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 
 
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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