GIFT OF Dr. Horace Ivie EDUCATION DEPT HAY'S MATHEMATICAL SERIES. SURVEYING NAVIGATION, : : v j : \ ; ;';, WITH A PRELIMINARY TREATISE ON TRIGONOMETRY AND MENSURATION, A. CHUYLER, M. A. Professor of Applied Mathematics and Logic in Baldwin University; Author of Higher Arithmetic, Principles of Loyic, and Complete Algebra. VAN ANTWERP, BRAGG & CO., 137 WALNUT STREET, 28 BOND STREET, CINCINNATI. NEW YORK. RAY'S SERIES, EMBRACING A Thorough and Progressive Course in Arithmetic, Algebra, the Higher Mathematics. Primary Arithmetic. Higher Arithmetic. /^InteHect^i&L Arithmetic. Test Examples in Arithmetic. Rntiiiiients 01 Arithmetic. New Elementary Algebra. Practical AHtiwiietic. New Higher Algebra. Plane and Noliri Geometry. BY ELI T. TAPPAN, A.M., Preset Kenyan College. 12wo, cloth, 276 pp. Geometry and Trigonometry By ELI T. TAPPAN, A.M. I- res' t Kenyan College. Svo, sheep, 420 pp. Analytic Geometry. By GEO. H. HOWISON, A.M., Prof, in Mass. Institute of Technology. Treatise on Analytic Geometry, especially as applied to the Properties of Conies : including the Modern Methods of Ahridged Notation. Elements of Astronomy. By S. H*"PEABODY, A.M., Prof, of Physics and Civil Engineering, Amherst College. Handsomely and profusely illustrated. 8vo, sheep, 336 pp. KEYS. Ray's Arithmetical Key (To Intellectual and Practical); Key to Ray's Higher Arithmetic ; Key to Ray's New Elementary and Higher Algebras, GIFT OF The Publishers furnish Descriptive Circulars of the above Mathe- matical Text-BooJtSf with Prices and other information concerning them. _ Entered according to Act of Congress, in the year 1864, by SARGENT, WILSON & HINKLE, in the Clerk's Office of the District Court of the United States for the Southern District of Ohio. PREFACE. Nearly twenty years ago the Publishers made the following announcement: "Surveying and Navigation; containing Survey- ing and Leveling, Navigation, Barometric Heights, etc." To redeem this promise, the present work now appears. It is customary to preface works on Surveying by a meager sketch of Plane Trigonometry, but it has been thought best to include in this work a thorough treatment of Plane and Spherical Trigonometry and Mensuration. These subjects have been treated in .view of the wants of our best High Schools and Colleges. Certain modern writers have defined the Trigonometric func- tions as ratios; for example, in a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse, etc. The historical method of considering the sine, co-sine, tan- gent, etc., as linear functions of the arc, explains the origin of these terms avoids the ambiguity of the word ratio; explains how the logarithm of the sine, for example, can reach the limit 10, which would be impossible if the limit of the sine itself is 1, and is much more readily apprehended by the student. The advantages in analytic investigations resulting from defining these functions as ratios have been secured in the principles relating to the Eight Triangle, Art. 64. Each of the circular functions has, in the first place, been considered by itself, and its value traced, for all arcs, from to 360. 924229 <) iv PREFACE. Then follows the solution of triangles, right and oblique, the general relations of the circular functions, the functions of the sum or difference of two angles, and a variety of interesting practical applications. It is hoped that Spherical Trigonometry has been made in- telligible to the diligent student. More than ordinary care has been given to the development of Napier's principles, and to the discussion of the species of the parts of both right and oblique spherical triangles, Arts. 126, 129, 145, 148, 151. Mensuration, a subject at once interesting and practically im- portant, has been discussed at length, and formulas have been developed instead of rules for the solution of the problems. In the Surveying, the instruments are first represented and described, and the methods of making the adjustments given in detail. The Author takes this opportunity to express his obligations to Messrs. W. & L. E. Gurley, Manufacturers of Surveying and Engineering Instruments, Troy, N. Y., who have kindly granted him the use of their Manual for the delineation and descrip- tion of the instruments. In consequence of this courtesy, much better drawings and descriptions have been made than would otherwise have been possible. The instruments themselves should, however, be accessible to the student, who should study them in connection with the descriptions in the book, and learn to use them in practical work, guided by a competent instructor. The Rectangular method of surveying the Public lands, now brought to great perfection under the direction of the Govern- ment, has been minutely explained, and illustrated by field notes of actual surveys. In this portion of the work, the United States Manual of Surveying Instructions has been taken as authority, and thus the authorized methods, which must form the basis for subsequent surveys, have been made accessible to the student. The methods of finding the true meridian and the variation of the needle have been given at length; also specific direc- PREFACE. V tions for finding corners, taking bearings, measuring lines, re- cording field notes, and plotting. In addition to the ordinary method of finding the area, a new method, developed by E. M. Pogue, of Kentucky, is given in Art. 304. This method has the merit of giving always a uniform result from the same field notes, and thus avoids dis- putes about the different results of the ordinary method, un- avoidably attending the various distribution of errors by differ- ent calculators. The methods of supplying omissions are explained and illus- trated by examples. Laying out and dividing land, operations admitting of an unlimited variety of applications, have been treated in view of the wants of the practical surveyor. The subject is also full of interest to the student, who can not fail to receive from it new views of the resources of mathematical science. Leveling, the construction of railroad curves, embankments and excavations, the method of making Topographical surveys, with the authorised conventional symbols, Barometric heights, etc., have been explained and illustrated by diagrams and examples. It has been thought best to give a clear, elementary treat- ment of Navigation, not only on account of those who may desire to pursue the subject further, but for the sake of grati- fying the wishes of intelligent persons who may desire to know something of Navigation. The limits of the work, however, for- bid the discussion of Nautical Astronomy. The examples in Navigation have been selected from the English work of J. R. Young. The tables of Logarithms, Natural and Logarithmic sines, etc., have been carried only to five decimal places, and for the pur- poses intended will be found practically better than tables to six or seven places. The Traverse table has been thrown into a new form, at once condensed and convenient. These tables have been compiled by Mr. Henry H. Vail, and VI PREFACE. by him compared with Babbage's and Wittstem's tables, then by the Author with Vega's tables to seven decimal places. It is hoped that by this double comparison perfect accuracy has been attained. The table of Meridional Parts, taken from " Projection Tables for the use of the United States Navy," prepared by the Bureau of Navigation, and issued from the Government Printing office, was calculated in the Hydrographic office for the terrestrial spheroid, compression u^.ir?- This table, now for the first time pub- lished in a text-book, is believed to be more correct than those in general use. The Author takes pleasure in acknowledging his obligations to Prof. E. H. Warner for critical suggestions and acceptable aid in reading proof and testing the accuracy of the answers. With the hope that the book will be attractive and useful to the student, teacher, and practical surveyor, it is sent forth to accomplish its work. A. SCHUYLER. BALDWIN UNIVERSITY. ) BEREA, 0., June 12, 1873. j INDEX. PAGE INTRODUCTION 9 Logarithms 9 Table of Logarithms . . 12 Multiplication by Logarithms ....... 18 Division by Logarithms 19 Involution by Logarithms . . . . . . . .21 Evolution by Logarithms ........ 22 TRIGONOMETRY 23 Plane Trigonometry .23 Trigonometrical Functions .27 Table of Natural Functions 41 Table of Logarithmic Functions 43 Right Triangles 47 Oblique Triangles '. . 55 Application to Heights and Distances . ... . .69 Relations of Circular Functions ....... 72 Applications 92 SPHERICAL, TRIGONOMETRY . 108 Right Triangles 109 Oblique Triangles . . 124 Mensuration . . . .150 Mensuration of ^Surfaces 150 Mensuration of Volumes ........ 174 SURVEYING 185 Instruments 185 Survey of Public Lands 216 Variation of the Needle .265 Field Operations 274 Preliminary Calculations . . 284 Area of Land 296 (vii) viii INDEX. PAGE Supplying Omissions 308 Laying Out Land . 313 Dividing Land 318 Leveling 340 Surveying Railroads ......... 351 Topographical Surveying . . . . . . . . 369 Barometric Heights 375 NAVIGATION 381 Preliminaries 381 Plane Sailing 385 Parallel Sailing 389 Middle Latitude Sailing 390 Mercator's Sailing . 393 Current Sailing 398 Plying to Windward . . . 400 Taking Departures 402 TABLES , 405 INTRODUCTION i V, Oi : LOGARITHMS. 1. Definition. A logarithm of a number is the exponent denoting the power to which a fixed number, called the base, must be raised in order to produce the given number. Thus, in the equation, b x = n, b is the base of the sys- tem, n is the number whose logarithm is to be taken, and x is the logarithm of n to the base 6, which may be written : x = log b n. Any positive number, except 1, may be assumed as the base, but when assumed, it remains fixed for a sys- tem ; hence, there may be an infinite number of sys- tems, since there may be an infinite number of bases. 2. Common Logarithms. Common logarithms are the logarithms of numbers in the system whose base is 10. 10 = lj. .'. by def., log 1=0.' 10 1 =10; .;. by def., log 10=1. 10 2 = 100 ; . . by def., log 100 = 2. 10 3 = 1000; . ' . by def., log 1000 = 3. Hence, In the common system, the logarithm of an exact power of 10 is the whole number equal to the exponent of the power. (9) 10 LOGARITHMS. 3. Consequences. 1. If the number is greater than 1 and less than 10, its logarithm is greater than and less than 1, or is a decimal. 2f'If thV nuuiber is greater than 10 and less than * '10$, its' logarithm is greater than 1 and less than 2, or is 1 -}- a decimal. 3. In general, if the number is not an exact power of 10, its logarithm, in the common system, will consist of two parts an entire part and a decimal part. The entire part is called the characteristic and the dec- imal part is called the mantissa. 4:. Problem. To find the laws for the characteristic. . Let (1) 10 Z = n then, by def., log n = x. But (2) lO^lO. (l)-(2) = (3) 10*-' = ^; then, by def, log ^ = x-l. * log ^ = log n 1. Hence, The logarithm of the quotient of any number by 10 -is less by 1 than the logarithm of the number. Let us now take the number 8979 and its logarithm 3.95323, as given in a table of logarithms, and divide the number successively by 10, and Tor each division subtract 1 from the logarithm of the dividend, then we have, Log 8979 ^3.95323. Log .8979 =T95323. " 897.9 ~ 2.95323. " .08979 =^95323. " 89.79 = L95323. u .008979 3795323. " 8.979 = 0.95323. THE CHARACTERISTIC. 11 The minus sign applies only to the characteristic over which it is placed. The mantissa is always positive, and is the same for all positions of the decimal point. An inspection of the above will reveal the following laws : 1. If the number is integral or mixed, the characteristic is positive and is one less than the number of integral figures. 2. If the number is entirely decimal, the characteristic is negative and is one greater, numerically, than the number of O's immediately following the decimal point. 5. Exercises on the Characteristic. 1. What is the characteristic of the logarithm of 7? 2. What is the characteristic of the logarithm of 465? 3. What is the characteristic of the logarithm of 4678? 4. What is the characteristic of the logarithm of 34.75? 5. What is the characteristic of the logarithm of .65? 6. What is the characteristic of the logarithm of .0789? 7. What is the characteristic of the logarithm of .00084? 8. If the characteristic of the logarithm of a num- ber is 2, how many integral places has that number? 9. If the characteristic of the logarithm of a num- ber is 5, how many integral places has that number? 10. If the characteristic of the logarithm of a num- ber is 1, how many integral places has -that number? 11. If the characteristic of the logarithm of a num- ber is 0, how many integral places has that number? 12 LOGARITHMS. 12. If the characteristic of the logarithm of a num- ber is negative, is the number integral, decimal, or mixed? 13. If the characteristic of the logarithm of a num- ber is 4, how many O's immediately follow the decimal point? 14. If the characteristic of the logarithm of a num- ber is 2, how many O's immediately follow the decimal point? 15. If the characteristic of the logarithm of a num- ber is 1, how many O's immediately follow the decimal point ? TABLE OF LOGARITHMS. 6. Description of the Table. The table of logarithms annexed gives the mantissa of the logarithm of every number from 1000 to 10900. The characteristic can be found by the preceding laws. It follows, from Art. 4, that the mantissa of the loga- rithm of a number is the same as the mantissa of the logarithm of the product or quotient of that number by any power of 10. Thus : Log 12 : = 1.07918. " 120 = 2.07918. " .012 = 2X)7918: Hence, we can determine from the table the log- arithm of aii number less than 1000. Thus, the mantissa of the logarithm of 8 is the same as that of the logarithm of 8000. In the table, the first three or four figures of each number are given in the left-hand column, marked X. The next figure is given at the head and foot of one of the columns of mantissas. TABLE OF LOGARITHMS. 13 The mantissas, in the column under 0, are given to five decimal places. The first and second decimal figures of this column are understood to be repeated in the spaces below, and to be prefixed, across the page, to the three figures of the remaining columns. When the third decimal digit changes from 9 to 0, the second is increased by the 1 carried; and the cor- responding mantissa, and all to the right, commence with a smaller figure, to indicate that the first two decimal figures, to be prefixed, are to be taken from the line below. The last column, marked D, contains the differ- ence of two successive mantissas, called the tabular difference. 7. Problem. To find the logarithm of a given number. 1. Find the logarithm of 3675. The characteristic is 3. Opposite 367, in the column headed N, and under the column headed 5, we find 526, to which prefix the two figures, 56, in the column headed 0, and we have for the. mantissa .56526. .'. log 3675 r= 3.56526. 2. Find the logarithm of 76. The characteristic is 1, and the mantissa is the same as that of 7600, which is .88081. . . log 76 = 1.88081. 3. Find the logarithm of .004268. The characteristic is 3, and the mantissa is the same as that of 4268. Looking opposite 426, and under 8, we find 022, of which the is a small figure. Prefixing 14 LOGARITHMS. 63, from the line below, in the column headed 0, we have for the mantissa .63022. .-. log .004268 ="3.63022. 4. Find the logarithm of 109684. The characteristic =5. The mantissa of log 1096 = .03981 Tab. diff. is 40; and 40 X .84 a 34 log 109684 = 5.04015 The reason for multiplying the tabular difference by .84 will be apparent from the following: log 109600 = 5.03981. log 109700 = 5.04021. The difference of the logarithms is 40 hundred- thousandths, and the difference of the numbers is 100; but the difference of 109600 and 109684 is 84, which is .84 of 100; hence, the difference of the logarithms of 109600 and 109684 is .84 of 40 hundred-thousandths, which is 40 hundred-thousandths X -84 = 34 hundred- thousandths, nearly. It is assumed that the difference of the logarithms of two numbers is proportional to the difference of the numbers, which is approximately true, especially if the numbers are large. 5. Find the logarithm of 123.613. The characteristic = 2. The mantissa of log 1236 = .09202 Tab. diff. is 35; and 3oX-13= 5 .;., log 123.613 = 2.09207 The tabular difference is .00035, and .00035 X .13 = .0000455. But since the logarithms in this table are taken only to five decimal places, the two last figures, EXAMPLES. 15 55, are rejected, and 1 is carried to .00004, making .00005 for the correction. In general, when the left-hand figure of the part rejected exceeds 4, carry 1. When the tabular difference is large, as in the first part of the table, there may be small errors. Accord- ingly, for numbers between 10000 and 10900, it will be better to use the last two pages instead of the first page. 8. Rule. 1. If the number, or the product of the number by any power of W, is found in the table, take the corresponding man- tissa from the table, and prefix the proper characteristic. 2. If the number, without reference to the. decimal point or O'.s on the right, is expressed by more than five figures, take from the table the mantissa corresponding to the first four or five figures on the left, multiply the corresponding tabular difference by the number expressed by the remaining figures, considered as a decimal, reject from the product as many figures on the right as are in the multiplier, carrying to the nearest unit, and add the result as so many hundred- thousandths to the mantissa before found, and to the sum prefix the proper characteristic. 9. Examples. 1. What is the logarithm of 2347 ? Ans. 3.37051. 2. What is the logarithm of 108457? Ans. 5.03526. 3. What is the logarithm of 376542? Ans. 5.57581. 4. What is the logarithm of 229.7052? Ana. 2.36117. 5. What is the logarithm of 1128737? Ans. 6.05260. 6. What is the logarithm of .30365? Ans. L48237. 7. What is the logarithm of .0042683? Ana. "3.63025. 8. What is the logarithm of 1245400? Ans. 6.09531. 16 LOGARITHMS. 10. Problem. To find the number corresponding to a given logarithm. 1. What number corresponds to logarithm 2.03262? The mantissa is found in the column headed 8, and opposite 107 in the column headed N. Hence, without reference to the decimal point, the number corresponding is 1078; but since the characteristic is 2, the number is entirely decimal, and one imme- diately follows the decimal point. Hence, the number corresponding is .01078. 2. What number corresponds to logarithm 2.83037? Since this logarithm can not be found in the table, take the next less, which is 2.83033, and the corre- sponding number, without reference to the decimal point, which is 6766. The difference between the given logarithm and the next less is 4, and the tabular difference is 6, which is the difference of the logarithms of the two numbers, 6766 and 6767, whose difference is 1. If the tabular . difference . of the logarithms, 6, cor- responds to a difference in the numbers of 1, the difference of the logarithms, 4, will correspond to a difference of -f of 1 ; which, reduced to a decimal, and annexed to 6766, will give for the number, without reference to the decimal point, 676666. But since the characteristic is 2, there will be three integral places; hence, 676.666 is the number required. 3. What number corresponds to logarithm 2.76398? The given log =2.76398 . . number = 580.737 Next less log = 2.76395 . . number = 580.7 Tab. difference ct= 8)300 = difference. 37 = correction. TABLE OF LOGARITHMS. 17 It is necessary to write only that part of the next less logarithm which differs from the given logarithm. Conceive O's annexed to the difference, and divide by the tabular difference; and annex the quotient to the number corresponding to the next less logarithm. In practical work abbreviate thus : Let I denote the given logarithm ; /', the next less logarithm ; n and n', the corresponding numbers; t, the tabular difference; dj difference of logarithms; c, the correction. 4. What number corresponds to logarithm 1.73048? I =1.73048 r ;. n == .537625 P=T.73Q46 .'. n'= .5376 t = 8) 2 dL n' is found first, then 25 = c. n by annexing c. 11. Rule. 1. If the given mantissa can be found in the table, take the number corresponding, and place the decimal point accord- ing to the law for the characteristic. 2. If the given mantissa can not be found in the table, take the next less and the corresponding number. Subtract this mantissa from the given mantissa, annex O's to the re- mainder, divide the result by the tabular difference, annex the quotient to the number corresponding to the logarithm next less than the given logarithm, and place the decimal point according to the law for the characteristic. 12. Examples. 1. What number corresponds to logarithm ,4.55703? Ans. 36060. 2. What number corresponds to logarithm 3.95147? Ans. 8942.8. 3. What number corresponds to logarithm 2.41130? Ans. .025781. S. N. 2. 18 LOGARITHMS. 4. What number corresponds to logarithm 1.48237? Ans. .30365. 5. What number corresponds to logarithm 3.63025? Ans. .0042683. MULTIPLICATION BY LOGARITHMS. 13. Proposition. The logarithm of the product of two numbers is equal to the sum of their logarithms. (1) b x = m; then, by def., log m .= x. Let (2) b y = n; then, by def., log n = y. (1)X(2) = (3) b* + y = mn; then, by def., log mn=x+y. . ' . log m n = log m -f- log n. 14. Rule. 1. Find the logarithms of the factors and take their sum, which will be the logarithm of the product. 2. Find the number corresponding which will be their product. 15. Examples. 1. Find the product of 57846 and .003927. log 57846 = 4.76228 log .003927 ="3.59406 log product = 2.35634, . . product = 227.16. 2. Find the product of 37.58 and 75864. Ans. 2851000. 3. Find the product of .3754 and .00756. Ans. .002838. DIVISION BY LOGARITHMS. 19 4. Find the product of 999.75 and 75.85. Ans. 75831.667. 5. Find the product of 85, .097, and .125. Ans. 1.03062. DIVISION BY LOGARITHMS. 16. Proposition. The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of tlie divisor. C (1) b*= m; then, by def., log m = x. Let ] ( (2) 6 y -- n; then, by def, log n = y. (1) -5- (2) =; (3) &*-=; then, by def., log- = x y. i m i . . log log m log n. 17. Rule. 1. Find the logarithms of the numbers, subtract the loga- rithm of the divisor from the logarithm of the dividend, and the remainder will be the logarithm of the quotient. 2. Find the number corresponding which ivill be the quotient. 18. Examples. 1. Divide 73.125 by .125. log 73.125=1.86407 log .125 = L09691 log quotient = 2.76716, . . quotient = 585. 2. Divide 7.5 by .000025. Ans. 300000. 3. Divide 87.9 by .0345. Ans. 2547.824. 4. Divide .34852 by .00789. Ans. 44.171. 5. Divide 85734 bv 12.7523. Ans. 6723. 20 LOGARITHMS. ARITHMETICAL COMPLEMENT.- 19. Definition. The arithmetical complement of a logarithm is the result obtained by subtracting that logarithm from 10. Thus, denoting the logarithm by /., and its arithmetical complement by a. c. /., we shall have the formula, a. c. I =10^-1. The arithmetical complement of a logarithm is most readily found by commencing at the left of the loga- rithm, and subtracting each digit from 9 till we come to the last numeral digit, which must be subtracted from 10. Thus, to find the a. c. of 3.47540, we say: 3 from 9, 6; 4 from 9, 5; 7 from 9, 2; 5 from 9, 4; 4 from 10, 6; from 0, 0. .-. a. c. of 3.47540 = 6.52460. 20. Proposition. The difference of two logarithms is equal to the minuend, plus the arithmetical complement of the subtrahend, minus 10. For, I Z'=Z + (10 I'} 10. It is convenient to use the a. c. in division when either the dividend or the divisor is the indicated product of two or more factors. Thus, let it be re- quired to find x in the proportion : 37.5: 678.5:: 27.56:,; ..., = 678.5 x2 7 . 56 o7.o , , log x = log 678.5 -f log 27.56 -f a. c. log 37.5 10. log 678.5 = 2.83155 log 27.56=1.44028 a. c. log 37.5 = 8.42597 log x = 2.69780 . . x = 498.656. INVOLUTION BY LOGARITHMS. 21 21. Examples. 1. Given 125.5 : .0756 : : x : .0034532, to find x. Ans. 5.7325. 2. Given 843 : x : : 732.534 : .759, to find x. Ans. .87346. 3. Given x : .034 : : .784 : .00489, to find x, Ans. 5.451125. 32.015 X-.874 4 1Ven X ** .000216X90257 ' t0 find * An ' L4358 ' 5. Given .753 X 12.234 : 87.5 X 3.7547 : : 56.5 : x, to find x. Ans. 2014.96. INVOLUTION BY LOGARITHMS. 22. Proposition. The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. Let (1) b* =n; then, by def., log n =x. (1)*:=(2) b px =n p ; then, by def., log n p =px. . ' . log n '' = p log n. 23. Rule. 1. Find the logarithm of the number and multiply it by the exponent of the power, and the product will be the loga- rithm of the power. 2. Find the number^ corresponding which will be the power. 24:. Examples. 1. Find the cube of .034. (1) log .034 = 2T53148 (1) X 3 = (2) log .034^ 5.59444 . . .034 3 .000039305. 2. Find the square of 25.7. Ans. 660.47. 22 LOGARITHMS. 3. Find the fourth power of .75. Ans. .3164. 4. Find the cube of 8.07. Ans. 525.55. 5. Find the fifth power of .9. Ans. .59047. EVOLUTION BY LOGARITHMS. 25. Proposition. The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root. Let (1) b x = n; then, by def., log n = x. 1/'(1) = (2) b r r = i r/ n/ then, by def., log V^n = ~ r,- log n . . log V n = - . 26. Rule. 1. Find the logarithm of the number, divide it by the index of the root, and the quotient will be the logarithm of the root. 2. Find the number corresponding which will be the root. 27. Examples. 1. Extract the square root of .75. (1) log .75=T.87506 (1) -s- 2 = (2) log V J5 =T93753 . * . VTft = .86602. Scholium. L87506 ~ 2 = (2 + 1.87506) ~- 2 = T93753. 2. Extract the cube root of 91125. Ans. 45. 3. Find the value of i/67 -4ns. .89443. 4. Extract the fifth root of .075. Ans. .59569. 5. Find the value of 3 /B7.5 X Q78) 2 Ans >6 76317. 12.5X5.9 PLANE TRIGONOMETRY. 23 TRIGONOMETRY. 28. Definition and Classification. Trigonometry is that branch of Mathematics which treats of the solution of triangles. Trigonometry is divided into two branches Plane and Spherical. PLANE TRIGONOMETRY. 29. Definition. Plane Trigonometry is that branch of Trigonometry which treats of the solution of plane triangles. 30. Parts of a Triangle. Every triangle has six parts three sides and three angles. If three parts are given, one being a side, the re- maining parts can be computed. If the three angles only are given, the triangle is indeterminate, since an infinite number of similar triangles will satisfy the conditions. 31. Sexagesimal Division of Angles and Arcs. The horizontal diameter, P, called the primary di- ameter, and the vertical diameter, 0' P', called the secondary diameter, divide the circumference into four equal parts, called quadrants. (7 is the first quadrant, 0' P the second, P P' the third, and P' the fourth. 24 TRIGONOMETRY. A degree is one-ninetieth of a right angle, or of a quadrant. A minute is one-sixtieth of a degree. A second is one-sixtieth of a minute. Thus, 25 34' 46" denote 25 degrees, 34 minutes, and 46 seconds. An angle, whose vertex is at the center, has the same numerical measure, or contains the same number of degrees, minutes, and seconds, as the arc of the circumference intercepted by its sides. 32. Centesimal Division of Angles and Arcs. A grade is one-hundreth of a right-angle, or of a quadrant. A minute is one-hundreth of a grade. A second is one-hundreth of a minute. Thus, I 9 24' 40" denotes 7 grades, 24 minutes, and 40 seconds. Z 9~' = 27~ 1 = ^8T' i! r_?r 1W __8T "10"' ~50' ~250' Let d, m, s, respectively, denote an angle expressed in degrees, sexagesimal minutes and seconds, and let' g, M, <r, respectively, denote the same angle expressed in grades, centesimal minutes and seconds, then ex- pressing the ratio of the angle to a right angle in each kind of units, we shall have : damn s * 90 "100 5400~ 10000' 324000 ~ 1000000 9 27 81 10 50 250 PLANE TRIGONOMETRY. 25 Let r denote the radius, and 71=3.14159265358979... T: r = a semi-circumference = 180 == 200" = two right angles. ^j-r = & quadrant PR 90 = 3 100 7 f== one right angle. 2 - r a circumference 360 : = 400" = four right, angles. If r1, the above expressions become, respectively, 33. Unit of Circular Measure. The unit of circular measure is that angle at the center whose intercepted arc is equal in length to the radius. Let u denote the unit of circular measure, and r the radius. Then, since * r = the semi-circumference, -KU 180 == 200'. -j CAO 900 ' u = -= 57. 29577951 .. = - -=63*. 6619772... 77 7T Let rf, g, Cj respectively, denote the number of degrees, grades, and units of circular measure in an angle; then, 180 200 ~ TT : __ C) 9=c, C ^l80 rf ' e= WO g ' 34. Origin, Termini and Situation of Arcs. The origin of an arc is the extremity at which it begins. The primary origin of arcs is at the right extremity of the primary diameter. The secondary origin of arcs is at the upper extremity of the vertical diameter. S. N. 3. 26 TRIG OXOMETR Y. The terminus of an arc is the extremity at which it ends. An arc is said to be situated in that quadrant in which its ter- minus is situated, thus : The arc OT is in the first quad- rant. The arc 00' T' is in the second quadrant. The arc OPT" is in the third quadrant. The arc OPT'" is in the fourth quadrant, 35. Positive and Negative Arcs. Positive arcs are those which are estimated in the direction contrary to that of the motion of the hands of a watch. Negative arcs are those which are estimated in the same direction as that of the motion of the hands of a watch. Thus, OT, OT', OT", OT'", estimated to the left, are positive, and OT'", OT", OT', OT, estimated to the right, are negative. 36. The Complement of an Arc. The complement of an arc or angle is 90 minus that arc or angle. If the arc or angle is less than 90, its comple- ment is positive. If the arc or angle is greater than 90, its comple-- ment is negative. The complement of an arc, geometrically considered, is the arc estimated from the terminus of the given arc to the secondary origin. Therefore, by the preced- ing article, the complement of an arc will be positive FUNCTIONS. 27 or negative, according as the arc is less or greater than 90. TO' is the complement of OT, and is positive. TO' is the complement of OT', and is negative. T"O' is the complement of OT", and is negative. T'"O' is the complement of OT'", and is negative. 37. The Supplement of an Arc. The supplement of an arc or angle is 180 minus that arc or angle. If the arc or angle is less than 180, its supple- ment is positive. If the arc or angle is greater than 180, its supple- ment is negative. The supplement of an arc, geometrically considered, is the arc estimated from the terminus of the given arc to the left-hand extremity of the primary diameter. Therefore, by article 35, the supplement of an arc will be positive or negative, according as the arc is less or greater than 180. TP is the supplement of OT, and is positive. T'P is the supplement of OT', and is positive. T"P is the supplement of OT", and is negative. T" f P is the supplement of OT"", and is negative. TKIGONOMETKICAL FUNCTIONS. 38. Preliminary Definitions and Remarks. 1. A function of a quantity is a quantity whose value depends on the given quantity. 2. The trigonometrical functions, called also circular functions, are auxiliary lines, which are functions of an arc or of the angle which has the same measure as that arc. 28 TRIG OSOMETR Y. 3. These functions are eight in number, and are called the sine, co-sine, versed-sine, co-ver sect-sine, tangent, co-tangent, secant and co-secant, which are abbreviated thus, sin, cos, vers, covers, tan, cot, sec, cosec. 4. The solution of triangles is accomplished by the aid of these functions, since they enable us to ascertain the relations which exist between the sides and angles of triangles. 5. The primary origin will be taken as the common origin of the arcs, unless the contrary is stated. 6. The origin of any arc, wherever situated, may be considered the primary origin of that arc ; and its sec- ondary origin is a quadrant's distance from the primary origin, in the direction of the positive or negative arcs, according as the given arc is positive or negative. 7. An arc will be considered positive unless the con- trary is stated. 8. The primary diameter passes through the primary origin ; and the secondary diameter, through the sec- ondary origin. 9. Lines estimated upward, toward the right, or from the center toward the terminus of the arc, are considered positive. 10. Lines estimated downward, toward the left, or from, the center and the terminus of the arc, are considered negative. 11. The limiting values of the circular functions are their values for the arcs 0, 90, 180, 270, 360. 12. The sign of a varying quantity, up to a limit, is its sign at the limit. 13. Point out positive arcs in the following diagram, and the origin and terminus of each. 14. Point out negative arcs, the origin, terminus and primary diameter of each. 15. Point out the positive lines, also the negative. FUNCTIONS. 39. The Sine of an Arc. The sine of an arc is the perpendicular distance of its terminus from the primary diameter. MT is the sine of the arc OT. M'T' is the sine or the arc OT'. M'T" is the sine of the arc OT". MT'" is the sine of the arc OT'". By the arcs OT" and OT'", we are to understand the positive arcs, and not the negative arcs designated by the same letters. The sine of an arc is the sine of the angle measured by that arc. Thus, MT, the sine of the arc OT, is the sine of the angle OCT, which is measured by the arc OT; and similarly for the other arcs and angles. The arcs OT and OT' are in the first and second quadrants, respectively, and their sines MT and M'T' are estimated upward, and are therefore positive; hence, The sine of an arc in the first or second quadrant is positive. The arcs OT" and OT'" are in the third and fourth quadrants, respectively, and their sines, M' T" and MT'", are estimated downward, and are there- fore negative; hence, The sine of an arc in the third or fourth quadrant is negative. Let the chord TT' be parallel to the primary diame- ter OP, then will M' T' be equal to MT, and the arc OT will be equal to the arc T' P; but the arc T' P is the supplement of the arc OT'; therefore, the arc OT is the supplement of the arc OT'; but M'.T', 30 TRIGONOMETRY. the sine of the arc 0T', is equal to MT J the sine oC the arc OT, the supplement of OT'; hence, The sine of an arc is equal to the sine of its supplement. The sine of is 0. As the arc increases from to 90, the sine increases from to +1. As the arc increases from 90 to 180, the sine decreases from ~f 1 to -fO.' As the arc increases from 180 to 270, the sine passes through 0, changes its sign from -f to , and increases numerically, but decreases algebraically from to 1. As the arc increases from 270 to 360, the sine decreases numerically, but increases al- gebraically from 1 to 0. Hence, for the limiting values of the sine, we have sin = 0, sin 90 = -f 1, sin 180 =. -f 0, sin 270 = 1, sin 360 = 0. 40. The Co-sine of an Arc. The co-sine of an arc is the perpendicular distance of its terminus from the secondary diameter. NT is the co-sine of the arc OT. NT' is the co-sine of the arc OT'. N'T" is the cosine of the arc OT". N'T'" is the co-sine of the arc OT'". The arcs OT and OT'" are in the first and fourth quadrants, respective- ly, and their co-sines NT and N'T'" are estimated toward the right, and are therefore posi- tive; hence, The co-sine of an arc in the first or fourth quadrant -is positive. The arcs OT' and OT" are in the second and third quadrants, respectively, and their co-sines, NT' and JVT", are estimated toward the left, and are therefore negative; hence, FUNCTIONS. 31 The co-sine of an arc in the second or third quadrant is negative. The word co-sine is an abbreviation of complementi sinus, the sine of the complement. In fact, NT, the co-sine of OT, is the sine of O'T, the complement of OT; hence, The co-sine of an arc is the sine of its complement. MT, the sine of OT, is the co-sine of O'T, the com- plement of OT; hence, The sine of an arc is the co-sine of its complement. Since the radius CO' is perpendicular to the chord TT'j NT and NT' are numerically equal; but since NT is estimated toward the right, and NT f toward the left, they have contrary signs; hence, NT= NT'; but NT is the co-sine of OT, and NT' is the co-sine of OT', the supplement of OT; hence, The co-sine of an arc is equal to minus the co-sine of its supplement. It is evident that CN is equal to the sine of OT, or of OT', and that CN' is equal to the sine of OT", or of OT'"; hence, The sine of an arc is equal to that part of the secondary diameter from the center to the foot of the co-sine. It is evident that CM is equal to the co-sine of OT, or of OT'", and that CM' is equal to the co-sine of OT' or of OT"; hence, The co-sine of an arc is equal to that part of the primary diameter from the center to the foot of the sine. ; The co-sine of is -f 1. As the arc increases from to 90, the co-sine decreases from -}- 1 to -f 0. As the arc increases from 90 to 180, the co-sine passes through 0, changes its sign from -f- to , and increases numerically, but decreases algebraically from -- to - 1. As the arc increases from 180 to 270, the co- sine decreases numerically, but increases algebraically 32 TRIG OSOMETR Y. from ---1 to 0. As the arc increases from 270 to 360, the co-sine passes through 0, changes its sign from - to -f, and increases from -|- to + 1. Hence, for the limiting values of the co-sine, we have cos =: + 1, cos 90 = + 0, cos 180 = 1, cos 270 = 0, cos 360 == + 1. 41. The Versed-Sine of an Arc. The versed-sine of an arc is the perpendicular dis- tance of the primary origin from the sine. MO is the versed-sine of the arc OT, and of the arc OT'". M'O is the versed-sine of the arc OT', and of the arc OT". The versed-sine of an arc, in any quadrant, is estimated to the right, and is therefore positive ; hence, The versed-sine is always positive. The versed-sine of is 0. As the arc increases from to 90, the versed-sine increases from to -+- 1. As the arc increases from 90 to 180, the versed-sine in- creases from -f- 1 to + 2. As the arc increases from 180 to 270, the versed-sine decreases from -f- 2 to -f 1- As the arc increases from 270 to 360, the versed-sine decreases from -f 1 to -f- 0. Hence, the limiting values of the versed-sine are vers =0, vers 90 ==4r 1, vers 180 == + 2, vers 270 ==--!, vers 360 =: -f 0. What are the least and greatest values of the sine, and what are the corresponding arcs? What are the least and greatest values of the co-sine, and what are the correspond! 17 g arcs? What are the least and greatest values of the versed- sine, and what are the corresponding arcs? FUNCTIONS. 33 42, The Co-versed-sine of an Arc. The co-versed-sine of an arc is the perpendicular dis- tance of the secondary origin from the co-sine. Thus, see diagram of the last article, NO' is the co- versed-sine of the arc OT, and of the arc OT'; N'O' is the co-versed-sine of the arc OT", and of the arc OT". The co-versed-sine of an arc in any quadrant is esti- mated upward, and is therefore positive; hence, The co-versed-sine is always positive. The word co-versed-sine is an abbreviation of comple- menti vcrsatus sinus, the versed or turned sine of the com- plement. In fact, NO', the co-versed-sine of OT, is the versed-sine of O'T, the complement of OT; hence, The co-versed-sine of an arc is the versed-sine of its com- plement. MO, the versed-sine of OT, is the co-versed-sine of O'T, the complement of OT; hence, The versed-sine of an arc is the co-versed-sinc of its com- plement. The co-versed-sine of is 1. As the arc increases from to 90, the co-versed-sine decreases from -j- 1 to } 0. As the arc increases from 90 to 180, the co- versed-sine increases from -f to -f 1. As the arc in- creases from 180 to 270, the co-versed-sine increases from -f 1 to -f 2. As the arc increases from 270 to 360, the co-versed-sine decreases from -f 2 to + 1. Hence, the limiting values of the co-versed-sine are, covers == -f 1, covers 90 === -f 0, covers 180 == -f 1, covers 270 = -f 2, covers 360 == -f 1. What are the least and greatest values of the co- versed-sine, and what are the corresponding arcs? Trace the arcs from to 360, and the changing functions. 34 TRIGONOMETRY. 43. The Tangent of an Arc. The tangent of an arc is the perpendicular to the primary diameter, produced from the primary origin, till it meets the prolongation of the diameter through the terminus of the arc. OR is the tangent of the arcs OT and OT". OR is the tangent of the arcs OT' and OT'". The arcs OT and OT" are in the first and third quadrants, respectively, and their tangent, OR, is estimated upward, and is therefore positive; hence, The tangent of an arc in the first or third quadrant is- positive. The arcs OT' and OT'" are in the second and fourth quadrants, respectively, and their tangent, OR', is es- timated downward, and is therefore negative; hence, The tangent of an arc in the second or fourth quadrant is negative. Let the arc OT 7 be equal to the arc T'P. Then, since T'P is the supplement of OT', OT will be the supplement of OT'; but the arc T'"0 is the sup- plement of OT 7 '; hence, OT=T'"0, and the angle OCT is equal to the angle OCT'". The angle COR is equal to the angle COR', since each is a right angle. Hence, the two triangles COR and COR have two angles, and the included side of the one equal to two angles and the included side of the other, each to each, and are therefore equal in all their parts. Hence, OR, opposite the angle OCR, is equal to OR, opposite the equal angle OCR. Since OR is esti- mated upward, and OR' downward, they have contrary signs ; hence, OR - - OR. But OR is the tangent FUNCTIONS. 35 of the arc OT, and OR is the tangent of the arc 07", the supplement of OT; hence, The tangent of an arc is equal to minus the tangent of its supplement. The tangent of is 0. As the arc increases from to 90, the tangent increases from to -f oo. As the arc increases from 90 to 180, the tangent passes through oc, changes its sign from -f- to , and de : creases numerically, but increases algebraically from -oo to 0. As the arc increases from 180 to 270, the tangent passes through 0, changes its sign from - to -f , and increases from -j- to -f oc. As the arc increases from 270 to 360, the tangent passes through oo, changes its sign from -|- to , and decreases nu- merically, but increases algebraically from oo to 0. Hence, for the limiting values of the tangent we have tan = 0, tan 90 = : -f oo, tan 180 = 0, tan 270 = -f oo, tan 360 = 0. 44. The Co-tangent of an Arc. The jo-tangent of an arc is the perpendicular to the secondary diameter, produced from the secondary origin, till it meets the prolongation of the diameter through the terminus of the arc. O^S is the co-tangent of OT and OT". O'S' is the co-tangent of OT' and OT". The arcs OT and OT" are in the first and third quadrants, respectively, and their co-tangent, 0'$, is estimated to the right, and is therefore positive; hence, The co-tangent of an arc in the first or third quadrant is positive. The arcs OT' and OT'" are in the second and fourth quadrants, respectively, and their co-tangent, O'S', is es- timated to the left, and is therefore negative; hence, 86 TRIG OSOMETR Y. Tlic co-tangent of an arc in the second or fourth quadrant is negative. The word co-tangent is an abbreviation of complementi tangens, the tangent of the complement. In fact, O'S, the co-tangent of OT, is the tangent of O'T, the com- plement of OT; hence, The co-tangent of an arc 'is the tangent of its complement. OR, the tangent of OT, is the co-tangent of O'T, the complement of OT; hence, The tangent of an arc is the co-tangent of its complement. Let the arcs OT and T'P be equal. Then, since T'P is the supplement of OT', OT will be the supple- ment of OT'. The arcs O'T and O'T' are equal, since they are complements of the equal arcs OT and T'P; hence, the angles O'CT and O'CT', measured by these equal arcs, are equal. The angles CO'S and GO'S' are equal, since each is a right angle. Hence, the two triangles CO'S and COS' have the common side CO', and the two adjacent angles equal, and are therefore equal in all their parts; and O'S, opposite the angle O'CS, is equal to O'S', opposite the equal angle O'CS'. Since O'S is estimated to the right, and O'S' to the left, they have contrary signs; hence, O'S = -O'S'. But O'S is the co-tangent of OT, and O'S' is the co- tangent of OT', the supplement of OT; hence, The co-tangent of an arc is equal to minus the co-tangent of its supplement. The co-tangent of is -foe. As the arc increases from to 90, the co-tangent decreases from -f GO to -J- 0. As the arc increases from 90 to 180, the co- tangent passes through 0, changes its sign from -f to , and increases numerically, but decreases algebra- ically from to -- x. As the arc increases from 180 to 270, the co-tangent passes through cc, changes FUNCTIONS. 37 its sign from to -f? an d decreases from -f- oo to -fO. As the arc increases from 270 to 360, the co-tangent passes through 0, changes its sign from -j- to , and increases numerically, but decreases algebraically from -0 to oo. Hence, the limiting values of the co-tangent are cot = -f- oo, cot 90 s= -f 0, cot 180 = oo, cot 270 = 4-0, cot 360 oo. 4:5. The Secant of an Arc. The secant of an arc is the line drawn from the center of. the circle to the terminus of . the tangent. CR is the secant of OT and OT". CR is the secant of OT' and OT'". The arcs OT and OT'" are in the first and fourth quadrants, respect- ively, and their secants, CR and CR' are estimated from the center toward the termini of the arcs, and are therefore positive ; hence, The secant of an arc in the first or .fourth quadrant is positive. -The arcs OT' and OT" are in the second and third quadrants, respectively, and their secants, CR' and CR, are estimated from the center, from the termini of the arcs, and are therefore negative; hence, The secant of an arc in the second or third quadrant is negative. Let the arcs OT and T'P be equal. Then, since T'P is the supplement of OT', OT is the supplement of OT'; but T'"0 is the supplement of OT'; therefore, T'"0 is equal to OT, and the angle T'"CO, measured by T'"0, is equal to the angle OCT, measured by the equal arc OT. The right angles COR and COR' are 38 TRIGONOMETRY. equal. Hence, in the triangles having the common side CO, and the two adjacent angles equal, CR is equal to CR'; but CR, the secant of OT, is positive; and CR', the secant of OT', the supplement of OT, is negative; hence, CR= CR; hence, The secant of an arc is equal to minus the secant of its supplement, The secant of is -f 1. As the arc increases from to 90, the secant increases from -}- 1 to -j- oo. As the arc increases from 90 to 180,^ the secant passes through GO, changes its sign from -f to , and de- creases numerically, but increases algebraically from oo to 1. As the arc increases from 180 to 270, the secant increases numerically, but decreases alge- braically from 1 to oo. As the arc increases from 270 to 360, the secant passes through oo, changes its sign 'from -- to -j-, and decreases from -f oo to -J- 1. Hence, for the limiting values of the secant we have sec P= -f 1, sec 90 fc: -f oo, sec 180 = 1, sec 270 = oo, sec 360 = -f 1. 46. The Co-secant of an Arc. The co-secant of an arc is the line drawn from the center of the circle to the terminus of the co-tangent. CS is the co-secant of OT and OT". CS' is the co-secant of OT' and OT'". The arcs OT and OT' are in the first and second quadrants, respect- ively, and their co-secants CS and CS' are estimated from the center toward the termini of the arcs, and are therefore positive; hence, The co-secant of an arc in the first or second quadrant is positive. FUNCTIONS. 39 The arcs OT" and OT'" are in the third and fourth quadrants, respectively, and their co-secants, CS and CS', are estimated from the center and the termini of the arcs, and are therefore negative; hence, The co-secant of an arc in the third or fourth quadrant is negative. The word co-secant is an abbreviation of complement secans, the secant of the complement. In fact, CS, the co-secant of OT, is the secant of O'T, the complement of OT; hence, The co-secant of an arc is the secant of its complement. CR, the secant of OT, is the co-secant of O'T, the complement of OT; hence, The secant of an arc is the co-secant of its complement. Let the arcs OT and T'P be equal. Then, since T'P is the supplement of OT', OT will be the supplement of OT'. 0'T = O'T', since they are complements of equal arcs. Hence, the angle O'CT, measured by the arc O'T, is equal to the angle OCT', measured by the equal arc O'T'. The right angles, COS and COS', are equal. Hence, in the triangles having the common side CO', and the two adjacent angles equal, CS is equal to CS'; but CS is the co-secant of OT, and positive, and CS' is the co-secant of OT', and positive; hence, The co-secant of an arc is equal to the co-secant of its supplement. The co-secant of is -j- oo. As the arc increases from to 90, the co-secant decreases from -|- oo to -f-1. As the arc increases from 90 to 180, the co-secant in- creases from + 1 to -f- oo. As the arc increases from 180 to 270, the co-secant passes through oo, changes its sign from + to , and decreases numerically, but increases algebraically from oo to 1. As the arc increases from 270 to 360, the co-secant increases 40 TRIGONOMETRY. numerically, but decreases algebraically from 1 to oo. Hence, the limiting values of the co-secant are cosec = 4- oo, cosec 90 == 4 1, cosec 180 =; -f ex, cosec 270 = 1, cosec 360 = - oo. To aid the memory, and for convenience of reference, we give the following tabular summaries: 47. Signs of the Circular Functions. Functions. Istq. 2d q. Uq. 4th q. sine. 4 4 co-sine. + 4 versed-sine. 4 4 4 + co- versed-sine. 4- - + t + tangent. 4 + co-tangent. 4 4 - secant. 4 H- co-secant. 4 Ht 48. Limiting Tallies of the Circular Functions. 90 180 270 360 sin = + sin -=4 1 sin ==40 sin = 1 sin = -0 cos -f- 1 cos - = 40 cos = - 1 cos = cos =4 "A vsin -{- vsin 4 1 vsin=4-2 vsin 4 1 vsin=r40 cvs =^ -j- 1 cvs -=40 cvs =41 cvs =42 cvs =4 1 tan = + tan =4 oc tan = tan ==4~. tan = -0 cot = 4 cot = = 40 cot = oo cot =+0 cot = co sec = + 1 sec =4~ ' sec = ~~ 1 sec = oo sec =4 A cose = -f oo cose =^4 1 1 cose=4 oc! cose = - 1 cose= oo NATURAL FUNCTIONS. 41 49. Problem. To find any function of an angle to the radius R, in terms of the corresponding function of the same angle to the radius 1, and the reverse. Let sin O l denote sin C to the radius CT=l, and sin C R denote sin C to the radius CT' = R. From similar triangles, CT : CT' : : MT : M'T', or 1 : R : : sin C l : sin O R . .-.(I) sinC JI = sinC' 1 X^. - ' (2) sin Let formulas for other functions be deduced; hence, 1. Any function of an angle to the radius R is equal to the corresponding function of the same angle to the radius 1, multiplied by R. 2. Any function of an angle to the radius 1 is equal to the corresponding function of the same angle to the radius R, divided by R. TABLE OF NATURAL FUNCTIONS. 50. Description of the Table. This table gives, to the radius 1, the values of the sine, co-sine, tangent, and co-tangent, to five decimal places, for every 10' from to 90. For sines and tangents, the degrees are given in the left column, and the minutes at the top. For co-sines and co-tangents, the degrees are given in the right-hand column, and the minutes at the bottom. S. N. 4. 42 TRIGONOMETRY. 51. Problem. To find the natural sine, co-sine, tangent, or co-tangent of a given arc or angle. Let us find the natural sine of 35 42' 24". The difference between the natural sines of 35 40' and 35 50', as given in the table, is .00236. Now 2' 24" = .24 of 10', which is found thus : 60 10 24 2.4 .24 Then take Nat sin 35 40'= .58307 Correction for 2' 24" = .00236 X .24 = .00057 .-. Nat sin 35 42' 24"= .58364 In case of co-sine or co-tangent, the correction must be subtracted, since, between and 90, the greater the angle, the less the co-sine and co-tangent. 52. Examples. 1. Find the natural sine of 75 45' 30". Ans. .96927. 2. Find the natural co-sine of 15 36' 12". Ans. .96315. 3. Find the natural tangent of 43 33' 18". Am. .95079. '4. Find the natural co-tangent of 84 28' 30". Ans. .09673. 53. Problem. To find the angle corresponding to a given natural sine, co-sine, tangent, or co-tangent. 1. Find the angle corresponding to the natural sine .50754. Looking in the table we find the angle 30 30'. LOGARITHMIC FUNCTIONS. 43 2. Find the angle whose natural sine .82468. The next less sine, sin 55 30' = .82413. Difference = 55 Difference corresponding to 10' = 164 . . Correction = 10' X 7^ = 3' 21". 164 . . Angle = 55 30' + 3' 21" = 55 33' 21". In case of co-sine and co-tangent, the angular differ- ence must be subtracted, since the greater the co-sine or o-tangent, the less the angle, for values between and 90. 54. Examples. 1. Find the angle whose sine is .75684. Ans. 49 11' 13". 2. Find the angle whose co-sine is .67898. Ans. 47 14' 10". 3. Find the angle whose tangent is 1.34567. Ans. 53 22' 59". 4. Find the angle whose co-tangent is .98765. ' Ans. 45 21' 22". TABLE OF LOGARITHMIC FUNCTIONS. 55. Description of the Table. The table of logarithmic functions gives to the radius 10,000,000,000 the logarithm of the sine, co-sine, tangent, and co-tangent, for every minute, from to 90. The expression, logarithmic sine, tangent, etc., is equiv- alent to the logarithm of the sine, of the tangent, etc. For sines and tangents, the degrees are given ut the top of the page, and the minutes in the left-hand column. 44 TRIO ONOMETR Y. For co-sines and co-tangents, the degrees are given at the bottom of the page, and the minutes in the right- hand column. The columns marked D 1" contain the difference for 1". 50. Problem. Find the logarithmic sine of 48 25' 30". log sin 48 25'= 9.87390. D 1" = .19. . . Correc. for 30" = .19 X 30 = 6 . . log sin 48 25' 30" =9.87396 In case of co-sine or co-tangent, the correction must be subtracted, since between and 90, the greater the angle, the less the co-sine and co-tangent. 57. Examples. 1. Find the logarithmic sine of 75 35'. Am. 9.98610. 2. Find the logarithmic sine of 25 40' 24". Ans. 9.63673. 3. Find the logarithmic co-sine of 29 55' 55". Ans. 9.93782. 4. Find the logarithmic tangent of 50 50' 50". Ans. 10.08927. 5. Find the logarithmic co-tangent of 65 45' 30". Am. 9.65349. 58. Problem. To find the angle corresponding to a given logarithmic sine, co-sine, tangent, or co-tangent. LOGARITHMIC FUNCTIONS. 45 Find the angle whose logarithmic sine = 9.84567 For next less we have sin 44 3(X = 9.84566 D 1" = .21 . . Correc. = 1" X = 5", .21)1.00(5. .". Angle = 44 30' 05". In case of co-sine and co-tangent, the correction for seconds must be subtracted, since the greater the co- sine or co-tangent, and consequently the greater the logarithm, the less the angle for values between and 90. 59. Examples. 1. Find the angle whose logarithmic sine is 9.98437. Ans. 74 43' 17". 2. Find the angle whose logarithmic co-sine is 9.78456. Ans. 52 29' 19". 3. Find the angle whose logarith. tangent is 10.12346. Ans. 53 02' 11". 4. Find the angle whose logarith. co-tangent is 9.99999. Ans. 45 00' 03". <>0. Problem. Given any natural function, to find the corresponding logarithmic function. 1st SOLUTION. Find from the natural function the corresponding angle; then, from, the angle, the corresponding loga- rithmic function. 2d SOLUTION. Let a denote any arc or angle, /()i any function of a to the radius 1, and /(a)* the corresponding 4 6 TRIG ONOMETR Y. function of a to the radius R. Then, by article 49 we have, /(a)*=/(o)i X R. Substituting the value of R in the second member, f(d) s =f(a) l X 10,000,000,000. log /(a)* = log /(a) i + 10. Hence, Add 10 ta the logarithm of the natural function. 61. Examples. 1. Given nat. sin a .98457, required a and iog sin a. Ans. a = 79 55' 25", log sin a = 9.99325. 2. Given nat. cos a .63878, required a and log cos a. Ans. a = 50 IT 52", log cos a = 9.80536. 3. Given nat. tan a = 1.68685, required a and log tan a. Ans. a = 59 20 23", log tan a = 10.22708. 4. Given nat. cot a = 1.41987, required a and log cot a. Ans. a == 35 09' 24", log cot a = 10.15225. 62. Problem. Given any logarithmic function, to find the corresponding natural function. 1st SOLUTION. Find from the logarithmic function the correspond- ing angle ; then, from the angle, the corresponding natural function. 2d SOLUTION. From article 49 we have, /wp^fp- .-. \oef(a), = \osf(a) K 10. RIGHT TRIANGLES. 47 Hence, Subtract 10 from the logarithmic function, and find the number corresponding to the resulting logarithm. 63. Examples. 1. Given log sin a 9.87654, required a and nat sin a, Ans. a =48 48' 44", nat. sin a = .75255 2. Given log cos a = 9.84877, required a and nat cos a. Am. a = 45 05' 41", nat. cos a =.70595 3. Given log tan a = 10.22708, required a and nat tan a. Ana. a = 59 20' 23", nat. tan a = 1.68685 4. Given log cot a = 10.15225, required a and nat cot a. Ans. a = 35 ' 24", nat. cot a = 1.41987. RIGHT TRIANGLES. 64. Principles. PB . PK : : HB : MK, or h : 1 : : p : sin P. BP : R : : HP : SR, or h : 1 : : b : sin B. .'. (2) sn = 1. Either side adjacent to the right angle is equal to the sine of the opposite angle multiplied by the hypotenuse. 2. The sine of either acute angle is equal to the opposite side divided by the hypotenuse. 48 TRIGONOMETRY. Since the angles P and B are complements of each other, sin P=cos B, and sin B cos P; . . (1) and (3) become, (3.) 4 ]- and (4) 3. Either side adjacent to the right angle is equal to the co-sine of the adjacent acute angle multiplied by the hypot- enuse. 4. The co-sine of either acute angle is equal to the adja- cent side divided by the hypotenuse. PR : PN : : HB : NL, or b : 1 : : p : tan P BH : BT : : HP : TQ, or p : 1 : : b : tan B. tan P = 2-. (6M t n tan B = . 5. Either side adjacent to the right angle is equal to the tangent of the opposite angle multiplied by the other side. 6. The tangent of either acute angle is equal to the oppo- site side divided by the adjacent side. Since the angles P and B are complements of each other, tan P=cot B, and tan B = cot P; .-. (5) and (6) become, 'cotB = . (7) ^ V and (8) ' P 7. Either side adjacent to the right angle is equal to the co-tangent of the adjacent acute angle multiplied by the other side. RI&HT TRIANGLES. 49 8. The co-tangent of either acute angle is equal to the adjacent side divided by the opposite side. BH : BT : : BP : BQ, or p : 1 : PH : PN : : PB : PL, or b : 1 : secP 9. Either side adjacent to the right angle is equal to the hypotenuse divided by the secant of the adjacent acute angle. 10 The secant of either acute angle is equal to the hypot- enuse divided by the adjacent side. Since the angles B and P are complements of each other sec B = cosec P, sec P cosec B; . ' . (9) and (10) become, V r p = - > cosec P= P I and (12) J cosec = -J- cosec B J b 11. Either side adjacent to the right angle is equal to the hypotenv.se divided by the co-secant of the angle opposite that side. 12. The co-secant of either acute angle is equal to the hypotenuse divided by the side opposite that angle. Scholium. By some authors, principles 2, 4, 6, 8, 10, and 12, have been given in the form of definitions. Introducing radius into these formulas, by substitut- ing for any function to the radius 1, the corresponding function to the radius R divided by R, and reducing, we have: S. N. 5. 50 (1) TRIGONOMETRY. h sin P (2) sin = sin B = (3) (4) cos D cos P = -= h (5) P = R (6) Rb tan E = (7) b cot B (8) cot = ^r- cot P= (9) (10) sec = cosec (12) cosec P= cosec =^ P Rh Applying logarithms to these formulas, we have: (1) f lo g P log A + log sin P 10. 1 I log b log h 4- log sin 10. J f log sin P=10-f log p log h. 1 I log sin B = 10 + log b log h. J f log p = log h -f log cos 5 10. ) I log 6 = log h -f log cos P 10. j RIGHT TRIANGLES. 51 ( ^ ( log cos B = 10 + log p log h. } \ log cos P = 10 -f log b log h. ( ^ ( log jp = log 6 -f log tan P 10. 1 I log 6 == log p + log tan JB 10. / (&) ( log tan P= 10 -f log JD - log 6. \ 1 log tan B = 10 + log b log p. J ^ ( log p = log 6 + log cot 5 10. ) (log b = log p + log cot P 10. J (8) ( log cot 5 = 10+ log j>-log b. ) ( log cot P = 10 + log b log p. J /9) f lo g ;> = = 10 + log h log sec B. ) 1 log 6 10 + log h log sec P. j (10) / log sec B ^ 10 + log h ~~ log p ' \ \ log sec P = 10 + log h log 6. J (11) / log ^ ~ 10 + log ^~~ lo S cosec | log 6 = 10 + log h log cosec B p. 1 b. f ^ og cosec P : =10+ log h -- log log cosec J5= 10 + log ^ -- log 65. Case I. Given the hypotenuse and one acute angle, required the remaining parts. B (B. 1. Given /^ = 365 - I RequirJp. lP^3312'.J 5 = 90 P= 90 33 12' = 56 48'. Either side adjacent to the right angle is equal to the sine of the opposite angle, multiplied by the hypotenuse. . - . = h sin P. 52 TRIGONOMETRY. i > T- h sin P Introducing radius, we have, p ^~ A Applying logarithms, we have, log p = log h -f log sin P 10. log h (365) = 2.56229 log sin P (33 12') = 9.73843 log p = 2.30072 . . jt> = 199.85. In like manner, from either formula, b = h sin B, or b = h cos P, we find b = 305.41. rP^ 40 47' 40". ir. I b = _ 2 G . ^ f 5 = 62 21' 10". 3. Given {p^'^ } RequiJ ^=1018.512. IP^27 3850J 1^1944.364. 66. Case II. Given the hypotenuse and one side adjacent to the right angle, required the remaining parts. r - 1. Given/ h = 112 ' I Required^ A I p. *7. J The sine of either acute angle is equal to the opposite side divided by the hypotenuse. .-. sin P=4' h Introducing radius, and multiplying by J?, we have, r> HP sin P=--. EIGHT TRIANGLES. 53 Applying logarithms, we have, log sin P 10 + log p log h. log p (97) =1.98677 log h (112) = 2.04922 log sin P =9.93755 .-. P= 60 00' 17". B = 90 P= 90 60 00' 17" ** 29 59' 43". b = h sin B, or b = h cos P, .'. 6 .= 55.991. We can also find b as follows: b = l/A 2 p 2 = 1/(A 4- p) (h p). log 6 = i [ log (A + P) 4- log (^ *>)] (5 = 25 47' 07". 2. Given { * =r 7269 ' 1 Required <{ P- 64^ 12' 53". I p = 6545. rP-=19 43' 36". 3. Given / h " : 44 ^ 4 1 Required 4cB^ 70 16' 24". i*=H j I 6= 418.33. 67. Case III. Given one side adjacent to the right angle and one angle, required the remaining parts. = 152.67. (B ' U. 90 _ p= 90 - 50 18' 32" = 39 41' 28". 54 TRIGONOMETRY. Either side adjacent to the right angle is equal to the tan- gent of the opposite angle multiplied by the other side. . . p = b tan P. Introducing radius and applying logarithms, as in the preceding cases, we find p = 183.95. Either side adjacent to the right angle is equal to the co-sine of the adjacent acute angle multiplied by the hypotenuse. b b = h cos P; . . h = cos P Introducing radius and applying logarithms, as above, we shall find h = 239.05. 2 Given f p== 3963 ' 35 miles = the earth's radius. I P= 57' 2.3"= the moon's horizontal parallax. Required A, the distance of the moon from the earth. Ans. h =238889 miles. 3 Q.- f P 3963.35 miles = the earth's radius. \ P= 8.9" = the sun's horizontal parallax. Required h, the distance of the sun from the earth. Ans. h = 91852000 miles. O Q Scholium. Sin 8.9" = sin 1' X . |jsr- . - log sin 8.9"= log sin I'-f log 8.9 + a.c. log 60 10. 68. Case IT. Given the two sides adjacent to the right angle, required the remaining parts. OBLIQUE TRIANGLES. 55 The tangent of either acute angle is equal to the opposite #ide divided by the adjacent side. Introducing radius and applying logarithms, we shall find that P=38 13' 28". B = 90 P = 90 38 13' 28" = 51 46' 32". Either side adjacent to the right angle is equal to the sine of the opposite angle multiplied by the hypotenuse^ . . p = h sin P. . * . h= . .. sm P Introducing radius and applying logarithms, we find h = 47.466. ( P=4 44' 37". 2. Given-/ * = ^73. 1 Required < 5-85 15' 23". to =oo72il. ) I , OA ^ I A =8401 3. Given/ p " J' } Required X B= 45 33' 43". (6 lUO. J \iiAA nr* (P^4739'0r. 4. Given J f: V Required^ 5^42 20' 53". I 6 = 15/5. J . rt000 , th= 144.253. I B = 42 2(X ^=2338.1. OBLIQUE TRIANGLES. 69. Case I. Given one side and two angles, required the remaining parts. Let ABC be an oblique triangle, and let the sides opposite the angles A, Bj and C be denoted respectively A by a, b and c. 56 TRIGONOMETRY. Let the angles A and B and the side a be given, and the angle C and the sides b and c be required. We find C from the formula, <7=180 (.4 + 5). Draw the perpendicular p from the vertex C to the side , thus forming two right triangles. There are two cases: 1st. When the perpendicular falls on the side c. From the principles of the right triangle we have, p = b sin A and p = a sin B. . ' . b sin A == a sin B. (1) sin A : sin B : : a : b. 2d. When the perpendicular falls on c produced. p = b sin A and p a sin CBD. But CBD is the supplement of CBA, or B of the triangle. Since the sine of an angle is equal to the sine of its supplement, sin CBD = sin B; .- . p= a sin B. .'. b sin A = a sin B. (1) sin A : sin B : : a : b. In like manner we may find, (2) sin A : sin C : : a : c. Hence, The sine of the angle opposite the given side is to the sine of the angle opposite the required side as the given side is to the required side. Introducing radius by substituting for the function to the radius 1, the corresponding function to the OBLIQUE TRIANGLES. 57 radius R divided by J?, and reducing, the proportions (1) and (2) will be of the same form as before substi- tution, and hence are true for any radius. From proportions (1) and (2), we find, *L , a sin B a sin C (3) b = r -r- (4) c = ^ -T- sin ^4 sin ^4 Applying logarithms to (3) and (4), we have, (5) log b log a --f- log sin B -j- a. r. log sin A 10. (6) log c = log a -f log sin C -\- a. c. log sin A 10. 70. Examples. ( ,4 = 35 45'. 1. Given < B = 45 28'. I a = 7985. C= ISO (A+ B) = 18Q 81 13' = 98 47'. Since the sine of the angle opposite the given side is to the sine of the angle opposite the required side as the given side is to the required side, we have the proportion, . _ j r a sin B sin A : sin B:\a\b, . ' . b = : sin A . . log b = log a -\- log sin B -f a. c. log sin A 10. log a (7985) = 3.90227 log sin B (45 28') == 9.85299 a.c. log sin A (35 45') == 0.23340 log b - 3.98866 . . b = 9742.25. In like manner we have the proportion, a sin C sin A : sin C : : a : c, . ' . c = - j- Sill & 58 TRIGONOMETRY. .'. log c = log a + log sin C -f <* log sin A 10. log a (7985; == 3.90227 log sin C (98 47') = 9.99488 a. c. log sin A (35 45') = 0.23340 log c = 4.13055 . . c = 13506.88. In finding log sin 98 47', take the supplement of 98 47, which is- 81 13', and find log sin 81 13'. f A = 50 30' 40". \ c C = 58 43' 50". 2. Given <B = 70 45' 30". V Req. < b = 585.2 yd. I a = 478.35 yd. ) ( c = 529.8 yd. f B = 65 25' 35". ^ ( A = 54 05' 51". 3. Given I C= 60 28' 34". V Req. < c = 11.72 miles. I b = 12.25 miles. J I a = 10.91 miles. 71. Case II. Gicen two sides and an angle opposite one of them, re- quired the remaining parts. 1. WHEN THE GIVEN ANGLE is ACUTE. Let the sides a and 6 and the angle A be given, and the remaining parts be required. Let the perpendicular p be drawn from C to the opposite side. Then we shall have, p = b sin A. l.sl If a > p and a < b, there will be two solutions. .For, if with C as a center and a as radius a circum- ference be described, it will intersect the side opposite C in two points, B and &, and either triangle, ABC or A B'C will fulfill the conditions of the problem, since OBLIQUE TRIANGLES. 59 it will have two sides and an angle opposite one of them the same as those given. Hence, there will be two solutions if a has any value between the limits p and b. 2d. If a = p, there will be but one solution. For, as a diminishes and approaches A- p, the two points B and B' approach ; and if a = p, B and B' will unite, the arc will be tan- gent to r, and the two triangles will become one, and there will be one solution. 3d If a 6, there will be but one solution. For, as a increases and approaches /;, the points B and B' separate, the triangle ABC increases, and the triangle AB'C decreases; and when a becomes equal to b, the triangle AB'C van- ishes, and there remains but one triangle, or there is but one solution. 4th. If a > b, there will be but one solution. For, although there are two tri- angles ABC and AB'C, the latter is excluded by the condition that the given angle A is acute, since CAB' is obtuse, and there remains but one triangle ABC which satisfies the conditions, or there is but one solution. 5th. If a < p, there will be no solution. For the arc described with C as center and a as radius will neither intersect the oppo- site side nor be tangent to it. The triangle can not be constructed, or there will be no solution. 60 TlilG OXOMETR Y. 2. WHEN THE GIVEN ANGLE is OBTUSE. 1st. If a > b there will be but one solution. For, although there are two triangles ABC and AFC, the latter is excluded by the condi- tions of the problem, since the angle CAB' is acute while the given angle is obtuse. There remains but one triangle, ABC, which satisfies all the conditions of the problem, or there is but one pos- sible solution. 2d. If a . = b there will be no solution. For as a diminishes and approaches 6, B will ap- proach A ; and when a becomes equal to 6, B will unite with J, and the triangle ABC will vanish. The tri- angle AB'C will remain, but will be excluded by the conditions of the problem, since the angle CAB' is acute while the given angle is obtuse. 3(1. If a << b there will be no solution If a > p there will be two tri- angles, ARC and AB"C, but both are excluded by the condition that the given angle is obtuse. If a=p the two triangles reduce to one, right-angled at B, which is excluded by the condition that the given angle is obtuse. If a << p no triangle can be con- structed with the given parts, and there will be no solution. OBLIQUE TRIANGLES. 61 72. Summary of Results. 1. When A < 90. Two Solutions, If a > p and a < ft. r Is*. If a=p. One Solution, -< 2rf. If a = b. I 3d. If a > 6. No Solution, If a < _p. 2. When A > 90. One Solution, If a > ft. / 1st. If a = ft. No Solution, | ^ Ifa <6> 73. Method of Computation. Reversing the order of the couplets of the proportion m Case I, we have (1) a : ft : : sin A : sin B. Hence, The side opposite the given angle is to the side opposite the required angle, as the sine of the given angle is to the sine of the required angle. ft sin A (1) gives (2) sin B = .-. (3) log sin B= log ft -f log sin A -\- a. c. log a 10. If there is but one solution, take from the table the angle B corresponding to log sin B; if there are two solutions, take B and its supplement 5', for both cor- respond to log sin B. We find C from the formula, C 180 -.(A + B) or C= 180 (A -f B'). 62 TRIGONOMETRY. We find c from the proportion, sin A : sin C :: a : r, ^ c = ^5-- sm J. . . log c = log a -j- log sin C -j- a. c. log sin A 10. 74. Examples. ( a = 9.25. ^ (B. 1. GivJ ft = 12.56. I Req.< C. I ,4 = 30 25'. J t A p = b sin A Introducing 7? and applying logarithms, we have log p = log 6 -f- log sin A 10. log b (12.56) = 1.09899 log sin A (30 25') = 9.70439 log p = 0.80338 . . p = 6.3589. Since a > p and a < 6, there are two solutions. Since the side opposite the given angle is to the side opposite the required angle as the sine of the given angle is to the sine of the required angle, we have the proportion, n n b sin A a : 6 : : sin A : sin B, . . sin B = a log sin B = log b -f log sin A -j- a. c. log a 10. log b (12.56) = 1.09899 log sin ^(30 25') = 9.70439 a. c. log a (9.25) = 9.03386 log sin B =9^3724 . .. 1 B = 43'25'41". 1 5'= 136 34' 19". OBLIQUE TRIANGLES. C== 180 (A + B) = 106 9' 19", C ' = 180 (A + F) = 13 0' 41". a sin 63 sin A : sin C sn log c = log a -f log sin C -f- a. c. log sin .4 10. Taking the value of (7, we have, log a (9.25) ~? 0.96614 log sin C (106 9' 19") =9.98250 o.c. log sin A (30 25') =0.29561 log c =1.24425 .-. = 17.549. Taking the value of C", we have, log a (9.25) = 0.96614 log sin C' (13 0' 41") = 9.35246 a. c. log sin A (30 25') = 0.29561 log c = 0.61421 /- a =? 20.35. 2. Given I b == 20.35. Req. 3. Given A = 52 35' 27". a = 645.8. 6 = 234.5. ^ = 48 35'. . c = 4.1135. '=52 35' 27". '=74 49' 06". == 24.725. ^ 15 48' 04". Req 0= 115 36' 56". I c = 776.53. | o = 17. 4. Given < 6 = 40.25. I ^. = 27 43' 15". r a = 94.26. 5. Given X 6 = 126.72. Req. Req. No Solution. 38 52' 46". i4i r 14". c f 113 17' 14". \ 11 2' 46". _ f 185.439- "t 38.682. 64 TRIG OSOMETR Y. \ (A = 57 O'.SO". 6. Given -J 6 = 2000. V Req. < C = 11 44' 10". 15 = 111 15'. ) I c = 436.49. 75. Case III. Given tivo sides and their included angle, required the remaining parts. Fj Let ABC be a triangle, and let the sides opposite the angles A, B, C, be de- noted, respectively, by a, b, c. Let a and 6, and their included angle C, be given, and the remaining parts, A, B, and c, required. The sum of the angles A and B is found from the formula, A + B = 180 C. With C as a center, and b, the shorter of the two given sides, as a radius, describe a circumference cut- ting a in./), a produced in E, and c in H. Draw AE, AD, CH, and DF parallel to AE. The angle DAE is a right angle, since it is inscribed in a semi-circle; hence, its alternate angle, ADF, is also a right angle. The angle ACE being exterior to the triangle ABC, is equal to A -f B. But ACE having its vertex at the center, is measured by the intercepted arc AE, The inscribed angle ADE is measured by one-half the arc AE; hence, ADE = | ACE = \ (A -j- B). CH == CAj since they are radii of the same circle ; hence, the angle CHA = A. The angle CHA being exterior to the triangle CHB is equal to HCB -f- B ; hence, OBLIQUE TRIANGLES. 65 But HCB, having its vertex at the center, is meas- ured by the intercepted arc DH; and DAF, being an inscribed angle, is measured by one-half the arc DH; hence, DAF = J HCB = (A B). In the right triangles ADE and ADF we have AE=AD tan ADE==--AD tan \(A + B). DF=AD tan DAF = AD tan $(A B). From the similar triangles, ABE and F.RD, we have BE : BD : : AE : DF. Since <7 = CA, BE = BC + CA == a -j- 6. Since CD=CA, BD= BCCA=a b. Substituting the values of BE, BD, AE, and DF in the above proportion, and omitting the common factor AD in the second couplet, we have a-t-6 : a b :: tan %(A + B) : tan %(A B). Hence, In any plane triangle, the sum of the sides in- chiding an angle is to their difference as the tangent of half the sum of the other two angles is to the tangent of half their difference. We find from the proportion, the equation . log tan i (AB) = log (a b) -f log tan -f a. r. log (a -f 6) 10. We have now found A + B and (A B. a sin C sin A : sin C : : a : r, . . c = : -r- sin A . log c = log-a -f- log sin C-f- a. c. log sin ^4 10. S. N. 6. 66 TRIGONOMETRY. 76. Examples. (a = 37.56. 1. Given < b = 23.75. lc=6825' A+B = 180 C= 111 35'. a + 6 : a b :: tan (^-|_) : tan ^_ .'. tan ^-B= ( " 6) te .'. log tan i(v4 B) = log (a 6) -f log tan + .a, c. log (a + b) 10. log (a 6) (13.81) : 1.14019 log tan $(A+B) (55 47' 30") = 10.16761 a. c. log (a+6) (61.31) =_8.21247 log tan i (^4 B) i 9^52027^ 5)^18 19' 55". 5) = 74 7' 25". B = ^ (4 + 5) - (4 - B) = 37 27' 35". sin ,4 : sinC:: a : c, .'. c = a * inC . sin A log c = log a -f log sin C -f- a. f . log sin A 10. log a (37.56) = 1.57473 log sin C (68 25') = 9.96843 a. c. log sin A (74 7' 25") = 0.01689 log c 1.56005, .-.c= 36.312. f a = 996.63. \ sA = 66 30' 37". 2. Given 1 b = 712.83. V Req. ^5 = 40 59' 35". I C= 72 29' 48". J I c = 1036.35. OBLIQUE TRIANGLES. 67 < b = 776.525. ^ f B = 115 36' 56". 3. Given 1 c = 234.5. V Req. 1 C = 15 48' 04". I .4 = 48 35'. J I a = 645.8. c a == 11.7209. \ fA = 60 25' 34". 4. Given j c = 10.9232. V Req. < C = 54 08' 51". ( B = 65 25' 35". J 1 6 == 12.256.- 77. Case IT. Given the three sides of a triangle, required the angles. Let ABC be a triangle, take the longest side for the base, and draw the perpendicular p from the vertex B to the base. Denote the segments of the base by s and s' respect- ively. Then,' (1) c 2 s' 2 =p 2 , and (2) a 2 s 2 =p 2 . . . (6) s -f- s' : a -f- c : : a c : s s'. Hence, The sum of the segments of the base is to the sum of the other sides as the difference of those sides is to the difference of the segments. (6)8iyeB(7).W=. (a + ?.< a - C) - .-. (8) log (s = log (a + c) + log (a c) -j- a. c. log (s -{- s') 10. In case the sides of the triangle are small, find s s' from (7); otherwise, it will be more convenient to em- ploy (8). 68 TRIG ONOMETR Y. Having s-\-s r and s s', we find s and s thus, (9) = .( + *' (11) cos ^= -, (12) cos C = - c C Introducing #, reducing, and applying logarithms, (13) log cos A =-- 10 -f log s' log c. (14) log cos C 10 -f- log s log a. From which we find A and C. Then, (15) B = 180 (4 + C). 1. Given < b = 150. U===ioa 78. Examples. .^| r^4. Req. < 5. a s -}- *' : a-j-c :: a c : s s'. , (a -f c) (a-c) 225 X 25 , ^"" = j (.5 -f- s') -f i (* s') = 75 + 18.75 ^r 93.75. s'r= I (x + s') i (s s') = 75 18.75 = 56.25. cos A ~ '- , or introducing 7?, cos A = -- c c . . log cos A = 10 -j- log -s-' log c. log s' (56.25) 4= 1.75012 log <r (100) |= 2.QQOOQ log cos ^l =9.75012 .;. ^4 = 55 46' 18". a IDn cos C= j or introducing 7?, cos (7= a a HEIGHTS AND DISTANCES. 69 . . log cos C = 10 -f log ,? log a. log s (93.75) == 1.97197 log a (125) = 2.09691 log cos C -9.87506 .-. C=41 24' 34". B = 180 (A + C) = 82 49' 08". ' ( a = 332.21. ^ fA = 66 30' 35". 2. Given < b = 345.46. V Required 1 B = 72 29' 53". I c = 237.61. 3 I C = 40 59' 32". ra = 864. 3. Given < b =^1308. f a = 251.25. 4. Given < 6 = 302.5. ( c = 342. rv4=,41 00' 38". Required^ 5 83 25' 14". lc=55 34' 08". (^^45 22' 41". Required ] B = 58 58' 20". ( C = 75 38' 59". APPLICATION TO HEIGHTS AND DISTANCES. 79. Definitions. 1. A horizontal plane is a plane parallel to the horizon. 2. A vertical plane is a plane perpendicular to a horizontal plane. 3. A horizontal line is a line parallel to a horizontal plane. 4. A vertical line is a line perpendicular to a hori- zontal plane. 5. A horizontal angle is an angle whose plane is horizontal. 70 TRIGONOMETRY. 6. A vertical angle is an angle whose plane is vertical. 7. An angle of elevation is a verticle angle, one of whose sides is horizontal, and the inclined side above the hori- zontal side. Thus, BAG. 8. An angle of depression is a vertical angle, one of whose sides is horizontal, and the inclined side below the horizontal side. Thus, DCA. 80. Problems. 1. Wishing to know the height of a tree standing on a horizontal plane, I meas- ured from the tree the hori- zontal line BA, 150 ft., and found the angle of elevation, BAC, to the top of the tree to be 35 20'. Required the height of the tree. Ans. 106.335 ft. 2. In surveying a tract of land, I found it impractic- able to measure the side AB on account of thick brush- wood lying between A and B. I therefore measured AE, 7.50 ch., and EB, 8.70 ch., and found the angle AEB = 38 46'. Required AB. Ans. 5.494 ch. 3. One side of a triangular field is double another, their included angle is 60, and the third side is 15 ch. Required the longest side. Am. 17.32 ch. HEIGHTS AND DISTANCES. 71 4. Wishing to know the width of a river, from the point A on one bank to the point C on the other bank, I measure the distance AB, 75 yd., and find the angle BAC = 87 28' 30", and the angle ABC = 47 38' 25". Required AC, the width of the river. Ans. 78.53 yd. 5. I find the angle of elevation, BA C, from the foot of a hill to the top to be 46 25' 30". Measuring back from the hill, AD = 500 ft., I find the angle of elevation ADC =25 38' 40''. Required BC, the vertical height of the hill. Ans. 441.87 ft. 6. From .the foot of a tower standing at the top of a declivity, I measured AB = 45 ft., and the angle ABD = 50 15'. I also measured, in a straight line with AB, BC=68 ft., and the angle BCD = 30 45'. Required AD, the height of the tower. Ans. 82.94 ft. 7. Wishing to know the height of a tower standing on a hill, I find the angle of elevation, BA C, to the top of the hill to be 44 35', and the angle of ele- vation to the top of the tower to be 59 48'. Measuring the horizontal line AE, 275 ft., I find the angle of eleva- 72 TRIGONOMETRY. tion to the top of the tower to be 46 25'. Required the height of the tower. Ans. 317.143 ft. DC = 24 ch. CDB = 45. 8. Given Required 9. Given DCA =48. ACB =60. = 38.61 ch. = 800 yd., 4C=600 yd ADC=ZZ 45', BDC =22 30'. Re- quired DA, DC, DB. Ans. 4=710.15 yd., DC= 1042.5 yd., DB = 934.28 yd. Remark. Describing the circumfer- ence through 4, B, D, and drawing AE and BE, EAB = BDC, EBA = ADC. RELATIONS OF CIRCULAR FUNCTIONS. 81. Fundamental Formulas. Let a = the angle OCT = the arc OT, and CO = = 1. Then, we have MT = CN= sin a, AT C3f = cos a, MO = vers a, JVO' = covers a, OR = tan a, O'S' = cot a, CR = sec a, CS=cosec a. By articles 3946, sin (90 a) = cos a, cos (90 a) = sin a, etc. From the diagram we have o' + CM =CT Substituting the values of MT, CM, and CT, we have (1) sin 2 o +cos 2 a = l. CIRCULAR FUNCTIONS. 73 Hence, The square of the sine of any arc plus the square of its co-sine is equal to 1. From (1) we have, by transposition, (2) sin 2 a= 1 cos 2 a, (3) cos 2 a = 1 sin 2 a. Hence, 1. The square of the sine of any arc is equal to 1 minus the square of its co-sine. 2. The square of the co-sine of any arc is equal to 1 minus the square of its sine. From the diagram we have MO = CO CM. Substituting the values of MO, CO, and CM, we have (4) vers a 1 cos a. Hence, The versed-sine of any arc is equal to 1 minus its co-sine. . . vers (90 a) = 1 cos (90 a). . . (5) covers a 1 sin a. Hence, The co-versed-sine of any arc is equal to 1 minvx its sine. From the. diagram we have CM : CO :: MT : OR, or cos a : 1 : : sin a : tan a. ,, sin a .' . (6) tan a = cos a Hence, The tangent of any arc is equal to its sine divided by its co-sine. S. N. 7. 74 TRIGONOMETRY. cos a . . (7) cot a = -r --- sin a Hence, The co-tangent of any arc is equal to its co-sine divided by its sine. (6) X (7) = (8) tan a cot a = 1. Hence, The tangent of any arc * into its co-tangent is equal to 1. 1 (8) -f- cot a = (9) tan a = cot a Hence, The tangent of any arc is equal to the reciprocal of its co-tangent. 1 (8) -^ tan a = (10) cot a = tan a Hence, The co-tangent of any arc is equal to the recip- rocal of its tangent. CM : CO :: CT : CR, or cos a : 1 : : 1 : sec a. 1 . . (11) sec a cos a Hence, The secant of any arc is equal to the reciprocal of its co-sine. .'. sec (90 a) = . * . (12) cosec a cos (90 a) sin a Hence, The co-secant of any arc is equal to the recip- rocal of its sine. .'. (13) sec 2 a = l -f tan 2 a. CIRCULAR FUNCTIONS. 75 Hence, The square of the secant of any arc is equal to 1, plus the square of its tangent. .-. sec 2 (90 a) == 1 + tan 5 (90 a). . . (14) cosec 2 a = l -\- cot 2 a. Hence, The square of the co-secant is equal to 1, plus the square of the co-tangent. 82. Summary of Fundamental Formulas. 1. sin 2 a + COS 2/7 1 9. 1 cot a 2. sin 2 a, \ cos 2 a. 10 1 3. cos 2 a = 1 sin 2 a. tan a 4. vers a -cos a.. 11. 1 cos a 5 covers a 1 sin a. 19 1 LL. cosec a - ~ sin a p. , sin a . cos a 7. cot a - cos a 13. sec 2 a 1 -+- tan 2 a. sin a 8. tan a cot a = l. 14. cosec 2 a = 1 -h cot 2 a. 83. Problems. 1. Prove that the above formulas become homogene- ous by the introduction of R. 2. Deduce formulas (5), (7), (12) and (14) from the diagram. 3. Prove that the above formulas are true if a is in the second, third ? or fourth quadrant. 76 TRIGONOMETRY. 84. Each Function in Terms of the Others. sin sin sin sin sin sin sin cos cos cos cos cos COS COS ""] a 1/1 cos 2 a. vers a=lV 1 sin 2 a. . vers a=l cos a. a =1/2 vers a vers 2 a. a 1 covers a. tan a vers a = 1 l/ 2 cvs a cvs 2 a. vers a 1 1/1+ tan 2 a a 1/1+ -tan 2 a vers a - 1 Cot a VI -f cot 2 a V 14- cot 2 a sec a 1 vers a V 7 sec 2 a 1 sec a 1 sec a V cosec 2 a 1 VPTS (1 1 cosec a cosec a covers a = l sin a. a =1/1 sin 2 a. a = 1 vers a. covers a=l 1/1 cos 2 a. a =1/2 cvs a cvs 2 a. 1 cvs a = l 1/2 vs a vs 2 a. tan a l/l -f tan 2 a cot a d 1/1+ tan 2 a onvpi"^ <7 1 1/1-j- cot 2 a 1 1/1-f-COt 2 ft V sec 2 a 1 sec a sec a cosec a 1 pnvpv^: /7 - -..- i I/ cosec 2 a 1 cosec a cosec a CIRCULAR FUNCTIONS. 77 84. Each Function in Terms of the Others. tan tan tan tan tan tan tan cot cot cot cot cot cot 1 cot sin a SPP n 1/1 sin 2 a 1/1 sin 2 a sec ft V 1 cos 2 a a= cos ft COS ft Sjpp /7 . 1/2 vs a vs- a 1 vers a Sf*P r/ _ ^ 1 vs a 1 cvs a V 2 cvs ft cvs 2 a 1 sec ft 1/1 + tan 2 ft. cot ft sen ,^l/l+cot 2 ft ft ;= I/ sec 2 ft 1. 1 cot ft cosec a V cosec ^ a 1 I 7 cosec 2 a 1 1 cosec ft : 1/1 sin 2 a sin a COS ft sin a 1 POCJPP (7 - - 1/1 cos 2 ft IXTO /] 1/1 cos 2 ft 1 1/2 vs ft vs 2 1/2 vs ft vs 2 a 1 1 CVS ft 1 1/1 + tan 2 a ft , tan ft 1 tan ft cosec a = -|/l-f- cot 2 ft. sec ft I/ sec 2 1 ft = 1/cosec 2 ft 1. I/ sec 2 ft 1 78 TRIGONOMETRY. 85. Functions of Negative Arcs. We first find the sine and co-sine of a, in terms of the functions of a from the diagram. Then, dividing the sine by the co-sine, the cosine by the sine, taking the reciprocal of the co- sine and the reciprocal of the sine, we have sin ( a) = tan ( a) = sec ( a) = sin a, cos ( a) = cos a, tan a, cot ( a) cot a, sec a, cosec ( a) cosec a. 86. Functions of (n 90 + a). 1. Let n be 1 and a be negative. From the figure of the last article, and by similar processes, sin (90 a)=cos a, cos (90 a) = sin a, tan (90 a) = cot a, cot (90 a) = tan a, sec (90 a) = cosec a, cosec (90 a) = sec a. These relations have already been found, articles 3946. 2. Let n be I and a be positive. sin (90 -f- a) = cos a, tan (90 + a) = cot a, sec (90 -f a) = cosec a, cos (90 -f a) cot (90 -f a) cosec (90 + a) 3. Let n be 2, and a be negative. sin (180 a) == sin a, cos (180 a) = tan (180 a) = tan a, cot (180 a) = sec (180 a) .= sec a, cosec (180 a) = sin o, tan o, sec a. cos a, cot a, cosec a. CIRCULAR FUNCTIONS. 79 4. Let n be 2, and a be positive. sin (180 -f a) = - sin a, cos (180 -f- a) = - cos a, tan (180+ a) = tan a, cot (180-f- a) ==i cot a, sec (180 -f a) = sec a, cosec (180 -j- a) cosec a. 5. Let ?i be 3, and a be negative. sin (270 a) = - cos a, cos (270 a) = sin a, tan (270 a) == cot a, cot (270 a) sM tan a, sec (270 a) cosec a, cosec (270 a) = - sec a. 6. Let 91 be 3, and a be positive. sin (270 -f a) = - cos a, cos (270+ a) = sin a, tan (270+ a) = - cot a, cot (270 -f- a) = tan a, sec (270 -f- ) cosec a, cosec (270 -(- a) = sec a. 7. Let w be 4, and a be negative. sin (360 a). = sin a, cos (360 a) = cos a, tan (360 a) = tan a, cot (360 a) = cot a, sec (360 a) sec a, cosec (360 a) = cosec a. 8. Let n be 4, and a be positive. sin (360+ a) = sin a, cos (360 -f a) ^ cos a, tan (360+ a) jes tan a, cot (360+ a) 5 cot a, sec (360 -f a) = sec a, cosec (360+ a) == cosec a. It will be observed that when n is even, the func- tions in the two members of the equations have the same name; and that when n is odd, they have con- trary names. The algebraic sign attributed to the sec- ond member is determined by the quadrant in which the arc is situated. Let this article be reviewed, and these principles applied in determining the names and algebraic signs of the second members. 80 TRIGONOMETRY. Hence, functions of arcs greater than 90 can be found in terms of functions of arcs less than 90. Thus, 1. sin 120 = sin ( 90 -f 30) = cos 30. 2. cos 290 = cos (270 + 20) = sin 20. 3. tan 165 = tan (180 15) = tan 15. If n is integral and positive, prove the following: 4. sin In 180 + ( 1)" a] = sin a. 5. cos (n 360 a) == cos a. 6. tan (n 180 + a) = tan a. 7. Any function of (n 360 -f ) the same func- tion of a, whatever be the value of a. 87. Values of Functions of Particular Arcs. 1. To find the functions of 30. Since 60 is one-sixth of the circumference, the chord of 60 is equal to one side of a regular inscribed hex- agon, which is equal to the radius or 1. But the sine of 30 is equal to one-half the chord of 60. .-. (1) sin 30= |, ./. (2) cos30=:l/l ^=1/3. Dividing (1) by (2), then (2) by (1), taking the reciprocals of (2) and (1), we have (3) tan 30 = L , (4) cot 30 == l/^ 1' o (5) sec 30 = -JL , (6) cosec 30 = 2. V o 2. To find the functions of 60. From article 40, sin 60 = sin (90 30) = cos 30, cos 60 = cos (90 30) = sin 30. Hence, CIRCULAR FUNCTIONS. 81 (1) sin 60 = il/~3; (2) cos 60'=$, (3) tan 60 = i/~3", (4) cot 6( (5) sec 60 = 2, V 7 3 (6) cosec 60 = -^L . 3. To find the functions of 45. From Art. 40, sin 45 = sin (90 45) = cos 45 ; but sin 2 45 + cos 2 45 == 1', 2 sin 2 45 =1, (1) sin 45 = -} 1/27 (3) tan 45 = 1, (5) sec 45 = 1/27 sin 2 45 = f Hence, (2) cos 45 == il/2T (4) cot 45 = 1, (6) cosec 45 = V 27 5. cosec 210 2. 240 = = . V 3 Prove the following : 1. sec 120 = 2. 2. cos 135= 1/27 3. sin 300^ -^1/37 4. tan 225= 1. 9. Construct an angle whose tangent is 1. 10. Construct an angle whose sine is \. 11. Find all the functions of 150. 6. cot 7. sin 390 = f 8. cos( 120 3 )= f 88. Inverse Trigonometric Functions. If x sin a, then a is the angle or arc whose sine is x, which is written a == sin" 1 x, and read a equals the arc whose sine is x. 82 TRIG ONOMETR Y. It must not be supposed that ~ ] is an exponent, and that sin" 1 x = ; this would be a grievous error. sm x Let the following be read : cos" 1 ^, tan" 1 .?, sec" 1 :*;, cosec" 1 ^, sin -1 (cosa:), sin(sin~'j}, sin^z^cosec" 1 , cos" 1 x = sec" 1 , tan" 1 x = cot" 1 . x x x The above notation is not altogether arbitrary; for let f(x) be any function of x, and let /[/(x)], or, nioro briefly, let f 2 (x) be the same function of /(a?), which notation denotes, not the square of /(#)> that is, not [/(a?)] 2 , but that the same function is taken of f(x) as of x. Thus, if f(x) = sin x, /[/(&)] = sin (sin x), then, in general, (1) /"/ (*)=/"*; 0). If n =0, (1) becomes, (2) /- / (*)=/"(*) .-. (3) /'(*) = *. If m 1, and n = - 1, (!) becomes, (4) ff- l (x)=f(x)=x. Hence, f~ l (x) denotes a quantity whose like func- tion is x. ' Hence, if y=^sin~ l x, sin y = sin (sin' } x)=x; that is, y or sin -1 x is an arc whose sine is x. It would follow from the above that sin 2 a ought to signify sin (sin a), and not (sin a) 2 ; but since we rarely have sin (sin a), it is customary to write sin 2 a for (sin a) 2 , as we are thus saved the "trouble of writing the parenthesis. CIRCULAR FUNCTIONS. Ifc would not, of course, do to write sin a 2 for (sin a) 2 , for then we should have the sine of the square of an arc for the square of the sine of an arc. Let the following equations be proved : 1. 2. sin~ 1 ^ = |tan~ 1 V / 3. 4. cos- 1 i = 2cot- 1 V / 3. 5.. sin- 1 1 = 2 tan- >1. 89. Problem. To find the sine and co-sine of the sum of two angles. Let a = the angle OCA, and b = the angle ACS. Draw BL perpendicular to CA, BP and LM perpendicular to CO, and LN parallel to CO. The triangles NBL and MCL are sim- ilar, since their sides are respectively perpendicular; hence, the angle NBL opposite the side NL equals the angle MCL opposite the homologous side ML. But MCL = a; hence NBL = a, From the diagram we find the following relations: (1) LB = sin 6. (2) CL = cos 6. (3) PB = ML + NB. (4) PB = sin OCB sin (a -f 6). (5) ML = sin MCL X CL = sin a cos b. (6) A T =35 cos NBL X LB = cos a sin b. Substituting the values of PB, ML, and NB, found in (4), (5), and (6), in (3), and denoting the formula by (a), we have (a) sin (a -f 6) = sin a cos b + cos a sin 6. 84 TRIGONOMETRY. Hence, The sine of the sum of two angles is equal to the sine of the first into the co-sine of the second, plus the co- sine of the first into the sine of the second. From the diagram we find the follow- ing relations: (1) CP = CM NL. (2) CP = cos OCB = cos (a + 6). (3) CM= cos MCL X CL = cos a cos b. (4) NL = sin NBL X LB = sin a sin b. Substituting the values of CP, CM, and NL, found in (2), (3), and (4), in (1), we have (6) cos (a -j- 6) = cos a cos b sin a sin b. Hence, The co-sine of the sum of two angles is equal to the product of their co-sines minus the product of their sines. 90. Problems. 1. Prove that formulas (a) and (b) become homogene- ous by introducing R. 2. Prove that formulas (a) and (b) are true when (a -f ft) is in the second quadrant, 3. Prove that formulas (a) and (6) are true when (a -j- 6) is in the third quadrant. 4. Prove that formulas (a) and (b) are true when (a -|- 6) is in the fourth quadrant. 5. Deduce formula (6) from formula (a) by substitu- ting 90 a for a, .and 6 for b, and reducing by articles 8586. 6. Develop sin (45-^ 30) by formula (a). i 7. Develop cos 105 by formula (b). CIRCULAR FUNCTIONS. 85 91. Problem. To find the sine and co-sine of the difference of two angles. Let a the angle OCA, and 6 = the angle EC A. Draw BL perpendicular to CA, LP and BM perpendicular to CO, and BN parallel to CO. The triangles NLB and PCX are sim- c __ ilar, since their sides are respectively perpendicular; hence, the angle NLB, opposite the side NB, equals the angle PCL opposite the homologous side PL. But the angle PCL = a ; hence, the angle NLB = a. Then we shall have (1) LB = sin ft. (2) CL = cos b. (3) MB = PL NL. (4) MB F= sin OCB = sin (a 6). (5) PL --= sin PC Y L X CL = sin cos b. (6) JVL =: cos NLB X LB = cos a sin 6. Substituting the values of MB, PL, and NL, found in (4), (5), and (6), in (3), we have (c) sin (a. 6) sin a cos 6 cos sin b. Hence, The sine of the difference of two angles is equal to the sine of the first into the co-sine of the second, minus the co-sine of the first into the sine of the second. From the diagram we find the following relations: (1) CM=CP+NB. (2) CM = cos OCB = cos (a 6). (3) CP = cos PCL XCL=-- cos a cos b. (4) NB = sin NLB X LB = sin a sin b. 86 TRIGONOMETRY. Substituting in (1) the values of CM, CP, and NB found in (2), (3), and (4), we have (d) cos (a 6) = cos a cos 6 + sin a sin b. Hence, The co-sine of the difference of two angles is equal to the product of their co-sines, plus the product of their sines. 92. Problems. 1. Prove that formulas (r) and (d) become homogene- ous by introducing R. 2. Deduce formulas (c) and (d) from (a) and (6), re- spectively, by substituting b for b, and reducing by article 85. 3. Prove that formulas (c) and (d) are true when (a 6) is in the second quadrant. 4. Prove that formulas (c} and (d) are true when (a 6) is in the third quadrant. 5. Prove that formulas (r) and (d) are true when (a b) is in the fourth quadrant. 93. Problem. To find the tangent and co-tangent of the sum or differ- ence of two angles. Dividing (a) by (6), we have sin (a -J- 6) sin a cos b -(- cos a sin b cos (a -f- b) cos a cos b sin a sin b Dividing both terms of the fraction in the second member by cos a cos 6, reducing, and recollecting that CIRCULAR FUNCTIONS. 87 the sine of an arc divided by its co-sine is equal to its tangent, we have tan a -f tan b (e) tan (a -f 6) = 1 tan a tan b Hence, The tangent of the sum of two angles is equal to the sum of their tangents, divided by 1 minus the product of their tangents. Dividing (6) by (a), and reducing, we have cot a cot b 1 (/) cot (a -f- 6) = cot a 4- cot b Hence, The co-tangent of the sum of two angles is equal to the product of their co-tangents, minus 1, divided by the sum of their co-tangents. Dividing (c) by (d), and reducing, we have tan a tan b (g) tan (a b} = 1 -4- tan a tan b Hence, The tangent of the difference of two angles is equal to the tangent of the first minus the tangent of the second, divided by 1 pliis the product of their tangents. Dividing (c/) by (e), and reducing, we have cot a cot b -f 1 (/O cot (a 6) = - r-y - cot b cot a Hence, The co-tangent of the difference of two angles is equal to the product of their co-tangents, plus 1, divided by the co- tangent of the second, minus the co-tangent of the first. 94. Problems. 1. Prove that (e), (/), (#), (K) become homogeneous by introducing R. 88 TRIGONOMETRY. 2. Deduce (g) from (e) by substituting 6 for b. 3. Deduce (h) from (/) by substituting 6 for 6. 4. Deduce (/) from (e) by taking the reciprocal of each member, substituting -- for tan o, , t for tan 6, cot a cot 6 and reducing. 5. Deduce, in like manner, (A) from (g). 6. Find the value of sin (a -\-b -{- c) by substituting b -{- c for b in (a). 7. Find the value of cos (a -f- b -f c) by substituting 6 -f- e for & in (6;. 8. Find the value of tan (a -\- b -\- c) by substituting b -j- c for 6 in (e). 9. Find the value of cot (a -f b -f- c) by substituting b + c for b in (/). 95. Functions of Double and Half Angles. Making b = a in (a), (6), (g), and (/), we have (1) sin 2 a = 2 sin a cos a. (2) cos 2 a cos 2 a sin 2 a. 2 tan a (3) tan 2 a = r - -^ - 1 tan 2 a cot 2 a 1 (4 cot 2 a = 2 cot a Substituting $ a for a in (1), (2), (3), (4), we have (5) sin a = 2 sin \ a cos J a. (6) cos a cos 2 J a sin 2 a. CIRCULAR FUNCTIONS. 89 2 tan i a 2 cot J a Substituting 1 sin 2 Ja for cos 2 a, then 1 cos 2 -Ja for sin 2 Ja, in (6), and reducing, we have (9) 1 cos a = 2 sin 2 % a. (10) 1 + cos a = ^ cos 2 ^ a. /UN : i /A cos a - (11) * 2 Dividing (11) by (12), then (12) by (11), we have (13) tan ^ a - \l a . * 1 + cos a (14) cota='J +cos a . * 1 cos a Dividing (5) first by (10), then by (9), and trans- posing, we have (15) tan J a = - : 8na 1 1 cos a Taking the reciprocal of (16), then of (15), we have 1 cos a (17) tan J a = (18) cot J a = sin a 1 -}- cos a sin a Let the formulas of this article be expressed in words. S. N. 8. 90 TRIGONOMETRY. 90. Consequences of (a), (b), (c), (d). Taking the sum and difference of (a) and (c), (d) and (6), we have (1) sin (a -j- 6) + sin (a b) = 2 sin a cos 6. (2) sin (a -f 6) sin (a b) = 2 cos a sin 6. (3) cos (a -f 6) + cos (a 6) 2 cos a cos 6. (4) cos (a b) cos (a -)- 6) 2 sin a sin 6. Let {a _ b = d j then ft = , (g __ Substituting the values of a + 6, 6, a, and 6, in (1), (2), (3), and (4), we have (5) sin s -\- sin d = 2 sin J (s + d) cos J (s d). (6) sin .$ sin d 2 cos ^ (s -f- d) sin % (s d). (7) cos s -\- cos d =? 2 cos ^ (s -f d) cos J (a d). (8) cos d cos 8 = 2 sin -J (s -j- d) sin J (8 d). By formula (5) of the preceding article we have (9) sin (8 + d) = 2 sin J (s -f d) cos \ (s -f- d). (10) sin (s d) = 2 sin i ( d) cos J (s d). Dividing each of these formulas by each of the fol- lowing, we have sin^-j-sind sin-|(s-t-d) cosj(s d) tanj(8-|-d) (11) sins sind cosj(8-f-d) sin(8 rf) tan^(s d) S _in^+sin_d smi^+d) tan cos s + cos a cos -J (s -f a) _ , sin. 5? + sine? cosj(s d) (16) ^-=COl-?(S cos a cos s sin J (8 d) sin s -f sin d cos -J (s d) sin (s -f d) " cos (s -f- d) CIRCULAR FUNCTIONS. 91 (10) (16) (17) (18) (19) (20) (21) (23) (24) (25) sin (a d) sin s sin d sin J(a c/) sin -J (a d) tnrI/e /7^ cos s -f cos d cos (a rf) cos d cos s sin s sin d sin 3- (s -j- d) sin(a d) cos J (a -f- d) sin (a -j- d) sin s sirfrf sin (a -|- d) sin (s d) cos a -f- cos d! cos J (a d) cot J (a + d) cos d cos a cos s -f cos d tan-J(s f?) cos J (s rf) cos s -f- cos d sin \ (a -f d) cos J (a -f d) sin (a f?) cos d cos a sin J(a f?) sin J (a d) sin (a -f- d) cos d cos s cos J (a -f- c?) sin J (a -f d) sin (s d) sin (s -f~ d) cos -J (a d) sin -| (a -I- d) sin (a^d) sin J (s d) cos J (s d) Formula (11) gives the proportion, sin s -f- sin d : sin s sin d : : tan -J (s -f- c?) : tan (s d). Hence, The sum of the sines of two angles is to their difference as the tangent of one-half the sum of the angles is to the tangent of one-half their difference. Let us apply this principle in solving triangles when two sides and their included angle are given. Article 75. 92 TRIGONOMETRY. a : b :: sin A : sin B. a-\-b : a 6:: sin A -{- sin B : sin A sin B. B sin A+ sin B : sin A sin B : : tau(A+B) : tan(A B}. .'. a + 6 : a 6 : : tan K^+) : tan %(AB). 97. Theorem.. sum The square of any side of a triangle is equal to the of the squares of the other sides, minus twice their product into the co-sine of their included angle. 1st. When the angle is acute. (1) m b 7i. B (I) 2 =(2) m* = 6*+n2 26n. Zj (3) P 2 =p*. * (2)+(3)=(4) mz-f p = 62 + n 2 + p 2 ._ 2 6w . But m 2 -f p 2 = a 2 and n 2 -f p 2 c 2 , . . (4) becomes (5) a 2 = b 2 -\- c 2 2 bn. But n c cos J, which substituted in (5) gives (6) a 2 = 6 2 + c 2 2 6c cos A. 2d. When the angle is obtuse. (1) m = b + n, B (3) p 2 =p 2 . ^""^ ^ r (2) + (3) = (4) m 2 +P 2 - CIRCULAR FUNCTIONS. 93 But m 2 -f j3 2 a 2 and n 2 -\-p 2 = c 2 , .*. (4) becomes (5) ft2^2_!_ c 2_|_2 6tt. But ?i = c cos iL4.D c cos 5^4 C = c cos A .-. (6) a 2 = 6 2 + c 2 2 6c cos A. 98. Problem. Tb find the angles of a triangle when the sides are given. From either formula (6) of the last article we have 7)2 l C 2 _ a 2 / \ A I ** (1) cog ^ = ___ Hence, The co-sine of any angle of a triangle is equal to the sum of the squares of the adjacent sides, minus the square of the opposite side, divided by twice the rectangle of the adjacent sides. Formula (1) gives the natural co-sine of A; hence, A can be found. But it is best to place the formula under such a form as to adapt it to logarithmic com- putation. Adding 1 to both members of (1) we have (I + g )2_ a 2 __ (a + 6-f- C )(6 + g a) Tbc~ 2 be But 1 + cos A = 2 cos 2 A. Article 95, (10). (a + b + c}(b + c a) Let +&+*=;>, then - -- Substituting these values in (2), and dividing by 2, we have 94 TRIO ONOMETE Y. In like manner, (5) cos B = Also, (6) cos $ C = - 06 Introducing R, applying logarithms, and reducing, (4) becomes log cos \A \ [log Jp+log typ a) -\-a.c. log b-\-a.c. log c]. In like manner introduce R and apply logarithms to (5) and (6). By subtracting ..both members of (1) from 1 and re- ducing we find be (8) sin J B = (9) sin^C = * (7) .,_ (4) = (10) tan \ A = (9) ^_ (6) = (12) tan i C = 99. Examples. r a =125. ^| r^ = 5546' 18". 1. Given 1 b = 150. > Required < 5= 82 49' 08". I c = 100. J I C = 41 24' 34". ra=864. -| ryl^41 00' 38". 2. Given 1 b'^ 1308. V Required < B ^'83 25' 14". U = 1086.J (c 55 34' 08". CIRCULAR FUNCTIONS. 95 100. Problem. To find the area of a triangle when two sides and their included angle are given. Let k denote the area of the tri- angle ABC, of which the two sides 6 and c and their included angle A are given. (1) 2 k = bp. (2) p c sin A. .'. (3) 2 k = be sin A. Introducing R, and applying logarithms, we have log (2 k) = log b -f log c -f log sin 4 10. 101. Examples. 1. Two sides of a triangle are 345.6 and 485, respect- ively, and their included angle is 38 45' 40"; what is the area? Am. 52468. 2. Two sides of a triangle are 784.25 and 1095.8, re- spectively, and their included angle is 85 40' 20"; what is the area. Ans. 428470. 102. Problem. To find the area of a triangle when the three sides are given. By the last problem we find (1) k = \ be sin A, (2) sin A = 2 sin J A cos J A. Article 95, (5). (3) sin J A = P c . Article 98, (7). be 96 ' TRIGONOMETRY. (4) cos \A = ^/my- Ify . Article 98, (4). (5) sin A = be . . (6) k = V \p(\p a) (ip 6) (ip r ). 103. Examples. 1. The sides of a triangle are 40, 45, 55, required the area. An*. 887.412. 2. The sides of a triangle are 467, 845, 756, required the area. Ans. 175508. 104. Problem. Given the perimeter and angles of a triangle, required the sides. . a sin A Adding and reducing by Articles 96, (5) and 95, (5), we have b c sin a sin J A cos J ^4 sin %(B -h C) cos 4, and sin J ^4 cos | (5 + C). a cc Adding 1 to both members, we have cos 4 (B C) a cos (J5 -f C) Let _p = a -j- 6 -f c, and reduce by 96, (7), we have 2 cos i 5 cos C5) -. sin \A . . (6) a = ^P sin -M cos ^5 cos & C ' CIRCULAR FUNCTIONS. 97 Introducing R and applying logarithms, we have log a = log \p + log sin J A -f a. c. log cos J- 5 -f- c. log cos J C 10. Similar formulas can be found for 6 and c. But, after a is found, 6 and c can be more readily found by article 69. 105. Examples. 1. Given p = 150, ,4 = 70, 72^60, (7=50, re- quired , 6, c. Ans. a = 54.81,. 6 = 50.51, <? = 44.68. 2. Given ;> == 31234.36, A == 35 45', 5 = 45 28', (7=98 47', required a, ,6, c. . a = 7985, 6 = 9742.5, c ^ 13506.86. 3. Given p = 375, A & 55 46' 18", B = 82 49' 08", 741 24' 34", required a, 6, c. Ans. a = 125, 6 = 150, c = 100. 106. Problem. Given tfie three sides of a triangle, to find the radius of the inscribed circle. (1) BOC+AOC+AOB = (2) (3) (4) AOE\cr. .'. (5) But (6) S. N. 9. 98 TRIG OS METE Y. .'. (7) \pr l/~ r -_ \ 107. Examples. 1. The three sides of a triangle are 20, 30, 40, re- spectively, required the radius of the inscribed circle. Am. 6.455. 2. The three sides of a triangle are 100, 150, 200, re- spectively, required the radius of the inscribed circle. Am. 32.275. 108. Problem. Given the three sides of a triangle to find the radius of the circumscribed circle. Let be the center of the circle, and R the radius. Let OD be perpendicular to 6, then , A A . , The angle = the angle B, since each is measured by one-half the arc AC. (1) AD = 4- = AO sin = R sin B. 2i .'. (2) R = 2 sin B 2 sin B = 2 sin ^B cos ^B = ac ,. R= abc = abr 4 V\p (4p a) (\p b} ( \ p c) 4 k Prove that the formula will be the same if the cen- ter is without the triangle. CIRCULAR FUXCTIOXS. ' 99 109. Examples. 1. The sides of a triangle are 7, 9, 10, respectively, required the radius of the circumscribed circle. Ans. 5.148. 2. The sides of a triangle are 50, 60, 70, respectively, required the radius of the circumscribed circle. Ans. 35.72. 110. Theorem. The perpendicular let fall on either side of a triangle from the vertex of the opposite angle is equal to that side into the product of the sines of the adjacent angles divided by the sine of the sum of those angles. (1) p = c sin A. (2) sin B : sin C : : b : c, .'. c=- =- sin jj b sin A sin C (4) sin B = sin [180 (A+ C)] == sin 04 + C). _ p ' 111. Problem. Given ^0 ^ree sid<?s o/ a triangle to find the radii of the escribed circles. The escribed circles are the three circles external to the triangle, each tangent to one side and to the pro- longation of the other sides. 100 TRIGONOMETRY.- The centers .of the escribed circles are the points of intersection of the lines bisecting the external angles. The radii r, r", r'", of the escribed circles, will be the perpendiculars let fall from their centers 0', 0", 0'", respectively, on the three sides a, 6, c. Hence, by the last ar- ticle, ,_ a sin (90 '~~ /ox > . . (2) r ' a COS COS Substituting the value of tan 4^4> article 98, we. have 1 1 '2. Examples. 1. Given the sides of a triangle, 6, 9, 11, required the radii of the three escribed circles. Am. 3.854, 6.745, 13.49. 2. Given p = 100, .4^55, = 60, 0=65, required the radii of the three escribed circles. [See (2), Art. 111.] Ans. 26.028. 28.867 : 31.854. CIRCULAR FUNCTIONS. 101 113. Theorem. The product of the radius of the inscribed circle and the radii of the three escribed circles is equal to the square of the area of the triangle. The product of (8), article 106, and (3), (4), (5), article 111, gives H 1.4 r rVV" _ __ __ , _ __ 1U2 ~ 114. Theorem. The reciprocal of the radius of the inscribed circle, the sum of the reciprocals of the radii of the escribed circles, and the sum of the reciprocals of the perpendiculars let fall from the vertices of the three angles on the opposite sides of a triangle are equal to each other. Taking the reciprocal of (8), article 106, we have m ! - p ~~ 2k' Taking the sum of the reciprocals of (3), (4), (5), article 111, 111 _p-2a p-2b p-2c p W -7- -- r - r n, - -gy- -gj- 2k ~ 2k - Let p', p", p'", respectively, be the perpendiculars let fall from the vertices of the three angles on the sides a, b, and c. Then we have In like manner, -^ = Also, -777- == - 102 TRIGONOMETRY. m a + b + c _ _P__ p' p" " p'" ~~ 2k ~~ 2 k ' ... (4) J-^J^ _i_ + 4.=^ + j_ + 4 115. Problem. To find the disfyrtce between the centers of the circum- and "instnfad circles of a triangle. ? 'iiid'r b<? thj& radii, and P and the centers of the circles, and let D = OP. Draw PE perpendicular to AC. The angle APE = B, since each is meas- ured by one-half the arc AC; but PAE = 90 APE, .-. PAE=90B. OAC=1>A. PAO=PAEOAC. PAO = 90 B \A = 4(C- (1) OP 2 =^4P 2 -M# 2 2 APxAO cos PAO. Art. 97. Substituting the values of OP, AP, AO, and PAO, we have 6 _ 6 _ ( 108, (2). ~ 2 sin B " 4 sin J cos JB ' 3 ' 1 95, (5). 6 sin \A sin \C b sin J^4 sin JC A - ~ r 2 4 J?r sin |P sin sin 2 W ~" sin I A CIRCULAR FUNCTIONS. 103 Substituting in (2), and reducing by article 91, (d), and 89, (6), we have Dt=Bf _ sm = ,-. (7) D = VRt 2 Rr. 116. Examples. 1. The sides of a triangle are 12, 13, 15; required the distance between the centers of the circumscribed and inscribed circles. Ans. 1.616. 2. Two sides of a triangle are 35 and 37, and their included angle is 50; required the distance between the centers of the circumscribed and inscribed circles. Am. 3.266. 3. The perimeter of a triangle is 120, the angles are 40, 60, and 80, respectively; required the distance between the centers of the circumscribed and inscribed circles. Ans. 8.353. 117. Problem. To find the distance between the centers of the circumscribed and escribed circles. Let /, r", r" be the radii of the escribed circles, and >', D", D'", be the distances of their centers, 0', 0", 0'", respectively, from P, the center of the circumscribed circle, whose radius is R. 104 TRIG ONOMETR Y. As in the last Problem, we find sin 2 \A sin \A (*) R = a - " Arts I 108 '' 2 sin ,4 4 sin \A cos A<4 ''I 95, (5). (3) ^co Substituting (4) in (1), and reducing by (d) and (6), we have (5) iy. = .-. (6) # = .-. (7) D" = .-. (8) D"'= 118. Examples. 1. The three, sides of a triangle are 21, 23, 26; re- quired the distances from the center of the circum- scribed circle to the centers of the three escribed circles. Ans. 25.19, 26.64, 29.73. 2. The angles of a triangle are 56, 60\ 64, the greatest side is 25 ; required the distances from the center of the circumscribed circle to the centers of the three escribed circles. Am. 26.96, 27.80, 28.65. 3. Given p == 100, A = 55, B =-- 60, C = 65, required U, D". D"'. Ans. 37.10,- 38.55, 40.01. CIRCULAR FUNCTIONS. 105 119. Problem. To find the distance between the centers of the inscribed, and escribed circles. Let Dj, D 2 , > 3 , be the distances. In the triangle OO'E, we have r' r (1) *>!=;- sin \A Substituting the values of r', r, and sin \A, we have 120. Examples. 1. The three sides of a triangle are 30, 50, 60; re- quired the distances between the centers of the in- scribed and escribed circles. Ans. 31.05, 56.69, 87.83. 2. The sides of a triangle are 500, 600, 700; required the sides of the triangle formed by jpining the centers of the inscribed and circumscribed circles and the center of the escribed circle, tangent to the sides 600 and 700 produced. Ans.- 540.06,- 104.58, 624,58, 10(5 r/,' /f.-n.YM v rrnv. is Kxercises. i. Prove bhfctsin ir) = ~ cos I5 = 21/2' 2 V 2 tan 16 = 2 V li^ cot 15 = 2 -f V ~Z, sec r> f} I 2. Find the sine and 00 line of 75. J,,, sin75 = i^li, cos 75-= *I ' 21 2 LM 2 8. Why is sin 76= cos 15, and cos 75=- sin 15 ? I How may the value* of tangent, co-tangent, se- t, and eo-seeant of 7-~> ' he found from the values of the sine and co-sine? r >. Find the functions of 150. Am. sin I/XT = 4, cos 150 = 0. (Jiven sin n + cos n 1 7 2, to find a. .l*w. 45, or 45 4- 300; or, in general, '' \ L> TH. 7. (Jiv<n sin 2 <f eos </, to find a. 7T l"- s '. i 2 ?m, or f tr j 2 TT?L S. Prove that the sum of the tangents of the thn-e anglea of a plane triangle is i><|ii:il to their produet. 1). Prve tliat the sum of the cotangents of one-half the angles of a plane triangle is e^ual to lluir product. 10. Prove that Mir is isoseeles if COS I 2 sin r 11. I'rove that the sum of the diameters of the in- serihed and eirenm-vnhed eireles of any plane tri- angle ABC is a cot A 4 l> cot B 4 c cot C. 107 12. If ft is ihe hase of the triangle Al\<\ '/>, the per- pendiCUl&T ti the h;ise IVom the vertex of the opposite anj.de, and *, the sum of the sides H. and r, prove that tan i* - 2 '"' 115. If /> is tli<> fasc of Uic triangle AIM', />, UK- )MT |.cndicidar to tin: hasc from (lie vertex of the oppo- site; an^lc, ;md J, the dillerenee of tin- sides a and r, {rove that . B _(6 + d)(ft-l) 2 /,,, 14. If a, />, and r le the sides <(' the triangle ABC, x, the sum of the flidew a and r, and r, the radius of the inscribed eirde, prove tliat o r 122. Computation of Natural Fiinctioiw. the length of the senii-eireiinifei-enee to the radius 1, which JH n = 3.14 \KWWM\WK . . . hy 1080, the niiiiiher of ininuteH in ISO 1 , the <jiiof,ienf, which is .000'2 ( .K)SSS-2 . . . , will he the length of the are T, and will differ insensihly from its sine. .', (1) sin 1' .(X0290SSS2. .-. (2) COM 1' ll Hin 54 1' WH9999577. Adding (a) and ((?), then < I, ) and (<l >, articles 89, 91, ' and transposing, (3) sin (a + b) -- 2 sin <> col // sin (a It). (4) cos (a + 6) ~ 2 cos a cos 6 cos (a &;. 108 TRIGONOMETRY. If in (3) and (4) 6 = 1, a = 1, 2, 3 . . . , in succession, we have sin 2' = 2 cos 1' sin 1' sin 0'= .0005817764. sin 3' = 2 cos 1' sin 2' sin 1' == .0008726646. sin 4'= 2 cos 1' sin 3' sin 2' = .0011635526. cos 2' = 2 cos 1' cos 1' cos 0' = .9999998308. cos 3' = 2 cos 1' cos 2' cos 1' = .9999996193. To facilitate computation, for 2 cos 1' = 1.9999999154, use its equal, 2 .0000000846. Then we have sin 2' = 2 sin V .0000000846 sin 1' sin 0'. sin 3' = 2 sin 2' - .0000000846 sin 2' sin 1'. After finding the sines and co-sines, the tangents and co-tangents can be calculated from the formulas: /K , sin a cos a (5) tan a = - (6) cot a = - cos a sm a It is not necessary to carry the computation beyond 45, since sin cos (90 a), etc. The logarithmic functions can be found from the corresponding natural functions by the method of article 60. SPHERICAL TRIGONOMETRY. 123. Definition and Remarks. Spherical Trigonometry is that branch of Trigonome- try which treats of the solution of spherical triangles. If any three of the six parts of a spherical triangle are given, the remaining parts can be computed. The radius of the sphere is taken equal to 1, and RIGHT TRIAKULES. 109 each side has the same numerical measure as the subtended angle whose vertex is at the center of the sphere. Thus, a L BOO, b = AOC, c = AOB. An angle of a spherical triangle is the angle included by the planes of its sides which is measured by the angle included by two lines, one line in one plane, the other in the other, both per- pendicular to the common intersection of the planes at the same point. Thus, if BE, in the plane AOB, is perpendicular to OA, and if ED, in the plane AOC, is perpendicular to OA, then the angle BED will measure the inclination of the planes AOB and AOC, and will be equal to the angle A of the spherical triangle. RIGHT TRIANGLES. 124. Napier's Circular Parts. Napier's circular parts are the two sides adjacent to the right angle, the complements of their opposite angles, and the complement of the hypotenuse. Thus, if HBP is a spherical triangle, right-angled at H, the circular parts are b, p, 90 B, 90 P, and 90 h. 9()0 _ p . Adjacent parts are those which are not separated by an intervening circular part. Thus, b and 90 P, 90 P and 90 /*, 9<P h and 90 B, 90 B and p, p and 6 are adjacent parts. The right angle H is not regarded as a circular part, nor as separating the parts 6 and p. 110 TRIG ONOMETR Y. Opposite parts are those which are separated by an intervening circular part. Thus, b and 90 A, 90 P and 90 P, 90 h and p, 90 # and 6, p and 90 P are opposite parts. Any one of these five circular parts is adjacent to two of the remaining parts, and opposite the other two parts. Of any three circular parts, one part is either adja- cent to both the others or opposite both. A middle part is that which is adjacent to two other parts, or opposite two other parts. 125. Exercises. Tell which is the middle part, and whether the other parts are adjacent to, or opposite, the middle in the following : 1. 90 B, 90 P, 9(T 2. 6, 90 h, p. 3. 90 h, 90-, p. 4. 90 P, 9(T 5, 6. 5. 6, 90 5, p. 6. 90 P, 90 7. b, 90 P, p. 9. 90 h, 90 P, b. 10. 90 P, 90 , p. 126. Napier's Principles. 1. The sine of the middle part ix equal to the product of the tangents of the adjacent parts. Draw BD and DE, respectively perpendicular to OH and OP, and draw BE. BDE is a right. angle, since the plane BOH is perpendicu- lar to the plane POH, and BD is perpendicular to OH. The angle BED is equal to P. EIGHT TRIANGLES. Ill EB = sin h, OE = cos 7i, DB = sin p, and OD = cos jy. ED OE ED "Fir X -7TTT' or cos P= cot /i tan 6. .-. (1) sin (90 P) == tan (90 h~) tan 6. ED DB ED or Bin 6 = tan > cot P. . . (2) sin b = tan p tan (90 P). By changing P, 6, p into B, p, b, (1) and (2) become (3) sin (90 5) = tan (90 A) tan p. (4) sin _p = tan b tan (90 5). Multiplying (2) by (4), member by member, we have sin b sin p = tan 6 tan p tan (90 B) tan (90 P). Dividing by tan b tan p, and reducing, we have cos b cos p = tan (90 5) tan (90 P). cos 6 cos p = cos EOD xOD=OE= cos h = sin (90 A). . . (5) sin (90 h) i tan (90 P) tan (90 P). 2. JTie siW of ffo middle part is equal to the product of the co-sines of the opposite parts. OE = cos EOD X -OD, or cos h = cos 6 cos p. . . (6) sin (90 h) = cos 6 cos p. DB=EB sin DEB, or sin p = sin h sin P. . . (7) sin p = cos (90 h) cos (90 P). ' sin (90' A) sin p (3) gives sin (90-*) = _ -^^-^ - 112 I RIG ONOMETR Y. This, by substituting cos 6 cos p for sin (90 A), cos (90 h) cos (90 P) for sin p, and reducing, gives (8) sin (90 R) ~ cos b cos (90 P). By changing p, P, B, b into 6, 5, P, p, (7) and (8) become (9) sin b = cos (90 h) cos (90 P). (10) sin (90 P) =cos p cos (90 > 5). These ten formulas are thus reduced to two princi- ples, from which the formulas can be written. The memory will be further aided by observing the common vowel a in the first syllables of the words tangent and adjacent of the first principle, and the common vowel o in the first syllables of the words co-sine and opposite of the second principle; that is, we take the product of the tangents of the parts adjacent to the middle, and the product of the co-sines of the parts opposite the middle. 127. Mauduit's Principles. If we take, as circular parts, the complements of the two sides adjacent to the right angles, their oppo- site angles, and the hypotenuse, we can readily deduce from the diagram, or from Napier's principles, the following principles: 1. The co-sine of the middle part is equal to the product of the co-tangents of the adjacent parts, 2. The co-sine of the middle part" is equal to the product of the sines of the opposite parts. Let the ten formulas be written and compared with those of the last article. RIGHT TRIANGLES. 113 128. Analogies of Plane and Spherical Triangles. The formulas which demonstrate Napier's principles may be placed under forms which will exhibit the analogies existing between Plane and Spherical Tri- angles, as in the subjoined table. Plane Right Triangles. Spherical Right Triangles. 1. sin P=-f-- ri 1. sin p = ^^P . sin h 2. sin B=*HJL 2. sin B== sin ^ sin h 3. cos P = 4-' A 3. cos p tan b tan h 4^^* D - 4. cos P tan p' . COo Jj j " n tan ^ 5. tan P=--- 5. tan p tanj) ~ sin b 6. tan =-.. > 7. sin P cos 5. 6. 7. tan sin tan b sin jo p _ Plj? j cos 6 8_f -. T"> 7~> 8. sin cos P . sin B cos r. COS JO 9. A. = 6^ + p. 9. cos ^ = cos b cos p. 10. 1 = cot 5 cot P. 10. cos A cot B cot P These formulas can be committed and applied in- stead of Napier's principles by those who prefer to do so. The analogies will assist the memory. S. N. 10. 1 14 TRIG ONOMETR Y. 129. Species of the Parts. Two parts of a spherical triangle are of the same species when both are less than 90 or both greater than 90. Two parts of a spherical triangle are of different species when one part is less than 90 and the other part greater than 90. We shall, at present, consider those triangles only whose parts do not exceed 180. Let it be remembered that the sine is positive from to 180, and that the co-sine, the tangent, and the co-tangent are positive from to 90, and negative from 90 to 180. Hence, if the co-sines, tangents, or co-tangents of two parts have like signs, these parts will be of the same species; if they have unlike signs, these parts will be of different species. cos B , . cos P sin P = r and sin B ~ - Art. 128, 7, 8. cos o cos p Since neither P nor B exceeds 180, sin P and sin B are both positive; hence, cos B and cos b have like signs, so also have cos P and cos p. Therefore, B and b are of the same species; so also are P and p. Hence, The sides adjacent to the right angle are of the same species as their opposite angles. cos h = cos b cos p. Art. 128, 9. If h < 90, cos h is positive; hence, cos b cos p is positive ; .- . cos 6 and cos p have like ' signs ; . . b and p are of the same species; . . B and Pare of the same species. Hence, If the hypotenuse is less than 90, the two sides adjacent to the right angle are of the same species; so also are their opposite angles. RIGHT TRIANGLES. 115 If h > 90, cos h is negative; hence, cos b cos p is negative ; . ' . cos 6 and cos p have unlike signs ; . . b and p are of different species ; . . B and P are of different species. Hence, If the hypotenuse is greater than 90, the two sides adjacent to the right angle are of different species; so also are their opposite angles. Let us now investigate the case lt ^JZ ^ in which a side adjacent to the p< right angle and its opposite angle are given. Let p and P be given. Produce the sides PH and PB till they meet in P'. The angles P and P' are equal, since each is the angle included by the plane of the arcs PHP' and PBP'. Take P'H' --=PH=b and P'B' = PB=h. The two triangles, PHB and P'H'B', have the two sides PH and PB and the included angle P of the one, equal to P'H' and P'B' and the included angle P' of the other; hence, they are equal in all their corresponding parts; .*. H' = H, B' = B, and H'B'-=HB. But H is a right angle; . . H' is a right angle. Hence, either triangle, PHB or PH'B', will answer to the given conditions. Since P'H' and PH are equal, and P'H' and PH' are supplements of each other, PH and PH' are supplements of each other. In like manner it can be shown that PB and PB' are supplements of each other. When, therefore, a side adjacent to the right angle and an opposite angle are given, there are apparently two solutions. The conditions of the problem, how- ever, may be such as to render the two solutions possible, reduce them to one, or render any solution impossible. 116 TRIG ONOMETRY. Let us now proceed to investigate these conditions. 1. When P < 90' and p < P. ~T\ T< \ r / p We have from Napier's princi- ^x/ vjx^ 1 TTT TLJ^ pies, sin b = tan p tan (90 P), or sin b = tan p cot P Since P < 90 and p < P, tan _p < tan P; but we have tan P cot P - 1 ; . . tan p cot P < 1 ; hence, sin b < 1; then b < 90 or 6 > 90; hence, b may be either of the supplementary arcs PH or PH' which have the same sine equal to tan p cot P If b < 90, since p < 90, h < 90 ; if 6 > 90, since p < 90, A > 90. Hence, if P < 90 and p < P, either triangle, PHB or P#'', will satisfy the conditions, and there will be two solutions. 2. When P < 90 and p = P. P We have sin b tan p cot P, as before. Since p = P, tan jo cot P= tan P cot P 1 ; there- fore, sin 6 = 1 ; . . 6 = 90, or PH = 90. From Napier's principles, we have sin (90 h) cos 6 cos jo, or cos h = cos b cos >. Since b 90, cos 6 ; . . cos b cos p = ; hence, cos h = 0; . . h = 90, or PB = 90. sin (90 B} = tan p tan (90 A), which reduces to cos B = tan p cot A. Since h = 90, cot A = 0; . . tan p cot h ^= 0; .-. cos B = 0; .-. B = 90. PH ' = 180 - P# = 90 ; . . PR' = PH. PB' = 180 PB = 90 ; . . PB' = PB. EIGHT: .TRIANGLES. 117 Hence, if P < 90 and p = P, b = 90, A *a 90, J3--90 , the two triangles. reduce, to the bi-rectangular triangle PHB, and there is but one solution. 3. When P < 90 and p > P. As before, we have sin b = tan p cot P. Since p and P are of the same species, p < 90. Then, if p > P, tan p > tan P; but tan P cot P = 1; . . tan p cot P ]> 1 ; . . sin b ]> 1, which is impossible. Hence, if P < 90 and p > P, no solution is possible. 4. When P > 90 and j> > P. K /V V We have sin b tan p cot P, as before, tan p and cot P are both negative, and tan p < tan P, numerically; but tan P cot P = 1 ; . . tan p cot P < 1 ; hence, sin b < 1 ; . . b < 90, or b > 90 ; hence, b may be either of the supplementary arcs PH or PH' which have the common sine equal to tan p cot P. If 6 < 90, since p > 90, h > 90 ; if 6 > 90, since p > 90, h < 90. Hence, if P*> 90 and p > P, either triangle, PHB or PH'B' will satisfy the conditions, and there will be two solutions. 5. When P > 90 and p = P. x^~~T"\ P \ / r We have sin b = tan cot P, as \. / jx^ ^""'^ / "" x ^ before. H . . sin b = tan P cot -P= 1 ; . . b = 90. . . cos b 0; . ' . cos h = cos b cos jo ; . . k = 90. = 0; .-. 5 = 90. 118 TRIGONOMETRY. Hence, if P > 90 and p = P, b = 90, h = 90, B = 90, the two triangles reduce to the bi-rectangular PHB, and there is but one solution. 6. When P > 90 and p < P. As before, we have sin b = tan p cot P. Since p and P are of the same species, and since P > 90, p > 90 ; hence, tan jo, cot P are both nega- tive, and tan p > tan P, numerically; but since tan P cot P = 1, tan p cot P > 1 ; . . sin b > 1, which is impossible. Hence, if P > 90 and p < P, there is no solution. 7. When P==90. sin b sin 6 . == ^Fp = IT =oo; ' ?=. . . cos jo = ; . . cos A, = cos b cos 7) = ; . * . h = 90. sin b = tan p cot P= oo x 0; . * . sin 6 is indeterminate. sin B = - = -pr- ; . . sin 5 is indeterminate. cos p Hence, if P=90, then JD = 90, h = 90, 6 and B are indeterminate ; the triangle is bi-rectangular, and there is an infinite number of solutions. Hence, the following results : ( p < P, Two solutions. P < 90 and | p = P, One solution. I p > P, No solution. r p > P, Two solutions. P > 90 and < JD = P, One solution. ( ^ < P, No solution. RIGHT TRIANGLES. '11.9 P g= 90, x /i = 90, f Infinite number ft indeterminate, f of solutions. 5 indeterminate, J By a comparison of these results, we find, 1. If jo differs more from 90 than P, there will be two solutions. 2. If p P, and P < 90 or P > 90, there will be one solution. 3. If p = P = 90, there will be an infinite number of solutions. 4. If p differs less from 90 than P, there will be no solution. 130. Remarks. 1. Napier's principles render it unnecessary to di- vide the subject of right-angled spherical triangles into cases. 2. Two parts will be given, and three required. 3. These parts or their complements will be circular parts. 4. Take the two given parts, if they are circular parts, otherwise their complements, and any one part required, if it is a circular part, otherwise its comple- ment, and observe which is the middle part, and whether the other parts are adjacent to, or opposite, the middle part : if adjacent, the first of Napier's principles will give the formula; if opposite, the second. 5. Introduce R and apply logarithms. 6. Apply the principles which determine the species of the required part. 120 ' TEIGONOMETR F. h = 110 30'. 1. Giv.1 5Q o 45 1. To fjnd b. From the second of Napier's principles, we have sin (90 K) cos b cos p, or cos h = cos b cos p. Finding cos b and introducing 7?, we have R cos A cos b cos p , ' . log cos b = 10 -{- log cos A log cos p. log cos h (110 3CX) '== 9.54433 - log cos p ( 50 45' ) == 9.80120 + log cos b = 9.74313 .-.&= 123 36' 31". Since the hypotenuse is greater than 90, the sides b and p are of different species ; but p < 90 ; . . b > 90. But log cos b corresponds to 56 23' 29", and to its supplement 123 36' 31" which must be taken, since b > 90. The species of b can also be determined by the form'ula, cos h cos b cos p Since h > 90, cos h is negative, and since p < 90, cos p is positive; .'. cos b is negative; . '. b > 90. The signs of the functions may be conveniently indi- cated by placing the signs after their logarithms. RIGHT TRIANGLES. 121 2. To find B. sin (90 B) = tan p tan (90 K), tan p cot h .-. CO SJ B= -^- . * . log cos 5 =^= log tan jp -|- log cot h 10. log tan p ( 50 45') = 10.08776 + log cot h (110 30') = 9.57274 ~ log cos B -- 9.66050- ..-. B= 117 14'. Since 6 and B are of the same species, and since b > 90, B > 90. The species of B can also be de- termined from the sign of cos B. 3. To find P. sin p = cos (90 h) cos (90 P), or sin p = sin h sin P. n P sin p . . gin *= . ,- ; sin h ' . . log sin P = 10 -J- log sin j9 log sin A. log sin j9 ( 50 45') == 9.88896 + log sin h (110 30') = 9.97159 -f- log sin P - 9.91737 + ' P=^ &' 57". P is of the same species as p, and since p < 90, P < 90. The species of P can not be determined by the sign of sin P, since the sign of sin P is plus from to 180. : 67 33 ' 27 "' 2. Given Re. B= 67 54' 47". = 99 5735". = 67 06' 44". 3. Given = ( h 94 05' I f : \ __ ^ ^/ \ Req. 1 B= 1 p = \P= r^ 110^46' 26"! ( 6 = 67 06' 44". { h = l1 ^ f ^ } Req. \p = 155 47' 05", I P= 153^ 58' 45". S. N. 11. 122 TRIGONOMETRY '. = . 990 , ,, f # = 54 3 01' 15". . 4. Given j p = 13?0 ^ 2r | Req. j *=^ 142 09' 12". ( jo = 155 27' 55". r , fi 1V , fh= 75 13' 01". 5. Given] ~ " Zf ReqJ * = 67 27' 01". lp = 58 25' 45". rP^-5^0'l . Giv. ] ^Lo-,,/ f H = 52 34' 31" or 127 25' 29'. B = 23 03' 06" or 156 56' 54". 8. If a line make an angle of 40 with a fixed plane, and a plane embracing this line be perpendicular to the fixed plane, how many degrees from its first posi- tion must the plane embracing the line revolve about it in order that it may make an angle of 45 with, the fixed plane? Ans. 67 22' 44" or 112 37' 16". 132. Polar Triangles. The polar triangle of a given triangle is the triangle formed by the intersection of three arcs of great circles described about the vertices of the given triangle as poles. If one triangle is the polar of an- other, the second is the polar of the first. Thus, if A'B'C' is the polar of the triangle ABC, then ABC is the polar of A'B'C'. Each angle in one of two polar triangles is the sup- plement of the side lying opposite to it in the other; RIGHT TRIANGLES. 123 and each side is the supplement of the angle lying opposite to it in the other. Thus, A = 180 a', B = 180 &', C = 180 c'. a = 180 A', 6 = 180 B', C = 180 C'. A'= 180 a, B'= ISO b, C'= 180 c. Cor. If a' = 90, A = 90 ; hence, if one side of a triangle is 90, one angle of its polar triangle is 90. 133. Quadrantal Triangles. A quadrantal triangle is a triangle one side of which is 90. By the corollary of the last article, it follows that the polar of a quadrantal triangle is a right-angled triangle. A quadrantal triangle is solved by passing to its polar triangle, which is solved as a right-angled tri- angle, then by passing back to the quadrantal triangle, which is the polar of the right-angled triangle. 134. Examples. rh'= 90. ^ fH'= 69 30'. 1. Given { F = 129 15'. V Req. \B' = 56 23' 30". I &' = 62 46' 01". ) I p' = 124 14' 03". Passing to the polar triangle, which is right-angled, we have fH= 90. ^ f h = 110 30'. Given \ p = 50 45': V . '. < b = 123 36' 30". I B = 117 13' 59". J I P = 55 45' 57". 124 TRIGONOMETRY. Passing back to the quadrantal triangle, we find f a'=: 90. ^ (A'= 74 26'. % Given < c' == 99 20'. V Req.K C" = 108 05' 26'. (B'= 30 12' 23". j I &' = 31 29' 14'. OBLIQUE TRIANGLES. 135. Proposition I. The sines of the sides of a spherical triangle are propor- tional to the sines of their opposite angles. Let ABC be a spherical tri- angle. From C draw p, the arc of a great circle perpendicular to the opposite side or to the opposite side produced. In the first case we have, by Napier's principles, sin p = cos (90 a) cos (90 5) &* sin a sin B. sin p = cos (90 ft) cos (90 A) = sin 6 sin A. . ' . sin a sin B sin b sin A. . ' . sin a : sin b : : sin ^4 : sin B. In the second case we have, by Napier's principles, sin p = cos (90 a) cos (90 B') == A sin a sin B'= sin a sin 5. sin p = cos (90 ft) cos (90 A) = sin 6 sin A. .'. sin a sin B = sin ft sin A . . sin a : sin ft : : sin A : sin B. OBLIQ UE TRIA NGL ES. 125 In like manner other proportions may be deduced, giving the group, (1) sin a : sin b :: sin A : sin B. (2) sin a : sin c : : sin A : sin C. (3) sin b : sin c : : sin B : sin C. 136. Proposition II. The co-sine of any side of a spherical triangle is equal to the product of the co-sines of the other sides, plus the product of their sines into the co-sine of their included angle. Let ABC be a spher- ical triangle, and the center of the sphere. Let CM be perpendic- ular to the plane AOB. Draw MD and ME, re- spectively perpendicu- lar to OB and OA, and draw CD and CE, which will be respectively perpen- dicular to OB and OA; hence, the angle OEM A. and CDM == B. Draw EF perpendicular to OB, and MN perpendicular to EF. Each of the angles MEN and EOF is the complement of OEF; . . MEN=EOF= c. OD = OF -f- NM. OD = cos a. OF = OE cos EOF = cos 6 cos c. NM= EM sin MEN = sin b cos A sin c. Substituting the values of OD, OF, and NM, we have cos a cos b cos c -f- sin 6 sin c cos A. TRIGONOMETRY. In like manner other formulas may be deduced, giv- ing the group, (1) cos a = cos b cos c -\- sin b sin c cos A. (2) cos b ~ cos a cos c -f- sin a sin c cos J?. (3) cos c = cos a cos 6 -j- sin a sin b cos (7. 137. Proposition III. The co-sine of any angle of a spherical triangle is equal to the product of the sines of the other angles into the co-sine of their included side, minus the product of the co-sines of these angles. The formulas for passing to the polar triangle are, a = 180 A', b = 180 ', c = 180 C'. A = 180 a', B = 180 6', C = 180 c'. Substituting these values in the formulas of the preceding article and reducing, we have cos A'= cos B' cos C" sin B' sin C' cos a'. cos B'= cos A' cos C' sin A' sin C" cos b'. cos C'= cos -4' cos B' sin A' sin B' cos c'. Changing the signs and omitting the accents, since the formulas are true for any triangle, we have (1) cos A sin B sin C cos a cos B cos C. (2) cos B = sin A sin C cos b cos A cos C. (3) cos C = sin A sin J5 cos c cos -4 cos B. 138. Proposition IV. 7Vi co-sine of one- half of any angle of a spherical tri- angle is equal to the square root of the quotient obtained by OBLIQUE TRIANGI.IW. 127 dividiinj tin' ,s///r <>f true-half th< xmit of (!/<' sides into tin- vine of o'lH'-hatf the .s/rwi minus the M</< o/^ax/Vr //// timjlr, by tJie product of flic xincx of tin- aa'jarcnf -svVfx. The first formula of article 130 gives cos a cos 6 cos c COS A : - : - ; sin b sin c Adding 1 to botli im-mlMTs, we have cos a -f sin 6 sin c cos b cos c 1 4- cos A = : 7 r sin /> sin c 1 + cos A = 2 cos 2 A Article 95, (10). sin' b sin c cos fr cos c = - cos (/> H c). Art. 89, (tf). . sin ft sm c But by article 96, (8), we have cos a cos (ft H- c) = 2 sin J(a -f ft -f- c) sin J(6 + c a). Substituting and dividing by 2, we have sin K<* 4- b H- c)* sin fr(ft + g o) COS 5^1 ; - ; - ; Sill ft Bill c Let 8 a -f ft -h r, then will s ^ J (a -f ft + <0> ^ 8 a J(ft 4- c a). Substituting in the value of cos 2 \A, and in the similar values for cos 2 \E and cos 2 JC T , and extracting the square root, we have (1) cos \A = (2) cos \E = ' s ' sn iH sin ft sin c s* 8n s- sin a sin c (3) cos JO .-=Jri'n forin . ! cj _ ^ sin a sin ft 128 TRIG ONOMETR Y. 139. Proposition Y. The sine of one-half of any side of a spherirnl triangle is equal to the square root of the, quotient obtained In/ dividing minus the co-sine of one-half the sum of the angles into the co-sine of one-half the sum minus the angle opposite the side, by the product of the sines of the adjacent angles. Taking the formulas of the last article, passing to the polar triangle, making S = A' -f B' -j- C", substitut- ing in these formulas, reducing, and omitting the ac- cents, we have (1) sin ^a = J CQS S cos QS A) ^ sin B sin C (2) sin i 6 = J costs cos (-) . * sin A sin (7 (3) sin l f = I-<X**8<X*QS- ^ sin A sin 5 140. Proposition VI. The sine of one-half of any angle of a spherical triangle is equal to the square root of the quotient obtained by dividing the sine of one-half the sum of the sdd.es minus one adjacent side into the sine of one-half the sum minus the other adjacent side, by the product of the sines of the adjacent sides. cos a cos 6 cos c cos A = - . r T - Article 136, (1). sin b sin c Subtracting both members from 1, we have cos b cos c 4- sin b sin c cos a 1 cos A -- - . i r-r - sin 6 sin c 1 cos A = 2 sin 2 JA Article 95, (9). OBLIQUE TRIANGLES. 129 cos b cos c -}- sin b sin c = cos (6 r). Article 91, (d).' .. sin o sin c But by article 96, (8), we have cos (ft 6-) cos a = 2 sin J(a -f c ft) sin J(a + 6 c). Substituting and dividing by 2, we have . a t , _ sin (q + c ft) sin |(a + ft c) Sin *^i ; j - ; - sin ft sin c But J(a -|- c ft) = s ft and J(a -f ft c) = %s c. Substituting in the value of sin 2 J^4, and in the similar values for sin 2 ^B and sin 2 J(7, and extract- ing the square root, we have (1) sin \A = sm ft s - ft) jmjjs-- c) . ^ sin ft sin c /sin (jg a) sin (jg c) (2) sin $B = * sin a sin c (3) sin iC = J sin (i - a) sin (j 8 -6) ^ sin a sin ft 141. Proposition VII. The co-sine of one-half of any side of a spherical triangle is equal to the square root of the quotient obtained by dividing the co-sine of one-half the sum of the angles minus one adja- cent angle into the co-sine of half the sum minus the other adjacent angle, by the product of the sines of the adjacent angles. 130 TRIGONOMETRY. Taking the formulas of the last article, passing to the polar triangle, making S = A' -f B r -j- C", substitut- ing, reducing, and omitting the accents, we have sin B sin C (1) (2) ~lh- cos QS-A) cos (jg= * sin A sin (7 (3) cos \c = / v cos - cos sin ^4 sin B U2. Proposition VIII. The tangent of one-half of any angle of a spherical tri- angle is equal to the square root of the quotient obtained by dividing the sine of one-half the sum of the sides minus one adjacent side into the sine of one-half the sum minus the other adjacent side, by the sine of one-half the sum of the sides into the sine of one-half the mm minus the opposite side. Dividing (1), (2), (3), article 140, respectively, by (1), (2), (3), article 138, we have sin Js sin (J* a) (1) tan \A = Jsin(i-6)8in(t*- g ) * sin s sin * a (2) tan JP - J sin \ si sn s sn s sn (3) tan iC = / sin^a) sin ($* ft) * sin ^s sin (-Js c) 143. Proposition IX. The tangent of one-half of any side of a spherical triangle is equal to the square root of the quotient obtained by dividing OBLIQUE TRIANGLES. 131 minus the co-sine of. one-half the sum of the angles into the co-sine of one-half the sum minus the angle opposite the side, by the co-sine of one-half the sum of the. angles minus one adjacent angle into the co-sine of one-half the sum minus the other adjacent angle. Dividing (1), (2), (3), article 139, respectively, by (1), (2), (3), article 141, we have ~ COS S cos ~ (1) ^ cos as B) cos as C) (2) tan Jfr==^ - cos cos A) COB asc (3) tan $c = J~ - co* $S cos as C) > cos as A) cos' as B) The reciprocals of (1), (2), (3), articles 142, 143, will give formulas for co-tangents, which may be written and expressed in words. 144. Napier's Analogies. Dividing (1), article 142, by (2), we have tan J A sin a s ) tan J B sin (J -9 ) This, as a proportion taken by composition and di- vision, gives tan J A + tan 4 -B sin ( s 6) + s i n (i s a ) tan \A ' tan \ B sin (-J s 6) sin (J a) sin %A sin \E tan i -4 -h tan ^ B _ cos J ^4 cos tan J ^4 tan \B~~ sin ^^4 _ sin cos J A cos 132 TRIG ONOMETR Y. Multiplying both terms of the second member ly cos \A cos J#, tan \A + tan \B _ sin \A cos \E -f cos \A sin \E tan J^4 tan \E sin \A cos J5 cos \A sin # Reducing the. second member by articles 89, (a), and 91, (c\ tan \A 4- tan ^7? _ sin \(A -f ) tan i^ tarT ~" sin AB ' sin (jg 6) -f sin tys a} tan ^g sin (is 6) -sin Qs a) ~tan"K 6) ' sin (^ + B) tan c ' sin \{A B} ~ tan ^(a 6) ; . (1) sin %(A-f-B) : sin|(^4 B) : : tan %c : tan |(a />). The reciprocal of (1) X (2), article 142, gives 1 _ sin Js tan \ A tan J B ~~ sin (J s c) By division and composition, we have 1 tan \ A tan ^5 __ sin \s sin (\s c) 1 -f tan %A tan J J5 sin -|s -j- sin (J c} Reducing both members as before, we have cos %(A + B) tan \c cos \{A B) ~^ tan J (a + 6) ' . . (2) cos JC^ + B) : cos tfAE) : : tan Jr : tan J( + &) Passing from (1) and (2) to the polar triangle, we have (3) sin J(a + 6) : sin $(ab) : : cot \G : tan \(A (4) cos \(a + 6) : cos Ma 6) : : cot \C : tan OBLIQUE TRIANGLES. 133 145. Proposition. In a right-angled spherical triangle, as b increases from to 90, from 90 to ISO 3 , /row 180 to 270, and from 270 to 360, if p < 90, h increases from p to 90, from 90 to 180 p, decreases from 180" p to 90 D , <md ;>om 90 to IV (/ P > 90, ^ decreases from p to 90, /rom 90 to 180 jo, increases from 180^ p to 90, and /rom 90 to p; if p = 90, h = 90 /or aM values of b. 1. p < 90; .'. cos p is positive, cos h = cos ft cos p. If 6 0, cos 6 = 1; therefore, cos h = cos >/ . . h = p. As b increases from to 90, cos 6 is positive, and diminishes from 1 to 0; ' .'. cos h is positive, and diminishes from cos p to 0; . . h increases from p to 90. As b increases from 90 to 180, cos b is negative, and increases numerically from to 1 ; . . cos h is negative, and increases numerically from to cos p; . . h increases from 90 to 180 jo, and the triangle becomes the lune HH'. As 6 increases from 180 to 270, cos b is negative, and decreases numerically from --- 1 to 0; .*. cos h is negative, and decreases numerically from cos p to 0; .'. h decreases from 180 p to 90. As b increases from 270 to 360, cos b is positive, and increases from to 1 ; . ' . cos h is positive, and increases from to cos p; ..'. h decreases from 90 to p, and the triangle becomes the hemisphere. 134 TRIGONOMETRY. 2. p > 90 ; . . cos p is negative. TJ cos h = cos 6 cos p. -= =! If 6 = 0, cos 6 = 1; therefore, cos h = cos p ; .' . h = p. As 6 increases from to 90, cos 6 is positive, and decreases from 1 to ; . * . cos h is negative, and de- creases numerically from cos p to ; . . h decreases from p to 90. As 6 increases from 90 to 180, cos 6 is negative, and increases numerically from to 1 ; . . cos h is positive, and increases from to cos p; . . h de- creases from 90^ to 180 p, and the triangle becomes the lune HH'. As 6 increases from 180^ to 270, cos 6 is negative, and decreases numerically from 1 to 0; . . cos h is positive, and decreases from cos p to 0; . . h in- creases from 180 p to 90. As 6 increases from 270 to 360,. cos 6 is positive, and increases from to 1; . *. cos h is negative, and increases numerically from to cos p; .'. h increases from 90 to p, and the triangle becomes the hemi- sphere. 3. p = 90 ; . . cos p = 0. . . cos h = cos 6 cos p = ; . . h = 90. Cor. Since B and 6 are of the same species, B may be substituted for 6 in the preceding proposition. In the application of these principles to the discus- sion of Case I, in which two sides and an angle oppo- site one of them are given, a corresponds to A, and HB to 6. OBLIQUE TRIANGLES. 135 146. Case I. Given two sides of a spherical triangle, and the angle oppo- site one of them; required the remaining parts. Let a and b be the given sides and A the given angle. A< P sin p sin b sin A. 1. a = p. B coincides with H, and the triangle ABC becomes the right triangle AHC. 2. a < 90 and a > p. By the last proposition the point B lies in the first or fourth quadrant, estimated from H. 3. a = 90. HB = 90 or 270, and HCB = 90 or 270. 4. a > 90 and . < 180 p. B lies in the second or third quadrant from H. 5. a = 180 p. HB == 180, and ABC = AHC -f J the hemisphere. 6. a == 180 b. Q= #.4' or 360 HA', and then the first tri- angle becomes the lune AA'. 7. a = 6. HB ^ AH or 360 ^#, and the second triangle becomes the hemisphere. 136 TRIG ONOMETR Y. 8. a < p or a > 180 p. The triangle is impossible, since p is the least, and 1 80 - p is the greatest value of a. II. yl > 90 5 ; .'. p > 90. - . , . A<(^ W \j^ sin p = sin 6 sin A. N. 7 I x - 1. a=p. B coincides with H, and ABC becomes AHC. 2. a > 90 and a < p. B lies in the first or fourth quadrant from H. 3. a = 90. # = 90 or 270, and HOB = 90 or 270. 4. a < 90 and a > 180 p. B lies in the second or third quadrant from H. 5. a = 180 p. HB = 180, and ABC = AHC + the hemisphere. 6. a == 180 6. HB = #4' or 360 /L4', and the first triangle be- comes the lune A A. 7. = 6. HB -= ,4# or 360 AH] and the second triangle becomes the hemisphere. 8. a > p or a < 180 ^. The triangle is impossible, since p is the greatest, and 180 p is the least value of a. III. A == 90. The triangle is right-angled, and is solved as in article 131. OBLIQUE TRIANGLES. 137 147. Examples. ( a = 60 D 20'. ^ f B. 1. Given I b = 80 35'. V Req. 1 C. A < 90; .-. p < 90. sin p = sin b sin A, . . p = 37 48' 26". Since a > p and a < 180 -- p, the triangle is possible. Since a < 6 and a < 180 6, B lies between If and A or H and .4'. sin p = sin a sin J5, . - . B = 44 52' 05". cos #<? = tan p cot a, . . HCB = 63 46' 18". cos a = cos p cos #, . . HB = 51 12' 41". cos ACH = tan p cot ft, . . ^ICTT == 82 36' 25". cos b = cos p cos AH, .'.AH = 78 02' 54". C = ACH HCB = 146 22' 43" or 18 50' 07". c = AH HB = 129 15' 35" or 26 50' 13". In ACB, ABC = 180 HBC = 135 07'' 55". We can also find B from the proportion, sin a : sin b : : sin A : sin B. C and c can be found from the proportions, sin i(6 + a) : sin J (6 a) : : cot JC : tan %(B A). sin ^4 : sin C : : sin a : sin c. 2. Given. Required. f a = 63 5<y. ^| f 5= 59 16' 00" or 120 44' 00". < b = 80 19'. V < C = 131 29' 42" or 24 37' 30". 1 A = 51 3(y. j I ? = 120 47' 50" or 28 32' 44". S. N. 12. 138 TRIG ONOMETR Y. 3. Given. Required. a = = 75 38'. -) rB= 65 28' or 114 32'. b = 104 22'. V l\q= 180 or 57 03' 32". A = = 65 28'. ) ( c = 180 or 63 3 20' 18". 4. Given. R equ'i [red. a -.- - 99 40 7 48". ^ (B = 114 26' 50" or 65 33' 10". b = 64 23' 15". V 1 C = 236 51' 27" or 97 27' 13". A = 95 38' W.) U = 236 or 51" or 100 49' 49". 5. Given. R equ\ a s= 100. ) (B= 50 47' 41" or 129 12' 19". b = 85. \ 1 (7=186 05' 16" or 342 o 03' 12". A == 50. } 1 c == 187 50' 09" or 336 o 39' 45". 6. If A < 90, what is the relation of a to p, or to 180 pj when there is no solution ? 7. If A > 90, what is the relation of a to p, or to 180 _p, when there is no solution ? 148. Proposition. In a right-angled spherical triangle, as B increases from to 90, from 90 to 180, from 180 to 270, and from 270 to 360; if p < 90, P decreases from 90 to p, in- creases from, p to 90 D , increases from 90 to 180 p, and decreases from 180 p to 90; if p > 90, P increases from 90 to p, decreases from p to 90, decreases from 90 to 180 p, and increases from 180 p to 90; ifp = 9Q, P=. 90, for all values of B. V~" ~~\H' */v . / H 1. jo < 90; . . cos p is positive. / ^^ / H( ^_^ XP cos P = cos j9 sin J5. If B =0, sin B = 0; . . cos P = 0; .-.P=90. OBLIQUE TRIANGLES. 139 As B increases from to 90, sin B is positive, and increases from to 1 ; . '. cos P is positive, and increases from to cos p; . . P decreases from 90 to p. As B increases from 90 to 180, sin B is positive, and decreases from 1 to 0; . ', cos P is positive, and decreases from cos p to ; . . P increases from p to 90, and the triangle becomes the lime HH'. As B increases from 180 to 270, sin B is negative, and increases numerically from to 1 ; . . cos P is negative, and increases numerically from to cos p; . . P increases from 90 to 180 p. As B increases from 270 to 360 sin B is negative, and decreases numerically from 1 to ; . . cos P is negative, and decreases numerically from cos p to 0; . . P decreases from 180 p to 90, and the triangle becomes the hemisphere. 2. p > 90 ; . . cos p is negative. cos P = cos p sin B. If B = 0, sin B = 0', .'.cosP=0; .-.P=W. As B increases from to 90, sin B is positive, and increases from to 1 ; . . cos P is negative, and in- creases numerically from to cos p; . . P increases from 90 to p. As B increases from 90 to 180, sin B is positive, and decreases from 1 to ; . . cos P is negative, and decreases numerically from cos p to ; . . P decreases from p to 90, and the triangle becomes the lune. As B increases from 180 to 270, sin B is negative, and increases numerically from to 1 ; . . cos P is positive, and increases from to -- cos p; .'. P de- creases from 90 to 180 p. 140 TRIGONOMETRY. As B increases from 270 to 360, sin B is negative, and decreases numerically from 1 to ; . . cos P is positive, and decreases numerically from cos p to 0; .'..P increases from 180 -- p to 90, and the triangle becomes the hemisphere. 3. p = 90; .'', cos p = 0. . -. cos P = cos p sin B = 0; . . P 90. Cor. Since b and B are of the same species, b may be substituted for B in the preceding proposition. 149. Case II. Given two angles of a spherical triangle and the side opposite one of them; required the remaining parts. Let A and B be the given angles, and b the given side. 3- - A \ /Iy> I. A < 90; . : p < 90. ^^Jzffi sin p sin b sin A. 1. B > p and B < 90. By the last proposition, the point B lies in the first or second quadrant estimated from H as origin. 2. B = p. The angle HCB = 90, and the arc HB == 90. 3. 5 < 180 p, and B > 90. 5 lies in the third or fourth quadrant from H. 4. B == 180 p. The angle HCB = 270, and the arc HB = 270. OBLIQUE TRIANGLES. 141 5. B = 90. HB = 0, 180, or 360, and the triangle becomes ACH, ACH -f- | of a hemisphere, or a hemisphere -f ACH. 6. B =- A. B lies in the first or second quadrant from H, and one of the triangles becomes the lune A A'. 7. B = 180-A. B lies in the third or fourth quadrant from H, and one of the triangles becomes the hemisphere. 8. B < p or B > 180 p. The triangle is impossible, since p is the least, and 180 p is the greatest value of B. II. A > 90 ; . . p > 90. sin p = sin b sin A. 1. B < p and B > 90. jB lies in the first or second quadrant from H. 2. B=p. The angle HCB = 90, and the arc HB = 90. 3. B > 180 p and B < 90. B lies in the third or fourth quadrant from H. 4. B = 180 ;>. The angle tfC = 270, and the arc HB = 270. 5. B = 90. HB = 0, 180, or 360, and the triangle becomes ACH, ACH -f of a hemisphere, or a hemisphere -f- 142 TRIGONOMETRY. 6. B = A. B lies in the first or second quadrant from H, and one of the triangles becomes the lune AA'. 7. B = 180 A. B lies in the third or fourth quadrant from H, and one of the triangles becomes the hemisphere. 8. B > p or B < 180 p. The triangle is impossible, since p is the greatest, and 180 p is the least value of B. III. A = 90. The triangle is right-angled, and is solved as in article 131. 150. Examples. ( A = 75 30'. ^ r a. 1. Giv. < B = 80 40'. V Req. < C. I 6 = 70 50'. J I c. A A < 90 ; . . p < 90. sin p = sin 6 sin A] . . p = 66 07' 56". Since B > j? and < 180 >, the triangle is pos- sible. Since B < 90 and > p, B lies in the first or second quadrant from H. ( 67 56'. sin p = sin a sin J5, . . a = < ^o 94' The second value of a, the supplement of the first, is taken when B lies in the second quadrant from H. OBLIQUE TRIANGLES. 143 f 23 37' 44" cos B = cosp sin HCB, . . HCB = <j 1560 / 21 48' 19". 158 11' 41". \ 156 22' 16". sin HB = tan p cot 5, , ; . J7J3 = | cos ACH=- tan p cot 6, . ' . ACH = 38 13' 36''. cos 6 = cos p cos v4#, .-. AH = 35 46'. C = ylO^/ + JEfO5 ='61 51' 20'' or 194 35' 52". c = AH + HB - 57 34' 19" or 193 57' 41". We can find a, r, and C from the proportions, sin B : sin A : : s4n ft : sin a. sin -J (B -}- ^4) : sin J (B ^4) : : tan J r : tan \ (b a). sin b : sin c : : sin J5 : sin C. 2. GrMtetl. ^ = : 33 15'. B = 31 34' 38". b = 70 10' 30". 3. Given. A = 132 16'. B = 139 44'. b = 127 30'. 4. Given. A = 48 50'. 5=131 10'. b= 75 48'. Required. a= 80 03' 25" or 99 56' 35". C = 161 24' 52" or 173 30' 52". c = 145 03' 13" or 168 18' 23". Required. a = 65 16' 30" or 114 43' 30". C = 165 41' 46" or 126 40' 44". c = 162 2(T 55" or 100 07' 25". Required. a= 75 48' C = 360 c = 360 or 104 12'. or 328 39' 28". or 317 56' 42". Scholium. In the two preceding cases some of the parts are found to be greater than 180 ; but the cor- responding triangles conform to the conditions of the problem, and are therefore true solutions. 144 TRIGONOMETRY. Parts greater than 180 are usually excluded, in which case the principles of the following article will aid in determining the species of the parts. The principles established in Geometry are given without demonstration. 151. Principles. 1. Each part of a spherical triangle is less than 180. 2. The greater side is opposite the greater angle, and con- versely. 3. Each side is less than the sum of the other sides. 4. The sum of the sides is less than 360. 5. The sum of the angles is greater than 180, and less than 540 \ 6. Each angle is greater than the difference between 180 and the sum of the other angles. For, A + B + C> 180. Principle 5. .-. A > 180- (B -j-<7). The last formula is always algebraically true; but in case B + C> 180, it might be doubted whether it is numerically true. Passing to the polar triangle, we have, by principle 3, a' < V + c'. or 180 A < 180 B + 180 C. or -A < 180 (B + C). .-. A > B + C ISO . 7. A side differing more from 90 than another side is of the same species as its opposite angle. OBLIQUE TRIANGLES. 145 By article 136, we have cos a = cos b cos c -f- sin 6 sin c cos A. cos a cos b cos c cos A = sn sn c But sin 6 sin c is positive, since 6 and c are each less than 180' If a differs^ more from 90 than ft or c, then we shall have cos a > cos 6, or cos a > cos c, numerically ; and since neither cos b nor cos c exceeds 1, we have cos a > cos ft cos c. . ' . cos A and cos a have the same sign, . . A and a are of the same species. 8. An angle differing more from 90 than another angle is of the same species as its opposite side. By article 137, we have cos A sin R sin C cos a cos B cos C. cos A -{- cos B cos C . ' . COS a - - : sin B sin C If ^4 differs more from 90 than B or C, then, as before, cos A and cos a have the same sign, or A and a are of the same species. 9. Two sides, at least, are of the same species as their oppo- site angles, and conversely. If each of two sides differs more from 90 than the remaining side, they will be of the same species as their opposite angles, as is evident from principle ,7, If the triangle is isosceles, and the equal sides less than 90, the perpendicular from the vertex to the third side will be less than 90, since one-half the S. N. 13. 146 TRIGONOMETRY. third side is less than 90, and the angles opposite this perpendicular will be less than 90, article 129, or of the same species as their opposite sides. If the equal sides are greater than 90, the perpen- dicular will be greater than 90, since one-half the third side is less than 90, and the angles opposite the perpendicular will be greater than 90, article 129, or of the same species as their opposite sides. If one side exceeds 90 by as much as 90 exceeds another side, and the third side is greater or less than each of the other sides, this third side is of the same species as its opposite angle by principle 7. If the greater of the two sides is of the same species as its opposite angle, then we shall have two sides of the same species as their opposite angles. If the greater of the two sides is not of the same species as its opposite angle, this angle will be of the same species as the other side, or less than 90 ; but the angle opposite this other side is less than the angle opposite the greater side, and hence less than 90, or of the same species as its opposite side, and again we have two sides of the same species as their opposite angles. 10. The sum of two sides is greater than, equal to, or less than, 180, according as the sum of their opposite angles is greater than, equal to, or less than, 180. tan K a + &) cos %(A-\-B) tan Jc cos %(AE). Art. 144. But c < 180, .-. Jc < 90, tan \c > 0, and AB < 180, . - . %(AB) < 90, cos %(AE) > 0. . . tan \c cos $(AB) > 0, tan |(+&) cos $(A+ E) > 0. . . tan J(a -f &) and cos %(A -f B) have like signs. OBLIQUE TRIANGLES. 147 If %(A + B) >, = or < 90, J(a-f 6) >, = or < 90. If 4 f B >, = or < 180, a + 6 >, = or < 180. 152. Case III. Given two sides and, the included angle of a spherical triangle; required the remaining parts. (a =85 W.\ 1. Given <^ b = 65 40'. V Req. { B. I C= 95 50'. J We have, article 144, cos i(a -f- 6) : cos (a 6) : : cot \C : tan \(A -f -B). sin ^ (a- -f 6) : sin J (a b) : : cot -J C : tan %(A B)* t$(A+'B) = 74 21' 49". ) f A = 83 29' 10''. \$(A B)== 9 07' 21". j *' \ B = 65 14' 28". We also have, article 144, sin %(A-{-E) : sin %(A ) : : tan J c : tan J (a b). . . J c = 46 43' 09", ' . . c = 93 26' 14". We can also find c from the proportion, sin A : sin C : : sin a : sin c. But the species of c is more readily determined from the proportion employed; for if we take the supple- ment of 46 43' 09", then c would be greater than 180. Again, all the known terms of the proportion are positive; hence, tan %c is positive, .*. ^c < 90. a = 120 30' 30". ^ fA= 135 05' 29". 2. Given < b =~70 ft 20' 20". V- Req. < J5 = 50 30' 09". 50-10<-10": ) H SP 69 34' 58". (a = I Given < b = (c'=^ 148 TRIGONOMETRY. 153. Case IV. Given two angles and the included side of a spherical triangle; required the remaining parts. fA= 62 54'. 1. Giv. IB = 48 30'. I c = 114 29' 58". c : tan | (a -j- b). c : tan J (a 6). f a == 83 12' 06". " *\ C * \ We also have, article 144, sin \ (a -J- 6) : sin J (a 6) : : cot j (7 : tan (A B). . . | C = 62 40', . . C = 125 20'. We have, article 144, -i cosion-*: > : cos J (4 B) : : tan sin JC4-f B: > : sin 1 (A 5) :' : tan (i( a j_ 6) = 69 55' 48". 1 ( A = 126 35' 02". 2. Given < B = 61 43' 58". I c = 57 30'. a = 115 19' 57". b = 82 27' 59". C = 48 31' 38". 154. Case V. Given the three sides of a spherical triangle; required the angles. fa == 100 49' 30".^| f A. 1. Giv. < b = 99 4(X 48". V Req. < B. U =^ 64 23' 15". J la By article 138, we have cos \A 'sin $s sin ^ a) sin b sin r OBLIQUE TRIANGLES. 149 Introducing R and applying logarithms, we have log cos \A = \ [ log sin \ s -}- log sin (-J s a) -f- a- c. log sin b 4- a. c. log sin c]. . . \A = 48 43' 14", . . A = 97 26' 28". , f B = 95 38' 00". In like manner we find < . a = 85 30'. ^ . f A = 83 29' 08". 2. Given < b = 65 40'. V ReqJ B == 65 14' 20". r a = < b = I c = 93 26' 18". == 95 50'. 155. Case VI. Given the three angles of a, spherical triangle; required the sides. r4 = 119 15'.} fa. f. Given <== 70 39'. V Req .1 b. ( C == 48 36'. ) I c. By article 139, we have cos >a = sin B sin C Introducing R and applying logarithms, we have log cos J a = J [log cos (-J& B) 4- log cos QSC) 4- .. 6-. log sin 5 -j- a. c. log sin C] . .-. 4 a = 56 11' 31", .-. a = 112 23' 02". / b = 89 16' 54". In like manner we find < ^o OQ/ QQ-/ r ^ . = 121 36' 24". ^ c a = X B =-- 42 15' 13". > ReqJ ft ^ I C = 34 15' 03". j I c = ^ 121 36' 24". ^ f a = 76 36' 00". 2. GivenX B i 42 15' 13". > Req. < ft bi 50 10' 40". 40 00' 20". 150 MENSURATION. MENSURATION. 156. Definition and Classification. Mensuration is the art of calculating the values of geometrical magnitudes. Mensuration is divided into two branches Mensu- ration of surfaces and Mensuration of volumes. MENSURATION OF SURFACES. 157. Unit of Superficial Measure. A unit of superficial measure is a square each side of which is a linear unit. Thus, according to the object to be accomplished, a square inch, a square foot, a square yard, an acre, etc., is the superficial unit taken. 158. Problem. To find the area of a rectangle. Let k denote the area, b the base, and a the altitude of a rectangle. There are a rows of b superficial units each. Since there are b superficial units in one row, in a such rows there will be a times b or ab superficial units. .-. (1) k = ab. The above demonstration applies only in case the base and altitude are commensurable, or have a com- mon .unit, ... SURFACES. 151 If the base and altitude are incommensurable, denote the area by ', the base by 6', and the altitude by a'. Then, since by Geometry any two rectangles are to each other as the products of their bases and alti- tudes, we have I- : // : : ab : a'V. But k = ab, .'. k' = a'U. 159. Problem. To find the area of a parallelogram. 1. When the base and altitude are given. Let k denote the area, b the base, and a the altitude of a parallelogram. Since a parallelogram is equal to a [/ rectangle, having the same base and altitude, and since the area of the rectangle is equal to the product of its base and altitude, the area of the parallelogram is equal to the product of its base and altitude. . . (1) k = ab. . _______ A 7 2. When two sides and their included angle are given. b a = c sin A. .' . (2) k = be sin A. 1GO. Problem. To find the area of a triangle. 1. When the base and altitude are given. Since a triangle is one-half the parallelogram having the same base and altitude, we have for the tri- angle, (1) lt=\ab. 152 MENSURATION. 2. When two sides and their included angle are given. Since a triangle is one-half the parallelogram, having an equal angle c / and equal adjacent sides, we have for / the triangle, (2) k == J be sin A. 3. When two angles and a side are given. The third angle is equal to 180 minus the sum of the given angles. Let, then, the angles and the side A b be given. By the last case, we have k = %bc sin A. But sin B : sin C : : b : c, . . . sin B Substituting this value of e, we have b 2 sin A sin C b sin C (3) *= 2 sin B 4. When two sides and an angle opposite one of them are given. IB Let a and c be the given sides, and A the given angle. In case of one or two solutions determined by article 72, find the value or values of C and B from the formulas, sin c = C Sm A , and B = 180 (A + C). Then, by (2), we have (4) . k = ac sin 1?. SURFACES. 153 5. When the three sides are given. Let p --- the perimeter = a -f b -f c. Then, by article 102, we have (5) k = V~ 6. When the perimeter and angles are given. Let p be the perimeter, and A, and C the angles. By article 98, (10), (11), (12), A ^ lp 2 tan \A tan \E tan \C=' ... (6) jk = Jp2 tan p tan tan JG 7. When the perimeter and radius of the inscribed circle are given. B Let p = (i + 6 + <> an d y be the radius of the inscribed circle. ABC = ~ ABC=--k, ... jb=: J(o + 6 + <?) r; but a .'. (7) fc = 161. Examples. 1. Find the area of a triangle whose base is 75 ft., and altitude is 24 ft. Arts. 900 sq. ft. 2. Two sides of a triangle are 25 yds. and 30 yds., respectively, and their included angle is 50 ; required the area. Ans. 287.2665 sq. yds. 154 MENSURATION. 3. In a triangle, b = 100 ft., A = 50, C 60: required the area. Ans. 3529.9 sq. ft. 4. In a triangle, a = 40 yds., c = 50 yds., ^4 = 40; required the area. Ans. 998.18, or 232.83 sq. yds. 5. In a triangle, a = 12 ft., b = 15 ft., c = 17 ft.; required k. Ans. 87.75 sq. ft. 6. In a triangle the perimeter is 20 ft., and the angles are 50, 60, and 70, respectively; required the area. Ans. 18.85 sq. ft. 7. In a triangle the perimeter is 60 ft., and the radius of the inscribed circle is 5 ft.; required the area. Ans. 150 sq. ft. 162. Problem. To find the area, of a quadrilateral. 1. When two opposite sides and the perpendiculars to these sides from the vertices of the angles at the extremities of a diagonal are given. Let b and b' be two opposite sides, and a and a' the perpendiculars to /i these sides from the vertices of the A L angles D and B. ABCD = ABD DCB. k = 06 Corollary 1. If b' is parallel to b, the quadrilateral becomes a trapezoid, a' = a, and (1) becomes .(2) k = $a SURFACES. 155 Corollary 2. If 6' 6, the trapezoid becomes a parallelogram, and (2) becomes (3) k = ab. Corollary 3. If U = 0, the trapezoid becomes a tri- angle, and (2) becomes (4) k = $db. 2. When a diagonal and the perpendiculars to the diagonal from the vertices of the opposite angles are given. B Let d denote the diagonal, and p and p' the perpendiculars. A ABCD = ABC + ADC. ABCD = k, ABC = J dp, ADC = $ dp'. .-. (5) * = id(p+j/). 3. When the sides and a diagonal are given. Let the areas of the triangles be de- noted by k' and k", which are found by article 160, (5). .-. (6) Jk = ik'.-f Ik". , 4. When the sides .and one angle are given. Draw the diagonal opposite the given angle, and call the areas of the tri- angles k' and k". A< In one triangle we have two sides and their included angle, from which we find the area and the diagonal. 156 MENSURATION. Then, in the other triangle, we have the three sides, from which we find the area. .-. (7) k = k' + k". 5. When the diagonals and their included angle are given. Let d and d r denote the diagonals p and q, r and s their segments, and A their included angle. The angles at A are equal or sup- plementary; hence their sines are equal. BODE ~= BAC -f CAD + DAE + EAB. BCDE = fc, BAC = %ps sin A, CAD = %qs sin A. DAE = \qr sin A, EAB = \pr sin A. . . k = % (ps -\-qs-\-qr-\- pr) sin A. .But ps -f 9.9 -f qr +'pr = (p -f q). (r + s) = <M. .'. (8) k = %dd' sin A. 6. When the angles and two opposite sides are given. Let a = EC, and b = AD. E = 180 (B + C). ..- E The angles at ^4 being supplementary, their sines are equal. The same is true of the angles at D. ABCD = BCE ADE, ABCD = k. a 2 sin B sin C b 2 sin A sin D (9) * = 2sin E 2sin a 2 sin sin C 6 2 sin A sin 2sin SURFACES. 157 7. When three sides and their included angles are given. Let a, 6, and c be the given sides, and A and B their included angles. ABCD = ABD + DBC. = k, ABD = \ ab sin A. Find B' and rf, B" = B B', DBC=%cd sin B". . . (10) k ^ a& sin ^4 -f~ J erf sin 5". 8. When the sides of a quadrilateral inscribed in a circle are given. Let a, b y c, rf be the given sides. ACBD=ACB -\-ADB. -^ A GBD = k, ACB = \ ab sin C. ADB = \ erf sin D -- \ erf sin (7, since D = 180 (7. . . fc = (a& -j- erf) sin C. ~ 2 = a 2 -f- 6 2 2 a6 cos (7, article 97. == c 2 + rf 2 2 erf cos 7) = e 2 -f rf 2 -f 2 erf cos C. + rf 2 -f 2 erf cos (7 = ft 2 + 6 2 - 2 6 cos (7. a 2 + /> 2 e 2 rf 2 cos = 2 (afe -f- erf) sin C =?V 1.7- cos 2 (7, Let s = a . -f 6 -f- e -f rf. 2.|/(|8 a)(|8 6)(t8 c)qg d) Sill L/ , ; ab -j- co (11) fc = i/(J_ ) (Js 6) (Js e) (J rf). 158 XfEXS URA TION. 163. Examples. 1. Two opposite sides of a quadrilateral are 35 rds. and 25 rds., and the perpendiculars to these sides from the extremities of the diagonal are, respectively, 12 rds. and 16 rds.; required the area. Ans. 410 sq. rds. 2. Find the area of a trapezoid whose bases are 15 rds. and 20 rds., and whose altitude is 18 rds. Ans. 315 sq. rds. 3. Two adjacent sides of a parallelogram are 30 rds. and 40 rds., and their included angle is 30 ; required the area. Ans. 600 sq. rds. 4. The diagonal of a quadrilateral is 40 rds., and the two perpendiculars to the diagonal from the vertices of the opposite angles are 10 rds. and 15 rds., respect- ively; required the area. Ans. 500 sq. rds. 5. The sides of a quadrilateral are 30 rds., 40 rds., 50 rds., and 60 rds., and the diagonal drawn from the intersection of the sides, whose lengths are 30 rds. and 40 rds., is 70 rds. ; required the area. Ans. 1874.22 sq. rds. 6. The sides of a quadrilateral are 25 rds., 35 rds., 45 rds., 55 rds., and the angle included by the sides, whose lengths are 35 rds. and 45 rds., is 50; required the area. A n s. 927.47 sq. rds. 7. The diagonals of a quadrilateral are 30 rds. and 40 rds., and their included angle is 30 ; required the area. Ans. 300 sq. rds. 8. The angles of a quadrilateral are 80, 110, 88, 82, the side included by the first and second of these angles is 25 rds., and the side included by the third and fourth angles is 45 rds. ; required the area. Ans. 4105.08 sq. rds. ' SURFACES. 159 9. Three sides of a quadrilateral are 20 rds., 30 rds., 40 rds., the angle included by the first and second is 60, and between the second and third, 80 ; required the area. Ans. 593.58 sq. rds. 10. The sides of a quadrilateral inscribed in a circle are 40 rds., 50 rds., 60 rds., 70 rds. ; required the area. Ans. 2898.28 sq. rds. 11. The area of a parallelogram is 47.055 sq. ft., the sides are 6 ft. and 8 ft.; required the diagonal. Ans. 9 ft., or 10.906 ft. 12. If the adjacent sides of a parallelogram are b and c, and their included angle A, find A and k when & is a maximum. Ans. A = 90, k = be. 13. The sides and angles being expressed as in the last example, find A and k when k is a minimum. Ans. A = or 180, k = 0. 14. If only two adjacent sides, b and e, of a paral- lelogram be given, prove that k is indeterminate be- tween the limits and be. 15. Prove that the diagonals of a parallelogram divide it into four equal triangles. 164. Problem. To find the area of an irregular polygon. 1. When the sides and diagonals from the same vertex are given. The diagonals divide the polygon into triangles whose sides are given. The areas of these triangles, &', fc", &"', . . . are found by article 160, (5). 160 MENSURATION. 2. When the diagonals from the same vertex, and the perpendiculars to these diagonals from the oppo- site vertices are given. (2) k = 3. When the perpendiculars to a diagonal from the vertices of the opposite angles and the segments of the diagonal made by these perpendiculars are given. The polygon is divided into right triangles and trapezoids, whose areas &', fc", &'", .... are found by article 162, (2), (4). (3) & = V + Jb" + F' + . . . 4. When one side of a figure is a straight line, and the opposite side is an irregular curve or broken line. vLet the straight line be divided into the parts , a', a", ...., and r \ let the perpendiculars be p, </, r, . . . dividing the figure into parts which may be con- sidered trapezoids. (4) k = 4 a (p + q) + i '(? + r) + } a"(r + ). If ' == a and a" : = a, (4) becomes, (5) fc = Ja(p-f27 + 2 r -4- s). 165. Examples. 1. Find the area of the annexed polygon if p = 10 rds., q = 6 rds., r = 6 rds., s = 7 rds., t = 15 rds., .d = 14 rds., d' = 16 rds. Ana. 119.86 sq. rds. SURFACES. 161 2. Find the area of the annexed polygon if p = 3 rds., d = 9 rds., p' = 4 rds., d r -- 12 rds., and p" == 5 rds. Am. 67.5 sq. rds. 3. Find the area of the annexed polygon if p = 3 ft., p' 5 ft., p"= 4 ft., a = 5 ft., 6 == 6 ft., c = 6 ft., d = 9 ft., e=8ft. Ans. 80.5 sq. ft. 4. Find the area of the annexed figure, p = 2 rds., q = 3 rds., r 4 rds., s = 3 rds., a = a' = a"= 5 rds. 47.5 sq. rds. 166. Problem. To find the area of a regular polygon. 1. When the perimeter and apothegm are given. Let p be the perimeter, a the apo- them, and s one side of the polygon. k = J as + J as -f J as -j- J as + ... .-. (1) fc = i 2. When the value of each side and the number of sides are given. Let s be one side, n the number of sides, a the apothem, and p the perimeter. 360 180 p = ns. DOB = " -?- 2 n OD = DB cot DOB, or a = S. N. 14. cot 180 C 162 MENSURATION, .-. (2) k = J ns 2 cot If s == 1, then (3) k = J w cot 180 180< From (3) calculate the areas of the regular poly- gons each of whose sides is 1, as given in the table subjoined 167. Table. Triangle = 0.4330127. Square = 1.0000000. Pentagon == 1.7204774. Hexagon = 2.5980762. Heptagon = 3.6339124. Octagon == 4.8284271. Enneagon = 6.1818242. Decagon 7.6942088. Hendecagon^ 9.3656399. Dodecagon % 11.1961524. 168. Application of the Table. Denoting the area of a regular polygon whose side is s by A*, and the area of a similar polygon whose side is 1, as given in the table by k', and apply- ing the principle that the areas of similar polygons are to each other as the squares of the homologous sides, we have the proportion, k : k' I 2 . k = k's 2 . 169. Examples. 1. What is the area of a regular hexagon each of whose sides is 6? Ans. 93.5307432. 2. What is the area of a regular pentagon each of whose sides is 10? Ans. 172.04774. SURFACES. 163 3. What is the area of a regular decagon each of whose sides is 20? Am. 3077.68352. 4. What is the area of a regular dodecagon each of whose sides is 100? An*. 111961.524. 5. What is the area of a regular enneagon each of whose sides is 30? Am. 5563.64178. 170. Formulas for the Circle. Let r be the radius, d the diameter, c the circum- ference, and k the area of a circle, then, by Geometry, we, have d =-- 2 r, c ird, k = \rc. From which verify the following table of formulas: 2 - y-jfc 3. r 4. d = -JI ~vr 5. d = -~ 7T 7. c = 2 :rr. 8. c = Tfd. 9. c = 2 l/Jbr! 10. Jk = Trr 2 . 11. fc 12. k = ^- 171. Examples. 1. Given the radius of a circle 10 rds. ; required d, c, and k. 2. Given the diameter of a circle 20 rds. ; required r, c, and k. 1 64 MEffiS URA TION. 3. Given the circumference of a circle =150 rds. ; required r, d, and k. 4. Given the area of a circle 1000 sq. rds.; re- quired r, d, and c. 5. Find the diameter of a circle whose area is equal to that of a regular decagon, each side of which is 10 ft. Ans. 31.3. 6. The radius of a circle is 10 ft., the diagonals of an equal parallelogram are 24 ft. and 30 ft.; required their included angle. Ans. 60 46' 17". 7. The radii of two concentric circles are r and r'; find the area of the ring included by their circum- ferences. Ans. TT (r + r') (r r). 172. Problem. To find the area of a sector of a circle. Let a be the arc of a sector, d the de- grees in the arc, r the radius, and k the area. By Geometry, (1) k = \ ra. -rrr = the semi-circumference, = the arc of 1. . - . = the arc of d. , k= 173. Examples. 1. Find the area of a sector whose arc is 40 and radius is 10 ft. Ans. 34.907 sq. ft. 2. Find the area of a sector whose arc is 60 24' 30" and radius is 100 rds. Ans. 5271 64 sq. rds. 'SURFACES...' 165 3. The area of a sector is 345 sq. ft., the radius is 20 ft.; required the arc. Am. 98 50' 06". 4. The area of a sector is 1000 sq. rds., the arc is 30 45'; required the radius. Ans. 61.04 rds. \U. Problem. To find the area of a segment of a circle. Let d be the degrees in the arc of the segment, r the radius, and k the area. By the last problem, .,.. the area of the sector. obU ^r 2 sin d = the area of the triangle. dfrr 2 ' If d is greater than 180, sin d is negative, and the second term in the value of k becomes positive, as it should, since, in this case,, the segment is equal to the corresponding sector plus the triangle. 175. Examples. 1. Find the area of the segment of a circle whose arc is 36 and radius 10 ft. Ans. 2.027 sq. ft. 2. Find the area of a segment whose chord is 36 ft. and radius 30 ft. Ans. 147.30 sq. ft. 3. Find the area of' a segment whose altitude w is 36 rds. and radius 50 rds. Ans. 2545.85 sq. rds. 166 4. The area of a segment is 2545.85 sq. rds., the radius is 50 rds.; required the number of degrees in the arc. . . 176. Problem. To find the area of an ellipse. Let a be the semi-major axis, and b the semi-minor axis! Then, Ray's Analytic Geometry, article 446, k = TTob. 177. Examples. 1. The semi-axes of an ellipse are 10 in. and 7 in. ; required the area. An#. 219.912 sq. in. 2. The area of an ellipse is 125 sq. rds.; find the axes if they are to each other as 3 is to 2. Am. 15.45; 10.30. 178. Problem. To find the area of the entire surface of a right prism Let p be the perimeter of the base, a the altitude, s one side of the base, k' the area of a polygon similar to the base, each side of which is unity, ar- ticle 167, and k the area of the entire .surface. ap = the convex surface. 2 k's* = the areas of the bases. Article 168. . . k = ap -f 2 'SURFACES: 167 179. Examples. 1. What is the entire surface of a right prism whose altitude is 20 ft., and base a regular octagon each side of which is 10 ft.? Ans. 2565.68542 sq. ft. 2. What is the entire surface of a right hexagonal prism whose altitude is 12 ft., and each side of the base is 6 ft.? Ana. 619.0614864 sq. ft. 3. What is the entire surface of a right prism whose altitude is 15 in., and base a regular triangle each side of which is 3 in.? Am. 142.7942286 sq. in. 180. Problem. To find the area of the surface of a regular pyramid. Let p be the perimeter of the base, a the slant height, s one side of the base, k' and k as in the last problem. ap the convex surface. k's 2 the area of the base. 181. Examples. 1. What is the entire surface of a regular pyramid whose slant height is 12 ft., and base a regular tri- angle each side of which is 5 ft. ? Ans. 100.82532 sq. ft. 2. What is the entire surface of a right pyramid whose slant height is 100 ft., and base a regular deca- gon each -side of which is 20 ft.? Am. 13077.68352 sq. ft. 168 MENSURATION. 182. Problem. To find the entire surface of a frustum of a right pyramid. Let p be the perimeter of the lower base, p' the perimeter of the upper base, a the slant height, s one side of the lower base, s' one side of the upper base, k f and k as in Art. 178. $a(p -f p') the convex surface. k's 2 = the area of lower base. k's' 2 = the area of upper base. 183. Examples. 1. What is the entire surface of a frustum of a pyramid whose slant height is 12 ft., an-d the bases regular decagons whose sides are 8 ft. and 5 ft., re- spectively? Ans. 1464.78458 sq. ft. 2. What is the entire surface of a frustum of a pyramid whose slant height is 15 ft., and the bases regular hexagons whose sides are 10 ft. and 6 ft., re- spectively? Ans. 1073.338 sq. ft. 184. Problem. To find the area of the entire surface of a cylinder. Let r be the radius of the cylinder, a its altitude, and k the area of the entire surface. 2 nra = the convex surface. 2 -rrr 2 = the area of the bases. .-. k = 2 TIT (a + r). SURFACES. 169 185. Examples. 1. What is the entire surface of a cylinder whose altitude is 6 ft. and radius 2 ft.? Am. 100.5312 sq. ft. 2. What is the entire surface of a cylinder whose altitude is 100 ft. and radius 20 ft.? Am. 15079.68 sq. ft. 186. Problem. To find the area of the entire surface of a rone. Let r be the radius of the base of the cone, a the slant height, and k the area of the entire surface. -rrra = the convex surface. Trr 2 = the area of the base. . . k = TIT (a -f r). 187. Examples. 1. What is the entire surface of a cone whose slant height is 10 ft. and radius 5 ft.? Am. 235.62 sq. ft. 2. What is the entire surface of a cone whose alti- tude is 100 ft. and radius 25 ft. ? Ans. 10059.1675 sq. ft. 188. Problem. To find the area of the entire surface of the frustum of a cone. Let r be the radius of the lower base, / be the S. N. 15. 170 MENSURATION. radius of the upper base, a the slant height, and k the area of the entire surface. na (r -j- r' ) the convex surface. 77T 2 = the area of the lower base. 7T/ 2 = the area of the upper base. .;. k == 7r[a(r-hr')-f-r 2 189. Examples. 1. Find the entire surface of the frustum of a cone of which the radius of the lower base is 10 ft., the radius of the upper base is 6 ft., and slant height is 20 ft. An*. 1432.5696 sq. ft. 2. Find the entire surface of the frustum of a cone of which the radius of the lower base is 25 in., the radius of the upper base 12 in., and the slant height 36 in. Ans. 45.8368 sq. ft. 190. Problem. To find the area of the surface of a sphere. Let r be the radius, d the diameter, c the circum- ference, and k the area. Then, by Geometry, (1) k = 4nr 2 . (2) k = nd 2 . (3) k = ^-- (4) k = cd. 191. Examples. 1. The radius of a sphere is 10 ft.; required the area. Ans. 1256.64 sq. ft. 2. The diameter of a sphere is 25 ft. ; required the area. Ans. 1963.5 sq. ft. SURFACES. 171 3. The circumference of a sphere is 100 in.; required the area. Ans. 3183.0914 sq. in. 4. The circumference of a sphere is 62.832, and di- ameter 20; required the area. Ans. 1256.64. 192. Problem. To find the area of a zone. By Geometry, the area of a zone is eq'ual to the cir- cumference of a great circle multiplied by the altitude of the zone. Let a denote the altitude of the zone, r the radius of the sphere, and k the area of the zone. . . k = 2 Trra, 193. Examples. 1. What is the area of the torrid zone, calling its width 46 56', and the earth a perfect sphere whose radius is 3956.5 mi.? Ans. 78333333. sq. mi. 2. What is the area of the tv/o frigid zones if the polar circles are 23 28' from the poles? Ans. 16270370. sq. mi. 3. What is the area of the two temperate zones? Ans. 102109933. sq. mi. 194. Problem. To find the area of a spherical triangle. Let s = A + B -f- <?, and ^Trr 2 = the tri-rectangular triangle. Then, by Geometry, I 774*2 ( _ O \ 90 Z) ' 1 72 MENS URA TIOX. In this formula, 7^ --2 is to be regarded as an yu abstract number. Minutes and seconds are to be re- duced to the decimal of a degree. 195. Examples. 1. Find the area of the spherical triangle whose angles are 60, 80, 100, and the radius 3956.5 mi. Am. 16392592 sq. mi. 2. Find the area of a spherical triangle whose sides are 70, 90, 100, respectively, and radius 100 in. Ans. 10942.1928 sq. in. 196. Problem. To find the area of a spherical polygon. Let s be the sum of the angles, n the number of sides, k the area of the polygon, and r the radius of the sphere. Then, by Geometry, 197. Examples. 1. The sum of the angles of a spherical hexagon is 800, the radius is 100 ft. ; required the area. Ans. 13963. sq. ft. 2. Each angle of a spherical pentago.n is 120, the radius is 50 ft. ; required the area. -4ns. 2618. sq. ft. SURFACES. 173 3. The angles of a spherical polygon are 90, 100, 110, 150, respectively, the radius is 10 ft.; required the area. Ans. 157.08 sq. ft. 4. Each angle of a spherical decagon is 150, the radius is 1 ft.; required the area. Ans. 1.0472 ft. 198. Problem. To find the area of the surface of a regular polyhedron. Let e be one edge, n the number of faces, k' the area of a polygon whose side is 1, and similar to one face, and k the area of the entire surface. k'e 2 the area of one face. Article 168. 199. Examples. 1. What is the area of the entire surface of a tetra- hedron whose edge is 10 ft.? Ans. 173.20508 sq.ft. 2. What is the area of the entire surface of a hexa- hedron whose edge is 5 ft.? Ans. 150 sq. ft. 3. What is the area of the entire surface of an octa- hedron whose edge is 20 ft.? Ans. 1385.64064 sq. ft. 4. What is the area of the entire surface of a dodec- ahedron whose edge is 15 in.? A ns. 32.25895 sq.ft. 5. What is the area of the entire surface of an icosa- hedron whose edge is 100 in. ? Ans. 601.4065 sq. ft. 174 MENSURATION. MENSURATION OF VOLUMES. 200. Problem. To find the volume of a prism. Let k be the area of the base, a the altitude, and v the volume. Then, % Geometry, v = ak. 201. Examples. 1. What is the volume of a regular hexagonal prism whose altitude is 20 ft., and each side of the base 10 ft.? Am. 5196.1524 cu. ft. 2. What is the volume of a triangular prism whose altitude is 6 ft., and the sides of its base 3 ft., 4 ft., and 5 ft., respectively? Ans. 36 cu. ft. 3. What is the volume of a regular octagonal prism whose altitude is 120 ft., and each side of the base 20 ft.? Ans. 231764.5008 cu. ft. 202. Problem. To find the volume of a pyramid. Let k be the area of the base, a the altitude, and v the volume. 203. Examples. 1. What is the volume of a pyramid whose altitude is 15 ft., and whose base is a regular heptagon each side of which is 5 ft.? Ans. 454.23905 cu. ft. 2. What is the volume of a pyramid whose altitude is 21 in., and whose base is a triangle each side of which is 30 in.? Ans. 2727.98 cu. in. VOLUMES. 175 204. Problem. To find the volume of the frustum of a pyramid. Let k and k l be the areas of the bases, a the alti- tude, and v the volume. Then, by Geometry, (1) v = $a(k + k l +V~kk;'). If the bases are regular polygons whose .sides are s and s', we shall have, by article 168, k = &'s 2 , and fcj = k's' 2 , in which k' is given in the table of article 167, and (1)' becomes (2) v = J- a (s 2 -f s' 2 + ss') k f . 205. Examples. 1. What is the volume of the frustum of a pyramid whose altitude is 9 ft., and whose bases are regular triangles, one side of the lower being 8 ft., and one side of upper, 5 ft.? Ann. 167.576 cu. ft. 2. What is the volume of the frustum of a pyramid whose altitude is 27 in., and the bases regular hexa- gons, the sides of which are 10 in. and 6 in., respect- ively? ' An*. 4583.0064 cu. in. 206. Problem. To find the volume of a 'cylinder. Let r represent the radius, a the altitude, and v the volume - 207. Examples. 1. What is the volume of a cylinder whose altitude is 50 in., and radius 15 in.? Ans. 20.453 cu. ft. 2. What is the volume of a cylinder whose altitude is 25 ft., and radius 4 ft.? Ans. 1256.64 cu. ft. 176 MENSURATION. 208. Problem. To find the volume of a cone. Let r be the radius of the base, a the altitude, and v the volume. v = arrr 2 . . Examples. 1. What is the volume of a cone whose altitude is 21 in., and radius 10 in.? Am. 2199.12 cu. in. 2. What is the volume of a cone whose altitude is 30 ft., and radius is 10 ft.? Ans. 31416. cu. ft. 210. Problem. To find the volume of the fnmtum of a rone. Let r and r be the radii of the bases, n the altitude, and v the volume. v = Jcwr(r 2 -|-r' 2 +rr f ). f 211. Examples. 1. What is the volume of the frustum of H cone whose altitude is 15 ft., and the radii of whose bases are 9 ft. and 4 ft, respectively? Am. 2089.164 cu. ft. How many barrels will that cistern contain wlmsc altitude is s ft., the diameter at the bottom 4 ft., and at the top (i ft.? Am. 37.8 bbl. 212. Formulas for the Sphere. Let r be the radiu>. <l the diameter, < the oircum- ference, k the area of the surface, and r the volume VOLUMES. 177 of a sphere, then, by Geometry, we have d 2 r, = nd, k = 4 Try 2 , v = ^rk. From which verify the following table of formulas: 1. r == $d. 2. r = J5_. 3. r = 4. r = \ ; _2 5. (2 := 2 r. 7. fc/?,^ 8. d = III 1 9. r ^ 2 rrr. 10. r- rr(/. 11. c ^ V-nk. 13. k = 4 7rr2. 14. k = rrrf2. 15. jfc^- 2 -. 7T 16. Jb = ^367ri 17. = t7ryS. 18. |7r=r|-7Trf 3 . 19. * = g^. 20. t; =="Jti/Z Examples. 1. Calling the diameter of the earth 7913 mi., and the diameter of the sun 856,000, find the ratio of their surfaces, also the ratio of their volumes. 2. What is the volume of the shell of a hollow >*phere whose radius is 8 ft. 4 in., and the thickness of the shell 3 ft. 6 in.? Am. 1951.1081 cu. ft. 178 MENSURATION. 214. Problem. To find the volume of a spherical sector. A spherical sector is the volume generated by the revolution of any circular sector, ABC, about any diameter, DE. By Geometry, the volume of a spherical sector is equal to the zone which forms its base, multiplied by one-third of the radius. Let a be the altitude of the zone, and r the radius. . . v = 7rr 2 a. 215. Examples. 1. The altitude of the zone which forms the base of a sector is 6 ft., the radius is 12 ft.; required the volume. Ans. 1809.5616 cu. ft. 2. The angle BCD, in the diagram of last article, is 20, ACS is 35, r = 20 ft. ; required the volume. Ans. 6134.25 cu. ft. 216. Problem. 'To find the volume of a spherical segment. A spherical segment is the portion of a sphere in- cluded between two parallel planes. Let r' - - BF perpendicular to DE, and r" AG perpendicular to DE. r = the radius, d'= CF, and d" = CG. v the vol. generated by ABFG. tf= the vol. generated by ABC=%Trr 2 a. VOLUMES. 179 v"= the vol. generated by BFC = t/"= the vol. generated by ,4GC = JrfW 2 . v = v'+v"^v m . The sign of v m is or -f according as AG is on the same or opposite side of the center as BF. . . v --= J- TT (2 ar 2 -f- dVa + d'Y" 2 ). 217. Examples. 1. r = 12 in., r' 3 in., r" = 10 in.; required v. 2. Two parallel planes divide a sphere whose diame- ter is 36 in. into three equal segments; required the altitude of each. An*. 13.93 in.; 8.14 in.; 13.93 in. 218. Problem. To find the volume generated by the revolution of a cir- cular segment about a diameter exterior to it. Let 'v = vol. generated by ADB. v' = vol. generated by ADBC. v"= vol. generated by ABC. 'v = v' v". Let a FGj c = AB, p C7, perpendicular to AB. v' = Trar 2 , v" = f nap 2 . .-. v'v"=% TTO (r 2 jo 2 ) = J Trac 3 . 219. Examples. 1. a - 5 in., c = 8 in.; find v. ^16-. 167.552 cu. In. 2. A sphere 6 in. in diameter is bored through the center with a 3-inch auger ; required the volume re- gaining. Ans. 73.457 cu. in. 180 MENSURATION. 3. Prove that the volume generated by the segment whose altitude is a and chord c is to the sphere whose diameter is c as a : c. 4. Prove that if c is parallel to the diameter about which it is revolved, the volume generated by the segment is equal to the volume of a sphere whose diameter is c. 2-20. Problem. To find the volume of a ivedge. The base is a rectangle, the sides are trapezoids, the ends, triangles. Let e be the edge, I the length of base, b the breadth of base, and a the altitude. VXl"" l X' Passing planes through the extremities of the edge perpendicular to the base, we have a triangular prism and two pyramids. These pyramids may fall within or without the wedge, or one or both of the pyramids may vanish. But in all cases the formula is the same. = the volume of the prism. a (I e) b = the volume of the pyramids. 221. Examples. 1. The edge of a wedge is 6 in., the altitude 12 in., the length of base 9 in., and the breadth of base 5 in.; what is the volume ? Ans. 240 cu. in. VOLUMES. 181 . The edge of a wedge is 20 ft., the altitude 24 ft., length of base 15 ft., the breadth of base 10 ft. ; what is the volume ? Ans. 2000 cu. ft. 22'2. Problem. To find the volume of a rectangular prismoid. The bases are parallel rect- angles, the other faces are trapezoids. Let / and b be the length and breadth of the lower base, /' and 6' the length and breadth of the upper base, and a the altitude. Passing the plane as . indicated, the prismoid is divided into t\vo wedges. J 06 (2 I + I') ~ the vol. of wedge whose base is bl. b'(2 I' + == the vol. of wedge whose base is bT. 223. Examples. 1. The length and breadth of the lower base of a rectangular prismoid are 25 ft. and 20 ft., the length and breadth of the upper base are 15 ft. and 10 ft., and the altitude is 18 ft.; what is the volume? Ans. 5550 cu. ft. 2. The length and breadth of the lower base of a rectangular prismoid are 15 yds. and 10 yds., the length and breadth of the upper base are 9 yds. and 6 yds,, and the altitude is 18 yds.; what is the volume? Ans. 1764 cu. yds. 182 MENSURA TIGS. 224. Problem. To find the dihedral angle included by the faces of a regular polyhedron. Conceive a sphere whose radius is 1 so placed that its center shall be at any vertex of the polyhedron. The faces of the polyhedral angle will intersect the surface of the sphere in a regular polygon^ whose sides measure the plane angles that include the polyhedral angle, and whose angles are each equal to the required dihedral angle. Let ABCD be such a polygon, P the pole of a small circle passing through A, B, C, D, E. Join P with the vertices and with the middle of AB by arcs of great circles. Let n denote the number of sides of the polygon, s = one side, and A = a dihedral angle. 360 180 . . APQ = - ==., and A Q ^ s. 2 n n By Napier's circular parts, we have sin (90 APQ) = cos AQ cos (90 PAQ). 1SO or sin (90 -) = cos * cos (90 $ A). n ' 180 or cos - = cos IM sin \A. n cos 1180 . '. sin \ A = - cos -J-s In the Tetrahedron, n == 3, and s = 60, * i COS Ov/ ~/~\n n-4/ inn ' Sln = : - ' A =' 31 42 ' VOLUMES. 183 In the Hexahedron, n = 3, and s = 90, i A cos 6Q . j _ ono In the Octahedron, n = 4, and s = 60, . . sin J yl = C M~ ' ^ = 109 28' 18". In the Dodecahedron, n = 3, and s = 108, .-. sin \A= C S ??. .-. ^ = 116 33 r 54". cos 54 In the Icosahedron, n = 5, and s = 60, .-, sin \ A = COS ^!- .-. 4= 138 11' 23 /r . 225. Problem. To find the volume of a regular polyhedron. If planes be passed through the edges of the poly- hedron and the center, they will bisect the dihedral angles and divide the polyhedron into as many pyra- mids as it has faces. The faces will be the bases of the pyramids, the center will be their common vertex, the line drawn from the center of the polyhedron to the center of any base will be perpendicular to the base, and will be the altitude of the pyramid. From the foot of the perpendicular draw a perpen- dicular to one side of the base, and join the foot of this perpendicular with the center. We thus have a right triangle whose perpendicular is the altitude of the pyramid, the base the apothem of one face of the polyhedron, the angle opposite the perpendicular one- half the dihedral angle of the polyhedron. 184 MENSURATION. Let p be the perpendicular, a the apothem of one face, J A one-half of a dihedral angle, n' the number of sides of one face, and e one edge. p = a tan $ A, a = \e cot -7- 180. Article 166. . . p = \ e cot ^180 tan \ A. Let &', w, and k be the same as in article 198. Then, ^pk = the volume of the polyhedron. .. v = J^fc'e 3 cot ^180 tan \A. Let c 1, and verify the table subjoined : 226. Table. Names. Surfaces. Volume. Tetrahedron 1.7320508 0.1178513 Hexahedron 6.0000000 1.0000000 Octahedron 3.4641016 0.4714045 Dodecahedron 20.6457288 7.6631189 Icosahedron 8.6602540 2.1816950 227. Application of the Table. Let i'' and v denote similar regular polyhedrons whose edges are 1 and e, respectively. Then we have r' : c : : I 3 : e 3 . .;. 228. Examples. 1. What is the volume of a tetrahedron whose edge is 10 ft.? -I/,*. 117.8513 cu. ft. 2. The volume of a hexahedron is 134217728 cu. in. what is its surface? Ana. 1572864 sq. in. SURVEYING. 229. Definition and Classification. Surveying is the art of laying out, measuring, and dividing land, and of representing on paper its bound- aries and peculiarities of surface. There are three branches Plane, Geodesic, and Topo- graphical. Plane surveying is that branch in which the por- tion surveyed is regarded as a plane, as is the case in small surveys. Geodesic surveying is that branch in which the curva- ture of the surface of the earth is taken into consider- ation, as is the case in all extensive surveying. Topographical surveying is that branch in which the slope and irregularities of the surface, the course of streams, the position and form of lakes and ponds, the situation of trees, marshes, rocks, buildings, etc., are considered and delineated. INSTRUMENTS. 230. Classification. The instruments employed in surveying may be classed as Field instruments and Plotting instruments. The principal field instruments are the chain and tally pins, marking tools, field-book and pencil, the magnetic S. N. 16. (185) 186 SURVEYING. compass, the solar compass, the transit compass, the level, and the theodolite. The principal plotting instruments are the dividers, the ruler and triangle, parallel rulers, the diagonal scale, the semicircular protractor. 231. The Chain and Tally Pins. The chain is 4 rods or 66 feet in length, and is di- vided into 100 links, each equal to 7.92 inches. After every tenth link from each end is a piece of brass, notched so as to indicate the number of links from the end of the chain, thus facilitating the count- ing of the links. A half-chain of 50 links is sometimes used, especially in rough or hilly districts. The tally pins are made of iron or steel, about 12 inches in length and one-eighth of an inch in thick- ness, heavier toward the point, with a ring at the top in which is fastened a piece of cloth of some con- spicuous color. These pins are conveniently carried by stringing them on an iron ring attached to a belt which is passed over the right shoulder, leaving the pins sus- pended at the left side. In Government surveys eleven tally pins are used. 232. Marking Tools. A surveying party will need an ax for cutting notches, cutting and driving stakes and posts; a spade or mattock for planting or finding corners; knives, or other tools, for cutting letters or figures; and a file and whetstone for keeping the tools in order. INSTRUMENTS. 187 233. Field-Book and Pencil. In ordinary practice one field-book will be sufficient; but in surveying the public lands, four different books are required one for meridian and base lines, another for standard parallels or correction lines, another for exterior or township lines, and another for subdivision or section lines, as designated on the title-page. A good pencil, number 2 or 3, well sharpened, should be used, so that the notes may be legible. A temporary book may be used on the ground, and the notes taken with a pencil. These notes can then be carefully transcribed with pen and ink into the permanent field-book. 234. The Magnetic Compass. The vernier magnetic compass is exhibited in the drawing on page 189. The needle turns freely on a pivot at the center, and settles in the magnetic meridian. The compass circle is divided, on its upper surface, to half-degrees, numbered from to 90 each side of the line of zeros. The sight standards are firmly fastened at right angles to the plate by screws, and have slits cut through nearly their whole length, terminated at in- tervals by apertures through which the object toward which the sights are directed can be readily found. Two spirit levels at right angles to each other are attached to the plate. Tangent scales are scales on the right and left edges of the north sight standard, the one on the right be- 188 SURVEYING. ing used in taking angles of elevation, and the one on the left in taking angles of depression. Eye-pieces are placed on the right and left sides of the south sight standard the one on the right near the bottom, the one on the left near the top each on a level, when the compass is level, with the zero of its tangent scale. These eye-pieces are centers of arcs tangent to the tangent scales at the zero point. The vernier is a scale movable by the side of another scale, and divided into parts each a little greater or a little less than a part of the other, and having a known ratio to it. In the drawing the vernier is represented on the plate near the south sight. The needle lifter is a concealed spring, moved from beneath the main plate, by which the needle may be lifted to avoid blunting the point of the pivot in transporting the instrument. The out-keeper is a small dial plate, having an index turned by a milled head, and is used in keeping tally in chaining. The ball spindle is a small shaft, slightly conical, to which the compass is fitted, having on its lower end a ball confined in a socket by a light pressure, so that the ball can be moved in any direction in leveling the instrument. The clamp screw is a screw in the side of the hol- low cylinder or socket, which fits to the ball spindle, by which the compass may be clamped to the spindle in any position. A spring catch, fitted to the socket, slips into a groove when the instrument is set on the spindle, and secures it from slipping from the spindle when carried. 190 SURVEYING. The Jacob staff is a single staff to support the com- pass, about 5J feet long, having at the upper end the ball and socket joint, and terminating at the lower end in a sharp steel point, so as to be set firmly in the ground. The tripod is a three-legged support sometimes used instead of the Jacob staff. 235. Adjustments of the Compass. 1. To adjust the level, Bring the bubbles to the cen- ter of the tubes by pressing the plates so as to turn the ball slightly in its sockets. Turn the compass half- way round, and if either bubble runs to one end of its tube, that end is the higher. Loose the screw un- der the lower end, and tighten the one at the higher end till the bubble is brought half-way back. Level the plate again, and repeat the operation till the bubble will remain in the center during an entire revolution of the compass. 2. To adjust the sights, Observe through the slits a fine thread made plumb by a weight. If both sights do not exactly range with the thread, file a little off the under surface of the highest side. 3. To adjust the needle. Bring'the eye nearly in the same plane with the graduated circle, move with a splinter one end of the needle to any division of the circle, and observe whether the other end corresponds with the division 180 from the first; if so, the needle is said to cut opposite degrees; if not, bend the center pin with a small wrench about one-eighth of an inch below the point, till the ends of the needle cut op- posite degrees. Hold the needle in the same direc- tion, turn the compass half-way round, and again see INSTRUMENTS. 1'91 whether the needle cuts opposite degrees; if not, cor- rect half the error by bending the needle, and the re- mainder by bending the center pin, and repeat the operation till perfect reversion is secured in the first position. Try the needle in another quarter, and correct by bending the center pin only, since the needle was. straightened by the previous operation, and repeat the operation in different quarters. The adjustments are made by the maker of the in- strument, but the instrument can be re-adjusted by the surveyor when necessary. 230. Nature of the Vernier. Let the arc or limb AB, on the main plate of the instrument, be graduated to one r half degrees or 30', numbered each way from at the mid- dle; and let the vernier CD, attached to the compass box, which is movable around the main plate, be so graduated that 30 spaces of the vernier shall be equal to 31 spaces of the limb, that is, equal to 31 X 30'; then 1 space of the vernier will be equal to 31', and the difference between one space of the vernier and one space of the limb will be 31' 30'= 1'. The vernier is numbered in two series: the lower, nearer the spectator, who is supposed to stand at the south end of the instrument, is numbered 5, 10, 15, each way from 0; the upper series has 30 above the 0, from the observer, and 20 each way above the 10 of the lower series. Let, now, the points of the vernier and limb co- incide; then, if the vernier be moved forward V to 192 SURVEYING. the right, which is done by means of a tangent screw, the first division line of the vernier at the left of its will coincide with the first division line of the limb at the left of its 0; if the vernier be moved for- ward 2' to the right, then the second division line of the vernier at the left of its will coincide with the second division line of the limb at the left of its 0. If the vernier be moved to the right so that its fifteenth division line at the left of its shall coin- cide with the fifteenth division line of the limb at the left of its 0, the vernier will have moved forward 15'. If the vernier be moved more than 15', the excess over 15' is found by reading the division line, in the vernier, which coincides with a division line of the limb, from the upper row of figures on the vernier, on the other side of 0, and so on, up to 30', when the of the vernier will coincide with the first division line from the of the limb. If the vernier is moved more than 30', the excess over 30', up to 15' and then to 30' is found as before. If the of the vernier coincides with a division line of the limb, the reading of the division line of the limb will be the true reading. If the of the vernier has passed one or more division lines of the limb, and does not coincide with any, read the limb from its point up to its divis- ion next preceding the of the vernier; to this add the reading of the vernier, and the sum will be the true reading. If the vernier be moved to the left, the minutes must be read off on the vernier scale to the right. Sometimes the spaces of the vernier are less than the spaces of the limb; then if the vernier be moved INSTRUMENTS. 193 either way, the imitates must be read off the same way from the of the vernier. Verniers may be so graduated as to read to any appreciable angle; but the graduation which reads to minutes is the most common. 237. Uses of the Vernier. 1. To turn off the variation. Let the instrument be placed on some definite line of an old survey, and the tangent screw be turned till the needle indicates the same bearing for the line as that given in the field notes of the original survey. Then will the reading of the limb and vernier indi- cate the variation. 2. To retrace an old survey. Turn off the variation as above, and screw up the clamping nut beneath, then old lines can be retraced from the original notes with- out further change of the vernier. 3. To run a true meridian. The absolute variation of the needle being known, not simply its change since a given date, move the vernier to the right or left, according as the variation is west or east, till the given variation is turned off, screw up the clamping nut beneath, and turn the compass till the needle is made to cut zeros, then will 'the line of sights indicate a true meridian. Such a change in the position of the vernier is necessary in subdividing the public lands, after the principal lines have been truly run with the solar compass. 4. To read the needle to minutes. Note the degrees given by the needle, then turn back the compass circle, with the tangent screw, till the nearest whole degree mark coincides with the point of the needle; the space S. N. 17. 194 SURVEYING. passed over by the vernier will be the minutes which, added to the degrees, will give the reading of the needle to minutes. This operation is simplified when the of the ver- nier is first made to coincide with the of the limb ; otherwise the difference of the two readings of the vernier must be taken. 238. Uses of the Compass. 1. To take the bearing of a line. Place the compass on the line, turn the north end in the direction of the course, and, standing at the south end, direct the sights to some well-defined object, as a flag-staff, in the course. Read the bearing from the north end of the needle, which can be clone accurately to quarter-degrees by observing the position of the point of the needle, since the compass circle is divided into half-degrees. It will be observed that the letters E and IF, on the face of the compass, are reversed from their true posi- tion. This is as it should be; for if the sights are turned toward the west, the north end of the needle is turned toward the letter W. If the north end of the needle is turned toward E, the sights will be turned toward the east. If the north end of the needle point exactly to cither letter E or IF, the sights will range east or west. In general, to guard against error, let the surveyor turn the letter , f > toward himself, and read the arc cut off by the north end of the needle from the nearest zero of the compass circle. If, for example, the near- est is at 5, and the north end of the needle is turned toward *, cutting off 25 from this 0, then the course is S 25 E INSTRUMENTS. 195 If it is desired to find the bearing to minutes, the vernier must be used. 2. To run from a given point a line having a given bearing. Place the compass over the point, and turn it so that the reading of the needle shall be the given bearing; the line of sights observed from the south end of the compass will be the required line. 3. To take angles of elevation, Level the compass, bring the south end toward you, place the eye at the eye-piece on the right side of the south sight, and, with the hand, fix a card on the front surface of the north sight, so that its top edge shall be at right angles to the divided edge and coincide with the zero mark ; then, sighting over the top of the card, note upon a flag-staff the height cut by the line of sight, move the staff up the elevation, and carry the card along the sight until the line of sight again cuts the same height on the staff, read off the degrees and half-degrees passed over by the card, and the result will be the angle required. 4. To take angles of depression. Proceed in the same manner, using the eye-piece and scale on the opposite sides of the sights, and reading from the top of the standard. 239. Surveyor's Transit. The Surveyor's transit exhibited in the drawing on page 197 is, in fact, a transit theodolite, combining the advantages of the ordinary transit and the theodolite. The vernier plate, carrying two horizontal verniers, two spirit levels at right angles, the telescope and attachments, moves around a circle graduated to half- degrees, so that, by the vernier, horizontal angles can be taken to minutes, and any variation turned off. 196 SURVEYING. The telescope and its attachments, the clamp and tangent, the vertical circle, the level, and the sights, give to this instrument a great advantage over the ordinary compass. The cross wires, two fine fibers of spider's web, ex- tending across the tube at right angles, intersect in a point which, when the wires are adjusted, determines' the optical axis or line of collimutioii of the telescope, and enables the surveyor to fix it upon an object with great precision. The clamp and tangent screw consist of a ring en- circling the axis of the telescope, having two project- ing arms the one above, slit through the middle, hold- ing the clamp screw ; the other, longer, connected be- low with the tangent screw. The ring is brought firmly around the axis by means of the clamp screw, and the telescope can be moved up or down by turning the tangent screw. The vertical circle, graduated to half-degrees, is at- tached to the axis of the telescope, and, in connection with the vernier, gives the means of measuring ver- tical angles to minutes with great facility. The level attached to the telescope enables the sur- veyor to run horizontal lines, or to find the difference of level between two points. Sights on the telescope are useful in taking back- sights without turning the telescope, and in sighting through bushes or woods. Sights for right angles attached to the plate of the instrument, or to the standards supporting the tele- scope, afford the means of laying off right angles, or running out offsets without changing the position of the instrument. SURVEYOR'S TRANSIT. (197) 198 SURVEYING. 240. Adjustments. 1. The levels are adjusted in the same manner as those of the compass, and when adjusted should keep their position if the two plates are clamped together and turned on a common socket. 2. The needle is adjusted as in the compass. 3. The line of the collimation is adjusted by bringing the intersection of the wires into the optical axis of the telescope, which is accomplished as follows : Set the instrument firmly on the ground and level it carefully, then, having brought the wires into the focus of the eye-piece, adjust the object glass on some well defined object, as the edge of a chimney, at a distance of from two to five hundred feet. Determine whether the vertical wire is plumb by clamping the instrument firmly to the spindle, and applying the wire to the vertical edge of a building, or observing if it will move parallel to a point a little -to one side; if it does not, loosen the cross-wire screws, and, by the pressure of the hand on the head outside the tube ? move the ring within the tube, to which the wires are attached, gently around till the error is corrected. The wires being thus made respectively horizontal and vertical, fix their- point of intersection on the object selected, clamp the instrument to the spindle, and, having revolved the telescope, find or place some object in the opposite direction, at about the same distance from the instrument as the first object. Great care should be taken in turning the telescope not to disturb the position of the instrument upon the spindle. Having found an object which the vertical wire bi- sects, unclamp the instrument, turn it half-way round, INSTRUMENTS. 199 and direct the telescope to the first ohject selected, and having bisected this with the wires, again clamp the instrument, revolve the telescope and note if the vertical wire bisects the second object observed; if so, the wires are adjusted, and the points bisected are, with the center of the instrument, in the same straight line. If the vertical wire does not bisect the second point, ithe space which separates this wire from that point is double the distance of that point from a straight line drawn through the first point and the center of the instrument, as is shown thus: Let A represent the center of the instrument, BC the line on whose extremities, B and (7, the line of collimation is to be adjusted, B the first object, and D the point which the wires bisected after the telescope was made to revolve on its axis. The side of the telescope which was up when the object glass was di- rected to B, is down when the object glass is turned toward D. When the telescope is undamped from its spindle and turned half-way round its vertical axis, and again directed to , the side of its tube which was down when the object glass was first directed to B will now be up. Then clamping the instrument, and revolving the telescope about its axis, and di- recting it toward D, the side of its tube which was down when the object glass was first turned toward D will now be up, or the telescope will virtually have revolved about its optical axis, and the vertical wire will appear at E as far on one side of C as D is on the other side. 200 SUJZVEYIXG. To move the vertical wire to its true position, turn the capstan head screws on the sides of the telescope, remembering that the eye -piece inverts the position of the wire, and, therefore, that in loosening one of the screws and in tightening the other the operator must proceed as if to increase the error. Having moved back the vertical wire, as nearly as can be judged, so as to bisect the space ED, unclamp the in- strument, direct the telescope as at first, so that the cross wires bisect 5, proceed as before, and continue the operation till the two points D and E coincide at C. 4. The standards must be of the same height, in order that the wires may trace a vertical line when the telescope is turned up or down. To ascertain this, and to make the correction, proceed as follows : Having the line of collimation previously adjusted, set the instrument in a position where points of ob- servation, such as the point and base of a lofty spire, can be selected, giving a long range in a vertical direction. Level the instrument, fix the wires on the top of the object, and clamp to the spindle; then bring the telescope down till the wires bisect some good point, either found or marked at the base; turn the instru- ment half around, fix the wires on the lower point, clamp to the spindle, and raise the telescope to the highest object, and if the wires bisect it, the vertical adjustment is effected. If the wires are thrown to one side, the standard opposite that side is higher than the other. The correction is made by turning a screw under- neath the sliding piece of the bearing of the movable axis. INSTRUMENTS. 201 e5. The vertical circle is adjusted thus : First care- fully level the instrument, bring the zeros of the wheel and vernier into line, and find or place some well defined point which is cut by the horizontal wire; then turn the instrument half-way around, revolve the telescope, fix the wire on the same point as be- fore, note if the zeros are again in line. If not, loosen the screws, move the zero over half the error, and again bring the zeros into coincidence, and proceed as before till the error is corrected. 6. The level on the telescope can be adjusted thus : First level the instrument carefully, and with the clamp and tangent movement to the axis make the telescope horizontal as nearly as possible with the eye. Then, having the line of collimation previously adjusted, drive a stake at a distance of from one to two hundred feet, and note the height cut by the horizontal wire upon a staff set on the top of the stake. Fix another stake in the opposite direction, at the same distance from the instrument, and, without dis- turbing the telescope, turn the instrument upon its spindle, set the staff upon the stake and drive in the ground till the same height is indicated as in the first observation. The top of the two stakes will then be in the same horizontal line, whether the telescope is level or not. Now remove the instrument to a point on the same side of both stakes, in a line with them, and from fifty to one hundred feet from the nearest one ; again level the instrument, clamp the telescope as nearly horizontal as possible, and note the heights indicated on the staff placed first on the nearest, then on the more distant stake. 202 SURVEYING. If both agree, the telescope is level ; if they do not agree, then with the tangent screw move the wire over nearly the whole error, as shown at the distant stake, and repeat the operation just described till the horizontal wire will indicate the same height at both stakes, when the telescope will be level. Bring the bubble into the center by the leveling nuts at the lend, taking care not to disturb the position of the telescope, and the adjustment will be completed. The adjustments above described are always made by the maker of the instrument, but the instrument may need re-adjusting. 241. Uses of the Transit. 1. The transit may be used for all the purposes for which the compass is employed, and, in general, with much greater precision. 2. Horizontal angles can be taken by the needle, or without reference to the needle, as follows : Level the plate, set the limb at zero, direct the telescope so that the intersection of the wires shall fall upon one of the objects selected, clamp the instrument firmly to the spindle, unclamp the vernier plate, turn it with the hand till the intersection of the wires is nearly upon the second object; then clamp to the limb, and with the tangent screw fix the intersection of the wires precisely upon the second object. The reading of the vernier will give the angle whose vertex is at the center of the instrument, and whose sides pass through the objects respectively. 3. Vertical angles can be measured thus: Level the instrument, fix the zeros of the vertical circle and vernier in a line, note the height cut upon the staff INSTRUMENTS. 203 by the horizontal wire, carry the staff up the eleva- tion or down the depression, fix the wire again upon the same point, and the angle will be read off by the vernier. Sometimes, of course, it will be impossible to carry the staff up the elevation, as in taking the angle of elevation of the top of a steeple from a given point in a horizontal plane. 4. Horizontal lines can be run, or the difference of level easily found, by means of the level attached to the telescope. 242. The Solar Compass. Burt's solar compass, represented in the drawing on page 205, includes the essential parts of the magnetic compass, together with the solar apparatus, which con- sists mainly of three arcs of circles by which the latitude of the place, the declination of the sun, and the hour of the day can be set off. The latitude arc, a, graduated to quarter-degrees and read to minutes by a vernier, has its center of motion in two pivots, one of which is seen at r/, and is moved by the tangent screw, /, up or down a fixed arc of similar curvature through a range of about 35. The decimation arc, fo, having a range of about 24, is graduated to quarter-degrees and read to minutes by the vernier, r, fixed to the movable arm, /<, which has its center of motion in the center of the declination arc at g. The vernier may be set to any reading by the tangent screw, k, and the arm clamped in any po- sition by a screw concealed in the engraving. A solar lens, set in a rectangular block of brass at each end of the arm, A, has its focus at the inside of 204 SURVEYING. the opposite block on the surface of a silver plate on which are drawn certain lines, as shown in the an- nexed figure. The lines bb, called hour lines, and the lines cc, called equatorial lines, inter- sect each other at right angles. The rectangular space between the lines is just sufficient to include the circular image of the sun formed by the solar lens on the op- posite end of the arm. The three other lines below the equatorial lines are five minutes apart, and are used in making allowance for refraction. An equatorial sight, used in adjusting the solar ap- paratus, is placed on the top of each rectangular block by a small milled head screw, so as to be detached at pleasure. The hour arc, c, supported by the pivots of the lati- tude arc, and connected with that arc by a curved arm, has a range of 120, graduated to half-degrees and figured in two series, designating both the hours and the degrees; the middle division being marked 12 and 90 on either side of the graduated lines. The polar -axis, p, consists of a hollow socket con- taining the spindle of the declination arc, around which this arc can be moved over the hour arc, which is read by the lower edge of the graduated side of the declination arc. The declination arc may be turned half round, if required, and the hour arc read by a point below y. The needle box, /,, with an arc of 36, graduated to half-degrees, and numbered from the center as zero, is attached by a projecting arm to a tangent screw, ?, by which it is moved about its center, and its needle w 3 O o "T3 I (205) 206 SURVEYING. set to any variation which may be read to minutes by the vernier at the end of the arm. The levels are similar to those of the ordinary com- pass. Lines of refraction are drawn on the inside faces of the sights, graduated and figured to indicate the amount allowed for refraction when the sun is near the horizon. The adjuster is an arm used in adjusting the instru- ment. It is not attached to the instrument, and is laid aside in the box when the adjustment is effected. 243. Adjustments. 1. The levels are adjusted by bringing the bubbles into the center of the tubes by the leveling screws of the tripod, reversing the instrument on the spindle, raising or lowering the ends of the tubes till the bubbles will remain in the center during a complete revolution. 2. The equatorial lines and solar lenses are adjusted as follows : First detach the arm, ^, from the decli- nation arc by withdrawing the screws shown in the drawing from the ends of the posts of the tangent screw, kj and also the clamp screw, and the conical pivot with its small screws by which the arm and declination arc are connected. Attach the adjuster in the place of the arm, /i, by replacing the conical pivot and screws, and insert the clamp screw so as to clamp the adjuster at any point on the declination arc. Now level the instrument, place the arm, h, on the adjuster, with the same side resting against the sur- face of the declination arc as before it was detached, INSTRUMENTS. 207 turn the instrument on its spindle, so as to bring the solar lens to be adjusted in the direction of the sun, raise or lower the adjuster on the declination arc till it can be clamped in such a position as to bring the sun's image, as near as may be, between the equatorial lines on the opposite silver plate, and bring the image precisely into position by the tan- gent of the latitude arc, or the leveling screws of the tripod. Then carefully turn the arm half-way over, till it rests upon the adjuster by the opposite faces of the rectangular blocks, and again observe the position of the sun's image. If it remains between the lines as before, the lens and plate are in adjustment; if not, loosen the three screws which confine the plate to the block, and move the plate under their heads till one-half the error in the position of the sun's image is removed. Again bring the image between the lines, and re- peat the operation till it will remain in the same situation in both portions of the arm, when the ad- justment will be complete. To adjust the other lens and plate, reverse the arm, end for end, on the adjuster, and proceed as in the former case. Remove the adjuster, and replace the arm, A, with its attachments. In tightening the screws over the silver plate, care must be taken not to move the plate. 3. The vernier of the declination arc is adjusted as- follows : Having leveled the instrument, and turned its lens in the direction of the sun, clamp to the spindle, and set the vernier, v, of the declination arc at zero, by means of the tangent screw, fc, and clamp to the arc. 208 SURVEYING. See that the spindle moves easily and truly in the socket, or polar axis, and raise or lower the latitude arc by turning the tangent screw, /, till the sun's im- age is brought between the equatorial lines on one of the plates; clamp the latitude arc by the screw, and bring the image precisely into position by the level- ing screws of the tripod or socket, and without dis- turbing the instrument carefully revolve the arm, //, till the opposite lens and plate are brought in the direction of the sun, and note if the sun's image comes between the lines as before. If the sun's image comes between the lines, there is no index error of the declination arc; if not, then with the tangent screw, k, move the arm till the sun's im- age passes over half the error, and again bring the image between the lines, and repeat the operation as before till the image will occupy the same position on both plates. We shall now find that the zero marks on the arc and the vernier do not correspond ; and to remedy this error, the little flat-head screws above the vernier must be loosened till it can be moved so as to make the zeros coincide, when the operation will be com- plete. 4. The solar apparatus is adjusted to the sights as follows: First level the instrument, then with the clamp and tangent screws set the main plate at 90 by the verniers and horizontal limb. Then remove the .clamp screw, and raise the latitude arc till the polar axis is by estimation very nearly horizontal, and, if necessary, tighten the screws on the pivots of the arc so as to retain it in this position. Fix the vernier of the declination arc at zero, and direct the equatorial sights to some distant and well- INSTRUMENTS. . 209 marked object, and observe the same through the com- pass sights. If the same object is seen through both, and the verniers read to 90 on the limb, the adjust- ment is complete ; if not, the correction must be made by moving the sights or changing the position of the verniers. The adjustments are all made by the maker of the instrument, and, ordinarily, need not concern the sur- veyor, as the instrument is very little liable to de- rangement. 244. Use of the Solar Compass. The declination of the sun, or its angular distance from the celestial equator, must be set off on the declination arc. The declination of the sun for apparent noon at Greenwich, England, is given from year to year in the Nautical Almanac, To determine the declination for another place and hour, allowance must be made for the difference of time arising from longitude, and for the change of declination from hour to hour. The longitude of the place can be determined with sufficient accuracy by reference to that of given promi- nent places which are situated nearly on the same meridian. The difference of longitude, divided by 15, will, by changing degrees, minutes, and seconds into hours, minutes, and seconds, give the difference of time, which is usually taken to the nearest hour, as it will be sufficiently accurate. In practice, surveyors in states just east of the Mis- sissippi allow a difference of 6 hours for longitude; S. N. 18. 210 SURVEYING. 7 hours for about the longitude of Santa Fe; 8 hours for California and Oregon; 5 hours for the eastern portions of the United States. Having found the hour at any place from its longi- tude when it is noon at Greenwich, the declination for noon at Greenwich will be the declination for the determined hour at the given place. To find the declination for the following hours of the day, add or subtract, for each succeeding hour, the difference of declination for 1 hour, as given in the almanac. Thus, let it be required to find the declination of the sun for the different hours of May 20th, 1873. W. Ion. 95. 95 = 6 h. 20 m., practically G h. Sun's dec., Greenwich, noon == 20 3' 14".6 .'. Sun's dec., Ion. 95, 6 A. M. == 20 3' 14".6 Add difference for 1 h. I A 1 ^ 3 Sun's dec. 7 A. M. -= 20~ 3 7 45".63 Add difference for 1 h. = _3J-^03 Sun's dec. 8 A. M. == 20 4' 16".66 In like manner proceed for the remaining hours. 'Such a calculation should be made before beginning the work of the day. Refraction, or the bending of the sun's rays as they pass obliquely through the atmosphere, affects its dec- lination by increasing its apparent altitude. The amount of refraction depends upon the altitude, being less as the altitude is greater. At the horizon the refraction is 35'; at the altitude of 45, 1'; at the zenith, 0. Meridional refraction, by increasing the apparent al- titude of the sun, when on the meridian, increases or INSTRUMENTS. 211 diminishes its apparent declination according as it is north or south of the equator. To find the amount of meridional refraction, we must first find the meridional altitude of the sun for the given latitude, which is equal to the comple- ment of the latitude, plus or minus the declination, according as the sun is north or south of the equator. The meridional altitude of the sun being given, the tables will give the refraction. The meridional refraction, being quite small, may be disregarded in practice except when great accuracy is required, as in running great standard meridians or base lines. Incidental refraction, as affected by the hour of the day and the state of the atmosphere, can not, in prac- tice, be determined by a precise calculation. It will about compensate for incidental refraction to keep the image of the sun square between the equi- noctial lines for the middle of the ,day, but toward morning or evening, to run the image, which is then hazy round the edge, so that the hazy edge shall over- lap one or two lines of the spaces below. To set off the latitude, find the declination of the sun for the given day at noon, and set it off on the decli- nation arc, and clamp the arm firmly to the arc. Find in the almanac the equation of time for the given day, in order to ascertain the time when the sun will reach the meridian. About twenty minutes before noon, set up the in- strument, level it carefully, fix the divided surface of the declination arc at 12 on the hour circle, and turn the instrument on its spindle till the solar lens is brought into the direction of the sun. 212 SURVEYING. Loosen the clamp screw of the latitude arc, raise or lower this arc with the tangent screw till the im- age of the sun is brought precisely between the equa- torial lines, and turn the instrument so as to keep the image between the hour lines on the plate. As the sun ascends, in approaching the meridian, its image will move below the lines, and the arc must be moved to follow it. Keep the image between the two sets of lines till it begins to pass above the equatorial, which is the moment after it passes the meridian. Read off the vernier of the arc, and we have the latitude of the place which is to be set off on the latitude arc. To run lines with the solar compass. Having adjusted the instrument and set off the declination and latitude, the surveyor places the instrument over the station, levels it carefully, clamps the plates at zero on the horizontal limb, and directs the sights north and south, approximately, by the needle. The solar lens is then turned toward the sun, and with one hand on the instrument, and the other on the revolving arm, both are moved from side to side till the image of the sun is made to appear on the silver plate, and is brought precisely within the equatorial lines, when the line of sights will indicate the true meridian. In running an east and west line, the verniers of the horizontal limb are set at 90, and the sun's im- age kept between the equatorial lines. The needle is made to indicate zero on the arc of the compass box by turning the tangent screw. Lines can then be run by the needle in the temporary dis- appearance of the sun. INSTRUMENTS. 213 The variation of the needle, which should be noted at every station, is read off to minutes on the arc along the edge of which the vernier of the needle box moves. Since the limb must be clamped at when the sun's image is in position, in order that the sights may indi- cate the meridian, it is evident that the bearing of any line may be found by the solar compass, as well as by the compass or transit. In running long lines, allowance must be made for the curvature of the earth. Thus, in running north or south the latitude changes V for 92.30 ch.; and six miles, or one side of a township, requires a change of 5' 12" on the latitude arc. In running east and west lines, the sights are set at 90 on the limb, and the line run at right angles to the meridian ; but this line, if sufficiently produced, would cross the equator. Hence, at the next station, a backsight is taken, and one-half the error is set off for the next foresight on the side toward the pole. The most favorable season for using the solar com- pass is the summer; and the most favorable time of day, between 8 and 11 A. M., and 1 and 5 P. M. A solar telescope compass is sometimes used; and, in this case, the telescope is placed at one side of the center. All error from this position of the telescope is avoided by an offset from the flag-staff. The solar compass, while indispensable in the survey of public lands, can be used, in common practice, with considerable advantage over ordinary needle instru- ments, since lines can be run by it without regard to the variation of the needle or local attraction, and the bearings being taken from the true meridian will re- main constant for all time. 214 SURVEYING. 245. Dividers and Pens. 1. Dividers with lead-pencil. 2. Hair dividers with one leg movable by screw. a, 6. Lengthening bar and pen which may be inserted together or the pen alone instead of pencil leg. 3. Bow pen with spring and adjusting screw. 4. Spacing dividers. 5. Drawing pen. 246. Parallel Rulers. 1. Parallel ruler for drawing parallel lines. 2. Sliding parallel ruler with scales. INSTRUMENTS. 215 247. Diagonal Scale. Let c?e be .1, then the distance from ad to ae on the first line above ab is .01, on the second line .02, etc. Let it be required to lay off on AB 4.63. Place one foot of the dividers at the intersection of the diagonal line, 6, and the horizontal line, 3. Extend the other foot till the horizontal line, 3, intersects the vertical line, 4, then will the distance from one point of the dividers to the other be 4.63. Now place one foot of the dividers at A, and the other at B, then AB will be 4.63. 248. Protractors. These protractors are used in laying off or measur- ing angles. The vertex of the angle is at the center, and one side is made to coincide with the horizontal line passing through the center; then, counting the degrees, from the horizontal line round the circumfer- ence till the required degree is reached, and drawing 216" SURVEYING. a line from this degree to the center, we shall have the angle required. The first of these protractors will give angles to quarter-degrees ; and the second, by means of a ver- nier, to 8'. Instruments may be multiplied indefinitely, but the manner of using them will be readily discovered by the ingenious operator. SURVEY OF PUBLIC LANDS. 249. Division into Townships. In the rectangular system of surveying the public lands, adopted by the government, two principal lines an east and west line, called a base line, and a north and south line, called a principal meridian are estab- lished before the survey of the townships. Six miles to the north of the base line another east and west line is run, and six miles to the north of this another, and so on. Every fifth parallel from the base is called a standard parallel, or correction line. Six miles to the west of the principal meridian, measured on the base line, another north and south line is run to the first standard parallel, and six miles to the west of this another, and so on. The intersection of the east and west with the north and south lines divides the tract into townships, which would be exactly six miles square were it not for the convergence of the meridians. To preserve as nearly as possible the form and size of the townships, the standard parallels before men- PUBLIC LANDS. 217 tioned are established, which serve as hase lines for the townships north up to the next standard parallel. Tiers of townships north and south arc called ranges, and are numbered east or west, as the case may be, from the principal meridian. Lines running north and south, bounding the town- ships on the east and west, are called range lines. Lines running east and west, bounding the townships on the north and south, are called township lines. A township marked thus, T. 5 TV., R. 4 W., read township five north, range four west, would be in the fifth tier north of the base line, and in the fourth tier west of the principal meridian. Townships are divided into sections, or square miles, containing 640 acres; each section into four quarter sections, each quarter section into two half-quarter sec- tions, and each half-quarter section into two quarter- quarter sections. These are called legal subdivisions, and are the only divisions recognized by the govern- ment, except pieces made fractional by water-courses or other natural agencies. On base lines and standard parallels two sets of corners are established. 1. Standard corners, established when these lines are run, embracing township, section, and quarter-section corners, common to two townships, sections, or quarter sections north of the base line or standard parallels. 2. Closing corners, established when exterior and sub- division lines close on them from the south, embracing township and section corners, common to two town- ships or sections south of the standard parallels. In consequence of the convergence of the meridians, the north and south lines, produced to the standard S. N. 19. 218 SURVEYING. parallels, will not close on the standard corners previ- ously established, but will strike the standard parallels to the east or west of the standard corners, making the closing corners east or west of the standard cor- ners, according as the field of operation is west or east of the principal meridian, The following diagram will illustrate the subject : AB is the base line. A' AC, the principal me- ridian. A'B', a standard paral- lel. ab, cd, etc., township lines. ijj klj etc., range lines. ,<?, ., Wj etc., standard corners. j, I, ?i, etc., closing cor- ners. The distances js, Zit, etc., are measured and recorded in the field book. The details of running lines will be given after describing the methods of perpetuating corners, the process of chaining, and the method of marking lines. Burt's improved solar compass is used in surveying standard and township lines, but the ordinary compass may be used in subdividing. 250. Methods of Perpetuating Corners. 1. Corner trees. A sound tree, five inches or more in diameter, standing exactly at a corner, is the best monument. PUBLIC LANDS. 219 2. Corner stones. A stone, at least 14 inches long and 6 inches square, set from two-thirds to three- fourths in the ground, is preferred to other monu- ments, except a tree. 3. Posts and witness trees. In the absence of corner trees and stones, when trees are near, a post may be planted and witnessed by taking the bearing and distance of two or more trees in different directions from the corner. These trees are marked by a blaze in which is marked the number of the township, range, and section. A notch is cut in the lower end of the blaze, under which another blaze is made in which are cut the letters B. T 7 ., signifying bearing tree. 4. Posts, mounds, and witness pits. When neither corner trees, stones, nor witness trees are available, corners may be marked by posts, mounds, and witness pits. The posts are planted 12 inches in the ground, and at the lower end, on the north or west side, accord- ing as the course is north or west, a marked stone, a small quantity of charcoal, or a charred stake must be deposited. Four pits are dug, 6 feet from the post, on opposite sides, 2 feet square and 1 foot deep, and the excavated earth packed round the post within 1 foot of the top. If sod is to be had, it is to be used in covering the mounds. The method of marking the corner is to be care- fully noted in the field book. 251. Township Corners. 1. Posts used in marking township corners must be 4 feet in length, and 5 inches, at least, in diameter. These posts are to be set 2 feet in the ground, and 220 SURVEYIXG. the upper part squared to receive the marks to be cut on them. T. 2 X. P. 31V. S. 31. T. 1 3f. P. 4 W. S.I. T. IN. E . 3 AN". S. 6. If the corner is common to four townships, the post is set so as to present the angles in the direction of the line; and the number of the township, range, and section must be marked on the side facing, and six notches cut on each of the four edges. If the township corner is on a base line or standard parallel, unless it is also on the principal meridian, it will be common to two townships only; and if these are on the north, the corner will be a standard corner. In this case, six notches are cut on the east, north, and west edges, but not on the south edge, and the letters S. (7., signifying standard corner, cut on the flat surface. If the corner is common to two townships on the south, but not on the north, it will be a closing cor- ner, and six notches are cut on the east, south, and west edges, but not oh the north edge, and the let- ters C. C., signifying closing corner, cut on the flat surface. 2. Township corner stones should be inserted at least 10 inches in the ground, with their sides facing the cardinal points of the compass, and small mounds of stones heaped against them. These corner stones are notched in the same manner as posts in similar circumstances, but are not otherwise marked. PUBLIC LANDS. 221 3. A tree of proper size on the corner is marked in the same manner as a post. The mounds, when made round the posts, must be 5 feet in diameter at the base, and 2J feet high. The posts, therefore, must be 4J feet long, so as to be 1 foot in the ground and 1 foot above the top of the mound. Witness pits for township corners must be 2 feet long, 1^ feet wide, and 1 foot deep. If the corner is common to four townships, there will be four pits placed lengthwise on the lines ; but if the corner is common to only two townships, only three pits are dug, and are placed lengthwise on the lines. Thus the kind of township corners are readily distinguished. These pits are made only in the absence of witness trees, which are to be selected, if possible, one from each township. 252. Section Corners. Section corners are established at intervals ,of 80 chains or 1 mile, and are perpetuated by the follow- ing methods : 1. Section corner posts are 4 feet in length, and at least 4 inches in diameter. They are planted 2 feet in the ground, and the part above the ground squared to receive the marks. If the corner is common to four sections, the post is set cornerwise to the lines, the number of the section is marked on the side facing it, and the number of the township and range on the north-east face. Mile-posts on township lines have as many notches on the corresponding edges as they are miles from the respective township corners. 222 SURVEYING. Section posts within a township have as many notches on the south and east edges as they are miles from the south and east boundaries of the township; but no notches are cut on the north and west edges. Section posts must be witnessed by trees, one in each section, or, in the absence of trees, by pits 18 inches square and 12 inches deep. 2. Section corner mounds are 4| feet in diameter at the base, and 2 feet high. The post must be 4 feet long, 1 foot in the ground, and 1 foot high above the mound, and at least 3 inches square. At corners common to four sections, the edges are in the direction of the cardinal points; but at cor- ners common only to two sections, the flattened sides face the cardinal points. Section posts in mounds are to be marked and wit- nessed in the same manner as the post without the mound. 3. Stones used to mark section corners on township lines are set with their edges in the direction of the line; but for interior sections they face the north. They are witnessed in the same manner as posts, but are not marked except by notches. 4. Section corner trees are marked and witnessed the same as posts. 253. Quarter Section Corners. Quarter section corners are established at intervals of 40 chains or half a mile, except in the north or west tiers of sections of a township. In subdividing these sections, the quarter post is placed 40 chains from the interior section corner, so PUBLIC LANDS. 223 that the excess or deficiency shall fall in the last half mile. Quarter section corners are not required to be estab- lished on base or standard parallel lines on the north. The methods of perpetuating these corners are the following : 1. Quarter section posts, 4 feet in length and 4 inches in diameter, are planted or driven 2 feet into the ground, and the part above the ground squared and marked J , signifying quarter section. These corners are witnessed by two bearing trees. 2. Quarter section mounds are, like section mounds, packed round the posts, and pits may be used in the absence of witness trees. 3. Quarter section stones have J- cut on the west side of north and south lines, and on the north side of east and west lines, and are witnessed by two bearing trees or pits. 4. A quarter section tree is marked and witnessed in the same manner as a post. 254. Meander Corners. Meander corners are the intersections of township or section lines with the banks of lakes, bayous, or navi- gable rivers. These corners are marked by the following methods : 1. Meander posts of the same size as section posts, are planted firmly in the ground, and witnessed by two bearing trees or pits, but are not marked. 2. Mounds of the same size as those for section cor- ners are, in the absence of witness trees, formed round 224 SURVEYING. the posts, and a pit dug exactly on the line, 8 links further from the water than the mound. 3. Stones or trees, witnessed in the same manner as posts, may be employed. 255. Chaining. Eleven tally pins are employed, ten of which are taken by the fore chainman, or leader, and the re- maining one by the hind chainman, or follower, who sticks it at the beginning of the course, and against it brings the handle at one end of the chain. The leader, holding the other handle of the chain and one pin in his right hand, draws out the chain to its full length in the direction of the course; both taking care that the chain is free from kinks. The leader standing to the left of the line, so as not to obstruct the range, with his right arm extended, draws the chain tight, brings the pin into line accord- ing to the order "right" or "left," from the follower, sticks it at the order "down" by pressing his left hand on the top of the pin, and replies " down." The follower then withdraws his pin, and both ad- vance, the leader drawing the chain in the direction of the course, but a little to one side to avoid drag- ging out the pin, till the follower comes up to the pin, against which he brings the handle at his end of the chain, and directs the sticking of another pin, as be- fore, and so on. When the leader has stuck his last pin, he cries "tally," which is repeated by the other, and each regis- ters the tally by slipping a ring on a belt. The follower then comes forward, and counting in presence of his fellow, to avoid mistake, the pins taken PUBLIC LANDS. 225 up, takes the foreward end of the chain and proceeds, as the leader, for another tally. If a whole chain is employed, a tally is ten chains; and accordingly four tallies make half a mile, and eight tallies a mile. If a half-chain is employed, a tally is five chains, eight tallies are half a mile, and sixteeen tallies a mile. In measuring up or down a hill, the chain must be kept horizontal, so that it is often necessary to use but a portion of the chain. The chain employed in the field must be compared, from day to day, with a standard chain furnished by the Surveyor-General, and any variation promptly cor- rected. 256. Marking Lines. Line trees, called also "station trees," or "sight trees," are marked by two notches on each side of the tree, in the direction of the line. The line is marked, so as to be easily followed, by blazing a sufficient number of trees near the line on two sides quartering toward the line. Saplings near the line are cut partly off by a blow from the ax, at the usual height of blazes, and bent at right angles to the line. Random lines are not marked by blazing trees; but to enable the surveyor to retrace the line on his re- turn, bushes are lopped and bent in the direction of the line, and stakes are driven every ten chains, which are pulled up when the true line is established. Insuperable objects, such as ponds, marshes, etc., are passed by making right-angled offsets, or by trigono- 226 SURVEYING. metrical operations, a complete record of which must be made in the field book. 257. Initial Point and Principal Lines. 1. The initial point, which is usually some perma- nent natural object, as the confluence of two rivers, or an isolated mountain, is first selected. 2. Principal meridians are run from the initial points due north or due south, and the quarter section, sec- tion, and township corners on these lines are accu- rately located and perpetuated. The following are the principal meridians already established : 1st. The first runs north from the mouth of the Great Miami river, between Ohio and Indiana, to the south line of 'Michigan. 2d. The second runs north from the mouth of the Little Blue river through the center of Indiana to its north line. 3d. The third runs north from the mouth of the Ohio river through Illinois to its north line. 4th. The fourth runs north from the Illinois river through the western part of Illinois and the center of Wisconsin to Lake Superior. 5th. The fifth runs north from the mouth of the Ar- kansas river through the eastern portion of Arkansas, Missouri, and Iowa, and regulates the surveys in Min- nesota west of the Mississippi river, and the surveys in Dakota east of the Missouri river. 6th. The sixth commences on the Arkansas river, in Kansas, and runs north through the eastern part of Kansas and Nebraska to the Missouri river. PUBLIC LANDS. 227 7th. Independent meridians. These are the Independent meridian of New Mexico, the Salt Lake meridian in Utah, the Willamette meridian of Oregon and Washington, and the Humboldt meridian, the ML Diablo meridian, and the St. Bernardino meridian of California. 3. Base lines are run from the initial points due east or due west, and the quarter section, section, and town- ship corners, for the land north of the line, are accu- rately located, at full measure, and perpetuated. 4. Standard parallels are also run due east or due west thirty miles north of the base line or other standard parallel, and the corners located and perpetu- ated as on the base li'ne. 5. Range lines are run between the ranges of town- ships due north from a base line or standard parallel to the next standard parallel. 258. Exterior or Township Lines. S JM P' 63 51 38 87 14 13 14 13 28 27 49 50 46 36 37 25 26 24 11 12 10 J2 11 10 26 26 24 46 47 45 34 35 22 23 21 8 9 7 9 8 7 23 22 21 43 44 42 32 33 31 19 1!0 18 5 6 4 6 5 4 20 1 18 40 il 39 29 US Iti IT 15 3 1 3 2 1 17 16 15 U P In the above diagram let P denote the initial point, PM the principal meridian, BL the base line, SP' the 228 SURVEYING. first standard parallel north, and let the squares denote townships. 1. For townships west of the meridian, begin at the first pre-established township corner on the base line west of the meridian. This is the S. W. corner of T. 1 N., R. 1 TT., and is marked 1 in the diagram. Measure thence due north 480 chains, establishing the quarter section and section corners, to 2, at which point establish the corner common to T.'s 1 and 2 N. and R.'s 1 and 2 W.; thence east on a random line, set- ting temporary quarter section and section stakes to 3. If the random line should overrun, or fall short, or intersect the meridian north or south of the true cor- ner, more than 3.50 chains, a material error has been committed, and the line^ must be retraced. If the random line should terminate within 3.50 chains of the corner, measure the distance at which the meridian is intersected north or south of the cor- ner, calculate a course which will run a true line back from the corner to the point from which the random line started, measure westward to 4, which is the same point as 2, establish the permanent corners, obliterate the temporary corners on the random line, and throw the excess or defect, if any, on the west end of the line. In like manner, measure from 4 to 5, from 5 to 6, from 6 to 7, and so on to 14, on the standard parallel, throwing the excess or deficiency on the last half mile. At the intersection with the standard, parallel, estab- lish the township closing corner, measuring and re- cording the distance to the nearest standard corner on said standard parallel. If from any cause the standard parallel has not been run, the surveyor will plant the corner. of the PUBLIC LANDS. 229 township in place, subject to removal north or south when the standard parallel shall have been run. The surveyor then proceeds to the S. W. corner of T. 1 N., R. 2 W., on the base line at 15, and proceeds in a similar manner with another range of townships, and so on. 2. For townships east of the meridian, begin at the S. E. corner of T. 1 A 7 "., R. 1 "., at 1 on the base line, and proceed as on the west of the meridian, except that the random lines are run west and the true lines east, throwing the excess over 480 chains, or the de- ficiency, on the west end of the line in measuring the first quarter section boundary on the north, the remain- ing distances will be exact half-miles and miles. With the field notes of the exterior or township lines, a plot of the lines, run on a scale of 2 inches to the mile, must be submitted, on which are noted all objects of topography, which will illustrate the notes, as the direction of streams, by arrow-heads pointing down stream, the intersection of the lines by lakes, streams, ponds, marshes, swamps, ravines., mountains, etc. 259. Subdivision or Section Lines. The deputy employed to run the exterior lines of a township is not allowed to subdivide it, but another is employed to do this work, that the one may be a check to the other, thus securing greater accuracy. Before subdividing a township, the surveyor must ascertain and note the change in the variation of the needle which has taken place since the township lines were run, and adjust his compass to a variation which will retrace the eastern boundary of the township. 230 SURVEYING. He must also compare his own chaining with the original by remeasuring tne first mile both of the south and east lines of the township, and note the discrepancies, if any. The following is a diagram of a township: H N 51 M 17 6 5 4 3 2 1 94 67 jC : J .3 16 93 92 90 sn 65 UG 48 40 31 32 14 15 7 8 9 10 II 12 89 64 47 30 u 8b b7 85 86 62 63 45 46 28 29 11 12 18 17 16 15 14 13 84 61 44 27 10 si 82 80 til 59 60 42 43 25 26 8 9 19 20 21 22 23 24 79 58 41 24 7 78 77 75 76 56 57 39 40 22 23 5 6 30 29 28 27 26 25 74 55 38 21 4 73 72 70 71 53 54 36 37 19 20 2 3 31 32 33 34 35 36 69 52 35 18 i The sections are designated by beginning at the N. E. corner and numbering west, 1, 2, 3, 4, 5, 6, then east on the next tier, 7, 8, . . . , then west, and so on. In running the subdivision lines, begin on the south i line of the township, at the first section corner west of the east line, numbered 1 in the diagram, and com- mon to sections 35 and 36. Measure thence due north 40 chains, at which point establish a quarter section corner; thence due north another 40 chains to 2, where establish a section cor- ner common to sections 25, 26, 35, and 3. PUBLIC LANDS. 231 Run a random line from 2 due east to the township line, setting up a temporary quarter section stake 40 chains from 2. If the random line intersect the township line pre- cisely at the pre-established section corner at 3, it may be established as the true line by blazing back and making the quarter section corner permanent. If the random line intersect the township line either north or south of the section corner, measure and note the distance of the intersection from said corner, and calculate a course which will run a true line from the corner back to 4, where the random line started. Let A correspond to sec- tion corner 2, B to 3, and n C to the intersection of the B township and random lines, and north, for example, of B the section corner. BC Then, tan A = --r=- A.JJ Let / the number of links in 5(7, and m the num- ber of minutes in A. Then, practically, we shall have, If AB = I mile, m= I } I If AB = 1 mile, m = l -& I. If AB ^ 3 miles, m = JJ. If AB = 6 miles, m = \ of \ I Let us suppose that we have found A -- 10J'. Now, as CA is west by the compass, BA is N. 89 49V W. Run this line and establish the quarter sec- tion at a point equidistant from the two section cor- ners, which will be, with sufficient accuracy, one-half the^ length of the random line from 2. Pull up the temporary quarter section stake on the random line. 232 SURVEYING. Proceed from 4 to 5, then on a random line to 6, and back on a true line to 7, and so on to 16. From 16- run due north on a random line to the north line of the township, setting up a temporary quarter section stake at 40 chains. If the random line intersect the north line of the township at the pre-established section corner, the' ran- dom line will be the true line, and is made permanent by blazing back, and making the quarter section cor- ner permanent. If the random line does not close exactly on the pre-established section corner, measure and note the distance of the intersection from said corner, calculate a course that will run a true line southward from the corner to 16, run this line, and establish the quarter section corner on it just 40 chains from 16, throwing the excess or deficiency, if any, on the last half mile. If the north township line is a base line or stand- ard parallel, no random line is run, but a true line due north, on which a quarter section post is established 40 chains from 16; and at the intersection with said base line or standard parallel, establish a closing cor- ner, measuring and noting its distance from the corre- sponding standard corner. Pass from 17 to 18, and survey the second tier of sections in the same manner as the first, closing on the interior section corners before established as upon those on the east line of the township. In running the line between the fifth and sixth tiers of sections, not only is a random line run east as before, but one is run west to the range line, and a true line run back, and the permanent quarter section corner established on it just 40 chains from the in- PUBLIC LANDS. 233 terior corner, throwing the excess or deficiency on the west half mile. The Surveyor-General furnishes the outline of the diagram, and the deputy fills it out, and makes the appropriate topographical sketches. 260. Meandering. Navigable rivers, lakes, and bayous, being public highways, are meandered and separated from the ad- joining land. Standing with the face down stream, the bank on the right hand is called the right bank; the bank on the left, the left bank. . If a river is navigable, both banks are meandered, care being taken not to mistake, in high water, the border of bottom-land for the true bank. Commence at a meander corner of the township line, take the bearing along the bank of the river, and measure the distance of the longest possible straight course to the nearest chain, if the distance exceeds 10 chains; otherwise, to the nearest ten links; and so on to the next meander corner on another boundary line of the township. Enter in the field book, after the township notes, keep- ing the notes separate through each fractional section, the date, the point of beginning, the bearings and dis- tances in order, the intersections with all intermediate meander corners, the height of falls, the length of rapids, the location and width at the mouth of streams run- ning into the water you are meandering, the location of springs on the banks, the nature of their waters, the location of islands, the elevation of banks, etc. S. N. 20. 234 SUE VEYING. If the river is not navigable, meander the right bank, unless it presents formidable obstacles not found on the left, bank; but the crossing of the stream, in meandering, must be made from a pre-established me- ander corner on one bank to the corner on the other bank, and the width of the river between the corners computed trigonometrically. Wide flats, whose area is more than 40 acres, per- manently covered with water, along rivers not navi- gable, are meandered on both banks. The position of islands in rivers is determined by measuring, on or near the bank, a base line, connected with the surveyed lines, and taking the proper bear- ings to a flag or other object on the island, and comput- ing the distance from the meander corners of the river to points on the bank of the island. The island can be meandered from such points. In meandering lakes, ponds, or bayous, commence at a meander corner of the township line, and proceed as in case of a river. If, however, the body of water is entirely within a township, begin at a meander cor- ner established in subdividing. In meandering a pond lying entirely within the boundaries of a section, run to the pond two lines from the nearest section or quarter section corners, on opposite sides of the pond, giving their bearings and distances, and at the intersection of these lines with the bank of the pond establish witness points by planting posts, witnessed by bearing trees or mounds and pits, then commence to meander at one of these points, and proceed around to the other, and thence to the point of beginning. No blazes or marks are made on meander lines be- tween established corners. PUBLIC LANDS'. 235 261. Swamp Lands. By the act of Congress approved Sept. 28th, 1850, swamp and overflowed lands, unfit for cultivation, are granted to the state in which they are situated. If the larger part of the smallest legal subdivision is swamp, it goes to the state; if not, it is retained by the Government. In order to determine what lands fall to the state under the swamp act, it is required that the field notes, beside other things required to be noted, should indicate the points where the public lines enter and leave all such land. The aforesaid grant does not embrace lands subject to casual inundation, but those only .where the over- flow would prevent the raising of crops without arti- ficial aid, such as levees, etc. The surveyor should therefore state whether such lands are continually and permanently wet, or subject to overflow so frequently as to render them totally unfit for cultivation. The depth of inundation is to be stated, as deter- mined from indications on the trees, and the frequency of inundation should be given as accurately as pos- sible, from the nature of the case or reliable testimony. The character of the timber, shrubs, plants, etc., growing on such lands, and on the land near rivers, lakes, or other bodies of water, should bo stated. The words "unfit for cultivation" should be em- ployed, in connection with the usual phraseology, in the notes, on entering or leaving such lands. If the margin of bottoms, swamps, or marshes, in which such uncultivable land exists, is not identical with the body of land unfit for cultivation, a separate entry must be made opposite the marginal distance. 236 SURVEYING. In case the land is overflowed by artificial means, such as dams for milling, logging, etc., such overflow will not be officially regarded, but the lines of the public surveys will be continued across the same with- out setting meander posts, stating particularly in the notes the depth of the water, and how the overflow was caused. 2G2. Field Books. The field books are the original and official records of the location and boundaries of the public lands, and afford the elements from which the plots are constructed. They should, therefore, contain an accurate record of every thing officially done by the surveyor, pursuant to instructions in running, measuring, and marking lines, and establishing corners, and should present a full topographical description of the tract surveyed. There are four distinct field books. 1. A field book for the meridian and base lines, ex- hibiting the establishment of the township, section, and quarter section corners on these lines, the crossing of streams, ravines, hills, and mountains, the character of the soil, timber, minerals, etc. 2. A field book for standard parallels or correction lines, showing the township, section, and quarter section cor- ners on the lines, and the topography of the country through which the lines pass. 3. A field book for exterior or township lines, showing the establishment of corners on the lines, and the to- pography. 4. A field book for subdivision or section lines, giving the corners and topography as aforesaid. PUBLIC LANDS. 237 The variations of the needle must be stated in a separate line, preceding the notes of measurement, which must be recorded in the order in which the work is done, and the date must immediately follow the notes of each day's work. The exhibition of every mile surveyed must be com- plete in itself, and be separated from the preceding and following notes by a line drawn across the paper. The topographical description must follow the notes for each mile, and not be mixed up with them. No abbreviations are allowed, except for words con- stantly occurring, as sec. for section, ch. for chains, ft. for feet, J- sec. cor. for quarter section corner. Proper names are never to be abbreviated. The field books must be so kept as to show the amount of work done in each fiscal year. The notes should be expressed in clear and precise language, and the writing legible. No record is to be obliterated, or leaf mutilated or taken out. The title-page of each book should designate the kind of lines run, giving prominently the name of the state or territory and surveyor, the dates of con- tract, and of commencing and completing the work. The second page should contain the names and duties of assistants; and whenever a new assistant is employed, or the duties of any of them changed, such facts, with the reason, should be stated in an appro- priate entry, immediately preceding the notes taken under such changed arrangements. An index, in the form of a diagram or plot of the survey, with number on each line, referring to the page of the field notes on which -is found the descrip- tion of the line, must accompany the notes. 238 SURVEYING. 263. Records in the Field Book. 1. General heading of the pages. The number of the township and range, and the name of the principal meridian of reference, stand at the head of each page. 2. Heading for each mile. The bearing, location, and kind of line run, whether random or true, must be stated in a line ; and the variation of the needle, in a separate line on the page at the head of the notes, for each mile run. 3. Courses and distances. The course and length of each line run, noting all necessary offsets therefrom, with the reason and mode thereof. 4. The method of perpetuating corners. If a tree, note the kind and diameter; if a stone, its dimensions, as factors in the order of length, breadth, and thickness; if a post, its dimensions, the kind of timber, the kind of memorial, if any, buried by its side, and .if surrounded by a mound, the material of which the mound is con- structed, whether of stones or earth ; the course and distance of the pits from the center of the mound where a necessity exists for deviating from the general rule of witness trees. 5. Bearing trees. The kind and diameter of all bear- ing trees, with the course and distance of the same from their respective corners, and the precise relative position of the witness corners with respect to the true corners. 6. Line trees. The kind, diameter, and distance on the line, from the corner, of all trees which the line intersects. 7. Intersection of land objects. The distance at which the line first intersects and then leaves every settler's claim and improvement, prairie, bottom-land, swamp, PUBLIC LANDS. 239 marsh, grove, or windfall, with the course of the same at both points of intersection; the distance at which a line begins to ascend, arrives at the top, or reaches the foot of all remarkable hills and ridges, with their courses and estimated height above the surrounding country. 8. Intersection of water objects. The distance at which the line intersects rivers, creeks, or other bodies of water, the width of navigable streams, and small lakes or ponds between the meander corners, the height of banks, the depth and nature of the water. 9. Surface. Level, rolling, broken, or hilly. 10. Soil. First, second, or third-rate; clay, sand, loam, or gravel. 11. Timber. Kind, in order of abundance, and un- dergrowth. 12. Bottom-lands. Wet or dry; whether subject to inundation, and to what depth. 13. Springs. Fresh, saline, or mineral; and course of their streams. 14. Improvements. Towns and villages, Indian vil- lages and wigwams, houses and cabins, fields, fences, sugar-tree groves, mill-seats, forges or factories. 15. Coal beds. Note the quality of coal beds, and their extent to the nearest legal subdivision. 16. Roads and trails. Whence, whither, and direc- tion. 17. Rapids, cascades, Length of rapids, height of falls in feet. 18. Precipices. Describe precipices, caves, ravines, sink-holes. 19. Quarries. Whether marble, granite, lime-stone or Band-stone. 240 SURVEYING. 20. Natural curiosities. Interesting fossils, ancient works, as mounds, fortifications, embankments, etc. 21. Change of variation. Any material change in the variation of the needle must be noted, and the exact points where such variation occurs. 22. Dates. State the date of each day's work in a separate line, immediately after the notes for that day. 23. General description. At the conclusion of the notes for the subdivisional work, taken on the line, the deputy must subjoin a general description of the township in the aggregate, in reference to the face of the country, its soil, timber, geological features, etc. 24. Verification of Deputy Surveyor. The deputy must append to each separate book of field notes his affidavit that all the lines therein described have been run, and all the corners established and perpetuated according to the instructions and laws, and that the foregoing notes are the true and original field notes of such survey. 25. Verification of Assistants. The compassman, flag- man, chainmen, and axman must also attest, under oath, that they assisted said deputy in executing said sur- veys, and that, to the best of their knowledge and be- lief, the work has been strictly performed according to the instructions furnished by the Surveyor-General. 26. Approval and certificate of the Surveyor-General. The Surveyor-General will attach his official approval to each of the original field books, and affix his of- ficial certificate to the copies of the field notes trans- mitted to the general land office, that they are true copies of the originals on file in his office. The following specimen pages of field notes, taken from the United States Manual of Surveying Instructions, will illustrate the subject : PUBLIC LANDS. 241 FIELD NOTES OF THE Exterior and Subdivision Twines OF TOWNSHIP 25 NORTH, RANGE 2 WEST, WILLAMETTE MERIDIAN, OREGON. Surveyed by Robert Acres, Deputy Surveyor, Under his contract, dated ? 18 . Survey commenced . Survey completed . S. TS. 2L 242 SURVEYING. 264. Index. Referring the lines to the pages of the field notes. T. 25 N., R. 2 W., Willamette Meridian. r J The lines numbered are described in the notes on the pages indicated by the numbers. NAMES OF SURVEYOR AND ASSISTANTS. Robert Acres, Surveyor. George Sharp, Axman. Peter Long, Chainman. Adam Dull, Axman. John Short, Chainman Henry Flagg, Compassman. PUBLIC LANDS. 243 265. Field Notes. South Boundary, T. 25 N., R. 2 W., Willamette Meridian. Chains. Begin at the post, the established corner to Townships 24 and 25 North, in Ranges 2 and 3 West. The witness trees all standing, and agree with the description furnished me by the office, viz : A Black Oak, 20 in. dia., N. 37 E. 27 links, A Burr-oak, 24 in. dia., N. 43 W. 35 links, A Maple, 18 in. dia., S. 27 W. 39 links, A White Oak, 15 in. dia., S. 47 E. 41 links. East on a random line on the South Bound- aries of sections 31, 32, 33, 34, 35, and 36. Variation by Burt's improved solar com- pass, 18 41' E. I set temporary half-mile and mile posts at every 40 and 80 chains, and at 5 miles, 74 chains 53 links, to a point 2 chains and 20 links north of the corner to Townships 24 and 25 North, Ranges 1 and 2 W. (Therefore, the correction will be 5 chains, 47 links West, and 37 links South per mile.) I find the corner post standing and the witness trees to agree with the description furnished me by the Surveyor-General's office, viz: A Burr-oak, 17 in. dia., bears N. 44 E. 31 links, A White Oak, 16 in. dia., bears N. 26 W. 21 links, A Linden, 20 in. dia., bears S. 42 W. 15 Iks., A Black Oak,- : 24 in. dia., bears S. 27 E. 14 links. 244 SURVEYING. (2) South Boundary, T. 25 N., R. 2 W., Willamette Meridian. Chains. From the corner to Townships 24 and 25 X., Ranges 1 and 2 W., I run (at a variation of 18 41' East) ' [See Arts. 258, 289.] N. 89 44' W., on a true line along the 40.00 South Boundary of section 36, set a post for quarter section corner, from which A Beech, 24 in. dia., bears N. 11 E. 38 links dist. A Beech, 9 in. dia., bears S. 9 E. 17 links dist. 62.50 A Brook, 6 links wide, runs North. 80.00 Set a post for corner to sections 35 and 36, 1 and 2, from which A Beech, 9 in. dia., bears N. 22 E. 16 links dist. A Beech, 8 in. dia., bears N. 19 W. 14 links dist. A White Oak, 10 in. dia., bears S. 52 W. 7 links dist. A Black Oak, 14 in. dia., bears S. 46 E. 8 links dist. Land level, good soil, fit for cultivation. Timber Beech, various kinds of Oak, Ash, Hickory. 40.00 N. 89 44' W., on a true line along the South Boundary of section 35, Variation 18 41' E. Set a post for quarter section corner, from which A Beech, 8 in. dia., bears N. 20 E. 8 links dist. No other tree convenient; made a trench around post. PUBLIC LANDS. 245 (3) South Boundary, T. 25 N., R. 2 W., Willamette Meridian. Chains. 65.00 80.00 Begin to ascend a moderate hill; bears N. and S. Set a post with trench, for corner of sections 34 and 35, 2 and 3, from which A Beech, 10 in. dia., bears N. 56 W. 9 links dist. A Beech, 10 in. dia., bears S. 51 E. 13 links dist. No other tree convenient to mark. Land level, or gently rolling, and good for farming. Timber Beech, Oak, Ash, and Hickory; som? Walnut and Poplar. 40.00 80.00 N. 89 44' W. on a true line along the South Boundary of section 34, Variation 18 41' E. Set a quarter section post with trench, from which A Black Oak, 10 in. dia., bears N. 2 E. 635 links dist. No other tree convenient to mark. To point for corner sections 33, 34, 3 and 4. Drove charred stakes, raised mounds with trenches, as per instructions, from which A Burr-oak, 16 in. dia., bears N. 31 E. 344 links. A Hickory, 12 in. dia., bears S. 43 W. 231 links. No other tree convenient to mark. Land level, rich, and good for farming. Timber some scattering Oak and Walnut. 246 SURVEYING. (4) Smith Boundary, T. 25 N., R. 2 W., Willamette Township. Chains. 37.51 40.00 62.00 80.00 N. 89 44' W. on a true line along the South Boundary of section 33, Variation 18 41' E. A Black Oak, 24 in. dia. Set a post for quarter section corner, from which A Black Oak, 18 in. dia., bears N. 25 E. 32 links dist. A White Oak, 15 in. dia., bears N. 43 W. 22 links dist. To foot of steep hill, bears N. E. and S.W. Set a post for .corner to sections 32, 33, 4 and 5, from which A White Oak, 15 in. dia., bears N. 23 E. 27 links dist. A Black Oak, 20 in. dia., bears N. 82 W. 75 links dist. A Burr-oak, 20 in. dia., bears S. 37 W. 92 links dist. A White Oak, 24 in. dia,, bears S. 26 E. 42 links dist. Land gently rolling ; rich farming land. Timber Oak, Hickory, and Ash. 37.50 40.00 N. 89 44' W. on a true line along the South Boundary of section 32, Variation 18 41' E. A Creek, 20 links wide, runs North. Set a granite stone, 14 in. long, 10 in. wide, and 4 in. thick, for quarter section corner, from which A Maple, 20 in. dia., bears N. 41 E. 25 links dist. A Birch, 24 in. dia., bears N. 35 W. 22 links dist. PUBLIC LANDS. 247 South Boundary, T. 25 JV., R. 2 W., Willamette Meridian. Chains. ! 76.00 80.00 To S. E. edge of swamp. As it is impossible to establish permanently the corner to sections 31, 32, 5 and 6, in the swamp, I therefore, at this point, 4.00 chains east of the true point for said section corner, raise a witness mound with trench, as per instructions, from which A Black Oak, 20 in. dia., bears N. 51 E. .115 links. * A point in deep swamp for corner to sec- tions 31, 32, 5 and* 6. Land rich bottom; west of creek, part wet; east of creek, good for farming. Timber good; Oak, Hickory, and Walnut. 11.00 40.00 54.00 57.50 61.00 70.00 N. 89 44' W. on a true line along the South Boundary of section 31, Variation 18 41' E. Leave swamp and rise bluff 30 feet high, bears N. and S. Set post for quarter section corner, from which A Sugar tree, 27 in. dia., bears S. 81 W. 42 links dist. A Beech, 24 in. dia., bears S. 71 E. 24 links dist. Foot of rocky bluff 30 feet high, bears N. E. and S. W. A spring branch comes out at the' foot of the bluff, 5 links wide ; runs N. W. into swamp. Enter swamp; bears N. and S. Leave swamp ; bears N. and S. 248 SURVEYING. (6) South Boundary, T. 25 N., R. 2 IF., Willamette Meridian. Chains. The swamp contains about 15 acres, the | greater part in section 31. 74.73 I The corner to Townships 24 and 25 N., Ranges 2 and 3 W. Land except the swamp, rolling, good, rich soil. Timber Sugar-tree, Beech, Swamp Maple. Jan. 25th, 1854. 8.56 Between Ranges 2 and 3 West, from corner to Townships 24 and 25 N., I run North, on the range line between sections 31 and 36, Variation 18 56' East. Set a post on the left bank of Chickeeles river, for corner to fractional sections 31 and 36, from which A Hackberry, 11 in. dia., bears N. 50 E. 11 links dist. A Sycamore, 60 in. dia., bears S. 15 W. 24 links-dist. I now cause a flag to be set on the right bank of the river, and in the line between sections 31 and 36. I now cross the river, and from a point on the right bank thereof, west of the corner just established on the left bank, I run North on an offset line, 25 chains and 94 links, to a point 8 chains and 56 links west of the flag. I now set a post in the place of the flag, for ' corner to frac- tional sections 31 and 36, from which A Beech, 10 in. dia., bears X. 2 E. 12 links dist. PUBLIC LANDS. 249 (7) Between Ranges 2 and 3 W., T. 25 N., Willamette Meridian. Chains. 34.50 40.00 43.41 80.00 A Black Oak, 12 in. dia., bears N. 80 W. 16 links dist. The corner above described. Set a post for J section corner, from which A Burr-oak, 20 in. dia., bears N. 37 E. 26 links dist. A Black Oak, 24 in. dia., bears N. 80 W. 16 links dist. A Black Walnut, 30 in. dia. Set a post for corner to sections 30, 31, 25, and 36, from which A Beech, 14 in. dia., bears N. 20 E. 14 links dist. A Hickory, 9 in. dia., bears N. 25 W. 12 links dist. A Beech, 16 in. dia., bears S. 40 W. 16 links dist. A White Oak, 10 in. dia., bears S. 44 E. 20 links dist. Land level; rich bottom; not inundated. Timber Oak, Hickory, Beech, and Ash. In like manner all the other Township lines are run. General Description. This township contains a large amount of first-rate land for farming. It is well timbered with Oak, Hick- ory, Sugar-tree, Walnut, Beech, and Ash. Chickeeles river is navigable for small boats in low water, and does not often overflow its banks, which are from ten to fifteen feet high. The township will admit of a large settlement, and should therefore be subdivided, 250 SURVEYING. (8) Field Notes of the Subdivision Lines and Meanders of Chickeeles River, in Towmhip 25 N., R. 2 W., Willamette Meridian. Chains. 40.05 80.09 9.19 29.97 40.00 51.00 76.00 To determine the proper adjustment of my compass for subdividing this township, I commence at the corner to Townships 24 and 25 N., R. 1 and 2 W., and run North, on a blank line along the East Boundary of section 36, Variation 17 51' East, To a point 5 links west of the quarter section corner. To a point 12 links west of the cottier to sections 25 and 36. To retrace this line, or run parallel thereto, my compass must be adjusted to a variation of 17 46' East. Subdivision commenced Feb. 1, 1854. From the corner to sections 1, 2, 35, and 36, on the South Boundary of the Township, I run North, between sections 35 and 36, Varia- tion 17 46' East, A Beech, 30 in. dia. A Beech, 30 in. .dia. Set a post for quarter section corner, from which A Beech, 8 in. dia., bears N. 23 W. 45 links dist. A Beech, 15 in. dia,, bears S. 48 E. 12 links dist. A Beech, 18 in. dia. A Sugar-tree, 30 in. dia. PUBLIC LANDS. (9) Township 25 JV., Range 2 TF., Willamette Meridian. 251 Chains. 80.00 Set a post for corner to sections 25, 26, 35, and 36, from which A Beech, 28 in. dia., bears N. 60 E. 45 links dist. A Beech, 24 in. dia., bears N. 62 W. 17 links dist. A Poplar, 20 in. dia., bears S. 70 W. 50 links dist. A Poplar, 36 in. dia., bears S. 66 E. 34 links dist. Land level, second-rate. Timber Poplar, Beech, Sugar-tree, and some Oak; undergrowth same, and Hazel. 9.00 15.00 40.00 55.00 72.00 80.00 40.00 East, on a random line between sections 25 and 36, Variation 17 46' East. A Brook, 20 links wide, runs north. To foot of hills, bears N. and S. Set a post for temporary quarter section corner. To opposite foot of hill, bears N. and S. A brook, 15 links wide, runs N. Intersected East Boundary at post corner to sections 25 and 36, from which corner I run West, on a true line between sections 25 and 36, Variation 17 46' East. Set a post on top of hill, bears N. and S., from which A Hickory, 14 in. dia., bears N. 60 E. 27 links dist. A Beech, 15 in. dia., bears S. 74 W. 9 links dist. 252 SURVEYING. (10) Township 25 A r ., Range 2 IF., Willamette Meridian. Chains. 80.00 The corner to sections 25, 26, 35, and 36. Land east and west parts, level, first-rate; middle part, broken, third-rate. Timber Beech, Oak, Ash, etc. ; under- growth same, and Spice in the bottoms. 7.00 17.20 18.05 23.44 40.00 60.15 80.00 North, between sections 25 and 26, Vari- ation 17 46' East. A Poplar, 40 in. dia. A Brook, 25 links wide, runs N. W. A Walnut, 30 in. dia. A Brook, 25 links wide, runs N. E. Set a post for J sec. corner, from which A Burr-oak, 36 in. dia., bears N. 42 E. 18 links dist. A Beech, 30 in. dia., bears S. 72 W. 9 links dist. A Beech, 30 in. dia. Set a post for corner to sections 23, 24, 25, 26, from which A White Oak, 14 in. dia., bears N. 50 E. 40 links. A Sugar-tree, 12 in. dia., bears N. 14 W. 31 links. A White Oak, 13 in. dia., bears S. 38 W. 32 links. A Sugar-tree, 12 in. dia., bears S. 42 E. 14 links. Land level on the line; high ridge of hills through the middle of section 25, run- ning N. and S. Timber Beech, Walnut, Ash, Maple, etc. PUBLIC LANDS. 253 (11) Township 25 N., Range 2 W., Willamette Meridian. Chains, j In like manner other subdivison lines are run. 24.00 Notes of the Meanders of a Small Lake in Section 26. Begin at the J sec. cor. on the line between sections 23 and 26, run thence South To the margin of the lake, where set a post for meander corner, from which A Beech, 14 in. dia., bears N. 45 E. 10 links dist. A Beech, 9 in. dia., bears N. 15 W. 14 links dist. Thence meander around the lake as follows : S. 53 E. 17.75. At 75 links, cross outlet to lake 10 links wide, runs N. E. S. 3 E. 13.00. S. 30' W. 8.00. S. 65 W. 12.00 to a point previously deter- mined 20.30 chains North of the quarter sec- tion corner on the line between sections 26 and 35. Set post meander corner, Maple, 16 in. dia., bears S. 15 W. 20 links dist. Ash, 12 in. dia,, bears S. 21 E. 15 links dist. ( In this vicinity we ! discovered remarkable N. 63 W. 10.00 ! foggil ' N. 13 W. 21.00 tention of naturalists. 254 SURVEYING. (12) Township 25 N., Range 2 TF., Willamette Meridian. Chains. N. 52 E. 17.30 to the place of beginning. This is a beautiful lake, with well-defined banks from 6 to 10 feet high. Land first-rate. Meanders of the left bank of Chickeeles River. Begin at the corner to fractional sections 4 and 33, in the North Boundary of the Township, and on the left and S. E. bank of the river, and run thence down the stream with the meanders of the left bank of said river, in fractional section 4, as follows : Remarks. To the corner to fractional sections 4 and 5 ; thence in section 5, Courses. Dist. S.76W. 18.50 S.61W. 10.00 S.59W. 8.30 S.54W. 10.70 S.40W. 5.60 S.50W. 8.50 S.37W. 17.00 S.44W. 22.00 S.38W. 26.72 S.21W. 16.00 S.10W. 13.00 South 8.50 S.9E. 5.00 S.17E. 20.00 S.10E. 12.00 S.22JE. 8.46 To the corner to fractional sections 5 and 8; thence in section 8, To the head of rapids. To the foot of rapids. To the corner to fractional sections 8 and 17. Land, along fractional section 8, PUBLIC LANDS. (13) Township 25 N., Range 2 W., Willamette Meridian. 255 Courses. Dist. Remarks. high, rich bottom; not inundated. The rapids are 37.00 chains long ; rocky bottom ; estimated fall, 10 feet. Meanders in Section 17. S.17E. 15.00 At 5 chains, discovered a vein of coal, which appears to be 5 feet thick, and may be readily worked. S.8E. 12.00 S.4W. 22.00 At 3 chains, the ferry across the river to Williamsburgh, on the oppo- site side of the river. S.25W. 17.00 S.78W. 12.00 S.71W. 9.55 To the corner to fractional sections 17 and 18; thence in section 18, S.65W. 15.00 S73fW. 15.93 To the corner to fractional sections 18 and 19. S.65W. 14.00 In section 19. S.60W. 23.00 S.42W. 10.00 S.20W. 10.00 S16JW. 13.83 JS^r* At 2 chains, cross outlet to pond and lake, 50 links wide, to the corner to fractional sections 19 and 24, on the range line, 32.50 chains North of the corner to sections 19, / 30, 24, and 25. The above selections will serve as specimens of the manner of taking the field notes. 256 SURVEYING. 266. General Description. The quality of the land in this township is con- siderably above the average. There is a fair propor- tion of rich bottom-land, chiefly situated on both sides of Chickeeles river, which is navigable, through the township, for steamboats of light draft, except over the rapids in Section 8. The uplands are generally rolling, good first and second rate land, etc. 267. Certificates. I, Robert Acres, Deputy Surveyor, do solemnly swear that, in pursuance of a contract with , Surveyor of the public lands of the United States, in the State [or Territory] of , bearing date the day of , 18 , and in strict conformity to the laws of the United States and the instructions furnished by the said Surveyor-General, I have faithfully surveyed the exterior boundaries [or subdivision and meanders, as the case may be] of Township number twenty-five North of the base line of Range number two West of the Willamette Meridian, in the aforesaid; and do further solemnly swear that the foregoing are the true and original field notes of such survey. ROBERT ACRES, Deputy Surveyor. Subscribed by said Robert Acres, Deputy Surveyor, and sworn to before me, a Justice of the Peace for the County, in the State [or Territory] of this day of , 18 . HENRY DOOLITTLE, Justice of the Peace. PUBLIC LANDS. 257 We hereby certify that we assisted Robert Acres, Deputy Surveyor, in surveying the exterior boundaries, and subdividing Township number twenty-five North of the base line of Range number two West of the Willamette Meridian, and that said Township has been, in all respects, to the best of our knowledge and belief, well and faithfully surveyed, and the boundary monu- ments planted according to the instructions furnished by the Surveyor-General. PETER LONG, Chainman. JOHN SHORT, Chainman. GEORGE SHARP, Axman. ADAM DULL, Axman. HENRY FLAGG, Compassman. Subscribed and sworn to by the above named per- sons, before me, a Justice of the Peace for the county of , in the State [or Territory] of , this day of , 18 . HENRY DOOLITTLE, Justice of the Peace. SURVEYOR'S OFFICE AT , 18 . The foregoing field notes of the Survey of [here de- scribe the survey], executed by Robert Acres, under his contract of the clay of , 18 , in the month of , 18 , having been critically examined, the necessary corrections and explanations made, the said field notes, and the surveys they describe, are hereby approved. A. B., Surveyor- General. To the notes of each Township, in the copies of the field notes transmitted to the seat of government, the Surveyor-General will append the following certificate: S. N. 22. 258 SURVEYING. I certify that the foregoing transcript of the field notes of the Survey of the [here describe the character of the surveys, whether meridian, base line, standard parallel, exterior township lines, or subdivision lines and meanders of a particular township], in the State [or Territory] of , has been correctly copied from the original notes on file in this office. A. B., Surveyor- General. 268. Corners and Boundaries Unchangeable. According to an act of Congress, entitled "An act concerning the mode of Surveying the Public Lands of the United States," approved February llth, 1805, and still in force, 1st. "All the corners marked in the surveys returned by the Surveyor-General, shall be established as the proper corners of sections or subdivisions of sections which they were intended to designate; and the cor- ners of half and quarter sections, not marked on said surveys, shall be placed, as nearly as possible, equi- distant from those two corners which stand on the same line." 2d. "The boundary lines actually run and marked in the surveys returned by the Surveyor-General, shall be established as the proper boundary lines of the sections or subdivisions for which they were intended; and the length of such lines, as returned by the Sur- veyor-General aforesaid, shall be held and considered as the true length thereof." If it is afterward found that a post is out of line, or that the line has been unequally subdivided, the general government only has the power of correction, and that only while it holds the title to the lands affected. PUBLIC LANDS. 259 Such boundaries only as "are established by the Sur- veyor-General, or the deputy, in the performance of his official duties, and in accordance with law, come under the above rules. 269. Restoring: Lost Boundaries. Lost boundaries must be restored in conformity with the laws under which they were originally established. At an early day, three sets of section corners were established on the range lines; later, two sets on all the township boundaries; at present, the section lines close on previously established corners on township corners, making one set of corners, except on the base lines and standard parallels, where double corners standard corners and closing corners are established. In order to restore lost boundaries correctly, the surveyor must know the manner in which townships were originally subdivided. In case of three sets of corners on the range lines, one set was planted when the exteriors were run. Corners on the east and west lines between two town- ships, belong to the sections of the township north. From these corners, section lines were run due north, which would not, in general, close on the corners of the township line on the north, thus making two sets of corners on the north and south boundaries of the township. The east and west lines were run due east and west from the last interior section corner, and new corners established at the intersections with the range lines. In case of two sets of corners, the subdivisions were made as &bove, except that the east and west lines 260 SURVEYING. were closed on the corners previously established on the east boundary, but were run due west from the last interior section corner to the range line, and new section corners established at the intersection with the range line. The method of making but one set of corners, ex- cept on the base line and standard parallels, is the one now in vogue, and has been sufficiently considered. 270. Restoring Lost Corners. Lost corners must be restored, if possible, to their exact original position. The surveyor should seek to accomplish this, first, by the aid of bearing trees, mounds, etc., described in the original field notes. If the corner can not be located in this way, good testimony may be taken. It often happens that in retracing lines, the meas- urements do not agree with the field notes. When such cases occur, from whatever cause, the surveyor must establish his corners at intervals proportional to those given in the original field notes. 1. To restore a lost corner common to four sections. Find the distances between the nearest noted line trees or well-defined corners, north and south, and east and west of the lost corner. Establish the corner be- tween them at a point intercepting distances propor- tional to those given in the original notes. 2. To restore one of a double corner when the other is standing. First ascertain to which sections the existing cor- ner belongs. Then re-establish the lost corner in the PUBLIC LANDS. 261 direction and at the distance stated in the original notes. Verify the work by chaining to noted line trees or corners, having previously compared your chaining with that of the United States deputy by rechaining between corners noted in the original sur- vey, and making all distances proportional. 3. To restore that one of a double corner established in run- ning the township lines when both are missing. Run a straight line between tttg nearest noted line trees or corners on the line, and, at the distance given in the notes, establish the corner which will be com- mon to two sections north or west of the line. Let the accuracy of the result be verified by measur- ing to the next section corner west or north. 4. To restore that one of a double corner established in subdi- viding the toivnship when both are missing. Retrace the section line which closed on the corner, and establish the section post at the intersection with the township line. Verify the result by measuring on the township line to noted objects. The restored corner will be common to two sections south or east of the line. 5. To restore one of a triple corner, on a range line when one at least remains standing. The one of the triple corner, established when the range line was run, is not a section corner. First identify the existing corners, then establish the lost corner, according to the field notes, north or south of the existing corner, on the line, and verify the result. 262 SURVEYING. If the field notes do not give the distances between the triple corners, retrace the section line closing on said corner. 6. To restore a triple corner when alt are lost. Rechain the range line, and retrace the section lines closing on the range line. 7. To restore lost quarter section corners. 1st. Except on those section lines which close on the north or west boundaries of a township, quarter section corners are equidistant between the two section corners. Hence, rechain the section line, then chain back one-half the distance. 2d. On township lines, where there may be double section corners, only one set of quarter section corners are actually marked in the field those established when the exteriors are run half-way between the section corners established at the same time. These are restored as above. The same will apply when there are triple corners. 3d. If the section line closes on the north or west boundary of a township, the quarter section corner must be established 40 chains of the original measure- ment from the last interior section corner. 8. To restore lost township corners. 1st. If the corner is common to four townships, re- trace the township and range lines, and establish trie corner at their intersection. 2d. If the corner is common only to two townships, as may be the case on the base line or standard paral- lels, retrace the base line or standard parallel from the PUBLIC LANDS. 263 last standing corner, if the lost corner is common to two townships north; but if the lost corner is common to two townships south, retrace also the range line. 9. To restore lost meander corners. Retrace the lines which close upon the banks in the direction they were originally run. Fractional section lines closing on Indian boundaries, private grants, etc., should be retraced, and the corners established in the same manner. Remark. If, in restoring a lost corner, the original corner is found by some unmistakable trace, it must stand, and the resurvey be made to correspond. 271. Subdividing Sections. The United States deputy runs only the exterior or section lines, and makes the section and quarter sec- tion corners. Lines joining the opposite quarter section corners divide the section into quarter sections of 160 acres each. These quarter sections are di- visible into half-quarters of 80 acres, and these into quarter- quarters of 40 acres. These are the legal subdivis- ions of a section, and are exhib- ited in the annexed diagram. If private parties wish the subdivision lines traced on the ground, they employ the county surveyor, or a private surveyor, who must be governed by the section and quarter section corners previously established. 40 A. 40 A. 80 A 80 A- 40 A. 40 A. 80 A. ) A. so A. 264 SURVEYING. The following rules will enable the surveyor to sub- divide a section in accordance with the laws of the United States : 1. The original section and quarter section earners must stand where they were established by the govern- ment surveyor. 2. The quarter-quarter corners must be established equidistant, and on the line between the section and quarter section corners of the exterior lines of the sec- tion, and equidistant and on the line between quarter section corners of internal lines of the section. 3. All subdivision lines must run straight from the proper corner in one exterior line of the section to the corresponding corner in the opposite exterior line. 4. In fractional sections, where no opposite corre- sponding corner has been established, the subdivision line must be run from the given corner due north and south, or east and west, to the exterior boundary of said fractional section. 5. Anomalous sections or sections larger than a mile, sometimes close on a previously established line, in finishing up a public survey. Quarter section and section corners are established 40 chains and 80 chains, respectively, from the previ- ously established corners, and posts are planted every 20 chains of the remaining distance. Anomalous sections are subdivided by running straight lines from the corners on the south line to the corresponding corners on the north, and east, and west lines, the same as in regular sections. VARIATION OF THE NEEDLE. 265 VARIATION OF THE NEEDLE. 272. Definitions and Illustrations. The variation of the needle is the angle which the magnetic meridian makes with the true meridian. The variation is east or west, according as the north end of the needle is east or west of the true meridian. The variation is different at different places, and it does not remain the same at the same place. The line of no variation is that line traced through those points on the surface of the earth where the needle points due north. At all places east of this line, the variation is west; and at all places west of this line, the variation is east. West variation is designated by the sign phis, and east variation by the sign minus. In the year 1840, at a point whose latitude is 40 53', and longitude 80 13', being a little 8. E. of Cleve- land, O., the variation was nothing. The line of no variation passed through this point N. 24 35' W., and S. 24 35' E. 273. Changes of Variation. 1. Irregular changes. The needle is subject to sud- den changes coincident, in time, with a thunder storm, an aurora borealis, solar changes, etc. 2. Diurnal changes, In the northern hemisphere, the north end of the needle moves from 10' to 15' west from about 8 A. M. to 2 P. M., and then gradu- ally returns to its former position. S. N. 23. 266 SURVEYING. 3. Annual changes. The diurnal changes vary with the season, being about twice as great in the summer as in the winter. 4. Secular changes. In addition to the above changes, there is a change of variation, in the same direction, running with considerable regularity through a period of about 234 years, as is indicated by observations at Paris. In the United States, the north end of the needle was moving east from the earliest recorded observa- tions till about the year 1810, since. which time the movement has been west, at the rate, on an average, of about 5' per annum. We give the following tables of places, their latitude and longitude, and variation as it was in 1840, and the annual change of variation, from the tables prepared by Professor Loomis for the 39th and 42d volumes of Silliman's Journal: Places near the Line of no Variation. Places. Lat. Lon. Var. An. Mo. \ A Point. 40 53' 80 13' 000' 4- 4'.4 Cleveland, O. 41 31' 81 45' -019' 4'.4 Mackinaw. 45 51' 84 41' 2 08' 3'.9 Charlottesville,Va. 39 02' 78 30' + 19' 3'.7 Assuming the annual motion uniform, and correctly found for 1840, the variation for any subsequent time can be found by multiplying the annual motion by the number of years since 1840, and taking the algebraic sum of the product and the variation at that date. VARIATION OF THE NEEDLE. Places where the Variation was West. 267 Places. Lat. Lon. Var. An. Mo. Point in Maine. 480(y 67 37' + 19 30' + 8'.8 Waterville, Me. 44 27' 69 32' 12 36' -5'.7 Montreal. 45 31' 73 35' 10 18' 5'.7 Burlington, Vt. 44 27' 73 10' 9 27' 5'.3 Hanover, N. H. 43 42' 72 14' 9 20' 5'.2 Cambridge, Mass. 42 22' 71 08' 9 12' 5'." Hartford, Conn. 41 46' 72 41' 6 58' 5'. Newport, R. I. - 41 28' 71 21' 7 45' 5'. Geneva, N. Y. 42 52' 77 03' 4 18' 4'.1 West Point. 41 25' 74 00' 6 52' 4'. New York City. 40 43' 71 01' 5 34' 3'.6 Philadelphia. 39 57' 75 11' 4 08' 3'.2 Buffalo, N. Y. 42 52' 79 06' 137' 4'.1 Places where the Variation was East. Places. Lat. Lon. Var. An. Mo. Jacksonville, 111. 39 43' 90 20' -8 28' -f 2'.5 St. Louis, Mo. 38 37' . 90 17' 8 37' 2'.3 Nashville, Tenn. 36 10' 86 52' 6 42' 2'. Louisiana. 29 40' 94 00' 8 41' 1'.4 Mobile, Ala. 30 42' 88 16' 7 05' 1'.4 Tuscaloosa, Ala. 33 12' 87 43' 7 26' 1'.6 Columbus, Ga. 32 28' 85 11' 5 28' 2'. Milledgeville, Ga. 33 07' 83 24' 5 07' 2'.4 Savannah, Ga. 32 05' 8112 ; 4 13' 2'.7 Tallahassee, Fa. 30 26' 84 27' 5 03' 1'.8 Pensacola, Fa. 30 24' 87 23' 5 53' 1'.4 Logansport, Ind. 40- 45' 86 22' 5 24' 2'.7 Cincinnati, 0. 39 06' 84 27' 4 46' 3/1 268 SURVEYING. 274. Methods of Ascertaining the Variation. First establish a true meridian, which may be done 1. By means of Burfs Solar Compass. 2. By observation of the North star, when on the meridian. The north star is about 1 22' from the true pole, around which it revolves in a siderial day, or 23 h., 56 in., 4 s. Twice in this period the star will be on the meridian. The exact moment of its passage can be determined very nearly, from the fact that it reaches the meridian almost at the same instant as Alioth in the tail of the Great Bear, or the first star in the handle of the Dipper. Suspend a plumb line a few feet in front of the telescope, and place ^ a faint light near the object glass of the telescope, so that the spider lines may be seen. Just 17 minutes after the plumb line, the North star, and Alioth all fall on the vertical spider line, the North star is on the meridian. The horizontal limb of the instrument is then firmly clamped, and the telescope is turned down horizontally. A light, shining through a small aperture in a board, at some distance, say ten rods, is moved by an assistant, according to signals, till it ranges with the intersection of the spider lines. A stake driven into the ground directly under the light, and another directly under the telescope, will mark, on the ground, the true meridian. VARIATION OF THE NEEDLE. 269 The season of the year may be such that Alioth may be above instead of below the North star, when both are on the meridian at night. With, the telescope, the stars can be seen in the day-time. 3. By the azimuth of the North star. When the North star is farthest from the meridian, east or west, it is said to be at its greatest eastern or western elongation. The azimuth of a star is the angle which a vertical plane, through the star, makes with the meridian plane. Let us now find the azimuth of the North star at its greatest elongation. Let Z be the zenith, P the pole, S the North star at its greatest elong- ation, ZP, ZS, and PS arcs of great circles. Then ZPS will be a spherical triangle, right-angled at , and the angle Z will be the azimuth, PS the greatest elongation, and ZP the com- plement of latitude; since the elevation of the pole above the horizon is equal to the latitude. Now, from Napier's principles, we have sin e = cos I cos (90 Z). sin e . . sin Z = r - cos I Introducing R and applying logarithms, we have log sin Z = 10 -f- log sin e log cos I, Hence, the azimuth is readily computed if we know the greatest elongation of the star and the latitude of the place. 270 SURVEYING. Greatest Elongation of Polaris. Date. Elongation. Date. Elongation. Date. Elongation. 1870 1 23' 01". 1880 1 19' 50".4 1890 1 16' 40".7 1871 122'41".9 1881 19' 31".4 1891 1 16' 21".8 1872 1 22' 22".9 1882 19' 12".5 1892 1 16' 03" 1873 1 22' 03".8 1883 18' 53".5 1893 1 15'44".l 1874 1 21' 44".8 1884 18' 34".5 1894 1 15' 25".3 1875 121"25".7 1885 1 18' 15".5 1895 1 15'06".4 1876 1 21' 06".6 1886 1 17' 56".6 1896 1 14' 47".6 1877 1 20' 47".6 1887 1 17' 37".6 1897 1 14' 28".7 1878 1 20' 28".5 1888 1 17' 18".6 1898 1 14'09".9 1879 1 2(X 09".5 1889 1 16' 59".7 1899 1 13' 51" The elongation in the table is given for the 1st of January of each year; but the elongation for any month of the year can be readily found. Thus, let us find the elongation for May 1st, 1873. Jan. 1st, 1873, Elongation == 1 22' 03".8 Jan. 1st, 1874, Elongation = 1 21' 44".8 Change for 12 months 19" Change for 4 months 6.3" . . Then, for May 1st, 1873, we shall have, Elongation = 1 22' 03".8 6".3 = 1 21' 57".5. 1. Find the azimuth of the North star at its greatest elongation, May 1st, 1873 latitude 40. Ans. 1 47'. 2. Find the azimuth of the North star at its greatest elongation, July 1st, 1875 latitude 42. Ans. 1 49J'. 3. Find the azimuth of the North star at its greatest elongation, Sept. 21st, 1880 latitude 45 45'. Ans. 1 54 '. VARIATION OF THE NEEDLE. 271 It will be necessary to know the times of the greatest elongation. These times are given in the following tables, for the 1st, llth, and 21st of each month of the year 1880, which will answer the purpose for the rest of the century, since the change of time is very slow, being only about 16 minutes in 50 years. Eastern Elongation. Month. 1st day. llth day.' 21st day. April. 6h. 40m. A.M. 6h. Olm. A.M. 5h. 22m. A.M. May. 4h. 42m. A.M. 4h. 03m. A.M. 3h. 24m. A.M. June. 2h. 41m. A.M. 2h. Olm. A.M. Ih. 22m. A.M. July. Oh. 43m. A.M. Oh. 00m. A.M. lib. 21m. P.M. August. lOh. 38m. P.M. 9h. 59m. P.M. 9h. 19m. P.M. Sept. 8h. 36m. P.M. 7h. 57m. P.M. 7h. 17m. P.M. Western Elongation. Month. 1st day. llth day. 21st day. Oct. 6h. 31m. A.M. 5h. 52m. A.M. 5h. 13m. A.M. Nov. 4h. 30m. A.M. 3h. 50m. A.M. 3h.llm. A.M. Dec. 2h. 31m. A.M. Ih. 52m. A.M. Ih. 13m. A.M. Jan. Oh. 28m. A.M. llh.44m.P.M. lib. 04m. P.M. Feb. lOh. 22m. P.M. 9h. 42m. P.M. 9h. 03m. P.M. March. 8h. 31m. P.M. 7h. 52m. P.M. 7h. 13m. P.M. About half an hour before the greatest eastern or western elongation, place the transit in a convenient position, and level it carefully. Paste white paper on a board about one foot square, and perforate the board through the center with a two- inch auger, and, on the lower edge, fix some contriv- ance for holding a candle. 272 SURVEYING. Let this board be fixed to a vertical staff, so as to slide freely up and down, and let it be placed about one foot in front of the telescope, so that the light reflected from the paper will render the spider lines visible. Slide the board up or down the staff till the North star is visible through the telescope and orifice in the board, and bring the vertical spider line in range with the star. As the star approaches its greatest elongation, move the telescope by a tangent screw, so as to keep the vertical line in range with the star. When the star reaches its greatest elongation, it will appear, for some time, to coincide with the spider line, and then leave it in the opposite direction. Clamp the horizontal limb, and turn the telescope down till it is horizontal. Let now a staff, with a light on its upper end, be carried ten or fifteen rods distant, toward the star, and placed so as to range, when vertical, with the vertical spider line of the telescope. Drive a stake at the foot of the staff, and another directly under the instrument, then will the line de- termined by the stakes make an angle with the true meridian, equal to the azimuth of the North star. The true meridian will lie west or east of the line of stakes, north of the telescope, according as the elongation was east or west, and may readily be located by the instrument. The location of the meridian can be verified thus: Let AB be the line of the stakes produced to a considerable distance, say from 20 to 40 A VARIATION OF THE NEEDLE. 273 chains, A the azimuth angle, AC the true meridian, and EC perpendicular to AB. BC can be found from the formula, BC = AB tan A. Then laying off BC on the ground, and driving a stake at (7, the stakes A and C will trace the true meridian. Having found the true meridian, the variation of the needle can be readily determined by turning the telescope or the sights of the compass in the direction AC. Without finding the true meridian, the bearing of AB being equal to the known azimuth of the North star at its greatest elongation, the variation of the needle can be found by directing the telescope or the sights of the compass in the direction AB. The following method may be resorted to by the surveyor who does not possess an instrument with a telescope. Fix a plank, firmly level, east and west, about three feet above the ground; then take a board about six inches square, and having detached one of the com- pass sights, fix it to the board, at right angles with its upper edge. Drive a nail obliquely a little way into the board, so that it can be tacked to the plank. About fifteen feet north of the plank suspend a plumb line, from the top of an inclined stake of such height that the North star, when seen through the sight while the board rests on the plank, will appear about one foot below the upper end of the plumb line. Suspend the plumb in a vessel of water to prevent the line from vibrating, and let an assistant hold a light near it, so that it can be seen through the sight, 274 SURVEYING. About half an hour before the time of the greatest elongation of the North star, place the board on the plank, and slide so that the star and plumb line shall range when seen through the sight. As the star ap- proaches its greatest elongation, move the board along the plank in the opposite direction, so as to keep the range. When the star reaches its greatest elongation, it will appear to keep the range for several minutes, then it will move slowly in the opposite direction. Tack the board to the plank, taking care not to change its position. Then let a staff with a light on its top be placed about ten rods farther to the north, so as to range, when vertical, through the sight, with the plumb line. Drive a stake at the foot of the staff, and one di- rectly under the plumb line, then will the line of the stakes make, with the meridian, an angle equal to the azimuth of the North star at its greatest elongation. The true meridian, and the variation of the compass, can then be found as above. FIELD OPERATIONS. 275. Finding Corners. In searching for a corner, first seek for the monu- ment, whether tree, post, stake, or stone, as given and witnessed in the original field notes, which, if found, must be considered decisive in establishing the corner. If no monument can be found, the corner can often be found by indirect methods, of which the following are the most available: FIELD OPERATIONS. 275 Thus, if a monument can be found at each of the cor- ners A, 0, />, but not at B, find the corners E and F, at each of which set up a flag- staff or high pole, and send the flag-man as near to B as possible, and let him stand facing D, so that he can see signals made both at A and C. .0 The observer at A can, by waving his hand, bring the flag-man in the line AE, and the observer at C can bring him in the line CF, and being in both lines, AE and OF, at the same time, he will be at their in- tersection B, the corner required. If the corner E can be found, but not F, measure AB the required distance in the line AE. If the dis- tance AB is not known, but it is simply known that AB is equal to DC, first measure DC. If neither E nor F can be found, run AB parallel to DC, and CB parallel to DA, and the intersection of these lines will determine B, if the field is a parallelogram. If the field is not a parallelogram, retrace one of the lines terminated lay known corners, and compare the bearing with the bearing in the original notes, which will give the variation of the needle. Then run the lines AB and CB from the notes, allowing for the vari- ation, and the intersection will determine B. In like manner two or more lost corners may be found. If the bearings and distances are given in the origi- nal notes, and but one corner can be found, retrace some established line in the neighborhood to find the variation, and, beginning at the known corner, run the lines from the notes, allowing for the variation. 276 .SURVEYING. The importance of allowing for the variation may be illustrated thus: Let the full lines bound the lot. If the surveyor should run this lot from the original notes, one corner being known, the dotted lines would mark the boundaries as run, and their intersections the corners, thus encroaching on one side, and leaving gaps on the other, which of course would never do. 276. Finding Bearings and Distances. After finding the corners, set a stake at each, and, beginning at any corner, place the compass or transit directly over the stake, and send the flag-man to the next corner, who must place the flag-staff' on the stake. Take the bearing, and measure the distance as here- tofore directed; and, in like manner, find the bearings and distances of the remaining sides. If obstacles should prevent the taking of the bear- ing of any line, measure the same distance from each corner, at right angles to the line, on the same side, so as to secure a line free from obstacles, and take the bearing of this line, which will be the bearing of the required line, since they are parallel. Lines are measured a little to one side when fences, ponds, or other obstacles, are in the line. Thus, if the perpendiculars AC and BD are equal, AB can be found by Trigo- nometry, if AE and EB and two angles be measured. FIELD OPERATIONS. 277 277. Offsets. Offsets are perpendiculars measured from a line to the angles of a neighboring broken line, or to the banks or centers of creeks, rivers, or other bodies of ^vater. Thus, a, 6, c. 278. Taking Field Notes. First Method. Second Method. Sta, Bearings. Dist. 1 N. 20 E. 15.50 2 E. 18.00 3 S. 20 E. 30,00 4 i W. 25.00 O , X. 32J W. 16.09 The first method is in the proper form for calcula- tion, and may be conveniently employed when it is not important to make a map of the lot surveyed. The second method, being a random outline with bearings and distances indicated, may be employed when it is desirable for the surveyor to keep before him, while at work, an outline of the lot. Third Method. 68.00 Station A. 57.60 Orchard fence 42.00 Oatfield fence 26.00 Meadow fence 14.00 S. Bank of Greek 13.20 INT. Bank of Creek: 10.80 Pasture fence Station E 4.80 A S. Bank of River S. Left Bank of River 18.40 ^%v IS". Bank of River 17.40 __ 9 _ep._1| Offset 16.40 _?-*5of Offset 10.50 Station D A s S. 32E. Left Bank of River f~ 30.00 Lei't Bank of River NN N ^ RigTit Bank of River tfL___!L.._ 26.00 \Offset C- 16.40 7.30 Orfset Offset N x Station C 4.80 A N. Line ot'SRoad JST. 63E. Right Bank^f B,iver 68.00 58.00 Road East x ^ j *. _ "Woods ^ 55.20 Pond 42.00 Pasture fence-"' ^ 26.00 Cornfield fence /^' 10.52 " Wheatfield fence Station B A"" IN. Middle of Turnpike 40.00 Lot Line . 31.20 Meadow fence 24.00 / Grove fence Station A 17.20 10.08/ x A'" X)ooryard fence Orchard fence "W. ^fiddle of Turnpike .4- (278) MAP OF FARM Scale 16 p. to 1 inch. A (279) 280 SURVEYING. 279. Remarks on the Third Method. The third method should be employed whenever a map, more or less perfect, is to be made. The notes should be placed on a left-hand page of the field book, and the map on the right page, facing. By referring to the notes and map illustrating this method, it will be observed that the survey began at A, the S. E. corner of the -farm, at the middle of the turnpike, and that we commenced to record the notes at the bottom of the page. . This will keep the notes of the objects, at the right or left of each line run, in their natural position on the page, at the right or left of the parallel lines in- closing the distance from the station at the beginning of the line to the objects worthy of record encountered in running the line. The character /\ denotes station, at the left of which stands the letter marking its position on the map, and at the right the bearing of the next course. A prominent object, such as the chimney of the house, a large tree standing in an open field, may be selected, and its bearings from the principal stations be taken. These bearings will serve as checks against errors in drawing the map, and may aid in finding the corners should they be lost. In the present example, a chestnut tree on the top of a hill, in the pasture at the left of the lane, is selected, and its bearing from A, B, and D given. 280. Surveying Creeks and Roads. 1. Creeks may be meandered as described under the head of Survey of the Public Lands. FIELD OPERATIONS. 281 2. They may also be surveyed by running straight lines connecting points on the bank, taking the bear- ings of these lines, the distances from the origin of these lines to the perpendicular offsets run from the lines to the bank of the river, and the length of the offsets, as exhibited in the following field notes and plot. Field Notes. Plot. Station C /\ 3.48 3.04 Station B A 6.19 4.39 3.14 2.84 2.24 1.08 .40 A The name of stations and the left-hand offsets are noted on the left of the parallels, the right- hand offsets and bearings on the right, the distance from the station to the offsets, and the sign for station, between the parallels. 3. In surveying an existing winding road, keep in the road, run straight lines as far as possible, without running out of the road, note the bearing of these lines, the distances to the offsets at different points to the sides of the road, the lengths of these offsets, and make an accurate plot of the road. 4. To survey a new road, find the bearing of the middle line from the origin to the next angle or in- tersection with another road, measuring the distance S. N. 24. 282 SURVEYING. from the origin to the lines of farms, creeks, etc., which it intersects. Set temporary stakes at the angles, and at convenient .distances along the middle line, to guide in making the 'road, and plant monuments at a given distance and bearing from the angular points, so that they will not be disturbed in making or working the road. Take notes, and make a correct plot of the road. 281. Surveying Towns. Commence at the intersection of principal streets, take their bearings, measure their lengths, noting the distances to the streets and alleys crossed, taking off- sets to corners of streets and prominent objects, as public buildings, etc., till a prominent cross-street is reached, which survey in the same manner, changing the courses at such stations as will lead back to the original station. Survey all the streets and alleys enclosed. Then sur- vey an adjoining district, and so on, till the entire town or city has been surveyed. Take notes, and make an accurate map of the town, on which locate not only the streets and alleys, but public buildings, parks, fountains, monuments, etc. 282. Reverse Bearing. Let AB be a line run from A to B, AN and BS meridians, then will NAB be the bearing of A.B, and SBA will be the reverse bearing. Since the meridians AN and BS may be regarded as parallel, the bearing and reverse FIELD OPERATIONS. 283 bearing are equal. Thus, if the bearing of AB is N. 30 E., the reverse bearing is S. 30 W. The bearing and reverse bearing agree in the value of the angle, and differ in both the letters which in- dicate the general direction of the line. In fact, the reverse bearing of a line is the bearing of the line if run in the opposite direction. Thus, SBA, the reverse bearing of the line AB, run from A to B, is the bear- ing of the line BA, run from B to A. Of the letters used in bearings, we shall call N and S latitude letters, and E and W departure letters. To guard against inaccurate observations, and the disturbance of the needle occasioned by local attraction, the reverse bearing should be taken at every station. If the bearing and reverse bearing agree in value, the bearing may be considered as correctly taken; if they differ materially, both should be taken again. If they still differ, the difference may be regarded as occasioned by local attraction. To ascertain at which station the local attraction exists, place the instrument at a third station, at a considerable distance from each of the doubtful stations, and sight to each, then from these back to the third station. The local attraction may be considered to exist at the station where the bearing of the third station disagrees with its bearing taken at the third station. If the error occurred in the foresight, correct it before entering the bearing in the field notes, and note the amount of disturbance; if the error occurred in the backsight, the next foresight will be affected, and should be corrected before entered. 284 SURVEYING. PRELIMINARY CALCULATIONS. 283. Angles between Courses. 1. If the latitude letters are alike, also the departure letters, the included angle is equal to the difference of the bearings. If AB bears N. 40 E., and AC N. 20 E., BAG = BAN CAN = 40 - 20 = 20. If AD bears S. 40 W., and AE S. 20 W., DAE = DAS EAS = 40 - 20 = 20. 2. If the latitude letters are alike, and the departure letters unlike, the included angle is equal to the sum of the bearings. If AB bears N. 38 E., and AC N. 18 W., BAC=BAN+NAC=S& H- 18 = 56. If AD bears S. 38 W., and AE S. 18 E., DAE => DAS + SAE = 38 + 18 = 56. 3. . If the latitude letters are unlike, and the departure letters alike, the included angle is equal to 180 minus the sum of the bearings. If AB bears N. 45 E., and AE S. 30 E., BAE = 180 (NAB +SAE) = 180 75= 105. If AD bears S. 45 W., and AC = 180 75 = 105. PRELIMINARY CALCULATIONS. 285 4. If the latitude letters are unlike, also the departure letters, the included angle is equal to 180 minus the difference of the N E If AB bears N. 45 E., and AC S. 15 W., BAG = 180 (NAB SAC) = 180 30 = 150. If AD bears S. 45 W., and AE N. 15 E., DAE = 180 (SAD - NAE) = 180 30 == 150. Remark. These principles apply when both courses run from or toward the vertex ; if one runs from the vertex, and the other toward it, reverse the bearing of one side before applying the principles. c s 284. Examples. 1. Find the angle A, if AB bears N. 78 E., and AC N. 24' E. Ans. 54. 2. Find the angle A, if BA bears S. 34 E., and AC S. 48 W. Ans. 98. 3. Find the angle A, if BA bears S. 70 W., and CA N. 25 E. Ana. 135. 4. Find the angles of the polygon ABODE, if AB bears N. 30 E. ; BC, N. 60 E. ; CD, S. 50 E.; DE, S. 40 W. ; EA, N. 78 W. ,4 = 72, 5=150, (7=110, > = 90, JE=118. 285. Problem. Given the bearings of the sides of a field, to find the bear- ings if the field be supposed to revolve, so as to cause one of the sides to become a meridian. 286 SURVEYING. In the following diagram let the full lines denote the original position of the sides of the field, a the side that is to become the meridian, and the dotted lines the revolved position of the sides. a, N. 30 E. 6, N. 60 E. c, N. 10 E. d, S.- 45 E. e, S. 75 E. /, S. g, S. 55 W. h, S. 20 W. t, W. j, N. 25 W. 'fc, N. 80 W. a', N. 6', N. 30 E. c'. X. 20 W. d', S. 75 E. e', X. 75 E. /', S. 30 E. #', S. 25 W. tf, S. 10 E. *', S. 60 W. /, X. 55 W. F, S. 70 W. From the above illustration we derive the following principles :" 1. If the letters which indicate the general direction of the side which is to be made a meridian are both alike or both unlike those of another side, then, 1st. If the bearing of the former is less than that of the latter, the difference of the bearings will be the bearing of the latter, the letters remaining the same as before. 2cL If the bearing of the former is greater than that of the latter, the difference of the bearings will be the bearing of the latter, the departure letter being changed. 2. If one of the letters which indicate the general direction of the side which is to be made a meridian is like and the other unlike the corresponding letter of another side, then, PRELIMINARY CALCULATIONS. 287 1st. The sum of the bearings, if less than 90, will be the bearing of that side, the letters remaining -the same as before. 2cL If the sum of the bearings is greater than 90, its supplement will be the bearing of that side, the latitude letter being changed. 286. Examples. 1-. The bearings of the sides of a field are as follows : 1st, N. 30 E. ; 2d, N. 60 E. ; 3d, S. 40 E. ; 4th, S. 30 W.; 5th, W.; 6th, N. 18J W. Find the bearings of the sides if the second side becomes a meridian. Ans. 1st, N. 30 W. ; 2d, N. ; 3d, N. 80 E.; 4th, S. 30 E.; 5th, S. 30 W. ; 6th, N. 78f W. <? Ov-* 2. The bearings of the sides of a field are as follows : 1st, N. 45 W.; 2d, N. 18 E.; 3d, E.; 4th, N. 32 E.; 5th, S. 42^ E.; 6th, S.; 7th, S. 65J W. Find the bearings if the first side be made a meridian. Ans. 1st, N.; 2d, N. 63 E.; 3d, S. 45 E.; 4th, N. 77 E.; 5th, S. 2| W. ; 6th, S. 45 W. ; 7th, N. 69f W. 3. The bearings of the sides of a field are as follows : 1st, N. 20 E.; 2d, N. 70 E. ; 3d, E.; 4th, S. 45 E. ; 5th, S. ; 6th, S. 45 W. ; 7th, W. ; 8th, N. 3| W. Find the bearings if the sixth side be made a meridian. Ans. 1st, N. 25 W. ; 2d, N. 25 E. ; 3d, N. 45 E. ; 4th, E.; 5th, S. 45 E. ; 6th, S. ; 7th, S. 45 W.; 8th, N. 48f W. 287. Latitude and Departure. The latitude of a course is the distance between the two parallels of latitude passing through the extremi- ties of the course. 288 SURVEYINQ. The departure of a course is the distance between the two meridians passing through the extremities of the course. Let AB be a course, AD and BC paral- lels of latitude, and ^Cand BD meridians. Then will AC or DB be the latitude of the course, and CB or AD its departure. But AC = AB X cos CAB, and CB = AB X sin CAB. Hence, latitude = course X cosine of bearing, and departure course X sine of bearing. If the line runs due east or west, its latitude is 0. If the line runs due north or south its departure is 0. Latitude north is considered plus; latitude south, minus. Departure east is considered plus; departure west, minus. For brevity let us designate the bearing by b, the course by c, the latitude by I, and departure by d, then we shall have the cases given in the following article: 288. Table of Cases. Given. Req. Formulas. 1 b, r, l,d. I c cos b, d = c pin ft. 2 M, c,d. r rl 1 tan A cos b 3 b,d, c,l. c d I d sin 6' tan b 4 5 c, I, c,d, b,d. cos 6=, d= i/c 2 P. C sin 6 = , / = 1/c 2 d 2 . 6 w b,c. ^ _ ^ PRELIMINARY CALCULATIONS. 289 289. Examples. 1. Given b = N. 53 20' E., and c == 26.50 ch. ; required I and d. Ans. I = 15.82 ch. N., d = 21.26 ch. E. 2. Given b = S. 75 47' W., and / = 22.04 ch. S. ; re- quired c and d. Ans. c = 89.75 ch., d ==-- 87 ch. W. 3. Given b = N. 35 W., and d - 1.55 ch.W. ; required c and /. Ans. c == 2.70 ch., J == 2.21 ch. N. 4. Given c * 35.35 ch.,. and I = 31 ch. N. ; required b and d. Ans. b = N. 28 44' E. or W., d == 16.99 ch. E. or W. 5. Given c = 31.30 ch., and d as 22.89 ch.W. ; required b and . 6 == N. or S. 47 W., and I = 21.35 ch. N. or S. 6. Given I = 7.02 ch. S., and d = 7.14 ch.W. ; required b and c. Ans. b = S. 45 29' W., c == 10.01 ch. 290. Traverse Table. The traverse table affords a ready method of finding the latitude and departure of a course whose distance and bearing are given. Let us find the I and d of a line whose b is N. 35 15' E., and c = 47.85 ch. Turning to the traverse table, under 35 15' we find c =-- 40* gives I = 32.67, d = 23.09. c = 7 gives I --= 5.72, d = 4.04. c= .8 gives I = .65, d = .46. c == .05 gives 1= .04, d = .03. . . c ^ 47.85 gives I = 39.08, d = 27.62. S. N. 25. 290 SURVEYING. The I and d for 40 are found from the I and d of 4, as given in the table, by multiplying by 10, or remov- ing the decimal point one place to the right. The I and d for the distance 7 are given in the table, but the right hand figure is dropped, and 1 is carried if the figure dropped exceeds 5. The I and d for the distance .8 are found from the I and d for the distance 8 by removing the decimal point one place to the left, rejecting the figures at the right of the second decimal place, carrying as above. For the distance .05, remove the decimal point two places to the left, reject and carry as before. If the bearing exceeds 45, the I and d will be found in columns marked at the bottom of the page. 291. Examples. 1. Given b = N. 28 45' E., and c = 35.35 ch. ; required I and d. Ans. I == 30.98 ch. N., d == 17 ch. E. 2. Given b = S. 36f E., and c= 19.36 ch. ; required I and d. Am. I = 15.51 ch. S., d = 11.59 ch. E. 3. Given b = N. 53 15' E., c = 11.60 ch.; required I and d. Am. I ~ 6.94 ch. N., d = 9.29 ch. E. 4. Given b = S. 74i E., c = 30.95 ch. ; required I and c?. Ans. I = 8.27 ch. S., d = 29.83 ch. E. 5. Given b = N. 33J W., c = 37 ch. ; required I and d. Ans. I === 30.94 ch. N., d = 20.29 ch. W. 6. Find the Z and d of the sides of a lot of which the following are the field notes: Commencing at the most westerly station, and running thence N. 52 E., 21.28 ch.; thence S. 29f E., 8.18 ch.; thence S. 31f W., 15.36 ch.; thence N. 61 W., 14.48 ch., to the point of beginning. PRELIMINARY CALCULATIONS. The work is written thus: 291 Sta. Bearings. Dist, tf. a*. ,9. Lat. . Dep. W.Dep. 1 N. 52 E. 21.28 13.10 16.77 2 S. 29f E. 8.18 7.11 4.06 r> I S. 31fW. N. 61 W. 15.36 14.48 7.02 13.06 8.08 12.67 292. Balancing the Work. It is evident that in passing around a field to the point of beginning, we have gone just as far north as south, and just as far east as west. Hence, the sum of the northings should be equal to the sum of the southings, and the sum of the eastings to the sum of the westings. In practice, however, this is seldom the case, owing to the fact that the bearings are taken only to quarter degrees, and that the chaining is not perfectly correct. It is not a settled point among surveyors how great an error in latitude or departure can be allowed with- out resurveying the lot. Some would admit an error of 1 link for every 10 chains in the sum of the courses; others, 1 link for every 3 chains. Each surveyor must settle this point for himself by ascertaining, by expe- rience, how nearly he can make his work balance. When an error is as likely to occur in one course as in another, the errors of latitude and departure are distributed among the courses in proportion to their length. It will not, in general, be necessary to make all the proportions, for after making one for latitude and one for departure, the remaining corrections can be made by a comparison of distances. 292 SURVEYING. Let us take example 6 of the last article. StoJ Bearings. DM. SLat. SLat. EDep. }\T)<p. CIV/,. CSX. CED. CWD. \ 1 j N.52E. 21.28 13.10 16.77 13.12 16.74 2 j S.29JE. 8.18 7.11 4.06 7.10 4.05 3 S.SlfW. 15.36 13.06 8.08 13.05 8.1C 4 JN.61\V. 14.48 7.02 12.67 7.03 12.69 59.30 20.12 20.17 20.83 20.75 20.15 20.15 120.79 20.79 Error in Lat. = Error in Dep. = Corrections for latitude. 59.30 : 21.28 : : .05 : .02. 59.30 : 8.18 :: .05 : .01. 59.30 : 15.36 : : .05 : .01. 59.30 : 14.48 : : .05 : .01. 20.17 20.12 = .05. 20.83 20.75 = . 08. Corrections for Departure. 59.30 : 21.28 : : .08 : .03. 59.30 : 8.18 :: .08 : .01. 59.30 : 15.36 : : .08 : .02. 59.30 : 14.48 : : .08 : .02. The corrections are made to the nearest link or hundredth. Since the north latitude is too small, and the south latitude too great, add to each north latitude the corre- sponding correction, and subtract from the south lati- tude. In a similar manner correct the departure. If one side is much more difficult to measure than the remaining sides, it is to be presumed that the error occurred chiefly in measuring that side, and the correc- tions should be made accordingly. If, in taking one bearing, the object could not be distinctly seen, the error probably occurred in that bearing; then correct mainly in the latitude and de- parture of that course. In practice it will not be necessary to make addi- tional columns for the corrected latitude and departure, since they may be written in the same columns, over the others, with different colored ink. PRELIMINARY CALCULATIONS. 293 293. Examples. 1. Find the I and d, and balance the work from the following notes: 1st, N. 34i E., 8.19 ch.; 2d, N. 85 E., 3.84 ch.; 3d, S. 56f E., 6.60 ch.; 4th, S. 34J W., 10.59 ch.; 5th, i\. 56 W., 9.60 ch. 2. Find the I and <i, and balance the work from the following notes : 1st, N. 5 E., 22.50 ch.; 2d, S. 83 E., 12.96 ch.; 3d, N. 50 E., 19.20 ch.; 4th, S. 32 E., 32.76 ch.; 5th, S. 41 W., 12.60 ch.; 6th, W., 16.86 ch.; 7th, N. 79 W., 21.84 ch. 3. Find the balanced I and d of the following: 1st, N. 30 E., 10 ch.; 2d, N. 60 E., 18.18 ch. ; 3d, S. 40 E., 20.10 ch.; 4th, S. 30 W., 24.50 ch. ; 5th, W, 15 ch. ; 6th, N. 18| W., 19.92 ch. 294. Double Meridian Distance. The double meridian distance of a course is double the distance of its middle point from a given meridian. Let AB be a given course, NS the given meridian, P the middle point of AB, PQ perpendicular to NS. Then will 2 QP be the double meridian Q " distance of AB. In the following illustration we shall as- sume that the meridian of reference passes through the most westerly station, which we shall call the principal station, that departures east are plus, and west, minus, that the lines were -run in the direction 294 SURVEYING. ABCD, so as to keep the field on , B E the right. The following relations can be A verified from the diagram: T Q ...... 1. 3. 2VU=2TR + FC-}- ( GD). v 4. 2XW=2VU+(GD) > x _H. 1. TTie double meridian distance of the first course is equal to its departure. 2. The double meridian distance of the second course is equal to the double meridian distance of the first course, plus the departure of the first course, plus the departure of the second course. 3. The double meridian distance of any course is equal to the double meridian distance of the preceding course, plus the departure of that course, plus the departure of the given course. 4. The double meridian distance of the last course is equal to its departure with its sign changed. Take the example of a preceding article, as balanced. Sta. Bearings. DM. NLat. SLat. EDep. WDep. DMD. 1 N.52E. 21.28 13.12 16.74 16.74 2 S. 29| E. 8.18 7.10 4.05 37.53 3 S.31fW. 15.36 13.05 8.10 33.48 4 N.61W. 14.48 7.03 12.69 12.69 Dep. of 1st course -f dep. of 1st course = -{- dep. of 2d course = 16.74 = D.M.D. of 1st course. 16.74 4.05 37.53 = D.M.D. of 2d course. PRELIMINARY CALCULATIONS. 295 -f dep. of 2d course & 4.05 41.58 -j- dep. of 3d course = - - 8.10 33.48 = D.M.D. of 3d course. -f dep. of 3d course = 8.10 25.38 -f dep. of 4th course == 12.69 12.69 = D.M.D. of 4th course. The principal or most westerly station is not always the first station in the field notes. It will be observed that the word plus, in the above principles and illustrations, is used in the algebraic sense, that east departure is considered plus and west departure minus] that plus, an east departure, is a plus quantity, and plus a west departure a minus quantity ; and that the double meridian distance of the last course is equal to its departure with its sign changed, which will serve as a verification of the work. The first station of the notes, in the preceding ex- ample, is the most westerly, and was therefore taken for the principal station. The most westerly station can readily be determined by inspecting the bearings of the courses as given in the field notes, and should be taken as the principal station, and the corresponding course as the first course in finding the double meridian distances. 295. Examples. 1. Given the following field notes : 1st, N. 30 E., 10 ch.; 2d, N. 60 E., 18.18 ch. ; 3d, S. 40 E., 20.10 ch,; 4th, S. 30 W., 24.50 ch. ; 5th, W., 15 ch. ; 6th, N. 18 45' W., 19.92 ch. : Required the 296 SURVEYING. latitude and departure ; balance the work, and find the double meridian distances. 2. Given the following field notes : 1st, N. 45 W., 20 ch.; 2d, N. 18 E., 12.25 ch. ; 3d, E., 12.80 ch.; 4th, N. 32 E., 6.50 ch. ; 5th, S. 42J E., 13.20 ch.; 6th, S., 14.75 ch.; 7th, S. 65J W., 16.30 ch. : Required the corrected latitude and departure, and the double meridian distances. AREA OF LAND. 296. Table of Linear Measure. Mi. Ch. Eds. 1 = 80= 320 = 1 = 4 = I - Yds. Ft. Lks. 7n. 1760 = 5280 = 8000 = 63360. 22 == 66 100 = 792. 5| 16^ r= 25 198. 1 = 3~ 4 T 6 , r = 36. 1 : 14": O i I = 12. 1 7| 297. Table of Superficial Measure. Mile. Acres. Roods. Chains. Perches. Links. I 640 = 2560 -. 6400 102400 = 64000000. 1 4 - 10 : 160 = 100000. 1 = 2J = 40 = 25000. 1 = 16= 10000. 1 = 625. Note 1. It should oe remembered that in finding the area of a tract of land the inequalities of its surface are not considered, but the tract is treated as a horizontal plane. AREA OF LAND. 297 Note 2. The area of a portion of land can, in a great variety of cases, be calculated by the rules already given for Mensuration of Plane Surfaces. 298. Problem. To find the area of a tract of land when the length and direction of the bounding lines are given. It is evident from the diagram that the area of A BCD is equal to the sum of the trapezoids EBCY and YCDH, minus the sum of the triangles AEB and ADH ; and that twice the sum of the trape- zoids, minus twice the sum of the triangles, is equal to twice ABCD. The following table will exhibit the general form of operation : Sta. Cour. NLat. SLat. DMD. Triangles. Trapezoids. 1 2 3 4 AB BC CD DA AE HA EY YH 2QP 2TR 2VU <2XW 2QPXAE 2XWXHA 2TRXEX 2VUXXH i It will be observed that we have taken the most westerly station for the principal station, and have multiplied the double meridian distance of each course by its latitude, and that the product is double the area of a triangle when the latitude is north, and double the area of a trapezoid when the latitude is south. 298 SURVEYING. If we had taken the most easterly station for the principal station, the reverse would be true. In the above we have supposed that the lines were run in such direction. as to keep the lot at the right. If the lines were run in the opposite direction, so as to keep the lot at the left, the reverse would be true. In any case, the sum of the double areas of the trape- zoids, minus the sum of the double areas of the tri- angles, is equal to double the area required. 299. Rule. Multiply the double meridian distance of each course by its latitude, placing the product in one column when the latitude is north, and in another column when the latitude is south, and divide the difference of the sums of the tivo columns by 2, and the quotient will be the area required. Take the example of a preceding article whose D. M. DSs have been found. Sta. Bearing*. Dint. NLat. SLat. EDep. WDep. DMD. Triany. Trait. 1 N.52E. 21.28 13.12 16.74 16.74 219.6288 2 S.29fE. 8.18 7.10 4.05 37.53 266.4630 3 S.312W. 15.36 13.05 8.10 33.48 436.9140 4 N.61W. 14.48 7.03 12.69 12.69 89.2107 Area = 19 A. 2 R. 36 P. Triangles. Trapezoids. 16.74X13.12 = 219.6288. 37.53 X 7.10 = 266.4630. 12.69 X 7.03= 89.2107. 33.48X13.05 = 436.9140. Divide double the area by 2, the result by 10 to reduce the chains to acres, multi- ply the decimal by 4 to reduce to roods, and the next decimal by 40 to reduce to perches. 308.8395 703.3770 308.8395 2)394.5375 10)197.26875 19.726875 4 2.907500 40 36.300000 AREA OF LAND. 299 300. Plotting. Plotting is the process of representing, to a given scale, the length, direction, and relative position of the bounding lines of a tract of land. 1st Method. By means of latitudes and departures, Take the example of the last article. Let NS represent the meridian passing through the principal station A. Select a scale whose unit shall represent 1 ch., and take AE = 13.12 ch., the lat. of first course. Through E draw a line perpen- dicular to NS; take EB = 16.74 ch., the dep. of first course, and draw AB. Through B draw a meridian, and take BF = 7.10, the lat. of second course. Through F draw a line perpendicular to BF; take FC = 4.05 ch., the dep. of second course, and draw BC. Through C draw a meridian, and take CG = 13.05, the lat. of third course. i Through G draw a line perpendicular to CG, and take GD = 8.10 ch., the dep. of third course, and draw CD. Through D draw a meridian, and take DI = 7.03 ch., the lat. of fourth course. Through / draw a line perpendicular to DI; take IA - 12.69 ch., the dep. of fourth course, and draw DA. Remark 1. If the departure of fourth course termi- nates at A, the w*ork will be verified. 300 SURVEYING. 2. It will be observed that N. lat. is laid off upward, S. lat. downward, E. dep. to the right, and W. dep. to the left. 3. The auxiliary lines can be drawn with a pencil and afterward erased. 4. If every scale in possession of the surveyor should make the diagram too large or too small, all the lati- tudes and departures can be divided or multiplied by the same number, and the results taken instead of the given latitudes and departures. 2d Method. By means of bearings and distances. Take the same example. Let NS represent the meridian passing through the principal station A. With a protractor lay off the angle NAB = 52, the bearing of first course, and take AB = 21.28 ch., the first course. Through B draw a meridian, and lay off &'BC-29%, the bearing of second course, and take BC = 8.18 ch., the second course. Through C draw a meridian, and lay off S"CD= 31f, the bearing of third course, and take CD = 15.36 ch., the third course. Through D draw a meridian, and lay off N'DA = 61, the bearing of fourth course, and take DA == 14.48 ch., the fourth course, which will terminate at A if the work is correct. Remark 1. The latitude and departure letters indicate the general direction of the lines, and the degrees the exact direction. AREA OF LAND. 301 2. Let the examples of the following article be care- fully plotted, and the area be found. 3. By a careful inspection of the bearings, the most westerly station can be found, which take for the principal station. 4. The distances are all given in chains. 1. 301. Examples. Sta. Bearings. Dist. 1 N. 30 E. 10. 2 N. 60 E. 18.18 3 8. 40 E. 20.10 4 S. 30 W. 24.50 5 W. 15. 6 N. 18| W. 19.92 2. Sta. Bearings. Dist. 1 N. 47 E. 15.65 2 S. 57 E. 10.55 3 S. 28|W. 17.67 4 S. 29JW. 1.11 5 S. 54 W. 1.04 6 N. 40JW. 15.90 Ans. 80 A. 1 R. 25 P. Ans. 23 A. R. 38 P. 3. 4, Sta. Bearings. Dist. 1 N. 45 W. 20. 2 N. 18 E. 12.25 3 E. 12.80 4 N. 32 E. 6.50 5 S. 42iE. 13.20 6 S. 14.75 7 S. 65} W. 16.30 Sta. Bearings. Dist. 1 N. 58 E. 12.97 2 S. 27fE. 3.30 3 S. 85JE. 11.65 4 S. 19 E. 15.56 5 S. 66iW. 14.03 6 N. 64 W. 14.86 7 N. 15iW. 11.23 Ans. 58 A. 3 R. 30 P. Ans. 45 A. 2 R. 5 P. 302 SURVEYING. 5. Sta. Bearings. Dist. 1 N. 20 E. 12.20 2 N. 70 E. 15.50 3 E. 18.25 4 S. 45 E. 20.00 5 S. 20.00 6 S. 45 W. 20.00 7 W. 18.25 8 N. 30|W. 36.66 Sta. Bearings. Dist, 1 S. 34 E. 4.56 2 S. 66JW. 13.84 3 N. 12JE. 12.15 4 N. 48J W. 12.30 5 N. 58fE. 9.92 6 N. 39iE. 5.22 7 S. 45iE. 18.63 8 S. 52iW. 10.76 An*. 188 A. 3 R. 20 P. Ans. 32 A. 2 R. 26 P. 7. 8. Stfi. Bearings. Dist. 1 N. 30 E. 15. 2 N. 60 E. 15. 3 E. 15. 4 S. 60 E. 15. 5 S. 30 E. 15. 6 S. 15. 7 S. 30 W. 15. 8 S. 60 W. 15. 9 W 15. 10 N. 60 W. 15. 11 N. 30 W. 15. 12 N. 15. Sta. Bearings. Dist, 1 S. 76JE. 6.69 2 S. 14JW. 5.96 3 S. 38 E. 9.82 4 N. 30JE. 8.63 5 S. 73iE. 9.43 6 S. !QfW. 15.70 7 S. 42JW. 13.06 8 N. 64 W. 11.93 9 S. 79JW. 10.45 10 N. 22JW. 11.60 11 N. 37JE. 14.37 12 N. 22f E. 10.79 Ans. 251.9 A.+ Ans. 76.14 A. AREA OF LAND. 303 302. Problem. To find the area when offsets are taken. Find the area of the tract of land bounded by the full lines and middle of the river, as shown in the annexed diagram. Having run the sta- tionary line CD, we have the following notes. For ABODE. For Offsets. Sta. Bearings. Dist. 1 N. 20 E. 15.50 2 E. 18.00 3 S. 20 E. 30.00 4 W. 25.00 5 N. 32JW. 16.09 Sta. Dist. Offsets. 1 0.00 2.50 2 7.00 6.00 3 12.20 4.00 4 22.25 7.00 5 30.00 2.55 Area = 70 A. 1 R. 33 P. +14 A. 3 R. 8 P. = 85 A. 1 R. 1 P. We find, as in the last article, ABODE = 70 A. 1 R. 33 P. To calculate the area included between the stationary line CD and the line passing along the middle of the river, we find Ca = 7, ab == Ob Oa == 12.20 7 = 5.20, etc., which gives the altitudes of the trapezoids. The parallel sides are given under the head of offsets. The altitude of a trapezoid multiplied by the sum of the parallel sides will give twice its area. The calculation is made as in the subjoined table, the letters, S., S. D., 0., I. D., S. 0., D. T., heading the 304 SURVEYING. columns of the table, denoting stations, station dis- tances or distances from (7, offsets, intercepted distances, sum of offsets, and double trapezoids. & S. D. 0. /. D. S.O. D. T. 1 0.00 2.50 2 7.00 6.00 7.00 8.50 59.5000 3 12.20 4.00 5.20 10.00 52.0000 4 22.25 7.00 10.05 11.00 110.5500 5 30.00 2.55 7.75 9.55 74.0125 Area, 14 A. 3 R. 8 P. 2)296.0625 10) 148.03125 14.803125 4 3.212500 40 8.500000 If the offsets fall within the stationary line, the sum of the trapezoids must be subtracted. In general, if the lines are run so as to keep the field on the right, the sum of the trapezoids must be added in case of left-hand offsets, and subtracted in case of right-hand offsets. In case of -navigable rivers, the bank is, in general, the boundary the first and last offsets become 0, and the first and last trapezoids become triangles, but the form of the computation is the same. 303. Examples. 1. Find the area of the lot of which the following are the field notes, and make a plot of the survey. AREA OF LAND. 305 Rectilinear Area. L.H. Offsets* R.H. Offsets.** & Bearings. Dist. St.Dist. Offsets. St.Dist. Offsets. 1 N. 45 E. 10.00 0.00 1.00 0.00 1.10 2 N. 10.00 6,50 4.25 5.62 4.00 3 N. 45 E. 10.00: 12.50 2.43 12.62 5.27 4 E. 10.00 17,50 5.17 17.07 1.13 5* S. 31.21 26.21 5.83 o** w. 17.07 31.21 1.25 7 N. 45 W. 10.00 55.774715 A. + 12.17075 A. 6.10160 A.= 61 A. 3 R. 15 P. The left-hand offsets were made from the fifth course, as indicated by the single star; and the right-hand offsets from the sixth course, as indicated by the double star. 2. Find the area of the lot of which the following are the field notes, and make a plot of the survey. Rectilinear Area. L.H. Offsets* R.H. Offsets** Sta, Bearings. Dist. St.Dist. Offsets. St.Dist. Offsets. 1 N. 30 E. 20. 0.00 0.00 0.00 0.00 2 E. 20. 6.00 3.00 6.00 4.00 3* S. 30 E. 20. 10.00 2.00 14.00 4.00 i 4** S. 30 W. 20. 15.00 3,50 20.00 0.00 5 W. 20. 20.00 0.00 6 N. 30 W. 20. Ans. 102 A. 1 R. 30 P. S. N. 26. 306 SURVEYING. 304. Pogue's Method of Finding the Area, This method is illustrated by the following example : 1. N. 20 2. N. 43 3. S. 70 4. S. 40 5. S. 65 6. S. 42 7. S, 8. S. 70 9. N. 36}: E., 24.50 ch. E., 22.40 ch. E., 25.50 ch. W., 16.58 ch. E., 25.10 ch. W., 13.50 ch. 14.20 ch. W., 32.15 ch. W., 34.55 ch. Make a plot from the field notes, draw meridians through the most easterly and westerly stations, and parallels of latitude through the most northerly and southerly, thus enclosing the whole figure in a rect- angle. Find, from the traverse table, the latitudes and de- partures as in diagram. To find xy, pass from the most westerly station, round the north, to the most easterly, taking the sum of the eastings minus the sum of the westings; and to find zw, pass from the most easterly station, round the south, to the most westerly, taking the sum of the westings minus the sum of the eastings, thus : xy ='8.38 + 15.27 -f- 23.96 10.66 + 22.75 == 59.70 zw = 9.03 + 30.21 + 20.44 == 59.68 2)119.38 i (py + zw ) the average base = 59.69 To find wx, pass from the most southerly station, round the west, to the most northerly, taking the sum of the northings minus the sum of the southings ; and to find AREA OF LAND. 307 2/z, pass from the most northerly station, round the east, to the most southerly, taking the sum of the southings minus the sum of the northings, thus : WB= 27.86+23.02+ 16.38 = 67.26 yz = 8.72+12.70+10.60+10.03+14.20+10.99 = 67.24 2)134.50" (wx + yz) = the average altitude = 67.25 Area of rectangle = 59.69 X 67.25 = 4014.1525. From the area of the rectangle we must deduct the area included between wxyz and abcdefghi, thus found. 8.72 (W = 1170X1066 = kyml --= (8.72 + 12.70) (22.75 10.66) =-- 258.9678 hnzi = 9.03+ 9.08 + 30.21 .^3X44X27.86 = ^ 1633.9631 abcdefghi = 4014.1525 sq. ch. 1633.9631 sq. ch. = 2380.1894 sq. ch. = 238.02 A. For additional exercises, work the examples of arti- cles 301 and 303, and compare the answers obtained by the two methods. 308 SURVEYING. SUPPLYING OMISSIONS. 305. Case I. When the bearing and length of one side are wanting. The wanting side must be such that its latitude and departure will make the work balance. Hence, its lati- tude must be the difference between the sum of the northings and the sum of the southings of the given sides, and of the same name as the less ; and its- de- parture must be the difference between the sum of the eastings and the sum of the westings of the given sides, and of the same name as the less. Having found the latitude and departure of the wanting side, construct a right-angle triangle by draw- ing on the paper, to represent the latitude, a line, up or down, according as the latitude is north or south ; and at the terminus of the line, draw, to represent the departure, a horizontal line, to the right or left, accord- ing as the departure is east or west, and join the ori- gin of the line representing the latitude with the ter- minus of the line representing the departure, and this last line will be the hypotenuse which will represent the course or length of the line sought, and the angle which it makes with the vertical line will be the bearing. Denote the latitude by I, the departure by d, the course by e, and the bearing by 6, then we have, Having found the bearing and distance, enter them in the notes and find the area. SUPPLYING OMISSIONS. 309 306. Examples. Supply the omissions in the following field notes, calculate the areas, and plot the surveys. 1. Sta. Bearings. Di*t. I N. 18 E. 9.25 2 N. 71 E. 8.33 3 S. 43JE. 12.37 4 S. 36JW. 16.00 5 Wanting. Want'g. N. 43 W., 14.18 ch. 23 A. 3 R. 32 P. 2. Sta. . Bearings. Dist. 1 N. 24 W. 15.50 2 N. 31 E. 17.07 3 E. 20. 4 5 Wanting. S. 56 W. Want'g. 30.30 f S. 12iE., 12.13 eh, Ans - \ 56 A. 3 R. P. 307. Case II. When the lengths of two sides are wanting. Revolve the field so that one of the sides whose bearing only is given shall become a meridian, and find, by article 285, the bearings of all the sides in their new position. The departure of the side made a meridian will then be 0, and the difference of the sums of the columns of the departures will be the departure, in the new position, of the other side whose distance is wanting. Knowing the bearing and departure of this side, we can find its distance and latitude. Then the differ- ence between the sums of the columns of latitudes will be the length of the side made a meridian. Revolve the field to its original position, calculate its area, and make a plot of it; or, if the area only 310 SURVEYING. is required after supplying omissions, it may be com- puted more readily without revolving the field to its original, position. 308. Examples. 1. 2. Sta. Bearings. Dist. 1 N. 30 E. 10.00 2 N.60 E. 18.18 3 S. 40 E. Want'g. 4 S. 30 W. Want'g. 5 W. 15.00 6 N. 18|W. - 19.92 Sta. Bearings. Dist. 1 N. 47 E. 15.65 2 S. 57 E. 10.55 3 S. 28JW. Want'g. 4 S. 29JW. 1.11 5 S. 54 W. 1.04 6 N. 40JW. Want'g. r3d. 20.08 ch. Ans.< 4th. 24.52 ch. ISO A. 1 R. 25 P. r3d. 17.69 ch. Aw. < 6th. 16.01 ch. 123 A. 1 R. 14 P. 309. Case III. When the bearings of two sides are loanting. If the sides whose bearings are wanting are separated from each other by one or more intervening sides, sup- pose one of these sides and a side adjacent to the other to change places, so as to bring the sides under con- sideration together without changing the bearings or lengths of the sides transposed. Then, throwing these sides out of consideration, find, by Case I, the bearing and length of the line joining the extremities of the sides whose bearings are wanting. This line with those sides form a triangle whose sides are known, from which the angles can be computed. Knowing the angles and the bearing of one side, the bearings of the other sides can be found. SUPPLYING OMISSTOSS. 311 Restore to their original position the sides which have changed places, if such is the fact, calculate the area, and make a plot of the field. 310. Examples. 1. 2. Sta. Bearings. Dist, 1 N. 45 W. 20.00 2 N. 18 E. 12.25 3 E. 12.80 4 N. 32 E. 6.50 5 S. 42JE. 13.20 6 Wanting. 14.75 7 Wanting. 16.30 Sta. Bearings. Dist. 1 N.58 E. 12.97 2 S. 27|E. 3.30 3 S. 85JE. 11.65 4 S. 19 E. 15.56 5 Wanting. 14.03 6 N. 64 W. 14.86 7 Wanting. 11.23 r6th. S. Ans. < 7th. S. 65J< 1 59 A. W. ( 5th. S. 66J W. . < 7th. N. 15J W. U5 A. 2 R. 5 P. 311. Case IV. Wlien the bearing of one side and the length of another are wanting. Revolve the field so that the side whose bearing only is given shall become a meridian. The departure of this side will then be 0, and the difference of the sums of the columns of departures will be the departure, in its new position, of the side whose bearing is wanting. Knowing the length and departure of this side, its bearing and latitude can be found. Then the difference of the sums of the columns of lati- tudes will be the length of the side made a meridian. . Revolve the field to its original position, compute the area and plot the work. 312 SURVEYING. Remark 1. In finding the bearing of the side whose distance only is given, though the angle can be readily found, the bearing, and consequently the latitude, may be either north or south, since either will comply with the condition. The length of the side whose bearing only is given will therefore be ambiguous, and there will be two solutions to the problem. If but one solution is admissible, the omission should be supplied by a remeasurement ; and if the lost bearing or dis- tance can not be taken directly, auxiliary lines may be run, and the omissions supplied by Trigonometry. 2. From the fact that two omissions can be supplied, the surveyor should not deem it unimportant to find all the measurements on the ground, since thus he can ascertain the correctness of his notes by balan- cing his work a test not applicable when omissions are supplied. 312. Examples. 1. Sta. Bearings. Dist. 1 N. 20 E. 12.20 2 N. 70 E. 15.50 3 E. 18.25 4 S. 45 E. 20.00 5 S. 20.00 6 Wanting. 20.00 7 W. Want'g, 8 N. 30| W. 36.66 Sta. Bearings. Dist. 1 S. 34 E. 4.56 2 S. 66|W. 13.84 3 N. 12| E. 12.15 4 Wanting. 12.30 5 N. 58} E. 9.92 6 N. 39JE. 5.22 l-r I S. 45JE. Want'g. 8 S. 52iW. 10.76 r6th. S. 45 W. Ans. < 7th. 18.25. 1 188 A. 3 R. 20 P. r 4th. N. 48J W. Ans. < 7th. 18.63. 132 A. 2 R. 26 P. LA YING UT LAND. 313 LAYING OUT LAND. 313. Laying out Squares. To lay out a given quantity of land in the form of a square. Let a be the area of the square, and x one side. Then, x 2 = a, . . x = \/~a~ Reduce the given area to square chains, extract the square root, and the result will be the length of one side. With the chain and transit lay out the square on the ground. EXAMPLES. 1. Lay out 12 A. 3 R. 20 P. in the form of a square. 2. Find the side of a square containing 1 A., and lay out the square on the ground. 314. Laying out Rectangles. 1. To lay out a given quantity of land in the form of a rectangle, one side of which is given. Let a be the area of the rectangle, 6 the given side, and x an adjacent side. Then, bx = a, . . x = -r- 2. To lay out a given quantity of land in the form of a rectangle whose length is to its breadth in a given ratio. Let a denote the area of the rectangle, x its length, y its breadth, and m : n the ratio of x to ?/. 1_ lam \1T' y^ f^-- m S. N. 27. 314 SURVEYING. 3. To lay out a given quantity of land in the form of a rectangle when the sum of its length and breadth is given. Let a be the area of the rectangle, x the length, y the breadth, and .9 the sum of x and y. = a. 4. To lay out a given quantity of land in the form of a rect- angle when the difference of the length and breadth is given. Let a denote the area of the rectangle, x its length, y its breadth, and d the difference of x and y. 315. Examples. 1. The area of a rectangle is 3 A., one side is 4 ch. Find an adjacent side and lay out the rectangle. 2. The area of a rectangle is 8 A.; the length is to the breadth as 3 is to 2. Find the sides and lay out the rectangle. Ans. 10.95 ch. and 7.30 ch. 3. The area of a rectangle is 4.8 A. ; the sum of the length and breadth is 14 ch. Find the sides and lay out the rectangle. Ans. 8 ch. and 6 ch. 4. The area of a rectangle is 18 A. ; the difference of the length and breadth is 3 ch. Find the sides and lay out the rectangle. Ans. 15 ch. and 12 ch. 316. Laying out Parallelograms. 1. To lay out a given quantity of land in the form of a parallelogram when the base is given. LAYING OUT LAND. 315 Let a be the area, b the base, and x the altitude. Then bx a. . . x = -=- o Measure the base, from any point of which erect a perpendicular equal to the calculated altitude. Through the extremity of the perpendicular run a line parallel to the base, any point of which may be taken for one extremity of the upper base, which may then be measured off' on this line. 2. When one side and an adjacent angle are given. Let a be the area, b the given side, A the given angle, and x the other side adjacent to this angle. Then bx sin A = a, . . x = 7- - - - - b sin A 3. When two adjacent sides are given. Let a be the area, b and c the given sides, and x their included angle. Then be sin x = a, . . sin x ^= -j be Remark. If be = a, then sin x = 1, x = 90, and the parallelogram becomes a rectangle. If be < a, the solution is impossible. 317. Examples. 1. The area of a parallelogram is 6 A., the base is 6 ch. Find the altitude and lay out the land. 2. The area of a parallelogram is 12 A., one side is 12 ch., and an adjacent angle is 60. Find the other side adjacent to the given angle and lay out the land. - '<- 316 SVEVEYISG. 3. The area of a parallelogram is 8 A., two adjacent sides are 8 ch. and 12 ch. Find their included angle and lay out the land. 318. Laying out Triangles. 1. To lay out a given quantity of land in the form of a triangle when the base is given. Let a denote the area, b the base, and x the altitude. Then, \ bx = a, . ' . x = -y- Measure the base, at any point of which erect a per- pendicular equal to the calculated altitude. Through the extremity of this perpendicular draw a line parallel to the base. This parallel will be the locus of the vertex, any point of which may be taken for the vertex. 2. When the base is to the altitude in a given ratio. Let a denote the area, x the base, y the altitude, and m : n the ratio of the base to the altitude. / 2 am x : y :: m : n. J2 an r \"^r 3. When the triangle is equilateral. Let a denote the area and x one side. Then, .4330127 * = a, . x = 4. When one side and an adjacent angle are given. Let a denote the area, b the given side, x the adja- cent side, and A the included angle. Then, bx sin A = a. . . x = = r. - b sin A L A YING O UT LAND, 317 5. When two sides arc given. Let a denote the area, b and c the given sides, and x their included angle. 2 a 1 hen, f oc sin x = a, . . sin x = - 319. Examples. 1. The area of a triangle is 3 A., the .base is 5 ch. Find the altitude and lay out the triangle on the ground. 2. The area of a triangle is 12 A., the base is to the altitude as 3 is to 2. Find the base and altitude and lay out the triangle on the ground. 3. The area of an equilateral triangle is 1 A. Find a side and lay out the triangle. 4. The area of a triangle is 1.2 A., one side is 2 ch., an adjacent angle is 45. Find the other side adjacent to the given angle and lay out the land. 5. The area of a triangle is 2 A., two sides are 6 ch. and 10 ch. Find the included angle and lay out the triangle. _ 320. Laying out Circles or Regular Polygons. 1. Let a be the area of the circle, and x the radius. Then, 3.1416 *t = a, .-. x = 2. Let a be the area of a regular polygon, x one side, y one angle, n the number of sides, and a' the area of a similar polygon whose side -is 1. Article 167. [ 180 (n 2) Then, a x 2 a, .*. x = \ -> y = - 318 SURVEYING. 321. Examples. 1. Find the radius of a circle whose area is 1 A. and lay out the circle. 2. Find the sides and angles of a regular hexagon containing 1 A. and lay out the hexagon. 3. Find the sides and angles of a regular octagon containing 1 A. and lay out the octagon. DIVIDING LAND. 322. Division of Rectangles or Parallelograms. 1 . To cut off a given area by a line parallel to a given side. Let a be the area, b the given side, x the distance to be cut off on the sides adjacent to ft, and A the acute angle of the parallelogram. For the rectangle, bx = a, .' . x = For the parallelogram, foe sin A = a, ,*. % = j -; 2. When the lot is to be divided into parts having a given ratio, by lines parallel to two of the sides, divide the other sides into parts having the same ratio. 323. Examples. 1. The sides of a rectangle are 15 ch. and 10 ch.; cut off 8 A. by a line parallel to the shorter sides. 2. The adjacent sides of a parallelogram >are 12 ch. and 20 ch., and their included angle is 65; cut off 10 A. by a line parallel to the shorter sides. 3. A man willed that his farm, which was 1 mile long and J mile wide, be divided among his three DIVIDING LAND. 319 sons, A, B, and C, aged 21 yrs.. 18 yrs.. 15 yrs., respect- ively, in proportion to their ages, by lines parallel to the shorter sides. Make the divisions. 324. Division of Triangles. 1. To find a point on a given side of a triangle from which a line drawn to the vertex of the opposite angle will 1 divide the triangle into parts having a given ratio. Let 6 = AC, the given side; D, the required point; x = AD. and ABD : DEC : : m : n. A By composition we have, ABC : ABD : : m -f n : m; but ABC : ABD : : b : x. bm Hence, m -\- n : m : : b : a-, . . x = m -f- n 2. Two sides of a triangle being given, to divide the triangle into parts having a given ratio by a line parallel to the third side. o Let a = BC, b '== AC, the given sides; s and DEC : ABED : : m : n. By composition we have, ABC : DEC :: m+n: m; but ABC : DEC : : a 2 : x 2 : : b 2 : y 2 . m 4- n : m :: a 2 : x 2 . If, for example, the triangle is to be divided into three equal parts by lines parallel to the third side, then, 320 SURVEYING. The distances cut off on a are a ] ^ a The distances cut off on b are b i/ b 3. Two cStV/es o/ a triangle being gire,n, to cut off, by a line intersecting the given sides, an isosceles triangle having a given ratio to the given triangle. Let b = AC, c ~ = AB, the two given sides; x = AE == AD, and ADE : ABC : : m : n. But, ^.E : ABC :: x 2 : be. Hence, m : n : : x 2 : be, . , n 4. Two sides o/ a triangle being given, to cut off a triangle having a given ratio to the given triangle by a line running from a given point in one of the given sides to the other given side. Let b = AC, c AB, the given sides; D, the given point; d=AD, x = AE, and AED : ABC :: m : n. But, AED : ABC : : dx : be. Hence, m : n : : dx : be, .'. x = , dn 5. The three sides being given, to divide the triangle into three equal parts by lines running from a g^ven point in one of the sides. Let a, b, c be the sides of the triangle, respectively, opposite the angles A, B, C; p r= AD, q = CD, x = AE, and y == CF. , ,- bc 3 : 1 : : be : px. ' I 3 : 1 : : ab : qv. J ab * I n t . \ Then " . ) DIVIDING LAND. 321 If x, thus found, is greater than c, both lines will intersect a. Then find 'y as above. Let x = CE. Then, 3 : 2 : : aft : qx, . ' . x = ~ If 27, found above, is greater than a, both lines will intersect c. Then find x as in first case. Let AF = y. Then, 3 : 2 : : be : py, .-. y = - 6p 6. To divide a, triangle into four equal triangles, join the middle points of the sides. The lines ED, EF, and DF are, respectively, parallel to J?(7, AC, and AB. EBF == EDF, since each is J the parallelogram ED. ADE = EDF, since each is J the parallelogram AF. CDF = EDF, since each is J the parallelogram CE. I^ence, the triangles are all equal, and each is ABC. 7. The bearing of two sides being given, to cut off a tri- angle having a given area by a line of a given bearing intersecting the sides whose bearings are given. Let ADE be the triangle cut off, a the area of ADE; x = AD and y = AE. The angles A, D, E can be determined from the bearings. ( \xy sin A = a. \ ' \ sin E : sin D : : x : y. ) ' ' j 2 a sin E x = rr\-\ j % M y " A " ** ** ii sin A sin u 2 a sin D _ $ \"si sin ^4 sin E 322 SURVEYING. 8. To divide a triangle into two equal parts by lines running from a point within. Let ABC be the given triangle, and P the given point. -A.'' Run a line from P to the vertex A, and another from P to D, the middle point of the opposite side BC. Run DE parallel to PA, and run PE. PD and PE will be the dividing lines, and CDPE will be \ ABC. For, draw the line AD, then we have, CDPE = CDE + PED, and ACD = ODE + AED. But PED = AED, . - . CDPE = ACD. .-. CDPE = 9. Through a given point, within a given tnangle, to draw a line which shall cut off a triangle having a given ratio to the given triangle. B Let ABC be the given triangle ; a, 6, c, the sides opposite the angles A, B, C, respectively; D the point given by knowing p = AF = ED, parallel to AC; q = A AE = FD, parallel to AB; x = AH, y = AG, and AGH : ABC :: m : n. Then, -v xy : be : : m : n. j bcm V b 2 c 2 m 2 4 bcmnpq y 2 bcmq bcm V b 2 c 2 m 2 4 bcmnpq Remark. If either x > b or y > c, the line cuts off the triangle from another angle; and the distances cut off from the vertex of this angle can be found in a manner similar to the above. DIVIDING LAND. 323 10. To find a point within a triangle from ivhich the lines drawn to the vertices will divide the triangles into three equal triangles. Let ABC be the triangle. Take AD = J AB; CE = J CB, and draw DE. Take BF^=% BA, CG = ^CA, and draw FG. P, the intersection of these lines, will be the point required. For AD : AB : : altitude of APC : altitude ABC. But AD = \AB, .- . altitude APC = $ altitude ABC. .-. APC = i ABC. In like manner, BPC = J ABC. Remark. If APC, BPC, and APE are to be to each other as p, q, r, take AD = P ^ of AB, CE = p -- i of CB, BF = -- - of 54, C6? = of OA, and draw D" and FG, their intersection will be the point required. Examples. 1. One side of a triangle is 15 ch.; from what point in this side must a line be drawn to the vertex of the opposite angle so as to divide the triangle into two triangles which are to each other as 2 to 3? Ans. 6 ch. from one extremity. 2. Two sides of a triangle are 10 ch. and 15 ch., respectively; find the distance from the vertex of the 324 SUEVEYJNO. angle included by these sides, cut off on each of these sides by a line parallel to the third side, dividing the triangle into a triangle and a trapezoid, so that the triangle cut off shall be to the trapezoid as 9 to 16. Ans. 6 ch. and 9 ch. 3. Two sides of a triangle are 4 ch. and 9 ch., re- spectively ; find the distance from the vertex cut off on each of these sides by a line cutting off an isos- celes triangle which shall be to the given triangle as 16 to 25. Ans. 4.80 ch. 4. Two sides of a triangle are 7 ch. and 9 ch., re- spectively. From a point in one side, 5 ch. from the vertex of the angle included by these sides, a line is run to the other given side, cutting off a triangle which is to the given triangle as 5 to 9. How far from the same vertex does this line intersect that side ? Ans. 7 ch. 5. The sides of a triangle, ABC, are a =- 6 ch., b = 12 ch., and c = 9 ch. From the middle point of 6 two lines are run, dividing the. triangle into three equal parts. To what points of what sides must the lines be run? Ans. To c, 6 ch. from A, and to <7, 4 ch. from C. 6. The sides of a triangle, ABC, are a = 10 ch., 6 = 12 ch., and c = 4 ch. From a point in fr, 3 ch. from A, two lines are run, dividing the triangle into three equal parts. To what points of what side must these lines be run? Ans. To a, 8.89 ch. from C, and to o, 4.44 ch. from C. 7. The sides of a triangle, ABC, are a = 5 ch., b = 18 ch., and c = 15 ch. From a point in 6, 12 ch. from A, two lines are run, dividing the triangle into three equal parts. To what points must these lines be run? Ans. To C, 7.50 ch. from A, and to B. DIVIDING LAND. 825 8. In the triangle ABC, the side AB runs N. 50 E., AC runs E. DE, running N. 10 W., intersects these lines in D and E, and cuts off ADE= 10 A. Required AD and AE. Ans. AD = 16.54, AE = 18.81. 9. In the 9th general problem of the last article, b = 10 ch., c = 12 ch., m = 1, n 4, jo 2 ch., q == 3 ch. Find a; and y. ^4ns. a; = 7.24 ch., y ^4.14 ch. 326. Division of Trapezoids. 1. Given the bases and a third side of a trapezoid, to divide it into parts having a given ratio by a line parallel to the bases. Let ABCD be the trapezoid, b=AB, G b' = CD, s = AD, x == AE, y = EF, the /\ dividing line, parallel to the bases, / \ and ABFE : EFCD : : m : n. Produce AD and BC to G. / ABG : DCG :: b 2 : V*. Then, . . . 2 . , 2 , J \ \ These proportions taken by division give, ABCD : DCG : : b 2 6' 2 : 6' 2 , EFCD : DCG : : y 2 ft' 2 : 6 /2 . Since the consequents are the same, we have, ABCD : EFCD : : 6 2 - 6' 2 : y 2 b' 2 . This proportion taken by division gives, ABFE : EFCD : : b 2 y* : y 2 b' 2 , But ABFE : EFCD : : 326 SURVEYING. Drawing DH parallel to BC, we have, AH : El : : AD : ED, or b b' : y V :: s : s x, .'. x = _^_ , (6 y). 771 -|- 71 2. Given a side and two adjacent angles of a tract of land, to cut off a trapezoid of a given area by a line parallel to the given side. 1st. When the sum of the two angles < 180. Let a = = ABCD = the area cut off, b = AB the given side, x = AD, y = BC, z = DC, E= 180 (A + B). (1) Area ABE= J EB X EA sin E. sin E : sin A : : b : EB, .'. EB= f 111 ^- sin E sin E : sin B : : b : EA, . . EA Sm r , sin L Substituting the values of EB and EA in (1), we have, b 2 sin A sin B (2) 2 sin E .'. (3) j >cg = r _ 2 sin But ^45^ : DCE : : 6 2 : z 2 . 6 2 sin A sin 6 2 sin A sin 2 sin E 2 sin sn sin ^4 sin B DIVIDING LAND. 327 Draw DF parallel to EB, then ADF= E and DFA = B. (b -z)sin sin E : sin B : : b z : x, . ' . x = ~ (b z) sin A In like manner we shall find y= . ., Dill -/-> Since z is known, x and y are known. 2d. When the sum of the two angles > 180. E and DC lie on opposite sides of ^ AB. Let a === .4C Y D === the area to be cut off, b = AB the given side, x = AD, y = BC, z = DC, E^A + 5 180. By a process similar to that em- ployed in first case, we find, 2 a sin E r sin A sin (g_ 6) sin B (z - 6) sin ^4 "sin E 3. To dm'de a trapezoid into proportional parts by a line joining the bases. Let A BCD be the trapezoid, 6 and b' the bases, a the altitude, m and n the ratio of the parts. p E Take ,4E= > then F = 328 SURVEYING. Then, AEFD = > , and JJ 5C F = tt " (6 + 6 '> 2 (m -f n) 2 (m -j- w) am (6 + 6') an (6 + 6') F c - 2 (m + n) 2 (m -\~ n) .-. AEFD : EBCF :: m : w. Remark. If the line is to be drawn from a given point P, in one base, first divide as^ above ; then, if P is on one side of ", take P' as far on the other side of* F, and draw PP'. This change in the dividing line does not affect the altitude of the parts, nor the sum of their bases, since one is increased as much as the other is diminished, nor, consequently, their area. A similar process can be employed whatever be the number of parts. 327. Examples. 1. A trapezoid whose bases are b = 15 ch. and b' = 12 ch., and third side s 10 ch., is divided by a line parallel to the bases into two parts, such that the part adjacent to b is to the part adjacent to b' as 3 to 2. Required the length of the dividing line, and the dis- tance from b cut off on s. Ans. 13.28 ch., and 5.73 ch. 2. Given a side 14.30 ch., and the two adjacent angles, 60 and 70, respectively, of a tract of land from which 10 A. are to be cut off by a line parallel to the given side. Required the length of the dividing line, and the respective distances from the given side cut off on the adjacent sides. Ans. 4.05 ch., 12.60 ch., and 11.61 ch. 3. Given a side 10 ch., and the two adjacent angles, 120 and 115, respectively, of a tract of land, from which 15 A. are to be cut off by a line parallel to the DIVIDING LAND. 329 given side. Required the length of the dividing line, and the respective distances from the given side cut off on the adjacent sides. Ana. 20.32 ch., 11.42 ch., 10.91 ch. 4. A trapezoid whose parallel sides are AB = 14 ch., and DC =- 7 ch., is divided by the line PP' into two parts which are to each other as 3 to 4 ; AP = 4 ch., find DP'. Am. 5 ch. 328. Division of Trapeziums. 1. Given a side, two adjacent angles, and the area of a trapezium, to divide it, by a line parallel to the given side, into parts having a given ratio. Let ABCD be the trapezium ; b = AB, the given side; A and B, the given angles; G == 180 (A + B), a = the area of ABCD, x = AE, y = BF, and ABFE : EFCD : : m : n. m -J- n m-\- n ABG = %BGX AGX sin G. D ~ 6 sin yl ft sin 7? .B(r = . ^ and AG = . 77-- sin G sin G 6 2 sin A sin B 2 sin G b 2 sin A sin I? ma iLf Or = ~ ;; "^ ~ 2 sin (r m -(- w JB6r : EFG :: AG 2 : 'EG 2 , ABG : EFG :: ^G 2 : TG 2 . Substituting, in the proportions, the values of ABG, EFG, AG and BG, find EG and FG, and substituting the values of AG, EG, BG and FG in the equations, x == ^G -- EG and y = BG FG, we have, S. N. 28. 330 SURVEYING. b sin B Ib 2 sin 2 B 2 ma sin B /v . ( ^ I sin G * sin 2 G (ra -j- n) sin J. sin G b sin ^4 Ib 2 sin 2 ^4 _ ^iVTTT " V sin 2 G 2 ma sin sin G * sin 2 G (m-\-n) sin 1? sin G D 2. 6rwm the bearings of three adjacent sides of a tract \of land, and the length of the middle side, to cut off, by a lline running a given course, a trapezium of a given area. Let a = ABCD, the area cut off; b = AB, the given side ; x = AD, y = BC, z = CD. From the given bearings, find the angles A, B, C, D, E. _, b sin A , A b sin B BE = . ^r and AE = ^- sin E sin E A _, _ b 2 sin A sin B ABE = ^BEX AEXsin E= ^. = z sin hi b 2 sin A sin B = s : pi a. 2 sin E z sin (7 z sin 7) sin (7 sin D 6 2 sin ^4 sin B 2 sin E 2 sin E b 2 sin ^4 sin B 2 a sin E sin C sin D sin C sin Z) Substituting the value of z in the values of DE and CE, then the values of AE, DE, BE and CE in the equations, DIVIDING LAND. 331 x = A E DE, and y = BE CE, we find, b sin B /6 2 sin ^4 sin .# sin 2 a sin C~ x = -V- sin E \ sin 2 !? sin D sin .D sin 6 2 sin .4 sin B sin Z) 2 a sin 6 sin .4 / E" ' " \ sin 2 E sin sin C sinE Remark. If .4-f-# > 180, the values of x and y are sin yl sin B sin C 2 ft sin C b sin sin 2 E sin D sin D sin E 1 sin E V6 2 sin A sin B sin D ' sin 2 E sin C si 2 a sin D b sin sin 2 E sin C sin C sin E 1 sin " 3. The bearings of several adjacent sides of a tract of land being given, and the length of each, except the first and last, to cut off a given area by a line of given bearing intersecting the first and last sides. Let the bearings and distances of AK, KL, LM, MN, NB be given, and the bearings of AD and EC; and let a be the area cut off' by CD. Draw ABj then, in the polygon, ABNMLK, -the bearings and dis- tances of all the sides are known, except AB, which can be computed, and the area of ABNMLK found. Subtract the area thus found from the area to be cut off by CD, and the remainder will be the area of A BCD. Then, by the last case, find AD and BC. 4. The bearings of the sides of any quadrilateral tract of land and the distances of two opposite sides being given, to divide it into parts having a given ratio by a line of a given course intersecting the other sides. 332 SURVEYING. Let b = AB, c = CD, x = .4F, ?/ == BF, z == FF, and. .4FF : EFCD :: m : w. Find the angles A, B, C, D, E, F, G. P ~ b sin A . b sin 1 sin G ' sin G _ e sin I> _ 2 sin F ^ z sin C- (JT . 77~ > f (jr . 7~ f ~ j . , _ . , Sill Lr Sill LT Sin Cr sin ^4 sin B c 2 sin (7 sin />) sin T sin .4 sin B z 2 sin E sin ~ : - 7i 2 sin G Equating these values of ABFE, we find, ??& 2 sin J sin B -f me 2 sin C sin D (m -f ) sin F sin F Substituting this value of z in the values of FG and FG; then the values of ^4G, FG, BG and FG in x = AG EG, and y = BG FG, we have, _ 6 sin B sin F Inb 2 sin A sin B -j- me 2 sin C sin /) sin G sin G ^ (m -+- ) sin F sin F _ & sin A sin F / n6 2 sin A sin # + me 2 sinC sin /) sin G sinG^/ (m + n) sin F sinF 5. !T^6 bearings and distances of the sides of any quad- rilateral tract of land being given, to divide it into parts having a given ratio by a line dividing two opposite sides proportionally. b AB, c --= CD, d ==-- AD, e =BC, x = AE, y = BF, ABFE : EFCD :: m : n, ex x : d x :: y : e y, . . y = DIVIDING LAND. 333 From the bearings find the angles A, B, C, Z), G. nri b sin A , 4ri b sin B BG = : 77- = p. and AG = . = q. sin G sin (r m (b 2 sin A sin B c 2 sin C sin D) 2 (m -f- n) sin (r 6 2 sin ^4 sin 7? m(b 2 sin ^4 sin B c 2 sinCsinD) 2 sin G 2 (w -f- ri) sin G nb 2 sin A sin B -f- ??ic 2 sin (7 sin D 2 (m + ri) sin Gr But EFG = %(q x)(p- --T-) sin G. c. . - x) ( p T-) sin G = s. Sdes sin G dp -f eg : Sdes smG 6. y^e bearings and distances of the sides of a quadrilat- eral being given, to cut off a given area by a line running through a point whose bearing and distance from the vertex of one of the angles are given. Let a be the area of ABFE, cut off by EF through P. b =AB, c = CD, u = EG, v=FG, x=AE, y = BF. The bearings give the angles A, JB, C, D, PCQ, PCD. n ~ b si n A . b sin B BG. ~-, AG = -. 77- sin Cr sin Or _ 6 2 sin A sin B 2 sinG b 2 sin A sin B - ; 2 sin G 334 SURVEYING. In the triangle DCP we have given f7), OP, and DCP; hence CDP and DP can be found ; then PDR = CDR CDP. PR = DP sin PDR = p, and PQ = CP sin PCQ = q. EPG = \pu, and FPG = % But sn X = sin G b sin A sin G 7. The bearings and distances of the sides of a quadrilat- eral being given, to divide it into four equal parts by two lines intersecting the pairs of opposite sides, respectively, one line being parallel to one side. Let EF, parallel to AB, and MN, parallel to BC, each divide A BCD into two equal parts ; and PQ, parallel to FC, divide EFCD into two equal parts. Find AE, BF, BM, CN, CP, and FQ, by problem 1 of this article. EF = AB AE cos A BF cos B. Likewise find MN and PQ. NP = CNCP. Produce MQ to /, draw NH parallel to IM, and draw HI; then will EF and HI be the lines required. The line EF is evidently one of the required lines. We are now to prove that HI is the other. -.- DIVIDING LAND. 335 The two triangles, HNI and HNM, are equal, since they have a common base, HN, and a common altitude, their vertices being in IM, parallel to the base. To each of these equal triangles add AHND, and we have ARID = AMND = %ABCD. We are now to prove that HI divides EFCD, and also ABFE into two equal parts. IMH : IQL :: IM 2 : ~IQ\ IMN : IQP ::IM 2 : 7Q 2 . . . IMH : IQL : : IMN : IQP. But IMH = IMN. . . IQL = IQP. To each add QFCI, and we shall have, LFCI = QFCP == J EFCD. Again, HBCI=AHID and LFCI = ELID. Subtracting the second from the first, member from member, we have, HBFL = AHLE. Hence, HI is the other division line required. Let us now find the situation of the points H and /, on the lines AB and CD, respectively. NM : PQ :: NP+PI : PI. (NM PQ) PI=PQX NP. PI ^Q X NP T nen nj np _ pj ~ NMPQ The bearing and length of JAf, and the area of ICBM, can be found by Art. 305. IMH == ICBH ICBM. If p be the perpendicular from /to AB, MH . BH = BM + ME. 336 SURVEYING. 329. Examples. 1. A trapezium, one side of which is 20 ch., the ad- jacent angles 60 and 80, respectively, and the area 10 A., is divided into two 'equal parts by a line paral- lel to the given side. Required the distance from the given side cut oft' on the adjacent sides. Ans. 3.04 ch., and 2.08 ch. 2. From a tract of land, the bearings of three of whose adjacent sides are S. 20 W., E, and N. 10 W., and the distance of the middle side is 10 ch., 5 A. are cut off by a line running S. 70 W, and intersecting the first and third of the above mentioned sides. Re- quired the distances cut off on these sides from the middle side. Am. 4.91 ch., and 7.29 ch. 3. From a tract of land, the bearings of whose sides are S. 38 E., S. 29f E., S. 31f W., N. 61 W., and N. 10 W., respectively, and the distance of the second, third, and fourth sides are 8.18 ch., 15.36 ch., and 14.48 ch., respectively, 39 A. 2 R, 36 P., are cut off by a line running N. 80 E., and intersecting the first and last sides. Required the distances cut off on these sides respectively. Ans. 7.01 ch., 16.19 ch. 4. A tract of land, the bearing and distances of whose sides are AB, E. 22.21 ch.; BC, N. ; CD, N. 56J W., 12. ch. ; DA, S. 24 W., is cut by EF running S. 76J E., intersecting AD and BC, and dividing the field so that ABFE : EFCD : : 5 : 3. Required AE and BF. Ans. AE=-~ 16.50. ch., BF = 11.34 ch. 5. A trapezium whose sides are AB -- 20.45 ch., BC == 21.73 ch., CD = 13.98 ch., DA a* 13.32 ch., and whose angles are A *= 97|, B == 64, C = 89J, D = 109, is divided into two equal parts by the line EF, DIVIDING LASD. 337 dividing AD and BC proportionally. Required AE and BF. Ans. AE = 6.22 ch., BF = 10.15 ch. 6. Within a tract of land whose sides are 1st. E. 45.58 ch.; 2d. X. 134 W., 40.86 ch.; 3d. S. 82 W., 30.40 ch., 4th. S. 9{> W., 36 ch. there is a spring whose bearing and distance from the 3d corner is S. 21 W., 15.80 ch. It is required to cut off 40 A. from the north side of this tract by a line running through the spring and intersecting the 2d and 4th sides. Required the distance from the 1st corner to the point of intersection on the 4th side. Ans. 26.73 ch. 7. A tract of land whose boundaries are 1st. E. 23.24 ch.; 2d. N. 11} W., 15.25 ch.; 3d. N. 5H W., 11.50 ch.; 4th. S. 27 W., 24.82 ch. is to be divided into four equal parts by two lines, one parallel to the first side, the other intersecting the first and third sides. Required the dis- tances cut off by the parallel from the first and second corners, measured on the fourth and second sides, respect- ively; also the distances cut off by the other line from the first and fourth corners, measured on the first and third sides, respectively. Ans. 8.57 ch., 7.79 ch., 10.66 ch., 3.15 ch. 330. Division of Polygons. 1. From a given point in the boundary of a tract of land, the bearings and distances of whose sides are given, to run a line which shall cut off a given area. c Let A be the point, and suppose it probable that the dividing line will terminate on DE. Suppose the closing line AD to be run, the bearing and distance of which can be found on the S. N. 29. 338 SURVEYING. ground by observation and measurement, or, as in sup- plying omissions, from the bearings and distances of AB, BC, and CD. Compu e the area of ABCD, which, if less than the -area to be cut off, subtract from that area, which gives the addition, a, to ABCD. The bearings of AD and DE give the angle ADE. The perpendicular, AG=AD sin ADG. Then, if AP is the dividing line, DP t= J^TTT - If DP > DE, run another closing line AE, and pro- ceed as before. If ABCD is greater than the area to be cut off, sub- tract the area to be cut off from ABCD and divide the difference by one-half the perpendicular from A to CD, and the quotient, if less than DC, will be the distance from D to the point on DC to which the division line is to be drawn. If the quotient is greater than DC, run another clos- ing line, AC, and proceed as before. 2. Through a given point within a tract of land, the bear- ings and distances of whose sides are given, to run a line which shall cut off a given area. Let P be the given point. Run a trial line, AB, and calculate the area which it cuts off. Let d -be the difference between this area, which we will suppose too small, and the area to be cut off. Let CD be the division line required. DIVIDING LAND. 339 Let m = AP, and n = PB, which measure; find the angle PAG, also PBD. We are to find the angle P. --=- 180 (A + P) and D == 180 (B + P). . . sin C-= sin (J. -f P) and sin Z) s= sin (5 -f- P). p m sin ^4 ??i sin P " ' * sin (?+ P) ' .p m 2 sin ^i sin P : . . BPD = /r, d = sin(-f-P)' 2 sin B sin P 2 sin m 2 sin A sin P n 2 sin B sin P 2 sin (A -f-P) 2 sin (P -f-P) cot P + cot A cot P -f cot B Use natural co-tangents, find cot P, and then P. 331. Examples. 1. The boundaries of a tract of land are : AB, W. 25 ch. ; PC, N. 32i W., 16.09 ch. ; CD, N. 20 E., 15.50 ch. ; DE, E. 25 ch. ; EF, S. 30 E. ; and FA, S. 25 W., to the point of beginning. A line is run from A, cutting- off 70 A. 1 R. 33 P. from the west side. Required the second point in which this line cuts the boundary. An?. The side 'DE, 18 ch. East of D. 2. It is required to run a line through a point, P, within a field, so as to cut off 10 A. A guess line through P, intersecting opposite sides in A and J5, cuts off 9 A. Required the angle which the true division line, CD, makes with AB, if AP = 12 ch., PB = 4 ch., PAC 90, PBD = 60. Ans. 8 48'. 340 SURVEYING. LEVELING. 332. The Y Level. The Y level, so called from the form of the support? in which the telescope rests, is exhibited in the ar, nexed engraving. The telescope is inclosed in rings, by which it can be revolved in the Y's or clamped in any position. The Y's have each two nuts, adjustable with the steel pin, and the rings are clamped in the Y's by bringing the clips firmly on them by means of tapering Y pins. The interior construction of the telescope is exhibited in the following figure. The rack and pinion, A A and C(7, are contrivances, the first for centering the eye-piece, and the second for insuring the accurate projection of the object-glass in a straight line. The level is a ground bubble tube, attached to the under side of the telescope, and furnished at each end with arrangements for the usual movements in both horizontal and vertical directions. The tripod head is similar to that in the transit. 333. Adjustments. 1. To adjust the line of collimation, set the tripod firmly, remove the Y pins from the clips, so that the telescope shall turn freely, clamp the instrument to (341) 342 SURVEYING. the tripod head, and by means of the leveling and tangent screws, bring either of the wires to bear on a clearly marked edge of an object, distant from two to five hundred feet. Turn the telescope half-way round, so that the same wire is brought to bear on the same object. Should the wire not range with the object, bring it half-way back by moving the capstan head screws, BB, at right angles to it, in the opposite direction, on account of the inverting property of the eye-piece, and repeat the operation till it will reverse correctly. Proceed in like manner with the other wire. Should both wires be much out, adjust the second cifter having nearly completed the adjustment of the first, then complete the adjustment of the first. To bring the intersection of the wires into the center of the field of view, slip off the covering of the eye- piece centering screws, shown at AA, and move, with a small screw-driver, each pair in succession, with a direct motion, as the inversion of the eye-piece does not affect this operation, till the wires are brought, as nearly as can be judged, into the required position. Test the correctness of the centering by revolving the telescope and observing whether it appears to shift the position of an object. If the position of the object is shifted by revolving the telescope, the centering is not perfectly accomplished. Continue the operation till the centering is perfect. 2. To adjust the level bubble, clamp the instrument over either pair of leveling screws, and bring the bubble to the middle. Revolve the telescope in the Y's so as to bring the level tube on either side of the center of the level bar. LEVELING. 843 Should the bubble run to one end, rectify the error by bringing it, as nearly as can be estimated, half-way back with the capstan screws in the level holder. Again bring the level over the center of the bar, and bring the bubble to the center; turn the level to one side, and, if necessary, repeat the operation till the bubble will keep its position when the tube is turned to either side of the center of the bar. Now bring the bubble to the center with the level- ing screws, and reverse the telescope in the Y's with- out jarring the instrument. Should the bubble run to either end, lower that end, or raise the other by turn- ing small adjusting nuts at one end of the level till, by estimation, half the correction is made. Again bring the bubble to the middle, and repeat the operation till the reversion can be made without caus- ing any change in the bubble. 3. To adjust the Y's, or to bring the level into a posi- tion at right angles with the vertical axis, so that the bubble will remain in the center during an entire revolution of the instrument, bring the level tube directly over the center of the bar, and clamp the tele- scope in the Y's, placing it, as before, over two of the leveling screws, unclamp the socket, level the bubble, and turn the instrument half-way around, so that the level bar may occupy the same position with respect to the leveling screws beneath. Should the bubble run to either end, bring it half- way back by the Y nuts on either end of the bar. Now move the telescope over the other set of level- ing screws, bring the bubble again into the center, and proceed as before, changing to each pair of screws, suc- cessively, till the adjustment is nearly completed, which may now be done over a single pair of screws. 344 SURVEYING 334. The Use of the Level. Set the legs firmly in the ground, test the adjust- ments, making corrections if necessary. Bring the wires precisely in the focus, and the object distinctly in view, so that the spider lines will appear fastened to the surface of the object, and will not change in position however the eye be moved. The bubble resting in the middle, the intersection of the spider lines will indicate the line of apparent level. 335. Leveling Rod. The New York Leveling Rod, represented in the engraving with a piece cut out of the middle, so that both ends may be exhibited, consists of two pieces, one sliding from the other. The graduation commences at the lower end, which is to rest on the ground, and is made to tenths and hundredths of a foot. A circular target, divided into quadrants of different colors, so as to be easily seen, moves on the front surface of the rod, which reads to six and one-half feet. If a greater height is required, the horizon- tal line of the target is fixed at 6J feet, on the front surface, and the upper part of the rod, which carries the target, is run out of the lower, and the reading is obtained on the graduated side up to an elevation of twelve ft. A clamp screw on the back is used to fasten the rods together in any position. LEVELING. 345 336. Definitions. A level surface is the surface of still water, or any surface parallel to that of still water. Such a surface is convex, and conforms to the sphe- roidal form of the earth. A level line is a line in a level surface. The difference of level of two places is the distance of one above or below the level surface passing through the other. . Leveling is the art of ascertaining the difference of level of two places. The apparent level of any place is the horizontal plane tangent to the level surface at that place. The line of apparent level of any place is a horizontal line, tangent to a level line at that place. The Y Level indicates the line of apparent level and not the true level, which is a curved line. The correction for curvature is the amount of devia- tion for a given distance of the line of apparent level from the line of true level to which it is tangent at the point from which the distance is measured. 337. Problem. To compute the correction for curvature. Let t denote the tangent, c the cor- rection for curvature, d the diameter of the earth. Then, by Geometry, we have, (<* + )* = ', .-.< = 346 SURVEYING. Since c is very small compared with d, it can be dropped from the denominator without sensibly affect- ing the result. t* The arc, which is the distance measured, will not differ perceptibly from the tangent, for all distances at which observations are, made, and may be substituted for it. Calling another distance. ^, and the corresponding correction, c', we have, /'2 J= .'. c : c' :: t 2 : t f *. a 1. The correction for curvature, for a given distance, is equal to the square of the distance divided by the diameter of the earth. 2. The corrections for different distances are to each other as the squares of the distances. Let us find the correction for the distance 100 chains, calling the diameter of the earth 7920 miles. 100 2 X66X 12 7920X80" ^ 12 - 5l ches - The correction for any other distance, for example, 5 ch., can be found from the proportion. 100 2 : 5 2 :: 12.5 : c, .-. c = .031 inches. For 1 mile, 100 2 : SO 2 : : 12.5 : c, .'. c= 8 inches. For m miles, I 2 : m 2 ::$:<?, . '. c = 8 m 2 in. A correction for refraction is sometimes made by di- minishing the correction for curvature by J of itself. If the leveling instrument is placed midway between the two places whose difference of level is to be found, the curvature and refraction on the two sides of the LEVELING. 347 instrument balance, and the difference of apparent level will be the difference of true level. 338. Problem. To find the difference of level of two places visible from a point midway between them or from each other, when the difference of level does not exceed twelve feet. Let A and B be the two places, and C the place mid- way from which both are visible. Place the level at (7, and let the rod-man set up the leveling rod at A, and slide the vane till he learns, by signals from the surveyor at the level, that its hori- zontal line is in the line of apparent level. Let the height be accurately observed and noted, and the rod be transferred to B, and the height observed, and noted as before. The difference of these heights will be the difference of level. If a gully intervene, so that the line of apparent level, from the intermediate station, would not cut the rod, place the instrument at one station, and take the height on the staff at the other station marked by the vane when in the line of apparent level, from which subtract the height of the instrument, and the differ- ence corrected for curvature and refraction will be the difference of level required. 348 SURVEYING. 339. Problem. To find the difference of level of two places which differ con- siderably in level, or which can not be seen from each other. Let A and D be the places whose difference of level is required. Place the level at the station L, midway between two convenient points, A and B. Take the backsight to J, and note the height of E. Send the rod to #, and note the height of the foresight at F. Remove the level to M, note the height of the backsight at G and the foresight at H. Remove the level to A 7 , note the height of the backsight at /, and the foresight at J. Then will the difference of the sum of the backsights and the sum of the foresights be the difference of level of A and D. For, we find for the sum of the backsights, AE + BO + CT ^ AE + BF + FG + CI. And, we find for the sum of the foresights, BF + OH + DJ=BF + 01+ IH + DJ = BF+CI+PG. The sum of the backsights, minus the sum of the fore- sights, = AE + FG PG = -AK = difference of level, which in the field notes is denoted by D. L. LEVELING. 349 If the sum of the foresights exceeds the sum of the backsights, the point D is below A; if the reverse were true, the point D would be above A, as indicated by the sign. It is not essential that the intermediate stations be directly between the places. 340. Field Notes. Stations. Backsights. Foresights. 1 2 3 5.40 3.12 2.40 1.50 5.25 8.16 Sums . .. D. L. = 10.92 14.91 14.91 = 3.99 341. Leveling for Section. Leveling for Section is leveling for the purpose of obtaining a section or profile of the surface along a given line. A Bench-mark is made to indicate the beginning of the line by drilling a rock or driving a nail into the upper end of a post. Such marks should be made at different points along the line, to serve as checks in case of a new survey. It is necessary also to measure the distance between the stations. The bearings of the lines should be taken in case a map or plot is to be made, representing the horizontal surface. 350 SURVEYING. In the following table of specimen field notes, S. de- notes stations; B., bearings; D., distances; B. S., back- sights; F. S.j foresights; B. S. F. , backsights minus foresights; T. D. L., total difference of level; R., remarks, and B. M., bench-mark. The numbers in the column headed B. S. F. S. are obtained by subtracting each foresight from the corre- sponding backsight, observing to write the proper sign. The numbers in the column headed T. D. L. are ob- tained by continued additions of the numbers in the column B. S. F. S., each being the sum of the back- sights minus the sum of the foresights, up to a given point, expresses the distance of that point above or below the bench-mark at the beginning of the line. The minus sign of a result indicates that the sum of the foresights exceeds the sum of the backsights, and hence, that the corresponding station is below the first station; the plus sign indicates the reverse. In order to bring out prominently the difference of level, the vertical distances are usually plotted on a much larger scale than the horizontal. Let us suppose the numbers in the column D. ex- press chains, and that the numbers in the following columns express feet. In the following profile section the horizontal dis- tances are plotted to the scale of 20 chains to an inch, and the vertical distances to the scale of 20 feet to an inch. The profile of the section is therefore distorted, the vertical distances being 66 times too great to exhibit their true proportion to the horizontal distances. The horizontal line, AG, through the point of begin- ning is called the datum line, RAILROADS. 342. Field Notes. 351 s. B. D. B.S. F.S. BS.FS. T.D.L. H. 1 N. 10.00 3.25 11.63 8.38 8.38 BM. on post. 2 N. 14.00 4.80 10.20 - 5.40 13.78 3 N. 8.25 12.00 1.40 + 10.60 3.18 BM. on rock. 4 N.10E. 12.00 10.80 2.30 -|- 8.50 + 5.32 5 N.10E. 10.75 1.18 12.00 - 10.82 5.50 6 N. 10.00 2.15 8.40 - 6.25 11.75 BM. on oak. 343. Profile of Section. JO. oo G SURVEYING RAILROADS. 344. General Plan. The surveys for the construction of railroads, appli- cable also to canals, graded pikes, dikes, etc., are made in the following order. 1. The reconnoissance, to locate the route. The ter- mini being agreed upon, sometimes several routes are examined, so that an approximate judgment can be formed in reference to the economy of construction and purchasing the right of way, the amount of stock taken at different towns along the route, and the profits from local business. 2. The transit survey, to determine definitely the 352 SURVEYING. middle line along the surface, after the route has been decided upon by the preliminary reconnoissance. 3. The section leveling, to determine the profile of the middle line along the surface. . 4. The cross-section work, to determine the position and slopes of the sides, so that the amount of earth to be removed or filled can be estimated. 345. Section Leveling. Section leveling is simply an application, with slight modifications, of leveling for section, before described. The first bench-mark is assumed at some convenient point near the beginning of the line, and its location described in the column of remarks. The datum line is generally assumed at such a depth below the first bench-mark for example, at mean high-tide water, in case one end of the route is in the vicinity of tide-water that its whole length shall be below the section line at the surface. The engineer's chain, 100 feet in length, is usually employed in taking the horizontal distance. A turning-point is a hard point chosen as far in advance as possible, but not necessarily in exact line, upon which the rod rests while a careful reading is taken just before it is necessary to change the position of the instrument, whose exact height above the datum line thus becomes known in the new position. The difference between a turning-point and a bench is this : A turning-point is merely a temporary point, neither marked nor recorded, used to determine the height of RAILROADS. 353 the instrument in a new position. A bench is both marked and noted, and thus made permanent. If, however, it is thought best to make a turning- point permanent, it is marked and recorded, and be- comes a bench. In order that a bench be not destroyed in construct- ing the road, it should be a little removed from the line surveyed. The location of the benches should be carefully noted, so that they may be readily found from the field notes. The plus sights are the first readings of the rod, made after each new position of the instrument, as the rod stands on a bench or turning-point, and are taken to thousandths of a foot. The minus sights are the other readings, and are taken to tenths, except the last minus sight, before the position of the instrument is changed, which, being taken as the rod stands on a turning-point or bench, is taken to thousandths. The height of the instrument above the datum line is equal to a plus sight, plus the height of the corre- sponding bench or turning-point. The height of the surfaco above the datum line, at any position of the rod, is equal to the height of the instrument, minus the corresponding backsight. These heights are taken at intervals of 1 chain, and at intermediate points where the irregularitj^ of the surface is deemed sufficient to render it important. In the following field notes D. denotes distance; B., bench ; T. P., turning-point ; -f S. 9 plus sight ; H. /., height of instrument ; - - S, minus sight ; S. H., sur- face height; G. H., grade height; Cl, cut; F., fill; R., remarks. S. N. 30. 354 SURVEYING. 346. Field Notes. D. -f #. /. & S.H. G. H. C. F. R. B. 2.911 32.911 30. B. 50 ft. 0. 3.4 29.5 29.5 E. of 1. 4.9 28.0 26.5 1.5 stake. 2. 12.7 20.2 23.5 3.3 3. 4.1 28.8 20.5 8.3 T.P. 2.243 23.755 11.399 21.512 19.2 2.3 4. 2.0 21.8 17.5 4.3 5. 12.5 11.3 14.5 3.2 5.6 4.6 19.2 12.7 6.5 0. 12.3 11.5 11.5 The numbers in the horizontal column, T. P., are found thus: The , 11.399, is obtained from the first position of the instrument by the reading of the rod on T. P. 21.512 = 32.911 11.399. The + , 2.243, is the reading of the rod from the new position of the instrument. 23.775 = 21.512 -f 2.243. The cutting or filling is the difference of S. H. and G. JI. 347. Profile of Section and Grade. 32.911 above Datum Line. RAILROADS. 355 348. Remarks. 1. The grade height at 0, minus the grade at 6, which is 29.5 11.5 == 18 === the descent from to 6. 18 -*- 6 = 3 = the descent for 1 chain, 29.5 3 = 26.5 = G. H. at 1 ; 26.5 - 3 = 23.5 = G. H. at 2, etc. . 2. The establishment of the grade is influenced by the object of the work, economy, the balance of cuttings and fillings, the points desirable for termini, etc. 3. The method exhibited above may be extended to any distance. 349. Example. Fill out the notes of the following table, and make a profile of section and grade from S. H. at to S. H. at 5. D. t# H.I. -S. S.H. G.Jf. c. F. E. B. 6.248 36.248 30 ?L ^ J5. 20 ft, 1 5.3 9.8 IS n S.ofO. 2 2.3 Sl> T. P. 10.718 _' 11.814 3 7.6 1%> 4 12.0 13-f 32A 5 2.1 ^ii 350. Cross-Section Work. Excavations and embankments are constructed with sloping sides, in order to prevent the sliding of earth down the surface. The ratio of slope is the vertical distance divided by the horizontal, and is therefore the tangent of the angle which the sloping surface makes with a hori- zontal plane. The usual ratio of slope is , and the angle 33 41'. 356 SURVEYING. Slope stakes are driven to mark where the sloping sides, whether of cutting or filling, will intersect the surface, and thus indicate the boundaries of the work. The rod used in cross-section leveling is 15 feet long, graded and plainly marked to feet and tenths, and is read by the leveler at the .instruments. The assistants of the leveler are the rodman, axman, and two tapemen. The Field book is ruled into four columns, headed D. for distance ; L. for left ; C. 0. for center-cut ; R. for right. The numbers in the columns D. and C. C. are, respect- ively, the distance and the corresponding cut, or fill marked minus cut, taken from the field book for sec- tion leveling. The fractions in the columns L. and R. have for their numerators the vertical distances of the cross-section, and for their denominators, the corresponding horizon- tal distances, from the center or from the vertex of the angle of slope, according as the vertical distance is taken within or without the limits of the horizontal portion of the road. 351. Cross-Section Excavations. We give the following profile of cross-section, the method of performing the field operations and record- ing the notes. Let us suppose the cross-section to be taken at the distance 3 of the field notes of article 343, where the center cut is 8.3; that the road bed is 20 feet wide, that the ratio of slope is , and that both horizontal and vertical distances are plotted to the scale of 20 feet to 1 inch. RAILROADS. 357 15.7 Take AA f for the datum line, and suppose the read- ing at the center stake to be 7.4. The height of the instrument above the datum line is therefore 8.3 + 7.4 = 15.7. The reading of the rod at the depression F, between the center and the angle A, is 8.5 ; hence, the cut is 15.78.5 = 7.2. The horizontal distance, CF, is 4 feet; hence, the record in the field notes, as seen in the next 7 2 article in the column L, is j- The reading of the rod, at the temporary stake , is o o 7.4; hence, the cut is 15.7 7.4 = 8.3, and the entry, '^- A. The point S, where the slope intersects the surface, is found by trial. Since the vertical distance of the slope is f of the horizontal, then ES, if horizontal, would be | of EA, which is 12.4; but, on account of the inclination of the surface, ES will be less, say 10 feet. Setting the rod 10 feet out from E, the reading is 8.3, and hence the cut = 15.7 8.3 == 7.4. Now, f of 7.4 is 11.1; hence, the assumed distance, 10 feet, is too small. For a second trial, take 11 feet out from J5", at which the reading of the rod is 8.4, and the cut 7.3. Now, f of 7.3 10.9, which lacks but .1 of 11, and is suf- ficiently accurate. The record for the slope stake, in the column I/, is -r~ 358 SURVEYING. The reading of the rod at the stake D is 6.9; hence, Q Q the cut is 8.8, and the record in the column R is ~ A The reading at the elevation G is 5.1; hence, the cut is 10.6. The horizontal distance, Z)G, is 9 feet; , . 10.6 hence, the record is -^ To find AS 1 ' where the slope intersects the surface, since, on account of the rising of the surface, it is more than |- of 8.8, which is 13.2, take, for a first trial, 18 feet out from D, at which point the reading of the rod is 4.5, and hence the cut 15.7 4.5 = 11.2. Now, f of 11.2 = 16.8; hence, 18 feet is too far out. For a second trial, take 17 feet out from D. The reading of the rod is 4.3, and the cut 15.7 4.3 11.4. Now, f of 11.4 17.1, which is sufficiently accurate; hence, the record for the slope stake <S", in the column 11.4 *' 1S 171' ' 352. Field Notes. D. L. c. a R. 3 7.3 8.3 A 7.2 4 \ 8.3 8.8 10.6 9 11.4 17.1 10.9 A' 353. Cross-Section Embankments. The following is the profile of the cross section drawn to a scale of 20 feet to 1 inch, taken at the distance 5 of the field notes of article 346, where the filling is 3.2, now called a minus cut, and written 3.2. Take AA', which is the horizontal top of the embank- ment 20 feet wide, for the datum line. RAILROADS. 359 6.3 A 10 " ___ The ratio of slope, in case of embankments, is f. The reading of the rod at the center stake is 6.6, and the height of the instrument, with reference to the datum line, is the algebraic sum of the reading of the rod and the minus cut, which is 6.6 3.2 3.4. If the instrument should be below the datum line, the reading of the rod would be numerically less than the minus cut, and the height of the instrument would be negative. The readings of the other points along the surface S&, subtracted from the height of the instrument, will give the corresponding minus cuts. The reading at A is 7.4, the cut, 4, and the record, -r - A The reading at G is 12.4, the cut, 9, the horizontal g distance FG. 6.3, and the record, -mr-' b.o To find the position of the slope stake S t take for the first trial 20 feet out from F, where the reading is 16, and the cut, 12.6. Now, -- 12.6 X f = 18.9; hence, 20 feet is too far out. Next try 18 feet out, where the reading is 15.5, and the cut, 12.1. Now, 12.1 X f == 18.1, which is sufficiently accurate; hence, the record for the slope stake S is ^r lo.l 360 SURVEYING. The reading at A' is 6.4, the cut, 3, and the record, -p- A To find the position of the slope stake 5", take for the first trial o feet out from P, where the reading is 6.2, and the cut, 2.8. Now, -- 2.8 X -- | === 4.2; hence, o feet is too far out. Next take 4 feet out, where the reading is 6.1, and the cut, 2.7. Now, --2.7 X f = 4; hence, the 2 7 record for the slope stake $' is - 354. Field Notes. D. L. a a R. 5 -12.1 -9 -4 32 3 -2.7 18.1 6.3 A A' 4 355. Remark. It sometimes occurs that an excavation will be re- quired on one side, and an embankment on the other. Guided by the stakes and' field notes, the excavations and embankments can be correctly made. 356. Computation of Earth-work. The computation of earth-work is the determination of the volume of excavation or embankment. The cross-sections, being taken, wherever necessary, at every 100 feet or less, divide the excavations or embankments into blocks, which may be regarded as frustums of pyramids. RAILROADS. 361 Denoting the areas of the sections regarded as bases of the frustum by b and 6', respectively, the length by /, and the volume by v, we have the formula, 357. Examples. 1. The length of an excavation is 100 feet; find the volume, the two ends being thus represented : The area required, in each case, is the area of the whole figure, regarded as a trapezoid, which is one-half the altitude multiplied by the sum of the parallel bases, minus the sum of the two triangles; hence, 6 = 28 X 24 (24 X 8 + 12 X 4) = 432. V = 19 X 12 - (12 X 4 + 6X2)--= 168. v = J- X 100 (432 + 168 -f V 432 X 168). v = 28980 cubic feet = 1073 cubic yards. 2. Compute the volume of the embankment whose horizontal breadth at the top is 16 feet, from the fol- lowing field notes : S. N. 31. 362 SURVEYING. D. L a a R. 5 -11.6 -10.5 10 9.5 8.6 6 17.4 A -17 A -15.5 1n A' 13 - 14.2 - 13 26.1 A A' 19.5 1607 cu. yds. 358. Remarks. 1. The above method of computing earth-work is called by engineers The mean average method. 2. The method known as The arithmetical mean method is easier than the above, though less accurate. The following is the formula : t> = i*(6-h&'). 3. The volume can also be computed as a rectangular prismoid. 4. Irregularities in the cross-section surface line, as elevations, depressions, or a curvature of this line, must be considered. Thus, the elevation may be regarded as a triangle, its area com- puted and added to the trapezoid before the area of the two triangles at the right and left be deducted. 359. Railroad Curves. In the preliminary survey of a railroad, any change in direction is made by an angle which must, in the final survey, be replaced by a curve, to which the sides of the angle are tangents. RAILROADS. 363 Let the annexed diagram represent such an angle and curve. Run out one of the tangents, as BA, to Ej and let A denote the ex- ternal angle EAD. Then we shall have C= A, since each is the supplement of BAD, the angles B and D being right angles. Let r BC, the radius of curvature, and t = AB, the tangent. Then, (1) t = r tan A, (2) r = rr-%-j The degree of curvature is the number of degrees in an arc whose length is 1 chain or 100 feet. 360. Problem. Given the degree of curvature, to find the radius; and, con- versely, given the radium of curvature, to find the degree. 2 rrr = the circumference, Q i)Q~ r^r = 1 of circumference, ' d of circumference. TT 1AA Hence, == 100. 18000 ' ~" Having found the radius of curvature, we can find t, the tangent, or the distance from the vertex of the angle to the point where the curve begins by formula (1) of the preceding article. 384 SURVEYING. 361. Examples. 1. Find r of 1 of curvature and t, if A = 40. Ans. r == 5729.58 ft., t = 2087.4 ft, 2. Find r of 2 of curvature and t, if A = 40. Ans. r = 2864.79 ft., t = 1043.7 ft. 3. Find r of 3 of curvature and t, if A = 50. Ans. r = 1909.86 ft., t = 890.6 ft. 4. Find r and d, if A = 35 and * = 1000 ft. Ans. r == 3171.6 ft., rf = 1 48' 23". 5. Find r and d, if A = 100 and t = 1 mile. . r = 4430.4 ft., d = 1 17' 35". 362. Location of the Curve. Method. Let each of the arcs, p, pq, qr, ... be 1 chain, then will the number of degrees in each, or in the corresponding angle at the center, be equal to d, the degree of curvature. The angle ABp, formed by a tangent and a chord, is measured by one-half the arc Bp, and is therefore equal to i^ Each of the inscribed angles, pBq, qBr, is measured by one-half the intercepted arc, and is therefore equal to ^d. Having determined the point B, where the curve begins, the transitman sets his instrument at this point, and directs it to A. He then turns it an angle equal to JrZ, on the side toward the curve. RAILROADS. 365 The chainmen then take the chain, the follower placing his end at B, and the leader drawing out"the chain at full length toward ^4, is directed by the trans- itman into line so as to locate the -point p, at which the axman drives a stake. The transitman again turns his instrument an angle equal to Jrf, the chainmen advance, the follower plac- ing his end of the chain at p, the leader again draw- ing out the chain at full length, is directed by the transitman so as to locate the point (/, at which the axman drives a stake, and so on. The last distance will usually not be 1 chain; but if ?i be the number of preceding deflections, the last angle of deflection, since the sum of all the deflections is equal to \C = T. A, will be equal to I A \dn. It is to be observed that the chord is made equal to 1 chain instead of the arc; but as the radius is much greater than the chord, the arc and chord will not differ materially, and no appreciable error arises in practice. Second Method. Points on the curve may be located by the use of two transits, without the use of the A chain, as may be desirable, in case the curve is to be located in marsh} 7 ground or shallow water. ?> Let one transit be placed at B and another at /), the extremities of the curve. Direct the transit at B to A, the one at D to B, then turn each to the right an angle equal to ^d. 366 SURVEYING. The intersection of the lines will -determine _p, where the axman, directed by both transitmen, drives a stake. .In like manner other points can be located. If A is visible from D, but not B, direct the transit at D to A; then, to locate p, turn it to the left an angle equal to \A Jd. To locate 7, turn the transit at D from p to the right an angle equal to d, or from A to the left an angle equal to \A d, and the transit at B to the right from p an angle equal to -Jd, or to the right from A an angle equal to cZ, and so on. Third Method. Let B be the point where the curve begins. Take Bm equal to 1 chain. Then, to find the length of the off- set mp, complete the circle, draw the diameter BE, let fall the perpendicular pn to BE, and draw pE. In the right triangle BpE, Bp is a mean proportional between BE and Bn ; hence, BExBn^Ttp 1 *; but BE = 2 r, Bp = l, and Bn = mp, To find q, produce Bp till ps = 1 chain, and draw tv, tangent to the curve at p. Then, *pv = tpB = mBp =-- ypq, For the first and second are vertical, and all the rest are included between tangents and equal chords. RAILROADS. 367 . . spq = 2 mBp, . ' . the arc sq = 2 arc mp, Or, the arcs being small, do not differ materially from their chords, . . sq = 2 mp = Hence, to locate a curve by this method without the transit, commence at B, where the curve is to begin, take Bm = 1 chain in the direction of the straight track, make the offset mp = , produce Bp till ps = 1 chain, make the offset sq equal to twice the first offset, produce pq till the produced part 1 chain, make an offset equal to the last, and so on. Fourth Method. It is evident from the diagram that But BC = r, and nC = Vr* t 2 . . . mp = r 1 r 2 t 2 . By giving to t different values, other points of the curve can be determined. Fifth Method,. It is evident from the diagram that mp = mC Cp. But mC = Vr 2 -\- t 2 , and Cp r. ' >P = Vr 2 -\-t 2 r. In this method the offset is not made at right angles to the tangent, but in a direction toward the center, which is supposed to be visible from m. 368 SURVEYING. The preceding methods apply to points of the curve 1 chain or 100 feet from each other, which will be sufficient for the excavations or embankments. Before laying the track, stakes are driven at points on the curve, distant from each other about 10 feet. 363. Problem. To locate inter mediate points on the curve. Let the diameter in the diagram be parallel to the chord, which is equal to 1 chain = 100 feet, the ordi- nates a, 6, e, el, e, /, e. d, c, fr, a be 10 feet from each other, / and v, w, or, y, z, y, x, w, v be / offsets from the chord to the curve, corresponding to the ordinates b, c, d, e,f, e, d, c, b. The square of an ordinate is equal to the rectangle of the segments into which it divides the diameter. a* = (r 50) (r + 50), a = 1 \r 50) (r b = [ (r 40)(r + 40), c = (r - c a. d = l (r 20) (r-H 20), x =da. e = \ (r T.O) (r~4^10), y = e a. f=r, z = f a. 364. Example. Find the radius of a 1 curvature, and the offsets from the chord of 100 feet to the curve. = 5729.58 ft., v == .08 ft., w -= .14 ft. = .19 ft., y = .21 ft., z .22 ft. (r = Ans. | TOPOGRAPHICAL. TOPOGRAPHICAL SURVEYING. 365. Definition and Method. Topographical surveying is that branch in which the form of the surface, the situation of ponds, streams, marshes, rocks, trees, buildings, etc., are considered and delineated. The surface is supposed to be intersected by hori- zontal planes equally distant from each other, and the curves formed b}^ the. "intersection of the planes and the surface projected on a horizontal plane. These projections will be nearer together or farther apart, according as the slope of the surface approaches a vertical or a horizontal position. The operations are of two kinds field operations and plotting. 366. Field Operations. Field operations consist in finding and recording points of the curves of intersection of the surface and the horizontal planes, the course of streams, and the situation of noteworthy objects on the surface. Range with the level, or transit theodolite, which is more convenient in topographical operations, stakes marked as in the annexed diagram, and cause them to be driven into the ground, at a horizontal distance from each other of 100 feet or less, varying with the inequality of the surface and the degree of accuracy with which it is desirable that the work be executed. Find by the eye, or by the instrument if necessary, the lowest point in the field, at which make a permanent bench-mark, and assume for the plane of reference the 370 SURVEYING. BI S 1)3 D * c, o a C 3 C, B 3 B 4 S' A, *! A., A 4 horizontal plane passing through this point, which we will suppose to be C\. Place the instrument at some convenient station, S, from which take the read- ing of the rod at C r 1? which suppose to he 10.378, and enter this as a backsight in the field notes. Take the readings of the rod at as many stakes as possible from the station S. Suppose these readings to be <7 2 , 6.481; C 8 , 1.214; Z> 1? 8.235; D 2 , 6.378; D 3 , 4.102; D 4 , 2.304, and enter these readings in the field notes as foresights, placing the smallest reading, (7 3 , last. At (7 3 drive a small stake for a check. Subtract the foresight O 2 6.481 from the backsight 10.378, and enter the difference in the column of differ- ence, headed D.; also in the column of total difference of level above C,, headed T. D. L. Subtract each of the remaining foresights from the next preceding one, and enter the results, with their proper signs, in the column D. Add each result to the previous total difference of level, and enter the results in the column T. D. L. The total difference of level for C 3 is also found by subtracting the foresight of O 3 from the backsight of C 15 which, compared with the result before found, will serve as a cheek. Move the instrument to S", and take a backsight to the check stake O 3 , and the foresights to as many of the remaining stakes as possible, suppose all of them and enter the readings in the field notes as before. TOPO GRA PHICA L. 871 Subtract the first of these foresights from the back- sight (?3, and add the result to the total difference of level for (7 3 , and enter the sum in the column T. D. L. Subtract each of the following foresights from the next preceding foresight, and enter the result, with its proper sign, in the column ZA, and add it to the next preceding difference of level, and enter the sum in the column T. I). L. As a check, subtract the foresight of Z? 3 from the backsight O 3 ; the difference will be the height of Z? 3 above C 3 , which add to the former check number, which is the difference of level of <7 3 and C^, and the sum will be the total difference of level of B s and C l . Compare the explanations of this article with the field notes of the following article. 367. Field Notes. B. S, F. S. D. T. I). L. R. P\ 0.000 Ci 10.378 C 2 6.481 + 3.897 C 2 3.897 DI 8.235 - 1.754 D, 2.143 D 2 6.378 + 1.857 D 2 4.000 D 3 4.102 + 2.276 D 3 6.276 #4 2.304 + 1.798 D* 8.074 c, 1.214 + 1.090 | C 3 9.164 Check 9.164 C 3 9.687 a< 12.000 - 2.313 c* 6.851 B, 11.845 + 0.155 B, 7.006 B 2 5.184 + 6.661 B 2 13.667 B 4 8.314 - 3.130 B. v 10.537 AI 12.000 3.686 A l 6.851 A z 11.321 + 0.679 A, 7.530 A 3 10.987 + 0.334 A 3 7.864 A 4 7.125 + 3.862 A, 11.726 B 3 0.132 4- 6.993 B 3 18.719 Check 9.555 18.719 372 SURVEYING. 308. Plotting. Let the annexed diagram be a plot of the ground on which is written, with red ink, the height to tenths, taken from the field notes, of the surface, at each stake, above the plane of reference passing througli C 1 . Let us suppose that the horizontal planes intersect- ing the surface are 4 feet apart. The intersection of the surface and the plane 4 feet above the plane of refer- ence crosses the line A l D l between the points B l C 19 at a point 4 feet above C^. To determine this point, observe that the rise from C l to B l is 7 feet. Then the distance on this -line from C\ to the point where the height above (7 t is 4 feet is found by the proportion, : 4 : : 100 : x, x - 57.1. This method assumes the ascent to be uniform be- tween B l and C^; but this point can be tested and other points of the curve found as follows : Set up the instrument at S, and make the backsight to C\ 10.378, the same as before ; then depress the vane on the rod 4 feet that is, to the reading 6.378. Now let the rodman set up the rod at the point be- tween C x and B l determined from the proportion, and let the surveyor observe whether the horizontal wire of the telescope ranges with the horizontal line of the vane ; if not, let the rod be moved a little toward B^ or TO PO G RA PHICA L. 373 C 1 till they do range, and at the point thus determined let a stake marked 4 be driven by the axman. An inspection of the plot will show that the curve passes between B 2 and C 2 at a distance from C 2 found from the proportion, 9.8 : .1 :: 100 : a-, .-. sf=l. Let the rodman advance toward this point, pausing at one or two intermediate points, and at this point, whose positions are definitely determined and marked. In a similar manner determine where the curve crosses (7 2 3 and trace it to Z) 2 . In like manner, trace the curves of intersection of the surface and planes, 8 feet, 12 feet, and 16 feet above the plane of reference, and let these curves be marked on the ground by stakes numbered 8, 12, and 16, re- spectively. The horizontal distance of each stake from two sides of a square can be measured and recorded. From this record the surveyor can draw the curves on the plot as exhibited above. 3G9. Shading. The slopes may be repre- sented to the eye by short lines drawn perpendicular to the curves, marking the intersection of the surface with the horizontal planes. These lines are heaviest and closest where the slopes are steepest, and lighter where the slopes are less abrupt. 374 SURVEYING. 370. Conventional Signs. The following conventional, though not altogether arbitrary signs, are used to indicate objects worthy of note : Pasture. Sand. Gardens. Meadow. Fields. Orchards. Swamp. '::: ftifiift -a^i^i^^: Cotton. Turnpike, Common Road, Toot Path, Rail Road, mum Stone Bridge, Suspension Bridge, Carriage Ford, Canal & Lock, D$ Water Mill. D /"Steam Mill. ^> Post Office. ti Hotel. Bushes. I 1 Railroad Station. $ Telegraph Station. 6 Church. A Monument. X Custom House, i Way mark. O Building, Wood. Mile Stone. m " Stone. J^ Lime Kiln. jv-lGold. y Silver, if Tin. -^ Lead. Vineyard. Hopfield. Hedges, Rail &nC Boardfence, Stone Wall, Wootl Bridge, Pontoon Bridge, Horse Ford, Stone Dam. $ Laud-mark, stone. Light-house, rev. ? " " wood. >&c -" " fixed. ^^ " " mouud. J.tX Beacons. ^ trees, ^-to Ancliorago. ^ Survey Station. P ra Buoys. > Rock bare. < 1 Current. *i|?'Sunken rocks. ^^ Nb Current. 9 Copper. C? Iron, $ Mercury. Coal. BAROMETRIC HEIGHTS. 375 371. Finishing a Map. The points of compass are indicated as is usual, the top of the map denoting the north, etc., etc. The meridian, both true and magnetic, should be drawn, and the variation of the needle indicated. The lettering should be executed with care, after printed models of various styles. The border may be made by a heavy line, relieved by a light parallel. The title, in ornamental letters, should occupy one corner of the map, with the name of the locality, the dates of the survey and drawing, and the names of the surveyor and draughtsman. The scale of horizontal distances, for finding and com- paring distances on the map, and the scale of construction, used in the smallest measurements required in project- ing dimensions in the drawing, should be accurately drawn in some convenient position within the border. Parallels of latitudes and meridians, in extended sur- veys, should be drawn in their true position. BAROMETRIC HEIGHTS. 372. Preliminary Remarks. The barometer affords an approximative method for finding the difference of level of two stations. To attain to as great a degree of accuracy as possible, it is important to employ two good barometers, one at the lower and the other at the upper station. Before using the barometers, they should be carefully compared by frequent trials, and the variation ascer- tained, which is to be allowed for in the observations. 376 SURVEYING. Increased accuracy is attained by making repeated observations, and taking the mean of the results. To guard against varying local conditions of the at- mosphere affecting pressure, beside difference of eleva- tion, the stations should not be distant from each other more than four or five miles; and the observation^ should be made when there is no wind. 373. Bailey's Formula. The subjoined formula requires a knowledge, at both stations, of the height of the column of mercury, its temperature as indicated by an attached thermometer, the temperature of the air as indicated by a detached thermometer, and the latitude of the locality. Let d denote the difference of level in feet; /, the latitude of the place in degrees; A, T, t, respectively, the height of the barometer, the temperature of the mercury, and the temperature of the air at the lower station ; A', T", t', respectively, the same at the upper station. Then, d = 60345.51 [1 + .001111 (t + *' 64)] h X (1 + .002695 cos 2 /) X log 77 Let ,4=, log 560345.51 [1+. 001111 (* + *' 64;]}, . B = log (1 -f- .002695 cos 2 I ), c = log [i -f .0001 (T r )], D=log A- (log A'-f<7). .-. log d = A+ + log D. This formula is applied by the aid of the tables: BAROMETRIC HEIGHTS. 377 374. Howlet's Tables. Table A, for Detached Thermometer. t + r A. t i A. t + tf A. t + r A. 1 4.74914 46 4.77187 91 4.79348 136 4.81407 2 .74966 47 .77236 92 .79395 137 .81452 3 .75017 48 .77285 93 .79442 138 .81496 4 .75069 49 .77335 94 .79489 139 .81541 5 .75120 50 .77384 95 .79535 140 .81585 6 .75172 51 .77433 96 .79582 141 .81630 7 .75223 52 .77482 97 .79628 142 .81674 8 .75274 53 .77530 98 .79675 143 .81719 9 .75326 54 .77579 99 .79721 144 .81763 10 .75377 55 .77628 100 .79768 145 .81807 : 11 .75428 56 .77677 101 .79814 146 .81851 12 .75479 57 .77725 102 .79861 147 .81896 13 .75531 58 .77774 103 .79907 148 .81940 14 .75582 59 .77823 104 .79953 149 .81984 15 .75633 60 .77871 105 .79999 150 .82028 16 .75684 61 .77919 106 .80045 151 .82072 17 .75735 62 .77968 107 .80091 152 .82116 18 .75786 63 .78016 108 .80137 153 .82160 19 .75837 64 .78065 109 .80183 154 .82204 20 .75888 65 .78113 110 .80229 155 .82248 21 .75938 66 .78161 111 .80275 156 .82291 22 .75989 67 .78209 112 .80321 157 .82335 23 .76039 68 .78257 113 .80367 158 .82379 24 .76090 69 .78305 114 .80413 159 .82423 25 .76140 70 .78353 115 .80458 160 .82466 26 .76190 71 .78401 116 .80504 161 .82510 27 .76241 72 .78449 117 .80550 162 .82553 28 .76291 73 .78497 118 .80595 163 .82597 I 29 .76342 74 .78544 119 .80641 164 .82640 30 .76392 75 .78592 120 .80686 . 165 .82684 31 .76442 76 .78640 121 .80731 166 .82727 32 .76492 77 .78687 122 .80777 167 .82770 33 .76542 78 .78735 123 .80822 168 .82814 34 .76592 79 .78782 124 .80867 169 .82857 35 .76642 80 .78830 125 .80913 170 .82900 ! 36 .76692 81 .78877 126 .80958 171 .82943 37 .76742 82 .78925 127 .81003 172 .82986 38 .76792 83 .78972 128 .81048 1 173 .83029 39 .76842 84 .79019 129 .81093 ! 174 .83072 40 .76891 85 .79066 130 .81138 175 .83115 41 .76940 86 .79113 131 .81183 176 .83158 42 .76990 87 .79160 132 .81228 177 .83201 43 .77039 88 .79207 133 .81273 178 .83244 44 .77089 89 .79254 134 .81317 179 .83287 45 .77138 90 .79301 ! 135 .81362 180 .83329 378 SURVEYING* Table B, for Latitude. I. B. 1. B. /. B. /. B. 0.00117 27 0.00069 50 1.99980 59 1.99945 3 .00116 30 .00058 51 , .99976 60 .99941 6 .00114 33 3 .00048 52 i .99972 63 .99931 9 .00111 36 ' .00036 53 i .99968 66 .99922 12 .00107 39 D ! .00024 54 .99964 69 .99913 15 .00101 42 .00012 i 55 .99960 75 .99899 18 .00095 45 3 .00000 i 56 .99956 80 j .99890 21 .00087 48 D 1.99988 i 57 .99952 85 .99885 ! 24 .00078 49 .99984 : 58 .99949 90 .99883 Table C, for an Attached Thermometer. TT / C. TT' C. T T'\ C. T T' a 0.00000 12' 0.00052 24 0.00104 36 0.00156 1 .00004 13 .00056 25 ! .00108 37 .00161 2 .00009 14 .00061 26 , .00113 38 .00165 3 .00013 15 .00065 27 I .00117 39 .00169 4 .00017 16 .00069 28 ! .00121 40 .00174 5 .00022 17 3 .00074 29 .00126 41 .00178 6 .00026 18 .00078 30 ! .00130 42 .00182 7 .00030 19 .00082 31 , .00134 43 .00187 8 .00035 20 .00087 32 | .00139 44 .00191 9 .00039 21 .00091 33 ! .00143 45 .00195 10 .00043 22 .00095 34 1 .00148 46 .00200 11 .00048 23 .00100 35 .00152 47 .00204 375. Examples. 1. At the mountain Guanaxuato, in Mexico, lat. 21, Humboldt made the following observations: Lower Station. Upper Station. Barometric column, h == 30.05, h' = 23.66. Attached thermometer, T = 77.6, T' = 70.4. Detached thermometer, t = 77. 6, t' = 70.4. log d = A + B 4- log D. BAROMETRIC HEIGHTS. 379 log h (30.05) =5= 1.47784 A = 4.81940 log A' (23.66) == 1737402 B = 0.00087 Table C gives C = 0.0003.1 log. D = 101498 log A'+C == L37433 log. d = 3^83525 7) log A (log h'-}- C) === 0.10351 . . d == 6843 ft. 2. Find the difference of level of two stations, lat. 42, from the following data: h = 30, T == 75.o, t = 75. 1 # = 25, 7" = 70.3, *' = 70. j ^ 71S ' 5195 ' ftl 3. Find the difference of level of two stations, lat. 45, from the following data : h = 29.2, T = 80.3, t = 80. i' = 27.1, T fcft; 77.4, t' = 77. J ~'"*' 2149 ' 9 4. Find d, lat. 50, from the following data : h 29, T == 60. 1, f 60. '= 28, r= 59M, t'= ' 973 ' 8 370. Leveling with one Barometer. Take the observations at the lower station, then pro- ceed to the upper station and take the observations there, and note the interval of time which has inter- vened, then go back to the lower station and at the expiration of an equal interval repeat the observations. Reduce the mercurial column of the second observa- tion at the lower station to what it would have been at the temperature of the first observation, on the principle that mercury expands or contracts .0001 of its volume for each degree of increase or diminution of temperature. Then take the arithmetical mean of this reduced height and the first observed height for the height at the lower station, the mean of the temperature denoted 380 SURVEYING. by the detached thermometer at the lower station for the temperature of the air at thai; station, and the temperature denoted by the attached thermometer at the first observation for the temperature of the mer- cury, then proceed as if the observations had been taken with two barometers. 377. Examples. ( 1st obv., h = 29.62, T = 56,5, t == 56. 1. J Lower sta - \ 2d obv., h r 29.63, T = 63, t = 61. \Lat. 41.4; upper sta., A' 28.94, 7"=57 .5, t'=57. Reducing h of 2d obv. from T= 63 to 7^56.5, we have, Reduced h = 29.63 (1 6,5 X .0001) ==, 29.611. , 29.62 + 29.611 . . Mean h - - = 29.6155. Mean t l = 56 + 61 = 5 8.5. .-. t + t' = 58.5+57 = : 115.5. andT r=--56.5 57.5 = -1. log h (29.6155) = 1.47152 A = 4.80481 log h' (28.94) L46150 B 0.00014 C = -0.00004 log D = 2J00260 logA'+C= i"46146 log d = 2.80755 D = \ogh (log A'+C)= 0.01006' .-. d == 642 feet. f 1st obv., A = 29.7, 7 1 == 60, t = 60. 2. J Lower sta> I 2d obv., h = 29.75, T = 66, t = 66. \Lat. 40; upper sta. &'= 28.6, ' T'= 62, - f= 62. i= 1077 ft. ( / 1st- obv., h --- 29.6, 7 r = 50, = 50. 3. ) Lowersta - \2d obv., A = 29.65, T = 46, Z = 46. (Lat. 50; upper sta. A'= 27.6, T'^45, *'= 45. Ans. d = 1909 ft. NAVIGATION. PRELIMINARIES. 378. Definition and Classification. Navigation is the art of ascertaining the place of a ship at sea, and of conducting it from port to port. There are two methods of finding the place of a ship: 1. By dead reckoning; that is, by tracing from the record the courses and distances sailed. 2. By Nautical Astronomy; that is, by deducing the latitude and longitude of the place of the ship from celestial observations. The first method is subdivided into the following: Plane sailing, parallel sailing, middle latitude sailing, Mercator's sailing, and current sailing. 379. The Mariner's Compass. The magnetic needle rests on a pivot, so as to turn freely. The compass box is suspended by gimbals or rings, turning on axes at right angles to each other, thus securing a horizontal position notwithstanding the roll- ing motion of the ship. A circular card, whose circumference is divided into thirty-two equal parts, called points, each of which is (381) 382 NAVIGATION. subdivided into four equal parts, called quarter points, rests upon the needle, with which it turns freely. N. b. E. is read north by east ; N. N. "., north north- east, etc. 380. Table of Points and Angles. North. South. Angles. 1 N.b.E. N.b.W. S.b.E. S.b.W. 11 15' 2 N.N..E. N.N.W. S.S.E. s.s.w. 22 30' 3 N E.b.N. N.W.b.N. S.E.b.S. S.W.b.S. i 33 45' 4 N.E. N.W. S.E. S.W. 45 (X 5 N.E.b.E. N.W.b.W. S.E.b.E. S.W.b.W. 56 15' 6 E.N.E. W.N.W. E.S.E. W.S.W. 67 30' 7 E.b.N. W.b.N. E.b.S. W.b.S. 78 45' 8 E. W. E. W. 90 0' Note 1. J point = 2 48'f, J point = 5 37'^, i point = 8 26'}. Note 2. The compass is placed near the helm, at the stern, and the line from the center of the compass to the ship's head indicates the track of the ship. PRELIMINARIES. 383 381. Variation and Deviation of the Compass. The variation of the compass is the angle included be- tween the magnetic meridian and the true meridian. The amount of variation is ascertained by Nautical Astronomy. The deviation of the compass is the deflection of the needle from the magnetic meridian, caused by the iron in the ship. The amount of deviation is ascertained by special experiments. 382. Course, Leeway, Rhumb Line. The compass course of a ship, at any point, is the angle which her track makes with the magnetic me- ridian at that point. The true course of a ship, at any point, is the angle which her track makes with the true meridian at that point. In the compass course, the deviation is supposed to be ascertained and allowed for, but not the variation ; but in the true course, both the deviation and variation. The leeway is the oblique motion of the ship, caused by a side wind driving the ship along a track oblique to the fore-and-aft line, and therefore not indicated by the compass. The amount of leeway, under a wind of a given obliquity and velocity, for each ship with a given freight, is best found by trial. A rhumb line is the track of a ship which continues to make the same angle with the meridians. It is also called a loxodromic curve. 384 NAVIGATION. Since the meridians converge, the rhumb line is a spiral curve. In what follows we shall suppose that proper allow- ances have been made for the variation and deviation of the compass, and, therefore, that the courses given are the true courses. 383. The Log and Log Line. The log, a drawing of which is annexed, is a board in the form of a quadrant whose radius is about six inches, the circular part of which is loaded with lead, sufficient to give it a vertical position and to cause it to sink so that the vertex shall be just above the surface. The log line is a line about 120 fathoms in length, and so attached to the log as to keep its face toward the ship, that it may, by the resistance it encounters from the water, unwind the line from a reel as the vessel advances. The log line is divided into equal parts called knots, each knot being T ^ of a nautical mile, or 50| feet. The time is measured by a sand glass, through which the sand passes in T J of an hour, or in J of a minute. Since the number of knots in a nautical mile is equal to the number of half-minutes in an hour, it follows that the number of knots run off in half a minute is equal to the number of miles the ship is sailing an hour. The divisions of the line are marked by strings pass- ing through the line and knotted, the number of knots in the string indicating the number of parts between PLANE SAILING. 385 it and that point of the line where the divisions com- mence at that end of the line next to the log. The stray line is about 10 fathoms of the end of the line from the log to the point where the divisions begin. This portion allows the log to settle in the water, clear of th*e ship, before the measurement of the rate begins. The termination of the stray line is marked by a piece of red cloth. The sand glass is turned the instant this cloth passes the reel, which is stopped the moment the sand has run out. The number of knots on the string which marks the last division run from the reel, indicates the rate of sailing. PLANE SAILING. 384. Single Courses. Let P be the pole of the earth ; RQ, r the equator; AD, a rhumb line divided into AB, BC, CD, etc., parts so small that we may regard them as straight lines; and the triangles ABE, BCF, CDG, plane triangles and similar, which give the continued proportions : AB : AE : : BC : BF :: CD : CG. AB : EB : : BC : FC : : .CD : GD. Since the sum of the antecedents is to the sum of the consequents as one antecedent is to its consequent, we have, AD : AE+BF + CG : : AB : AE. AD : EB f FC +'GD :: AB : EB. S. N. 33. 386 NAVIGATION. A d B Now let a right triangle, ABC, be con- structed, in which C is the course or the angle which the rhumb line makes with the meridian, r = CB = AD, the rhumb line of the first figure; I -- CA == AE + HF _|_ CG =-- difference of latitude ; d ^ AB = EB + FC -\- GD == the sum of the elementary departures. We may now, without supposing the ship to sail on* a plane, replace the surface on which it actually sails by a plane surface, and hence the name plane sailing. 385. Table of Cases. 1 Given. Req. Formulas. 1 r, C, I, d. I == r cos (7, d r sin C. J 2 r, I, C,d. cos C= , d=i > 2 -^. r j 3 A r, d, ai C,l. v fl v- 1 & T , ' O 7 O sin C= , 1 = I r 2 a". r ,7 / fin P * 5 > l i C,d, r, a. r, I. cos C d d ~ sm~C' "fanTc" ^ 6 I, d, r, C. ? =:l//2_|_^2^ fom C= y 1. / .in miles may be reduced to degrees by dividing by 60. Note 2. Examples in case I. may be solved by the Traverse table. PLANE SAILING. 387 386. Examples. 1. A ship sails 105 miles N. E. by N., from latitude 50 ; required the latitude in which the ship then is, and the departure made. Ans. 51 27'.3 N., d == 58.34 mi. 2. A ship sailed between S. and W. 148 miles, mak- ing the difference of latitude 114.4; required the course and the departure made. Ans. 3i pts. W. of S., d = 93.9 mi. 3. A ship in latitude 3 52' S. sails between N. and W. 1065 miles, making a departure of 939 miles; re- quired the course and the latitude in which she then is. Ans. N. W. b. W. JW., lat. 4 30' N. 4. A ship ran from latitude 38 32' N. to latitude 36 56' N. on a course S. E. by S. JE. ; required the distance sailed and the departure made. Ans. r == 129.56 mi., d = 87.009 mi. 5. A ship sailed S. 56 47' E. from latitude 50 13' N. till her departure was 82 miles; required r and lati- tude in. Ans. r = 98 mi., lat. 49 19' N. 6. A ship from latitude 36 12' N. sails between S. and W. till she is in latitude 35 1' N., having made 76 miles of departure ; required r and C. Ans. r =- 104 mi., C = S. 46 57' W. 387. Compound Courses. A compound course or traverse is the zigzag course which a ship usually, takes in a voyage of consider- able length. Working the traverse is the computation of a single course and distance from the place of departure to the place of destination. 388 NAVIGATION. To do this, find by the Traverse . table the latitude and departure of each course. The difference of the sum of the northings and the sum of the southings will be the latitude of the single course required, and the difference of the sum of the eastings and the sum of the westings will be the departure, both of the name of the greater. Then proceed as in last article. 388. Examples. 1. A ship sailed from latitude 51 24' N. as follows : S. E. 40 miles, N. E. 28 miles, S. W. by W. 52 miles, N. W. by W. 30 miles, S. S. E. 36 miles, S. E. by E. 58 miles; required the latitude in, and the single equiv- alent course and distance. Solution. Courses. Did. \ A T . L. S. L. E.D. W. D. S. E. 40 28.3 28.3 N. E. 28 19.8 19.8 43.2 S. W. b. W. 52 28.9 24.9 N.W. b. W. 30 16.7 S. S. E. 36 33.3 13.8 S. E. b. E. 58 32.2 48.2 36.5 122.7 110.1 68.1 36.5 68.1 86.2 42 42 / = 86.2 mi. = 1 26'. . . C = 25 59'. ! = 95.87 mi. 51 24' 1 26' 49 58' N. 2. Given the following courses and distances: S. W. W. 62 miles, S. by W. 16 miles, W. J S. 40 miles, S.W. PARALLEL SAILING. }W. 29 miles, S. by E. 30 miles, S. f E. 14 miles; re- quired /, (7, and r. Am. 1 = 1 55' S., (7= S. 43 14' W., r L 158 mi. 3. A ship, from latitude 1 12' S., has sailed as fol- lows: E. by N. JN. 56 miles, N. }E. 80 miles, S. by E. JE. 96 miles, N. JE. 68 miles, E. S. E. 40 miles, N. N. W. $}V. 86 miles, E. by S. 65 miles; required the lati- tude in, C, and r. Ans. Lat. in, 48' N., C= 51 47' E., r = 193.8 mi. PARALLEL SAILING. 389. Definition and Principles. Parallel sailing is that case of sailing in which the track is on a parallel of latitude. Let EFQ be the equator; GAB, the parallel of the track; r AB = the distance sailed ; L FQ = the difference of longitude ; / = QB the latitude of the track. Since similar arcs are to each other as their radii, (1) DB : CQ :: AB : FQ. Consider the radius CQ as the unit of the first couplet, then DB will be the natural co-sine of latitude ; and take 1 mile as the unit in the second couplet, put r for AB, L for FQ, then (1) becomes, (2) COS l:l::r:L, .: (3) L = -^ -. We can compute L in (3) by taking nat. cos Z, or by introducing R and taking log. cos I. In either case L will be found in miles, since r is given in miles; but L can be reduced to degrees by dividing by 60. 390 NAVIGATION. Let r and r', measured on the parallels whose latitudes are / and /', respectively, be the distances between two meridians whose difference of longitude is L. cos I : 1 : : r : L, : cos /' : : r : /. cos i : i : : r ' : L, ) Hence, The distances between two meridians, measured on different parallels, are as the co-sines of the latitudes of tho$c parallels. To find the length of a degree of longitude on any parallel, observe that at the equator 1 of Ion. = 60 nautical miles, and that cos / = 1, then we shall have, 1 : cos/' :: 60 : /, .-. / = 60 cos /'. 390. Examples. 1. A ship in -latitude 49 32' N., and longitude 10 16' W., sails due W. 118 miles; required the longitude arrived at. Ans. 13 18' W. 2. A ship in latitude 53 36' N., and longitude 10 18' E., sails due W. 236 miles; required the longitude arrived at. Ans. 3 40' E. 3. A ship in latitude 32 N. sails 6 24' due W. ; re- quired d. Ans. d = 325.6 mi. 4. A ship sails 310 miles from longitude 81 36' W. to longitude 91 50' W. ; required the latitude of the track. Ans. 59 41'. MIDDLE LATITUDE SAILING. 391. Definition and Principles. Middle latitude sailing is a combination of plane sail- ing and parallel sailing, on the supposition that the departure in plane sailing is equal to the distance MIDDLE LATITUDE SAILING. 391 between the meridians passing through the extreme points of the rhumb line, measured on the middle parallel between these points. Let AD be a rhumb line; IK, the middle parallel ; w, the latitude of IK; then d = EB + FC + GD = IK. For . r, formula (3), parallel sailing, ,substitude d or its value as found in plane sailing; and for cos I substitute cos ?, then we shall have, T d - r sin C l'> 2 J 2 I tan C cos m cos m cos m cos m Note 1. Remember that in these formulas I denotes the difference of latitude ; L , the difference of longitude in miles; d, the departure; r, the distance run or the rhumb line; C, the course, and m, the middle latitude. Note 2. The middle latitude is the half sum of the extreme latitudes ; or the less latitude, plus the half difference of latitude ; or the greater latitude, minus the half difference of latitude. Note 3. That the departure is not strictly equal to the middle-latitude distance between the meridians, through the extremities of the rhumb line, is thus shown : Suppose a ship to sail on this middle latitude from one of the meridians to the other, then the distance {sailed w r ill be the departure; but if a second ship were jjto sail from a lower latitude on the first meridian, imd a third ship, from a higher, to the same place, Ithe departure of the second would be greater, and the departure of the third woulcfc be less than that of the first. It is necessary, therefore, to make the correction for middle latitude as found in the table for such corrections. 392 NAVIGATION. The following is the rule for correcting the middle latitude : Add to the unconnected middle latitude the correc- tion found in the table under the difference of latitude, and opposite the middle latitude the sum m' is the corrected middle latitude. 7- - d _ rsin < 1 _ i V2ZT^2 _ / tan C cos m' cos m' cos m f cos m f 392. Examples. 1. A ship from latitude 51 18' N., longitude 9 50' W., sails S. 33 8' W. 1024 miles ; required the latitude and longitude in. I ------ r cos C, .'. 1 = 857.4 mi. M 14 17'. .-. 51 18' 14 17= 37 1', the hit. in. 4(51 18'+ 37 I')' = 44 9J'=mid. lat., correction = 27'. 44 9J' -f- 27' = 44 36J' = m' = corrected mid. lat. y m 11 \ L = - ^ , .-. L = 786.3 mi. == 13 6'. cos m 9 50' + 13 6' --= 22 56' W., the Ion. in. 2. A ship, from latitude 52 6' N., and longitude 35 6' W., sails N. W. by W. 229 miles; required the lati- tude and longitude arrived at. Am. Lat. 54 13' N. and Ion. 40 23' W. 3. A ship from latitude 49 57' N., and longitude 5 11' W., sails between S. and W. till she is in latitude 38 27' N., when she has made 440 miles departure; required (7, r, and the longitude in. An8. C = S. 32 32' W. ; r & 818 mi. ; Ion. in, 15 28' W. 4. A ship from latitude 37 N., longitude 22 56' W., sails N. 33 19' E. till she is in latitude 51 18' N. What longitude is she in?- Ans. 9 45' W. MERCATOR'S SAILING, 393 5. A ship from latitude 40 41' N., longitude 16 37' W., sails between N. and E. till she is in latitude 43 57' N., and finds that she has made 248 miles departure ; required (7, r, and longitude in. Ans, C= 51 41' E. ; r = 316 mi.; Ion. in, 11 W. MERCATOR'S SAILING. 393. Definitions and Principles. Mercator's chart, so called from its originator, Gerrard Mercator, a Fleming, who first published it in 1556, is a representation of the surface of the earth on the sup- position that the earth is a cylinder. The meridians are thus represented parallel and every-where too far apart except at the equator. To guard as much as possible against distortion, the distances between the parallels are proportionally in- creased. The surface is thus relatively magnified more and more toward the poles. Mercator's sailing is the method of computing the difference of longitude from the principle on which Mercator's chart is projected. The mathematical theory of this method was devel- oped, and the Table of Meridional Parts, necessary to its application, computed by Edward Wright, an English- man, in 1599. A' Let CA and AB. respectively, be the dif- ference of latitude and departure corre- sponding to the rhumb line (75, and let CA be produced to A' till A'B', the corre- sponding departure, is equal to the differ- c 394 NAVIGATION. ence of longitude of C and B. CA' is called the 'merid- ional difference of latitude, which is simply the proper difference of latitude increased till the corresponding departure is equal to the difference of longitude corre- sponding to the proper departure. To find the meridional difference of lati- tude, let Cb, bd, df, ... be indefinitely small portions of the rhumb line CB. Ca, be, de, . . . corresponding differences of latitude ; ab, cd, ef,... corresponding differences of departure ; Ca', be', de', . . . corresponding meridional differences of latitude ; a'b', c'd', ef, . . . differences of longitude corresponding to the departures ab, cd, ef, . . . whose latitudes are /, I', /", . . . Then, as found in Parallel sailing, ab : a'b' : : cos I : 1. but ab : a'b' : : Ca : Ca'. . cos I : 1 : : On : Ca', . . Ca' = but 7 = sec /, cos / In like manner. Ca cos I . Ca' = Ca sec /. 6^ = be sec /', dJ = rfg sec r. But CA' = Ca' -1- be' + de' + ... Substituting the values of Ca', be', de', . . . and making Ca . = be = de = . . . = V, we have, CA = sec I -j- sec I' -f sec l"-\- . . . Commencing at the equator, and putting m. p. for meridional parts, and taking natural secants, we -have, m. p. of r = sec 1'. m. p. of 2' = sec 1' -(- sec 2'. MERC A TOR'S SAILING. 395 w. p. of 3' sec V -f sec 2' -{- sec 3'. m. p. of 4' == sec 1' -)- sec 2' -f- sec 3' -f- sec 4'. By substituting and condensing, we have, m.p. of 1' = 1.0000000 = 1.0000000 m.y, of 2' 4: 1.0000000 + 1.0000002 s= 2.0000002 m. p. of 3' = 2.0000002 -f 1.0000004 = 3.0000006 m.p. of 4'= 3.0000006 + 1.0000007 = 4.0000013 The accuracy of the result is increased by taking the parts still smaller, as %. Having found the meridional latitude corresponding to C, and also to J, their difference will be the merid- ional difference of latitude found from the table; and the corresponding departure, A'B', will be the differ- ence of longitude. Denoting the proper difference of latitude CA by /, the meridional difference of latitude by /', the departure AB by d, and the difference of longitude A'B' by L, the triangles CAB and CAB' give, 1 : tan C : : /' : -L, . . L = I' tan C. I : d :: V : L, .-. L=- 394. Examples in Single Courses. 1. A ship from latitude 52 6' N., and longitude 35 & W., sails N. W. by W. 229 miles; required the lati- tude and longitude in. I = r cos C = 229 cos 56 15', .', 1= 127.3 mi.-= 2 7'. lat. in == 52 6' N. + 2 7' N. = 54 13' N. 396 NAVIGATION. m.p. of 54 13' == 3868 w..of52 6' = 3657 But L = I' tan <7, . . L = 211 tan 56 15'. or L = 315.8 mi. = 5 16'. , . Ion. in - 35 6' W. -f 5 16' W. = 40 22' W. 2. A ship from latitude 51 18' N., and longitude 9 50' W., sails S. 33 8' W. 1024 miles ; required the lati- tude and longitude in. Ans. Lat. in 37 1' N. ; Ion. in 22 50' W. 3. Required the course and distance from Ushant, latitude 48 28' N., longitude 5 3' W., to St. Michael's, latitude 37 44' N., longitude 25 40' W. Ans. S. 54 3(X W., r = 1106 mi. 4. A ship from latitude 51 9' N. sails S. W. b. W. 216 miles; required the latitude in, and the difference of longitude made. Ans. Lat. 49 9' N., 1=4 39'. 5. A ship sails from latitude 37 N., longitude 22 56' W., on the course N. 33 19' E., till she arrives at latitude 51 18' N. ; required the distance sailed and the longitude arrived at. Ans. 1027 mi., Ion. 9 47' W. 6. A ship sails N. E. b. E. from latitude 42 25' N., and longitude 15 6' W., till she finds herself in latitude 46 20' N. ; required the distance sailed and the longi- tude in. Ans. Dist., 423 mi. ; Ion. 6 55' W. 395. Examples in Compound Courses. 1. A ship from latitude 60 9' N., and longitude 1 7' W., sailed as follows: N. E. b. N., 69 miles; N. N. E., 48 miles; N. b. W. W., 78 miles; N. E., 108 miles; S. E. b. E., 50 miles; required the latitude and longi- tude in, and the direct course and distance. MERCA TOP'S SAILING. 397 Courses. Did. JV. L. -v. /,. Lot. m. p. m.d.l. E.L. W.L. N. E. b. N. 69 57.4 609 X 4525 N. N. E. 48 44.4 61 6 X 4641 116 77.5 N.b.W.^W. 78 74.6 WtW 4733 92 38.1 N. E. 108 76.4 635 / 4895 162 49. S. E. b. E. 50 27.8 6421 / 5067 172 172.0 ' 252.8 6353 X 5003 64 95.8 27.8 Dif. hit. = I = 225 mi. = 3 45' N. 383.4 Lat. Left = 60 9' N. Dif. Lat. = 3 45' N. Lat. in =s 63 54' N. Dif. Ion. = L == 334.4 mi. == Dif. Ion. = 5 34' E. Lon. left = 1 7' W. Lon. in : ^4~27 T ET m. p. of lat. in (63 540 : - 5005. m. p. of lat. left (60 9') = 4525. Meridional dif. lat. =1' = -- ' 480. n L 334.4 tan C .3= r r 480 I 225 cos C cos 34 53' = N. 34 53' E. . . r = 273 mi. 2. A ship from latitude 38 14' N., and longitude 25 56' W., has sailed the following courses : N. E. b. N. JE., 56 miles; N. N. W., 38 miles; N. W. b. W., 46 miles; S. S. E., 30 miles; S. b. W., 20 miles; N. E. b. N., 60 miles; required the latitude and longitude in, and the direct single course and distance. Ana Lat. in, 40 2'. 3 N. ; Ion. in, 25. 30' W. ; (7= N. 10 33' E., r =i 110.2 mi. 390. Correction for Middle Latitude. We are now prepared to understand how the correc- tion for middle latitude, before used, is found. NAVIGATION. I denotes the proper difference of latitude ; I', the meridional difference of latitude ; .L, the difference of longitude ; m, the middle latitude uncorrected; c, the correction; ?>i', the middle latitude corrested. Then, by Plane, Middle latitude, and Mercator's sailing, d L cos m' L I tan C = -j- = -, = p- , . ' . cos m = y From which m' is readily found. Then, c = in' m. . . m' = m -(- c. CURRENT SAILING. 397. Definition and Principles. Current sailing is the sailing of a ship as affected by a current. Irrespective of the current the ship would move, in a certain time, a certain course and distance. The current alone would carry the ship, in the same time, a certain other course and distance. The actual track of the ship, which is the resultant of the two, will bring her to the same position as if she had sailed separately the two tracks. Current sailing may therefore be treated as Plane sailing, compound courses. The set of the current is its direction. The drift of the current is its velocity. The set and drift of a current may be ascertained by taking, a short distance from the ship, a boat, which is kept from being carried by the current by letting CURRENT SAILING. 399 down, to a considerable depth, a heavy weight, which is attached by a rope to the stern of the boat. The log being thrown from the boat into the current, the direction in which it is carried, or set of the cur- rent, is determined by the boat compass, and the rate at which it is carried, or drift of the current, by the number of knots of the log line run out in half a minute. 398. Examples. 1. A ship sails N. W. a distance, by the log, of 60 miles, in a current that sets S. S.W., drifting 25 miles in the same time; required the course and distance. Courses. Dist. N.L. 8.'L KD. W.D. N. W. 60 42.4 42.4 S. S. W. 25 23.1 9.6 I = 19.3. d = 52. "^Jff.., ... c= N. 69 38' W. r = = V (19.3) 2 + (52) 2 == 55.5. 2. A ship, sailing 7 knots an hour, is bound to a port bearing S. 52 W., through a current S. S. E., 2 miles an hour; required the course. N Let AB be the direction of the port. AE, the direction of the current, _ o AD, the required direction, = 7. Complete the parallelogram, DBA = BAE = 52 + 22 3(X = 74 30'. Then we have, 400 NAVIGATION. AD : DE : : sin DBA : sin DAB. 2 sin DBA - .'. (715 59' + 52 = 67 59'. 3. A ship runs N. E. by N. 18 miles in 3 hours, in a current W. by S. 2 miles an hour; required the course and distance. Ans. C = N. b. E. JE., r === 14 mi. 4. In a current S. E. by S. 1J miles an hour, a ship sails 24 hours as follows: S.W., 40 miles; W. S.W., 27 miles ; S. by E., 47 miles; required the direct course and the distance. Ans. C = S. 11 50' W., r = 117 mi. 5. The port bears due E., the current sets S. W. by S. 3 knots an hour, the rate of sailing is 4 knots an hour; required the -course steered. Ans. N. 51 E. 6. A ship sailing in a current has, by her reckoning, run S. by E. 42 miles, and, by observations, is found to have made 55 miles of difference of latitude, and 18 miles of departure; required the set and drift of the current. Ana. Set, S. 62 12' W. ; whole drift, 30 mi. PLYING TO WINDWARD. 399. Definitions. Plying to windward is the zigzag course which a ship makes by tacking when she encounters a foul wind. Starboard signifies the right side. Larboard signifies the left side. The starboard tacks are aboard when a ship plies with the wind on the right. The larboard tacks are aboard when a ship plies with the wind on the left. PLYING TO WINDWARD. 401 A ship i^ said to be dose-hauled when she sails as nearly as poppibJ.e toward the point from which the wind is blowing. 400. Examples. 1. Being within sight of my port bearing N. by E. ^E., distant 18 miles, a fresh gale sprung up from the N. E. With my larboard tacks aboard, and close-hauled within six points of the wind, how far must I run be- fore tacking about, and what will be my distance from the port on the second board ? N Let A be the place of the ship ; P, the port; AB y the distance of the first board; BP, that of the second; WA or W'B y the direction of the wind. Then, WA B '=-. W'BC = W'BP = 6 points. . . ABP --16 points 12 points = 4 points. PA W-= NA W NAP = 4 points \\ points =3 2 points. PAB = PAW -f- WAB = 24 points -f 6 points == 8J points. APB == 16 points (PAB + ABP) = 3J points, sin ABP : sin APB : : AP : AB, . . AB = 16.15 mi. sin ABP : sin BAP : : AP : BP, . *. BP = 25.23 mi. 2. If a ship can lie within 6 points of the wind on the larboard tack, and within 5J points on the star- board tack; required her course and distance on each tack to reach a port lying S. by E. 22 miles, the wind being at S. W. ( Starboard tack, S. b. E. JE. 23.66 mi. - Ans ' \ Larboard tack, W. N. W.~2.79 mi. S. N. 34. 402 NAVIGATION. 3. A ship is bound to a port 80 miles distant, and directly to windward, which is N. E. by N. -J^., and proposes to reach her port at two boards, each within 6 points of the wind, and to lead with the starboard tack ; required her course and distance on each tack. ( Starboard tack, X. N.W. iW., 104.5 mi. Am ' \ Larboard tack, E. S. E. iE., 104.5 mi. 4. Wishing to reach a point bearing N. N. W. 15 miles, but the wind being at W. by N., I was obliged to ply to windward the ship, close-hauled, could make way within 6 points of the wind; required the course and distance on each tack. f Larboard tack, N. b. W. 17.65 mi. Ans ' I Starboard tack, S.W. b. S. 4.138 mi. TAKING DEPARTURES. 401. Explanation. Before losing sight of land, at the beginning of % a voyage, the bearing and distance of some well-known object, as a light-house or headland, is taken, the re- verse bearing and distance of which are entered as the first course and distance on the log board. The. bearing is taken by the compass; but the dis- tance is sometimes estimated by the eye, as can be done with considerable accuracy by navigators of ex- perience. A more correct method of taking a departure is by means of data, obtained by taking the bearing at two different positions of the ship, the distance between these positions being measured by the log. TAKING DEPARTURES. 403 402. Examples. 1. Sailing down the channel, the Eddystone bore N.W. by N., and after running W. S. W. 18 miles, it bore N. by E.; required the course and distance from the Eddy- stone to the place of the last obser- E vation. E = NAE -f N'BE = 4 points. A = 16 points (NAE + BAS) = 7 pts. sin E : sin A : : AB : BE, . . BE= 24.97. 2. At 3 o'clock P. M. the Lizard bore N. by W. and after sailing 7 knots an hour, W. by N. JN., till 6 o'clock, the Lizard bore N. E. f E. ; required the course and distance from the Lizard to the place of the last observation. Ans. S.W. |W., 19.35 mi. 3. In order to get a departure, I observe a headland of known latitude and longitude to bear N. E. by N., and after sailing E. by N. 15 miles, the same headland bore W. N.W. ; required my distance from the headland at each place of observation. Ans. 8.5 mi. and 10.8 mi. Remark. To find the latitude and longitude of a ship by means of celestial observations, requires a knowledge of Nautical Astronomy; but a thorough discussion of this subject would require an amount of space far ex- ceeding our limits. TABLES. I. LOGARITHMS OF NUMBERS, II. NATURAL SINES AND CO-SINES, III. NATURAL TANGENTS AND CO-TANGENTS, IV. LOGARITHMIC SINES AND TANGENTS, V. TRAVERSE TABLE, VI. MISCELLANEOUS TABLE, . VII. MERIDIONAL PAHTS, . VIII. CORRECTIONS TO MIDDLE LATITUDES, . PAOK. . 1 . 24 . 26 . 28 . 73 . 84 . 85 87 Logarithms of Numbers to 100. 1 00000 21 1.32222 41 161278 61 1.78533 81 1.90849 2 0.30103 22 1.34242 42 1.62325 62 1 79239 82 1.91381 3 047712 23 .36173 43 1.63347 63 1.79934 83 1.91908 4 0.60206 24 .38021 44 .64345 64 1.80618 84 1.92428 a 0.69897 25 .39794 45 .65321 65 1.81291 85 1.92942 6 0.77815 26 .41497 46 .66276 66 1.81954 86 1.93450 7 0.84510 27 .43136 47 .67210 67 1.82607 87 .93952 8 0.90309 28 .44716 48 .68124 68 1.83251 88 .94448 9 0.95424 29 .46240 49 .69020 69 1 83885 89 .94939 10 .00000 30 .47712 50 .69897 70 1.84510 90 .95424 11 1.04139 31 .49136 51 .70757 71 -1.85126 91 .95904 12 .07918 32 .50515 52 71600 72 1.85733 92 1.96379 13 .11394 33 .51851 53 .72428 73 1.86332 93 .96848 14 .14613 34 .53148 54 .73239 74 1.86923 94 .97313 15 .17609 35 ,54407 55 .74036 75 1.87506 95 .97772 16 .20412 36> .55630 56 .74819 76 1.88081 96 .98227 17 .23045 37 .56820 57 .75587 77 1.88649 97 .98677 18 .25527 38 .57978 58 .76343 78 1.89209 98 .99123 19 .27875 39 .59106 59 .77085 79 1.89763 99 1.99564 20 .30103 40 1.60206 60 .77815 80 1.90309 100 2.00000 (1) 100-144 LOGARITHMS. 00000-16107 I, 1 2 3 4 5 ! '6 78 9 D. 100 00000 043 087 130 173 217 260 303 346 389 43 101 432 475 518 561 604 647 689 732 775 817 43 102 860 903 945 988 o30 o72 i!5 i57 i99 242 42 103 01284 326 368 410 452 494 536 578 620 662 42 104 703 745 787 828 870 912 953 995 o36 o78 42 105 02119 160 202 243 284 325 366 407 449 490 41 106 531 572 612 653 694 735 776 816 857 898 41 107 938 979 o!9 o60 i()0 i41 18! 222 262 302 40 108 03342 383 423 463 503 54:) 583 623 663 703 40 109 743 782 822 862 902 941 981 o21 o60 lOO 40 110 04139 179 218 258 297 336 i 376 415 454 493 39 111 532 571 610 650 689 727 766 805 844 883 39 112 922 961 999 o38 o77 .i!5 1-34 i92 231 2 69 39 113 05308 346 385 423 461 500 538 576 614 652 38 114 690 729 767 805 843 881 918 956 994 o32 38 115 06070 108 145 183 221 258 296 333 371 408 38 116 446 483 521 558 595 633 670 707 744 781 37 117 819 856 893 930 967 o()4 o41 o78 i!5 iol 37 118 07188 225 262 298 335 372 408 445 482 518 37 119 555 591 628 664 700 737 773 809 846 882 36 120 918 954 990 o27 o63 o99 i35 i71 207 2 43 36 121 08279 314 350 386 422 458 493 529 565 600 36 122 636 672 707 743 778 814 849 884 920 955 35 123 991 o26 06 1 o96 1 32 i67 2 02 237 2 72 s07 35 124 09342 377 412 447 482 517 552 587 621 656 35 125 691 726 760 795 830 864 899 934 968 o03 35 126 10037 072 106 140 175 209 243 278 312 346 34 127 380 415 449 483 517 551 585 619 653 687 34 128 721 755 789 823 857 890 924 958 992 o25 34 129 11059 093 126 160 193 227 261 294 327 361 34 130 394 428 461 494 528 561 594 628 661 694 33 131 727 760 793 826 860 893 926 959 992 o24 33 132 12057 090 123 156 189 222 254 287 320 352 33 133 385 418 450 483 516 548 581 613 646 678 33 134 710 743 775 808 840 872 905 937 969 oOl 32 135 13033 066 098 130 162 194 226 258 290 322 32 136 354 386 418 450 481 513 545 577 609 640 32 137 672 704 735 767 799 830 862 893 92f) 956 32 138 988 o!9 o51 o82 i!4 i45 i76 2 08 239 270 31 139 14301 333 364 395 426 457 489 520 551 582 31 140 613 644 675 706 737 768 799 829 860 891 31 141 922 953 983 o!4 o45 (.76 i06 i37 168- i98 31 142 15229 259 290 320 351 381 412 442 473 503 31 143 534 564 594 625 655 685 715 746 776 806 30 144 836 866 897 927 957 987 o!7 o47 o77 i07 30 y, 1 2 3 4 5 6 7 8 9 D. 145-189 LOGARITHMS. 16137-27852 I, 1 2 3 ' 4 5 6 7 8 9 D, 145 16137 167 197 227 256 286 316 346 376 406 30 146 435 465 495 524 554 584 613 643 673 702 30 147 732 76.1 791 820 850 879v 909 938 967 997 29 148 17026 056 085 n4 143 173 202 231 260 289 29 149 319 348 377 406 435 464 493 522 551 580 29 150 609 638 667 696 725 754 782 811 840 869 29 51 898 926 955 984 o!3 o41 o70 o99 i27 i56 29 152 18184 213 241 270 298 327 355 384 412 441 29 1 53 469 498 526 554 583 611 639 667 696 724 28 154 752 780 808 837 865 893 921 949 977 o05 28 155 19033 061 089 117 145 173 201 229 257 285 28 156 312 340 368 396 424 451 479 507 535 562 28 157 590 618 645 673 700 728 756 783 811 838 28 158 866 893 921 948 976 o03 o30 o58 o85 i!2 27 159 20140 167 194 222 249 276 303 330 358 385 27 160 412 439 466 493 520 548 575 602 629 656 27 161 683 710 737 763 790 817 844 871 898 925 27 162 952 978 o05 o32 o59 o85 i!2 i39 i65 i92 27 163 21219 245 272 299 325 352 378 405 431 458 27 164 484 511 537 564 590 617 643 669 696 722 26 165 748 775 801 827 854 880 906 932 958 985 26 166 22011 037 063 089 115 141 167 194 220 246 26 167 272 298 324 350 376 401 427 453 479 505 26 168 531 557 583 608 634 660 686 712 737 763 26 169 789 814 840 866 891 917 943 968 994 o!9 26 170 23045 070 096 121 147 172 198 223 249 274 25 171 300 325 350 376 401 426 452 477 502 528 25 72 553 578 603 629 654 679 704 729 754 7T9 25 73 805 830 855 880 905 930 955 980 o05 o30 25 174 24055 080 105 130 155 180 204 229 254 279 25 75 304 329 353 378 403 428 452 477 502 527 25 76 551 576 601 625 650 674 699 724 748 773 25 177 797 822 846 871 895 920 944 969 993 o!8 25 178 25042 066 091 115 139 164 188 212 237 261 24 179 285 310 334 358 382 406 431 455 479 503 24 180 527 551 575 600 624 648 672 696 720 744 24 181 768 792 816 840 864 888 912 935 959 983 24 182 26007 031 055 079 102 126 150 174 198 221 24 183 245 269 293 316 340 364 387 411 435 458 24 184 482 505 529 553 576 600 623 647 670 694 24 185 717 741 764 788 811 834 858 881 905 928 23 186 951 975 998 o2l o45 fr68 o91 i!4 1 38 16! 23 187 27184 207 231 254 277 300 323 346 370 393 23 188 416 439 462 485 508 531 554 577 600 623 23 189 646 669 692 715 738 761 784 807 830 852 23 IT. 1 2 3 4 5 6 7 8 9 D, 190-234 LOGARITHMS. 27875-3708^ N, 1 2 3 4 5 6 7 8 9 D. 190 27875 898 921 944 967 989 o!2 o35 o58 08! 23 191 28103 126 149 171 194 217 240 262 285 307 23 192 330 353 375 398 421 443 466 488 511 533 23 193 556' '578 601 623 646 668 691 713 735 758 22 194 780 803 825 847 870 892 914 937 959 981 '22 195 29003 026 048 070 092 115 137 159 181 203 22 196 226 248 270 292 314 336 358 380 403 425 22 197 447 469 491 513 535 5o< 0/9 601 623 645 22 198 667 688 710 732 7.') 4 776 798 820 842 863 22 199 885 907 929 951 973 994 o!6 (.38 oGO 08! 22 200 30103 125 146 168 190 211 233 255 276 298 22 201 320 341 363 384 406 428 449 471 492 514 22 202 535 557 578 600 621 643 664 685 707 728 21 203 750 771 792 814 835 856 878 899 920 942 21 204 963 984 o06 o27 o48 o69 o91 i!2 i33 i54 21 205 31175 197 218 239 260 281 302 323 345 366 21 206 387 408 429 450 471 492 513 534 555 576 21 207 597 618 639 660 681 702 723 744 765 785 21 208 806 827 848 869 890 911 931 952 973 994 21 209 32015 035 056 077 098 118 139 160 181 201 21 210 222 243 263 284 305 325 346 366 387 408 21 211 428 449 469 490 510 531 552 572 593 613 20 212 634 654 675 695 715 736 756 777 797 818 20 213 838 858 879 899 919 940 960 980 oOl .21 20 214 33041 062 082 102 122 143 163 183 203 224 20 215 244 264 284 304 325 345 365 385- 405 425 20 216 445 465 486 506 526 546 566 586 606 626 20 217 646 666 686 706 726 746 766 786 806 826 20 218 846 866 885 905 925 945, 965 985 o05 o25 20 219 34044 064 084 104 124 143 163 183 203 223 20 220 242 262 282 301 321 341 361 380 400 420 20 221 439 459 479 498 518 537 557 577 596 616 20 222 635 655 674 694 713 733 753 772 792 811 19 223 830 850 869 889 908 928 947 967 986 o05 19 224 35025 044 064 083 102 122 141 160 180 199 19 225 218 238 257 276 295 315 334 353 372 392 19 226 . 411 430 449 468 488 507 526 545 564 583 19 227 603 622 641 660 679 698 717 736 755 <4 19 228 793 813 832 851 870 889 908 927 946 965 19 229 984 o03 o21 o40 o59 o78 o97 i!6 i35 i54 19 230 36173 192 211 229 248 267 286 305 324 342 19 231 361 380 399 418 436 455 474 493 511 530 19 232 549 568 586 605 624 642 661 680 698 717 19 233 736 754 773 791 810 829 847 866 884 903 19 234 922 940 959 977 996 o!4 o33 o51 o70 088 18 N, 1 \ 2 3 4 5 6 7 8 9 D, 235-279 LOGARITHMS. 37107-44700 I, 1 2 3 4 5 6 ' c 8 9 D, 235 37107 125 144 162 181 199 218 236 254 273 18 236 291 310 328 346 365 383 401 420 438 457 18 237 475 493 511 530 548 566 585 603 621 639 18 238 658 676 694 712 731 749 767 785 803 822 18 239 840 858 876 894 912 931 949 967 985 o03 18 240 38021 039 057 075 093 112 130 148 166 184 18 241 202 220 238 256 274 292 310 328 346 364 18 242 382 399 417 435 453 471 489 507 525 543 18 243 561 578 596 614 632 650 668 686 703 721 18 244 739 757 775 792 810 828 846 863 881 899 18 245 917 934 952 970 987 o05 o23 o41 o58 o76 18 246 39094 111 129 146 164 182 199 217 235 252 18 247 270 287 305 322 340 358 375 393 410 428 18 248 445 463 480 498 515 533 550 568 585 602 18 249 620 637 655 672 690 707 724 742 759 777 17 250 794 811 829 846 863 881 898 915 933 950 17 251 967 985 o02 o!9 o37 o54 o71 088 i06 i23 17 252 40140 157 175 192 209 226 243 261 278 295 17 253 312 329 346 364 381 398 415 432 449 466 17 254 483 500 518 535 552 569 586 603 620 637 17 255 654 671 688 705 722 739 756 773 790 807 17 256 824 841 858 875 892 909 926 943 960 976 17 257 993 olO o27 o44 06! o78 o95 ill i28 i45 17 258 41162 179 196 212 229 246 263 280 296 313 17 259 --330 347 363 380 397 414 430 447 464 481 17 260 497 514 531 547 564 581 597 614 631 647 17 261 664 681 697 714 731 747 764 780 797 814 17 262 830 847 863 880 896 913 929 946 963 979 16 263 996 o!2 o29 o45 o62 o78 o95 ill i27 i44 16 264 42160 177 193 210 226 243 259 275 292 308 16 265 325 341 357 374 390 406 423 439 455 472 16 266 488 504 521 537 553 570 586 602 619 635 16 267 651 667 684 700 716 732 749 765 781 797 16 268 813 830 846 862 878 894 911 927 943 959 16 269 975 991 o08 o24 o40 o56 o72 088 i04 i20 16 270 43136 152 169 185 201 217 233 249 265 281 16 271 297 313 329 345 361 377 393 409 425 441 16 272 457 473 489 505 521 537 553 569 584 600 16 273 616 632 648 664 680 696 712 727 743 759 16 274 775 791 807 823 838 854 870 886 902 917 16 275 933 949 965 981 996 o!2 (.28 o44 o59 o75 16 276 44091 107 122 138 154 170 185 201 217 232 16 277 248 264 279 295 311 326 342 358 373 389 16 278 404 420 436 451 467 483 498 514 529 545 16 279 560 576 592 607 623 638 654 669 685 700 16 If, 1 2 3 4 5 6 7 8 9 D, S. N. 35. 280-324 LOGARITHMS. 47716-51175 V, 1 2 3 4 5 6 7 8 9 D. 280 44716 731 747 762 778 793 809 824 840 855 15 281 871 886 902 917 932 948 963 979 994 olO 15 282 45025 040 056 ^ 071 086 102 117 133 148 163 15 283 179 194 209 225 240 255 271 286 301 317 15 284 332 347 362 378 393 408 423 439 454 469 15 285 484 500 515 530 545 561 576 591 606 621 15 286 637 652 667 682 697 712 728 743 758 773 15 287 788 803 818. 834 849 864 879 894 909 924 15 288 939 954 969 984 oOO o!5 o30 o45 o60 o75 15 289 46090 105 120 135 150 165 180 195 210 225 15 290 240 255 270 285 300 315 330 345 359 374 15 291 389 404 419 434 449 464 479 494 509 523 15 292 538 553 568 583 598 613 627 642 657 672 15 293 687 702 716 731 746 761 776 790 805 820 15 294 835 850 864 879 894 909 923 938 953 967 15 295 982 997 o!2 o26 o41 o56 o70 o85 lOO i!4 15 296 47129 144 159 173 188 202 217 232 246 261 15 297 276 290 305 319 334 349 363 378 392 407 15 298 422 436 451 465 480 494 509 524 538 553 15 299 567 582 596 611 625 640 654 669 683 698 15 300 712 727 741 756 770 784 799 813 828 842 14 301 857 871 885 900 914 929 943 958 972 986 14 302 48001 015 029 044 058 073 087 101 116 130 14 303 144 159 173 187 202 216 230 244 259 273 14 304 287 302 316 330 344 359 373 387 401 416 14 305 430 444 458 473 487 501 515 530 544 558 14 306 572 586 601 615 629 643 657 671 686 700 14 307 714 728 742 756 770 785 799 813 827 841 14 308 855 869 883 897 911 926 940 954 968 982 14 309 996 olO o24 o38 o52 066 o80 o94 i08 i22 14 310 49136 150 164 178 192 206 220 234 248 262 14 311 276 290 304 318 332 346 360 374 388 402 14 312 415 429 443 457 471 485 499 513 527 541 14 313 554 568 582 596 610 624 638 651 665 679 14 314 693 707 721 734 748 762 776 790 803 817 14 315 831 845 859 872 886 900 914 927 941 955 14 316 969 982 996 nlO o:24 o37 o51 o65 o79 o92 14 317 50106 120 133 147 161 174 188 202 215 229 14 318 243 256 270 284 297 311 325 338 352 365 14 319 379 393 406 420 433 447 461 474 488 501 14 320 515 529 542 556 569 583 596 610 623 637 14 321 651 664 678 691 705 718 732 745 759 772 14 322 786 799 813 826 840 853 866 880 893 907 13 323 920 934 947 961 974 987 oOl o!4 o28 o41 13 324 51055 068 081 095 108 121 135 148 162 175 13 N, 1 2 3 4 5 6 7 8 9 D, 325-369 LOGARITHMS. 51188-58808 I, 1 2 3 4 5 6 7 8 g D. 325 51188 202 215 228 242 255 268 282 295 308 13 326 322 335 348 362 375 388 402 415 428 441 13 327 455 468 481 495 508 521 534 548 561 574 13 328 587 601 614 627 640 654 667 680 693 706 13 329 720 733 746 759 772 786 799 812 825 838 13 330 851 865 878 891 904 917 930 943 957 970 13 331 983 996 o09 o22 o35 o48 06! o75 088 lOl 13 332 52114 127 140 153 166 179 192 205 218 231 13 333 244 257 270 284 297 310 323 336 349 362 13 334 375 388 401 414 427 440 453 466 479 492 13 335 504 517 530 543 556 569 582 595 608 621 13 336 634 647 660 673 686 699 711 724 737 750 13 337 763 776 789 802 815 827 840 853 866 879 13 338 892 905 917 930 943 956 969 982 994 o07 13 339 53020 033 046 058 071 084 097 110 122 135 13 340 148 161 173 186 199 212 224 237 250 263 13 341 275 288 301 314 326 339 352 364 377 390 13 342 403 415 428 441 453 466 479 491 504 517 13 343 529 542 555 567 580 593 605 618 631 643 13 344 656 668 681 694 706 719 732 744 757 769 13 345 782 794 807 820 832 845 857 870 882 895 13 346 908 920 933 945 958 970 983 995 o08 o20 13 347 54033 045 058 070 083 095 108 120 133 145 13 348 158 170 183 195 208 220 233 245 258 270 12 349 283 295 307 320 332 345 357 370 382 394 12 350 407 419 432 444 456 469 481 494 506 518 12 351 531 543 555 568 580 593 605 617 630 642 12 352 654 667 679 691 704 716 728 741 753 765 12 353 777 790 802 814 827 839 851 864 876 888 12 354 900 913 925 937 949 962 974 986 998 oil 12 355 55023 035 047 060 072 084 096 108 121 133 12 356 145 157 169 182 194 206 218 230 242 255 12 357 267 279 291 303 315 328 340 352 364 376 12 358 388 400 413 425 437 449 461 473 485 497 12 359 509 522 534 546 558 570 582 594 606 618 12 360 630 642 654 666 678 691 703 715 727 739 12 361 751 763 775 787 799 811 823 835 847 859 12 362 871 883 895 907 919 931 943 955 967 979 12 363 991 o03 o!5 o27 o38 o50 o62 o74 086 o98 12 364 56110 122 134 146 158 170 182 194 205 217 12 365 229 241 253 265 277 289 301 312 324 336 12 366 348 360 372 384 396 407 419 431 443 455 12 367 467 478 490 502 514 526 538 549 561 573 12 368 585 597 608 620 632 644 656 667 679 691 12 369 703 714 726 738 750 761 773 785 797 808 12 N. 1 2 3 4 5 6 7 8 9 D, 370-414 LOGARITHMS. 56820-61794 N. 1 2 3 4 5 6 7 8 9 D. 37CT 56820 832 844 855 867 879 891 902 914 926 12 371 937 949 961 972 984 - 996 o08 o!9 o31 o43 12 372 57054 066 078 089 101 113 124 136 148 159 12 373 171 183 194 206 217 229 241 252 264 276 12 374 287 299 310 322 334 345 357 368 380 392 12 375 403 415 426 438 449 461 473 484 496 507 12 376 519 530 542 553 565 576 588 600 611 623 12 377 634 646 657 669 680 692 703 715 726 738 11 378 749 761 772 784 795 807 818 830 841 852 11 379 864 875 887 898 910 921 933 944 955 967 11 380 978 990 oOl o!3 o24 o35 o47 o58 o70 08! 11 381 58092 104 115 127 138 149 161 172 184 195 11 382 206 218 229 240 252 263 274 286 297 309 11 383 320 331 343 354 365 377 388 399 410 422 11 384 433 444 456 467 478 490 501 512 524 535 11 385 546 557 569 580 591 602 614 625 636 647 11 386 659 670 681 692 704 715 726 737 749 760 11 387 771 782 794 805 816 827 838 850 861 872 11 388 883 894 906 917 928 939 950 961 973 984 11 389 995 o06 o!7 o28 040 o51 o62 o73 o84 o95 11 390 59106 118 129 140 151 162 173 184 195 207 11 391 218 229 240 251 262 273 284 295 306 318 11 392 329 340 351 362 373 384 395 406 417 428 11 393 439 450 461 472 483 494 506 517 528 539 11 394 550 561 572 583 594 605 616 627 638 649 11 395 660 671 682 693 704 715 726 737 748 759 11 396 770 780 791 802 813 824 835 846 857 868 11 397 879 890 901 912 923 934 945 956 966 977 11 398 988 999 olO o21 o32 o43 o54 060 o76 086 11 399 60097 108 119 130 141 152 -163 173 184 195 11 400 206 217 228 239 249 260 271 282 293 i 304 11 401 314 325 336 347 358 369 379 390 401 412 11 402 423 433 444 455 466 477 487 498 509 520 11 403 531 541 552 563 574 584 595 606 617 627 11 404 638 649 660 670 681 692 703 713 724 735 11 405 746 756 767 778 788 799 810 821 831 842 11 406 853 863 874 885 895 906 917 927 938 949 11 407 959 970 981 991 o02 o!3 o23 o34 o45 o55 11 408 61066 077 087 098 109 119 130 140 151 162 11 409 172 183 194 204 215 225 236 247 257 268 11 410 278 289 300 310 321 331 342 352 363 374 11 411 384 395 405 416 426 437 448 458 469 479 11 412 490 500 511 521 532 542 553 563 574 584 11 413 595 606 616 627 637 648 658 669 679 690 11 414 700 711 721 731 742 752 763 773 784 794 10 N. 1 2 3 4 5 6 7 8 9 D. 415-459 LOGARITHMS. 61805-662G6 N. 1 2 3 4 5 6 7 8 9 D, 415 61805 815 826 836 847 857 868 878 888 899 10 416 909 920 930 941 951 962 972 982 993 o03 10 417 62014 024 034 045 055 066 076 086 097 107 10 418 118 128 138 149 159 170 180 190 201 211 10 419 221 232 242 252 263 273 284 294 304 315 10 420 325 335 346 356 366 377 387 397 408 418 10 421 428 435 449 459 469 480 490 500 511 521 10 422 531 542 552 562 572 583 593 603 613 624 10 423 634 644 655 665 675 685 696 706 716 726 10 424 737 747 757 767 778 788 798 808 818 829 10 425 839 849 859 870 880 890 900 910 921 931 10 426 941 951 961 972 982 992 o02 o!2 o22 o33 10 427 63043 053 063 073 083 094 104 114 124 134 10 428 144 155 165 175 185 195 205 215 225 236 10 429 246 256 266 276 286 296 306 317 327 337 10 430 347 357 367 377 387 397 407 417 428 438 10 431 448 458 468 478 488 498 508 518 528 538 10 432 548 558 568 579 589 599 609 619 629 639 10 433 649 659 669 679 689 699 709 719 729 739 10 434 749 759 769 779 789 799 809 819 829 839 10 435 849 859 869 879 889 899 909 919 929 939 10 436 949 959 969 979 988 998 o08 o!8 o28 o38 10 437 64048 058 068 078 088 098 108 118 128 137 10 438 147 157 167 177 187 197 207 217 227 237 10 439 246 256 266 276 286 296 306 316 326 335 10 440 345 355 365 375 385 395 404 414 424 434 10 441 444 454 464 473 483 493 503 513 523 532 10 442 542 552 562 572 582 591 601 611 621 631 10 443 640 650 660 670 680 689 699 709 719 729 10 444 738 748 758 768 777 787 797 807 816 826 10 445 836 846 856 865 875 885 895 904 914 924 10 446 933 943 953 963 972 982 992 o02 oil o21 10 447 65031 040 050 060 070 079 089 099 108 118 10 448 128 137 147 157 167 176 186 196 205 215 10 449 225 234 244 254 263 273 283 292 302 312 10 450 321 331 341 350 360 369 379 389 398 408 10 451 418 427 437 447 456 466 475 485 495 504 10 452 514 523 533 543 552 562 571 581 591 600 10 453 610 619 629 639 648 658 667 677 686 696 10 454 706 715 725 734 744 753 763 772 782 792 9 455 801 811 820 830 839 849 858 868 877 887 9 456 896 906 916 925 935 944 954 963 973 982 9 457 992 oOl oil o20 o30 o39 o49 o58 068 o77 9 458 66087 096 106 115 124 134 143 153 162 172 9 459 181 191 200 210 219 229 238 247 257 266 9 N, 1 2 3 4 5 6 7 8 9 D, 430-504 LOGARITHMS. 66276-70321 N, 1 2 3 4 5 6 7 8 9 D. 460 66276 285 295 304 314 323 332 342 351 361 9 461 370 380 389 398 408 417 427 436 445 455 9 462 464 474 483 492 502 511 521 530 539 549 9 463 558 567 577 586 596 605 614 624 633 642 9 464 652 661 671 680 689 699 708 717 727 736 9 465 745 755 764 773 783 792 801 811 820 829 9 466 839 848 857 867 876 885 894 904 913 922 9 467 932 941 950 960 969 978 987 997 o06 o!5 9 468 67025 034 043 052 062 071 080 089 099 108 9 469 117 127 136 145 154 164 173 182 191 201 9 470 210 219 228 237 247 256 265 274 284 293 9 471 302 311 321 330 339 348 357 367 376 385 9 472 394 403 413 422 431 440 449 459 468 477 9 473 486 495 504 514 523 532 541 550 560 569 9 474 578 587 596 605 614 624 633 642 651 660 9 475 669 679 688 697 706 715 724 733 742 752 9 476 761 770 779 788 797 806 815 825 834 843 9 477 852 861 870 879 888 897 906 916 925 934 9 478 943 952 961 970 979 988 997 o06 o!5 o24 9 479 68034 043 052 061 070 079 088 097 106 115 9 480 124 133 142 151 160 169 178 187 196 205 9 481 215 224 233 242 251 260 269 278 287 296 9 482 305 314 323 332 341 350 359 368 377 386 9 483 395 404 413 422 431 440 449 458 467 476 9 484 485 494 502 511 520 529 538 547 556 565 9 485 574 583 592 601 610 619 628 637 646 655 9 486 664 673 681 690 699 708 717 726 735 744 9 487 753 762 771 780 789 797 806 815 824 833 9 488 842 851 860 869 878 886 895 904 913 922 9 489 931 940 949 958 966 975 984 993 o02 oil 9 490 69020 028 037 046 055 064 073 082 090 099 9 491 108 117 126 135 144 152 161 170 179 188 9 492 197 205 214 223 232 241 249 258 267 276 9 493 285 294 302 311 320 329 338 346 355 364 9 494 373 381 390 399 408 417 425 434 443 452 9 495 461 469 478 487 496 504 513 522 531 539 9 496 548 557 566 574 583 592 601 609 618 627 9 497 636 644 653 662 671 679 688 697 705 714 9 498 723 732 740 749 758 767 775 784 793 801 9 499 810 819 827 836 845 854 862 871 880 888 9 500 897 906 914 923 932 940 949 958 966 975 9 501 984 992 oOl olO o!8 o27 o36 o44 o53 o62 9 502 70070 079 088 096 105 114 122 131 140 148 9 503 157 165 174 183 191 200 209 217 226 234 9 504 243 252 260 269 278 286 295 303 312 321 9 N. 1 2 3 4 5 6 7 8 9 D, 505-549 LOGARITHMS. 70329-74028 N. 1 2 3 4 5 6 7 8 9 D, 505 70329 338 346 355 364 372 381 389 398 406 9 506 415 424 432 441 449 458 467 475 484 492 9 507 501 509 518 526 535 544 552 561 569 578 9 508 586 595 603 612 621 629 638 646 655 663 9 509 672 680 689 697 706 714 723 731 740 749 9 510 757 766 774 783 791 800 808 817 825 834 9 511 842 851 859 868 876 885 893 902 910 919 9 512 927 935 944 952 961 969 978 986 995 o03 9 513 71012 020 029 037 046 054 063 071 079 088 8 514 096 105 113 122 130 139 147 155 164 172 8 515 181 189 198 206 214 223 231 240 248 257 8 516 265 273 282 290 299 307 315 324 332 341 8 517 349 357 366 374 383 391 399 408 416 425 8 518 433 441 450 458 466 475 483 492 500 508 8 519 517 525 533 542 550 559 567 575 584 592 8 520 600 609 G17 625. 634 642 650 659 667 675 8 521 684 692 700 709 717 725 734 742 750 759 8 522 767 775 784 792 800 809 817 825 834 842 8 523 850 858 867 875 883 892 900 908 917 925 8 524 933 941 950 958 966 975 983 991 999 o08 8 525 72016 024 032 041 049 057 066 074 082 090 8 526 099 107 115 123 132 140 148 156 165 173 8 527 181 189 198 206 214 222 230 239 247 255 8 528 263 272 280 288 296 304 313 321 329 337 8 529 346 354 362 370 378 387 395 403 411 419 8 530 428 436 444 452 460 469 477 485 493 501 8 531 509 518 526 534 542 550 558 567 575 583 8 532 591 599 607 61-6 624 632 640 648 656 665 8 533 673 681 689 697 705 713 722 730 738 746 8 534 754 762 770 779 787 795 803 811 819 827 8 535 835 843 852 860 868 876 884 892 900 908 8 536 916 925 933 941 949 957 965 973 981 989 8 537 997 o06 o!4 o22 o30 n 3 8 o46 o54 o62 o70 8 538 73078 086 094 102 111 119 127 135 143 151 8 539 159 167 175 183 191 199 207 215 223 231 8 540 239 247 255 263 272 280 288 296 304 312 8 541 320 328 336 344 352 360 368 376 384 392 8 542 400 408 416 424 432 440 448 456 464 472 8 543 480 488 496 504 512 520 528 536 544 552 8 544 560 568 576 584 592 600 608 616 624 632 8 545 640 648 656 664 672 679 687 695 703 711 8 546 719 727 735 743 751 759 767 775 783 791 8 547 799 807 815 823 830 838 846 854 862 870 8 548 878 886 894 902 910 918 926 933 941 949 8 549 957 965 973 981 989 997 o05 o!3 o2() o28 8 N, 1 2 3 4 5 6 7 8 9 D. 11 550-594 LOGARITHMS. 74036-77444 N. 1 2 3 4 5 6 7 8 9 D. 550 74036 044 052 060 068 076 084 092 099 107 8 551 115 123 131 139 147 155 162 170 178 186 8 552 194 202 210 218 225 233 241 249 257 265 8 553 273 280 288 296 304 312 320 327 335 343 8 554 351 359 367 374 382 390 398 406 414 421 8 555 429 437 445 453 461 468 476 484 492 500 8 556 507 515 523 531 539 547 554 562 570 578 8 557 586 593 601 609 617 624 632 640 648 656 8 558 663 671 679 687 695 702 710 718 726 733 8 559 741 749 757 764 772 780 788 796 803 811 8 560 819 827 834 842 850 858 865 873 881 889 8 561 896 904 912 920 927 935 943 950 958 966 8 562 974 981 989 997 o05 o!2 o20 o28 o35 o43 8 563 75051 059 066 074 082 089 097 105 113 120 8 564 128 136 143 151 159 166 174 182 .189 197 8 565 205 213 220 228 236 243 251 259 266 274 8 566 282 289 297 305 312 320 328 335 343 351 8 567 358 366 374 381 389 397 404 412 420 427 8 568 435 442 450 458 465 473 481 488 496 504 8 '569 511 519 526 534 542 549 557 565 572 580 8 570 587 595 603 610 618 626 633 641 648 656 8 571 664 671 679 686 694 702 709 717 724 732 8 572 740 747 755 762 770 778 785 793 800 808 8 573 815 823 831 838 846 853 861 868 876 884 8 574 891 899 906 914 921 929 937 944 952 959 8 575 967 974 982 989 997 o05 o!2 o20 o27 o35 8 576 76042 050 057 065 072 080 087 095 103 110 8 577 118 125 133 140 148 155 163 170 178 185 8 578 193 200 208 215 223 230 238 245 253 260 8 579 268 275 283 290 298 305 313 320 328 335 8 580 343 350 358 365 373 380 388 395 403 410 8 581 418 425 433 440 448 455 462 470 477 485 7 582 492 500 507 515 522 530 537 545 552 559 7 583 567 574 582 589 597 604 612 619 626 634 7 584 641 649 656 664 671 678 686 693 701 708 7 585 716 723 730 738 745 753 760 768 775 782 7 586 790 797 805 812 819 827 834 842 849 856 7 587 864 871 879 886 893 901 908 916 923 930 7 588 938 945 953 960 967 975 982 989 997 o04 7 589 77012 019 026 034 041 048 056 063 070 078 7 590 085 093 100 107 115 122 129 137 144 151 7 591 159 166 173 181 188 195 203 210 217 225 7 592 232 240 247 254 262 269 276 283 291 298 7 593 305 313 320 327 335 342 349 357 364 371 7 594 379 386 393 401 408 415 422 430 437 444 7 N, 1 2 3 4 5 6 7 8 g D, 12 595-639 LOGARITHMS. 77452-80611 N. 1 2 3 4 5 6 7 8 9 D, 595 77452 459 466 474 481 488 495 503 510 517 7 596 525 532 539 546 554 561 568 576 583 590 7 597 597 605 612 619 627 634 641 648 656 663 7 598 670 677 685 692 699 706 714 721 728 735 7 599 743 750 757 764 772 779 786 793 801 808 7 600 815 822 830 837 844 851 859 866 873 880 7 601 887 895 902 909 916 924 931 938 945 952 7 602 960 967 974 981 988 996 o03 olO o!7 o25 7 603 78032 039 046 053 061 068 075 082 089 097 7 604 104 111 118 125 132 140 147 154 161 168 7 605 176 183 190 197 204 211 219 226 233 240 7 606 247 254 262 269 276 283 290 297 305 312 7 607 319 326 333 340 347 355 362 369 376 383 7 608 390 398 405 412 419 426 433 440 447 455 7 609 462 469 476 483 490 497 504 512 519 526 7 610 533 540 547 554 561 569 576 583 590 597 7 611 604 611 618 625 633 640 647 654 661 668 7 612 675 682 689 696 704 711 718 725 732 739 7 613 746 753 760 767 774 781 789 796 803 810 7 614 817 824 831 838 845 852 859 866 873 880 7 615 888 895 902 909 916 923 930 937 944 951 7 6J6 958 965 972 979 986 993 oOO o07 o!4 o21 7 617 79029 036 043 050 057 064 071 078 085 092 7 618 099 106 113 120 127 134 141 148 155 162 7 619 169 176 183 190 197 204 211 218 225 232 7 620 239 246 253 260 267 274 281 288 295 302 7 621 309 316 323 330 337 344 351 358 365 372 7 622 379 386 393 400 407 414 421 428 435 442 7 623 449 456 463 470 477 484 491 498 505 511 7 624 518 525 532 539 546 553 560 567 574 581 7 625 588 595 602 609 616 623 630 637 644 650 7 626 657 664 671 678 685 692 699 706 713 720 7 627 727 734 741 748 754 761 768 775 782 789 7 628 796 803 810 817 824 831 837 844 851 858 7 629 865 872 879 886 893 900 906 913 920 927 7 630 934 941 948 955 962 969 975 982 989 996 7 631 80003 010 017 024 030 037 044 051 058 065 7 632 072 079 085 092 099 106 113 120 127 134 7 633 140 147 154 161 168 175 182 188 195 202 7 634 209 216 223 229 236 243 250 257 264 271 7 635 277 284 291 298 305 312 318 325 332 339 7 636 346 353 359 366 373 380 387 393 400 407 7 637 414 421 428 434 441 448 455 462 468 475 7 638 482 489 496 502 509 516 523 530 536 543 7 639 550 557 564 570 577 584 591 598 604 611 7 N. 1 2 3 4 5 6 7 8 9 D, 13 649-684 LOGARITHMS. 80618-83563 t. 1 2 3 4 5 6 7 8 g D. 640 80618 625 632 638 645 652 659 665 672 679 7 641 686 693 699 706 713 720 726 733 740 747 7 642 754 760 767 774 781 787 794 801 808 814 7 643 821 828 835 841 848 855 862 868 875 882 7 644 889 895 902 909 916 922 929 936 943 949 7 645 956 963 969 976 983 990 996 o03 olO o!7 7 646 81023 030 037 043 050 057 064 070 077 084 7 647 090 097 104 111 117 124 131 137 144 151 7 648 158 164 171 178 184 191 198 204 211 218 7 649 224 231 238 245 251 258 265 271 278 285 7 650 291 298 305 311 318 325 331 338 345 351 7 651 358 365 371 378 385 391 398 405 411 418 7 652 425 431 438 445 451 458 465 471 478 485 7 653 491 498 505 511 518 525 531 538 544 551 7 654 558 564 571 578 584 591 598 604 611 617 7 655 624 631 637 644 651 657 664 671 677 684 7 656 690 697 704 710 717 723 730 737 743 750 7 657 757 763 770 776 783 790 796 803 809 816 7 658 823 829 836 842 849 856 862 869 875 882 7 659 889 895 902 908 915 921 928 935 941 948 7 660 954 961 968 974 981 987 994 oOO o07 o!4 7 661 82020 027 033 040 046 053 060 066 073 079 7 662 086 092 099 105 112 119 125 132 138 145 7 663 151 158 164 171 178 184 191 197 204 210 7 664 217 223 230 236 243 249 256 263 269 276 7 665 282 289 295 302 308 315 321 328 334 '341 7 666 347 354 360 367 373 380 387 393 400 406 7 667 413 419 426 432 439 445 452 458 465 471 7 668 478 484 491 497 504 510 517 523 530 536 7 669 543 549 556 562 569 575 582 588 595 601 7 670 607 614 620 627 633 640 646 653 659 666 7 671 672 679 685 692 698 705 711 718 724 730 6 672 737 743 750 756 763 769 776 782 789 795 6 673 802 808 814 821 827 834 840 847 853 860 6. 674 866 872 879 885 892 898 905 911 918 924 6 675 930 937 943 950 956 963 969 975 982 988 6 676 995 oOl o08 o!4 o20 o27 o33 o40 o46 o52 6 677 83059 065 072 078 085 091 097 104 110 117 6 678 123 129 136 142 149 155 161 168 174 181 6 679 187 193 200 206 213 219 225 232 238 245 6 680 251 257 264 270 276 283 289 296 302 308 6 681 315 321 327 334 340 347 353 359 366 372 6 682 378 385 391 398 404 410 417 423 429 436 6 683 442 448 455 461 467 474 480 487 493 499 6 684 506 512 518 525 531 537 544 550 556 563 6 N. 1 2 3 4 5 6 7 8 9 D, 14 685-729 LOGARITHMS. 83569-86323 N. 1 2 3 4 5 6 7 8 g D. 685 83569 575 582 588 594 601 607 613 620 626 6 686 632 639 645 651 658 664 670 677 683 689 6 687 696 702 70S 715 721 727 734 740 746 753 6 688 759 765 771 778 784 790 797 803 809 816 6 689 822 828 835 841 847 853 860 866 872 879 6 690 885 891 897 904 910 916 923 929 935 942 6 691 948 954 960 967 973 979 985 992 998 o04 6 692 84011 017 023 029 036 042 048 055 061 067 6 693 073 080 -086 092 098 105 111 117 123 130 6 694 136 142 148 155 161 167 173 180 186 192 6 695 198 205 211 217 223 230 236 242 248 255 6 696 261 267 273 280 286 292 298 305 311 317 6 697 323 330 336 342 348 354 361 367 373 379 6 698 386 392 398 404 410 417 423 429 435 442 6 699 448 454 460 466 473 479 485 491 497 504 6 700 510 516 522 528 535 541 547 553 559 566 6 701 572 578 584 590 597 603 609 615 621 628 6 702 634 640 646 652 658 665 671 677 683 689 6 703 696 702 708 714 720 726 733 739 745 751 6 704 757 763 770 776 782 788 794 800 807 813 6 705 819 825 831 837 844 850 856 862 868 874 6 706 880 887 893 899 905 911 917 924 930 936 6 707 942 948 9"54 960 967 973 979 985 991 997 6 708 85003 009 016 022 028 034 040 046 052 058 6 709 065 071 077 083 089 095 101 107 114 120 6 710 126 132 138 144 150 156 163 169 175 181 6 711 187 193 199 205 211 217 224 230 236 242 6 712 248 254 260 266 272 278 285 291 297 303 6 713 309 315 321 327 333 339 345 352 358 364 6 714 370 376 382 388 394 400 406 412 418 425 6 715 431 437 443 449 455 461 467 473 479 485 6 716 491 497 503 509 516 522 528 534 540 546 6 717 552 558 564 570 576 582 588 594 600 606 6 718 612 618 625 631 637 643 649 655 661 667 6 719 673 679 685 691 697 703 709 715 721 727 6 720 733 739 745 751 757 763 769 775 781 788 6 721 794 800 806 812 818 824 830 836 842 848 6 722 854 860 866 872 878 884 890 896 902 908 6 723 914 920 926 932 938 944 950 956 962 968 6 724 974 980 986 992 998 o04 oJO o!6 o22 o28 6 725 86034 040 046 052 058 064 070 076 082 088 6 726 094 100 106 112 118 124 130 136 141 147 6 727 153 159 165 171 177 183 189 195 201 207 6 728 213 219 225 231 237 243 249 255 261 267 6 729 273 279 285 291 297 303 308 314 320 326 6 N. 1 2 3 4 5 6 7 8 9 D, 15 730-774 LOGARITHMS. 86332-88925 N. 1 2 3 4 5 6 7 8 9 D. 730 86332 338 344 350 356 362 368 374 380 386 6 731 392 398 404 410 415 421 427 433 439 445 6 732 451 457 463 469 475 481 487 493 499 504 6 733 510 516 522 528 534 540 546 552 558 564 6 734 570 576 581 587 593 599 605 611 617 623 6 735 629 635 641 646 652 658 664 670 676 682 6 736 688 694 700 705 711 717 723 729 735 741 6 737 747 753 759 764 770 776 782 788 794 800 6 738 806 812 817 823 829 835 841 847 853 859 6 739 864 870 876 882 888 894 900 906 911 917 6 740 923 929 935 941 947 953 958 964 970 976 6 741 982 988 994 999 o05 oil o!7 .o23 o29 o35 6 742 87040 046 052 058 064 070 075 081 087 093 6 743 099 105 111 116 122 128 134 140 146 151 6 744 157 163 169 175 181 186 192 198 204 210 6 745 216 221 227 233 239 245 251 256 262 268 6 746 274 280 286 291 297 303 309 315 320 326 6 747 332 338 344 349 355 361 367 373 379 384 6 748 390 396 402 408 413 419 425 431 437 442 6 749 448 454 460 466 471 477 483 489 495 500 6 750 506 512 518 523 529 535 541 547 552 558 6 751 564 570 576 581 587 593 599 604 610 616 G 752 622 628 633 639 645 651 656 "662 668 674 6 753 679 685 691 697 703 708 714 720 726 731 6 754 737 743 749 754 760 766 772 777 783 789 6 755 795 800 806 812 818 823 829 835 841 846 6 756 852 858 864 869 875 881 887 892 898 904 6 757 910 915 921 927 933 938 944 950 955 961 6 758 967 973 978 984 990 996 oOl o07 oJ3 o!8 6 759 88024 030 036 041 047 053 058 064 070 076 6 760 081 087 093 098 104 110 116 121 127 133 6 761 138 144 150 156 161 167 173 178 184 190 6 762 195 201 207 213 218 224 230 235 241 247 6 763 252 258 264 270 275 281 287 292 298 304 6 764 309 3J5 321 326 332 338 343 349 355 360 G 765 366 372 377 383 389 395 400 406 412 417 6 766 423 429 434 440 446 451 457 463 468 474 6 767 480 485 491 497 502 508 513 519 525 530 6 768 536 542 547 553 559 564 570 576 581 587 6 769 593 598 604 610 615 621 627 632 638 643 G 770 649 655 660 666 672 677 683 689 694 700 G 771 705 711 717 722 728 734 739 745 750 756 6 772 762 767 773 779 784 790 795 801 807 812 6 773 818 824 829 835 840 846 852 857 863 868 6 774 874 880 885 891 897 902 908 913 919 925 6 N. 1 2 3 4 5 6 7 8 9 D. 1G 775-819 LOGAKITHMS. 88939-91376 I, 1 2 3 4 5 6 7 8 9 D. 775 88930 936 941 947 953 958 964 969 975 981 6 776 986 992 997 o03 o09 o!4 o20 o25 o31 o37 6 777 89042 048 053 059 064 070 076 081 087 092 6 778 098 104 109 115 120 126 131 137 143 148 6 779 154 159 165 170 176 182 187 193 198 204 6 780 209 215 221 226 232 237 243 248 254 260 6 781 265 271 276 282 287 293 298 304 310 315 6 782 321 326 332 337 343 348 354 360 365 371 6 783 376 382 387 393 398 404 409 415 421 426 6 784 432 437 443 448 454 459 465 470 476 481 6 785 487 492 498 504 509 515 520 526 531 537 6 786 542 548 553 559 564 570 575 581 586 592 6 787 597 603 609 614 620 625 631 636 642 647 6 788 653 658 664 669 675 680 686 691 697 702 6 789 708 713 719 724 730 735 741 746 752 757 6 790 763 768 774 779 785 790 796 801 807 812 5 791 818 823 829 834 840 845 851 856 862 867 5 792 873 878 883 889 894 900 905 911 916 922 5 793 927 933 938 944 949 955 960 966 971 977 5 794 982 988 993 998 o04 o09 o!5 o20 o26 o31 5 795 90037 042 048 053 059 064 069 075 080 086 5 796 091 097 102 108 113 119 124 129 135 140 5 797 146 151 157 162 168 173 179 184 189 195 5 798 200 206 211 217 222 227 233 238 244 249 5 799 255 260 266 271 276 282 287 293 298 304 5 800 309 314 320 325 331 336 342 347 352 358 5 801 363 369 374 380 385 390 396 401 407 412 5 802 417 423 428 434 439 445 450 455 461 466 5 803 472 477 482 488 493 499 504 509 515 520 5 804 526 531 536 542 547 553 558 563 569 574 5 805 580 585 590 596 601 607 612 617 623 628 5 806 634 639 644 650 655 660 666 671 677 682 5 807 687 693 698 703 709 714 720 725 730 736 5 808 741 747 752 757 763 768 773 779 784 789 5 809 795 800 806 811 816 822 827 832 838 843 5 810 849 854 859 865 870 875 881 886 891 897 5 811 902 907 913 918 924 929 934 940 945 950 5 812 956 961 966 972 977 982 988 993 998 o04 5 813 91009 014 020 025 030 036 041 046 052 057 5 814 062 068 073 078 084 089 094 100 105 110 5 815 116 121 126 132 137 142 148 153 158 164 5 816 169 174 180 185 190 196 201 206 212 217 5 817 222 228 233 238 243 249 254 259 265 270. 5 818 275 281 286 291 297 302 307 312 318 323 5 819 328 334 339 344 350 355 360 365 371 376 5 I, 1 2 3 4 5 6 7 8 9 D, 17 820-864. LOGARITHMS. 91381-93697. N. 1 2 3 4 5 6 7 8 9 D, 820 91381 387 392 397 403 408 413 418 424 429 5 821 434 440 445 450 455 461 466 471 477 482 5 822 487 492 498 503 508 514 519 524 529 535 5 823 540 545 551 556 561 566 572 577 582 587 5 824 593 598 603 609 614 619 624 630 635 640 5 825 645 651 656 661 666 672 677 682 687 693 5 826 698 703 709 714 719 724 730 735 740 745 5 827 751 756 761 766 772 777 782 787 793 798 5 828 803 808 814 819 824 829 834 840 845 850 5 829 855 861 866 871 876 882 887 892 897 903 5 830 908 913 918 924 929 934 939 944 950 955 5 831 960 965 971 976 981 986 991 997 o02 o07 5 832 92012 OJ8 023 028 033 038 044 049 054 059 5 833 .065 070 075 080 085 091 096 101 106 111 5 834 117 122 127 132 137 143 148 153 158 163 5 835 169 174 179 184 189 195 200 205 210 215 5 836 221 226 231 236 241 247 252 257 262 267 5 837 273 278 283 288 293 298 304 309 314 319 5 838 324 330 335 340 345 350 355 361 366 371 5 839 376 381 387 392 397 402 407 412 418 423 5 840 428 433 438 443 449 454 459 464 469 474 5 841 480 485 490 495 500 505 511 516 521 526 5 842 531 536 542 547 552 557 562 567 572 578 5 843 583 588 593 598 603 609 614 619 624 629 5 844 634 639 645 650 655 . 660 665 670 675 681 5 845 686 691 696 701 706 711 716 722 727 732 5 846 737 742 747 752 758 763 768 773 778 783 5 847 788 793 799 804 809 814 819 824 829 834 5 848 840 845 850 855 860 865 870 875 881 886 5 849 891 896 901 906 911 916 921 927 932 937 5 850 942 947 952 957 962 967 973 978 983 988 5 851 993 998 o03 o08 o!3 o!8 o24 o29 o34 o39 5 852 93044 049 054 059 064 069 075 080 085 090 5 853 095 100 105 110 115 120 125 131 136 141 5 854 146 151 156 161 166 171 176 181 186 192 5 855 197 202 207 2J2 217 222 227 232 237 242 5 856 247 252 258 263 268 273 278 283 288 293 5 857 298 303 308 313 318 323 328 334 339 344 5 858 349 354 359 364 369 374 379 384 389 394 5 859 399 404 409 414 420 425 430 435 440 445 5 860 450 455 460 465 470 475 480 485 490 495 5 861 500 505 510 515 520 526 531 536 541 546 5 862 551 556 561 566 571 576 581 586 591 596 5 863 601 606 611 616 621 626 631 636 641 646 5 864 651 656 661 666 671 676 682 687 692 697 5 N. 1 2 3 4 5 6 7 8 9 D. 18 865-909 LOGARITHMS. 93702-95899 N. 1 2 3 4 5 6 7 8 9 D, 865 93702 707 712 717 722 727 732 737 742 747 5 866 752 757 762 767 772 777 782 787 792 797 5 867 802 807 812 817 822 827 832 837 842 847 5 868 852 857 862 867 872 877 882 887 892 897 5 869 902 907 912 917 922 927 932 937 942 947 5 870 952 957 962 967 972 977 982 987 992 997 5 871 94002 007 012 017 022 027 032 037 042 047 5 872 052 057 062 067 072 077 082 086 091 096 5 873 101 106 111 116 121 126 131 136 141 146 5 874 151 156 161 166 171 176 181 186 191 196 5 875 201 206 211 216 221 226 231 236 240 245 5 876 250 255 260 265 270 275 280 285 290 295 5 877 300 305 310 315 320 325 330 335 340 345 5 878 349 354 359 364 369 374 379 384 389 394 5 879 399 404 409 414 419 424 429 433 438 443 5 880 448 453 458 463 468 473 478 483 488 493 5 881 498 503 507 512 517 522 527 532 537 542 5 882 547 552 557 562 567 571 576 581 586 591 5 883 596 601 606 611 616 621 626 630 635 640 5 884 645 650 655 660 665 670 675 680 685 689 5 885 694 699 704 709 714 719 724 729 734 738 5 886 743 748 753 758 763 768 T73 778 783 787 5 887 792 797 802 807 812 817 822 827 832 836 5 888 841 846 851 856 861 866 871 876 880 885 5 889 890 895 900 905 910 915 919 924 929 934 5 890 939 944 949 954 959 963 968 973 978 983 5 891 988 993 998 o02 o07 o!2 o!7 o22 o27 o32 5 892 95036 041 046 051 056 061 066 071 075 080 5 893 085 090 095 100 105 109 114 119 124 129 5 894 134 139 143 148 153 158 163 168 173 177 5 895 182 187 192 197 202 207 211 216 221 226 5 896 231 236 240 245 250 255 260 265 270 274 5 897 279 284 289 294 299 303 308 313 318 323 5 898 328 332 337 342 347 352 357 361 366 371 5 899 376 381 386 390 395 400 405 410 415 419 5 900 424 429 434 439 444 448 453 458 463 468 5 901 472 477 482 487 492 497 501 506 511 516 5 902 521 525 530 535 540 545 550 554 559 564 5 903 569 574 578 583 588 593 598 602 607 612 5 904 617 622 626 631 636 641 646 650 655 660 5 905 ' 665 670 674 679 684 689 694 698 703 708 5 906 713 718 722 727 732 737 742 746 751 756 5 907 761 766 770 775 7SO 785 789 794 799 804 5 908 809 813 818 823 828 832 837 842 847 852 5 909 856 861 866 871 875 880 885 890 895 899 5 N. 1 2 3 4 5 6 7 8 9 D, 19 910-954 LOGARITHMS. 95904-97996. ft 1 2 3 4 5 6 7 8 9 D. 910 95904 909 914 918 923 928 933 938 942 947 5 911 952 957 961 966 971 976 980 985 990 995 5 912 999 o04 o09 o!4 o!9 o23 o28 o33 o38 o42 5 913 96047 052 057 061 066 071 076 080 085 090 5 914 095 099 104 109 114 118 123 128 133 137 5 915 142 147 152 156 161 166 171 175 180 185 5 916 190 194 199 204 209 213 218 223 227 232 5 917 237 242 246 251 256 261 265 270 275 280 5 918 284 289 294 298 303 308 313 317 322 327 5 919 332 336 341 346 350 355 360 365 369 374 5 920 379 384 388 393 398 402 407 412 417 421 5 921 426 431 435 440 445 450 454 459 464 468 5 922 473 478 483 487 492 497 501 506 511 515 5 923 520 525 530 534 539 544 548 553 558 562 5 924 567 572 577 581 586 591 595 600 605 609 5 925 614 619 624 628 633 638 642 647 652 656 5 926 661 666 670 675 680 685 689 694 699 703 5 927 708 713 717 722 727 731 736 741 745 750 5 928 755 759 764 769 774 778 783 788 792 797 5 929 802 806 811 816 820 825 830 834 839 844 5 930 848 853 858 862 867 872 876 881 886 890 5 931 895 900 904 909 914 918 923 928 932 937 5 932 942 946 951 956 960 965 970 974 979 984 5 933 988 993 997 o02 o07 oil o!6 o21 o25 o30 5 934 97035 039 044 049 053 058 063 067 072 077 5 935 081 086 090 095 100 104 109 114 118 123 5 936 128 132 137 142 146 151 155 160 165 169 5 937 174 179 183 188 192 197 202 206 211 216 5 938 220 225 230 234 239 243 248 253 257 262 5 939 267 271 276 280 285 290 294 299 .304 308 5 940 313 317 322 327 331 336 340 345 350 354 5 941 359 364 368 373 377 382 387 391 396 400 5 942 405 410 414 419 424 428 433 437 442 447 5 943 451 456 460 465 470 474 479 483 488 493 5 944 497 502 506 511 516 520 525 529 534 539 5 945 543 548 552 557 562 566 571 575 580 585 5 946 589 594 598 603 607 612 617 621 626 630 5 947 635 640 644 649 653 658 663 667 672 676 5 948 681 685 690 695 699 704 708 713 717 722 5 949 727- 731 736 740 745 749 754 759 763 768 5 950 772 777 782 786 791 795 800 804 809 813 5 951 818 823 827 832 836 841 845 850 855 859 5 952 864 868 873 877 882 886 891 896 900 905 5 953 909 914 918 923 928 932 937 941 946 950 5 954 955 959 964 968 973 978 982 987 991 996 5 N. 1 2 3 4 5 6 7 8 9 D, 20 955-999 LOGARITHMS. 98000-99996 N, 1 2 3 4 5 6 7 8 9 D. 955 98000 005 009 014 019 023 028 032 037 041 5 956 046 050 055 059 064 068 073 078 082 087 5 957 091 096 100 105 109 114 118 123 J27 132 5 958 137 141 146 150 155 159 164 168 173 177 5 959 182 186 191 195 200 204 209 214 218 223 5 960 227 232 236 241 245 250 254 259 263 268 5 961 272 277 281 286 290 295' 299 304 308 313 5 962 318 322 327 331 336 340 345 349 354 358 5 963 363 367 372 376 381 385 390 394 399 403 5 964 408 4J2 417 421 426 430 435 439 444 448 5 965 453 457 462 466 471 475 480 484 489 493 4 966 498 502 507 511 516 520 525 529 534 538 4 967 543 547 552 556 561 565 570 574 579 583 4 968 588 592 597 601 605 610 614 619 623 628 4 969 632 637 641 646 650 655 659 664 668 673 4 970 677 682 686 691 695 700 704 709 713 717 4 971 722 726 731 735 740 744 749 753 758 762 4 972 767 771 776 780 784 789 793 798 802 807 4 973 811 816 820 825 829 834 838 843 847 851 4 974 856 860 865 869 874 878 883 887 892 896 4 975 900 905 909 914 918 923 927 932 936 941 4 976 945 949 954 958 963- 967 972 976 981 985 4 977 989 994 998 o03 o07 o!2 o!6 o21 o25 o29 4 978 99034 038 043 047 052 056 061 065 069 074 4 979 078 083 087 092 096 100 105 109 114 118 4 980 123 127 131 136 140 145 149 154 158 162 4 981 167 171 176 180 185 189 193 198 202 207 4 982 211 216 220 224 229 233 238 242 247 251 4 983 255 260 264 269 273 277 282 286 291 295 4 984 300 304 308 313 317 322 326 330 335 339 4 985 344 348 352 357 361 366 370 374 379 383 4 986 388 392 396 401 405 410 414 419 423 427 4 987 432 436 441 445 449 454 458 463 467 471 4 988 476 480 484 489 493 498 502 506 511 515 4 989 520 524 528 533 537 542 546 550 555 559 4 990 564 568 572 577 581 585 590 594 599 603 4 991 607 612 616 621 625 629 634 638 642 647 4 992 651 656 660 664 669 673 977 682 686 691 4 993 695 699 704 708 712 717 721 726 730 734 4 994 739 743 747 752 756 760 765 769 774 778 4 995 782 787 791 795 800 804 808 813 817 822 4 996 826 830 835 839 843 848 852 856 861 865 4 997 870 874 878 883 887 891 896 900 904 909 4 998 913 917 922 926 930 935 939 944 948 952 4 999 957 961 965 970 974 978 983 987 991 996 4 N. 1 2 3 4 5 6 7 8 9 D. 1000-1044 LOGARITHMS. 00000-01907 N. 1 2 3 4 5 6 7 8 9 D. 1000 00000 004 009 013 017 022 026 030 035 039 4 1001 043 048 052 056 061 065 069 074 078 082 4 1002 087 091 095 100 104 108 113 117 121 126 4 1003 130 134 139 143 147 152 156 160 165 169 4 1004 173 178 182 186 191 195 199 204 208 212 4 1005 217 221 225 230 234 238 243 247 251 255 4 1006 260 264 268 273 277 281 286 290 294 299 4 1007 303 307 312 316 320 325 329 333 337 342 4 1008 346 350 355 359 363 368 372 376 381 385 4 1009 389 393 398 402 406 411 415 419 424 428 4 1010 432 436 441 445 449 454 458 462 467 471 4 1011 475 479 484 488 492 497 501 505 509 514 4 1012 518 522 527 531 535 540 544 548 552 557 4 1013 561 565 570 574 578 582 587 591 595 600 * 1014 604 608 612 617 621 625 629 634 638 642 4 1015 647 651 655 659 664 668 672 677 681 685 4 1016 689 694 698 702 706 711 715 719 724 728 4 1017 732 736 741 745 749 753 758 762 766 771 4 1018 775 779 783 788 792 796 800 805 809 813 4 1019 817 822 826 830 834 839 843 847 852 856 4 1020 860 864 869 873 877 881 886 890 894 898 4 1021 903 907 911 915 920 924 928 932 937 941 4 1022 945 949 954 958 962 966 971 975 979 983 4 1023 988 992 996 oOO o05 o09 o!3 o!7 o22 o26 4 1024 01030 034 038 043 047 051 055 060 064 068 4 1025 072 077 081 085 089 094 098 102 106 111 4 1026 115 119 123 127 132 136 140 144 149 153 4 1027 157 161 166 170 174 178 182 187 191 195 4 1028 199 204 208 212 216 220 225 229 233 237 4 1029 242 246 250 254 258 263 267 271 275 280 4 1030 284 288 292 296 301 305 309 313 317 322 4 1031 326 330 334 339 343 347 351 355 360 364 4 1032 368 372 376 381 385 389 393 397 402 406 4 1033 410 414 418 423 427 431 435 439 444 448 4 1034 452 456 460 465 469 473 477 481 486 490 4 1035 494 498 502 507 511 515 519 523 528 532 4 1036 536 540 544 549 553 557 561 565 569 574 4 1037 578 582 586 590 595 599 603 607 611 616 4 1038 620 624 628 632 636 641 645 649 653 657 4 1039 662 666 670 674 678 682 687 691 695 699 4 1040 703 708 712 716 720 724 728 733 737 741 4 1041 745 749 753 758 762 766 770 774 778 783 4 1042 787 791 795 799 803 808 812 816 820 824 4 1043 828 833 837 841 845 849 853 868 862 866 4 1044 870 874 878 883 887 891 895 899 903 907 4 N, 1 2 3 4 5 6 7 8 9 D. 1045-1089 LOGARITHMS. 01912-03739 N. 1 2 3 4 5 6 7 8 9 D. 1045 01912 916 920 ( ,24 928 932 937 941 945 949 4 1046 953 957 961 %6 970 974 978 982 986 991 4 1047 995 999 o03 o07 oil o!5 o20 o24 o28 o32 4 1048 02036 040 044 049 053 057 061 065 069 073 4 1049 078 082 086 090 094 098 102 107 111 115 4 1050 119 123 127 131 135 140 144 148 152 156 4 1051 160 164 169 173 177 181 185 189 193 197 4 1052 202 206 210 214 218 222 226 230 235 239 4 1053 243 247 251 255 259 263 268 272 276 280 4 1054 284 288 292 296 301 305 309 313 317 321 4 1055 325 329 333 338 342 346 350 354 358 362 4 1056 366 371 375 379 383 387 391 395 399 403 4 1057 407 412 416 420 424 428 432 436 440 444 4 1058 449 453 457 461 465 469 473 477 481 485 4 1059 490 494 498 502 506 510 514 518 522 526 4 1060 531 535 539 543 547 551 555 559 563 567 4 1061 572 576 580 584 588 592 596 600 604 608 4 1062 612 617 621 625 629 633 637 641 645 649 4 1063 653 657 661 666 670 674 678 682 686 690 4 1064 694 698 702 706 710 715 719 723 727 731 4 1065 735 739 743 747 751 755 759 763 768 772 4 1066 776 780 784 788 792 796 800 804 808 812 4 1067 816 821 825 829 833 837 841 845 849 853 4 1068 857 861 865 869 873 877 882 886 890 894 4 1069 898 902 906 910 914 918 922 926 930 934 4 1070 938 942 946 951 955 959 963 967 971 975 4 1071 979 983 987 991 995 999 o03 o07 oil o!5 4 1072 03019 024 028 032 036 040 044 048 052 056 4 1073 060 064 068 072 076 080 084 088 092 096 4 1074 100 104 109 113 117 121 125 129 133 137 4 1075 141 145 149 153 157 161 165 169 173 177 4 1076 181 185 189 193 197 201 205 209 214 218 4 1077 222 226 230 234 238 242 246 250 254 258 4 1078 262 266 270 274 278 282 286 290 294 298 4 1079 302 306 310 314 318 322 326 330 334 338 4 1080 342 346 350 354 358 362 366 371 375 379 4 1081 383 387 391 395 399 403 407 411 415 419 4 1082 423 427 431 435 439 443 447 451 455 459 4 1083 463 467 471 475 479 483 487 491 495 499 4 1084 503 507 511 515 519 523 527 531 535 539 4 1085 543 547 551 555 559 563 567 571 575 579 4 1086 583 587 591 595 599 603 607 611 615 619 4 1087 623 627 631 635 639 643 647 651 655 659 4 1088 663 667 671 675 679 683 687 691 695 699 4 1089 703 707 711 715 719 723 727 731 735 739 4 TS. 1 2 3 4 5 6 7 8 9 D, 23 II. NATURAL SINES. Deg, 0' 10' 20' 30! 40' 50' Deg, 00000 00291 00582 00873 01164 01454 01745 89 1 01745 02036 02327 02618 02908 03199 03490 88 2 03490 03781 04071 04362 04653 04943 05234 87 3 05234 05524 05814 06105 06395 06685 06976 86 4 06976 07266 07556 07846 08136 08426 08716 85 5 08716 09005 09295 09585 09874 10164 10453 84 6 10453 10742 11031 11320 11609 11898 12187 83 * 7 12187 12476 12764 13053 13341 13629 13917 82 8 13917 14205 14493 14781 15069 15356 15643 81 9 15643 15931 16218 16505 16792 17078 17365 80 10 17365 17651 17937 18224 18509 18795 19081 79 11 19081 19366 19652 19937 20222 20507 20791 78 12 20791 21076 21360 21644 21928 22212 22495 77 13 22495 22778 23062 23345 23627 23910 24192 76 14 24192 24474 24756 25038 25320 25601 25882 75 15 25882 26163 26443 26724 27004 27284 27564 74 16 27564 27843 28123 28402 28680 28959 29237 73 17 29237 29515 29793 30071 30348 30625 30902 72 18 30902 31178 31454 31730 32006 32282 32557 71 19 32557 32832 33106 33381 33655 33929 34202 70 20 34202 34475 34748 35021 35293 35565 35837 69 21 35837 36108 36379 36650 36921 37191 37461 68 22 37461 37730 37999 38268 38537 38805 39073 67 23 39073 39341 39608 39875 40141 40408 40674 66 24 40674 40939 41204 41469 41734 41998 42262 65 25 42262 42525 42788 43051 43313 43575 43837 64 26 43837 44098 44359 44620 44880 45140 45399 63 27 45399 45658 45917 46175 46433 46690 46947 62 28 46947 47204 47460 47716 47971 48226 48481 61 29 48481 48735 48989 49242 49495 49748 50000 60 30 50000 50252 50503 50754 51004 51254 51504 59 31 51504 51753 52002 52250 52498 52745 52992 58 32 52992 53238 53484 53730 53975 54220 54464 57 33 54464 54708 54951 55194 55436 55678 55919 56 34 55919 56160 56401 56641 56880 57119 57358 55 35 57358 57596 57833 58070 58307 58543 58779 54 36 58779 59014 59248 59482 59716 59949 60182 53 37 60182 60414 60645 60876 61107 61337 61566 52 38 61566 61795 62024 62251 62479 62706 62932 51 39 62932 63158 63383 63608- 63832 64056 64279 50 40 64279 64501 64723 64945 65166 65386 65606 49 41 65606 65825 66044 66262 66480 66697 66913 48 42 66913 67129 67344 67559 67773 67987 68200 47 43 68200 68412 68624 68835 69046 69256 69466 46 44 69466 69675 69883 70091 70298 70505 70711 45 Deg, 5<y 4(X 3(y 20' 10' (X .Deg, NATURAL COSINES. II. NATURAL SINES. Deg, 0' 10' 20' 30' 40' 50' Deg- 45 70711 70916 71121 71325 71529 71732 71934 44 46 71934 72136 72337 72537 72737 72937 73135 43 47 73135 73333 73531 73728 73924 74120 74314 42 48 74314 74509 74703 74896 75088 75280 75471 41 49 75471 75661 75851 76041 76229 76417 76604 40 50 76604 76791 76977 77162 77347 77531 77715 39 51 77715 77897 78079 78261 78442 78622 78801 38 52 78801 78980 79158 79335 79512 79688 79864 37 53 79864 80038 80212 80386 80558 80730 80902 36 54 80902 81072 81242 81412 81580 81748 81915 35 55 81915 82082 82248 82413 82577 82741 82904 34 56 82904 83066 83228 83389 83549 83708 83867 33 57 83867 84025 84182 84339 84495 84650 84805 32 58 84805 84959 85112 85264 85416 85567 85717 31 59 85717 85866 86015 86163 86310 86457 86603 30 60 86603 86748 86892 87036 87178 87321 87462 29 61 87462 87603 87743 87882 88020 88158 88295 28 62 88295 88431 88566 88701 88835 88968 89101 27 63 89101 89232 89363 89493 89623 89752 89879 26 64 89879 90007 90133 90259 90383 90507 90631 25 65 90631 90753 90875 90996 91116 91236 91355 24 66 91355 91472 91590 91706 91822 91936 92050 23 67 92050 92164 92276 92388 92499 92609 92718 22 68 92718 92827 92935 93042 93148 93253 93358 21 69 93358 93462 93565 93667 93769 93869 93969 20 70 93969 94068' 94167 94264 94361 94457 94552 19 71 94552 94646 94740 94832 94924 95015 95106 18 72 95106 95195 95284 95372 95459 95545 95630 17 73 95630 95715 95799 95882 95964 96046 96126 16 74 96126 96206 96285 96363 96440 96517 96593 15 75 96593 96667 96742 96815 96887 96959 97030 14 76 97030 97100 97169 97237 97304 97371 97437 13 77 97437 97502 97566 97630 97692 97754 97815 12 78 97815 97875 97934 97992 98050 98107 98163 11 79 98163 98218 98272 98325 98378 98430 98481 10 80 98481 98531 98580 98629 98676 98723 98769 9 81 98769 98814 98858 98902 98944 98986 99027 8 82 99027 99067 99106 99144 99182 99219 99255 7 83 99255 99290 99324 99357 99390 99421 99452 6 84 99452 99482 99511 99540 99567 99594 99619 5 85 99619 99644 99668 99692 99714 99736 99756 4 86 99756 99776 99795 99813 99831 99847 99863 3 87 99863 99878 99892 99905 99917 99929 99939 2 88 99939 99949 99958 99966 99973 99979 99985 1 89 99985 99989 99993 99996 99998 99999 1.0000 Deg. 50' 40' 30' 20' 10' 0' Deg, 25 NATURAL COSINES. Ill NATURAL TANGENTS. Deg, 0' 10' 20' 30' 40' 50' Deg, 00000 00291 00582 00873 01164 01455 01746 89 1 01746 02036 02328 02619 02910 03201 03492 88 2 03492 03783 04075 04366 04658 04949 05241 87 3 05241 05533 05824 06116 06408 06700 06993 86 4 06993 07285 07578 07870 08163 08456 08749 85 5 08749 09042 09335 09629 09923 10216 10510 84 6 10510 10805 11099 11394 11688 11983 12278 83 7 12278 12574 12869 13165 13461 13758 14054 82 8 14054 14351 14648 14945 15243 15540 15838 81 9 15838 16137 16435 16734 17033 17333 17633 80 10 17633 17933 18233 18534 18835 19136 19438 79 11 19438 19740 20042 20345 20648 20952 21256 78 12 21256 21560 21864 22169 22475 22781 23087 77 13 23087 23393 23700 24008 24316 24624 24933 76 14 24933 25242 25552 25862 26172 26483 26795 75 15 26795 27107 27419 27732 28046 28360 28675 74 16 28675 28990 29305 29621 29938 30255 30573 73 17 30573 30891 31210 31530 31850 32171 32492 72 18 32492 32814 33136 33460 33783 34108 34433 71 19 34433 34758 35085 35412 35740 36068 36397 70 20 36397 36727 37057 37388 37720 38053 38386 69 21 38386 38721 39055 39391 39727 40065 40403 68 22 40403 40741 41081 41421 41763 42105 42447 67 23 42447 42791 43136 43481 43828 44175 44523 66 24 44523 44872 45222 45573 45924 46277 46631 65 25 46631 46985 47341 47698 48055 48414 48773 64 26 48773 49134 49495 49858 50222 50587 50953 63 27 50953 51319 51688 52057 52427 52798 53171 62 28 53171 53545 53920 54296 54673 55051 55431 61 29 55431 55812 56194 56577 56962 5734$ 57735 60 30 57735 58124 58513 58905 59297 59691 60086 59 31 60086 60483 60881 61280 61681 62083 62487 58 32 62487 62892 63299 63707 64117 64528 64941 57 33 64941 65355 65771 66189 66608 67028 67451 56 34 67451 67875 68301 68728 69157 69588 70021 55 35 70021 70455 70891 71329 71769 72211 72654 54 36 72654 73100 73547 73996 74447 74900 75355 53 37 75355 75812 76272 76733 77196 77661 78129 52 38 78129 78598 79070 79544 80020 80498 80978 51 39 80978 81461 81946 82434 82923 83415 83910 50 40 83910 84407 84906 85408 85912 86419 86929 49 41 86929 87441 87955 88473 88992 89515 90040 48 42 90040 90569 91099 91633 92170 92709 93252 47 43 93252 93797 94345 94896 95451 96008 96569 46 44 96569 97133 97700 98270 98843 99420 1.00000 45 Deg. 50' 4<y 30' 20' 10' 0' Deg. NATURAL COTANGENTS. III. NATURAL TANGENTS. Deg, 0' 10' 20' 30' 40' 50' Deg, 45 1.00000 1.00583 1.01170 1.01761 1.02355 1.02952 1.03553 44 46 1.03553 1.04158 1.04766 1.05378 1.05994 1.06613 1.07237 43 47 1.07237 1.07864 1.08496 1.09131 1.09770 1.10414 1.11061 42 48 1.11061 1.11713 1.12369 1.13029 1.13694 1.14363 1.15037 41 49 1.15037 1.15715 1.16398 1.17085 1.17777 1.18474 1.19175 40 50 1.19175 1.19882 1.20593 1.21310 1.22031 1.22758 1.23490 39 51 1 .23490 124227 1.24969 1.25717 1.26471 1.27230 1.27994 38 52 1.27994 1.28764 1.29541 1.30323 1.31110 1.31904 1.32704 37 53 1.32704 1.33511 1.34323 1.35142 1.35968 1.36800 1.37638 36 54 1.37638 1.38484 1.39336 1.40195 1.41061 1.41934 1.42815 35 55 1.42815 1.43703 1.44598 1.45501 1.46411 1.47330 1.48256 34 56 1.48256 1.49190 1.50133 1.51084 1.52043 1.53010 1.53987 33 57 1.53987 1.54972 1.55966 1.56969 1.57981 1.59002 1.60033 32 58 1.60033 1.61074 1.62125 1.63185 1.64256 1.65337 1.66428 31 59 1.66428 1.67530 1.68643 1.69766 1.70901 1.72047 1.73205 30 60 1.73205 1.74375 1.75556 1.76749 1.77955 1.79174 1.80405 29 61 1.80405 1.81649 1.82906 1.84177 1.85462 1.86760 1.88073 28 62 1.88073 1.89400 1.90741 1.92098 1.93470 1.94858 1.96261 27 63 1.96261 1.97680 1.99116 2.00569 2.02039 2.03526 2.05030 26 64 2.05030 2.06553 2.08094 2.09654 2.11233 2.12832 2.14451 25 65 2.14451 2.16090 2.17749 2.19430 2.21132 2.22857 2.24604 24 66 2.24604 2.26374 2.28167 2.29984 2.31826 2.33693 2.35585 23 67 2.35585 2.37504 2.39449 2.41421 2.43422 2.45451 2.47509 22 68 2.47509 2.49597 2.51715 2.53865 2.56046 2.58261 2.60509 21 69 2.60509 2.62791 2.65109 2.67462 2.69853 2.72281 2.74748 20 70 2.74748 2.77254 2.79802 2.82391 2.85023 2.87700 2.90421 19 71 2.90421 2.93189 2.96004 2.9886<S 3.01783 3.04749 3.07768 18 72 3.07768 3.10842 3.13972 3.17159 3.20406 3.23714 3.27085 17 73 3.27085 3.30521 3.34023 3.37594 3.41236 3.44951 3.48741 16 74 3.48741 3.52609 3.56557 3.60588 3.64705 3.68909 3.73205 15 75 3.73205 3.77595 3.82083 3.86671 3.91364 3.96165 4.01078 14 76 4.01078 4.06107 4.11256 4.16530 4.21933 4.27471 4.33148 13 77 4.33148 4.38969 4.44942 4.51071 4.57363 4.63825 4.70463 12 78 4.70463 4.77286 4.84300 4.91516 4.98940 5.06584 5.14455 11 79 5.14455 5.22566 5.30928 5.39552 5.48451 5.57638 6.67128 10 80 5.67128 5.76937 5.87080 5.97576 6.08444 6.19703 6.31375 9 81 6.31375 6.43484 6.56055 6.69116 6.82694 6.96823 7.11537 8 82 7.11537 7.26873 7.42871 7.59575 7.77035 7.95302 8.14435 7 83 8.14435 8.34496 8,55555 8.77689 9.00983 9.2553C 9.51436 6 84 9.51436 9.78817 10.0780 10.3854 10.7119 11.0594 11.4301 5 85 11.4301 11.8262 12.2505 12.7062 13.1969 13.7267 14.3007 4 86 14.3007 14.9244 15.6048 16,3499 17.1693 18.0750 19.0811 3 87 19.0811 20.2056 21.4704 22.9038 24.5418 26.4316 28.6363 2 88 28.6363 31.2416 34,3678 38.1885 42.9641 49.1039 57.2900 1 89 57.2900 68.7501 85.9398 114.589 171.885 343.774 00 Deg 50' 40' 30' 20' 10' 0' Deg, 27 NATURAL COTANGENTS. TABLE IV. LOGARITHMIC 31. Sine. Dl" THUS. Dl" M. M. Sine. Dl" | Tang. Dl" j M. o 00 00 60 S.24186^ ft 8.24192 120 60 1 2 3 4 5 6 7 6.46373 76476 94085 7.06579 16270 24188 30882 502 293 208 162 132 112 6.46373 76476 94085 7.06579 16270 24188 30882 502 293 208 162 132 112 59 58 57 56 55 54 53 1 2 3 4 5 6 7 24903 25609 26304 2t)9b8 27661 28324 28977 11.8 11.6 11.4 11.2 11.0 10.9 24910 25616 26312 26996 27669 28332 28986 11.8 11.6 11.4 11.2 11.0 10.9 59 58 57 56 55 54 53 8 9 10 11 12 36682 41797 46373 7.50512 54291 96.7 85.2 76.3 69.0 63.0 36682 41797 46373 7.50512 54291 85.2 76.3 69.0 63.0 57 9 52 51 50 49 48 8 9 10 11 12 29621 30255 30879 8.31495 32103 10.6 10.4 10.3 10.1 29629 30263 30888 8.31505 32112 10.7 10.6 10.4 10.3 10.1 52 51 50 49 48 13 14 57767 60985 53.6 57767 60986 53.6 47 46 13 14 32702 33292 9.85 32711 33302 9.85 47 46 15 16 17 18 19 20 21 63982 66784 69417 71900 74248 76475 7.78594 49.9 46.7 43.9 41.4 39.1 37.1 35.3 63982 667*85 69418 71900 74248 76476 7.78595 49.9 46.7 43.9 41.4 39.1 37.1 35.3 45 44 43 42 41 40 39 15 16 17 18 19 20 21 33875 34450 35018 35578 36131 36678 8.37217 9.71 9.59 9.46 9.34 9.22 9.10 8.99 33886 34461 35029 35590 36143 36689 8.37229 9.72 9.59 9.47 9.35 9.22 9.11 9.00 45 44 43 42 41 40 39 22 23 80615 82545 33.7 32.2 80615 82546 66. ( 32.2 38 37 22 23 37750 38276 8.77 37762 38289 .00 8.78 38 37 24 25 26 27 28 29 30 31 32 33 84393 86166 87870 89509 91088 92612 94084 7.95508 96887 98223 30.8 29.5 28.4 27.3 26.3 25.4 24.5 23.7 23.0 22.3 84394 86167 87871 89510 91089 92613 94086 7.95510 96889 98225 30.8 29.5 28.4 27.3 26.3 25.4 24.5 23.7 23.0 22.3 36 35 34 33 32 31 30 29 28 7.1 24 25 26 27 28 29 30 31 32 33 38796 39310 39818 40320 40816 41307 41792 8.42272 42746 43216 8.67 8.56 8.46 8.37 8.27 8.18 8.09 8.00 7.91 7.S2 38S09 39323 39832 40334 40830 41321 41807 8.42287 42762 43232 8.67 8.57 8.47 8.37 8.28 8.18 8.09 8.0t) 7.91 7.83 36 35 34 33 32 31 30 29 28 27 34 35 36 37 38 39 40 99520 8.00779 02002 03192 04350 05478 06578 21.6 21.0 20.4 19.8 19.3 18.8 18.3 99522 8.00781 02004 03194 04353 05481 06581 21.6 21.0 20.4 19.8 19.3 18.8 18.3 26 25 24 23 22 21 SO 34 35 36 37 38 39 40 43680 44139 44594 45044 45489 45930 46366 7.74 7.66 7.58 7.50 7.42 7.35 7.27 43696 44156 44611 45061 45507 45948 46385 7.75 7.66 7.58 7.50 7.43 7.35 7.28 26 25 24 23 22 21 20 41 42 8.07650 08696 17.9 17.4 8.07653 08700 17.9 17.4 19 18 41 42 8.46799 47226 f'f J 8.46817 7ftJ 47245 7.21 7.13 19 18 43 44 45 46 47 48 49 50 51 09718 10717 11693 12647 13581 14495 15391 16268 8.17128 17.0 16.6 16.3 15.9 15.6 15.2 14.9 14.6 14.3 09722 10720 11696 12651 13585 14500 15395 16273 8.17133 17.0 16.6 16.3 15.9 15.6 15.2 14.9 14.6 14.3 17 16 15 14 13 12 11 10 9 43 44 45 46 47 48 49 50 51 47650 48069 48485 48896 49304 49708 50108 50504 8.50897 6.99 6.92 6.86 6.79 6.73 6.67 6.61 6.55 ft 4Q 476C9 48089 48505 48917 49325 49729 50130 50527 8.50920 7.00 6.93 6.87 6.80 6.74 6.68 6.62 6.55 6P,n 17 16 15 14 13 12 11 10 9 52 53 54 55 56 57 58 59 60 17971 18798 19610 20407 21189 21958 22713 23456 24186 13.8 13.5 13.3 13.0 12.8 12.6 12.4 12.2 17976 18804 19616 20413 21195 21964 22720 23462 24192 13.8 13.5 13.3 13.0 12.8 12.6 12.4 12.2 8 7 6 5 4 3 2 1 52 53 54 55 56 57 58 59 60 51287 51673 52055 52434 52810 53183 53552 53919 54282 6.43 6.37 6.32 6.26 6.21 6.16 6.11 6.05 51310 51696 52079 52459 52835 53208 53578 53945 54308 6.44 6.38 6.33 6.27 6.22 6.17 6.11 6.06 8 7 6 5 4 3 2 1 M. Cosine. Dl" Cotang. Dl" M. M. Cosine. Dl" Cotang. Dl" M 89 SINES AND TANGENTS. M. Sine. | Dl" THUS. Dl" M. .M Sim-. Dl" Tang. Dl" M. 8.54282 8.54308. fi 0] 60 18.71881) 8.71940 1 54642.?'^ 54669 r'fw, 59 1 72120 *]" 72181 H? 59 2 549991 TO , 55027 ;^; 58 2 7285Q 3 72420 r": 58 3 55354 55382 I?-*} 57 3 72597 |*Jjj 72659 5.yn r y Q Q<S h ' 4 5 55705| .'Q? 56054 1 j?yl 55734 56083 5.82 ^ 77 56 55 4 5 73069 \t 72896 73132 o. yo 3.93 Q-l 56 55 6 56400 1 56429 "'I' 54 6 73303 rXX 73366 d91 54 7 567431 M? 56/73-12 53 7 73535 73600 3.89 53 8 57084 a.o/ 571141 Hg 52 8 73767 3.86 73832 3.87 52 9 10 57421 57757 5.63 5.59 57452 Z'XJ 57788 X'?: 51 50 9 10 73997 74226 3.84 3.82 74063 74292 3.85 3.83 51 50 11 8.58089 5.54 8.58121 49 11 8.74454 3.80 8.74521 3.81 49 12 58419 5.50 t* A A 58451 ir;?' 48 12 74680 3.78 9 7 . 74748 3.79 3T7 48 13 58747 D.4O 58779 J1 47 13 74906 .7o 74974 .77 47 14 59072 5.42 59105 r*JJ 46 14 75130 3.74 75199 3.75 46 15 59395 5.38 59428 ir",r 45 15 7i353 3.72 75423 3.73 45 16 59715 5.34 59749 .', 44 16 75575 3.70 75645 3.71 44 17 18 19 20 21 22 23 24 25 26 27 60033 60349 60662 60973 8.61282 61589 61894 62196 62497 62795 63091 5.30 5.26 5.22 5.19 5.15 5.11 5.08 5.04 5.01 4.97 4.94 60068 60384 60698 61009 8.61319 61626 61931 62234 62535 62834 63131 5.27 5.23 5.19 5.16 5.12 5.08 5.05 5.02 4.98 4.95 43 42 41 40 39 38 37 36 35 34 33 17 18 19 20 21 22 23 24 25 26 27 75795 76015 76234 76451 8.76667 76883 77097 77310 77522 77733 77943 3.68 3.66 3.64 3.62 3.61 3.59 3.57 3.55 3.53 3.52 3.50 75867 76087 76306 76525 8.76742 76958 77173 77387 77600 77811 78022 3.69 3.67 3.65 3.64 3.62 3.60 3.58 3.57 3.55 3.53 3.51 43 42 41 40 39 38 37 36 35 34 33 28 29 30 31 32 33 34 35 37 38 39 41) 41 42 43 44 45 46 47 48 49 50 51 52 53 63385 63678 63968 8.64256 64543 64827 65110 65391 65670 65947 66223 66497 66769 8.67039 67308 67575 67841 68104 68367 68627 68886 69144 69400 8.69654 69907 70159 4.90 4.87 4.84 4.81 4.78 4.74 4.71 4.68 4.65 4.62 4.59 4.56 4.53 4.51 4.48 4.45 4.42 4.40 4.37 4.34 4.32 4.29 4.27 4.24 4.22 4.19 4*1 7 63426 63718 64009 8.64298 64585 64870 65154 65435 65715 65993 66269 66543 66816 8.67087 67356 67624 67890 68154 68417 68678 68938 69196 69453 8.69708 69962 70214 4.91 4.88 4.85 4.82 4.78 4.75 4.72 4.69 4.66 4.63 4.60 4.57 4.54 4.52 4.49 4.46 4.43 4.41 4.38 4.35 4.33 4.30 4.28 4.25 4.23 4.20 A 1 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 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 78152 78360 78568 8.78774 78979 79183 79386 79588 79789 79990 80189 80388 80585 8.80782 80978 81173 81367 81560 81752 81944 82134 82324 82513 8.82701 82888 83075 3.48 3.47 3.45 3.43 3.42 3.40 3.39 3.37 3.35 3.34 3.32 3.31 3.29 3.28 3.26 3.25 3 23 3.22 3.20 3.19 3.18 3.16 3.15 3.13 3.12 3.11 78232 78441 78649 8.78855 79061 79266 79470 79673 79875 80076 80277 80476 80674 8.80872 81068 81264 81459 81653 81846 82038 82230 82420 82610 8.82799 82987 83175 3.50 3.48 3.46 3.45 3.43 3.42 3.40 3.38 3.37 3.35 3.34 3.32 3.31 3.29 3.28 3.26 3.25 3.23 3.22 3.20 3.19 3.18 3.16 3.15 3.14 3.12 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 54 70409 /t 1 A 70465 4* lo 41 *S 6 54 83261 3.10 _ no 83361 3.11 31 A 6 55 70658 1"' 1 " 70714 It) 4 1 3 5 55 83446 o.Uo 83547 .10 5 56 57 58 59 60 70905 *" 71151 J'i? 'tt*1Ioi niasfJJ 71880 4 ' 03 70962 71208 71453 71697 71940 .10 4.11 4.08 4.06 4.04 4 3 2 1 56 57 58 59 60 83630 83813 83996 84177 84358 3.07 3.06 3.04 3.03 3.02 83732 83916 84100 84282 84464 3.08 3.07 3.06 3.04 3.03 4 3 2 1 M. f'osine. Dl" Cotang.j Dl" M. M. Cosine. Dl" Ootang. DJ" j M. S. N. 37. TABLE IV. LOGARITHMIC M. Bine. 1)1" Tang. Dl" 31. .M. Sim-. IM" Tans. Dl" M. 8.84358 g t\ i x 84404 3 Q2 60 Is. 94030 tn 8.94195 CO 2 84539 84718 O.U 1 2.99 2.98 * 84826 H,) 59 58 1 2 94174 **" 94317 J-JJ 9434 f^ 94485 }{ 59 58 3 84897 85006 57 3 1)4461 f JJ 94630 ^.1 1 57 4 85075 2.97 2O A 85185 Hf 56 4 94603 f Jj 94773 2.40 20ft 56 5 852.32 .yo 85363 I'll 55 5 94746 ";,". 94917 .ov 55 6 85429 2.94 85540 fir 54 6 94887 "'"*' 95060 2.38 54 7 8 9 10 85(i05 85780 85955 86128 2.93 2.92 2.91 2.90 85717 :/;:! 85893 ! ^ 92 8G243 r/f, 1 . 53 52 51 50 7 8 9 10 95029 95170 95310 95450 Z.dU 2.35 2.34 2.33 95202 9 5: ',4 4 95486 95627 2.37 2.37 2.36 2.35 53 52 51 50 11 12 8.86301 86474 2.88 2.87 2QA 8.86417 fJJ 86591 fS 49 48 11 12 8.95589 95728 |;^f 8.95767 95908 2.34 2.34 49 48 13 14 15 86645 86816 86987 .oO 2.85 2.84 2 83 86763 ;- 86935 fS| 8710(i fJJJ 47 45 13 14 15 960? ^30 %?43 :>;;;; 96047 96187 96325 2.32 2.31 9 Ml 47 46 45 16 87156 Z.oO 209 87277 fj! 44 16 96280 1;-;^ 96464 230 44 17 87325 O^j ' s 1 87447 - *:; 43 17 D6417 --'_ 96602 Z-oU 4 29Q ! 18 87494 A.Ol > TO 87616 fjf 42 18 9655;; :,--! 96739 *2V 29Q 42 19 20 21 87661 87829 8.87995 atv 2.78 2.77 2.76 87785 fjj 87953 f JJj 8.88120 f ' 41 40 39 19 20 21 9668<> --;. 9^825 --!; 96877 97013 8.97150 ~Zv 2.28 2.27 OA 41 40 39 22 88161 27 ^ 88287 r/l_ 38 22 '97095 :~*. 97285 *fi .ZQ 20 A 38 23 88326 . to 88453 H; 37 23 9722H 97423 20 37 24 25 8S490 88654 2.74 2.73 2*7O 88618 2-l b 88783 fij 36 35 24 25 97363 9749(5 Z.23 8.22 2 no 97556 97091 2.25 2.24 2nj 36 35 26 27 28 88817 88980 89142 . i t 2.71 2.70 > an 88948 f'* 89111 fi'f 89274 fif 34 33 32 26 27 28 97629 97762 97894 .11 2.21 2.21 t>) on 97825 97959 98092 ./4 2.23 2.22 299 34 33 32 29 89304 z*ov 89437 r' 31 29 98026 -l./U 98225 ./z 31 30 89464 j 9598 z</ " 30 30 98157 2.19 98358 2.21 30 31 8.89625 O (*(* 8.89760 J-JJ 29 31 8.98288 2.18 21 Q 8.9841)0 2.20 2nn 29 32 89784 Z.OO 89920 i f J" 28 32 98419 .18 98622 .zU 28 33 34 89943 90102 2.65 2.64 90080 J'JJ 27 26 33 34 98549|ffJ 98671' J-JJ 98753 98884 2.19 2.18 O 1Q 27 26 35 90260 9* '9 90399 ^ 25 35 98808 21 ^ 99015 /lo 21 7 25 36 37 90417 90574 2^61 2fiO j$g8 24 23 36 37 98937 99066 .10 2.14 214 99145 99275 1 / 2.16 1A 24 23 38 39 90730 90885 DU 2.59 2r Q 90872 2'62 ld39|fi: 22 21 38 39 991 94 99322 i 4 2.13 2*1 'J 99405 ;;" 99534 J-}j| 22 21 40* 41 91040 8.91195 .Oo 2.58 91185 ';"" 8.91340 jrri 20 19 40 41 99450 8.99577 . J *> 2.12 99662 8.99791 .JJ 2.14 20 19 42 91349 2.57 9 Sfi 91495 ! ;'? 18 42 99704 2.11 911 m *i! 18 43 44 45 46 47 48 49 50 51 52 53 91502 91655 91807 91959 92110 92261 92411 92561 8.92710 92859 93007 Z.OO 2.55 2.54 2.53 2.52 2.51 2.50 2.49 2.49 2.48 2.47 9 4fi 91650 91803 f?J 91957 frr 92110 -" 92262 f?! 92414 J'g 92565 f J* 92716 f?J 8.92866 J-JJ 93016 2-*9 93165 fJJ 17 16 15 14 13 12 11 10 g 8 7 43 44 45 47 48 49 50 51 52 53 99830 99956 9.00082 00207 00332 00456 00581 00704 9.00828 0095] 01074 ^.11 2.10 2.09 2.09 2.08 2. (>s 2. (17 2.06 2.0P 2.05 2.05 9.00046 00174 00301 00427 00553 00679 00805 00930 9.01055 01179 01303 2.12 2-12 2.11 2.10 2.10 2.09 2.08 2.08 2.07 2.07 o (\a 17 16 15 14 13 12 11 10 9 8 7 54 55 56 57 58 59 60 93154 93301 93448 93594 93740 93885 94030 j2.^tO 2.45 2.44 2.43 2.43 2.42 2.41 93313 2-47 93009 ;;[!? 93756 jj 9390:; 94049 fJJ 94195 - 4 " 6 5 4 3 2 1 54 55 56 57 58 59 60 01196 01318 01440 01561 01682 01803 01923 2^03 2.03 2.02 2,02 2.01 2.01 01427 01550 01673 01796 01918 02040 02162 zuo 2.06 2.05 2.05 2.04 2.03 2.03 6 5 4 3 2 1 M. Cosine. Dl" CotaW?- Dl" M. M. Cosine. Dl" Cot;ins. PI" M. 30 SINES AND TANGENTS. M. Sine. Dl" Tang. 1)1" M. M. Siue. Dl" Tang. Dl" M. 1 2 9.01923 02043 02163 2.00 2.00 9.02162 02283 02404 2.02 2.02 60 59 58 1 2 9.08589 08692 08795 1.71 1.71 9.08914 09019 09123 1.74 1.73 60 59 58 3 4 5 02283 02402 02520 1.99 1.98 1.98 02525 02645 02766 2.01 2.01 2.00 57 56 55 3 4 5 08897 08999 09101 1.70 1.70 1.70 09227 09330 09434 .73 1.73 1.72 57 56 55 6 7 02639 02757 ^.97 1.97 02885 03005 1.99 1.99 54 53 6 7 09202 09304 1.69 1.69 09537 09640 .72 1.72 54 53 8 9 10 02874 02992 03109 1.96 1.96 1.95 03124 03242 03361 1.98 1.98 1.97 52 51 50 8 9 10 09405 OU506 09606 1.68 1.68 1.68 09742 09845 09947 . 71 1.71 1.70 1 ^- A 52 51 50 11 9.03226 1 .95 9.03479 1.97 49 11 9.09707 1.67 9.10049 I. IV 49 12 13 03342 03458 1.94 1.94 03597 03714 1.96 1.96 48 47 12 13 09807 09907 l.Ol 1.67 10150 10252 1.69 1.69 48 47 14 03574 1.93 03832 i}-rr 46 14 10006 1.66 10353 L.69 46 15 16 03690 03805 1.93 1.92 03948 -r? 04065 1 94 45 44 15 16 10106 10205 1.66 1.65 10454 10555 1.68 1.68 45 44, 17 03920 1.92 Ini 43 17 10304 1.65 1AA 10656 L.68 1A7 43 18 04034 .y L 1 Q 1 04297s r'^ 42 18 10402 .04 1.64 10756 .D/ I 67 42 19 20 04149 04262 L t? L 1.90 04413; * 04528 i 41 40 19 20 10501 10599 L64 10856 10956 L67 .41 40 21 9.04376 1.89 9.04643 J'^f 39 21 9.10697 1.63 9.11056 1.66 39 22 04490 1.89 1 88 04758 J'JJ 38 22 10795 1.63 ICO 11155 1.66 1fi5 38 23 04603 1 .00 04873 J'J* 37 23 10893 .Do 11254 .Ow 37 24 04715 1.88 04987 !} 36 24 10990 1.62 11353 1.65 36 25 04828 1.87 1Q7 05101 \'H 35 25 11087 1.62 1A*> 11452 1.65 1C A 35 26 04940 .of 1Q7 05214 I'll 34 26 11184 .OZ 1/1-1 11551 .0-4 34 27 05052 .87 05828 {-3 33 27 11281 .ol 11649 1.64 33 28 29 05164 05275 K86 05441 ' 88 05553 . L8 32 31 28 29 11377 11474 1.61 1.61 11747 11845 1.64 1.63 32 31 30 31 05386 J.05497 L85 05666 };JJ 9.05778 | l -*i 30 29 30 31 11570 9.11666 1.60 1.60 11943 9.12040 1.63 1.62 30 29 32 33 05607 05717 1.84 1.84 05890 | '*' 06002 { ; JJ 28 27 32 33 11761 11857 1.59 1.59 12138 12235 1.62 1.62 28 27 34 05827 1.83 061131 JQ? 26 34 11952 1.55 12332 1.62 26 35 05937 1.83 1QO 06224 j 1 -^ 25 35 1 2047 1.58 ICQ 12428 1.61 Ir*~\ 25 36 06046 .oZ 06335 MjT 24 36 12142 .Oc 12525 .ol 24 37 06155 1.82 06445 l.tt-4 23' 37 12236 1.58 12621 1.60 23 38 06264 1.81 06556 1.84 22 38 12331 1.57 12717 1.60 22 39 06372 1 .81 06666 1.83 21 39 12425 1.57 12813 1.60 21 40 06481 1.80 06775 1.83 20 40 12519 1.57 12909 1.59 20 41 9.06589 1.80 9.06885 1.82 19 41 9.12612 1.56 9.13004 1.59 19 42 06696 1.79 06994 1.82 18 42 12706 1.56 13099 1.59; ,Q 43 06804 1.79 1 7Q 07103 1 w 1 17 43 12799 1.56 1 ^ 13194 1.58 ,_ ICQ 1 44 45 46 47 48 49 06911 07018 07124 07231 07337 07442 L. i VI 1.78 1.78 1.77 1.77 1.76 07320 !' 8 J 07428 ill? 07536 I 79 07643 J ' 07751 iH 9 16 15 14 13 12 11 44 45 46 47 48 49 12892 12985 13078 13171 13263 13355 l.Ot 1.55 1.55 1.54 1.54 1.53 13289 13384 13478 13573 13667 13761 1 .Do 1.58 1.57 1.57 1.57 1.56 16 15 14 13 12 11 50 07548 1.76 07858 l.i a 10 50 13447 1.5.' 1 3854 1.56 10 51 9.07653 1.75 9.07964 l.7 9 51 9.13539 1.51 9.13948 1.56 9 52 07758 1.75 08071 \'ll 8 52 13630 1.52 14041 1.55 8 53 07863 1.75 08177 I'ml i 7 53 13722 1.52 14134 1.55 7 54 07968 1.74 OS283 1.71 6 54 13813 1.52 14227 1.55 6 55 08072 1.74 08389 1.7( 5 55 1 3904 1.52 14320 1.55 5 56 08176 1.72 08495 1.7( 4 56 13994 1.51 14412 1.54 4 57 08280 1 .72 08600 1.75 3 57 14085 1.51 14504 1.54 3 58 08383 1.72 08705 1.75 2 58 14175 1.51 14597 1.54 2 59 08486 1.75 08810 1.75 1 59 14266 1.56 14688 1.53 1 60 08589 1.72 08914 1.74 60 14356 1.50 14780 1.53 IT Cosine. Dl" Cotang. Dl" M. M. CnsilK'. Dl" Cotang. Dl" M. 31 82 8 TABLE IV. LOGARITHMIC 9 M. Sine. 1)1'' Tang. 1)1" M. 31. Sine. Dl" Tang. Dl" 31. 9.14356 9.14780 60 9.19433 9.19971 60 1 14445 1.50 14872 1.53 59 19513 .33 20053 1.36 59 2 14535 1.49 14963 1.52 58 2 19592 .33 20134 1.36 58 3 14624 1.49 15054 1.52 57 3 19672 .32 20216 1.36 57 4 5 14714 14803 1.49 1.48 15145 15236 1.52 1.51 56 55 4 5 19751 19830 .32 .32 20297 20378 1.35 1.35 56 55 6 7 14891 14980 1.48 1.48 15327 15417 1.51 1.51 54 53 6 7 19909 19988 .32 .31 20459 20540 1.35 1.35 54 53 8 15069 1.48 15508 1.50 52 8 20067 .31 20621 1.35 52 9 15157 1.47 15598 1.50 51 9 20145 .31 20701 1.34 51 10 15245 1.47 15688 .50 50 10 20223 .31 20782 1.34 50 11 9.15333 1.47 9.15777 .50 49 11 9.20302 .30 9.20862 1.34 49 12 15421 1.46 15867 .49 48 12 20380 .30 20942 1.33 48 13 15508 1.46 15956 .49 47 13 20458 .30 21022 1.33 47 14 15 15596 15683 1.46 1.45 16046 16135 .49 .48 46 45 14 15 20535 20613 .30 .29 21102 21182 1.33 1.33 46 45 16 15770 1.45 14 ti 16224 .48 A U 44 16 20691 .29 oo 21261 1.33 1QO 44 17 15857 .40 16312 .4o 43 17 20768 .zy 21341 ,oZ 43 18 15944 1.45 16401 .48 1 7 42 18 20845 .29 oo 21420 1.32 Too 42 19 20 16030 16116 1.44 1.44 16489 16577 ,4/ .47 41 40 19 20 20922 20999 Zo .28 21499 21578 .0.6 1.32 41 40 21 9.16203 1.44 9.16665 .47 39 21 9.21076 .28 9.21657 1.32 39 22 16289 1.43 16753 .46 38 22 21153 .28 21736 1.31 38 23 16374 1.43 16841 .46 37 23 21229 .27 21814 1.31 37 24 25 26 16460 16545 16631 1.43 1.42 1.42 16928 17016 17103 .46 .46 .45 36 35 34 24 25 26 21306 21382 21458 .27 .27 .27 21893 21971 22049 1.31 1.31 1 .30 36 35 34 27 16716 1.42 17190 .45 33 27 21534 .27 OC 22127 1.30 33 28 29 16801 16886 1.42 .41 17277 17363 .45 .44 32 31 28 29 21610 21685 .ZO .26 22205 22283 1 .30 1.30 32 31 30 16970 1.41 17450 .44 30 30 21761 .26 22361 1.29 30 31 9.17055 1.41 9.17536 .44 29 31 9.21836 .26 9.22438 1 .29 29 32 33 34 35 36 17139 17223 17307 17391 17474 11.40 1.40 1.40 1.40 1.39 17622 17708 17794 17880 17965 .44 .43 .43 .43 .42 28 27 26 25 24 32 33 34 35 36 21912 21987 22062 22137 22211 !25 .25 .25 22516 22593 22670 22747 22824 1.29 1.29 1.29 1.28 1.28 28 27 26 25 24 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 17558 17641 17724 17807 9.17890 17973 18055 18137 18220 18302 18383 18465 18547 18628 9.18709 18790 18871 18952 1.39 1.39 1.39 1.38 1.38 1.38 1.37 1.37 1.37 1.37 1.36 1.36 1.36 1.36 1.35 1.35 1.35 1.35 18051 18136 18221 18306 9.18391 18475 18560 18644 18728 18812 18896 18979 19063 19146 9.19229 19312 19395 19478 .42 .42 .42 .42 1.41 1.41 1.41 1.40 .40 1.40 1.40 1.39 .39 1.39 1.39 1.38 1.38 1.38 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 22286 22361 22435 22509 9.22583 22657 22731 22805 22878 22952 23025 23098 23171 23244 9.23317 23390 234(52 23535 .24 .24 .24 .24 .24 .23 1.23 1.23 1.23 1.22 1.22 1.22 1.22 1.22 1.21 1.21 1.21 1.21 22901 22977 23054 23130 9.23206 23283 23359 23435 23510 23586 23661 23737 23812 23887 9.23962 24037 24112 24186 1.28 1.28 1.28 1.27 1.27 1.27 1.27 1.27 1.26 1.26 1.26 1.26 1.25 1.25 1.25 1.25 1.25 1.24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 55 19033 1.34 19561 1.38 IOT 5 55 23607 .20 9ft 24261 1.24 194 5 56 57 58 59 60 19113 19193 19273 19353 19433 1.34 1.34 1.34 1.33 1.33 19643 19725 19807 19889 19971 Of 1.37 1.37 1.37 1.36 4 3 2 1 56 57 58 59 60 23679 23752 23823 23895 23967 .zu .20 .20 .20 .20 24335 24410 24484 24558 24632 ./* 1.24 1.24 1.23 1.23 4 3 2 1 M. Cosine. Dl" Cotang. Dl" M. M. Cosine. Dl" Cotang. Dl" M. 81 C 32 SINES AND TANGENTS. 11 M. Sine. Dl" Tiing. Dl" M. M. Sine. Dl" Tang. Dl" M. 9.23967 11 9.24632 1.) .. 60 9.28060 9.28865 11 O 60 1 24039 . i y 24706 .20 59 1 28125 1.08 28933 .12 59 2 24110 1.19 24779 1.23 58 2 28190 1.08 29000 1.12 58 3 24181 1.19 11 u 24853 1.23 100 57 3 28254 1.08 29067 1.12 1 o 57 4 24253 . i y 24926 .22 56 4 28319 1.08 29134 .1 z 56 5 6 7 24324 24395 24466 1.18 1.18 1.18 25000 25073 25146 1.22 1.22 1.22 55 54 53 5 6 7 28384 i 28448 1 28512 1.08 1.07 1.07 29201 29268 29335 .12 .12 .11 55 54 53 8 24536 1.18 25219 1.22 52 8 28577 1.07 29402 .11 52 9 24607 1.18 25292 1.22 51 9 28641 1.07 29468 .11 51 10 24677 1.17 25365 1.21 50 10 28705 1.07 29535 .11 50 11 9.24748 1.17 9.25437 1.21 49 11 9.28769 1.07 9.29601 .11 49 12 24818 1.17 25510 1.21 48 12 28833 1.06 29668 .11 48 13 14 15 16 24888 24958 25028 25098 1.17 1.17 1.17 1.16 25582 25655 25727 25799 1.21 1.20 1.20 1.20 47 46 45 44 13 14 15 16 28896 28960 29024 29087 1.06 1.06 1.06 .06 29734 29800 29866 29932 .10 .10 .10 .10 47 46 45 44 17 25168 1.16 25871 1.20 43 17 29150 .06 29998 .10 43 18 25237 1.16 25943 1.20 42 18 29214 .05 30064 .10 42 19 25307 1.16 26015 1.20 41 19 29277 .05 30130 .10 41 20 25376 1.16 26086 1.19 40 20 29340 .05 30195 .09 40 21 22 9.25445 25514 1.15 1.15 9.26158 26229 1.19 1.19 39 38 21 22 9.29403 29466 .05 .05 9.30261 30326 .09 .09 39 38 23 25583 1.15 26301 1.19 37 23 29529 .05 30391 .09 37 24 25 25652 25721 1.15 1.15 26372 26443 1.19 1.18 36 35 24 25 29591 29654 .04 .04 30457 30522 .09 .09 36 35 26 25790 1.14 26514 1.18 34 26 29716 .04 30587 .08 34 27 25858 1.14 11 \ 26585 1.18 1 1 Q 33 27 29779 .04 30652 .08 0,8 33 28 25927 . IT: 26655 1 . lo 32 28 29841 .U4 30717 .Uo 32 29 30 25995 26063 1.14 1.14 26726 26797 1.18 1.18 31 30 29 30 29903 29966 .04 .04 30782 30846 .08 .08 31 30 31 9.26131 1.13 9.26867 1.17 29 31 9.30028 .03 9.30911 .08 29 32 26199 1.13 110 26937 1.17 117 28 32 30090 .03 AO 30975 .07 28 33 26267 1 . IO 11 O 27008 1 . 1 1 27 33 30151 .Uo 31040 .07 27 34 26335 .id 27078 1.17 26 34 30213 .03 31104 .07 26 35 26403 1.13 27148 1.17 25 35 30275 .03 31168 .07 25 36 26470 1.13 27218 1.17 11 G 24 36 30336 .03 31233 .07 24 37 38 26538 26605 1.12 1.12 27288 27357 .16 1.16 23 22 37 38 30398 30459 .03 .02 31297 31361 .07 .07 23 22 39 40 26672 26739 1.12 1.12 27427 27496 1.16 1.16 21 20 39 40 30521 30582 .02 .02 31425 31489 .07 .06 21 20 41 42 9.26806 26873 1.12 1.12 9.27566 27635 1 .16 1.15 19 18 41 42 9.30643 30704 .02 .02 9.31552 31616 .06 .06 19 18 43 26940 1.11 27704 1.15 17 43 30765 .02 31679 .06 17 44 V 27007 1.1 1 ill 27773 1.15 11 E 16 44 30826 .01 Al 31743 .06 16 45 27073 1.11 27842 .10 15 45 30887 .U 1 31806 .06 15 46 27140 1.11 1-i t 27911 1.15 14 46 30947 .01 A 1 31870 .05 A 14 47 27206 .1 L 11 (\ 27980 1.15 11 ^ 13 47 31008 .U 1 Al 31933 ,uo A & 13 48 49 27273 27339 .1 U 1.10 28049 28117 .10 1.14 12 11 48 49 31068 31129 .Ul .01 31996 32059 .UO .05 12 11 50 27405 1.10 28186 1.14 10 50 31189 .01 32122 .05 10 51 9.27471 1.10 11 {\ 9.28254 1.14 9 51 9.31250 .00 AA 9.32185 1.05 j\f 9 52 27537 .1U 28323 1.14 8 52 31310 .uu 32248 .UO 8 53 54 55 56 57 27602 27668 27734 27799 27864 1.10 1.09 1.09 1.09 1.09 28391 28459 28527 28595 28662 1.14 1.13 1.13 1.13 1.13 7 6 5 4 3 53 54 55 56 57 31370 31430 31490 31549 31609 .00 .00 .00 .00 1.00 32311 32373 32436 32498 32561 .05 .04 .04 .04 .04 7 6 5 4 3 58 59 27930 27995 1.09 1.09 28730 28798 1.13 1.13 2 1 58 59 31669 31728 .99 .99 32623 32685 .04 1.04 2 1 60 28060 1.08 28865 1.12 60 31788 .99 32747 1.04 M. Cosine. Dl" Cotangr. Dl" M. M. Cosine. Dl" Cotang. Dl" M. 79 78 C TABLE IV. LOGARITHMIC M. Sine. Di" Tang. 1)1" M. M. Sine. 1)1" Taug. Dl" M. 1 9.31788 31847 0.99 9.32747 32810 1.03 60 59 1 9.35209 35263 0.91 9.36336 36394 0.96 60 59 2 31907 .99 on 32872 1.03 In ! 58 2 35318 .91 ni 36452 .96 58 3 31966 .yy 32933 jVo 57 3 35373 .y i 3C.509 .96 57 4 5 32025 32084 .99 .98 32995 33057 1.03 1.03 56 55 4 5 35427 35481 .91 .91 36566 36624 .96 .96 56 55 6 7 32143 32202 .98 .98 33119 33180 1.03 1.03 1AO 54 53 6 7 35536 35590 .91 .90 36681 36738 .95 .95 f\c 54 53 8 9 10 11 12 13 32261 32319 32378 9.32437 32495 32553 !98 .98 .98 .97 .97 33242 33303 33365 9.33426 33487 33548 .06 1.02 1.02 1.02 1.02 1.02 52 51 50 49 48 47 8 9 10 11 12 13 35644 35698 35752 9.35806 35860 35914 !QO .90 .90 .90 .90 36795 36852 36909 9.36966 37023 37080 .yo .95 .95 .95 .95 .95 52 51 50 49 48 47 14 15 16 32612 32670 32728 .97 .97 .97 33609 33670 33731 1.02 1.02 1.01 46 45 44 14 15 16 35968 36022 36075 .90 .89 .89 37137 37250 .95 .94 .'.14 46 45 44 17 18 19 32786 32844 32902 .97 .97 .97 Qfi 33792 33853 33913 1.01 1.01 1.01 1A 1 43 42 41 17 18 19 36129 36182 36236 89 89 .89 CO 37306 37363 37419 .94 .94 43 42 41 20 21 32960 9.33018 yo .96 33974 9.34034 .01 1.01 40 39 20 21 36289 9.36342 by .89 37476 9.37532 .94 40 39 22 33075 34095 1.01 AI .38 22 36395 on 37588 .94 n i 38 23 33133 Qii 34155 .0 1 37 23 36449 by 37644 .,14 no 37 24 33190 .yo 34215 .00 nn 36 24 36502 88 QO 37701) .y,> 36 25 33248 OA 34276 .00 35 25 36555 OO 37756 .y.i 35 26 33305 .yo n \ 34336 .00 An 34 26 36608 88 QO 37812 .93 34 27 33362 .y o 34396 .Uu 33 27 36660 OO 37868 .93 33 28 33420 _ 34456 .00 32 28 36713 88 37924 .93 32 29 33477 9o 34516 .00 no 31 29 36766 88 QO 37980 .93 31 30 33534 .95 34576 .UU on 30 30 36819 OO QQ 38035 os 30 31 9.33591 9.34635 UU 29 31 9.36871 oo 9.38091 * * 29 32 33647 95 34695 .99 28 32 36924 88 87 38147 .93 28 33 33704 34755 .99 27 33 36976 O 4 38202 JZ 27 34 33761 n i 34814 .99 26 34 37028 87 38257 .92 26 35 33818 94 94 34874 .99 qq 25 35 37081 87 38313 .92 25 36 33874 34933 .yy 24 36 37133 o< 38368 "* 24 37" 33931 .94 34992 .99 23 37 37185 87 38423 ' .92 23 38 33987 94 35051 ".99 qq 22 38 37237 87 07 38479 '.92 22 39 34043 35111 .yy 21 39 37289 O i 38534 V* 21 40 34100 94 35170 .98 20 40 37341 87 38589 .92 20 41 9.34156 94 9.35229 .98 19 41 9.37393 87 9.38644 .92 19 42 34212 **' 35288 .98 no 18 42 37445 86 Qrt 38699 .92 18 43 34268 '*:* 35347 .9o 17 43 37497 OO 38754 .yi 17 44 34324 !> 35405 .98 16 44 37549 .86 38808 .91 16 45 34380 93 93 35464 .98 15 45 37600 .86 QC 38863 .91 Q1 15 46 34436 35523 yo 14 46 37652 .00 38918 ' 1 14 47 34491 93 93 35581 .98 13 47 37703 .86 Qi! 38972 .91 Q1 13 48 34547 35640 .9o 12 48 37755 oO Q/> 39027 .y i fit 12 49 34602 J6 35698 .97 11 49 37806 .OO 39082 .yi 11 50 34658 no 3575.7 .97 10 50 37858 .86 39136 .91 10 51 9.34713 31 qo 9.35815 .97 n*7 9 51 9.37909 .85 QC 9.39190 .91 01 9 52 34769 y Zi 35873 y t 8 52 37960 OO 39245 .y i 8 53 34824 92 35931 .97 7 53 38011 .85 39299 .90 7 54 34879 yz 35989 .97 6 54 38062 .85 39353 .90 6 55 34934 92 no 36047 .97 5 55 381 1 3 85 39407 .90 5 56 34989 92 36105 .97 4 56 38164 .85 39461 .90 4 57 35044 92 36163 .96 3 57 38215 .85 39515 .90 3 58 59 35099 35154 92 91 36221 36279 .96 .96 2 1 58 59 38266 38317 .85 .85 39569 39623 .90 .90 2 1 60 35209 .9] 36336 .96 60 38368 .85 39677 .90 M. Cosine. Dl" Cotang. Dl" M. M. Cosine. Dl" CutitllU. Dl" M. 77 ; 14 C SINES AND TANGENTS. I5 C M. Sine. Dl" Tan-. 1)1" | M. M. Sine. Dl" Tang. 1)1" M. 1 9.38368 38418 0.84 y.:',w>77 39731 0.90 un M 59 1 9.41300 41347 0.79 .78 9.42805 42S50 0.84 .84 60 59 2 38469 .84 39785 .yu Qf\ 58 2 4 1 394 42906 58 3 38519 .84 39838 .89 f>7 ;; 41441 .78 42957 .84 57 4 5 fi 38570 38620 38670 .84 .84 .84 39892 39945 39999 !89 .89 50 5 5 54 4 5 41488 41535 41582 .78 .78 .78 43007 43057 43108 M [ 7 3S721 .84 Q 1 40052 .89 0(1 53 7 41628 .78 170 43158 .84 84 53 8 9 38771 38821 .o4 .84 QQ 40106 40159 . o J .89 52 51 8 9 41675 41722 . t o .78 7ft 43208 43258 .O^r .83 QO 52 51 10 38871 .OO 00 40212 on 50 10 41768 . 1 O 70 43308 .OO QO 50 11 12 9.38921 38971 .OO .83 9.40266 40319 .oy .89 49 48 11 12 9.41815 4 1 SO 1 I O .78 9.43358 43408 .OO .83 49 48 13 14 39021 39071 .83 .83 80 40372 40425 .88 .88 00 47 46 13 14 41908 41954 .77 .77 77 43458 43508 .83 .83 QO 47 46 15 39121 QQ 40478 .So QQ 45 15 42001 I 43558 OO QO 45 10 17 39170 39220 . OO .83 QO 40531 40584 . OO .88 QQ 44 43 16 17 42047 42093 [77 43607 43657 oo .83 83 44 43 18 19 39270 39319 .80 .83 89 40036 40689 . OO .88 QQ 42 41 18 19 42140 42186 !77 77 43707 43756 !83 QO 42 41 20 21 39369 9.39418 & .82 40742 9.40795 . oo .88 40 39 20 21 42232 9.42278 i .77 . 43806 9.43855 .OO .83 40 39 22 39467 .82 40847 .88 QQ 38 22 42324 .77 43905 89 38 23 39517 o 40900 .OO 37 23 42370 . i ( 43954 ,oZ 37 24 39566 .82 no 40952 .87 8*7 36 24 42416 .76 7fi 44004 .82 36 25 39615 .04 41005 1 35 25- 42461 . i 44053 .bZ 35 26 27 39664 39713 .82 .82 41057 41109 .87 .87 34 33 26 ' 27 42507 42553 .76 .76 44102 44151 .82 .82 34 33 28 39762 .82 41161 87 32 28 42599 .76 >-/ 44201 .82 32 29 39811 .82 41214 .87 31 29 42644 . 1 O 44250 .82 31 30 39860 .81 41266 .87 30 30 42690 .76 44299 .82 30 31 9.39909 .81 9.41318 .87 29 31 9.42735 '!?i 9.44348 .82 29 32 39958 .81 41370 .87 28 32 42781 7o 44397 .82 28 33 40006 .81 41422 .87 27 33 42826 .76 44446 .82 27 34 40055 .81 41474 .87 26 34 42872 .76 44495 .81 26 35 40103 .81 41526 .86 25 35 42917 .76 44544 .81 25 36 40152 .81 41578 .86 24 36 42962 .75 44592 .81 24 37 40200 .81 41629 .86 23 37 43008 .75 44641 .81 23 38 40249 .81 41681 .86 22 38 43053 .75 44690 81 22 39 40297 .81 41733 .86 21 39 43098 .75 44738 .81 21 40 40346 .81 41784 .86 20 40 43143 .75 44787 .81 20 41 42 43 9.40394 40442 40491) .80 .80 .80 9.41836 41887 41939 .86 .86 .86 19 18 17 41 42 43 9.43188 43233 43278 .75 .75 .75 9.44836 44884 44933 .81 .81 .81 19 18 17 44 40538 .80 41990 .86 16 44 43323 .75 44981 .81 16 45 40586 .80 Q A 42041 .86 15 45 43367 .75 45029 .81 Q 1 15 46 40634 .Oil 42093 .85 14 46 43412 .75 45078 ol 14 47 48 40682 40730 .80 .80 42144 42195 .85 .85 13 12 47 48 43457 43502 .75 .74 45126 45174 .80 .80 13 12 49 40778 i '*!! 42246 .85 11 49 43546 .74 45222 .80 11 50 40825 .( 42297 .85 10 50 43591 .74 45271 .80 10 51 9.40873 .79 9.42348 .85 9 51 9.43635 .74 9.45319 .80 9 52 40921 .79 42399 .85 8 52 43680 .74 45367 .80 8 53 40968 .79 42450 .85 7 53 43724 .74 45415 .80 7 54 41016 .79 42501 .85 6 54 43709 .74 45463 .80 6 55 41063 .79 42552 .85 5 55 43813 .74 45511 .80 5 56 41111 .79 42603 .85 4. 56 43857 .74 45559 .80 4 57 41158 .79 42653 .85 3 57 43901 .74 45606 .80 3 58 41205 .79 42704 .84 2 58 43946 .74 45654 .80 2 59 41252 .79 42755 .84 1 59 43990 .74 45702 .80 1 60 41300 .79 42805 ' 60 44034 .73 45750 .80 M. Cosine. Dl" Cotansr. Dl" M. M. Cosine. IM" Cotansr. DP M. 74 C 16 C TABLE IV. LOGARITHMIC 17 M. Sine. Dl" Tuna. Dl" .M. M. Sine. Dl" Tmitr. Dl" 11. 1 9.44034 44078 0.73 70 9.45750 45797 1 60 59 1 9.46594 46635 0.69 | />< il. 48534 48579 0.75 60 59 2 44122 . id 70 45845 70 58 2 46676 .0,' ft 11 48624 .7 ) 58 3 44160 . I <l 7'J 45892 70 57 3 45717 .0 *J rfq 48669 .75 57 4 44210 .IO 45940 T A 56 4 46758 OV4 48714 .7 56 5 44253 70 45987 '11 55 5 46800 fiO 48759 .7 55 6 44297 1 > 7. 46035 7Q 54 6 46841 Do CO 48804 'I 54 7 44341 / > 46082 * 53 7 46882 .Do 48849 .<0 53 8 9 44385 4442* .73 .73 46130 46177 .79 .79 52 51 8 9 46923 46964 .68 .6* 48894 48939 .75 .75 52 51 10 44472 .73 tjn 46224 .79 TO 50 10 47005 .68 48984 .75 50 11 9.44516 "" 9.46271 . i y 49 11 9.47045 .68 9.49029 .75 49 12 44559 .7.1 79 46319 .79 48 12" 47086 .68 49073 .75 48 13 44602 46366 .<y 47 13 47127 .68 49118 .74 47 14 44646 .72 46413 .78 46 14 47168 '* 49163 .74 46 15 44689 .72 TO 46460 .78 45 15 47209 49207 .74 45 16 44733 .12, 46507 .78 44 16 47249 i ' 49252 .74 44 17 44776 .72 46554 .78 43 17 47290 .GO 49296 .74 43 18 44819 .72 46601 .78 TO 42 18 47330 .68 49341 .74 42 19 44862 .72 46648 .7o 41 19 47371 .68 49385 .74 41 20 44905 .72 46694 .78 40 20 47411 .67 49430 .74 40 21 22 9.441)48 44992 .72 .72 9.4(5741 46788 .78 .78 39 38 21 22 9.47452 '[?i 9.49474 47492i '"ll 49519 .74 .74 39 38 23 45035 .72 46835 .78 rrO 37 23 47533 Ji 49563 .74 37 24 45077 .72 46881 .78 36. 24 47573 1 49607 .74 36 25 45120 .71 46928 .78 35 25 47613 J- 49652 .74 35 26 45163 .71 46975 .78 34 26 47654 J; 4U696 .74 34 27 45206 .71 47021 .78 33 27 47694 .67 49740 .74 33 28 45249 .71 47068 .77 32 28 47734: 'JJ 49784 .74 32 29 45292 .71 47114 .77 31 29 47774 '" 49828 .73 31 30 45334 .71 47160 .77 30 30 47814 .07 49872 .73 30 31 9.45377 .71 9.47207 .77 29 31 9.47854 .67 9.49916 .73 29 32 45419 .71 47253 .77 28 32 47894 5?l 49960 .73 28 33 34 45462 45504 .71 .71 47299 47346 .77 .77 27 26 33 34 47934 -JJ 50004 479741 5il 50048 .73 .73 27 26 35 45547 .71 47392 .77 25 35 48014 JJ 50092 .73 25 36 45589 .71 47438 .77 24 36 48054 JJ 50136 .73 24 37 45632 .71 47484 .77 23 37 48094 JJ 50180 .73 23 38 45674 .71 47530 .77 22 38 48133 JJ 50223 .73 22 39 45716 .70 47576 .77 21 39 48173 JJ 50267 .73 21 40 45758 .70 47622 .77 20 40 48213 ( ! b 50311 JJ 20 41 9.45801 .70 9.47668 .77 19 41 9.48252 JJi 9.50355 .73 19 42 45843 .70 47714 .77 18 42 48292 .66 50398 .73 ]8 43 45885 .70 47760 .76 17 43 48332 .66 50442 .73 ,. 44 45927 .70 47806 .76 16 44 48371 .66 50485 .73 16 45 45969 .70 47852 .76 15 45 48411 .66 50529 " 15 46 46011 .70 47897 .76 14 46 48450 .66 50572 11 14 47 46053 .70 47943 .76 13 47 48490 .66 50616 ' - 13 48 46095 .70 47989 .76 12 48 48529 .66 50659 '~-l 12 49 46136 .70 48035 .76 11 49 48568 .66 50703 50 51 46178 9.46220 .70 11 48080 9.48126 .76 .76 10 9 50 51 48607 9.48647 JJ 50746 JJ 9.50789 '-2 10 52 46262 .69 48171 .76 8 52 48686 JJ 50833 ''11 8 53 46303 .69 48217 .76 7 53 48725 .65 50876 .72 7 54 46345 .69 48262 .76 6 54 4S764 .65 50919 .72 6 55 46386 .69 48307 .76 5 55 48803 .65 50962 .72 5 56 46428 .69 48353 .76 4 .56 48812 .65 51005 .72 4 57 46469 .69 48398 .76 3 57 48881 .65 51048 .72 3 58 46511 .69 48443 .75 2 58 48920 .65 51092 .72 2 59 46552 .69 48489 .75 1 59 48959 .65 51135 .72 1 60 46594 .69 4-.-..1; .75 60 48998 .65 51178 .72 M. Cosine. Dl" CtititllJ.'. Dl" M. M. Uorfne. Dl" Cotang. Dl" M. 73 73= SINES AND TANGENTS. 19 M. i Sine. Dl" Tang. 1)1" M. M. Sine. ; Dl" Tang. Dl" M. 1 9.48998 49037 0.65 9.51178 51221 0.72 60 59 1 9.51264 51301 0.61 9.53697 53738 0.68 60 59 2 49076 .65 51264 .72 58 2 51338 .61 A1 53779 .68 AQ 58 3 49115 .65 51306 .71 57 3 51374 .0 J 53820 .00 57 4 49153 .65 51349 .71 56 4 51411 .61 /> i 53861 .68 AQ 56 5 49192 .65 51392 .71 55 5 51447 .01 53902 .OO 55 6 49231 .64 51435 .71 54 6 51484 .61 53943 .68 54 7 8 49269 49308 .64 .64 RA 51478 51520 .71 .71 53 52 7 8 51520 51557 .61 .61 A1 53984 54025 .68 .68 AQ 53 52 9 49347 .o4 51563 51 9 51593 .01 54065 .Do 51 10 49385 .64 /> j 51606 .71 50 10 51629 .61 Al 54106 .68 AQ 50 11 9.49424 .04 9.51648 49 11 9.51666 .01 A A 9.54147 .Oo AO 49 12 49462 .64 51691 .71 48 12 51702 .ou 54187 .00 48 13 49500 .64 a A 51734 .71 *7l 47 13 51738 .60 AA 54228 .68 AQ 47 14 49539 .04 51776 7 1 46 14 51774 .OU 54269 .Oo 46 15 49577 .64 A/I 51819 .71 45 15 51811 .60 I'M 54309 .68 AQ 45 16 49615 .64 51861 .71 44 16 51847 .OU i A 54350 .Oo AQ 44 17 49654 .64 51903 .71 43 17 51883 .00 54390 .OO 43 18 19 49692 49730 .64 .64 51946 51988 .71 .71 42 41 18 19 51919 51955 .60 .60 54431 54471 .68 .67 42 41 20 49768 .64 52031 .71 40 20 51991 .60 54512 .67 40 21 9.49806 .64 9.52073 .70 39 21 9.52027 .60 9.54552 .67 AT 39 22 49844 .63 />*> 52115 70 38 22 52063 .60 Art 54593 .07 A>7 38 23 24 49882 49920 .00 .63 52157 52200 70 .70 37 36 23 24 52099 52135 .OU .60 54633 54673 .07 .67 37 36 25 49958 .63 52242 70 35 25 52171 .60 54714 .67 35 26 49996 .63 52284 .70 34 26 52207 .60 54754 .67 34 27 50034 .63 52326 .70 33 27 52242 .60 54794 .67 33 28 50072 .63 52368 .70 32 28 52278 .60 54835 .67 32 29 50110 .63 52410 .70 31 29 52314 .60 54875 .67 A If 31 30 50148 .63 52452 70 30 30 52350 .5^) 54915 .07 30 31 9.50185 .63 9.52494 .70 29 31 9.52385 .5S 9.54955 .67 29 32 50223 .63 52536 .70 28 32 52421 .5S 54995 .67 28 33 50261 .63 52578 .70 27 33 52456 .5S 55035 .67 27 34 50298 .63 52620 .70 26 34 52492 .5S 55075 .67 26 35 36 50336 50374 .63 .63 52661 52703 .70 .70 25 24 35 36 52527 52563 .55 .59 55115 55155 .67 .67 25 24 37 50411 .63 52745 .70 23 37 52598 .5 55195 .67 23 38 50449 .63 52787 .70 22 38 52634 .5 55235 .67 22 39 50486 .62 52829 .70 21 39 52669 .5 55275 .66 21 40 50523 .62 52870 .69 20 40 52705 .54 55315 .66 20 41 9.50561 .62 9.52912 .69 19 41 9.52740 .59 9.55355 .66 19 42 50598 .62 52953 .69 18 42 52775 .5J 55395 .66 18 43 50635 .62 52995 .69 17 43 52811 5t 55434 .66 17 44 50673 .62 AO 53037 .69 16 44 52846 .59 r ft 55474 .66 A A 16 45 50710 OJ 53078 .69 15 45 52881 .01 55514 .00 15 46 50747 .62 53120 .69 14 46 5291 6 .59 55554 .66 14 47 50784 .62 53161 .69 13 47 52951 .5$ 55593 .66 13 48 50821 .62 /o 53202 .69 12 48 52986 .58 C Q 55633 .66 AA 12 49 50858 0.6 53244 .69 11 49 53021 .Dc 55673 .01 11 50 50896 .62 53285 .69 10 50 53056 .5 55712 .66 10 51 9.50933 .62 9.53327 .69 9 51 9.53092 .58 9.55752 .66 9 52 50970 .62 53368 .68 8 52 53126 .58 55791 .6( 8 53 51007 62 6C 53409 .69 rtfl 7 53 53161 .58 C Q 55831 .66 /. 7 54 51043 4 53450 .oV: 6 54 53196 .OO 5587( .Of 6 55 51080 .61 53492 .6? 5 55 53231 .58 55910 .66 5 56 51117 .61 A1 53533 .69 Aft 4 56 53266 .58 e Q 55949 .66 AA 4 57 51154 VI 53574 .Ot 3 57 53301 .UO 55989 .OO 3 58 51191 .61 /*-! 53615 .69 QQ 2 58 53336 .58 CQ 56028 .66 AA 2 59 51227 .OJ 53656 .OO 1 59 53370 .OO 56067 .01 1 60 51264 .61 53697 .68 60 53405 .58 56107 .66 M. Cosine. Dl" Cotsmgr. Dl" M. M. Cosine, j Di" Cotang Dl" j M. 71 s 37 20 TABLE IV. LOGARITHMIC Al Sine. Dl" Tang. D," M, 31. Sin*. Dl" Tug. Di'' M. 1 9.53405 53440 0.58 9.56107 1. R . 56146 U *^? 60 59 1 9.55433 55466 0.55 9.584 Ib 58455 0.63 60 59 2 3 5M475 53509 !58 CO 56185 'JJ 56224| 'X 58 57 2 3 55499 55532 .55 .55 58493 58531 .63 AO 58 57 4 53544 .Do 5.6264 .Oil 56 4 555C.4 .55 58569 .OO 56 5 53578 .58 56303 .65 55 5 55597 .55 58606 .63 55 6 53613 .58 CO 56342 .65 A c. 54 6 55630 .55 C C 58644 .63 A*? 54 7 8 53647 53682 .Do .57 57 56381 56420 .00 .65 Afv 53 52 7 8 55663 55695 DO .55 58681 58719 Otj .63 /> 53 52 9 53716 i 56459 .00 51 9 55728 .54 58757 .00 51 10 53751 .57 56498 .65 50 10 55761 .54 58794 .63 50 11 9.53785 .57 9.56537 .65 49 11 9.55793 .54 9.58832 .63 49 12 53819 .57 56576 .65 48 12 55826 .54 58869 .oz 48 13 53854 .57 56615 .65 47 13 55858 .54 58907 .62 47 14 53888 .57 56654 .65 46 14 55891 .54 58944 .62 46 15 53922 .67 c 7 56693 .65 A C. 45 15 55923 .54 58981 .62 AO 45 16 53957 ,91 C7 56732 .00 A ^ 44 16 55956 .54 59019 .OZ AO 44 17 53991 .0 1 56771 .OO 43 17 55988 .54 59056 .OZ 43 18 54025 .57 56810 .65 42 18 56021 54 59094 .62 42 19 54059 56849 .65 41 19 56053 .54 59131 .62 41 20 54093 .57 56887 .65 40 20 56085 .54 59168 .62 40 21 9.54127 .57 c 7 9.56926 .65 A ^ 39 21 9.56118 .54 9.59205 .62 AO 39 22 54161 91 56965 .OO 38 22 56150 .54 59243 .OZ 38 23 54195 .57 57004 .65 37 23 56182 .54 59280 .62 37 24 54229 .57 57 57042 .64 A/i 36 24 56215 .54 59317 .62 AO 36 25 54263 1 57081 .04 35 25 56247 54 59354 .OZ 35 26 54297 .57 57120 .64 34 26 56279 54 59391 .62 34 27 54331 .56 cc 57158 .64 A 1 33 27 56311 .54 59429 .62 AO 33 28 54365 OO KA 57197 .04 AJ 32 28 56343 54 K.A 59466 .OZ AO 32 29 30 54399 54433 DO .56 57235 57274 .04 .64 31 30 29 30 56375 56408 04 53 59503 59540 .Oz .62 31 30 31 9.54466 Do 9.57312 .64 29 31 9.56440 53 9.59577 .62 29 32 33 54500 54534 .56 57351 57389 .64 A/I 28 27 32 33 56472 56504 .53 .53 CO 59614 59651 .62 .62 AO 28 27 34 35 54567 54601 .56 57428 57466 .04 .64 26 25 34 35 5653C 56568 OO .53 59688 59725 02 .62 26 25 36 54635 .56 57504 .64 24 36 56599 .53 59762 .62 24 37 38 54668 54702 .56 57543 57581 .64 .64 23 22 37 38 56631 56663 53 .53 59799 59835 .61 23 22 39 54735 .56 57619 .64 .64 21 39 56695 .53 CO 59872 .61 Al 21 40 54769 57658 20 40 56727 DO 59909 .O 1 20 41 9.54802 .56 r 9.57696 .64 19 41 9.56759 .53 9.59946 .61 19 42 54836 .DO c t> 57734 .64 18 42 56790 .53 59983 '!!! 18 43 54869 .DO 57772 64 17 43 56822 53 60019 .01 17 44 54903 .56 .56 57810 .64 16 44 56854 53 c>> 60056 .61 Al 16 45- 54936 57849 .04 15 45 56886 DO 60093 01 15 46 54969 .56 57887 .64 14 46 56917 .53 6013(1 .61 14 47 55003 .55 .55 57925 .63 13 47 56949 .53 CO 60166 .61 fii 13 48 49 55036 55069 .55 K.Z. 57963 58001 .63 12 11 48 49 56980 57012 Do .53 6020?, 60240 .01 .61 12 11 50 51 55102 9.55136 .00 .55 58039 9.58077 .63 10 9 50 51 57044 9.57075 .53 .53 60276 9.60313 .61 .61 10 9 52 55169 .55 r r 58115 .68 8 52 57107 .52 60349 .61 8 53 55202 '"? 58153 .63 A -J 7 53 57138 .52 60386 .61 7 54 55235 JO 58191 .00 6 54 57109 .52 60422 .61 6 55 55268 .55 r - 58229 .63 5 55 57201 .52 60459 .61 5 56 55301 5o f r 58267 .63 4 56 57232 .52 60495 .61 4 57 55334 .Do 58304 .63 3 57 57264 .52 60532 .61 3 58 55367 .55 r r 58342 .63 2 58 57295 .52 60568 .61 2 59 60 55400 55433 .OD .55 58380 58418 .63 .63 1 59 60 57326 57358 .52 .52 60605 60641 .61 .61 1 M. Cosine. D"l Ootaner. Dl" M. M. Cosine. Dl" Cotang. Dl" | M. SINES AND TANGENTS, 23 M. Sine, j Di" Tung. Dl" M. M. Sine. 1)1" Tang, j D." 31. o y.:>7:;.~>s 9.60641 C,!) T 9.5UloS n , n 9. 62785 L .,, 60 l 57389 0.52 60677 0.61 A1 59 i 59218 ; U ' 62820 u.oy CO 59 2 57420 .oz 60714 .01 C 1 58 2 59247 .49 62855 ,08 e,8 58 3 57451 ro 60750 .0 1 /* A 57 3 59277 62890 .00 CO 57 4 57482 q 60786 ,OU en 56 4 59307 .49 62926 .08 CO 56 5 57514 J, 60823 . OU 55 5 59336 62961 Oo 55 6 57545 ~ 608.)9 .60 54 6 59366 .49 62996 .58 54 7 57576 60895 .60 An 53 7 59396 .49 63031 .58 CO 53 8 9 57607 57638 '.52 FLO 61)9-51 60967 .OU .60 fift 52 51 8 9 59425 59455 ^49 63066 63101 .08 .58 eo 52 5] 10 57669 OZ c o 61004 .OU Aft 50 10 59484 j 63135 .JO eo 50 11 12 13 14 9.57700 57762 57793 .o/ .52 .52 .52 r 1 9.61040 61076 61112 61148 .OU .60 .60 .60 AH 49 48 47 46 11 12 13 14 9.59514 59543 59573 59602 !49 .49 .49 9.63170 63205 63240 63275 .08 .58 .58 .58 eo 49 48 47 46 15 57824 .01 T 1 61184 .OU 60 45 15 59632 '*? 6H310 .08 .58 45 16 57855 .0 1 r 1 61220 An 44 16 59661 JJ ! 63345 CO 44 17 57885 .51 r I 61256 .OU An 43 17 59690 A U 63379 .08 eo 43 18 57916 .01 61292 .OU 42 18 59720 4iy 63414 .08 CO 42 19 57947 el 61328 An 41 19 59749 * 63449 .08 F.Q 41 20 57978 01 T I 61364 .OU 60 40 20 59778 49 63484 OO .58 40 21 9.58008 .0 i r | 9.61400 An 39 21 9.59808 4Q 9.63519 CO 39 22 23 24 58039 ft SI) 70- 58101 .01 .51 .51 el 61436 61472 61508 .OU .60 fid 38 37 36 22 23 24 59837 59866 59895 !49 6 ? 55 * .49 4O ! 63bZo .08 .58 .58 RQ 38 37 36 25 58131 01 61544 .Oil 60 35 25 59924 1 63657 ,0o 35 26 58162 61579 60 34 26 59954 .49 63692 *?Q 34 27 58192 61615 33 27 59983 4Q 63726 '- 33 28 58223 . .51 61651 'g0 32 28 60012 ^rt7 .48 63761 .58 32 29 58253 61687 31 29 60041 /I U 63796 , CO 31 30 58284 51 61722 .OU 30 30 60070 4o 63830 ' 30 31- 9.58314 .51 9.61758 .60 An 29 31 9.60099 48 9.63865! .0 29 32 58345 .51 61794 .00 28 32 60128 63899 .00 28 58375 .51 el 61830 .59 5Q 27 33 60157 '4! 63934 .57 (^7 27 34 58406 .01 61865 .oy 26 34 60186 'iO .48 63968 .0 ( .57 26 35 36 58436 58467 'SI C 1 61901 61936 169 F;Q 25 24 35 36 60215 60244 .48 64003 64037 '.57 25 24 37 38 58497 58527 1 .51 61972 62008 ,O i7 .59 23 22 37 38 60273 60302 : 1? 64072 64106 !57 23 22 39 58557 .50 en 62043 .59 C.Q 21 39 603311 64140 .57 .57 21 40 58588 OU 62079 .oy 20 40 60359 TO 64175 20 41 9.58618 .50 en 9.62114 .59 19 41 9.60388 .48 48 9.64209 57 19 42 58648 OU 62150 'f!u 18 42 60417 .48 48 64243 O I .57 18 43 58678 ft 62185 en 17 43 60446 64278 17 44 58709 en 62221 .Oy 16 44 60474 '48 64312 .0 / 57 16 45 58739 . OU 62256 .Oa 15 45 60503 64346 15 46 58769 .50 62292 ,59 14 46 60532 .48 64381 .57 14 47 58799 .50 en 62327 .5S G 13 47 60561 .48 .48 64415 .57 13 48 58829 OU 62362 O c 12 48 60589 64449 _ 12 49 58859 .50 62398 .5 11 49 606181 '^ 64483 .57 11 50 58889 .50 62433 .5? 10 50 60646 !48 64517 .57 10 51 9.58919 .50 9.62468 .59 en 9 51 9.60675 A Q 9.64552 .57 e 7 9 52 .58949 .50 en 62504 .Oi 'er 8 52 60704 4o 64586 .0 / .57 8 53 58979 O'J en 62539 .OV c r 7 53 60732 40 64620 57 7 54 59009 .OU en 62574 .oy er 6 54 60761 ^to 40 64654 .0 1 .57 6 55 59039 .OU r A 62609 .OV 5 55 i 60789 rto 4' 7 64688 ^7 5 56 59069 .OU 62645 '** 4 56 60818 I 64722 .0 1 4 57 59098 .50 e A 62680 1 { 3 57 1 60846 .47 64756 .57 1 P.7 3 58 59128 .OU CA 62715 e r 2 58 60875 .47 A7 64790 .91 K7 2 59 59158 OU CA 62750 .oy r n 1 59 60903 rt ( 64824 1 e.7 1 60 59188 .OU 62785 .Ov 60 60931 .47 64858 .O/ M. Cosine. | PI" Cot an ar. Dl" M. M. Cosine. Dl" Ootang. Dl" M. 67 C TABLE IV. LOGARITHMIC 31. Sine. Di" Tang. Dl" 31. M. .Sine. Dl" Tang. Dl" II. 1 9.60931, 609M -l; 9.64858 64892 0.57 60 59 1 9.62595 62622 0.45 9.66867 66900 0.55 60 59 2 60988 ,1 64926 'IL 58 2 62649 .45 66933 .55 c c 58 3 61016 .*< 64960 57 3 62676 .45 66966 .00 57 4 61045 .47 64994 .57 56 4 62703 .45 66999 .55 56 5 61073 .47 A 7 65028 .57 t 7 55 5 62730 .45 A Z* 67032 .55 t c 55 6 61101 .4< 65062 .0 / 54 6 62757 .40 67065 .00 54 7 61129 .47 A 7 65096 .56 53 7 62784 .45 A f^ 67098 .55 c r 53 8 9 61158 61186 .4< .47 65130 65164 !56 52 51 8 9 62811 62838 .40 .45 67131 67163 .00 .55 52 51 10 61214 .47 65197 .56 50 10 62865 .45 67196 .55 50 11 9.61242 .47 9.65231 .56 49 11 9.62892 .45 9.67229 .55 49 12 13 61270 61298 .47 .47 65265 65299 .56 .56 48 47 12 13 62918 62945 .45 .45 67262 67295 .55 .55 48 47 14 61326 .47 47 65333 s 46 14 62972 .45 A f\ 67327 .55 46 15 61354 .4 1 65366 '"" 45 15 62999 40 4S 67360 R Cv 45 16 61382 A 7 65400 '2 44 16 63026 .40 A K. 67393 Ot> C r 44 17 61411 .4* A 7 65434 . 43 17 63052 .40 A R 67426 .00 c A 43 18 61438 .4< 65467 _ 42 18 63079 .40 67458 .00 42 19 61466 .47 A *7 65501 -/ 41 19 63106 .45 67491 .54 41 20 61494 .4* 65535 .00 40 20 63133 .44 67524 : 40 21 9.61522 A T 9.65568 ' 39 21 9.63159 t! |9.67556 .54 39 22 61550 .47 65602 '{[X 38 22 63186 ** 67589 38 23 61578 4fi 65636 *2 37 23 63213 44 67622 Rl 37 24 25 61606 61634 .40 .46 A A 65669 '2 65703 '2 36 35 24 25 83239 63266 44 .44 A A 67654 67687 .0-4 .54 f\A 36 35 26 61662 .40 4fi 65736 '2 34 26 63292 .44 A A 67719 .04 34 27 61689 .40 A A 65770 n. A 33 27 63319 .44 A A 67752 C ,1 33 28 61717 .40 A A 65803 .00 32 28 63345 .44 A A 67785 .04 32 29 30 61745 61773 .40 .46 A a 65837 65870 ^56 FvA 31 30 29 30 63372 63398 .44 .44 A A 67817 67850 .54 31 30 31 32 9.61800 ' 61828 ! "JJ 9.65904 65937 .00 .56 56 29 28 31 32 9.63425 63451 .44 .44 44 9.67882 67915 .54 .54 29 28 33 61856 A A 65971 jJU 27 33 63478 .44 A 1 67947 C | 27 34 61883 .40 4fi 66004 .56 26 34 63504 .44 44 67980 .04 .54 26 35 61911 4O 66038 25 35 63531 44 68012 25 36 61939 .46 66071 fit* 24 36 63557 .44 1 A 68044 .54 KJ 24 37 61966 46 66104 .00 flfi. 23 37 6S583 .44 44 68077 .04 .54 23 38 61994 46, 66138 i '"" 22 38 63610 44 A A 68109 22 39 40 62021 62049 .40 .46 661711 '2 66204 '2 21 20 39 40 63636 63662 .44 .44 68142 68174 .54 21 20 41 9.62076! 'JJ 9.66238 TI 19 41 9.63689 .44 9.68206 .54 19 42 62104 ' 66271 - 18 42 63715 .44 A A 68239 .54 18 43 62131 66304 TT 17 43 63741 .44 68271 17 44 62159 ' 66337 'rr 16 44 63767 .44 A A 68303 16 45 -62186! 1 66371 'J? 15 45 63794 .44 44 68336 15 46 62214 Af\ 664041 '2 14 46 63820 4^t 44. 68368 " 14 47 48 62241 62268 .40 .46 A A 66437 '2 66470 :2 13 12 47 48 63846 63872 .44 .44 A A 68400 68432 1J '* 49 62296 .40 66503 '2 11 49 63898 .44 68465 .04 ! i , 50 62323 .46 4^ 66537 10 50 8*924 .44 49 68497 11 10 51 9.62350 40 4i 9.66570 '2 9 51 9.63950 .40 4S 9.68529 M^ 52 62377 .40 66603 ?? 8 52 63976 rt'> 68561 Q fid " 53 62405 'H 66636 '?! 7 53 64002 .43 A *-J 68593 .04 7 54 62432 .4.) 66669 '2 6 54 64028 .4o 68626 6 55 62459 .45 A & 66702 "2 5 55 64054 }?! 68658 .54 5 56 62486 .40 66735 TI 4 56 64080 JJ i 68690 4 57 62513 .45 66768 3 57 64106 *J 68722 .54 K. O 3 58 62541! '[? 66801 *?? 2 58 64132 J:? 68754 2 59 62568 ** 66834 .'- 1 59 64158 JJ 68786 .0.} 1 60 62595 - 4o 66867 60 64184 68818 .53 M. Cosine. Dl" Cotang. Dl" 31. M. Cosine. Dl" Cotang.j Dl" M. SINES AND TANGENTS. 27 M. Sine. Dl" Tang. Dl" M. M. Sine. Dl" Tang. Di" M. 1 9.64184 64210 ..J 9.68818 * 68850 0.53 CO 60 59 1 9.65705 65729 0.41 A 1 9.70717 70748 0.52 60 59 2 64236 JJ 68882 .Oo 58 2 65754 .41 70779 .52 58 3 4 64262 64288 .43 .43 68914 68946 .53 .53 57 56 3 4 65779 65804 .41 .41 70810 70841 .52 .52 57 56 5 64313 .43 A 9 68978 .53 CO 55 5 65828 .41 A 1 70873 .52 CO 55 6 64339 .4o 69010 Oo 54 6 65853 .41 70904 .02 54 7 64365 .43 69042 .53 CO 53 7 65878 .41 A 1 70935 .52 co 53 8 64391 .43 ,1 O 69074 .OO CO 52 8 65902 .41 A 1 70966 .02 en 52 9 61417 A6 A O. 69106 .00 r 51 9 65927 .41 70997 02 CO 51 10 11 12 64442 9.64468 64494 .4o .43 .43 A 'i 69138 9.69170 69202 OO .53 .53 C O 50 49 48 10 11 12 65952 9.65976 66001 .41 .41 .41 A 1 71028 9.71059 71090 OZ .52 .52 CO 50 49 48 13 14 64519 64545 ,4o .43 A 9 69234 69266 .Oo .53 C O 47 46 13 14 66025 66050 41 .41 A 1 71121 71153 .02 .52 CO 47 46 15 64571 .4o A . 69298 .00 CO 45 15 66075 .41 A 1 71184 02 CO 45 16 17 64596 64622 .4o .43 69329 69361 .00 .53 44 43 16 17 66099 66124 41 .41 71215 71246 .OZ .52 44 43 18 19 64647 64673 .43 .43 69393 69425 .53 .53 CO 42 41 18 19 66148 66173 .41 .41 41 71277 71308 .52 .52 42 41 20 21 64698 9.64724 .42 69457 9.69488 .Do .53 CO 40 39 20 21 66197 9.66221 41 .41 .41 71339 9.71370 '.52 .52 40 39 22 64749 49 69520 .00 CO 38 22 66246 41 71401 CO 38 23 64775 42 69552 .Do C ) 37 23 66270 rxl 41 71431 OZ .52 37 24 64800 .^tz 69584 Do co 36 24 66295 .rt 1 41 71462 CO 36 25 26 27 64826 64851 64877 .42 .42 69615 69647 69679 .Do .53 .53 CO 35 34 33 25 26 27 66319 66,343 66368 .41 .41 .41 41 71493 71524 71555 Oz .51 .51 ci 35 34 33 28 64902 49 69710 Do CO 32 28 66392 rtl 40 71586 .01 Kl 32 29 64927 49 69742 .Do 31 29 66416 .^iv 40 71617 OX C1 31 30 64953 4Z 49 69774 CQ 30 30 66441 4U 71648 01 30 31 32 33 9.64978 65003 65029 *Z .42 .42 4') 9.69805 69837 69868 Do .53 .53 cq 29 28 27 31 32 33 9.66465 66489 66513 .'40 .40 40 9.71679 71709 71740 .51 .51 P.1 29 28 27 34 65054 A** .42 69900 .Do cq 26 34 66537 A" 40 71771 01 26 35 65079 69932 Do CO 25 35 66562 rrv 40 71802 M 25 36 37 65104 65130 .'42 69963 69995 .Do .53 24 23 36 37 66586 66610 .4v .40 71833 71863 01 .51 r 1 24 23 38 65155 49 70026 .53 co 22 38 66634 40 71894 .oJ Cl 22 39 65180 42 70058 .Do CO 21 39 66658 4" 40 71925 J e.1 21 40 41 42 43 65205 9.65230 65255 65281 .*Z .42 .42 .42 70089 9.70121 70152 70184 OZ .52 .52 .52 20 19 18 17 40 41 42 43 66682 9.66706 66731 66755 rrV .40 .40 .40 71955 9.71986 72017 72048 01 .51 .51 .51 20 19 18 17 44 65306 .42 70215 .52 CO 16 44 66779 .40 40 72078 .51 .51 16 45 65331 70247 .Oz 15 45 66803 Av 72109 15 46 65356 .42 49 70278 .52 CO 14 46 66827 .40 4ft 72140 .51 14 47 65381 .4z 1 O 70309 .Oz 13 47 66851 .4U 72170 C -I 13 48 65406 .4z 70341 .52 12 48 66875 .40 72201 .01 12 49 65431 .42 70372 .52 CO 11 49 66899 .40 .40 72231 .51 KI 11 50 65456 A ) 70404 .Oz 10 50 66922 72262 .ul c -i 10 51 52 9.65481 65506 AL .42 .42 9.70435 70466 .52 .52 9 9 8 51 52 9.66946 66970 .40 .40 9.72293 72323 .01 .51 .51 9 8 53 65531 70498 .OZ 7 53 66994 72354 7 54 65556 .42 70529 .52 6 54 67018 .40 72384 .51 6 55 65580 .41 1 1 70560 .52 5 55 67042 A A 72415 .61 C I 5 56 65605 .4 1 A 1 70592 .52 4 56 67066 .4U A A 72445 .01 c T 4 57 58 65630 65655 .41 .41 A 1 70623 70654 .52 3 2 57 58 67090 67113 .40 .40 A A 72476 72506 01 .51 K1 3 2 59 65680 .4 1 70685 .52 1 59 67137 .40 72537 .01 1 60 65705 .41 70717 .52 60 67161 .40 72567 .51 M. Cosine. | Dl" CotHllg. Dl" M. M. Cosine. Dl" Cotang. Dl" M. 62 TABLE IV. LOGARITHMIC M. Sine. 1)1" Tang. 1)1" M. 31. Sine. Dl" Tang. 1)1" 0. 1 2 9.l>7161 67185 67208 0.40 .40 9.72567 72598 72628 0.51 .51 K-l 60 59 58 1 2 9.68557 68580 68603 0.38 .38 QQ 9.74375 744D5 74485 0.50 .50 fiO 59 58 3 67232 on 72659 .01 C -I 57 3 68625 .GO Oc 7440,3 r f , 57 4 5 67256 67280 ov .39 qq 72689 72720 .0 I .51 RI 56 55 4 5 68648 68671 Go .38 .38 74494 74524 OU .50 .50 56 55 6 67303 oy 72750 Ol r ] 54 6 68694 oo 74554 r f\ 54 7 67327 .39 72780 .01 53 7 68716 .00 qo 74583 .OU C A 53 8 67350 39 72811 52 8 68739 .OO 7461., .OU 52 9 67374 39 72841 - .51 51 9 68762 .38 OQ 74643 .50 51 10 67398 39 qq 72872 .51 si 50 10 68784 .00 .38 74673 '49 50 11 9.67421 oy 9.72902 .01 49 11 9.68807 >s 9.74702 * 49 12 67445 oy 72932 si 48 12 68829 oo .38 74732 "do .49 48 13 67468 39 72963 O 1 47 13 68S52 74762 47 14 67492 39 72993 .51 C -| 46 14 68875 .38 <>Q 74791 49 46 15 67515 39 73023 .01 50 45 15 68897 .OO .38 748-21 49 4.0 45 16 17 18 67539 67562 67586 39 39 on 73054 73084 73114 !so .50 SO 44 43 42 16 17 18 68920 68942 68965 .38 .38 .38 74851 74880 74910 .^V' 49 .49 /in 44 43 42 19 67609 ft) 9 73144 .OU 41 19 68987 74939 ' 41 20 67C33 .39 73175 .50 40 20 69010 .37 37 74969 ' ' 40 21 9.67656 .39 9.73205 .50 39 21 9.69032 i 07 9.74998 39 22 23 67680 - 67703 .39 .39 73235 73265 .50 .50 38 37 22 23 69055 69077 Ol .37 O 7 75028 75058 .49 .49 38 37 24 67726 .39 73295 .50 36 24 69100 o4 07 75087 .49 36 25 26 67750 67773 .39 .39 73326 73356 .50 .50 35 34 25 26 69122 69144 O/ .37 75117 75146 .49 49 35 34 27 67796 .39 73386 .50 33 27 69167 i 75176 49 33 28 67820 .39 73416 .50 32 28 69189 3i 75205 49 32 29 67843 .39 73446 .50 31 29 69212 o 7 75235 .49 31 30 67866 .39 73476 .50 30 30 69234 > / O*7 75264 .49 30 31 9.67890 .39 9.73507 .50 CA 29 31 9.69256 61 37 9.75294 .49 29 32 33 67913 67936 .39 73537 73567 .OU .50 28 27 32 33 69279 69301 .37 75323 75353 .49 28 27 34 35 36 67959 67982 68006 .39 .39 .39 73597 73627 73657 .50 .50 .50 26 25 24 34 35 36 69323 69345 69368 r 37 37 q7 75382 75411 75441 .41) .49 .49 26 25 24 37 68029 .39 73687 .50 23 37 69390 O 4 07 75470 49 23 38 68052 .39 73717 .50 C A 22 38 69412 6i 37 75500 .49 4Q 22 39 68075 39 73747 .OU 21 39 69434 75529 4y 21 40 41 68098 9.68121 !:',8 73777 9.73807 .50 .50 20 19 40 41 69456 9.69479 37 .37 q7 75558 9.75588 .49 .49 20* 19 42 68144 g 73837 .50 18 42 69501 o t 37 75617 .49 18 43 68167 73867 .50 17 43 69523 i 75647 .49 17 44 68190 .38 73897 .50 16 44 69545 07 75676 49 16 45 68213 .38 qo 73927 .50 so 15 45 69567 o< 37 75705 49 15 46 47 68237 68260 oo .38 73957 73987 .OU .50 14 13 46 47 69589 69611 .37 75735 75764 ^49 14 13 48 68283 38 74017 .50 12 48 69633 37 37 75793 .49 12 49 68305 38 74047 .50 11 49 69655 i *>7 75822 .49 -, 50 68328 .38 74077 .50 10 50 69i>77 07 75852 .49 10 51 9.68351 .38 9.74107 .50 9 51 9.69699 "o- 9.75881 .49 9 52 68374 .38 741E7 .50 8 52 69721 !s7 7591 .49 8 53 68397 .38 74166 .50 7 53 69743 07 75939 49 7 54 68420 .38 74 1 '.16 .50 6 54 69765 Ol 75969 .49 6 55 68443 .38 qo 74226 .50 5 55 697S7 37 9,7 75998 .49 5 56 68466 oo 742.36 .OU 4 56 69809 O 4 76027 4y 4 57 68489 .38 74286 .50 3 57 69831 .37 76056 .49 3 58 68512 .38 74316 .50 2 58 69853 37 3- 76086 .49 2 59 68534 .38 74345 .50 1 59 69875 < 76115 .49 1 60 68557 .38 74375 .50 60 69897 .36 76144 .49 M. Cosine. Dl" Cotang. Dl" M. M. Cosine. Dl" CntaiiLr. Dl" M. 6O SINES AND TANGENTS. 31 M. Sine. Dl" Taiig. Di" 11. M. Sine'. 1)1" Taiig. Dl" M. 1 9.69897 69919 0.3(5 / 9.76144 76173 0.49 A O 60 59 1 9.71184 71205 0.35 o r 9.77877 77906 0.48 A Q 60 59 2 69911 .00 76202 .4 l , 58 2 71226 .OO 77935 .48 58 3 69963 .36 >/ 76231 .49 57 3 71247 .35 3x 77963 .48 A Q 57 4 69984 .00 76261 .49 56 4 71268 77992 .48 56 5 6 70006 70028 .36 .36 76290 7631 9 .49 .49 55 54 5 6 71289 71310 .35 .35 78020 78049 .48 .48 55 54 7 8 9 70050 70072 70093 .36 .36 .36 76348 76377 76406 .48 .48 53 52 51 7 8 9 71331 71352 71373 .35 .35 .35 78077 78106 78135 .48 .48 .48 53 52 51 10 70115 .36 no 76435 .48 A Q 50 10 71393 .35 C 78 i 63 .48 A U 50 11 9.70137 .OO ifi 9.76464 4o 40 49 11 9.71414 .00 .35 9.78192 .48 .48 49 12 70159 ..DO 76493 4o 48 12 71435 78220 48 13 70180 .36 76522 .48 4Q 47 13 71456* 9C 78249 -.48 47 47 14 15 16 70202 70224 70245 .36 .36 76551 76580 76609 4o .48 .48 4U 46 45 44 14 15 16 - 71477 71498 71519 .0 j .35 .35 .35 78277 78306 78334 .^t < .47 .47 .47 46 45 44 17 70267 .fi, 76639 4o 43 17 71539 or 78363 A 7 43 18 70288 .00 76668 4k 42 18 71560 O J .35 78391 ! . .47 42 19 70310 .00 76697 4o 41 19 71581 78419 41 20 70332 .36 Of> 76725 .48 4Q 40 20 71602 .35 OF, 78448 .47 40 21 9.70353 .00 oc 9.76754 .48 AQ 39 21 9.71622 oo .35 9.78476 .47 39 22 70375 .00 76783 .'iO 38 22 71643 78505 38 23 70396 .36 O/; 76812 .48 A Q 37 23 71664 .35 78533 .47 37 24 70418 .OO Q A 76841 ,4o A Q 36 24 71685 .4 78562 A ^ 36 25 70439 .OO Ofi 76870 4o A Q 35 25 71705 o4 04 78590 .4 1 35 26 70461 .OO Ofi 76899 ,4o 40 34 26 71726 O* 04 78618 A7 34 27 28 70482 70504 OO .36 Of* 76928 76957 4:0 .48 33 32 27 28 71747 71767 O^r .34 78647 78675 ^47 A 7 33 32 29 70525 .00 Ofi 76986 .48 4ft 31 29 71788 ** 78704 41 31 30 70547 .00 Of 77015 *4o A Q 30 30 71809 78732 47 30 31 9.70568 .00 Ofl 9.77044 .48 10 29 31 9.71829 9.78760 4 4 47 29 32 33 34 70590 70611 70633 .OO .36 .36 O A 77073 77101 77130 .48 .48 .48 A Q 28 27 26 32 33 34 71850 71870 71891 '.34 .34 0,1 78789 78817 78845 ! t .47 .47 4 fr 28 27 26 35 70654 .00 OA 77159 .4o 40 25 35 71911 ..14 .34 78874 4* 47 25 36 o 7 70675 70697 .00 .36 Ofi, 77188 77217 'io .48 4ft 24 23 36 37 71932 71952 .'34 04 78902 78930 4: i .47 24 23 38 70718 OO 77246 .48 22 38 71973 OT: .34 78959 j" 22 39 70739 .00 . 77274 .48 21 39 71994 78987 .4 / 21 40 70761 .36 C 77303 .48 A Q 20 40 72014 O 1 79015 .47 20 41 9.70782 .00 9.77332 .48 19 41 9.72034 .o4 9.79043 7 19 42 43 70803 70824 .35 .35 3 ^ 77361 77390 .48 .48 4ft 18 17 42 43 72055 72075 .34 .34 79072 79100 .47 .47 18 17 44 70846 ..30 ox 77418 .rro A Q 16 44 72096 '[. 79128 47 16 45 70867 .OO 77447 .48 4ft 15 45 72116 79156 .4 1 47 15 46 70888 "v 77476 .48 14 46 72137 .34 79185 .4 1 47 14 47 70909 ~ 77505 A Q 13 47 72157 o A 79213 47 13 48 49 70931 70952 .03 .35 77533 77562 .48 .48 12 11 48 49 72177 72198 .o4 .34 79241 79269 I .47 12 11 50 51 70973 9.70994 .35 .35 77591 9.77619 .48 .48 10 9 50 51 72218 9.72238 .34 .34 .34 79297 9.79326 .47 .47 .47 10 9 52 71015 .OU 77648 .48 8 52 72259 79354 8 53 54 55 71036 71058 71079 .35 .35 .35 77677 77706 77734 .48 .48 .48 7 6 5 53 54 55 72279 72299 72320 .34 .34 .34 79382 79410 79438 .47 .47 .47 7 6 5 56 71100 .35 77763 .48 4 56 72340 34 79466 .47 4 57 71121 .35 77791 .48 3 57 72360 .34 79495 .47 3 58 71142 .35 r 77820 .48 A Q 2 58 72381 .34 0/1 79523 .47 2 59 71163 .OD 77849 .48 ] 59 72401 .o4 79551 .47 1 60 71184 .35 77877 .48 60 72421 .34 79579 .47 Mb Cosine. Dl" Cotans. Dl" M. 31. Cosine. Dl" Cotanji. Dl" M. 59 43 58 32 TABLE IV. LOGARITHMIC 38 M. Sine. Dl" Tang. Dl" M. M. Sine. Dl" Tang. Dl" M. 1 9.72421 72441 0.34 9.79579 79607 0.47 60 59 1 9.73611 73630 0.32 9.81252 81279 0.46 60 59 2 72461 j ft 79635 .47 58 2 73650 .32 QO 81307 .46 58 a 4 72482 ** 72502 * 79663 79691 !47 57 56 3 4 73669 73689 62 .32 81335 81362 .46 .46 57 56 5 72522 '4 79719 .47 55 5 73708 .32 81390 .46 55 6 72542 .34 79747 .47 54 6 73727 .32 81418 .46 54 7 72562 .34 79776 .47 53 7 73747 .32 81445 .46 53 8 72582 .34 79804 .47 52 8 73766 .32 81473 .46 52 9 72602 .34 79832 .47 51 9 73785 .32 81500 .46 51 10 72622 .33 79860 .47 50 10 73805 32 81528 .46 50 11 9.72643 .33 9.79888 .47 49 11 9.73824 .32 9.81556 .46 49 12 72663 .33 79916 .47 48 12 73843 .32 81583 .46 48 13* 14 72683 72703 .33 .33 79944 79972 .47 .47 47 46 13 14 73863 73882 .32 .32 81611 81638 .46 .46 47 46 15 16 17 72723 72743 72763 .33 .33 .33 qq 80000 80028 80056 .47 .47 .47 4,7 45 44 43 15 16 17 73901 73921 73940 .32 .32 .32 QO 81666 81693 81721 .46 .46 .46 AR 45 44 43 18 72783 60 OQ 80084 .4 f 42 18 73959 62 81748 .40 42 19 20 72803 72823 66 33 80112 80140 .47 .47 41 40 19 20 73978 73997 .32 .32 81776 81803 .46 .46 41 40 21 22 23 9.72843 72863 72883 33 .33 .33 9.80168 80195 80223 !47 .47 39 38 37 21 22 23 9.74017 74036 74055 .32 .32 .32 9.81831 81858 81886 .46 .46 .46 39 38 37 24 25 72902 72922 *33 .33 00 80251 80279 .47 .47 4- 36 35 24 25 74074 74093 .32 .32 OO 81913 81941 .46 .46 36 35 26 72942 OO oo 80307 7 34 26 74113 OZ QO 81968 A A 34 27 28 72962 72982 OO .33 80335 80363 !46 33 32 27 28 74132 74151 02 .32 81996 82023 40 .46 33 32 29 30 31 32 33 73002 73022 9.73041 73061 7303 1 O CO CO CO CO o co co eo co < 80391 80419 9.80447 80474 80502 .46 .46 .46 .46 .46 31 30 29 28 27 29 30 31 32 33 74170 74189 9.74208 74227 74246 32 .32 .32 .32 .32 82051 82078 9.82106 82133 82161 .46 .46 .46 .46 .46 31 30 29 28 27 34 73101 66 80530 .46 26 34 74265 32 82188 .46 26 35 73121 33 80558 .46 25 35 74284 32 82215 .46 25 36 73140 33 80586 .46 24 36 74303 32 82243 .46 24 37 73160 .33 QQ 80614 .46 A 23 37 74322 .32 QO 82270 .46 A A 23 38 73180 OO OQ 80642 .4o 22 38 74341 62 oo 82298 4o 22 39 73200 66 80669 .46 21 39 74360 62 82325 .46 21 40 73219 33 80697 .46 20 40 74379 .32 oo 82352 .46 20 41 9.73239 -33 qq 9.80725 .46 A a 19 41 9.74398 62 qo 9.82380 .46 AO 19 42 73259 oo OQ 80753 .40 A a 18 42 74417 62 QO 82407 .40 A A 18 43 73278 oo 80781 .46 17 43 74436 .64 82435 .40 17 44 73298 33 OQ 80808 .46 16 44 74455 .32 oo 82462 .40 16 45 73318 66 80836 '* 15 45 74474 .62 82489 .46 15 46 73337 33 80864 14 46 74493 '%* 82517 .46 14 47 73357 33 80892 .46 13 47 74512 ** 82544 .46 13 48 73377 33 QQ 80919 .46 A - 12 48 74531 6i O1 82571 .46 i A 12 49 73396 OO OQ 80947 .4} A t* 11 49 74549 31 82599 .40 11 50 73416 66 QQ 80975 .46 A a 10 50 74568 'Jf 82626 .46 10 51 9.73435 66 9.81003 .4o 9 51 9.74587 } 9.82653 .46 9 52 73455 33 OQ 81030 .46 8 52 74606 } 82681 .46 8 53 73474 00 81058 .46 7 53 74625 *} 82708 .45 7 54 73494 33 QQ 81086 .46 A & 6 54 74644 *{ 82735 .45 A EL 6 55 73513 oo 81113 .40 5 55 74662 ! *} 82762 .40 5 56 73533 33 81141 .46 4 56 74681 *} 82790 .45 4 57 73552 32 oo 81169 .46 3 57 74700 {] 82817 .45 3 58 73572 62 81196 .46 2 58 74719 ^; 82844 .45 2 59 73591 32 oo 81224 .46 1 59 74737 *} 82871 .45 1 60 73611 62 81252 .46 60 74756) ' 82899 .45 M. Cosine. Dl" Cotang. Dl" M. M. Cosine. Dl" Cotang. Dl" M. 57 44 56 SINES AND TANGENTS. M. Sine. Dl" Tang. IK" M. M. bine. Dl" Tung. Dl" M. 1 9.74756 74775 0.31 9.82899 82926 0.45 60 59 1 9.75859 75877 0.30 9.84523 84550 0.45 60 59 2 74794 .31 82953 .45 58 2 75895 .30 O A 84576 .45 .1 - 58 3 74812 .31 82980 .45 57 3 75913 .oU 84603 .40 57 4 74831 .31 83008 .45 56 4 75931 .30 84630 .45 56 5 74850 .31 83035 .45 55 5 75949 .30 84657 .45 55 6 74868 .31 83062 .45 54 6 75967 .30 84684 .45 54 7 74887 .31 83089 .45 53 7 75985 .30 8471 1 .45 53 8 74906 .31 Q1 83117 .45 A C 52 8 76003 .30 A 84738 .45 AZ 52 9 74924 .OJ 83144 .40 51 9 76021 .60 84764 .40 51 10 74943 .31 83171 .45 A E*. 50 10 76039 .30 O A. 84791 .45 AZ. 50 11 9.74961 .31 9.83198 .40 49 11 9.76057 0(1 9.84818 .40 49 12 74980 .31 83225 .45 1JL 48 12 76075 .30 on 84845 .45 A *\ 48 13 14 15 74999 75017 75036 .31 .31 .31 83252 83280 83307 .40 .45 .45 47 46 45 13 14 15 76093 76111 76129 .oU .30 .30 84872 84899 84925 .40 .45 .45 47 46 45 16 75054 .31 83334 .45 44 16 76146 .30 84952 .45 44 17 75073 .31 83361 .45 43 17 76164 .30 84979 .45 43 18 75091 .31 83388 .45 42 18 76182 .30 85006 .45 A ^ 42 19 75110 .31 1 83415 * 41 19 76200 .30 >n 85033 .40 45 41 20 75128 Ol 83442 A ^ 40 20 76218 oU A 85059 A ^\ 40 21 9.75147 .31 9.83470 .40 39 21 9.76236 .30 9.85086 .40 39 22 75165 .31 83497 .45 38 22 76253 .30 85113. .45 38 23 75184 .31 88524 .45 37 23 76271 .30 85140 .45 A ^ 37 24 25 26 75202 75221 75239 .31 .31 .31 83551 83578 83605 .45 .45 .45 36 35 34 24 25 26 76289 76307 76324 .30 .30 .30 85166 85193 85220 .40 .45 .45 36 35 34 27 75258 .31 9.1 83632 .45 33 27 76342 .30 on 85247 .45 45 33 28 75276 ol 01 83659 45 32 28 76360 oU on 85273 .45 32 29 75294 Ol 83686 31 29 76378 .0" 85300 31 30 31 32 75313 9.75331 75350 .31 .31 .31 Q1 83713 9.83740 83768 .45 .45 .45 30 29 28 30 31 32 76395 9.76413 76431 .30 .30 .29 oft 85327 9.85354 85380 !45 .45 45 30 29 28 33 75368 01 O 1 83795 4^ 27 33 76448 Z 7 OQ 85407 27 34 75386 .ol O 1 83822 .40 26 34 76466 .z 85434 44 26 35 75405 6 I q 1 83849 A* 25 35 76484 9Q 85460 .^-t .44 25 36 75423 .6 L O 1 83876 At. 24 36 76501 9Q 85487 24 37 75441 .ol 83903 .40 23 37 76519 zy 85514 A i 23 38 75459 .30 on 83930 .45 45 22 38 76537 .29 85540 .44 .44 22 39 75478 .60 83957 21 39 76554 .^y 85567 21 40 41 75496 9.75514 .30 .30 on 83984 9.84011 .45 .45 .45 20 19 40 41 76572 9.76590 .29 .29 29 85594 9.85620 .44 .44 .44 20 19 42 75533 OU on 84038 A 18 42 76607 on 85647 AA. 18 43 44 75551 75569 .60 .30 OA 84065 84092 .40 .45 4^ 17 16 43 44 76625 76642 .zy .29 85674 85700 .44 .44 .44 17 16 45 46 75587 75605 .60 .30 84119 84146 rtO .45 15 14 45 46 76660 76677 .29 85727 85754 !44 A I 15 14 47 75624 .30 84173 .45 .45 13 47 76695 .29 85780 .44 .44 13 48 75642 OA 84200 12 48 76712 OQ 85807 44 12 49 75660 .oU on 84227 '* 11 49 76730 zy 85834 .44 .44 11 50 75678 Ov 84254 * 10 50 76747 9Q 85860 44 10 51 9.75696 O A 9.84280 A f\ 9 51 9.76765 n 9.85887 .44 A A 9 52 75714 .60 OA 84307 .40 .45 '8 52 76782 9Q 85913 .44 .44 8 53 75733 .OU O A 84334 A P 7 53 76800 Q 85940 A A 7 54 55 56 75751 75769 75787 .oU .30 .30 on 84361 84388 84415 .40 .45 .45 A P* 6 5 4 54 55 56 76817 76835 76852 .29 .29 85967 85993 86020 .44 .44 .44 A A 6 5 4 57 58 59 75805 75823 75841 .oU .30 .30 84442 84469 84496 *40 .45 .45 3 2 1 57 58 59 76870 76887 76904 .29 .29 86046 86073 86100 .44 .44 .44 3 2 1 60 75859 .30 84523 .45 60 76922 .29 86126 .44 M. Cosine. Dl" Cotang.! Dl" M. M. Cosine. Dl" Cotang. Dl" M. 55 C 54' 36= TABLE IV. LOGARITHMIC 37 31. Sine. Di" Tang. Dl" 31. 31. Sine. Dl" Tang. Dl" 31. 1 9.76922 76939 0.29 9.86126 86153 0.44 60 59 1 9.77946 7796:; 0.28 9.87711 877:18 "- '! 2 STB957 .29 86179 .44 58 2 77980 ,2s 877114 3 76974 .29 86206 .44 57 3 77997 .28 87790 '11 57 4 76991 .29 86232 .44 56 4 78013 .28 87817 11 56 5 77009 .29 86259 .44 55 5 78030 .28 87843 .44 , - 6 77026 .29 86285 .44 54 6 78047 .28 87869 .44 54 7 77043 .29 86312 .44 53 7 78063 .2S 87895 .44 53 8 77061 .29 86338 .44 52 8 78080 .2s 87922 .44 52 9 77078 .29 86365 .44 51 9 78097 .28 8794S .44 51 10 77095 2r\ 86392 .44 50 10 78113 .28 87974 .44 50 11 9.77112 9 9.86418 .44 49 11 9.78130 .Zo 9.88000 .44 49 12 77130 .29 86445 .44 48 12 78147 .28 88027 .44 48 13 77147 .29 86471 .44 47 13 78163 .28 88053 .44 47 14 15 77164 77181 .29 .29 86498 8524 .44 .44 46 45 14 15 78180 78197 .28 .28 88079 88105 .44 .44 46 45 16 77199 .29 86551 .44 44 16 78213 .28 88131 .44 44 17 77216 29 86577 .44 43 17 78230 .28 88158 .44 43 18 77233 .29 86603 .44 42 18 78246 .28 88184 .44 42 19 77250 .29 86630 .44 41 19 78263 .2s 88210 .44 41 20 21 77268 9.77285 29 .29 86656 9.86683 .44 .44 40 39 20 21 782SO 9.78296 .28 .28 88236 9.88262 .44 .44 40 39 22 77302 29 86709 .44 38 22 78313 .28 88289 .44 38 23 77319 29 86736 .44 37 23 78329 .28 88315 .4-4 37 24 77336 29 86762 .44 36 24- 78346 .28 88341 .44 36 25 26 77353 77370 29 29 86789 86815 .44 .44 35 34 25 26 78362 78379 .28 .28 88367 88393 .44 .44 35 34 27 77387 29 86842 .44 33 27 78395 .27 88420 .44 32 28 77405 28 86868 .44 32 28 78412 .27 88446 .44 32 29 77422 28 86894 .44 31 29 78428 .27 88472 .44 31 30 77439 28 OQ 86921 .44 A 4 30 30 78445 .27 O7 88498 .44 30 31 9.77456 Zo 9.86947 .44 29 31 9.78461 .Zl 9.88524 .44 29 32 77473 28 86974 .44 28 32 78478 .27 88550 .44 28 33 77490 28 87000 .44 27 33 78494 .27 88577 .44 27 34 77507 28 87027 .44 26 34 78510 .27 88603 .44 26 35 77524 28 87053 .44 25 35 78527 .27 88629 .44 25 36 77541 28 87079 .44 24 36 78543 .27 88655 .44 24 37 77558 28 87106 .44 23 37 78560 .27 88681 .44 23 38 77575 *! 87132 .44 22 38 78576 .27 88707 .44 22 39 40 77592 77609 28 87158 87185 .44 .44 21 20 39 40 78592 78609 .27 .27 88733 88759 .44 .44 21 20 41 42 9.77626 77643 28 28 9.87211 87238 .44 .44 19 18 41 42 9.78625 78642 .27 .27 9.88786 88812 .44 .44 19 18 43 77660 .28 87264 .44 17 43 78658 .27 88838 .44 17 44 77677 28 87290 .44 16 44 78674 .27 o7 88864 .44 16 45 77694 28 87317 .44 15 45 78691 .Zl 88890 .43 15 46 47 77711 77728 28 .28 87343 87369 .44 .44 14 13 46 47 78707 78723 .27 .27 88916 88942 .43 .43 14 13 48 77744 28 87396 .44 12 48 78739 .27 o- 88968 .4:; 12 49 77761 28 87422 .44 11 49 78756 .Zt 0*7 88994 .4., 11 50 77778 .28 87448 .44 10 50 78772 .Zt 89020 .4:; 10 51 9.77795 .28 9.87475 .44 9 51 9.78788 .27 9.89046 .43 9 52 77812 .28 87501 .41 8 52 78805 .27 89073 .43 8 53 77829 .28 87527 .44 7 53 78821 .27 89099 .43 7 54 77846 .28 87554 .44 6 54 78837 .27 89125 .43 6 55 77862 28 87580 .44 5 55 78853 .27 89151 .43 5 56 57 77879 77896 .28 .28 87606 87633 .44 .44 4 3 56 57 78869 78886 .27 .27 89177 89203 .43 .43 4 3 58 77913 .28 87659 . 1 1 2 58 78902 .27 89229 .43 2 59 77930 .28 87685 .44 1 59 78918 .27 89255 .43 1 60 77946 .28 87711 .44 60 78934 .27 89281 .43 M. Cosine. Cotnng. Dl" 31. 31. Cosine. ~D1" Ootang. Dl" M. 53= 88 SINES AND TANGENTS. 30 At. Sine. Di" Taug. Di" M. M. ; Mac. Dl" Tafag. D," M. 1 9.78934 78950 0.27 9.89281 89307 0.43 6!) 59 1 9*9887 79903 0.26 9.90837 90S63 0.43 60 59 2 78967 .27 89333 .4:1 58 2 79918 J26 908S9 .43 58 3 78983 .27 89359 .43 57 3 79934 .26 90914 .43 57 4 78999 .27 89385 .43 56 4 79950 .26 90940 .43 56 5 79015 ''I 89411 .43 55 5 79965 .26 90966 .43 55 6 79031 .ZY 89437 .43 4>> 54 6 79981 .26 cw 90992 .43 54 7 79047 .27 89463 .> 53 7 79996 .ZO 91018 ! 53 8 79063 .27 89489 .43 52 8 80012 .26 91043 .43 52 9 10 79079 79095 .27 .27 89515 89541 .43 .43 51 50 9 10 80027 80043 .*26 91069 91095 A3 51 50 11 12 9.79111 79128 .27 .27 07 9.89567 89593 .43 .43 49 48 11 12 9.80058 80074 .26 .26 9.91121 91147 .43 .43 .43 49 48 13 14 79144 79160 .1 i .27 89619 89645 .43 47 46 13 14 80089 80105 .26 91172 91198 '.43 47 46 15 79176 2 i 89671 4.-> 45 15 80120 .zo 91224 40 45 16 79192 .z7 89697 44 16 80136 nn 91250 *~rG 44 17 18 19 20 79208 79224 79240 79256 .27 .27 .27 .27 89723 89749 89775 89801 .43 .43 .43 .43 43 42 41 40 17 18 19 20 80151 80166 80182 80197 ZU .26 .26 .26 91276 91301 91327 913D3 143 .43 .43 43 42 41 40 21 22 9.79272 7928S .27 .27 9.S9S-J7 89853 .43 .43 39 38 21 22 9.80213 80228 .26 .26 .26 9.91379 91404 !43 .43 39 38 23 79304 2/ 89879 .4.5 37 23 80244 91430 A O 37 24 25 79319 79335 .27 .27 89905 89931 .43 .43 36 35 24 25 80259 80274 .26 91456 91482 .4o .43 36 35 26 79351 .27 89957 .43 34 26 80290 .26 91507 .43 34 27 79367 .27 89983 .43 33 27 80305 .26 91533 .43 A 9 33 28 79383 .26 90009 .43 32 28 80320 .26 91559 .4d 32 29 79399 .26 90035 .43 31 29 80336 .26 91585 .43 A O 31 30 31 32 79415 9.79431 79447 .26 .26 .26 90061 9.90086 90112 .43 ..43 .43 30 29 28 30 31 32 80351 9.80366 80382 .26 .26 .26 91610 9.91636 91662 .43 .43 .43 A O 30 29 28 33 34 35 36 37 38 39 40 41 42 79463 79478 79494 79510 79526 79542 79558 79573 9.79589 79605 .26 .26 .26 .26 .26 .26 .26 .26 .26 .26 90138 90164 90190 90216 90242 90268 90294 90320 9.90346 90371 .43 .43 .43 .43 .43 .43 .43 .43 .43 .43 27 26 25 24 23 22 21 20 19 18 33 34' 35 36 37 38 39 40 41 42 80397 80412 80428 80443 80458 80473 80489 80504 9.80519 80534 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 91683 91713 91739 91765 91791 91816 91842 91868 9.91893 91919 A6 .43 .43 .43 .43 .43 .43 .43 .43 .43 27 26 25 24 23 22 21 20 19 18 43 44 45 79621 79636 79652 .26 .26 .26 90397 90423 90449 .43 .43 .43 17 16 15 43 44 45 80550 80565 80580 25 .25 .25 (1C 91945 91971 91996 .43 .43 17 16 15 46 47 48 79668 79684 79699 26 .26 .26 90475 90501 90527 .43 .43 .43 14 13 12 46 47 48 80595 80610 80625 /O .25 .25 92022 92048 92073 !43 .43 14 13 12 49 50 51 52 53 54 79715 79731 9.79746 79762 79778 79793 26 .26 .26 .26 .26 .26 90553 90578 9.90604 90630 90656 90682 '.43 .43 .43 .43 .43 A O 11 10 9 8 7 6 49 50 51 52 53 54 80641 80656 9.80671 80686 80701 80716 .25 .25 .25 .25 .25 92099 92125 9.92150 92176 92202 92227 !43 .43 .43 .43 .43 11 10 9 8 7 6 55 56 79809 79825 !26 90708 90734 .TEG .43 5 4 55 56 80731 80746 .25 92253 92279 !43 5 4 57 79840 o/> 90759 A 9 3 57 80762 OR 92304 A Q 3 58 59 79856 79872 .zo .26 90785 90811 Ao .43 40 2 1 58 59 80777 80792 00 .25 92330 92356 4o .43 2 1 60 79887 ' 90837 G 60 80807 92381 ' M. Cosine. Dl" Cotang. Dl" M. M. Cosine. DI" Cotanj*. Dl" M. 47 5O 40 TABLE IV. LOGARITHMIC M. Situ,. Dl" Tang. Dl" M. M. Sine. Dl" Tang. Dl" M. 1 9.80807 80822 0.25 9.92381 92407 O.f3 60 59 1 9.8 1694 81709 G.24 9.93916 93942 0.4.3 60 59 2 80837 .25 92433 .43 58 2 81723 .24 93967 .43 58 3 80852 .25 92458 .43 57 3 81738 .24 93993 .43 57 4 80867 .zo 92484 .43 56 4 81752 .24 94018 .43 56 5 80882 .25 92510 .43 55 5 81767 .24 94044 .43 55 6 80897 .25 92535 .43 54 6 81781 .24 94069 .43 54 7 80912 .25 92561 .43 53 7 81796 .24 94095 .43 53 8 80927 .25 92587 .43 52 8 81810 .24 94120 .42 52 9 10 80942 80957 .25 .25 92612 92638 .43 .43 51 50 9 10 81825 81839 .24 .24 94146 94171 J2 51 U 50 11 9.80972 .25 9.92663 .43 49 11 9.81854 .24 9.94197 .42 49 12 80987 .25 92689 .43 48 12 81868 .24 94222 .42 48 13 81002 .25 92715 .43 47 13 81882 .24 94248 .42 47 14 81017 .25 92740 .43 46 14 81897 .24 94273 .42 46 15 81032 .25 92766 .43 45 15 81911 .24 94299 .42 45 16 81047 .25 92792 .43 44 16 81926 .24 94324 .42 44 17 81061 .25 OK 92817 .43 1 O 43 17 81940 .24 94350 .42 43 18 81076 ftJEO 92843 .43 42 18 81955 .Z4 94375 .42 42 19 81091 .25 92868 .43 41 19 81969 .24 94401 .42 41 20 81106 .25 92894 .43 40 20 81983 .24 94426 .42 40 21 9.81121 .25 9.92920 .43 39 21 9.81998 .24 9.94452 .42 39 22 23 81136 81151 .25 .25 92945 92971 .43 .43 38 37 22 23 82012 82026 Hi 94477 fj. 94503 .42 .42 38 37 24 25 81166 81180 .25 .25 92996 93022 .43 .43 36 35 24 25 82041 82055 .24 .24 94528 94554 .42 ' 1? S 26 81195 .25 93048 .43 34 26 82069 .24 94579 '42 34 27 81210 .25 93073 .43 33 27 82084 .24 94604 An 33 28 81225 .25 93099 .43 32 28 82098 .24 94630 .42 32 29 81240 .25 93124 .43 31 29 82112 .24 94655 .42 31 30 31 32 81254 9.81269 81284 .25 .25 .25 93150 9.93175 93201 .43 .43 .43 30 29 28 30 31 32 82126 9.82141 82155 .24 .24 .24 94681 9.94706 94732 .42 .42 .42 30 29 28 33 81299 .25 93227 .43 27 33 82169 .24 94757 .42 27 34 81314 .25 93252 .43 26 34 82184 .24 94783 .42 26 35 81328 .25 93278 .43 25 35 82198 .24 94808 .42 25 36 81343 .25 93303 .43 24 36 82212 .24 94834 .42 24 37 81358 .25 93329 .43 23 37 82226 .24 94859 .42 23 38 81372 .25 93354 .43 22 38 82240 .Z4 94884 .42 22 39 81387 .25 93380 .43 21 39 82255 .24 94910 .42 21 40 81402 .25 93406 .43 20 40 82269 .24 94935 .42 20 41 9.81417 .25 9.93431 .43 19 41 9.82283 .24 9.94961 .42 19 42 81431 .24 93457 .43 18 42 82297 .24 94986 .42 18 43 81446 .24 93482 .43 17 43 82311 .24 95012 .42 17 44 81461 .24 93508 .43 16 44 82326 .24 95037 .42 16 45 81475 .24 93533 .43 15 45 82340 .z4 95062 .42 15 46 81490 .24 93559 .43 14 46 82354 .24 95088 .42 14 47 81505 .24 93584 .43 13 47 82368 .24 95113 .42 13 48 81519 .24 93610 .43 12 48 82382 .24 95139 .42 12 49 81534 .24 93636 .43 11 49 82396 .24 95164 .42 11 50 81549 .24 93661 .43 10 50 82410 .24 95190 .42 10 51 9.81563 .24 9.93687 .43 9 51 9.82424 .24 9.95215 .42 9 52 81578 .24 93712 .43 8 52 82439 .23 95240 .42 8 53 81592 .24 93738 .43 7 53 82453 .23 95266 .42 7 54 81607 .24 93763 .43 6 54 82467 .23 95291 .42 6 55 81622 .24 93789 .43 5 55 82481 .23 95317 .42 5 56 81636 .24 93814 .43 4 56 82495 .23 95342 .42 4 57 58 81651 81665 .24 .24 93840 93865 .43 .43 3 2 57 58 82509 82523 .23 .23 95368 ' 95393' 3 2 59 81680 .24 93891 .43 1 59 82537 .23 95418 .1Z 1 60 81694 .24 93916 .43 60 82551 .23 95444 .42 M. Cosine. D!" Ootari!?. Dl" M. M. Cosine. Dl" Cotang. Dl" M. 49 48 48* SINES AND TANGENTS. 43 M. Sine. Dl" Tang. : Dl" M. M. Sine. Dl" Tang. Dl" M. 1 2 9.82551 82565 82579 0.23 .23 9.95444 65469 95495 0.42 60 59 58 2 9.83378 83392 83405 0.23 .23 9.96966 96991 97016 0.42 .42 60 59 58 3 82593 .23 or) 95520 A O 57 3 83419 .23 oo 97042 .42 ( n 57 4 82607 .ZO 95545 .4z 56 4 83432 .zo 97067 .4z 56 5 82621 .23 oo 95571 .42 55 5 83446 .23 oo 97092 .42 A O 55 6 7 82635 82649 Zo .23 95596 95622 .42 54 53 6 7 83459 83473 ZO .22 97118 97143 AZ .42 54 53 8 82663 .23 oo 95647 .42 A O 52 8 83486 .22 oo 97168 .42 A O 52 9 82677 .zo 95672 .4z 51 9 83500 .zz 97193 .4z 51 10 82691 .23 95698 .42 A O 50 10 83513 .22 OO 97219 .42 A O 50 11 12 9.82705 82719 .23 9.95723 95748 AZ .42 49 48 11 12 9.83527 83540 .ZZ .22 9.97244 97269 AZ .42 49 48 13 14 82733 82747 .23 .23 95774 95799 .42 .42 47 46 13 14 83554 83567 .22 .22 97295 97320 .42 .42 47 46 15 16 82761 82775 .23 .23 95825 95850 .42 .42 45 44 15 16 83581 83594 .22 .22 97345 97371 .42 .42 45 44 17 82788 .23 95875 .42 43 17 83608 .22 97396 .42 43 18 82802 .23 95901 .42 42 18 83621 .22 97421 .42 42 19 82816 .23 95926 .42 41 19 83634 .22 97447 .42 41 20 82830 .23 95952 .42 A O 40 20 83648 .22 97472 .42 A O 40 21 9.82844 .23 9.95977 4z 39 21 9.83661 .zz 9.97497 .4z 39 22 82858 .23 96002 .42 38 22 83674 .22 97523 .42 38 23 82872 .23 960281 A * 37 23 83688 .22 97548 .42 37 24 82885 .23 96053 1 ;; 36 24 83701 .22 97573 .42 36 25 26 82899 82913 .23 .23 96078 96104 35 34 25 26 83715 83728 .22 .22 97598 97624 .42 .42 35 34 27 28 82927 82941 .23 .23 96129 96155 .42 .42 33 32 27 28 83741 83755 .22 .22 97649 97674 .42 .42 33 32 29 82955 .23 96180 42 31 29 83768 .22 97700 .42 31 30 82968 .23 96205 .42 30 30 83781 .22 97725 .42 30 31 9.82982 .23 9.96231 42 29 31 9.83795 .22 9.97750 .42 29 32 82996 .23 oo 96256 42 28 32 83808 .22 oo 97776 .42 A O 28 33 83010 zo 96281 .42 27 33 83821 .zz 97801 .4z 27 34 83023 .23 96307 .42 26 34 83834 .22 97826 .42 26 35 83037 .23 O'-J 96332 .42 25 35 83848 .22 oo 97851 .42 A O 25 36 83051 zo 96357 .42 24 36 83861 zz 97877 .4z 24 37 83065 23 96383 .42 23 37 83874 .22 97902 .42 23 38 83078 23 96408 .42 22 38 83887 22 97927 .42 22 39 83092 23 96433 .42 21 39 83901 22 97953 .42 21 40 83106 .23 96459 .42 20 40 83914 .22 97978 .42 20 41 9.83120 23 9.96484 .42 19 41 9.83927 .22 9.98003 .42 19 42 43 83133 83147 23 .23 96510 96535 .42 .42 18 17 42 43 83940 83954 .22 .22 98029 98054 .42 .42 18 17 44 83161 23 96560 .42 16 44 83967 22 98079 42 16 45 83174 23 96586 .42 15 45 83980 22 98104 .42 15 46 83188 23 96611 .42 14 46 83993 22 98130 42 14 47 83202 23 96636 .42 13 47 84006 .22 98155 .42 13 48 83215 23 oo 96662 .42 12 48 84020 22 00 98180 .42 49 12 49 83229 zo 96687 4z 11 49 84033 ZZ 98206 4Z 11 50 83242 23 96712 .42 10 50 84046 22 98231 .42 10 51 9.83256 23 9.96738 .42 9 51 9.84059 22 9.98256 42 9 52 83270 23 96763 .42 8 52 84072 22 98281 .42 8 53 8328:'. 23 oo 96788 .42 7 53 84085 22 98307 .42 7 54 83297 Zo 96814 42 6 54 84098 .22 98332 .42 6 55 83310 23 96839 .42 5 55 84112 22 98357 42 5 56 83324 .23 96864 .42 4 56 84125 .22 98383 .42 4 57 83338 .23 96890 .42 3 57 84138 .22 98408 .42 3 58 83351 23 96915 .42 2 58 84151 22 98433 .42 2 59 83365 .23 96940 .42 1 59 84164 .22 98458 .42 1 60 83378 .23 96966 .42 60 84177 .22 98484 .42 M. Cosine. Dl" Cotang. "DP M. M. C/osine. Dl" OotRllg. Dl" M. 47 C 49 46 44 TABLE IV. LOGARITHMIC 45 M. Sine. Dl" Tane. Dl" M. M. Hue. Dl" Tan-. Dl" M. 1 9.84177 84190 0.22 9.98484 9509 0.42 60 59 1 9.84949 84961 0.21 10.00000 00025 0.42 60 59 2 84203 .22 98534 .42 58 2 84974 .21 00051 .42 58 3 84216 .22 98560 .42 57 3 84986 .21 06078 .42 57 4 84229 .22 98585 .42 56 4 84999 .21 00101 .42 56 5 84242 .22 98610 .42 55 5 85012 .21 00126 .42 55 6 84255 .22 98635 .42 54 6 85024 .21 00152 .42 54 7 84269 .22 98661 .42 53 7 85037 .21 00177 .42 53 8 84282 .22 98686 .42 52 8 85049 .21 00202 .42 52 9 84295 .22 98711 .42 51 9 85062 .21 00227 .42 51 10 84308 .22 98737 .42 50 10 85074 .21 00253 .42 50 11 9.84321 .22 .1.) 9.98762 .42 o 49 11 9.85087 .21 01 10.00278 .42 49 12 84334 .zz 98787 4z 48 12 85100 .zl 00303 .42 48 13 14 15 84347 84360 84373 .22 .22 .22 98812 98838 98863 .42 .42 .42 47 46 45 13 14 15 85112 85125 85137 .21 .21 .21 00328 ** 00354 ** 00379 ** 47 46 45 16 84385 .22 98888 .42 44 16 85150 00404 '-J; 44 17 84398 .22 93913 .42 43 17 85162 00430 ** 43 18 84411 .22 98939 .42 42 18 85175 ;} 00455 '**! 42 19 84424 .22 98964 .42 41 19 85187 00480 .42 41 20 S4437 .zz 98989 .42 40 20 85200 '*\ \ 00505 .42 40 21 9.84450 .22 99 9.99015 .42 39 21 9.852121 '^| 10. 00531 .42 39 22 84463 .ZZ 99040 Ai 38 22 85225 * 1 00556 .4z 38 23 84476 .22 99065 .42 37 23 85237 } 00581 .42 37 24 84489 .21 01 99090 .42 A O 36 24 85250 'i\ \ 00606 .42 36 25 84502 .zl 99116 .4z 35 25 85262 00632 .42 35 26 84515 .21 99141 .42 34 26 85274 "i\\ 00657 '^ 34 27 28 29 30 84528 84540 84553 84566 .21 .21 .21 .21 99166 99191 99217 99242 .42 .42 .42 .42 33 32 31 30 27 28 29 30 85287 85299 85312 85324 5 006S2 '- 1 1 707 "42 1 7: " *42 J'l 007581 -JJ 33 32 31 30 31 9.84579 .21 9.99267 .42 29 31 9.85337 i\: 10.00783 '?f 29 32 84592 .21 O 1 99293 .42 1 O 28 32 85349 ! 00809J ** 28 33 84605 .2 1 01 99318 .4z 27 33 8536 IS i{ 00834 27 34 84618 .zl 99343 .42 26 34 85374 *! 00859 '^! 26 35 84630 .21 99368 .42 25 35 85386 *\ 00884 ** 25 36 84643 .21 99394 .42 24 36 85399 ; 00910 24 37 38 39 84656 84669 84682 .21 .21 .21 99419 99444 99469 .42 .42 .42 23 22 21 37 38 39 854 1 1 85423 85436 .21 .21 1! 00935 *JJ 00960 00985 J 23 22 21 40 84694 .21 99195 .42 20 40 85448 .zl 01011 I *: 20 41 9.84707 .21 9.99520 .42 19 41 9.85460 r,! 10.01036! - q .l 19 42 84720 .21 99545 .42 18 42 85473 I 1061 1? 18 43 84733 .21 99570 .42 17 43 85485 \\ olosr ?; 17 44 84745 .21 99596 .42 16 44 85497 .21 Ct 1 01112 ** 16 45 84758 .21 99621 .42 15 45 85510 't\ 01137 42 ,, 4 ' * " 46 84771 .21 99646 .42 14 46 85522 20 1162 .42 14 47 84784 .21 99672 .42 13 47 85534 OA 01188 .42 13 48 84796 .21 99697 .42 12 48 85547 .20 01213 .42 12 49 84809 .21 99722 .42 11 49 85559 1o 01238 .42 11 50 84822 .21 99747 .42 10 50 85571 'on 01263 ' '" 10 51 9.84835 .21 9.99773 .42 9 51 9.85583 ! 1 10. 01 289 -JJ 9 52 84847 .21 99798 .42 8 52 85596 01314 A1 8 53 84860 .21 99823 .42 7 53 85608 JJ 111 33ii -I 1 ; 7 54 84873 .21 99848 .42 6 54 85620 |J 01365' r, 6 55 84885 .21 99874 .42 5 55 85632 .Z() 01390) "} 5 56 84898 .21 99899 .42 4 56 85645 .20 01415! 'f 4 57 84911 .21 99924 .42 3 57 85657 .20 01440 3 84923 .21 99949 .42 2 58 85669 .20 01466 'J* 2 59 84936 .21 999^(5 .42 59 85681 .20 01 -I'M 'JJ 1 60 84949 .21 10.00000 .42 60 85693 .20 01 5 Mi: ' M. Cosine. Dl" Cntnnsc. Dl" M. M. Cosino. Dl" ' ('otaim. I'i" M. 44 C SINES AND TANGENTS. 47 31. Sine. Dl" Tang. Dl" M. M. Sine. Dl" Tang. Dl" M 1 9.85693 85708 0.20 10.01516 01542 0.42 ,1 O 60 59 i 9.86413 86425 0.20 on 10.03034 03060 0.42 A O 60 59 2 85718 .20 01567 AL 58 2 86436 .20 03085 AL 58 3 4 85730 85742 .20 .20 01592 01617 .42 .42 57 56 3 4 86448! 'f 864601 ^JJ 03110 03U6 .42 .42 57 56 5 85754 .20 90 01643 .42 4.9 55 5 86472 '2 03161 .42 4.9 55 6 85766 ZU 01668 AL 54 6 86483 03186 AL 54 7 85779 .20 90 01693 .42 53 7 86495 .20 on 03212 .42 4.9 53 8 85791 .zu 01719 AL 52 8 86507 '*" 03237 AL 52 9 85803 .20 01744 .42 51 9 86518 -;J 03262 .42 51 10 85815 .20 01769 .42 50 10 86530 !! 03288 .42 50 11 9.85827 .20 .20 fO.01794 .42 49 49 11 9.86542 .ZU on 10.03313 .42 4.9 49 12 85839 01820 AL 48 12 86554 '* v n - 03338 AL 48 13 85851 .20 90 01845 J.42 A O 47 13 86565 03364 .42 47 14 85864 01870 AL 46 14 86577 19 'in 03389 .42 46 15 85876 .20 01896 .42 A O 45 15 86589 .19 03414 .42 45 16 85888 .20 01921 AL 44 16 86600 .19 03440 .4- 44 17 85900 .20 01946 .42 43 17 8661.2! '}j! 03465 .42 43 18 85912 .20 01971 .42 42 18 86624! : 03490 .42 42 19 85924 .20 01997 .42 41 19 86635 .jy 03516 .42 41 20 21 22 85936 9.85948 85960 .20 .20 .20 on 02022 10.02047 02073 .42 .42 .42 40 39 38 20 21 22 86647 9.866.59 86670 .19 .19 i n 03541 10.03567 03592 .42 .42 40 39 38 23 24 85972 85984 .2" .20 02098 '** 02123 .* 37 36 23 24 86682 86694 .19 .19 03617 03643 .42 .42 37 36 25 26 85996 86008 .20 02149 02174 A-L .42 35 34 25 26 86705 86717 .19 .19 03668 03693 .42 35 34 27 86020 .20 02199 .42 33 27 86728 .19 03719 .42 33 28 86032 .20 02224 .42 32 28 86740 .19 03744 .42 32 29 86044 .20 02250 .42 31 29 86752 '}J 03769 .42 31 30 86056 .20 90 02275 A O 30 30 86763 .19 1 03795 .42 A 30 31 9.86068 10.02300 AL 29 31 9.86775 :r. 10.03820 AL 29 32 33 86080 86092 .20 .20 02326 02351 .42 .42 28 27 32 33 86786 86798 03845 03871 .42 .42 28 27 34 86104 .20 02376 .42 26 34 86809 .19 03896 .42 26 35 S61 1(1 .20 90 02402 .42 A O 25 35 86821 .19 -i o 03922 .42 A O 25 36 ' 86128 02427 AL 24 36 86832 .1 .' 03947 AL 24 87 86140 .20 02452 .42 23 37 86844 .19 03972 .42 23 38 39 86152 86164 .20 .20 on 02477 02503 .42 .42 22 21 38 39 86855 86867 .19 .19 03998 04023 .42 .42 22 21 40 86176 .20 02528 .42 20 40 86879 04048 .42 20 41 9.86188 .20 10.02553 .42 19 41 9.86890 'Jo! 10.04074 .42 19 42 86200 .20 on 02579 .42 18 42 86902 .19 04099 .42 18 43 86211 .zU on 02604 .42 17 43 86913 .19 04125 .42 17 44 86223 .20 ' 02629 .42 16 44 86924 .19 04150 .42 16 45 86235 .20 02655 .42 15 45 86936 .19 04175 42 15 46 86247 .20 02680 .42 14 46 86947 .19 04201 .42 14 47 48 86259 86271 .20 .20 OA 02705 02731 .42 .42 13 12 47 48 86959 86970 .19 .19 04226 04252 .42 .42 13 12 49 50 86283 86295 .ZO .20 02756 02781 .42 .42 11 10 49 50 86982 86993 .19 .19 04^7 04302 .42 .42 11 10 51 9. 863 06 .20 10.02807 .42 9 51 9.87005 .19 10.04328 .42 9 52 86318 OA 02832 .42 8 52 87016 .19 04353 .42 8 53 86330 .20 02857 .42 7 53 87028 .19 04378 42 7 54 86342 .20 02882 .42 6 54 87039 .19 04404 .42 . 6 55 86354 .20 02908 .42 5 55 87050 .19 04429 42 5 56 86366 .20 02933 .42 4 56 87062 .19 04455 .42 4 57 86377 .20 02958 .42 3 57 87073 .19 04480 .42 3 58 86389 .20 02984 .42 2 58 87085 .19 04505 .42 2 59 86401 .20 03009 .42 1 59 87096 .19 04531 .42 1 60 86413 .20 03034 .42 60 87107 .19 04556 .42 M. Cosine. Dl" Cotang. Dl" 31. M. Cosi7ie. Dl" Cotang. Dl" M. 43 42 TABLE IV. LOGARITHMIC M. Sino. l)i" Tan?. PI" M. M. Sine. 1)1" Tang. 1)1" It. 1 9.87107 87119 0.19 10.0455(3 04582 0.42 60 59 1 9.87778 87789 0.18 10.06084 06109 0.43 60 59 2 87130 .19 04607 '42 58 2 87800 .18 06135 .43 58 3 4 87141 87153 .19 .19 04632 04658 ^42 57 56 3 4 87811 87822 .18 .18 1 Q 06160 06186 .43 .43 A o 57 56 5 87164 .19 04683 .42 55 5 87833 . 1 06211 .4o 55 6 87175 .19 04709 .42 54 6 87844 .18 1 o 06237 .43 40 54 7 8 87187 87198 !l9 04734 04760 !42 53 52 7 8 87855 87866 lo .18 06262 06288 .40 .43 53 52 9 87209 19 04785 *42 51 9 87877 .18 I O 06313 .43 AQ 51 10 11 87221 9.87232 J9 1Q 04810 10.04836 !42 50 49 10 11 87887 9.87898 . 1 o .18 1 o 06339 10.06364 TO .43 A<\ 50 49 12 13 14 87243 87255 87266 . 1 7 .19 .19 .19 04861 04887 04912 ^42 .42 48 47 46 12 13 14 87909 87920 87931 . lo .18 .18 I Q 06390 06416 06441 4o .43 .43 48 47 46 15 16 17 18 19 87277 87288 87300 87311 87322 !l9 .19 .19 .19 1 Q 04938 04963 04988 05014 05039 .42 .42 .42 .42 45 44 43 42 41 15 16 17 18 19 87942 87953 87964 87975 87985 .Jo .18 .18 .18 .18 1 8 06467 06492 06518 06543 06569 !43 .43 .43 .43 45 44 43 42 41 20 21 87334 9.87345 .1*7 .19 05065 10.05090 '.42 40 39 20 21 87996 9.88007 .1 o .18 06594 10.06620 '.43 40 39 22 87356 .19 .19 05116 .42 38 22 88018 .18 1 u 06646 .43 38 23 87367 05141 40 37 23 88029 1O 1 ft 06671 A'\ 37 24 87378 1Q 05166 - 36 24 88040 . 1 O 1 o . 06697 .4o 36 25 87390 .19 05192 49 35 25 88051 1 O I Q 06722 A*t 35 26 27 87401 87412 !l9 1 Q 05217 05243 !42 34 33 26 27 88061 88072 1O .18 I Q 06748 06773 .4*-* .43 4, 34 33 28 87423 i y .19 05268 *49 32 28 88083 . 10 1 o 06799 A'J 32 29 87434 05294 ** 31 29 88094 1 o 06825 .4o 31 30 87446 .19 05319 .42 30 30 88105 18 1 Q 06850 .43 30 31 32 33 9.87457 87468 87479 .19 .19 .19 1 Q 10.05345 05370 05396 .42 .42 .42 JO 29 28 27 31 32 33 9.88115 88126 88137 .lo .18 .18 18 10.06876 06901 06927 .43 .43 .43 29 28 27 34 35 36 87490 87501 87513 i y .19 .19 1 Q 05421 05446 05472 42 .42 .42 49 26 25 24 34 35 36 88148 88158 88169 lo .18 .18 .18 06952 06978 07004 !43 .43 26 25 24 ' 37 38 87524 87535 i y .19 1 Q 05497 05523 ~LZ .42 23 22 37 38 88180 88191 .18 1 S 07029 07055 !43 23 22 39 40 41 87546 87557 9.87568 - iy .19 .19 i q 05548 05574 10.05599 .42 .42 21 20 19 39 40 41 88201 88212 9.88223 1 o .18 18 18 07080 07106 10.07132 !43 .43 1H 21 20 19 42 87579 iy 05625 Ati 18 42 88234 .10 07157 .iO 18 43 87590 .18 05650 .42 17 43 88244 .18 1 Q 07183 .43 17 44 87601 .18 1 Q 05676 '.42 A O 16 44 88255 .lo I 07208 .43 A *J 16 45 46 47 87613 87624 87635 . lo .18 .18 1 Q 05701 05727 05752 42 .42 .42 4 O 15 14 13 45 46 47 88266 88276 88287 .10 .18 .18 1 o 07234 07260 07285 .4o .43 .43 A O. 15 14 13 48 87646 .lo 05778 Ai 12 48 88298 .lo -< rt 07311 A6 12 49 876^- .18 05803 .42 11 49 883081 '! 07337 .43 11 50 S7668 .18 1 8 05829 .42 10 50 88319! 'JJ 07362 .43 40 10 51 9.87679 . 10 1 Q 10.05854 49 9 51 9.88330 1 o 10.07388 43 9 52 87690 lo 1 Q 05880 . 8 52 88340 *1O -I Q 07413 A O 8 53 87701 .13 1 ft 05905 7 53 88351 lo 18. 07439 A6 7 54 87712 lo 05931 .4o 6 54 88362 10 07465 .4o 6 55 87723 .18 i ft 05956 .43 JO 5 55 88372 .18 1Q 07490 .43 5 56 87734 .10 05982 .4o 4 56 88383 '!" 07516 ** 4 57 87745 .18 06007 .43 3 57 88394 *!Q 07542 .43 3 58 87756 .18 06033 .43 2 58 88404 'I* 07567 .43 2 59 87767 .18 06058 .43 1 59 88415 '\l 07593 .43 1 60 87778 .18 OfiO4 .43 60 88425 i ' 07619 .43 M. Cosine. Dl" CotniiS. PI" M. M. Cosine. PI" <"otanc. PI" M. 41 C 52 50 SINES AND TANGENTS. M bine. Dl" TaiiK. Dl" | M. M. Sine. Dl" Tang. D!" M. 1 9.88425 88436 0.18 10.07619 07644 0.43 60 59 1 9.89050 89060 0.17 10.09163 09189 0.43 60 59 2 88447 .18 07670 .43 58 2 89071 .17 09215 .48 58 3 88457 .18 07696 .43 57 3 89081 .17 09241 .43 57 4 88468 .18 07721 .43 56 4 89091 .17 09266 .43 56 5 88478 .18 07747 .43 55 5 89101 .17 09292 .43 55 6 88489 .18 07773 .43 54 6 89112 .17 09318 .43 54 7 88499 .18 07798 .43 53 7 89122 .17 09344 .43 53 8 88510 .18 07824 .43 52 8 89132 .17 09370 .43 52 9 88521 .18 07850 .43 51 9 89142 .17 09396 .43 51 10 88531 .18 07875 .43 50 10 89152 .17 09422 .43 50 11 9.88542 .18 10.07901 .43 49 11 9.89162 .17 10.09447 .43 49 12 88552 .18 07927 .43 48 12 89173 .17 09473 .43 48 13 88563 .18 07952 .43 47 13 89183 .17 09499 .43 47 14 88573 .18 07978 'X 46 14 89193 .17 09525 .43 46 15 88584 .18 1 7 08004 .43 45 15 89203 .17 09551 .43 A O 45 16 88594 . 1 / 08029 .43 44 16 89213 .17 09577 .4o 44 17 88605 .17 08055 .43 43 17 89223 .17 09603 .43 43 18 88615 .17 08081 .43 42 18 89233 .17 09629 .43 42 19 88626 .17 08107 s 41 19 89244 .17 09654 .43 41 20 88636 .17 08132 .43 40 20 89254 .17 09680 .43 40 21 9.88647 .17 1 7 10.08158 .43 A > 39 21 9.89264 }l 110.09706 .43 A O 39 22 88657 .1 1 08184 .43 38 22 89274 . 1 1 09732 .4o 38 23 8866* .17 08209 .43 37 23 89284 .17 09758 .43 37 24 88678 .17 08235 .43 36 24 89294 .17 09784 .43 36 25 88688 .17 08261 .43 35 25 89304 17 09810 .43 35 26 27 88699 88709 .17 .17 08287 08312 .43 .43 34 33 26 27 89314 89324 '}l\ 09836 ';: 09862 .43 .43 34 33 28 88720 .17 08338 .43 32 28 89334 .1 i 09888 .43 32 29 88730 .17 08364 .43 31 29 89344 .17 09914 .43 31 30 88741 .17 08390 .43 30 30 89354 .17 09939 .43 30 31 9.88751 .17 10.08415 .43 29 31 9.89364 17 10.09965 .43 29 32 88761 08441 .43 28 32 89375 .17 09991 .43 28 33 88772 .17 08467 .43 27 33 89385 .17 10017 .43 27 34 88782 08493 .43 26 34 89395 17 10043 .43 26 35 88793 .17 08518 .4.3 25 35 89405 17 10069 .43 25 36 88803 .17 1 7 08544 .43 24 36 89415 .17 10095 .43 24 37 88813 .1 / 08570 .43 23 37 89425 .17 10121 .43 23 38 88824 08596 .43 22 38 89435 17 10147 .43 22 39 88834 .17 1 7 08621 .43 21 39 89445 17 10173 .43 21 40 88844 .17 Ni 08647 .43 20 40 89455 17 10199 .43 20 41 9.88855 10.08673 s 19 41 9.89465 17 10.10225 .43 19 42 88865 17 -1 7 08699! '*!! 18 42 89475 17 10251 .43 18 43 88875 .] / 1 7 08724 .6 17 43 89485 17 i - 10277 .43 17 44 88886 .1 I 08750 .43 16 44 89495 17 10303 .43 16 45 88896 .17 1 7 08776 .43 15 45 89504 17 10329 .43 15 46 88906 . I I 08802 .43 14 46 89514 .17 10355 .43 14 47 88917 .17 1 7 08828 .43 13 47 89524 .17 10381 .43 13 48 49 88927 88937 .!< .17 08853 08879 .43 .43 12 11 48 49 89534 89544 17 .17 10407 10433 .43 .43 12 11 50 88948 .17 08905 .43 10 50 89554 17 10459 .43 10 51 9.88958 17 10.08931 .43 y|O 9 51 9.89564 17 1 7 10.10485 .43 9 52 88968 i t 08957 .43 8 52 89574 LI 10511 .43 8 53 88978 .17 1^ 08982 .43 7 53 89584 .17 10537 .43 7 54 88989 i 09008 .43 6 54 89594 17 10563 .43 6 55 88999 .17 09034 .43 5 55 89604 .17 10589 .43 5 56 89009 .17 09060 .43 4 56 89614 .16 10615 .43 4 57 89020 09086 .43 3 57 89624 .16 10641 .43 3 58 89030 .17 09111 .43 2 58 89633 .16 10667 .43 2 59 89040 .17 1 7 09137 .43 A 1 59 89643 .16 10693 .43 1 60 89050 .17 09163 /*' 60 89653 .16 10719 .43 M. Cosine. Dl" GotaiiKi i Dl" M. M. Cosine. Dl" Cot;uiK. Dl" M. 39 S. N. 39. 53 38 52 TABLE IV. LOGARITHMIC M. Miie. pr 'fa ti jr. !,' M. M. Muf. 1>." 'lituir. In" M. 9.89653 10.10719 60 \ 9.90235 L 10.122hH| ,,60 1 89663 0.16 10745 0.43 59 1 90244 , }j| 123151: 59 2 89673 .16 10771 .43 58 2 90254 'JJ 12341 44 58 3 69683 .16 10797 .43 57 3 90263 '? 12367 57 4 89693 .16 1 0823 .43 56 4 902731 4* 12394 .44 56 5 89702 .16 10849 .43 55 5 90282 .10 12420 .44 55 6 7 89712 89722 .16 .16 10875 10901 .43 .43 54 53 6 7 90292 90301 .16 .16 12446 12473 .44 .44 54 53 8 89732 .16 10927 .43 52 8 90311 '? 12499 .44 52 9 89742 .16 10954 .43 51 9 90320 '}J 12525 .44 51 10 89752 .16 10980 .43 50 10 90330! '*2 12552 .44 50 11 9.89761 .16 10.11006 .43 49 11 9.90339 .10 10.12578 .44 49 12 89771 .16 11032 .43 48 12 90349 .16 12604 .44 48 13 89781 .16 11058 .43 47 13 90356 .16 12631 .44 47 14 89791J 'IJ 11084 .43 A *J 46 14 90368 .16 1 C 12657 .44 4.1 46 15 89801 JU 11110 .4o 45 15 90377 JO . rt 1 2683 .4-4 45 16 89810 .16 11136 .44 44 16 90386 .10 12710 11 4 4 17 89820 .16 11162 .44 43 17 90396 .16 12736 AA 43 18 89830 .16 11188 .44 42 18 90405 .16 i / 12762 11 i 42 J9 89840 .16 11214 .44 41 19 90415 .lo 12789 '5 41 20 89849 .16 rt 11241 .44 40 20 90424 .16 12815 *S 40 21 9.89859 1 \ J " 10.11267 .44 39 21 9.90434 ]j| 10.12842 39 22 89869 11293 .44 38 22 90443 . 1 12868 11 38 23 89879 .16 11319 .44 37 23 90452 .16 i / 12894 .44 o 7 ii ** 24 89888 .16 11345 .44 36 24 90462 .]b 12921 .44 36 25 89898 .16 11371 .44 35 25 90471 .16 12947 .44 35 26 89908 .16 11397 .44 34 26 90480 .16 12973 .44 34 27 89918 .16 11423 .44 33 27 90490 'JJ 13000 .44 33 28 89927 .16 11450 .44 32 28 90499 ,J 13026 .44 32 29 89937 .16 i c 11476 .44 31 29 90509 j '{5 13053 ..44 A 1 31 30 89947 Jo 11502 .44 30 30 905181 4" 13079 .44 A 4 30 31 32 0.89956 89966 .16 .16 10.11528 11554 2 29 28 31 32 9.90527 90537 .10 .16 10.13106 13132 .44 .44 29 28 34 89976 89985 .16 .16 11580 11607 .44 .44 27 26 33 34 90546 90555 .16 .16 i f* 13158 13185 .44 .44 4 A 27 26 35 89995 .16 1 A 1 1 633 .44 25 35 90565 .1 13211 .44 4 1 25 36 37 90005 90014 .lo .16 11659 11685 .44 .44 24 23 36 37 90574 90583 .16 1 1 3238 1 3264 .44 .44 1 1 24 23 38 39 90024 90034 .16 .16 11711 11738 .44 .44 22 21 38 39 90592 90602 * 1 .15 1 . 13291 13317 .44 .44 22 21 40 41 42 90043 9.90053 90063 .16 .16 .16 11764 10.11790 11816 .44 .44 .44 20 19 18 40 41 42 90611 9.90620 90630 .10 .15 .15 1 . 13344 10.13370 13397 .44 .44 .44 20 19 18 43 90072 .16 11842 .44 17 43 90639 10 1 *v 134231 '** 17 44 45 90082 90091 .16 .16 11869 11895 .44 .44 16 15 44 45 90648 90657 10 .15 1 ^ 1 3449 13476 .44 16 15 46 90101 .16 11921 .44 14 46 90667 JO 1 ^ 13502 .44 11 14 47 48 90111 90120 .16 .16 11947 11973 .44 .44 13 12 47 48 90676 90685 10 .15 1 13529 1 3555 .44 .44 A A 13 12 49 90130 .16 12000 .44 11 49 90694 .10 1 13582 .44 11 11 50 51 90139 9.90149 .16 .16 12026 10.12052 .44 .44 10 9 50 51 90704 9.90713 . 1 .15 1 13608 10.13635 .44 .44 11 10 9 52 53 54 90159 90168 90178 .16 .16 .16 12078 12105 12131 .44 .44 .44 8 7 6 52 53 54 90722 90731 90741 .10 .15 .15 i n 13662 13688 13715 .44 .44 .44 8 7 6 55 56 90187 90197 .16 .16 12157 12183 .44 .44 5 4 55 56 90750 907f>9' 10 .15 13741 13768 .44 5 4 57 58 59 90206 90216 90225 .16 .16 .16 12210 12236 12262 .44 .44 .44 3 2 1 57 58 59 90768 90777 90787 .15 .15 .15 13794 13821 13847 .44 .44 A A 3 2 1 60 90235 .16 10.12289 .44 fill 90796 .15 13874 *" M. Cosine. ; PI" ; CotHiiir. 1)1" M. M C(i<ino. PI" Cotan-r. I>1" 1H . 54 54 SINES AND TANGENTS. 55' M. Sine. Dl" Tung. Dl" M. M. Sine. Dl" THUS. . Dl" M. 1 9.90796 90805 0.15 1 S 10.13874 13900 0.44 A A 60 59 1 9.91336 91345 0.15 1 S 10.15477 15504 0.45 60 59 2 90814 * JLO 13927 '4 58 2 91354 1 15531 .40 58 3 4 90823 90832 .15 .15 13954 13980 .44 .44 57 56 3 4 91363 91372 .15 .15 15558 15585 .45 .45 57 56 5 90842 .15 14007 .44 55 5 91381 .15 15612 .45 55 6 90851 .15 1 E 14033 .44 A A 54 6 91389 .15 i ^ 15639 .45 A f\ 54 7 90860 .10 14060 .44 53 7 91398 . 10 15666 .40 53 8 90869 .15 -| r 14087 .44 A A 52 8 91407 .15 1 c 15693 .45 A fi 52 9 90878 .10 1 Pi 14113 .44 A -1 51 9 91416 .10 1 1 15720 .40 A f\ 51 10 90887 .10 14140 .44 50 10 91425 .10 15746 .40 50 11 9. 90896 .15 1 c 10.14166 .44 A A 49 11 9.91433 .15 i R 10.15773 .45 A Pi 49 12 90906 . 10 14193 .44 48 12 9!442 . I 15800 .40 A F; 48 13 90915 .15 14220 .44 47 13 91451 .15 15827 .40 47 14 90924 .15 14246 .44 46 14 91460 .15 15854 .45 46 15 90933 .15 14273 .44 45 15 91469 .15 15881 .45 45 16 90942 .15 i ^ 14300 .44 A A 44 16 91477 .15 15908 .45 A Pi 44 17 90951 .10 14326 .44 43 17 91486 .15 15935 .40 43 18 90960 .15 14353 .44 42 18 91495 .15 15962 .45 42 19 90969 .15 1 Pi 14380 .44 A A 41 19 91504 .15 -| e 15989 .45 A *\ 41 20 90978 .10 1 Pi 14406 .44 A t 40 20 91512 .10 16016 .40 A Pi 40 21 9.90987 .10 10.14433 .44 39 21 9.91521 .15 10.16043 .40 39 22 90996 .15 1 r 14460 .44 38 22 91530 .15 16070 .45 38 23 24 91005 91014 .10 .15 14486 14513 .44 .44 37 36 23 24 91538 91547 .15 .15 16097 16124 .45 .45 37 36 25 91023 .15 14540 .44 35 25 91556 .15 16151 .45 35 26 91033 .15 14566 .44 34 26 91565 .15 16178 .45 34 27 91042 .15 14593 .45 33 27 91573 .14 16205 .45 33 28 91051 .15 14620 .45 32 28 91582 .14 16232 .45 32 29 91060 .15 14646 .45 31 29 91591 .14 16260 .45 31 30 91069 .15 14673 .45 30 30 91599 .14 16287 .45 30 31 9.91078 .15 10.14700 .45 29 31 9.91608 .14 10.16314 .45 29 32 91087 .15 i \ 14727 .45 A c 28 32 91617 .14 16341 .45 A Pi 28 33 34 91096 91105 .10 .15 14753 14780 .40 .45 27 26 33 34 91625 91634 .14 .14 16368 16395 .40 .45 27 26 35 91114 .15 1 *\ 14807 .45 A Pi 25 35 91643 .14 16422 .45 A Pi 25 36 91123 .10 i 14834 .40 24 36 91651 .14 16449 .40 24 37 91132 .10 1 14860 .45 A Ci 23 37 91660 .14 16476 .45 23 38 91141 .10 14887 .40 22 38 91669 .14 16503 .45 22 39 91149 .15 14914 .45 21 39 91677 .14 16530 .45 21 40 91158 .15 14941 .45 20 40 91686 .14 16558 .45 20 41 9.91167 .15 1 Pi 10.11967 .45 t r- 19 41 9.91695 .14 10.16585 .45 19 42 91176 .10 14994 .40 18 42 91703 .14 16612 .45 18 43 91185 .15 i fi 15021 .45 17 43 91712 .14 16639 .45 17 44 91194 .10 15048 .45 16 44 91720 .14 16666 .45 16 45 91203 .15 i Pi 15075 .45 15 45 91729 .14 16693 .45 15 46 91212 .1 15101 .45 14 46 91738 .14 16720 .45 14 47 91221 .15 1 ^ 15128 .45 13 47 91746 .14 16748 .45 13 48 91230 .10 f r 15155 .45 12 48 91755 .14 16775 .45 12 49 91239 .10 15182 .45 11 49 91763 .14 16802 .45 11 50 91248 .15 1 c 15209 .45 10 50 91772 .14 16829 .45 10 51 9.91257 .10 10.15236 .45 9 51 9.91781 .14 10.16856 .45 9 52 91266 .15 15262 .45 8 52 91789 .14 16883 .45 8 53 91274 .15 i *i 15289 .45 7 53 91798 .14 16911 .45 7 54 91283 . 10 15316 .45 6 54 91806 .14 16938! '*? 6 55 91292 .15 15343 .45 5 55 91815 .14 16965 .40 5 56 91301 .15 1 e 15370 .45 4 56 91 823 .14 16992 .45 4 57 58 91310 91319 .10 .15 i Pi 15397 15424 .45 .45 3 2 57 58 91832 91840 .14 .14 17020 17047 45 .45 3 2 59 91328 .10 15450 .45 1 59 91849 .14 17074 .45 1 60 91336 .15 15477 .45 60 91857 .14 17101 .45 M. Cosine. Dl" | Cotang. Dl" M. M. Cosine. Dl" Cot a MS. Dl" M. 55 34 56' TABLE IV. LOGARITHMIC 57 31. Si no. i Dl" Tans,'. ; Dl" M. M. Sine. Dl" Tung. 1)1" M. 1 9.91857' 91866 0.14 10.17101 17129 0.45 60 59 1 9.92359 92367 0.14 10.18748 18776 0.46 60 59 2 91874 .14 17156 .45 A C 58 2 92376 .14 1 A 18804 .46 A 58 3 91883 17183 .40 57 3 92384 . 14 18831 .40 57 4 91891 .14 17210 .45 56 4 92392 .14 18859 .46 56 5 91900 .14 17238 .45 55 5 92400 .14 18887 .46 55 6 91908 .14 17265 .45 54 6 92408 .14 18914 .46 54 7 91917 .14 17292 .45 53 7 92410 .14 18942 .46 53 8 91925 .14 17319 .45 52 8 92425 .14 18970 .46 52 9 91934 .14 17347 .46 51 9 92433 .14 18997 .46 51 10 91942 .14 i * 17374 .46 40 50 10 92441 .14 19025 .46 A / 50 11 9.91951 .1-4 10.17401 .40 49 11 9.92449 10.19053 .40 49 12 91959 .14 1 4 17429 .46 1 A 48 12 92457 .14 19081 .46 A a 48 13 14 91968 91976 .14 .14 17456 17483 .-40 .46 47 46 13 14 92465 '* 92473 '!* 19108 19136 .40 .46 47 46 15 91985 .14 17511 .46 45 15 92482 .14 19164 .46 45 - 16 91993 .14 17538 .46 \ A. 44 16 92490 .14 19192 .46 A A 44 17 92002 17565 .40 43 17 92498 19219 .40 43 18 19 92010 92018 .14 .14 17593 17620 .46 .46 A t 42 41 18 19 92506 92514 .14 .14 19247 19275 .46 .46 42 41 20 92027 .14 17648 .40 40 20 92522 .13 19303 .46 40 21 9.92035 .14 10.17675 .46 .1 & 39 21 9.92530 .13 -I 10.19331 .46 A 39 22 92044 17702 .40 38 22 92538 .10 19358 .40 38 23 92052 .14 17730 .46 A A 37 23 92546 .13 1 o 19386 .46 37 24 92060 17757 .40 \ fi 36 24 92555 . 1 6 1 o 19414 .46 A C 36 25 92069 17785 .40 dfi 35 25 92563 lo 1 Q 19442 .40 A A 35 26 92077 17812 .40 34- 26 92571 lo 19470 .40 34 27 92086 .14 1 A 1 7839 .46 40 33 27 92579 .13 1 o 19498 .46 A 33 28 29 92094 92102 14 .14 17867 17894 .40 .46 32 31 28 29 92587 92595 . 1 O .13 19526 19553 .4o .46 32 31 30 92111 .14 17922 .46 30 30 92603 .13 19581 .46 30 31 9.92119 .14 10.17949 .46 j.fi 29 31 9.92611 .13 1 Q 10.19609 .46 A A 29 32 92127 17977 .40 28 32 92619 1 > 19637 .4o 28 33 92136 .14 18004 .46 1 A 27 33 92627 .13 -i > 19665 .47 27 34 92144 18032 .40 4fi 26 34 92635 .1 o 1 9 19693 .47 A 7 26 35 92152 18059 .40 25 35 92643 13 19721 .4( 25 36 92161 .14 18087 .46 i A 24 36 92651 .13 1 O 19749 .47 24 37 92169 18114 .40 23 37 92659 lo 19777 .47 23 38 92177 .14 18142 .46 22 38 92667! 'JJ 19805 .47 22 39 92186 .14 18169 .46 1 A 21 39 92675! >J* 19832 .47 A 7 21 40 92194 18197 .40 20 40 92683] '{J 19860 .47 20 41 9.92202 .14 10.18224 .46 A A 19 41 9.92691 i *fj 10.19888 .47 19 42 92211 18252 .40 18 42 926991 *' 199 Hi .47 18 43 92219 .14 18279 .46 17 43 92707 .Id 19944 .47 17 44 92227 .14 18307 .46 16 44 92715 .13 19972 .47 16 45 46 92235 92244 .14 .14 18334 18362 .46 .46 i 15 14 45 46 92723 92731 .13 .13 1 *.} 20000 20028 .47 .47 15 14 47 92252 .14 18389 .40 1 f* 13 47 92739 . l > i } 20056 .47 A 7 13 48 92260 18417 .40 iO 12 48 92747 .lo i S 20084 A I A *7 12 49 92269 18444 .40 11 49 92755 . l > 20112 .4i 11 50 92277 j ' 18472 .46 4 10 50 92763 .13 1 3 20140 .47 47 10 51 52 9.92285 92293 . i- .14 10.18500 18527 TcO .46 9 8 51 52 9.92771 92779 .10 .13 10.20168 20196 4 1 .47 9 8 53 92302 .14 18555 .46 to 7 53 92787 .13 1 *? 20224 .47 A 7 7 54 55 92310 92318 .14 18582 18610 .40 .46 4 A 6 5 54 55 92795 92803 * I O .13 1 9 20253 20281 4< .47 I *? 6 5 56 92326 .14 1 8638 .40 4 56 92810 lo 20309 .4< 4 57 92335 14 18665 .4fi 3 57 92818 .13 20337 .47 3 58 9234'! .14 18693 .4(i 2 58 92826 .13 20365 .47 2 59 92351 .14 18721 .4fi 1 59 92834 i '** 2039., .47 1 80 92359 .14 18M8 .46 80 92842 * 20421 .47 M. Cosine. nr Cot;tll2. T)l" M. M. cosinf. ni" ('otanz. Dl" M. 32 SINES AND TANGENTS. 59* 31. Sine. 1)1" Tang. Dl" M. M. Sine. Dl" Tang. Dl" M. 1 9.92842 02850 0.13 10.20421 20449 0.47 60 59 1 9.93307 93314 0.13 10.22123 22151 0.48 60 5'J 2 92858 .13 20477 .47 58 2 93322 .13 22180 .48 58 3 92866 .13 1 o 20505 .47 57 3 93329 .13 1 9 22209 .48 A Q 57 4 92874 .10 20534 .47 56 4 93337 . lo 22237 .48 56 5 6 92881 92889 .13 .13 20562 20590 .47 .47 55 54 5 6 93344 93352 .13 .13 22266 22294 .48 .48 55 54 92897 .13 20618 .47 53 7 93360 .13 22323 .48 53 8 92905 .13 20646 .47 52 8 93367 .13 22352 .48 52 9 92913 .13 20674 .47 51 9 93375 .13 22381 .48 51 10 92921 .13 20703 .47 50 10 93382 .13 22409 .48 50 11 9.92929 .13 10.20731 .47 49 11 9.93390 .13 10.22438 .48 49 12 92936 '! 20759 .47 48 12 93397 .13 22467 .48 48 13 92944 .lo 20787 .47 47 13 93405 .13 22495 .48 47 14 92952 .13 20815 .47 46 14 93412 .13 22524 .48 46 15 92960 .13 20844 .47 45 15 93420 .13 22553 .48 45 16 92968 .13 20872 .47 44 16 93427 .13 22582 .48 44 17 92976 .13 20900 .47 43 17 93435 .13 22610 .48 43 18 92983 .13 20928 .47 42 18 93442 .13 22639 .48 42 19 92991 .13 20957 .47 41 19 93450 .12 22668 .48 41 20 92999 .13 20985 .47 40 20 93457 .12 22697 .48 40 21 9.93007 .13 10.21013 .47 39 21 9.93465 .12 10.22726 .48 39 22 93014 .13 21041 .47 38 22 93472 .12 22754 .48 38 23 93022 .13 21070 .47 37 23 93480 .12 22783 .48 37 24 93030 .13 21098 .47 36 24 93487 .12 22812 .48 i Q 36 25 93038 .13 21126 .47 35 25 93495 .12 22841 .48 35 26 93046 .13 21155 .47 34 26 93502 .12 22870 .48 A Q 34 27 93053 .13 21183 .47 33 27 93510 .12 22899 .48 33 28 93061 .13 21211 .47 32 28 93517 .12 22927 .48 32 29 93069 .13 21240 .47 31 29 93525 .12 22956 .48 A Q 31 30 93077 .13 21268 .47 30 30 93532 .12 22985 .48 30 31 32 9.93084 93092 .13 .13 10.21296 21325 .47 .47 29 28 31 32 9.93539 93547 .12 .12 10.23014 23043 .48 .48 29 28 33 93100 .13 1 o 21353 .47 A *7 27 33 93554 .12 23072 .48 A Q 27 34 93108 la 21382 .47 26 34 93562 .12 23101 4o 26 35 93115 .13 21410 .47 25 35 93569 .12 23130 .48 25 36 93123 .13 21438 .47 24 36 93577 .12 23159 .48 24 37 93131 .13 21467 .47 23 37 93584 .12 23188 .48 23 38 93138 .13 21495 .47 22 38 93591 .12 23217 .48 22 39 93146 .13 21524 .47 21 39 93599 .12 23246 .48 21 40 93154 .13 1 O 21552 .47 20 40 93606 .12 23275 .48 20 41 9.93161 .lo 10.21581 .47 19 41 9.93614 .12 10.23303 .48 19 42 93169 .13 1 Q 21609 .47 47 18 42 93621 12 1 n 23332 .48 40 18 43 93177 J 21637 7 17 43 93628 LZ 23361 .48 17 44 93184 .13 1 Q 21666 .47 4*- 16 44 93636 .12 1 Q 23391 .48 .JO 16 45 93192 .lo 21694 j 15 45 93643 Iz 23420 .48 15 46 93200 .13 1 Q 21723 .47 /I T 14 46 93650 .12 i o 23449 .48 A Q 14 47 93207 .lo 21751 .47 13 47 93658 .12 23478 .48 13 48 93215 .13 1 O 21780 .48 A Q 12 48 93665 .12 1 rt 23507 .48 1 Q 12 49 50 93223 93230 Jo .13 21808 21837 4o .48 11 10 49 50 93673 93680 I 2 .12 23536 23565 .4o .48 11 10 51 9.93238 .13 10.21865 .48 9 51 9.93687 .12 10.23594 .48 9 52 93246 .13 21894 .48 8 52 93695 .12 23623 .48 8 53 93253 .13 10 21923 .48 40 7 53 93702 .12 1 9 23652 .48 4Q 7 54 93261 .1 o 21951 rto 6 54 93709 . 1 Zt 23681 .4y 6 55 93269 .13 1 o 21980 .48 5 55 93717 .12 23710 .49 5 56 93276 .13 22008 .48 4 56 93724 .12 23739 .49 4 57 93284 .13 1 3 22037 .48 40 3 57 93731 .12 1 9 23769 .49 AQ 3 58 93291 lo 1 q 22065 :8 AQ 2 58 93738 1 4 1 o 23798 .4 y A O 2 59 93299 .lo 1 O 22094 .48 A Q 1 59 93746 \ 1 1 O 23827 .4y 1 60 93307 .lo 22123 .4c 60 93753 .12 23856 .49 M. Cosine. Dl" Gotang. Dl" M. M. Cosine. Dl" Cot:uur. Dl" M. Sl c 57 6O TABLE IV. LOGARITHMIC 61 M. Sine, i Dl" Tang. Dl" M. M. Sine. 1)1" Tang. Dl" M. 9.93753L L 10. 23856 1 60 9.94182 10.25625 60 1 93760 U.iZ 23885 '*l 59 1 94189 0.12 In 25655 0.50 r ,i 59 2 93768 .12 23914 -Jr 58 2 94196 2 25684 .OU 58 3 93775 .12 23944! -JJ 57 3 94203 .12 25714 .50 57 4 93782 .12 1 O 23973 i *! 56 4 94210 .12 1 o 25744 .50 C A. 56 5 93789 .12 24002 .IV 55 5 94217 . 1 2 25774 .OU 55 6 7 93797 93804 .12 .12 24031 24061 .49 .49 54 53 6 7 94224 94231 .12 .12 25804 25834 .50 .50 C A 54 53 8 938 U .12 24090 .49 52 8 94238 .12 25863 .50 52 9 93819 .12 1 O 24119 .49 51 9 94245 .12 25893 .50 en 51 10 11 93826 9.93833 . IZ .12 1 A 24148 10.24178 !49 50 49 10 11 94252 9.94259 !l2 i a 25923 10.25953 .oU .50 50 49 12 93840 .12 24207 .49 48 12 94266 " 25983 .50 48 13 93847 .12 24236 .49 47 13 94273 '" 26013 .50 47 14 93855 .12 24265 .49 46 14 94279 '** 26043 .50 46 15 16 17 93862 93869 93876 .12 .12 .12 24295 24324 24353 .49 .49 .49 45 44 43 15 16 17 94286 '" 94293 ' ; 94300 ';* 26073 26103 26133 .50 .50 .50 45 44 43 18 93884 .12 24383 .49 42 18 94307! " 26163 .50 42 19 93891 .12 1 O 24412 .49 1 O 41 19 94314 l Tf 26193 .50 41 20 93898 .12 24442 .4 40 20 i 943211 *ff 26223 .50 40 21 9.93905 .12 10.24471 .49 39 21 9.94328 '{, 10.26253 .50 39 22 93912 .12 24500 .49 38 22 i 94335, ,! 26283 .50. 38 23 93920 .12 24530 .49 37 23 94342 '} 26313 .50 37 24 93927 .12 24559 A A 36 24 j 94349 ''! 26343 .50 36 25 93934 .12 24589 .49 35 25 94355 .11 26373 .50 35 26 93941 .12 24618 .49 34 26 94362 .11 26403 .50 r A 34 27 93948 .12 24647 4 A 33 27 94369 -Jl 26433 OU 33 28 93955 1 " 24677 .49 32 28 94376 '}; 26463 .50 32 29 93963 'Jo 24706 .49 31 29 94383) '}! 26493 .50 tn 31 30 93970 1 9 24736 A O 30 30 94390 '" 26524 Ol/ c A 30 31 9.93977 12 10.24765 * 29 31 9.943971 ' 10.26554 .OU 29 32 93984 .12 1 rt 24795 .49 28 32 94404 26584 .50 C A 28 33 34 93991 93998 .!/ .12 1 9 24824 24854 '.49 27 26 33 34 94410 94417 .11 26614 26644 .OU .5-0 CA 27 26 35 94005 12 24883 Ay 25 35 94424 n 1 1 26674 .OU 25 36 94012 1 O 24913 A A 24 36 94431 .11 26705 .50 24 37 94020 .12 24942 .49 23 37 94438 .11 26735 .50 23 38 39 94027 94034 .12 .12 24972 25002 ^49 22 21 38 39 94445 94451 .11 .11 26765 26795 .50 .50 22 21 40 94041 .12 25031 A A 20 40 94458 .11 26825 .50 20 41 9.94048 .12 19 10.25061 .49 4Q 19 41 9.94465 .11 10.26856 .50 EA 19 42 94055 .12 1 A 25090 .4 18 42 94472! "'I 2688 OU C A 18 43 94062| '!A 25120 in 17 43 94479 .1 1 26916 .OU 17 44 94069 .1^ 25149 .49 16 44 94485 .11 26946 .50 16 45 94076 .12 -I A 25179 .49 \(\ 15 45 94492 .11 26977 .50 e T 15 46 94083 .12 1 A 25209 .4W 14 46 94499 27007 01 14 47 94090 .12 1 A 25238 .49 13 47 94506 .11 27037 .51 c-i 13 48 94098 .12 25268 A (i 12 48 94513 -t -i 27068 .01 c -i 12 49 94105 1 O 25298 .4 A O 11 49 94519 .11 27098 .01 r 1 11 50 94112 . 12 25327 .4y 10 50 94526 .11 27128 .01 r i 10 51 9.94119 .12 10.25357 -49 9 51 9.94533 . 10.27159 .01 9 52 94126 .12 25387 .50 8 52 94540 .11 27189 .51 8 53 94133 .12 1 9 25417 .50 C A 7 53 94546 .11 27220 .51 7 54 94140 .12 25446 .OU 6 54 94553 27250 . 6 55 94147 .12 25476 .50 5 55 94560 .11 27280 .51 5 56 94154 .12 25506 .50 4 56 94567 .11 27311 .51 4 57 94161 .12 25535 .50 3 57 94573 .11 27341 .51 3 58 94168 .12 1 9 25565 .50 2 58 94580 .11 27372 .51 L1 2 59 94175 .12 1 n 25595 .OU 1 59 94587 27402 .01 1 60 94182 .12 25625 .50 60 94593 .11 27433 .51 M. Cosine. Dl" Cotang. Dl" M. M. Cosine. Dl" Cotang. D\" M. 29 SINES AND TANGENTS. 63 C M. Sine. D." Tang. Dl" M. M. Sine. Dl" Tang. DI" M. 1 9.94593 94600 0.11 10.27433 27463 0.51 60 59 1 9.94988 94995 0.11 10.29283 29315 0.52 60 59 2 94607 .11 27494J '?! 58 2 95001 .11 29346 .52 58 3 94614 .11 27524 .01 57 3 95007 .11 29377 .52 57 4 94620 .11 27555 .51 56 4 95014 .1 1 29408 .52 co 56 5 94627 .11 27585 .51 55 5 95020 .11 29440 .OZ 55 6 94634 .11 27616 .51 54 6 95027 .11 29471 .52 54 7 94640 .11 27646 .51 53 7 95033 .11 29502 .52 53 8 94647 .11 27677 .51 52 8 95039 .11 29534 .52 52 9 94654 .11 27707 .51 51 9 95046 .11 29565 .52 51 10 94660 .11 27738 .51 50 10 95052 .11 29596 .52 50 11 9.94667 .11 10.27769 .51 49 11 9.95059 .11 10.29628 .52 co 49 12 94674 .11 27799 .51 48 12 95065 .11 29659 .OZ 48 13 94680 .11 27830 .51 47 13 95071 .11 29691 .52 47 14 94687 .11 27860 .51 46 14 95078 .11 29722 .52 46 15 94694 .11 27891 .51 45 15 95084 .11 29753 .52 45 16 94700 .11 27922 .51 44 16 95090 .11 29785 .52 44 17 94707 .11 27952 .51 43 17 95097 .11 29816 .52 co 43 18 94714 .11 27983 .51 42 18 95103 .11 29848 .OZ 42 19 94720 .11 28014 .51 41 19 95110 .11 29879 .52 41 20 94727 .11 28045 .51 40 20 95116 .11 29911 .52 40 21 9.94734 .11 10.28075 -JJI 39 21 9.95122 .11 10.29942 .53 39 22 94740 .11 28106 '} 38 22 95129 .11 29974 .53 r o 38 23 94747 .11 28137 '{J 37 23 95135 .11 30005 .5o CO 37 24 94753 .11 28167 'r, 1 36 24 95141 .11 30037 .Oo 36 25 94760 .11 28198 .01 35 25 95148 .11 30068 .53 35 26 94767 .11 28229 .51 34 26 95154 .11 30100 .53 34 27 94773 .11 28260 '?! 33 27 95160 .11 30132 .53 CO 33 28 94780 .11 28291 .01 32 28 95167 .11 30163 .00 32 29 94786 .11 28321 .51 31 29 95173 .11 30195 .53 31 30 94793 -!! 28352 .51 30 30 95179 .11 30226 .53 30 31 9.94799 .ii 10.28383 .51 29 31 9.95185 .10 10.30258 .53 CO 29 32 94806 .11 28414 .51 28 32 95192 '}J 30290 .00 28 33 94813 .11 28445 .51 27 33 95198 .1U 30321 .53 27 34 94819 .11 28476 .51 26 34 95204 .10 30353 .53 26 35 94826 .11 28507 .51 25 35 95211 .10 30385 .53 25 36 94832 .11 28538 .52 24 36 95217 .10 30416 .53 24 37 94839 .11 28569 .52 23 37 95223 .10 30448 .53 23 38 94845 .11 28599 .52 CO 22 38 95229 .10 -I A 30480 .53 c o 22 39 94852 28630 .oz 21 39 95236 .11) 30512 .0.1 21 40 94858 .11 28661 .52 20 40 95242 .10 30543 .53 20 41 9.94865 .11 i -i 10.28692 .52 19 41 9.95248 .10 1 A 10.30575 .53 C 19 42 94871 .11 28723 .52 18 42 95254 1U 30607 .Oo 18 43 94878 .11 28754 j '? 17 43 95261 .10 1 A 30639 .53 r o 17 44 94885 28785! *?* 16 44 95267 1U 30671 .Do 16 45 94891 .11 28816 .02 15 45 95273 .10 30702 .53 15 46 94898 .11 28847 .52 14 46 95279 .10 30734 .53 14 47 94904 .11 28879 .52 13 47 95286 .10 in 30766 .53 CO 13 48 94911 28910 .52 12 48 95292 lu 30798 .Oo 12 49 94917 .11 28941 .52 11 49 95298 .10 30830 .53 11 50 94923 .11 28972 .52 10 50 95304 .10 30862 .53 10 51 9.94930 .11 10.29003 .52 c n 9 51 19.95310 .10 1 A 10.30894 .53 CO 9 52 94936 29034 02 8 52 95317 .1U 30926 .Do 8 53 94943 .11 29065 .52 co 7 53 95323 .10 30958 .53 CO 7 54 94949 29096 .OZ 6 54 95329 .10 30990 .Do 6 55 94956 .11 29127 .52 5 55 95335 .10 31022 .53 5 56 94962 .11 29159 .52 co 4 56 95341 .10 1 t\ 31054 .53 CO 4 57 94969 29190 .OZ 3 57 95348 .10 31086 .03 3 58 94975 .11 29221 .52 2 58 95354 .10 31118 .53 2 59 60 94982 949S8 .11 .11 29252) ** 29283 ' 1 59 60 95360 95366 .10 .10 31150 31182 .53 .53 1 M. j Cosine. Dl" Ootang. Dl" M. M. Cosine. Dl" Cotang. Dl" M. 59 20 TABLE IV. LOGARITHMIC M. Sine. Dl" Tang. Di" M. M. Sine. Dl" Tung, j Dl" M. 9.95366 A , n 10.31182 ' .., 60 |9.95728 . 10.33133 60 1 95372 u< {" 31214 U.Od 59 1 95733 33166 0.55 , g 2 95378 .10 31246 .53 58 2 95739 ' 33199 ?? 58 3 95384 .10 31278 .53 57 3 95745 .10 33232 '55 57 4 95391 . .10 31310 .54 56 4 95751 .10 33265 56 5 95397 .10 31342 .54 55 5 95737 .10 33298 .55 55 6 95403 .10 31.] 74 .54 54 6 95763 .10 33331 .55 54 7 95409 .10 31407 .54 53 7 95769 .10 33364 .55 53 8 95415 .10 31439 .54 52 8 95775 .10 33397 .55 52 9 95421 .10 31471 .54 51 9 95780 .10 33430 .55 51 10 95427 .10 31503 .54 50 10 95786 .10 33463 .55 50 11 12 9.95434 95440 .10 .10 10.31535 31568 .54 .54 49 48 11 12 9.95792 95798 .10 .10 10.33497 ?? 335301 * 49 48 13 95446 .10 31600 .54 47 13 95804 .10 33563 ?'? 47 14 15 95452 95458 .10 .10 31632 31664 .54 .54 46 45 14 15 95810 95815 .10 .10 33596 33629 .00 .55 46 45 16 95464 .10 31697 .54 44 16 95821 .10 33663 .55 44 17 95470 .10 31729 .54 43 17 95827 .10 33696 ?? 43 18. 95476 .10 1 A 31761 .54 -A 42 18 95833 .10 i n 33729 '*? 42 19 95482 . IU 1 31794 .04 41 19 95839 .10 1 A 337621 '2 41 20 95488 1 v 31826 40 20 95844 . J U 33796 .uu 40 21 9.95494 .10 10.31858 .54 39 21 9.95850 .10 10.33829 .56 39 22 95500 .10 31891 .54 38 22 95856 .10 33862 .56 38 23 95507 .10 31923 .54 37 23 95862 .10 33896! '?? 37 24 95513 .10 31956 .54 36 24 958(58 -J" 33929 .06 36 25 95519 .10 31988 .54 35 25 95S73 \'!. 33962 .56 35 26 95525 .10 32020 .54 34 26 95879 .IU 33996 .56 34 27 95531 .10 32053 .54 33 27 95885 .10 34029 .56 33 28 95537 .10 32085 .54 32 28 95891 .10 34063 .56 32 29 95543 .10 32118 .54 31 29 95897 .10 34096 .56 31 30 95549 .10 32150 .54 30 30 95902 .10 34130 .56 30 31 9.95555 .10 1 A 10.32 13 .54 29 31 9.95908 .10 i n 10.34163 .56 p n 29 32 95561 .10 1 A 32215 .54 28 32 95914 . 1 1 n 34197 .00 c rt 28 33 95567 . 1 32248 27 33 95920 .10 34230 .00 27 34 95573 .10 32281 .54 26 34 95925 .10 34264 .56 26 35 95579 .10 1 A 32313 .54 25 35 95931 .10 1 34297 '!?!? 25 36 95585 .1 U 32346 24 36 95937 . 1 U 34331 'JJ 24 ;>7 95591 .10 1 f\ 32378 .54 23 37 95942 .10 1 A 34364 -J5 23 38 95597 .10 32411 .54 22 38 95948 . 10 34398 .00 22 39 95603 .10 32444 .54 21 39 95954 .10 34432 .56 21 40 95609 .10 32476 .54 20 40 95960 .10 34465 .56 20 41 9.95615 .10 10.32509 .54 19 41 9.95965 .1 10.34499 .56 19 42 95621 .10 32542 .54 18 42 95971 .10 34533 .56 18 43 95627 .10 1 A 32574 .55 5e 17 43 95977 1 It 1 34566 .56 c / 17 44 95633 . 10 32607 16 44 95982 .oy 34600 .00 16 45 95639 .10 32640 .55 15 45 95988 .09 34634 .56 15 46 95645 .10 32673 .55 14 46 95994 .09 34667 .56 14 47 95651 .10 32705 .55 13 47 96000 .09 Afk 34701 .56 13 48 95657 .10 32738 .55 12 48 96005 .09 34735 .56 12 49 95663 .10 32771 .55 11 49 96011 .09 34769 .56 11 50 51 95668 9.95674 .10 .10 32804 10.32837 .55 .55 10 9 50 51 96017 9.96022 .09 .09 34803 10.34836 .56 .56 10 9 52 95680 .10 32S69 .55 8 52 96028 .09 34870 .56 8 53 95686 .10 1 fi 32902 .55 cc 7 53 96034 .09 .09 34904 .56 .56 7 54 95692 J U 32935 .00 6 54 96039 34938 6 55 95698 .10 32968 .55 5 55 96045 .09 A A 34972 .57 KT 5 56 95704 .10 33001 .55 4 56 96050 .09 35006 .07 4 57 95710 .10 33034 .55 3 57 96056 .09 35040 .57 3 58 95716 .10 33067 .55 2 58 96062 .09 35074 .57 2 59 95722 .10 33100 .55 1 59 96067 .09 35108 o< 1 60 95728 .10 1 0.33133 .55 60 96073 .09 35142 .57 M. Cosine. Dl" Cotnnt. Dl" M. M. Cosiiic. Dl" Cotang. Di" M. 25= 66 C SINES AND TANGENTS. 67 C M. Sine. Dl" Tang. Dl" M. M. Si no. Dl" Tang. Dl" M. 1 9.96073 96079 0.09 10.35142 35176 0.57 c 7 60 59 1 9.96403 96408 0.09 AQ J 0.372151 37250 0.59 CQ 60 59 2 3 96084 96090 .09 .09 An 35210 35244 .Of .57 57 58 57 2 3 96413 96419 .uy .09 no 37285 37320 .oy .59 K.O 58 57 4 96095 .09 Aft 35278 t c 7 56 4 96424 .uy OQ 37355 .oy CQ 56 5 96101 .uy 35312 .0 1 55 5 96429 .uy 37391 .oy 55 6 7 8 96107 96112 96118 .09 .09 .09 35346 35380 35414 .57 .57 .57 K7 54 53 52 6 7 8 96435 96440 96445 .09 .09 .09 OQ 37426 37461 37496 .59 .59 .59 .59 54 53 52 9 96123 .uy 35448 1 51 9 96451 .uy 37532 51 10 96129 .09 Aft 35483 .57 r 7 50 10 96456 .09 Aft 37567 .59 C Q 50 11 9.96135 .09 Aft 10.35517 .07 c n 49 11 9.96461 .uy OQ 10.37602 .oy CQ 49 12 96140 .uy 35551 .07 48 12 96467 .uy 37638 .oy 48 13 96146 .09 OQ 35585 .57 47 13 96472 .09 OQ 37673 .59 fjQ 47 14 96151 .uy 35619 .57 46 14 96477 .uy Aft 37708 t>y eft 46 15 16 96157 96162 .09 .09 35654 35688 .57 .57 S7 45 44 15 16 96483 96488 .uy .09 .09 37744 37779 .oy .59 .59 45 44 17 96168 no 35722 .0 < t\7 43 17 96493 OQ 37815 P.O 43 18 96174 uy AQ 35757 Of 42 18 96498 .uy AQ 37850 .oy .59 42 19 20 96179 96185 .uy .09 OQ 35791 35825 .57 C7 41 40 19 20 96504 96509 .uy .09 AQ 37886 37921 159 F.Q 41 40 21 9.96190 .uy OQ 10.35860 Of 39 21 9.96514 .uy .09 10.37957 .oy .59 39 22 96196 .uy AQ 35894 M 38 22 96520 OQ 37992 cq 38 23 96201 .uy Aft 35928 r 7 37 23 96525 Uy OQ 38028 .oy CQ 37 24 96207 .uy OQ 35963 .0 R7 36 24 96530 .uy .09 38064 .oy .59 36 25 96212 .uy Aft 35997 O i C 7 35 25 96535 Aft 38099 C Q 35 26 96218 .uy OQ 36032 .07 C7 34 26 96541 .uy .09 38135 .oy .59 34 27 96223 .uy OQ 36066 i CO 33 27 96546 38170 33 28 96229 .uy Aft 36101 .Oo c.Q 32 28 96551 Aft 38206 An 32 29 96234 .uy Aft 36135 .OO 31 29 96556 .uy Aft 38242 .OU 31 30 96240 .Uy 36170 .58 30 30 96562 .uy 38278 .60 30 31 9.96245 .09 OQ 10.36204 .58 K.O 29 31 9.96567 .09 OQ 10.38313 .60 (ill 29 32 96251 .uy OQ 36239 "" 28 32 96572 .uy AQ 38349 .ou fiO 28 33 96256 .uy no 36274 | *rj[ 27 33 96577 .uy Aft 38385 .OU An 27 34 96262 .uy .09 36308 *r2 26 34 96582 .uy AQ 38421 .oU .60 26 35 96267 36343; 'r 25 35 96588 .uy 38456 25 36 37 38 39 96273 96278 96284 96289 .09 .09 .09 .09 09 56377 36412 36447 36481 .08 .58 .58 .56 24 23 22 21 36 37 38 39 96593 96598 96603 96608 .09 .09 .09 .09 nn 38492 38528 38564 38600 !eo .60 .60 fiO 24 23 22 21 40 96294 36516 "" 20 40 96614 .uy 38636 .ou 20 41 9.96300 no 10.36551 'g 19 41 9.96619 .09 AQ 10.38672 .60 AA 19 42 43 96305 96311 .uy .09 AQ 36586 1 -J5 36621 j 'JJ 18 17 42 43 96624 96629 091 38708 38744 .OU .60 CA 18 17 44 45 96316 96322 .uy .09 no 36655 36690 .f O .58 16 15 44 45 96634 96640 .uy .09 nn 38780 38816 ,OU .60 16 15 46 96327 .uy An 36725 CO 14 46 96645 38852 14 47 96333 .uy Aft 36760 'J 13 47 96650 38888 fiO 13 48 96338 .uy .09 36795 :* 12 48 96655 OQ 38924 .ou fiO 12 49 96343 36830 ?: 11 49 96660 .uy 38960 .ou 11 50 96349 .09 36865 * 10 50 96665 .09 38996 .60 10 51 9.96354 .09 OQ 10. 36899 ! '2 9 51 9.96670 J J j 10.39033 .60 An 9 52 96360 .uy 36934 .1^0 8 52 96676 .uy 39069 .OU 8 53 96365 .09 36969 .5 7 53 96681 .09 39105 .60 7 54 96370 .09 37004 '?* 6 54 96686 .09 39141 .60 6 55 56 96376 96381 .09 .09 37039 37074 .*5E 5 4 55 56 96691 96696 .09 .09 39177 39214 .60 .60 5 4 57 96387 .09 37110 .5 3 57 96701 .09 39250 .60 3 58 96392 .09 37145 .5 2 58 96706 .09 39286 .61 2 59 96397 .09 37180 .Ofc 1 59 96711 ^ 39323 .61 1 60 96403 .09 37215 .59 1 60 96717 .uy 39359 .61 M. Cosiue. Dl" Cotang. 1 Dl" 31. 31. Cosine. Dl" Cotang. Dl" M. 61 TABLE IV. LOGARITHMIC 69* If. Siue. Dl" Tang. Dl" M. M. Siue. | Dl" Taug. Dl" M. 9.96717 10.39359 60 9.97015 ! n M 10.41582 60 1 96722 O.OJ 39395 0.61 fil 59 1 97020 , U ' n 41620 0.63 60 59 2 3 4 96727 96732 96737 'M .08 39432 39468 39505 .01 .61 .61 58 57 56 2 3 4 97025 97030 97035 !08 .08 41658 41696 41733 O .63 .63 58 57 56 5 96742 .Oj 39541 .61 55 5 97039 .08 41771 vK 55 6 96747 .Oc AC 39578 .61 fil 54 6 97044 .08 no 41809 .63 6 54 7 8 96752 96757 Uc .08 AQ 39614 39651 O I .61 fil 53 52 7 8 97049 97054 .UO .08 no 41847 41885 t .63 6e 53 52 9 96762 Uc no 39687 .0 1 fil 51 9 97059 .Uo AQ 41923 60 51 10 96767 .Uo 39724 .01 50 10 97063 .Uo 41961 f, 50 11 9.96772 AC 10.39760 .61 A 1 49 11 9.97068 .08 AQ 10.41999 .62 49 12 96778 .Uo .Of 39797 .O 1 61 48 12 97073 .Uo AQ 42037 .62 63 48 13 96783 39834 A 1 47 13 97078 *UO AO 42075 47 14 96788 AC 39870 .ol fil 46 14 97083 .Uo ()Q 42113 .62 46 15 96793 Uc no 39907 Ol fil 45 15 97087 Uo AQ 42151 A /I 45 16 96798 .Uc nc 39944 .0 I A 1 44 16 97092 Uo AO 42190 .04 44 17 96803 .Uo .08 39981 .01 61 43 17 97097 Uo AQ 42228 .64 a\ 43 18 96808 40017 42 18 97102 Uo 42266 .0-4 42 19 20 96813 96818 .08 .08 nc 40054 40091 .61 .61 fi l 41 40 19 20 97107 97111 .08 .08 42304 42342 .64 .64 A 1 41 40 21 9.96823 .Uc 10.40128 .0 L 39 21 9.97116 Jf 110.42381 .O4 39 22 96828 .08 08 40165 .61 38 22 97121 08 i 42419 .64 fid 38 23 96833 .Uc 40201 .0 I 37 23 97126 no 42457 .04 37 24 25 26 96838 96843 96848 .08 .08 .08 40238 40275 40312 .62 .62 .62 36 35 34 24 25 26 97130 97135 97140 .08 .08 .08 42496 42534 42572 .64 .64 .64 36 35 34 27 96853 .08 no 40349 .62 33 27 97145 .08 AC 42611 .64 33 28 96858 .Uo .08 40386 62 32 28 97149 Uo AQ 42649 .64 32 29 96863 AQ 40423 AO 31 29 97154 Uo 42688 /> t 31 30 96868 .Uo 40460 ,02 30 30 97159 .08 42726 .o4 30 31 9.96873 .08 AQ 10.40497 62 29 31 9.97163 .08 AQ 10.42765 .64 29 32 96878 UO Act 40534 AO 28 32 97168 Uo AO 42803 A i 28 33 96883 .Uo no 40571 .OZ 27 33 97173 .Uo AQ 42842 .o4 RA 27 34 96888 .Uo no 40609 62 26 34 97178 Uo AQ 42880 .04 26 35 96893 .uo AQ 40646 25 35 97182 Uo AQ 42919 fi4 25 36 96898 UO no 40683 fi9 24 36 97187 Uo AQ 42958 .04 24 37 96903 .Uo no 40720 .62 23 37 97192 .Uo AQ 42996 fi 23 38 39 96907 96912 .uo .08 08 40757 40795 [62 22 21 38 39 97196 97201 Uo .08 AQ 43035 43074 !65 22 21 40 96917 .UO 40832 OZ 20 40 97206 Uo 43113 .00 20 41 9.96922 .08 no 10.40869 .62 62 19 41 9.97210 .08 no 10.43151 .65 AC 19 42 96927 .uo no 40906 AO 18 42 97215 Uo AQ 43190 .00 Ax 18 43 96932 .uo 40944 .0^ 17 43 97220 .UO 43229 DO 17 44 96937 .08 no 40981 .62 AO 16 44 97224 .08 no 43268 .65 Ax 16 45 96942 .Uo no 41019 OJ5 15 45 97229 UO AQ 43307 .00 Ax 15 46 96947 .uo AQ 41056 AO 14 46 97234 .UO 43346 OO AX 14 47 96952 .Uo AQ 41093 .oZ AO 13 47 97238 .08 no 43385 .00 13 48 49 96957 96962 .UO .08 AO 41131 41168 OZ .62 12 11 48 49 97243 97248 .UO .08 AO 43424 43463 !e5 - 12 11 50 51 52 96966 9.96971 96976 .Uo .08 .08 41206 10.41243 41281 .63 .63 10 9 8 50 51 52 97252 9.97257 97262 43502 '" 110.43541 08 i 4358 .00 .65 .65 ax 10 9 8 53 54 55 96981 96986 96991 !08 .08 41319 41356 41394 !63 .63 7 6 5 53 54 55 97266 97271 97276 43619 ol 43658 S 43697 DO .65 .65 A " 7 g 5 58 57 96996 97001 .08 .08 41431 41469 .63 .63 4 3 56 57 97280 -jjjj 43736 972851 '55 43776 .00 .65 4 3 58 59 97005 97010 .08 .08 41507 41545 .63 .63 2 1 58 | 97289 59 97294 .08 43815 43854 .65 .65 2 1 60 97015 .08 41582 .63 60 i 97299' ' 43893 .65 M. Cosine. Dl" C'ot:in.a. Dl" .M. M. | Cosine. Dl" Cotuiig. Dl" M 70 SINES AND TANGENTS. M. Sine. Dl" Tang. Dl' M. M. Sine. Dl" Taug. Dl" If. 9.97299 97303 0.08 10.43893 43933 0.66 60 59 1 9.97567 97571 0.07 ] 0.46303 46344 0.68 60 59 2 97308 .08 43972 .66 58 2 97576 .07 46385 .68 58 3 97312 .08 44011 .66 57 3 97580 .07 46426 .69 57 4 97317 .08 44051 .66 56 4 97584 .07 46467 .69 56 5 97322 .08 44090 .66 55 5 97589 .07 46508 .69 55 6 97326 .08 44130 .66 54 6 97593 O? 46550 .69 54 7 97331 .08 44169 .66 53 7 97597 .07 46591 .69 53 8 97335 .08 AQ 44209 .66 52 8 97602 .07 0-7 46632 .69 - An 52 9 97340 .Uo 44248 .00 51 9 97606 7 46673 .oy 51 10 11 97344 9.97349 .08 .08 44288 10.44327 .66 .66 50 49 10 11 97610 9.97615 .07 .07 46715 10.46756 .69 .69 50 49 12 97353 .08 44367 .66 48 12 97619 .07 46798 .69 48 13 14 97358 97363 .08 .08 44407 ' 44446 'JJ 47 46 13 14 97623 97628 .07 .07 46839 46880 .69 .69 47 46 15 97367 .08 444861 '55 45 15 97632 .07 46922 .69 45 16 97372 .08 AQ 44526 .00 AA 44 16 97636 .07 AT 46963 .69 44 17 97376 .Uo 44566 .00 43 17 97640 .U7 47005 .OJ 43 18 97381 .08 AQ 44605 .66 //> 42 18 97645 .07 47047 .69 A A 42 19 97385 .Mo 44645 .On 41 19 97649 .07 47088 .oy 41 20 21 97390 9.97394 .08 .08 AQ 44685 10.44725 .66 .66 AT 40 39 20 21 97653 9.97657 .07 .07 47130 10.47171 .69 .69 40 39^ 22 97399 .Uo 44765 .Of 38 22 97662 .07 47213 .70 38 23 97403 .08 44805 .67 37 23 97666 .07 47255 ./O 37 24 25 97408 97412 .07 .07 44845 44885 .67 .67 31) 35 24 25 97670 97674 .07 .07 47297 47339 .70 .70 36 35 26 97417 .07 07 44925 .67 67 34 26 97679 .07 47380 .70 tf A 34 27 97421 U t 44965 I 33 27 97683 .07 47422 . 1 U 33 28 97426 .07 45005 .67 32 28 97687 .07 47464 .70 32 29 97430 .07 45045 .67 6^ 31 29 97691 .07 A*7 47506 .70 31 30 97435 .07 45085 i 30 30 97696 .07 47548 .70 30 31 9.97439 .07 10.45125 .67 29 31 9.97700 .07 10.47590 .70 29 32 33 97444 97448 .07 .07 AT 45165 45206 .67 .67 28 27 32 33 97704 97708 .07 .07 47632 47674 .70 .70 28 27 34 97453 VI 45246 "' 26 34 97713 t 47716 .70 26 35 97457 .07 45286 .67 25 35 97717 .07 47758 .70 25 36 37 97461 97466 .07 .07 45327 '"' 45367 '!; 24 23 36 37 97721 97725 .07 .07 47800 47843 .70 .70 24 23 38 97470 .07 07 45407 'r' 22 38 97729 07 47885 .70 70 22 39 97475 " / 45448 'H 21 39 97734 .VI 47927 . l() 21 40 97479 07 45488 J 20 40 97738 .07 /IT 47969 .70 20 41 9.97484 U t 10.45529 rJJ 19 41 9.97742 .U I 10.48012 .71 19 42 97488 .07 455691 'JJ 18 42 97746 .07 48054 .71 18 43 97492 .07 45610! -Jo 17 43 97750 .07 48097 .71 17 44 97497 .07 456501 16 44 97754 .07 48139 .7] 16 45 97501 .07 AT 45691 .00 15 45 97759 .07 48181 .71 15 46 97506 Ml AT 45731 AQ 14 46 97763 .07 07 48224 .71 71 14 47 97510 .11 4 AT 45772 Oo 13 47 97767 .U t 48266 . 1 13 48 97515 .0* AT 45813 .68 12 48 97771 .07 48309 17 1 12 49 97519 .U 1 45853 .68 11 49 97775 .07 48352 .71 11 50 97523 .07 07 45894 .68 AQ 10 50 97779 J. 48394 .71 7"1 10 51 9.97528 U t 10.45935 .Oo 9 51 9.97784 '\ 10.48437 1 i 9 52 97532 .07 AT 45975 .68 8 52 97788 'AT 48480 .71 8 53 97536 .0 t AT 46016 .68 AQ 7 53 97792 .07 48522 .71 7 54 55 97541 97545 .U t .07 46057 46098 .Oo .68 6 5 54 55 97796 97800 t .07 48565 48608 .71 .71 6 5 56 57 97550 97554 .07 .07 46139 46180 .68 .68 4 3 56 57 97804 97808 .07 .07 48651 48694 .71 .71 4 3 58 97558 .07 46221 .68 2 58 97812 .07 48736 .72 2 59 97563 .07 07 46262 ' 1 59 97817 .07 ft 7 48779 .72 1 60 97567 *U ( 46303 ' 60 97821 U t 48822 ^ 31. Cosine. Dl" Cotang. I Dl" 31 M. Cosine. Dl" Cotang. Dl" ~M7 19 C 63 72 TABLE IV. LOGARITHMIC M. Sine. D." Tang. 1)1" M. 31. Sine. Dl" Tan*. Di" 31. 1 9.97821 97825 0.07 10.48822 48865 0.72 60 59 9.98060 98063 0.06 10.51466 51511 0.75 60 59 2 97829 .07 48908 .72 58 2 98067! - 51557 .75 58 4 5 97833 97837 97841 .07 .07 .07 48952 48995 49038 .72 .72 .72 57 56 55 3 4 5 98071 98075 98079 .ub .06 .06 51602 51647 51693 .75 .76 .76 57 56 55 6 97845 .07 .07 4906! .72 54 6 98083 .06 OH 5 1738 .76 54 7 97849 49124 . ^ 53 7 98087 .UO 51783 ' 53 8 97S53 .07 .07 49167 .72 7.) 52 8 98090 .06 or. 51829 .76 7fi 52 9 10 97857 97861 .*07 07 49211 49254 . I Z .72 TO 51 50 9 10 98094 98098 UO .06 AC 51874 51920 . i .76 ft G 51 50 11 9.97866 / .07 10.49297 ./ L 79 49 11 9.98102 Do M, 10.51965 .7o 49 12 97870 49341 . i Z 48 12 98106 UO 52011 .10 48 13 97874 .07 (17 49384 .72 7O 47 IS 981 10 .06 52057 .76 7 A 47 14 97878 .0 t 49428 ,t L 46 14 98113 .06 52103 . / 46 15 97882 .07 07 49471 .72 45 15 98117 .06 AA 52148 .76 7fi 45 16 97886 .U i 49515 ,73 44 16 98121 .UO 52194 . * 44 17 97890 .07 .07 49558 43 17 98125 .06 52240 .76 76 43 18 97894 .07 49602 73 42 18 98129 OH 52286 i O .77 42 19 97898 07 49645 41 19 98132 UO (\K 52332 41 20 97902 .U t 49689 .' 40 20 98136 UO 52378 'l- 40 21 9.97906 .07 .07 10.49733 .73 73 39 21 9.98140 .06 06 10.52424 .'' i 39 22 97910 49777 38 22 98144 .UO 52470 ' 38 2.5 97914 .07 07 49820 .73 37 23 98147 .06 06 52516 .77 37 24 97918 U i 49864 .' * 36 24 98151 52562 . it 36 25 97922 .07 49908 .73 35 25 98155 .06 52608 .77 35 26 97926 .07 49952 .73 34 26 98159 .06 52654 .77 34 27 979:50 .07 49996 .73 S3 27 98162 .06 A A 52701 .77 77 33 28 29 97934 97938 .*07 07 50040 50084 !73 32 31 28 29 98166 98170 .UO .06 n/ 52747 52793 . I ' .77 77 32 31 30 31 97942 9.97946 i .07 50128 10.50172 '.?* 30 29 30 31 98174 9.98177 Do .06 52840 10.52886 t .77 30 29 32 97950 .07 A7 50216 .74 28 32 98181 .06 AA 52932 .77 77 28 33 97954 .0 1 50260 .74 27 33 98185 .lit) 52979 t 27 35 36 37 97958 97962 97966 97970 .07 .07 .07 .07 50304 50348 5039:5 50437 .74 .74 .74 .74 26 25 24 23 34 35 36 37 98189 98192 98196 98200 .06 .06 .06 .06 53025 53072 53U9 53165 '.78 .78 .78 26 25 24 23 38 97974 .07 50481 .74 22 38 98204 :::: 53212 '" 22 39 97978 .07 50526 .74 21 39 98207 .UO 53259 .78 21 40 41 97982 9.97986 .07 .07 50570 10.50615 .74 .74 20 19 40 41 98211 9.98215 .06 .06 53306 10.53352 !78 20 19 42 97989 .07 50659 .74 18 42 98218 .06 53399 .to 18 43 97993 .07 50704 .74 17 43 98222 "" 53446 17 44 97997 .07 50748 .74 16 44 98226 '}!!! 53493 .78 16 45 46 98001 98005 .07 .07 50793 50837 .74 .74 15 14 45 46 98229 98233 .UO .06 63540 63587 .78 .78 15 14 47 98009 .07 50882 .74 13 47 98237 ;;;; 53634 .7M 13 48 98013 .07 50927 .75 12 48 98240 ' 53681 .79 12 49 98017 .07 50971 'I.' 11 49 98244 J 53729 .79 11 50 98021 .07 51016 .< j 10 50 98248, ",! 53776 .79 10 51 9.98025 .07 10.51061 .75 9 51 9.98251 .UO t\t\ 10.53823 .79 70 9 52 98029 .06 51106 . t ,> 8 52 98255 UO 53870 . i y 8 53 98032 .1)6 51151 .75 7 53 98259 .06 53918 .79 ~ t 7 54 98036 .06 51196 .75 6 54 98262 .06 53965 .79 6 55 98M40 .06 51241 .75 5 55 98266 .06 54013 .79 5 56 98044 .06 51286 .75 4 56 98270 .06 54060 .79 4 57 98048 .06 51331 .75 3 57 98273 .06 54108 "I 3 58 98052 .06 51376 .75 2 58 98277 .06 54155 . t y rrf\ 2 59 98056 .06 51421 .75 1 59 98281 .06 5420H .79 it\ 1 U 98060 .06 51461 .75 60 98284 .06 54250 . < y M. Cosine. HI" Cotang. w> M. M. Cosine. ~D~i" Cotang "DT M. 17 SINES AND TANGENTS. 75 C M. Sine. I Dl" Tang. I Dl" M. M. |- Sine. Dl" Tanx. Dl" M. 1 J.98284 98288 0.06 A(> 10,54250 54298 0.80 60 59 1 1 9.98494 98498 0.06 06 10.57195 57245 0.84 fiO 59 2 3 4 5 98291 98295 98299 98302 .uo .06 .06 .06 54346 54394 54441 54489 .80 .80 .80 OA 58 57 56 55 2 3 4 5 98501 98505 98508 98511 .*06 .06 .06 Ofi 57296 57347 57397 57448 M .85 .85 85 58 57 56 55 6 7 98306 98309 '.06 54537 54585 oU .80 54 53 6 7 98515 98518 .uo .06 57499 57550 !85 QC 54 53 8 98313 .06 54633 .80 52 8 98521 .06 AC 57601 .00 52 9 10 98317 98320 !o6 54681 54729 '.80 51 50 9 10 98525 98528 .uo .06 57652 57703 !85 OC 51 50 11 9.98324 .06 10.54778 .80 P.O 49 It 9.98531 06 10.57754 .80 85 49 12 98327 .Uo 54826 8U 48 12 98535 .Uv) 57805 48 13 98331 .06 AC 54874 .80 47 13 98538 .06 06 57856 85 47 14 98334 .UO 54922 .80 46 14 98541 UO 57907 46 15 98338 .06 54971 .81 1 45 15 98545 06 57959 *86 45 16 17 98342 98345 !06 55019 55067 .0 I .81 - 1 44 43 16 17 98548 98551 UO .06 ofi 58010 58061 !86 86 44 43 18 19 98349 98352 .06 55116 55164 O I .81 42 41 18 19 98555 98558 UO .06 AC 58113 58164 [86 CO 42 41 20 98356 .06 Ofi 55213 .81 Q I 40 20 98561 .Uo nfi 58216 .80 86 4 21 9.1)8359 Uo 10.55262 .8 I 39 21 9.98565 UO 10.58267 QC 39 22 98363 .06 55310 .81 Q 1 38 22 98568 .06 A t 58319 .80 38 23 98366 .uo ~ 55359 .8 1 Q 1 37 23 98571 .UO AR 58371 *86 37 24 98370 .06 55408 .81 36 24 98574 .UO 58422 36 25 98373 .06 55456 .81 Q 1 35 25 98578 .05 At 58474 86 35 26 27 98377 98381 '.06 Ofi 55505 55554 .01 .81 34 33 26 27 98581 98584 .UO .05 .05 58526 58578 .00 .87 87 34 33 28 98384 .uo 55603 ** 32 28 98588 58630 OT 32 29 98388 .06 Ofi 55652 .82 31 29 98591 .05 .05 58682 .87 87 31 30 31 32 33 34 98391 9.98395 98398 98402 98405 .uo .06 .06 .06 .06 55701 10.55750 55799 55849 55898 !82 .82 .82 .82 QO 30 29 28 27 26 30 31 32 33 34 98594 9.98597 98601 98604 98607 .05 .05 .05 .05 58734 10.58786 58839 58891 58943 '.87 .87 .87 .87 07 30 29 28 27 26 35 98409 Ofi 55947 .06 OO 25 35 98610 .05 58995 .01 25 36 98412 .uo Ofi 55996 ' 24 36 98614 .05 59048 .87 24 37 98415 .uo 56046 .V* 23 37 98617 t\~ 59100 DO 23 38 39 98419 98422 .1)6 56095 56145 .82 .82 22 21 38 39 98620 98623 .Uo .05 - 59153 59205 .88 .88 QO 22 21 40 98426 .06 Ofi 56194 ) 20 40 98627 *? 59258 .OO QQ 20 41 42 43 9.98429 98433 98436 UO .06 .06 10.56244 56293 56343 .00 .83 .83 19 18 11 41 42 43 9.98630 98633 98636 J-J 10.59311 f\ 59364 J? 59416 .00 .88 .88 QQ 19 18 17 44 9844oi -;; 56393 *? 16 44 98640 J? 59469 .80 16 45 98443 i ' 56442 .00 15 45 98643 59522 .88 UCJ 15 46 98447 ' 56492 .83 14 46 98646 At 59575 .88 00 14 47 48 98450; ' 98453 ' 56542' '^ 56592 rj 13 12 47 48 98649 98652 .UO .05 59628 59681 .88 .88 w ( . 13 12 49 98457 .uo Ofi 56642 i ?:: 11 49 98656 .05 59734 .ay on 11 50 98460 .uo Oft 56692 'JJ 10 50 98659 A?! 597881 -^ 10 51 9.98464 IJt 10.56742 *' 9 51 9.98662 JJ 10.59841 Q{\ 9 52 98467 .06 56792 '* 8 52 98665 II . 59894 .sy 8 53 9847 .Oe 56842 **} 7 53 98668 "? 59948 .89 7 54 9S474 .06 56892 i ~1 6 54 98671 '? 60001 .8.9 Oft 6 55 98477 .06 56943 *} 5 55 9S675 ~f. 60055 .8iJ 5 56 98481 .06 56993 ~j 4 56 98678 J 60108 ,8S 4 57 98484 .Of AC 57043 ! -] 3 57 98681 >Jr 60162 .8^ 3 58 98488 .Uo 57094 ~1 2 58 98684 J? 60215 .8^ 2 59 98491 .Ut 57144 *". 1 59 98687 JJ 60269 iVt 1 60 98494 .06 57195 .84 60 98690 fi0323 .9( .M. Cosine. Dl" Cotmig. Dl" M. M. Cosiiu'. Dl" Cotang. Dl" M. 13 C 76 C TABLE IV. LOGARITHMIC 77 M. Sine. Dl" Tang. Di': M. M. Sine. Dl" Tang. Dl" M. 1 9.98690 98694 0.05 10.60323 60377 0.90 60 59 1 9.98872 98875 0.05 A' 10.63664 63721 0.96 60 59 2 98697 .05 60431 .90 58 2 98878 .Oo 63779 .96 58 3 98700 .05 60485 .90 57 3 98881 .05 63837 .96 57 4 98703 .05 60539 .90 56 4 98884 .05 63895 .96 56 5 98706 .05 60593 .90 55 5 98887 .05 63953 .97 55 6 98709 .05 60647 .90 54 6 9S890 .05 64011 .97 54 7 98712 .05 AX 60701 .90 /\A 53 7 98893 .05 Ar 64069 .97 53 8 9 10 98715 98719 98722 .Uo .05 .05 60755 60810 60864 .yu .91 .91 52 51 50 8 9 10 98896 98898 98901 I.Oo .05 .05 64127 64185 64243 .97 .97 .97 52 51 50 11 12 9.98725 98728 .05 .05 10.60918 60973 .91 .91 49 48 11 12 9.98904 98907 .05 .05 10.64302 64360 .97 '11 49 48 13 14 98731 98734 .05 .05 61028 61082 .91 .91 47 46 13 14 98910 98913 .05 .05 64419 64477 ,98 .98 47 46 15 98737 .05 61137 .91 45 15 98916 .05 64536 .98 45 16 98740 .05 A " 61192 .91 44 16 98919 .05 64595 .98 44 17 98743 .Uo 61246 .91 43 17 98921 .05 64653 .98 43 18 98746 .05 A c 61301 .91 42 18 98924 .05 64712 .98 42 19 98750 .U5 61356 .92 41 19 98927 .05 64771 .98 41 20 98753 .05 61411 .92 40 20 98930 .05 64830 .98 40 21 22 9.98756 98759 .05 .05 A - 10.61466 61521 .92 .92 no 39 38 21 22 9.98933 98936 .05 .05 A - 10.64889 64949 .98 .99 39 38 23 24 25 98762 98765 98768 UO .05 .05 61577 61632 61687 .2 .92 .92 37 36 35 23 24 25 98938 98941 98944 .1)0 .05 .05 65008 65067 65126 .99 .99 .99 37 36 35 26 98771 .05 61743 .92 34 26 98947 .05 65186 .99 34 27 98774 .05 61798 .92 33 27 98950 .05 65245 .99 33 28 98777 .05 61853 .93 32 28 98953 .05 65305 .99 32 29 98780 .05 61909 .93 31 29 98955 .05 65365 .99 31 30 98783 .05 61965 .93 30 30 98958 .05 65424 .00 30 31 9.98786 .05 Ar 10.62020 .93 no 29 31 9.98961 .05 A r. 10.65484 .00 29 32 33 98789 98792 .UO .05 AX 62076 62132 .\?6 .93 QO 28 27 32 33 98964 98967 .UO .05 A P. 65544 65604 .00 .00 28 27 34 98795 .Uo 62188 .V6 26 34 98969 .Uo 65664 .00 26 35 98798 .05 Ar 62244 .93 25 35 98972 .05 65724 .00 25 36 98801 .Uo 62300 .93 24 36 98975 .05 65785 .00 24 37 98S04 .05 A- 62356 .93 o 4 23 37 98978 .05 A C 65845 .00 A1 23 38 98807 .UO A H 62412 .4 22 38 98980 .UO 65905 Ul 22 39 98810 .UO A Z. 62468 .94 f 1 4 21 39 98983 .05 A P. 65966 .01 A1 21 40 ' 98813 .UO A " 62524 .V4 20 40 98986 .Uo 66026 .01 20 41 9.98816 .Uo 10.62581 .94 19 41 9.98989 .05 10.66087 .01 19 42 98819 .05 62637 .94 18 42 98991 .05 66147 .01 18 43 44 98822 98825 .05 .05 62694 62750 .94 .94 17 16 43 44 98994 98997 .05 .05 66208 66269 .01 .01 17 16 45 98828 .05 0- 62807 .94 CIA 15 45 99000 .05 AT 66330 .01 AO 15 46 47 98831 98834 j .05 A 62863 62920 .4 .95 14 13 46 47 99002 99005 .UO .05 f\- 66391 66452 .02 .02 AO 14 13 48 98837 .1)0 AC 62977 .95 Ar 12 48 99008 Uo A C 66513 .02 no 12 49 50 51 52 98840 98843 9.98846 98849 .MO .05 .05 .05 63034 63091 10.63148 63205 .yo .95 .95 .95 11 10 9 8 49 50 51 52 90011 99013 9.99016 99019 *UD .05 .05 .05 66574 66635 10.66697 66758 .02 .02 .02 .02 11 10 9 8 53 98852 .05 A - 63262 .95 f\ - 7 53 99022 .05 A X 66820 .03 AO 7 54 55 98855 98858 .Uo .05 63319 63376 .yo .95 6 5 54 55 99024 99027 .UO .05 66881 60943 .Uo 1.03 6 5 56 98861 .05 A 63434 .96 4 56 99030 .05 A - 67005 .03 AO 4 '57 98864 .UO 63491 3 57 99032 .UO 67067 Uo 3 58 98867 .05 63548 .96 2 58 99035 .04 67128 .03 2 59 60 98869 98872 .05 .05 63606 63G64 .96 .96 1 59 60 99038 99040 .04 .04 67190 67253 1.03 1.03 1 M. Cosine. Dl" Cotans?. Dl" M. .M. Cosine-. Dl" Cotanjr. Dl" II. 13 G6 12 78 SINES AND TANGENTS. 79 31. Siue. Dl" Tang. Dl" 31. 31. Sine. Dl" Tang. Dl" 31. 1 9.99040 99043 0.04 10.67253 67315 1.04 60 59 1 9.99195 99197 0.04 10.71135 71202 .13 60 59 2 99046 .04 67377 1.04 58 2 99200 .04 71270 .13 58 3 99048 .04 67439 1.04 57 3 99202 .04 71338 .13 57 4 99051 .04 67502 1.04 56 4 99204 .04 71405 1.13 56 5 99054 .04 67564 1.04 55 5 99207 .04 71473 1.13 55 6 99056 .04 67627 1.04 54 6 99209 .04 71541 1.13 54 7 99059 .04 67689 1.04 53 7 99212 .04 71609 1.13 53 8 99062 04 67752 1.05 52 8 99214 .04 71677 1.14 52 9 99064 .04 67815 .05 A 51 9 99217 .04 A 1 71746 1.14 11 \ 51 10 99067 .04 67878 .00 50 10 99219 .04 71814 .14 50 11 9.99070 .04 10.67941 .05 49 11 9.99221 .04 10.71883 1.14 49 12 99072 .04 68004 .05 48 12 99224 .04 71951 1.14 48 13 99075 .04 68067 .05 47 13 99226 .04 72020 1.14 47 14 99078 .04 68130 .05 46 14 99229 .04 72089 1.15 46 15 99080 .04 68194 .06 45 15 99231 .04 72158 1.15 45 16 99083 .04 68257 .06 44 16 99233 .04 72227 1.15 44 17 99086 .04 68321 .06 43 17 99236 .04 72296 1.15 43 18 99088 .04 68384 1.06 42 18 99238 .04 72365 1.15 42 19 99091 .04 68448 1.06 41 19 99241 .04 72434 1.16 41 20 21 22 99093 9.99096 99099 .04 .04 .04 68511 10.68575 68639 1.06 1.06 1.07 40 39 38 20 21 22 99243 9.99245 99248 .04 .04 .04 A A 72504 10.72573 72643 1.16 1.16 1/16 40 39 38 23 99101 .04 68703 .07 37 23 99250 .04 72712 1.16 37 24 25 99104 99106 .04 .04 68767 68832 1.07 1.07 1A>7 36 35 24 25 99252 99255 .04 .04 A/I 72782 72852 1.16 1.17 36 35 26 99109 .04 68896 .07 34 26 99257 .04 72922 .17 34 27 99112 .04 68960 1.07 33 27 99260 .04 A/I 72992 1.17 33 28 99114 04 69025 1.07 1 AQ 32 28 99262 .04 A f 73063 1.17 11 *7 32 29 99117 .04 69089 .08 31 29 99264 .04 73133 .17 31 30 31 32 33 ."54 99119 9.99122 99124 99127 99130 .04 .04 .04 .04 .04 69154 10.69218 69283 69348 69413 1.08 1.08 1.08 1.08 1.08 30 29 28 27 26 30 31 32 33 34 99267 9.99269 99271 99274 99276 .04 .04 .04 .04 .04 73203 10.73274 73345 73415 73486 1.17 1.18 1.18 1.18 1.18 30 29 28 27 26 35 99132 .04 69478 1 .08 25 35 99278 .04 73557 1.18 25 36 37 38 39 99135 99137 99140 99142 04 .04 .04 .04 69543 69609 69674 69739 1.09 1.09 1.09 1.09 24 23 22 21 36 37 38 39 99281 99283 99285 99288 .04 .04 .04 .04 73628 73699 73771 73842 1.19 1.19 1.19 1.19 24 23 22 21 40 41 42 43 44 99145 9.99147 99150 99152 99155 .04 .04 .04 .04 .04 A /I 69805 10.69870 69936 70002 70068 1.09 1.09 1.10 1.10 1.10 11 1\ 20 19 18 17 16 40 41 42 43 44 99290 9.99292 99294 99297 99299 .04 .04 .04 .04 .04 A 1 73914 10.73985 74057 74129 74201 1.19 1.19 1.20 1.20 1.20 1*iii 20 19 18 17 16 45 99157 04 70134 .10 15 45 99301 .04 74273 .20 15 46 99160 .04 A 1 70200 1.10 14 46 99304 .04 A 1 74345 1.20 101 14 47 99162 04 70266 1 A 13 47 &9306 .04 74418 .21 13 48 99165 .04 70332 .10 12 48 99308 .04 74490 1.21 12 49 99167 .04 70399 .11 11 49 99310 .04 74563 1.21 11 50 99170 .04 A/l 70465 .1 ] 1 -1 10 50 99313 .04 A/I 74635 1.21 101 10 51 9.99172 .04 10.70532 .1 1 9 51 9.99315 .04 10.74708 .21 9 52 99175 .04 f>4 70598 .11 8 52 99317 04 74781 1.21 199 8 53 99177 U4 A/I 70665 1 1 7 53 99319 .04 A ,1 74854 ..2.6 7 54 99180 .04 .04 70732 . 1 1 .12 6 54 99322 .04 04 74927 1 99 6 55 99182 70799 5 55 99324 U-i 75000 1.2 5 56 57 99185 99187 .04 70866 70933 .12 .12 4 3 56 57 99326 9932r : .04 .04 75074 75147 1.22 1.22 4 3 58 99190 .04 A/I 71000 .12 11 > 2 58 99331 .04 A/I 75221 1.23 100 2 59 60 99192 99195 .04 .04 71067 71135 .1 L L12 1 59 60 99333 99335 .04 .04 75294 75368 20 1.23 1 M. Cosine. Dl" Cotang. Dl" M. M. Cosine. Dl" Cotang. Dl" M. 11 67 10 80 TABLE IV. LOGARITHMIC 81 M. Sine. Dl" Tang. Dl" M. M. Sine. Dl" Tang. Dl" M. 1 9.99335 99337 0.04 10.75368 75442 1.23 60 59 1 9.99462 99464 0.03 10.80029 80111 1.36 60 59 2 99340 .04 75516 .23 58 2 99466 .03 80193 1.37 58 3 99342 .04 75590 .24 57 3 994G8 .03 80275 1.37 57 4 99344 .04 75665 .24 56 4 99470 .03 80357 1.37 56 5 99346 .04 n 4 75739 .24 55 5 99472 .03 no 80439 1.37 1.JQ 55 6 99348 .04 75814 .24 54 6 99474 .Uo 80522 .OO 54 7 8 99351 99353 .04 .04 75888 75963 .24 .25 53 52 7 8 99476 99478 .03 .03 80605 80688 1.38 1.38 53 52 9 99355 .04 (\A 76038 .25 o ^ 51 9 99480 .03 no 80771 1.38 51 10 99357 .04 76113 .20 50 10 99482 .Uo 80854 1 .39 50 11 9.99359 .04 10.76188 .25 49 11 9.99484 .03 10.80937 1.39 49 12 99362 .04 f\4 76263 .25 48 12 99486 .03 MO 81021 1.39 Ion 48 13 14 15 99364 99366 99368 .04 .04 .04 76339 76414 76490 !26 .26 47 46 45 13 14 15 99488 99490 99492 .Uo .03 .03 81104 81188 81272 .oy 1.40 1.40 47 46 45 16 99370 .04 76565 1.26 44 16 99494 .03 81356 1.40 44 17 18 99372 99375 .04 .04 76641 76717 1.26 1.27 43 42 17 18 99495 99497 .03 .03 81440 81525 1.40 1.41 43 42 19 99377 .04 76794 1.27 41 19 99499 .03 81609 1.41 41 20 21 22 99379 9.99381 99383 .04 .04 .04 76870 10.76946 77023 1.27 1.27 1.28 40 39 38 20 21 22 99501 9.99503 99505 .03 .03 .03 81694 10.81779 81864 1.41 1.41 1.42 40 39 38 23 24 99385 99388 .04 .04 77099 77176 1.28 1.28 37 36 23 24 99507 99509 .03 .03 81949 82035 1.42 1.42 37 36 25 26 27 99390 99392 99394 .04 .04 .04 77253 77330 77407 1.28 1.28 .29 35 34 33 25 26 27 99511 99513 99515 .03 .03 .03 82120 82206 82292 1.43 1.43 1.43 35 34 33 28 99396 .04 77484 .29 32 28 99517 .03 82378 1.43 32 29 30 99398 99400 .04 .04 77562 77639 .29 .29 31 30 29 30 99518 99520 .03 .03 82464 82550 1.44 1.44 31 30 31 9.99402 .04 10.77717 : .29 29 31 9.99522 .03 10.82637 1.44 29 32 99404 .04 n l 77795 .30 on 28 32 99524 .03 no 82723 1.44 14 C 28 33 99407 .U4 77873 .oU 27 33 99526 Uo 82810 .40 27 34 35 36 99409 99411 9941 3 .04 .03 .03 77951 78029 78107 .30 .30 .31 26 25 24 34 35 36 99528 99530 99532 .03 .03 .03 82897 82984 83072 1.45 1.45 1.46 26 25 24 37 38 99415 99417 .03 .03 78186 78264 .31 .31 23 22 37 38 99533 99535 .03 .03 83159 83247 1.46 1.46 23 22 39 99419 .03 78343 .31 21 39 99537 .03 83335 1.46 21 40 41 99421 9.99423 .03 .03 78422 10.78501 .31 .32 20 19 40 41 99539 9.99541 .03 .03 83423 10.83511 1.47 1.47 20 19 42 99425 .03 78580 .32 18 42 99543 .03 83599 ] .47 18 43 99427 .03 78659 .32 17 43 99545 .03 83688 1.48 14 Q 17 44 99429 .03 78739 .32 16 44 99546 .03 83776 .48 16 45 99432 .03 no 78818 .33 OO 15 45 99548 .03 OO 83865 1.48 1 4ft 15 46 47 99434 99436 .Uo .03 78898 78978 .00 .33 14 13 46 47 99550 99552 o .03 83954 84044 1.48 1.49 14 13 48 99438 .03 79058 .33 12 48 99554 .03 84133 1.49 12 49 50 99440 99442 .03 .03 79138 79218 .34 .34 11 10 49 50 99556 99557 .03 .03 84223 84312 1.50 1c n 11 10 51 9.99444 .03 no 10.79299 .34 9 51 9.99559 .03 .03 10.84402 .00 1.50 9 52 99446 Uo 79379 o4 8 52 99561 84492 8 53 99448 .03 no 79460 .34 Q r 7 53 99563 .03 no 84583 1.51 -i ci 7 54 99450 .Uo no 79541 .00 6 54 99565 .Uo .03 84673 1.01 1.51 6 55 99452 .Uo 79622 .00 5 55 99566 84764 5 56 57 99454 99456 .03 .03 79703 79784 .35 .35 4 3 56 57 99568 99570 .03 .03 84855 84946 1.51 1.52 4 3 58 99458 .03 no 79866 .36 i ^fi 2 58 99572 .03 no 85037 1.52 1 i2 2 59 99460 .Uo 79947 i .00 1 59 99574 .UO no 85128 J.. J-6 1 60 99462 .03 80029 1 .36 60 99575 .Uo 85220 1.53 ~M7 Cosine. Dl" Coiling. Dl" M. M. Cosine. "BT 7 Col R rig. Dl" M. 9 SINES AND TANGENTS. 88 M. Sine. Dl" Tang, i Dl" BI. M. Sine. Dl" Tang. Dl" M. 1 2 3 4 5 6 9.99575 99577 99579 99581 99582 99584 99586 0.03 .03 .03 .03 .03 .03 10.85220 85312 85403 85496 i 85588 | 85680 85773 1.53 1.53 1.54 1.54 1.54 1.55 60 59 58 57 56 55 54 1 2 3 4 5 6 9.99675 99677 99678 99680 99681 99683 99684 0.03 .03 .03 .03 .03 .03 10.91086 91190 91295 91400 91505 91611 91717 1.74 1.75 1.75 1.76 1.76 1.76 60 59 58 57 56 55 54 7 99588 .03 85866 1.55 53 7 99686 .03 91823 1.77 53 8 99589 .03 AO 85959 1.55 Ire 52 8 99687 .03 AO 91929 L.77 1 *7Q 52 9 10 99591 99593 .Uo .03 86052 86146 .00 1.56 -I C I* 51 50 9 10 99689 99690 .Uo .03 AO 92036 92142 L.7o 1.78 1 *7Q 51 50 11 9.99595 .03 10.86239 i'?" 49 11 9.99692 .Ui 10.92249 L.7o 49 12 13 14 99596 99598 99600 .03 .03 .03 86333 86427 86522 l.OO 1.57 1.57 48 47 46 12 13 14 99693 99695 99696 .03 .03 .03 92357 92464 92572 L.79 1.79 1.80 48 47 46 15 99601 .03 86616 If'?' 45 15 99698 .02 92680 1.80 45 16 99603 .03 86711 rjg 44 16 99699 .02 92789 1.81 44 17 18 99605 99607 .03 .03 868061} ; 86901 ! !'_ 43 42 17 18 99701 99702 .02 .02 92897 93006 1.81 1.81 43 42 19 20 21 99608 99610 9.99612 .03 .03 .03 no, 86996 87091 10.87187 i.oy 1.59 1.59 i n 41 40 39 19 20 21 99704 99705 9.99707 .02 .02 .02 93115 93225 10.93334 1.82 1.82 1.83 1 ft^ 41 40 39 22 99613 .UO 87283 ;"'" 38 22 99708 .uz 93444 ioO 38 23 99615 .03 no 87379 i}' 37 23 99710 .02 09 93555 1.84 1 84. 37 24 25 99617 99618 .UO .03 AO 87475 jj-jr 87572 L 1 '^ 36 35 24 25 99711 99713 UZ .02 An 93665 93776 1 .OTC 1.85 In c 36 35 26 27 99620 99622 .Uo .03 87668 Hi 87765 |H1 34 33 26 27 99714 99716 .02 .02 93887 93998 3 A 1.86 | 28 99624 .03 03 87862 1.02 32 28 99717 .02 94110 1.86 IQfi 32 29 30 31 99625 99627 9.99629 '.03 .03 A-J 87960 88057 10.88155 1.63 1.63 1AO 31 30 29 29 30 31 99718 99720 9.99721 !02 .02 On 94222 94334 10.94447 .00 1.87 1.87 Ion 31 30 29 32 99630 .Uo 88253 .Do 28 32 99723 2 94559 .00 28 33 99632 .03 no 88351 1.64 l fid 27 33 99724 .02 09 94672 1.88 1QQ 27 34 35 99633 99635 .Uo .03 88449 88548 1 .04 1.64 26 25 34 35 99726 99727 2 .02 94786 94899 .oy 1.89 26 25 36 99637 .03 .03 88647 1.65 1C X 24 36 99728 .02 09 95013 1,90 1 Qft. 24 37 99638 88746 00 23 37 99730 / 95127 l.VU 23 38 39 99640 99642 .03 .03 03 88845 88944 1.65 1.66 Ififi 22 21 38 39 99731 99733 .02 .02 09 ^95242 95357 1.91 1.91 1 Q2 22 21 40 99643 89044 .DO 20 40 99734 L 95472 20 41 9.99645 .03 AO 10.89144 1.67 IrtW 19 41 9.99736 .02 On 10.95587 1.92 Too 19 42 99647 Uo AO 89244 .o/ 18 42 99737 i 95703 .y6 18 43 99648 .Uo .03 89344 1.67 1 fift 17 43 99738 ,02 09 95819 1.93 17 44 99650 89445 l.Oo 16 44 99740 z 95935 1.V4 16 45 99651 .03 89546 1.68 15 45 99741 .02 96052 1.94 15 46 99653 .03 89647 1.68 14 46 99742 .02 96168 1.95 14 47 99655 .03 v 89748 1.69 13 47 99744 .02 96286 1.95 13 48 99656 .03 A-> 89850 1.69 12 48 99745 .02 96403 1.96 12 49 9658 .Uo 89951 1.70 11 49 99747 .02 96521 1.96 11 50 99659 .03 90053 1.70 10 50 99748 ,02 96639 1.97 10 51 9.99661 .03 10.90155 1.70 9 51 9.99749 .0$ 10.96758 1.97 9 52 99663 .03 90258 1.71 8 52 99751 .02 96876 1.98 8 53 99664 .03 90360 1.71 7 53 99752 .02 96995 1.98 7 54 99666 .03 AO 90463 1.71 1- . 6 54 99753 .02 On 97115 1.99 6 55 99667 .Uo 90566 .72 5 55 99755 L 97234 2.00 5 56 99669 .03 90670 1.72 4 56 99756 .02 97355 2.00 4 57 99670 .03 90773 1,73 3 57 99757 .02 97475 2.01 3 58 99672 .03 90877 1.73 2 58 99759 .02 97596 2.01 2 59 60 99674 99675 .03 .03 90981 91086 1.73 1.74 1 59 60 99760 99761 .02 .02 97717 97838 2.02 2.02 1 IT Cosine. Dl" Cotang. Tvr~ M. M. Cosine. DP Cotang. DV M. 7 S. N. 40. TABLE IV. LOGARITHMIC M. Sine. Dl" Tang. Dl" M. 11. Sine. Dl" Tang. Dl" M. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 9.99761 99763 99764 99765 99767 99768 99769 99771 99772 99773 99775 9.99776 99777 99778 99780 99781 99782 0.02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 10.97838 97960 98082 98204 98327 98450 98573 98697 98821 98945 99070 10.99195 99321 99447 99573 99699 99826 2.03 2.03 2.04 2.04 2.05 2.06 2.06 2.07 2.07 2.08 2.09 2.09 2.10 2.10 2.11 2.12 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 J5 16 9.99834 99836 99837 99838 99839 99840 99841 99842 99843 99844 99845 9.99846 99847 99848 99850 99851 99852 0.02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 11.05805 05951 06097 06244 06391 06538 06687 06835 06984 07134 07284 11.07435 07586 07738 07890 08043 08197 2.43 2.44 2.45 2.45 2.46 2.47 2.48 2.49 2.50 2.50 2.51 2.52 2.53 2.54 2.55 2.56 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 17 18 19 99783 99785 99786 .02 .02 .02 99954 11.00081 00209 2.12 2.13 2.13 43 42 41 17 18 19 99853 99854 99855 .02 .02 .02 08350 08505 08660 2.56 2.57 2.58 43 42 41 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 99787 9.99788 99790 99791 99792 99793 99795 99796 99797 99798 99800 9.99801 99802 99803 99804 99806 99807 99808 99809 99810 99812 9.99813 99814 99815 99816 99817 99819 99820 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 09 00338 11.00466 00595 00725 00855 00985 01116 01247 01378 01510 01642 11.01775 01908 02041 02175 02309 02444 02579 02715 02850 02987 11.03123 03261 03398 03536 03675 03813 03953 2.14 2.15 2.15 2.16 2.16 2.17 2.18 2.18 2.19 2.20 2.20 2.21 2.22 2.22 2.23 2.24 2.24 2.25 2.26 2.26 2.27 2.28 2.29 2.29 2.30 2.31 2.31 2.32 200 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 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 99856 9.99857 99858 99859 99860 99861 99862 99863 99864 99865 99866 9.99867 99868 99869 99870 99871 99872 99873 99874 99875 99876 9.99877 99878 99879 99879 99880 99881 99882 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 0* 08815 11.08971 09128 09285 09443 09601 09760 09920 10080 10240 10402 11.10563 10726 10889 11052 11217 11382 11547 11713 11880 12047 11.12215 12384 12553 12723 12894 13065 13237 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82 2.83 2.84 2.85 2.87 2QO 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 48 99821 04092 12 48 99883 09 13409 2 on 12 49 50 99822 99823 .02 09 04233 04373 2.34 2 OR 11 10 49 50 99884 99885 .02 02 13583 13757 2.90 2Q1 11 10 51 52 9.99824 99825 .02 09 11.04514 04656 2.36 9 8 51 52 9.99886 99887 .02 02 11.13931 14107 2.92 2Q-J 9 8 53 99827 09 04798 9 ^7 7 53 99888 02 14283 2 95 7 54 55 56 57 99828 99829 99830 99831 .02 .02 .02 04940 05083 05227 05370 2.38 2.39 2.40 6 5 4 3 54 55 56 57 99889 99890 99891 99891 .02 .01 .01 01 14460 14637 14815 14994 2.96 2.97 2.98 6 5 4 3 58 59 60 99832 99833 99834 .02 .02 05515 05660 05805 .41 2.41 2.42 2 1 58 59 60 99892 99893 99894 .01 .01 15174 15354 15536 3.01 3.02 2 1 BI. Cosine. Dl" Cotang. Dl" 31. M. Cosine. Dl" Cotang. Dl" M. 70 86 C SINES AND TANGENTS. 87 M. | Sine. Dl" Tang. Dl" M. M. Sine. Dl" Tan-. Dl" M. 1 9.99894 99895 0.01 11.15536 15718 3.03 60 59 1 9.99940 99941 0.01 11.28060 28303 4.04 59 2 99896 .01 15900 J 58 2 99942 .01 28547 4.06 58 3 99897 .01 n i 16084 |:M 57 3 99942 .01 I1 1 28792 4.09 4-1 1 57 4 5 6 99898 '99898 99899 .U 1 .01 .01 16268 1 6453 16639 ).U 1 3.08 3.10 56 55 54 4 5 6 99943 99944 99944 .U I .01 .01 29038 29286 29535 .1 1 4.13 4.15 56 55 54 7 99900 .01 01 16825 ?'!J 53 7 99945 .01 01 29786 4.18 490 53 8 ' 99901 . U 1 17013 r!. 52 8 99946 .U 1 30038 .^U 52 9 99902 .01 17201 51 9 99946 .01 30292 4.23 51 10 99903 .01 1 7390 3.15 50 10 99947 .01 30547 4.25 50 11 9.99904 .01 11.17580 3.16 49 II 9.99948 .01 11.30804 4.28 49 12 99904 .01 A 1 17770 3.18 31 O 48 12 99948 .01 A 1 31062 4.30 4QQ 48 13 99905 .(> I 17962 . 1 ,1 47 13 99949 .0 1 31322 .GO 47 14 15 99906 99907 .01 .01 18154 18347 3.21 3.22 46 45 14 15 99949 99950 .01 .01 31583 31846 4.35 4.38 46 45 16 99908 .01 18541 3.23 44 16 99951 .01 32110 4.41 44 17 18 19 99909 99909 99910 .01 .01 .01 01 18736 18932 19128 3.25 3.26 3.28 o 90 43 42 41 17 18 19 99951 99952 99952 .01 .01 .01 01 32376 32644 32913 4.43 4.46 4.49 43 42 41 20 21 22 99911 9.99912 99913 V 1 .01 .01 1 19326 11.19524 19723 3^31 3:S 40 39 38 20 21 22 99953 9.99954 99954 U 1 .01 .01 01 33184 11.33457 33731 4!54 4.57 J. fiO 40 39 38 23 24 25 26 27 99913 99914 99915 99916 99917 !oi .01 .01 01 19924 20125 20327 20530 20734 3.35 3.37 3.38 3.40 > 41 37 36 35 34 33 23 24 25 26 27 99955 99955 99956 99956 99957 .(' 1 .01 .01 .01 .01 (II 34007 34285 34565 34846 35130 4:.OU 4.63 4.66 4.69 4.72 47^ 37 36 35 34 33 28 99917 20939 .> .4 I 32 28 99958 U 1 35415 . t 4*rn 32 29 30 31 32 99918 99919 9.99920 99920 '.01 .01 .01 01 21145 21351 11.21559 21768 3.43 3.45 3.46 3.48 ^50 31 30 29 28 29 30 31 32 99958 99959 9.99959 99960 .01 .01 .01 .01 01 35702 35991 11.36282 36574 .78 4.82 4.85 4.88 4Q-I 31 30 29 28 33 34 99921 99922 !oi 21978 22189 3.51 27 26 33 34 99960 99961 ."J 1 .01 36869 37166 . ' 1 4.95 27 26 35 36 37 38 39 40 41 42 99923 99923 99924 99925 99926 99926 9.99927 99928 .01 .01 .01 .01 .01 .01 .01 .01 01 22400 22613 22827 23042 23258 23475 11.23694 23913 3.53 3.55 3.57 3.58 3.60 3.62 3.64 3.65 25 24 23 22 21 20 19 18 35 36 37 38 39 40 41 42 99961 99962 99962 99963 99963 99964 9.99964 99965 .01 .01 .01 .01 .01 .01 .01 .01 01 37465 37766 38069 ^38374 38681 38991 11.39302 39616 4.98 5.02 5.05 5.09 5.12 5.16 5.19 5.23 t 97 25 24 23 22 21 20 19 18 43 44 45 99929 99929 99930 !oi .01 A1 24133 24355 24577 3^69 3.71 17 16 15 43 44 45 99966 99966 99967 V I .01 .01 39932 40251 40572 O.Z i 5.31 5.35 17 16 15 46 99931 .Ul 24801 *'? 14 46 99967 .01 40895 5.39 14 47 99932 .01 25026 IH* 13 47 99967 .01 41221 5.43 13 48 99932 .01 25252 6.1 I 12 48 99968 .01 41549 5.47 12 49 50 99933 99934 .01 .01 25479 25708 3.79 3.81 11 10 49 50 99968 99969 .01 .01 41879 42212 5.51 5.55 11 10 51 9.99934 .01 A 1 11.25937 3.83 30 r 9 51 9.99969 .01 11.42548 5.59 5R.A 9 52 99935 .U 1 26168 .oD 8 52 99970 .01 42886 .04 8 53 99936 .01 26400 3.87 7 53 99970 .01 43227 5.68 7 54 99936 .01 26634 3.89 6 54 99971 .01 43571 5.73 6 55 56 99937 99938 .01 .01 26868 27104 3.91 3.93 5 4 55 56 99971 99972 .01 .01 43917 44266 5.77 5.82 5 4 57 99938 .01 27341 3.95 3 57 99972 .01 44618 5.87 3 58 99939 .01 27580 3.97 2 58 99973 .01 44973 5.91 2 59 99940 .01 27819 3.40 1 59 99973 .01 45331 5.96 1 60 99940 .01 28060 4.02 60 99974 .01 45692 6.01 M. Cosine. Dl" Cotang. Dl" M. M. Cosine. Dl" Cotans. 1)1" M. 71 2 TABLE IV. SIXES AND TANGENTS. 89 M. Sine. Dl" Tang. Dl" M. M. Sine. Dl" Tang. 1)1" M. 9.99974 OA1 11.45692 C A 60 9.99993 i ftn . 11.75808 too 60 1 99974 .01 46055 b.Uo 59 1 99994 '" 76538 \L.L 59 2 3 99974 99975 .01 .01 46422 J' l i 46792 I'H 58 57 2 3 99994 99994 .UU4 .004 77280 )** 78036 J;- 58 57 4 99975 .01 47165. Jq: 56 4 99994 .003 78805. lir 56 5 99976 .01 47541 Sis 55 5 99994 .003 79587 10. U 55 6 99976 .01 47921 i R '.,Q 54 6 99995 .003 80384 13.3 54 7 99977 .01 A 1 48304 J*J5 53 7 99995 .003 81196 13.5 ]} Q 53 8 99977 .u L 48690 i J'jTJ 52 8 99995 .003 82024 >.O 52 9 99977 .01 49080 !^r 51 9 99995 .003 82867 14.1 51 10 99978 .01 49473 iJ'J? 50 10 99995 .003 83727 14.3 50 11 9.99978 .01 11.49870 49 11 9.99996 .003 11.84605 14.6 49 12 99979 .01 50271 MJ? 48 12 99996 .003 85500 14.9 48 13 14 15 99979 99979 99980 .01 .01 .01 50675 51083 51495 D./4 6.80 6.87 47 46 45 13 14 15 99996 99996 99996 .003 .003 .003 86415 87349 88304 15.2 15.6 15.9 47 46 45 16 99980 .01 51911 2' 44 16 99996 .003 89280 16.3 i a 44 17 99981 .01 52331 r22 43 17 99997 .003 90278 Ib.b 43 18 19 20 99981 99981 99982 .01 .01 .01 52755 53183 53615 42 41 40 18 19 20 99997 99997 99997 .003 .003 .002 91300 {!" 92347 \ ltA 93419 'I': 42 41 40 21 22 9.99982 99982 .01 .01 A 1 11.54052 '' 54493 ;?; 39 38 21 22 9.99997 99997 .002 .002 11.94519 |j|; 95647 \' 39 38 23 99983 .Ul 54939 \1 A * 37 23 99997 .002 96806 1 W.O 37 24 25 26 27 99983 99983 99984 99984 .01 .01 .01 .01 01 55389 '?" 55844 J'J 56304 ^ 56768 ,00 36 35 34 33 24 25 26 27 99998 99998 99998 99998 .002 .002 .002 .002 ffcJIQ 97996 |iJ-J W219;"' 4 12.00478 J{'" 9im-! 36 35 34 33 28 29 30 31 32 33 34 35 36 99984 99985 99985 9.99985 99986 99986 99986 99987 99987 U 1 .01 .01 .01 .01 .01 .01 .01 .01 01 57238 57713 58193 11.58679 59170 59666 60168 60677 61191 7.91 8.00 8.09 8.18 8.28 8.37 8.47 8.57 o f7 32 31 30 29 28 27 26 25 24 28 29 30 31 32 33 34 35 36 99998 99998 99998 9.99998 99999 99999 99999 99999 99999 UU^J .002 .002 .002 .002 .002 .002 .002 .002 An i 03111 04490 05914 12.07387 08911 10490 12129 13833 15606 4.O 23.0 23.7 24.5 25.4 26.3 27.3 28.4 29.5 on Q 32 31 30 29 28 27 26 25 24 37 38 39 40 41 42 43 44 45 99987 99988 99988 99988 9.99989 99989 99989 99989 99990 Ul .01 .005 .005 .005 .005 .005 .005 .005 61711 62238 62771 63311 11.63857 64410 64971 65539 66114 o.O i 8.78 8.88 8.99 9.11 9.22 9.34 9.46 9.59 23 22 21 20 19 18 17 16 15 37 38 39 40 41 42 43 44 45 99999 99999 99999 99999 9.99999 99999 99999 10.00000 00000 UU1 .001 .001 .001 .001 .001 .001 .001 .001 17454 19385 21405 23524 12.25752 28100 30582 33215 36018 Ou.o 32.2 33.7 35.3 37.1 39.1 41.4 43.9 46.7 23 22 21 20 19 18 17 16 15 46 47 99990 99990 .005 .004 66698 ig'85 14 13 46 47 00000 00000 .001 .001 39014 42233 49.9 53.6 14 13 48 49 50 51 52 53 54 55 56 57 58 59 99990 99991 99991 9.99991 99992 99992 99992 99992 99992 99993 99993 99993 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 67888 68495 69112 11.69737 70371 71014 71668 72331 73004 73688 74384 75090 S3 10.3 10.4 10.6 10.7 10.9 11.1 11.2 11.4 11.6 11.8 12 11 10 9 8 7 6 5 4 3 2 1 48 49 50 51 52 53 54 55 56 57 58 59 00000 00000 00000 10.00000 00000 00000 00000 00000 00000 00000 00000 00000 .001 .001 .001 .001 .000 .000 .000 .000 .000 .000 .000 45709 49488 53627 12.58203 63318 69118 75812 83730 93421 13.05915 23524 53627 57.9 63.0 69.0 76.3 85.3 96.7 112 132 162 208 294 502 12 11 10 9 8 7 6 5 4 3 2 1 60 99993 .004 75808 12.0 60 00000 Infinite. M. Cosine. Dl" Cot a n g. Dl" 31. M. Cosine. Dl" 1 Cotang. Dl" 31. 72 O 0-345' TRAVERSE TABLES. 8615'-90 C D, Lat, Dep, Lat, Dep. Lat. Dep. Lat. Dep. D. O' O 3 15' O SO' O 45' 1 2 3 4 5 6 7 8 9 10 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 1 .000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 .004 .009 .013 .018 022 !026 .031 .035 .039 .044 1 .000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 .009 .018 .026 .035 .044 .052 .061 .070 .079 .087 1.000 2.000 3.000 4.000 5.000 5.999 6.999 7.999 8.999 9.999 .013 .026 .039 .052 .065 .079 .092 .105 .118 .131 1 2 3 4 5 6 7 8 9 10 9O O' 89 45' 89 30' 89 15' 1 O' 1 15' 1 3<y 1 45' 1 2 3 4 5 6 7 8 9 10 1.000 2.000 3.000 3.999 4.999 5.999 6.999 7.999 8.999 9.999 .017 .035 .052 .070 .087 .105. .122 .140 .157 .174 1.000 2.000 2.999 3.999 4.999 5.999 6.998 7.998 8.998 9.998 .022 .044 .065 .087 .109 .131 .153 .175 .196 .218 1 .000 1.999 2.999 3.999 4.998 5.998 6.998 7.997 8.997 9.997 .026 .052 .079 .105 .131 .157 .183 .209 .236 .262 1 .000 1.999 2.999 3.998 4.998 5.997 6.997 7.996 8.996 9.995 .031 .061 .092 .122 153 .183 .214 .244 .275 .305 1 2 3 4 5 6 7 8 9 10 89 J O' 88 45' 88 3O' 88 15' 2 0' 2 15' 2 30' *> 45' 1 2 3 4 5 6 7 8 9 10 .999 1.999 2.998 3.998 4.997 5.996 6.996 7.995 8.995 9.994 .035 .070 .105 .140 .174 209 .244 .279 .314 .349 .999 1.999 2.998 3.997 4.996 5.995 6.995 7.994 8.993 9.992 .039 .079 .118 .157 .196 .236 .275 .314 .353 .393 .999 1 .998 2.997 3.996 4.995 5.994 6.993 7.99'2 8.991 9.990 .044 .087 .131 .174 .218 .262 .305 .349 .393 .436 .999 1.998 2.997 3.995 4.994 5.993 6.992 7.991 8.990 9.988 .048 .096 .144 .192 .240 .288 ,336 .384 .432 .480 1 2 3 4 5 6 7 8 9 10 88 w 87 45' 87 30' 87 15' 3 O' 3 15' 3 n 3O' 3 45' J 2 3 4 5 6 7 8 9 10 .999 1.997 2.996 3.995 4.993 5.992 6.990 7.989 8.988 9.986 .052 .105 .157 .209 .262 .314 .366 .419 .471 .523 .998 1.997 2.995 3.994 4.992 5.990 6.989 7.987 8.986 9.984 .057 .113 .170 .227 .283 .340 .397 .454 .510 .567 .998 1.996 2.994 3.993 4.991 5.989 6.987 7.985 8.983 9.981 ,061 .122 .183 .244 .305 .366 .427 .488 .549 .610 .998 1.996 2.994 3.991 4.989 5.987 6.985 7.983 8.981 9.979 .065 .131 .196 .262 .327 .392 .458 .523 .589 .654 1 2 3 4 5 6 7 8 9 10 87 O' 86 45' 86" 30' 86 e 15' D. Dep, Lat. Dep. Lat. Dep. Lat. Dep. Lat. D. 73 4-745' TRAVERSE TABLES. 8215'-86 D, Lat. Dep, O r ^ Lat, Dep. Lat, Dep. Lat. Dep. D. 1 2 3 4 5 6 7 8 9 10 40 4 15' 4 30' 4 45' .998 1.995 2.993 3.990 4.988 5.986 6.983 7.981 8.978 9.976 .070 .140 .209 .279 .349 .418 .488 .558 .628 .698 .997 1.995 2.992 3.989 4.986 5.984 6.981 7.978 8.975 9.973 .074 .148 .222 .296 .371 .445 .519 .593 .667 .741 .997 1.994 2.991 3.988 4.985 5.981 6.978 7.975 8.972 9.969 .078 .157 .235 .314 .392 .471 .549 .628 .706 .785 .997 1.993 2.990 3.986 4.983 5.979 6.976 7.973 8.969 9.966 .083 .166 .248 .331 .414 .497 .580 .662 .745 ' .828 1 2 3 4 5 6 7 8 9 10 S O' 85 45' 85 3O' 85 15' 5 0' 5 15' 5 3O' 5 45' 1 2 3 4 5 6 7 8 9 10 .996 1.992 2.989 3.985 4.981 5.977 6.973 7.970 8.966 9.962 .087 .174 .261 ,349 .436 ,523 .610 .697 .784 .872. .996 1.992 2.987 3.983 4.979 5.975 6.971 7.966 8.962 9.958 .092 .183 .275 .366 .458 .549 .641 .732 .824 .915 .995 1.991 2.986 3.982 4.977 5.972 6.968 7.963 8.959 9.954 .096 .192 .288 .383 .479 ,575 .671 .767 .863 .958 .995 1.990 2.985 3.980 4.975 5.970 6.965 7.960 8.955 9.950 .100 .200 .301 .401 ,501 .601 .701 .802 .902 1.002 1 2 3 4 5 6 7 8 9 10 85 O' 84 45' 84 SO' 84 C 15' ~F 2 3 4 5 6 7 8 9 10 ~T 2 8 4 5 6 7 8 9 10 6 O' 6 15' 6 30' 6 45' T 2 3 4 5 6 7 8 9 10 .995 1.989 2.984 3.978 4.973 5.967 6.962 7.956 8.951 9.945 .105 .209 .314 .418 .523 .627 .732 .836 .941 1.045 .994 1.988 2.982 3.976 4.970 5.964 6.958 7.952 8.947 9.941 .109 .218 .327 .435 .544 .653 .762 .871 .980 1.089 .994 1.987 2.981 3.974 4.968 5.961 6.955 7.949 8.942 9.936 .113 .226 .340 .453 ,566 .679 .792 .906 1.019 1.132 .993 1.986 2.979 3.972 4.965 5.958 6.952 7.945 8.938 9.931 .118 .235 .353 .470 ,588 .705 .823 .940 1.058 1.175 84 O' 83 45' 83 30' 83 15' 7 0' 7 15' 7 SO' 7 D 45' .993 1.985 2.978 3.970 4.963 5.955 6.948 7.940 8.933 9.925 .122 .244 ,366 .487 .609 .731 .853 .975 1.097 1.219 .992 1.984 2.976 3.968 4.960 5.952 6.944 7.936 8.928 9.920 .126 .252 .379 .505 .631 .757 .883 1.010 1.136 1.262 .991 1.983 2.974 3.966 4.957 5.949 6.940 7.932 $.923 9.914 .131 .261 ,392 ,522 .653 .783 .914 1.044 1.175 1 .305 .991 1 .982 2.973 3.963 4.954 5.945 6.936 7.927 8.918 9.909 .135 .270 .405 ,539 .674 .809 .944 1.079 1.214 1.349 1 2 3 4 5 6 7 8 9 10 IT IT 83 0' 82 45' 82 30' 82 = 15' Dep. Lat, Dep. Lat. Dep, Lat. Dep. Lat. 74 TRAVERSE TABLES. 7815'-82 D. Lat, Dep, Lat. Dep. Lat. Dep. Lat. Dep. D, 8 <y 8 15' 8 30' 8 45' 1 2 3 4 5 6 7 8 9 10 .990 1.981 2.971 3.961 4.951 5.942 6.932 7.922 8.912 9.903 .139 .278 .418 .557 .696 .835 .974 1.113 1 .253 1.392 .990 1.979 2.969 3.959 4.948 5.938 6.928 7.917 8.907 9.897 .143 .287 .431 .574 .717 .861 1.004 1.148 1.291 1.435 .989 1.978 2.967 3.956 4.945 5.934 6.923 7.912 8.901 9.890 .148 .296 .443 ,591 .739 .887 1 .035 1.182 1,330 1.478 .988 1.977 2.965 3.953 4.942 5.930 6.919 7.907 8.895 9.884 .152 .304 .456 .608 .761 .913 1 .065 .217 1 .369 1,521 1 2 3 4 5 6 7 8 9 10 82"' O' 81 45' 81" SO' 81 15 9 o 9 15' 9 30' 9 45' 1 2 3 4 5 6 7 8 9 10 .988 1.975 2.963 3.951 4.938 5.926 6.914 7.902 8.889 9.877 .156 .313 .469 .626 .782 .939 1.095 1.251 1.408 1 .564 .987 1.974 2.961 3.948 4.935 5.922 6.909 7.896 8.883 9.870 .161 .321 .482 .643 .804 .964 1.125 .286 .447 1.607 .986 1.973 2.959 3.945 4.931 5.918 6.904 7.890 8.877 9.863 .165 .330 .495 .660 .825 .990 1.155 1.320 1 .485 1.650 .986 1.971 2.957 3.942 4.92S 5.914 6.899 7.884 8.870 9.856 .169 .339 ,508 .677 .847 .016 .185 .355 ,524 .693 1 2 3 4 5 6 7 8 9 10 81 ? O' M> 45 so 30 8O 15' NT O' 1O 15 10 3O 10- 45' 1 2 3 4 5 6 7 8 9 10 .985 1.970 2.954 3.939 4.924 5.909 6.894 7.878 8.863 9.848 .174 .347 .521 .695 .868 1.042 1.216 1.389 1,563 1.736 .9*4 1.968 2.952 3.936 4.920 5.904 6.888 7.872 8.856 9.840 .178 .356 .534 .712 .890 1.068 1.246 1.424 1.601 1.779 .983 1.967 2.950 3.933 4.916 5.900 6.883 7.866 8.849 9.833 .182 ,364 ,547 729 .911 1.093 1.276 1.458 1.640 1.822 .982 1 .965 2.947 3.930 4.912 5.89.5 6.877 7.860 8.842 9.825 .187 .373 ,560 .746 .933 .119 .306 .492 .679 .865 1 2 3 4 5 6 7 8 9 10 80 o 79 45' 79^ 30' 79" 15 11" 0' 11 15' 11 30' 11 45' 1 2 3 4 5 6 7 8 9 10 .982 1,963 2.945 3.927 4.908 5.890 6.871 7.853 8.835 9.816 .191 .382 .572 .763 .954 1.145 1.336 1,526 1.717 1.908 .981 1.962 2.942 3.923 4.904 5.885 6.866 7.846 8.827 9.808 .195 .390 .585 .780 .976 .171 .366 .561 .756 .951 .980 1.960 2.940 3.920 4.900 5.880 6.860 7.839 8.819 9.799 .199 .399 .59* .797 .997 1.196 1.396 1,595 1.794 1 .994 .979 1.958 2.937 3.916 4.895 5.874 6.853 7.832 8.811 9.790 .204 .407 .611 .815 .018 .222 .426 .629 .833 2.036 1 2 3 4 5 6 7 8 9 10 79 O' 78 45' 78 30' 78 15' D. Dep. Lat. Dep. Lat. Dep. Lat. De P . Lat. D. 75 12-1545' TRAVERSE TABLES. 7415'-78< D, Lat. Dep. Lat, Dep. Lat, Dei). Lat. Dep, D. 12 O' 12 15' 12 3W 12 45' 1 2 3 4 5 6 7 8 9 10 .978 1.956 2.934 3.913 4.891 5.869 6.847 7.825 8.803 9.781 .208 .416 .624 .832 1.040 1.247 1.455 1.663 1.871 2.079 .977 1.954 2.932 3.909 4.886 5.863 6.841 7.818 8.795 9.772 .212 .424 .637 .849 1.061 1.273 1.485 1.697 1.910 2.122 .976 1.953 2.929 3.906 4.882 5.858 6.834 7.810 8.787 9.763 .216 .433 .649 .866 1.082 1.299 ,515 .731 .948 2.164 .975 1.951 2.926 3.901 4.877 5.852 6.827 7.803 8.778 9.753 .221 .441 .662 .883 .103 .324 ,545 1.766 .986 2.207 1 2 3 4 5 6 7 8 9 10 78 e o 77 45' 77 3O' 77 15' 13 o 13 15' 13 3O' 13 45' 1 2 3 4 5 6 7 8 9 10 .974 1.949 2.923 3.897 4.872 5.846 6.821 7.795 8.769 9.744 .225 .450 .675 .900 1.125 1.350 1.575 1.800 2.025 2.250 .973 1.947 2920 3.894 4.867 5.840 6.814 7.787 8.760 9.734 .229 .458 .688 .917 1.146 1.375 1.604 1.834 2.063 2.292 .972 1.945 2.917 3.889 4.862 5.834 6.807 7.779 8.751 9.724 .233 .467 .700 .934 1.167 1.401 1.634 1.868 2.101 2.334 .971 1.943 2.914 3.885 4.857 5.828 6.799 7.771 8.742 9.713 .238 .475 .713 .951 1.188 1.426 1.664 1.901 2.139 2.377 1 2 3 4 5 6 7 8 9 10 77 O' 76 45' 70 3O' 7tt- 15' 14 0' 14 15' 14 3O' 14 45' 1 2 3 4 5 6 7 8 9 10 .970 1.941 2.911 3.881 4.851 5.822 6.792 7.762 8.733 9.703 .242 .484 .726 .968 1.210 1.452 1.693 1 .935 2.177 2.419 .969 r.938 2.908 3.877 4.846 5.815 6.785 7.754 8.723 9.692 .246 492 .738 .985 1.231 1.477 1.723 1.969 2.215 2.462 .968 1.936 2.904 3.873 4.841 5.809 6.777 7.745 8.713 9.681 .250 .501 .751 1.002 1.252 1.502 1.753 2.003 2.253 2,504 .967 1.934 2.901 3.868 4.835 5.802 6.769 7.736 8.703 9.670 .255 .509 .764 1.018 1.273 1 ,528 1.782 2.037 2.291 2.546 1 2 3 4 5 6 7 8 9 10 70 O' 75 45' 75 30' 75 15' 15 O' 15 15' 15 3O' 15 45' 1 2 3 4 5 6 7 8 9 10 .966 1.932 2.898 3.864 4.830 5.796 6.761 7.727 8.693 9.659 .259 .518 .776 1.035 1 .294 1,553 1.812 2.071 2.329 2,588 .965 1.930 2.894 3.859 4.824 5.789 6.754 7.718 8.683 9.648 .263 ,526 .789 1.052 1.315 1,578 1.841 2.104 2.367 2.631) .964 1.927 2.891 3.855 4.818 5.782 6.745 7.709 8.673 9.636 .267 ,534 .802 1.069 1.336 1.603 1.871 2.138 2.405 2.672 .962 1.925 2887 3.850 4.812 5.775 6.737 7.700 8.662 9.625 .271 ,543 .814 1.086 1,357 1.629 1 .900 2.172 2.443 2.714 1 2 3 4 5 6 7 8 9 10 75 <y 74 45' 74 3O' 74" 15' D, Dep. Lat, Dep. Lat. Dep, Lat. Dep. Lat. D. 16-1945' TRAVERSE TABLES. 70*15-74* D, "I 2 3 4 5 6 7 8 9 10 ~T 2 3 4 5 6 7 8 9 10 Lat, Dei>. Lat, Dep. Lat, Dep, Lat, Dep. D. 1 2 3 4 5 6 7 8 9 10 16 O' 16 15' 16 30' 16 45' .961 1.923 2.884 3.845 4.806 5.768 6.729 7.690 8.651 9.613 .276 .551 .827 1.103 1.378 1.654 1.929 2.205 2.481 2.756 .960 1.920 2.880 3.840 4.800 5.760 6.720 7.680 8.640 9.600 .280 .560 .839 1.119 1.399 1.679 1.959 2.239 2.518 2.798 .959 1.918 2.876 3-.835 r4.794 5.753 6.712 7.671 8.629 9.588 .284 .568 .852 1.136 1.420 1.704 1.988 2.272 2.556 2.840 .958 1.915 2.873 3.830 4.788 5.745 6.703 7.661 8.618 9.576 .288 .576 .865 1.153 1.441 1.729 2.017 2.306 2.594 2.882 is 5 74 O' 73 45' 73 3O' 73 3 17~ O' 17 15' 17 30' 17 45' T 2 3 4 5 6 7 8 9 10 .956 1.913 2.869 3.825 4.782 5.738 6.694 7.650 8.607 9.563 .292 .585 .877 1.169 1.462 1.754 2.047 2.339 2.631 2.924 .955 1.910 2.865 3.820 4.775 5.730 6.685 7.640 8,595 9.550 .297 .593 .890 1.186 1.483 1.779 2.076 2.372 2.669 2.965 .954 1.907 2.861 3.815 4.769 5.722 6.676 7.630 8.583 9.537 .301 .601 .902 1.203 1.504 1.804 2.105 2.406 2.707 3.007 .952 1.905 2.857 3.810 4.762 5.714 6.667 7.619 8.572 9.524 .305 .610 .915 1.219 1.524 1.829 2.134 2.439 2.744 3.049 73 0' 73-' 45' 72-' 3O' 72 15' ~T 2 3 4 5 6 7 8 9 10 ~T 2 3 4 5 6 7 8 9 10 D: 18 O' 18 15' 18 30' 18 45' .951 1.902 2.853 3.804 4.755 5.706 6.657 7.608 8,559 9^511 .30J .618 .927 1.236 1.545 1.854 2.163 2.472 2.781 3.090 .950 1.899 2.849 3.799 4.748 5.698 6.648 7.598 8.547 9.497 .313 .626 .939 1.253 1.566 1.879 2.192 2.505 2.818 3.132 .948 1.897 2.845 3.793 4.742 5.690 6.638 7.587 8.535 9.483 .317 .635 .952 1.269 1.587 1.904 2.221 2.538 2.856 3.173 .947 1.894 2.841 3.788 4.735 5-.6S2 6.628 7.575 8.522 9.469 .321 .643 .964 1.286 1.607 1.929 2.250 2.572 2.893 3.214 1 2 3 4 5 6 7 8 9 10 72 O' 71 45' 71 30' 71 15' 19 O' 19 15' 19 30* 19 45' T 2 3 4 5 6 7 8 9 10 .946 1.891 2.837 3.782 4.728 5.673 6.619 7,564 8.510 9.455 .326 .651 .977 1.302 1.628 1.953 2.279 2.605 2.930 3.256 .944 1.888 2.832 3.776 4.720 5.665 6.609 7.553 8.497 9.441 .330 .659 .989 1.319 1.648 1.978 2.308 2.638 2.967 3.297 .943 1 .885 2.828 3.771 4.713 5.656 6.598 7.541 8.484 9.426 .334 .668 1.001 1.335 1.669 2.003 2.337 2.670 3.004 3.338 .941 1.882 2.824 3.765 4.706 5.647 6.588 7.529 8.471 9.412 .338 .676 1.014 1.352 1.690 2.027 2.365 2.703 3.041 3.379 71 0' 7O 45' 70 30' 70 15' Dep. Lat, Dep. Lat. Dep, Lat. Dep. Lat. D, S.N.41. 77 20-2345' TRAVERSE TABLES. 15'-70 D. Lat, Dep. o 1 Lat. Dep. Lat, Dep. Lat. Dep. D. ~T 2 3 4 5 6 7 8 9 10 i 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 20 20- 15' 2O 3O' 2O 45' 1 2 3 4 5 6 7 8 9 10 .940 1.879 2.819 3.759 4.698 5.638 6.578 7,518 8.457 9.397 .342 .684 1.026 1.368 1.710 2.062 2.394 2.736 3.078 3.420 .938 1.876 2.815 3.753 4.691 5.629 6.567 7.506 8.444 9.382 .346 .692 1.038 1/384 1.731 2.077 2.423 2.769 3.115 3.461 .937 1.873 2.810 3.747 4.683 5.620 6.557 7.493 8.430 9.367 ,350 .700 1.051 1.401 1.751 2.101 2.451 2.802 3.152 3.502 .935 1.870 2.805 3.740 4.676 5.611 6.546 7.481 8.416 9,351 .354 .709 1.063 1.417 1.771 2.126 2.480 2.834 3.189 3.543 70 69 45 69 30' 69 15' 21 21 3 15' 21 30' 21 45' "I 2 3 4 5 6 7 8 9 10 .934 1.867 2.801 3.734 4.668 5.601 6,535 7.469 8.402 9,336 .358 .717 1.075 1.433 1.792 2.15!) 2.509 2.867 3.225 3.584 ,932 1.864 2.796 3.728 4.660 5.592 6.524 7.456 8.388 9.320 .362 .725 1.087 1.450 1.812 2.175 2.537 2.900 3.262 3.624 .930 1.861 2.791 3.722 4.652 5.582 6,513 7.443 8.374 9,304 ,367 .733 1.100 1.466 1.833 2.199 2.566 2.932 3.299 3.665 .929 1.858 2.786 3.715 4.644 5.573 6.502 7.430 8.359 9.288 .371 .741 1.112 1.482 1.853 2.223 2.594 2.964 3.335 3.706 1F O' 68 45' 68 3O' 68 15' 22 O' 22 15' 22 SO' 22 45' T 2 3 4 5 6 7 8 9 10 .927 1.854 2.782 3.709 4.636 5.563 6.490 7.418 8.345 9.272 .375 .749 1.124 1.498 1.873 2.248 2.622 2.997 3.371 3.746 .926 1.851 2.777 3.702 4.628 5.553 6.479 7.404 8.330 9.255 .379 -:>- 1.136 1.515 1.893 2.272 2.651 3.029 3.408 3.786 .924 1.848 2.772 3.696 4.619 5.543 6.467 7,391 8.315 9.239 .383 .765 1.148 1.531 1.913 2.296 2.679 3.062 3.444 3.827 .yj 1.844 2.767 3.689 4.611 5.533 6.455 7.378 8.300 9.222 .387 .773 1.160 1.547 1.934 2.320 2.707 3.094 3.480 3 867 68 O' 67 45' 67 30' 67 C 15' 23 O' 23 15 23 3O' 23= 45' 1 2 3 4 5 6 7 8 9 10 IT .921 1.841 2.762 3.682 4.603 5.523 6.444 7.364 8.285 9.205 .391 .781 1.172 1,563 1.954 2.344 2.735 3.126 3.517 3.907 .919 1.838 2.756 3.675 4.594 5.513 6.432 7.350 8.269 VI. 1 SS .395 .789 1.184 1.579 1.974 2.368 2:763 3.158 3.553 3.947 .917 1.834 2.751 3.668 4,585 5.502 6.419 7.336 8.254 9.17L .399 .797 1.196 1.595- 1.994 2.392 2.791 3.190 3.589 3.987 .915 1.831 2.746 3.661 4.577 5.492 6.407 7.322 8.238 9.153 .403 .805 1.208 1.611 2.014 2.416 2.819 3.222 3.625 4.027 1 2 3 4 5 6 7 8 9 10 "DT 6 7 Dep. O' Lat. 66 Dep. 45' 66^ 3O' 66 ' 15' Lat. Dep, Lat. Dep. | Lat. 78 24-2745' TRAVERSE TABLES. 6215'-66 D, Lat, Dep, Lat, Dep. Lat, Dep. Lat, Dep, D. ~T 2 3 4 5 6 7 8 9 10 T 2 3 4 5 6 7 8 9 10 ~T 2 3 4 5 6 7 8 9 10 24 <y 24 15' 24 3O' 24 45' .914 1.827 2.741 3.654 4.5G8 5.481 6.395 7.308 8.222 9.135 .407 .813 1.220 1.627 2.034 2.440 2.847 3.254 3.66L 4.067 .'Jl2 1.824 2.735 3.647 4.559 5.471 6.382 7.294 8.206 9.118 .411 .821 1.232 1.643 2.054 2.464 2.875 3.286 3.696 4.107 .910 1.820 2.730 3.640 4.550 5.460 6.370 7.280 8.190 9.100 .415 .829 1.244 1.659 2.073 2.488 2.903 3.318 3.732 4.147 .908 1.816 2.724 5.633 4.541 5.449 6.357 7.265 8.173 9.081 .419 .837 1.256 1.675 2.093 2.512 2.931 3.349 3.768 4.187 6 w 05 45' 65 30' 65 15' ~T 2 3 4 5 G 7 8 9 10 "T 2 3 4 5 6 7 8 9 10 25 O' 25" 15' 25 so' 25 45' .9U6 1.813 2.719 3.625 4.532 5.438 6.344 7.250 8.157 9.063 .423 .845 1.268 1.690 2.113 2.536 2.958 3.381 3.804 4.226 .904 1.809 2.713 3.618 4.522 5.427 6.331 7.236 8.140 9.045 .427 .853 1.280 1.706 2.133 2,559 2.986 3.413 3.839 4.266 .903 1.805 2.708- 3.610 4.513 5.416 6.318 7.221 8.123 9.026 .431 .861 1.292 1.722 2.153 2,583 3.014 3.444 3.875 4.305 .901 1.801 2.702 3.603 4,504 5.404 6,305 7.206 8.106 9.007 .434 .869 1.303 1.738 2.172 2.607 3.041 3.476 3.910 4.344 65 a O' 64 45' 64 3<y 64 15' 26 O' 26 15' 26 SO 7 26 45' "T 2 3 4 5 6 7 8 9 10 T 2 3 4 5 6 7 8 9 10 .899 1.798 2.696 3.595 4.494 5.393 6.292 7.190 8.089 8.988 .438 .877 1.315 1.753 2.192 2.630 3.069 3.507 3.945 4.384 .897 1.794 2.691 3.587 4.484 5.381 6.278 7.175 8.072 8.969 .442 .885 1.327 1.769 2.211 2.654 3.096 3,538 3.981 4.423 .895 1.790 2.685 3.580 4.475 5.370 6.265 7.159 8.054 8.949 .446 .892 1.339 1.785 2.231 2.677 3.123 3,570 4.016 4.462 .893 1.786 2.679 3.572 4.465 5.358 6.251 7.144 8.037 8.930 .450 .900 1.350 1.800 2.250 2.701 3.151 3.601 4.051 4.501 64 0' OS' 1 45' 63 3O' 63 15' ~T 2 3 4 5 6 7 8 9 10 DT 27 0' 27 r 15' 27 SO' 27 45' .891 1.782 2.673 3.564 4.455 5.346 6.237 7.128 8.019 8.910 .454 .908 1.362 1.816 2.270 2.724 3.178 3.632 4.086 4,540 .889 1.778 2.667 3,556 4.445 5.334 6.223 7.112 8.001 8.890 .458 .916 1.374 1.831 2.289 2.747 3.205 3.663 4.121 4,579 .887 1.774 2.661 3,548 4.435 5,322 6.209 7.096 7.983 8.870 .462 .923 1.385 1.847 2.309 2.770 3.232 3.694 4.156 4.617 .885 1.770 2.655 3.540 4.425 5.310 6.195 7.080 7.965 8.850 .466 .931 1.397 1.862 2.328 2.794 3.259 3.725 4.190 4.656 63 O' 62 45' 62 30' 62 15' Dep, Lat. Dep, Lat. Dep. Lat. Dep, Lat, D, 79 28-3145' TRAVERSE TABLES. 5815'-62 ( D, T c 4 tJ 4 P e 7 8 9 10 Lat, Dep, Lat, Dep. Lat. Dep. Lat. Dep. D, ~f 2 3 4 5 6 7 8 9 10 ~T 2 3 4 5 6 7 8 9 10 T 2 3 4 5 6 7 8 9 10 ~T 2 3 4 5 6 7 8 9 10 28 O* 28 15' 28 30' 28 45' .883 1.766 2.649 3.532 4.415 5.298 6.181 7.064 7.947 8.829 .469 .939 1.408 1.878 2.347 2.817 3.286 3.756 4.225 4.695 .881 1.762 2.643 3.524 4.404 5.285 6.166 7.047 7.928 8.809 .473 .947 1.420 1.S93 2.367 2.840 3.313 3.787 4.260 4.733 .879 1.758 2.636 3.515 4.394 5.273 6.152 7.031 7.909 8.788 .477 .954 1.431 1.909 2.386 2.863 3.340 3.817 4.294 4.772 .877 1.753 2.630 3.507 4.384 5.260 6.137 7.014 7.890 8.767 .481 .962 1.443 1.924 2.405 2.886 3.367 3.848 4.329 4.810 62 0' 61 45' 61 3O' 61- 15' 29 0' 29 15' 29 3O' 29 45' 1 2 3 4 5 6 7 8 9 10 .875 1.749 2.624 3.498 4.373 5.248 6.122 6.997 7.872 8.746 .485 .970 1.454 1.939 2.424 2.909 3.394 3.878 4.363 4.848 .872 1.745 2.617 3.490 4.362 5.235 6.107 6.980 7.852 8.725 .489 .977 1.466 1.954 2.443 2.932 3.420 3.909 4.398 4.886 .870 1.741 2.611 3.481 4.352 5.222 6.092 6.963 7.833 8.704 .492 .985 1.477 1.970 2.462 2.954 3.447 3.939 4.432 4.924 .868 1.736 2.605 3.473 4.341 5.209 6.077 6.946 7.814 8.682 .496 .992 1.489 1.985 2.481 2.977 3.473 3.970 4.466 4.962 61 O' >o 45' 6O 3O' 6O 15' ~T 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 30 0' 30 15' 3O 3O> 30 45' .866 1.732 2.598 3.464 4.330 5.196 6.062 6.928 7.794 8.660 .500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000 .864 1.728 2.592 3.455 4.319 5.183 6.047 6.911 7.775 8.638 .504 1.008 1.511 2.015 2.519 3.023 3.526 4.030 4.534 5.038 .862 1.723 2.585 3.446 4.308 5.170 6.031 6.893 7.755 8.616 .508 1.015 1.523 2.030 2,538 3.045 3,553 4.060 4.568 5.075 .859 1.719 2.578 3.438 4.297 5.156 6.016 6.875 7.735 8,594 .511 1.023 1.534 2.045 2.556 3.068 3.579 4.090 4.602 5.113 6O O' 59- 45' 59 3O' 59 15' 31 W 31 15' 31 30' 31 45' .857 1.714 2.572 3.429 4.286 5.143 6.000 6.857 7.715 8,572 .515 1.030 1.545 2.060 2.575 3.090 3.605 4.120 4.635 5.150 .855 1.710 2.565 3.420 4.275 5.129 5.984 6.839 7.694 8.549 .519 1.038 1.556 2.075 2.594 3.113 3.631 4.150 4.669 5.188 .853 1.705 2.558 3.411 4.263 5.116 5.968 6.821 7.674 8,526 ,522 1.045 1,567 2.090 2.612 3.135 3.657 4.180 4.702 5.225 .850 1.701 2,551 3.401 4.252 5.102 5.952 6.803 7.653 8,504 .526 1.052 1.579 2.105 2.631 3.157 3.683 4.210 4.736 5.262 59 O' 58 45' 58 30* 58 15' "oT D. Dep. Lat, Dep, Lat. Dep, Lat. Dep, Lat. 80 32-3545' TRAVERSE TABLES. 5415 / -58 D, Lat, Dep. Lat, Dep. Lat, Dep, Lat. Dep, D, 32 O' 32 15' 32 3O' 32 45' 1 2 3 4 5 6 7 8 9 10 ~7 2 3 4 5 6 7 8 9 10 .848 1.696 2.544 3.392 4.240 5.088 5.936 6.784 7.632 8.480 .530 1.060 1.590 2.120 2.650 3.180 3.709 4.239 4.769 5.299 .846 1.691 2.537 3.383 4.229 5.074 5.920 6.766 7.612 8.457 .534 1.067 1.601 2.134 2.668 3.202 3.735 4.269 4.802 5.336 .843 1.687 2.530 3.374 4.217 5.060 5.904 6.747 7.591 8.434 .537 1.075 1.612 2.149 2.686 3.224 3.761 4.298 4.836 5.373 .841 1.682 2.523 3.364 4.205 5.046 5.887 6.728 7.569 8.410 .541 1.082 1.623 2.164 2.705 3.246 3.787 4.328 4.869 5.410 1 2 3 4 5 6 7 8 9 10 58 W 57 45' 57 SO' 57 15' ~J 2 3 4 5 6 7 8 9 10 33 O' 33 15' 33 30' 33 45' .839 1.677 2.516 3.355 4.193 5.032 5.871 6.709 7.548 8.387 .545 1.089 1.634 2.179 2.723 3.268 3.812 4.357 4902 5.446 .836 1.673 2.509 3.345 4.181 ,5.018 5.854 6.690 7.527 8.363 .548 1.097 1.645 2.193 2.741 3.290 3.838 4.386 4.935 5.483 .834 1.668 2.502 3.336 4.169 5.003 5.837 6.671 7.505 8.339 .552 1.104 1.656 2.208 2.760 3.312 3.864 4.416 4.967 5.519 .831 1.663 2.494 3.326 4.157 4.989 5.820 6.652 7.483 8.315 .556 1.111 1.667 2.222 2.778 3.333 3.889 4.445 5.000 5.556 57 W 5 45' 56 30' 56 15' 34 0' 34 15' 34 &W 34 45' 1 2 3 4 5 6 7 8 9 10 .829 1.658 2.487 3.316 4.145 4.974 5.803 6.632 7.461 8.290 .559 1.118 1.678 2.237 2.796 3.355 3.914 4.474 5.033 5.591 .827 1.653 2.480 3.306 4.133 4.960 5.786 6.613 7.439 8.266 .563 1.126 1.688 2.251 2.814 3.377 3.940 4.502 5.065 5.628 .824 1.648 2.472 3.297 4.121 4.9.45 5.769 6.593 7.417 8.241 .566 1.133 1.699 2.266 2.832 3.398 3.965 4.531 5.098 5.664 .822 1.643 2.465 3.287 4.108 4.930 5.752 6.573 7.395 8.216 .570 1.140 1.710 2.280 2.850 3.420 3.990 4.560 5.130 5.700 1 2 3 4 5 6 7 8 9 10 56 0' 55 45' 55 30' 55 15' ~T 2 3 4 5 6 7 8 9 10 35 0' 35 15' 35 30' 35 45' 1 2 3 4 5 6 7 8 9 10 IT .819 1.638 2.457 3.277 4.096 4.915 5.734 6.553 7.372 8.192 .574 1.147 1.721 2.294 2.868 3.441 4.015 4.589 5.162 5.736 .817 1.633 2.450 3.267 4.083 4.900 5.716 6.533 7.350 8.166 .577 1.154 1.731 2.309 2.886 3.463 4.040 4.617 5.194 5.771 .814 1.628 2.442 3.256 4.071 4.885 5.699 6.513 7.327 8.141 .581 1.161 1.742 2.323 2.904 3.484 4.065 4.646 5.226 5.807 .812 1.623 2.435 3.246 4.058 4.869 5.681 6.493 7.304 8.116 .584 1.168 1.753 2.337 2.921 3.505 4.090 4.674 5.258 5.842 55 Dep, 0' 54 45' 54 30' 54 15' Lat, Dep. Lat. Dep, Lat. Dep, Lat, D, 81 >-3945' TRAVERSE TABLES, 5015'-54 D, Lat. Dep. Lat. Dep. Lat, Dep. Lat. Dep. D. 36 O 7 36 15' 36 3O' 36 45' 1 2 3 4 5 6 7 8 9 10 .809 1.618 2.427 3.236 4.045 4.854 5.663 6.472 7.281 8.090 .588 1.176 1.763 2.351 2.939 3.527 4.115 4.702 5.290 5.878 .806 1.613 2.419 3.226 4.032 4.839 5.645 6.452 7.258 8.064 .591 1.183 1.774 2.365 2.957 3.548 4.139 4.730 5.322 5.913 .804 1.608 2.412 3.215 4.019 4.823 5.627 6.431 7.235 8.039 .595 1.190 1.784 2.379 2.974 3.569 4.164 4.759 5.353 5.948 .801 1.603 2.404 3.205 4.006 4.808 5.609 6.410 7.211 8.013 .598 1.197 1.795 2.393 2.992 3.590 4.188 4.787 5.385 5.983 1 2 3 4 5 6 7 8 9 10 54 O' 53 45' 53 30' 53 15' 37 O' 37 15' 37 30' 37 45' 1 2 3 4 5 6 7 8 9 10 .799 1.597 2.396 3.195 3.993 4.792 5.590 6.389 7.188 7.986 .602 1.204 1.805 2.407 3.009 3.611 4.213 4.815 5.416 6.018 .796 1.592 2.388 3,184 3.980 4.776 5.572 6.368 7.164 7.960 .605 1.211 1.816 2.421 3.026 3.632 4.237 4.842 5.448 6.053 .793 1.587 2.380 3.173 3.967 4.760 5.553 6.347 7.140 7.934 .609 1.218 1.826 2.435 3.044 3.653 4.261 4.870 5.479 6.088 .791 1.581 2.372 3.163 3.953 4.744 5.535 6.326 7.116 7.907 .612 1.224 1.837 2.449 3.061 3.673 4.286 4.898 5.510 6.122 1 2 3 4 5 6 7 8 9 10 53 cy 52 45' 52 30' 52 15' 38 <y 38 15' 38 :$o 38 45' 1 2 3 4 5 6 7 8 9 10 .788 1.576 2.364 3.152 3.940 4.728 5.516 6.304 7.092 7.880 .616 1.231 1.847 2.463 3.078 3.694 4.310 4.925 5.541 6.157 .785 1.571 2.356 3.141 3.927 4.712 5.497 6.283 7.068 7.853 .619 1.238 1.857 2.476 3.095 3.715 4.334 4.953 5.572 6.191 .783 1.565 2.348 3.130 3.913 4.696 5.478 6.261 7.043 7.826 .623 1.245 1.868 2.490 3.113 3.735 4.358 4.980 5.603 6.225 .780 1.560 2.340 3.120 3.899 4.679 5.459 6.239 7.019 7.799 .626 1.252 1.878 2.504 3.130 3.756 4.381 5.007 5.633 6.259 1 2 3 4 5 6 7 8 9 10 52 0' 51 45' 51 30' 51 15' 39 w 39 15' 39 30' 39 45' 1 2 3 4 5 6 7 8 9 10 .777 1.554 2.331 3.109 3.886 4.663 5.440 6.217 6.994 7.771 .629 1.259 1.888 2.517 3.147 3.776 4.405 5.035 5.664 6.293 .774 1.549 2.323 3.098 3.872 4.646 5.421 6.195 6.970 7.744 .633 1.265 1.898 2.531 3.164 3.796 4.429 5.062 5.694 6.327 .772 1.543 2.315 3.086 3.858 4.630 5.401 6.173 6.945 7.716 .636 1.272 1.908 2.544 3.180 3.816 4.453 5.089 5.725 6.361 .769 1.538 2.307 3.075 3.844 4613 5.382 6.151 6.920 7.688 .639 1.279 1.918 2.558 3.197 3.837 4.476 5.116 5.755 6.394 1 2 3 4 5 6 7 8 9 10 51 0' 50 45' 5O 30' 50 15' D, Dep, Lat, Dep, Lat, Dep, Lat. Dep. Lat D, 40-4345' TRAVERSE TABLES. 4615'-50 D, Lat, Dep. Lat, Dep. Lat, Dep. Lat. Dep. D. 4O <y 40 15' 40 30' 40^ 45' 1 2 3 4 5 6 7 8 9 10 .766 1.532 2.298 3.064 3.830 4.596 5.362 6.128 6.894 7.660 .643 1.286 1.928 2.571 3.214 3.857 4.500 5.142 5.785 6428 .763 1.526 2.290 3.053 3.816 4.579 5.343 6.106 6.869 7.632 .646 1.292 1.938 2.584 3.231 3.877 4.523 5.169 5.815 6.461 .760 1.521 2.281 3.042 3.802 4.562 5.323 6.083 6.844 7.604 .649 1.299 1.948 2.598 3.247 3.897 4.546 5.196 5.845 6.494 .758 1.515 2.273 3.030 3.788 4.545 5.303 6.061 6.818 7.576 .653 1.306 1.958 2.611 3.264 3.917 4.569 5.222 5.875 6.528 1 2 3 4 5 6 7 8 9 10 50 O' 49 45' 49 30' 49 15' 41 w 41 15' 41 3O' 41 45' 1 2 3 4 5 6 7 8 9 10 .755 1.509 2.264 3.019 3.774 4.528 5.283 6038 6.792 7.547 .656 1.312 1.968 2.624 3.280 3.936 4.592 5.248 5.905 6.561 .752 1.504 2.256 3.007 3.759 4.511 5.263 6.015 6.767 7.518 .659 1.319 1.978 2.637 3.297 3.956 4.615 5.275 5.934 6.593 .749 1.498 2.247 2.996 3.745 4.494 5.243 5.992 6.741 7.490 .663 1.325 1.988 2.650 3.313 3.976 4.638 5.301 5.964 6.626 .746 1.492 2.238 2.984 3.730 4.476 5.222 5.968 6.715 7.461 .666 1.332 1.998 2.664 3.329 3.995 4.661 5.327 5.993 6.659 1 2 3 4 5 6 7 8 9 10 49 <y 48 45' 48 3O' 48 15' 42 O' 42 15' 42 30' 42 45' 1 2 3 4 5 6 7 8 9 10 .743 1.486 2.229 ,2.973 3.716 4.459 5.202 5.945 6.688 7.431 .669 1.338 2.007 2.677 3.346 4.015 4.684 5.353 6.022 6.691 .740 1.480 2.221 2.961 3.701 4.441 5.182 5.922 6.662 7.402 .672 1.345 2.017 2.689 3.362 4.034 4.707 5.379 6.051 6.724 .737 1.475 2.212 2.949 3.686 4.424 5.161 5.898 6.636 7.373 .676 1.351 2.027 2.702 3.378 4.054 4.729 5.405 6.080 6.756 .734 1.469 2.203 2.937 3.672 4.406 5.140 5.875 6.609 7.343 .679 1.358 2.036 2.715 3.394 4.073 4.752 5.430 6109 6.788 ] 2 3 4 5 6 7 8 9 10 48 0' 47 45' 47 30' 47 15' 43 0' 43 15' 43 30' 43 45' 1 2 3 4 5 6 7 8 9 10 .731 1.463 2.194 2.925 3.657 4.388 5.119 5.851 6.582 7.314 .682 1.364 2.046 2.728 3.410 4.092 4.774 5.456 6.138 6.820 .728 1.457 2.185 2.913 3.642 4.370 5.099 5.827 6.555 7.284 .685 1.370 2.056 2.741 3.426 4.111 4.796 5.481 6,167 6.852 .725 1.451 2.176 2.901 3.627 4.352 5.078 5.803 6.528 7.254 .688 1.377 2.065 2.753 3.442 4.130 4.818 5.507 6.195 6.884 .722 1.445 2.167 2.889 3.612 4.334 5.057 5.779 6.501 7.224 .692 1.383 2.075 2.766 3.458 4.149 4.841 5.532 6.224 6.915 1 2 3 4 5 6 7 8 9 10 47 O' 46 45' 46 30' 46 15' D, Dap. Lat, Dep. Lat. Dep, Lat. Dep. Lat. D, 83 44<>-45 TRAVERSE TABLES. 45-46 D. Lat, Dep, Lat, Dep. Lat, Dep, Lat. Dep, D. 44 3 (y 44 a 15' 44 3^ 44 45' 1 .719 .695 .716 .698 .713 .701 .710 .704 1 2 1.439 1.389 1.433 1.396 1.427 1.402 1.420 1.408 2 3 2.158 2.084 2149 2.093 2.140 2.103 2.131 2.112 3 4 2.877 2.779 2.865 2.791 2.853 2.804 2.841 2.816 4 5 3.597 3.473 3.582 3.489 3.566 3.505 3.551 3.520 5 6 4.316 4.168 4.298 4.187 4.280 4.205 4.261 4.224 6 7 5.035 4.863 5.014 4.885 4.993 4.906 4.971 4.928 7 8 5.755 5.557 5.730 5.582 5.706 5.607 5.682 5.632 8 9 6.474 6.252 6.447 6.280 6.419 6.308 6.392 6.336 9 10 7.193 6.947 7.163 6.978 7.133 7.009 7.102 7.040 10 46 3 <K 45 45> 45 3W 45 15' 45 (X 45 15' 45 3O' 45 45' 1 .707 .707 .704 .710 .701 .713 .698 .716 1 2 1.414 1.414 1.408 1.420 1.402 1.427 1.396 1.433 2 3 2.121 2.121 2.112 2.131 2.103 2.140 2.093 2.149 3 4 2.828 2.828 2.816 2.841 2.804 2.853 2.791 2.865 4 5 3.536 3.536 3.520 3.551 3.505 3.566 3.489 3.582 5 6 4.243 4.243 4.224 4.261 4.205 4.280 4.187 4.298 6 7 4.950 4.950 4.928 4.971 4.906 4.993 4.885 5.014 7 8 5.657 5.657 5.632 5.682 5.607 5.706 5.582 5.730 8 9 6.364 6.364 6.336 6.392 6.308 6.419 6.280 6.447 9 10 7.071 7.071 7.040 7.102 7.009 7.133 6.978 7.163 10 45^ <y 44 45' 44 3W 44 15' D. Dep, Lat. Dep, Lat, Dep, Lat, Dep, Lat, D, MISCELLANEOUS TABLE. DIAMETER = 1. LOG'. Circumference of circle, IT, 3.14159 0.49715 Area of circle, 78540 9.89509-10 Contents of sphere, 52360 9.71900-10 Earth's equatorial radius, in miles, . . .3962.57 3.59798 Earth's polar radius, in miles, 3949.324 3.59652 Compression, 1 -<- 299. 1528, 0.00334 7.52411-10 EQUIVALENTS. American, mile = .86756 nautical miles, . . . 9.93830-10 " =1609.40831 meters, 3.20667 " " == .21689 German geosraph. miles, 9.33624-10 " " = 1.50866 Russian versts, . . . 0.17859 yard .91444 meters, 9.96115-10 " = .48217 Vienna klafter, . . . 9.68320-10 foot= .30481 meters, 9.48403-10 " = .15639 toises, 9.19421-10 " = .93835 Parisian feet, .... 9.97236-10 " .96435 Vienna feet, .... 9.98423-10 " = 1.09395 Spanish feet, .... 0.03900 84 MERIDIONAL PARTS. Deg, 0' 10' 20' 30' 40' 50' 0.0 9.9 19.9 29.8 39.7 49.7 1 59.6 69.5 79.5 89.4 99.3 109.3 2 119.2 129.2 139.1 149.0 159.0 168.9 3 178.9 188.8 198.8 208.7 218.7 228.6 4 238.6 248.6 258.5 268.5 278.4 288.4 5 298.4 308.4 318.3 328.3 338.3 348.3 6 358.3 368.3 378.2 388.2 398.2 408.2 7 418.3 428.3 438.3 448.3 458.3 468.3 8 478.4 488.4 498.4 508.5 518.5 528.6 9 538.6 548.7 458.8 568.8 578.9 589.0 10 599.1 609.2 619.3 629.4 639.5 649.6 11 659.7 669.8 680.0 690.1 700.2 710.4 12 720.5 730.7 740.9 751.0 761.2 771.4 13 781.6 791.8 802.0 812.2 822.5 832.7 14 842.9 853.2 863.4 873.7 884.0 894.2 15 904.5 914.8 925.1 935.4 945.7 956.1 16 966.4 976.7 987.1 997.5 1007.8 1018.2 17 1028.6 1039.0 1049.4 1059.8 1070.2 1080.7 18 1091.1 1101.6 1112.0 1122.5 1133.0 1 143.5 19 1154.0 1164.5 1175.1 1185.6 1196.1 1206.7 20 1217.3 1227.9 1238.5 1249.1 1259.7 1270.3 21 1281.0 1291.6 1302.3 1313.0 1323.7 1334.4 22 1345.1 1355.8 1366.6 1377.3 1388.1 1398.9 23 1409.7 1420.5 1431.3 1442.1 1453.0 1463.8 24 1474.7 1485.6 1496.5 1507.4 1518.4 1529.3 25 1540.3 1551.3 1562.3 1573.3 1584.3 1595.4 26 1606.4 1617.5 1628.6 1639.7 1650.8 1661.9 27 1673.1 1684.3 1695.5 1706.7 1717.9 1729.1 28 1740.4 1751.7 1762.9 1774.3 1785.6 1796.9 29 1808.3 1819.7 1831.1 1842.5 1854.0 1865.4 30 1876.9 1888.4 1899.9 1911.4 1923.0 1934.6 31 1946.2 1957.8 1969.4 1981.1 1992.8 2004.5 32 2016.2 2028.0 2039.7 2051.5 2063.3 2075.2 33 2087.0 2098.9 2110.8 2122.7 2134.7 2146.7 34 2158.6 2170.7 2182.7 2194.8 2206.9 2219.0 35 2231.1 2243.3 2255.5 2267.7 2279.9 2292.2 36 2304.5 2316.8 2329.2 2341.5 2353.9 2366.4 37 2378.8 2391.3 2403.8 2416.3 2428.9 2441.5 38 2454.1 2466.8 24795 2492.2 2504.9 2517.7 39 2530.5 2543.3 2556.2 2569.1 2582.0 2594.9 40 2607.9 2621.0 2634.0 2647.1 2660.2 2673.3 41 2866.5 2699.7 2713.0 2726.3 2739.6 2752.9 42 2766.3 2779.8 2793.2 2806.7 2820.3 2833.8 85 MERIDIONAL PARTS. Deg. 0' 10' 20' 30' 40' 50' 43 2847.4 2861.1 2874.8 2888,5 2902.2 2916.0 44 2929.9 2943.7 2957.6 2971.6 2985.6 2999.6 45 3013.7 3027.8 3042.0 3056.2 3070.4 3084.7 46 3099.0 3113.4 3127.8 3142.3 3156.8 3171.3 47 3185.9 3200.5 3215.2 3230.0 3244.7 3259.6 48 3274.5 3289.4 3304.3 3319.4 3334.4 3349.6 49 3364.7 3380.0 3395.2 3410.6 3425.9 3441.4 50 3456.9 3472.4 3488.0 3503.7 3519.4 3535.1 51 3550.9 3566.8 3582.8 3598.7 3614.8 3630.9 52 3647.1 3663.2 3679.6 3696.0 3712.4 3728.9 53 3745.4 3762.0 3778.7 3795.4 3812.2 3829.1 54 3846.0 3863.1 3880.1 3897.3 3914.5 3931.8 55 3949.1 3966.6 3984.1 4001.7 4019.3 4037.0 56 4054.8 4072.7 4090.7 4108.7 4126.9 4145.1 57 4163.3 4181.7 4200.2 4218.7 4237.3 4256.0 58 4274.8 4293.7 4312.7 4331.7 4350.9 4370.1 59 4389.4 4408.9 4428.4 4448.0 4467.7 4487.5 60 4507.5 4527.5 4547.6 4567.8 4588.1 4608.6 61 4629.1 4649.8 4670,5 4691.4 4712.4 4733.5 62 4754.7 4776.0 4797.5 4819.0 4840.7 4862.5 63 .4884.5 4906.5 4928.7 4951.0 4973.5 4996.0 64 5018.8 5041.6 5064.6 5087.7 5111.0 5134.4 65 5158.0 5181.7 5205.5 5229.5 5253.7 5278.0 66 5302.5 5327.1 5351.9 5376.9 5402.1 5427.4 67 5452.8 5478.5 5504,3 5530.3 5556.5 5582.9 68 5609.5 5636.3 5663.2 5690.4 5717.7 5745.3 69 5773.1 5801.1 5829.3 5857.7 5886.3 5915.2 70 5944.3 5973.6 6003.2 6033.0 6063.1 6093.4 71 6124.0 6154.8 6185.9 6217.2 6248.9 6280.8 72 6313.0 6345.5 6378.2 6411.3 6444.7 6478.4 73 6512.4 6546.8 6581.5 6616,5 6651.8 6687.6 74 6723.6 6760.1 6796.9 6834.1 6871.7 6909.7 75 6948.1 6987.0 7026.2 7065.9 7106.1 -7146.7 76 7187.8 7229.3 7271.4 7313.9 7357.0 7400.6 77 7444.8 7489.5 7534.8 7580.7 7627.0 7674.3 78 7722.1 7770,5 7819.6 7869.4 7919.9 7971.1 79 8023.1 8075.9 8129.5 8184.0 8239.3 8295.4 80 8352.5 8410.6 8469.6 8529.7 8590.8 8653.0 81 8716.3 8780.9 8846.6 8913.6 8981.9 9051.6 82 9122.7 9195.3 9269.4 9345.2 9422.7 9501.9 83 9583.0 9666.0 9751.1 9838.3 9927.8 10019.6 84 10114.0 10211.0 10310.8 10413.6 10519.6 10628.8 85 10741.7 10858.4 10979.2 11104.3 11234.2 11369.1 CORRECTIONS FOR MIDDLE LATITUDE. DIFFERENCE OF LATITUDE. Mid. Lat, 2 ? 3 r 5 6 7 C 8 9 10 11" 12 n 13' 14 15' 16 C 17 18 19 20 Mid. Lat. 15 , 2' 3! 5' 7' 9' 12' 15' 18' 22' 26' 31' 36' 41' 47' 52' 59' 65' 72' 15. 16 2 8 4 6 9 11 14 18 21 25 30 34 39 44 50 56 62 69 16 17 2 s 4 6 8 11 14 17 20 24 28 33 38 43 48 54 60 66 17 18 1 3 4 6 8 10 13 16 20 23 27 32 36 41 46 52 58 64 18 19 1 8 4 6 8 10 13 16 19 22 26 30 35 40 45 50 56 61 19 20 1 2 4 5 7 10 12 15 18 22 25 29 34 38 43 48 54 60 20 21 1 2 4 5 7 9 12 15 18 '21 25 29 33 37 42 47 52 58 21 22 1 2 4 5 7 9 12 14 17 21 24 28 32 36 4L 46 51 56 22 23 1 2 3 6 7 9 11 14 17 20 23 27 31 35 40 45 50 55 23 , 24 1 2 3 5 ? 9 11 14 16 20 23 27 31 35 39 44 49 54 24 25 1 2 3 5 7 9 11 13 16 19 23 26 30 34 39 43 48 53 25 -. 26 1 3 5 8 11 13 16 19 22 26 30 34 38 42 47 52 26 27 1 2 3 5 6 8 11 13 16 19 22 25 29 33 37 42 47 52 27 28 1 2 8 5 6 8 10 13 16 18 22 25 29 33 37 41 46 51 28 29 1 2 3 5 6 8 10 13 15 18 21 25 28 32 37 41 46 51 29 30 1 2 3 5 6 8 10 13 15 18 21 25 28 32 36 41 45 50 30 31 1 2 3 6 6 8 10 12 15 18 21 24 28 32 36 40 45 50 31 32 1 2 3 4 6 8 10 12 15 18 21 24 28 32 36 40 45 50 32 33 1 2 3 1 8 8 10 12 15 18 21 24 28 32 36 40 45 49 33 34 1 2 :. 4 8 8 10 12 15 18 21 24 28 32 36 40 45 49 34 85 1 2 3 4 6 8 10 12 15 18 21 24 28 32 36 40 45 49 35 3o 1 2 3 4 6 8 10 12 15 18 21 24 28 32 36 40 45 49 36 ' 37 1 2 3 4 6 8 10 12 15 18 21 24 28 32 36 40 45 49 37 38 1 2 3 4 I) 8 10 12 15 18 21 24 28 32 36 40 45 50 38 39 1 2 3 4 (i S 10 12 15 18 21 24 28 32 36 40 45 60 39 40 1 2 3 5 B 8 10 13 15 18 21 25 28 32 36 41 45 50 40 41 1 2 3 .) 8 8 10 13 15 18 21 25 JH 32 37 41 46 51 41 42 1 2 3 5 6 8 10 13 15 18 22 25 2!) 33 37 41 46 51 42 43 1 2 3 5 (i 8 10 13 16 18 22 25 2!) 33 37 42 46 -.52 43 44 1 2 3 5 ti S 10 13 16 19 22 25 20 33 38 42 47 &i 44 45 1 2 3 5 (i 8 11 13 16 19 22 26 i 30 34 38 43 48 53 45 46 1 2 3 5 (i S 11 13 16 19 22 26 30 34 38 43 48 53 46 47 1 2 3 5 7 9 11 13 16 19 23 2<i 30 35 39 44 1 49 54 47 48 1 2 3 5 - !) 11 14 17 20 23 27 31 35 40 44 I 50 55 48 49 1 2 3 5 " 9 11 14 17 20 23 27 31 36 40 45 50 56 49 50 1 2 4 5 - 9 11 14 17 20 24 28 32 36 41 46 51 57 50 5L 1 >) 1 5 1) 12 14 17 21 24 as 32 37 42 47 52 58 51 ' 52 1 2 4 5 - 9 12 15 18 21 25 29 3:i 38 43 .48 53 59 52 53 1 2 4 5 7 10 12 15 18 21 25 29 31 38 43 >49 54 60 53 : 54 1 2 4 5 7 10 12 15 18 22 26 30 34 39 44 |50 56 62 34 j * 55 1 2 4 6 8 10 13 16 19 22 26 31 35 40 45 51 57 63 55 56 1 3 4 6 8 10 13 16 19 23 27 31 36 41 46 52 58 65 56 ; 57 1 3 4 6 8 10 13 16 20 24 28 32 37 42 48 54 60 66 57 i 58 2 3 4 6 8 11 14 17 20 24 28 33 38 43 49 55 61 <58 58 ' 59 2 3 4 6 8 11 14 17 21 25 29 34 39 45 50 57 63 70 50 60 2 3 4 6 9 11 14 18 22 26 30 35 40 46 52 58 65 72 60 61 1 3 5 7 12 15 18 22 26 31 36 42 47 53 60 67 75 61 , 62 2 3 5 7 9 12 15 19 23 27 32 37 4:', 49 55 62 70 77 62 63 2 3 5 7 10 12 It) 20 24 28 33 89 44 51 57 64 72 80 63 64 .) 3 5 7 10 13 10 20 24 29 34 40 46 52 59 67 75 83 64 65 2 a 5 7 10 13 17 21 25 30 36 41 48 54 62 69 78 86 65 66 2 3 5 8 11 14 18 22 26 32 37 50 57 64 72 81 90 66 . 67 i 2 4 6 S 11 14 IS 23 28 33 39 45 52 59 07 76 85 94 67 68 i 2 4 6 8 12 15 1!) 24 29 34 40 47 51 62 70 79 89 99 68 69 1 2 4 6 9 12 w 20 25 30 36 42 49 57 65 74 83 93 104 69 70 1 2 4 6 9 13 16 21 26 32 38 44 52 60 68 7S 88 98 110 70 87 ECLECTIC EDUCATIONAL SERIES. Published by VAN ANTWERP, BRAGG & CO., Cincinnati and New York. THALHEIMER'S HISTORICAL SERIES. By M. E. THALHEIMER, Teacher of History and Composition in Packer Collegiate Institute. For Graded Schools, High Schools, Academies, and Colleges. These books furnish to Teachers, stu- dents and general readers the best brief and economical course in Ancient, Modern and English History. ECLECTIC HISTORY OF THE UNITED STATES. lamo., half roan, 392 pp. Copiously illustrated with Maps, Portraits, etc. Contains reliable References and Explanatory Notes ; Declaration of Indepen- dence ; Constitution and Questions on the same ; Synopses of Presidential Ad- ministrations, etc. THALHEIMER'S HISTORY OF ENGLAND. 121110., 288 pp. A compact volume, comprehensive in scope, but sufficiently brief to be completed in one school term. Its statements of historical facts are based upon the studies of the most recent and reliable authorities. Reliable Maps and pictorial illustrations. THALHEIMER'S GENERAL HISTORY, I2mo., 355 pp. Maps and pictorial illustrations. The wants of common schools, and those of higher grade unable to give much time to the study of history, are here exactly met. The teacher is aided by Revierv Questions at the end of each principal division of the book, and by references to other works in which each subject will be found more fully treated. THALHEIMER'S ANCIENT HISTORY. A Manual of Ancient History from the Earliest Times to the fall of the Western Empire, A. D. 476. 8vo., full cloth, 365 pp., with Pronouncing Vo- cabulary and Index. Illustrated with Engravings, Maps and Charts. In compliance with a demand for separate Histories of the Early Eastern Monarchies, of Greece and Rome, an edition 0/THALHEIMER's MANUAL OF ANCIENT HISTORY** three Parts has been published, viz: 1. Thalheimer's History of Early Eastern Monarchies. 2. Thalheimer's History of Greece. 3. Thalheimer's History of Rome. The First embraces the pre-classical Period and that of Persian Ascendency. The Second, Greece and the Macedonian Empires. The Third, Rome as Kingdom, Republic and Empire. Each part sufficiently full and comprehensive for the Academic and Univer- sity Course, Liberally illustrated with accurate Maps. Large 8vo., full cloth, For convenience the numbering of pages and chapters corresponds with that of Thalheimer's Ancient History, so that these separate volumes can be used in classes partially supplied with the complete work. THALHEIMER'S MEDIAEVAL AND MODERN HISTORY. A Manual of Mediaeval and Modern History. 8vo., cloth, uniform with Thalheimer's Ancient History. 455 pp., and very full Index. Numerous double-page Maps. A sketch of fourteen centuries, from the fall of one empire at Ravenna to the establishment of another at Berlin. 36159 924229 THE UNIVERSITY OF CALIFORNIA LIBRARY