 5.32 6.08 6.84 7.60

11 12 13 14 15 16 17 18 19 20

8.43 9.19 9.96 10.72 11.49 12.26 13.02 13.79 14.55 15.32

7.07 7.71 8.36 9.00 9.64 10.28 10.93 11.57 12.21 12.86

8.40 9.16 9.92 10.69 11.45 12.21 12.97 13.74 14.50 15.26

7.11 7.75 8.40 9.05 9.69 10.34 10.98 11.63 12.28 12.92

21 22 23 24 25 26 27 28 29 30

16.09 16.85 17.62 18.39 19.15 19.92 20.68 21.45 22.22 22.98

13.50 14.14 14.78 15.43 16.07 16.71 17.36 18.00 18.64 19.28

16.03 16.79 17.55 18.32 19.08 19.84 20.61 21.37 22.13 22.90

31 32 33 34 35 36 37 38 39 40

23.75 24.51 25.28 26.05 26.81 27.58 28.34 29.11 29.88 30.64

19.93 20.57 21.21 21.85 22.50 23.14 23.78 24.43 25.07 25.71

. 41 42 43 44 45 46 47 48 49 50

31.41 32.17 32.94 33.71 34.47 35.24 36.00 36.77 37.54 38.30 Dep.

« c

J s m

Dep.

Dep.

3 O

0.76 1.52 2.27 3.03 3.79 4.55 5.30 6.06 6.82 7.58

0.65 1.31 1.96 2.61 3.26 3.92 4.57 5.22 5.87 6.53

1 2 3 4 5 6 7 8 9 10

8.36 9.12 9.89 10.65 11.41 12.17 12.93 13.69 14.45 15.21

7.14 8.33 7.79 9.09 8.44 9.85 9.09 10.61 9.74 11.36 10.39 , 12.12 11.04 12.88 11.69 13.64 12.34 14.39 12.99 15.15

7.18 7.83 8.49 9.14 9.79 10.44 11.10 11.75 12.40 13.06

11 12 13 14 15 16 17 18 19 20

13.57 14.21 14.86 15.51 16.15 16.80 17.45 18.09 18.74 19.38

15.97 16.73 17.49 18.25 19.01 19.77 20.53 21.29 22.05 22.81

13.64 14.29 14.94 15.59 16.24 16.89 17.54 18.18 18.83 19.48

15.91 16.67 17.42 18.18 18.94 19.70 20.45 21.21 21.97 22.73

13.71 14.36 15.01 15.67 16.32 16.97 17.62 18.28 18.93 19.58

21 22 23 24 25 26 27 28 29 30

23.66 24.42 25.19 25. §5 26.71 27.48 28.24 29.00 29.77 30.53

20.03 20.68 21.32 21.97 22.61 23.26 23.91 24.55 25.20 25.84

23.57 24.33 25.09 25.85 26.61 27.37 28.13 28.90 29.66 30.42

20.13 20.78 21.43 22.08 22.73 23.38 24.03 24.68 25.33 25.98

23.48 24.24 25.00 25.76 26.51 27.27 28.03 28.79 29.54 30.30

20.24 20.89 21.54 22.19 22.85 23.50 24.15 24.80 25.46 26.11

31 32 33 34 35 36 37 38 39 40

26.35 27.00 27.64 28.28 28.93 29.57 30.21 30.85 31.50 32.14

31.29 32.06 32.82 33.58 34.35 35.11 35.87 36.64 37.40 38.16

26.49 27.14 27.78 28.43 29.08 29.72 30.37 31.01 31.66 32.31

31.18 31.94 32.70 33.46 34.22 34.98 35.74 36.50 37.26 38.02

26.63 27.28 27.93 28.58 29.23 29.87 30.52 31.17 31.82 32.47

31.06 31.82 32.58 33.33 34.09 34.85 35.61 36.36 37.12 37.88

26.76 27 42 2b. 07 28.72 29.37 30.03 30.68 31.33 31.99 32.64

41 42 43 44 45 46 47 48 49 50

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.


50 Deg.

491 Deg.

0.65 1.30 1.95 2.60 3.25 3.90 4.55 5.20 5.84 6.49

Lat.

So"

49} :Deg.

49} Deg.

3 tn

s

83

TRAVERSE TABLE. Distance.

g ET s o

40 Deg.

40} Deg. Lat.

Dep.

38.78 1' 33.12 32.95 39.54 33.77 33.60 34.24 40.30 34.42 34.89 1 41.06 35.07 35.54 41.82 35.72 36.18 42.58 36.37 43.34 37.02 36.83 37.48 44.10 37.67 44.86 38.32 38.12 38.77 45.62 38.97

38.64 39.39 40.15 40.91 41.67 42.42 43.18 43.94 44.70 45.45

33.29 33.94 34.60 35.25 35.90 36.55 37.21 37.86 38.51 39.17

51 52 53 54 55 56 57 58 59 60

46.56 47.32 48.08 48.85 49.61 50.37 51.14 51.90 52.66 53.43

39.41 40.06 40.71 41.35 42.00 42.64 43.29 43.94 44.58 45.23

46.38 47.15 47.91 48.67 49.43 50.19 50.95 51.71 52.47 53.23

39.62 40.27 40.92 41.56 42.21 42.86 43.51 44.16 44.81 45.46

46.21 46.97 47.73 48.48 49.24 60.00 50.76 51.51 52.27 53.03

39.82 40.47 41.12 41.78 42.43 43.08 43.73 44.39 45.04 45.69

61 62 63 64 65 66 67 68 69 70

45.64 46.28 46.92 47.57 48.21 48.85 49.49 50.14 50.78 51.42

54.19 54.95 55.72 56.48 57.24 58.01 58.77 59.53 60.30 61.06

45.87 46.52 47.17 47.81 48.46 49.11 49.75 50.40 51.04 51.69

53.99 54.75 55.51 56.27 57.03 57.79 58.55 59.31 60.07 60.83

46.11 ! 63.79 46.76 1 54.54 55.30 47.41 56.06 48.06 56.82 48.71 57.57 49.36 58.33 50.01 59.09 50.66 51.31 59.85 60.61 51.96

46.35 47.00 47.65 48.30 48.96 49.61 50.26 50.92 51.57 52.22

71 72 73 74 75 76 77 78 79 80

52.07 52.71 53.35 53.99 54.64 55.28 55.92 56.57 57.21 57.85

61.82 62.59 63.35 64.11 64.87 65.64 66.40 67.16 67.93 68.69

52.34 52.98 53.63 54.27 54.92 55.57 56.21 56.86 >7.5C 58.15

61.59 62.35 63.11 63.87 64.63 65.39 66.16 66.92 67.68 68.44

52.61 53.25 53.90 54.55 55.20 55.85 56.50 57.15 57.80 58.45

61.36 62.12 62.88 63.64 64.39 65.15 65.91 66.67 67.42 68.18

52.87 53.53 54.18 54.83 55.48 56.14 56.79 57.44 58.10 58.75

81 82 83 84 85 86 87 88 89 90

69.20 69.96 70.72 71.48 72.24 73.00 73.76 74.52 75.28 76.04

59.10 59.75 60.40 61.05 61.70 62.35 63.00 63.65 64.30 64.94

68.94 69.70 70.45 71.21 71.97 72.73 73.48 74.24 75.00 75.76

59.40 91 60.05 92 60.71 93 61.36 94 62.01 95 62.66 96 63.32 97 63.97 98 64.62 99 65.28 100

Dep.

Lat.

Dep.

9

Lat.

Dep.

Lat.

51 52 53 54 55 56 57 58 59 60

39.07 39.83 40.60 41.37 42.13 42.90 43.66 44.43 45.20 45.96

32.78 33.42 34.07 34.71 35.35 36.00 36.64 37.28 37.92 38.57

38.92 39.69 40.45 41.21 41.98 42.74 43.50 44.27 45.03 45.79

61 62 63 64 65 66 67 68 69 70

46.73 47.49 48.26 49.03 49.79 50.56 51.32 52.09 52.86 53.62

39.21 39.85 40.50 41.14 41.78 42.42 43.07 43.71 44.35 45.00

71 ' 72 73 74 75 76 77 78 79 80

54.39 55.16 55.92 56.69 57.45 58.22 58.99 59.75 60.52 61.28

81 62.05 82 62.82 83 • 63.58 84 64.35 85 65.11 86 65.88 87 66.65 88 67.41 89 68.18 . 90 68.94

Lat.

Dep.

91 92 93 94 95 96 97 98 99 100

69.71 70.48 71.24 72.01 72.77 73.54 74.31 75.07 75.84 76.60

58.49 59.14 59.78 60.42 61.06 61.71 62.35 62.99 63.64 64.28

69.45 70.22 70.98 71.74 72.51 73.27 74.03 74.80 75.56 76.32

58.80 59.44 60.09 60.74 61.38 62.03 62.67 63.32 63.97 64.61

6

Dep.

Lat.

Dep.

Lat.

o

40£ Deg.

40} Deg.

j i ! !} j! !i 1 1 | ;

1 Dep-

Lat.

<2

-5

5

50 Deg.

49} Deg.

49$ Deg.

49} Deg.

« o

1 m

3

TRAVERSE TABLE,

84 Distance.

o

c/>' c+

41 Deg.

41^ Deg.

41 £ Deg.

411 Deg.

as

3 O OB

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

0.75 1.50 2.26 3.01 3.76 4.51 5.26 6.01 6.77 7.52

0.66 1.32 1.98 2.64 3.30 3.96 4.62 5.27 5.93 6.59

0.75 1.50 2.25 3.00 3.74 4.49 5.24 5.99 6.74 7.49

0.66 1.33 1.99 2.65 3.31 3.98 4.64 5.30 5.96 6.63

0.75 1.49 2.24 2.98 3.73 4.48 5.22 5.97 6.71 7.46

0.67 1.33 2.00 2.66 3.33 4.00 4.66 5.33 5.99 6.66

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 3 9 10

0.75 1.51 2.26 3.02 3.77 4.53 5.28 6.04 6.79 7.55

0.66 1.31 1.97 2.62 3.28 3.94 4.59 5.25 5.90 6.56

11 12 13 14 15 16 17 18 19 20

8.30 9.06 9.81 10.57 11.32 12.08 12.83 13.58 14.34 15.09

7.22 7.87 8.53 9.18 9.84 10.50 11.15 11.81 12.47 13.12

8.27 9.02 9.77 10.53 11.28 12.03 12.78 13.53 14.23 15.04

7.25 7.91 8.57 9.23 9.89 10.55 11.21 11.87 12.53 13.19

8.24 8.99 9.74 10.49 11.23 11.98 12.73 13.48 14.23 14.98

7.29 7.95 8.61 9.28 9.94 10.60 11.26 11.93 12.59 13.25

8.21 8.95 9.70 10.44 11.19 11.94 12.68 13.43 14.18 14.92

7.32 7.99 8.66 9.32 9.99 10.65 11.32 11.99 12.65 13.32

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

15.85 16.60 17.36 18.11 18.87 19.62 20.38 21.13 21.89 22.64

13.78 14.43 15.09 15.75 16.40 17.06 17.71 18.37 19.03 19.68

15.79 16.54 17.29 18.04 18.80 19.55 20.30 21.05 21.80 22.56

13.85 14.51 15.16 15.82 16.48 17.14 17.80 18.46 19.12 19.78

15.73 16.48 17.23 17.97 18.72 19.47 20.22 20.97 21.72 22.47

13.91 14.58 15.24 15.90 16.57 17.23 17.89 18.55 19.22 19.88

15.67 16.41 17.16 17.91 18.65 19.40 20.14 20.89 21.64 22.38

13.98 14.65 15.32 15.98 16.65 17.31 17.98 18.64 19.31 19.98

21 22 23 24 25 26 27 . 28 29 30

31 32 33 34 35 36 37 38 39 40

23.40 24.15 24.91 25.66 26.41 27.17 27.92 28.68 29.43 30.19

20.34 20.99 21.65 22.31 22.96 23.62 24.27 24.93 25.59 26.24

23.31 24.06 24.81 25.56 26.31 27.07 27.82 28.57 29.32 30.07

20.44 21.10 21.76 22.42 23.08 23.74 24.40 25.06 25.71 26.37

23.22 23.97 24.72 25.46 26.21 26.96 27.71 28.46 29.21 29.96

20.54 21.20 21.87 22.53 23.19 23.85 24.52 25.18 25.84 26.50

23.13 23.87 24.62 25.37 26.11 26.86 27.60 28.35 29.10 29.84

20.64 21.31 21.97 22.64 23.31 23.97 24.64 25.30 25.97 26.64

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

30.94 31.70 32.45 33.21 33.96 34.72 35.47 36.23 36.98 37.74

26.90 27.55 28.21 28.87 29.52 30.18 30.83 31.49 32.15 32.80

30.83 31.58 32.33 33.03 33.83 34.58 35.34 36.09 36.84 37.59

27.03 27.69 28.35 29.01 29.67 30.33 30.99 31.65 32.31 32.97

30.71 31.46 32.21 32.95 33.70 34.45 35.20 35.95 36.70 r.45

27.17 27.83 28.49 29.16 29.82 30.48 31.14 31.81 32.47 33.13

30.59 31.33 32.08 32.83 33.57 34.32 35.06 35.81 36.56 37.30

27.30 27.97 28.63 29.30 29.97 30.63 31.30 31.96 32.63 33.29

41 42 43 . 44 45 : 46 47 48 49 50 j

«S O

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

8 tn

3

49 Deg.

48J Deg.

48$ Deg.

|

Lat.

6 \ I fl 1 a I o

484 Deg. p !'

TRAVERSE TABLE g

41 Deg;.

41} Deg.

41 $ Deg.

05

.

41} Deg.

g MJ '

ST o ?

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

§ a et>

51 52 53 54 55 56 57 58 59 60

38.49 39.24 40.00 40.75 41.51 42.26 43.02 43.77 44.53 45.28

33.46 34.12 34.77 35.43 36.08 36.74 37.40 38.05 38.71 39.36

38.34 39.10 39.85 40.60 41.35 42.10 42.85 43.61 44.36 45.11

33.63 34.29 34.95 35.60 36.26 36.92 37.58 38.24 38.90 39.56

38.20 38.95 39.69 40.44 41.19 41.94 42.69 43.44 44.19 44.94

33.79 34.46 35.12 35.78 36.44 37.11 37.77 38.43 39.09 39.76

38.05 38.79 39.54 40.29 41.03 41.78 42.53 43.27 44.02 44.76

33.96 34.63 35.29 35.96 36.62 37.29 37.96 38.62 39.29 39.95

51 52 53 54 55 56 57 58 59 60

61 62 63 64 - 65 66 67 68 69 70

46.04 46.79 47.55 48.30 49.06 49.81 50.57 51.32 52.07 52.83

40.02 40.68 41.33 41.99 42.64 43.30 43.96 44.61 45.27 45.92

45.86 46.61 47.37 48.12 48.87 49.62 50.37 51.13 51.88 52.63

40.22 40.88 41.54 42.20 42.86 43.52 44.18 44.84 45.49 46.15

45.69 46.44 47.18 47.93 48.68 49.43 50.18 50.93 51.68 52.43

40.42 41.08 41.75 42.41 43.07 43.73 44.40 45.06 45.72 46.38

45.51 46.26 47.00 47.75 48.49 49.24 49.99 50.73 51.48 52.22

40.62 41.28 41.95 42.62 43.28 43.95 44.61 45.28 45.95 46.61

61 62 63 64 65 . 66 67 68 69 70

71 72 73 74 75 76 : 77 78 79 80

53.58 54.34 55.09 55.85 56.60 57.36 58.11 58.87 59.62 60.38

46.58 47.24 47.89 48.55 49.20 49.86 50.52 51.17 51.83 52.48

53.38 54.13 54.88 55.64 56.39 57.14 57.89 58.64 59.40 60.15

46.81 47.47 48.13 48.79 49.45 50.11 50.77 51.43 52.09 52.75

53.18 53.92 54.67 55.42 56.17 56.92 57.67 58.42 59.17 59.92

47.05 47.71 48.37 49.03 49.70 50.36 51.02 51.68 52.35 53.01

52.97 53.72 54.46 55.21 55.95 56.70 57.45 58.19 58.94 59.68

47.28 47.94 48.61 49.28 49.94 50.61 51.27 51.94 52.60 53.27

71 72 73 74 75 76 77 78 79 80

81 61.13 82 61.89 83 62.64 84 63.40 85 64.15 86 64.90 87 65.66 88 66.41 89 -' 67.17 90 6T.92

53.14 53.80 54.45 55.11 55.76 56.42 57.08 57.73 58.39 59.05

60.90 61.65 62.40 63.15 63.91 64.66 65.41 66.16 66.91 67.67

53.41 54.07 54.73 55.38 56.04 56.70 57.36 58.02 58.68 59.34

60.67 61.41 62.16 62.91 63.66 64.41 65.16 65.91 66.66 67.41

53.67 54.33 55.00 55.66 56.32 56.99 57.65 58.31 58.97 59.64

60.43 61.18 61.92 62.67 63.41 64.16 64.91 65.65 66.40 67.15

53.94 54.60 55.27 55.93 56.60 57.27 57.93 58.60 59.26 59.93

81 82 83 84 ' 85 86 87 88 89 90

68.68 69.43 70.19 70.94 71.70 72.45 73.21 73.96 74.72 75.47

59.70 60.36 61.01 61.67 62.33 62.98 63.64 64.29 64.95 65.61

68.42 69.17 69.92 70.67 71.43 72.18 72.93 73.68 74.43 75.18

60.00 60.66 61.32 61.98 62.64 63.30 63.96 64.62 65.28 65.93

68.15 68.90 69.65 70.40 71.15 71.90 72.65 73.40 74.15 74.90

60.30 60.96 61.62 62.29 62.95 63.61 64.27 64.94 65.60 66.26

67.89 68.64 69.38 70.13 70.88 71.62 72.37 73.11 73.86 74.61

60.60 91 61.26 92 61.93 93 62.59 , 94 63.26 95 63.92 96 64.59 97 65.26 98 65.92 99 66.59 100

Dep.

Lat

Dep.

Lat.

Dep.

Lat.

Dep.

91 92 93 94 95 96 ' 97 98 99 ? 100
o

is o

Lat.

0 o

ri0 45 Deg. :

48} Deg.

48$ Deg.

48} Deg.

Q

TRAVERSE TABLE,

86 g OD*

42 Deg.

42i Deg.

42$ Deg

42| Deg.

g ST

P

s o ?

Lat.

Dep.

Lat.

1 Z 3 4 5 6 7 8 9 10

0.74 1.49 2.23 2.97 3.72 4.46 5.20 5.95 6.69 7.43

0.67 1.34 2.01 2.68 3.35 4.01 4.68 5.35 6.02 6.69

0.74 1.48 2.22 2.96 3.70 4.44 5.18 5.92 6.66 7.40

0.67 1.34 2.02 2.69 3.36 4.03 4.71 5.38 6.05 6.72

0.74 1.47 2.21 2.95 3.69 4.42 5.16 5.90 6.64 7.37

0.68 1.35 2.03 2.70 3.38 4.05 4.73 5.40 6.08 6.76

0.73 1.47 2.20 2.94 3.67 4.41 5.14 5.87 6.61 7.34

11 12 13 14 15 16 17 18 19 20

8.17 8.92 9.66 10.40 11.15 11.89 12.63 13.38 14.12 14.86

7.36 8.03 8.70 9.37 10.04 10.71 11.38 12.04 12.71 13.38

8.14 8.88 9.62 10.36 11.10 11.84 12.58 13.32 14.06 14.80

7.40 8.07 8.74 9.41 10.09 10.76 11.43 12.10 12.77 13.45

8.11 8.85 9.58 10.32 11.06 11.80 12.53 13.27 14.01 14.75

7.43 8.11 8.78 9.46 10.13 10.81 11.48 12.16 12.84 13.51

8.08 8.81 9.55 10.28 11.01 11.75 12.48 13.22 13.95 14.69

7.47 8.15 8.82 9.50 10.18 10.86 11.54 12.22 12.90 13.58

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

15.61 16.35 17.09 17.84 18.58 19.32 20.06 20.81 21.55 22.29

14.05 14.72 15.39 16.06 16.73 17.40 18.07 18.74 19.40 20.07

15.54 16.28 17.02 17.77 13.51 19.25 19.99 20.73 21.47 22.21

14.12 14.79 15.46 16.14 16.81 17.48 18.15 18.83 19.50 20.17

15.48 16.22 16.96 17.69 18.43 19.17 19.91 20.64 21.38 22.12

14.19 14.86 15.54 16.21 16.89 17.57 18.24 18.92 19.59 20.27

15.42 16.16 16.89 17.62 18.36 19.09 19.83 20.56 21.30 22.03

14.25 14.93 15.61 16.29 16.97 17.65 18.33 19.01 19.69 20.36

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

23.04 23.78 24.52 25.27 26.01 26.75 27.50 28.24 28.98 29.73

20.74 21.41 22.08 22.75 23.42 24.09 24.76 25.43 26.10 26.77

22.95 23.69 24.43 25.17 25.91 26.65 27.39 28.13 28.87 29.61

20.84 21.52 22.19 22.86 23.53 24.21 24.88 25.55 26.22 26.89

22.86 23.59 24.33 25.07 25.80 26.54 27.28 28.02 28.75 29.49

20.94 21.62 22.29 22.97 23.65 24.32 25.00 25.67 26.35 27.02

22.76 23.50 24.23 24.97 25.70 26.44 27.17 27.90 28.64 29.37

21.04 21.72 22.40 23.08 23.76 24.44 25.12 25.79 26.47 27.15

31 32 33 34 35 36 37 38 39 40

41 42 43 . 44 45 46 47 48 49 50

30.47 31.21 31.96 32.70 33.44 34.18 34.93 35.67 36.41 37.16

27.43 28.10 28.77 29.44 30.11 30.78 31.45 32.12 32.79 33.46

30.35 31.09 31.83 32.57 33.31 34.05 34.79 35.53 36.27 37.01

27.57 28.24 28.91 29.58 30.26 30.93 31.60 32.27 32.95 33.62

30.23 30.97 31.70 32.44 33.18 33.91 34.65 35.39 36.13 36.86

27.70 28.37 29.05 29.73 30.40 31.08 31.75 32.43 33.10 33.78

30.11 30.84 31.58 32.31 33.04 33.78 34.51 35.25 35.98 36.72

27.83 28.51 29.19 29.87 30.55 31.22 31.90 32.58 33.26 33.94

41 42 43 44 45 46 47 48 49 50

Dep.

Lat.

Dep.

Lat.

Lat.

Dep.

Lat

V

o eg ■ 3

48 Deg.

Dep.

47} Deg.

Lat.

Dep. 1

Dep.

474 Deg.

Lat.

Dep.

D O

to

0.68 ! i 1.36 i 2 2.04 3 2.72 4 3.39 5 4.07 6 4.75 7 . 5.43 8 6.11 9 6.79 10

6

O

rta 47$ Deg.

1

87

TRAVERSE TABLE. g o

42 Deg.

42$ Deg.

42f Deg.;

42$ Deg.

! g 3>' 1: P

®

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

O »

51 52 53 54 55 56 57 58 59 60

37.90 38.64 39.39 40.13 40.87 41.62 42.36 43.10 43.85 44.59

34.13 34.79 35.46 36.13 36.80 37.47 38.14 38.81 39.48 40.15

37.75 38.49 39.23 39.97 40.71 41.45 42.19 42.93 43.67 44.41

34.29 34.96 35.64 36.31 36.98 37.65 38.32 39.00 39.67 40.34

37.60 38.34 39.08 39.81 40.55 41.29 42.02 42.76 43.50 44.24

34.46 35.13 35.81 36.48 37.16 37.83 38.51 39.18 39.86 40.54

37.45 38.18 38.92 39.65 40.39 41.12 41.86 42.59 43.32 44.06

34.62 35.30 35.98 36.66 37.33 38.01 38.69 39.37 40.05 40.73

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68* 69 70

45.33 46.07 46.82 47.56 48.30 49.05 49.79 50.53 51.28 52.02

40.82 41.49 42.16 42.82 43.49 44.16 44.83 45.50 46.17 46.84

45.15 45.89 46.63 47.37 48.11 48.85 49.59 50.33 51.07 51.82

41.01 41.69 42.36 43.03 43.70 44.38 45.05 45.72 46.39 47.07

44.97 45.71 46.45 47.19 47.92 48.66 49.40 50.13 50.87 51.61

41.21 41.89 42.56 43.24 43.91 44.59 45.26 45.94 46.62 47.29

44.79 45.53 46.26 47.00 47.73 43.47 49.20 49.93 50.67 51.40

41.41 42.09 42.76 43.44 44.12 44.80 45.48 46.16 46.84 47.52

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

52.76 53.51 54.25 54.99 55.74 56.48 57.22 57.97 58.71 59.45

47.51 48.18 48.85 49.52 50.18 50.85 51.52 52.19 52.86 53.53

52.56 53.30 54.04 54.78 55.52 56.26 57.00 57.74 58.48 59.22

47.74 48.41 49.08 49.76 50.43 51.10 51.77 52.44 53.12 53.79

52.35 53.08 53.82 54.56 55.30 56.03 56.77 57.51 58.24 58.98

47.97 48.64 49.32 49.99 50.67 51.34 52.02 52.70 53.37 54.05

52.14 52.87 53.61 54.34 55.07 55.81 56.54 57.28 58.01 58.75

48.19 48.87 49.55 50.23 50.91 51.59 52.27 52.95 53.63 54.30

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

60.19 60.94 61.68 62.42 63.17 63.91 64.65 65.40 66.14 66.88

54.20 54.87 55.54 56.21 56.88 57.55 58.21 58.88 59.55 60.22

59.96 60.70 61.44 62.18 62.92 63.66 64.40 65.14 65.88 66.62

54.46 55.13 55.81 56.48 57.15 57.82 58.50 59.17 59.84 60.51

59.72 60.46 61.19 61.93 62.67 63.41 64.14 64.88 65.62 66.35

54.72 55.40 56.07 56.75 57.43 58.10 58.78 59.45 60.13 60.80*

59.48 60.21 60.95 61.68 62.42 63.15 63.89 64.62 65.35 66.09

54.98 55.66 56.34 57.02 57.70 58.38 59.06 59.73 60.41 61.09

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

67.63 68.37 69.11 69.86 70.60 71.34 72.08 72.83 73.57 74.31

60.89 61.56 62.23 62.90 63.57 64.24 64.91 65.57 66.24 66.91

67.36 68.10 68.84 69.58 70.32 71.06 71.80 72.54 73.28 74.02

61.19 61.86 62.53 63.20 63.87 64.55 65.22 65.89 66.56 67.24

67.09 67.83 68.57 69.30 70.04 70.78 71.52 72.25 72.99 73.73

61.48 62.15 62.83 63.51 64.18 64.86 65.53 66.21 66.88 67.56

66.82 67.56 68.29 69.03 69.76 70.49 71.23 71.96 72.70 73.43

61.77 91 62 45 92 63.13 93 94 63.81 64.49 95 65.16 96 65.84 97 66.52 98 67.20 99 67.88 100

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

V

a

Lat.

rt

5

48 Deg*.

47| Deg.


rtG 47$ Deg. °

47$ Deg. i

Q

88

TRAVERSE TABLE. 43£ Deg. Dep.

43£ Deg. Dep.

43} Deg.

Lat.

Dep.

Lat.

1 , 2 3 4 5 6 7 8 9 10

0.73 1.46 2.19 2.93 3.66 4.39 5.12 5.85 6.58 7.31

0.68 1.36 2.05 2.73 3.41 4.09 4.77 5.46 6.14 6.82

0.73 1.46 2.19 2.91 3.64 4.37 5.10 5.83 6.56 7.23

0.69 1.37 2.06 2.74 3.43 4.11 4.80 5.48 6.17 6.85

0.73 1.45 2.18 2.90 3.63 4.35 5.08 5.80 6.53 7.25

0.69 1.38 2.07 2.75 3.44 4.13 4.82 5.51 6.20 6.88

0.72 1.44 2.17 2.89 3.61 4.33 5.06 5.78 6.50 7.22

11 12 13 14 15 16 17 13 19 20

8.04 8.78 9.51 10.24 10.97 11.70 12.43 13.16 13.90 14.63

7.50 8.18 8.87 9.55 10.23 10.91 11.59 12.28 12.96 13.64

8.01 8.74 9.47 10.20 10.93 11.65 12.38 13.11 13.84 14.57

7.54 8.22 8.91 9.59 10.28 10.96 11.65 12.33 13.02 13.70

7.98 8.70 9.43 10.16 10.88 11.61 12.33 13.06 13.78 14.51

7.57 8.26 8.95 9.64 10.33 11.01 11.70 12.39 13.08 13.77

7.95 8.67 9.39 10.11 10.84 11.56 12.28 13.00 13.72 14.45

7.61 11 8.30 12 8.99 13 9.68 14 10.37 15 11.06 16 11.76 17 12.45 • 18 13.14 19 13.83 20

21 22 23 24 25 26 27 28 29 30

15.36 16.09 16.82 17.55 18.28 19.02 19.75 20.48 21.21 21.94

14.32 15.00 15.69 16.37 17.05 17.73 18.41 19.10 19.78 20.46

15.30 16.02 16.75 17.48 18.21 18.94 19.67 20.39 21.12 21.85

14.39 15.07 15.76 16.44 17.13 17.81 18.50 19.19 19.87 20.56

15.23 15.96 16.68 17.41 18.13 18.86 19.59 20.31 21.04 21.76

14.46 15.14 15.83 16.52 17.21 17.90 18.59 19.27 19.96 20.65

15.17 15.89 16.61 17.34 18.06 18.78 19.50 20.23 20.95 21.67

14.52 15.21 15.90 16.60 17.29 17.98 18.67 19.36 20.05 20.75

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

22.67 23.40 24.13 24.87 25.60 26.33 27.06 27.79 28.52 29.25

21.14 21.82 22.51 23.19 23.87 24.55 25.23 25.92 26.60 27.28

22.58 23.31 24.04 24.76 25.49 26.22 26.95 27.68 28.41 29.13

21.24 21.93 22.61 23.30 23.98 24.67 25.35 26.04 26.72 27.41

22.49 23.21 23.94 24.66 25.39 26.11 26.84 27.56 28.29 29.01

21.34 22.03 22.72 23.40 24.09 24.78 25.47 26.16 26.85 27.53

22.39 23.12 23.84 24.56 25.28 26.01 26.73 27.45 28.17 28.89

21.44 22.13 22.82 23.51 24.20 24.89 25.59 26.28 26.97 27.66

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

29.99 30.72 31.45 32.18 32.91 33.64 34.37 35.10 35.84 36.57

27.96 28.64 29.33 30.01 30.69 31.37 32.05 32.74 33.42 34.10

29.86 30.59 31.32 32.05 32.78 33.51 34.23 34.96 35.69 36.42

28.09 28.78 29.46 30.15 30.83 31.52 32.20 32.89 33.57 34.26

29.74 30.47 31.19 31.92 32.64 33.37 34.09 34.82 35.54 36.27

28.22 28.91 29.60 30.29 30.98 31.66 32.35 33.04 33.73 34.42

29.62 30.34 31.06 31.78 32.51 33.23 33.95 34.67 35.40 36.12

28.35 29.04 29.74 30.43 31.12 31.81 32.50 33.19 33.88 34.58

41 42 43 44 45 46 47 48 49 50

6

Dep.

Lat.

Dep.

Lat.

Dep.

Lat%

Dep.

Lat.

o

Lat.

Distance.

Distance.

43 Deg.

Lat.

Dep. 0.69 1.38 2.07 2.77 3.46 4.15 4.84 5.53 6.22 6.92

1 2 3 4 5 6 7 8 9 10

0) o S3 C6 4-»

«

s

47 Deg.

46| Deg.

46£ Deg.

464 Deg. i

1 \ l' i

*1

TRAVERSE TABLE.

89

Dep.

Lat.

Dep.

Lat.

Dep.

Lat.

Dep.

Distance.

Distance.

Lat.

51 52 53 54 55 56 57 58 59 60

37.30 38.03 38.76 39.49 40.22 40.96 41.69 42.42 43.15 43.88

34.78 35.46 36.15 36.83 37.51 38.19 38.87 39.56 40.24 40.92

37.15 37.88 38.60 39.33 40.06 40.79 41.52 42.25 42.97 43.70

34.94 35.63 36.31 37.00 37.69 38.37 39.06 39.74 40.43 41.11

36.99 37.72 38.44 39.17 39.90 40.62 41.35 42.07 42.80 43.52

35.11 35.79 36.48 37.17 37.86 38.55 39.24 39.92 40.61 41.30

36.84 37.56 38.29 39.01 39.73 40.45 41.17 41.90 42.62 43.34

35.27 35.96 36.65 37.34 38.03 38.72 39.42 40.11 40.80 41.49

51 52 53 54 55 56 » 57 58 59 60

61 62 63 64 65 66 67 68 69 70

44.61 45.34 46.08 46.81 47.54 48.27 49.00 49.73 50.46 51.19

41.60 42.28 42.97 43.65 44.33 45.01 45.69 46.38 47.06 47.74

44.43 45.16 45.89 46.62 47.34 48.07 48.80 49.53 50.26 50.99

41.80 42.48 43.17 43.85 44.54 45.22 45.91 46.59 47.28 47.96

44.25 44.97 45.70 46.42 47.15 47.87 48.60 49.33 50.05 50.78

41.99 42.68 43.37 44.05 44.74 45.43 46.12 46.81 47.50 48.18

44.06 44.79 45.51 46.23 46.95 47.68 48.40 49.12 49.84 50.57

42.18 42.87 43.57 44.26 44.95 45.64 46.33 47.02 47.71 48.41

61 ' 62 63 1 64 65 66 67 68 69 70

71 72 73 74 75 76 ; 77 78 79 80

51.93 52.66 53.39 54.12 54.85 55.58 56.31 57.05 57.78 58.51

48.42 49.10 49.79 50.47 51.15 51.83 52.51 53.20 53.88 54.56

51.71 52.44 53.17 53.90 54.63 55.36 56.08 56.81 57.54 58.27

48.65 49.33 50.02 50.70 51.39 52.07 52.76 53.44 54.13 54.81

51.50 52.23 52.95 53.68 54.40 55.13 55.85 56.58 57.30 58.03

48.87 49.56 50.25 50.94 51.63 52.31 53.00 53.69 54.38 55.07

51.29 52.01 52.73 53.45 54.18 54.90 55.62 56.34 57.07 57.79

49.10 49.79 50.48 51.17 51.86 52.55 53.25 53.94 54.63 55.32

71 72 73 74 ■ 75 76 77 1 78 79 80

81 82 83 84 85 86 87 88 89 90

59.24 59.97 60.70 61.43 62.17 62.90 63.63 64.36 65.09 65.82

55.24 55.92 56.61 57.29 57.97 58.65 59.33 60.-02 60.70 61.38

59.00 59.73 60.45 61.18 61.91 62.64 63.37 64.10 64.82 65.55

55.50 56.18 56.87 57.56 58.24 58.93 59.61 60.30 60.98 61.67

58.76 59.48 60.21 60.93 61.66 62.38 63.11 63.83 64.56 65.28

55.76 56.45 57.13 57.82 58.51 59.20 59.89 60.58 61.26 61.95

58.51 59.23 59.96 60.68 61.40 62.12 62.85 63.57 64.29 65.01

56.01 56.70 57.40 58.09 58.78 59.47 60.16 60.85 61.54 62.24

81 82 1 83 84 85 86 > 87 88 89; 90

66.55 62.06 67.28 . 62.74 68.02 63.43 68.75 64.11 69.48 64.79 70.21 65.47 70.94 66.15 71.67 66.84 72.40 67.52 73.14 68.20

66.28 67.01 67.74 68.47 69.20 69.92 70.65 71.37 72.11 72.84

62.35 63.04 63.72 64.41 65.09 65.78 66.46 67.15 67.83 68.52

66.01 66.73 67.46 68.19 68.91 69.64 70.36 71.09 71.81 72.54

62.64 63.33 64.02 64.71 65.39 66.08 66.77 67.46 68.15 68.84

65.74 66.46 67.18 67.90 68.62 69.35 70.07 70.79 71.51 72.24

62.93 91 63.62 92 93 64.31 65.00 94 65.69 95 66.39 96 67.08 97 67.77 98 68.46 99 69.15 100

Dep.

Dep.

Lat.

Dep.

Lat.

Dep.

1

91 92 93 94 95 96 97 98 99 100 flj

o

43 Deg.

Lat.

43£ Deg.

431 Deg.

43} Deg.

Lat.

r<

do

rt

s

47 Deg.

46£ Deg.

46} Deg.

3A

464 Deg.

s

TRAVERSE TABLE

90 5 g o ?

'

O U)' p S o

44 Deg.

44$ Deg.

44 h:Deg.

44} Deg.

45 Deg.

Lat. Dep.

Lat. Dep.

Lat. Dep.

Lat. Dep.

Lat. Dep.

0.72 1.43 2.15 2.87 3.58 4.30 5.01 5.73 6.45 7.16

0.70 1.40 2.09 2.79 3.49 4.19 4.88 5.58 6.28 6.98

0.71 1.43 2.14 2.85 3.57 4.28 4.99 5.71 6.42 7.13

0.70 1.40 2.10 2.80 3.50 4.21 4.91 5.61 6.31 7.01

0.71 1.42 2.13 2.84 3.55 4.26 4.97 5.68 6.39 7.10

0.71 1.41 2.11 2.82 3.52 4.22 4.93 5.63 6.34 7.04

0.71 1.41 2.12 2.83 3.54 4.24 4.95 5.66 6.36 7.07

0.71 1.41 2.12 2.83 3.54 4.24 4.95 5.66 6.36 7.07

7.64 7.88 7.68 8.34 8.60 8.37 9.03 9.31 9.07 9.73 10.03 9.77 10.42 10.74 10.47 11.11 11.46 11.16 11.81 12.18 11.86 12.50 12.89 12.56 13.20 13.61 13.26 13.89 14.33 13.96

7.85 8.56 9.27 9.99 10.70 11.41 12.13 12.84 13.55 14.26

7.71 8.41 9.11 9.81 10.51 11.21 11.92 12.62 13.32 14.02

7.81 8.52 9.23 9.94 10.65 11.36 12.07 12.78 13.49 14.20

7.74 8.45 9.15 9.86 10.56 11.26 11.97 12.67 13.38 14.08

7.78 8; 49 9.19 9.90 10.61 11.31 12.02 12.73 13.43 14.14

7.78 8.49 9.19 9.90 10.61 11.31 12.02 12.73 13.43 14.14

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.69 1.39 2.08 2.78 3.47 4.17 4.86 5.56 6.25 6.95

1 2 3 4 5 6 7 ' 8 9 10

0.72 1.44 2.16 2.88 3.60 4.32 5.04 5.75 6.47 7.19

11 12 13 14 15 16 17 13 19 20

7.91 8.63 9.35 10.07 10.79 11.51 12.23 12.95 13.67 14.39

21 22 23 24 25 26 27 28 29 30

15.11 14.59 15.83 15.28 16.54 15.98 17.26 16.67 17.98 17.37 18.70 18.06 19.42 18.76 20.14 19.45 20.86 20.15 21.58 20.84

15.04 14.65 15.76 15.35 16.47 16.05 17.19 16.75 17.91 17.44 18.62 18.14 19.34 18.84 20.06 19.54 20.77 20.24 21.49 20.93

14.98 14.72 15.69 15.42 16.40 16.12 17.12 16.82 17.83 17.52 18.54 18.22 19.26 18.92 19.97 19.63 20.68 20.33 21.40 21.03

14.91 14.78 14.85 14.85 15.62 15.49 15.56 15.56 16.33 16.19 16.26 16.26 17.04 16.90 16.97 16.97 17.75 17.60 17.68 17.68 18.46 18.30 18.38 18.38 19.17 19.01 19.09 19.09 19.89 19.71 19.80 19.80 20.60 20.42 20.51 20.51 21.31 21.12 21.21 21.21

31 32 33 34 35 36 37 38 39 40

22.30 21.53 23.02 22.23 23.74 22.92 24.46 23.62 25.18 24.31 25.90 25.01 26.62 25.70 27.33 26.40 28.05 27.09 28.77 27.79

22.21 21.63 22.92 22.33 23.64 23.03 24.35 23.72 25.07 24.42 25.79 25.12 26.50 25.82 27.22 26.52 27.94 27.21 28.65 27.91

22.11 21.73 22.82 22.43 23.54 23.13 24.25 23.83 24.96 24.53 25.68 25.23 26.39 25.93 27.10 26.63 27.82 27.34 28.53 28.04

22.02 21.82 22.73 22.53 23.44 23.23 24.15 23. $4 24.86 24.64 25.57 25.34 26.28 26.05 26.99 26.75 27.70 27.46 28.41 28.16

41 42 43 44 45 46 47 48 49 50

29.49 28.48 30.21 29.18 30.93 29.87 31.65 30.56 32.37 31.26 33.09 31.95 33.81 32.65 34.53 33.34 35.25 34.04 35.97 34.73

29.37 28.61 30.08 29.31 30.80 30.00 31.52 30.70 32.23 31.40 32.95 32.10 33.67 32.80 34.38 33.49 35.10 34.19 35.82 34.89

29.24 28.74 29.96 29.44 30.67 30.14 31.38 30.84 32.10 31.54 32.81 32.24 33.52 32.94 34.24 33.64 34.95 34.34 35.66 35.05

29.12 28.86 28.99 28.99 29.83 29.57 29.70 29.70 30.54 30.27 30.41 30.41 31.25 30.98 31.11 31.11 31.96 31.68 31.82 31.82 32.67 32.38 32.53 32.53 33.38 33.09 33.23 33.23 34.09 33.79 33.94 33.94 34.80 34.50 34.65 34.65 35.51 35.20 35.36 35.36

« o a ■2

Dep. Lat.

Dep. Lat.

Dep. Lat.

Dep. Lat.

Dep. Lat.

p u

46 Deg.

451:Deg.

45 i Deg.

45$ Deg.

45 Deg.

5

s

1 2 3 4 5 6 7

8 9 10 ; i

21.92 21.92 31 22.63 22.63 32 23.33 23.33 33 24.04 24.04 34 24.75 24.75 35 25.46 25.46 36 26.16 26.16 37 26.87 26.87 38 27.58 27.58 . 39 28.28 28.28 40: 41 42 43 44 45 46 47 48 • 49 50

fl

&

i

TRAVERSE TABLE, g «5 * ST

44 Deg.

44$ Deg.

44£ Deg.

44$ Deg.

o

to

L^ta Dep.

Lat. Dep.

Lat. Dep.

Lat. Dep.

51 52 53 54 55 56 57 58 59 60

36.69 35.43 37.41 36.12 38.12^36.82 38.84 37.51 39.56 38.21 40.28 38.90 41.00 39.60 41.72 40.29 42.44 40.98 43.16 41.68

36.53 35.59 37.25 36.29 37.96 36.98 38.68 37.68 39.40 38.38 40.11 39.08 40.83 39.77 41.55 40.47 42.26 41.17 42.98 41.87

91 45 Deg. '! Lat. Dep.

36.38 35.75 36.22 35.90 36.06 36.06 37.09 36.45 36.93 36.61 36.77 36.77 37.80 37.15 37.64 37.31 37.48 37.48 38.52 37.85 38.35 38.02 38.18 38.18 39.23 38.55 39.06 38.72 38.89 38.89 39.94 39.25 39.77 39.42 39.60 39.60 40.66 39.95 40.48 40.13 40.31 40.31 41.37 40.65 41.19 40.83 41.01 41.01 42.08 41.35 41.90 41.54 41.72 41.72 42.79 42.05 42.61 42.24 42.43 42.43

61 43.88 42.37 43.69 42.57 43.51 42.76 62 44.60 43.07 44.41 43.26 44.22 43.46 63 45.32 43.76 45.13 43.96 44.93 44.16 64 46.04 44.46 45.84 44.66 ,45.65 44.86 65 46.76 45.15 46.56 45.36 46.36 45.56 66 47.48 45.85 47.23 46.05 47.07 46.26 67 48.20 46.54 47.99 46.75 47.79 46.96 68 48.92 47.24 48.71 47.45 48.50 47.66 69 49.63 47.93 49.42 48.15 49.21 48.36 70 50.35 48.63 50.14 48.85 49.93 49.06

43.32 42.94 43.13 43.13 44.03 43.65 43.84 43.84 44.74 44.35 44.55 44.55 45.45 45.06 45.25 45.25 46.16 45.76 45.96 45.96 46.87 46.46 46.67 46.67 47.58 47.17 47.38 47.38 48.29 47.87 48.08 48.08 49.00 48.58 48.79 48.79 49.71 49.28 49.50 49.50

g in' 57 53 O ®

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

71 51.07 49.32 50.86 49.54 50.64 49.76 50.42 49.98 72 51.79 50.02 51.57 50.24 51.35 50.47 51.13 50.69 73 52.51 50.71 52.29 50.94 52.07 51.17 51.84 51.39 74 53.23 51.40 53.01 51.64 52.78 51.87 52.55 52.10 75 53.95 52.10 53.72 52.33 53.49 52.57 53.26 52.80 76 54.67 52.79 54.44 53.03 54.21 53.27 53.97 53.51 77 55.39 53.49 55.16 53.73 54.92 53.97 54.68 54.21 78 56.11 54.18 55.87 54.43 55.63 54.67 55.39 54.91 79 56.83 54.88 56.59 55.13 56.35 55.37 56.10 55.62 80 57.55 55.57 57.30 55.82 57.06 56.07 56.81 56.32

50.20 50.20 50.91 50.91 51.62 51.62 52.33 52.33 53.03 53.03 53.74 53.74 54.45 54.45 55.15 55.15 55.86 55.86 56.57 56.57

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

58.27 56.27 58.99 56.96 59.71 57.66 60.42 58.35 61.14 59.05 61.86 59.74 62.58 60.44 63.30 61.13 64.02 61.82 64.74 62.52

58.02 56.52 58.74 57.22 59.45 57.92 60.17 58.61 60.89 59.31 61.60 60.01 62.32 60.71 63.03 61.41 63.75 62.10 64.47 62.80

57.77 56.77 58.49 57.47 59.20 58.18 59.91 58.88 60.63 59.58 61.34 60.28 62.05 60.98 62.77 61.68 63.48 62.38 64.19 63.08

57.52 57.03 58.24 57.73 58.95 58.43 59.66 59.14 60.37 59.84 61.08 60.55 61.79 61.25 62.50 61.95 63.21 62.66 63.92 63.36

57.28 57.28 57.98 57.98 58.69 58.69 59.40 59.40 60.10 60.10 60.81 60.81 61.52 61.52 62.23 62.23 62.93 62.93 63.64 63.64

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

65.46 63.21 66.18 63.91 66.90 64.60 67.62 65.30 68.3465.99 69.06 66.69 69.78 67.38 70.50 68.08 71.21 68.77 71.93 69.47

65.18 63.50 65.90 64.20 66.62 64.89 67.33 65.59 68.05 66.29 68.76 66.99 69.48 67.69 70.20 68.38 70.91 69.08 71.63 69.78

64.91 65.62 66.33 67.05 67.76 68.47 69.19 69.90 70.61 71.33

64.63 64.07 65.34 64.77 66.05 65.47 66.76 66.18 67.47 66.88 68.18 67.59 68.89 68.29 69.60 68.99 70.31 69.70 71.02 70.40

64.35 64.35 91 65.05 65.05 92 65.76 65.76 93 66.47 66.47 94 67.18 67.18 95 67.88 67.88 96 68.59 68.59 97 69.30 69.30 98 70.00 70.00 99 70.71 70.71 100

»

Dep, Lat.

Dep. Lat.

Dep. Lat.

Dep. Lat.

Dep. Lat.

.a

46 Deg.

45} Deg.

45$ Deg.

45$ Deg.

45 Deg.

o rtfl

Q

63.73 64.48 65.18 65.89 66.59 67.29 67.99 68.69 69.39 70.09

v o a m

3

A TABLE OP

LOGARITHMS, FROM

Note.

1

TO

10,000.

The index of the logarithm of every integer

number consisting of only one figure is 0, of two figures 1, of three figures 2, of four figures 3; being always a unit less than the number of figures contained in the integer number.

In this table, as is generally the case,

the index to the logarithm of every number above 100 is omitted; yet in the operation must be prefixed according to this remark; so the logarithm of 700 is 2,84510, and of 7000 is 3,84510, and so of the rest. No.

Log.

No.

Log.

No.

Log.

No.

Log.

No.

Log.

1 2 3 4 5 6 7 8 9 10

0.00000 21 0.30103 22 0.47712 23 0.60206 ' 24 0.69897 25 0.77315 26 0.84510 27 0.90309 28 0.95424 29 1.00000 30

1.32222 1.34242 1.36173 1.38021 1.39794 1.41497 1.43136 1.44716 1.46240 1.47712

41 42 43 44 45 46 47 48 49 50

1.61278 1.62325 1.63347 1.64345 1.65321 1.66276 1.67210. 1.68124 1.69020 1.69897

61 62 63 64 65 66 67 68 69 70

1.78533 1.79239 1.79934 1.80618 1.81291 1.81954 1.82607 1.83251 1.83885 1.84510

81 82 83 84 85 86 87 88 89 90

1.90849 1.91381 1.91908 1.92428 1.92942 1.93450 1*93952 1.94448 1.94939 1.95424

11 12 13 14 15 16 17 18 19 20

1.04139 1.07918 1.11394 1.14613 1.17609 1.20412 1.23045 1.25527 1.27875 1.30103

1.49136 1.50515 1.51851 1.53148 1.54407 1.55630 1.56820 1.57978 1.59106 1.60206

51 52 53 54 55 56 57 58 59 60

1.70757 1.71600 1.72428 1.73239 1.74036 1.74819 1.75587 1.76343 1.77085 1.77815

71 72 73 74 75 76 77 78 79 80

1.85126 91 1.85733 92 1.86332 93 1.86923 94 1.87506 95 1.88081 96 1.88649 97 1.89209 98 1.89763 99 1.90309 100

1.95904 1 96379 1.96848 1.97313 1.97772 1.98227 1.98677 1.99123 1.99564 2.00000

31 32 33 34 35 36 37 38 39 40

J j

Logarithms from I to 10,000.

93

No.

0

1

2

3

4

5

6

7

8

9

100 101 102 103 104 105 106 107 103 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159

00000 00432 00860 01234 01703 02119 02531 02933 03342 03743 04139 04532 04922 05308 05690 06070 06446 06819 07188 07555 07918 08279 08636 08991 09342 09691 10037 10380 10721 11059 11394 11727 12057 12385 12710 13033 13354 13672 13938 14301 14613 14922 15229 15534 15836 16137 16435 16732 17026 17319 17609 17898 18184 18469 18762 19033 19312 19590 19866 20140

00043 00475 00903 01326 01745 02160 02572 02979 03383 03782

00087 00518 00945 01368 01787 02202 02612 03019 03423 03822 04218 04610 04999 05385 05767 06145 06521 06893 07262 07628 07990 08350 08707 09061 09412 09760 10106 10449 10789 11126 11461 11793 12123 12450 12775 13098 13418 13735 14051 14364 14675 14983 15290 15594 15897 16197 16495 16791 17085 "7377 17667 17955 18241 18526 18808 19089 19368 19645 19921 20194

00130 00561 00988 01410 01828 02243 02653 03060 03463 03862 04258 04650 05038 05423 05805 06183 06558 06930 07298 07664 08027 08386 08743 09096 09447 09795 10140 10483 10823 11160 11494 11826 12156 12483 12808 13130 13450 13767 14082 14395 14706 15014 15320 15625 15927 16227 16524 16820 17114 17406 17696 17984 18270 18554 18837 19117 19396 19673 19948 20222

00173 00604 01030 01452 01870 02284 02694 03100 03503 03902 04297 04689 05077 05461 05843 06221 06595 06967 07335 07700 08063 08422 08778 09132 09482 09830 10175 10517 10857 11193 11528 11860 12189 12516 12840 13162 13481 13799 14114 14426 14737 15045 15351 15655 15957 16256 16554 16850 17143 17435 17725 18013 18298 18583 18865 19145 19424 19700 19976 20249

00217 00647 01072 01494 01912 02325 02735 03141 03543 03941

00260 00689 01115 01536 01953 02366 02776 03181 03583 03981 04376 04766 05154 05538 05918 06296 06670 07041 07408 07773 08135 08493 08849 09202 09552 09899 10243 10585 10924 11261 11594 11926 12254 12581 12905 13226 13545 13862 14176 14489 14799 15106 15412 15715 16017 16316 16613 16909 17202 17493 17782 18070 18355 18639 18921 19201 19479 19756 20030 20303

00303 00732 01157 01578 01995 02407 02816 03222 03623 04021

00346 00775 01199 01620 02036 02449 02857 03262 03663 04060 04454 04844 05231 05614 05994 06371 06744 07115 07482 07846 08207 08565 08920 09272 09621 09968 10312 10653 10992 11327 11661 11992 12320 12646 12969 13290 13609 13925 14239 14551 14860 15168 15473 15776 16077 16376 16673 16967 17260 17551 17840 18127 18412 18696 18977 19257 19535 19811 20085 20358

00389 00817 01242 01662 02078 02490 02898 03302 03703 04100 04493 04883 05269 05652 06032 06408 06781 07151 07518 07882 08243 08600 08955 09307 09656 10003 10346 10687 11025 11361

04179 04571 04961 05346 05729 06108 06483 06856 07225 07591 07954 08314 08672 09026 09377 09726 10072 10415 10755 11093 11428 11760 12090 12418 12743 13066 13386 13704 14019 14333 14644 14953 15259 15564 15866 16167 16465 16761 17056 17348 17638 17926 18213 18498 18780 19061 19340 19618 19893 20167

32

04336 04727 05115 05500 05881 06258 06633 07004 07372 07737 0$®99 08458 08814 09167 09517 09864 10209 10551 10890 11227 11561 11893 12222 12548 12872 13194 13513 13830 14145 14457 14768 15076 15381 15685 15987 16286 16584 16879 17173 17464 17754 18041 18327 18611 18893 19173 19451 19728 20003 20276

04415 04805 05192 05576 05956 06333 06707 07078 07445 07809 08171 08529 08884 09237 09587 09934 10278 10619 10958 11294 11628 11959 12287 12613 12937 13258 13577 13893 14208 14520 14829 15137 15442 15746 16047 16346 16643 16938 17231 17522 17811 18099 18384 18667 18949 19229 19507 19783 20058 20330

11694 12024 12352 12678 13001 13322 13640 13956 14270 14582 14891 15198 15503 15806 16107 16406 16702 16997 17289 17580 17869 18156 18441 18724 19005 1928£ 19562 19838 20112 2038A

1

91

Logarithms from 1 to 10,000.

No.

0

1

2

3

4

5

6

7

8

160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219

20412 20683 20952 21219 21484 21748 22011 22272 22531 22789 23045 23300 23553 •23805 24055 24304 24551 24797 25042 25285 25527 25768 26007 26245 26482 26717 26951 27184 27416 27646 27875 28103 28330 28556 28780 29003 29226 29447 29667 29885 30103 30320 30535 30750 30963 31175 31387 31597 31806 32015 32222 32428 32634 32838 33041 33244 33445 33646 33846 34044

20439 20710 20978 21245 21511 21775 22037 22298 22557 22814 23070 23325 23578 23830 24080 24329 24576 24822 25066 25310 25551 25792 26031 26269 26505 26741 26975 27207 27439 27669 27898 28126 28353 28578 28803 29026 29248 29469 29688 29907

20466 20737 21005 21272 21537 21801 22063 22324 22583 22840 23096 23350 23603 23855 24105 24353 24601 24846 25091 25334

20493 20763 21032 21299 21564 21827 22089 22350 22608 22866 23121 23376 23629 23880 24130 24378 24625 24871 25115 25358

20548 20817 21085 21352 21617 21880 22141 22401 22660 22917

25575 25816 26055 26293 26529 26764 26998 27231 27462 27692 27921 28149 28375 28601 28825 29048 29270 29491 29710 29929 30146 30363 30578 30792 31006 31218 31429 31639 31848 32056 32263 32469 32675 32879 33082 33284 33486 33686 33885 34084

2560g 25840 26079 26316 26553 26788 27021 2?254 27485 27715 27944 28171 28398 28623 28847 29070 29292 29513 29732 29951 30168 30384 30600 30814 31027 31239 31450 31660 31869 32077

20520 20790 21059 21325 21590 21854 22115 22376 22634 22891 23147 23401 23654 23905 24155 24403 24650 24895 25139 25382 25624 25864 26102 26340 26576 26811 27045 27277 27508 27738 27967 28194 28421 28646 28870 29092 29314 29535 29754 29973 30190 30406 30621 30835 31048 31260 31471 31681 31890 32098 32305 32510 32715 32919 33122 33325 33526 33726 33925 34124

20575 20844 21112 21378 21643 21906 22167 22427 22686 22943 23198 23452 23704 23955 24204 24452 24699 24944 25188 25431 25672 25912 26150 26387 26623 26858 27091 27323 27554 27784

20602 20871 21139 21405 21669 21932 22194 22453 22712 22968 23223 23477 23729 23980 24229 24477 24724 24969 25212 25455 25696 25935 26174 26411 26647 26881 27114 27346 27577 27807

28012 28240. 28466 28691 28914 29137 29358 29579 29798 30016 30233 30449 30664 30878 31091 31302 31513 31723 31931 32139 32346 32552 32756 32960 33163 33365 33566 33766 33965 34163)

28035 28262 28488 28713 28937 29159 29380 29601 29820 30038 30255 30471 30685 30899 31112 31323 31534 31744 31952 32160 32366 32572 32777 32980 33183 33385 33586 33786 33985 34183

20629 20898 21165 21431 21696 21958 22220 22479 22737 22994 23249 23502 23754 24005 24254 24502 24748 24993 25237 25479 25720 25959 26198 26435 26670 26905 27138 27370 27600 27830 28058 28285 28511 28735 28959 29181 29403 29623 29842 30060 30276 30492 30707 30920 31133 31345 31555 31765 31973 32181 32387 32593 32797 33001 33203 33405 33606 33806 34005 34203

30125 30341 30557 30771 30984 31197 31408 31618 31827 32035 32243 32449 32654 32858 33062 33264 33465 33666 33866 34064

32284 32490 32695 32899 33102 33304 33506 33706 33905 34104 1

23172 23426 23679 23930 24180 24428 24674 24920 25164 25406 25648 25888 26126 26364 26600 26834 27068 27300 27531 27761 27989 28217 28443 28668 28892 29115 29336 29557 29776 29994 30211 30428 30643 30856 31069 31281 31492 31702 31911 32118 32325 32531 32736 32940 33143 33345 33546 33746 33945 34143

1

9

20656 20925 21192 21458 21722 ' 21985 22246 22505 22763 23019 23274 23528 23779 24030 24279 24527 24773 25018 25261 25503 25744 25983 26221 26458 26694 26928 27161 27393 27623 27852 28081 28307 28533 28758 28981 29203 29425 29645 29863 30081 30298 30514 30728 30942 31154 31366 31576 31785 31994 32201 32408 32613 32818 33021 33224 33425 33626 33826 34025 34223

95

Logarithms from 1 to 10,000, No.

0

1

2

3

4

5

6

7

8

220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 . 275 276 i <277 278 279

34242 34439 34635 34830 35025 35218 35411 35603 35793 35984 36173 36361 36549 36736 36922 37107 37291 37475 37658 37840 38021 38202 38382 38561 38739 38917 39094 39270 39445 39620 39794 39967 40140 40312 40483 40654 40824 40993 41162 41330 41497 41664 41830 41996 42160 42325 42488 42651 42813 42975 43136 43297 43457 43616 43775 43933 44091 44248 44404 44560

34262 34459 34655 34850 35044 35238 35430 35622 35813 36003 36192 36380 36568 36754 36940 37125 37310 37493 37676 37858 38039 38220 38399 38578 38757 38934 39111 39287 39463 39637 39811 39985 40157 40329 40500 40671 40841 41010 41179 41347

34282 34479 34674 34869 35064 35257 35449 35641 35832 36021 36211 36399 36586 36773 36959 37144 37328 37511 37694 37876 38057 38238 38417 38596 38775 38952 39129 39305 39480 39655

34301 34498 34694 34889 35083 35276 35468 35660 35851 36040 36229 36418 36605 36791 36977 37162 37346 37530 37712 37894 38075 38256 38435 38614 38792 38970 39146 39322 39498 39672 39846 40019 40192 40364 40535 40705 40875 41044 41212 41380 41547 41714 41380 42045 42210 42374 42537 42700 42862 43024 43185 43345 43505 43664 43823 43981 44138 44295 44451 44607

34321 34518 34713 34908 35102 35295 35488 35679 35870 36059 36248 36436 36C24 36810 36996 37181 37365 37548 37731 37912 38093 38274 38453 38632 38810 38987 39164 39340 39515 ; 39690 1 39863 40037 40209 40381 40552 40722 40892 41061 41229 41397 41564 41731 41896 42062 42226 42390 42553 42716 42878 43040 43201 43361 43521 43680 43838 43996 44154 44311 44467 44623

34341 34537 34733 34928 35122 35315 35507 35698 35889 36078 36267 36455 36642 36829 37014 37199 37383 37566 37749 37931

34361 34557 34753 34947 35141 35334 35526 35717 35908 30097

34380 34577 34772 34967 35160 35353 35545 35736 35927 36116

36286 36474 36661 36847 37033 37218 37401 37585 37767 37949 38130 38310 38489 38668 38846 39023 39199 39375 39550 39724

36305 36493 36680 36866 37051 37236 37420 37603 37785 37967 38148 38328 38507 38686 38863 39041 39217 39393 39568 39742

39898 40071 40243 40415 40586 40756 40926 41095 41263 41430 41597 41764 41929 42095 42259 42423 42586 42749 42911 43072 43233 43393 43553 43712 43870 44028 44185 44342 44498 44654

39915 40088 40261 40432 40603 40773 40943 41111 41280 41447

34400 34596 34792 34986 35180 35372 35564 35755 35946 36135 36324 36511 36698 36884 37070 37254 37438 37621 37803 37985 38166 38346 38525 38703 38881 39058 39235 39410 39585 39759 39933 40106 40278 40449 40620 40790 40960 41128 41296 41464 41631 41797 41963 42127 42292 42455 42619 42781 42943 43104 43265 43425 43584 43743 43902 44059 44217 i 44373| 44529 44685 I

41514 41681 41847 42012 42177 42341 42504 42667 42830 42991 43152 43313 43473 43632 43791 43949 44107 44264 44420 44576

39829 40002 40175 40346 40518 40688 40858 41027 41196 41363 41531 41697 41863 42029 42193 42357 42521 42684 42846 43008 43169 43329 43489 43648 43807 43965 44122 44279 44436 44592

38112 38292 38471 38650 38828 39005 39182, 39358 39533 39707 39881 40054 40226 40398 40569 40739 40909 41078 41246 41414 41581 41747 41913 4207b 42243 42406 42570 42732 42894 43056 43217 43377 43537 43696 43854 44012 44170 44326 44483 44638

41614 41780 41946 42111 42275 42439 42602 42765 42927 43088 43249 43409 43569 43727 43886 44044 44201 44358 44514 44669

9 34420 34616 34811 35005 35199 35392 35583 35774 35965 36154 36342 26530 36717 36903 37088 37273 37457 37639 37822 38003 38184 38364 38543 38721 ' 38899 39076 39252 39428 39602 39777 39950 40123 40295 40466 40637 40807 409.76 41145 41313 41481 41647 41814 41979 42144 42308 42472 42635 42797 42959 13120 43281 43441 43600 43759 43917 44075 44232 44389 44545 44700

96

Logarithms from 1 to 10,000.

No.

0

1

2

3

4

5

6

7

* 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 ■295 296 297 298 299 300 ‘301 , 302 303 304 305 306 307 308 '309 310 311 312 313 ■ 314 315 ■ 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339

44716 44871 45025 45179 45332 45484 45637 45788 45939 46090 46240 46389 46538 46687 46835 46982 47129 47276 47422 47567

44731 44886 45040 45194 45347 45500 45652 45803 45954 46105 46255 46404 46553 46702 46850 46997 47144 47290 47436 47582 47727 47871 48015 48159 48302 48444 48586 48728 48869 49010 49150 49290 49429 49568 49707 49845 49982 50120 50256 50393 50529 50664 50799 50934 51068 51202 51335 51468 51601 51733

44747 44902 45056 45209 45362 45515 45667 45818 45969 46120 46270 46419 46568 46716 46864 47012 47159 47305 47451 47596 47741 47885 48029 48173 48316 48458 48601 48742 48883 49024 49164 49304 49443 49582 49721 49859 49996 50133 50270 50406 50542 50678 50813 50947 51081 51215 51348 51481 51614 51746 51878 52009 52140 52270 52401 52530 52660 52789 52917 53046

44762 44917 45071 45225 45378 45530 45682 45834 45984 46135 46285 46434 46583 46731 46879 47026 47173 47319 47465 47611 47756 47900 48044 48187 48330 48473 48615 48756 48897 49038 49178 49318 49457 49596 49734 49872 50010 50147 50284 50420 50556 50691 50826 5flQ61 51095 51228 51362 51495 51627 51759 51891 52022 52153 52284 52414 52543 52673 52802 52930 53058

44778 44932 45086 45240 45393 45545 45697 45849 46000 46150 46300 46449 46598 46746 46894 47041 47188 47334 47480 47625 47770 47914 48058 48202 48344 48487 48629 48770 48911 49052 49192 49332 49471 49610 49748 49886 50024 50161 50297 50433

44793 44948 45102 45255 45408 45561 45712 45864 46015 46165 46315 46464 46613 46761 46909 47056 47202 47349 47494 47640 47784 47929 48073 48216 48359 48501 48643 48785 48926 49066

44809 44963 45117 45271 45423 45576 45728 45879 46030 46180

49206 49346 49485 49624 49762 49900 50037 50174 50311 50447

50569 50705 50840 50974 51108 51242 51375 51508 51640 51772 51904 52035 52166 52297 52427 52556 52686 52815 52943 53071

50583 50718 50853 50987 51121 51255 51388 51521 51654 51786 51917 52048 52179 52310 52440 52569 52699 52827 52956 53084

44824 44979 45133 45286 45439 45591 45743 45894 46045 46195 46345 46494 46642 46790 46938 47085 47232 47378 47524 47669 47813 47958 48101 48244 48387 48530 48671 48813 48954 49094 49234 49374 49513 49651 49790 49927 50065 50202 50338 50474 50610 50745 50880 51014 51148 51282 51415 51548 51680 51812 51943 52075 52205 52336 52466 52595 52724 52853 52982 53110

47712 47857 48001 48144 48287 48430 48572 48714 48855 48996 49136 49276 49415 49554 49693 49831 49969 50106 50243 50379 50515 50651 50786 50920 51055 51188 51322 51455 51587 51720 51851 51983 52114 52244 52375 52504 52634 52763 52892 l 53020

51865 51996 52127 52257 52388 52517 52647 52776 52905 53033

46330 46479 46627 46776 46923 47070 47217 47363 47509 47654 47799 47943 48087 48230 48373 48515 48657 48799 48940 49080 49220 49360 49499 49638 49776 49914 50051 50188 50325 50461 50596 50732 50866 51001 51135 51268 51402 51534 51667 51799 51930 52061 52192 52323 52453 52582 52711 52840 52969 53097

8

9

I

44840 44855 44994 45010 45148 45163 45301 45317 45454 45469 45606 45621 45758 45773 45909 45924 46060 46075 46210 46225 46359 46374 46509 46523 40657 46672 46805 46820 46953 46967 47100 47114 47246 47261 47392 47407 47538 47553 47683 47698 47828 47842 47972 47986 48116 48130 48259 48273 48401 48416 48544 48558 48686 48700 48827 48841 48968 48982 49108 49122 49248 49262 49388 49402 49527 49541 49665 49679 49803 49817 49941 49955 50079 50092 50215 50229 50352 50365 50488 50501 50623 50637 50759 50772 50893 j! 60907 51028 j 51041 51162. 51175 51295 51308 51428 51441 51561 51574 51693 ! 51706 51825 51838 51957 61970 52088 52101 52218 52231 52349 52362 52479 52492 52608 52621 52737 52750 52866 52879 52994 53007 53122 53135

Logarithms from 1 to 10,000.

97

No.

0

1

2

3

4

5

6

7

8

9

340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 . 367 368 369

53148 53275 53403 53529 53656 53782 53908 54033 54158 54283 54407 54531 54654 54777 54900 55023 55145 55267 55388 55509

53131 53288 53415 53542 5366° 53794 53920 54045 54170 54295 54419 54543 54667 54790 54913 55035 55157 55279 55400 55522

53173 53301 53428 53555 53681 53807 53933 54058 54183 54307 54432 54555 54679 54802 54925 55047 55169 55291 55413 55534

53186 53314 53441 53567 53694 53820 53945 54070 54195 54320 54444 54568 54691 54814 54937 55060 55182 55303 55425 55546

53212 53339 53466 53593 53719 53845 53970 54095 54220 54345 54469 54593 54716 54839 54962 55084 55206 55328 i 55449 55570

53224 53352 53479 53605 53732 53857 53983 54108 54233 54357

53250 53377 53504 53631 53757 53882 54008 54133 54258 54382

53263 53390 53517 53643 53769 53895 54020 54145 54270 54394

54481 54605 54728 54851 54974 55096 55218 55340 o5461 55582

53237 53364 53491 53618 53744 53870 53995 54120 54245 54370 54494 54617 54741 54864 54986 55108 55230 55352 55473 55594

55630 55751 55871 55991 56110 56229 56348 56467 56585 56703 56820 56937 57054 57171 57287 57403 57519 57634 57749 57864

55654 55775 55895 56015 56134 56253 56372 56490 56608 56726 56844 56961 57078 57194 57310 57426 57542 57657 57772 57887 58001 58115 58229 58343 58456 58569 58681 58794 58906 59017 59129 59240 59351 59461 59572 59682 59791 59901 60010 60119

55666 55787 55907 56027 56146 56265 56384 56502 56620 .56738 56855 56972 57089 57206 57322 57438 57553 57669 57784 57898 58013 581&7 58240 58354 58467 58580 58692 58805 58917 59028 59140 59251 59362 59472 59583 59693 59802 59912 60021 60130

55691 55811 55931 56050 56170 56289 56407 56526 56644 56761 56879 56996 57113 57229 57345 57461 57576 57692 57807 57921 58035 58149 58263 58377 58490 58602 58715 58827 58939 59051 59162 59273 59384 59494 59605 59715 59824 59934 60043 60152

55703 55823 55943 56062 56182 56301 56419 56533 56656 56773 56891 57008 57124 57241 57357 57473 57588 57703 57818 57933 58047 58161 58274 53388 58501 58614 58726 58838 58950 59062 59173 59284 59395 59506 59616 59726 59835 59945 60054 60163

55715 55835 55955 56074 56194 56312 56431 56549 56667 56785 56902 57019 57136 57252 57368 57484 57600 57715 57830 57944 58058 58172 53286 58399 58512 58625 58737 58850 58961 59073 59184 59295 59406 59517 59627 59737 59846 59956 60065 60173

54506 54630 54753 54876 54998 55121 55242 55364 55485 55606 55727 55847 55967 56086 56205 56324 56443 56561 56679 56797

370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399

55642 55763 55883 56003 56122 56241 56360 56478 56597 56714 56832 56949 57066 57183 57299 57415 57530 57646 57761 57875 57990 58104 58218 58331 58444 58557 58670 58782 58894 59006 59118 59229 59340 59450 59561 59671 59780 59890 59999 60108

53199 53326 53453 53580 53706 53832 53958 54083 54208 54332 54456 54580 5470^ 54827 54949 55072 55194 55315 55437 55558 55678 55799 55919 56038 56158 56277 56396 56514 56632 56750 56867 56984 57101 57217 57334 57449 57565 57680 57795 57910 58024 58138 58252 58365 58478 58591 58704 58816 58928 59040 59151 59262 59373 59483 59594 59704 59813 59923 60032 60141

54518 54642 54765 54888 55011 55133 55255 55376 55497 55618 55739 55859 55979 56098 56217 56336 56455 56573 56691 56808 56926 57043 57159 57276 57392 57507 57623 57738 57852 57967 58081 58195 58309 58422 58535 58647 58760 58872 58984 59095 59207 59318 59428 59539 59649 59759 59868 59977 60086 60195



,

■ ’

57978 58092 58206 58320 58433 58546 58659 58771 58883 58995 *59106 59218 59329 59439 59550 5966U 59770 59879 59988 60097

23*

31*

56914 57031 57148 57264 57380 57496 57611 57726 57841 57955 58070 58184 58297 58410 58524 58636 58749 58861 58973 59084 59195 59306 59417 59528 59638 59748 59857 59966 60076 60184

98

Logarithms from 1 to 10,000.

No.

0

1

2

3

4

5

C

7

8

400 401 402 403 404 405 j 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 .431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459

60206 60314 60423 60531 60638 60746 60853 60959 61066 61172

60217 60325 60433 60541 60649 60756 60863 60970 61077 61183

60249 60358 60466 60574 60681 60788 60895 61002 61109 61215 61321 61426 61532 61637 61742 61847 61951 62055 62159 62263 62366 62469 62572 62675 62778 62880 62982 63083 63185 63286 63387 63488 63589 63689 63789 63889 63988 64088 64187 64286

60260 60369 60477 60534 60692 60799 60906 61013 61119 61225 61331 61437 61542 61648 61752 61857 61962 62066 62170 62273 62377 62480 62583 62685 62788 62890 62992 63094 63195 63296 63397 63498 63599 63699 63799 63899 63998 64098 64197 64296 64395 64493 64591 64689 64787 64885 64982 65079 65176 65273 65369 65466 65562 65658 65753 65849 65944 66039 66134 66229 1

60271 60379 60487 60595 60703 60810 60917 61023 61130 61236 61342 61448 61553 61658 61763 61868 61972 62076 62180 62284 62387 62490 62593 62696 62798 62900 63002 63104 63205 63306 63407 63508 63609 63709 63809 63909 64008 64108 64207 64306 64404 64503 64601 64699 64797 64895 64992 65089 65186 65283 65379 65475 65571 65667 65763 65858 65954 66049 66143 66238 1

60293 60401 60509 60617 60724 60831 60938 61045 61151 61257

61289 61395 61500 61606 61711 61815 61920 62024 62128 62232 62335 62439 62542 62644 62747 62849 62951 63053 63155 63256 63357 63458 63558 63659 63759 63859 63959 64058 64157 64256 64355 64454 64552 64650 64748 64846 64943 65040 65137 65234 65331 65427 65523 65619 65715 65811 65906 66001 66096 66191

60239 60347 60455 60563 60670 60778 60885 60991 61098 61204 61310 61416 61521 61627 61731 61836 61941 62045 62149 62252 62356 62459 62562 62665 62767 62870 62972 63073 63175 63276 63377 63478 63579 63679 63779 63879 63979 64078 64177 64276 64375 64473 64572 64670 64768 64865 64963 65060 65157 65254 65350 65447 65543 65639 65734 65830 65925 66020 66115 66210

60282 60390 60498 60606 60713 60821 60927 61034 61140 61247

61278 61384 61490 61595 61700 61805 61909 62014 62118 62221 62325 62428 62531 62634 62737 62839 62941 63043 63144 63246 63347 63448 63548 63649 63749 63849 63949 64048 64147 64246

60228 60336 60444 60552 60660 60767 60874 60981 61087 61194 61300 61405 61511 61616 61721 61826 61930 62034 62138 62242 62346 62449 62552 62655 62757 62859 62961 63063 63165 63266 63367 63468 63568 63669 63769 63869 63969 64068 64167 64266 64365 64464 64562 64660 64758 64856 64953 65050 65147 65244 65341 65437 65533 65629 65725 65820 65916 66011 66106 66200

64345 64444 64542 64640 64738 64836 64933 65031 65128 65225 65321 65418 65514 65610 65706 65801 65896 65992 66087 66181

64385 64483 64582 64680 64777 64875 64972 65070 65167 65263 65360 65456 65552 65648 65744 65839 65935 66030 66124 66219

61352 61458 61563 61669 61773 61878 61982 62086 62190 62294 62397 62500 62603 62706 62808 62910 63012 63114 63215 63317 63417 63518 63619 63719 63819 63919 64018 64118 64217 64316 64414 64513 64611 64709 64807 64904 65002 65099 65196 65292; 65389 65485 65581 65677 65772 65868 65963 66058 66153 66247

9

60304 60412 60520 60627 60735 60842 60949 61055 61162 61268 61363 61374 61469 61479 61574 61584 61679 61690 61784 61794 61888 61899 61993 62003 62097 62107 62201 62211 62304 62315 62408 62418 62511 62521 62613 62624 62716 62726 62818 62829 62921 62931 63022 63033 63124 63134 63225 63236 63327 63337 63428 63438 63528 63538 63629 63
99

Logarithms from 1 to 10,000, No.

0

1

460 461 462 463 464 465 466 467 468 469

66276 66370 66464 66558 66652 66745 66839 66932 67025 67117

66285 66380 66474 66567 66661 66755 66848 66941 67034 67127

470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491* 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 >509 510 511 512 513 514 515 516 517 518 ! 519

67210 67302 67394 67486 67578 67669 67761 67852 67943 68034 68124 68215 68305 68395 68485 68574 68664 68753 63842 68931

67219 67311 67403 67495 67587 67679 67770 67861 67952 68043 68133 68224 68314 68404 68494 68583 68673 68762 68851 68940 69028 69117 69205 69294 69381 69469 69557 69644 69732 69819 69906 69992 70079 70165 70252 70338 70424 70509 70595 70680 70766 70851 70935 71020 71105 71189 71273 71357 71441 ! 71525

69020 69108 69197 69285 69373 69461 69548 69636 69723 69810 69897 69984 70070 70157 70243 70329 70415 70501 70586 70672 70757 70842 70927 71012 71096 71181 71265 71349 71433 71517

2 66295 66389 66483 66577 66671 66764 66857 66950 67043 67136 67228 67321 67413 67504 67596 67688 67779 67870 67961 68052 68142 68233 68323 68413 68502 68592 68681 68771 68860 68949 69037 69126 69214 69302 69390 69478 69566 69653 69740 69827 69914 70001 70088 70174 70260 70346 70432 70518 70603 70689 70774 70859 70944 71029 71113 71198 71282 71366 71450 71533

3 66304 66398 66492 66586 66680 66773 66867 66960 67052 67145 67237 67330 67422 67514 67605 67697 67788 67879 67970 68061 68151 68242 68332 68422 68511 68601 68690 68780 68869 68958 69046 69135 69223 69311 69399 69487 69574 69662 69749 69836 69923 70010 70096 70183 70269 70355 70441 70526 70612 70697 70783 70868 70952 71037 71122 71206 71290 71374 71458 71542

4 66314 66408 66502 66596 66689 66783 66876 66969 67062 67154 67247 67339 67431 67523 ,67614 67706 67797 67888 67979 68070 68160 68251 68341 68431 68520 68610 68699 68789 68878 68966 69055 69144 69232 69320 69408 69496 69583 69671 69758 69845 69932 70018 70105 70191 70278 70364 70449 70535 70621 70706 70791 70876 70961 71046 71130 71214 71299 71383 71466 71550

5

6

7

8

9

66323 66417 66511 66605 66699 66792 66885 66978 67071 67164 67256 67348 67440 67532 67624 67715 67806 67897 67988 68079 68169 68260 68350 68440 68529 68619 68708 68797 68886 68975 69064 69152 69241 69329 69417 69504 69592 69679 69767 69854 69940 70027 70114 70200 70286 70372 70458 70544 70629 70714 70800 70885 70969 71054 71139 71223 71307 71391 71475 71559

66332 66427 66521 66614 66708 66801 66894 66987 67080 67173 67265 67357 67449 67541 67633 67724 67815 67906 67997 68088 68178 68269 68359 68449 68538 68628 68717 68806 68895 68984

66342 66436 66530 66624 66717 66811 66904 66997 67089 67182 67274 67367 67459 67550 67642 67733 67825 67916 68006 68097 68187 68278 68368 68458 68547 68637 68726 68815 68904 68993 69082 69170 69258 69346 69434 69522 69609 69697 69784 69871

66351 66445 66539 66633 66727 66820 66913 67006 67099 67191 67284 67376 67468 67560 67651 67742 67834 67925 68015 68106 68196 68287 68377 68467 68556 68646 68735 68824 68913 69002 69090 69179 69267 69355 69443 69531 69618 69705 69793 69880 69966 70053 70140 70226 70312 70398 70484 70569 70655 70740 70825 70910 70995 71079 71164 71248 I /1332 71416 71500 71584

66361 66455 66549 66642 66736 66829 66922 67015 67108 67201 67293 67385 67477 67569 67660 67752 67843 67934 68024 68115 68205 68296 68386 68476 68565 68655 68744 68833 68922 69011 69099 69188 69276 69364 69452 69539 69627 69714 69801 69888;

69073 69161 69249 69338 69425 69513 69601 69688 69775 69862 69949 70036 70122 70209 70295 70381 70467 70552 0638 70723 70808 70893 70978 71063 71147 71231 71315 71399 71483 71567

69958 70044 70131 70217 70303 70389 70475 70561 70646 70731 70817 70902 70986 71071 71155 71240 71324 71408 71492 71575

699751 70062 70148 70234 70321 70406 70492 70578 70663 70749 70834 70919 71003 71088 71172 71257 71341 71425 71508 71592

100

Logarithms from 1 to 10,000.

No.

0

1

2

520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549

71600 71684 71767 71850 71933 72016 72099 72181 72263 72346

71609 71692 71775 71858 71941 72024 72107 72189 72272 72354 72436 72518 72599 72681 72762 72843 72925 73006 73086 73167 73247 73328 73408 73488 73568 73648 73727 73807 73886 73965 74044 74123 74202 74280 74359 74437 74515 74593 74671 74749 74827 74904 74981 75059 75136 75213 75289 75366 75442 75519 75595 75671 75747 75823 75899 75974 76050 76125 76200 76275

71617 71700 71784 71867 71950 72032 72115 72198 72280 72362 72444 72526 72607 72689 72770 72852 72933 73014 73094 73175 73255 73336 73416 73496 73576 73656 73735 73815 73894 73973 74052 74131 74210 74288 74367 74445 74523 74601 74679 74757 74834 74912 74989 75066 75143 75220 75297 75374 75450 75526 75603 75679 75755 75831 75906 75982 76057 76133 76208 76283

550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 , 569 570 571 572 573 574 575 576 577 578 579

72428 72509 72591 72673 72754 72835 72916 72997 73078 73159 73239 73320 73400 73480 73560 73640 73719 73799 73878 73957 74036 74115 74194 74273 74351 74429 74507 74586 74663 74741 74819 74896 74974 75051 75128 75205 75282 75358 75435 75511 75587 75664 75740 75815 75891 75967 76042 76118 76193 76268

3 71625 71709 71792 71875 71958 72041 72123 72206 72288 72370 72452 72534 72616 72697 72779 72860 72941 73022 73102 73183 73263 73344 73424 73504 73584 73664 73743 73823 73902 73981 74060 74139 74218 74296 74374 74453 74531 : 74609 74687 74764 74842 74920 74997 75074 75151 75228 75305 75381 75458 75534 75610 75686 75762 75838 75914 75989 76065 76140 76215 76290



4

5

6

71634 71717 71800 71883 71966 72049 72132 72214 72296 72378 72460 72542 72624 72705 72787 72868 72949 73030 73111 73191 73272 73352 73432 73512 73592 73672 73751 73830 73910 73989 74068 74147 74225 74304 74382 74461 74539 74617 74695 74772 74850 74927 75005 75082 75159 75236 75312 75389 75465 75542 75618 75694 75770 75846 75921 75997 76072 76148 76223 76298

71642 71725 71809 71892 71975 72057 72140 72222 72304 72387 72469 72550 72632 72713 72795 72876 72957 73038 73119 73199 73280 73360 73440 73520 73600 73679 73759 73838 73918 73997 74076 74155 74233 74312 74390 74468 74547 74624 74702 74780 74858 74935 75012 75089 75166 75243 75320 75397 75473 75549 75626 75702 75778 75853 75929 76005 76080 76155 76230 76305 1

71650 71734 71817 71900 71983 72066 72148 72230 72313 72395 72477 72558 72640 72722 72803 72884 72965 73046 73127 73207 73288 73368 73448 73528 73608 73687 73767 73846 73926 74005 74084 74162 74241 74320 74398 74476 74554 74632 74710 74788 74865 74943 75020 75097 75174 75251 75328 75404 75481 75557 75633 75709 75785 75861 75937 76012 76087 76163 76238 76313 1

7K69 71742 71825 71908 71991 72074 72156 72239 72321 72403 72485 72567 72648 72730 72811 72892 72973 73054 73135 73215 73296 73376 73456 73536 73616 73695 73775 73854 73933 74013 74092 74170 74249 74327 74406 74484 74562 74640 74718 74796 74873 74950 75028 75105 75182 75259 75335 75412 75488 75565 75641 75717 75793 75868 75944 76020 76095 76170 76245 76320

8

9

71667 71750 71834 71917 71999 72082 72165 72247 72329 72411 72493 72575 72656 72738 72819 72900 72981 73062 73143 73223 73304 73384 73464 73544 73624 73703 73783 73862 73941 74020 74099 74178 74257 74335 74414 74492 74570 74648 74726 74803 74881 74958 75035 75113 75189 75266 75343 75420 75496 75572 75648 75724 75800 75876 75952 76027 76103 76178 76253 76328

71675 71759 71842 71925 72008 72090 72173 72255 72337 72419 72501 72583 72665 72746 72827 72908 72989 73070 73151 73231 73312 73392 73472 73552 73632 73711 73791 73870 73949 74028 74107 74186 74265 74343 74421 74500 74578 74656 74733 74811 74889 74966 75043 75120 75197 75274 75351 75427 75504 75580 75656 75732 75808 75884 75959 76035 76110 76185 76260 76335

101

Logarithms from 1 to 10,000. No.

0

1

2

3

4

5

6

7

8

9

580 581 58*2 583 584 585 586 581 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 ; 638 639

76343 76418 76492 76567 76641 76716 76790 76864 76938 77012 77085 77159 77232 77305 77379 77452 77525 77597 77670 77743

76350 76425 76500 76574 76649 76723 76797 76871 76945 77019 77093 77166 77240 77313 77386 77459 77532 77605 77677 77750 77822 77895 77967 78039 78111 78183 78254 78326 78398 78469 78540 78611 78682 78753 78824 78895 78965 79036 79106 79176 79246 79316 79386 79456 79525 79595 79664 79734 79803 79872 79941 80010 80079 80147 80216 80284 80353 80421 80489 80557

76358 76433 76507 76582 76656 76730 76805 76879 76953 77026 77100 77173 77247 77320 77393 77466 77539 77612 77685 77757

76365 76440 76515 76589 76664 76738 76812 76886 76960 77034 77107 77181 77254 77327 77401 77474 77546 77619 77692 77764 77837 77909 77981 78053 78125 78197 78269 78340 78412 78483

76373 76448 76522 76597 76671 76745 76819 76893 76967 77041 77115 77188 77262 77335 77408 77481 77554 77627 77699 77772 77844 77916 77988 78061 78132 78204 78276 78347 78419 78490 78561 78633 78704 78774 78845 78916 78986 79057 79127 79197 79267 79337 79407 79477 79546 79616 79685 79754 79824 79893 79962 80030 80099 80168 80236 80305 80373 80441 80509 80577

76380 76455 76530 76604 76678 76753 76827 76901 76975 77048 77122 77195 77269 77342 77415 77488 77561 77634 77706 77779 77851 77924 77996 78068 78140 78211 78283 78355 78426 78497

76388 76462 76537 76612 76686 76760 76834 76908 76982 77056 77129 77203 77276 77349 77422 77495 77568 77641 77714 77786

76395 76470 76545 76619 76693 76768 76842 76916 76989 77063 77137 77210 77283 77357 77430 77503 77576 77648 77721 77793

76403 76477 76552 76626 76701 76775 76849 76923 76997 77070 77144 77217 77291 77364 77437 77510 77583 77656 77728 77801

76410 76485 76559 76634 76708 76782 76856 76930 77004 77078

77859 77931 78003 78075 78147 78219 78290 78362 78433 78504 78576 78647 78718 78789 78859 78930 79000 79071 79141 79211

77866 77938 78010 78082 78154 78226 78297 78369 78440 78512

77873 77945 78017 78089 78161 78233 78305 78376 78447 78519 78590 78661 78732 78803 78873 78944 79014 79085 79155 79225 79295 79365 79435 79505 79574 79644 79713 79782 79851 79920 79989 80058 80127 80195 80264 80332 80400 80468 80536 80604

77815 77887 77960 78032 78104 78176 7^247 78319 78390 78462 78533 78604 78675 78746 78817 78888 78958 79029 79099 79169 79239 79309 79379 79449 79518 79588 79657 79727 79796 79865 79934 80003 80072 80140 80209 80277 80346 80414 {10482 80550

77830 77902 77974 78046 78118 78190 78262 78333 78405 78476 78547 78618 78689 78760 78831 78902 78972 79043 79113 79183 79253 79323 79393 79463 79532 79602 79671 79741 79810 79879 79948 80017 80035 80154 80223 80291 80359 80428 80496 80564

78554 78625 78696 78767 78838 78909 78979 79050 79120 79190 79260 79330 79400 79470 79539 79609 79678 79748 79817 79886 79955 80024 80092 80161 80229 80298 80366 80434 80502 80570

78569 78640 78711 78781 78852 78923 78993 79064 79134 79204 79274 79344 79414 79484 79553 79623 79692 79761 79831 79900 79969 80037 80106 80175 80243 80312 80380 80448 80516 80584

79281 79351 79421 79491 79560 79630 79699 79768 79837 79906 79975 80044 80113 80182 80250 80318 80387 80455 80523 80591

78583 78654 78725 78796 78866 78937 79007 79078 79148 79218 79288 793/ 8 79428 79498 I 79567 79637 79706 79775 79844 79913 79982 80051 80120 80188 80257 80325 80393 80462 80530 80598

77151 77225 77298 77371 77444 77517 77590 77663 77735 77808 77880 77952 78025 78097 78168 78240 78312 78383 78455 78526 78597 78668 78739 78810 78880 78951 79021 79092 79162 79232 79302 79372 79442 79511 79581 79650 79720 79789 79858 79927 79996 80065 , 80134 80202 80271 80339 80407 80475 80543 80611

Logarithms irom 1 to 10,000.

102 No.

0

1

2

3

4

5

6

7

640 641 642 643 644 645 646 647 643 649 650 651 652 653 654 655 656 657 653 659 660 661 662 663 664 665 666 667 663 660 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 f 607 ] 698 1699

80618 80686 80754 80821 80889 80956 81023 81090 81158 81224 81291 81358 81425 81491 81558 81624 81690 81757 81823 81889

80625 80693 80760 80828 80895 80963 81030 81097 81164 81231 81298 81365 81431 81498 81564 81631 81697 81763 81829 81895 81961 82027 82092 82158 82223 82289 82354 82419 82484 82549 82614 82679 82743 82808 82872 82937 83001 83065 83129 83193 83257 83321 83385 83448 83512 83575 83639 83702 83765 83828 83891 83954 84017 84080 84142 84205 84267 84330 84392 84454

80632 80699 80767 80835 80902 80969 81037 81104 81171 81238 81305 81371 81438 81505 81571 81637 81704 81770 81836 81902 81968 82033 82099 82164 82230 82295 82360 82426 82491 82556 82620 82685 82750 82814 82879 82943 83008 83072 83136 83200 83264 83327 83391 83455 83518 83582 83645 83703 83771 83835 83897 83960 84023 84086 84148 84211 84273 84336 84398 84460

80638 80706 80774 80841 80909 80976 81043 81111 81178 81245 81311 81378 81445 81511 81578 81644 81710 81776 81842 81908 31974 82040 82105 82171 82236 82302 82367 82432 82497 82562 82627 82692 82756 82821 82885 82950 83014 83078 83142 83206 83270 83334 83398 83461 83525 83588 83651 83715 83778 83841 83904 83967 84029 34092 84155 84217 84280 84342 84404 84466

80645 80713 80781 80848 80916 80983 81050 81117 81184 81251 81318 81385 81451 81518 81584 81651 81717 81783 81849 81915 81981 82046 82112 82178 82243 82308 82373 82439 82504 82569 82633 82698 82763 82827 82892 82956 83020 83085 83149 83213 83276 83340 83404 83467 83531 83594 83658 83721 83784 83847 83910 83973 84036 84098 84161 84223 84286 84348 81410 84473 1

80652 80720 80787 80855 80922 80990 81057 81124 81191 81258 81325 81391 81458 81525 81591 81657 81723 81790 81856 81921 81987 82053 82119 82184 82249 82315 82380 82445 82510 82575 82640 82705 82769 82834 82898 82963 83027 83091 83155 83219 83283 83347 83410 83474 83537 83601 83664 83727 83790 83853 83916 83979 84042 84105 84167 84230 84292 84354 84417 84479

80659 80726 80794 80862 80929 80996 81064 81131 81198 81265 81331 81398 81465 81531 81598 81664 81730 81796 81862 81928 81994 82060 82125 82191 82256 82321 82387 82452 82517 82582 82646 82711 82776 82840 82905 82969 83033 83097 83161 83225 83289 83353 83417 83480 83544 83607 83670 83734 83797 83860 83923 83985 84048 84111 84173 84236 84298 84361 84423 84485

80665 80733 80801 80868 80936 81003 81070 81137 81204 81271 81338 81405 81471 81538 81604 81671 81737 81803 81869 81935 82000 82066 82132 82197 82263 82328 82393 82458 82523 82588 82653 82718 82782 82847 82911 82975 83040 83104 83168 83232 83296 83359 83423 83487 83550 83613 83677 83740 83803 33366 83929 83992 84055 84117 84180 84242 84305 84367 84429 84491

81954 82020 82036 82151 82217 82282 82347 82413 82478 82543 82607 82672 82737 82802 82866 82930 82995 83059 83123 83187 83251 83315 83378 83442 83506 83569 83632 83696 83759 83822 83885 83943 84011 84073 84136 84198 84261 84323 84386 84448

8

9

80672 80679 80740 80747 80808 , 80814 80875 80882 80943 80949 81010 81017 81077 81084 81144 81151 81211 81218 81278 81285 81345 81351 81411 81418 81478 81485 81544 81551 81611 81617 81677 81684 81743 81750 81809 81816 81875 81882 81941 81948 82007 82014 82073 82079 82138 82145 82204 82210 82269 82276 82334 82341 . 82400 82406 82465 82471 82530 82536 82595 82601 82659 82666 82724 82730 82789 82795 82853 82860 82918 82924 82982 82988 83046 83052 83110 83117 83174 83181 83238 83245 83302 83308 83366 83372 83429 83436 83493 83499 83556 83563 83620 83626 83683 83689 83746 83753 83809 83816 83872 83879 83935 83942 83998 1 84Qf)4 84061 84067 84123 84130 84186 84192 84248 84255 84311 84317 84373 84379 84435 84442 84497 84504

Logarithms from 1 to 10,000, No. 700 701 702 703 704 ' 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 i721 722 723 7.24 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 J 759

103

0

1

2

3

4

5

6

7

8

9

84510 84572 84634 84696 84757 84819 84880 84942 85003 85065 85126 85187 85248 85309 85370 85431 85491 85552 85612 85673 85733 85794 85854 85914 85974 86034 86094 86153 86213 86273 86332 86392 86451 86510 86570 86629 86688 86747 86806 86864 86923 86982 87040 87099 87157 87216 87274 87332 87390 87448 87506 87564 87622 87679 87737 87795 87852 87910 87967 88024

84516 84578 84640 84702 84763 84825 84887 84948 85009 85071 85132 85193 85254 85315 85376 85437 85497 85558 85618 85679 85739 85800 85860 85920 85980 86040 86100 86159 86219 86279 86338 86398 86457 86516 86576 86635 86694 86753 86812 86870 86929 86988 87046 87105 87163 87221 87280 87338 87396 87454 87512 87570 87628 87685 87743 87800 87858 87915 87973 88030

84522 84584 84646 84708 84770 84831 84893 84954 85016 85077 85138 85199 85260 85321 85382 85443 85503 85564 85625 85685 85745 85806 85866 85926 85986 86046 86106 86165 86225 86285 86344 86404 86463 86522 86581 86641 86700 86759 86817 86876

84528 84590 84652 84714 84776 84837 84899 84960 85022 85083 85144 85205 85266 85327 85388 85449 85509 85570 85631 85691

84535 84597 84658 84720 84782 84844 84905 84967 85028 85089 85150 85211 85272 85333 85394 85455 85516 85576 85637 85697 85757 85818 85878 85938 85998 86058 86118 86177 86237 86297 86356 86415 86475 86534 86593 86652 86711 86770 86829 86838 86947 87005 37064 87122 87181 87239 87297 87355 87413 87471

84541 84603 84665 84726 84788 84850 84911 84973 85034 85095 85156 85217 85273 85339 85400 85461 85522 85582 85643 85703 85763 85824 85884 85944 86004 86064 86124 86183 86243 86303 86362 86421 8648,1 86540 86599 86658 86717 86776 86835 86894 86953 87011 87070 87128 87186 87245 87303 87361 87419 87477 87535 87593 87651 87708 87766 87823 87881 87938 87996 88053

84547 84609 84671 84733 84794 84856 84917 84979 85040 85101 85163 85224 85285 85345 85406 85467 85528 85588 85649 85709 85769 85830 85890 85950 86010 86070 86130 86189 86249 86308

84553 84615 84677 84739 84800 84862 84924. 84985 85046 85107

84559 84621 84683 84745 84807 84868 84930 84991 85052 85114 85175 85236 85297 85358 85418 85479 85540 85600 85661 85721 85781 85842 85902 85962 86022 86082 86141 86201 86261 86320 86380 86439 86499 86558 86617 86676 86735 86794 86853 86911

84566 84628 84689 84751 84813 84874 84936 84997 85058i 85120 85181 85242 85303 85364 85425 85485 85546 . 85606 85667 85727

86935 86994 87052 87111 87169 87227 87286 87344 87402 87460 87518 87576 87633 87691 87749 87806 87864 87921 87978 88036

85751 85812 85872 85932 85992 86052 86112 86171 86231 86291 86350 86410 86469 86528 86587 86646 86705 86764 86823 86882 86941 86999 87058 87116 8^175 87233 87291 87349 87408 87466 87523 87581 87639 87697 87754 87812 87869 87927 87984 88041

87529 87587 87645 87703 87760 87818 87875 87933 87990 88047

86368 86427 86487 86546 86605 86664 86723 86782 86841 86900 86958 87017 87075 87134 87192 87251 87309 87367 87425 87483 87541 87599 87656 87714 87772 87829 87887 87944 88001 88058

85169 85230 85291 85352 85412 85473 85534 85594 85655 85715 85775 85836 85896 85956 86016 86076 86136 86195 86255 86314 86374 86433 86493 86552 86611 86670 86729 86788 86847 86906 86964 87023 87081 87140 87198 87256 87315 87373 87431 87489 87547 87604 87662 87720 87777 87835 87892 87950 88007 88064

86970 87029 87087 87146 87204 87262 87320 87379 87437 87495 87552 87610 87668 87726 87783 87841 87898 87955 88013 88070

85788 85848 85908 85963 86028 86088 86147 86207 86267 86326 86386 86445 86504 86564 86623 86682 86741 86800 86859 86917 86976 87035 37093 87151 87210 87268 87326 87384 87442 87500 87558 j 876161 87674 877311 87789 87846 87904 87961 88018 88076 I

104

Logarithms from 1 to 10,000.

i i

No.

0

760 761 762 763 764 765 766 ~«7 766 769 770 771 772 773 774 775 776 777 773 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809

88081 88138 88195 88252 88309 88366 88423 88480 88536 88593

8808* 88144 88201 88258 88315 88372 88429 88485 88542 88598

88649 88705 88762 88818 88874 88930 88986 89042 89098 89154

88655 83711 88767 88824 88880 88936 88992 89048 89104 89159 89215 89271 89326 89382 89437 89492 89548 89603 89658 89713 89768 89823 89878 89933 89988 90042 90097 90151 90206 90260 90314 90369 90423 90477 90531 90585 90639 90693 90747 90800 90854 90907 90961 91014 91068 91121 91174 91228 91281 91334

mo 811 812 813 814 815 816 817 818 819 1

89209 89265 89321 89376 89432 89487 89542 89597 89653 89708 89763 89818 89873 89927 89982 90037 90091 90146 90200 90255 90309 90363 90417 90472 90526 90580 90634 90687 90741 90795 90849 90902 90956 91009 91062 91116 91169 91222 91275 91328

2

3

4

5

6

7

88093 88150 88207 88264 88321 88377 88434 88491 88547 88604 88660 88717 88773 88829 88885 88941 88997 89053 89109 89165 89221 89276 89332 89387 89443 89498 89553 89609 89664 89719 89774 89829 89883 89938 89993 90048 90102 90157 90211 90266 90320 90374 90428 90482 90536 90590 90644 90698 90752 90806

88098 88156 88213 88270 88326 88383 88440 88497 88553 88610 88666 88722 88779 88835 88891 88947 89003 89059 89115 89170 89226 89282 89337 89393 89448 89504 89559 89614 89669 89724

88104 88161 88218 88275 88332 88389 88446 38502 88559 88615 88672 88728 88784 88840 58897 88953 89009 89064 89120 89176 89232 89287 89343 89398 89454 89509 89564 89620 89675 89730 89785 89840 89894 89949 90004 90059 90113 90168 90222 90276 90331 90385 90439 90493 90547 90601 90655 90709 90763 90816 90870 90924 90977 91030 91084 91137 91190 91243 91297 91350

88110 88167 83224 88281 88338 88395 88451 88508 88564 88621 88677 88734 88790 88846 88902 88958 89014 89070 89126 89182 89237 89293 89348 89404 89459 89515 89570 89625 89680 89735 89790 89845 89900 89955 90009 90064 90119 90173 90227 90282 90336 90390 90445 90499 90553 90607 90660 90714 90768 90822 90875 90929 90982 91036 91089 91142 91196 91249 91302 91355 1

88116 88173 88230 88287 88343 88400 88457 88513 88570 88627

88121 88178 88235 88292 88349 88406 88463 88519 88576 88632 88689 88745 88801 88857 88913 88969 89025 89081 89137 89193 89248 89304 89360 89415 89470 89526 89581 89636 89691 89746

90859 90913 90966 91020 91073 91126 91180 91233 91286 91339

89779 89834 89889 89944 89998 90053 90108 90162 90217 90271 90325 90380 90434 90488 90542 90596 90650 90703 90757 90811 90865 90918 90972 91025 91078 91132 91185 91238 91291 91344

88683 88739 88795 88852 88908 88964 89020 89076 89131 89187 89243 89298 89354 89409 89465 89520 89575 89631 89686 89741 89796 89851 89905 89960 90015 90069 90124 90179 90233 90287 90342 90396 90450 90504 90558 90612 90666 90720 90773 90827 90881 90934 90988 91041 91094 91148 91201 91254 91307 91360

89801‘ 89856 89911 89966 90020 90075 90129 90184 90238 90293 90347 90401 90455 90509 90563 90617 90671 90725 90779 90832 90886 90940 90993 91046 91100 91153 91206 91259 91312 91365

1

8

9

88127 88184 88241 88298 88355 88412 88468 88525 88581 88638 88694 88750 88807 88863 88919 88975 89031 89087 89143 89198 89254 89310 89365 89421 89476 89531 89586 89642 89697 89752 89807 89862 89916 89971 90026 90080 90135 90189 90244 90298 90352 90407 90461 90515 90569 90623 90677 90730 90784 90838 90891 90945 90998 91052 91105 91158 91212 91265 91318 91371

88133 88190 88247 88304 88360 88417 88474 88530 88587 88643 88700 88756 88812 88868 88925 88981 89037 89092 89148 89204 89260 89315 89371 89426 89481 89537 89592 89647 89702 89757 89812 89867 89922 89977 90031 90086 90140 90195 90249 90304 90358 90412 90466 90520 90574 90628 90682 90736 90789 90843 90897 90950 91004 91057 91110 91164 91217 91270 91323 91376

105

0

2

3

4

5

6

7

8

9

91387 91440 91492 91545 91598 91651 91703 91756 91808 91861 91913 91965 92018 92070 92122 92174 92226 92278 92330 92381

91392 91445 91498 91551 91603 91656 91709 91761 91814 91866 91918 91971 92023 92075 92127 92179 92231 92283 92335 92387

91397 91450 91503 91556 91609 91661 91714 91766 91819 91871 91924 91976 92028 92080 92132 92184 92236 92288 92340 92392

91403 01455 91508 91561 91614 91666 91719 91772 91824 91876 91929 91981 92033 92085 92137 92189 92241 92293 92345 92397

91413 91466 91519 91572 91624 91677 91730 91782 91834 91887

91429 91482 91535 91587 91640 91693 91745 91798 91850 91903

92438 92490 92542 92593 92645 92696 92747 92799 92850 92901

92443 92495 92547 92598 92650 92701 92752 92804 92855 92906 92957 93008 93059 93110 93161 93212 93263 93313 93364 93414 93465 93515 93566 93616 93666 93717 93767 93817 93867 93917 93967 94017 94067 94116 94166 94216 94265 94315 94364 94414

92449 92500 92552 92603 92655 92706 92758 92809 92860 92911

91418 91471 91524 $1577 91630 91682 91735 91787 91840 91892 91944 91997 92049 92101 92153 92205 92257 92309 92361 92412 92464 92516 92567 92619 92670 92722 92773 92824 92875 92927 92978 93029 93080 93131 93181 93232 93283 93334 93384 93435 93485 93536 93586 93636 93687 93737 93787 93837 93887 93937 93987 94037 94086 94136 94186 94236 94285 94335 94384 94433

91424 91477 91529 91582 91635 91687 91740 91793 91845 91897 91950 92002 92054 92106 92158 92210 92262 92314 92366 92418

92433 92485 92536 92588 92639 92691 92742 92793 92845 92896 92947 92998 93049 93100 93151 93202 93252 93303 93354 93404

91408 91461 91514 91566 91619 91672 91724 91777 91829 91882 91934 91986 92038 92091 92143 92195 92247 92298 92350 92402 92454 92505 92557 92609 92660 92711 92763 92814 92865 92916 92967 93018 93069 93120 93171 93222 93273 93323 93374 93425 93475 93526 93576 93626 93676 93727 93777 93827 93877 93927 93977 94027 94077 94126 94176 94226 94275 94325 94374 94424

7

!

f

Logarithms from 1 to 10,000,

;820 821 ! 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879

91381 91434 91487 9154091593 91645 91698 91751 91803 91855 91908 91960 92012 92065 92117 92169 92221 92273 92324 92376 92428 92480 92531 92583 92634 92686 92737 92788 92840 92891 92942 92993 93044 93095 93146 93197 93247 93298 93349 93399 93450 93500 93551 93601 93651 93702 93752 93802 93852 93902 93952 94002 94052 94101 94151 94201 94250 94300 94349 94399

93455 93505 93556 93606 93656 93707 93757 93807 93857 93907 93957 94007 94057 94106 94156 94206 94255 94305 94354 94404

92952 93003 93054 93105 93156 93207 93258 93308 93359 93409 93460 93510 93561 93611 93661 93712 93762 93812 93862 93912 93962 94012 94062 94111 94161 94211 94260 94310 94359 94409

92962 93013 93064 93115 93166 93217 93268 93318 93369 93420 93470 93520 93571 93621 93671 93722 93772 93822 93872 93922 93972 94022 94072 94121 94171 94221 94270 94320 94369 94419

3 (j

91939 91991 92044 92096 92148 92200 92252 92304 92355 92407 92459 92511 92562 92614 92665 92716 92768 92819 92870 92921 92973 93024 93075 93125 93176' 93227 93278 93328 93379 93430 93480 93531 93581 93631 93682 93732 93782 93832 93882 93932 93982 94032 94082 94131 94181 94231 94280 94330 94379 94429

92469 92521 92572 92624 92675 92727 92778 92829 92881 92932 92983 93034 93085 93136 93186 93237 ; 93288 93339 93389 93440 93490 93541 93591 93641 93692 93742 93792 93842 93892 93942 93992 94042 94091 94141 94191 94240 94290 94340 94389 94438

91955 92007 92059 92111 92163 92215 92267 92319 92371 92423 92474 92526 92578 92629 92681 92732 92783 92834 92886 92937 92988 93039 93090 93141 93192 93242 93293 93344 93394 93445 93495 93546 93596 93646 93697 93747 93797 93847 93897 93947 93997 94047 94096 94146 94196 94245 94295 94345 94394 94443

t

106

Logarithms from 1 to 10,000.

No.

0

1

880 881 882 . 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 . 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939

94448 94498 94547 94596 94645 94694 94743 94792 94841 94890 94939 94988 95036 95085 95134 95182 95231 95279 95328 95376 95424 95472 95521 95569 95617 95665 95713 95761 95809 95856 95904 95952 95999 96047 96095 96142 96190 96237 96284 96332 96379 96426 96473 96520 96567 96614 96661 96708 96755 96802 96848 96895 96942 96988 97035 97081 97128 97174 97220 97267

94453 94503 94552 94601 94650 94699 94748 94797 94846 94895 94944 94993 95041 95090 95139 95187 95236 95284 95332 95381 95429 95477 95525 95574 95622 95670 95718 95766 95813 95861

2

94458 94507 94557 94606 94655 94704 94753 94802 94851 94900 94949 94998 95046 95095 95143 95192 95240 95289 95337 95386 95434 95482 95530 95578 95626 95674 95722 95770 95818 95866 95909 95914 95957 95961 96004 96009 96052 96057 96099 96104 96147 96152 96194 96199 96242 96246 96289 96294 96336 96341 96384 96388 96431 96435 96478 96483 96525 96530 96572 96577 96619 96624 96666 96670 96713 96717 96759 96764 96806 96811 96853 96858 96900 96904 96946 96951 96993 96997 97039 97044 97086 97090 97132 97137 97179 97183 97225 97230 97271 97276

3

4

5

6

7

8

9

94463 94512 94562 94611 94660 94709 94758 94807 94856 94905 94954 95002 95051 95100 95148 95197 95245 95294 95342 95390 95439 95487 95535 95583 95631 95679 95727 95775 95823 95871

94468 94517 94567 94616 94665 94714 94763 94812 94861 94910 94959 95007 95056 95105 95153 95202 95250 95299 95347 95395 95444 95492 95540 95588 95636 95684 95732 95780 95828 95875 95923 95971 96019 96066 96114 96161 96209 96256 96303 96350 96398 96445 96492 96539 96586 96633 96680 96727 96774 96820 96867 96914 96960 97007 97053 97100 97146 97192 97239 97285

94473 94522 94571 94621 94670 94719 94768 94817 94866 94915 94963 95012 95061 95109 95158 95207 95255 95303 95352 95400 95448 95497 95545 95593 95641 95689 95737 95785 95832 95880 95928 95976 96023 96071 96118 96166 96213 96261 96308 96355 96402 96450 96497 96544 96591 96638 96685 96731 96778 96825 96872 96918 96965 97011 97058 97104 97151 97197 97243 97290

94478 94527 94576 94626 94675 94724 94773 94822 94871 94919 94968 95017 95066 95114 95163 95211 95260 95308 95357 95405 95453 95501 95550 95598 95646 95694 95742 95789 95837 95885 95933 95980 96028 96076 96123 96171 96218 96265 96313 96360 96407 96454 96501 96548 96595 96642 96689 96736 96783 96830 96876 96923 96970 97016 97063 97109 97155 97202 97248 97294

94483 94532 94581 94630 94680 94729 94778 94827 94876 94924 94973 95022 95071 95119 95168 95216 95265 95313 95361 95410 95458 95506 95554 95602 95650 95698 95746 95794 95842 95890 95938 95985 96033 96080 96128 96175 96223 96270 96317 96365 96412 96459 96506 96553 96600 96647 96694 96741 96788 96834 96881 96928 96974 97021 97067 97114 97160 97206 97253 97299

94488 94537 94586 94635 94685 94734 94783 94832 94880 94929 94978 95027 95075 95124 95173 95221 95270 95318 95366 95415 95463 95511 95559 95607 95655 95703 95751 95799 95847 95895 95942 95990 96038 96085 96133 96180 96227 96275 96322 96369 96417 96464 96511 96558 96605 96652 96699 96745 96792 96839 96886 96932 96979 97025 97072 97118 97165 97211 97257 97304

94493 94542 94591 94640 94689 94738 94787 94836 94885 94934 94983 95032 95080 95129 95177 95226 95274 95323 95371 95419 95468 95516 95564 95612 95660 95708 95756 95804 95852 95899 95947 95995 96042 96090 96137 96185 96232 96280 96327 96374 96421 96468 96515 96562 96609 96656 96703 96750 96797 96844 96890 96937 96984 97030 97077 97123 97169 97216 97262 97308

95918 95966 96014 96061 96109 96156 96204 96251 96298 96346 96393 96440 96487 96534 96581 96628 96675 96722 96769 96816 96862 96909 96956 97002 97049 97095 97142 97188 97234 97280

107

Logan thins from 1 to 10,000, No.

0

1

2

3

4

5

940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999

97313 97359 97405 97451 97497 97543 97589 97635 97681 97727

97317 97364 97410 97456 97502 97548 97594 97640 97685 97731 97777 97823 97868 97914 97959 98005 98050 98096 98141 98186 98232 98277 98322 98367 98412 98457 98502 98547 98592 98637 98682 98726 98771 98816 98860 98905 98949 98994 99038 99083 99127 99171 99216 99260 99304 99348 99392 99436 99480 99524 99568 99612 99656 99699 99743 99787 99830 99874 99917 99961

97322 97368 97414 97460 97506 97552 97598 97644 97690 97736 97782 97827 97873 97918 97964 98009 98055 98100 98146 98191 98236 98281 98327 98372 98417 98462 98507 98552 98597 98641

97327 97373 97419 97465 97511 97557 97603 97649 97695 97740 97786 97832 97877 97923 97968 98014 98059 98105 98150 98195 98241 98286 98331 98376 98421 98466 98511 98556 98601 98646 98691 98735 98780 98825 98869 98914 98958 99003 99047 99092 99136 99180 99224 99269 99313 99357 99401 99445 99489 99533 99577 99621 99664 99708 99752 99795 99839 99883 99926 99970

97331 97377 97424 97470 97516 97562 97607 97653 97699 97745 97791 97836 97882 97928 97973 98019 98064 98109 98155 98200

97336 97382 97428 97474 97520 97566 97612 97658 97704 97749 97795 97841 97886 97932 97978 98023 98068 98114 98159 98204 98250 98295 98340 98385 98430 98475 98520 98565 98610 98655 98700 98744 98789 98834 98878 98923 98967 99012 99056 99100 99145 99189 99233 99277 99322 99366 99410 99454 99498 99542 99585 99629 99673 99717 99760 99804 99848 99891 99935 99978

97772 97818 97864 97909 97955 98000 98046 98091 98137 98182 98227 98272 98318 98363 98408 98453 98498 98543 98588 98632 98677 98722 98767 98811 98856 98900 98945 98989 99034 99078 9912*3 99167 99211 99255 99300 99344 99388 99432 99476 99520 99564 99607 99651 99695 99739 99782 99826 99870 99913 99957

98686 98731 98776 98820 98865 98909 98954 98998 99043 99087 99131 99176 99220 99264 99308 99352 99396 99441 99484 99528 99572 99616 99660 99704 99747 99791 99835 99878 99922 99965

98245 98290 98336 98381 98426 98471 98516 98561 98605 98650 98695 98740 98784 98829 98874 98918 98963 99007 99052 99096 99140 99185 99229 99273 99317 99361 99405 99449 99493 99537 99581 99625 99669 99712 99756 99800 99843 99887 99930 99974

6

7

97340 97345 97387 97391 97433 97437 97479 97483 97525 97529 97571 97575 97617 97621 97663 97667 97708 97713 97754 97759 97800 97804 97845 97850 97891 97896 97937 97941 97982 97937 98028 98032 98073 98078 98118 93123 98164 98168 98209 98214 98254 98259 98299 98304 98345 98349 98390 98394 98435 98439 98480 98484 98525 98529 98570 98574 98614 98619 98659 98664 98704 98709 98749 98753 98793 98798 98838 98843 98883 98887 98927 98932 98972 98976 99016 99021 99061 99065 99105 99109 99149 99154 99193 99198 99238 99242 99282 99286 99326 99330 99370 99374 99414 99419 99458 99463 99502 99506 99546 99550 99590 99594 99634 99638 99677 99682 99721 99726 99765 99769 99808 99813 99852 99856 99896 09900 99939 99944 99983 99987

8

9

97350 97396 97442 97488 97534 97580 97626 97672 97717 97763 97809 97855 97900 97946 97991 98037 98082 98127 98173 98218 98263 98308 98354 98399 98444 98489 98534 98579 98623 98668 98713 98758 98802 98847 98892 98936 98981 99025 99069 99114 99158 99202 99247 99291 99335 99379 99423 99467 99511 99555 99599 99642 99686 99730 99774 9981" 99861 99904 99948 99991

97354 97400 97447 97493 • 97539 97585 97630 97676 97722 97768 97813 97859 97905 97950 97996 98041 98087 98132 98177 98223 98268 98313 98358 98403 98448 98493 98538 08583 98628 98673 98717 98762 98807 98851 98896 98941 98985 99029 99074 99118, 99162 f 99207 99251 99295 99339 99383 99427 99471 99515 99559 99603 99647 99691 99734 99778 99822 99865 99909 99952 99996

tOt> M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 J6 57 58 59 60

Artificial Sines, Tang, and Sec. 0 Degree, Sine. Inf. Neg. 6.46373 76476 94085 7.06579 16270 24188 30882 36682 41797 46373 7.50512 54291 57767 60985 63982 66704 69417 71900 74248 76475 7.78594 80615 82545 84393 86166 87870 89509 91088 92612 94084 7.95508 96887 98223 99520 8.00779 02002 03192 04350 05478 06578 8.07650 08696 09718 10717 11693 12647 13581 14495 15391 16268 8.17128 17971 18798 19610 20407 21189 21958 *22713 23456 24186 1 Co-sine.

Co-sine. Tangent. 10.00000 Inf. Neg. 10.00000 6.46373 00000 76476 00000 94085 00000 7.06579 00000 16270 00000 24188 00000 30882 00000 36682 00000 41797 00000 46373 10.00000 7.50512 00000 54291 00000 57767 00000 60986 00000 63982 00000 66785 9.99999 69418 99999 71900 99999 74248 99999 76476 9.99999 7.78595 99999 80615 99999 82546 99999 84394 99999 86167 99999 87871 99999 89510 99999 91089 99998 92613 99998 94086 9.99998 7.95510 99998 96889 99998 98225 99998 99522 99998 8.00781 99998 02004 99997 03194 99997 04353 99997 05481 99997 06581 9.99997 8.07653 99997 08700 99997 09722 99996 10720 99996 11696 99996 12651 99996 13585 14500 99996 99996 15395 99995 16273 8.17133 9.99995 17976 99995 18804 99995 19616 99995 20413 99994 99994 21195 21964 99994 22720 99994 23462 99994 24192 99993 Co-tang. 1 Sine.

Co-tang. Infinite. 13.53627 23524 05915 12.93421 83730 75812 69118 63318 58203 53627 12.49488 45709 42233 39014 36018 33215 30582 28100 25752 23524 12.21405 19385 17454 15606 13833 12129 10490 08911 07387 05914 12.04490 03111 01775 00478 11.99219 97996 96806 95647 94519 93419 11.92347 91300 90278 89280 88304 87349 86415 85500 84605 83727 11.82867 82024 81196 80384 79587 78805 78036 77280 76538 75808 Tangent.

Secant. Co-secant 10.00000 Infinite. 60 10.00000 13.53627 59 00000 23524 58 00000 05915 57 00000 12.93421 56 00000 83730 55 75812 54 00000 69118 53 00000 00000 63318 52 00000 58203 51 53627 50 00000 10.00000 12.49488 49 00000 45709 48 42233 47 00000 00000 39015 46 00000, 36018 45 00000 33216 44 00001 30583 43 00001 28100 42 00001 25752 41 00001 23525 40 10.00001 12.21406 39 00001 19385 38 00001 17455 37 00001 15607 36 00001 13834 35 00001 12130 34 00001 10491 33 00001 08912 32 00002 07388 31 00002 05910 30 10.00002 12.04492 29 00002 03113 28 00002 01777 27 00002 00480 26 00002 11.99221 25 00002 97998 24 00003 96808 23 00003 95650 22 00003 94522 21 00003 93422 20 10.00003 11.92350 19 91304 18 00003 00003 90282 17 00004 89283 16 88307 15 00004 87353 14 00004 86419 13 00004 85505 12 00004 84609 11 00004 83732 10 00005 9 10.00005 11.82872 8 82029 00005 7 81202 00005 6 80390 00005 5 79593 00006 4 78811 00006 3 78042 00006 77287 2 00006 76544 1 00006 75814 0 00007 M. Secant. Co-secant

Artificial Sines, Tang, and Sec. 1 Degree. \1. Sine. Co-sine. Tangent. 8.24192 9.99993 0 8.24186 24910 1 99993 24903 25616 2 99993 25609 26312 99993 3 26304 26996 4 99992 26988 27661 99992 27669 5 6 28332 28324 99992 7 99992 28986 28977 8 29621 99992 29629 9 99991 30263 30255 10 1.30879 9.99991 8.30888 31495 99991 11 31505 12 32103 99990 32112 13 32702 99990 32711 99990 14 33292 33302 99990 33875 33886 15 16 34450 99989 34461 17 35018 99989 35029 99989 18 35578 35590 36131 19 99989 36143 8.36678 ■ 9.99988 20 8.36689 21 37217 99988 37229 22 37750 99988 37762 23 99987 38276 38289 24 38796 99987 38809 25 39310 99987 39323 26 39818 99986 39832 27 40320 ‘ 99986 40334 28 40816 99986 40830 29 41307 99985 41321 30 8.41792 9.99985 8.41807 31 42272 99985 42287 32 42746 99984 42762 33 43216 99984 43232 34 43680 99984 43696 35 44139 99983 44156 36 44594 99983 44611 37 45044 99983 - 45061 38 45489 9S982 45507 39 45930 99982 45948 8.46366 40 9.99982 8.46385 41 46799 99981 46817 42 47226 99981 47245 43 47650 99981 47669 44 48069 99980 48089 45 48485 99980 48505 46 48896 99979 48917 47 49304 99979 49325 48 49708 99979 49729 49 50108 99978 50130 50~ 8.50504 9.99978 8.50527 50897 51 99977 50920 51287 52 99977 51310 53 51673 99977 51696 54 52055 99976 52079 52434 55 99976 52459 56 1| 52810 99975 52835 57 53183 99975' 53208 53552 58 99974 53578 59 53919 99974 53945 60 54282 99974 54308 Co-sine. Sine. Co-tang.

83*

Co-tang.

Secant.

11.75808 10.00007 00007 75090 00007 74384 00007 73688 73004 00008 72331 00008 00008 71668 71014 00008 70371 00008 69737 00009 11.69112 10.00009 68495 00009 67888 00010 67289 00010 66698 00010 66114 00010 65539 00011 64971 00011 64410 00011 63857 00011 11.63311 10.00012 62771 00012 62238 00012 61711 00013 61191 00013 60677 00013 60168 00014 59666 00014 59170 00014 58679 00015 11.58193 10.00015 57713 00015 57238 00016 56768 00016 56304 00016 55844 00017 55389 00017 54939 00017 54493 00018 54052 00018 11.53615 10.00018 53183 00019 52755 00019 52331, 00019 51911 00020 51495 00030 51083 00021 50675 00021 50271 00021 49870 00022 11.49473 10.00022 49080 00023 48690 00023 48304 00023 47921 00024 47541 00024 47165 00025 46792 00025 46422 00026 46055 00026 45692 00026 Tangent. (Co-secant

88 Degrees.

109

Co-secant 11.75814 60 75097 59 74391 58 73696 57 73012 56 72339 55 71676 54 71023 53 70379 52 69745 51 11.69121 50 68505 49 67897 48 67298 47 66708 46 66125 45 65550 44 64982 43 64422 42 63869 41 11.63322 40 62783 39 62250 38 61724 37 61204 36 60690 35 60182 34 59680 33 59184 32 58693 31 11.58208 30 57728 29 57254 28 56784 27 56320 26 55861 25 55406 24 54956 23 54511 22 54070 21 11.53634 20 53201 19 52774 18 52350 17 51931 16 51515 15 51104 14 50696 13 50292 12 49892 11 11.49496 10 49103 9 48713 8 48327 7 47945 6 47566 5 , 4 47190 46817 3 46448 2 1 46081 45718 0 Secant. M.

110 M. 0 1 2 3 4 5 6

7

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 43 49 50 51 52 ,53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 2 Degrees. Sine. 3.54282 54642 54999 55354 55705 56054 56400 56743 57034 57421 8.57757 58089 58419 58747 59072 59395 59715 60033 60349 60662 8.60973 61282 61589 61894 62196 62497 62795 63091 63385 63678 8.63968 64256 64543 64827 65110 65391 65670 65947 66223 66497 8.66769 67039 67308 67575 67841 68104 68367 68627 68886 69144 8.69400 69654 69907 70159 70409 70658 70905 71151 71395 71638 71880 Co-sine.

(Jo-sine. 9.99974 99973 99973 99972 99972 99971 99971 99970 99970 99969 9.99969 99968 99968 99967 99967 99967 99966 99966 99965 99964 9.99964 99963 99963 99962 99962 99961 99961 99960 99960 99959 9.99959 99958 99958 99957 99956 99956 99955 99955 99954 99954 9.99953 99952 99952 99951 99951 99950 99949 99949 99948 99948 9.99947 99946 99946 99945 99944 99944 99943 99942 99942 99941 99940 Sine.

1 angent.

Co-tang. Secant. Co-secant 8.54308 11.45692 10.00026 11.45718 60 54G69 45331 00027 45358 59 55027 44973 00027 45001 58 55382 44618 00028 ' 44646 57 55734 44266 00028 44295 56 56083 43917 00029 43946 55 56429 43571 00029 43600 54 56773 43227 00030 43257 53 57114 42886 00030 42916 52 57452 42548 00031 42579 51 8.57788 11.42212 10.00031 11.42243 50 58121 41879 00032 41911 49 58451 41549 00032 41581 48 58779 41221 00033 41253 47 59105 40895 00033 40928 46 59428 40572 00033 40605 45 ‘ 59749 40251 00034 40285 44 60068 39932 50034 39967 43 60384 39616 90035 39651 42 60698 39302 00036 39338 41 8.61009 11.38991 10.00036 11.39027 40 61319 38681 00037 38718 39 61626 38374 00037 38411 38 61931 38069 00038 38106 37 62234 37766 00038 37804 36 62535 37465 00039 37503 35 62834 37166 00039 37205 34 63131 36869 00040 36909 33 63426 36574 00040 . 36615 32 63718 36282 00041 36322 31 8.64009 11.35991 10.00041 11.36032 30 64298 35702 00042 35744 29 64585 35415 00042 35457 28 64870 35130 00043 35173 27 65154 34846 00044 34890 26 65435 34565 00044 34609 25 65715 34285 00045 34330 24 65993 34007 00045 34053 23 66269 33731 00046 33777 22 66543 33457 00046 33503 21 8.66816 11.33184 10.00047 11.33231 20 67087 32913 00048 32961 19 67356 32644 00048 32692 18 67624 32376 00049 32425 17 67890 32110 00049 32159 16 68154 31846 00050 31896 15 68417 31583 00051 31633 14 68678 31322 00051 31373 13 68938 31062 00052 31114 12 69196 30804 00052 30856 11 8.69453 Tl .30547 10.00053 11.30600 10 69708 30292 00054 30346 9 69962 30038 00054 30093 8 70214 29786 29841 7 00055 70465 29535 29591 6 00056 70714 29286 29342 5 00056 4 : 70962 29038 00057 29095 28849 3 71208 28792 00058 28547 2 71453 00058 28605 28362 71697 28303 00059 28060 28120 0 00060 71940 Secant. M. Co-tansr- Tangent. Co-secant

T

07 Doajro.fis,

Artificial Sines, Tang, and Sec. 3 Degrees. lM. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 , 52 53 54 55 56 57 58 59 60

Sine. 8.71880 72120 72359 72597 72834 73069 73303 73535 73767 73997 8.74226 74454 74680 74906 75130 75353 75575 75795 76015 76234 8.76451 76667 76883 77097 77310 77522 77733 77943 78152 78360 8.78568 78774 ■ 78979 79183 79386 79588 79789 79990 80189 80388 8.80585 80782 80978 81173 81367 81560 81752 81944 82134 82324 8.82513 82701 82888 83075 83261 83446 83630 83813 83996 84177 84358 Co-sine.

Co-sine. 9.99940 99940 99939 99938 99938 99937 99936 99936 99935 99934 9.99934 99933 99932 99932 99931 99930 99929 99929 99928 99927 9.99926 99926 99925 99924 99923 99923 99922 99921 99920 99920 9.99919 99918 99917 99917 99916 99915 99914 99913 99913 99912 9.99911 99910 99909 99909 99908 99907 99906 99905 99904 99904 9.99903 99902 99901 99900 99899 99898 99898 99897 99896 99895 99894 Sine.

Ill

Co-secant Secant. Co-tang. 8.71940 11.28060 10.00060 11.28120 60 27880 59 00060 27819 72181 27641 58 00061 27580 72420 27403 57 00062 27341 72659 27166 56 00062 27104 72896 26931 55 00063 26868 73132 26697 54 00064 26634 73366 26465 53 00064 26400 73600 26233 52 00065 26168 73832 26003 51 25937 00066 74063 8.74292 1 ’.25708 10.00066 11.25774 50 £5546 49 00067 25479 74521 00068 25320 48 25252 74748 25094 47 00068 74974 25026 24870 46 00069 75199 24801 24647 45 24577 00070 75423 00071 24425 44 24355 75645 00071 24205 43 75867 24133 00072 23913 23985 42 76087 00073 23766 41 23694 76306 8.76525 11.23475 10.00074 11.23549 40 23333 39 00074 23258 76742 23117 38 00075 23042 76958 ' 22903 37 22827 00076 77173 00077 22690 36 77387 22613 00077 22478 35 22400 77600 22267 34 00078 77811 22189 22057 33 00079 78022 21978 00080 21848 32 21768 78232 00080 21640 31 78441 21559 8.78649 11.21351 10.00081 11.21432 30 00082 21226 29 78855 21145 21021 28 20939 00083 79061 20817 27 j 20734 00083 79266 00084 20614 26 79470 20530 20327 00085 20412 25 79673 00086 20211 24 79875 20125 00087 19924 20010 23 80076 00087 19811 22 80277 19723 19524 19612 21 80476 00088 8.80674 11.19326 10.00089 11.19415 20 00090 19218 19 19128 80872 00091 19022 18 18932 81068 18827 17 00091 81264 18736 18633 16 18541 00092 81459 00093 18440 15 81653 18347 00094 18154 18248 14 81846 00095 18056 13 82038 17962 00096 17866 12 82230 17770 00096 82420 17580 17676 11 8.82610 11.17390 10.00097 11.17487 10 17201 00098 82799 9 17299 82987 17013 00099 8 17112 16825 00100 83175 16925 7 83361 16639 00101 6 16739 83547 16453 00102 16554 5 83732 16268 00102 16370 4 16084 83916 00103 16187 3 84100 15900 00104 16004 2 84282 15718 00105 1 15823 84464 15536 00106 0 15642 Co-tang. Tan tre rit. Co-secant Secant. M.

Tangent.

86 Decrees.

112 M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 1 35 36 37 38 39 40 41 42 43 : 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 4 Degrees, Sine. Co-sine. Tangent. Co-tang. Secant. Co-secant 9.99894 8.84464 11.15536 10.00106 11.15642 60 8.84358 99893 84646 84539 15354 00107 15461 59 99892 84826 84718 15174 00108 15282 58 84897 99891 85006 14994 00109 15103 57 99891 85185 85075 14815 00109 14925 56 85252 99890 85363 14637 00110 14748 55 99839 85540 85429 14460 00111 14571 54 99888 85717 85605 14283 00112 14395 53 99887 85893 85780 14107 00113 14220 52 85955 99886 86069 13931 00114 14045 51 8.86128 9.99885 8.86243 11.13757 10.00115 11.13872 50 99884 86301 86417 13583 00116 13699 49 86474 99883 86591 13409 00117 13526 48 99882 86645 86763 13237 00118 13355 47 86816 99881 86935 13184 46 13065 00119 86987 99880 87106 12894 00120 13013 45 87156 99879 87277 12723 00121 12844 44 99879 87447 00121 87325 12553 12675 43 99878 87494 87616 12384 00122 12506 42 99877 87661 87785 00123 12215 12339 41 9.99876 8.87829 8.87953 11.12047 10.00124 11.12171 40 99875 87995 88120 11880 00125 12005 39 88161 99874 88287 11713 00126 11839 38 99873 11547 00127 11674 37 88326 88453 99872 88490 88618 11382 00128 11510 36 99871 88654 11217 00129 11346 35 88783 11183 34 88817 99870 00130 88948 11052 89111 00131 11020 33 88980 99869 10889 99868 00132 10858 32 89274 89142 10726 10696 31 99867 89437 00133 10563 89304 8.89464 9.99866 8.89598 11.10402 10.00134 11.10536 30 10375 29 00135 99865 10240 89625 89760 00136 10216 28 99864 89784 10080 89920 10057 27 00137 99863 89943 09920 90080 09898 26 00138 99862 90102 09760 90240 09740 25 00139 99861 09601 90260 90399 09583 24 00140 90557 90417 99860 09443 00141 09426 23 90574 99859 90715 09285 09270 22 00142 99858 09128 90730 90872 00143 09115 21 99857 08971 90885 91029 9.99856 8.91040 8.91185 11.08815 10.00144 11.08960 20 08805 19 00145 91195 99855 91340 08660 08651 18 00146 99854 08505 91349 91495 08498 17 00147 99853 08350 91502 91650 08345 16 00148 08197 91655 99852 91803 08193 15 00149 91807 99851 08043 9195" 08041 14 00150 91959 99850 92110 07890 07890 13 00152 99848 92262 07738 92110 07739 12 00153 99847 07586 92261 92414 07589 11 00154 07435 92411 99846 92565 8.92561 “9799845 8.92716 11.07284 10.00155 11.07439 10 07290 9 00156 07134 92710 99844 92866 07141 8 00157 06984 92859 99843 93016 7 06993 00158 06835 93007 99842 93165 06846 6 00159 06687 93154 99841 93313 5 06699 00160 93301 06538 99840 93462 4 06552 00161 06391 93448 93609 99839 3 06406 00162 06244 93594 93756 99838 2 06260 00163 06097 93740 99837 93903 06115 00164 05951 93885 99836 94049 05970 0 00166 05805 94030 99834 94195 Secant. M. Co-tang. Tangent. Co-secant Co-sine. Sine.

T

85 Degrees.

Artificial Sines, Tang, and Sec. 5 Degrees. M. 0 1 2 * 3 4 5 6 7 8 9 10 11 12 33 14 35 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 ' 51 52 53 54 55 56 57 58 59 60

8 me. 8.94030 94174 94317 94461 94603 94746 94887 95029 95170 95310 8.95450 95589 95728 95867 96005 96143 96280 96417 96553 96689 8.96825 96960 97095 97229 97363 97496 97629 97762 97894 98026 8.98157 98288 98419 98549 98679 98808 98937 99066 99194 99322 8.99450 99577 99704 99830 99956 9.00082 00207 00332 00456 00581 9.00704 00828 00951 01074 01196 013 6 01440 01561 01682 01803 01923 Co-sine.

Secant. 1 1 anient. Co-tang. Co-sine. 8.94195 11.05805 10.00166 9.99834 00167 05660 94340 99833 00168 05515 94485 99832 00169 05370 94630 99831 05227 00170 94773 99830 00171 94917 05083 99829 00172 04940 95060 99828 00173 04798 95202 99827 00175 95344 04656 99825 04514 00176 95486 99824 8.95627 11.04373 10.00177 9.99823 95767 00178 04233 99822 04092 00179 99821 95908 00180 96047 03953 99820 96187 99819 03813 00181 99817 00183 03675 96325 00184 96464 03536 99816 00185 96602 03398 99815 00186 03261 99814 96739 00187 96877 03123 99813 8.97013 11.02987 10.00188 9.99812 00190 97150 02850 99810 00191 99809 97285 02715 97421 00192 99808 02579 99807 00193 97556 02444 00194 99806 97691 02309 00196 99804 02175 97825 00197 02041 99803 97959 99802 00198 98092 01908 99801 01775 00199 98225 9.99800 8.98358 11.01642 10.00200 99798 98490 00202 01510 99797 98622 01378 00203 99796 01247 98753 00204 99795 98884 01116 00205 00207 99793 99015 00985 99792 00208 99145 00855 99791 99275 00209 ; 00725 99790 99405 00210 00595 99788 99534 00466 00212 9.99787 8.99662 11.00338 10.00213 99786 99791 00209 00214 99785 99919 * 00081 00215 99783 00217 9.00046 10.99954 99782 00174 00218 99826 99781 00301 00219 99699 99780 00427 99573 00220 99778 00553 99447 00222 99777 00679 99321 00223 99776 00805 00224 99195 9.99775 9.00930 10.99070 10.00225 99773 01055 00227 98945 99772 01179 98821 00228 • 99771 01303 98697 00229 99769 01427 98573 00231 99768 01550 98450 00232 99767 01673 98327 00233 99765i 01796 98204 00235 99764 01918 98082 00236 99763 02040 97960 00237 99761 02162 97838 00239 Sine. Co-tang. Tangent. Co-secant

84 Degrees.

3D

Co-secant 11.05970 05826 05683 05539 05397 05254 05113 04971 04830 04690 11.04550 04411 04272 04133 03995 03857 03720 03583 03447 03311 11.03175 03040 02905 02771 02637 02504 02371 02238 02106 01974 11.01843 01712 01581 01451 01321 01192 01063 00934 00806 00678 11.00550 00423 00296 00170 00044 10.99918 99793 99668 99544 99419 10.99296 99172 99049 98926 98804 98682 98560 98439 98318 98197 98077 Secant.

113 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 M.

114

1

Al. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 53 59 60

Artificial Sines, Tang, and Sec. 6 Degrees Sine. Co-sine. Tangent. Co-tang. Secant. 9.01923 9.99761 9.02162 , 10.97838 10.00239 02043 99760 02283 97717 00240 02163 99759 02404 97596 00241 02283 99757 02525 97475 00243 02402 99756 02645 97355 00244 02520 99755 02766 97234 00245 02639 99753 02885 97115 00247 02757 99752 03005 96995 00248 02874 99751 03124 96876 00249 02992 99749 03242 96758 00251 9.03109 9.99748 9.03361 10.96639 10.00252 03226 99747 03479 96521 00253 03342 99745 03597 96403 00255 03458 99744 03714 96286 00256 03574 99742 03832 96168 00258 03690 99741 03948 . 96052 00259 03805 99740 04065 95935 00260 03920 99738 04181 95819 00262 99737 04034 04297 95703 00263 04149 99736 04413 95587 00264 9.04262 9.99734 9.04528 10.95472 10.00266 04376 99733 04643 95357 00267 04490 99731 04758 95242 00269 04603 99730 95127 04873 00270 04715 99728 04987 95013 00272 04828 99727 05101 94899 00273 04940 99726 05214 94786 00274 05052 99724 05328 94672 00276 05164 99723 05441 94559 00277 05275 99721 94447 05553 00279 9.05386 9.99720 9.05666 10.94334 TO.00280 05497 99718 05778 94222 00282 05607 99717 05890 94110 00283 05717 99716 06002 93998 00284 05827 99714 06113 93887 00286 05937 99713 00287 06224 93776 06046 99711 06335 93665 00289 06155 99710 06445 93555 00290 06264 99708 06556 93444 00292 06372 99707 06666 93334 00293 9.06481 9.99705 9.06775 10.93225 10.00295 06589 99704 06885 93115 00296 06696 99702 06994 93006 00298 06804 99701 07103 92897 00299 06911 07211 99699 92789 00301 07018 99698 07320 00302 92680 07124 99696 07423 00304 92572 07231 99695 07536 92464 00305 07337 99693 07643 92357 00307 07442 99692 07751 00308 92249 9.07548 9.99690 9.07853 10.92142 10.00310 07653 99689 00311 07964 92036 07758 08071 99687 00313 91929 07863 08177 99686 91823 00314 07968 99684 08283 00316 91717 08072 00317 99683 08389 91611 08176 99681 08495 00319 91505 08280 00320 99680 08600 91400 08383 00322 99678 08705 91295 08486 99677 00323 91190 08810 08589 00325 99675 08914 91086 Co-sine. Sine. Co-tang. | Tangent. [ Co-secant P.3 Decr-'r*

Co-secant — 10.98077 60 97957 59 97837 58 97717 57 97598 56 97480 55 97361 54 97243 53 97126 52 97008 51 10.96891 50 96774 49 96658 48 96542 47 96426 46 96310 45 96195 44 96080 43 95966 42 95851 41 10.95738 40 95624 39 95510 38 95397 37 95285 36 95172 35 95060 34 94948 33 94836 32 94725 31 10.94614 30 94503 29 94393 28 94283 27 94173 26 94063 25 93954 24 93845 23 93736 22 93628 21 10.93519 20 93411 19 93304 18 93196 17 93089 16 92982 15 92876 14 92769 13 92663 12 92558 11 10.92452 10 92347 9 8 92242 7 1 92137 6 92032 91928 5 91824 4 3 91720 91617 2 91514 1 91411 0 Secant. 1 "M7|

Artificial Sines, Tang, and Sec. 7 Degrees. 115 AI. 0 1 2 3 4 5 6 7 3 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.08589 08692 08795 08897 08999 09101 09202 09304 09405 09506 9.09606 09707 09807 09907 10006 10106 10205 10304 10402 10501 9.10599 10697 10795 10893 10990 11087 11184 11281 11377 11474 9.11570 11666 11761 11857 11952 12047 12142 12236 12331 12425 9.12519 12612 12706 12799 12892 12985 13078 13171 13263 13355 9.13447 13539 13630 13722 13813 13904 13994 14085 14175 14266 14356 Co-sino.

Co-secant Secant. Co-sine. Tangent. (Jo-tang. 9.08914 10.91086 10.00325 10.91411 60 9.99675 91308 59 00326 90981 99674 09019 91205 58 00328 90877 09123 99672 91103 57 00330 90773 092.27 99670 91001 56 00331 90670 99669 09330 90899 55 00333 90566 99667 09434 90798 54 00334 90463 09537 99666 90696 53 00336 99664 90360 09640 00337 90595 52 90258 99663 09742 90494 51 00339 90155 99661 09845 9.99659 9.09947 10.90053 10.00341 10.90394 50 90293 49 89951 00342 99658 10049 90193 48 00344 99656 89850 10150 90093 47 89748 00345 99655 10252 89647 00347 89994 46 99653 10353 89546 00349 89894 45 99651 10454 00350 89445 99650 89795 44 10555 89344 00352 89696 43 99648 10656 99647 89244 00353 89598 42 10756 00355 99645 89144 89499 41 10856 9.99643 9.10956 10.89044 10.00357 10.89401 40 88944 89303 39 99642 11056 00358 00360 89205 38 99640 88845 11155 89107 37 99638 88746 00362 11254 99637 88647 00363 89010 36 11353 88548 99635 00365 88913 35 11452 88449 00367 88816 34 * 99633 11551 88351 00368 99632 11649 88719 33 88253 11747 00370 88623 32 99630 00371 88155 88526 31 99629 11845 9.99627 9.11943 10.88057 10.00373 10.88430 30 87960 88334 29 99625 12040 00375 99624 12138 87862 88239 28 00376 88143 27 99622 87765 12235 00378 87668 • 88048 26 99620 12332 00380 87953 25 99618 12428 87572 00382 99617 87858 24 12525 87475 00383 87764 23 99615 12621 87379 00385 87283 99613 12717 00387 87669 22 87187 99612 12813 00388 87575 21 9.99610 9.12909 10.87091 10.00390 10.87481 20 99608 87388 19 13004 00392 86996 99607 86901 87294 18 13099 00393 99605 13194 86806 00395 87201 17 99603 86711 00397 13289 87108 16 99601 13384 86616 00399 87015 15 99600 13478 86522 00400 86922 14 99598 86427 86829 13 13573 00402 99596 13667 86333 00404 86737 12 99595 13761 86239 00405 86645 11 9.99593 9.13854 10.86146 10.00407 10.86553 10 99591 13948 86461 86052 00409 9 99589 14041 00411 8 85959 86370 99588 14134 i 85866 00412 86278 99586 14227 85773 00414 86187 6 99584 14320 85680 00416 86096 5 99582 14412 85588 00418 86006 4 99581 14504 85496 00419 3 85915 99579 14597 00421 85403 2 85825 99577 14688 85312 00423 1 85734 99575 14780 0 85220 00425 85644 Sin-. Co-tan
82 Decrees

116 M. 0 1 , 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 8 Degrees. Sine. 9.14356 14445 14535 14624 14714 14803 14891 14980 15069 15157 9.15245 15333 15421 15508 15596 15683 15770 15857 15944 16030 9.16116 16203 16289 16374 16460 16545 16631 16716 16801 16886 9.16970 17055 17139 17223 17307 17391 17474 17558 17641 17724 9.17807 17890 17973 18055 18137 18220 18302 18383 18465 18547 9.18628 18709 18790 18871 18952 19033 19113 19193 19273 19353 19433 Co-sine.

Co-sine. Tangent. Co-tang. Secant. Co-secant 9.99575 9.14780 10.85220 10.00425 10.85644 99574 14872 85128 00426 85555 99572 14963 85037 00428 85465 99570 15054 84946 00430 85376 99568 15145 84855 00432 85286 99566 15236 84764 00434 85197 99565 15327 84673 00435 85109 99563 15417 84583 00437 85020 99561 15508 84492 00439 84931 99559 15598 84402 00441 84843 9.99557 9.15688 10.84312 10.00443 10.84755 15777 99556 84223 00444 84667 15867 99554 84133 00446 84579 99552 15956 84044 00448 84492 99550 16046 83954 00450 84404 99548 16135 83865 00452 84317 99546 16224 83776 00454 84230 99545 16312 83688 00455 84143 99543 16401 83599 00457 84056 99541 16489 83511 00459 83970 9.99539 9.16577 10.83423 10.00461 10.83884 99537 16665 83335 00463 83797 16753 99535 83247 00465 83711 16841 99533 00467 83159 83626 99532 16928 83072 00468 83540 17016 99530 82984 00470 83455 99528 17103 82897 00472 83369 99526 17190 82810 00474 83284 17277 99524 82723 00476 83199 17363 82637 99522 00478 83114 9.99520 9.17450 10.82550 10.00480 10.83030 99518 17536 . 82464 00482 82945 99517 17622 00483 82861 82378 99515 17708 00485 82777 82292 99513 17794 00487 82693 82206 99511 17880 00489 82609 82120 99509 00491 82526 17965 82035 99507 18051 00493 82442 81949 99505 18136 00495 82359 81864 99503 00497 18221 82276 81779 9.99501 9.18306 10.81694 10.00499 10.82193 99499 18391 00501 82110 81609 82027 99497 18475 00503 81525 99495 18560 81945 81440 00505 99494 18644 81863 81356 00506 81780 99492 81272 00508 18728 18812 81698 99490 81188 00510 81617 18896 99488 81104 00512 81021 00514 81535 99486 18979 00516 80937 81453 99484 19063 9.19146 10.80854 10.00518 10.81372 9.99482 81291 80771 00520 19229 99480 81210 00522 19312 99478 80688 81129 00524 19395 80605 99476 81048 00526 19478 80522 99474 80967 00528 80439 19561 99472 80887 00530 80357 19643 99470 80807 00532 19725 80275 99468 80727 00534 19807 80193 99466 80647 00536 80111 99464 19889 80567 00538 80029 19971 99462 Co-tang. Tangent. Co-secant Secant. Sine.

81

I>n«:rep.!*.

60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 | 36 l 35 34 33 32 31 30 29 28 27 : 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

urA

Artificial Sines, Tang, and Sec. 9 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 . 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 ! 00

Sine. 9.19433 19513 19592 19672 19751 19830 19909 19988 20067 20145 9.20223 20302 20380 20458 20535 20613 20691 20768 20845 20922 9.20999 21076 21153 21229 21306 21382 21458 21534 21610 21685 9.21761 21836 21912 21987 22062 22137 22211 22286 22361 22435 9.22509 22583 22657 22731 22805 22878 22952 23025 23098 23171 9.23244 23317 23390 23462 23535 23607 23679 23752 23823 23895 23967 Co-sine.

34

117

Co-secant Secant. Tangent. Co-tang. Co-sine. 9.19971 10.80029 10.00538 10.80567 60 9.99462 80487 59 00540 79947 20053 99460 80408 58 00542 79866 20134 99458 80328 57 00544 79784 20216 99456 80249 56 00546 79703 20297 99454 80170 55 00548 79622 20378 99452 80091 54 00550 79541 20459 99450 80012 53 00552 79460 20540 99448 79933 52 00554 79379 20621 99446 79855 51 00556 79299 20701 99444 9.20782 10.79218 10.00558 10.79777 50 9.99442 79698 49 00560 20862 79138 99440 79620 48 00562 79058 20942 99438 79542 47 00564 21022 78978 99436 79465 46 00566 21102 78898 99434 79387 45 00568 21182 78818 99432 00571 79309 44 78739 21261 99429 79232 43 00573 21341 78659 99427 79155 42 00575 21420 78580 99425 00577 79078 41 78501 21499 99423 9.21578 10.78422 10.00579 10.79001 40 9.99421 78924 39 00581 21657 78343 99419 78847 38 00583 78264 21736 99417 78771 37 00585 21814 78186 99415 00587 78694 36 78107 99413 21893 78618 35 00589 21971 99411 78029 78542 34 00591 22049 77951 99409 00593 78466 33 22127 99407 77873 00596 78390 32 77795 99404 22205 00598 78315 31 77717 22283 99402 9.22361 10.77639 10.00600 10.78239 30 9.99400 00602 78164 29 99398 22438 77562 00604 78088 28 77484 99396 22516 78013 27 ■ 00606 77407 99394 22593 00608 77938 26 77330 99392 22670 00610 77863 25 22747 77253 99390 00612 77789 24 77176 99388 22824 00615 77714 23 99385 22901 77099 00617 77639 22 99383 22977 77023 00619 77565 21 99381 23054 76946 9.99379 9.23130 10.76870 10.00621 10.77491 20 . 77417 19 00623 99377 23206 76794 77343 18 00625 99375 23283 76717 00628 77269 17 99372 76641 23359 00630 77195 16 99370 23435 76565 77122 15 99368 00632 23510 76490 00634 77048 14 99366 23586 76414 99364 00636 76975 13 23661 76339 76902 12 23737 00638 99362 76263 99359 00641 76829 11 23812 76188 9.99357 9.23887 10.76113 10.00643 10.76756 10 9 99355 76683 23962 76038 00645 8 00647 99353 24037 76610 75963 7 99351 24112 00649 76538 75888 6 99348 00652 24186 75814 76465 99346 . 00654 5 24261 76393 75739 4 99344 00656 76321 24335 75665 99342 3 24410 75590 00658 76248 2 99340 24484 75516 00660 76177 99337 1 24558 75442 00663 76105 0 99335 24632 76033 75368 00665 Sine. M. Co-tang. 7’angent. Co-secant Secant. ■80 Degrees.

118 M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 10 Degrees, Sine. Co-sine. Tangent. Co-tang. Secant. Co-secant 9.23967 9.99335 9.24632 10.75368 10.00665 10.76033 60 99333 24039 24706 75294 00667 75961 59 99331 24110 24779 75221 00669 75890 58 24181 99328 24853 75147 00672 75819 57 24253 99326 24926 75074 00674 75747 56 24324 99324 25000 75000 00676 75676 55 24395 99322 25073 74927 00678 75605 54 24466 99319 25146 74854 00681 75534 53 24536 99317 25219 74781 00683 75464 52 24607 99315 25292 74708 00685 75393 51 9.24677 9.99313 9.25365 10.74635 10.00687 10.75323 50 24748 99310 25437 74563 00690 75252 49 24818 99308 25510 74490 00692 75182 48 24888 99306 25582 74418 00694 75112 47 24958 99304 25655 74345 00696 75042 46 25028 99301 25727 74273 00699 74972 45 25098 99299 25799 74201 00701 74902 4425168 99297 25871 74129 00703 74832 43 25237 99294 25943 74057 00706 74763 42 25307 99292 26015 73985 00708 74693 41 9.25376 9.99290 9.26086 10.73914 10.00710 10.74624 40 25445 99288 73842 26158 00712 74555 39 25514 99285 73771 26229 00715 74486 38 99283 25583 26301 73699 00717 74417 37 99281 25652 26372 73628 00719 74348 36 25721 99278 26443 73557 00722 74279 35 , 99276 25790 26514 73486 00724 74210 34 25858 99274 26585 73415 00726 74142 33 99271 25927 26655 73345 00729 74073 32 25995 73274 99269 26726 00731 74005 31 9.26063 9.99267 9,26797 10.73203 10.00733 10.73937 30 26131 99264 26867 73133 00736 73869 29 26199 99262 26937 73063 73801 28 00738 26267 99260 27008 72992 73733 27 00740 26335 99257 27078 72922 73665 26 . 00743 26403 73597 25 72852 99255 27148 00745 26470 73530 24 99252 72782 00748 27218 26538 99250 27288 72712 73462 23 00750 26605 99248 27357 72643 00752 73395 22 26672 73328 21 99245 27427 72573 00755 9.26739 9.99243 9.27496 10.72504 10.00757 10.73261 20 26806 99241 72434 73194 19 27566 00759 73127 18 26873 99238 72365 00762 27635 26940 72296 00764 73060 17 99236 27704 72227 00767 72993 16 27007 99233 27773 99231 72158 72927 15 00769 27073 27842 00771 72860 14 99229 27140 72089 27911 72794 13 72020 00774 99226 27206 27980 72727 12 9$224 71951 00776 27273 28049 72661 11 71883 00779 99221 28117 27339 9.27405 9.99219 9.28186 10.71814 10.00781 10.72595 10 72529 9 27471 71746 00783 99217 28254 27537 8 71677 00786 72463 99214 28323 7 00788 72398 27602 71609 99212 28391 6 72332 00791 27668 . 99209 71541 28459 72266 5 00793 27734 99207 71473 28527 72201 4 00796 27799 99204 71405 28595 3 72136 27864 00798 71338 99202 28662 72070 2 00800 27930 99200 71270 28730 72005 00803 1 27995 71202 99197 23798 71940 0 28060 00805 i 71135 99195 28865 M. | Co-sine. Sine. Co-tang. Tangent. Co-secant Secant.

79 Degrees.

Artificial Sines, Tang, and Sec. 11 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

tjme. 9.28060 28125 28190 28254 28319 28384 28448 28512 28577 28641 9.28705 28769 28833 28896 28960 29024 29087 29150 29214 29277 9.29340 29403 29466 29529 29591 29654 29716 29779 29841 29903 9.29966 30028 30090 30151 30213 30275 30336 30398 30459 30521 9.30582 30643 30704 30765 30826 30887 30947 31008 31068 31129 9.31189 31250 31310 31370 31430 31490 31549 31609 31669 31728 31788 Co-sine.

Co-secant Secant. Co-sine. Tangent. Co-tang. 9.28865 10.71135 10.00805 10.71940 9.99195 00808 71875 71067 28933 99192 71810 00810 71000 29000 99190 71746 00813 29067 70933 99187 71681 00815 70866 29134 99185 71616 00318 29201 70799 99182 71552 00820 70732 29268 99180 71488 00823 70665 99177 29335 71423 00825 70598 29402 99175 71359 00828 70532 29468 99172 9.99170 9.29535 10.70465 10.00830 10.71295 71231 29601 00833 99167 70399 71167 70332 99165 29668 00835 71104 29734 70266 00838 99162 99160 71040 29800 70200 00840 70134 99157 70976 29866 00843 70068 99155 29932 00845 70913 70002 00848 70850 99152 29998 70786 30064 69936 00850 99150 69870 00853 70723 99147 30130 9.99145 9.30195 10.69805 10.00855 10.70660 70597 00858 99142 30261 69739 69674 70534 99140 30326 00860 70471 99137 00863 30391 69609 99135 30457 69543 70409 00865 70346 99132 30522 00868 69478 69413 70284 99130 30587 00870 99127 69348 70221 30652 00873 70159 99124 69283 00876 30717 70097 99122 30782 69218 00878 9.99119 9.30846 10.69154 10.00881 10.70034 99117 30911 00883 69972 69089 99114 00886 69025 30975 69910 99112 31040 68960 00888 69849 99109 68896 00891 69787 31104 99106 31168 68832 00894 69725 99104 68767 69664 31233 00896 99101 31297 68703 00899 69602 99099 69541 31361 68639 00901 99096 68575 00904 31425 69479 9.99093 9.31489 10.68511 10.00907 10.69418 99091 69357 00909 31552 68448 99088 31616 69296 68384 00912 99086 31679 68321 00914 69235 99083 31743 68257 00917 69174 99080 31806 68194 69113 00920 99078 31870 68130 69053 00922 99075 31933 68067 . 00925 68992 99072 31996 68004 00928 68932 99070 32059 67941 00930 68871 9.99067 9.32122 10.67878 10.00933 10.68811 99064 32185 67815 00936 68750 99062 32248 67752 00938 68690 99059 32311 00941 67689 68630 99056 32373 67627 00944 68570 99054 32436 67564 00946 68510 99051 32498 67502 00949 68451 99048 32561 67439 00952 68391 99046 32623 67377 00954 68331 99043 32685 67315 00957 68272 99040 32747 67253 00960 68212 Sine. Co-tang. Tangent. Co-secant Secant.

78 Degrees,

119 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 M.

120 M. 0 l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 . 26 , 27 , 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 12 Degrees, Sine. 9.31788 31847 31907 31966 32025 32084 32143 32202 32261 32319 9.32378 32437 32495 32553 32612 32670 32728 32786 32844 32902 9.32960 33018 33075 33133 33190 33248 33305 33362 33420 33477 9.33534 33591 33647 33704 33761 33818 33874 33931 33987 34043 9.34100 34156 34212 34268 34324 34380 34436 34491 34547 34602 9.34658 34713 34769 34824 34879 34934 34989 35044 35099 35154 35209 l Co-sine.

Co-sine. Tangent. Co-tang. 9.99040 9.32747 10.67253 99038 32810 67190 99035 32872 67128 99032 32933 67067 99030 32995 67005 99027 33057 66943 99024 33119 66881 99022 33180 66820 99019 33242 66758 99016 33303 66697 9.99013 9.33365 10.66635 99011 33426 66574 99008 33487 66513 99005 33548 66452 99002 33609 66391 99000 33670 66330 98997 33731 66269 98994 33792 66208 98991 33853 66147 98989 66087 33913 9.98986 9.33974 10.66026 98983 34034 65966 98980 34095 65905 98978 34155 65845 98975 34215 65785 98972 65724 34276 93969 34336 65664 98967 34396 65604 98964 65544 34456 98961 65484 34516 9.98958 9.34576 10.65424 65365 98955 34635 98953 65305 34695 98950 65245 34755 98947 34814 65186 98944 65126 34874 98941 65067 34933 98938 65008 34992 98936 35051 64949 35111 64889 98933 9.98930 9.35170 10.64830 64771 98927 35229 64712 98924 35288 98921 64653 35347 98919 64595 35405 • 98916 64536 35464 64477 98913 35523 64419 98910 35581 64360 98907 35640 64302 98904 35698 9.98901 9.35757 10.64243 64185 98898 35815 64127 98896 35873 35931 64069 98893 64011 98890 35989 98887 63953 36047 63895 98884 36105 63837 98881 36163 63779 98878 36221 63721 98875 36279 63664 1 98872 36336 Sine. Co-tang. Tangent. 1

77 Degrees.

Secant. Co-secant 10.00960 10.68212 60 00962 68153 59 00965 68093 58 00968 68034 57 00970 67975 56 00973 67916 55 00976 67857 54 00978 67798 53 00981 67739 52 00984 67681 51 10.00987 10.67622 50 00989 67563 49 00992 67505 48 00995 67447 47 00998 67388 46 01000 67330 45 01003 67272 44 01006 67214 43 01009 67156 42 01011 67098 41 10.01014 10.67040 40 01017 66982 39 01020 66925 38 01022 66867 37 66810 36 01025 01028 66752 35 01031 66695 34 01033 66638 33 01036 66580 32 01039 66523 31 10.01042 10.66466 30 01045 66409 29 01047 66353 28 66296 27 01050 01053 66239 26 66182 25 01056 66126 24 01059 01062 66069 23 01064 66013 22 65957 21 01067 10.01070 10.65900 20 65844 19 01073 65788 18 01076 65732 17 01079 65676 16 01081 65620 15 01084 65564 14 01087 65509 13 01090 65453 12 01093 65398 11 01096 10.01099 10.65342 10 9 65287 01102 8 65231 01104 7 01107 65176 6 65121 OHIO 5 65066 01113 4 65011 01116 3 64956 01119 2 64901 01122 64846 1 01125 0 64791 01128 M. Secant. Co-secant

Artificial M. 0 1 2 3 4 5 6

7

8 9 10

n

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 1 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 1 00

1

Sine. 9.35209 35263 35318 35373 35427 35481 35536 35590 35644 35698 9.35752 35806 35860 35914 35968 36022 36075 36129 36182 36236 9.36289 36342 36395 36449 36502 36555 36608 36660 36713 36766 9.36819 36871 36924 36976 37028 37081 37133 37185 37237 37289 9.37341 37393 37445 37497 37549 37600 37652 37703 37755 37806 9.37858 37909 37960 38011 38062 38113 38164 38215 38266 38317 38368

i i Co-sine.

Sines, Tang, and Sec. 13 Degrees.

121

Secant. Co-secant Co-sine. Tangent. Co-tang. 9.36336 10.63664 10.01128 10.64791 60 9.98872 64737 59 01131 63606 36394 98869 64682 58 01133 63548 98867 36452 64627 57 01136 63491 98864 36509 64573 56 01139 63434 98861 36566 64519 55 01142 63376 36624 98858 64464 54 01145 63319 98855 36681 64410 53 01148 63262 36738 98852 64356 52 01151 63205 98849 36795 64302 51 01154 63148 98846 36852 9.98843 9.36909 10.63091 10.01157 10.64248 50 64194 49 01160 63034 98840 36966 01163 64140 48 98837 62977 37023 01166 64086 47 98834 37080 62920 64032 46 37137 01169 98831 62863 62807 01172 63978 45 37193 98828 01175 63925 44 98825 37250 62750 63871 43 01178 62694 98822 37306 63818 42 01181 37363 62637 98819 01184 63764 41 62581 98816 37419 9.98813 9.37476 10.62524 10.01187 10.63711 40 01190 63658 39 37532 62468 98810 01193 98807 63605 38 . 62412 37588 01196 63551 37 98804 37644 62356 01199 63498 36 98801 37700 62300 01202 37756 63445 35 62244 98798 01205 63392 34 98795 37812 62188 01208 63340 33 98792 37868 62132 01211 63237 32 98789 37924 62076 01214 63234 31 98786 37980 62020 9.98783 9.38035 10.61965 10.01217 10.63181 "30 01220 63129 29 38091 98780 61909 01223 63076 28 98777 38147 61853 01226 63024 27 98774 38202 61798 98771 01229 62972 26 38257 61743 01232 98768 61687 62919 25 38313 01235 62867 24 98765 38368 61632 98762 01238 61577 82815 23 38423 01241 98759 62763 22 61521 38479 01244 62711 21 98756 38534 61466 9.98753 9.38589 10.61411 10.01247 10.62659 20 98750 01250 62607 19 38644 61356 ! 98746 01254 38699 62555 18 61301 01257 98743 38754 61246 62503 17 98740 38808 01260 62451 16 61192 98737 01263 38863 61137 62400 15 98734 38918 01266 62348 14 61082 98731 38972 01269 62297 13 61028 98728 39027 01272 60973 62245 12 98725 39082 01275 62194 11 60918 9.98722 9.39136 10.60864 10.01278 10.62142 10 98719 39190 01281 62091 9 60810 98715 39245 01285 8 60755 62040 98712 39299 60701 01288 7 61989 98709 39353 60647 01291 61938 6 98706 39407 60593 01294 61887 5 98703 39461 01297 60539 4 61836 98700 39515 01300 60485 3 61785 98697 39569 60431 01303 2 . 61734 98694 39623 60377 01306 1 61683 98690 39677 60323 01310 61632 0 Sine. j Co-tang. Tangent. 1 Co-secant M. , Secant. 76 Decrees.

3E

122 JVl. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 \7 18 19 20 21 22 23 , 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ' 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 14 Degrees. Sine. 9.38368 38418 38469 38519 38570 38620 38670 38721 38771 38821 9.38871 38921 38971 39021 39071 39121 39170 39220 39270 39319 9.39369 39418 39467 39517 39566 39615 39664 39713 39762 39811 9.39860 39909 39958 40006 40055 40103 40152 40200 40249 40297 9.40346 40394 40442 40490 40538 40586 40634 40682 40730 40778 9.40825 40873 40921 40968 41016 41063 41111 41158 41205 41252 41300 Co-sine.

Co-sine. Tangent. Co-tang. 9.98690 9.39677 10.60323 98687 39731 60269 98684 39785 60215 98681 39838 60162 98678 39892 60108 98675 39945 60055 98671 39999 60001 98668 40052 59948 98665 40106 59894 98662 40159 59841 9.98659 9.40212 10.59788 98656 40266 59734 98652 40319 59681 98649 40372 59628 98646 40425 59575 98643 40478 59522 98640 40531 59469 98636 40584 59416 98633 40636 59364 98630 40689 59311 9.98627 9.40742 10.59258 98623 40795 59205 98620 40847 59153 98617 40900 59100 98614 40952 59048 98610 41005 58995 98607 41057 58943 98604 41109 58691 41161 98601 58839 98597 41214 58786 9.98594 9.41266 10.58734 98591 41318 58682 98588 41370 58630 98584 41422 58578 98581 41474 58526 98578 41526 58474 98574 41578 58422 98571 58371 41629 41681 98568 58319 98565 41733 58267 9.98561 9.41784 10.58216 98558 41836 58164 98555 41887 58113 98551 58061 41939 98548 41990 58010 42041 98545 57959 98541 42093 57907 98538 42144 57856 98535 42195 57805 98531 42246 57754 9.98528 9.42297 10.57703 98525 42348 57652 98521 42399 57601 98518 42450 57550 98515 42501 57499 98511 42552 57448 57397 98508 42603 98505 57347 42653 98501 57296 42704 98498 57245 42755 98494 57195 42805 Sine. Co-tang. Tangent. I in Dftgrett*

Secant. 10.01310 01313 01316 01319 01322 01325 01329 01332 01335 01338 10.01341 01344 01348 01351 01354 01357 01360 01364 01367 01370 10.01373 01377 01380 01383 01386 01390 01393 01396 01399 01403 10.01406 01409 01412 01416 01419 01422 01426 01429 01432 01435 10.01439 01442 01445 01449 01452 01455 01459 01462 01465 01469 10.01472 01475 01479 01482 01485 01489 01492 01495 01499 01502 01506 Co-secant 1

Co-secant 10.61632 60 61582 59 61531 58 61481 57 61430 56 61380 55 61330 54 61279 53 61229 52 61179 51 10.61129 50 61079 49 61029 48 60979 47 60929 46 60879 45 60830 44 60780 43 60730 42 60681 41 10.60631 40 60582 39 60533 38 60483 37 60434 36 60385 35 60336 34 60287 33 60238 32 60189 31 10.60140 30 60091 29 60042 28 59994 27 59945 26 59897 25 59848 24 59800 23 59751 22 59703 21 10.59654 “20 59606 19 59558 18 59510 17 59462 16 59414 15 59366 14 59318 13 59270 12 59222 11 10.59175 10 9 59127 8 59079 7 59032 6 58984 5 58937 4 58889 3 58842 2 58795 58748 1 58700 0 | M. } Secant.

Artificial Sines, Tang, and Sec. 15 Degrees. M. 0 1 2 3 4 5 6 7 8 9 ■ 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.41300 41347 41394 41441 41488 41535 41582 41628 41675 41722 9.4176*3 41815 41861 41908 41954 42001 42047 42093 42140 42186 9.42232 42278 42324 42370 42416 42461 42507 42553 42599 42644 9.42690 42735 42781 42826 42872 42917 42962 43008 43053 ' 43098 9.43143 43188 43233 43278 43323 43367 43412 43457 43502 43546 9.43591 43635 43680 43724 43769 43813 43857 43901 43946 43990 44034 Co-sine.

Co-stne. Tangent. Co-tang. 9.42805 10.57195 9-98494 57144 42856 98491 57094 42906 98488 57043 42957 98484 56993 43007 98481 56943 98477 43057 56892 98474 43108 56842 98471 43158 56792 43208 98467 56742 98464 43258 9.98460 9.43308 10.56692 98457 56642 43358 98453 56592 43408 56542 98450 43458 98447 56492 43508 98443 43558 56442 43607 56393 98440 56343 98436 43657 56293 98433 43707 56244 98429 43756 9.98426 9.43806 10.56194 98422 43855 56145 98419 43905 56095 98415 43954 56046 98412 44004 55996 98409 44053 55947 98405 44102 55898 98402 *44151 55849 98398 44201 55799 98395 44250 55750 9.98391 9.44299 10.55701 98388 55652 44348 98384 44397 55603 98381 44446 55554 98377 55505 44495 98373 55456 44544 98370 55408 44592 98366 55359 44641 98363 44690 55310 98359 55262 44738 9.98356 9.44787 10.55213 98352 55164 44836 98349 44884 55116 98345 55067 44933 98342 44981 55019 98338 54971 45029 98334 54922 45078 98331 45126 54874 98327 54826 45174 98324 45222 54778 9.98320 9.45271 10.54729 98317 45319 54681 98313 45367 54633 98309 45415 54585 98306 45463 54537 98302 45511 54489 98299 45559 54441 98295 45606 54394 98291 45654 54346 98288 45702 54298 98284 45750 54250 Sine. Co-tang. Tangent. |

74 Degrees

123

Secant. Co-secant 10.01506 10.58700 60 01509 58653 59 01512 58606 58 58559 57 01516 58512 56 01519 58465 55 01523 58418 54 01526 58372 53 01529 58325 52 01533 58278 51 01536 10.01540 10.58232 50 01543 58185 49 01547 58139 48 58092 47 01550 58046 46 01553 01557 57999 45 57953 44 01560 57907 43 01564 01567 57860 42 01571 57814 41 10.01574 10.57768 40 57722 39 01578 01581 57676 38 57630 37 01585 57584 36 01588 01591 57539 35 57493 34 01595 57447 33 01598 01602 57401 32 57356 31 01605 10.01609 10.57310 30 01612 57265 29 01616 57219 28 01619 57174 27 01623 57128 26 01627 57083 25 01630 57038 24 01634 56992 23 01637 56947 22 01641 56902 21 10.01644 10.56857 20 01648 56812 19 01651 56767 18 01655 56722 17 56677 16 01658 01662 56633 15 01666 56588 14 01669 56543 13 01673 56498 12 01676 56454 11 10.01680 10.56409 10 01683 56365 9 > 01687 8 56320 01691 7 56276 01694 56231 6 01698 56187 5 01701 4 56143 01705 3 56099 01709 56054 2 01712 56010 1 01716 55966 0 Co-secant M. Secant.

124 Artificial Sines, Tang, and Sec. 16 Degrees, M. 0

1 2

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 . 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.44034 44078 44122 44166 44210 44253 44297 44341 44385 44428 9.44472 44516 44559 44602 44646 44689 44733 44776 44819 44862 9.44905 44948 44992 45035 45077 45120 45163 45206 45249 45292 9.45334 45377 45419 45462 45504 45547 45589 45632 45674 45716 9.45758 45801 45843 45885 45927 45969 46011 46053 46095 46136 9.46178 46220 46262 46303 46345 46386 46428 46469 46511 46552 46594 Co-sine.

Co-sine. Tangent. Co-tang. Secant. Co-secant 9.98284 9.45750 10.54250 10.01716 10.55966 , 60 98281 45797 54203 01719 55922 59 98277 45845 54155 01723 55878 58 98273 45892 54108 01727 55834 57 98270 45940 54060 01730 55790 56 98266 45987 54013 01734 55747 55 98262 46035 53965 01738 55703 54 98259 46082 53918 01741 55659 53 98255 46130 53870 01745 55615 52 98251 46177 53823 01749 55572 51 9.98248 9.46224 10.53776 10.01752 10.55528 50 98244 46271 53729 01756 55484 49 98240 46319 53681 01760 55441 48 98237 46366 53634 01763 55398 47 98233 46413 53587 01767 55354 46 98229 46460 53540 01771 55311 45 98226 46507 53493 01774 55267 44 98222 46554 53446 01778 55224 43 98218 46601 53399 01782 55181 42 98215 46648 53352 01785 55138 41 9.98211 9.46694 10.53306 10.01789 10.55095 40 98207 46741 53259 01793 55052 39 98204 46788 53212 01796 55008 38 98200 46835 53165 01800 54965 37 46881 98196 53119 01804 54923 36 98192 46928 53072 01808 54880 35 98189 46975 53025 01811 54837 34 47021 98185 52979 01815 54794 33 98181 47068 52932 01819 54751 32 98177 47114 52886 01823 54708 31 9.98174 9.47160 10.52840 10.01826 10.54666 30 98170 47207 52793 01830 54623 29 98166 52747 47253 01834 54581 28 98162 47299 52701 01838 54538 27 47346 98159 52654 01841 54496 26 47392 98155 52608 01845 54453 25 98151 47438 52562 01849 54411 24 98147 47484 52516 01853 54368 23 98144 47530 52470 01856 54326 22 98140 47576 52424 01860 54284 21 9.98136 9.47622 10.52378 10.01864 10.54242 20 98132 47668 52332 01868 54199 19 98129 01871 47714 52286 54157 18 98125 47760 52240 01875 54115 17 98121 52194 54073 16 47806 01879 98117 54031 15 01883 47852 52148 98113 53989 14 47897 52103 01887 53947 13 98110 52057 47943 01890 98106 52011 01894 53905 12 47989 53864 11 98102 01898 48035 51965 9.98098 9.48080 10.51920 10.01902 10.53822 10 9 98094 53780 48126 01906 51874 8 53738 48171 98090 51829 01910 7 53697 98087 48217 01913 51783 6 01917 53655 48262 98083 51738 5 53614 48307 51693 01921 98079 4 53572 51647 01925 98075 48353 3 53531 98071 51602 01929 48398 2 53489 01933 98067 51557 48443 1 53448 01937 98063 51511 48489 0 53406 51466 01940 98060 48534 Secant. ~M. [ Sine. Co-tang. Tangent. Co-secant 73 Degrees

Artificial Sines, Tang, and Sec. 17 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 T6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 , 58 59 60

Smei 9.46594 46635 46676 46717 46758 46800 46841 46882 46923 46964 9.47005 47045 47086 47127 47168 47209 47249 47290 47330 47371 9.47411 47452 47492 47533 47573 47613 47654 47694 47734 47774 9.47814 47854 47894 47934 47974 48014 48054 48094 48133 48173 9.48213 48252 48292 48332 48371 48411 48450 48490 48529 48568 9.48607 48647 48686 48725 48764 48803 48842 48881 48920 48959 48998 Co-sine.

Secant. Co-secant Co-sine. 1 angent. Co-tang. 9.98060 9.48534 10.51466 10.01940 10.53406 01944 53365 51421 98056 48579 01948 53324 48624 51376 98052 53283 51331 01952 98048 48669 53242 98044 01956 48714 51286 98040 51241 01960 53200 48759 98036 01964 53159 48804 51196 98032 53118 48849 01968 51151 98029 48894 01971 53077 51106 98025 48939 51061 01975 53036 9.98021 9.48984 10.51016 10.01979 10.52995 98017 49029 50971 01983 52955 98013 49073 50927 01987 52914 98009 49118 50882 01991 52873 98005 49163 50837 01995 52832 98001 49207 50793 01999 52791 97997 49252 50748 02003 52751 97993 49296 50704 02007 52710 97989 49341 50659 02011 52670 97986 49385 50615 02014 52629 9.97982 9.49430 10.50570 10.02018 10.52589 97978 49474 50526 02022 52548 97974 49519 50481 02026 52508 97970 49563 50437 52467 02030 97966 49607 50393 02034 52427 97962 49652 50348 02038 52387 97958 49696 50304 02042 52346 97954 49740 50260 02046 52306 97950 49784 50216 02050 52266 97946 49828 50172 02054 52226 9.97942 9.49872 10.50128 10.02058 10.52186 97938 49916 50084 02062 52146 97934 49960 50040 02066 52106 97930 50004 49996 02070 52066 97926 50048 49952 02074 52026 97922 50092 ' 49908 02078 51986 97918 50136 49864 02082 51946 97914 50180 49820 02086 51906 97910 50223 49777 02090 51867 97906 50267 49733 02094 51827 9.97902 9.50311 10.49689 10.02098 10.51787 97898 50355 49645 02102 51748 97894 50398 49602 02106 51708 97890 50442 49558 02110 51668 97886 50485 49515 02114 51629 97882 50529 49471 02118 51589 97878 50572 49428 02122 51550 97874 50616 49384 02126 51510 97870 50659 49341 02130 51471 97866 50703 49297 02134 51432 9.97861 9.50746 10.49254 10.02139 10.51393 97857 50789 49211 02143 51353 97853 50833 49167 02147 51314 97849 *50876 49124 02151 51275 97845 50919 49081 02155 51236 97841 50962 49038 02159 51197 97837 51005 48995 02163 51158 97833 51048 48952 02167 51119 97829 51092 48908 02171 51080 97825 51135 48865 02175 51041 97821 51178 48822 02179 51002 Sine. Co-tang-. Tangent. Co-secant Secant. 72 Degrees.

125 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 l 0 M.

126 M".

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 , 23 24 25 26 ■ 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 , 60

Artificial Sines, Tang, and °ec. 13 Degrees. bine. 9.48998 49037 49076 49115 49153 49192 49231 49269 49308 49347 9.49385 49424 49462 49500 49539 49577 49615 49654 49692 49730 9.49768 49806 49844 49882 49920 49958 49996 50034 50072 50110 9.50148 50185 50223 50261 50298 50336 50374 50411 50449 50486 9.50523 50561 50598 50635 50673 50710 50747 50784 50821 50858 9.50896 50933 50970 51007 51043 51080 51117 51154 51191 51227 51264 Co-sine.

Co-sine. Tangent. Co-tang. Secant. 9.97821 9.51178 10.48822 10.02179 97817 51221 48779 02183 97812 51264 48736 02188 97808 51306 48694 02192 97804 51349 48651 02196 97800 51392 48608 02200 97796 51435 48565 02204 97792 51478 48522 02208 97788 51520 48480 02212 97784 51563 48437 02216 9.97779 9.51606 10.48394 10.02221 97775 51648 48352 02225 97771 51691 48309 02229 97767 51734 48266 02233 97763 51776 48224 02237 97759 51819 48181 02241 97754 51861 48139 02246 97750 51903 48097 02250 97746 51946 48054 02254 97742 51983 48012 02258 9.97738 9.52031 10.47969 10.02262 97734 52073 47927 02266 9772* 52115 47885 02271 97725 52157 47843 02275 97721 52200 47800 02279 97717 52242 47758 02283 97713 52284 47716 02287 97708 52326 47674 02292 97704 52368 47632 02296 97700 52410 47590 02300 9.97696 9.52452 10.47548 10.02304 97691 52494 47506 02309 97687 52536 47464 02313 97683 52578 47422 02317 97679 52620 47380 02321 97674 52661 47339 02326 97670 47297 52703 02330 97666 52745 47255 02334 97662 52787 47213 02338 97657 47171 52829 02343 9.97653 9.52870 10.47130 10.02347 97649 - 52912 47088 02351 97645 52953 47047 02355 97640 52995 47005 02360 97636 53037 46963 02364 97632 53078 46922 02368 97628 53120 46880 02372 97623 53161 46839 02377 97619 46798 53202 02381 97615 53244 46756 02385 9.97610 9.53285 10.46715 10.02390 97606 53327 46673 02394 97602 46632 53368 02398 97597 53409 46591 02403 97593 02407 46550 53450 97589 02411 46508 53492 97584 46467 02416 53533 97580 02420 53574 46426 97576 02424 53615 46385 97571 02429 53656 46344 97567 46303 02433 53697 Sine. Co-tang. Tangent. Co-secant J

71 Degrees.

Co-secant 10.51002 60 50963 59 50924 58 50885 57 50847 56 50808 55 50769 54 50731 53 50692 52 50653 51 10.50615 50 50576 49 50538 48 50500 47 50461 46 50423 45 50385 44 50346 43 50308 42 50270 41 10.50232 40 50194 39 50156 38 50118 37 50080 36 50042 35 50004 34 49966 33 49928 32 49890 31 10.49852 30 49815 29 49777 28 49739 27 49702 26 . 49664 25 49626 24 49589 23 49551 22 49514 21 10.49477 20 ' 49439 19 49402 18 49365 17 49327 16 49290 15 49253 14 49216 13 49179 12 49142 11 10 49104 10 9 49067 8 49030 7 48993 6 48957 5 48920 4 48883 3 48846 2 48809 48773 1 0 48736 M. | Secant.

Artificial Sines, Tang, and Sec. 19 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.51264 51301 51338 51374 51411 51447 51484 51520 51557 51593 9.51629 51666 51702 51738 51774 51811 51847 51883 51919 51955 9.51991 52027 52063 52099 52135 52171 52207 52242 52278 52314 9.52350 52385 52421 52456 52492 52527 52563 52598 52634 52669 9.52705 52740 52775 52811 52846 52881 52916 52951 52986 53021 9.53056 53092 53126 53161 53196 53231 53266 53301 53336 53370 53405 Co-sine.

127

Co-secant Secant. Tangent. Co-tang. Co-sine. 9.53697 10.46303 10.02433 10.48736 60 9.97567 48699 59 02437 46262 53738 97563 48662 58 02442 46221 53779 97558 48626 57 02446 46180 53820 97554 48589 56 02450 46139 53861 97550 48553 55 02455 46098 53902 97545 48516 54 02459 46057 97541 53943 48480 53 02464 46016 53984 97536 48443 52 02468 97532 54025 - 45975 48407 51 02472 45935 97528 54065 9.97523 9.54106 10.45894 10.02477 10.48371 50 48334 49 02481 54147 45853 97519 48298 48 02485 54187 45813 97515 48262 47 54228 45772 02490 97510 48226 46 02494 97506 54269 45731 48189 - 45 97501 54309 45691 02499 48153 44 45650 02503 97497 54350 48117 43 : 02508 97492 54390 45610 48081 42 54431 02512 97488 45569 48045 41 54471 02516 45529 97484 9.97479 9.54512 10.45488 10.02521 10.48009 40 47973 39 02525 97475 54552 45448 47937 38 97470 54593 45407 02530 02534 97466 45367 47901 37 54633 97461 47865 36 54673 45327 02539 97457 54714 02543 47829 35 45286 47793 34 97453 02547 54754 45246 47758 33 97448 54794 02552 45206 97444 02556 47722 32 54835 45165 97439 47686 31 54875 02561 45125 9.97435 9.54915 10.45085 10.02565 10.47650 30 97430 47615 29 54955 02570 45045 97426 02574 47579 28 54995 45005 97421 47544 27 55035 02579 44965 97417 47508 26 55075 02583 44925 97412 47473 25 55115 44885 02588 97408 47437 24 55155 44845 02592 97403 55195 02597 44805 47402 23 97399 55235 02601 47366 22 44765 97394 55275 47331 21 02606 44725 9.97390 9.55315 10.44685 10.02610 10.47295 20 97385 55355 02615 44645 47260 19 97381 55395 44605 02619 47225 18 97376 55434 02624 44566 47189 17 97372 55474 44526 02628 47154 16 97367 55514 02633 44486 47119 15 97363 55554 02637 44446 47084 14 97358 55593 44407 02642 47049 13 97353 55633 44367 02647 47014 12 97349 55673 44327 02651 46979 11 9.97344 9.55712 10.44288 10.02656 10.46944 10 97340 55752 44248 02660 9 46908 97335 55791 44209 02665 8 46874 97331 55831 44169 02669 7 46839 97326 55870 44130 02674 46804 6 97322 55910 44090 02678 46769 5 97317 55949 44051 02683 4 46734 97312 55989 44011 02688 3 46699 97308 56028 43972 02692 2 46664 97303 56067 43933 02697 1 46630 97299 56107 43893 02701 0 46595 Sine. Co-tang. Tangent. Co-secant M. Secant.

70 Degrees

128 M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 . 36 37 38 39 40 41 42 43 44 1 45 46 47 48 49 50 51 52 53 54 55 1 56 57 58 59 60

Artificial Sines, Tang* and Sec. 20 Degrees, Sine. 9 53405 53440 53475 53509 53544 53578 53613 53647 53682 53716 9.53751 53785 53819 53854 53888 53922 53957 53991 54025 54059 9.54093 54127 54161 54195 54229 54263 54297 54331 54365 54399 9.54433 54466 54500 54534 54567 54601 54635 54668 54702 54735 9.54769 54802 54836 54869 54903 54936 54969 55003 55036 55069 9.55102 55136 55169 55202 55235 55268 55301 55334 55367 55400 55433 Co-sine.

Co-sine. Secant. Tangent. Co-tang. Co-secant [ 9.56107 10.43893 10.02701 10.46595 9.97299 56146 97294 43854 02706 46560 97289 02711 56185 43815 46525 56224 97285 43776 02715 46491 v 97280 43736 56264 02720 46456 97276 43697 56303 02724 46422 97271 56342 43658 2729 46387 56381 97266 43619 £734 46353 97262 .6420 02738 43580 46318 97257 56459 43541 02743 46284 9.56498 10.43502 10.02748 10.46249 9.97252 97248 56537 43463 02752 46215 97243 56576 43424 02757 46181 97238 46146 56615 43385 02762 97234 56654 43346 02766 46112 02771 97229 56693 43307 46078 97224 46043 56732 43268 02776 56771 43229 46009 97220 02780 43190 02785 45975 97215 56810 45941 43151 02790 97210 56849 9.97206 9.56887 10.43113 10.02794 10.45907 97201 43074 45873 56926 02799 45839 43035 02804 97196 56965 45805 42996 02808 97192 57004 45771 02813 97187 42958 57042 45737 02818 57081 42919 97182 45703 42880 02822 97178 57120 45669 02827 42842 97173 57158 45635 02832 42803 97168 57197 02837 45601 42765 97163 57235 9.97159 9.57274 10.42726 10.02841 10.45567 45534 02846 42688 97154 57312 45500 02851 42649 57351 97149 45466 42611 02855 97145 57389 45433 02860 42572 97140 57428 45399 02865 42534 57466 97135 45365 02870 42496 57504 97130 45332 42457 02874 97126 57543 45298 02879 42419 57581 97121 45265 02884 42381 97116 57619 9.97111 9.57658 10.42342 10.02889 10.45231 45198 02893 42304 97107 57696 45164 02898 42266 97102 57734 45131 02903 42228 97097 57772 45097 02908 42190 97092 57810 45064 02913 42151 97087 57849 45031 02917 42113 57887 97083 44997 02922 42075 97078 57925 44964 02927 42037 97073 57963 44931 02932 41999 97068 58001 9.58039 10.41961 10.02937 10.44898 9.97063 44864 02941 41923 58077 97059 44831 02946 41885 58115 97054 44798 02951 41847 58153 97049 44765 02956 41809 58191 97044 44732 02961 41771 58229 97039 44699 02965 41733 58267 97035 44666 02970 41696 58304 97030 44633 41658 r 02975 58342 97025 44600 02980 41620 97020 58380 44567 02985 41582 58418 97015 Secant. Sine. Co-tang. Tangent. Co-secant

69 Degrees

1 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 , 8 7 6 5 4 3 2 1

O' M.

Artificial Sines, Tang, and Sec. 21 Degrees. 129 i\l. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.55433 55466 55499 55532 55564 55597 55630 55663 55695 55728 9.55761 55793 55826 55858 55891 55923 55956 55988 56021 56053 9.56085 56118 56150 56182 56215 56247 56279 56311 56343 56375 9.56408 56440 56472 56504 56536 56568 56599 56631 56663 56695 9.56727 56759 56790 56822 56854 56886 56917 56949 56980 57012 9.57044 57075 57107 57138 57169 57201 57232 57264 57295 57326 57358 Co-sine.

Co-secant Secant. (Jo-sme. Tangent. Co-tang. 9.58418 10.41582 10.02985 10.44567 60 9.97015 44534 59 02990 41545 58455 97010 44501 58 02995 41507 58493 97005 44468 57 02999 41469 97001 58531 44436 56 03004 41431 96996 58569 44403* 55 41394 03009 96991 58606 44370 54 03014 41356 96986 58644 44337 53 03019 41319 96981 58681 44305 52 03024 41281 96976 • 58719 44272 51 03029 96971 58757 41243 9.96966 9.58794 10.41206 10 03034 10.44239 50 44207 49 03038 96962 58832 41168 44174 48 96957 41131 03043 58869 58907 41098 03048 96952 44142 47 96947 58944 44109 46 , 41056 03053 96942 58981 41019 03058 44077 45 96937 40981 59019 03063 44044 44 40944 96932 03068 59056 44012 43 96927 59094 40906 03073 43979 42 ■ 96922 59131 40869 03078 43947 41 9.96917 9.59168 10.40832 10.03083 10.43915 40 96912 59205 40795 03088 43882 39 96907 40757 59243 03093 43850 38 96903 59280 40720 03097 43818 37 96898 59317 40683 03102 43785 36 96893 59354 40646 03107 43753 35 96888 40609 59391 03112 43721 34 40571 96883 59429 03117 43689 33 96878 40534 59466 03122 43657 32 96873 40497 59503 03127 43625 31 9.96868 9.59540 10.40460 10.03132 10.43592 30 96863 59577 40423 03137 43560 29 96858 59614 40386 03142 43528 28 96853 59651 40349 03147 43496 27 96848 59688 40312 03152 43464 26 96843 59725 40275 03157 43432 25 96838 59762 40238 03162 43401 24 96833 59799 40201 03167 43369 23 96828 59835 40165 03172 43337 22 96823 59872 40128 03177 43305 21 9.96818 9.59909 10.40091 10.03182 10.43273 20 96813 40054 59946 03187 43241 19 96808 ' 59983 40017 03192 43210 18 96803 60019 39981 03197 43178 17 96798 60056 39944 03202 43146 16 96793 60093 39907 03207 43114 15 96788 60130 39870 03212 43083 14 96783 60166 39834 03217 43051 13 96778 60203 39797 03222 43020 12 96772 60240 39760 03228 42988 11 9.96767 9.60276 10.39724 10 03233 10.42956 10 96762 60313 39687 03238 42925 9 96757 60349 39651 03243 42893 8 96752 60386 39614 03248 7 42862 96747 60422 39578 03253 42831 6 96742 60459 39541 03258 12799 5 96737 60495 39505 03263 4 42768 96732 60532 39468 03268 3 42736 96727 60568 39432 03273 2 42705 96722 60605 39395 03278 1 42674 96717 60641 39359 03283 0 42642 Sine. Co-tang. Tangent. Co-secant Secant. M.

68 Degrees 3 F

130 M.

Artificial Sines, Tang, and Sec. 22 Degrees.

Sine. 0 “5757358 1 ' 57389 2 57420 3 57451 4 57482 5 57514 6 57545 7 57576 8 57607 9 57638 10 9.57669 11 57700 12 57731 13 ✓ 57762 14 57793 15 57824 16 57855 17 57885 18 57916 19 57947 20 9.57978 21 58008 22 58039 23 58070 24 58101 25 58131 26 58162 27 58192 28 58223 29 58253 30 9.58284 31 58314 32 58345 33 58375 34 58406 35 58436 36 58467 37 58497 38 58527 39 58557 40 9.58588 41 58618 42 58648 43 58678 44 58709 45 58739 46 58769 47 58799 48 58829 49 58859 50 9.53889 51 58919 52 58949 53 58979 54 59009 55 59039 56 59069 57 59098 58 59128 59 59158 60 59188 Co-sine.

Co-sine. Tangent. Co-tang. 9.96717 9.60641 10.39359 96711 60677 39323 96706 60714 39286 96701 60750 39250 9t696 60786 39214 96691 60823 39177 96686 60859 39141 96681 60895 39105 96676 60931 39069 96670 60967 39033 9.96665 9.61004 10.38996 96660 61040 38960 96655 61076 38924 96650 61112 38888 96645 61148 38852 96640 61184 38816 96634 61220 38780 96629 61256 38744 96624 61292 38708 96619 61328 38672 9.96614 9.61364 10.38636 96608 61400 38600 96603 61436 38564 96598 61472 38528 96593 61508 38492 96588 61544 38456 96582 61579 38421 96577 61615 38385 96572 61651 38349 96567 61687 38313 9.96562 9.61722 10.38278 96556 61758 38242 96551 61794 38206 96546 61830 38170 96541 61865 38135 96535 61901 38099 96530 61936 38064 96525 61972 38028 96520 62008 37992 96514 62043 37957 9.96509 9.62079 10.37921 96504 62114 37886 96498 62150 37850 96493 62185 37815 96488 62221 37779 96483 62256 37744 96477 62292 37708 96472 62327 37673 96467 62362 37638 96461 62398 37602 9.96456 9.62433 10.37567 96451 62468 37532 96445 62504 37496 96440 62539 37461 96435 62574 37426 96429 62609 37391 96424 62645 37355 96419 62680 37320 96413 62715 37285 96408 62750 37250 96403 62785 37215 Sine. Co-tan"*. Tansrent. I

67 Degrees.

Secant. Co-secant 10 03283 10.42642 60 C8289 42611 59 03294 42580 58 03299 42549 67 03304 42518 56 03309 42486 65 03314 42455 54 03319 42424 53 ! 03324 42393 52 03330 42362 51 10.03335 10.42331 I’ 50 03340 42300 49 ; 03345 42269 48 03350 42238 47 03355 42207 46 03360 42176 45 03366 42145 44 03371 ‘ 42115 43 03376 42084 42 03381 42053 41 10.03386 10.42022 40 03392 41992 39 03397 41961 38 03402 41930 37 03407 41899 36 03412 41869 35 03418 41838 34 03423 41808 33 03428 41777 32 1 03433 41747 31 . 10.03438 10.41716 30 f 03444 41686 29 03449 41655 28 03454 41625 27 . 03459 41594 26 . 03465 41564 25 03470 41533 24 03475 41503 23 03480 41473 22 03486 41443 21 10.03491 10.41412 20 03496 41382 19 03502 41352 18 03507 41322 17 41291 16 ' 03512 03517 41261 15 41231 14 03523 03528 41201 13 41171 12 03533 03539 ’ 41141 11 10.03544 10.41111 10 9 1 03549 41081 8 , 03555 41051 7 41021 03560 6 40991 03565 5 03571 40961 4 03576 ' 40931 3 40902 03581 2 40872 03587 1 40842 03592 0 40812 03597 M. Co-srcant Recant.

Artificial Sines, Tang, and Sec. 23 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Co-sine. Tangent. Co-tang. Sine. 9.62785 10.37215 9.96403 9.59188 37180 96397 62820 59218 37145 52855 59247 96392 37110 62890 59277 96387 37074 96381 62926 59307 37039 96376 62961 59336 37004 62996 59366 96370 63031 36969 96365 59396 36934 63066 96360 59425 96354 36899 59455 63101 9.96349 9.63135 10.36865 9.59484 59514 96343 63170 36830 59543 96338 36795 63205 96333 36760 59573 63240 96327 59602 63275 36725 96322 36690 63310 59632 36655 59661 96316 63345 96311 36621 59690 63379 59720 36586 96305 63414 59749 36551 96300 63449 9.59778 9.96294 9.63484 10.36516 36481 59808 96289 63519 59837 36447 96284 63553 96278 36412 59866 63588 59895 96273 63623 36377 96267 63657 59924 36343 59954 96262 63692 36308 59983 96256 63726 36274 60012 63761 96251 36239 60041 96245 63796 36204 9.60070 9.96240 9.63830 10.36170 60099 96234 63865 36135 60128 96229 63899 36101 60157 96223 63934 36066 60186 96218 63968 36032 60215 96212 35997 64003 60244 96207 64037 35963 60273 96201 64072 35928 60302 96196 64106 35894 60331 96190 64140 35860 9.60359 9.96185 9.64175 10.35825 60388 96179 64209 35791 60417 96174 64243 35757 60446 96163 64278 35722 60474 96162 64312 35688 60503 96157 64346 35654 60532 96151 64381 35619 60561 96146 64415 35585 60589 96140 64449 35551 60618 96135 64483 35517 9.60646 9.96129 9.64517 10.35483 60675 96123 64552 35448 60704 96118 64586 35414 60732 96112 64620 35380 60761 96107 64654 35346 60789 96101 64688 35312 60818 96095 64722 35278 60846 96090 64756 35244 60875 96084 64790 35210 60903 96079 64824 35176 60931 96073 64858 35142 Sine. j C o-tang. Tangent. 1 Co-sine.

6G Degrees.

131

Secant. Co-secant 10.03597 10.40812 60 40782 59 03603 40753 58 03608 40723 57 03613 40693 56 03619 40664 55 03624 40634 54 03630 40604 53 03635 40575 52 03640 40545 51 03646 10.03651 10.40516 50 40486 49 03657 40457 48 03662 40427 47 03667 40398 46 03673 40368 45 03678 03684 40339 44 03689 40310 43 03695 40280 42 03700 40251 41 10.03706 10.40222 40 03711 40192 39 03716 40163 38 03722 40134 37 03727 40105 36 , 03733 40076 35 03738 40046 34 03744 40017 33 03749 39988 32 03755 39959 31 10.03760 10.39930 30 39901 29 03766 03771 39872 28 03777 39843 27 39814 26 03782 03788 39785 25 03793 39756 24 39727 23 03799 03804 39698 22 03810 39669 21 10.03815 10.39641 20 03821 39612 19 03826 39583 18 03832 39554 17 03838 39526 16 03843 39497 15 03849 39468 14 03854 39439 13 03860 39411 12 03865 39382 11 10.03871 10.39354 10 9 03877 39325 8 03882 39296 7 03888 39268 6 03893 39239 03899 39211 | 5 4 03905 39182 3 03910 39154 2 03916 39125 1 03921 39097 0 03927 39069 Co-secant Secant. M.

132

1

M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 ' 45 46 47 48 49 50 51 52 - 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 24 Degrees. Sine. 9.60931 60960 60988 61016 61045 61073 61101 61129 61158 61186 9.61214 61242 61270 61298 61326 61354 61382 61411 61438 61466 9.61494 61522 61550 61578 61606 61634 61662 61689 61717 61745 9.61773 61800 61828 61856 61883 61911 61939 61966 61994 62021 9.62049 62076 62104 62131 62159 62186 62214 62241 62268 62296 9.62323 62350 62377 62405 62432 62459 62486 62513 62541 62568 62595 < o-sine.

Co-sine. Tangent. 9.64858 9.96073 96067 648S2 96062 64926 96056 64960 96050 64994 96045 65028 96039 65062 96034 65096 96028 65130 96022 65164 9.96017 9.65197 96011 65231 96005 65265 96000 65299 95994 65333 95988 65366 95982 65400 95977 65434 95971 65467 95965 65501 9.95960 9.65535 95954 65568 95948 65602 95942 65636 95937 65669 95931 65703 95925 65736 65770 95920 95914 65803 65837 95908

Co-tang. 10.35142 35108 35074 35040 35006 34972 34938 34904 34870 34836 10.34803 34769 34735 34701 34667 34634 34600 34566 34533 34499 10.34465 34432 34398 34364 34331 34297 34264 34230 34197 34163

9.95902 9.65870 10.34130 95897 65904 34096 65937 34063 95891 65971 95885 34029 66004 33996 95879 95873 66038 33962 66071 95868 33929 66104 33896 95862 66138 33862 95856 66171 33829 95850 9.95844 9.66204 10.33796 33762 95839 66238 66271 95833 33729 95827 66304 33696 66337 33663 95821 66371 33629 95815 66404 95810 33596 66437 95804 33563 66470 33530 95798 33497 95792 66503 9.66537 10.33463 9.95786 66570 33430 95780 66603 33397 95775 66636 33364 95769 33331 66669 95763 66702 33298 95757 66735 33265 95751 66768 33232 95745 66801 33199 95739 66834 33166 95733 33133 66867 95728 Sine. Co-tang. Tangent.

P>5 Degrees.

Secant. Co-secant 10.03927 10.39069 60 03933 39040 59 03938 39012 58 . 03944 38984 57 03950 38955 56 38927 55 03955 03961 38899 54 38871 53 03966 03972 38842 52 03978 38814 51 10.03983 10.38786 50 03989 38758 49 03995 38730 48 04000 38702 47 38674 46 04006 04012 38646 45 04018 38618 44 04023 38589 43 38562 42 04029 38534 41 04035 10.04040 10.38506 40 04046 38478 39 04052 38450 38 38422 37 04058 38394 36 04063 38366 35 04069 38338 34 04075 38311 33 04080 38283 32 04086 04092 38255 31 10.04098 10.38227 30 04103 38200 29 38172 28 04109 38144 27 04115 38117 26 04121 38089 25 04127 38061 24 04132 38034 23 04138 38006 22 • 04144 37979 21 04150 10.04156 10.37951 20 37924 19 04161 37896 18 04167 37869 17 04173 37841 16 04179 37814 15 04185 37786 14 04190 37759 13 04196 37732 12 04202 37704 11 04208 10.04214 10.37677 10 9 37650 04220 8 37623 04225 7 37595 04231 6 37568 04237 37541 5 04243 37514 4 04249 37487 3 04255 37459 2 * 04261 37432 1 04267 0 37405 04272 Secant. M. Co-secant

Artificial Sines, Tang, and Sec. 25 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ,r~

Sine. 9.62595 62622 62649 62676 62703 62730 62757 62784 62811 62838 9.62865 62892 62918 62945 62972 62999 63026 63052 63079 63106 9.63133 63159 63186 63213 63239 63266 63292 63319 63345 63372 9.63398 63425 63451 63478 63504 63531 63557 63583 63610 63636 9.63662 63689 63715 63741 63767 63794 63820 63846 63872 63898 9.63924 63950 63976 64002 64028 64054 64080 64106 64132 64158 64184 Co-sine.

35

*

Co-secant Secant. Co-sine. .Tangent. Co-tang. 9.66867 10.33133 10.04272 10.37405 9.95728 37378 04278 33100 66900 95722 37351 04284 33067 66933 95716 37324 04290 33034 66966 95710 37297 04296 33001 66999 95704 37270 04302 32968 95698 67032 37243 04308 32935 95692 67065 37216 04314 95686 32902 67098 37189 04320 67131 32869 95680 37162 32837 04326 95674 67163 9.95668 9.67196 10.32804 10.04332 10.37135 37108 95663 32771 04337 67229 95657 04343 37082 67262 32738 95651 37055 67295 32705 04349 67327 95645 32673 37028 04355 95639 67360 32640 04361 37001 95633 32607 04367 67393 36974 95627 32574 04373 67426 36948 32542 04379 36921 95621 67458 67491 04385 95615 32509 36894 9.95609 9.67524 10.32476 10.04391 10.36867 95603 04397 32444 67556 36841 95597 32411 04403 36814 67589 95591 67622 32378 04409 36787 95585 67654 04415 32346 36761 67687 04421 95579 32313 36734 95573 67719 32281 04427 36708 95567 67752 32248 04433 36681 95561 67785 04439 32215 36655 67817 95555 32183 04445 36628 9.95549 9.67850 10.32150 10.04451 10.36602 95543 04457 67882 32118 36575 95537 67915 04463 32085 36549 95531 67947 32053 04469 36522 95525 67980 32020 04475 36496 95519 68012 31988 04481 36469 95513 68044 31956 04487 36443 95507 68077 31923 04493 36417 95500 68109 31891 04500 36390 95494 68142 31858 04506 56364 9.95488 9.68174 10.31826 10.04512 10.36338 95482 68206 31794 36311 04518 95476 68239 31761 04524 36285 95470 68271 31729 04530 36259 95464 68303 31697 04536 36233 95458 68336 31664 04542 36206 95452 68368 31632 04548 36180 95446 68400 31600 04554 36154 95440 68432 31568 04560 36128 95434 68465 31535 04566 36102 9.95427 9.68497 10.31503 10.04573 10.36076 95421 68529 31471 04579 36050 95415 68561 31439 04585 36024 95409 68593 31407 04591 35998 95403 68626 04597 31374 35972 95397 68658 31342 04603 35946 95391 68690 31310 04609 35920 95384 68722 31278 04616 35894 95378 68754 31246 04622 35868 95372 68786 31214 04628 35842 95366 68818 31182 04634 35816 Sine. Co-tan®-. Tangent. Co-secant Secant.

G4 Degrees.

133 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 3ft 3‘, 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 3\T.

134 M. 0 1 2 3 4 5 6 7 8 9 JO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 I 60

|

Artificial Sines, Tang, and Sec. 26 Degrees, Sine. 9.64184 64210 64236 64262 64288 64313 64339 64365 64391 64417 9.64442 64468 64494 64519 64545 64571 64596 64622 64647 64673 9.64698 64724 64749 64775 64800 64826 64851 64877 64902 64927 9.64953 64978 65003 65029 65054 65079 65104 65130 65155 65180 9.65205 65230 65255 65281 65306 65331 65356 65381 65406 65431 9.65456 65481 65506 6553 6555* 6558C 65605 65630 65655 65680 65705 Co-sine.

Co-sine. Tangent. Co-tang. Secant. Co-secant 9.95366 9.68818 10.31182 10.04634 10.35816 60 95360 68850 31150 04640 35790 59 95354 68882 31118 04646 35764 58 95348 68914 31086 04652 35738 57 95341 68946 31054 04659 35712 56 95335 68978 31022 04665 35687 55 95329 69010 30990 04671 35661 54 95323 69042 30958 04677 35635 53 | 95317 69074 30926 04683 35609 52 95310 69106 30894 04690 35583 51 9.95304 9.69138 10.30862 10.04696 10.35558 50 95298 69170 30830 04702 35532 49 95292 69202 30798 04708 35506 48 95286 69234 30766 04714 35481 47 95279 69266 30734 04721 35455 46 95273 69298 04727 30702 35429 45 95267 69329 30671 04733 35404 44 95261 69361 30639 04739 35378 43 95254 69393 30607 04746 35353 42 95248 69425 30575 04752 35327 41 9.95242 9.69457 10.30543 10.04758 10.35302 40 95236 69488 30512 04764 35276 39 04771 95229 69520 35251 38 ’ 30480 95223 69552 04777 30448 35225 37 95217 69584 35200 36 30416 04783 95211 35174 35 69615 30385 04789 69647 95204 30353 04796 35149 34 35123 33 95198 69679 30321 04802 35098 32 69710 04808 95192 30290 69742 04815 30258 35073 31 95185 9.95179 9.69774 10.30226 10.04821 10.35047 30 95173 04827 35022 29 30195 69805 95167 69837 34997 28 30163 04833 34971 27 95160 04840 69868 30132 34946 26 95154 04846 69900 30100 34921 25 04852 95148 69932 30068 34896 24 95141 30037 04859 69963 34870 23 95135 69995 30005 04865 04871 34845 22 95129 70026 29974 95122 04878 34820 21 70058 29942 9.95116 9.70089 10.29911 10.04884 10.34795 20 04890 34770 19 95110 70121 29879 04897 34745 18 95103 70152 29848 04903 34719 17 95097 70184 29816 34694 16 04910 95090 70215 29785 04916 34669 15 95084 29753 .70247 04922 34644 14 95078 29722 70278 34619 13 04929 95071 29691 70309 34594 12 04935 70341 29659 95065 34569 11 04941 29628 95059 70372 9.95052 9.70404 10.29596 10.04948 10.34544 10 34519 9 04954 29565 95046 70435 8 34494 04961 29534 95039 70466 7 34469 04967 29502 95033 70498 6 34444 04973 95027 29471 70529 5 34420 04980 95020 29440 70560 4 34395 04986 29408 95014 70592 3 34370 04993 29377 95007 70623 34345 2 04999 29346 95001 70654 34320 1 05005 29315 94995 70685 0 34295 05012 29283 70717 94988 Secant. M. Sine. Co-tans:. 'Tangent. Co-secant

63 Degrees.

Artificial Sines, Tang, and Sec. 27 Degrees. iM. 0 ! 1 ! 2 ! 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

j

Sine. 9-65705 65729 657.54 65779 65804 65828 65853 65878 65902 65927 9.65952 65976 C6001 66025 66050 66075 66099 66124 66148 66173 9.66197 66221 66246 66270 66295 66319 66343 66368 66392 66416 9.66441 66465 66489 66513 66537 66562 66586 66610 66634 66658 9.66682 66706 66731 66755 66779 66803 66827 66851 66875 66899 9.66922 66946 66970 66994 67018 •67042 67066 67090 67113 67137 67161 Co-sine.

Co-sine. 9.94988 94982 94975 94969 94962 94956 94949 94943 94936 94930 9.94923 94917 94911 94904 94898 94891 94885 94878 94871 94865 9.94858 94852 94845 94839 94832 94826 94819 94813 94806 94799 9.94793 94786 94780 94773 94767 94760 94753 94747 94740 94734 9.94727 94720 94714 94707 94700 * 94694 94687 94680 94674 94667 9.94660 94654 94647 94640 94634 94627 94620 94614 94607 94600 94593 Sine.

Tangent. Co-tang. | 9.70717 10.29283 70748 29252 70779 29221 70810 29190 70841 29159 29127 70873 29096 70904 29065 70935 70966 29034 70997 29003 9.71028 10.28972 28941 71059 71090 28910 71121 28879 71153 28847 71184 28816 71215 28785 71246 28754 71277 28723 71308 28692 9.71339 10.28661 71370 28630 71401 28599 71431 28569 71462 28538 28507 71493 71524 28476 71555 28445 71586 28414 71617 28383 9.71648 10.28352 71679 28321 .71709 28291 71740 28260 71771 28229 71802 28198 71833 28167 71863 28137 71894 28106 71925 28075 9.71955 10.28045 71986 28014 72017 27983 72048 27952 72078 27922 72109 27891 72140 • 27860 72170 27830 72201 27799 72231 27769 9.72262 10 27738 27707 72293 72323 27677 72354 27646 72384 27616 72415 27585 72445 27555 72476 27524 72506 27494 72537 27463 72567 27433 Co-tang. Tangent.

62 Degrees.

Secant. Co-secant 10.05012 10.34295 34271 05018 34246 05025 34221 05031 34196 05038 34172 05044 34147 05051 34122 05057 34098 05064 34073 05070 10.05077 10.34048 34024 05083 33999 05089 05096 33975 05102 33950 33925 05109 33901 05115 05122 33876 05129 33852 33827 05135 10.05142 10.33803 05148 33779 33754 05155 05161 33730 33705 05168 33681 05174 33657 05181 33632 05187 23608 05194 33584 05201 10.05207 10.33559 33535 05214 33511 05220 33487 05227 33463 05233 33438 05240 05247 33414 33390 05253 33366 05260 33342 05266 10.05273 10.33318 33294 05280 33269 05286 05293 33245 33221 05300 33197 05306 33173 05313 33149 05320 33125 05326 05333 33101 10.05340 10.33078 05346 33054 05353 33030 05360 33006 05366 32982 05373 32958 05380 32934 05386 32910 32887 05393 32863 05400 05407 32839 Secant. Co-secant

135 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 M.

136 Artificial Sines, Tang, and Sec. 28 Degrees M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 , 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.67161 67185 67208 67232 67256 67280 67303 67327 67350 67374 9.67398 67421 67445 67468 67492 67515 67539 67562 67586 67609 9.67633 67656 67680 67703 67726 67750 67773 67796 67820 67843 9.67866 67890 67913 67936 67959 67982 68006 68029 68052 68075 9.68098 68121 68144 68167 68190 68213 68237 68260 68283 68305 9.68328 68351 68374 68397 68420 68443 68466 68489 68512 68534 68557 Co-sine.

Co-sine. Tangent. Co-tang. Secant. J Co-secant 9.94593 9.72567 10.27433 10.05407 i 10.32839 60 94587 72598 27402 05413 32815 59 94580 72628 27372 05420 32792 58 94573 27341 05427 72659 32768 57 94567 05433 1| 32744 56 27311 72689 94560 27280 05440 32720 55 72720 94553 05447 27250 32697 54 72750 94546 27220 05454 72780 32673 53 94540 72811 27189 05460 32650 52 05467 94533 72841 27159 32626 51 9.94526 9.72872 10.27128 10.05474 f 10.32602 50 05481 94519 72902 27098 32579 49 94513 27068 05487 72932 32555 48 27037 94506 05494 72963 32532 47 27007 94499 05501 72993 32508 46 94492 ' 26977 05508 73023 32485 45 26946 05515 32461 44 94485 73054 05521 94479 73084 26916 32438 43 26886 05528 32414 42 94472 73114 05535 26856 32391 41 94465 73144 9.94458 9.73175 10.26825 10.05542 10.32367 40 05549 32344 39 94451 26795 73205 26765 05555 32320 38 94445 73235 05562 32297 37 26735 94438 73265 32274 36 05569 94431 73295 26705 73326 26674 05576 32250 35 94424 32227 34 05583 94417 73356 26644 32204 33 05590 73386 26614 94410 05596 32180 32 26584 94404 73416 32157 31 Q5603 26554 94397 73446 9.73476 10.26524 10.05610 10.32134 30 9.94390 32110 29 05617 73507 26493 94383 32087 28 05624 73537 26463 94376 32064 27 05631 73567 26433 94369 32041 26 05638 26403 73597 94362 32018 *25 05645 26373 73627 94355 31994 24 05651 26343 73657 94349 31971 23 05658 26313 73687 94342 31948 22 05665 26283 73717 94335 05672 31925 21 26253 73747 94328 9.94321 9.73777 10.26223 10.05679 10.31902 20 31879 19 05686 26193 73807 94314 31856 18 26163 ’ 05693 94307 73837 31833 17 05700 26133 73867 94300 31810 16 05707 26103 94293 73897 31787 15 ■057*4 26073 73927 94286 31763 14 05721 26043 73957 94279 31740 13 05727 26013 94273 73987 31717 12 05734 25983 74017 94266 05741 1i 31695 11 25953 74047 94259 9.74077 10.25923 10.05748 10.31672 10 9.94252 9 31649 05755 25893 74107 94245 8 31626 05762 25863 74137 94238 7 31603 05769 25834 94231 74166 6 31580 05776 25804 94224 74196 31557 5 05783 25774 94217 74226 4 31534 05790 25744 94210 74256 3 31511 05797 25714 94203 74286 31488 2 05804 25684 94196 74316 31466 1 05811 25655 94189 74345 31443 0 05818 25625 94182 74375 M. Co-tang. Tangent. Co-secant / Secant. Sine.

61 Degrees.

Artificial Sines, Tang, and Sec. 29 Degrees. 137 M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 ‘39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sme. 9.68557 68580 68603 68625 68648 68671 68694 68716 68739 68762 9.68784 68807 68829 68852 68875 68897 68920 68942 68965 68987 9.69010 69032 69055 69077 69100 69122 69144 69167 69189 69212 9.69234 • 69256 69279 69301 69323 69345 69368 69390 69412 69434 9.69456 . 69479 69501 69523 69545 69567 69589 69611 69633 69655 9.69677 69699 69721 69743 69765 69787 69809 69831 69853 69875 69897 Co-sine.

Co-sine. Tangent. 9.74375 9.94182 74405 94175 74435 94168 74465 94161 94154 74494 74524 94147 74554 94140 74583 94133 74613 94126 74643 94119 9.94112 9.74673 94105 74702 94098 74732 94090 74762 94083 74791 94076 74821 94069 74851 94062 74880 94055 74910 * 94048 74939 9.94041 9.74969 94034 74998 94027 75028 94020 75058 94012 75087 94005 75117 93998 75146 93991 75176 93984 75205 93977 75235 9.75264 9.93970 93963 75294 93955 75323 93948 75353 93941 75382 93934 75411 93927 75441 93920 75470 93912 75500 93905 75529 9.93898 9.75558 93391 75588 93884 75617 93876 75647 93869 75676 - 93862 75705 93855 75735 93847 75764 93840 75793 93833 75822 9.93826 9.75852 93819 75881 93811 75910 93804 75939 93797 75969 93789 75998 93782 76027 93775 76056 93768 76086 93760 76115 93753 76144 Sine. Co-tang. 1

Co-tang. 10.25625 25595 25565 25535 25506 25476 25446 25417 25387 25357 10.25327 25298 25268 25238 25209 25179 25149 25120 25090 25061 10.25031 25002 24972 24942 24913 24883 24854 24824 24795 24765 10.24736 24706 24677 24647 24618 24589 24559 24530 24500 24471 10.24442 24412 24383 24353 24324 24295 24265 24236 24207 24178 10.24148 24119 24090 24061 24031 24002 23973 23944 23914 23885 23856 Tangent.

60 Degrees,

Secant. Co-secant 10.05818 10.31443 60 05825 31420 59 05832 31397 58 05839 31375 57 31352 56 05846 05853 31329 55 05860 31306 54 05867 31284 53 05874 31261 52 05881 31238 51 10.05888 10.31216 50 05895 31193 49 05902 31171 48 05910 31148 47 . 05917 31125 46 05924 31103 45 05931 31080 44 05938 31058 43 05945 31035 42 05952 31013 41 10.05959 10.30990 40 05966 30968 39 05973 30945 38 30923 37 05980 05988 30900 36 05995 30878 35 06002 30856 34 30833 33 06009 30811 32 06016 06023 30788 31 10.06030 10.30766 30 06037 30744 29 06045 30721 28 06052' 30699 27 06059 30677 26 06066 30655 25 06073 30632 24 06080 30610 23 06088 30588 22 06095 30566 21 10.06102 10.30544 20 06109 30521 19 06116 30499 18 06124 30477 17 06131 30455 16 06138 30433 15 30411 14 06145 06153 30389 13 06160 30367 12 06167 30345 11 10.06174 10.30323 10 9 06181 30301 06189 30279 8 7 06196 30257 6 06203 30235 06211 5 30213 4 06218 30191 3 06225 30169 2 06232 30147 1 06240 30125 0 06247 • 30103 M. Co-secant Secant

8G

138

Artificial Sines, Tang, and Sec. 30 Degreeo.

J Al. bine. 0 9.69397 1 69919 2 69941 3 69963 4 69984 5 70006 6 70028 7 70050 8 70072 9 70093 10 9.70115 70137 11 12 70159 13 70180 14 70202 15 70224 16 70245 17 70267 18 70288 19 70310 20 9.70332 21 70353 22 70375 23 70396 24 70418 25 70439 26 70461 27 70482 28 70504 29 70525 30 9.70547 31 70568 32 70590 33 70611 34 70633 35 70654 36 70675 37 70697 38 70718 39 70739 40 9.70761 41 70782 42 70803 43 70824 44 70846 45 70867 46 70888 47 70909 48 70931 49 70952 50 9.70973 51 * 70994 52 71015 53 71036 1 54 71058 55 71079 56 71100 57 71121 58 71142 59 71163 60 71184 Co-sine.

Co-sine. Tangent. Co-tang. fcecant. 9.93753 9.76144 10.23856 10.06247 93746 76173 23827 06254 93738 76202 23798 06262 93731 76231 23769 06269 93724 76261 23739 06276 93717 76290 23710 06283 93709 76319 23681 06291 93702 76348 23652 06298 93695 76377 23623 06305 93687 76406 23594 06313 9.93680 9.76435 10.23565 10.06320, 93673 76464 23536 06327 93665 76403 23507 06335 93658 76522 23478 06342 93650 76551 23449 06350 93643 76580 23420 06357 93636 76609 23391 06364 93628 76639 23361 06372 93621 76668 23332 06379 93614 76697 23303 06386 9.93606 9.76725 10.23275 10.06394 93599 76754 23246 06401 93591 76783 23217 06409 93584 76812 23188 06416 93577 76841 23159 06423 93569 76870 06431 23130 93562 76899 23101 06438 93554 76928 23072 06446 93547 76957 23043 06453 93539 76986 23014 06461 9.93532 9.77015 10.22985 10.06468 93525 77044 22956 06475 93517 77073 22927 06483 93510 77101 22899 06490 93502 22870 06498 77130 93495 22841 77159 06505 93487 06513 77188 22812 93480 77217 22783 06520 93472 22754 77246 06528 93465 22726 06535 77274 9.93457 9.77303 10.22697 10.06543 06550 93450 77332 22668 93442 22639 06558 77361 93435 77390 06565 22610 06573 93427 22582 77418 93420 77447 06580 22553 93412 22524 06588 77476 93405 22495 06595 77505 93397 22467 06603 77533 22438 06610 93390 77562 9.93382 9.77591 10.22409 10.06618 06625 93375 22381 77619 06633 93367 22352 77648 22323 06640 93360 77677 22294 06648 93352 77706 06656 93344 77734 22266 06663 93337 22237 77763 06671 93329 77791 22209 . 06678 22180 93322 77820 06686 93314 22151 77849 06693 93307 77877 22123 Sine. Co-tang. Tangent. Co-secant

59 Degrees

Cc -secant |I 11 301031 30081 11 30059 1 30037 30016 29994 29972 29950 29928 29907 U 29885 29863 29841 29820 29798 29776 29755 29733 29712 29690 V 29668 29647 29625 29604 29582 29561 29539 29518 29496 29475 1C 29453 29432 29410 29389 29367 29346 29325 29303 29282 29261 10.29239 29218 29197 29176 29154 29133 29112 29091 29069 29048 10.29027 29006 28985 28964 28942 28921 28900 28879 28858 28837 28816 Secant.

60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 .36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 i 4 I 3 ‘ 2 1 1 0 M

Artificial Sines, Tang, and Sec. 31 Degrees. Al. 0 1 2 3 4 5 6 . 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 . 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.71184 71205 71226 71247 71268 71289 71310 71331 71352 71373 9.71393 71414 71435 71456 71477 71498 71519 71539 71560 71581 9.71602 71622 71643 71664 71685 71705 71726 71747 71767 71788 9.71809 71829 71850 71870 71891 71911 71932 71952 71973 71994 9.72014 72034 72055 72075 72096 72116 72137 72157 72177 72198 9.72218 72238 72259 72279 72299 72320 72340 72360 72381 72401 72421 Co-sine.

139

Co-sine. Tangent. Co-tang. . Secant. Co-secant 9.77877 10.22123 10.06693 10.28816 60 9.93307 28795 59 06701 22094 77906 93299 28774 58 06709 22065 93291 77935 28753 57 06716 22037 93284 77963 28732 56 06724 22008 77992 93276 28711 55 06731 21980 78020 93269 28690 54 06739 21951 93261 78049 28669 53 06747 21923 78077 93253 28648 52 06754 21894 93246 78106 28627 51 21865 " 06762 93238 78135 9.78163 10.21837 10.06770 10.28607 50 9.93230 28586 49 21808 06777 93223 78192 28565 48 21780 93215 78220 06785 28544 47 93207 21751 78249 06793 21723 28523 46 93200 78277 06800 21694 93192 78306 06808 28502 45 93184 21666 78334 28481 44 06816 21637 93177 06823 28461 43 78363 21609 93169 06831 28440 42 78391 93161 21581 78419 06839 28419 41 9.93154 9.78448 10.21552 10.06846 10.28398 40 21524 93146 78476 06854 28378 39 93138 21495 28357 38 78505 06862 21467 93131 78533 28336 37 06869 93123 21438 78562 06877 28315 36 93115 21410 78590 06885 28295 35 93108 21382 78618 06892 28274 34 93100 21353 78647 06900 28253 33 93092 21325 06908 28233 32 78675 93084 21296 78704 06916 28212 31 9.93077 9.78732 10.21268 1C.06923 10.28191 30 93069 21240 78760 06931 28171 29 93061 21211 78789 06939 28150 28 93053 78817 21183 06947 28130 27 93046 21155 78845 06954 28109 26 93038 21126 78874 28089 25 06962 93030 21098 78902 28068 24 06970 93022 21070 78930 06978 28048 23 93014 21041 78959 06986 28027 22 93007 78987 21013 06993 28006 21 9.92999 9.79015 10.20985 10.07001 10.27986 20 92991 79043 20957 07009 27966 19 92983 79072 20928 07017 27945 18 92976 79100 20900 07024 27925 17 92968 79128 20872 07032 27904 16 92960 79156 20844 07040 27884 15 92952 79185 20815 07048 27863 14 92944 79213 20787 07056 27843 13 92936 79241 20759 07064 27823 12 92929 79269 20731 07071 27802 11 9.92921 9.79297 10.20703 10.07079 10.27782 10 92913 79326 20674 07087 9 27762 92905 79354 20646 07095 8 27741 92897 79382 20618 07103 7 27721 92889 79410 20590 07111 6 27701 92881 79438 20562 07119 5 27680 92874 79466 20534 07126 4 27660 92866 i 79495 20505 07134 3 27640 92858 79523 20477 07142 2 27619 92850 . 79551 20449 07150 1 27599 92842 79579 20421 07158 0 27579 Sine. Co-tang. Tangent. Co secant Secant. M.

58 Degrees.

140 M. O' 1 , 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ' 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 32 Degrees, Sine. 9.72421 72441 72461 72482 72502 72522 72542 72562 72582 72602 9.72622 72643 72663 72683 72703 72723 72743 72763 72783 72803 9.72823 72843 72863 72883 72902 ‘ 72922 72942 72962 72982 73002 9.73022 73041 73061 73081 73101 73121 73140 73160 73180 73200 9.73219 73239 73259 73278 73298 73318 73337 73357 73377 73396 9.73416 73435 73455 73474 73494 73513 73533 73552 73572 73591 73611 Co-sine.

Co-sine. Tangent. Co-tang. Secant. Co-secant 9.92842 9.79579 10.20421 10.07158 10.27579 92834 79607 20393 07166 27559 92826 79635 20365 07174 27539 92818 79663 20337 07182 27518 92810 79691 20309 07190 27498 92803 79719 20281 07197 27478 92795 79747 20253 07205 27458 92787 79776 20224 07213 27438 92779 79804 07221 27418 : 20196 92771 79832 20168 07229 27398 9.92763 9.79860 10.20140 10.07237 10.27378 92755 79888 20112 07245 27357 92747 79916 20084 07253 27337 92739 79944 20056 07261 27317 92731 79972 20028 07269 27297 92723 80000 20000 07277 27277 92715 80028 19972 07285 27257 92707 80056 19944 07293 27237 92699 80084 19916 07301 27217 92691 80112 19888 07309 27197 9.92683 9.80140 10.19860 10.07317 10 27177 92675 80168 19832 27157 07325 92667 80195 19805 27137 07333 92659 19777 80223 07341 27117 92651 80251 19749 07349 27098 92643 19721 80279 07357 27078 92635 80307 19693 07365 27058 92627 80335 19665 07373 27038 92619 80363 19637 27018 07381 92611 80391 19609 26998 07389 9.92603 9.80419 10.19581 10.07397 1C.26978 92595 19553 80447 07405 26959 92587 07413 26939 80474 19526 92579 07421 26919 80502 19498 92571 26899 19470 07429 80530 92563 07437 26879 19442 80558 92555 19414 26860 07445 80586 26840 92546 07454 19386 80614 26820 19358 07462 92538 80642 26800 92530 19331 07470 80669 9.80697 10.19303 10.07478 10.26781 9.92522 26761 07486 92514 19275 80725 26741 19247 07494 92506 80753 26722 07502 92498 19219 80781 26702 19192 07510 92490 80808 26682 07518 19164 92482 80836 26663 07527 92473 19136 80864 26643 07535 19108 92465 80892 26623 07543 92457 19081 80919 26604 07551 80947 19053 92449 9.92441 9.80975 10.19025 10.07559 10.26584 26565 07567 92433 18997 81003 26545 07575 18970 92425 8103G 26526 07584 18942 92416 81058 26506 07592 18914 92408 81086 26487 07600 18887 92400 81113 26467 07608 i1 92392 18859 81141 26448 07616 18831 j| 92384 81169 26428 07624 ;| 52376 18804 81196 26409 07633 18776 92367 81224 26389 07641 18748 92359 81252 Sine. Co-tang. Tangent. Co-secant Secant. |

57 Degrees.

60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 MT

Artificial Sines, Tang, and Sec. 33 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 . 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 , 55 56 57 58 59 60

Sine. 9.73611 73630 73650 73669 73689 73708 73727 73747 73766 73785 9.73805 73824 73843 73863 73882 73901 73921 73940 73959 73978 9.73997 74017 74036 74055 74074 74093 74113 74132 74151 74170 9.74189 74208 74227 74246 74265 74284 74303 74322 74341 74360 9.74379 74398 74417 74436 74455 74474 74493 74512 74531 74549 9.74568 74587 74606 74625 74644 74662 74681 74700 74719 74737 74756 Co-sine. ft

Co-secant Secant. Tangent. Co-tang. Co-sine. 9.81252 10.18748 10.07641 10.26389 9.92359 26370 07649 18721 81279 92351 26350 07657 18693 81307 92343 26331 07665 18665 81335 92335 26311 07674 18638 81362 92326 26292 07682 18610 81390 92318 26273 18582 07690 81418 92310 26253 07698 18555 81445 92302 26234 07707 18527 81473 92293 26215 18500 07715 81500 92285 9.81528 10.18472 10.07723 10.26195 9.92277 18444 26176 81556 07731 92269 81583 18417 26157 92260 07740 92252 81611 18389 26137 07748 92244 81638 18362 26118 07756 81666 92235 18334 07765 26099 92227 81693 18307 07773 26079 81721 92219 18279 07781 26060 81748 92211 18252 26041 07789 92202 81776 18224 07798 26022 9.92194 9.81803 10.18197 10.07806 10.26003 92186 81831 18169 25983 07814 92177 81858 18142 07823 25964 92169 81886 18114 07831 25945 92161 81913 18087 07839 25926 92152 81941 18059 07848 25907 92144 81968 18032 25887 07856 92136 81996 18004 07864 25868 92127 82023 17977 07873 25849 92119 82051 17949 07881 25830 9.92111 9.82078 10.17922 10.07889 10.25811 92102 82106 17894 07898 25792 92094 82133 17867 07906 25773 92086 82161 17839 07914 25754 92077 82188 17812 07923 25735 92069 82215 17785 07931 25716 92060 82243 17757 07940 25697 92052 82270 17730 07948 25678 92044 82298 17702 07956 25659 92035 82325 17675 07965 25640 9.92027 9.82352 10.17648 10.07973 10.25621 92018 82380 17620 07982 25602 92010 82407 17593 07990 25583 92002 82435 17565 07998 25564 91993 82462 17538 08007 25545 91985 82489 17511 08015 25526 91976 82517 17483 08024 25507 91968 82544 17456 08032 25488 91959 82571 17429 08041 25469 91951 82599 17401 08049 25451 9.91942 9.82626 10.17374 10.08058 10.25432 91934 82653 17347 08066 25413 91925 82681 17319 08075 25394 91917 82708 17292 08083 25375 91908 82735 17265 08092 25356 91900 82762 17238 08100 25338 91891 82790 17210 08109 25319 91883 82817 17183 08117 25300 91874 82844 17156 08126 25281 91866 82871 17129 08134 25263 91857 82899 17101 08143 25244 Sine. Co-tang. Tangent. Co-secant Secant.

56 Degrees.

141 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 M. |

142 M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ‘ 24 25 26 27 28 29 30 31 32 33 i 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 34 Degrees, Sine. 9.74756 74775 74794 74812 74831 74850 74868 74887 74906 74924 9.74943 74961 74980 74999 75017 75036 75054 75073 75091 75110 9.75128 75147 75165 75184 75202 75221 75239 75258 75276 75294 9.75313 75331 75350 75368 75386 75405 75423 75441 75459 75478 9.75496 75514 75533 75551 75569 75587 75605 75624 75642 75660 9.75678 75696 75714 75733 75751 75769 75787 75805 75823 75841 75859 Co-sine.

Co-sine. Tangent. 9.91857 9.82899 91849 82926 82953 91840 91832 82980 83008 91823 83035 91815 83062 91806 83089 91798 83117 91789 83144 91781 9.83171 9.91772 91763 83198 83225 91755 83252 91746 83280 91738 83307 91729 83334 91720 83361 91712 83388 91703 83415 91695 9.83442 9.91686 83470 91677 83497 91669 83524 91660 83551 91651 83578 91643 91634 83605 83632 91625 83659 91617 83686 91608 9.83713 9.91599 91591 83740 83768 91582 83795 91573 83822 91565 83849 91556 91547 83876 83903 91538 83930 91530 83957 91521 9.83984 9.91512 84011 91504 84038 91495 84065 91486 91477 84092 84119 91469 84146 91460 84173 91451 84200 91442. 84227 91433 9.84254 9.91425 84230 91416 84307 91407 84334 91398 84361 91389 84388 91381 84415 91372 84442 91363 84469 91354 84496 91345 84523 91336 Co-tang. Sine.

Co-tang. 10.17101 17074 17047 17020 16992 16965 16938 16911 16883 16856 10.16829 16802 16775 16748 16720 16693 16666 16639 16612 16585 10.16558 16530 16503 16476 16449 16422 16395 16368 16341 16314 10.16287 16260 16232 16205 16178 16151 16124 16097 16070 16043 10.16016 15989 15962 15935 15908 15881 15854 15827 15800 15773 10.15746 15720 15693 15666 15639 15612 15585 15558 15531 15504 15477 ' ; ngent. 1

55 Decrees

Secant. Co-secant 10.08143 10.25244 60 08151 25225 59 08160 25206 58 08168 25188 57 08177 25169 56 08185 25150 55 08194 25132 54 08202 25113 53 08211 25094 52 08219 25076 51 10.08228 10.25057 50 08237 25039 49 08245 25020 48 08254 25001 47 08262 24983 46 08271 24964 45 24946 44 08280 08288 24927 43 08297 24909 42 08305 24890 41 10.08314 10.24872 40 24853 39 08323 08331 24835 38 24816 37 08340 24798 36 08349 08357 24779 35 24761 34 08366 24742 33 08375 24724 32 08383 24706 31 08392 10.08401 10.24687 30 24669 29 08409 24650 28 08418 24632 27 08427 24614 26 : 08435 24595 25 08444 24577 24 z 08453 24559 23 08462 24541 22 08470 24522 21 08479 10.08488 10.24504 20 24486 19 08496 24467 18 08505 24449 17 08514 24431 16 08523 24413 15 08531 24395 14 08540 24376 13 08549 24358 12 08558 24340 11 08567 10.08575 10.24322 10 9 24304 08584 8 24286 08593 7 24267 08602 6 24249 08611 5 24231 08619 4 24213 08628 3 24195 08637 2 24177 08646 1 24159 08655 0 24141 08664 !V1. , Secant. Co-secant

Artificial Sines, Tang, and Sec. 35 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 . 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 ; 60

Sme. 9.75859 75377 75895 75913 75931 75949 75967 * 75985 76003 76021 9.76039 76057 76075 76093 76111 76129 76146 76164 76182 76200 9.76218 76236 76253 76271 76289 76307 76324 76342 76360 76378 9.76395 76413 76431 76448 76466 76484 76501 >6519 76537 76554 9.76572 76590 76607 76625 76642 76660 76677 76695 76712 76730 9.76747 70765 76782 76800 76817 76835 76852 76870 76887 76904 76922 Co-sine.

143

Secant. Co-secant Co-sine. Tangent. Co-tang. 9.91336 9.34523 10.15477 10.08664 10.24141 60 08672 15450 91328 84550 24123 59 15424 08681 84576 24105 58 91319 15397 24087 57 84603 08690 91310 91301 84630 15370 08699 24069 56 91292 84657 15343 08708 24051 55 91283 84684 15316 08717 24033 54 91274 84711 15289 08726 24015 53 84738 15262 91263 08734 23997 52 91257 84764 15236 08743 23979 51 9.91248 9.84791 10.15209 10.08752 10.23961 50 91239 84818 15182 08761 23943 49 91230 84845 15155 08770 23925 48 91221 84872 15128 08779 23907 47 91212 84899 15101 08788 23889 46 91203 84925 15075 08797 23871 45 91194 84952 15048 08806 23854 44 91185 84979 15021 08815 23836 43 9ir6 85006 14994 08824 23818 42 91167 85033 14967 08833 23800 41 9.91158 9.85059 10.14941 10.08842 10.23782 40 91149 85086 14914 08851 23764 39 91141 85113 14887 08859 23747 38 91132 85140 14860 08868 23729 37 91123 85166 14834 08877 23711 36 91114 85193 14807 08886 23693 35 91105 85220 14780 08895 23676 34 91096 85247 14753 08904 23658 33 91087 85273 14727 08913 23640 32 91078 85300 14700 08922 23622 31 9.91069 9.85327 10.14673 10.08931 10.23605 30 91060 85354 14646 08940 23587 29 91051 85380 14620 08949 23569 28 91042 85407 14593 08958 23552 27 91033 85434 14566 08967 23534 26 91023 85460 14540 08977 23516 25 91014 85487 14513 08986 23499 24 91005 85514 14486 08995 23481 23 90996 85540 14460 09004 23463 22 90987 85567 14433 09013 23446 21 9.90978 9.85594 10.14406 10.09022 10.23428 20 90969 85620 14380 09031 23410 19 90960 85647 14353 09040 23393 18 90951 85674 11326 09049 23375 17 90942 85700 14300 09058 23358 16 90933 85727 14273 09067 23340 15 90924 85754 14246 09076 23323 14 90915 85780 14220 09085 23305 13 90906 85807 14193 09094 23288 12 90896 85834 14166 09104 23270 11 9.90887 9.85860 10.14140 10.09113 10.23253 10 90878 85887 14113 09122 9 23235 90869 85913 14087 09131 8 23218 90860 85940 14060 09140 7 23200 90851 85967 14033 09149 6 23183 90842 85993 14007 5 09158 23165 90832 86020 13980 4 09168 23148 90823 86046 13954 3 09177 23130 90814 86073 13927 09186 23113 2 90805 86100 13900 1 09195 23096 90796 86126 13874 09204 23078 , 0 Sine. Co-tan"-. Tangent. Co-secant Secant. | ~w\

54 Degree*

144 i\I. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 , 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 36 Degrees. Sme. 9.76922 76939 76957 76974 76991 77009 77026 77043 77061 77078 9.77095 77112 77130 77147 77164 77181 77199 77216 77233 77250 9.77268 77285 77302 77319 77336 77353 77370 77387 77405 77422 9.77439 77456 77473 77490 77507 77524 77541 77558 77575 77592 9.77609 77626 77643 77660 77677 77694 77711 77728 77744 77761 9.77778 77795 77812 77829 77846 77862 77879 77896 77913 77930 77946 Co-sine.

Co-sine. 9.90796 90787 90777 90768 90759 90750 90741 90731 90722 90713 9.90704 90694 90685 90676 90667 90657 90648 90639 90630 90620 9.90611 90602 90592 90583 90574 90565 90555 90546 90537 90527 9.90518 90509 90499 90490 90480 90471 90462 90452 90443 90434 9.90424 90415 90405 90396 90386 90377 90368 90358 90349 90339 9.90330 90320 90311 90301 90292 90282 90273 90263 90254 90244 90235 Sine.

Tangent. Co-tang. becant. Co-secant 9.86126 10.13374 10.09204 10.23078 60 86153 13847 09213 23061 59 86179 13821 09223 23043 58 86206 13794 09232 23026 57 86232 13768 09241 23009 56 86259 13741 09250 22991 65 86285 13715 09259 22974 54 86312 13688 09269 ' 22957 53 86338 13662 09278 22939 52 86365 13635 09287 22922 51 9.8(5392 10.13608 10.09296 10.22905 50 86418 13582 09306 22888 49 86445 13555 09315 22870 48 86471 13529 09324 22853 47 86498 13502 09333 22836 46 86524 13476 09343 22819 45 86551 13449 22801 44 09352 86577 13423 09361 22784 43 86603 13397 22767 42 09370 86630 13370 09380 22750 41 9.86656 10.13344 10.09389 10.22732 40 86683 13317 09398 22715 39 86709 13291 09408 22698 38 86736 13264 09417 22681 37 i 86762 13238 22664 36 09426 13211 22647 35 86789 09435 86815 22630 34 13185 09445 86842 13158 09454 22613 33 13132 86868 09463 22595 32 86894 . 13106 22578 31 09473 9.86921 10.13079 10.09482 10.22561 30 86947 13053 09491 22544 29 13026 22527 28 86974 09501 13000 87000 09510 22510 27 87027 12973 22493 26 09520 12947 22476 25 87053 09529 22459 24 12921 09538 87079 12894 22442 23 09548 87106 09557 12868 22425 22 87132 09566 . 22408 21 12842 87158 9.87185 10.12815 10.09576 10.22391 20 22374 19 09585 87211 12789 22357 18 12762 09595 87238 22340 17 09604 12736 87264 22323 16 09614 12710 87290 22306 15 12683 09623 87317 22289 14 09632 12657 87343 22272 13 12631 09642 87369 22256 12 09651 12604 87396 22239 11 09661 12578 87422 9.87448 10.12552 10.09670 10.22222 10 9 22205 09680 12525 87475 8 22188 09689 12499 87501 7 22171 09699 12473 87527 6 22154 09708 12446 87554 5 22138 09718 12420 87580 4 ■ 22121 09727 12394 87606 3 09737 22104 12367 87633 2 22087 09746 12341 87659 1 22070 09756 12315 87685 0 22054 09765 12289 87711 Secant. M. Co-tang. Tangent. Co-secant

63 Degrees.

Artificial Sines, Tang, and Sec. 37 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 . 25 26 27 28 29 30 31 32 33 34 35 36 1 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

145

Co-secant Secant. Co-tang. Sine. Co-sme. Tangent. 9.87711 10.12289 10.09765 10.22054 60 9.77946 9.90235 22037 59 09775 12262 87738 77963 90225 22020 58 09784 12236 87764 77980 90216 22003 57 09794 77997 90206 87790 12210 21987 56 09803 12183 90197 87817 78013 21970 55 09813 90187 12157 87843 '’’8030 21953 54 78047 09822 12131 90178 87869 21937 53 78063 09832 90168 12105 87895 21920 52 78080 09841 90159 87922 12078 21903 51 78097 90149 09851 87948 12052 9.90139 9.78113 9.87974 10.12026 10.09861 10.21887 50 21870 49 78130 90130 88000 12000 09870 78147 21853 48 88027 11973 90120 09880 78163 90111 21837 47 88053 11947 09889 90101 78180 88079 . 21820 46 11921 09899 78197 90091 88105 21803 45 11895 09909 78213 90082 88131 21787 44 09918 11869 78230 90072 88158 11842 09928 21770 43 78246 90063 88184 09937 21754 42 11816 78263 90053 21737 41 88210 09947 11790 9.78280 9.90043 9.88236 10.11764 10.09957 10.21720 40 78296 90034 88262 21704 39 11738 09966 78313 90024 21687 38 88289 11711 09976 78329 90014 88315 ' 21671 37 11685 09986 78346 90005 88341 21654 36 11659 09995 78362 89995 88367 21638 35 11633 10005 78379 89985 88393 11607 21621 34 10015 78395 89976 88420 10024 11580 21605 33 78412 89966 88446 11554 10034 21588 32 78428 89956 88472 11528 10044 21572 31 9.78445 9.89947 9.88498 10.11502 10.10053 10.21555 30 78461 89937 88524 11476 10063 21539 29 89927 78478 88550 11450 10073 21522 28 78494 89918 88577 11423 10082 21506 27 78510 89908 88603 11397 10092 21490 26 78527 89898 88629 11371 10102 21473 25 78543 89888 88655 11345 10112 21457 24 78560 89879 88681 11319 10121 21440 23 78576 89869 88707 11293 10131 21424 22 78592 89859 88733 11267 10141 21408 21 9.78609 9.89849 9.88759 10.11241 10.10151 10.21391 20 78625 89840 88786 11214 10160 21375 19’ 78642 89830 88812 11188 10170 21358 18 78658 89820 88838 11162 10180 21342 17 78674 89810 88864 11136 21326 16 10190 78691 89801 88890 11110 10199 21309 15 78707 89791 88916 11084 10209 21293 14 78723 89781 88942 11058 10219 21277 13 78739 89771 88968 11032 10229 21261 12 78756 89761 88994 11006 10239 21244 11 9.78772 9.89752 9.89020 10.10980 10.10248 10.21228 10 78788 89742 89046 10954 10258 9 21212 78805 89732 89073 10927 10268 8 21195 78821 89722 89099 10901 10278 7 21179 78837 89712 89125 10875 10288 6 21163 78853 i 89702 89151 10849 10298 21147 5 78869 89693 89177 10823 10307 4 21131 78886 89683 89203 10797 10317 3 21114 78902 89673 89229 10771 10327 2 21098 78918 89663 89255 10745 10337 1 21082 78934 89653 89281 10719 10347 0 21066 Co-sine. Sine. Co-tang. Tangent. Co-secant Secant. M. *>/* *

52 Degrees

3 11

146 i\l.

0 1

2 3 4 5 G 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Artificial Sines, Tang, and Sec. 33 Degrees, Sine. 9.78934 78950 78967 78983 78999 79015 79031 79047 79063 79079 9.79095 79111 79123 79144 79160 79176 79192 79208 79224 79240 9.79256 79272 79288 79304 79319 79335 79351 79367 79383 79399 9.79415 79431 79447 79463 79478 79494 79510 79526 79542 79558 9.79573 79589 79605 79621 79636 79652 79668 79684 79699 79715 9.79731 79746 79762 79778 79793 79809 79825 79840 79856 79872 79887 Co-siite.

Tangent. (Jo-sme. 9.89653 9.89281 89643 89307 89633 89333 89624 89359 89614 89385 89604 89411 89594 89457 89584 89463 89574 89489 89564 89515 9.89554 9.89541 89544 89567 89534 89593 89524 89619 89514 89645 89504 89671 89495 89697 89485 89723 89475 89749 89465 89775 9.89455 9.89801 89445 89827 89435 89853 89425 89879 89415 89905 89405 89931 89395 89957 89385 89983 89375 90009 89364 90035 9.89354 9.90061 89344 90086 89334 90112 89324 90138 89314 90164 89304 90190 89294 90216 89284 90242 89274 90268 89264 90294 9.89254 9.90320 89244 90346 89233 90371 89223 90397 89213 90423 89203 90449 89193 90475 89183 90501 89173 90527 89162 90553 9.89152 9.90578 89142 90604 89132 90630 89122 90656 89112 90682 89101 90708 89091 90734 89081 90759 89071 90785 89060 90811 90837 89050 Co-tnns*. i Sine.

Co-tang. Secant. Co-secant 10.10719 10.10347 10.21066 60 10693 10357 21050 59 10667 10367 21033 68 10641 10376 21017 57 10615 10386 21001 56 10589 10396 20985 55 10563 10406 20969 54 10537 10416 20953 53 10511 10426 20937 52 10485 10436 20921 51 10.10459 10.10446 10.20905 50 10433 10456 20889 49 10407 10466 20872 48 10381 10476 20856 47 10355 10486 20840 46 10329 10496 20824 45 10303 10505 20808 44 10277 10515 20792 43 10251 10525 20776 42 10225 10535 20760 41 10.10199 10.10545 10.20744 40 10173 10555 20728 39 10147 20712 38 10565 10121 10575 20696 37 10095 20681 36 10585 10069 20665 35 10595 10043 20649 34 10605 10017 20633 33 10615 09991 20617 32 10625 09965 20601 31 10636 10.09939 10.10646 10.20585 30 09914 205C9 29 10656 09888 20553 28 10566 20537 27 09862 10676 09836 20522 26 10686 20506 25 09810 10696 09784 20490 24 10706 09758 20474 23 10716 20458 22 09732 10726 20442 21 09706 10736 10.09680 10.10746 10.20427 20 20411 19 09654 10756 10767 20395 18 09629 09603 10777 20379 17 20364 16 09577 10787 20348 15 09551 10797 20332 14 10807 09525 10817 20316 13 09499 20301 12 . 09473 10827 20285 11 09447 10838 10.09422 10.10848 10.20269 10 9 20254 10858 09396 8 20238 10868 09370 7 20222 10878 09344 6 20207 10888 09318 5 20191 10899 09292 4 20175 10909 09266 3 20160 10919 09241 Q 20144 10929 09215 1 20128 10940 09189 0 20113 10950 09163 M. Taturent. Co-secant Secant.

51 Degrees.

Artificial Sines, Tang, and Sec. 39 Degrees. M. ; o i I i '2

\

3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 ‘20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.79887 79903 79918 79934 79950 79965 79981 79996 80012 80027 9.80043 80058 80074 80089 80105 80120 80136 80151 80166 80182 9.80197 80213 80228 80244 80259 80274 80290 80305 80320 80336 9.80351 80366 80382 80397 80412 80428 80443 80458 80473 80489 9.80504 80519 80534 80550 80565 80580 80595 80610 80625 80641 9.80656 80671 80686 80701 80716 80731 80746 80762 80777 80792 80807 Co-sine.

Co-sine. Tangent. 9.90337 9.89050 90863 89040 90889 89030 90914 89020 90940 89009 90966 88999 88989 90992 88978 91018 88968 91043 88958 91069 9.88948 9.91095 88937 91121 88927 91147 88917 91172 88906 91198 88896 91224 88886 91250 88875 91276 88865 91301 88855 91327 9.88844 9.91353 88834 91379 88824 91404 88813 91430 88803 91456 88793 91482 88782 91507 88772 91533 88761 91559 88751 91585 9.88741 9.91610 88730 91636 88720 91662 88709 91688 88699 91713 88688 91739 88678 91765 88668 91791 88657 91816 88647 91842 9.88636 9.91868 88626 91893 88615 91919 88605 91945 88594 91971 88584 91996 88573 92022 88563 92048 88552 92073 88542 92099 9.88531 9.92125 88521 92150 88510 92176 88499 92202 88489 92227 88478 92253 88468 92279 88457 92304 88447 92330 88436 92356 88425 92381 Sine. Co-tang.

Co-secant Secant. Co-tang. 10.09163 10.10950 10.20113 20097 10960 09137 20082 10970 09111 20066 10980 09086 20050 10991 09060 20035 11001 09034 20019 11011 09008 20004 11022 08982 19988 11032 08957 19973 11042 08931 10.08905 10.11052 10.19957 19942 11063 08879 19926 08853 11073 19911 11083 08828 19895 08802 11094 19880 08776 11104 19864 08750 11114 08724 19849 11125 19834 08699 11135 19818 08673 11145 10.08647 10.11156 10.19803 19787 11166 08621 19772 11176 08596 19756 11187 08570 19741 08544 11197 19726 08518 11207 19710 08493 11218 19695 08467 11228 19680 08441 11239 19664 08415 11249 10.08390 10.11259 10.19649 19634 08364 11270 19618 08338 11280 19603 08312 11291 08287 19588 11301 08261 19572 11312 19557 08235 11322 19542 08209 11332 19527 08184 11343 19511 08158 11353 10.08132 10.11364 10.19496 08107 19481 11374 08081 19466 11385 03055 19450 11395 08029 11406 19435 08004 11416 19420 07978 11427 19405 07952 11437 19390 07927 11448 19375 07901 11458 19359 10.07875 10.11469 10.19344 07850 11479 19329 07824 11490 19314 07798 11501 19299 07773 11511 19284 07747 11522 19269 07721 11532 19254 07696 11543 19238 07670 11553 19223 07644 11564 19208 07619 11575 19193 Tangent. Co-secant Secant.

50 Degrees,

147 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7

6 5 4 3 2

1 0 M.

1 <£8 M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 GO

Artificial Sines, Tang, and Sec. 40 Degrees Sine. 9.80807 80822 80837 80852 80867 80882 80897 80912 80927 80942 9.80957 80972 80987 81002 81017 81032 81047 81061 81076 81091 9.81106 81121 81136 81151 81166 81180 81195 81210 ’ 81225 81240 9.81254 81269 81284 81299 81314 81328 81343 81358 81372 81387 9.81402 81417 81431 81446 81461 81475 81490 81505 81519 81534 9.81549 81563 81578 81592 81G07 81622 81636 81651 81665 81680 81694 Co-sine.

Co-sine. Tangent. Co-tang. Secant. 1 9.83425 9.92381 10.07619 10.11575 88415 92407 07593 11585 88404 07567 92433 11596 88394 92453 07542 11606 88383 92484 07516 11617 88372 92510 07490 11628 88362 92535 07465 11638 88351 92561 07439 11649 88340 92587 07413 11660 88330 92612 07388 11670 9.88319 9.92638 10 07362 10.11681 88308 07337 92663 11692 88298 07311 92689 11702 88287 92715 07285 11713 88276 92740 11724 07260 88266 92766 07234 11734 92792 88255 07208 11745 88244 92817 07183 11756 88234 92843 07157 11766 88223 92868 07132 11777 9.88212 9.92894 10.07106 10.11788 88201 11799 92920 07080 88191 92945 11809 07055 92971 88180 07029 11820 88169 11831 92996 07004 88158 93022 11842 06978 88148 93048 11852 06952 88137 93073 11863 06927 88126 11874 93099 06901 88115 93124 11885 06876 9.88105 9.93150 10.06850 10.11895 88094 93175 11906 06825 11917 88083 93201 06799 93227 88072 11928 06773 88061 93252 11939 06748 88051 93278 11949 06722 93303 11960 88040 06697 11971 06671 88029 93329 11982 88018 93354 06646 88007 06620 11993 93380 9.93406 10.06594 10.12004 #.87996 93431 87985 06569 12015 93457 06543 12025 87975 12036 87964 93482 06518 12047 87953 93508 06492 12058 87942 93533 06467 06441 87931 12069 93559 12080 93584 87920 06416 12091 93610 87909 06390 12102 93636 87898 06364 9.87887 9.93661 10.06339 10.12113 12123 87877 93687 06313 12134 93712 06288 87866 12145 93738 06262 87855 12156 06237 93763 87844 12167 93789 06211 87833 12178 93814 06186 87822 12189 87811 06160 93840 12200 06135 93865 87800 12211 06109 93891 87789 12222 06084 93916 87778 Sine. Co-tang. Tangent. Co-secant.

49 Decrees*

Co-secant 10.19193 60 19178 59 19163 58 19148 57 19133 56 19118 55 19103 54 19088 53 19073 52 19058 51 10.19043 50 19028 49 19013 48 18998 47 18983 46 18968 45 18953 44 18939 43 18924 42 18909 41 10.18894 40 18879 39 18864 38 18849 37 18834 36 18820 35 18805 34 18790 33 18775 32 18760 31 10.18746 30 18731 29 18716 28 18701 27 18686 26 18672 25 18657 24 18642 23 18628 22 18613 21 10.18598 20 ' 18583 19 18569 18 ' 18554 17 18539 16 18525 15 18510 14 18495 13 18481 12 18466 11 10.18451 10 9 18437 8 18422 7 18408 6 18393 5 18378 4 18364 3 18349 2 18335 1 18320 0 18306 M. Secant.

Artificial Sines, Tang, and Sec. 41 Degrees.

*

:

i 1 1

|

' . .

M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.81694 81709 81723 81738 81752 81767 81781 81796 81810 81825 9.81839 31854 81868 81882 81897 81911 81926 81940 81955 81969 9.81983 81998 82012 82026 82041 82055 82069 82084 82098 82112 9.82126 82141 82155 82169 82184 82198 82212 82226 82240 82255 9.82269 82283 82297 82311 82326 8234C 82354 82368 82382 82396 9.8241C 82424 82439 82453 82467 82481 82495 82509 82523 82537 82551 (’o-sine.

Co-sine. Tangent. Co-tang. 9.87778 9.93916 10.06084 87767 93942 06058 93967 06033 87756 06007 93993 87745 87734 05982 94018 87723 94044 05956 87712 94069 05931 87701 05905 94095 87690 94120 05880 05854 87679 94146 9.87668 9.94171 10.05829 87657 94197 05803 87646 94222 05778 87635 94248 05752 87624 94273 05727 87613 94299 05701 87601 94324 05676 87590 94350 05650 87579 94375 05625 87568 94401 05599 9.87557 9.94426 10.05574 94452 87546 05548 87535 94477 05523 87524 94503 05497 87513 94528 05472 87501 94554 05446 87490 94579 05421 87479 ' 94604 05396 87468 94630 05370 87457 94655 05345 9.87446 9.94681 10.05319 87434 94706 05294 87423 94732 05268 •87412 94757 05243 87401 94783 05217 87390 94808 05192 87378 94834 05166 87367 94859 05141 87356 94884 05116 87345 94910 05090 9.87334 9.94935 10.05065 87322 94961 05039 87311 94986 05014 87300 95012 04988 87288 95037 04963 87277 95062 04938 87266 95088 04912 87255 95113 04887 87243 95139 04861 87232 95164 04836 9.87221 9.95190 10.04810 87209 95215 04785 87198 95240 04760 87187 95266 04734 87175 95291 04709 87164 95317 04683 87153 95342 04658 87141 95368 04632 87130 95393 04607 87119 95418 04582 87107 95444 04556 Sine. Co-tang. Tangent, i

43 Degrees

149

Co-secant Secant. 10.12222 10.18306 60 18291 59 12233 18277 58 12244 18262 57 12255 18248 56 12266 12277 18233 55 ; 18219 54 12288 18204 53 12299 18190 52 12310 18175 .11 12321 10.12332 10.18161 50 18146 49 12343 18132 48 12354 18118 47 12365 18103 46 12376 12387 18089 45 18074 44 12399 12410 18060 43 12421 18045 42 18031 41 12432 10.12443 10.18017 40 12454 18002 39 12465 17988 38 12476 17974 37 12487 17959 36 12499 17945 35 12510 17931 34 12521 17916 33 12532 17902 32 12543 17888 31 10.12554 10.17874 30 12566 17859 29 12577 17845 28 12588 17831 27 12599 17816 26 12610 17802 25 12622 17788 24 12633 17774 23 12644 17760 22 12655 17745 21 10.12666 10.17731 20 12678 17717 19 12689 17703 18 12700 17689 17 12712 17674 16 12723 17660 15 12734 17646 14 12745 17632 13 12757 17618 12 12768 17604 11 10.12779 10.17590 10 12791 9 17576 12802 8 17561 12813 7 17547 12825 6 17533 12836 5 17519 12847 4 17505 12859 3 17491 12870 2 17477 12881 1 17463 12893 0 17449 Co-Tenant Secant. M. .

160 M. 0 1 2 3

4 5 6 7 8 9 10 11 i 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 43 49 50 51 52 53 51 55 56 57 58 59 60 i

Artificial Sines, Tang, and Sec. 42 Degrees. Sine. 9.82551 82565 82579 82593 82607 82621 82635 82649 82663 82677 9.82691 82705 82719 82733 82747 82761 82775 82788 82802 82816 9.82830 82844 82858 82872 82885 82899 82913 82927 82941 - 82955 9.82968 82982 82996 83010 83023 83037 83051 83065 83078 83092 .83106 83120 83133 83147 83161 83174 83188 83202 83215 83229 ...83242 83256 83270 83283 83297 83310 83324 83338 83351 83365 83378 Co-sine.

Co-sine. 9.87107 87096 87085 87073 87062 87050 87039 87028 87016 87005 9.86993 86902 86970 86959 86947 86936 86924 86913 86902 86890 9.86879 .86867 86855 86844 86832 86821 86809 86798 86786 86775 9.86763 86752 86740 86728 86717 86705 86694 86682 86670 86659 9.86647 86635 86624 86612 86600 86589 86577 86565 86554 86542 9.86530 86518 86507 86495 86483 8.6472 86460 86448 86436 86425 86413 SinT.

Tangent. 9.95444 95469 95495 95520 95545 95571 95596 95622 95647 95672 9.95698 95723 95748 95774 95799 95825 95850 95875 95901 95926 9.95952 95977 *' 96002 96028 96053 96078 96104 96129 96155 96180 9.96205 96231 96256 96281 96307 96332, 96357 96383 96408 96433 9.96459 96484 96510 96535 96560 96586 96611 96636 96662 96687 9.96712 96738 96763 96788 96814 96839 96864 96890 96915 96940 96966 -tang.

Co-tang. Sec ant. Co-3ecanl i 10.04556 10.12893 10.17449 60* 04531 12904 17435 59 04505 17421 58 12915 04480 12927 17407 57 04455 12938 1 <393 56 04429 12950 17379 55 04404 12961 17365 54 04378 17351 53 12972 04353 17337 52 12984 04328 17323 51 12995 10.04302 10.13007 10.17309 50 04277 13018 17295 49 04252 17281 48 13030 04226 13041 17267 47 04201 17253 46 13053 04175 13064 17239 45 04150 13076 17225 44 13087 04125 17212 43 04099 13098 17198 42 04074 17184 41 13110 10.04048 10.13121 10.17170 40 04023 17156 39 13133 03998 17142 38 13145 17128 37 03972 13156 03947 17115 36 13168 03922 17101 35 13179 17087 34 03896 13191 03871 17073 33 13202 : O*, | 17059 OO 03845 13214 17045 -111 03820 13225 10.03795 10.13237 10.17032 80 17018 29 13248 03769 13260 17004 28, 03744 03719 13272 16990 27 16977 26 03693 13283 16963 25 13295 03668 16949 24 13306 03643 16935 23 . 13318 03617 16922 . t 03592 13330 16908 21 13341 03567 10.03541 10.13353 10.16894 20 16880 19 13365 03516 16867 18 , 13376 03490 16853 17 13388 03465 16839 16 13400 03440 16826 i5 13411 03414 16812 14 13423 03389 16798 13 13435 03364 13446 16785 12 03338 16771 11 13458 03313 10.03288 10.13470 10.16758 10 9 16744 03262 13482 8 * 16730 03237 13493 7 ; 16717 13505 03212 6 16703 13517 03186 5 16690 13528 03161 4 16676 13540 03136 3 16662 13552 03110 2 16649 13564 03085 1 16635 03060 13575 0 16622 03034 13587 Secant. ' M. Tangent. Co-secant

4‘i Degrees.

Artificial Sines, Tang, and Sec. 43 Degrees. M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1 46 47 48 ! 49 50 ! 51 52 , 53 54 55 56 57 58 59 60

feme. 9.83378 83392 83405 83419 83432 83446 83459 83473 83486 83500 9.83513 83527 83540 83554 83567 83531 83594 83608 83621 83634 9.83648 83661 83674 83688 83701 83715 83728 83741 83755 83763 9.83781 83795 83808 83321 83334 83848 83861 83874 83887 83901 9.83914 83927 83940 83954 83967 83980 83993 84006 84020 84033 5.84046 84059 84072 84085 84098 84112 84125 84138 84151 84164 84177 Co-pine.

Co-sine. 9.86413 86401 86389 86377 86366 86354 86342 86330 86318 86306 9.86295 86283 86271 86259 86247 86235 86223 86211 86200 86188 9.86176 86164 86152 86140 86128 86116 86104 86092 86080 86068 9.86056 86044 86032 86020 86008 85996 85984 85972 85960 85948 9.85936 85924 85912 85900 85888 85876 85864 85851 85839 85827 9.85815 85803 85791 85779 85766 85754 85742 85730 85718 85706 85693 Sine.

151

Co-secant | Tangent. Co-tang. | feecant. 1 9.96966 10.03034 10.13587 10.16622 60 16608 1 59 13599 03009 96991 16595 58 13611 02984 97016 16581 57 13623 02956 97042 16568 56 13634 02933 97067 16554 55 13646 02908 97092 16541 54 13658 02882 97118 16527 53 13670 02857 97143 16514 52 13682 02832 97168 16500 51 13694 02807 97193 9.97219 10.02781 10.13705 10.16487 50 16473 49 13717 02756 97244 16460 48 13729 02731 97269 16446 47 13741 97295 02705 16433 46 13753 02680 97320 16419 45 13765 97345 02655 13777 16406 44 97371 02629 13789 16392 43 02604 97396 13800 16379 42 97421 02579 13812 16366 41 97447 02553 9.97472 10.02528 10.13824 10.16352 40 13836 16339 39 97497 02503 02477 13848 16326 38 97523 13860 16312 37 97548 02452 02427 13872 16299 36 97573 13884 16285 35 02402 97598 13896 02376 16272 34 97624 13908 16259 33 ; 02351 97649 13920 16245 32 97674 02326 16232 31 02300 13932 97700 9.97725 10.02275 10.13944 10.16219 30 13956 16205 29 02250 97750 13968 16192 28 02224 97776 16179 27 97801 02199 13980 16166 2f 02174 13992 97826 14004 16152 2/ 97851 02149 97877 02123 14016 16139 24 97902 02098 14028 16126 23 97927 02073 14040 16113 22 02047 14052 16099 2. 97953 9.97978 10.02022 10.14064 10.16086 20 01997 14076 16073 19 98003 01971 14088 98029 16060 18 98054 01946 14100 16046 17 01921 98079 14112 16033 16 98104 01896 14124 16020 15 98130 01870 14136 16007 14 98155 01845 14149 15994 13 14161 98180 01820 15980 12 98206 01794 14173 15967 11 i 9.98231 10.01769 10.14185 10.15954 10 98256 01744 14197 15941 9 98281 01719 14209 8 15928 98307 01693 14221 7 15915 98332 01668 14234 6 15902 98357 01643 14246 15888 5 98383 01617 14258 4 15875 98408 01592 14270 3 15862 98433 01567 14282 2 15849 98458 01542 14294 15836 1 98484 01516 14307 0 15823 Co-tang. Tangent. Co-secant , Secant. M.

46 Degrees.

152

Artificial Sines, Tang, and Sec. 44 Degrees.

M. 0 ! 1 ! 2 ' 3 ! 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sine. 9.84177 84190 84203 84216 84:29 84242 84255 84269 84282 84295 9.84308 84321 84334 84347 84360 84373 84385 84398 84411 84424 9.84437 84450 84463 84476 84489 84502 84515 84528 84540 84553 9.84566 84579 84592 84605 84618 84630 84643 84656 84669 84682 9.84694 84707 84720 84733 84745 84758 84771 84784 84796 84809 9.84822 84835 84847 84860 84873 84885 84898 84911 84923 84936 84949 Co-sine.

Co-sine. Tangent. 1 Co-tang, j Secant, i Co-secant 9.85693 9.98484 j 10.01516 i 10.1430? ! ~~KM5823 98505} 1 " - 01491 :' 14319 85681 15810 85669 98534 01j■ 14331 15797 155657 “ 98560 01440 14343 15784 85645 98585 01415 14355 15771 85632 98610 01390 14368 15758 85620 98635 01365 14380 15745 85608 98661 01339 14392 15731 85596 98686 01314 14404 15718 ! 98711 85583 01289 14417 15705 9.85571 9.98737 10.01263 10.14429 10.15692 85559 98762 01238 14441 15679 85547 98787 01213 14453 15666 85534 98812 01188 14466 15653 85522 98838 01162 14478 15640 85510 98863 01137 14490 15627 85497 98888 01112 14503 15615 98913 85485 01087 14515 15602 85473 98939 01061 14527 15589 85460 98964 01036 14540 15576 9.85448 9.98989 10.01011 10.14552 10.15563 85436 99015 00985 14564 15550 85423 99040 00960 14577 15537 85411 99065 00935 14589 15524 99090 85399 00910 14601 15511 85386 99116 00884 14614 15498 85374 99141 00859 14626 15485 85361 99166 00834 14639 15472 85349 99191 00809 14651 15460 85337 99217 00783 14663 15447 9.85324 9.99242 10.00758 10.14676 10.15434 99267 85312 00733 15421 14688 99293 00707 85299 14701 15408 85287 99318 00682 14713 15395 99343 00657 85274 14726 15382 99368 00632 85262 15370 14738 99394 85250 00606 ! 14750 15357 85237 99419 00581 1 14763 15344 85225 99444 00556 15331 14775 00531 85212 99469 15318 14788 9.85200 9.99495 10.00505 10.14800 10.15306 85187 00480 . 15293 99520 14813 85175 99545 00455 14825 15280 15267 85162 99570 00430 14838 00404 85150 99596 14850 15255 85137 15242 99621 00379 14863 99646 00354 15229 14875 85125 15216 85112 99672 00328 14888 15204 99697 00303 14900 85100 15191 85087 99722 00278 14913 9.99747 10.00253 10.14926 10.15178 9.85074 00227 14938 15165 85062 99773 15153 14951 00202 85049 99798 14963 15140 00177 85037 99823 15127 14976 00152 85024 99848 14988 15115 00126 85012 99874 15102 15001 00101 84999 99899 15089 15014 99924 00076 84986 15077 15026 00051 84974 99949 15064 15039 00025 84961 99975 15051 15051 84949 10.00000 10.00000 Co-tang. Tangent. Co-secant Secant. Sine.

45 Degrees

feO 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 M.

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plane trigonometry. - The University of Chicago Library

12Gi *. .j,.., GUic&'Sk TIlEATilSE- R ON SURVEYING, CONTAINING THE THEORY AND PRACTICE: TO WHICH IS PREFIXED A PERSPICUOUS SYSTEM OF PLANE TR...

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