QA531 CORNELL UNIVERSITY LIBRARY FROM The C.E. College QA 531.B87 me " UniVersi,y Ubrary P'?"e and spherical trigonometry. 3 1924 004 124 685 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004124685 JAMES THOMAS, SIGMA NU HOUSE, Ttwata Y Y PLANE AND SPHERICAL TRIGONOMETRY. EDWARD BROOKS, A. M., Ph. D., Superintendent of Public Instruction in Philadelphia; Author of "Normal Series of Mathematics," "Normal Methods of Teaching," "Mental Science and Culture," "Philosophy of Arithmetic," etc. PHILADELPHIA: CHRISTOPHER SOWER COMPANY, 614 Arch Street. Copyright, 1891, By EDWARD BKOOKS. Westcott & Thomson, Shekman & Co. Sereotupen and Blectrolypers, Phila. Printers, Phila. PREFACE. The little work on Plane Trigonometry, written by the Author and published, in connection with his Elementary Geometry, some twenty years ago, served to introduce the subject of Trigonometry into many schools not prepared to use the larger works on that subject. This work is still well adapted to the wants of many institutions, but other schools using the writer's series of mathematics desire a more complete work on the subject. To meet this demand, the present treatise has been pre- pared. It aims to furnish just so much of the subject as is taught in our best schools and colleges. Great care has been taken to give clearness and simplicity to the treat- ment, and to so grade the difficulties as to make the path- way of the student - smooth and easy. In treating the subject the method of ratios, now gen- erally adopted by mathematicians, has been employed. The old method of lines is presented in the latter part* of the work, but care has been taken not to mix the two methods in the development of the principles. Some of the more difficult and less practical parts of the subject are printed in smaller type, and may be omitted by 4 PREFACE. students desiring a shorter course. A large number of carefully graded problems are given to aid the student in fixing the principles and understanding their applica- tion. These exercises can be used at the option of the teacher and the requirements of the student. In preparing the work, the best American and English works on the science have been consulted, and some valuable material has been derived from Casey and Tod- hunter, especially in the Exercises for the application of the principles. Philadelphia, \ EDWARD BROOKS. May 10, 1891. i CONTENTS. PAGE Introduction 9 PLANE TRIGONOMETRY. SECT. I. Measurement op Angles 23 II. Trigonometrical Functions 27 III. Functions of Angles in General 36 IV. The Sum and Difference of Two Angles 51 V. The Theorems or Trigonometry 59 VI. Numerical Value of Sines, Tangents, etc 63 VII. The Solution of Triangles 72 VIII. Practical Applications 85 IX. Supplement 91 SPHERICAL TRIGONOMETRY. X. Introductory Definitions 107 XI. The Right Spherical Triangle 110 XII. The Oblique Spherical Triangle 125 XIII. Supplement . . ; 144 5 HISTORY OF -TRIGONOMETRY AND LOGARITHMS. Trigonometry is believed to have originated with the Greek astronomers of Alexandria. The foundations of the science seem to.have been laid by Hipparchus and Ptolemy. The first step in the science was the use of a table of chords, which served the same pur- pose as our table of sines. Ptolemy's celebrated work, the Almagest, contains a table of chords expressed in terms of the radius ; and also the equivalent of several of our present formulas of trigonometry. Its treatment of spherical triangles was much more complete than that of plane triangles, which is natural, since the science was de- veloped in the interests of astronomy. " The Indians at a very early date are known to have been familiar with the elements of the science, which they probably obtained from the Greeks. They introduced tables of half-chords, or sines, instead of chords, understood the relation between the sines and cosines of an arc and its complement, and could find the sine of half an arc from the sine and cosine of the whole arc. The Arabs were acquainted with the Almagest, and probably learned from the Indians the use of the sine. Albategnius (930 a. d.) used the sine regularly, and was the first to calculate sin (j> from the formula sin ^ -r cos f — k. He was acquainted with the formula cos tt= cos b cos a -4- sin 6 sin c cos A for a spherical triangle ABC. Abu l'Wafa of Bagdad (b. 940) was the first to introduce the tangent as an independent function. Gheber of Seville, in the 11th century, wrote an astronomy, the first book of which contains an article on trigonometry, much in advance of that of the Almagest. He gave proofs of the formulas for right spherical triangles, and presented for the first time the formula cos B = cos b sin A, cos c = cot A cot B. He, however, made no advance m plane trigonometry^ Johannes Mttller (1536-1476), known as Regiomontanus, wrote a treatise on the Almagest, in which he reinvented the tangent, and calculated a table of tangents for each degree, though he made no use of it and did not use formula involving the tangent. This is 6 HISTORY OF TRIGONOMETRY AND LOGARITHMS. 7 said to have been the first complete European treatise on trigo- nometry ; but its methods were in some respects behind those of the Arabs. Copernicus (1473-1543) gave the first simple demonstration of the fundamental formulae of spherical trigonometry. George Joachim, known as Rheticus (1514-1576), wrote a work which con- tains tables of sines, tangents, and secants of arcs at intervals of 10" from 0° to 90°. He found the formulae for the sines of the half and third of an angle in terms of the sine of the whole angle. Vieta (1540-1603) gave formulas for the chords of multiples of a given arc in terms of the chord of the simple arc. Albert Girard (1590-1634) published a work containing theorems which gave the areas of spherical triangles, and also employed the principle of sup- plementary triangles. He used the notation sin, tan, sec, for the sine, tangent, and secant of an arc. Newton gave the series for an arc in terms of its sine, from which he obtained the series for the sine and cosine in powers of the arc. James Gregory in 1670 discovered the series for the arc in powers of the tangent, and for the tangent and secant in powers of the arc. Leibnitz published in 1693 the- series for the sine of an arc in powers of the arc. The greatest advance in the science was made by Euler (1707-1783), who really reduced the subject to its present condition. He intro- duced the present notation into general use, and made the transition from the geometrical conception of trigonometrical functions as lines to the analytical conception of functions of angles. The exponential values of sines and cosines, De Moivre's theorem, etc., are all due to Euler. History of Logarithms. — Logarithms were invented by Lord Na- pier, Baron of Merchiston, in Scotland. His first work upon the sub- ject, entitled Mirifica Logarithorum Canonis, was published in 1614, and gave an account of the nature of logarithms, and a table of natural sines and their logarithms for every minute of the quadrant to seven and eight figures. A second work, published after Napier's death by his son in 1619, explained the method of constructing his table.' These works did not contain the logarithms of numbers, but of sines ; and he called his numbers, not logarithms, but artificials. Napier's system of logarithms was afterward improved by Henry Briggs, a contemporary and friend of the inventor. Assuming 10 for the basis, he constructed a system of logarithms corresponding to our system of numeration, which is much more convenient for the ordi- nary purposes of calculation. Briggs's first work, a small octavo tract of 16 pages, was published in 1617, and contains the first published 8 HISTORY OF TRIGONOMETRY AND LOGARITHMS. table of decimal or common logarithms. It gave the logarithms of numbers from unity to 1000 expressed to 14 places of decimals. A copy of the tract, now very rare, is found in the British Museum. In 1624, Briggs published a second work, entitled Arithmetica Loga- rithmica, which contains the logarithms of numbers from 1 to 20,000, and from 90,000 to 100,000, calculated to 14 decimal places. In 1628, Adrian Vlacq, a native of Holland, published a work containing the logarithms of all numbers from 1 to 100,000. Vlacq's table is that from which all the hundreds of tables since published have been derived. It contained many errors, which have gradually been dis- covered and corrected; but, with one or two exceptions, no fresh calculations have ever been made. The first publication of the common logarithms of trigonometrical functions was made in 1620 by Gunter, a colleague of Briggs in Gresham College. This work contained logarithmic sines and tan- gents for every minute of the quadrant to 7 decimal places. In 1633, Vlacq published a work by Briggs, entitled Trigonometrica Britan- nica, which contained logarithmic sines and tangents at intervals of a hundredth of a degree. In the same year Vlacq published his Trigo- nometrica Artificialis, giving logarithmic sines and tangents for every 10 seconds of the quadrant to 10 decimal places. These were calcu- lated from the natural sines, etc., of the Opus Palatinum of Rheticus. This work fixed the method of applying logarithms to minutes and seconds, and it has never been superseded. Napier's system of logarithms is not nOw in use. A modification of this system is called the Napierian, or Hyperbolic, system. It is called Hyperbolic because the logarithms represent the area of a rectangular hyperbola between its asymptotes. The base of the Napierian system is 2.718, and is denoted by the letter e. The first logarithms to the base e were published by John Speidell in 1619, in a work entitled New Logarithms. It contains hyperbolic log. sines, etc., for every minute of the quadrant to 5 places of decimals. * For information on centesimal logarithms, antilogarithms, logistic numbers, Gaussian logarithms, etc., see Encyclopaedia Britannica, from which most of the above history is collated. INTRODUCTION. / THE LOGARITHMS OF NUMBERS. 1. Logarithms are a species of numbers used to abbreviate multiplication, division, involution, and evolution. 2. The Logarithm of a number is the exponent denoting the power to which a fixed number must be raised to produce the first number. Thus, if B x = N, then x is called the logarithm of K. 3. The Base of the system is the fixed number which is raised to the different powers to produce the numbers. Thus, in B x = N, x is the logarithm of iVto the base B ; so in 4 3 = 64, 3 is the logarithm of 64 to the base 4. 4. The term logarithm, for convenience, is usually written log. The expressiens above may be written log N= x ; and log 64 = 3. 5. In the Common System of logarithms the base is 10, and the nature of logarithms is readily seen with this base. Thus, 10 1 =10; hence log 10 =1. 10 2 = 100 ; hence log 100 = 2. 10" = 1000 ; hence log 1000 = 3. 10 2.369 _ 234 . hence log 234 = 2.369. 6. We shall first derive the general principles of logarithms, the Iiase being any number, and then explain the common numerical system. ,,,, Principles of Logarithms. Prin. 1. The logarithm of 1 is 0, whatever the base. For, let B represent any base, then B° ~ 1 ; hence by the definition of a logarithm, is the log of 1, or log 1 = 0. 9 10 INTRODUCTION. Prin. 2. The logarithm of the base of a system of logarithms is unity. For, let B represent any base, then B 1 = B ; hence 1 is the log of B, or log B — 1. Prin. 3. The logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers. For, let m = log M, and n = log N. Then, B™ = M, Bn = N. Multiplying, , £™ + ™ = if x N. Hence, m + « = log (if X N). Or, log (if X JV) = log if + log JV". Prin. 4. The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor. For, let m = log M, and n = log N. Then, B™ — M, Bn = N. Dividing, Bf ~ n = M -f N. Hence, log (if -r- N) = m — n. Or, log (if -h JV) =■ log if— log iV. Prin. 5. The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent vf fhe power. For, let m = log M. Then, J5m = if. Raising to nth power, B"- x ™ = M % . Whence, log M n = n X m. Or, log M n — n X log if. J 'J -\" Prin. 6. 27«e logarithm of the root of a number is equal to the logarithm, of the number divided by the index of tiie root. For, let m = log M. Then, j5'« = if. Taking nth root, .6* ' = if* Whence, log if" = —• Or, W if A = M". INTRODUCTION. 11 7. These principles are illustrated by the following exercises, ' which the pupil will work. EXERCISES. Prove each of the following : 1. Log (a.&.c.) = log a + log 6 + log c. 2. Log | ■ — J = log a + log 6 — log c. 3. Log a B = n log a. 4. Log (aPbv) = x log a + y log 6. 5. Log — — = x Jog-a +yiog b — z log c. 6. Log l/ft6 = i log a + £ log 6. 7. Log (a 2 — a 2 ) = log (a + x) + log (a — x). 8. Log \/a* — x 2 = £ log (a + a) + J log (a — x). 9. Log a 2 i^ar 2 = •$ log a. 10. Log ^f +x y - = i [log (a-*) -3 log (a + e )|. Common Logarithms. 8. In Common Logarithms the base is 10. This base is most convenient for numerical calculations, because our numerical systeai^ is decimal. 9. In this system every number is conceived to be some power of 10, and by the use of fractional and negative exponents may be thus, approximately, expressed. 10. Raising 10 to different powers, we have— 10° = 1 s hence = log 1. 10 1 = 10 ; hence 1 = log 10. 10* — 100 ; hence 2 = log 100. 10 3 - 1000 ; hence 3 = log 1000. etc. etc. Also, 10 -1 = .1 ; hence — 1 = log .1. 10-* == .01 ; hence — 2 ^ log .01. 10- 3 = .001 ; hence — 3 = log .001. 12 INTRODUCTION. 11. Hence the logarithms of all numbers between 1 and 10 will be + a fraction ; between 10 and 100 will be 1 + a fraction ; between 100 and 1000 will be 2 + a fraction ; between 1 and .1 will be — 1 + a fraction ; between .1 and .01 will be — 2 + a fraction ; between .01 and .001 will be — 3 + a fraction. 12. Thus it has been found that the log of 76 is 1.8808, and the log of 458 is 2.6608. This means that 10 1 - 8808 = 76, and 10 2 - 6608 = 458. 13. When the logarithm consists of an integer and a decimal, the integer is called the characteristic, and the decimal part the mantissa. Thus, in 2.660865, 2 is the characteristic, and .660865 is the mantissa. 14. The logarithm of a number less than 1 is negative ; but is written in such a .form that the fractional part is always positive. Thus, log .008 == log (8 X .001) = log 8 + log .001 = 0.903090 .— 3. Now, this may be written 3.903090. The minus sign is written over the characteristic to show that it only is negative. Principles of Common Logarithms. Piiin. 1. The characteristic of the logarithm of a number is one less than the number of integral places in the number. . For, from Art. 10, log 1=0 and log 10 = 1 ; hence the logarithm of numbers from 1 to 10 (which consist of one integral place) will have for the characteristic. Since log 10=1 and log 100 = 2, the logarithm of numbers from 10 to 100 (which consist of two integral places) will have one for the characteristic, and so on ; hence the characteristic is always one less than the number of integral places. Pbin. 2. The characteristic of the logarithm of a decimal is neg- ative, and is equal to the number of the place occupied by the first significant figure of the decimal. For, from Art. 10, log .1 =— 1, log .01 = — 2, log .001 — — ?>; hence the logarithms of numbers from .1 to 1 will have — 1 for a characteristic : the logarithms of numbers between .01 and .1 will INTRODUCTION. 13 have — 2 for a characteristic, and so on ; hence the characteristic of a decimal is always negative, and equal to the number of the place of the first significant figure of the decimal. Prin. 3. The logarithm of the product of any number multiplied by 10 is equal to the logarithm of the number increased by tj For, suppose log M =m ; then, by Prin. 3, Art. 6, log (M X 10) = log M+ log 10 ; but log 10 = 1 ; Hence, log (M X 10) = m + 1. Thus, log (76 X 10) = 1.880814 + 1 ; or log 760 = 2.8808 14. Prin. 4. The logarithm of the quotient of any number divided by 10 is equal to the logarithm of the number diminished by 1. For, suppose log M = m ; then, by Prin. 4, Art. 6, log (M -r- 10) = log M — log 10 ; Henee, log (M — 10) = m — 1. Thus, log (458 -7- 10) = 2.660865 — 1 ; or log 45.8 = 1.660865. Prin. 5. In changing the decimal point of a number we change the characteristic, but do not change the mantissa of its logarithm. This follows from Principles 3 and 4. To illustrate : log 234 = 2.360216. log .234 = 1.369216. log 23.4 = 1.369216. log .0234 = 2.369216. log 2.34 = 0.369216. We thus see that when we change the place of the decimal point of a number we change the characteristic, but do not change the decimal part of the logarithm. 15. Negative logarithms are sometimes written with 10 or a mul- tiple of 10 after them, and a positive characteristic equal to the differ- ence between its real characteristic and 10 or the given multiple of 10. Thus, 2.369216 may be written 8.369216 — 10; and 13.369216 may be written 7.369216 — 20. 14 INTRODUCTION. Table of Logarithms. 16. A Table of Logarithms is a table by means of which we can find the logarithms of numbers, or the numbers corresponding to given logarithms. 17. In the annexed table the entire logarithms of the numbers up to 100 are given. For numbers greater than 100 the mantissa, alone is given; the characteristic being found by Prin. 1, page 12. 18. The numbers are placed in the column on the left, headed N ; their logarithms are opposite, on the same line/ The first two figures of the mantissa are found in the first column of logarithms. 19. The column headed D shows the average differences of the ten logarithms in the same horizontal line. This difference is found by subtracting the logarithm in column 4 from that in column 5, and is very nearly the mean or average difference. Note. — The logarithms given in the Table are complete to six places. They can be readily changed to five-place logarithms by omitting the ' sixth figure, and when the sixth figure is 5 or more, increasing the fifth figure by 1. Similarly, we find four-place and three-place logarithms. To Find the Logarithm of any Number. 20. To find the logarithm of a number of one or two figures. . Look on the first page of the table, in the column headed N, and' opposite the given number will be found its logarithm. Thus, the logarithm of 25 is 1.397940, " " 87 is 1.939519. 21. To find the logarithm of a number of three figures. Look in the table for the given number ; opposite this, in column headed 0, will be found the decimal part of the logarithm, to which we prefix the characteristic 2, Prin. 1. Thus, the logarithm of 325 is 2.511883, ' " " 876 is 2.942504. 22. To find the logarithm of a number of four figures. Find the three left-hand figures in the column headed N, and opposite to these, in the column headed by the fourth figure, will be INTRODUCTION. 15 found four figures of the logarithm, to -which two figures from the column headed are to be prefixed. The characteristic is 3, Prin. 1. Thus,* the logarithm oT"34o6 is 3.53S574. 743$ is 3.S71456. 23. In some of the columns small dots are found in the place of figures : these dots mean xeros. and should be written xeros. If the four figures of the logarithm fall where xeros occur, or if, in passing back from the four figures found to the xero column, any of these dots are passed over, the two figures to be prefixed must be taken from the line just below. Thus, the logarithm of 173$ is 3.240050, 263$ is 3.421275. 24. Ibfitd Me logarithm of a number q/" more than tov* figures. Place a decimal point after the fourth figure from the left hand, thus changing the number into an integer and a decimal. ~ Find the mantissa of the entire part by the method just given. Then from the column headed D take the corresponding tabular difference, mul- tiply it by the decimal part, and add the product to the mantissa already found: the result will be the mantissa of the given number. The characteristic is determined by Prin. 1. If the decimal part of the product exceeds .5. we add 1 to the entire part ; if less than .5, it is omitted. EXERCISES. 1. Find the logarithm of 234567. Solitiox. — The characteristic is 5. Prin. 1. Placing a decimal point after the fourth figure from the left, we have 2345.67. The decimal part of the logarithm of 234-5 is .370143 ; the number in column D is 1S5 : and 1 v* X -67 = 123.95. and since .95 exceeds .5. we have 124. which, added to .370143, gives .370267: hence, log 234567 = 5,370267. 16 INTRODUCTION. Find the logarithm 2. Of 4567. Ans. 3.659631. 3. Of 3586. 4. Of 11806. 5. Of .4729. 6. Of 29.337. Ans. 3.554610. Ans. 4.072102. Ans. T.674769. Ans. 1.467416. 7. Of 734582. Ans. 5.866040. 8. Of 704.307. Ans. 2.847762. 9. Of .000476. Ans. ¥.677607. 10. Of— ■ Ans. 1.960233. 11. Of 400 375 463 Ans. 1.908450. Note. — To find the logarithm of a common fraction, subtract the log- arithm of the denominator from the logarithm of the numerator. 25. To find the number corresponding to a given logarithm. 1. Find the two left-hand figures of the mantissa in the column headed 0, and the other four, if possible, in the same or some other column on the same line ; then, in column N, opposite to these latter figures, will be found the three left-hand figures, and at the top of the page the other figure of the required number. 2. When the exact mantissa is not given in the table, take out the four figures corresponding to the next less mantissa in the table ; sub- tract this mantissa from the given one ; divide the remainder, with ciphers annexed, by the number in column D, and annex the quo- tient to the four figures already found. 3. Make the number thus obtained correspond with the character- istic of the given logarithm, by pointing off decimals or annexing ciphers. EXERCISES. 1. Find the number whose logarithm is 5.370267. Solution. — The mantissa of the given logarithm is . . .370267 The mantissa of the next less logarithm of the table is . . .370143 and its corresponding number is 2345. Their difference is , . . 124 The tabular difference is 185 The quotient is 185)124.00(.67 Hence the required number is 234567 Note. — If the characteristic had been 2, the number would have been 234.567 ; if it had been 7, the number would have been 23456700 ; if it had been 2, the number would have been .0234567, etc. INTRODUCTION. 17 Find the number whose logarithm 2. Is 3.659631. 3. Is 3.563125. 4. Is 2.554610. 5. Is 1.072102. 6. Is 4.883150. Ans. 4567. Ans. 3657. Ans. 358.6. Ans. 11.806. Ans. 76410. 7. Is 4.790285. Ans. 61700. 8. Is 2.674769. Ans. .04729. 9. Is 3.065463. Ans. .0011627. 10. Is 3".514548. Ans. .00327. 11. Is 1.84674-1. Ans. .00070265. Multiplication by Logarithms. 26. From Prin. 3, for the multiplication of numbers by means of logarithms, we have the following Rule. — Find the logarithms of the factors, take their sum, and find the number corresponding to the result; this number will, be the re- quired product. Note. — The term sum is used iu its algebraic sense. Hence, when any of the characteristics are negative, we take the difference between the sums of the positive and negative characteristics, and prefix to it the sign of the greater. If anything is to be " carried " from the addition of the mautissa, itnnust be added to a positive characteristic or subtracted from a negative one. When any of the characteristics are negative, we can write them as suggested in Art. 15, and proceed accordingly. EXERCISES. 1. Multiply 35.16 by .815. log 35.16 = 1.546049 log .815 — T.911158 Solution.- 28.6554 1.457207 457125 152)82.00(.54 Product, Find the product 2. Of .7856, 31.42. Ans. 24.6835. 3. Of 0.3854 by 0.0576. Ans. .022199. 4. Of 31.42, 56.13, and 516.78. Ans. 911393.7. 5. Of 31.462, .05673, and .006785. Ans. .01211168. 6. Of .06517, 2.16725, .000317, and 42.1234. Ans. .001886. 7. Of 2.3456, .00314, 123.789, .00078, and 67.105. Ans. .04772076. 2 18 INTRODUCTION. Division by Logarithms. 27. From Prin. 4, to divide by means of logarithms, we have the following Rule.— Find the logarithms of the dividend and divisor, subtract the latter from the former, and find the number corresponding to the result; this number will be the required quotient. Note.— The term subtract is here used in its algebraic sense : hence, when any of the.characteristies are negative we must subtract according to the principles of algebra. Negative characteristics may be written as in Art. 15, for subtraction. EXERCISES. 1. Divide 783.5 by .625. First Solution. log 783.5 = 2.894039 log .625 = 1.795830 3.098159 Quo. 1253.6 .097951 346)208(6 2. Divide 272.636 by 6.37. . 3. Divide 50.38218 by 67.8. 4. Divide 155 by .0625. 5. Divide 1.1134 by 0.225. 6. Divide 0.10071 by 0.00373. 7. Divide 435 X 684 by 583 X 760. The Cologarithm of a Number. 28. The Cologarithm of a number is the result arising from sub- tracting the logarithm of the number from 10. Thus, eolog N*= 10 — log N, and colog 40 = 10 — log 40, or 10— 1.60206 = 8.39794. 29. The cologarithm of a number may be written directly from the table by subtracting each term of the logarithm from 9, except the right-hand term, which must be taken from 10. 30. The cologarithm is used to simplify the operation of division Second Solution. log 783.5 = 12.894039 — 10 log .625 = 9.795880 — 10 Dif. = . 3.098159 Quo. 1253.6 Ans. 42.8. Ans. .7431. Ans. 2480. Ans. 5.04. Ans. 27. Ans. 671524. IXTRODUCTION. 19 when it is combined with multiplication. • Thus, suppose we wish to divide M by X. Now, log (M-i- N) = log M — log X. But, log X = 10 — colog N. Art. 2S. Substituting, log -1/— log N= log M-\- colog X — 10. Hence, instead of subtracting log X, we may add colog aY, and then deduct 10 from the sum. 31. Hence, to divide by means of the cologarithm of a number we have the following Rule. — Add the cologarithm of the dirisor to the logarithm of the dividend, subtract 10, and find the number corresponding to the result. Note. — The cologarithm is sometimes defined as the logarithm of the reciprocal of tlte number, and the rule for its use deduced accordingly. The cologarithm as defined above is usually known as the Arithmetical Complement. EXERCISES. 1. Divide 256.3 by 4.">.32. Solution.— log 856.3 2.932626 colog 45.32 8.343710 Quotient, 18.8945 1.276336 2. Divide 0.3156 by 78.35. log 0.3156 T.499137 colog 78.35 8.105961 Quotient, .004028 . 3.605098 3. Divide 3.7521 by 18.346. Ans. .204519. 4. Divide 483.72 by .30731. Ans. 1573.02. 5. Find value of 32.16 x 7.856 -r 45.327. Ans. 5.574. 6. Of 31.57 X 123.4 divided by 316.2 X .0316. Ans. 389.8884. 7. Of x, given x : 73.15 = 40.16 : 3167. Ans. 1.11237. 8. Of x, given 72.34 . 2. J19 = 357.48 : x. Ans. 12.448. 20 INTR OD UCTION. Involution by Logarithms. 32. From Prin. 5, to raise a number to any power, we have the following Rule. — Find the logarithm of the number, multiply it by the expo- nent of the power, and find the number corresponding to the result. EXERCISES. 1. Find the 4th power of 45. Solution.— log 45 = 1.653213 4 Power, 4100625 6.612852 2. Find the cube of 0.65. Ans. 0.2746. 3. Find the 6th power of 1.037. Ans. 1.243. 4. Find the 7th power of .4797. Ans. 0.005846. 5. Find the 30th power of 1.07. Am. 7.6123. Evolution by Logarithms. 33. From Prin. 6, to extract from any root of a number, we have the following Rule. — Find the logarithm, of the number, divide it by the index of the root, and find the number corresponding to the result. Note. — If the characteristic is negative, and not divisible by the index of the root, add to it the smallest negative number that will make it divisible, prefixing the same number with a plus sign to the mantissa. EXERCISES. 1. Find the square root of 576. Solution.— log 576 = 2.760422 2.760422 -=-2 = 1.380211 Hence, the root is 24. 2. Find the fourth root of .325. Solution.— log .325=1.511883 — 3 +3 4) — 4 + 3.511883 1.877971 Hence, the root is, .75504. INTRODUCTION. 21 3. Find the cube root of 7. Ans. 1.9129.. 4. Find the fifth root of 5. Ans. 1.3797. 5. Find the fifth root of .0625. Ans. .574348. 6. Find the seventh root of 7. Ans. 1.32047. 7. Find the tenth root of 8764.5. Ans. 2.479. Calculation of Logarithms. The pupil will naturally desire to know how these logarithms are calculated. While this is not the place to enter into a detailed ex- planation of the method of calculating logarithms, a general idea of the subject can be presented. In computing logarithms it is necessary to calculate only the log- arithms of prime numbers, since the logarithms of composite numbers may be obtained by adding the logarithms of their prime factors. The logarithms of the prime numbers were first computed by com- paring the geometrical and arithmetical series, 1, 10, 100, etc., and 0, 1, 2, etc., and finding geometrical and arithmetical means ; the arith- metical mean being the logarithm of the corresponding geometrical mean. This method was exceedingly laborious, involving so many multiplications and extractions of roots. The method now generally used is that of series, by which the. computations are much more easily made. The following formula is derived by algebraic reasoning : l 0gi l +x) = A(f-f + (-f + (-eK.). In this series the quantity A is called the modulus, which in the Napierian system is unity. The series, when A is one, put in a more convenient form, becomes l0 " (z+1) - l0gZ = 2 (^ + 3(^W + 5^W + etC -)- From which, knowing the logarithm of any number, we readily find the logarithm of the next larger number. The student will be interested in finding logarithms by this formula. Begin with 2, in which 2 = 1. 22 INTRODUCTION. The logarithm found will be the Napierian logarithm, and this multiplied by 0.434294 will give the common logarithm. Logarithms were invented by Lord Napier of Scotland, and are regarded as among the most useful inventions ever made. His sys- tem was subsequently improved by Henry Briggs, a cotemporary of Napier's, who, assuming 10 for a basis, constructed a system much more convenient for the ordinary purposes of computation. Napier's system was also modified by John Speidell, whose logarithms are now known as the Napierian or Hyperbolic logarithms. Briggs' logarithms are known as the Briggean or Common logarithms. It is generally believed that the so-called " Napierian logarithms " are identical with those first computed by Napier ; but this is not the case. For a more detailed statement of the origin of logarithms, see the History of Logarithms given in the Introduction of this work, page 6. PLANE TRIGONOMETRY. SECTION I. THE MEASUREMENT OP ANGLES. •1. Trigonometry is the science which investigates the relation of the sides and angles of triangles. 2. Plane Trigonometry treats of plane angles and tri- angles ; Spherical Trigonometry treats of spherical angles and triangles. 3. In every triangle there are six parts, three sides and three angles. These parts are so related that when certain ones are given, the others may be found. 4. .In Geometry the triangle can be constructed when a sufficient number of parts are given. In Trigonometry the unknown parts are computed from the known parts. 5. In order to subject a triangle to computation, we must be able to express its sides and angles by numbers. For this purpose proper units must be adopted. 6. The units of measure for the sides are straight lines of a fixed length, as the inch, foot, yard, etc. The units of measure for angles are degrees, minutes, and seconds. Note. — Trigonometry is really a numerical way of treating trian- gles in distinction from the geometrical way of treating them. The science extends also to the investigation of angles in general, and is then called Angular Analysis. 23 24 PLANE TRIGONOMETRY. Measures of Angles. 7. An angle is measured, as shown in geometry, by the arc intercepted between its sides, the centre of the circle being at the vertex of the angle. 8. The units of the arc are equal parts of the circum- ference called degrees, minutes, and seconds. A degree (marked °) is -j^ of the circumference ; & minute (marked ') is -fa of a degree ; and a second (marked ") is -fa of a minute.* 9. A Quadrant is one-fourth of the circumference of a circle. Each quadrant contains 90°, and is the measure of a right angle. 10. Reckoning from A, the arc AB is called the first quadrant; the arc BC the second quadrant; the arc CD the third quadrant ; the arc DA the fourth quadrant. The term quadrant is also applied in the same manner to the four equal parts of the circle. 11. Any arc AE, less than 90°, is said to be in the first quadrant; any arc AF, between 90° and 180°, is said to be in the second quadrant; any arc AG, between 180° and 270°, in the third quadrant, etc. 12. The Complement of an angle or an arc is the re- mainder obtained by subtracting the angle or arc from 90°/ Thus, the complement of arc AE is arc BE. 13. The Supplement of an angle of an arc is the re- mainder obtained by subtracting the angle or arc from 180°- Thus, the supplement of arc AF is arc CF. SUMER1CAL LEXGTHS OF ARCS. 25 14. According to these definitions, the complement of an arc greater than 90° is negative, and the supplement of an arc greater than 180° is negative. Thus, the complement of 120° is 90° — 120° = — 30°; and the supplement of 200° is 180° - 2(0° = - 20°. Numerical Lengths of Arcs. 15. The units of the circle — that is, degrees, minutes, and seconds — express equal parts of the circumference. An arc may also be expressed in numerical units cor- responding to a straight line. I. To find a numerical expression fan- an arc of a given num- ber of degrees, minutes, etc. The circumference of a circle is 2zR (B. V., Th. 8). Supposing J? = 1, we find the semi-circumference equal to jt = 3.14159265. Hence, Arc 180° = 3.14159265. Arc 1' = 0.00O29OSSS. Arc 1° = 0.01745329. Arc 1" = 0.000004848. II. To find the number of degives, minutes, etc.. in an arc equal to the radius. Since, 2,Ti2=360°, -£ = 180°. „ D 180° 180° __ — or Hence, R = — = Bum2Q5 = o, .29o, , 95. = 3437'.74677= 206264".806. 16. The angle at the centre measured by an arc equal to the radius, it is thus seen, is an invariable angle, whatever the length of the radius ; hence it is often taken as the unit of angular measure. 17. Since when the radius is unity, 2;? = 360°, t: is often used to express two right angles. Then 5 equals a right 26 PLANE TRIGONOMETRY. ' It angle; 2tt equals four right angles; j = an angle of 45°, etc. 18. This method of measuring an angle is called the cir- cular measure of an angle. The method by degrees, etc. is called the sexagesimal method. Note. — A third method, called the centesimal method, was pro- posed by the French at the introduction of the metric system. In this system the right angle was divided into 100 parts, called grades, each grade into 100 parts called minutes, etc. EXERCISES I. 1. How many degrees in an angle denoted by 2tt? By tt? By Jtt? By 3*-? ByfTr? By £71-? By^jr? mr? 2. Express in terms of it an angle of 180° ; of 90° ; of 60° ; of 45° ; of 30° ; of 70° ; of 80° ; of 63° ; of 67° 30' ; of 52° 30'. 3. How many degrees in an arc whose length is equal to the diam- eter of the circle 1 Ans. 1 14°. 59 +. 4. How many degrees in an arc whose length is 0.6684031 ? Whose length is 2.0052093 ? Ans. 38° 17' 48" ; 114° 53' 24". 5. Express ^ of a right angle in degrees and minutes ; also in cir- cular measure. Ans. 28° 7'.5 ; &n. 6. What is the length of an arc of 60° when the radius is 8? When the radius is 12 ? Radius 20 ? 7. When the radius is 8, required the length of an arc of 45° ; of 75° ; of 22° 30' ; of 52° 30' ; of 33° 45'. 8. Find the diameter of a globe when an arc of a great circle of 25° measures 4 feet. Ans. 18.3346. 9. Find the number of degrees in a circular arc 30 inches in length, the radius being 25 inches. Ans. 68° 45' 17" +! TRIGONOMETRICAL FUNCTIONS. 27 SECTION II. TRIGONOMETRICAL FUNCTIONS. 19. In Trigonometry, instead of comparing the angles of triangles or the arcs which measure them, we compare certain lines or ratios of lines called the functions of the angles. 20. A Function of a quantity is something depending on the quantity for its value. These functions in Trigonometry are the sine, cosine, tangent, cotangent, secant, and cosecant. 21. Since every oblique triangle can be resolved into two right triangles by drawing a perpendicular from one of its angles to the opposite side, the solution of all triangles can be made to depend on that of right triangles. 22. The functions, sine, cosine, etc., are used to express the relation of the sides of the right triangle.' These terms will now be defined and illustrated. 23. In the right triangle ABC, let AC be denoted by b, BCby a, and AB by c; then we have the following defi- nitions : 1. The Sine of an angle is the ratio of the opposite side to the hypotenuse. . BC a . v AC b Thus, smA = j E =-;smB=j2 = c- 2. The Tangent of an angle is the ratio of the opposite side to the adjacent side. Thus, tan^-^=5;tan J B=g^=-- 28 PLANE- TRIGONOMETRY. 3. The Secant of an angle is the ratio of the hypotenuse to the adjacent side. _,, . AB c _ AB c lhus, sec A = —r-z = t ■ sec B = -=— = - • ^40 b ' BC a 4. The Cosine of an angle is the sine of the complement of the angle. Thus, cos A = sin B = - ; hence cos A = - • Also, cos £ = sin 4 = - ; hence cos B = - ■ c ' c 5. The Cotangent of an angle is the tangent of the comple- ment of the angle. Thus, cot A = tan B = - ; cot B = tan A = -■ a' b 6. The Cosecant of an angle is the secant of the complement of the angle. C G Thus, esc A = sec B = - ; esc B = sec A = r ■ a' b 24. If A denotes any angle or arc, then we have from the above explanations, sin A = cos (90° — A) ; cos_ A = sin (90° — A). tan A = cot (90° — A); cot A = tan (90° -=- A). sec A = esc (90° — A) ; esc vl = sec (90° — ^4). Note. — The above definitions of cosine, cotangent, and cosecant show their true relation to the sine, tangent, and secant. We may, however, define them independently, as follows : 1. The cosine of an angle is the ratio of the adjacent side to the hypotenuse. . 2. The cotangent of an angle is the ratio of the adjacent side to the opposite side. 3. The cosecant of an angle is the ratio of the hypotenuse to the opposite side. ., TRIGONOMETRICAL FUNCTIONS. 29 .25. The sine, cosine, tangent, cotangent, etc. are called Trigonometrical Functions or Ratios. A large part of Trigo- nometry consists in the investigation of the properties and relations of these functions of an angle. Note. — If the cosine of an angle is subtracted from unity, the re- mainder is called the- versed-sine of an angle ; if the sine of an angle is subtracted from unity, the remainder is called the coversed-sine of the angle. Thus, vers A =J — cos A ; covers A = 1 — sin A. EXERCISES II. 1. Find the values of the trigonometrical functions of A, when a = 3, 6 = 4, and c = 5. Solution. — By Art. 23, . a 3 . b 4 . a 3 sin A = - = — ; cos A = — = — ; tan A = , = -7 : etc. e 5 c 5 6 4 2. Find the values of the trigonometrical functions of A when a = 5, 6 = 12, and c = 13. When a = 8, 6 = 15, and c = 17. 3. Write all the functions of B in the triangle of Fig. 2. 4. Find the functions of A and of B when a = 10 and 6 = 24. When a = 18 and c = 82. When 6 = 75 and c = 85. 5. Find the functions of A and of B when a =m and c = mi/2. When 6 = \/2mn and c = m + n. 6. Find a, if sin A = f and c = 5. Find 6, if tan A = ^ and c=13. Find c, if sec A = 2 and 6 = 5. 7. Compute the functions of A when as = |5. When6=:|c. When a + 6 = f c. When a — 6 = £c. 8. Construct a right triangle when sin A = f and a = 9. When tan .4 = f and 6 = 9. When esc A = 3.5 and c = 4J. 9. Compute the legs of a right triangle when sin A = 0.4, cos A = 0.6, and c = 4.5. Construct the triangle. 10. Given A + B = 90° ; to prove the following : ■ sin A = cos B. tan .4 = cot B. sec .4 -- esc B. cos .4 = sin B, cot .4 = tan B. esc .4 = sec B. 30 PLANE TRIGONOMETRY. Fundamental Formulas. 26. We now proceed to derive some fundamental form- ulas expressing the relations of trigonometrical functions. I. Formulas expressing the relation of sine and cosine. 1. In the right triangle ABC, by geometry, We have, a 2 + b 1 = c\ Whence, -; + -: = 1. ' t re 1 Substituting the values of sin A and cos A (Art. 23), We have, sin 2 A + cos 2 A = 1. [1] That is : The sum of the squares of the sine and cosine of an angle is equal to unity. Note. — We write sin 2 A for (sin Af and cos 2 A for (cos Ay as a matter of convenience, and similarly with the powers of the other trigonometrical functions. 2. From the above formula we have sin 2 A = l — cos 2 A = (1 + cos A) (1 — cos A). cos 2 A= 1 — sin 2 A = (1 + sin A) (1 — sin A). II. Formulas expressing the relation of tangent and cotangent. 1. Prom Art. 23, (1) and (4), we have sin A a b a . , a ^A = c^c-b' hnttanA = b- Art - 23 >( 2 )- * Whence, tan A = r ■ T21 oosA L J That is : The tangent of an angle is equal to the sine divided by the cosine. FUNDAMENTAL FORMULAS. 31 2. From Art. 23, (5), we have , . b . , cos A b a b cot A = - ; but 7 = - -7- - = - • a sin A c c a Whence. cot A = — — - • f31 sin A L J That is : The cotangent of an angle is equal to the cosine divided by the sine. 3. Taking the product of [2] and [3], we have tan A x cot A = 1. Whence, tan A =^ ; or At A- ^ [4] That is : The tangent and the cotangent of an angle are re- ciprocals of each other. III. Formulas expressing the relation of secant and cosecant to the other functions. a c 1. From Art. 23, sin A == - and esc A = -• c a Whence, sin A = ; or esc A = — — r • T51 esc A' sm A L J That is : The sine and cosecant of an angle are reciprocals of each other. b c 2. From Art. 23, cos A = - and sec A = r • ' c b Hence, cos A = — and sec A = r- ■ [61 sec A cos A J That is : The cosine and secant of an angle are reciprocals of each other. 3. In the right triangle ABC We have \ Q. Fig. 10. 39. Suppose two lines, as MN and PQ, to intersect each other at right angles in the point A. Then, 1. AU lines estimated upward from wl — — -> ~r~ W MN are positive ; and all lines esti- mated downward from MN are nega- tive. 2. All lines.estimated from the ver- tical line PQ toward the right are positive ; and all lines esti- mated from PQ toward the left -are negative. Thus, in Fig. 10, .4Cand BC are plus; jB,G is plus; AC is minus ; B 2 C 2 is minus ; and B a C s is minus. The sign of AB is not supposed to change for the different positions AB lt AB 2 , etc. ' 40. These principles, combined with the definitions of Art. 37, enable us to determine the algebraic sign of the trigonometrical functions of angles in the four quadrants. Thus, in the expressions of Art. 36 : 1. Since BC and AC are both positive in the first quad- rant, the sine, cosine, tangent, etc. of A are plus. 2. Since B^C is positive, and AC is negative, in the second quadrant the sine is plus, the cosine minus, the tangent (= sine -s- cosine) is minus ; etc. 3. Since B 2 C 2 - and AC 2 are both negative in the third quadrant, the sine and cosine are both minus, the tangent (= sine -»- cosine) is plus ; etc. 4. Since B 3 C 3 is negative and AC 3 is positive in the fourth quadrant, the sine is minus, the cosine plus, the tangent minus; etc. 41. If we let A, Ai, A 2 , and A 3 denote respectively the angles CAB, CAB M etc., as in Art. 36, we have the following : 40 PLANE TRIGONOMETRY. 1. Sin A is + ; cos A is -f- ; tan A is + ; cot A is + ; sec A is + ; etc. 2. Sin Ax is -f ; cos A x is — ; tan Ax is — ; cot A, is — ; sec Ai is — ; etc. 3. Sin j4 2 is — ; cos A 2 is — ; tan A 2 is + ; cot A 2 is + ; sec ^ 2 is — ; etc. 4. Sin .<4 S is — ; cos A 3 is + ; tan A 3 is — ; cot -4 3 is — ; sec A 3 is + ; etc. • 42. These values of the trigonomatrical functions in the different quadrants are concisely represented in the accom- panying table : Quadrants .... I. + II. III. IV. + - Cosine and Secant . . + _. - + Tangent and Cotangent . . . + - + - 43. The following exercises will serve to fix these prin- ciples clearly in the mind. Let the student illustrate them with a diagram. EXERCISES V. Show the sign of each function of an angle, 1. Of 60°- 3. Of 100°. 5. Of 200°. 7. Of 300°. 2. Of 75°. 4. Of 150°. 6. Of 250°. 8. Of 320°. 9. In what quadrant is 127°? 256°? 295°? 470°? 510°? |n-? }7T? 3tT? 7T + 40°? 7T+100? 27r+60 O ? M7T + 45? 10. Give the quadrant of angle A, if sin A is +, and tan A is — . 11. Give the quadrant of angle A, if cos A is — , and cot A is +. 12. Give the limits of angle A, if tan A is — , and osc A is +. 13. Give the limits of angle A, if tan A is +, and sec A is — . FUNCTIONS OF NEGATIVE ANGLES.. 41 14. If tan A = — f , and cos A is negative, tell the quadrant of A, and find values of all the functions of A. 15. If sin A = — f, and tan A is positive, tell the limits of A, and find the values of all the functions of A. 16. In a triangle, which functions may be negative, and when ? 17. In a triangle, which functions will determine the angle, and which will not ? 18. For what angle in each quadrant are the absolute values of the sine and cosine the same? Functions of Negative Angles. 44. Angles may also be regarded as positive and nega- tive when reckoned in opposite directions. Thus, in Fig. 7, if we regard angles reckoned from OA around in the direc- tion of AAi as positive, angles reckoned in the opposite direction towards AA 3 may be regarded as negative. 45. Suppose, in Fig. 11, BCB 3 is perpendicular to CC h and BC= B 3 C; then the sides and angles of the two triangles BAC and l? 3 ^Care respect- ively equal. Let A denote angle CAB, Fig. 11. then — A will denote angle CAB 3 . 1. Now, sin A = -j= ', an( i sin. (— A) = -j= ■ and BC and BiC are numerically equal, but have opposite signs (Art. 41) ; hence the results are equal with opposite signs. Therefore, sin (— A) = — sin A. 2. Also, cos A = -ju ; and cos (— A) = -j~n ! and the lines are equal with the same signs. 42 PLANE TRIGONOMETRY. Therefore, cos (= A) = cos A. „ .. , ., sin (—A) —sin A , , 3. Also, tan (— A) = — 7 — 77 = 7- = — tan A. ' v ' cos (— A) cos A . ., , ,■ „ cos (—-4) cos vl , , 4. Also, cot (— .4) = -r— ; j< = : — 7 = — cot A ' v J ■ sm (— ^4) — sin A 5. Also, sec (— 4) = z tt = -. = sec 4. ' v J cos (— 4) cos A 6. Also, esc (— A) = - — 7 7t = — - — r = — esc A. ' K ' sin (— A) — sin A Extension of Fundamental Formulas. 46. The Fundamental Formulas of Art. 26 were derived for acute angles, but it may be readily shown that they apply- to angles of any magnitude. 1. Thus, For. [1], Art. 26, sin 2 A + cos 2 4 = 1, holds for all values of 4 ; for whether a and 6 are plus or minus, a 2 and 6 2 are always plus, and & is plus j therefore the formula is .always true. 2. So also For. [2], Art. 26, . sin 4 tan A = 7, cos A holds for all values of A ; for both members equal a-r-b (see Art. 26), and hence will be equal whatever be the signs of a and b. 3. In the same manner all the formulas of Table J., Art. 27, are shown to be true for any value of angle A. Note. — It will be interesting to the student to derive these form- ulas for angles terminating in each one of the four quadrants. REDUCTION OF FUNCTIONS, ETC. 43 Reduction of Functions to the First Quadrant. 4 7 . Having seen that the trigonometrical functions apply to angles of any magnitude, we shall now show that the functions of all angles greater than a right angle can be reduced to functions of angles less than a right angle. 48. Suppose, in Fig. 12, that the diameters BB 2 and B,B d are drawn, making equal angles with MN. Then the triangles BAC, B,AC„ BzAC-i, and B S AC 3 are equal. De- note the angle NAB by A. Then angle NAB, = 180°- A. b,c, 1. Now, sin NAB, = and A = AB,' """ " AB But BC and B,C, are equal in magnitude and have the same sign. Hence, sin (180° — A) = sin A. 2. Also, cos NAB, = AC, AB,' and A AC cos A = —r^z : AB Now, AC, and AC are equal in magnitude, but have opposite signs, Hence, cos (180° — A) = — cos A. 49. The signs of the other trigonometrical functions may be found in a similar manner from the figure ; but a sim- pler method is to use the results already obtained, as given in Art, 26. Thus, sin (180° - A) _ s in A 3. tan (180° - A) = 4. cot (180° - A) = cos (180° — A) — cos A cos (180° — A) _ — cos ^4 sin (180° - A) ~ sin A = —tan A. — cot A. 44 PLANE TRIGONOMETRY. 5. sec (180° -A) = 6. esc (180° - A) = 1 1 cos (180° 1 ■A) — cos ^4 1 = — sec A. = esc A. sin (180° - A) sin A 7. vers (180°.— A) = 1 — cos (180° — ,4) = 1 + cos A. 50. From Fig. 12, we see that the angle NAB., = 180° + A, and angle NAB* = 360° — ^4 ; hence we can find the trig- onometrical functions of 180° + A and 360° — A, as we found them for 180° — A in Arts. 48 and 49. These are given in Table II. 51. Again, in Fig. 13, suppose p the radii so drawn that the angles NAB, PAB h QAB,, and QAB a are all equal; then the triangles BAG, M B,AC U B 2 AC 2) etc., are equal. De- note the angle NAB by A ; then reckoning from N around toward the left, Q Fig'. 13. NAB, = 90° + A ; NAB, = 270° — A. NAB 3 = 270° + A. B,G AC sin NAB, ■■ 1. Now, ^^- ABi - AB Hence, sin (90° + A) = cos A AC 2. Also, cos NAB, = — —^ Hence, cos (90° + A) = — sin A 3. Also, tan (90° + A) = — cot A and cot (90° + A) = — tan A = cos A. BO ' AB' ■ sin A. B xr ^\fi 1 ':c, 1 :c. A \ ,,c 1 BJ\. _-^». EXTENSION OF FUNDAMENTAL FORMULAS. 45 52. In a similar manner we may find the values of the functions of 270° — A, and 270° + A. All of these are em- braced in the following table : Table II. Angle = 90° + A. sin = cos A, cot = — tan A, cos = — sin A, sec = — esc A, tan = — cot A, esc = sec A. Angle = 180° — A. sin = sin A, cot = — cot A, cos = - — cos A, sec = — sec A, tan = — tan A, esc = esc A. ■■ 180° + A. cot=- cot ^4, Angle = sin = — sin A, cos = — cos A, sec = — sec A, tan = tan A, esc = — esc A. Angle = 270° — A. sin = — cos A, cot = tan A, cos = — sin A, sec = — csc^4, tan = cot A, esc = — sec A. Angle = 270° + A. sin = — cos A, cot = — tan A, cos = sin A, sec = esc A, tan = — cot A, esc = — sec A. Angle = 360° — A. sin = — sin A, cot = — cot A, cos = cos A, sec = sec A, tan = — tan A, csc = — cacA. Note. — It will be well to have students derive all the values in the above table. These values can be easily remembered by observing that when the angle is connected with 180° or 360°, the functions in both columns have the same name ; but when connected with 90° or 270°, they have different names. 53. From what has now been presented, we see that the trigonometrical functions of angles of any magnitude may be expressed in functions of angles less than 45°. The same is also readily shown to be true of negative angles. Thus, sin 120° = sin (90° + 30°) = cos 30°. tan 223° = tan (180° + 43°) = tan 43°. cot 304° = cot (270° + 34°) = - tan 34°. 46 PLANE~ TRIGONOMETRY. 54. The functions of 360° + x, it is readily seen, are the same as those of x, since the moving radius has the same position in both cases. In general, if n denotes any pos- itive whole number, The functions of (n x 360° + x) are the same as those of x. §5. Hence, when the angle is greater than 360°, we may subtract 360° one or more times until we obtain an angle less than 360° ; and the trigonometrical functions of this remainder will be the same as that of the given angle. This remainder being less than 360°, its functions can be expressed in functions of an angle less than 45°. 56. From the principle that the functions of all angles can be expressed in function of angles less than 45°, in the tables of sines and cosines we have only positive angles or arcs, and those of less than 45°. EXERCISES VI. 1. Express the sine and cosine of 145° in functions of an angle less than 45°. Solution.— Sin 145° = sin (180° — 35°) = sin 35°. Also, cos 145° = cos (180° — 35°) = — cos 35°. Express the following in functions of positive angles less than 45° : 2. Sin 170°. 9. Sec 246°. 16. Tan fjr. 3. Cos 105°. 10. Csc 395°. 17. Cot f tt. 4. Tan 125°. 11. Sin 412°. 18. Sin (— 60°). 5. Cot 204°. • 12. Cos 846°. 19. Cos (— 130°). 6. Tan 300°. 13. Sin (—35°). 20. Tan (— 200°). 7. Sin(7r + a). 14. Sin (2s- + a). 21. Cot (— 250°). 8. Cos (tt + a). 15. Cos (2 7T — a). 22. Sec (2» it f a). EXTENSION OF FUNDAMENTAL FORMULAS. 47 23. Derive the fpllowing table of values : Angle 30° 45° 60° 120° 135° 150° 210° 225° Sine . . . 1 2 1 1/2 V3 2 1/3 2 1 1/2 1 2 1 2 1 >/2 Cosine . . 1/3 2 1 1/2 1 2 1 2 1 1/2 1/3 2 V3 2 1 1/2 Tangent . 1 V* 1 1/3 -1/3 — 1 1 1/3 1 1/3 1 1/3 1/3 Cotangent /3 1 1 1/3 1 1/3 — 1 -V3 •3 • Secant . . 2 •2 2 — 2 -1/2 2 V3 2 1/3 2 V3 Cosecant . 2 1/2 1 2 1/3 ^2 2 — 2 — 2 Extension of Formulas of Table II. 57. The formulas of Table II. were derived on the supposition that the angle A is less than 90° ; but they are true whatever is the value of A. In order to prove this we will show first that they are true for 90° -f A, when A is obtuse. Let the angle NAB, be denoted by A. Draw BB 2 perpendicular to BiB 3 ; then angle NAB 2 = 90° + A. B,C 2 AC, Fig. 14. Now, sin (90° + A) = AB, AB, Hence, sin (90° + A) = cos A. Similarly, it may be shown that cos (90° + A) = — sin A. — "TTT = COS A. 48 PLANE TRIGONOMETRY. These two formulas correspond with those of Table II., and the other formulas drawn from these will also corre- spond with those of the Table. Hence all the formulas of 90° + A, when A is obtuse, are the same as those when A is acute. Similarly, it can be shown that they are true when A terminates in the third or fourth quadrant. Therefore they are universally true. 58. In a similar manner it may be shown that all the other formulas of Table II. are true for any value t of the angle A. EXERCISES VII. 1. Express sine and cosine of 257° in functions of an angle less than 45°. Solution.— Sin 257° = — cos (270° — 257°) = — cos 17°. Note. — The difference between this method of solution and that ,of Art. ^56 will be readily seen. Find the functions of the following in angles less than 45° : 2. 108°. 5. 196°. 8. 240°. 11. —125°. 3. 136°. 6. 215°. 9. 318°. 12. —265°. 4. it — 30°. 7. 7r — Itt. 10. 2ir — |7r. 13. 30° — 2tt. Find the functions of the following in terms of the functions of x : 14. x — 90°. 17. x — 360°. , 20. x + 450°. 15. x — 180°. 18. x + 360°. 21. a;— 540°. 16. x — 270°. 19. x — 450°. 22. x + 540°. # Limiting Values of Trigonometrical Functions. 59. The Limiting Values of trigonometrical functions are their values at the beginning and end of the different quadrants. > These values are determined by the principle that the LIMITING VALUES OF FUNCTIONS. 49 value of a variable up to the limit is equal to its value at the limit. 60. In Art. 23 we have sin A = - ; and cos A = - ■ c c Now, if A = 0, a = , and b = c. Hence, sin = - = ; and cos = - = 1. c ' c 61. In Art. 23 we have , sin A . , , _ cos A tan A = 7 ; and cot A ■■ Hence, cos A ' tan = r- = ; sin A and cot = t: = oo . 62. In Art. 53 we have sin (90° + A) = cos A ; and cos (90° + A) = — sin A. Hence, supposing A = 0°, we have sin 90° = cos 0° = 1 ; and cos 90° = — sin = — 0. 63. Proceeding in a similar manner, we find the limiting values of all the functions as expressed in the following table : Table III. Arc = 0. Arc = 90°. Arc = 180°. Arc = 270°. Arc = 360°. sin = sin = 1 sin = sin = ^- 1 sin = ■ — cos = 1 cos = cos = — 1 cos = — cos = 1 tan= tan= oo tan = — tan = oo tan = — cot = oo cot = cot = — oo cot = cot = — oo sec = 1 sec = oo sec =■ — 1 sec = — oo sec = 1 esc= oo CSC = 1 esc = 00 CSC = 1 esc = — CO 64. From the principles now explained we can often determine the angle from the trigonometrical functions by inspection. 4 50 PLANE TRIGONOMETRY. EXERCISES VIII. 1 . Given sin' a — cos 2 a = 0, to find a. Solution. — Transposing, we have sin 2 a = cos 2 a ; whence sin a = cos a ; hence a = 45° or 225°. Find the angle a in the following : 2. tan o = 1. 9. cos a = — J. 3. sin a = 1. 10. sec 2 a = 2. 4. cosa = — 1. 11. csc 2 a = $. 5. sin 2 a + cos' a = 0. 12. sin 2 a = 3 cos 2 a. 6. tan a +t;ot a = 0. 13. sin a + cos a = 1. 7. cot a — 2 cos a = 0. 14. sin 2 a — 2 cos a + £ = 0. 8. 3 sin 2 a + 2 cos 2 a = 3. 15. 3 sec* a + 8 = 10 sec 2 a. Prove the following : 16. sin A = tan A cos .4. on 1 +sin4 ., 1 + sec A . 22. -± -Xr7 -=tan^. 17. tan A+ cot A = sec ^ esc A. 1 + cosA 1 + cseA 18. tan A Bin A = sec 4 — cos A. 23. = cos A. esc .4 19. cot A cos 4 = esc A — sin A. ■ . OA sin A . , , ■ ^ i ^ 24 - 7 = sin 2 A on sin A + cos .4 . . . esc A 20. —■ = sin A cos A. sec 4 + csc 4 cscA 25. = sec A. „, sin 4 + tan A ... . cot A 21. , - = sin j4 tan A. cot 4 + esc ^ 26. seoM+tan 2 ^l=sec 4 .4— tanM. THE SUM AND DIFFERENCE OF TWO ANGLES. 51 SECTION IV. THE SUM AND DIFFERENCE OF TWO ANGLES. 65. We shall now find formulas for the trigonometrical functions of the sum and difference of two angles. Let the angle AOB he denoted by A and the angle BOG by B ; then the angle AOC=A + B. On OG take any point G, draw CD _L to OA, ON _L to OB, MN _1_ to CD, and NE _L to OA. Then the ' Fig. 15. -angle CNM is the complement of MNO, or NO A; therefore angle NCM = angle A. 1. Now, CD = NE+ CM. Hence, OCsin {A + B) =. OiV sin A + CN cos A Art. 23. = 0Ccos.Bsin.4+0Osini?cos:A Whence sin (A + B) = sin A cos B + cos A sin B. [9] 2. Again, 0D= OE— MN. Hence, OCcos {A + B) = OiVcos A — NO sin A. = OG cos B cos A— OCsin B sin A. Whence, cos (A + B) = cos A cos B — sin A sin B. [10] 66. These two formulas express the value of the sine and cosine of the sum of two angles in terms of the sines and cosines of the single angles. Enunciated in a theorem, the first gives The sine of the sum of .two angles is equal to the sine of the first into the cosine of the second, plus the cosine of the first into the sine of the second. 52 PLANE TRIGONOMETRY. 67. Again, in Fig 16, let the angle AOB be denoted by A, and the angle BOO by B; then angle NCM = angle ENC = A (B. 1. Th. 15). 3. Now, OD=NE-MO. Fi ,, 1(i Hence, 0(7 sin (yl — B) = OZVsin .4 — NO cos 4. = OCcos B sin J.— OCsin £ cos A. , Whence, sin (A — > B) = sin A cos B — cos A sin B. [11] 4. Again, OD = OE + MN. Hence, OCcos U— -B) = ON cos 4 + M7sin A. = OCcos i? cos A + OCsin £ sin A. Whence, cos (A — B) = cos A cos B + sin A sin B. [12] 5. From Table I., For. 4, and formulas [9] and [10], sin (A + B) tan (A + B) = sin A cos B + cos ^4 sin I? cos .4 cos B — sin A sin B cos (.4' + £) Dividing both terms of last member by cos A cos B, we have tan (A + B) = sin ^4 cos B cos ^4 sin B cos ^4 cos £ cos A cos 5 sin A sin £ cos .4 cos B Cancelling common factors, and reducing, we have j. r a i t>\ tan A + tan B tan (A + B) - [13] 6. Substituting - we have 1 — tan A tan B ' B for B in formula [13], and reducing, tan (A — B) = tan A — tan B 1 + tan A tan B [14] FORMULAS FOB DOUBLE AND HALF ANGLES. 53 7. Dividing formula [10] by [9], and reducing as in [13], we have W A , T^ cot A cot B — 1 ri ., cot(A+B) = cotB + cotA - [15] 8. Substituting — B for B in formula [15], and reducing, we have ... _.. cot A cot B + 1 r cot(A-B)= cotB-cotA " ^ 68. These eight formulas may be considered as the Fundamental Theorems of Trigonometry. Note. — For [14] can be derived like [13], and [16] like [15]. Formulas for Double and Half Angles. 69. We now proceed to derive from these fundamental theorems the trigonometrical formulas for double and half angles. 1. Making A = B in formulas [9], [10], [13], and [15], we have sin 2 A = 2 sin A cos A [17] cos 2 A = cos 2 A — sin 2 A [18] , „ . 2 tan A ri ... , _ . cot 2 A — 1 rrinn tan2A = 1 _ tan3A [19] oot2A— ^-j- [20] 2. If in [18] we put 1 — sin 2 A for cos 2 A, and 1 — cos 2 A for sin 2 A, we have cos 2 A = 1 — 2 sin 2 A. cos 2 A = 2 cos 2 A — 1. Whence, • a ^ /I — cos 2 A rr>H , . 11 + cos 2 A r „„_, sin A = \ - [21] cos A = -^/— ^-jj ■" [22] 54 PLANE TRIGONOMETRY. Dividing [21] by [22], and then [22] by [21], multiplying numerator and denominator by the denominator, and re- ducing, we have , . sin 2 A rrio - , . sin 2 A ro ., tanA== l + cos2A ^ C ° tA= l^^s-2A^ 3. Substituting \A for A in [21], [22], [23], and [24], we haye sin | A = + g) sin | (j> — g), [32] cos_p + cos q = 2 cos | Qj + 2) cos | (_p — q), [33] cos 2 — cos # = 2 sin £ {p + 2 ) sin £Q» — «)• [34] FORMULAS FOR TWO ANGLES GENERALIZED. 55 3. Now dividing [31] by [32], sinjH-sinj = sin ^ (p+q) cos '£ (p—q) _ tan Kl> + g) roc-i sinj> — sing cos $ (p+q) sin %(p — q) ta,n^(p — q) L J In a similar manner, we obtain surging 2 sin j (j,+ g ) cos j (?-g) tan i ( + } [36] cos^ + cosg 2 cos % (p+q) cos %(p — q) cosj)+cosg 2cos|(j)+g , )cos|(j> — #) sin j?+sin g ^ 2 sin ^ ( p+q) cos | Q-g) _ cos | (p—q) p 3g -. sin(j>+g) 2sin|(2)+g)cos|(i>+3) cos£(>+g) sin_p— sin g _ 2 sin | (p—q) cos j (p +g) _ sin fr (p—q) „„ sinO+g) 2sin^(j)+3)cos|(_p+g) sin£ (jp+g)' L J sin (j? — g) _ 2 sin \ (p—q) cos \ (p—q) _ cos \ (p—q) ^ Q ~. sinp — sing 2sin£(j) — q) cos $(p+q) cos ^ (p+q)''- J 71. These formulas may be enunciated in propositions; thus formula [35] gives The sum of the sines of two arcs is to the difference of their sines as the tangent of one-half of the sum of the arcs is to the tangent of one-half of their difference. Formulas for Two Angles Generalized. 72. In the demonstration of Arts. 65 and 67 both A and B, and also their sum, are assumed to be acute angles. These formulas, however, are entirely general, as may be readily seen. 1. If the sum A + B is obtuse, A and B being acute, as in Fig. 17, the proof is the same as in Art 65, except that the sign of OD will be negative, as NM is greater than OE. The formulas for sin (A + B) and.cos (A + B) are 56 PLANE TRIGONOMETRY. therefore true for all acute angles. Formulas [11] and [12] may be readily derived from formulas [9] and [10] by sub- stituting — B for B ; hence these formulas are also true for all acute angles. 2. Again, let AOB = A and BOC = B be both obtuse angles. Draw ON J_to BO produced, .EN _L to OA, CD _L to OA', and MN parallel to AA'. Then CD = NE+ CM; whence, Art, 36, Fig. 18. 0Gsm(A + B — 180°) = ON sin (180° - A) + CN cos (180° — A) = 00 cos (180°— E) sin (180° - A) + 00 sin (180° — 5) cos (180° — A). Whence, sin (A + B) = sin A cos -B + cos A sin 5. 3. Again, considering the algebraic signs, we have 0D= OE — MN; whence 0Ccos(^ + 5—180°) = ON cos (180° — A) — MMn (180° — A) = OG cos (180° — B) cos (180° - A) — 00 sin (180° — B) sin (180° — A). Whence, cos (A + B) = cos .4 cos B — sin A sin B. Substituting — B for B in each of the above formulas, we obtain sin (A — B) and cos (A — B), as in Art. 67. Hence, in the formulas of Arts. 65 and 67, and in all the formulas derived from them, A and B may be either acute "or obtuse. Note. — Another method of proving the universality of these form- ulas is given in the Supplement, Art. 149. OTHER FORMULAS. 57 EXERCISES IX. 1. Given sin A — % ; find sin \ A ; find cos \ A. 2. Given cos A = \ ; find cos 2 A ; find tan 2 4. 3. Given tan \ A = 1 ; find sin .4 ; find cos A. 4. Given cot \ A = j/2 ; find sin A ; find cos A. 15. Find the trigonometrical function of an angle of 15°. Solution.— Sin 15° = sin (45° — 30°) = sin 45° cos 30° — cos 45° sin 30°. Substituting the values of sin 45°, cos 30°, etc., as given in Art. 56, and reducing, we have an expression for sin 15°. Similarly, we find all the values given below. 6. sin 15° = y^ — X . g, ^j 150 _ 2 4. 1/3, 7. cos 15° = 1/ - 8 + 1 • 10. sec 15° = -^~" ■ 2j/2 1/3 + 1 8. tan 15° = 2 — t/3. 11. esc 15°= fj^ 2 . - ■J/3 — 1 Find sine, cosine, tangent, and cotangent of 12. 75°. 16. 90° + J. 20. 360° — X 13. 105°. 17. 180° + ^. 21. .4 — 180°. 14. 195°. 18. 18° — ^. 22. 450° + A. 15.240°. 19.27° + ^. 23. 30° — A. Other Formulas. 73. The student may now exercise his skill in demon- strating the following formulas. The Greek letter 8 (theta) is used by many writers to denote any angle. EXERCISES X. Prove the following : 1. Sin (30° +.0) + sin (30° — 0) = cos 6. 58 PLANE TRIGONOMETRY. 2. cos (60° + 0) + cos (60° — 0) = cos 0. 3. sin (60° + 0) — sin (60° — 0) = sin 6. 4. sin 31° + sin 29° = cos 1°. 5. sin 62° — sin 58° = sin 2°. 6. tan (45° + 0) — tan (45° — 0) = 2 tan 0. 7. tan + cot = 2 esc 0. 8. esc — cot = tan J 0. 9. esc + cot = cot £ 0. 10. cot \ — tan £ = 2 cot 0. 11. tan (0 + 45°) = * + tant> - v ^ ' 1— tan0 12. tan (0—45°) = tan(? — * . v ' tan + 1 , , cot — tan „ „ 13. — ■ = cos 2 0. cot + tan 14. sin 3 = 3 sin — 4sin»0. 15. cos30 = 4cos 3 — 3cos0. 16. cos (f tt + 0) + cos (f tt — 0) = — cos 0. 17. cos 55° + cos 65°+ cos 175° = 0. 18. sin (n + 1) a+ sin (» — 1) a = 2 sin na cos a. 19. If vl + B + C = 180°, prove tan A + tan JS + tan C = cot .4 cot B cot C. 20. If A + 5 + C -— 90°, prove cot A + cot B + cot C = cot ^1 cot 5 cot C. Note.— Iu 19th, tan (A + B) = tan (180° — C) ; develop and simplify. Similarly, iu 20th. THE THEOREMS OF TRIGONOMETRY. 59 SECTION V. THE THEOREMS OP TRIGONOMETRY. 74. The Theorems of Trigonometry express the rela- tion between the sides and trigonometrical functions of the .angles of a triangle. 75. These theorems are designed for the solution of tri- angles. By the solution of a triangle is meant the finding of the unknown parts from certain known parts. Theorem I. In any plane right triangle each side is equal to the product of the hypotenuse into the sine of the opposite angle. Let ABC be a right triangle, right angled at C; then (Art. 23), we have sin A = - : and sin B = -. c ' c Hence, a = c sin A ; and b = c sin B. Cor. — In a -plane right triangle each- side is equal to the product of the hypotenuse into the cosine of the adjacent angle. Theorem II. In any plane right triangle each side is equal to the product of the tangent of the opposite angle into the other side. In the triangle ABC we have (Art. 23), tan A ■= - ; and tan B = -. b a Hence, a = b tan A ; and b = a tan J5. 60 PLANE TRIGONOMETRY. Fig. 20. Cor. — In a plane right triangle each side is equal to the product' of the cotangent of the adjacent angle into the other side. Note. — These two theorems enable us to solve the different cases of right triangles. Theorem HI. In any plane triangle the sides are proportional to the sines of the opposite angles. Let ABC be a plane triangle whose angles are A, B, and C, and sides opposite these angles a, b, and c. From C draw CD perpendicular to AB. Then in the right triangle ADC we have (Art. 23), CD = AC sin A and CD = BC sin B. Hence, AC sin A = BC sin B, and AC : BC= sin" B : sin A. In a similar manner it may be shown that AC: AB= sin B : sin C, BC:AB= sin A : sin C. If the angle B is obtuse, as in Fig. 21, we have A' CD = AC sin A, Fi s-2i. And CD = BC sin (180° — B) = BC sin B. Hence, AC sin A = BC sin B. Scholium. — This theorem enables us to solve a triangle when we have two angles and one side, or two sides and one angle not included by the sides. THE THEOREMS OF TRIG 02k Q2IETRT. 61 Theorem IV. In any plane triangle the sum of any two sides is to their difference as tlie tangent of half tlie sum of the opposite angles is to the tangent of half their differenee. Let ABC be any plane triangle whose angles are A, B, and 0, and sides opposite these angles a, 6, and c. Then, Th. HI., a : b = sin A : sin B. Whence, Kg. 28. But, Hence, a+b:a — b = sin A + sin B : sin A — > sin B. sin A + sin B tan $ (A + B) [35] sin A — sin B tan^(yi — B) a + b:a — b — tan^ u* + ^) : tan ^ (_4 — £). Scholium. — This theorem enables us to solve a triangle when we have two sides and the included angle. Theorem V. In any plane triangle, if aline is drawn from Hie vertical angle perpendiciUa-r to the base, tlien the whole base wili be to tlie sum of the oilier tiro sides as the differenee of those sides is to tlie difference of the segments of the base. Let ABC be any plane triangle, and CD a line drawn perpendicular to the base. Then Th. XL, Book IV., ~AC*=AD i + DC i , and TiC* = BD* + DU 1 . Subtracting, AC* — SC» = AD 1 — BD\ 62 PLANE TRIGONOMETRY. Hence (B. IV,, Th. X., C), (AC + BC)(AC—BC) = {AD + BD){AD — BD). Whence, AD+DB: A0+ BC = AC— BO: AD — DB. Scholium. — This theorem enables us to solve a triangle when the three sides are given. Theorem VI. In any plane triangle the square of any side is equal to the sum of the squares of the other two sides, diminished by twice the product of the two sides and the cosine of the included angle. Let ABC be any plane triangle, and CD a line perpendicular to the base. Then Th. IV., Book 13., AC 2 = AB 2 +BC* — 2AB x BD; Also, BD = a cos B (Th. I. C). Hence, b 2 = c 2 + a* — 2 ac cos B Cor.— From Th. VI., a' + c' — b 2 2ao b'+c'—a Fig. 24. COS B = Similarly, cos A = 26c cos (7 = a 2 + b 2 — c 2 2ab Scholium.— These formulas can also be used to find the angles of a triangle when the three sides are given. 76. The formulas of Theorem VI. may be put in a more convenient form. Now, cos A = b 2 + c'- 2bc Th. VI. NUMERICAL VALUE OF SINES, TANGENTS, ETC. 63 Also, 1 — cos A = 2 sin 2 1 A. [25] Whence, 2 sin 2 J 4 = 1 — — "^ ■ '' [41] 26c _ (a + & — c)(a — & + c) ~ 26c Let 2s = « + 6 + c; Then, a + b — c = 2(s — c) and a — 6 4- c = 2(s — 6). Whence, sm 2 i A = ^=-^p^ ■ Similarly, cos 2 \ A = r ' ■ Hence, tan 2 1 A = (s ~ b)(S ~ °> ■ 2 s(s — a) By changing the letters we have » S inHB = (s - a)(S - c) ; Bin 2 iO= (s - a) l S - b) l * ac ' 2 ab 21 s(s — b) ,,_ s(s — o) cos 2 |i? = — *-; cos 2 £C=- — -• 2i-o (s— a)(s — c) 21 _ (s — a )(s — b) tan 2 AB = i — . .. — -: ta,n 2 lG = - — -r-^ — r — - * s(s — b) ' 2 s(s— c) f[42] SECTION VI. NUMERICAL VALUE OF SINES, TANGENTS, ETC. 77. The theorems now presented show the relation be- tween the sides of a triangle and the trigonometrical func- tions of its angles. These sides are expressed in numbers ; hence, to. solve a triangle we must find the numerical value of these trigonometrical functions for any given angle. 78. When the radius of the circle is unity, the sine of the angle NAB (Fig. 8) is equal to the straight line BO. 64 PLANE TRIGONOMETRY. When the angle is very small, the line BC is very nearly equal to the arc AB ; hence the sine of a very small angle is very nearly equal to the arc which measures the angle. By dividing n = 3.1415926 by the number of minutes in 180°, we find the length of an arc of 1' to be 0.0002908882. This arc is so small that it does not differ materially from the sine of the angle of which it is the measure ; hence, we may assume sin 1' = 0.0002908882. We then find the cosine of 1' by For. [3], Table I. Thus, cos 1' = l/l— sin'T = .9999999577, etc. 79. To find the sine of the other arcs, we take the for- mula under Art. 70, putting it in the form sin (a + 6) = 2 sin a cos 6 — sin (a — 6). Now, make b = 1', and then in succession, a equal to 1', 2' ( 3', etc. and we have sin 2' = 2 sin 1' cos 1' — sin = 0.0005817764. sin 3' = 2 sin 2' cos 1' — sin 1' = 0.0008726646. sin 4' = 2 sin 3' cos 1' — sin 2' = 0.0011635526. ~sin 5' = etc. Substituting in a similar manner in the formula cos (a + 6) = 2 cos a cos b — cos (a — 6). We find cos 2' = 2 cos 1' cos 1' — cos 0' = 0.9999998308. cos 3' = 2 cos 1' cos 2' — cos 1' = 0.9999996193. cos 4' = 2 cos 1' cos 3' — cos 2' = 0.9999993232. etc. etc. 80. We may thus obtain the sines and cosines of angles of any number of degrees and minutes up to 45°. Then, NUMERICAL VALUE OF SINES, TANGENTS, ETC. 65 since the sine and cosine of an angle are equal respectively to the cosine and sine of its complement, the sines and cosines of angles between 45° and 90° are immediately- derived from those between 0° and 45°. 81. The tangents and cotangents may be found from the sines and cosines by the formulas, sin a , cos a tan a = ; and cot a = — — ; cos a sin a ' and the secants and cosecants by the formulas, sec a = ; and esc a = ■ cos a ' sin a 82. These numerical values of the sines, cosines, tan- gents, etc. of angles from 0° to 45°, arranged in a table, consti- tute what is called a Table of Natueal Sines, Cosines, etc. Notes. — 1. In actual practice it is not necessary to continue the process of computation beyond 30°; for by Art. 70 we have, reducing, sin (30° + a) = cos a — sin (30° — a), . cos (30° + a) = cos (30° — a) — sin a ; so that the table may be continued above 30° by simply subtracting the sines and cosines under 30° previously found. 2. The values of the sines, cosines, etc. thus computed are very nearly but not absolutely correct. The equation, arc a = sin as = tan a, is true for the natural functions of 30° as far as six decimal places, and for 1° as far as five decimal places. For any arc a it has been shown that sin a lies between a and a — \ a? ; the values found above for large angles must therefore be corrected. 3. The results can be verified and corrected by means of independ- ent calculations. Thus, cos 45° = \Z$, Art 42 ; from which, by For. 22 and 23, we can find sine and cosine of 22° 30', 1J° 15', etc. So also from cos 30° = ^y / Z, we can find sine and cosine of 15°, 7° 30', 3° 45', etc. 5 66 PLANE TRIGONOMETRY. S3. By means of these natural signs the sides and angles of triangles can be readily determined. Thus, suppose in the triangle ABC, page 60, we have given a = 100 ft., an- gle A = 45°, and angle C = 60°, to find b. By Th. III., sin A : sin B = a : b. ,,., , a sin B Whence, b = — : — — • sin A Now, a = 100 ; sin B = sin 60° = ^3 ; sin A = sin 45°= 1- Hence, 6 = 100 X' ii/3 = 50 j/ 3. 84. In this example the numbers are small and the cal- culation easily made. In general, however, the sines, co- sines, etc. are expressed in large decimals, and the calcula- tion is exceedingly tedious. To avoid this labor, log- arithmic sines, cosines, etc. are used, which we shall now explain. * Logarithmic Sines, Cosines, Tangents, etc. 85. A Logarithmic Sine, Cosine, Tangent or Cotan- gent is the logarithm of the natural sine, cosine, tangent or cotangent. 86. The logarithmic sine, cosine, etc. of an angle is readily computed from the natural sine, cosine, etc., as follows : 1. We first find the logarithm of the natural sine or cosine. Then, since the sines and cosines of angles are less than unity, their log- arithms would have negative characteristics. In order to avoid these negative quantities, it has been found convenient to increase the logarithm by 10,' so we make the characteristic 9 instead of — 1,8 instead of — 2, etc. 2. The tangents of angles under 45° are also less than unity, and LOGARITHMIC TABLES AND THEIR USE. C7 the characteristics of logarithmic tangents are also increased by 10. The same principle applies to logarithmic cotangents, secants, etc. 87. In using these logarithmic functions, therefore, we have the rule that for each logarithmic function added in forming a sum, we must deduct 10 from that sum. 88. The logarithmic tangent and cotangent are readily derived from the logarithmic sine and cosine by subtract- ing the one from the other. Thus, tan A — ; • ' cos A Hence, log tan A = log sin A — log cos A. Similarly, log cot A = log cos A — log sin A. EXERCISES XI. 1. Given sin 36° 24' = .59342, find log sin. Am. 9.773361. 2. Given cos 64° 30' = .43051, find log cos. • Ans. 9.633984. 3. Given log cos 65° 24' = 9.619386, find cosine. Ans. .41628. 4. Given log tan 59° 44' = 10.233905, find tangent. Ans. .8639. 5. Find log cos 36° 24' from Ex. 1. Ans. 9.905739. 6. Find log tan 36° 24' from Ex. 1 and 5. Ans. 9.867622. Logarithmic Tables and their Use. 89. A Table of Logarithmic Sines, etc. is a table con- taining the logarithmic sine, cosine, tangent and cotangent of angles, increased by 10. (See Appendix, p. 17.) 90. In the Table the degrees are given at the top and bottom of the page, and the minutes at the sides, in the column headed M. 91. The column headed D contains the increase or de- crease for 1 second. This difference is found by subtract- ing the logarithmic sine, cosine, etc. of any angle from that 68 PLANE TRIGONOMETRY. of the angle next exceeding it by 1 minute, and dividing the result by 60. Note.— This use of the difference is based on the principle of pro- portional parts, which though not rigidly correct is nearly enough so for practical purposes. 92. We shall now explain the method of using the Tables of Logarithmic Functions. 93. To find the logarithmic sines, cosines, etc. of angles or arcs. 1. When the angle is expressed in degrees, or in degrees and minutes. If the angle is less than 45°, look for the degrees at the top of the page, and for the minutes in the Ze/Miand column ; then, opposite to the minutes, on the same horizontal line, in the columns headed Sine, will be found the logarithmic sine; in that headed Cosine will be found the logarithmic cosine, etc. Thus, log sin 23° 35' 9.602150 log tan 23° 35' 9.640027 If the angle exceeds 45°, look for the degrees at the bottom of the page, and for the minutes in the right-hand column ; then, opposite to the minutes, in the same horizontal line, in the column marked at the bottom Sine, will be found the logarithmic sine, etc. Thus, log cos 65° 24' 9.619386 log tan 65° 24' 10.339290 2. When the angle contains seconds. — Find the logarithmic sine, etc. as before ; then multiply the corresponding number found in column D by the number of seconds, and add the product to the pre- ceding logarithm for the sines or tangents, and subtract it for cosines or cotangents. "We subtract for cosine and cotangent, because the greater the an- gle the less the cosine or cotangent. In multiplying the tabular difference by the number of seconds, we observe the same rule for the decimal point as in logarithms. If the angle is greater than 90°, we find the sine, cosine, etc. of its supplement. LOGARITHMIC TABLES AND THEIR USE. 69 EXERCISES XII. 1. Find the logarithmic sine of 36° 24' 42". Solution. log sin 36° 24', Tabular difference, 2.85 No. of seconds, 42 Product, 119.70 to be added, log sin 36° 24' 42", 2. Find the logarithmic cosine of 64° 30' 30". Solution. log cos 64° 30', Tabular difference, 4.41 No. of seconds, 30 Product log cos 64° 30' 30", 9.773361 120 9.773481 9.633984 132.30 to be subtract ed, 132 9.633852 Find the logarithmic tangent of 120° 15' 24" Solution. 180° 00' 00" 120° 15' 24" The given angle, Supplement, log tan 59° 44', Tabular difference, No. of seconds, Product, log tan 120° 15' 24", Find the log sine of 40° 40' 40". Find the log cos of 140° 30' 20". Find the log tan of 85° 25' 45". 59° 44' 36" 4.84 36 174.24 , to be added 10.233905 174 10.234079 Ans. 9.814117. Ans. 9.887441. Ans. 11.097200. Ans. 10.150603. 7. Find the log cot of 144° 44' 28". 94. To find the angle corresponding to any logarithmic sine, cosine, tangent, or cotangent. 1. Look in the proper column of the table for the given logarithm ; if found there, and the name of the function is at the head of the column, take the degrees at the top, and the minutes on the left; but 70 PLANE TRIOONOMETRY. if the name of the function is at the foot of the column, take the de- grees at the bottom, and the minutes on the right. 2. If the given logarithm is not exactly given in the table, then take the next less logarithm, subtract it from the given logarithm, and divide the remainder by the corresponding tabular difference ; the quotient will be seconds, which must be added to the degrees and minutes corresponding to the logarithm taken from the table, for sines and tangents, and subtracted for cosines and cotangents. EXERCISES XIII. 1. Find the angle whose logarithmic sine is 9.617033. Solution. Given log sine, 9.617033 Next less in table, 9.616894 Tabular difference, 4.63) 139.00(30, to be added. Hence the angle is 24° 27' 30". 2. Find the angle whose logarithmic cosine is 9.704682. Solution. Given log cosine, 9.704682 Next less in table, 9.704610 Tabular difference, 3.58) 72.00(20, to be subtracted. Hence, the angle is 59° 33' 40". 3. Find the angle whose log sine is 9.438672. Ans. 15° 56' 14". 4. Whose log cosine is 9.634520. Ans. 64° 27' 47"- 5. Whose log tangent is 10.753246. Ans. 79° 59' 24". 6. Whose log cotangent is 11.449852. Ans'. 2°-V 40". 95. The secants and cosecants are omitted in the table, since they are easily derived from the sines and cosines. Thus, by Art. 26, sec A = 7, and esc A = — : — j • cos A sin A Whence, sec A cos A = 1 ; and esc A sin A= 1. Taking the logarithm and observing to add 10 to each logarithm, we have log sec A = 20 — log cos A. log esc A = 20 — log sin A. LOGARITHMIC TABLES AND THEIR USE. 71 Hence, the logarithmic secant is found by subtracting the log- arithmic cosine from 20, and the logarithmic cosecant is found by subtracting the logarithmic sine from 20. EXERCISES XIV. 1. Find the log cse of 24° 27' 34". Ans. 10.382949. 2. Find the log see of 54° 12' 40". Ans. 10.232992. 3. Prove that the log cot of an angle equals 20 minus the log tan of the angle, and conversely. Notes. — 1. The sine of an angle near 90° varies much more slowly than the sine of an angle near 0°, while the opposite is true of their cosines. Hence, in finding an angle near 90° it is better to avoid the use of its sine, and in finding an angle near 0° it is better to avoid the use of its cosine. The tangent varies with the arc more rapidly than either its sine or cosine, and may be used equally well with any angle. 2. The Tables of Logarithmic sines, cosines, etc. extend to six decimal places. They can be easily changed in use to five-place logarithms by omitting the sixth decimal and adding one to the fifth decimal when the figure omitted is greater than 5. Thus, for log tan 23° 35' = 9.640027, we may write log tan 23° 35' = 9.64003. In a similar way six-place logarithms may be reduced to four-place and three-place logarithms. Some mathematicians prefer five-place tables, and for work not requir- ing great accuracy even four-place and three-place tables are used. 72 PLANE TRIGONOMETRY. SECTION VII. THE SOLUTION OP TRIANGLES. 96. The Solution of Triangles is the process of finding the unknown parts when a sufficient number of the parts are given. 97. There are six parts in a plane triangle, and three of these, one of the three being a side, must be given to find the other parts. 98. If the angles alone were given, it is clear that the sides could not be determined, since there could be an indefinite number of triangles having their angles respect- ively equal. Solution of Plane Right Triangles. 99. In the solution of right triangles we have the four following cases: 1 When the hypotenuse and one acute angle are given. 2. When the hypotenuse and a leg are given. 3. When one leg and either acute angle are given. 4. When the two legs are given. . Case I. 100. Given the hypotenuse o and one acute angle •A, to find the other parts. Method.— Let ABC denote the tri- angle. Then, to find a, we have, Th. I., • a a sin A = - • c Whence, log a = log c -f- log sin A. SOLUTION OF PLANE BIGHT TRIANGLES. 73 Hence, to find a, we add log c to log sin A, and find the number corresponding to the resulting logarithm. Similarly we find b ; and B = 90° — A. EXERCISES XV. 1. In a right triangle ABC, given the hypotenuse c = 475, and angle A = 36° 34'; find the remaining parts. Solution. — From the method given above we have the following operation : log c (475) = 2.676694 log sin A (36° 34') = 9.775070 log a = 2.451764 a = 282.985 log c (475) = 2.676694 log sin B (53° 26') = 9.904804 log 6 = 2.581498 6 = 381.503 Note. — In adding log sin A to log c, 10 is rejected from the sum to correct for the 10 which was added to the log. of the sine (Art. 87). 2. Given the hypotenuse c= 45.36, A = 45° 36'; find a = 32.408, b = 31.736, and B = 44° 24'. 3. Given c = 250, and B = 37°, 30'; find A = 52° 3C, a = 198.338, and. b = 152.19. 4. Given c = 251.4, A = 75° 12'; find B = 14° 48', a = 243.06, and 6 = 64.22. Case II. lOl. Given the hypotenuse o and one of the legs M, to find the remaining parts. Method.— Let ABC denote the tri- angle. Then, 'to find the angle A, We have sin A = - • Th. I. c Whence, log sin A = log a — log c. Similarly, log 6 = log sin B + 1 og e. From these A and b are readily found. 90° - A. And B = 74 PLANE TRIGONOMETRY. EXERCISES XVI. 1. Given the hypotenuse c = 125, and the side a = 76.095 ; to find the remaining parts. Solution. — From the method indicated above we have the follow- ing operations : log a (76.095) = 1.881357 log c (125) = 2.096910 log sin A = 9.784447 A = 37° 30' B = 52° 30' log c (125) = 2.096910 log sin R (52° 30°) = 9.899467 log b = 1.996377 b = 99.169 Note. — After subtracting it is necessary to add 10 to the result to give log sin A. In practice we add 10 to the minuend before subtracting. 2. In a right triangle ABC, given c = 400, and a = 240; find b = 320, A = 53° 7 49", and B = 36° 52' 11". 3. Iri a right triangle ABC, given c = 396, and b = 218 ; find A = 56° 35' 54", B = 33° 24' 6", and a = 330.59. 4. In a right triangle ABC, given c =*126.206, and 6 = 97.72 ; find a = 82.507, A = 40° 10' 30", and B = 49° 49' 30". Case III. 102. Given one leg, as b, and either acute angle, • as A, to find the remaining parts. Method.— From Th. II., we have tan A = r ; b Whence, log a = log tan A + log b. ■ Also, log c = log a — log sin A. Pig. 27. SOLUTION OF PLASE RIGBT TRIANGLES. 75 EXERCISES XVII. 1. In a right triangle ABC, given the side b = 200, and the angle A = 34° 45' • to find the other parts. Solution. — From the method indicated above, we have the follow- ing operations : log tan A (34° 45Q = 9.841187 I log a (138.74) = 2.142217 log 6 (200) = 2.301030 log sin A (34° 45') = 9.755S72 log = 2.142217 | log c = 2.386345 a=13S.74 | c = 243.41 2. In a right triangle ABC, -given a = 364.3, A = 50° 45'; find b = 297.645, c = 470.433, and B = 39° 15'. 3. In a right triangle ABC, given b = 90.5, and A = 50° Stf ; find o = 109.7S, c = 142.27, and B = 39° Stf. 4. In a right triangle, given a = 305.34, and B = 50° 18' 32" ; find 6 = 367.9, c = 47S.1, and A = 39° 41' 2S". Case IV. 103. Given th-e two sides, a and o, about tJie rigM angle, to find the remaining parts. Method.— AVe have tan A = -, Th. II. o Whence, log tan A = log a — log b. Also, log c = log a — log sin A, EXERCISES XVIII. 1. In a right triangle, the side a = 239, side 6 = 188 ; find the angles and hypotenuse. Solutiox. log tan A = log a — log 6 log a (239) = 2.37S39S log 6 (188) = 2.274158 log tan .1 = 10.104240 ^ = 51° 48 '40" .B = 3S°11'20" log c = log a — log sin A logo (239) = 2.37S39S log sin A (51° 48', etc)= 9.895409 log c = 2.482989 c = 304.08 76 PLANE TRIGONOMETRY. 2. In a right triangle, given a = 99.98, b = 152.71 ; find c = 182.5, A = 33° 12', B = 56° 48'. 3. In a right triangle, given a = 515, b = 505 ; find A = 45° 33' 42", B = 44° 26' 18", and c = 721.28. 4. In a right triangle, given a = 29.37, b = 37.29 ; find c = 47.467, 4 = 38° 13' 28", B = 51° 46' 32". Solution of Plane Oblique Triangles. 104. In the solution of oblique triangles there are four cases, as follows : Given 1. Two angles and a side. 2. Two sides and an angle opposite to one of them. 3. Two .sides and the included angle. 4. The three sides. Note. — In the solution, let A, B, and C denote the angles of the triangle, and a, 6, c denote the sides opposite these angles respectively. Case I. 105. Given two angles A and B and, one side a, to find the remaining parts. Method.— Let ABC be the tri- angle. 1. Then to find 0, subtract the sum of A and B from 180°. Fig 28 2. To find b we have (Th. III.) a : b = sin A : sin B, a sin B Whence, b ■■ sin A 3. To find c we have a : c = sin A : sin 0, „., a sin C Whence, c = — — 7— sin A SOLUTION OF PLANE OBLIQUE TRIANGLES. 77 EXERCISES XIX. 1. In the triangle ABC, given A = 32° 24', B = 40° 32', a = 240 ; find the remaining parts. Solution. — Applying the logarithms to the formulas given above, and substituting the numerical values, we have the following oper- ations : a= 240 log a = 2.380211 A = 32° 24' log sin B = 9.812840 B = 40° 32' colog sin A = 0.270976 log a = 2.380211 log sin C= 9.980442 colog sin A = 0.270976 .4 + 5=72° 56' log 6 = 2.464027 log c = 2.631629 (7=107° 04' , 6 = 291.09 e = 428.182 2. In the triangle ABC, given 4 = 27° 40', C = 65° 45', e = 625 ; find B = 86° 35', a = 318.29, & = 684.266. 3. In A ABC, given A = 30° 20', B = 50° 10', and c = 186.74; find <7 = 99° 30', a = 95.62, and b = 145.39. 4. In A ABC, given jB = 51° 15' 33", C= 37° 21', 25",* and a = 305.296 ; find A = 91° 23', & = 238.197, c = 185.3. Case II. 106. Given two sides a and b, and the angle A opposite to the side a, to find the remaining parts. Method.— In this case we proceed as follows : 1. To find B, we have (Th. II.), a : o = sin A : sin B ; whence, sin'i? = b sin A 2. To find C, we have, C= 180° — (A + B). 3. To find c, we have, Th. III., , a sin C a : c — sin A : sin c ; whence, c = — : — — • ,' sin A Discussion. — Here the angle B is determined from its sine ; and since the sine of an angle equals the sine of its supplement, the 78 PLANE TRIGONOMETRY. angle B admits of two values, supplements of each other. We must therefore examine the problem to see which of the two angles (or if both) must be taken. Let AB C denote the triangle; then from the principles of Geometry we have the fol- lowing conclusions : 1. If a > b, then A^> B, and B must be acute ; hence there is only one value of B, and one, and only one, triangle that will satis- fy the given conditions. 2. If a = 6, then A = B, and both A and B are acute, s.ince their sum is less than 180°, and there is only one value of B, and only one triangle, and that is isosceles. 3. If a 1, which is impossible, and the 'le is impossible. So also if a < b and A = 90°. Notes. — 1. In practice the number of solutions can be usually deter- mined by the circumstances of the problem. When there is any doubt, compute the value of b sin A, and compare it with o, according to Art. 5. 2. Or find the value of log sin B. Then, if log sin B <^ 10, there is one solution when a > b, and two solutions if a < b. If log sin B > 10, the triangle is impossible. EXERCISES XX. 1. In the triangle ABC, given a = 75.5, b = 98.5, A = 37° 37' ; find B, C, and c. Solution.— Applying logarithms to the formulas given above, and substituting the numerical values, we have the following operations : SOLUTION OF PLANE OBLIQUE TRIANGLES. 79 o = 75.5 6 = 98.5 A s= 37° 37' Here a > 6 and log sin £ < 10. . ■ . one solution. log 6 = 1.993436 log sin A = 9.785597 colog a = 8.122053 log sin B = 9.901086 £ = 52° 46' 48" C= 89° 36' 12" log a = 1.877947 log sin C= 9.999989 eolog sin A = 0.214403 log c = 2.092339 c = 123.69 Note.— By constructing the triangle and examining it geometrically, it will be seen that there is but one solution. 2. In the triangle ABC, given a = 150, b = 200, A = 44° 26'; ftn&B, C, and c. Solution. — Substituting in the formulas given above, we have the following operations : o= 150 log 5 = 2.301030 log a = 2.176091 2.176991 6 = 200 log sin A = 9.845147 log sin C = 9,962692 9.618456 A = 44° 26' colog a = 7.823909 colog sin A = 0.154853 0.154853 Here a <^b log sin B = 9.970086 log c = 2.293636 1.949400 log sin £ <; 10.' B ■= 68° 58' 38" c = 196.623 ' . two solutions. or 111° 01' 22" C = 66° 35' 22" or c = 89.002 t or 24° 32' 38" Note. — By constructing the triangle and examining it geometrically, it will be seen that there are two solutions. In Fig. 29, AB' C — 68° 58' 38", and ABC = 111° 01' 38" ; ACB' = 66° 35' 22", ACB = 24° 32' 38" ; AB = 89.002, AB 1 = 196.623. 3. In A ABC, given a = 62.50, e = 45.96, A = 79° 21'; -find B = 54° 22' 22", C= 46° 16' 38", b = 51.69. 4. In the triangle ABC, given a = 15.71, b = 21.12, A = 27° 50'; find the other parts. Am. B = 38° 52' 47"; (7= 113° 17' 13"; e = 30.90G; or B = 141° 7' 13"; C= 11° 2' 47" ; c = 6.447. 5. In the triangle ABC, given a = 94.26, b =126.72, and .4 = 27° 5C ; find the values of c, 5, and C. ' 80 PLANE TRIGONOMETRY. 6. Given a = 40, b = 80, and A = 30° ; find the other parts of the triangle. 7. Find the other parts of a triangle, given a = 80, b = 100, and ^ = 60°. Case III. 107. Given two sides, a and b, and the included angle C, to find the remaining parts A, B, and e. Method.— In this case we pro- ceed as follows : 1. To find A and B, we subtract C from 180° and divide by 2, which gives us the value of % (A + B). We then find \ (A — B) from Th. IV., which gives Fig. 30. tan $ (A - B) = ^-| x tan \{A + B). Then, | (A + B) plus $(A - E) = A. And i(A + B) minus \ {A — B) = B. 2. To find c, we apply Th. II., which gives c = a sin C sin J. ' or c = b sin C sin 5 EXERCISES XXI. 1. In the triangle ABC, given a = 680, & =460, and C = 84° ; find the other parts of the triangle. Solution. — Following the method stated above, we have the fol- lowing work : SOLVTION OF PLANE OBLIQUE TRIANGLES. 81 a + ft = 1140 u — b = 220 A+B = 96° IU+B) = 48° i (A-B)=12° 5' 49" .4=60° 5' 49" 5=35° 54 j xv , log (a - b) = 2.342423 colog {a + b) = 6.943095 log tan i (4+.B)=10.045563 log tan i {A-B) = 9.331081 i(A-S)= 12° 5' 49" log a =2.832509 log sin C =9.997614 colog sin A =9.062046 log c =2.892169 c =780.134 2. In the triangle ABC, given a = 240, 6 = 360, C= 68° 36'; find A = 39° 21' 34", B =72° 02' 26", e = 352.349. 3. In the triangle ABC, given a = 320, 6 = 562, C = 128° 04'; find A =18° 21' 21", B = 33° 34' 39", c = 800. 4. In the triangle ABC, given b = 50.24, c = 43.25, A = 40° 15' ; find B = 81° 24' 25", (7= 58° 20' 35", a = 32.829. 5. If two sides of a triangle are each equal to 60 ft., and the included angle is 60°, what is the third side? 6. If two sides of a triangle are each equal to 120 ft., and the included angle equals 120°, what is the third side? Case IV. 108. Given the three sides, a, b, and c, of a plane triangle, to find the angles A, B. and C. Method. — Let fall a perpendicular upon the greater side' from the angle opposite, dividing the triangle into two right triangles. Find the difference of the segments of the base by Theorem V. ; half this difference added to half the base gives the greater segment, and subtracted from half the base gives the less. We shall then have two sides and the right angle of two right -triangles, from which we can find the acute angles by Theorem I. 82 PLANE TRIGONOMETRY. EXERCISES XXI!. 1. In a triangle ABC, given AB = 60, ^40=50, and BC== 40, to find the angles. Solution. — Let ABC denote the tri- angle ; then AB = 60, AC= 50, BC — 40. Then, by Th. V., AB : A C+B C=A C—B C : AD—BD, or, 60: 90 = 10 -.AD—BD, Hence, AD — BD = 90 X 10 -r- 60 = 15; Then, AD = £(60 + 15) = 37.5, And BD — i (60 — 15) = 22.5. Then, in the triangle A CD, to find the angle BCD, Fig. 31. colog^C (50) = 8.301030 log AD (37.5) = 1.574031 log sin ACD = 9.875061 ^OZ> = 48° 35' 25" colog BC (40) = 8.397940 logj8X> (22.5) = 1.352183 log sin BCD = 9.750123 .-. BCD = 34° 13' 44" Hence, A = 90° — 48° 35' 25" = 41° 24' 35", B = 90° — 34° 13' 44" = 5.5° 46' 16", C = 48° 35' 25" + 34° 1 3' 44" = 82° 49' 09". 2. In a triangle ABC, given a = 1005, b = 1210, c = 1368 ; find the angles. Am. 45° 22' 34" ; 58° 58' 19" ; 75° 39' 7". 3. In a triangle ABC, given a = 340, b = 280, and c = 460 ; find the angles. Ans. A = 47° 23' 16"; B = 37° 18' 31" ; C= 95° 18' 13". Another Method. 109. The angles of a plane triangle may also he found by means of the formulas given in Art. 76. SOLUTION OF PLANE OBLIQUE TRIANGLES. 83 EXERCISES XXIII. 1. In the triangle ABC, the side c = 800, the side 6 = 600, and the side a = 400 ; required the three angles. Solution. — By Art. 76 we have am? A- bc s = £ (800 + 600 + 400) = 900. s _ 6 = 900 — 600 = 300 ; s — c= 900 — 800 = 100. We then find log (s — 6), log (s — c), colog 6, and eolog c; their sum will be log sin 2 ^ A. Divid- ing by 2, we have log sin J A = 9.397940; from the Table, we find % A = 14° 28' 39", whence, A = 28° 57' 18". OpEitATION. log (s — 6) (300) = 2.477121 log (* — c) (100) = 2.000000 colog b (600) = 7.221849 colog c (800) = 7.096910 log sin 2 J A 2)18.795880 log sin i A = 9.397940 i A = 14° 28' 39" A = 28° 57' 18" The Other angles may be obtained in a similar manner from the formulas for sin 2 J B and sin 2 \ C. The other two formulas of Art. 76, which may be used in this case, are 2 1 4 s ( s cos^ \ A = -*- be s(s — a) Note. — Either of these three formulas may be used ; but sin" £ A is less accurate when the half angle is near 90° ; and cos" i A, when the half an- gle is near 0° ; while tan* 4 A is applicable for any angle. 2. The three sides of a plane triangle are 20, 30, and 40 ; required the three angles. Ans. 28° 57' 18"; 46° 34' 03"; 104° 28' 39". 3. In the triangle ABO, a = 200, b = 250, and c = 300; required the three angles. Ans. 41° 24' 35" ; 55° 46' 16" ; 82° 4& 09". 84 PLANE TRIGONOMETRY. Find the angles — 4. Given a = 10, b = 24, c = 26. 5. Given a = 12, 6 = 12, c = 12. 6. Given a = 7, 6 = 8, c = 16. 7. Given a = 2, b = t/6, e = j/3 + 1. Solve the following without the use of logarithms : 8. If b = 3, c = 2^3, and A = 30° ; prove C= 90°. 9. If a = 2^/3, 6 = 3— j/3, and e = 3y'2 ; prove 0= 120°. 10. If a = 2, & = 1 + i/3, and c = yQ; prove C= 60°. 11. If a = 12, b = 2££, and A = 45°; prove £ = 36°. 12. Find the angles of a triangle whose sides are in the ratio of 1, 2, and 3. Remark. — All three angles may be computed by the formulas, E,nd the accuracy of the results tested by seeing whether their sum equals 180°. For this method the formulas for the tangent may be put in a more convenient form. Thus, tan 2 \ A may be written : (s — g)(s — b)(s — c) 1 / (s — a)(s — b)(s — c) \ s(s — a) 2 (s — a) 2 \ s ] If we put (s-a)(s-b)(s-c) = ,, haTeton , M = ^. s 2 s — a Similarly, tan | B = and tan i C = V : — c In applying these formulas we may find the value of log r, and use it in each one of the formulas in the computation, and thus slightly abridge the labor of computation. HEIGHTS AND DISTANCES. 85 SECTION VIII. PEAOTIOAL APPLICATIONS. HEIGHTS AND DISTANCES. HO. A Horizontal Plane is a plane which is parallel to the plane of the horizon. 111. A Vertical Plane is a plane which is perpendic- ular to a horizontal plane. 112. A Horizontal Line is any line in a horizontal plane. A vertical line is a line perpendicular to a horizon- tal plane. 113. A Horizontal Angle is an angle in a horizontal plane. A Vertical Angle is an angle in a vertical plane. 114. An Angle of Elevation is a vertical angle having one side horizontal, and the inclined side above the horizontal side; as BAD. 115. An Angle of Depression is a vertical angle having one side horizon- 'tal, and the inclined side under the horizontal side ; as CD A. . Flg ' 32 - 116. Distances upon the ground are usually measured by a chain, called Gunter's Chain. This chain is 4 rods or 66 feet long, and consists of 100 links. Sometimes a half chain is used, consisting of 50 links. 117. Angles are measured by various instruments. Horizontal angles are measured by an instrument called O T) i i i fH- 1, A- i V- 86 PLANE TRIGONOMETRY. Fig. 33. The Compass. Horizontal and vertical angles are both measured by the Theodolite, or, what is still better for gen- eral use, a Transit-Theodolite. Case I. 118. To determine the height of a vertical object standing upon a horizontal plane. Method. — Measure from the foot of the object any con- venient horizontal distance AB; at the point A take the angle of eleva- tion BAC; then, in the triangle ABO we have a side and an acute angle ; hence, we can readily find the alti- tude. 1. From the foot of a tower I meas- ure a horizontal line 120 feet, and at its extremity find the angle of elevation to be 48° 36' ; what was the height of the tower? Ans. 136.113 feet.. Case II. 119. To find the distance of a vertical object whose height is known. Method.— Measure the angle of ele- vation to the top of the object, as before; we will then have a right triangle in which, we know the per- pendicular and an acute angle ; hence, we can readily find the base. 1. I took the angle of elevation to the top of a flagstaff whose height I knew to be 160 feet, and found it be 20° ; how far was I from the staff? Ans. 439.60 feet. Fig. 34 HEIGHTS AND DISTANCES. 87 Case III. 120. To find the distance of an inaccessible object. Method. — Measure a horizontal base-line AB, and then take the an- gles formed by this line and lines from the object to the extremities of this base-line, as CAB and ABC; the distance AC ov BC can then be readily found. 1. I am on one side of a river, and wish to know the distance to a tree on the other side. I measure 300 yards •by the side of the river, and find that the two angles formed by this line and the lines from its extremities to the tree, are 72° 40' and 45° 36' respectively ; required the distance from each extremity of the base-line to the tree. Am. 243.362 yards ; 325.15 yards. Case IV. 121. To find the distance between two objects sepa- rated by an impassable barrier. Method. — Select any convenient station, as C, and measure the distance from it to each of the objects A and B and the an- gle C included between these lines. We can then readily find the dis- tance AB. 1. The distance between two trees cannot be directly measured : I therefore take a third posi- tion, from, which each of the trees can be seen, and find the distances from it to the trees to be 300 and 250 yards re- 88 PLANE TRIGONOMETRY. spectively, and the included angle 43° 16' ; required the distance between the trees. Am. 208.025 yards. Case V. 122. To find the height of a vertical object stand- ing upon an inclined plane. Method.— Measure any convenient distance DC on a line from the foot of the object, and at the point D measure the angles of elevation, EDA and EDB, to foot and top of the tower. By means of the two triangles DEA and DEB we can find the height of AB. 1. Wishing to determine the height of a tower situated upon a hill, I measured a distance down the slope of the hill 400 feet, and found the angles of elevation to the foot of the tower 42° 28', and to the top of the tower 68° 42' ; required the height of the tower. Am. 486.747. Case VI. 123. To find the height of an inaccessible object above a horizontal plane. First Method.— Measure any I convenient horizontal line AB directly toward the object, and | take the angles of elevation at A and B; we will then have conditions sufficient to find DC. \ 1. Wishing to find the alti- tude of a hill, I measured the angle of elevation at the bottom 60° 37', and 460 feet from the foot, in a right line HEIGHTS AND DISTANCES. 89 from the top of the hill and the point at the foot, and in the same horizontal plane as the foot, I measured the angle of elevation 36° 52'; required the height of the hill. Ana. 597.092. Second Method. — If it is not convenient to measure a horizontal base-line toward the' object, we may measure any line AB, and also measure the horizontal angles BAD, ABD, and the angle of elevation DBC. Then, by means of the two triangles ABD and CBD, the height CD can be found. Case VII. 124. To find the distance between two inaccessible objects when points can be found from which both objects can be seen. Method. — The method of meas- urement is indicated in the follow- ing problem. The method of solu- tion we prefer'leavingto the ingenuity of the pupil, that he may learn to think for himself. 1. Wishing to know the horizontal distance between a tree and house on the opposite side of a river, I took the following meas- urement : ,18 = 400; GAD = 56° 30', BAD = 42° 24' ; ABC = 44° 36', and DBC= 68° 50'. Required the distance CD. Am. 747.913. 90 PLANE TRIGONOMETRY. Fig. 41. Case VIII. 125. To find the distance between two inaccessible objects when no points can be found from which both objects can be seen. Method. — The method is indicated in the following' problem and figure. This case and the following one may be omitted with j'oung students. 1. Wishing to know the horizontal distance between two inaccessible objects when no point can be found from which both objects can be seen, two objects C and D are taken, 600 feet apart, from the former of which A can be seen, from the latter B. From C we measure the dis- tance CF, not in the direction DC, equal to 600 feet, and from D a distance DE equal to 600 feet. We then measure the following angles. ' GFA = 80° 16', BED = 86° 25'. ACF= 52° 24', BDE = 60° 24', ACD= 56° 36', BDC*= 150° 30'. Required the distance AB. Ans. 1117.44 feet. Case IX. 126. To find the distances from a given point to three objects whose distances from each other are known. Method. — The method is indicated in the problem and figure. SUPPLEMENT. 91 1. I wish to locate three buoys, A, B, and C, in a harbor, so that the distance between A and B is 800 yards, between A and C 600 yards, between B and C 400 yards, and from a fixed point on shore the angle APC shall" equal 33° 45' and BPC 22° 30'; required the distances PA, PC, and PB. Fig. 42. ^ns. P^4 = 710.193; PC= 1042.522; PB = 934.291. SECTION XI. SUPPLEMENT. 127. The Supplement presents additional matter for those who wish to pursue the subject further. Some Properties of Triangles. The Right Triangle. 128. In the right triangle ABC, let 6 denote the base, a the alti- tude, c the hypotenuse, and M the area. Then, Geometry, B. IV., Th. 6, M-- But, a = b tan A, and b = a tan B. Art. 26. Hence S = %b 2 tan A, and M—% a 2 tan B, [43] Hence, we can find the area from A and 6 or from a and B. From Ex. 1 and 2 below we can find the area having a and A or b and B. From Ex. 3 and 4 we can find the area, having given c and A or 5. 92 PLANE TRIGONOMETRY. EXERCISES XXIV. Prove the following : 1. Jf=£a 2 eot A. 3. M=\& sin 2 A. 2. M=l&cotB. 4. M=ic?sin2B. Find the other three parts : 5. Given B and c. 7. Given 5 and a. 6. Given B and b. 8. Given 6 and c. The Isosceles Triangle. 129. In the isosceles triangle ABC, let h denote the altitude, a the equal sides, and c the base. Then we readily derive the rela- tions given in the following exercises: EXERCISES XXV. In an isosceles triangle find the other parts — 1. Given a and A. 4. Given c and C. 2. Given a and C. 5. Given h and A. 3. Given c and A. 6. Given A and C. Find the area — 7. Given a and A. 9. Given a and c. 8. Given a and C. 10. Given A and C. The General Triangle. 130. In any triangle ABO, let c denote the base, a and 6 the two sides opposite the angles A and B respectively, and h the altitude. Then, M = ^ ch; but A = a sin 5. Henee, M = J ae sin 5. Similarly, 4^ = 4 a0 s ' n C> and Jf=J5csin^. Hence, the area of a triangle is equal to one-half the product of any two of its sides into the sine of the included angle. 131. A formula may also be found for the area when a side and two angles are given, the third angle being then known". J> sin A b sin C .} [44] From Th. III., a = sin B sin B THE RADIUS OF AN INSCRIBED CIRCLE. 93 Substituting these values of a and c in M=^ac sin B, „ , ,, 6 2 sin A sin C We have M = — - — • ■ 2 sin B Hence, the area of a triangle is equal to the product of the sines of any two angles into the square of their included sides, divided by twice the sine of the third angle. 132. A formula may also be derived for the area of a triangle when the three sides are given. By For. [18], sin B = 2 sin J B cos %B. Substituting the values of sin J B and cos £ B, as given in Art. 76, 2 We have sin B = — \/s (s — a)(s — 6)(s — s). Substituting this value of sin B in [45], We have M= V* (s — a)(s — b)(s — c). [45] Hence, to find the a.rea of a triangle when the three sides are given, we subtract each side from the half sum of the sides, take the product of these differences and the half sum, and extract the square root of the product. EXERCISES XXVI. Find the area of a triangle — 1. Given a = 20, 6 = 30, ,C= 60°. 2. Given b = 30, c = 40, 'A= 45°. 3. Given a = 30, c = 40, B = 115°. 4. Given a = 40, b = 80, C = 48°. 5. Given a = 60, 6 = 80, c = 150°. 6. Given b = 100, A = 30°, B = 40°. The Radius of an Inscribed Circle. 133. Let ABC be any triangle whose sides are a, 6, and c, and r the radius of the inscribed circle ; then dividing the triangle into three triangles by drawing lines from the centre to the vertices of the three angles, 94 PLANE TRIGONOMETRY. We have M = %ar -\- % br -\- § cr = Jr(a + 6 + c). Let 2s denote the sum of the three sides, and We have M = rs; whence, * [46] Hence, the radius of the inscribed circle is equal to the area of the triangle divided by one-half the sum of the sides. Cor. — Substituting in For. [46] the value of M given in [45], and reducing, we have r — Ks — a)(s — b)(s — c) , hence r as used on page 84 is equal to the radius of the inscribed circle. Radius of a Circumscribed Circle. 134. Let ABC be circumscribed by a cir- cle whose centre is O ; and let R denote the radius. Draw OD J_ to B O ; then, BD = D C. By Geometry, B. III., Th. 18, the angle BOC=2A; hence, angle BOD = A, and BD — R sin BOD, or %a = R sin A. Hence, a = 2 R sin A. 2M ' be abc 4M' Therefore, the radius of the circumscribed circle is equal to the product of the three sides of the triangle divided by four times its area. Cob. — From Art. 134 we have 2R = -2— = JL- = -2_. sin A sin B sin C Hence, the diameter of the circumscribed circle is equal to the ratio of any side of a triangle to the sine of the opposite angle. From For. [44], Whence, sin A ■■ R = GENERALIZATION OF ANGLES. 95 EXERCISES XXVII. Find the radius of an inscribed circle, 1. Givena = 4, 6 = 5, c = 6. 4. Given a = 45, 5 = 45°, M= 24. 2. Given a=10, 6=20, .4=40°. 5. Given a = 30, C = 60°, M = 40. 3. Given o=30, 6=35, C=30°. 6. Giveno = 6 = c. 7. Find the radius of a circumscribed circle in each of the above cases. 8. Find the angles of a right triangle if the hypotenuse is equal to four times one of the legs. 9. Find the legs of a right triangle if the hypotenuse is 12, and one acute triangle is twice the other. 10. Derive a formula for the area of a parallelogram, given two ad- jacent sides o and 6 and the included angle A. 11. Derive a formula for the area of an isosceles trapezoid, given the two parallel sides a and 6 and acute angle A. Generalization of Angles. 135. We have Art. 54, sin A = 360° + A, or sin A = 2 X 180° + A. If we add any number of times 360°, as n times 360°, the sine is still the same ; hence sin A = 2 n X 1 80° + A or sin A = 2 n tt -\- A. 136. Also sin A =*sin (180° — ^1) or sin (*• — A). If we add any number of times 360°, as n times 360°, the sine is still the same ; hence sin ^ = sin(n360°+ 180°— ^) = sin(2rex 180° + 180° — J 1) = sin {(2re + 1) 180° — A) = sin {(2 n + 1) it — A). Therefore, if A z denotes the general value of an angle A whose sine is a, we have A I = 2nir+ A, and A z = (2n+ 1)tt — A. 137. From this we infer that if two angles have the same sine, either their difference is an even multiple of it, or their sum is an odd multiple of n. 138. Similarly, we may show that if A, denotes the general value of an angle A whose cosine is a, we have ^ x = 2nl80°±^, or A 1 = 2nir±A. 1 . From this it is seen that, if two angles have the same cosine, either their sum or their difference must be an even multiple of ir. 96 PLANE TRIGONOMETRY. 2. Similarly we may prove that, if two angles have the same tan- gent, their difference must- be some multiple of w. EXERCISES XXVIII. What is the general value of an angle A 1. When sin A = |? 6. When tan A = 1 ? 2. When sin A = 1 ? 7. When sec A = 2 ? 3. When cos A = 1 ? 8. When cot 3 ,4 =—3^3? 4. When sin 2 .4 = \ 1 9. When esc 2 A = J ? 5. When tan 2 .4 = £ ? 10. When tan 4 .4 = 9? 11. Find the general values of A in the equation sin 3 A = sin A cos 2.4. Solution. — We have sin (A + 2.4) — sin J. cos 2 A = ; whence, cos A sin 2.4 = ; hence, either cos A = 0, or sin 2 A = 0. From the former we get .4 = some odd multiple of Jir, and from the latter we get 2.4 = any multiple of vr. Hence, both are included in the equation A = \mr. Find the general value of A in the following equations : 12. cos A = cos 2 A. 16. sin 4 A + sin 6 A = 0. 13. sin 5 t1 = 16 sin 6 A. 17. tan A + cot .4 = 2. 14. sin 4. + cos .4 = 18. sec A = 2 tan 4. l/2 . 15. sip 9.4 — sin A = sin 4X 19. 080,4 cot A = 2y 3. 20. sin ^ — cos .4 = 4 sin .4 cos 2 A. 21. tan (Jff + 4) = 1 + sin 24. Inverse Trigonometric Functions. 139. The expressions sin A = n, cos A = n, etc., may also be expressed thus: A = sin -1 ?»»and A = cos -1 k. To read these, notice that A = sin -1 n is read, A equalB the angle whose sine is n. A = cos -1 n is read, A equals the angle whose cosine is n. A = tan -1 n is read, A equals the angle whose tangent is n. 140. These are called inverse trigonometric functions. They are often found to be convenient in trigonometry. Note. — The student will be careful to notice that in the expression sin -1 , the ( — 1) is not to be regarded as an exponent. INVERSE TBIGOMETRIC FUNCTIONS. 97 141. Any relation which exists among trigonometrical functions may be expressed by means of the inverse notation. EXERCISES XXIX. I. What is the value of sin -1 J? Solution. — Evidently sin -1 £ equals 30°, since sin 30° = \. Find A, given 2. A = sin-' | t/2. 6. A = cos^— J) 3. A = cos- 1 § i/3. 7. A = cot- 1 — i j/3. A. A = tan- 1 t/3. 8. ^ = sec - 1 |/2. 5. .4 = cos- 1 £. 9. .4 = esc- 1 f i/ 3 - 10. Given A = tan (cos -1 f ), to find the value of A. Solution. — This means, what is the tangent of the arc whose cosine is f ? Let x denote the angle ; then cos x = f , sin x = j/1 — $ = ^ j/5 ; hence, tan x = \ j/5 -r| = | j/5. II. Given sin as =^?, and sin b = q, to express sin (a + 6) in- versely. Solution. — -'Since sin a=p, cos a= |/1 — p' ; also cos 6 = |/1 — g* ; hence sin (a + 6) =j» l/l — 2 2 + ffT/l — i?. Therefore, o + 6 = sin" 1 (p VI — q' + q VI— p'). Or, sin -1 p -(- sin- 1 q = sin -1 (^> j/1 — q' + 2 I'l — p'). Express similarly the inverse functions 12. Of sin (a — 6). 14. Of cos (a — 6). 16. Of tan (a — 6). 13. Of cos (a + 6). 15. Of tan (a + b). 17. Of cot (a + b). Prove the following: 18. tan- 1 f = 2tan~ l \. 23. sin-^ + sin-'A =sin- 1 -|f. 19. sin-^y^ + tan- 1 J = 45°. 20. 2 tan- 1 6 = tan" 1 — - 1—6 24. sin- 1 ti = cos- 1 j/1 — « 2 . 25. sin- i e+cos- 1 6» = -- 21. 3 tan- 1 = tan" 1 j^ 2 ^ ™ .',.,, , ,^7 , 26. tan- 1 + cot" 1 » = -• 22. sin (sin- 1 J + cos- 1 \) = 1. T 2 7 98 PLANE TRIGONOMETRY. 27. sin ( tan-' — ) = cos ( cot -1 — I • V ml \ in I 28. am- 1 — — = tan" 1 • a + b a — o i/2 4-1 1 ir 29. tan- 1 Y ^ — tan"' — - = -• y"2 — 1 i/2 4 Solve the following equations : 30. sin- 1 z +-sin~ 1 - = -■ 31. tan- 1 2x4- tan" 1 3a; = j- 2 4 4 » 32. sin- 1 2k — sin - ' xy / 3 = sin- 1 a;. 33. sin 2x cos- 1 cot 2 tan" 1 x = 0. Extension of Functions of Two Angles. 142. In Art. 72 it was shown that the formulas for the sum of two angles hold when both angles are obtuse. We now show that they are true for all angles, positive or negative. 1. To do this we will show that they are true when one of the angles is increased by 90°. Thus, suppose A 1 — 90° + A ; then by Art. 52, sin (A 1 + B) = sin (90 4- A 4- B) = cos (A 4- B). And, cos (A 1 4- B) = cos (90 4- A 4- B) = — sin (a 4- B). Hence, sin (A 1 + B) = cos A cos B — sin A sin B. cos (A 1 4- B) = — sin .4 cos jB — cos .4 sin B. Now, cos .1 = sin (90° 4- A) = sin A 1 . Art. 52. sin A = — cos (90° + A) — — cos -4'. Substituting, sin (.l 1 4- -8) = sin .4 1 cos B + cos .4 1 sin B, cos^ 1 + B) = cos .4 1 cos B — sin A 1 sin B. It is thus seen that Formulas [9] and [10] are true when one of the angles is increased by 90° ; and in a similar way it may be shown that they are true for each increase of either or both angles by 90°, and therefore true for the sum of any two angles. 2. In a similar way it may be shown that the formulas for sin (A — B) and cos {A — B) are universal. Their universality may also be shown by deriving them from sin (A + B) and cos {A + B), which we have just shown are universal. EXTENSION OF FUNCTIONS OF TWO ANGLES. 99 Thus, write (4 — B) + B = A. Then by Art. 65, sin A = sin [(A — By+B']^ sin (A — J5) cos B + cos(A — B) sin B. cos A = cos [(4 — -S)+-B] = cos (^1 — E) cos -B — sin (A — B) sin 5. Multiply the first equation by cos B, and the second by sin B, sin A cos B = sin (J. — B) cos 2 5 + cos (A — E) sin B cos 5. cos .4 sin B = — sin {A — B) sin 2 B -f cos {A — B) sin B cos -B. Subtracting, we have sin A cos B — cos A sin B = sin (4 — i?)(sin 2 .8 + cos 2 5). But, sin 2 B + cos 2 2? = 1 ; hence, transposing , sin {A — ■ B) = sin J. cos B — cos A sin B. Hence the formula for sin (A — B), Art. 65, is universal. 3. In a similar manner, if we multiply the first equation by sin B, and the second by cos B, and add the results, and reduce, we shall obtain cos (A — B) = cos A cos B + sin A sin B. It is thus seen that Formulas [9], [10], [11], and [12] are gen- eral for all positive values of A and B ; and therefore all the formulas derived from these are also general for all positive values of A and A. 4. Lastly, it may also be shown that these formulas are true for any negative values of A and B. First, suppose C is negative, and less than A ; then A + B becomes A — B, and A — B becomes A + B, and the formulas are true, as already shown. If A is negative, it merely changes the order of the letters. Second, suppose B is negative and greater than A ; then A — B is negative. By Art. 44, sin (A — B) = — sin (B — A) = — (sin B cos A — cos B sin A). Whence, sin (A — B) = sin A cos B — cos A sin B. Also, cos {A — B) = cos (B — A) = cos B cos A -\- sin B sin A. Whence, cos (A — B) = cos A cos B + sin A sin B. The same is also true if A is negative and greater than B. 100 . PLANE TRIGONOMETRY. Third, suppose A and B are both negative ; and let A = — A 1 and B = — B\ sin {A + B) = sin (— A 1 — B 1 ) = — sin (A 1 + 5 1 )- = — (sin A 1 cos B 1 + cos A 1 sin B l ). = sin (— A 1 ) cos (— .B 1 ) + cos (— A 1 ) sin (— B 1 ). = sin .4 cos B -\- cos .4 sin B. 5. In the same manner Formulas [10], [11], and [12] may be shown to be true when both the angles are negative ; hence all the formulas derived from these are also true. It is thus seen that the Formulas [9], [10], etc. are true for every value of A and B, positive or negative. < Application to the Circle. 143. We have regarded sine, cosine, tangent, etc. of an angle as a ratio of the sides of a right triangle, formed by the moving radius and its projection. This is the modern method of treating trigonometry ; but the old method was to consider these functions as lines represented in a circle, and thus pri- marily as functions of arcs. 144. Thus, in the diagram, by the old system, Fi^Tii. 1. The line BC is the sine of the arc AB. 2. The line OC is the cosine of the arc AB. 3. The line AD is the tangent of the arc AB. 4. The line OD is the secant of the arc AB, etc. 145. In the old system the length of the sine, cosine, tangent, etc. depended upon the length of the radius of the circle ; in the new system it is a fixed numerical value. 146. If the radius of the circle is taken as unity, the trigonomet- rical functions of angles according to the new system correspond with the trigonometrical functions of the arcs of a circle in the old system. Thus, denoting the angle AOC, or arc AB, by Z (see Fig. 44), we have, regarding OB = 1 : APPLICATION TO THE CIRCLE. 101 (1) sin Z = '-^ = BC. (2) cos ^ = -° C OH (3) tan Z= || = ^ = ^D. (4) sec Z= OC. OD = 0D. Fig 45. OB OB OC'"~ OA 147. This graphic representation of the functions in the old sys- tem is for many minds simpler than the more abstract conception of ratios, and students should be familiar with it. The following definitions are given : 1. The Sine of an arc is the perpen- dicular drawn from its extremity to the diameter passing through its origin. 2. The Cosine of an arc is the dis- tance between the foot of the sine of the arc and the centre of the circle. 3. The Tangent of an arc is the perpendicular to the radius at its origin, and limited by the radius pro- duced passing through its extremity. 4. The Secant of an arc is a line drawn from the centre of the circle through the extremity of the arc and limited by a tangent at its origin. 5. The Cotangent and Cosecant are respectively the tangent and secant of the complement of the arc. 148. The functions of arcs terminating in the different quadrants are represented in Fig. 45. Thus, sin A ON is NP; cos A ON is OP ; tan ^OiVis AT; sec AONia OT; etc. sin A ON' is N'P'; cos A ON' is OP' ; tan AON' is AT'" ; sec AON' is OT'" ; etc. sin AON" is N"P" ; cos AON" is OP" ; tan AON" is AT; sec AON" is OT; etc. Note. — The student may be required to point out the lines of the cir- cle which correspond to the formulas of Tables I., II., and III. 149. The relation of the values of the trigonometrical functions in the two systems is shown as follows. Let R denote the radius of the circle ; 102 PLANE TRIGONOMETRY. BO Then, sin angle BOC '= — =- ; and BC=R X sin angle BOO. Also, sin arc AB = BC= R X sin angle BOC. sin of the arc sin Hence, sin angle BOC = radius of circle R 150. Similar results hold for all the other trigonometrical func- tions of the two systems. Hence for any formula of the modern sys- tem which involves functions of angles we can readily deduce the corresponding formulas in the ancient system depending on the ares, and vice versd. Notes. — 1. The modern method was introduced by Dr. Peacock, and has almost entirely superseded the ancient method. 2. The old definitions give some indications of the origin of the terms sine, cosine, etc. The word sine seems to have been derived from the Latin word simus, a bosom. The arc is supposed to represent a bow, and thus gets its name; and the string, half of which represents the sine of half the arc, would come against the breast of the archer. The words tangent and secant are naturally derived from their definitions in Geometry. EXERCISES XXX. 1. Construct the functions of an arc in quadrant II. Show their signs. 2. Construct the functions of an arc in quadrant III. Show -their signs. 3. Construct the function of an arc in quadrant IV. Show their signs. 4. Required the signs of the functions of 250° ; 320° ; 400° ; 450° = 600°; 800°. 5. Construct the angles less than 360° which have their sine equal to -fy; their cosine equal to -J-f? 6. Construct the angles less than 270° which have sin A = £; cos A = —%; tan A = — f ; cot A = f 7. Limit the angle when sine and cosine are both positive or both negative. When cosine and tangent are- both negative or both positive. APPLICATION TO THE CIRCLE. 103 Miscellaneous Exercises. . Additional Formulas. Note. — In' 92-29, *, h, e denote the sides of a triangle opposite to the respective angles A, £, and 0; and S denotes the half sum of the sides. . sin 40° + sin 20° = cos 10°. j 5. tan.4 + cot.4 = 2cse2 J. 2. sin 80° — sin 40° = sin 20°. . . i 6. tan.4 — cot J = 2 cot 2.1. 911- 1 pos 1 J. 3. 14-smJ=- : — -. 1 — sin .4 '7. tan 8 A— sin* J=tanM sin' J. 1 ! 9. sin 3 A 4- sin A = 2 sin 2 A cos A. 10. sin 3 A — sin A — 2 cos 2.4 sin A. 11. cos 2 A + cos 4 J = 2 cos ZA cos A. , .-> . o . 3 tan J — tan* J l_tanSJ= j_ JtaIJ • 13. tan ,1 + tan £ = ^#±^1. cos .4 cos B ■, • . t> sin (-4 — -B) 14. tan A — tan B = »- £■ cos J cos is , - , , . d sin* .4 — sin s B la. tan* J — tan- ,8 = — — — — cos 1 J cos' 5 .,. 1 — cos 2 A , , 1»>. t-t- 5-7= tan 5 A. 1 + cos 2A j_ tan A + tan B _ sin (A + B) tan A — tan B sin {A — B) , o tan A -4- tan 2? . . „ lt>. '■ — = tan A tan B. cot A + cot jB .„ cos .4 — sin -4 .... . 19. = see 2 J — tan 2 A. cos A -4- sin A ~. cos .4 + cos B cos ^(^4 — B) * sin(.4 + .B) — sinA(.4-r-.B)" 21 cos -B — cos .4 _ sin 1(^4 — B) ' sin(.4 + B) eosi(A + B)' 24. sin % (A — B) a — b sin \ (A + B) c 27. 25. cos^(A-B) a + b cos \ (A + B) c 28. 26. sin ^ A sin \ B s — c sin J C c 29. 104 PLANE TRIGONOMETRY. 22. * * an ^ = Bec 2^ — tan2X 1 + ta° A sin (A - B) _ (a +b)(a~b) sin (A + B) c> _ cos ^ .4 cos ^ B » sin J C c sin ]j -4 cos \ B s — 6 cos £ C c cos -jr .i sin j B s — a cos \ C c Functions of Special Angles. Prove the following, remembering sin 15° = sin (45° — 30°). 1. sin 15° = V 3 — 1 . 3. tan 150 = 2 — \/Z. 2. cos 15° = ■ V 3 + 1 4. cot 15° = 2 + x/Z. 2i/2 v 5. Find sin 75° ; cos 75° ; tan 75° ; cot 75°". 6. Find sine of 18°. Solution.— Let A = 18°; then 2 A =, 36°, and 3 A = 54°, and since 36° and 54° are complementary, we have sin 2^4 = cos Z A. Now, sin 2^1 = 2 sin A cos A ; and we can find cos 3^1 = 4 cos 3 A — 3 cos A. Substituting and reducing, we find sin A = \ ( j/5 — 1) = sin 18°. Prove the following : 7. cos l 8 ° = l/aO + 2 T /5) _ 36°=i+-^- 5 . 4 4 9. sin 36°= 1/(10- 2 T/5). 4 10. sin 9°= 1/(3+^) -1/(5 -1/5) . 4 11. cos 9°= V(3+V?)+T/(5-V5) 4 ADDITIONAL EXERCISES. 105 yt — 1 Note. — Similarly, since 18° — 15° = 3°, we can find the functions of 3°, and from this of 6°, etc. Find the value of a; in the following equations : 13. sin 2 x = cos a;. 16. tan x + tan (45° + a;) =2. 14. sin x -\- cos x = \/% 17. 2 sin 2 x + sin 2 - x = 2. 15. esc a; = csc %x. 18. tan (45° -* x) + cot (45° — x) = 4. 19. tan (45° 4- x) = 3 tan (45° — a:). 20. sin x sin 3 x = £. 25. 6 cot 2 a; — 4 cos 2 x = 1. 21. sin x + cos a; = \ l/2. 26. sin3a: + sin2a; + sina: = 0. 22. sin a; + sin 4 a; = 0. 27. cos 3 x + cos 2 a; + cos a; =0. 23. cos3a: — sin3a: = i|/2. 28. sin 2 2 x — sin 2 x = sin 2 30°. 24. |/3 sin a: — cos x = ^2- Additional Exercises. 1. Given sin x (sin x — cos x) = ^ ; find x. 2. Given tan a; + cot x = '2; fidd a:. 3. Given sin x + cos 2 a; = J y'S, to find sin .r. 4. Given 4 sin a: sin 3 x = 1, to find sin x. 5. Given a sin a; + 6 cos a; = c, to find sin x. 6. Given a cos a; = tan x, to find cos x. Change to forms for logarithmic computation : • 7. tan x + cot x. 11. 1 + tan x tan y. 8. cot. x — tan a;. 12. 1 — tan x tan y. 9. tan ar + cot y. 13. cot x cot y + 1. 10. cot a; — tan y. 14. cot x cot y — 1. Demonstrate the following : 15. I + sin " = tan 2 (45° — \ V). 1 — sin 106 PLANE TRIGONOMETRY. 16. sin 4 = 4 sin 8 cos — 8 sin s cos 0. 17. cos 4 = 1 — 8 cos 2 + 8 cos 4 0. 18. sin 3 = 4 sin sin (60° — 0) sin (60° + 0). 19. cos 3 = 4 cos cos (60° — 0) cos (60° + 0). 20. sin (a + 6) sin 3 (a — b) = sin 2 (2a — b) — sin 2 (2b — a). 21. Find the value of cos (sin -1 £ + cos -1 J). 22. Find the value of tan (tan -1 x + cot" 1 x). 23. Prove that tan" 1 \ + tan" 1 % + tan" 1 f + tan" 1 i = £ • o 24. Prove that tan (2 tan -1 a) = 2 tan (ta» -1 a -4- tan -1 a 3 ). 25. Prove tan -1 (J tan 2 a;) + tan" 1 (cot x) + tan -1 (cot 3 x) = 0. 26. Find *, given tan" 1 \ + 2 tan" 1 £ + tan" 1 £ + tan-' - = 7 • 27. Prove tan- 1 (a; — 1) + tan~ l ar + tan -1 (a; + 1) = tan" 1 3 a;. 28. If sec a — esc a = $ , prove that a = J sin -1 f . 29. If sin A = sin B and cos .4 = cos B, then either .4, and B are equal, or they differ by some multiple of four right angles. 30. If cos A = cos B and sin A = — sin B, then A + B is zero, or a multiple of four right angles, positive or negative. 31. The sum of the tangents of the three angles of a plane triangle is equal to their product. • 32. In any plane triangle, cot J A + cot \ B + cot \ C = cot J A cot \ B cot £ (7. 33. In a plane triangle, if 6 = a sin C and c = a cos 5, then the triangle is right-angled at A. 34. If the angles A, B, and C of a plane triangle are to each other as 2, 3, and 4, prove that 2 cos \ A= ~T • 35. In any plane triangle ABC, if the angle made by a line drawn from the vertex C to the middle of the base c'is denoted by Z, then 2 cot Z = cot 4 — cot B. INTRODUCTORY DEFINITIONS, 107 SPHEEIOAL TEIGONOMETET. SECTION XII. INTRODUCTORY DEFINITIONS. 151. Spherical Trigonometry treats of the solution of spherical triangles. 152. A Spherical Triangle is a portion of the surface of a sphere hounded by three arcs of great circles of the sphere. 153. The sides of a spherical triangle, being arcs of great circles, measure the plane angles formed by radii of the sphere drawn to the vertices of the triangle. 154. Each angle of a spherical triangle has the same measure as the dihedral angle included by the planes of its sides. 155. The sides of a spherical triangle may have any values from 0° to 360° ; but in this treatise only sides less than 180° will be considered. The angles may have any values from 0° to 180°. 156. If two parts of a spherical triangle are both greater or both less than 90°, they are said to be in the same quadrant; but if one part is greater and the other less than 90°, they are said to be in different quadrants. 157. Spherical triangles are divided into two general classes — right spherical triangles and oblique spherical triangles, 108 SPHERICAL TRIGONOMETRY. the same as plane triangles. A right spherical triangle may have one, two, or even three right angles. 158. A spherical triangle having two right angles is called a bi-rectangular triangle. A spherical triangle having, three right angles is called a tri-rectangular triangle. A spherical triangle having one or more sides equal to a quadrant is called a quadrantal triangle. 159. The nature of a spherical triangle will be seen by examining the diagram in the margin. The side AB measures the plane angle A OB ; the side BG measures the plane angle BOC, and the side AC measures the plane angle AOO. The spherical angle B is measured .by the dihedral Fig. 46. angle formed by the two planes AOB and COB; the spherical angle C is measured by the dihedral angle formed by the two planes AOC and BOC, etc. 160. It will be remembered, as shown in Geometry, that in every spherical triangle we have the following truths : 1. The sum of the sides is less than 360°. 2. The sum of the angles is greater than 180° and less than 540°. 3. If two angl.es of a spherical triangle are equal, the opposite sides are equal. 4. If one angle of a spherical triangle is greater than an- other, 'the side opposite the greater angle is greater than the side opposite the less angle — and conversely. INTRODUCTORY DEFINITIONS. 109 5. The sides and angles of any spherical triangle are re- spectively the supplements of the angles and sides of the polar triangle. 161. Thus, if the angles of a spherical triangle are denoted by A, B, C, and the sides opposite these angles respectively by a, b, c, and the corresponding angles and sides of the polar triangle by A', B", C", a', b', and ° r smb = cotAtan a . I ^ Similarly, sin a = cot B tan b. J Taking the product of the two formulas [50], we have sin a sin & , . ^ _ 7 ; r = COt A COt B. tan a tan b Whence [47], cos c = cot A cot B." [51] Multiply the first formula in [48] by the second in [49], we have sin a cos B = sin A sin c tan a cot c. „ T , t- . . tan a , . cos c Whence, cos B = sin A — x sm c cot c = sm A sm a cos a Or, [47], cos B = sin A cos b. *) [ [52] Similarly, cos A = sin B cos a. J 163. In deriving the formulas under Art. 162, it was assumed in the construction of the figure that all the parts of the triangle, except the right angle, are less than 90°. The formulas are, however, true for any right spherical triangle, as is readily seen. 112 SPHERICAL TRIGONOMETRY. Suppose one leg a to be greater than 90°. Then construct a fig- ure (Fig. 49), as in Art. 162. OE OE OD JNow ' OF = OD X OF' Or, cos (180°— e)=cos b cos (180°— a) Whence, cos c=cos a cos b • and this is the same as Formula [47J. 'Again, suppose that both legs a and b are greater than 90°. Construct the figure as before (see Fig. 50). Fig. 49. Then, OE OF OE OD OD X OF' Or, cos c = cos (180° — b) cos (180° —a). Whence, cos c = cos a cos b. and this is the same as Formula [47]. Therefore the formulas of Art. 162 are universally true. EXERCISES XXXII. 1. If c = 90°, what may be inferred in respect to the other parts? Solution. — In For. [47] eos c = cos a cos 6. If c = 90°, cos e = 0; hence, either cos a or cos b is 0, and either a or b is equal to 90°. If a = 90°, then A = 90°, and B—b. If b = 90°, then R = 90°, "and A = a. 2. If a =* 90°, what may be inferred in respect to the other parts? If a = 90° and c = 90° ? If a = 90° and 6 = 90° ? 3. What may be inferred in respect to the other parts if b = 90° ? Ifa = ^? If 6 = B, orc= CI 4. What will each of the formulas in Art. 162 become when ap- plied to the polar triangle ? NAPIER'S RULES. 113 Formulas of Plane and Spherical Compared. 164. The six formulas of Art. 162, comprising ten equations, enable us to solve every case of right spherical triangles. Put in another form, as below, they may be remembered by their analogy to the corresponding formu- las for plane triangles : In Plane Right Triangle. • A a sin A = - • Bin B = - • c A b cos A = - • c cos B = - • c tan A = r • tan B = - • a sin A = cos B. sin B = cos A & = a? + b\ 1 = cot A cot B. In Spherical Right Triangle. sin A = sin a sin c sin B = smb sine . tan b „ tana cos^4=t • C0SjH=- tan c tan e . tana , _ tan&' tan A = - — r- tan B = —. sin o sin a . cosB . „ sin A = =~ sin .6 cos A cos 6 cos « cos c = cos a cos 6. cos c = cot A cot 5. Napier's Rules. 165. The ten formulas of Art. 162, by a very ingenious device, may all be embraced in two general rules, easily remembered and applied. These rules are due to Baron Napier, the distinguished inventor of logarithms. 166. In this device, five of the parts of the triangle are considered, the two sides about the right angle, the comple- ments of their opposite angles, and the complement of the hypotenuse. These are called Napier's Circular Parts. These parts are represented thus : a, b, co. A, co. B, and co. C. Notice that co. A equals 90° — A, etc. 167. Any one of these parts may be taken as the middle 8 114 SPHERICAL TRIGONOMETRY. part; and then the two parts adjacent to it are called adjacent parts, and those which are separated from it are called opposite parts. eo. B Thus, if CO. O is taken as the middle part, then CO. A and CO. B are adjacent parts, and a and b are opposite parts, as is seen in the figure. Wi ^ It will be noticed that the right angle does not enter as one of the parts, and that the two sides including it are regarded as adjacent. 168. The two rules of Napier are as follows : , Rule I. The sine of the middle part is equal to the product of the tangents of the adjacent parts. Rule II. The sine of the middle part is equal to the product of the cosines of the opposite parts. Note. — It will aid the memory to notice that the vowel O occurs in cosine and opposite, while a occurs in tangent and adjacent. 169. The correctness of these rules may be shown by taking eaah of the five parts as the middle part, and com- paring the resulting equations with the formulas of Art. 162. Thus, let co. c (see Fig. 51) be taken as the middle part; then co. A and co. B are the adjacent parts, and a and b are the opposite parts. Then by Napier's Rules we have sin (co. c) = tan (co. A) tan (co. B). Whence, cos c = cot A cot B. Also, sin (co. c) = cos a cos 6. Whence, cos c = cos a cos b. These results, it will be seen, correspond with Formulas TBB AMBIGUOUS CASES. 115 [51] and [47] ; and in a similar manner all the formulas may be derived from the two roles. Nors. — The rales were originally derived from the formulas, and may be so derived by substituting for A, B, and C in die formulas, their complements. EXERCISES XXXIII. 1. Derive Formulas [43] from Napier's Rules, i Derive Formulas [50] from Napier's Rules. 3. Derive Formulas [49] and [52] from Napier's Rules. 4. If we take for the five parts of the triangle the hypotenuse, the two oblique angles, and the complements of the legs, what for- mulas will Napier's Rules give? a. On this supposition, what rales should we have to give the same results as Napier's Rules? Note. — The rules thus derived are known as Mmmdmifs Rules. The Ambiguous Cases. 170. In applying Napierfs Roles, or the formulas of Art. 162, where the part sought is to be determined by the sate — the same sine corresponds to two different angles or arts, supplements of each other — it becomes necessary to discover soch a relation between the parts as will enable us to determine which of the two angles or arcs is to be taken. 171. For this purpose we shall prove the following principles: Pros. 1. J* a right spherical triangle, a side and its opposite angle are always ta the same quadrant For we have [52], , cos J? sin A = r* cos 6 116 SPHERICAL TRIGONOMETRY. Now A is always less than 180° ; hence sin 'A is always plus ; therefore cos B and cos b must always have the same sign, and hence must both be greater or both less than 90°. Pbin. 2. If the two sides of a right spherical triangle, in- cluding the right angle, are in the same quadrant, the hypote- nuse is less than 90° ; but if the two sides are in different quad- rants, the hypotenuse is greater than 90°. For we have [47], cos c = cos a cos b. Now, if cos a and cos b have the same sign, cos c is positive, and hence c is less than 90° ; but if cos a and cos b have different signs, cos e is negative, and hence c is greater than 90.° Note. — These two principles enable us to determine the nature of the part to be found in every case, except when an oblique angle and an opposite side are given to find the other parts. In that case there may be two solutions, one solution, or no solution, as will be shown in the treatment of the case. Solution of Right Spherical Triangles. 1T2. In the solution of right spherical triangles there are six cases, as follows. Given, 1. The two legs. 3. The hypotenuse and one leg. 2. The two angles. 4. The hypotenuse and one angle. 5. One leg and its adjacent angle. 6. One leg and its opposite angle. 173. In solving these several cases the formulas given under Art. 162 may be taken from the book, or these formulas may be readily derived from Napier's Rules. 1. In applying Napier's Rules to obtain the formulas, it will be readily seen which of the three parts — the two given and the one SOLUTION OF RIGBT SPHERICAL TRIANGLES. 117 00. B co. A required — is to be taken as the middle part. Thus, if the three parts are all adjacent to one another, the middle one of the three is the middle part, and the other two are adjacent parts ; if one is separated from the other two parts, then the part which stands by itself is the middle part, and the other two parts are opposite parts. 2. Thus, suppose we have a and 6 given to find the other parts, then to find r. we write the terms, a, 6, co. c, and since co. c is separated from a and b (see Fig. 52), we take co. c for the middle part, and have by Rule I., sin (co. c) = cos a cos 6, or, cos c = cos a cos 6. To find A we write a, b, and co. A, and since the terms are not separated (see Fig. 52), we take 6 for the middle part, and by Rule II. have sin b = tan a tan (co. A) = tan a cot A. Whence, cot A= cot a sin b. Students are advised to derive the formulas from Napier's Rules. Case I. 174. Given, the two legs a and b of the triangle ABC. 1. In the spherical triangle ABC, right angled at C, a = 59° 38' and b = 48° 24' ; find A, B, and c. Fig. 52. Solution. — The formulas for the solution, derived by Napier's Rules or taken from Art. 162, are, cos c = cos a cos 6. [47] cot A"= cot a sin 6. [50] cot B = sin a cot b. [50] Here C, A, and B are deter- mined by the cosine, and there is no ambiguity. Operation. log cos a (59° 38') = 9-703749 log cos 6 (48° 24') = 9.822120 log cos c = 9.525869 c = 70° 23' 20" log cot a (59° 38') = 9.767834 log sin 6 (48° 24') = 9.873784 log cot A = 9.641618 A = 66° 20' 23" Similarly, B = 52° 32' 48" 118 SPHERICAL TRIGONOMETRY. EXERCISES XXXIV. 2. In the right spherical triangle, given a — 75° IS' and b = 120° 15' ;' find A = 77° 11' 14", B = 119° 25' 17", c = 97° 22' 9". 3. In the right spherical triangle ABC, given a = 155° 27' 54", and b = 29° 46' 8" ; find c = 142° 9' 13", A = 137° 24' 21", B = 54° 1' 16". Case II. 175. Given the two oblique angles A and B of the triangle ABC. 1. In the spherical triangle, right angled .at C, A = 62° 15', and B = 56° 30'; find a, b, and c. Operation. log cot A (62° 15') = 9.721089 log cot B (56° 300 = 9.820783 log cos c = 9.541872 « = 69° 37' 14" log cos A (62° 15') = 9.668027 log sin B (56° 30') = 9.921107 log cos a = 9.746920 a =56° 3' 25" Similarly, 6 = 51° 24' 56" Solution, — The formulas for the solution, taken from Art. 162 or derived by Napier's Rules are, cos c = cot A cot B cos A = cos a sin B cos B = cos 6 sin A From the second and third we have, cos a = cos A -.- sin J5 cos b = cos B -r- sin A, which we use to find a and 6. EXERCISES XXXV. 2. In the right spherical triangle ABC, given A — 69° 20', 5=58° 16'; find a= 65° 28' 58", 6 = 55° 47' 46", 7( SOLUTION OF RIGHT SPHERICAL TRIANGLES. 119 3. In the right spherical triangle ABC, given A = 47° 13' 43", B = 126° 40' 24"; find a =32° 08' 56", b = 144° 27' 03", c = 133° 32' 26". Case III. 176. Given the hypotenuse c and either leg a or b. 1. In the spherical triangle ABC, right angled at C, c — 56° 13', a = 48° 30' ; find A, B, and 6. Operation. Solution. — The formulas for the solution taken from Art. 162, or derived by Napier's Rules, are, cos c = cos a cos 6 sin a = sin A sin c cos.B== tan a cot a Whence, cos b = cos c -r- cos a and sin A = sin a -r- sin c log cos c (56° 13') = 9.745117 log cos a (48° 30') = 9.821265 log cos b = 9.923852 6 = 32° 56' 49" log sin a (48° 30') = 9.874456 log sin c (56° 13') = 9.919677 log sin A = 9.954779 ^ = 64° 18' 17" Similarly, B = 40° 52' 14" Note. — Two angles correspond to sin A, but since a is less than 90°, the angle A must also be less than 90° (Art. 170). EXERCISES XXXVI. 2. In the right spherical triangle ABC, given 6 = 37° 48' and c = 66° 32' ; find B = 41° 55' 34", A = 70° 19 7 18", and a = 59° 44' 13". 3. In the right spherical triangle ABC, given a = 95° 22' 30", c = 91° 42" ; find A = 95° 6', B = 71° 36' 45", and b = 71° 32' 12". 120 SPHERICAL TRIO ONOMETR Y. Case IV. 177. Given the hypotenuse c and either angle A or B. 1. Given c = 86° 50' and A = 58° 30'; find a, b, and B. log Bin A (58° 30") = 9.930766 Solution. — The formulas for the solution taken from Art. 162, or derived by Napier's Rules, are, sin a = sin A sin c tan 6 = cos A tan c cos c = cot A cot B Whence, cot B = cos c tan A log sin c (86° 50') = 9.999336 log sin a — 9.930102 a = 58° 21' 27" log cos A (58° 30') = 9-718085 log tan c (86° 50') = 11.257078 log tan 6 = 10.975163 6 = 83° 57' 21" Similarly, B = 84° 50' 56" EXERCISES XXXVII. 2. Given c = 115° 35' 20", and B = 110° 26' 30"; find a = 36° 6' 13", & = 122° 18' 54", and A = 40° 47' 35". 3. Given c = 70° 23' 42" and A = 66° 2C 40'' ; find a = 59° 38' 26", b = 48° 24' 15", and B = 52° 32' 55". Note. — In Ex. 1, two values of a correspond to sin a; but by Prin. I., a must be less than 90°, since A is less than 90°. Similarly, in Ex. 2, b must be greater than 90°. Case V. 178. Given one leg a and its adjacent angle B. 1. Given a = 102° 3^ and B = 43° 24' ; to find b, c, and A. Solution. — The formulas for the solution derived by Napier's Rules or taken from Art. 162, arc, tan b = sin a tan B cot c = cot a cos B cos A = cos a sin B log sin a (102° 30') = 9.989582 log tan B (43° 24 / ) = 9.975732 log. tan 6 =9.965314 6 = 42° 42' 52" Similarly, c = 99° 9' 2" and * 4 = 98° 33' 9" SOLUTION OF BIGHT SPHERICAL TRIANGLES. 121 EXERCISES XXXVIII. 2. Given b = 42° 4ff 24", and A = 116° 36' 20"; find a = 126° 27' 47", c = 115° 54' 35", and £ = 48° 54'. 3. Given a = 29° 46' 8", and B = 137° 24' 21"; find A = 54° 1' 16", b = 155° 27' 54", and c = 142° 9' 13". Note. — In Ex. 1, since cot a is negative, cot c is negative, and hence c is greater than 90°. In Ex. 2, a is greater than 90°, since A is greater than 90°. Case VI. 179. Given one leg a and its opposite angle A. 1. Given a= 110° 32' 25" and A = 98° 48' 50"; find b, c, and B. log tan a (110° 32' 25") = 10.426332 log cot A (98° 48' 50") = 9.190490 log sin b = 9.616822 . 6 = 24° 26' 44" or 155° 33' 16" Similarly, c = 71° 22' 23" or 108° 37' 37" And, B = 25° 53' 38" or 154° 6' 22" Solution. — The formulas for the solution are sin b = tan a cot A sin c = sin a — sin .4 sin B = cos .4 -f- cos a Note. — In this case, since all the required parts are determined by their sines, there are always two solu- tions. Thus, if in the triangle ABC, AB and AC are produced to meet in A', ABA' and AC A' are semi-circumferences, and the angle A = A' '. The two triangles, ABC and A'BC, botli have the two given parts a and A; but V , c' ', and B' in the second triangle are respectively the supplements of 6, c, and B in the first triangle. Fig. 53. 122 SPHERICAL TRIGONOMETRY. EXERCISES XXXIX. 2. Given, A = 102° and a = 120° ; find the other parts. Ans. b =21° 36' 08" ) ' ( b = 158° 23' 52" e = 62° 17' 51" I or J c = 117° 42' 09" £ = 24° 34' 16" J [ B = 155° 25' 44" 3. Given B = 80° and b = 75° ; solve the triangle. ^4ns. a = 41° 09' 18" \ ( a = 138° 50' 52" c = 78° 45' 45" I or j c = 101° 14' 15" A = 42° 08' 18" J [A= 137° 51' 42" Quadrantal Spherical Triangles. 180. A Quadrantal Spherical Triangle is one in which one side is equal to 90°. It is the polar triangle of some right spherical triangle. 181. To solve a quadrantal spherical triangle we pass to its polar triangle by subtracting each side and angle from 180°. The resulting polar triangle will be right angled, and may be solved as already explained. The parts of the given triangle may then be found by sub- tracting the parts of the polar triangle from 180.° EXERCISES XL. 1. Given the quadrantal triangle ABC, in which c = 90°, B = 42° 10' and C= 115° 20'. Solution. — Passing to the polar triangle A'B'C, we have C = 90°, c' = 64° 40', and // = 137° 50'. Solving this triangle by the method for right triangles, we find A' = 115° 23' 20", B' = 132° 2' 13", and a' = 125° 15' 30". Subtracting each of these from 180°, we find the required parts of FUNDAMENTAL FORMULAS. 123 the quadrantal triangle are BC = 64° 36' 40", AC = 47° 57' 4-7", and A = 54° 44' 24". 2. Let ABC be a quadrantal triangle in which c = 90°, -4 = 75° 42', and 6 = 18° 37"; find C = 103° 34' 49", B = 18° 04' 40", a = 85° 28' 39". 3. In the spherical triangle ABC, given a, 6, and c, each equal to 90°, to find the angles. 182. An Isosceles Triangle is readily solved by divid- ing it into two right triangles by drawing an arc of a great circle from the_ vertex perpendicular to the base. SECTION XIV. THE OBLIQUE SPHERICAL TRIANGLE. Fundamental Formulas. 183. We now proceed to find the relation of the func- tions of the sides and angles of an oblique spherical triangle. I. To find the relation of the sines of the sides and angles. 184. Let ABC, Pig. 54, be an oblique spherical triangle, A, B, C its three angles and a, b, c its three sides. From C draw an arc CD of a great circle perpendicular to the side AB, meeting AB in D; and denote CD by p. 124 SPHERICAL TRIGONOMETRY. [53] In the right triangles BCD and ACD, we have (Art. 162), ' sin p = sin a sin B sin p = sin b sin A Hence, sin a sin B = sin b sin A Similarly, sin a sin C = sin o sin A And sin b sin C = sin c sin B These equations may be written in the form of propor- tions; as, sin a : sin b = sin A : sin B. Hence, we have the following theorem : 1. The sines of the sides of a spherical triangle are propor- tional to the sines of their opposite angles. 185. If in Fig. 54, the perpendioular CD cuts the side AB produced, we must have in place of sin A, sin B, or sin C, sin (180° - A), sin (180° - B) or sin (180° - (J). But these sines are equal to sin A, sin B, and sin C, re- spectively (Art. 52) ; hence the formulas [53] are true for all cases. II. To find an expression for the cosines of the sides. 186. In the triangle ABC, CD being perpendicular to the base as before, let AD = m and BD = n. Now, in the right triangle BCD we have (Art. 162), cos a — cos p cos n = cos p cos (c — m) Or, cos a — cos p cos c cos m + cos p sin e sin m. 'FUNDAMENTAL FORMULAS. 125 But, Art. 162, cos p cos m = cos b. Whence, cos p = cos b sec m. And, cos p sin m = cos 6 tan m. Or, Art. 162, = cos b tan b cos A. = sin 6 cos A. Substituting these values of cosp cos m and cosp sin m in the second expression above, we obtain cos a = cos b cos c -f- sinb sin c cos A ~| Similarly, cos b = cos a cos c + sin a sin c cos B I [54] And, cos c = cos a cos b + sin a sin b cos C J These formulas give the following theorem : 2. In any spherical triangle the cosine of each side is equal to the product of the cosines of the other two sides plus the product ■ of the sines of these sides and the cosine of the included angle. HI. To find an expression for the cosines of the angles. 187. Let A'B'C be the polar triangle of ABC, and denote its angles by A', B', and C", and its sides by a', V, c'. Then from Art. 186, we have cos a' = cos V cos d + sin V sin d cos A'. Now by Art. 161, A' =180° — a, F = 180° — b, C'=180°—C. a?=180° — A, b'=180°—B, Sm( , S - 6) - - „ sin a sin o [56] 191. If we add unity in Art. 190, and reduce as before, we may derive the following formulas : , , , sin 8 (sins — a) cos 2 \ A = . , . J ■ sin o sm c , , -r, sins (sins — b) COS 2 \ B = r 1 : ' • ■* sin c sin a „ , „ sin s sin (s — e) cos 2 A C= : ~ — t — • * sin a sm o [57] 192. Dividing the corresponding formulas of Arts. 190 and 191, we have, For. [2], tan 2 i-A = sin(g ~ & ) sin 0- c ) . 2 sin s sin (s — a) tan sJnis-clsmXs-^). sin s sin (s — o) tan 2 lC= sln ! :s - a)s ; n(s ~ &) - A sin s sin (s — c) 193. Again, from the first equation of [55], we have cos B cos C + cos A [58] cos a = ■ sin B sin C ___, * cos A + cos (J+C) Whence, 1 — cos a = : — ^—. — ^ sin B sm C Hence, sin 2 * a — ™ s ^ + *+ S — B) Similarly, sin * 6 = sm (7sinl " , , . ,. — cos S cos (S— 0) And, sin 2 \c = r— ; — r — ~ ■ * sin A sin £ [59] 194. In a similar manner we may find the following : 2 cos(g-F)cos(£-(7) COS ■§■ w — , _ . sv ' 2 sin £ sin C cog2 ft = cos(S-C)cos(S-A) , sin C sin ^4 cos * , c= cos (S- J) cos (S- 2?) ¥ sin A Sin £ [60] 195. And from .[59] and [60] we derive, by For. [2], tan f«- cos(s __g )cos(/S _ ) + nT1 » xh- —cozScos(S— B) tan i o - cog (&l _ Q) cos (S _ A y . 2 , _ — cos £ cos (iS — C) tun ■* c — [61] cos (S — A) cos (S—B) Notes. — 1. The second members of Formulas [59] and [61] must be essentially positive, though their algebraic sign is negative ; for since 2 £> 180°, £> 90° and cos £ is negative ; hence, — cos S is positive. Also, the positive sign must be given to the radical, since 2 - is less than a right angle. 2. Formulas [59], [60], and [61] might have been deduced by applying Formulas [56], [57], and [58] to the polar triangle. GAUSS'S EQUATIONS. 129 Gauss's Equations. 196. From For. [9] we have sin \ (A + B) = sin £ A cos \ B + cos \ A sin \ B. Substituting the values of sin £ A, cos \ B, cos \ A, and sin | 5, derived from [56] and [57], and reducing by com- bining factors and extracting root, we have sinK^+£)= J Sin ( T? Sln (S-C) X J Sin * Sln ( *~ 6) • \ sin o sin c \ sin c sin a ! I sin s sin (s — a) I sin (g— c)sin (s — a) \ sin & sine \ sine sin a _ sin (s — a) + sin (s—b) I sin -s sin (s — c) sin c \ sin a sin b Now, sin c = 2 sin £ c cos £ e [17], and sin (s — a) + sin (« — &) = 2 sin \ c cos ^ (6 — a) [31] ; and the quantity under the radical equals cos \ C [57], hence, • , , a ■ n\ 2 sin A e cos A (6 — a) , n sin A 04 + B) = . T . 2 - L - ; -X cos ^ (7. "* v 2 sin \ c cos A c Cancelling, multiplying by cos \ c, and reducing, sin \ {A + B) cos A c = cos A (a — &) cos A, C. Operating in the same way with the values of sin \ (A — B), cos \{_A + B~), and cos A ( A — B), we have the four equations, sin A (A + B) cos \ c = cos \ (a — 6) cos A C. cos \ (A + B) cos A c = cos £ (a + V) sin A C. r L"2] sin A 04 — 5) sin |c= sinA(a — &) cos A 0. cos A (^4 — 2?) sin £ c = sin J (a + b) sin £ C. 130 SPHERICAL TRIGONOMETRY. These four formulas are'cailed Gauss's Equations, though, as Todhunter remarks, they are really due to Delambre. Napier's Analogies. •197. By dividing the first of Gauss's Equations by the second, the third by the fourth, the fourth by the second, and the third by the first, we obtain the following equa- tions : tan i (A + B) = """U^S co H C. * v J cos \ (a + b) 2 T , . _. sin \ (a — b) . , n tan I (A — B) = - — ^7 — r~d cot i G - 1 J sin \(a + V) cos \ (A—B) j tanH«+&) = d0S , ( ^ +jB) tani C . , . ,. sin I (A—B) , , tanH«-5) = — f^-^tanlc. 198. Writing these equations in the form of proportions, we have, [63] sin \ (a+b) : sin | (a— 5) = cot -| C : tan \ (A — B). ■ cos|(a+&) : cos-|-(a— 6) = cot£ C: tan%(A+B). sin \ ( A + B) : sin £ (A — B) = tan $ c : tan \ (a — b). cos j (A + B) : cos^(A — B) = tan \c: tan % (a+b). [64] These proportions are called, from their inventor, Napier's Analogies. Note. — As is seen, there is a very intimate relation between Gauss's Equations and Napier's Analogies. We have derived the Analogies from the Equations ; but the Analogies may be derived by an inde- pendent process, and the Equations deduced from the Analogies. SOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 131 199. By examining ,the formulas [63] we reach the fol- lowing conclusions : 1. In the first formula the factors cos \ (a — &) and cot \ C are always positive ; hence tan J (A -\- B) and cos \ (a + b) must always have the same sign. Therefore, if a + b < 180°, and consequently cos J (a + 6) > 0, then it follows that tan £ {A + B) > 0, and therefore A + B < 180°. Similarly," it follows that if a + 6 > 180°, then also A + B > 180°. 2. Also, if a + 6 = 180°, and consequently cos £ (a + 6) = 0, then tan J (^ + -B) = °° ; whence, J (4 + B) = 90°, and A + B = 180°. 3. Conversely, it may be shown from the third formula that a + & is less than, greater than, or equal to 180°, according as A + B is less than, greater than, or equal to 180°. Solution of Oblique Spherical Triangles. 200. In the solution of oblique spherical triangles, there are six cases, as follows. Given, 1. Two sides and their included angle. 2. Two angles and their included side. 3. Two sides and an angle opposite to one of them. 4. Two angles and a side opposite to one of them. 5. The three sides. 6. The three angles. Case I. 201. Given two sides, a and b, and the included angle C. Method.— We find the angles A and B by the first and second of Napier's Analogies, viz. : 2 v J cos \ (a + V) 2 i / a m sin 1 (o — b) . , n 132 SPHERigAL TRIG ONOMETR Y. The side e may then be found by [53], or by the third or fourth of Napier's Analogies. It is better, however, to find c from one of Gauss's Equations, since they involve functions of the same angles that are used in the two formulas of Napier's Analogies. We can use any one of the formulas ; thus from the second we have , cos \ (a + 6) . , „ cos \ c = ; * . , , ' sin * C. J cos % (A+B) 2 EXERCISES XLII. 1. In a spherical triangle, given a = 72° 36', & = 40° 44', and C= 54° 40'; find the other parts. Solution.— a = 72° 36'. 6 = 40° 44'. C=54°40'. log cos i (a — 6) = 9.982986 colog cos i (a + b) — 0.260025 log cot J C =10.286614 log tan J (A+B) =10.529625 $ (A+B) = 73° 32' 39" log cos i (a + b) = 9.730975 colog cos i (A+B) = 0.547790 log sin \ C = 9.661970 log cos J c = 9.949735 \a =27° 02' 16" hence, \(a — V) = 15° 56'. i (a + b) = 56° 40'. £ C =27° 20' log sin J (a — 6) = 9.438572 colog sin $ (a + b) = 0.078060 log cot £ O = 10.286614 log tan $ (A—B) = 9.803246 $(A — B)= 32° 26' 37" $(<* + £)= 73° 32' 39" A =105° 59' 16" B = 41° 06' 02" c = 54° 04' 32" 2. Given a = 80° 32' 40", 6 = 120° 27' 18", (7= 48° 12' 21" ; find A = 57° 9' 4", 5 = 132° 45' 46", C= 61° 5' 4". 3. Given a = 124° 50' 48", c = 75° 35' 50", B = 56° 36' 26"; find A = 134° 10' 34", C= 57° 49' 36", 6 = 72° 49' 18". SOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 133 Case II. 202. Given two angles, A and B, and the in- cluded side c. Method. — We find the sides a and b by the third and fourth of Napier's Analogies : , , , , N eos 4- ( A — B) , , tanl( ffl + 5) = cos |^ + ^ tani C . . , , . sin I (A — E) , , tanH«-ft) = sin ?^ + ^ tan| C . The angle C may then be found by the first or second of Napier's Analogies, or by one of Gauss's Equations. Thus, the. first gives , n sinj (A + B) , cos \ C= Vt rr cos i e. J cos \ (a — o) z EXERCISES XLIII. 1. In a spherical triangle, given A = 108° 36' 45", B = 40° 38' 28", c = 56° 42' 22" ; find the other parts. Solution.— ^4 = 108° 36' 45". B= 40° 38' 28". c = 56° 42' 22". log cos i (A—B) = 9.918647 colog cos i (A+B) = 0.576582 log tan i e = 9.732103 log tan \ (a + b) =10.227332 j(a + b) = 59° 21' 16" log sin i (A+B) = 9.984176 colog cos \ (a — 6) = 0.020276 log cos J c = 9.944501 \(A — 5) = 33° 59' 8$"- H^ + -B) = 74°37'36J". \ c = 28° 21' 11". log sin $ (A—B) = 9.747401 colog sin i (A+B) = 0.015824 log tan J c = 9.732103 log tan i (a — b) = 9.495328 J (a — b) = 17° 22' 19" J (a + b) = 59° 21' 16" a =76° 43' 35" 6 =41° 58' 57" C = 54° 28' 40" I6g cos $ C = 9.948953 £ C =27° 14' 20" 2. Given 4 = 130° 27' 38", 5 = 110° 43' 20", c = 124' 134 . SPHERICAL TRIGONOMETRY. 26' 37" ; find a = 125° 55' 41", b == 84° 30' 55", C= 129° 12' 22". 3. Given B = 148° 24' 36", C = 86° 38' 42", a = 88° 30' 47" ; find 6 = 148° 21' 3", c = 89° 30' 25", A = 86° 21' 50". Case III. 203. Given two sides a and b, and the angle A opposite one of them. Method. — The angle B is found from [53], from which we have . _ sin 6 sin A sin B = : sin a Then C and c may be found from the fourth and second of Napier's Analogies, which give tan \c= . ? , . — tan i (a — b). z sm^(A — B) 2y J Note. — In this case, since B is found from the sine, there will sometimes he two solutions. If it is seen in the problem that B 1, the problem is impossible. The following truths may be readily deduced : 1st. When A = 90°, there is only one solution, and may be no solution. 2d. When A < 90°, there are two solutions when a + 6 < 180°, and a 90°, there are two solutions when a+ 6 >180°, and a > 6. SOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 135 EXERCISES XLIV. 1. Given a = 53° 25', b = 34° 26', and A = 106° 35' ; find B, c, and C. Solution.— In this problem we have A > 90°, and a + 6 < 180°, henee, A + B < 180° ; ■whenoe, B < 90°, and there is only one solution. a + b= 87° 51' . a — 6 = 18° 59' A + B = 149° OF 43" .4 — 5= 64° 08' 17" log sin J (A + B) = 9.983941 log tan i (a — 6) = 9.223218 colog sin % (A — B) = 0.274954 log tan £ c = 9.482113 \ c = 16° 52' 53" c = 33° 45' 46" log sin ^ (106° 35') = 9.981549, log sin b ( 34° 36') = 9.752392 colog sin a ( 53° 25') = 0.095289 log sin B = 9.829230 B = 42° 26' 43" J (a + b) = 43° 55' 30" I (a — b)= 9° 29' 30" J (A+B) = 74° 30' 51 J" J 04—5) = 32° 04' 08J" log sin. i (a+b)= 9.841181 log tan i (A—B) = 9.796953 colog sin i (a — b) = 0.782768 log cot i C =10.420902 i C=20°46' 36" C = 41°33'12" 2. Given a = 75° 27' 40", & = 118° 45' 36", A = 84° 52' 34"; find B= 115° 34' 27", c= 111° 45' 16", C= 107° 07' 24". 3. Given b = 40° 16', e = 47° 44', B = 52° 30' ; find (7=65° 16' 35", a = 53° 19' 20", .4=79° 52' 22"; or, C= 114° 43' 25", a = 14° 18' 22", A = 17° 39' 22". 4. Given a = 40° 20', b = 60° 30', and A = 50° 45' ; show- that the solution is impossible. 136 SPHERICAL TRIGONOMETRY. .Case IV. 204. Given two angles,- A and B, and, the side a opposite one of them. Method. — The side a is found from [53], from which we have sin a sin B sin o = sin A Then e and C may be found from the fourth and second of Napier's Analogies, which give . sin A (A + B) , , , , . tan \c= . 'f ; . ^ tan \ (a — V). 2 sin \ (A — E) 3 v J , , „ sin i (a + 6) , . , . _. cot \ C= - — H — —t{ tan i (.4 — E). 2 sin i (a — &) 2 ' Note. — In this case, since b is found from the sine, there will sometimes be two solutions, and may be no solution. If it is seen in the problem that 6 < 90°, there will be but one solution. If in the calculation we find sin 6 > 1, there will be no solution. The following truths may be readily deduced : 1st. When a = 90°, there is only one solution, and may be no solution. 2d. When a •< 90°, there are two solutions when A + B < 180°, and A < B. 3d. When a > 90°, there are two solutions when A + B > 1 80°, and A > B. EXERCISES XLV. 1. Given A = 112° 5ff, B = 135° 25', a = 150° 36'; find b = 158° 2' 40", c = 30° 45' 26", (7= 73° 46' 46". .2. Given ^4 = 114° 36' 40", 5 = 82° 27' 18", 6 = 86° 2C 30"; find a = 113° 45' 44", c = 82° 7' 18", (7= 79° 44' 2". SOLUTION OF OBLIQUE SPHERICAL TRIAXGLES. 137 3. Given A = 132° 16', B = 139° 44', b = 127° 3(7 ; find « = 65° 16' 35", (7= 165° 41' 3S", c = 162° 2C 38" ; or, a = 114° 4? 25", C= 126° 40' 40", c = 100° 7' 38". 4. Given ^L = 60° 3C, B = 40° 24', b = 50° 36'; show that the solution is impossible. Case V. 205. Given the three sides, a, b, and c. Method. — The angles may be found by the Formulas [56] or [57] or [58]. The formulas for the tangent, however, are, generally, to be preferred. The formulas for the tangent may be put in a still more convenient form by making sin ( s — a) sin ( s — 6) sin (s — c) * ^ - -^ = tan 2 r, sin s and substituting this in each and reducing, by which we obtain tan I A = tan r -=- sin (s — a). 1 tan | B = tan r -r- sin (s — ft). > [65] tan + C= tanr -f- sin (s — c). J Notes. — 1. When only one angle is to be found, use For's. [56], [57], or [58] ; when all three angles are required, use For's. [65]. 2. Xo ambiguity can arise in this case, since the half angles must be less than 90°. EXERCISES XLVI. 1. Given a = 60° 34' 20", 6 = 48° 45' 26", c = 76° 48' 53" ; find A, B, and C. 138 SPHERICAL TRIGONOMETRY. Solution. — The solution by Formula [58] is as follows : a = 60° 34' 20" b = 48° 45' 26" c = 76° 48' 53" 2s =186° 08' 39" * = 93° 04' 19J' s — a = 32° 29' 59 J' s^-b = 44° 18' 53$' s _ c = 16° 15' 26JK log sin (a — 6) = 9.844229 log sin (a — c) = 9.447084 colog sin (a — a) = 0.269786 colog sin a = 0.000625 2)19.561724 log tan i A = 9:780862 i A = 31° 07' 18" ^ = 62° 14' 36" The angles J? and C are found in a similar manner. Solving the same problem by the three formulas [58], we have log tan % A = 9.780862 log tan J B = 9.666847 log tan i C = 10.063992 M = 31° 07' 18" ' J B = 24° 54' 28" i C = 49° 12' 22" A = 62° 14' 36" £ = 49° 48' 56" C= 98° 24' 44" log sin (a — a) = 9.730214 log sin (s — b)— 9.844229 log sin (a — e) = 9.447084 colog sin s = 0.000625 log tan 2 ?- =19.022152 log tan r = 9.511076 Note. — To find log tan J A, we need not rewrite log tan r and log sin (a — a), but can subtract log sin a — a from log tan r as they stand in the first column ; and similarly for tan J B, and tan J C. 2. Given a = 120° 45' 28", 5 = 62° 27' 40", e = 108° 23' 40" ; find A = 115° 44' 52", 5 = 68° 20' 24", C= 95° 57' 32". 3. Given a = 135° 16' 40", b = 110° 55' 30", e = 86° 32' 16"; find .4 = 137° 38' 32", B = 116° 34' 34", C = 107° 06' 36". * 4. Given a = 25° 24' 23", b = 48° 38' 28", c = 76° 46' 36" ; show that this is impossible. SOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 139 Case VI. 206. Given the three angles, A, B, and C. Method. — The sides may be found by the Formulas [59], [60], or [61]. The formulas for the tangent are usually preferred, since they require fewer logarithms and give accurate results in every part of the quadrant. These formulas for the tangent may be put in a still more convenient form by substituting in each tan 3 R = — cos S sec (S — A) sec (S — B) sec (S — C), which, when reduced, gives us tan % a = tan R -r- sec (S — ^4). \ tan £ 6 = tan R -f- sec (S — B). L [66] tan £ c = tan R -r- sec (S — C). S Note. — "When only one side is required, use For's. [59, 60, 61] ; ■when all the sides are required, use For's. [66]. EXERCISES XLVII. 1. Given A = 106° 36', B = S7° 45', C b, and c. Solution. — Since the three sides are required, we solre by Form- ulas [66]. A = 106° 36' B= S7°45' C= 96° 4S r 2 5=291° 09' log cosS= 9.916384 (n) log sec (S — A) = 10.109344 log sec (S— B) = 10.273674 log sec [S— C) = 10.181103 log tan' R = 40.480305 log tan R = 20.240252 45' , C=96 c 48' ; find a, £=145 °34' 30" S- -A= 38 °58 / 30" s- -5= 57 049/ 30" s- -C= 48 °46 y 30" log tan i a = 10.11 0908 log tan J 6 = 9.966578 log tan $ c = 10.059149 $ a = 53° 3C 26" ib = 42° 47' 51" ic = a = 48° 53' 23" 107° 00' 52" = 85° 35' 42" c = 97° 46' 46" 140 SPHERICAL TRIGONOMETRY. Note. — To find log tan J a, we need not rewrite log tan R and log see (S — A), but can subtract them as they stand in the first column ; and similarly for log tan J & and log tan $ c. 2. Given/ = 130° 46', B = 110° 50', C= 80° 30'; find a = 140° 32' 18", b = 128° 20' 40", c = 55° 51' 28". 3. Given A = 60° 25' 40", B = 87° 26' 32", C= 60° 25' 40"; find a= 53° 36' 16", b = 67° 36' 20", c = 53° 36' 16". 4. Given A = 90°, B = 90°, 0= 90° ; find a = 90°, & = 90°, and c = 90°. Solution by Means of a Perpendicular. 207. Oblique spherical triangles may be readily solved, also, by dividing them into right triangles and applying Napier's Rules. Thus, let CD be a perpendicular drawn from O to the base AB. 1. Then, Rule II. cos a = cos m cos h. Whence, cos h = cos a h- cos m. cos 6 = cos n cos h. Whence, cos h = cos b -5- cos n. Whence, cos a : cos m= cos& : cos n. That is, the cosines of the sides are proportional to the cosines of the segments of the base. 2. Again, by Rule II. cos A*= cos h sin ACD. cos B = cos h sin BCD. Whence, cos A : cos B = sin ACD : sin BCD. SOLUTION BY MEANS OF A PERPENDICULAR. 141 That is, the cosines of the angles at the base are 'proportional to the sines of the segments of the vertical angle. 3. Again, by Rule I. sin n = tan h cot A = tan h -j- tan A. sin m = tan h cot B = tan A -*- tan B. Whence, sin m : sin to = tan A : tan 5. That is, the sines of the segments of the base are inversely proportional to the tangents of the angles at the base. 4. Again, by Rule I. cos ACD = tan h cot b. cos BCD = tan h cot a. Whence, cot a : cot b = cos BCD : cos A CD. That is, the cotangents of the two sides are proportional to the cosines of the segments of the vertical angle. 5. Again, by Art. 207, we have cos a : cos b = cos m : cos n. Whence, cos 5+cos a : cos b— cos a=cos n-fcos m : cos n— cos m. But, Art. 70, cos 6-f cos a : cos b — cos a=cot \ (a+b) : tan \ (a — &■). cosn+cosm : cosn— cosm=cot£(m+«) : tan^-(m— n). Whence, cot \ (a+&) : cot \ (m+n)=tan \ (a — 6) : tan |- (m — n). And since tangents are inversely proportioned to cotan- gents, cot \ {a+b) : cot \ (m+n)=tan \ (m+n) '■ tan -£■ (a+b). Whence, tan \ (m-f-n) : tan \ (a+6)=tan^- (a — b) : tan | (m — n). 142 SPHERICAL TRIGONOMETRY. That is, the tangent of half the sum of the segments of the base is to the tangent of half the sum of the sides, as the tangent of half the difference of the sides is to the tangent of half the difference of the segments of the base. 208. These five principles, derived immediately from Napier's Rules, enable us to. solve every case of the oblique triangle. They are more easily remembered than the formulas previously used, and are preferred by some authors in solving these triangles. EXERCISES XLVIII. 1. In the spherical triangle ABC, given AC = %0° 30', BC = 80° 36', and the angle A = 35° 24'; required the other parts. Solution. — Let ABC denote the triangle. Draw CD J_ to AB. First, we have, Art. 183, sin BC : sin A C = sin A : sin B. Whence, B = 33° 36' 23". Then in triangle ACT), Rule I., cos A C = cot A cot A CD. Whence, ACD = 76° 39" 17". Also in triangle BCD, cos BC= cot B cot BCD. Whence, BCD = 83° 48' 19". Therefore, ACB = 160° 37' 36" Finally, we have sin A : sin C= sin BC : sin AB. Whence, AB = 145° 16' 33". SOLUTION BY MEANS OF A PERPENDICULAR. 143 2. In the spherical triangle ABC, given A =- 114° 36' 40", B = 82° 27' 18", and AC= 86° 20' 30"; find BC = 113° 45/ 44" ( ^j? = 82° 7' 18", and AGB = 79° 44' 3". 3. In the spherical triangle ABC, given AB = 72° 36', AC= 40° 44', and A = 54° 40' ; find B = 41° 06' 02", C = 105° 59' 16", and BC =.54° 04' 32". 4. In the spherical triangle .4.BC, given A = 108° 36' 45", C= 40° 38' 28", and AC = 56° 42' 22"; find BC = 76° 43' 35", £ = 54° 28' 40", and AB = 41° 58' 57". 5. In the spherical triangle ABC, given AB = 112° 25', AC= 60° 20', BC= 81° 10' ; find A = 64° 46' 36", B = 52° 42' 12", and C= 122° 11' 06"! 6. In the spherical triangle ABC, given A = 106° 36', B = 87° 45', C= 96° 48' ; find AB = 97° 46' 46", BC = 107° 00'' 55", AC= 85° 35'" 42". Notes. — The following suggestions will aid the student with the above examples. 1. In Ex. i, since the value of AB is found from the sine, we determine its quadrant by Art. 160. 2. In Ex. 2, we first find BC by Art. 184; then by Rule I. find AD and BD ; then take their sum and find A CB by Art. 1 84. 3. In Ex. 3, we first find AD by Rule I. ; then find B by For. 3, Art. 207 ; then BC by Rule I., then BCD by Rule I., and then ACD by Rule I., from which we find C. The latter part of this solution prevents ambiguity. 4. In Ex. 4, first find ACD by Rule L, from which find BCD; then find BC by For. 4, Art. 207 ; then find B by Rule I. ; then find AB by Art. 184. 5. In Ex. 5, first find AD and BD by Art. 184 ; then find A by Rule I. ; then find B by Rule I. ; then find C by Art. 188. 6. In Ex. 6, pass to the polar triangle ; find its angles as in Ex. 5 ; the supplements of these angles will be the sides of the given triangle. 144 SPHERICAL TRIGONOMETRY. SECTION XV. SUPPLEMENT. Area of a Spherical Triangle. 209. We now proceed to show how to find the area of a spherical triangle. I. When the three angles, A, B, and C, are given. Let R = the radius of the sphere. E — the spherical excess = A + B + C— 180°. S = the area of the triangle. Then by Geometry, B. IX., Th. XXVII., S= 180. ' II. Wlien the three sides are given. ■ 210. Take E= A + B+ 0—19,0° as above ; then tanji?= si " i ^ + i? + C - 180 > ¥ cos i (A + B + C— 180) _ sin $ (A + B) — sin j (180 — C) cos i(A+B) + cos J (180 — C) _ sin $ (A + B) — cos j C ~ cos I (A + B) + sin £ O _ cos j (a — b) — cos j c cos \ _ cos £ (a + 6) + cos i c sin i O For. [62] sin \ (c-\-a — b) sin j (c + 6 — a) I ( sin s sin (s — e) ) sin J (a-\-b-\-c) cos J (a + 6 — c) \ 1 sin (* — a) sin (s — b) j . For. [56, 57]. _ sin \ (s — b) sin j (s — ■ o) I f sin n sin (,; — c) ") cos $ s cos £ (s — c) \ \ sin (s — a) sin (s — b) ) Art. 70. CIRCUMSCRIBED AND INSCRIBED CIRCLES. 145 _ |/sins sin£(s — b) |/gjn.(j — e) sin £ (.9 — q) cos £ s |/sin(* — b) cos £ (s — c) |/sin (s — a) Substituting for cos £s, sin £ (* — 6), etc., their values of [26] and [25], and reducing, we have tan J E=j/{tan J s tan \ (s— a) tan J (s— b) tan J (s — c). [67] 211. This elegant formula is known as L'Huiller's Theorem. By means of it the value of E may be found from the three sides, and then the area of the triangle may be found from Art. 209. 212. In a similar manner we can find Cagnoli's Theorem, which is sin \ E = l/{sin s sin (s — a) sin (s — ft) sin (s — c)} 2 cos £ a cos £ b cos £ c Circumscribed and Inscribed Circles. I. To find the radial arc of a circumscribed circle. 213. Let be the pole of the small circle circumscribed about the spherical triangle ABC. Draw the radial arcs OA, OB, and OC, and draw OD per- pendicular to BC. The triangles OBC, AOC, and A OB are isosceles, andS-D = J a. Denote the angles by A, B, C. Denote the radial " arc of the circum- scribed circle by R. Then in the right triangle BOD Fig. 59. we have "Whence, Now, And, 10 cos OBD = cot R tan Ja, . „ p _ tan ja tan R = Tr^-fT cos OBD OBD = B — ABO = B — BAO. OBD= OCD= C—ACO=C—OAC. 146 SPHEKICAL TRIGONOMETRY. Whence, 2 OBD = ■ B + C- -(BAO+OAC) -B + C- -A = 2(S—A), ince, OBD = --S—A. Whence, tan R = tan J a c- ., , r, tan hl> , n tan J c similarly, tan ic = — f — =r, and tan R = — — ■ - — — • •" cos (S—B)' cos (S — 0) ,„, , „ tan i a tan I b tan i c Whence, tan* 1 ic = r ; — — ^ — ^ ■ ' cos (,S— .4) cos (S— B) cos (S— C) The product of the three formulas, [61], gives cos 3 S tan 2 i a tan 2 J 6 tan 2 i c = • cos (£— ^l)cos (S — B) cos (£— (?) Substituting this in the value of tan 3 R and reducing tan R = -1 / ^^ T681 Vcos (S—A) cos (S—B) cos (S— C) L J II. To find the radial arc of an inscribed circle. 214. Let be the pole of the circle in- scribed in the spherical triangle ABO. Draw the radial arcs OD, OE, and OF perpendicular to BC, AC, and AD respect- ively. Draw also the arc EGF. Now, since 0E= OF, the triangle EOF B ( is isosceles. Then ZOFG = ZOEG and ZGFA=ZGEA. Hence, A FAE is isosceles ; and A F= AE. Draw the arc OG perpendicular to EF at its middle point ; it will bisect the angle EOF, and will also pass through the point A and bisect the angle A. Similarly, the arcs OB and OC will bisect the angles B and C respectively. Denote the radial arc of the inscribed circle by r. CIRCUMSCRIBED ASD ISSCRIBED CIRCLES. 147 Then in the right triangle AOF sin JJ"= tut r cot J A. Sow, AF= AE; BF= BD; CE= CD. Also. J/*= e — BF= e—BD. And, AE — b — CE = b — CD. Adding. AF+ AE= 2 A F= b + c —(BD + CD). Or, 2JJ"=6+c— «. = 2 (*-.). AF=s— «. Substituting this value in sin XFand reducing, We have tan r = sin (i — «i)tan^J. Similarly, tan r = sin (s — o) tan | 22. tan r = sin (* — c) tan J l*. The product of these three formulas gives tan* r= sin (s — •) sin (s — b) sin (* — c) tan J -1 tan J B tan | C. Finding the mine oftan£~l.tanfJ3tan£C from For's. pSJ and substituting and reducing;, ire hate fan r = gj" (*~ *> sin jf — *> si" (■*—<•) j. pa] \ sin « J EXERCISES XLIX. 1. Find the area of a spherical triangle -whose angles are SOP 30% TO" 40*. and SO* 50'. i. Find the area, of a spherical triangle whose sides are W 30*", TflP-WandS^aO'. 3. Find the radios of the circumscribed circle in the triangle of Exercise 1. 4. Find the radius of die inscribed circle in the triangle of ExerciseS. 148 SPHERICAL TRIGONOMETRY. Miscellaneous Exercises. 1. In any spherical triangle, if A = a, show that B = b, and C=c, or that they are respectively supplementary. 2. When does the .polar triangle coincide with the primitive triangle ? 3. If B = A + and D is the middle point of b, prove that b = 2 AD. 4. If D is the middle point of c, prove that cos 6 + cos a = 2 cos J c cos CD. 5. If b + c = n, prove that sin 2 jB + sin 2 C = 0. 6. In an equilateral spherical triangle, prove that 2 cos J a sin J A = \. 7. In an equilateral spherical triangle, prove that tan 2 J a = 1 — 2 cos A. 8. In an equilateral spherical triangle, prove that sec A = 1 + sec a. 9. If b = c = 2a, prove that csc J A =; 4 cos a cos J a. Right Spherical Triangle. 1. Derive two rules similar to those of Napier for the direct solu- tion of quadrantal triangles. If ABC is a right triangle, C being the right angle, then 2. Prove sin 2 \ c = sin 2 £ a cos 2 \ b + cos 2 J a sin 2 ^ 6. 3. Prove tan \ (c + o) tan £ (c — a) = tan 2 J 6. 4. Prove sin (c — b) = tan 2 J .4 sin (c + &)• 5. Prove sin a tan J A — sin b tan \B = sin (a — 6). 6. Prove sin (r. — a) = sin 6 cos a tan J JS. sin (c — o) = tan 6 cos c tan J i?. OBLIQUE SPHERICAL TRIANGLES. 149 7. Prove sin (c -\- a) = sin b cos a cot i B. sin (c + a) = tan 6 cos c cot J 5. 8. In a right spherical triangle, C the right angle, if D is the middle point of AB, prove that sin' a -(- sin 2 6=4 cos 2 \ c sin 2 CD. 9. In a right triangle, if d is the length of the arc from C perpen- dicular to the hypotenuse, prove that cot 2 a + cot 2 b = cot 2 d. 10. If ABC is a right spherical triangle, A not being the right angle, prove that if A = a, then b and c are quadrants. In a right triangle, C being the right angle, prove 11. tan 2 i A = Sia( f~ b ) ■ 13. tan 2 (45°-M = tan * < c ^ 4 sin (c + o) tan i (c + «) ,„ , « , cos (.4 + /?) , , cos a sin 2 A 12. tan 2 ic = , AB \ - u - r = • ■> p - cos (^4 — B) cos o sin 2 B Oblique Spherical Triangles. 1. If the area of an equilateral triangle is one-fourth of the area of the sphere, what are its sides and angles ? 2. In a spherical triangle, if c = 90° and d denotes the perpendic- ular from C to c, then cos 2 d = cos 2 a -\- cos 2 b. 3. In a spherical triangle, if A = B = 2 C ; then 8 sin (o + J c) sin 2 J c cos i c = sin 3 a. 4. In a spherical triangle, if A = B = 2 C ; then 8 sin 2 J C (cos s + sin J C) cos J c = cos a. 5. In any equilateral triangle, R and r denoting respectively the radii of circumscribed and inscribed circles, prove tan R = 2 tan r. 6. In any spherical triangle, E denoting the spherical excess, prove sin J E = sin C sin J a sin J 6 sec J c. 150 SPHERICAL TRIGONOMETRY. 7. In any spherical triangle, E denoting tho spherical excess, prove eos J E = { cos i a cos J 6 -|- sin ^ ti sin J 6 cos C } sec i c. 8. If the angle C of a spherical triangle is a right angle, prove i sin \ E= sin J a sin i b sec i c ; cos i E= cos J a cos J 6 sec J e. 9. If the angle Cis a right angle, prove that sih 2 c, r, sin' a . sin' b cos E = 1 • cos c cos a cos o 10. If a = b and O = -, prove that 77 = 12 ^ 2 cos a 11. If the angles of a spherical triangle are together equal to four right angles, prove cos 2 % a + cos 2 i b + cos 2 J r = 1. 12. If ^IjBC is an equilateral spherical triangle, P the polo of the circumscribed circle, and Q any point on the sphere, prove that cos A Q + cos BQ + cos CQ = 3 cos R cos P5g 65i3 •698, 2544 4382 6212 8o34 9849 i656 3436 5249 7034 8811 •082 2345 4101 - 585o 1419.-7391 9134 9323 4196 6157 8110 ••541 0. 583, 7744i 9646 |539 3424 53oi| 7169 9o3o •883 2728 4563 6394 8216 •»3o 1837 3636 5428 7212 ;b 2321 I1I4I «l_. 2796' 2964 1 44721 463o| 6141 1 63o8i 7804 7970 9460 9623 1110 1275 2734! 2018 4392: 4555 6023 6186' 7648 78111 9263 9429 ■8811 1042, 2488 2649 4090 4249 5685 3844 1 7275 7433 8839 9017 •43 7 , <^i, 2009; 2 1 66, »76( 3-32 5i37i 5293 6692 1 6848, 602: TP\ 9301 1228 2949 4663 «370 8070 97*4 I45i 3iJx 4806 6474 8iJ5 W 1439 3o82 47i8 6J49 7.r"3 9^1 ■ 2o3 2K09 4409I 6op4 7392 9173 *t5; 2}j3 3449 7003, «7 67 66 65 65 64 64 63 62 62 61 61 60 39 3 53 56 »»' A TABLE jjpP LOGARITHMS FROM 1 TO 10,000. N. a8o | 1 y 2 | 3 | 4 | 5 1 6 7 8 9 D. 447158. i3i3 8706! 8861 7468 7623, 77781 7g33, 8088 82412 8397 8552 1 55 *8r 901J 9170, 9324 9478' 9633 9787 9941 •»g5 i633 1 54 282 450249 04c3 o55t 07111 0863] 1018 1172 i326 1479 1 54 283 1786 1940 2093 22471. 2 4ooj 2553, 2706 285g 3oi2, 3i65 i53 284 33i8i 3471 3624 3777' 3o3o, 4082! 4235 438 7 4340 4692 i53 285 4845 1 4997 5i5oi 53o2| 54541 56o6 5758 5gio 6062 6214 132 • 286 6366, 65i8 66701 6821I 6 97 3 "M25. 7276 7428 8940 7% 7 7 3i 132 287 288 7882; 8o33 8i84| 8336, 8487 8638, 8789 9091 9242 i5l 9392 9543 9694 9845! 9995 *U6 e 2g6 •447 1048 •5 97 •748 i5i S89 460898 1048 1198I 1348 1499! 1649' 1799 2098 2248 i5o 2go 462398 2548 2697) 2847 2997I 31.46 3206 3445 4490 1 4639 4188 4g36 35g4 5o85 3744 i5o 291 33 9 3 4042 4191: 434o 568o 5829 5234 149 292 5383 5532 5g77 6126 6274 7460 7608' 77D6 6423 6571 6719 US '9.3 6868 -7016 84o5 9969 I164' 73'ra 8643, '8790 •n6[ »263 7904 g38o *85i 8o52 8200 294 295 8347 9822 8g38 go85 9233 .•410! »557! V°4 9527 ^998 967J 1 145 148 147 146 296 47 1 292 1438 i585i n32' 1878 2025 2171 23i8 2464 2610 298 2756 2go3 4362 3o4q 3ig5 45o8; 4653 3341 3487 1 3633 oK 3925 538i 4071 146 4216 6*252 4o44 ! 5090 5526 146 299 5671 58i6 5g62i 6ro7 63 9 7 ; 6542! 6687 7844! 7989! 8i33 6832 6976 US 3 00 477>2> 8566 7266 8711 7411 7555 8855' 8999 7700 8278 8422 145 3oi 9143, 9287, 943i! 9575 97'9 g863 144 3o2 480007 oi5i| 02q4' 0438! o582 0725, 0869 1 1012 11S6 1299 144 3o3 1443 1 586 1729, 1872 20161 2i5g! 23o2l 2445 2588 2731 143 3o4 2874 3oi6 3i5o! 33o2 -*S85i 4727 3445! 3587, 373o| 3872 40i5 4i5 7 143 3o5 ^nn -444* 48691 5oii' 5i53i 5295 5437 5579 142 3o6 5l2I 5863 6oo5j 6147 6289 643o! 6572 1 6714 £855 6997 142 3oi 3o8 713.8 855! 7280 8692" 74? 1 -8333 7563 8974 •38o 7704 7845; 7936 1 8127 9255' 9496, 9537 8269 8410 141 9114 9677 1081 9818 141 3e>9 9 o58 49i362 "99 •23g •520 •661! •8011 »94i 2062; 2201' 2341 1222 140 3io l5o2 1642 1782 1922 2481 2621 140 3n ' 2760 2900 3o4o 3mi 33ig 3458 35o7 3 7 3 7 4089! 5i28 6370I 65i5 7.769 7897 38 7 6 4oi5 13, 3l2 4i55 4294 4433 4572 47" 485o 5267 54o6 l V> 3i3 5544 5683 5822! 5g6o 6099 7506] 7344 7483 8586! 8724 8862 6238 6653 6791 1 39 138 3i4 6g3o 83 u •7068 8448 7621 8999 8o35 8173 g55o 3i5 9137 g275 9412 i33 3i6 « 9 6 |7 9824 9962! £§159' •236 •374 •5n .•6148 "■785 •922 «?7 3l I 3i8 501009 1 196 i333j 1470 1607 1744 1S80 2017 2i54 2291 l3 J ■■' 2427 2564 2700: 2837 2973 4335 3iog 3246 3382 35i8 3655 i36 319- OODlSO 3927 4o63 i% 4471 4607 4743 4878 5oi4 i36 320 5286 5421 56 9 3 5828 5g64 7 3i6 8664 6099 6234 6370 i36 32) 65o5 6640 6776 6911 7046 7181 853o 7451 8799 •143 7586 8g34 ■HK i35 322 7856 9u 7328' 7460 8646! 8777 7592 7734 7855 .7.9-87j.8'!9 8?5i 8382 1 32 33o 8909 9040 9171 •48i q3o3I 9434 g566 9697 i3i 33i 9828^ 9959I ••90 "221 •353 •6i5| »745, '876 1007] i3i 332 5ji t36, 1269' 1400 i53o 1 66 1 1792 19221 2353, 2i83 23i4, '3i 333 2444! 2575 2705 2835; 2966 3og6 3226, 3356, 3486 36i6: i3o 33 i ; 3746' 3876; 4006, 4i36 ; 4266 43g6 4526, 46061 4780 491 5[ i3o 33.5 5o45] 5i74i 53o4l 5434 5563 569.3 698.5 5822, ,5g5r 60S1 6210 129 336 633 9 ! 6467 65g8| 6727 6856 7114, 7243, 7372' 7001, 129 33 7 763o, 7709 7888, 8016' 8i45 89J7I 9u45--9-i44i-93ajU^o 53o2oo o328 04561 o584 0712 8?74 8(02, 853 1, 8660 1 8788 I sg 338 339 9559 0840 96871 gRiSi 9943 1 "72 1,28 0968 1096 1223, i35T 128 | N. | | 1 | 2 j 3 | 4 | 5 ] 6 | 7 | 8 1 9 I D. A TAI1LE OF LOGARITHMS FROM 1 TO 10,000. N. , 3 3 4 | 5 | 6 7 8 1 9 D. 138 34o 53 1 475 37S; 'SS'* 1734 186s 19901 2117 22 4- 3372 25ool 2627 34i 2882 3009 3i36 3264I 3391! 35i8 3645 3772 3899 127 J42 4026 41 53 428o 4407 4534 4661 4787 4914 3041 316- 127 343 6294 5421 5547 56 7 4 58oo 6927; 6o5i 6180 63o6 643; 126 344 655> 6685 68ll 6937 7063 7189 7313 -8-448 8374 744i 7567 8 9 5 1 126 -Ji5- - -fiU) -7945 -80.7.1 -&io-7 -8322 . 8699 8825 12b 346 9076 9202 9 32 7 9432 9378 o83o 9703] 9829 9934 ••79 •204 125 3i7 540329 0455 o53o 070; 0955 1080 1203 i33o 1454 125 348 '^9 1704 .829 1953 2078 22o3| 2327 2452 2576 2701 135 349 2820 29D0 3074 3i 99 3323 34471 3571 36 9 6 3823 3944 124 3Jo 544p6f '4102 5431 43 16 4440 4564 4688! 4812 4986 5o6o 5i83 124 3ji 53o 7 654; 5555 5678 5802 6925! 6049 6172 6296 6419 124 352 6b66 6789 6913 7036 8267 7159 7282 83S9 83 12 74o5 . 8635 •7529 8738 7652 8881 123 333 7775 7898 8021 8144 123 334 900^ 9126 9249 9371 9494 9616! 9739 9S61 9984 •106 123 3J5 550228 o35i 0473 0J95 0717 oiJ40| 0962 1084 1206 1328 123 356 I45o 1572 1694 1816 1938 2060, 2181 23o3 2423 2347 122 35 7 2668 2790 2911 3o33 3i55 3276 3398 3319 364o 3762 121 338 3883 4004 •4-1-26 --4-24-7 --4368 -4-489 ~3o70 -473il 4S52 4973 6182 121 359 5094 52 1 a 5336 6437 5578 6785 5699 6900 5820 5940 6061 121 36o 5563o3 6423 6544 6664 7026 8228 7146 1 8349! 7267 7387 8589 120 36i 7507 8709 7627 8829 •°26 7748 8948 7868 7988 8108 8469 120 362 9068 9,88 93o8 9428 9348. 9667 •863 9787 120 363 9907 •146 •265 •385 •5o4 •624 e 7-'.3| "982 119 364 56iioi 1221 1 34o 1459 1578 1698I 1817 1 9361 2o55 2174 II9 365 2293 3481 2412 253 1 265o 2769 J933 2887 3oo6 . 3 1 251 3244 3362 119 366 36oo 3 7 i8 3837 4074 4192 43li! 4429 4548 !.'8 367 4666 4784 4903 5o2I 5l3g 5237 5376 5494 56 12 5730 368 5848 5966 6084 6202 6320 6437 6J5J 6673I 6791 6909 :i8 369 7026 560202 7144 83i 9 7262 8436 7^79 7497 7"i4 7 7 32 7S49; 7967 8084 118 370 8334 8671 1788 8905 9023 9140 9257 "7 3 7 i •9374 9491 9608 9726 9842 99 5 9 ••76 •193I °3o9 "426 "7 3 7 2 570543 0660 0776 oS § 3 1010 1 1 26 1243 1339 1476 l5o2 2765 1 17 V] 3872 4o3i 1825 1942 2038 2174 2291 3452 2407 23231 2639 116 374 2988 3 1 04 3220 3336 3568 3684! 38oo 3 9 i5 116 I 1 ! 4U7 4263 4379 4494 565o 4610 4726 484i( 4o5- 5072 .116 p b 5i83 .53o3 5419 5534 5 7 65 5u8o 5996 61 1 1 6226 1.5 Vl 6341 6437 6J72 6687 6802 691J Soto 7032 7i47i 7262 7377 8525 n5 378 8M9 7607 8754 9898 1039 7722 8868 7836 8 9 83 7 9 5i 0181 8293, 8410 li5 379 9097 9212 9326 9441 : 9555 9669 114 3oo 579784 •°12 •126 •241 •305 •469 •583 •697 1836 •811 114 38i 580925 n53 I267 i38i i4o5 263i 1608 1722 1950 114 332 2o63 2177 2291 2404 25i8 2745 2838 2972 3o85 114 .383 3 '?9 33i2 3/,26 333 9 3652 3763 3879 3 9?2 4103 4218 n3 334 433 1 4444 4537 4670 4783 4896 5009 5l22 5235 5348 n3 383 546i 5074 5686 6$ 6912 6024 6i3 7 6230 6362 6473 n3 336 6537 6700 7823 8944 63i2 7037 8160 7M9 8272 7262 73 7 4 8496 7486 7?99 112 337 77" 7 9 35 8047 8384 8608 8720 112 338 8832 9036 9167 9 2 79 9391 95o3 9613 9726 9'-138 113 3,! 9 99 5o •»6 1 "173 1287 •284 •396 •307 •619 °73o •842 •953 112 3 jo 59 1 060 1 176 1 3 99 i5io 1621 1732 1843 1955 J066 III 1 l>' 2177 2288 2399 25io 2621 2732 2843 2934 3o64 3 173 HI J;2 3286 3397 4393. 43o3 35o8 36i8 3729 3840! 395o 4061 4171 4282 in ! 3g3 | 4614 4724 4834 4945; 5o55 5i63 3276 5386 no 1 3q4 5496; 56o6 'iV 6817 6827 5937! 6047J 6157 6267 6377 6487 no | 3g5 | 6397' 6707 7^93; 7803 8791. 8900 6927 7037' 7146! 7236 81 34 8243, 8353 7 366 7476 7586 ■ 10 3^6 79'4 8024 8462 1 8372 868 1 110 1 3 9 7 9009 9"9 92281 9337! 9446 9556 9665J 9774 109 109 109 3 9 8 ; 9883 9992 £00973 1082 °101 •210 •3io "428 »537 1408 i5i7 1625 •646 °755i »864 18431 i 9 5i 399 1 IIQI 1299 J 734 N. ! ; I ] 2 ! 3 4 . 5 | 6,~~ 7 8 j 9 D. k 7ABLTC OF LOGARITHMS FKOM 1 TO 10,000. H. > 1 » 3 | 4 | 5 | 6 | 7 | 8 j o | D. 400 602060J 2169 1 227- 2386!, 2494 26o3 271 1! 2819' 2928' 3o36 108 401 3l44l 3253. 336i 3469!' 3577 3686 37941 3902; 4010' 4118 108 402 ' 4226 4334 4442 455o 4658. 4766 4874 4982I 5089 5197; 108 4o3 53o5 54i3 5521 5628 5736! 5844, 5g5i 60591 6 166 6274' 108 404 638i 64891 6596 6704 681 1 6919; 7026 7i33 820: 7241 8312 7348, 107 S419! 107 94881 101 4o5 7455 7062 8633 7669 8740 9808 7777 8847 7884 89J4 799 > 8098 406 8526 9061 9167 9274 g38i 407 9594 9701 9914 ••21 •128 •234 •341 •447 •354| 107 408 610660 0767 1829 o8 7 3 0979 1086 1192 1298 140! 1 5 1 1 1617, 106 2678I 106 409 , ' 7 2 3 1 936 2042 2148 2234 236o 2466 2572 410 612784 2S90 2996 3 1 02 3207 33i3 3419 4470 35aC 363o 3-S6' 106 411 3842 *>V 4oo3 41 O9 5 2 i3 4264 4370 458i 46S6 4702i 106 412 5930 5oo3 5io8 53i9 5424 5329 5634 57401 584 "1! io5 4i3 6o55 6160 6265 6370 6476 658 1 6686 6790 68 9 5 io5 414 41 5 7000 8048 7io5 81 53 7210 825 7 73i5 8362 7420 8466 $■ 7629 8676 V 3/ 878c 78J9 88S4 7943 8989 io5 Io5 416 90g3 9198 9302 9406 95n 9615 97'9 9824 9928 ••32 104 417 620136 0240 0344 0448 o552 0656 0760 0I6: 0968 1072 10/, 418 1 176 1280 1 384 1488 1592 i6o5 27J2 '7.99 190^ 2007 2110 104 419 2214 23i8 2421 2525 2628 2835 2939 3973 3o42 3i46 104 420 623249. 3353 3456 3559 3663 3 7 66 386 9 4076 4179 io3 421 4282 4385 4488 4591 46 9 5 4798 4901 5oO; 5107 5210 io3 422 53i2 541 5 55i8 5621 5724 5827 5929 ^>o3: 6i35 6238 lo3 423 6340 6443 6346 6648 6 7 5 1 6853 6936 7o58 8082 7161 8i85 7263 io3 424 7360 838 9 7468 8491 7571 85g3 7673 869 5 7775 8797 98'7 7878 8900 7980 8287 102 425 9002 910/ 9206 93o8 102 426 9410 9512 9613 97i5 9919 ••21 •123 •224 •326 102 427 630428 o53o o63 1 0733 o835 0936 io38 1139 2l53 3.65 1241 i342 102 428 429 __1444 1,5 ',5 1647 2660 1748 2761 1849 1951 2963 2o5? 3o64 2255 "3555 2356 336 7 101 101 2437 633468 2559 43o 3569 4578 -5584 3670 3771 3973 4074 4175 5i82 6187 1189 8190 9188 4276 5-i83 6287 7290 8290 9287 •283 4376 100 43 1 432 _JiS4 1 6488 4679 568o 6789 4880 4981 5o8i 5383 6388 7390 838 9 9 38 7 100 433 6588 6688 To 6989 7089 8090 9088 lco 434 435 7490 '8489 8589 7690 86S9 7790 8789 97*5 8888 $8 99 99 436 94S6 9586 9686 9 885 9984 ••84 •|83 •382 99 437 438 640481 o58i 0680 0779 0879 0978 1077 "77 2168 1276 2267 |3 7 5 99 »474 1573 1672 '77' 1871 1970 2060 "3o58 2366 99 43 9 " ■ — 2465 2563 2662 2761 2860 2959 3i56 &55 3354 2 44o 643453 355i 365o 3749 3847 3946 4044 4143 4242 4340 441 443 9 4537 4636 4734 4332 4g3i 5029 5127 5226 5324 98 442 5422 552i 5&19 5717 6698 58i5 5gl3 6894 601 1 6110 6208 63o6 98 443 6404 65o2 6600 6796 6992 7089 8067 9043 ••f6 7187 8i65 9140 7285 98 444 445 7383 836o 748i 8458 m 7676 8653 7774 8 7 5o 7872 8848 2$ 8262 9237 98 97 446 9 335 9432 9530 9627 9724 9821 9019 •n3 •210 97 * 4 I 448 65o3o8 04o5 0502 0299 0696 0793 0890 0987 1084 ii8i| 97 1278 1375 1472 1 D69 1666 1762 i8d 9 19J6 2o53 2i5oj 97 44 9 2246 2343 2440 2536 2633 273o 2826 2923 3888 3019J 3i:6l 07 400 653213 33oo 427J 34o5 35o2 35 9 8 36 9 5 3791 4754 3984I 40801 96 45i 4177 5r38 4369 4465, 4562 46D8 485o 4946' 5o42| 96 4)2 5235 533i 1 5427 1 5523 5619 57i5 58io 5oo6| 60021 96 453 6098 70D6 6194 6290J 6386J 6482 65 77 6673 6769 6864 1 6960, 96 454 7i52 7247 7343, 7438 7534 8488 7629 8584 7725: 7820, 79161 96 8679 87741 8870; 9 5 455 801 1 8107 8202 8298, 8393 456 _ 8 9 65 9060 -9-t55 -92D0, 9346j 9441 9536 96*1- 97261 98211 95 43? 9016 66o865 ••11 0960 °io6 Io55 •201 1 "296 e 3oi ii5o! 1245 1339, »486 1434 •58i 1 529 •676, »77ij 95 16231 17181 95 2569! 2663. 95 45g i8i3 1907 2002 2096' 2191 2286 1 238o 2473 l|j|3|4l5|6i7|8|9|D. A TABLE OP LOGARITHMS FROM 1 TO 10,000. N. 1 | 2 13 | 4 | 5 | 6 | 7 | 8 j 9 O. 94 460 662758 2852- 2947I 3o4i! 3i35. 32Jo r 3324; 34i8j 35iai 3607 3790; 3889 3?83 4078: 4172 4266, 436oj 4454| 4548 461 3701 94 462 4642 4736! 483o 4924 5oi8i 5u2| 32061 5299 53 9 3 5487 94 463 558 1 5675 5769 5362 5936! 6o5ol 6i43| 623" 6331 6424 94 464 65i8 6612 670D 6799 6892] 6986! 70791 7173 7266I 7360 81991 °293 '94 465 7453 8386 7546 7640 - 7 3J 8665 7826) 7920! 8oi3' 8106 8759 8852| 8943: 9038 9 i 466 8479 85 7 2 9 i3i 9224 9? 467 < 9 3 »] 9410 95oj 959ft 96S9I 9782; 9875; 9967 ••60 1 °|53 9 i 468 070246 0339 1265 043 1 o524 0617 07101 0M02 1 0895 0988 1080 9 i 469 n 7 3 i338 145 1 1 54; 1636 1728 1821 I9i3 2000 9 3 470 672098 2190 2283 2375 2467 2560' 265a 2744 2836 5929 92 471 3021 3n3 32o5 3297 _iai& 5i3i bo5; 3390 3482 1 35 7 4 3666 3 7 58 385o ■ 92 473 --J942 4861 -4o34 4953 58 7 o •6 7 85 -41 26 5o45 ti\o 4402 4494 ._458£|_4&?74-^69 55o3 5590 5687 92 92 5228 '5320 5412 474 6694 ■ 5962 6876 7789 8700 6145 623ff| 6328 6419 7333 8245 65n 6602 92 473 6968 7881 8791 7o5 g 7i5i 8o63 7242 81 54 7424I 75i6 8336 8427 9' 476 itu 7698 m 9' 477 478 8609 8 97 3 9882 9064 9 i55 9246I 9337 9» 9438 9 5l 9 9610 9700 000*] i5i3 979' 9973 •°63 •l 54| •245 9' 479 68o336 0426 o5n 0698 0789 0879 0970 1060 1131 9' 480 681241 i332 "T427 1600 1693 1784 1874 1964 2867 2055 90 481 2145 2235 2326 2416 25o6 2596I 2686 2777 2937 90 482 3047 3i37 3827 33i7 3407 3497 3587 36 7 7 3767 3857 90 483 3947 4845 4o3i 4935 583 1 4127 4217 43o7 4396 4486 4576 1 4666 4756 90 484 5o25 5ii4 5204 5294 5383 547'3: 5563 5652 r 9 485 5742 5921 681 5 6010 6100 6189 7083 6279 6368 6458 6547 486 6636 6726 6904 6994 7886 8776 7172 8064 8o53 7261 1 735i 8i53 8242 9042I 91 3i 744o $ 487 488 7529 8420 7618 85o 9 V°l 85o8 228? 9575 8865 833 1 9220 8, 89 489 9309 9 3o? 9486 9664 9753 9841 .9930' "19 •107 $ 490 690196 1081 0285 o373 0462 o55o o63g 0728 0816 ogoS •VJ93 S8 491 1170 1258 1 347 1435 1524 1612 1700 1789 1877 492 1965 2o53 2142 2230 23i8 2406 2494 2583 2671 355i 2759 363$ 88 493 2847 2 9 35 3o23 3i 1 1 3i 9 o 4078 3287 •3375 3463 88 494 3 t 2 1 38i5 3oo3 3991 4166 4204 4342 443o 45i7 88 495 46o5 4693 478i 4868 4956 5o44 5i3i 5219 ■53o7 5J 9 4 88 496 5432 5569 5637 5744 5832 5 9 i 9 6007 6094 6968 7839 8709 9078 6r82 6269 87 497 498 499 6356 6444 633 1 6618 6706 6 79 3 6880 7o55 7142 8014 8883 87 7229 8101 8188 7404 8275 7491 8362 75 7 8 8449 7665 8535 7752 8622 7926 9664 87 87 5oo 698970 9838 9057 9144 923 [ 93i7 9404 0491 1*358 975 1 87 5oi 9924 ••11 ••98 •184 •271 '444 •53 1 •617 87 5o2 700704 1568 243l 0790 0877 0963 IOJO n36 I22'2 IJ09 i3g5 1482 86 5o3 5o4 1634 25i7 26o3 1BT7J-113+3- 2689! 2775 18? -ee6& S947 ■»»I72 3o33 2208 3 r 19 2344 3205 86 86 5o5 3291 41 5i 3377 3463 3540 3635 4408 4494 5265 535o 3721 3807 38o3 4751 3979 4837 4o65 86 5o6 4236 4322 4570 5436 4665 4922 86 507 5oo8 5094 5179 5522 56071 56g3 5778 6632 86 5o8 5864 5 9 4o 68o3 6o35 6120 6206 6291 63 7 6 6462 6547 85 609 6718 68SS 6974 7° 5 9 7>44 7229 73 1 5 7400 7485 85 5io 707570 8421 7655 7740 7S26 8676 7911 79961 808 i 8166 825i 8336 85 5n 85o6 8091 8761 8846, 8 9 3 i 90i5 91001.9185 85 5l2 9270 9355 9440 9524 96091 96941 9779 o456j o540| 0623 9863, 99481 "33 85 5i3 710117 0202 oo63| 1048 1807I 1892 265o 27J4 0287. 0J71 0710 0794 0879 85 5i4 Il32 1217 i3oi l385j 1470 i554. i63q 172$ 84 5i5 1976 2060 21441 22291 23i3 2397 24811 2566 84. S16 2818 2902 29861 30701 3 1 54 3238 33 2 3 1 3407 84 5, z 5i6 3491 4J3o 3575 3659 3742 3826, 3910 3994 4665[ 4749' 4833 5502. 5586i 566 9 4078 4162I 42j6 84 44i4 4497 4681 4916 5ooo 5o84 84 519 5.6 7 5a5i 5335 54i8 5753 5836 5920 -M. N. 1 2 3 4 1 5 | 6 7 1 a ! 9 i O. A TABLE OP LOGARITHMS FROM 1 TO 10,000. N. lila 3 i i 5 6 ] 7 1 8 J 9 1 D. 530 716003 60871 6170 6254 6337 °42i 65o4| 6588 667 1 ' 6704, 83 521 6838 6921 ] 7004 7088. 7171, 7254 7338 8169 7421 75o4' 7587I 83 83361 8419 83 9163, 9248] 83 522 7671! 7754' 7837 8D02 1 8585| 8668 792CI 8oo3 ! 8086 8 7 5 1 , 8834* 8917 8253 523 9000 9083 524 933i 94141 9497 9580 9663 9745 9828 9911 99941 ••771 83 523 720159 02421 0323 0407 ' 0490 J233i i3i6 0573 0655 o 7 38 082 il 09031 83 52b 0986 1068 n5i 1 3 9 8 1481 1563 1646I 1728! 82 52 7 1811 1893 1975 2o58! 2140 2222 23o5 2387 2469I 25521 82 528 2634 2716 2798 3620 2881 2 9 63 3045 3127 3948 3209 3291 3374. 82 529 3456 3538 3702 1 3784 3866 4o3o 4H2| 41941 82 53o 724276 4358 4440 4522' 4604 4685 4767 4849 4q3 [ 1 5oi3 32 53 1 5095 5i 7 6 5258 534o 5422 55o3 5585 566-7J 5748; 583o 82 5J2 5912 5993 6073 61 56 623E 632o 6401 6483 6564i 6646 82 533 6727 6809 7623 8435 6890 6972 7053 7134 7216 8029 72971 7379! 7460 81 5J4 7341 7704 85|6 ll* 5 8d 9 7 7866 8678 7 9 48 8 7 5 9 8110 8i'9i 8273 81 535 8354 8841 8922 9003 1 9084 81 536 9165 9246 9327 ••55 ' »i36 9408 948Q 9570 g65 1 9732 9813! 9893 81 538 9974 730782 •217 •296 •378 1186 •45q •540 •621 •702 81 o863 0944 1024 IioS 1266 ■ 347 1428 i5o8 81 539 1589 1669 1 1750 2474J 2555 3278] 3358 i83o 1911 1991 2072 2l52 2233 23i3 81 54o 732394 2635 271 5 2796 2876 2956 3o37 ■ 31:7 80 54i 3197 3438 35i8 3598 3679 4480 3 7 5 9 383 9 3919 80 543 3999 4079 4160 4240 4320 4400 456o 4640 4720 80 543 4800 4880I 4q6o 5o4o 5l20 52O0 5279 5359 5439 55io 80 544 5099 56 79 5 7 59 5838 5 9 i8 5998 6078 6157 6237 63i7 80 545 6397 1 6476 6556 6635 6 7 i5 6795 6874 6954 70 3 i 71 13 80 545 547 7io3 79«7 8781 7272 8067 7352 8146 743i 8225 75n 83o5 7590 8384 7670 8463 854? 7S29 8622 7908 8701 79 79 548 8860 8939 9018 9097 9177 9256 9 335 9414 9493 79 549 9 5 7 2 9631 9?3 ' 9810 9889 9968 "47 •126 •2o5 •284 79 55o 740363 0442 0021, 0600 0678 0737 1546 0836 0915 0994 1073 79 DO 1 ll52 I 2 Jo i3o9 1 388 1467 1624 1703 1782 i860 79 552 >9 3 9 20l8 2096 2175 2254 2332 2411 2489 2568 2647 n 553 2720 2804 2882 2961 3o3 9 3823 3u8 3196 3 9 8o 3270 3353 343 1 554 35io 3588 3667 3745 3902 4o58 4i36 42 1 5 78 555 42 9 3 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 6075 5i53 5 2 3i 5309 5387 5465 5543 5621 5699 5777 78 557 5855 5 9 33 60 1 1 6089 6868 6167 6245 6323 6401 6479 6556 78 558 6634 6712 679O 6945 7023 7101 79 55 873i 7256 8o33 8808 7334 8110 8885 78 55 9 5bo 7412 748188 7489 8266 iW 7645 8421 8498 7800 85 7 6 7878 8653 78 77 56 1 8963 9040 9118 9 i 9 5 9272 935o 9427 9 5o4 9582 9659 77 562 9736 9814 9891 9968 ••45 •123 •200 •277 1048 •354 •43 1 77 563 75o5o8 o586 0663 0740 0817 0894 0971 1 1 25 1202 77 564 1279 2048 1356 1433 i5io 1 587 1664 1741 1818 i8 9 5 1972 77 565 2125 2202 2279 2356 2433 2509 2586 2663 2740 77 566 2816 2893 2970 3o47 3ii3 3200 3277 3353 343o 35o6 77 56t_ 5587 4348 -366a 4425 45oi — oSM 1 4578 3889 4654 3g66 473o 4042 4807 ittb 4883 4193 4960 4272 5o36 77 76 56 9 5lI2 5189 5265 5341 5417 5494 6256 5570 5646 5722 5799 76 570 755875 595i 6027 6788 6io3 6180 6332 6408 6484 656o 76 5 7 i 6636 6712 6864 6940 7016 7092 785 1 8609 7168 7244 7320 8079 76 572 7396 7472 823o 7548 83o6 7624 7700 8458 7775 7927 8oo3 76 573 8i53 8382 8533 8685 8761 8836 -6 574 8912 8988 9063 9J3o 9214 9290 9366 944" 9517 9592 76 575 9668 9743 98.9 0D73 9894 9970 • c 45 ti2i •196 •272 •347 75 5 7 6 760422 0498 0649 0724 0759 0875 0950 1025 1:01 75 in 1 176 I2DI 1326 1402 1477 22:8 \i^2\ 1627 23o3l 2378 1702 1778! i853 73 578 1928 2679 2003 20781 21 53 2453 2529 2604! 75 i>79 H. 2754 1 2829I 2904 2978I 3o53' 3128 32o3 3278! 33531 75 ~\ 3 1 4 5 6 7 ~n 9 1 D. 10 A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. 58o | 1 2 3 | 4 | 5 6 | 7 8 4027 9 4101 D. 75 763428 35o3 3578 3653| 37271 3802 3877 1 3 9 5J 58 1 4176I 425i 4326 44oo' 44751 455o 4624 4699 ir 4 4848 582 4923 499 8 5072 5U7 5221 1 5296 5370! 5443 5520 5594 7 5 583 566o 641 3 5743 58i8 58 9 2 6636 5966! 6041 6ii5 6190 6 9 33 6264 6338 74 584 6487 6562 6710! 6 7 85 685 9 7007 7082 74 585 586 7i56 7898 8638 723o 7972 ?3o4 8046 73 79 8120 7453 8i 9 4 7527 8268 7601 8342 7& 7 5 8416 7749 8490 7823 8564 74 74 58 7 588 8712 8786 8860 8 9 34 9008 9082 <}i56 92301 930J 74 9 3 77 9401 9 525 9 5 99 o336 9673 9746 9820 9894 9968I "42 74 58 9 770116 0189 0263 0410 0484 0557 o63i| 0705 077B 74 5qo 770862 0926 0999 1073 1 146 1220 1293 1367 1440 i5i4 74 5 9 i i58 7 "1661 1734 1808 1881 1955 2028 2102I 21 7 5 2248 73 592 2322 2395 2468 25/ ( 2 26i5 2688 2765 2835| 2908 2981 73 5 9 3 3o55 3i28 3201 3274 3348 3421 3494 3567| 3640 37i3 73 5q4 3786 45i7 386o 3 9 33 4006 4079 4i52 4225 4298! 4371 4444 73 5o5 45 9 o 4663 4736 4809 4882 4955 5o28 3100 5i73 73 5 9 6 III 5246 53i 9 5392 5465 5538 56io 5683 5 7 56 5829 5902 73 5974 6047 6120 6193 6265 6338 641 1 6483 6556 6629 73 6701 6774 6846 6919 6992 7064 7 J? 7 7209 7282 8006 8 7 3o 7354 8079 8802 73 f99 000 7427 778i5i 7499 8224 7572 8296 7644 8368 7717 8441 11% 7862 8585 7 9 34 8658 72 72 601 8874 8947 9019 9°9i 9 i63 9236 9 3o8 $3 80 9452 9524 72 602 9 5 9 6 9669 9741 9 8i3 9 885 9957 ••29 •101 •n 3 •245 72 6o3 780317 o38 9 0461 0533 o6o5 0677 1396 0749 1468 0821 o8 9 3 0965 72 604 1037 1 109 1181 1253 1324 1540 1612 1684 72 6o5 1755 1827 1899 1971 2688 2042 2114 2186 2258 2329 2401 72 606 2473 3i8 9 2544 2616 2 7 5o 34 7 5 283 1 2902 2974 368o 44o3 3046 3ii7 7a 607 608 326o 3332 34o3 3546 36i8 3761 3832 7' 3 9 o4 3975 4046 4118 4189 4261 4332 4475 4546 7> 609 4617 4689 4760 483 1 4902 4974 5o45 5n6 6187 5259 7« 610 78533o 54oi 5472 6i83 5543 56i5 5686 5 7 5 7 5828 58 99 5 97 o 7« 611 6041 6112 6254 6325 63 96 6467 6538 6609 6680 7> 612 6751 6822 68 9 3 6964 7o35 7106 7177 7885 85 9 3 7248 7319 8027 73go 8098 7« 6i3 7460 8168 753i 823 9 7602 7 6 7 3 838i 7744 8451 7815 8522 7 9 56 8663 7» 614 83io 8734 8804 7« 6i5 8875 8 9 46 9016 9087 91 57 922E 9299 9369 9440 9 5io 7i 616 9 58i 9 65i 9722 9792 9863 993^ •••4 ••74 •144 •2l5 70 6l I 618 7 9 0285 o356 0426 0496 0567 0637 0707 0778 0848 0918 7" 0988 io5 9 1 1 29 1 199 1269 i34o 1410 1480 1 55o 1620 70 Oxg 1691 1761 i83i 1901 1971 2041 21 11 2181 2252 2322 70 620 792392 2462 2532 2602 2672 2742 2812 2882 2952 3022 70 621 3o 9 2 3i62 323i 33oi 3371 3441 35n 358i 365i 3721 70 622 3700 386o 3930 4000 4070 4i 39 4209 4279 4349 5o45 44l8 7° 623 4488 4558 4627 4697 4767 4836 4906 4976 5n5 70 624 5i85 5254 5324 53 9 3 5463 5532 56o2 5672 5741 58u 7° 625 588o 5 9 4 9 6019 6 7 i3 6088 •61 58 6227 62 9 7 6366 6436 65o5 6 9 626 65 7 4 6644 6782 6852 6921 6990 7683 8374 7060 7129 7198 69 627 628 7268 i960 865 1 7337 8029 7406 8098 8789 9478 7, 4 7 5 8167 8858 7545 8236 7614 83o5 r 5 ? 8443 7821 85i3 lit 6 9 629 8720 8927 8996 9685 9 o65 9134 9 2o3 9272 6 9 63o 799341 S 000 29 9400 0098 9547 9616 9754 98231 9892 9061 69 63i 0167 0236 o3o5 037; 0442 o5ii! o58o 0648 69 632 0717 6786 o854 OQ23 1009 22 9 5 0992 1 061 1 1 29 I8i5 no8 : 1266 1335 69 633 1404 1472 1 54 ■ 1678 '747 1884 i 9 52 2021 69 634 2089 2 1 58 2226 2363 2432 ,25oo 2568 2637 2705 61 635 2774 2842 2010 35 9 4 2979 3o47 3u6 3i84 3252 3321 338g 636 3457 3525 3662 373o 3798 4480 -3867 4548 3 9 35 4°o3 407 1 68 637 638 4i3 9 4208 4276 4344 4412 4616. 4685 4753 68 4821 488q 4 9 57 5o25 5o93 r 5 1 61 5229 5908 5297 5365 5433 68 63 9 55oi 556 9 5637 5705 5 7 73i 5841 5 97 6 6044 6112 68 D K. I 2 3 4 | 3 6 7 8 9 A TABLE OF LOGARITHMS FROM 1 TO 10,000. 11 N. 1 1 2 3 4 5 6 7 8 | 9 D. 64o 1 8o6i8o| 6248 63 16 6384 645 1 6519 6587 6655 6723 6790 68 641 6838 6926 6994 7061 7129 7197 7264 7332 1400 7467 68 642 7535 760JI 7670 7738 7806 7873 794' 8008 8076 8i43 68 643 -— 82JjJ-82ij9 L - a l46 -8414 8481 8349 8616 8684 875i 8818 67 644 8886! 8 9 53 9O21 9088 gi 56 9223 9290 9 358 9425 9492 67 643 95601 9627 9694 9762 9829 9S96 9964 ••3 1 ••98 •i65 67 646 8io233 o3oo 0367 043/, o5oi o569 o636 0703 0770 0837 67 647 0904 097 1 1039 1 106 i. 7 3 1240 i3o7 1 374 1 441 1308 67 648 1373 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 649 2245 23 1 2 2379 2445 2312 2379 2646 2713 2780 2847 67 630 8i2 9 i3 2980 3o47 3ii4 3l8l 3247 33i4 338i 3448 35i4j 6i 631 358 1 3648 .3714 3 7 8i -3848 _.3 ? i4 3g8i 404S 4114 4i'8i 67 632 4248, 43 1 4 438i 4447 45i4 438i 4647 47U 4780 4847 67 653 49i3, 4980 5o46 5u3 5no 5246 53.12 ■ 53 7 8 5445 5Sn 66 604 5578, 5644 57 1 1 5777 5843 5910 65 7 3 5976 6042 6109 6175 66 655 6241 63o8 && 6440 65o6 663g 6705 6771 6838 66 656 6904 7565 6970 7o36 7102 7169 ^ 3 ? 73oi 8028 7433 8094 7499 8160 66 607 658 763i 7698 8338 7764 8424 J83o 8490 7896 8556 7962 8622 66 8226 8292 8931 8688 8734 8820 66 639 8885 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819544 9610 9676 9741 9807 9873 9g3o 0395 •••4 ••70 •i36 66 661 820201 0267 o333 0399 0464 o53o 0661 0727 0792 66 662 o858 0924 1579 2233 0989 1643 1035 1120 1 186 1231 i3i7 i3S2 U48 66 663 l5l4 1710 1775 1841 1906 1972 2037 2103 65 664 2 1 68 2299 2364 243o 2495 2360 2626 2691 2 7 56 65 665 2822 2887 2932 3oi8 3o83 3i48 32i3 3279 3344 •3409 65 666 3474 3539 36o5 3670 3 7 35 38oo 3865 3g3o 3996 4061 65 667 4126 4191 4256 432i 4386 445 1 45i6 438i 4646 4711 65 668 4776 -J84T - 4oo6 -4W _5o36 -J>iSI 5*66 Slit. — 5jq6 -536i 65 669 5426 5491 5356 562i 5686 5751 58i5 588o 5945 65 9 3 6010 65 670 826075 6140 6204 6269 ,6334 6399 6464 6528 66'58 65 671 6 7 23 6787 6852 6917 7563 6981 7046 7>n 7175 7240 73o5 65 672 7369 8oi5 7434 7499 7628 8273 7692 7757 8402 7821 7886 7 ? 5 1 65 673 8080 8l44| 820g 8338 8467 853 1 83 9 5 64 674 8660 8724 8789 8853 8918 9361 8982 9046 9111 9175 9239 64 675 9304 9 368 9432 9497 •• 7 5, •i3 9 9623 2& 9754 9818 9882 64 676 9947 ••11 •204 •268 •3o6 1037 1678 •460 •525 64 677 830389 o653 07171.0781 o845 0909 0973 1 102 1 166 64 678 1230 1294 l358i 1422 i486 I35o 1614 1742 1806 64 679 1870 1934 25 7 3 19981 2062 2126 2189 2253 23i7 238i 2445 64 680 832509 2637 2700 2764 2828 2892 353o 2956 3020 3o83 64 681 3i47 321 1 3275! 3338 3402 3466 3593 3657 3721 64 682 3784 3848 3912! 3975 4039 4io3 4'06 423o 4294 4357 64 683 4421, 4484 4348 46 1 1 4673 4739 4802 1 4866 4029 4993 64 684 5o56j 5 120 5i83 5247 53 10 53 7 3 5437' 55oo 5364 5627 63 685 5691 5734 58i7 58Si 5944 6007 6071I 6i34 61971 6261 63 680 6324 6387 6431 6314 6377 6641 6704I 6767 683o! 6894 63 687 6957! 7020 7083 7146 7210 7273 7336 7399 7462! 7325 8093 8 1 56 63 688 7388 7632 8219 8282 77i5 8343 7778 7841 mi 7967 8o3o 63 689 840S 847' 8597 8660 8723 8786 63 690 838840 8912 8975, 9038 9101 9164 9227 9289 9352 9415 63 691 9478, 9 3 4i 9604 1 9667 9729 9792 9855, 9918 9981 "43 63 692 840106, 0169 0232 0294 o357 0420 0482 o545 0608 0671 •63 693 0733 i 0796 0859 0921 0984 1046 1 logi 1172 1234 1297 63 694 l35g' 1422 1483 1347 1610 1672 17331 1797 i860 1922 2484 2647 63 693 1983J 2047! 21 10 2172 2609 26721 2734 2796 3233i 3293 3357 3420 2235, 2297 236o; 2422 62 696 2859 2921 3482 3544 2983; 3o46 3 1 08 3 1 70 62 697 36o6 3669 373 r 3793 62 698 3855 3918 3980 4042 4104 4166 4229 4291 4353 44 1 5 62 699 4477 4339| 4601 4664 4726 4788 485o 4912 4974 5o36' 62 N. ■ » 3 4 1 5 1 6 1 7 "T~i"~^ _ rDr 11 12 A TABLE OP LOGARITHMS FROM 1 TO 10,000. s. I 2 3 4 | 5 | 6 7 8 9 -5656 D. 700 846098 5i6o 5222 5284 5346' 54081 5470 5532 5594 ■701 5718 5780 5842 5904 6523 5966' 6028 6090 6i5i 62i3 6273 62 702 6337 6 ? 55 6399 6461 6585! 6646! 6708 6770 6832 68941 62 7o3 7017 7°79 7141 7202 7264 7326 7388 7449 8066 7311 8128 62 704 73.73 7634 82S1 7696 -7-738 7S19 8433 .7881 S497 It 8004 '62 7o5 §189 83i2 8374 8620 8682 8743 62 706 88o5 8866 8928 8989 9031 9112 9174 9738 9235 9297 9 358 61 707 708 • 9419 9481 9542 9604 9665 9726 9849 99" 921? o585 61 85oo33 0095 oi56 0217 0279 o34o 0401 0462 0324 61 709 0646 0707 0769 o83o 0891 0952 1 564 1014 1075 n36 1 1 97 6i • 710 , 851258 l320 i38i 1442 i5o3 I&23 1686 1747 2358 1809 61 711 1870 1931 2341 1992 2o53 2114 2175 2236 2297 2419 61 712 2480 2602 2663 2724 2 7 85 2846 2907 2968 302O 61 7 i3 3090 3i5o 3211 3272 3333 33 9 4 3455 33i6 3377 4i85 3637 61 7>4 3698 3759 3820 3881 3941 4002 4o63 4124 4245 61 7i5 43o6 4367 4428 4488 4549 4610 4670 473i 4792 4852 61 716 4913 497*4 5o34 5093 5i56 52i6 5277 5337 5398 5459 6c ?a 5319 558o 5640 5701 5 7 6i 5822 5882 5943 6548 6oo3 6064 61 6124 6i85 6245 63o6 6366 6427 6487 6608 666 b 60 719 6729 6789 685o 6910 6970 703 1 7091 7132 7212 7272 60 720 857332 7 3 9 3 7453 73i3 7374 7634 7694 7755 7815 8417 9018 7875 60 721 79 35 79 9 5 8597 8o56 -8jj6 8176 8236 8297 8357 8477 60 722 8537 865 7 8718 8778 8838 8898 8 9 58 9359 9078 60 723 9 ,38 9198 92581 9318 9370 9978 9439 9499 9619 9679 60 724 9739 9799 98391 9918 ••38 ••98 •i58 •218 •278 60 725 86.338 o3 9 8 0458 1 03 18 0378 0637 0697 0757 0817 oS 71 60 726 0937 0996 i5 9 4 1036: 11 16 1176 1236 1293 1355 1413 1475 60 728 1534 i654; 1714 1773 1 833 l8n3 1932 2012 2072 60 2l3l 2191 225l 23l0 2370 243o 2489 2349 2608 2663 60 729 2728 2787 2S47 2906 35oi 2966 3o25 3o83 3 144 3204 3263 60 73o 863323 3382 3442 336i 3£>20 368o 373o 4333 3799 3858 5 9 73i 3917 3977 4370 4o36 4096 4i55 4214 42-4 4392 4452 5g 732 4311 463o 4689 4748 4S08 4867 4926 4985 5o45 5 9 733 5io4 5i63 5222 5282 5341 -5933 5400 5459 5319 5378 5637 59 734 3676- "5T55- -48*4- -6874; -8s 6o5i 6110 6169 62^8 59- 735 6287 6346 64o5 6463 6324 6642 6701 6760 6819 59 736 63 7 8 6937 6906 7385 8174 7o55 7114 7173 7232 7201 735o 7409 5 9 7?7 738 7467 8o56 7326 8n5 7644 8233 7703 8292 7762 835o 7821 8409 7880 8468 8327 ml 59 73 9 8644 8703 8762 9349 9933 0321 8821 8879 8 9 38 8997 9036 9114 9173 5 9 740 869232 9290 9408 9466 9325 93S4 9642 9701 9760 39 74i 9818 9877 9994 •=53 •m • 170 •228 ••287 •345 u 742 870404 0462 0379 0638 0696 o 7 55 o8i3 0872 0930 i5i5 743 0989 I3 7 3 2i56 1047 1 106 1164 1223 1281 '33? 1 3 g 8 1981 J456 58 744 i63i 169O 1748 1806 1 865 1923 2306 2040 2098 2681 58 745 22l5 2273 2855 233 1 2389 2448 2364 2622 53 746 2739 2797 2913 2972 3o3o 3o88 3i46 3204 3262 58 ]il 3321 3379 3437 3495 3353 3611 3669 2727 43o8 3 7 85 38.« 58 3902 3g6o 4340 4018 4076 4i34 4192 425o 4366 4424 58 74? 4482 4598 4656 47U 4772 483o 4888 4943 5oo3 58 730 8 7 5o6i 5no 5 '77 5235 6293 535i 5409 5466 5324 5582 58 7 5i 564o 5698 5756 58i3 58 7 i olo? 5987 6045 6102 6160 58 702 6218 6276 6333 63gi 6449 6564 6622 6680 6 7 3 7 58 ■J53 6795 6853 6910 6968 7026; 7083 7141 7'99 7256 7 3i4 58 754 7371 7429 7487 7344 7602 7639 7717 8292 7774 7832 7889 8464 58 755 8522 8004 8062' 81 19 8177 8234 8349 8407 57 756 8579 8637. 8694 8732 8809 8S66 8924 8981 9039 57 }ll 9096 9i53 921 1 J 9268 9784' 9841 9325 9383 9440 9497 9335 9612 57 „„96°9 9726 9898 9936 | »"i3 0471 0328! o585 ••70 •127 •i85 57 759 880242 0299 o356 041.3 ~3~ 0642 0699 o 7 56 9 57 O. K. 1 » 4 5 6 7 8 A TABLE OP LOGARITHMS FROM 1 TO 10,000 13 H. 1 1 | i 4 5 6 7 8 9 1328 D. ^ 7 760 880814 0871 0928! 0985 1042 1 099 Ii56 I2l3 1271 761 1385 1442 1499! i556 i6i3 1670 1727 1784 i84j 1898 i 1 762 io55 2012 .2069 21261 2i83 2240 2297 2354 241 1 2468 i 763 2525 J58i 2638 2695 2752 2809 2866 2923 =9 8 ° 3o37 5t 764 - 3093 3i5o 3207 3264 332i 33 77 3434 3491 3348 36o5 57 76D 366 1 3718 3 77 5 3832 3888 3943 4002 4o5o 41 13 4625j 4682 4171 57 766 4229 4285 4342 4399 4455 4312 4069 4739 V 1 6 l 768 4795 536i 4852 4909 496D 5o22 6078 5i33 5ig2 5248 53o5 57 5418 5474 553i 5587 5644 5700 5757 58 1 3 5870 II 769 5926 5a83 6039 6096 6i52 6209 67,73 6265 6321 6378| 6434 770 886491 6547 6604 6660 6716 6829 6885 6942 j 6998 56 77' 7054 7m 7167 7223 7280 ^ 7 3 9 2 7449 801 1 73o5 8067 756i 56 772 7617 8179 7674 773o 8292 8853 IliB 7842 8404 7898 8460 itl 8i23 56 173 •8236 85 7 3 8629 8685 56 774 8741 8 797 9 3 d8 8909 8 9 65 9021 9°I2 9"34 9190 975o 9246 56 TJ5 9302 9414 9470 9326 9582 9038 9694 9806 56 776- 9862 9918 9974 ••3o ••86 •141 •197 •253 •3oo •365 56 777 778 890421 0477 0333 6589 o645 0700 0736 0812 0868 0924 56 0980 1 537 io3d 1091 1 147 1203 1259 i3i4 1370 1426 1482 56 779 i5q3 21 jo 1649 I7o5 1760 1816 1872 1928 io83 2340 2039 56 780 892005 ?6di 2206 2262 23i7 2373 2429 2484 2593 56 781 2707 2762 2818 2873 2929 2980 3o4o 3096 3i5i 56 782 3207 3262 33i8 33 7 3 3429 3484 35-10 33a5 4130 3631 3706 56 783 3762 3817 3873 3928 3 9 84 4039 4094 42o5 4261 55 784 43i6 4371 4427 4482 4538 4593 4648 4704 4739 4814 55 780 4870 4925 49S0 5533 5o36 5091 5u6 5201 5257 53 1 2 5367 55 786 5423 5478 55S8 5644 5699 6231 5754 5809 5864 5920 55 787 5975 6526 6o3o 6o85 6140 6195 63o6 636 1 6416 6471 55 788 658 1 6636 6692 6747 6802 6857 6912 6967 7022 55 789 7077 7i32 7187 7242 7297 7352 7407 7462 8067 86i5 75 7 2 8122 8670 55 79° 79' 897627 8176 7682 823i 7737 8286 V? 2 8341 7847 83 9 6 7902 845l 85o6 8012 856i 55 55 792 8 7 25 8780 8835 889O 9437 8944 8999 9054 9109 9164 9218 55 7 9 3 OT3 9 328 9 383 9492 ••39 9547 9602 9636 97 U 9766 55 794 9S21 9 8 7 5 99 3 ° 9? S5 ••94 •l4a •203 •258 •3l2 55 79D 900367 0422 0476 oj3i o5S6 0640 0695 0749 0804 o85g 55 796 0913 0968 1022 1077 ii3i 1 186 1240 1293 1349 1404 55 797 U58 i5i3 1 567 1622 1676 i 7 3i I7S5 1840 1894 2433 1948 54 798 2oo3 2o57 2112 2166 2221 2273 2320 2384 2492 54 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 3o36 54 800 903090 3i44 3199 3253 3307 336i 3416 3470 3524 3578 54 801 3633 3637 3741 3 7 o5 4337 3849 3904 3g53| 4012 4066 4120 54 802 4174 4229 42S3 43oi 4932 4445 4499: 45531 460] 5o4o 5094! 5]3 4766 822 4872 4925 4977 5o3o 5o83 5i36 5189 524i 5294 5347 53 823 54oo 5453 1 53o5 5558 56 n 5664 5716 6769 5822 3873 33 824 5927 5o8o' 6o33 6007! 6559 7033 7083 6o85 6i38 6191 6243 6296 6349' 6401 53 823 64D4 6612 6664 6717 6770 6822 6873I 6927 53 826 6980 7i38 7190 7243 7293 7348 7400I 7453 53 827 7306 8o3o 7 558 7611 8o83 8i35 7 663 8188 7716 8240 7768 7820 7873 83g 7 7925. 7978 52 828 82 9 3 8343 845o! 85o2 52 829 8555 8607, 865o 9i3o 9183 8712 8764 8S16 8869 8921 89-3J 9026 52 83o 919078 9235 9.287 9340 9 3 9 2 9444 94961 9549 52 52 52 83 1 9601 9653 9706 9738 9810 9862 9914 9967 ••19 "71 832 920123 0176^ 0228 0280 o332 0384 o436 0489 o54i 0593 833 o645 0697I 0749 1218; 1270 0801 o853 0906 0938 1010 1062 1114 52 834 1166 ■ 322 1 374 1426 1478 l53o 1582 1 634 52. 835 1686 ,738| 1790 1842 1894 1946 1998 25i8 2o5o 2102 2i54 52 836 2206 2258, 23lO 2362 2414 2466 2570 2622 2674 ' 52 83i 838 2725 27771 2829 2881 2933 2985 35o3 3o37 3o8 9 3 1 40 3192 52 3244 3296 3348 3399 345i 355d 3607 3658 3710 52 83 9 3762 38i4 3865 3917 3969 4021 4072 4i24 4176 4228 52 840 924279 433 1 1 4383 4434 4486 4538 4589 4641 4693 4744 52 841 4796 4848 . 4899 5364 64 1 5 495i 5oo3 5o54 5io6 5i57 6209 526i 52 842 53i2 5467 55i8 5570 5621 1 5673 5723 5776 52 843 5828 5879 5 9 3i 5982 6o34' 6o85 6i37j 6188 6240 6291 5i 844 6342 6394 6443 6497 6548, 6600 665 1 6702 6754 68o5 5i 845 6857 6908. 6939 74221 7473 701 1 7062] 71 14 7i65 i 7216 7268 73i9 5i 846 7370 7883 83g6 7524 8o3; 75 7 6| 7627 8088- 8140 7678, i 7 3o 9191; 8242 8703 8 7 54 7781 7.832 5i 847 848 7935, 7986 8447; 8498 8293 8345 5i 8549 860 1 1 8652 88o5, 8357 5i 849 8908 8959 9010 9061 9112; 9163 92i5 9266 9317' 9368 5i 85o 929419 9 4jo 9521 9981! "32 9572 ••83 9623! 9674 9725 9776 9827: 9879 5i 85i 9g3o •i34 •i85 •2361 '287 •333! «38o 0847! 0898 5i 852 930440 0491 o542 0592 o643 0694 0745 i 0796 31 853 0949 1000; io5i 1 102 1 1 53 1204 I254 1 >3o5 i356' 1407 5i 854 i458 1 509 i56o 1610 1661 1712 i 7 63 1814 iS65 I 9 i5 5i 855 1966 2017, 206S 2118 2169 2220 2271 2322 2372 2423 5i 856 2474 2981 2524 25i5 3o3i! 3oS2 2626 267- 3i83 2727 277? 2829 2879 2930 5i 837 858 3i33 3234 3283 3335 3386 3437 5i 3487 3538 3589 363g 4143 3690 3740 379' 3841 38 9 2 3943 5i 859 3993 4044; 4094 4195 4246 4296 4347 43 9 7 4448 5i 860 93449E 4549! 4599 465o 4700 475i 4801 4852 4902 4953 5o 861 5oo3 5o54 5 10^ 5i 54 52o5 5255 53o6 5356 5406 5457 5o 862 55o-; 5558, 56o8 5658 5709 5759 5809 5S6o 63i3l 6363 5910 5960 5o 863 6011' 606 1 1 61 11 6162 6212 6262 64i3 6463 5o 864 65i4 6564 6614 6665 6715, 6763 681 5 6865 6916 6966 5o 865 7016 ^7066 711' vki _7*ijt3Z6l 77i8| 776c ^.17 -^ ^ma -7468 5o 866 7318 8019 ■7566 806c 7618 -7668 816c 7819 8320 7869 8370 7919 7969 So 867 863 8119 8219, 8269 8420; 8470 5o 8520 807c 862C 867c 8720 877c 8820 887c 8920 8970 5o 869 9020 907c 912C 917c 922c 927c 9320 9 36 S 9415 9469 996S 5o 870 93951c 9 56c 96IC 966c o\bt 97'$ 021! 9769 9819 9869 9918 5o 871 9400 if 0068, 01 it 026- o3i' 081 5 o367 0417 0467 5o 872 o5i6 o566i 0616 o66f 07if 076: 086: 0915 096; 5o 873 1014 1064 1 1 14 n6C 121C 126; i3i3 i36a 1 412 1462 DO 874 i5n i56i 161 1 166c 171c 176c 1801; i85c 1909 2403 195E 5o 875 200S 2058| 210" 2i5- 220' 223( 23o£ 235: 245: 5o 876 2504 2334. 260; 265C 270 273' 2801 2831 2061 33q€ 2g5c 5o 878 3ooc 3o4o' 3ooc 3ue 3l 9 8! 324" 329- 334< 344: 59 349 5 3544, 339: 364: 36o2| 374 379 384 38oc 4334 3 9 3c 443. ^9 879 3989' 4o38 4o88j 4i3- 4186' 423( > 42851 433: 39 N. Ills 1_3 4 | 5 | 6 | 7 | 8 9 D. A TABLE OF LOGARITHMS FItOM 1 TO 10,000. 15 N. I 2 3 4 | 5 | 6 7 8 | 9 D. 88o 944483 4532 458 1 463 1 4680! 47291 4779 4828 4877; 4927 49 88 1 4976 5o25 5074 5i24 51731 52221 5272 5321 5370 5419 49 882 5469 55i8 5567 56i6 5665 57i5 5764 58i3| 5862 5oi2 49 883 5 9 6i 6010 6059 610S 6157 6207 6256J 63o5l 6354 64o3 49 884 6402 65oi 655i 6600 6649! 6698 6747! 6796I 6845 6894 49 885 69431 6992 7434| 7483 7041 7090 7140 7189 7238: 7287! 7336 7385 49 886 7532 758 1 763o 7679 7728; 7777' 7826 7873 49 88 I 888 7924 7973 8022 8070 81 19 1 8168 8217: 8266 83 1 5 8364 49 84i3 8462 85 11 856o 8609! 8657 8706, 87551 8804 8853 49 889 8902 949J90 8951 8999 9048 9°97 9146 9195I 9244 1 9292 9341 49 89a 9439 9488 9336 9385 9634 9 683 973 1 9780 9829 49 891 987f 9926 997 5 ••24 •073 •121 •170 •219 •267; »3i6 49 891 930363 04 14 0462 o5n o56o 0608 0657 0706 0754 o8o3 49 893 o85i 090a 0949 0997 1483 1046 1095 H43 1 192 1240 1289 49 894 1338 1 386 1433 1532 l58o 1629 1677 1726 1773 49 893 1823 1872 1920 1969 2017 2066 2114 2i63 2211 2260 48 896 23o8 2356 24o5 2453 2502 255o 2599 3o83 2647 2696] 2744 48 897 Sgi 2792 2841 28S9 2938 2986 3o34 3i3i 3i8a 3228 48 32,76 3325 3373 342i 3470 35i8 3566 36i5 3663i 37H 48 899 3760 38o8 3856 3905 3 9 53 4001 4049 4098 4146: 4194 48 900 954243 4291 4339 438 7 4435 4484 4532 458o 4628! 4677 5uoi 5i58 48 901 4723 4773 4821 4869 4918 4966 5oi4 5o62 48 902 5207 5255 53o3 535i 588*0 5447 5495 5543 55g2 5640 48 903 5688 5736 5784 5832 5928 5976 6024 6072 6120 48 904 6168 6216 6265 63i3 636i 6409 6437 65o5 6553 6601 48 905 6649 7128 6697 6745 6793 6840 6888 6 9 36 6984 7032, 7080 48 906 7176 •7224 7272 7320 7368 7416 7464 75i2| 7539 48 9°7 7607 8086 7655 8i34 7703 8181 775i 7799 7847 7894 7942 8421 7990; 8o38 48 908 8229 8277 8325 83 7 3 84681 85 1 6 48 909 8564 8612 8639 8707 8755 88o3 885o 8898 8946 8994 48 910 959041 9089 9 i37 9185 9232 9280 9328 9375 9423 9471 48 911 9618 9 566 9614 9661 9709 9757 9804 9852 9900 9947 48 912 9995 ••42 ••90 •i38 •i85 •233 •280 •328 •376 •423 48 913 960471 o5i8 o566 o6i3 0661 0709 0756 0804 o85i 0899 48 914 0^46 0994 1041 1089 1 563 n36 1 184 I23l 1279 1753 1326 1374 47 91 5 1421 1469 l5i6 161 1 1638 1706 1801 1848 47 916 i8 9 5 1943 1990 2o38 2o85 2l32 2180 2227 2275 2322 47 918 2369 2843 2417 2464 25 I I 2559 2606 2653 2701 2748 2795 47 2890 2937 2985 3o32 3079 3i26 3i74 3221 3268 47 919 33i6 3363 34io 3457 3504 3552 3599 3646 36o3 3741 47 920 963788 3835 3882 3929 3 977 4024 4071 41 18 4103 4212 47 921 4260 4307 4354 4401 4448 44 9 5 4542 4590 4637 4684 47 922 473 1 4778 4825 4872 4919 4966 56i3 5o6i 5io8 5i55 47 923 5202 5249 52o6 5343 5390 5437 5484 553i 55 7 8 5625 47 924 5672 5719 5766 58i3 586o 5907 5 9 54 6001 604S 6og5 47 923 6142 6189 6658 6236 6283 6329 6376 6423 6470 65i 7 6564 47 926 661 1 6705 6752 6799 6845 6892 6939 7408 6986 7033 • 47 III 7080 7127 7 i 7 3 7220 7267 7 3i4 736i 7454 •poi 47 7548 8016 7595 8062 7642 8109 7688 7735 82o3 7782 8249 7829 8296 7 8 7 5 7922 8390 7069 8436 47 929 8i56 8343 47 930 968483 853o 85 7 6 8623 8670 8716 8763 8810 8856 8903 47 931 8930 8996 9043 9090 9i36| 9183 922^ 9276 9')2J 9369 47 932 94l6 9463 95oc 9556 9602 1 9649 969C 9742 9789 9 835 47 933 .9882 OJ93 9975 ••21 ••68 •114 •161 •201 •254 •3oo 47 934 970347 044a 0486 o533 0579 0626 0672 ° 7 J2 u83 0765 46 935 0812 0838 0904 1369 0931 0997 1044 1090 n3 7 I22C 46 936 1276 ■ 322 I4i5 1461 i5o8 1 534 1601 1647 169; 2137 46 *ll 1740 1786 i832 1879 1925 2-388 1971 2018 2064 sno 46 938 220J 2249 229! 2342 2434 2481 252-; 2573 2619 46 939 2666 2713 2708 2804 285i 2897 2943 2989 3o35 3o82 46 N. I 2 3 4 5 6 7 8 9 " D. 16 A TABLE OF LOGARITHMS FROM 1 TO 10,000. H. 1 2 3 4 5 | 6 34o5 7 345 1 8 3497 3959 9 3543 D. 46 940 g4 1 973128 3i74 3220 3266 33 1 3 3359> 35oo 4o5i 3636 3682 3728 3774I 382o| 3866 3qi3 4374 4oo5 46 942 943 4097 4558 4143 4189 4235 4281 1 4327 4420! 440b 46 45i2 46o4 465a 4696 5i56 4742 4788 4834 488o| 4926 46 944 945 4972 5oi8 5o64 5no 5202 5248 5294 534o DJoo 46 5432 5478 6J96 6854 5524 5570 56i6 5662I 6707 5 7 53 5799 6258 6717 5845 46 946 947 948 949 5891 6330 5g83 6442 6029 6488 6075 6533 6121] 6167 65791 6625 7037 7083 6212 6671 63 04 6 7 63 46 46 6808 6900 6946 6992 T-l? 7175 7220 46 7266 73i2 7 J58 74o3 7449 7495 7541 7586 7632 8089 8546 7678 8i35 85 9 i 46 900 951 952 9 53 977724 8181 7769 8226 7 8i5 8272 7861 83 17 oJod 7952 8409 It 8o43 85oo 46 46 863i 9093 8683 8728 8774 8819 8865 8911 8g56 9002 9047 46 9i38 9184 923o 9270, 9321 9366 9412 9457 95o3 46 954 9 548 9394 9639 9685 973o| 9776 oi85j o23i 9821 9867 9 Q!2 0367 99,53 46 9 55 980003 0049 o5o3 0094 0140 0276 0322 0412 45 9 56 $1 959 0458 o549 0594 064c o685 0730 0776 0821 0867 45 0912 1 366 °9 5 7 ioo3 1048 1093 1139 1 184 I229 1683 1275 1320 45 1411 i456 i5oi 1 547 1592 1637 1728 , 17 3 45 1819 1864 19091 1954 2362 2407 2000 2045 2090 2i35i 2181 2226 45 960 982271 23i6 2452 2497 2543 2588 2633 2678 45 961 2723 2769 2814 2859 2904 3356 2949 2994 3o4o 3o85 3i3o 45 962 9 63 3i75 3220 3265 33 1 3401 3446 3491 3536 358i 45 3626 3671 3716 3762 3807 3852 38 9 7 3o42 43 9 2 3987 4o32 45 964 4077 4122 4167 4212 4257 4302 4347 4437 4482 45 9 65 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 5382 45 966 967 968 4977 5022 5067 5lI2 5i57 5202 5247 5292 5337 45 6426 5471 55i6 556i 56o6 565i 56 9 6 5741 5786 583o 45 5875 6920 636 9 5965 6010 6o55 6100 6144 6189 6234 6279 45 969 6324 64i 3 6458 65o3 6548 6593 6637 6682 6727 7175 45 970 986772 6817 6861 6906 7 353 6951 73 9 f 6996 7040 7o85 7i3o 45 97 ' 7219 7264 7309 7443 7488 7532 7577 8024 8470 7622 40 972 97J 7666 81 13 8i57 7736 8202 7800 8247 7845 8291 8336 iti 2'7? 8425 8068 85i4 43 45 974 855o 900O 8604 8643 8693 8737 9i83 8782 8826 8871 8916 9361 8960 45 975 9049 9094 9 i38 9227 9272 93i6 94o5 45 976 9450 9494 qOJq 9 583 9628 9672 9717 9761 9806 9850 44 977 978 9895 990339 ' 0783 99J0 o3S>3 59 83 e°28 ee 7 2 °H7 •161 •206 «25o •2 9 4 o 7 38 44 0428 0472 o5i6 o56i o6o5 o65o 0694 1 1 37 44 979 0827 0871 0916 l359 0960 1004 1049 1093 1182 44 980 991226 1270 l3i5 i4o3 1448 1492 1536 i58o 1625 44 981 1669 J713 n58 1802 1846 1890 2333 19 2377 1979 2023 2067 44 982 2111 21 56 2200 2244 2288 2421 2465 25o9 44 9 83 2554 ft 3o3^ 2642 2686 273o 2774 2819 2863 2907 3348 3392 3833 44 984 2995 3436 3o83 3127 3568 3172 32i6 3260 33o4 44 9 85 3480 352; 36i3 365 7 3701 3745 3789 44 986. 38 77 3o2I 436 1 396: 4009 4o53 4097 4537 4i4i 4i85 4229 42 7 3 44 $ 43i7 44o5 4449 449; 4o3* 5372 4081 4625 4669 5io8 47i3 44 5igc 995635 4801 4843 4889 4977 5021 5o65 5i52 44 989 5240 5284 532t 5416 5460 55o4 5547 55gi 6o3c 44 99c 5679 5723 5 7 6- 58n 5854 5898 6942 638c 5 9 86 44 99' 6074 6117 6161 620^ 62-49 629; 6 7 3i 6337 6424 6466 44 992 65i2 6555 65oc 664: 668 ; 6774 681S 6862 6906 7343 44 993 6949 7386 699; 743o 703- 708c 712/ 716S 7212 7255 7209 7736 8172 44 994 747^ 7 5i- 7 56i 76o5 8041 764E 8o85 7692 777? 82l{ 44 990 7823 7867 83o3 83> 79 5 ' 839c 7998 8434 812c 44 996 825o 869* 91)1 847- 8521 856^ 8608 865: 44 997 998 8739 878; 882f 8869 93o. 8oi3 934^ 8 9 5e 939: 900c 904C 908- 44 9'7. Cosine | D. Tang. L). Cotang. I 10-0000001 0-000000 Infinite. 60 I 6 '463726 5oi7-n 000000 -00 6-463726 5017 '7 i3 536274 u 2 764756 2934 85 000000! 00 764756 2934 83 235244 3 940847 2082 3i 000000 00 940847 2082 3i 059153 57 4 7.063786 I6i5 68 000000 00 7.063786 i6i5 '7 12-934214 56 5 162696 1319 000000 00 162696 i3i ? 1 831304 768122 55 6 ? 4 i 8 ™ 1 1 1 5 75 9.999999 01 241878 1110 54 I 308824 966 852 53 999999 01 3o8825 It 691175 53 3668i6 54 999999 01 366817 54 633i83| 52 9 417968 762 63 999999 01 417970 762 63 582o3o 5i 10 463723 689 88 999998 01 463727 689 88 536273 5o ii 7-5o5n8 629 81 9-999998 01 7-5o5i20 629 81 12-494880 407091 3 12 542906 679 36 999997 CI 542909 5 79 33 13 577668 536 41 999997 01 677672 536 42 422328 47 14 609853 499 38 999996 01 609857 499 3 ? 390143 46 i5 63g8i6 467 438 14 999996 01 639820 467 438 ID 36oi8o 45 16 667845 8i 999995 01 667849 82 332i5i 44 \l . 694173 4i3 11 999995 01' 694179 4i3 73 3o582i 43 718997 391 999994 01 719004 391 36 280997 42 ■9 742477 3 7 i 27 999993 01 742484 3 7 i 28 257616 41 20 '76475«i 353 i5 999993 01 764761 35i 36 235239 40 21 7-785943 8&6146 336 72 9-999992 01 7-785951 8o6i55 336 73 12-214049 3 9 22 321 75 999991 01 321 76 193845 38 23 82545i 3o? o5 999990 999989 01 825460 3o8 06 174540 36 24 843934 295 283 47 02 843944 295 283 49 1 56o56 25 861662 88 999988 02 '861674 90 13832b] 35 26 878690 895085 2 7 3 11 999988 02 878708 2 7 3 18 121292 34 3 263 999987 02 895099 263 25 1 0490 1 33 910879 253 99 999986 02 910894 926134 254 01 089106 073866 009142 32 ? 9 926119 245 38 999985 02 245 40 3i 3o 940842 23 7 33 999983 02 • 940858 23 7 35 3o ■ 3i 7-955082 229 80 9-999982 02 7-955100 229 81 1 2 • 044900 2: 32 968870 222 73 999981 02 968889 982253 222 75 o3iiii 33 982233 216 08 999980 02 216 10 017747 11 34 995108 209 203 81 999979 02 995219 209 203 83 004781 35 8-007787 90 999977 02 8-007809 n 11 -992191 970955 23 36 020021 198 3i 999976 02 020045 198 24 32 o3ioi9 o43ooi io3 02 999975 02 o3io45 193 o5 968055 23 188 01 999973 02 043527 188 o3 956473 22 3 9 054781 1 83 25 999972 02 054809 183 37 945191 21 40 065776 8-076000 086965 .78 72 999971 02 o658o6 178 74 934194 20 41 174 41 9-999969 02 8-07653i 174 44 11 •933469 9i3oo3 \l 42 170 3i 999968 02 086997 170 34 43 097183 166 It 999966 02 097217 166 42 002783 IT 44 107167 162 999964 o3 107202 162 68 °9=797 883o37 16 45 116926 159 1 55 08 999963 o3 116963 i5o 10 i5 4^ 126471 66 999961 o3 126010 i55 68 873490 U 47 i358io l52 38 999909 999958 o3 i3585i I 52 41 864149 i3 48 144953 ,149 24 o3 144956 1539S2 149 27 855oo4 12 49 153907 146 22 999956 o3 146 27 846048 11 5o 162681 143 33 999954 o3 162727 143 36 83 7 2 7 3 10 5i 8-171280 1 4o 54 9-999952 o3 8-171328 140 57 11-828672 8 52 1797.3 187985 i3 7 86 999950 o3 179763 i88o36 137-90 135-32 820237 53 135 2 9 999948 o3 811964 I 54 196102 132 80 999946 o3 196156 132-84 8o3844 55 204070 i3o 41 999944 o3 204 1 26 i3o-44 795874 5 56 2ii8o5 128 10 099942 04 21 1953 128-14 788047 4 u 219581 125 87 999940 04 219641 125-90 78o35o 772800 3 227134 123 72 999938 04 227195 123-76 3 5g 234557 121 64 - 9999 36 i 04 234621 121-68 765379 1 60 241855 119-63 999934I -04 241921 119-67 758079 Cosine D. Sine |89° Cotang. IX Tang. M. SINES AND TANGENTS (1 DEGREE.) 19 M. Sine ; D. Cosine D. Tang. D. Cotang. 8- J4i 855 119 63 9-999934 • 04 8-241921 119-67 11-758079 750898 743835 60 i 249033 n5 68 999932 • 04 249192 117-72 H5-84 % 3 256094 80 099929 999927 • 04 256i65 3 263o42 n3 98 .04 263n5 114-02 736885 57 4 269881 276614 283243 112 21 999923 • 04 269936 112-25 730044 56 5 no 5o 999922 .04 276691 no-54 723309 55 6 108 83 999920 ■ 04 283323 108-87 716677 54 I 9 289773 107 21 999918 .04 289856 107-26 103-70 104-18 710144 53 296207 302346 103 104 65 i3 999913 999913 • 04 • 04 296292 302634 703708 697366 5a 5i 10 308794 102 66 999910 • 04 3o8884 102-70 691 1 16 5o 11 8-314904 101 22 9-999907 • 04 8-3i5o46 101-26 11.684954 % 12 321027 3 82 999903 • 04 32II22 99.87 678078 672886 606975 i3 327016 47 999902 • 04 327II4 98.51 47 14 332924 ll U 999899 •03 333025 97.19 95-90 46 i5 338p3 344504 86 999897 •o5 338856 661 144 45 16 . 94 60 999894 • o5 344610 94-65 655390 44 \l 35oi8i 9 3 38 999891 -o5 360289 3558o5 36i43o 93-43 649711 43 355783 92 0? 9998S8 •03 92-24 644io5 42 •9 36i3i5 % 999883 •o5 91 -08 89-95 88 -85 638570 41 20 366777 90 999882 • o5 3'668 9 5 633io5 40 21 8-372171 88 80 9.999879 -o5 8-372292 11-627708 ll 22 377499 87 72 999876 -o5 377622 382889 87-77 86-72 622378 23 382762 86 67 999873 • o5 617111 37 H 387962 85 ■64 999870 ■ o5 388oo2 3932J4 85-70 611908 36 25 393101 84 •64 999867 • o5 84-70 606766 35 26 398179 83 -66 999864 •03 3 9 83i5 83-71 6oi685 34 3 403199 82 ■7' 999861 • o5 4o3338 82-76 81-82 596662 33 408161 8i 11 999858 • o5 4o83o4 59-1696 586787 32 ?° 4i3o68 80 999834 •o5 4i32i3 80-91 3i 3o 417919 79 • 9 5 99985 k • 06 418068 80 -02 58i 9 32 3o 3! 8-422717 3 :3 9.999848 .06 8-422869 79-U 78-3o 11-577131 3- 32 427462 999844 • 06 427618 572382 33 432i56 77 •40 999841 • 06 4323i5 77-45 76-63 56 7 685 ll 34 4368oo 76 57 999838 • 06 436962 44i56o 563o38 35 44i394 73 77 999834 •06 7 5- 83 558440 25 36 445941 74 99 999831 ■06 4461 10 75 -o5 5538;o 549387 24 ll 45o44o 74 22 999827 •06 45o6i3 74-28 23 454893 73 46 999823 •06 453070 459481 463849 73-52 544o3o 540319 22 3 9 459301 463665 V 73 999820 ■06 72-79 72-06 21 40 V 00 999816 .06 536i5i 20 41 8-467985 7i 2 9 9.999812 .06 8-468172 71-35 11.531828 \t 42 472263 7° 60 999809 999805 • 06 473454 70-66 69-98 6o-3i 527546 43 476498 480693 69 91 ■06 476693 480892 485o5o 5233o7 519108 ■ 7 44 ol 24 999801 • 06 16 45 484848 39 999797 .07 68-65 5i4 9 5o 5io83o i5 46 488963 67 Sf 999793 .07 489170 49J25o 68-oi 14 47 493040 67 999790 • 07 67. 38 506750 13 48 497078 5oio8o 66 3 999786 •07 497293 66-76 502707 12 49 66 999782 •07 5oi2q8 505267 66-i5 498702 11 5o 5o5o45 65 48 999778 ■07 65-55 494733 10 5i 52 8-508974 512867 64 64 89 3i 9-999774 999769 999765 •°7 .07 8-5oo200 513098 64-96 64-3q 11-490800 486902 8 53 516726 63 75 ■°l 5 16961 63-82 483o39 7 54 520551 63 ■9 999761 •07 520790 524586 63-26 479210 473414 6 55 524343 62 64 999737 •07 62-72 5 56 528102 62 11 999753 •07 528349 62-18 47i65i 4 n 5)i828 61 58 999748 ■07 532o8o 6i- 65 467920 3 535523 61 06 999744 •07 535779 53 9 #47 61 -13 464221 2 5 9 539186 60 55 999740 •07 60-62 46o553 1 6o 542819I 60 04 999735 ■07 543o84 60-12 456916 | Cosine i D Sine i8° Cotang. D. Tang 1 M. 20 (2 DEGREES.) A TABLE OF LOGARITHMIC M. , Sine D. | Cosine D. Tang. IX Cotang. I o 8-542819 60-04 1 9-999735 • 07 8-543o84 60-12 11-456916! 60 1 546422 5 9 .55 999731 ■ 07 546691 59-62 453309I 5q 44973 2 1 58 2 549995 553539 69-06 999726 • 07 550268 69-14 3 58-58 999722 .08 553817 58-66 446183! 57 4 567034 58-n 999T7 • 08 557336 58- 19 57-73 442664I 56 5 56o54o 57-65 999713 • 08 660828 439172 55 433709I 54 4322 7 3! 53 428863! 52 6 563999 56743i 67-19 999708 ■08 564291 67-27 I 56-74 999704J -08 9996991 -08 567727 56-82 570836 56 -3o 571 1 37 56-38 9 574214 55-87 999694 •08 574520 55-c)5 426480' 5i 10 577566 8.580892 55-44 999689 • 08 577877 8-58i2o8 55-52 4221 23] 5o 11 55-02 9-999685 •08 55-io 11-418792) 49 41 54861 48 12 684193 64-60 999680 •08 5345U 54-68 i3 687469 64-19 999676 •08 587796 5910JI 64-27 4l22o5| 47 14 59072 1 53-79 999670 .08 33-87 408949] 46 i5 593948 53- 3 9 999665 •08 594283 53-47 4057 171 45 16 697152 53-oo 999660 •08 697492 53-o8 402608! 44 K 6oo332 62-61 999655 •08 6006-'-' 62-70 3 99 323 43 603480 606623 . 52-23 999000 .08 6o383g 52-32 396161 1 42 •9 5i-86 999645 • 09 606978 5 1 -94 5i-58 5l-2I 3930221 41 20 21 609734 8-612823 5i-49 5l-I2 999640] -09 9- 99963 y -09 610094 8-613189 3899061 40 1 i-3868i 1 3 9 22 615S91 618937 5o-7& 999629 • 09 616262 5o-85 383738 38 23 5o-4i 999624 .09 6i93i3 5o-5o 380687 ll 24 621962 5o-o6 999619 .09 622343 5o- 1 5 377667 374648 2D 624966 49-72 999614 •09 625352 49-81 35 26 627948 49-38 999608 .09 628340 49-47 371660 34 27 6309 1 1 633854 49.04 48-71 999603 .09 63i3o8 49- 13 368692 33 28 999597 .09 634256 48-80 365744 362816 32 29 636776 48-3 9 999502 999686 .09 637184 48-48 3i 3o 63 9 68o 48-06 .09 . 640003 8-642982 645853 48-16 359907 11-357018 3o 3i 8-642563 47-75 9.999581 ■09 47-84 3 32 645428 47-43 999575 ■09 47-53 354147 33 648274 47-12 999570 .09 648704 47-22 351296 *7 34 65uo2 46-82 999664 .09 65i537 46-91 348463 26 35 65391 1 46-52 999558 • 10 654352 46-6i 345648 25 36 636702 46-22 999553 • 10 ' 657149 66992c 46-3i 34285i 24 M 659475 662230 43-92 999547 • 10 46-02 340072 23 45-63 999641 • 10 662689 665433 45-73 33 7 3u 22 3 9 664968 45-35 999535 •10 45-44 334567 21 4o 667689 45- 06 999529 ■ 10 668160 45-26 33i84o 20 41 8-670393 44-79 9-999624 • 10 8-670870 44-88 11 -329130 lo 42 673080 44-51 999516 • 10 673563 44-6i 326437 43 676751 44-24 999612 •10 676239 44-34 323761 17 44 678405 681043 43-97 999306 • 10 678900 44-17 321100 16 45 43-70 99950c • 10 681644 43- 80 3 1 8456 l5 46 683665 43-44 999493 999487 •10 684172 43-54 3i5828 14 % 686272 43- 18 •10 686784 43-28 3 1 3 2 1 6 13 688863 42-92 999481 -10 689381 43 -o3 310619 12 49 691438' 42-67 999476 • 10 69 1 o63 69432c. 42-77 3o8o37 11 5o 693998 42-42 999469 9-999463 • 10 42-52 3o547i 10 5i 8-696343 42-17 8-697081 42-28 11-302919 i 52 699073 701689 41-92 999456 699617 42 -o3 3oo383 53 41-68 99945o 702139 41-79 297861 I 54 704090 41-44 4 999443 704646 41-53 295354 55 706577 41-21 999437 707140 41-32 292860 5 56 709049 40-97 999431 709618 41 -08 2oo382 4 32 711607 40-74 999424 712083 40-85 287917 3 713952 716383 4o-6i 999418 7U534 40-62 285465 2 59 40-29 %994i 1 716972 719396 40-40 283028 1 60 718800 40 06 999404 40-17 280604 Cosine D. Sine 187° Cotang. D. Tang. M. BINES AND TANGENTS. (3 DEGREES.) 21 M. Sine D. Cosine D. Tang. D. Cotang. | 8-718800 40 06 9.999404 •11 8-719396 4o 17 11-280604! 60 l 721204 3 9 84 999398 -ii 721806 39 9 5 278194I 59 2 723595 3g 62 999391 •11 724204 39 74 275796 58 3 725972 728W7 3 9 41 999384 •11 726588 39 52 273412I 57 4 38 3 999378 •11 728959 •11 73i3i7 2' 3o 271041 56 5 73o688 999371 38 09 268683 55 6 733027 38 77 999364 .12 733663 89 266337 54 7 735354 38 57 999357 • 1 2 735996 38 68 264004 53 8 737667 38 36 999350 •12 738317 38 48 26i683 52 9 739969 38 16 999343 •12 740626 38 27 2593741 5i »o 742259 37 9 6 999336 •12 742922 38 2 7 257078 So n 8*744536 37 76 9-999329 •12 8-745207 -I 7 ll 11-254793 49 ii 746802 37 56 999322 ■12 747479 3 7 252521 48 i3 74go55 37 37 99931 5 • 12 749740 3 S 49 25o26o 47 U 751297 37 17 999308 •12 751989 37 29 2480 1 1 46 15 7535-28 36 98 999301 •12 754227 37 10 a45773 45 ^ 755747 36 79 999294 •12 756453 36 92 243547 44 17 757955 36 61 999286 •12 758668 36 73 24i332 43 i a 760161 36 42 999279 •12 760872 36 55 239128 42 '9 762337 36 24 999272 •12 763065 36 3d 236g35 41 20 76451 1 36 06 999265 ■12 765246 36 i3 234754 40 21 8-766675 35 88 9-999257 •12 8-767417 •13 769578 36 00 1 1 -232583 3! 22 768828 35 70 999250 35 83 230422 23 770970 35 53 999242 •i3 771727 35 65 228273 37 24 ^773ioi 35 35 999235 •i3 773866 35 48 226134 36 25 775223 35 18 999227 ■i3 775995 35 3i 224005 35 26 777333 35 01 999220 •i3 778114 35 14 221886 34 27 779434 34 84 999212 •i3 780222 34 80 219778 217680 33 28 781524 34 67 999205 •i3 782320 34 32 29 7836o5 34 5i 999-97 •i3 784408 34 64 215592 3i 3o 7856 7 5 34 3i 999189 •i3 786486 34 47 2i35i4 3o 3i 8-787736 34 18 9-999181 •i3 8-788554 34 3i 11-211446 21 32 789787 791828 34 02 999174 •i3 790613 34 i5 209387 207338 33 33 86 999166 • i3 792662 33 8 27 34 793859 33 • 70 99915S •i3 794701 33 206299 20 35 795881 33 ■54 999 1 5o •i3 796731 33 68 203269 25 36 797894 33 .2I 999142 •i3 798752 •i3 800763 33 52 201248 24 37 799897 801892 33 999134 33 37 199237 23 38 33 • 08 999126 -i3 ' 802765 33 22 197235 22 39 803876 32 - 9 3 9991 i£ •i3 8o4758 33 07 195242 21 4o 8o5852 32 -78 9991 10 •i3 806742 32 9' 193258 20 4i 8-807819 32 •63 9-999102 •i3 8-808717 32 £ 11 -101283 189317 !8 42 809777 32 49 999094 .14 8io683 32 43 811726 32 •34 999086 .14 812641 32 48 187359 n 44 8i366-> 32 ll 999077 .14 8i458 9 32 33 i854n 16 45 815599 32 999069 .14 816529 32 ll 183471 i5 46 817522 3i 9 1 999061 .14 818461 32 i8i53g 14 % 819436 3i 77 999053 ■14 820384 3i 91 179616 ■ 3 82i343 3i 63 999044 .14 822298 3i ll 177702 12 49 823240 3i n 999036 .14 824206 3i 175795 11 5o 825i3o 3i 999027 .14 826103 3i 5o 173897 10 5i 8-827011 3i 22 9-999019 .14 8-827992 •14 829874 3i 36 I I • 172008 8 52 828884 3i 08 999010 3i 23 170126 53 830749 3o i 999002 •14 83i748 3i 10 168252 I 54 832607 3o 998993 998984 .14 8336i3 3o 8^ 166387 55 834456 3o 69 •14 835471 3o 164520 b 56 836297 3o 56 998976 ■14 837321 3o 70 162679 4 Si 838 1 3o 3o 43 998967 •i5 83gi63 3o 57 160837 3 839956 3o 3o 998958 •i5 840998 • i5 843^25 3o 45 1 59002 3 59 841774 3o •'7 998950 3o 32 157175 155356 I 6o 843585 3o 00 998941 •i5 844644 30-19 Cosine T) Sine 86° Cotang. D. Ttmg. M. 22 (1 msertEEs.) a table OF LJGARIThMIC M. Sine D. Cosine D. Tang. D. Cotang. 60 8-843585 3o-o5 9.998941 ■i5 8-844644 3o '9 m55356 i • 845387 29-92 19-80 998932 •|5 840455 3o °7 153545 5 9 2 847183 998923 ■i5 848260 29 95 151740 58 3 848971 29 1l 29-55 998914 -i5 85oo57 29 82 149943 $ 4 85075 1 998905 998896 -i5 85 1846 29 70 U8i54 56 5 852525 29-43 -i5 853628 29 58 146372 55 6 854291 29-31 998887 ■i5 8554o3 29 46 144597 54 I 856o49 29-19 998878 •|5 857171 29 35 142829 141068 53 85 7 8oi 29-07 998869 -i5 858 9 32 39 23 52 9. 8D9546 28-96 28-84 998860 ■ i5 860686 29 11 i393i4 5i 10 861283 9988D 1 • i5 862433 It 00 137567 5o li 8-863oi4 28-73 9-998841 • 15 8-864173 88 11 -135827 49 13 864738 28-61 998832 •i5 865 9 o6 28 00 134094 48 i3 866455 28 -5o ■998823 •16 867632 28 132368 47 14 868 1 65 28-39 28-28 998813 • 16 869351 28 54 ■ 1 30649 46 i5 869868 998804 • 16 871064 28 43 128936 45 16 871565 28-17 998795 998785 • 16 872770 28 32 127230 44 \l 873255 38- 06 -16 874469 28 21 I2553I 43 874938 27-o5 998776 • 16 876162 28 11 123838 42 "P 876615- 27-86 998766 ■16 877849 28 00 I22l5l 41 20 878285 ' 27-73 998757 .16 879529 27 89 1 2047 1 40 31 8-879949 37-63 9-998747 .16 8-881202 27 11 II H8798 39 23 881607 37-53 998738 .16 882869 27 H7i3i 38 23 883258 37-43 998728 ■16 88453o 27 58 1 1 5470 ll 24 884903 886542 37-3i 998718 •16 886i85 27 47 n38i5 25 37-31 998708 ■16 887833 27 37 112167 35 26 888174 37-11 998699 •16 889476 27 27 uo524 34 11 889801 27-00 998689 •16 891112 27 17 108888 33 891421 26-90 26-80 998679 •16 892742 27 "7 107258 32 ?9 8 9 3o35 998669 894366 26 % io5634 3i 3o 894643 26-70 998659 895984 8-897596 26 104016 3o 3i 8-896246 36-60 9-998649 26 77 11-102404 29 32 897842 36-5i 998639 899203 26 67 100797 28 33 899432 36-41 998629 900803 26 58 099197 2 34 901017 26-3i 998619 902398 903987 905070 26 48 097602 35 902596 26-22 998609 26 38 096013 25 36 904169 26-12 998599 26 29 094430 24 U 90D736 26 o3 998589 907147 26 20 092853 23 907297 9 o8853 25-93 25-84 998578 908719 910285 26 10 091281 32 39 998568 26 01 089715 088 1 54 21 4o 910404 25- 7 5 998558 91 1846 25 f 3 20 4i 8-911949 913488 25-65 9-998548 8-913401 25 11-086599 !8 43 25-56 998537 914951 25 74 o85o4o o835oo 43 915022 25-47 25-38 998527 916495 918034 25 65 \l 44 9i655o 998516 ■18 25 56 081966 45 918073 25-29 998506 ■18 919568 25 n 080432 i5 46 919591 25-20 998495 .18 921096 25 078904 077381 14 % 921103 25-12 998485 •18 932619 25 3o l3 922610 25-o3 998474 ■18 924i36 25 21 075864 12 49 924112 34-9! 34-86 998464 .18 923649 25 12 074351 11 5o 925609 998453 •18 927156 25 o3 072844 10 5i 8-927100 928587 930068 24-77 9-998442 ■18 8-928658 24 £ II-07I342 8* 52 34-69 99843 1 .18 9301 55 24 069845 53 34- 60 99842 1 •18 931647 24 78 068353 I 54 931 544 34-53 998410 •18 933 1 34 24 70 066866 55 933oi5 24-43 998399 998388 .18 9'346i6 24 61 o65384 5 56 , 934481 24-35 .18 936093 24 53 063907 4 57 935942 24-27 998377 .18 9 3 7 565 24 45 062435 3 56 937398 93885o 24-19 998366 • 18 939082 24 37 060968 2 5o 24-11 998355 •18 940494 941902 24 3o o5a5o6 058048 1 6o 940296 24 -o3 998344 •18 34 21 Cosine B. Sine 85» Cotansr. T). Timer. M. SINES AND TANGENTS (5 DEGREES.} 2. ! u - Sine D. Cosine D. Tanff. D. Cotangr. 1 o 3-940296 941738 24 03 9-998344 •19 8941952 24-21 1 1 • o58o48; 60 I 23 t 998333 •19 943404 24 i3 056596! 5o o55i48! 58 2 943i74 23 998322 •19 944852 24 o5 3 944606 23 79 9983 1 1 • 19 946295 947734 23 97 . o537o5' 57 4 946034 23 7' 998300 • 19 23 8 o52266 56 5 947456 948874 23 63 998289 ■19 949168 ii o5o832 55 6 2'3 55 998277 .19 950597 23 -4 049403' 54 7 960287 23 48 99S266 • 19 952021 23 66 047979- ?3 8 961696 23 40 998255 ■19 953441 23 60 046559 52 9 953ioo 23 32 998243 •19 9 54856 23 5i 045 1 44| 5 1 10 954499 23 25 998232 • 19 956267 23 44 043733 5o ii 8.955bo4 957284 958670 23 '7 9.998220 •19 8-957674 23 37 II -042326' 49 12 23 10 998209 •19 959075 23 3 040925! 48 i3 23 02 998197 99S186 • 19 960473 , 23 0393271 47 14 960032 22 t •19 961866 23 U o38i34! 46 i5 961429 22 998174 .19 963255 23 07 o36745i 45 16 962801 22 80 998163 .19 964639 23 00 03536 i, 44 \l •964170 22 73 998151 • 19 966019 22 93 03398 i 1 43 965534 22 66 998139 998128 • 20 9U7394 22 86 032606, 42 "9 966893 22 59 •20 968766 22 79 o3i234' 41 20 968249 22 D2 998116 •20 970133 22 7' 029867 40 21 8-969600 22 44 9-998104 •20 8-071496 972835 22 65 n-o285o4; 39 22 970947 22 33 998092 ' 998080 • 20 22 V 027145, 38 23 972289 22 3i ■20 974209 22 5i 025791 3i 024440* 36 =4 97362a 22 24 998068 •20 975560 22 44 25 974962 22 '7 998056 •20 976906 22 37 O23oo4: 35 021732 34 26 976293 22 10 998044 •20 978248 22 3o s 977619 978941 22 o3 998032 • 20 979586 980921 22 23 020414; 33 21 97 998020 ■20 22 17 019079; 32 ? 9 980259 981573 8-982883 21 90 998008 •20 982251 22 10 017749' 3 1 3o 21 S3 997996 ■20 983577 22 04 016423 3o 3i 21 77 9 ■ 997985 • 20 8-984899 21 97 11-ClJlOI' 29 32 984189 21 7° 997972 • 20 986217 21 9' 013783 28 33 985491 986789 9 88o83 21 63 997939 ■20 987532 988842 21 84 012468 27 34 21 57 997947 •20 21 78 0iu58' 26 35 21 5o 997933 •21 990 1 49 21 7" ooo85i 25 008549 24 36 989374 21 44 997922 •2i 991451 21 65 ll 990660. 21 33 997910 ■21 992750 21 58 007230 23 991943 21 3i 997897 997883 •21 994o45 21 52 005955 22 3 9 993222 21 25 •21 995337 21 46 004663 2 1 4o 994497 8-995768 2, « '9 997872 •21 996624 21 40 003376 20 41 21* 12 9-997860 ■21 8-997908 21 34 11*002092 10 000812 18 42 997036 998299 21 06 95-847 •21 999188 21 27 43 21 00 997835 ■21 9.000465 21 21 10-9995351 17 44 999560 20 g 997822 •21 001738 21 i5 998262' 16 45 9-000816 20 997809 •21 003007 21 3 996993 1 5 46 002069 20 82 997797 •21 004272 21 995728I 14 % oo33i8 20 76 9977»4 •21 oo5534 20 9'' 994466' i3 004563 20 70 997771 997758 •21 006792 20 91 993208! 12 f> oo58o5 20 64 •21 008047 009298 20 85 991953! 11 5o 007044 9-008278 20 53 997745 •21 20 80 990702 10 5i 20 52 9-997732 -21 9-010546 20 H 10-989454,- 9 988210 8 52 009510 20 46 997T9 997706 ■21 011790 oi3o3i 20 68 53 010737 20 40 •21 20 6a 986969 7 54 011962 20 34 997693 997680 •22 014268 20 56 985732 6 55 oi3i82 20 3 ■22 oi55o2 20 5i 984498 5 56 014400 20 997667 ■22 016732 20 45 983268 4 n oi56i3 20 17 997664 •22 017959 019183 20 40 982041 3 016824 20 12 997641 •22 20 33 980817 2 5 9 oi8o3i 20 06 997628 •22 020403 20 28 979697 1 978380 6o 019235 20 00 997614: •22 021620 20 23 Cosine r>. Sine '84° Cotang. P- Tang. »• 24 (6 DEGREES.) A TABLI OF L0GAK1THMIC M. Sine E . Cosine D. Tang. D. Cotanjf. 60 Q'OI0235 20 00 9-997614 •22 9*021620 20'23 10-978380 I 020435 19 9 5 997601 ■22 022834 20.17 977166 973956 ll 2 021632 19 89 997588 •22 02404. 20-11 3 022825. 19 84 997574 •22 02525] 20- 06 974749 973545 Hi 4 024016 19 78 997561 •22 026455 20-00 56 5 025203 . 19 73 997547 •22 027655 028852 "9-95 972345 55 6 026386 19 67 997534 •23 19-00 97114" 54 I 027567 19 028744 1 9 62 57 997520 997507 •23 •23 o3oo46 o3i23- 032425 .9-85 19-79 969954 968763 53 52 9 029918 19 5i 997493 997480 •23 19-74 967575 5i 10 o3 1 089 1 9 47 ■23 o336og 19-69 966391 5o ii 9-032257 19 41 9-997466 ■23 9-034791 19-64 10-965209 3 12 o3342i 19 36 997452 •23 035969 i 9 -58 96403 1 13 o34582 19 3o 99743o 997425 •23 037 1 i'i 19-53 962856 2 14 o3574i 19 036896 19 25 •23 o383i6 19-48 961684 i5 20 997411 •23 039485 19-43 9605 1 5 45 ■ 6 o38o48 19 i5 997397 •23 04065 1 i 9 -38 9593491 44 958f87| 43 \l 039197 19 10 997383 •23 04181; 19-33 040342 19 o5 997369 •23 04297; 19-28 9570271 42 9558ioj 41 '9 04U85 18 99 99735:) ■23 044 1 3o 19-23 20 042625 18 n 997341 •23 045284 19-18 954716' 40 21 9-043762 18 9-997327 9973i3 ■24 9-046434 19-13 10-953566' 39 952418! 38 22 044895 18 84 ■24 047582 048727 19-08 23 046026 1 8 ]i 697299 9972bD •24 19-03 18-98 951273 37 24 o47>54 18 •24 04906c 95oi3i 36 2D 048279 18 7" 997271 997257 : 24 o5ioot i8- 9 3 948992 947836 35 26 049400 18 65 •24 o52i44 ■ 8.| 9 34 ll o5o5m 18 o5i635 18 60 997242 • 24 05327- 18-84 946723 945393 33 55 997228 • 24 054401 18-79 32 ?9 052749 18 053859 18 5o 997214 •24 055535 18-74 944465 3i 3o 45 997199 •24 o56659 18-70 943341 3o 3i 9.054966 18 41 9-997185 •24 9-057781 058900 18-65 10-942219 29 32 056071 18 36 997170 ■24 18-69 18.55 941100 28 33 057176 18 058271 18 3i 997 1 56 •24 060016 939984 938870 27 34 27 997 14 1 •24 061 i3o 18. 5i 26 35 059367 18 22 997 I2 7 •24 062240 18-46 937760 25 36 060460 18 17 997" 2 •24 063348 18-42 9366521 24 ll o6i55i 18 1 3 997098 ■24 064453 18-37 935547 23 062639 18 08 997083 •25 065556 i8-38 934444 22 3 9 063724 18 064806 1 7 04 997068 •25 066655 18-28 933345 21 4o 99 997053 •25 067752 9-068846 18-24 18-19 i8-i5 932248 20 4i 9-065885 17 9i 9-997039 •25 10-931154 \l 4a 066962 1 7 ll 997024 •25 069938 930062 43 o68o36 1 7 997009 ■25 071027 072113 18-10 928973 927887 \l 44 069107 17 8i 996994 •25 18.06 45 070176 17 77 996979 H 073197 074278 18-02 926803 15 46 071242 17 ll 996964 •25 17-97 925722 14 4-> 072306 17 996949 •". 075356 17.93 924644 |3 48 49 073366 17 074424 17 63 5, 996934 996919 •23 •25 076432 077505 078576 17-89 17-84 923568 12 9224951 ii 00 * 075480 17 5j 996904 •25 17-80 921424' 10 5i • 9-076533 17 5o 9-996889 •25 9.079644 0807 1 17-76 10.9203561 9 52 077583 17 078631 17 46 996874 9 9 6858 ■25 17-72 919290 8 918227! 7 53 42 •25 0S1773 o¥:833 17.67 17-63 54 079676 17 080719 17 38 996843 •25 9.17167! 6 1 55 33 996828; •25 083891 ■7-5 9 916109' 5 I 56 081769 17 29 9968 1 2 • 26 084947 17-53 91 5o53| 4 U 082797 17 083832 17 23 996797 •26 086000 .7-5, 914000' 3 21 996782 • 26 087050 '7-47 912950 2 I 9 084864 17 n i3 996766 ■ 26 088098 17-43 911002 1 60 085894 17 996751 .26 33° 089144 Cotan^. 17-38 D. 9io856| I Ooaine E . I Sine Tang. J M. HNBS AND TANGENTS. (V DKGRKES. 1 26 |'M. Sine D. Co3ine D. Tang. D. Cotang. [ o 9-086894 i 7 -i3 9-996751 -26 9-089144 17-38 10 .940856 60 i 086922 17-09 996730 •26 09018- 17-34 909813 5 9 2 087947 088970 17-04 99672c • 26 , 09122! i7-3o 908772 58 3 17-00 996704 • 26 092266 17-27 907734 57 4 089990 16-96 99668? •26 093302 17-22 906698 5t> 5 091008 16-92 99667c • 26 094336 17-19 17- id 906664 55 6 092024 16-88 99665- •26 09536- 904633 54 7 093037 16-84 996641 ■ 26 09639; 17-11 9o36o5 53 8 094047 16-80 996625 •26 097422 17-07 902678 52 9 095o56 16-76 99661a ■26 098446 17 -o3 901554 5i 10 096062 16-73 99659^ •26 099468 1699 16-96 ooo532 5o ii 9- 097065 098066 16-68 9 -996578 •27 9-100487 1 • 8995 1 3 % 12 16-65 996562 • 27 101 ~>04 16-91 898496 ' i3 099065 16-61 996546 •27 I025i(; 16-87 897481 47 U 100062 16-57 996530 •27 io3532 16-84 896468 46 15 ioio56 16-53 9965 m •27 104542 16-80 896458 45 16 102048 16-49 996498 -27 io555o 16-76 894450 44 11 Io3o37 10402 5 16-40 996482 •27 106556 16-72 893444 43 16-41 996465 •27 107559 16-69 16. 65 892441 42 >9 io5oio 16-38 996449 99643J •27 io856o 891440 41 20 105992 16-34 •27 109559 16-61 800441 10-889444 4o 21 9-106973 i6-3o 9- 99 6 4I7 •27 9-no556 i6-58 3g 22 107951 16-27 996400 •27 1 1 mi 16 -54 888449 38 23 108927 16-23 996384 •27 112543 i6-5o 887457, 3 7 886467! 36 24 1 0990 1 16-19 9 9 6368 •27 1 13533 16-46 25 1 10373 16-16 996351 • 27 H452I i6-43 8854701 35 8844931 34 26 1 1 1842 16-12 996335 ■27 n55o7 16-39 27 112809 16-08 9 9 63i8 :S 1 16491 16-36 883509] 33 28 113774 i6-o5 996302 1 17472 1 18452 16-32 8825281 32 2 9 J 14737 1 1 5698 ' 16-01 996285 ■28 16-29 16-25 881548J 3i 3o 15-97 996269 •28 1 19429 880671] 3o 3i 9-H6656 15-94 9-996252 .28 9-120404 16-22 10-879596! 29 878623! 28 32 117613 15-90 15.87 996235 .•28 121377 16-18 33 118067 996219 •28 122348 16- 1 5 8776521 27 , 34 I 10 5 19 15-83 996202 ■28 I233I7 1 6 - 1 1 8766831 26 35 I20469 i5-8o 99&i85 •28 124284 16-07 8757 1 61 25 36 I2I4I7 i5- 7 6 996168 •28 125249 16-04 87475i 1 24 32 122362 15-73 996151 •28 126211 16-01 873789 23 I233o6 15-69 996134 •28 127172 i28i3o 15-97 872828! 22 39 124248 15-66 9961 17 ■28 15-94 87187O! 21 40 125187 i5-62 996100 •28 1 29087 15-91 15.87 870913! 20 ■ 41 9-126125 1 5; 5 9 9-996083 •29 9*i3oo4i 10.869959 19 42 127060 15-56 996066 ■29 130994 15-84 869006 f 18 868o56i 17 43 127993 128923 1 29854 i5-52 996049 •29 1 3 1944 i5-8i 44 16-49 996032 •29 132893 133839 15.77 867 1071 16 45 i5-40 996015 •29 15-74 866161' i5 46 130781 15-42 995998 •29 134784 15-71 i5-o 7 8652i6l 14 % 131706 i5-3 9 995980 •29 135726 864274I i3 i3263o i5-3o 990963 •29 136667 15-64 863333| 12 49 i3355i 15-32 995946 •29 137605 i5-6i 862395I n 86i458i 10 5o 134470 15-29 995928 •29 138542 15-58 5i 52 9 135387 i363o3 l5-20 1 j-22 9-990911 995894 •29 ■29 9, 139476 140409 15-55 i5-5i 10-8606241 9 85969 1 1 8 85866o| 7 53 137216 138128 I0-I9 995876 ■291 , 141340 15-48 54 l5-)6 995859 •29! 142269 15-45 85773 1 1 6 55 139037 15-12 995841 •29 143196 i5-42 ' 8568o4J 5 56 139944 140800 15-09 995823 •29 144121 10-39 8558791 4 11 i5o6 995806 •29J J45o44, 15-35 8049561 3 1 41 754 i5o3 995788 •291 145966 146885, 15-32 854o34' 1 5 9 U2655 i5-oo 995771 •29! i5- 29 853n5 I _*L 143555 1496 995753 •2 9 | 147803 1 i5 26 862197 1 Coeinb D. 1 Sine S2°l Cotang. 1 x>. Tan?. M. | 26 (8 DEGREES.) A TABLE OP LOGARITHMIC 3TJ 'Sine D. Oosine D. 1 Tarn-. i B. ' Cotang. o 9-143^55 14-96 9-99575; •3o 9-14780; 148718 i5-26 I0-8521Q7 85i28a 6c I 144453 14-93 995735 •3o i5-23 % 2 145349 14-90 995717 ■3o 149632 i5--.ni 85o368 3 146243 14-87 995699 •3o i5o544 *i5.i 7 849466 848546 n 4 I41i36 148026 14-84 9 9 568i ■3o i5i454 15-14 5 14-81 995664 •3o 152363 i5-ii 847637 55 6 14891.5 149802 14-78 995646 •3o 153269 i5-o8 846731 845826 54 I 14-75 995628 •3o 154174 i5-o5 53 i5o686 14-72 14-69 14.66 993610 •3o 1 55077 l5-02 844923 5a 9 161369 995391 ■3o 155978 166877 9-157775 14-99 844022 5i 10 1 52411 995573 •3o 14-96 843123 5o ii 9-i5333o 14-63 9-995555 •3o 14-93 10-842225 49 12 154208 14-60 995537 -3o 1 586} 1 1 59565 14-00 841329 48 i3 i55o83 14-57 995519 -3o 14-87 840433 4 7 14 155957 14-54 9955oi • 3i 160457 14-84 83 9 543 838653 46 ■ 5 1 5683o I45i 993482 • 3i 161 347 162236 14.81 45 16 157700 158569 U-48 995464 •3i 14 -79 837764 836877 44 \l U-45 995446 ■3i 163123 14-76 43 1 59433 14-42 995427 •3i 164008 ■ 4-73 8.35992 42 '9 160J01 i4-3 9 995409 • 3i 164892 14.70 835io8 41 20 161 164 U-36 995390 ■3i ,165774 9- 166634 14-67 834226 40 21 9'l62023 U-33 9-993372 -3i 14-64 10-833346 3 9 22 162885 i4-3i 995353 • 3i 167532 168409 . 14-61 832468 38 23 I63-J43 164600 14-27 995334 • 3i U-58 83i5 9 i ll 24 14-24 9953i6 -3i 169284 14-55 83oti6 829843 828971 828101 25 165454 14-22 ' 995297 995278 995260 • 3i 170167 14-53 35 26 u 166307 167 1 5o 168008 14-19 14-16 • 3i • 3i 171029 171899 172767 173634 U-5o U'- 47 34 33 14-13 995241 •32 U-44 827233 826366 3a '9 168856 14-10 995222 •32 14-42 3i 3o 169702 14-07 14-03 993203 •32 174499 9-175362 U-3g 8255oi 3o 3i 9-I70547 9-995184 ■32 14-35 10-824638 3 32 1713CJ9 14-02 995i65 ■32 176224 U-33 823776 33 172230 13-99 i3- 9 6 993146 •32 177084 i4-3i 822916 27 34 173070 995127 995108 •32 177942 14-28 822058 26 33 173908 13-94 •32 173799 179655 14-25 821201 .25 36 174744 i3-§8 995089 •32 14-23 820345 24 u 1 7 55 7 8 995070 •32 i8o5o8 14-20 8 1 9492 818640 23 17641 1 13-86 995081 ■32 i8i36o 14-17 14-15 22 3 9 177242 13-83 995o32 -32 182211 817789 21 4o 178072 i3-8o 99501 3 -32 iS3o59 14-12 8 1 69 4 1 30 4i 9*178900 13.77 9-994993 -32 9 ■ 1 83907 14-09 10-816093 !8 42 179726 ioo55i 13-74 994974 994955 -32 181752 14-07 8i5248 43 ■ 3-72 -32 1 8 5597 1864J9 14-04 8i44o3 n 44 181374 i3-6 9 13-66 994935 -32 14-02 8i356i 16 45 182196 994916 994896 .33 187280 188120 llll 812720 i5 46 i83oi6 13-64 .33 811880J U % 183834 i3-6i 994877 994857 994838 .33 l88 9 58 13-93 811042 i3 1 8465 1 i3-5o 13-56 ■ 33 189794 13-91 i3-Bo 13-85 810206 12 49 185466 •33 191629 80937 ii 11 5o 186280 13-53 994S18 •33 . 191462 8o853S 10 5l 9 ■ 1 87092 l3-31 9-994798 • 33 9-192294 i3-8i 10-807106! 9 806876, 8 52 187903 188712 189319 190323 13-48 994779 994759 .33 193 124 •3-8i 53 i3-46 -33 193953 -3-79 13.76 806047 1 54 i3-43 994739 •33 i9',78o 8o5220 6 55 i3-4i 994719 •33 195606 13-34 804394 5 56 191130 13-38 994700 -33 196430 i3- 7I 803570 4 u 191933 13-36 994680 -33 197253 198074 i3-6 9 802747 3 "?n2 4 13-33 994660 ■33 13-66 801926 2 5, 193534 i3-3o 994640 •33 198894 13-64 801106 1 6o 194332 13-28 994620 • 33 199713 Cotunij. 13-61 800287 Cosine D. Sine ^1° D. Tbiib. J K. , SINKS AND TANMENTS (9 DEGREES.] 21 ITT Sine D. Cosine D. Tang. D. Cotang. "60" o y 194337 13-28 9-994620 ■33 9-199713 ■ 3 61 10-800287 I I95l 29 13-26 994600 ■33 20052^ i3 3 799471 5o a 195923 i3-23 994580 •33 201343 i3 56 798655 58 3 196719 IJ-21 994560 ■34 202159 i3 54 797841 Hi 4 197511 i3-i8 994540 ■34 202971 203782 i3 52 797029 56 5 198302 i3-i6 9945i9 •34 i3 49 796218, 55 6 199091 ■ 3-i3 .994499 ■34 204592 i3 47 795408 j 54 7 199879 i3ii 994479 •34 205400 i3 43 794600' 53 8 200666 i3o8 99445o 994438 ■34 206207 207013 i3 42 793793 32 9 201431 i3-o6 ■34 i3 40 792987 5i 10 202234 ■ 3-04 994418 •34 207817 9.208619 i3 38 792183 5o li 9-203017 i3-oi 9-994397 •34 i3 35 10-791381 % 12 2^3797 1299 994377 ■3 4 209420 13 33 790580 789780 13 204377 12-96 994357 ■34 210220 13 3i 47 14 205354 12.94 994336 ■34 21IOl8 i3 28 788982 46 i5 2o6i3i '12-92 1289 9943 1 6 •34 2Il8l5 i3 26 788185 45 16 206906 994295 ■34 2I26ll i3 24 787389 44 !o 207679 12-87 994274 ■35 2i34o5 i3 21 7865 9 3 43 208432 12-85 994254 •35 214198 13 ■9 785802 42 >9 209222 12-82 994233 ■35 214989 i3 17 785ou 41 30 209992 12-80 9942 i? ■ 35 215780 i3 i5 784220 40 21 9*210760 12-78 9-994191 •35 9-216568 i3 12 10-783432 31 22 211326 12-75 994'7i •35 217356 i3 10 782644 23 212291 12-73 994 1 5o •35 218142 i3 08 78i858 37 34 213033 12-71 994129 994108 •35 218926 i3 o5 781074 36 25 2i38i8 12-68 •35 219710 i3 o3 780290 35 26 214379 215338 12-66 994087 •35 220492 i3 01 779508 34 3 12-64 994066 •35 221272 12 99 778728 33 216097 216834 12-61 994045 •35 222052 12 97 777948 32 5° 12-59 994024 ■35 222830 12 94 777I7 3i 3o 217609 12-57 " 994003 ' •35 2236o6 12 92 776394 3o 3i 9-218363 12-55 9.993981 ■ 35 9-224382 12 90 10-775618 3 32 219116 12-53 993960 • 35 223156 12 88 774844 33 219868 12 -5o 9 9 3 9 3o 993918 993896 993875 ■ 35 223929 12 86 774071 ll 34 220618 12-48 •35 2267OO 12 84 7733oo 35 221367 12-46 • 36 227471 228239 12 81 772529 25 36 222Il5 12-44 • 36 12 79 771761 24 ll 222861 12-42 993854 • 36 229OO7 229773 12 77 770993 23 2236o6 l2-3g 9 9 3832 ■36 12 V 770227 22 3 9 224349 l2-3 7 993811 •36 23o539 12 73 7694&1 21 4o 2230Q2 9-225833 12-35 993789 ■36 23l302 12 7i 768698 1 20 10-767935! 10 7671741 18 41 12-33 9-993768 ■36 9- 232o65 12 69 42 226573 I2-3I 993746 •36 232826 12 67 43 227311 228048 12-28 99372D ■36 233586 12 65 766414 17 44 12-26 993703 ■ 36 234345 12 62 765655 16 45 228784 12-24 993681 •36 235 io3 12 60 764897 i5 46 229518 12-22 993660 ■36 235859 12 58 764141 14 % 230252 1 1 2 • 20 993638 ■36 2366 1 4 12 56 763386 i3 230984 12-18 993616 •36 237368 12 54 762632 12 4 9 231714 12-16 993594 •'37 238120 12 52 761880 11 5o 232444 1214 993572 •37 238872 12 5o 761 1 28 10 5i 9-233172 12-12 9-99355o •37 9-239622 12 48 10-760378 8 5j 233895 1209 993528 •37 24037 1 12 46 759629 758882 52 234623 1207 9935o6 •37 241 1 18 12 44 I 54 235349 236073 12-05 993484 •37 241865 12 42 738i35 55 I2-o3 993462 •37 242610 12 40 757390 5 56 236795 12-01 993440 1< 243354 12 38 756646 4 u 237315 n-99 99341 s •37 244097 244839 12 36 755903 3 238235 11-97 993396 •37 12 34 755i6i 2 5o j38o53 239670 11 -95 . 993374 •^ 245579 12 32 754421 1 6o 11-93 99335i •37 246319 12 3o 753681 1 Conine D. Sine S0° Cotang. I>. ' Tana-. M.. 12 28 (10 DEGREES.) A rABLB OF LOGARITHMIC M. Sine D. Cosine D. Tang. D. Cotang. u 9> 239670 n- 9 3 9*99335i ■3 7 9-246319 12- 3o io-75368i 60 I 240386 ::$ 993329 •3 7 247057 12-28 752943 59 a 241101 993307 -3 7 247794 248530 12-26 •J522o6j 58 3 241814 11. 87 99 3285 •37 12-24 75l4 7 0, 5l 75o 7 36l 56 4 242526 n-85 993262 •37 249264 12-22 5 243237 n-83 993240 :ll 249998 2507J0 12- 20 75ooo2 55 6 243947 11. 81 993217 I2-I8 749270 748539 54 J 244656 n-79 993195 ■38 25i46i 12-17 I2-|5 53 245363 n-77 993172 ■ 38 252191 747809 52 9 246069 246773 11-75 993149 •38 252920 12-13 747080! 51 7463521 5o 10 M- 7 3 993127 • 38 253648 12-11 II Q- 247478 ' 24Sl8l 11-71 9.993104 •38 9-254374 12 09 10-7456261 49 13 11-69 993081 • 38 255ioo 12 • 07 12 o5 744900 48 i3 248883 11 -67 993059 •38 255824 744176 743453 47 14 249583 n-65 993o36 ■38 256547 12-03 46 i5 25o2fi2 n-63 9930 1 3 •38 257269 12-01 742731 45 16 250980 u-6i 992990 •38 257990 258710 12-00 742010 44 \l 251677 252373 li.5o 992967 •38 I..98 741290 43 n-58 992044 ■38 259429 II- 9 6 74o57i 42 >9 253067 n-56 992921 ■38 260146 n-94 739854 41 20 233761 n-54 992898 ■ 38 260863 11-92 739137 40 21 9-254453 11-52 9-992875 992852 ■38 9-261578 11 -Q0 10-738422 It 22 255144 n-5o • 38 262292 11-89 737708 23 255834 J 1-48 992829 .39 263oo5 II.Sl n-85 736995 37 24 256523 n-46 992806 ■ 3 9 263717 264428 736283 36 25 25721 1 11 -44 992783 .39 n-83 735572 35 26 257898 258583 11.42 992759 -3 9 2 65i38 H-8l 734862 34 11 u-4i 992736 .3, 265847 266555 :i':a 734153 33 209268 li-3 9 992713 992690 •39 733445 32 ?9 259951 n-3 7 -3 9 267261 n- 7 6 732739 732033 3i 3o 26o633 11-35 992666 -3 9 267967 n-74 3o 3i 9-261314 n-33 9-992643 19 9-268671 11-72 10.731329 730626 3 32 261994 n-3i 992619 -3 9 269375 11-70 33 262673 n-3o 992596 ■39 270077 11-69 729923 V 34 263351 11-28 992572 ■39 270779 11-67 729221 728521 26 35 264027 26470J 265377 11-26 992549 992523 ■39 27U79 1 1-65 25 36 11-24 .39 272178 n-64 727822 24 ii 11-22 992501 • 3 9 272876 11-62 727124 23 266o5i 11-2(1 992478 .40 273573 n-6o 726427 22 3 9 266p3 267J95 II-I 9 992454 .40 274269 n-58 7 25 7 3i 21 40 ;:::? 992430 .40 274964 ii-5 7 725o36 20 41 9-268065 9.992406 • 40 9-275658 n-55 10-724342 19 42 268734 n-i3 992382 ■40 27635i n-53 723649 18 43 269402 11-11 992359 ■40 2--7043 n-5i 722957 ■7 41 270060 II-IO 99233D •40 277734 n-5o 722266 16 45 270730 11.08 9923i 1 •40 278424 n-48 721576 720887 i5 46 271400 Ii-c6 992287 992263 ■40 2791 i3 1 1 -47 14 % 272064 Ii-o5 ■ 40 279801 . n-45 720199 i3 272726 n-o3 . 992239 .40 280488 1 1-43 719512 12 i 9 273388 II-OI 992214 •40 281174 11-41 718826 11 00 274040 9 274708 275367 10.99 992190 .40 28i858 II-40 718142 10 5i 10-98 9-992166 • 40 9-282542 n-38 10 717458 I 52 10-96 992142 .40 283225 n-36 716775 53 276024 10-94 992117 •41 283907 n-35 7 1 6093 1 54 276681 10-92 992093 •41 o84588 n-33 715412 6 55 277337 io-gi 10-89 992069 ■41 285268 n-3i 714732 5 56 277991 992044 •41 285947 n-3o 714053 4 278644 10-87 992020 ■41 286624 11-28 713376 3 279297 279948 280099 Coeir.s io-86 991996 ■41 287301 II-26 71269c) i 10-84 991971 •41 288652 11-25 7 j 202 j 1 60 1082 ri>. 991947 Si no •41 11-23 711348 79o Ootung. D. -^"'(fc- .HI SINES AND TANGENTS. (11 DEGREES. ) '29 [Jk Sine 9 ■ 280599 281248 D. Cosine D. Tang. D. Cotang. 1 o 10-82 9-991947 •4i 9-288652 11-23 10-71134? 60 1 io- 81 991922 •41 289326 11-22 710674 58 1 2 281897 10-79 991897 991873 •41 289999 11-20 710001 3 282544 io-77 10-76 •41 290671 II-18 709329 7o8658 57 4 283190 283836 991848 •41 291342 .1-17 II-IO 56 5 10-74 991823 ■4i 292013 707987 707318 55 6 284480 10-72 991799 •41 292682 11-14 54 I 285124 10-71 991774 ■42 293350 n-12 7o665o 53 285766 10-69 991749 •42 294017 11 -11 7o5n83 7o53 1 6 52 9 286408 10-67 991724 •42 294684 11-09 5i 10 287048 10-66 991699 •42 295349 9-296013 11-07 70465 1 5o II 9-287687 10-64 9-991674 •42 II-o6 10-703 98- 703323 it 12 2S8326 io-63 991649 •42 296677 1 1 -04 i3 288964 10-61 991624 •42 297339 11 -o3 702661 47 U 289600 io-5 9 991599 •42 298001 11-01 701999 70i33t 46 i5 290236 10-58 991574 •42 298662 11-00 45 i& 290870 10-56 991549 •42 299322 10-98 700676 44 12 291504 10-54 99i524 •42 299980 10-96 700020 699362 43 292137 292768 10-53 99U98 •42 3oo638 10-95 42 19. io-5i 991473 •42 3oi2o5 301901 10-93 698705 41 20 293399 io-5o 99U48 ■42 10-92 69804c 10-697393 696739 40 21 9-294029 10-48 9-991422 ■42 9-302607 10-90 3 9 22 294658 10-46 991397 •42 3o326i 10-89 38 23 295286 10-45 991372 •43 3o3oi4 304067 10-87 696086 37 24 295913 10-43 991346 •43 10-86 695433 36 25 296539 10-42 991321 •43 3o52t8 10-84 694782 35 26 297164 10-40 991295 •43 3o586g 10-83 694131 34 27 297788 10-39 991270 •43 3o65io 307168 io-8i 693481 33 28 298412 10-37 991244 ■43 10-80 692832 '32 59 299034 10-36 991218 ■43 307815 10-78 692185 3i 3o 299655 10-34 991 193 ■43 3o8463 10-77 691537 3o 3i 9-300276 10-32 9-991167 •43 9-309109 10-75 10-690891 29 32 300895 io-3i 99i i4i ■43 309754 10-74 690246 28 33 3oi5i4 1029 991 1 i5 •43 310398 10-73 689602 27 34 3o2l32 10-28 991090 •43 3 1 10-12 10-71 688908 26 35 302748 1026 991064 •43 3u685 10-70 6883 1 5 25 36 3o3364 10-25 99io38 •43 3i2327 10.68 687673 24 ll 3o3o7Q 10-23 991012 •43 312967 10-67 68 7 o33 23 304093 10-22 990986 •43 3i36o8 10-65 686392 22 3 9 3o5207 10-20 990960 ■ 43 3i4247 10-64 685 7 53 21 40 3o58i9 10-19 990934 •44 314885 10-62 685n5 20 41 9-3o643o 10-17 9-990908 990882 •44 9-3i5523 10-61 10-684477 ;g 42 307041 10-16 ■44 3i6i5o 316790 3i743o io-6o 683841 43 3o765o 10-14 990855 •44 10-58 683205 \i 44 308259 io-i3 990829 990803 •44 10-57 682570 45 308867 10-11 •44 3i8o64 10-55 681936 i5 46 309474 10-10 990777 •44 318697 10-54 68i3o3 14 3 3 1 0080 10-08 990750 •44 319329 10-53 680671 i3 3 io685 10-07 990724 •44 319961 320092 io-5i 68oo3o 679408 12 49 311289 311893 io-o5 990697 ■44 io-5o 11 00 10-04 990671 •44 321222 10-48 678778 10 , 5i 9-312495 Io-o3 9-990644 ■44 9-32i85i 10-47 1.0-678149 3 52 '3 1 3097 10-01 990618 •44 322479 10-45 677521 53 313698 10-00 990591 •44 323 1 06 10-44 676894 7 54 314297 9-98 99o565 •44 323733 10-43 676267 6 55 314897 9.97 99o538 •44 324358 10-41 675642 5 56 3i5495 9-9° 99031 1 •45 324983 io-4'o 675017 4 E3 3i6oo2 9-94 990485 • 45 325607 10-39 674393 3 316689 993 990458 • 45 32623i 10-37 673769 2 h 317284 9-91 99043 1 • 45 326853 10-36 673147 I 60 317879 9.90 990404 • 4 5 327475 10-35 6725a5 O 1 . _ Cosine D. Sine T8° Cotang. 1 D. Tan*. M.I 30 (12 DEGREES.) A TABLE OP LOGARITHMIC M. | Sine D. | Cosine [ D. Tang. D. 1 Cotang.l 9-317879 9.00 9-990404! -45 9-327474 10-35 10-672526' 60 i 3i8473 9.88 990378J -45 328095 10-33 671905 5o 671285 58 2 319066 9-87 99o35i .45 328713 10-32 3 319658 9-86 990324 -45 329334 io-3o 67066C 57 4 320249 9-84 990207I -45 32995. 10-29 67004- 56 5 320840 9-83 99027c ) -45 33o5-70 10 28 66943c 55 6 32i43o 9-82 9902 c ■45 33u8- 10-26 6688i3 54 I 322019 9.80 99021: •4! 33 180; 10-25 668197 66 7 583 53 322607 9-79 99018? •4! 3324iE 10-24 52 9 323ig4 9-77 990161 ■At 333o3C 10-23 666967 66635/s 5i 10 323780 9-76 990134 ■4i 33364C 10-21 5o ■ i 9-324366 9- 7 5 9-99010- ■46! 9-334255 10-20 10-665741 49 12 32495o 9-73 990079 .46 334871 10-19 665 129 6645i8 48 [3 325534 9-72 99oo5: .46 335482 10-17 47 • 4 326117 9-70 99002s •46 336093 10-16 66390'; 46 i5 326700 9.69 989997 •46 336702 io-i5 663 208 662689 45 16 327281 9.66 989970 •46 3373 1 1 io- 13 44 \l 327862 9-66 989942 •46 337919 3385a-- 10-12 662081 43 328442 9-65 989915 .46 10-11 661473 42 «9 32902! 9.64 989887 •46 31gi33 10-10 660867 41 20 329599 9-62 989860 ■46 339739 10-08 660261 40 21 9-330176 9.61 9-989832 • 46 9 -340344 10-07 10-659656 3 9 22 33oi53 9. 60 989804 .46 340948 34i552 io-o6 65oo52 658448 38 23 33i329 9-58 989777 • 46 io-o4 37 24 33i9o3 9-57 989749 •47 342i55 10 -o3 65 7 845 36 25 332478 9-56 989721 •47 342757 10-02 657243 35 26 333o5i 9-54 989693 •47 343358 10-00 656642 34 11 333624 9-53 989665 •47 . 343o58 344558 9-99 9-98 656o42 33 334195 9-52 989637 •47 655442 32 ?9 334766 9-5o 989609 •47 345 1 5i 345755 9-97 654843 3i 3o 335337 9-49 989582 •47 9.96 654245 3o '3i 9.335906 9-48 9.989553 •47 9-346353 9-94 10-653647 29 32 336475 9.46 989525 ■47 346949 9-93 653o5i 28 33 337043 9.45 989497 •47 347545 9-92 652455 27 34 33i6io 338i 7 6 9.44 989469 •47 34S141 99' 65i85q 26 35 9.43 989441 •47 •47 348735 fts 651265 25 36 338742 9-41 989413 349329 650671 24 ll 339306 9.40 989384 •47 \ 349922 9-87 650078 23 339871 9.39 989356 •47 35o5i4 9-86 649486 22 3 9 340434 9-37 989328 •47 35uo6 9-85 648894 21 40 340906 9.341558 9-36 989300 •47 351697 9-83 6483o3 20 41 9-35 9-989271 •47 9-352287 9-82 10-647713 ;g 42 3421 19 9.34 989243 •47 352876 9-81 647124 43 342679 9-32 9S9214 •47 353465 9-80 646535 \i 44 343239 9>3i 989186 •47 354003 9-79 645947 64536o 45 343797 344355 9-3o 989157 989128 ■47 354640 9-77 9-76 i5 46 929 ■48 355227 3558i3 644773 14 % 344912 9-27 989100 ■48 9-75 644187 i3 ! 345469 9-26 989071 •48 3563o8 9-74 643602 12 i 9 346024 9.25 989042 ■48 356982 357566 9-358149 9-73 643oi8 1 1 5o 346579 9-24 989014 •48 9-71 642434 10 5i ?-347i34. 9.22 9.988985 9 88 9 56 -48 9-70 o-64i85i I 52 347687 348240 9-21 •48 358731 t% 641 269 • 53 9 -20 988927 •48 3593i3 640687 7 54 348792 9.19 988898 • 48 359893 9.67 640107 S 55 349343 9-17 988869 •48 360474 9.66 63o526 5 56 349893 ()■ 16 988840 •48 36io53 9-65 638947 4 ll 35o443 9.1s 988811 •49 36i632 9-63 638368 3 350992 35 1540 9-U 988782 •49 362210 9-62 637790 a I 9 9. i3 988753 •49 362787 9-6. 6372i3 1 60 352088 9-u 988724 •49 363364 9-60 636636 - Cosino D. Sine 7T° Cotunpt. 1 r>. Tang. 1 Ml. SINES AND TANGENTS. (13 DEGREES. ) 31 M. Sine D. Cosine t>. Tang. D. Cotang. 3.352088 9-1 1 9-988724 •49 9-363364 9-60 lo- 636636 60 l 352635 9-10 988695 •49 363940 9 5 9 63 6060 5 9 3 353i8i 9.09 9-08 988666 .49 3643 1 5 9 58 635485 58 3 353726 9 88636 •49 365090 9 57 634910 634336 57 4 354271 9.01 9-o5 988607 •49 365664 9 55 56 5 3548i5 988578 •49 366237 9 54 633763 55 6 355358 9.04 988548 .49 3668io 9 53 633mo 54 I 355901 9-o3 988519 .49 367382 9 52 632618 53 356443 9-02 988489! •49 36 79 53 368524 9 5i 632047 52 9 356o84 357624 9-01 §.09 988460 •49 9 5o 631476 5i 10 988430 •49 369094 9 a 630906 5o II 9 . 358o64 8-98 9-988401 ■49 9*369663 9 io-63o337 2 12 3586o3 8-97 988371 •49 370232 9 40 629768 13 359141 8-96 988342 •49 370799 9 45 629201 47 14 359678 8- 9 5 988312 • 50 371367 9 44 628633 46 i5 36o2i5 8- 9 3 9R8282 ■ 50 371933 9 43 628067 45 16 360752 8-92 9R8252 .50 372499 9 42 627301 44 \l 361287 8-91 988223 ■50 373o64 9 41 626936 626J71 43 361822 8-90 8-89 8-83 988193 •50 373629 374193 9 40 42 >9 362356 9881 63 •So 9 n 623807 41 20 362889 9X8133I ■5o 374736 9 625244 40 21 9-363422 8-87 9-988103; •50 9-373319 9 n 10-624681 ll 22 363g54 8-85 988073! •So 373881 9 6241 10 623558 23 364485 8-84 988043 ' -50 376442 9 34 37 24 365oi6 8-83 988013 1 ■5o 377003 9 33 622997 622437 36 25 365546 8-82 987983 ■50 37]563 378122 9 32 35 26 366075 8-8i 987953 ■5o 9 3i 621878 34 11 3666o4 8-8o 987022 987892 •50 378681 9 3o 621319 33 367i3i 8-79 •50 379239 9 3 620761 32 2 9 367659 368 1 85 8-77 987862 ■5o 379797 38o354 9 620203 3i 3o 8-76 987832 • Ji 9 27 619646 3o 3i 9-368711 8-75 9-987801 • 5i 9-380910 9 26 10-619090 6i8534 29 32 36g236 8-74 987771 ■ 51 ■ 38i466 9 25 28 33 369761 8-73 987740 •5'. 382020 9 24 617980 27 34 370285 8.72 9S7710 •51 382575 9 23 617425 26 35 370808 8-71 987679 •51 383i2 9 9 22 616871 25 36 37i33o 8-70 987649! 987618 •51 383682 9 21 6i63i8 24 ll 371852 8-69 •5i 384234 9 20 615766 23 372373 8-67 987588 •5i 384786 9 13 6i52i4 22 3 9 372894 8-66 987557 ■51 ' 385337 385888 9 614663 21 40 373414 8-65 987526 •51 9 17 614112 20 41 9*373933 8-64 9-987496 •5i 9-386438 9 i5 io-6i3562 19 42 374452 8-63 987465; 987434! •5i 386987 387536 388084 9 U 6i3oi3 l8 43 374970 375487 376003 8-62 • 51 9 i3 612464 17 44 8-6i 987403J • 52 9 12 611916 61 1369 16 45 8-6o 987372 •52 388631 9 11 15 46 376519 377035 8-5 9 98-1341 •52 389178 9 10 610822 14 47 8-58 987310 •52 389724 9 3 610276 i3 48 377549 378063 8-57 987279! 9872481 •52 390270 9 609730 12 49 8-56 -52 390810 9 07 6oni85 11 5o 378577 8-54 987217 •52 391360 9 06 600640 IC 5i 9-379089 8-53 9-987186! ■52 9-391903 9 o5 10.608097 607563 3 52 379601! 8-52 987155, •52 392447 9 04 53 38on3| 8-5i 987124 •52 392589 9 o3 607011 7 54 38o624| 8-5o 987092: •52 3g3a3i 9 02 606469 6 55 38n34l 8-49 38i643! 8-48 987061] 987030 •52 394073 9 01 605927 6o5386 5 56 •52 394614 I 00 4 u 382132 8-47 986998 •52 395154 38 604846 3 382661 8-46 986967, -52 395694 396233 8 6o43o6 2 59 383 1 68; 8-45 986936 •52 8 97 603767 1 J>0 382670 Coaiue 8-44 986904! •52 396771 8-96 6o3:'.'9 L>. Sine IT6° Cotang. D. Tang . Vu 32 Q14 DEGREES.) A 1ABLE OF LOGARITHMIC M. Bine <>• 3836-5 D. Cosine D. Tail};. D. Cotnng. "So 8-44 9-986904 986873 •52 9.396771 397309 8- 9 6 10-603229 i 384182 S-43 •53 8-96 602601 602134 It 2 384687 8-42 986841 •53 397846 8. 9 5 3 385192 8-41 986809 986778 •53 398383 g'94 601617 n 4 385697 8-40 •53 398919 399455 8-93 601081 5 386201 8-3 9 8-38 8-37 986746 • 53 8-92 6oo545 55 6 I 386704 387207 986714 9 86683 •53 ■53 399990 4oo524 8-91 8-90 8-89 600010 599476 598942 54 53 387709 388210 8-36 986651 •53 4oio5£ 53 9 8-35 986619 •53 401691 8-88 598409 5i 10 388 7 i 1 8-34 986587 •53 402124 8-87 8-S6 697876 5o ii 9-389211 8-33 9-986555 •53 9-402656 10-591344 3 12 38971 1 8-32 986523 •53 4o3i87 8-85 5 9 68 1 3 i3 390210 8-3i 986491 986459 •53 403718 8-S4 596282 % U 390708 8-3o •53 404249 404770 8-G3 595751 i5 391206 8-28 86427 98639S 9 86363 •53 8-S2 595222 45 16 391703 8-27 8-26 •53 4o53o8 8-8i 594692 44 \l 392199 392690 •54 405836 8- 80 594164 43 8-25 9 S633i ■54 4o6364 H' 7 2 5 9 3636 42 •9 393191 3 9 3685 8-24 986299 986266 •54 406892 M 593108 41 20 8-23 -54 407419 8:?Z .592581 40 21 9-394170 39467J 8-22 9-986234 •54 9-407943 io-592o55 ll 22 8-21 986202 •54 40847 1 8-75 591529 23 395i66 8-20 986169 ■54 40S997 409321 8-74 591003 ll 24 3 9 5658 8- 10 818 986137 •54 8-74 590479 589953 25 396150 986104 -54 410045 8-73 35 26 396641 8-17 986072 •54 • 410369 8.72 68943 1 34 n 397132 8-17 8-16 986039 •54 41 1092 8-71 688908 588385 33 397621 3981 1 1 986007 •54 4i(6i5 8-70 32 29 8i5 985974 •54 4l2l37 412658 8-69 8-68 58 7 863 3i 3o 398600 8-U 985942 ► 54 587342 3o 3l 9-399088 8-i3 9-985909 • 55 9-413179 8-67 ' 8-66 10. 586821 ll 32 3995^5 400062 8-12 985876 • 55 413699 5863oi 33 811 985843 • 55 414219 41473S 8-65 585 7 8i ll 34 400549 4oio33 8-io 9 858u • 55 8-64 585262 35 8-09 8-o8 985778 • 55 4i5257_ 8-64 584743 25 36 401 520 985745 • 55 4i5773 8-63 584225 24 u 4o2oo5 8-07 8-o6 985712 • 55 416293 8-62 583707 23 402489 985679 9 85646 •55 416S10 8-6i 583190 22 39 402972 8-o5 • 55 41^326 8>6o 582674 21 40 4o3455 • 8-o4 9856i3 • 55 417842 8-39 582158 20 41 9 '4o3938 8-o3 9 - 9 8558o • 55 9-418358 8.58 10-581642 10 42 404420 8-02 985547 • 55 418873 8-57 581127 IB 43 404901 4o5382 8-oi 9855i4 • 55 419387 8-56 58o6i3 lo 44 8-oo 985480 • 55 419901 8-55 580009 57 9 585 45 4o5862 7-99 935447 • 55 42o4i5 8-55 i5 46 4o634i 7-98 985414 • 56 420927 8-54 579073 578560 ■4 2 406820 7-97 9 8538o • 56 421440 8-53 l3 407299 7-96 985347 • 56 421952 8-52 578048 12 49 407777 408254 7-93 985314 • 56 422463 8-5i 577537 11 5o 7-94 985280 ■56 422974 8-5o 677026 10 5i 9 4o873"i 7-94 9-985247 •56 9-423484 8-4o 10.576316 8 52 409207 7- 9 3 985213 • 56 423993 424303 8.48 576007 53 409682 7-92 985180 •56 8 48 575497 *i 54 410157 f- 9 i 985146 •56 42101 T .8-47 8-46 574989 6 55 4io632 7-00 985i i3 •56 4255l9 574481 5 56 41 1 106 985079 983045 •56 426027 8-45 573973 4 573466 3 n 41 1 579 412052 •56 426534 8 44 7.87 985011 • 56 427041 8-43 572950 2 57 2453 1 1 59 412524 7.86 984978 ■56 42«47 428032 8-43 60 412996 7-85 984944 56 8.42 571948 ?nsiue J). Sine T5°| CotanR. I>. t«S£tI m7, SINES AND TANRENTS. (15 DEGREES. ) 3? to. Sine D. Cosine | D. Taug. D. Cotang 1 10-5719481 60 o 9-41299° 7 85 9- 984944 •:>7 9-428o32 8-42 I 413467 7 84 984910 ■57 428557 841 571443I 59 57093s! 58 2 413938 7 83 984876 1 •37 429065 8-40 3 414408 .7 83 984842 ■37 429366 8-3 9 8-38 570434I 5 7 4 414878 7 82 984808: ■37 430070 569930! 56 5 415347 7 81 984774] ■37 43o373 8-38 569427 < 55 568923 54 6 4i58i5 7 80 984740 •57 43l075 8-37 I 4i62&3 7 $ 084706 •57 43 i 577 8-36 568423 53 416731 7 984672 •57 43207c, 8-35 567921 52 9 417217 7 77 984637 •57 43258o 8-34 567420 5i to 417684 0-4i8i5o 7 76 984603 •37 433o8o 8-33 56692a 5o ii 7 75 9-984369 •37 9-43358o 8-32 lo- 566420 49 48 12 4i86i5 7 74 984535 •57 434o8o 8-32 565920 i3 419079 7 73 9845001 ■57 434579 435o78 8-3i 565421 47 14 419344 7 73 984466 :U 8-3o 564922 46 13 420007 7 72 084432 433376 8-29 564424 45 10 420470 7 7' 984397 •58 436073 8-28 56392-; 44 \l 42oo33 42i3cp 7 I q84363 -58 436570 8-28 56343c 43 7 984328 • 58 43706] 437563 8-27 562933 42 ■9 421837 422318 7 984294 984239 ■53 8-26 562437 41 20 7 67 •58 43 8059 8-25 561941 40 2! 9-422778 7 67 9-984224 ■58 9-438554 824 IO- 561446 3 37 22 23 423238 4:^697 7 7 66 65 984190 984135 •58 •58 43904S 439543 8-23 8-23 560952 56o45'] 24 424136 7 64 984120 •58 44oo36 8-22 559964 36 23 4246i5 7 63 ' 984083 •58 440329 8-21 559471 558978 558486 35 26 425o73 7 62 984060 •58 441022 8-20 34 11 2 9 42553o 7 61 984015 ■58 44i5i4 8- 19 33 422987 426443 7 7 60 60 983981 983946 •58 442006 •58| 442407 8- 19 8- 18 557994 5575o3 32 3i 3o 426899 9-427354 7 39 983911 •58j 442988 8-i 7 557012 3o 3i 7 58- 9-983075 •?8| g-443479 8-16 io- 556521 2 9 32 427809 42826J 7 57 983840 .39 443968 8-16 556o32 28 33 7 56 9 83 8o5 -5 9 444438 8-i5 555542 2 7 34 428717 7 55 983770 i 9 444947 8- 14 555o53 26 35 429170 7 54 983735 09 445435 8-i3 554565 25 36 429623 7 53 980700 .39 445923 8-12 554077 553589 24 ll ' 430075 7 32 983664 •3 9 44641 1 8-12 23 43o52T 430978 7 52 983629 ■3 9 446898 8-11 553 102 22 39 7 5l 983594 o83558 ■39 447384 8-10 5526i6 21 40 431429 7 5o ■°9> 447S70 8-09 552 i3o 20 41 9-431879 7 49 9-983523 ■39 9-448356 809 io-55i644 19 42 432329 432778 7 49 983487 •39| 448841 808 561169 18 43 7 48 983432 ■59 449326 8-07 55o674| 17 44 433226 7 47 983416 -39 449810 806 55oiqo, 16 45 433675 7 46 g8338i •39 450294 806 549706 ! l5 46 434122 7 45 983345 ■09 430777 8-o5 549223: 14 47 434569 7 44 • 983309 .39 451260 804 548740 1 3 48 435oi6 7 44 983273 ■60 45i743 8-o3 548207! 12 49 435462 7 43 9832J8 • 60 452225 8-02 547773! 11 5c "435908 7 42 983202 •60 452706 802 5472941 10 5i 9-4363 53 7 41 9-983166 •60; 9.453187 •60 453668 801 io-5468i3i 9 54oo3:! 8 52 436798 7 40 03 1 Jo 8-oo 53 437242 7 40 .983094 •6o| 454148 7-99 545852] 7 54 437686 7 3 9 o63o58 •6o| 454628 345372 6 55 438129 7 38 983022 60 433107 7-98 544893I 5 56 438372 7 37 9629.S6 60 455586 7-97 544414! 4 57 439014 7 36 902960! -6o, 456064 7-96 543936, 3 56 439456 7 36 982914 •00, 456542 7-96 543458 2 5? 439897 7 35 902.-178 •°°, 437019 7- 9 5 542981 542604 - I 60 441)338 7-34 982S42 •60 457496 7-94 D. *• 1— Cottiue D. biuc r 4 o Co taug. 34 (16 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine 1). Tan?. D. Cotang. o 9-44o338 7-34 9-982842 • 60 9-457496 7 94 lo-5425o4 60 I 440778 7 33 982305 • 60 457973 7 9 3 542027 51 2 441218 7 32 982769 •61 458449 7 9 3 54i55i 3 44i658 7 3i 9R2733 • 61 458920 7 92 541075 57 4 442096 7 3i 982696 •61 459400 7 91 540600 56 5 442535 7 3o 982660 •61 459875 7 90 540125 55 6 442973 7 29 982624 •61 460349 7 90 53965i 54 I 443410 7 28 982587 ■61 460823 7 it 530177 538703 53 443847 7 27 982551 ■61 461297 7 52 9 444284 7 27 982514 •61 461770 7 88 53823o 5i 10 444720 7 26 982477 •61 462242 7 52 537758 5o n 9-445i55 7 25 0-982441 •61 9-462714 7 86 10-537286 % 12 445590 7 24 982404 •61 - 463 1 86 7 85 5368i4 13 446028 7 23 982367 •61 463658 7 85 536342 47 U 44645o 446893 7 23 982331 •61 464129 7 84 5358 7 i 46 i5 7 22 982294 982207 •61 464599 7 83 535401 45 •i6 447326 7 21 •61 465069 .7 83 534931 44 \l 447759 448191 7 20 982220 • 62 46553g 7 82 53446i 43 7 20 982183 •62 466008 7 81 533992 533524 42 >9 448623 7 ]o 982146 •62 466476 7 80 41 20 449004 ■ 7 10 982109 •62 466945 7 80 533o55 40 21 9-449485 7 17 9-982072 •62 9-467413 7 3 10-532387 M 22 4499>5 7 16 982035 •62 467880 7 532120 23 45o345 7 16 . 981998 •62 468347 7 78 53 1 653 37 24 450775 7 i5 981961 •62 468814 7 77 53n86 36 25 451204 7 14 981924 981886 •62 469280 7 76 530720 35 26 45i632 7 i3 ■62 469746 7 ^ 530254 34 *1 452o6o 7 i3 981849 • 62 47021 1 7 75 529789 33 28 452488 7 12 981812 •62 470676 7 74 529324 32 ?9 452915 7 u 981774 •62 471 141 7 73 52885 9 5283 9 5 3l 3o 453342 7 10 981737 •62 471606 7 73 3o 3i 9-453768 7 10 9-981699 ■63 9-472068 - 7 72 10-527932 3 32 454194 7 3 981662 ■63 472532 7 71 527468 33 454619 7 981625 ■63 472995 473407 7 71 527005 27 34 455o44 7 07 9 8i58 7 ■63 7 70 526543 26 35 45546o 4558g3 7 07 981549 ■63 473919 474J81 7 69 526081 25 36 7 06 981512 ■63 7 69 5256iQ 24 ill 4563 1 6 7 o5 981474 •63 474842 7 68 525i58 23 456739 7 04 981436 ■63 4753o3 7 67 524697 22 ) 39 457162 7 04 981399 •63 475763 7 "7 524237 21 40 457584 7 o3 981361 ■63 476223 7 66 523777 20 4i 9- 458oo6 7 02 9-981323 •63 9-476683 7 65 10.523317 ■9 42 458427 7 01 981285 •63 477142 7 65 522858 i a 43 458848 7 01 981247 ■63 477601 478059 7 64 522399 n 44 459268 7 00 981209 •63 7 63 521941 16 45 459688 6 2 981171 ■63 478517 7 63 521483 i5 46 460108 6 98n33 •64 478975 7 62 521025 14 % 460527 6 98 981095 981037 ■64 479432 7 61 52o568 i3 460946 6 97 • 64 479889 48o345 7 61 5201 11 12 49 461 364 6 96 981019 •64 7 60 5 i 9 655 11 5o 461782 6 95 980981 •64 480801 7 I 9 519199 io-5i8743 10 5i 9-462199 6 9 5 9-980942 •64 9-481257 7 59 9 52 462616 6 94 980904 •64 481712 7 58 518288 8 53 463o32 6 9 3 980866 •64 482167 7 F 517833 7 54 463448 6 9 3 980827 ■ 64 482621 7 57 5i 7 3 7? 51692a 6 55 463864 6 92 980789 ■64 483075 7 56 5 56 464279 6 91 980750 •64 483529 7 55 516471 A tl 464694 6 90 9807 1 2 •64 483982 7 55 5i6oi8 3 465 1 08 6 % 980673 ■64 484435 7 54 5 1 5565 2 5 9 465522 6 980635 •64 484887 7 53 5i5u3 1 6o 46O935 6 88 980596 ■64 485339 7 53 514661 Tang. O ' Cosine D. Sine ra° Cotung. D. -*J SINES AND TANGENTS. (17 DROKEES. ) 33 IE Sine D. Cosine D. Tung. r». Oot&ng. 9-465o35 466348 6-88 9-980596 980538 64 9-485339 7 55 Io-5i466l 6o~ 1 6 88 64 485791 7 5j 514209 5i3758 51 2 466761 6 sz 980519 65 486242 7 5i 3 467173 467585 6 ■ 980480 65 486693 7 5i 5 1 3307 s 4 6 85 980442 65 487143 7 30 512-857 56 5 467996 '6 85 980403 65 487593 488043 7 49 512407 55 6 468407 6 84 980364 65 7 49 511957 5ii5o8 34 I 468817 6 83 980325 65 488492 7 48 53 469227 6 83 980286 65 488941 7 47 5no59 52 9 10 469637 470046 6 6 82 81 980247 980208 65 65 489390 48 9 838 7 7 47 46 610610 5lOI02 5i 5o ii 9-470455 6 80 9-980169 65 9-490286 7 46 lo- 509714 3 12 470863 6 80 980130 65 490733 7 45 509267 i3 47I27I 6 3 980091 65 491 180 7 44 5o882o 47 U 471679 6 980032 65 491627 7 44 5o8373 46 13 472086 6 78 980012 65 492073 7 43 607927 •45 16 472492 6 77 979973 65 492619 492965 7 43 507481 44 \l 472898 6 76 979934 66 7 42 5o7o35 43 4733o4 6 76 979895 979855 66 493410 7 41 606590 42 '9 473710 6 75 66 493854 7 40 5o6u6 41 20 «74"5 6 74 979816 66 494299 9-494743 7 4° 5o570i 40 21 9-4745ig 474023 475327 6 74 9-979776 • 979737 66 7 40 io-5o5257 3 9 22 6 73 66 493186 7 39 5o48i4 38 23 6 72 979697 979658 66 49563o 7 38 504370 37 24 475730 6 72 66 496073 7 37 503927 36 23 476133 6 7' 979618 66 4965i5 7 37 5o3435 35 26 476536 6 70 979579 66 496967 7 36 5o3o43 34 2 7 476938 477340 6 69 979539 66 497399 7 36 5o26oi 33 .28 6 8 979^99 979459 66 497841 7 35 5o2 1 5o 501718 32 ?9 477741 478142 6 65 498282 7 34 3i 3o 6 67 979420 65 49S722 7 34 501278 3o 3i 9-478542 6 67 9 -9793 So 66 9-499163 7 33 io-5oo837 2 2 32 33 478942 479342 6 6 66 65 979340 979300 66 67 499603 600042 7 7 33 32 5oo3o7 499938 499319 28 27 34 . 479741 480140 6 65 979260 67 5oo48i 7 31 26 35 6 64 979220 67 500920 7 3i 499080 25 36 480539 6 63 979180 '■'I 5oi359 7 3o 498641 24 ii 480937 48i334 6 63 979140 67 501797 5o2235 7 3o 498203 23 6 62 979100 67 7 29 497766 22 3 9 48n3i 6 61 979059 67 502672 7 to 497328 21 4o 482128 6 61 970019 9-978979 67 5o3i09 7 28 4-;68oi 10-496454 20 41 9-482525 6 60 67 9 ■ 5o3546 7 27 10 42 482921 6 59 978939 67 603982 7 27 496018 10 43 4833 1 6 6 is 978898 978868 67 5o44i8 7 26 495582 17 44 483713 6 67 5o4854 7 = 5 495146 16 45 484107 6 57 978817 67 606289 7 25 494711 l5 46 4845oi 6 57 97^777 67 5o5724 7 24 494276 14 41 4848o5 6 56 978736 67 506159 7 24 493841 i3 48 485289 6 55 97S696 68 506593 7 23 493407 12 49 485682 6 55 978635 68 607027 7 22 492973 11 DO 486075 6 54 978615 68 507460 7 12 492340 10 5i 9-486467 6 53 9-978574 63 9-507893 5o8326 7 21 10-492107 8 52 486860 6 53 978533 68 7 21 491674 53 487251 6 52 978493 978432 68 508739 7 20 491241 7 54 487643 488o34 6 5i 68 509191 7 '9 49o3oo 490378 489946 6 55 6 5i 978411 68 509622 7 !« 5 56 488424 6 5o 978370 68 5ioo54 7 4 •32 488814 6 5o 9783=9 68 5 1 0485 , 7 18 48 9 5 1 5| 3 489204 6 % 978288 68 510916 7 n 489084 488654 2 5 9 489353 489982 6 978247 5n346 7 16 1 oo 6 48 978206 S.LlU 681 511776 7 16 488224 a CortineJ_ 0. 7 20 Coluiy. . 0. lung. M. 36 (1 3 DEGREES.) A TABLE OF LOGARITHMIC ~M." Sine D. Cosine 1). Tang. D. Cotaujf. 9-489982 49037 1 6-48 9-978206 .68 9-511776 7-16 10-488224 60 1 6 •48 978165 • 68 612206 7- lb 487794 3 i 490759 6 :S 978'24 -68 5i2635 7-16 487366 3 491 147 6 978083 .69 5 1 3064 7-U 486n36 486507 57 4 ■491535 6 •46 978042 .69 5 1 3493 7-14 56 5 491922 492J08 6 45 978001 .69 5i3o2i 5i4349 7-i3 486079 65 6 6 44 977959 .69 7-i3 485651 64 7 492695 6 44 977918 .69 514777 7-12 485223 53 8 493081 6 43 977835 .69 5 1 5204 7-12 484796 484369 483o43 io-4835i6 5a 9 493466 6 42 .69 5i563i 7-U 5i 10 ' 49385i 6 42 977794 9-977752 .69 516067 7-10 5o II 9*494236 6 4i •6, 9-516484 7-10 4 2 12 494621 6 41 9777" ■69 5i6oio 5i7335 7.09 483090 48 i3 495oo5 6 40 977669 •69 7-09 482665 47 U 495388 6 3 9 977628 .69 5i77 6 i 5i8i85 7-08 482239 481810 46 i5 495772 6 39 977586 •69 7-08 45 16 496154 6 38 977544 •70 5i86i 7-07 48i3oo 480966 44 IT 496537 6 37 9775o3 •70 519034 7-06 43 i8 496919 497301 6 ll 977461 •70 519468 7-06 480042 42 '9 6 977419 •70 519882 7-o5 4801 18 41 20 497682 6 36 977377 •70 52o3o5 7-o5 479695 io 21 9 ■ 498064 6 35 9-977335 -70 9-520728 7-04 10-479272 39 22 498444 6 34 977293 -70 52it6i 7-o3 418849 38 2j 498825 6 34 977261 •70 521673 7-o3 4;842 J 37 24 499204 6 33 977209 •70 521995 7-o3 478000 36 25 499584 6 32 977167 977125 •70 522417 522833 7-02 477583 35 26 499963 6 32 •70 7-02 477162 34 =7 5oo342 6 3i 977083 •70 52325g 7-01 476741 33 28 500721 6 3i 977041 •7° 52368o 7-ci 4-J6320 3» ?9 501099 6 3o 976999 976957 .70 524100 T-00 6.99 475900 3i 3o 601476 6 29 •70 624520 476480 3o 3i 9-5oi854 6 2I 9-976914 976872 97683o •70 9-524939 52535n 626778 6.99 10-476061 29 32 502231 6 ■71 6 98 474641 28 33 502607 6 28 ' "71 6- 9 3 474222 27 34 502984 6 27 976787 •T 626197 6-97 4738o3 26 3b 5o336o 6 26 976745 •71 526615 6-97 473385 25 36 5o3735 6 26 976702 •7 1 527033 6.96 472967 472649 24 u 5o4iio 6 25 976660 ■7" 627451 6.96 23 5o4485 6 25 976617 •71 527868 528285 6- 9 5 472132 22 3 9 504860 6 24 976674 •71 6-96 , 471715 21 4o 5o5234 6 23 976532 •71 628702 694 471298 20 41 9-5o56o8 6 23 9-976489 •7' 9-629119 629535 ^ 10-470881 \l 42 5o5o8i 5o6354 6 22 976446 •7 1 6- 9 3 470466 43 6 22 976404 •71 529950 53o366 6- 9 3 470o5o '7 44 506727 6 21 976361 •'/I 6.92 469634 16 45 507099 6 20 6 7 63i8 •7' 630781 6-91 469219 i5 46 O07471 6 20 976275 ■7' 531196 6-91 468804 14 a 507843 6 ] 9 976232 .72 53i6ii 6-90 46838a 467976 i3 5o82i4 6 \l 976189 •72 532025 6-90 12 49 5o8585 6 976146 •7 2 53243g 532853 6-09 467561 11 5o 5o8a56 9-509326 6 18 976103 ■72 6-89 467147 10 5i 6 17 9.976060 ■72 9-533266 6-88 10-466734 2 02 509696 6 16 976017 ■72 533679 6-88 466321 53 5ioo65 6 16 975974 •72 534092 6-87 465908 7 54 510434 6 i5 975o3o 975887 •72 534604 6-87 466496 465o84 6 55 ; 5 10803 6 15 •72 534916 535328 6-86 5 56 511172 6 U 975844 •72 6-86 464672 4 h 5u54o 6 >3 975800 1 -72 535 7 3g 6-85 464261 3 68 5i 1907 6 i3 975767 -72 536 i5o 6S5 46385o 2 $9 612275 6 12 975714 72 53656i 6-84 46343a 463028 1 bo 512642 6 12 975670 Sine ■72 536972 6-84 ..__ Cosine ) ). ( Cotang. 1). Tang. M. BINES AND TANGENTS. (19 DEGRESS. ) 3 o Sine 1 »■ Cosine D. Tang. D. Cotang. | 9-512642 6-12 9-975670 ■73 9-536972 6*84 10*463028 60 i 5i3ooc 1 6-U 9756=7 •73 537382 6*83 46261S 5o 58 2 5i3370 1 6-U 97 5583 ■73 537792 6*83 462206 3 613741 610 975539 ■73 538202 6*82 461798 461389 57 4 514107 6-09 975496 ■73 5386ii 6*82 56 5 5i4472 &.09 6-o8 97 54:>2 •73 539020 6*8i 460980 460071 55 6 5 1 4837 975408 ■73 539429 539837 6*8i 54 I 5l5202 6-o8 97 5365 ■73 6-8o 46016; 53 5i5566 6-07 975321 ■73 540245 6*8o 459755 52 9 613930 6-07 6*o6 975277 •73 540653 6-79 456347 5i 10 516294 975233 •73 541061 6*70 458939 10*458535 5o ii 9-516607 6-o5 9-975189 9i5i45 •73 9-54U68 6-78 % 12 617020 6-o5 •73 541875 6-78 458ii5 i3 517382 6-o4 975101 •73 542281 6-77 457719 4573i2 47 U 5i7745 518107 5i8468 6-o4 975057 975oi3 •73 542688 6-77 46 i5 6-o3 •73 543094 6-76 456006 456501 45 16 6-o3 974969 974925 ■74 543499 6-76 44 11 518829 6-o2 •74 54390c 544310 6-75 456095 43 519100 6-oi 974880 •74 6-75 455690 455285 42 •9 5i955i 6-oi 974836 ■74 5447i5 6-74 4i 20 5 1 99 1 1 6-oo 97479 2 •74 545119 6-74 45488i 4o 21 9-520271 6-oo 9-974748 ■74 9-645524 6- 7 3 10*454476 3 9 22 52063 1 '5-99 974703 •74 545928 6- 7 3 454072 38 23 520990 5-99 974609 ■74 54633 1 6-72 453669 453265 37 24 521349 f-9» 974614 ■74 546735 6-72 36 25 521707 5- 9 8 974570 •74 547138 6-71 452862 35 26 522066 5-97 974525 ■74 547540 6-71 452460 34 11 522424 5-96 974481 •74 547943 548345 6-70 452057 33 522781 5- 9 6 974436 ■74 6-70 45i655 32 ?9 523i38 IhH 974391 ■74 548747 6*69 45i253 3i 3o 523495 5- 9 5 974347 - 7 5 549149 6*69 6*68 45o85i 3o 3i 9-523852 5-94 9-974302 •75 9*549550 io«45o45o It 32 524208 5-94 974257 •75 549951 55o352 6*68 450049 449648 33 524564 5- 9 3 974212 •T> 6*67 27 34 524920 5- 9 3 974167 ■75 550752 6*67 449248 26 35 523275 5-92 974122 •75 55n52 6*66 448848 25 36 52563o 5-91 974077 ■75 55i552 6*66 448448 24 11 525 9 84 5-91 974032 ■75 551952 55235i ,6*65 448048 23 526339 5-90 973987 ■75 6*65 447649 22 3 9 A 526693 5-90 973942 •75 552 7 5o 6*65 44725o 21 4o 527046 973897 9-973852 ■75 553149 6*64 44685 1 20 41 9-527400 5.89 ■75 9-553548 6*64 10*446452 \l 42 527753 528io5 5.88 973807 .75 553o46 554344 6*63 446o54 43 5-83 973761 •75 6*63 445656 17 44 528458 5-8, 973716 ■76 554741 6*62 445259 16 45 528810 5-8 7 973671 •76 555i3g 6*62 444861 i5 46 529161 5-86 973625 •76 555536 6*6i 444464 14 % 529513 5-86 97358o ■76 555 9 33 556320 6*6i 444067 i3 529864 5-85 973535 ■76 6*6o 443671 12 P 53o2i5 5-85 973489 •76 55672D 6*6o 443275 11 5o 53o565 5-84 973444 ■76 557121 6*59 442879 10-442483 10 Si o-53o9i5 5-84 9-973398 973352 •76 9-55*7517 6*5g I 52 531265 5-83 •76 55TOI3 5583o8 6*5 9 6*58 442087 53 53i6i4 5-82 973307 •76 441692 I 54 53i 9 63 5323 1 2 5-82 973261 ■76 558702 6*58 441298 55 5-8i 973215 -76 559097. 6*57 44ooo3 440309 4401 i5 5 56 53266i 5-8i 973169- •76 559491 6*57 4 57 533oo9 5-8o 973124 .76 55 9 8H5, 6*56 3 58 533357 5. So 973078 •76 560279 560673 6*56 439721 2 59 533704 Z-n 973o32 •77 6*55 439327 438934 ■ 6o 534652 5- 7 § 972986 •77 56 1 066 6*55 mT Cosine ]). Bine 1 0° Cotang. D. Tuug. 38 (20 DEGREES.) A fABLE OP LOGARITHMIC M. Sine D. Cosine | D. Tang. | V. Uotang. I 9-534002 5- 7 8 9-972986 -77 9.561066! 6 55 10-438934! 60 1 438D41 5q I 534399 5 77 972940 77 5614D9 6 54 9 034745 5 77 972894 77 56i85i 6 54 438149 00 3 535092 53543H 5 77 972.848 77 562244 6 53 437756 57 4 5 76 972802 77 562636 6 53 437364 56 5 535783 5 76 972 7 55 77 563028 6 53 436972 55 6 536129 5 75 .972709 972663 77 563419 6 52 436d8i 54 I 536474 5 74 77 5638ii 6 52 436189 53 5368i8 5 74 972617 77 564202 6 5i 435798 52 9 537163 5 73 972570 77 564592 564o83 9.565373 6 5i 4354o8 5i 10 537007 5 ?3 972524 77 6 5o 435oi7 5o ii 9-537851 538io4 538538 5 72 9-972478 77 6 5o 10-434627 49 12 5 72 972431 lti o 565 7 63 6 49 434237 48 i3 5 7' 972385 t 566 1 53 6 49 433847 433458 47 U 538880 5 7 1 972338 78 566542 6 49 46 i5 U9223 5 70 972291 78 566o32 567320 6 48 433o68 45 16 539565 5 7° 97224? 78 6 48 432680 44 IS W9907 5 69 972198 9721D1 78 567709 568098 6 47 432291 43 540249 5 28 78 6 47 431902 43i5i4 42 '9 54o5oo 5409J 1 5 972105 78 568486 6 46 41 20 5 68 972o58 78 5688i3 6 46 431127 40 21 9-541272 5 67 9-972011 78 9.569261 6 45 10-430739 31 22 54i6i3 5 67 971964 78 669648 6 45- 43o352 23 541953 5 66 97'9<7 971870 73 570035 6 45 429965J 37 24 5422o3 542632 5 66 78 570422 6 44 429678 36 25 5 65 971823 78 570809 57119D 6 44 429191 35 26 542971 5433io 5 65 97'776 78 6 43 4288o5 34 3 5 64 97>729 79 57i58i 6 43 428419 33 543649 5 64 971682 79 571967 572352 6 42 428o33 32 29 543987 5 63 971635 79 6 42 427648 3i 36 544325 5 63 971588 79 572738 6 42 427262 3o 3i 0-544663 5 62 9-971540 79 9.573123 6 41 10-426877 29 32 545ooo 5 62 971493 79 573507 6 41 426493 28 33 545338 5 61 97U46 79 573892 6 40 426108 27 34 545674 j 61 971398 97i35i 79 574276 6 40 425724 26 35 54601 1 3 60 79 574660 6 3 9 425340 25 36 546347 5 60 97i3o3 79 575o44 6 3 9 424g56 424573 24 32 546683 5 5g 971256 79 575427 6 3o 23 547019 5 59 971208 79 57D810 6 33 424190 22 3, S47354 5 58 971161 79 576193 6 33 4238g7 4234*4 21 40 D47689 9-548024 5 58 97Ui3 79 576576 6 37 20 41 5 57 9-971066 80 9.576958 577341 6 37 io*423o4i \l 42 5483 5g 548693 5 57 971018 80 6 36 422659 43 5 56 970970 80 577723 578104 6 36 422277 n 44 549027 5 56 970922 970874 80 6 36 421896 16 45 549360 5 55 80 578486 6 15 42i5i4 i5 46 549693 5 55 970827 80 578867 6 35 42u33 14 % 55oo26 5 04 97°779 80 579248 6 34 420752 i3 55o359 5 54 97073 1 80 579629 6 34 420371 12 ic 550692 5 53 970683 80 580009 6 34 419991 11 DO 55io24 5 53 970635; 80 58o38 9 6 33 419611 10 5i 9 -55i356 5 52 9-970386, 80 9.580769! 6 33 10-419231 4i885i 1 02 55i687 5 52 970538 80 581149! 6 32 53 5520i8 5 52 970490, 80 58ID28; 6 32 418472 54 E 523^9 5 5i 970442 80 "181907 6 32 418093 55 552680 5 5i 970394 80 582286' 6 3i 417T4 5 56 553oio 5 5o 970345 81 582665 6 3i 417335 4 57 553341 c DO 970297 81 583o43j 6 3o 416957 3 58 553670 5 49 >7°249 81 583422 6 3o . 416:178 2 5g 554ooo 5 49 970200 8! 583800 1 6 29 416200 1 60 554329 ' 5 48 970152 -8l 584177J 6.29 Cotang. | I). 4i5823 1 — « - ■ Cosine b. Sine 6 S)° Tang. [M. SINES AND TANGENTS. (21 DEOREES.] 39 M. Sine 1). Cosine D. Tang. D. Gotang. 9-55432Q 554658 5-48 9-970152 .81 9-584177 6-29 io-4i5823 60 i 5 48 970103 .81 584555 6 3 4i5445 59 2 554987 5 47 970000 .81 584932 585309 6 4i5o68 58 3 5553 i J 5 47 970006 .81 6 28 414691 57 4 555643 5 46 969957 .81 585686 6 2 7 4U3i4 56 5 55597 1 5 46 969909 .81 586o6s 6 27 -4i3938 55 6 556299 5 45 969860 • 81 58643o 5868i5 6 27 4i356i 54 I 556626 5 45 9698 1 1 .81 6 26 4i3i85 53 556953 5 44 969762 • 81 587190 587566 6 26 41 2810 52 9 557280 5 44 969714 .81 6 25 412434 5i 10 557606 5 43 969665 • 81 HiS? 4 , 1 6 25 412059 5o II 9-557932 5 43 9-969616 • 82 9-5883i6 6 25 10-411684 40 12 558258 5 43 969567 969518 ■ 82 588691 6 24 41 IJ09 48 13 5*583 5 42 .82 589066 6 24 410934 410060 47 14 558909 5 42 969469 .82 589440 6 23 46 i5 55o234 5 41 969420 ■ 82 589814 6 23 410186 45 16 55o558 5 41 969370 • 82 590188 6 23 409812 44 \l 55 9 883 5 40 969321 • 82 590562 1 22 409438 43 560207 5 40 969272 .82 590935 22 4opo65 408692 42 ■9 56o53i 5 39 969223 • 82 591J08 6 22 41 20 56o855 5 & 969173 • 82 591681 6 21 408319 40 21 9-561178 5 38 9-969124 .82 9-592054 6 21 10 •407946 u 22 56i5oi 5 38 969075 • 82 592426 6 20 407374 23 561824 5 37 969025 • 82 592798 6 20 407202 ll 24 562146 5 ll 968976 • 82 593170 6 '9 406829 406458 36 25 562468 5 968926 ■83 593542 6 '9 35 26 562790 5 36 968877 •83 593914 6 ■ 8 406086 34 a' 563U2 5 36 968827 •83 5942S5 6 18 405715 33 563433 5 35 968777 968728 •83 5g4656 6 18 4o5344 32 2 9 563755 5 35 •83 595027 595398 6 17 404973 3i 3o 564075 5 34 968678 •83 6 17 404602 3o 3i 9-564396 5 34 9-968628 •83 9-595768 6 17 10-404232 ll 32 564716 5 33 9 685 7 8 •83 5 9 6i38 6 16 4o3862 33 565o36 5 33 96S52S •83 5 9 65o8 6 16 403492 27 34 565356 5 32 968479 •83 596878 6 16 4o3i22 26 35 565676 5 32 968439 •83 597247 6 15 402753 402384 25 36 560995 5663 1 4 5 3i 968379 •83 597616 6 i5 24 31 5 3i 968329 •83 597985 5 9 8354 6 i5 4020 1 5 23 566632 5 3i 963278 ■83 6 14 401646 22 39 566951 5 3o 968228 •84 598722 6 14 401278 21 4o 567269 5 3o 968178 •84 599091 6 i3 400909 10-400041 20 41 9-567587 5 29 9-968128 .84 9-699459 6 i3 18 42 567904 568222 5 2 2 968078 • 84 699827 .6 i3 400173 43 5 28 968027 .84 600194 6oo5&2 6 12 399806 \l 44 56853 9 5 28 967977 •84 6 12 3 99 438 45 568856 5 28 967927 • 84 600929 6 11 399071 i5 46 569172 • 5 27 967876 • 84 601296 6 11 398704 14 % 569488 5 ll 967826 • 84 601662 6 11 3 9 8338 i3 569804 5 967775 • 84 602029 602390 6 10 397971 12 49 570120 5 26 967725 •84 6 10 397605 11 5o 570435 5 25 967674 • 84 602761 6 10 397239 10 5i 9-57075I 1 5 25 9-967624 .84 9-6o3i27 6 09 10-396873 g 52 571066 5 24 967573 •84 603493 6o3858 6 09 396007 53 571380 5 24 967522 • 85 6 09 396142 7 54 5*71695 5 23 967471 • 85 6o4223 6 08 395777 6 55 572009 ■ 572323 5 23 967421 •85 604688 6 08 395412 5 56 5 23 967370 •85 6o4o53 6o53i7 6 07 395047 4 a 572636 5 21 ■ 967319 •85 6 °7 3 9 4683 3 572o5o 5 22 967268 •85 6o5682 6 00 3 9 43i8 2 5 9 5 7 3263 5 21 967217 ■85 606046 6 393954 1 6o 573575 5-21 967166 ■85 606410 6 06 3930901 % Cosine D. Sine 38° Cotang. D. — Tanpr. 1 M. 40 (22 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosino | 1). Tang. D. Cotang. 1 o 9-573575 5 7 3888 5-21 9-967166] .85 9-606410 6-06 10-393590 60 ! I 5-20 9671 i5 •85 606773 6-o6 393227 is 2 574200 5-20 967064 -85 607137 6-o5 3 9 2863 3 574512 5- 19 967013 -85 607500 6-o5 392500 ii 4 574824 5-i9 96696 1 •85 607863 608225 6-04 392137 391775 56 5 570136 5-19 9669 1 966839 966808 • 85 6-04 55 6 575447 5-i8 • 85 6o8588 6-04 391412 54 I 575758 5-i8 .85 6o8o5o 609312 6-o3 391050 53 576069 5-17 966756 .86 6-o3 390688; 52 9 10 576379 .576689 5-i6 966705 9 66653 • 86 • 86 609674 610036 6-o3 6-02 3go326 5i 389964I 5o II 9 "576995 57730c 5 16 9-966602 • 86 9-6io3q7 610709 6-02 10-389603 % 12 5-i6 966500 • 86 6-02 389241 l3 577618 5-i5 966499 • 86 611120 6-oi 368880 40 U 577927 578236 5-1.5 966447 .86 61 1480 6-oi 388520 15 5-14 966395 • 86 611841 6-oi 388i5 9 45 16 578545 5-14 966344 .86 612201 6-oo 387799 387439 44 \l 578853 5-i3 966292 .86 6i256i 6 -.oo 43 579162 5-i* 966240 .86 612921 6-oo 387079 42 19 579470 5-i3 066188 .86 613281 5-99 386719 41 20 9-58oo85 5-12 9 66i36 .86 6i364i 5-99 38635 9 40 21 5-12 9-966085 •87 9.614000 5- 9 3 io-386ooo ll 22 580392 '•II 966033 .87 614359 5-98 385641 23 580699 58ioo5 5. II 965981 .87 614718 5- 9 8 385282 37 24 5. II 965928 • 87 616077 5-97 384923 36 25 58i3i2 5-io 965876 >°7 61 5435 5-97 384565 35 26 58i6i8 5-io $65824 .87 615793 6i6i5i {# 384207 34 20 581924 5.09 965772 .87 383849 33 582229 582535 5-og 965720 .87 616609 5- 9 6 383491 383 1 33 32 29 5.09 9 65668 .87 616867 5.96 3i 3o 582840 5.o8 9656i5 .87 617224 5-g5 362776 3o 3i 9-583U5 5-o8 9-965563 .87 9-617582 5-g5 10-382418 ll 32 583449 5.07 9655i 1 .87 '618295 6i8652 5- 9 5 382061 33 583754 5-07 5.06 965458 .87 5-94 38no5 ll 34 584058 965406 .87 5,94 38i348 35 58436i 5.06 9 65353 .88 619008 5 '- 9 4 380992 38o636 25 36 584665 5.06 9653oi .88 619364 5- 9 3 24 ll 584968 5-o5 965248 .88 619721 5. 9 3 380279 23 585272 5-o5 966195 .88 620076 620432 5- 9 3 379924 379068 22 3 9 585574 5- 04 965143 .88 5.92 21 4o 5858 77 5-o4 965090 9-965037 .88 620787 5.92 379213 10-378858 20 4i 9"586i79 586482 5-o3 .88 9-621142 5.92 18 42 5-o3 964984 .88 621497 621802 5-91 3785o3 43 586783 5-o3 964931 .88 5.91 378148 \l 44 587085 5-02 964879 .88 622207 5.90 377793 377439 377080 45 587386 5-02 964826 .88 622661 5.90 i5 46 587688 5-01 964773 .88 622915 5.90 • 5.1, 14 % 587989 588289 5-01 964719 964666 .88 623269 376731 i3 5.oi .89 623623 5.89 376377 12 49 588590 5-oo 964613 .89 623976 624330 5-89 376024 11 5o 688890 5-oo 964660 .89 5-88 375670 10 5i 9-589190 4-99 9-964507 .89 9-624683 5.88 10-375317 8 52 589489 4-99 964454 .89 625o36 . 5-88 374964 374612 53 589789 590088 4-99 964400 .89 625388 5.87 54 4-98 964347 .89 625741 H 7 374259 55 1 590387 4.98 964294 .89 626093 5.87 5-86 373907 373555 5 56 590686 4-97 964240 -8q 626445 4 n 590984 4-97 964187 .89 626797 5-86 373203 3 591282 4-97 4-96 964133 -89 627149 5-86 372861 2 5o 59i58o 964080 .89 627501 5-85 372499 372148 1 1° 591878 4-96 964026 ■89 627862 5.85 Cosine D. Bins 37° Gotang. D. Tang. 1 .Mi SINES AND TANGENTS. (23 DEGREES. 1 41 Tl" Sine D. Cosine | D. Tang. 1 D> Colang. 1 60 9-591878 4.96 9-964026 .89 9-627855 5-85 10-372148 I 592176 4-95 963972 .89 62820; 5-85 371797 % 3 692473 4-g5 963919 .89 628664 5-85 371446 3 592770 4-95 963865 .90 62890; 5-84 37109: 57 4 59306: 5 9 336; 4.94 963811 .90 62925; 5-84 370745 56 5 4-g4 963767 .90 629606 5-83 370394 55 6 59365c 59390. 4.93 96370*1 .90 629956 5-83 370044 54 2 4.93 963660 • 9 c 63o3o6 5-83 369694 53 594251 4-93 963696 .90 63o656 5-83 369344 52 9 594547 4-92 963542 .90 63ioo5 5-82 36S995 5i 10 594842 4-92 * 963488 .9c 63i355 5-82 • 368645 5o II 9.595137 4-91 9*963434 .90 9-63170* 5-82 10-368296 % 13 595432 4-91 96337c 963325 .90 632o5; 5-8i 367947 i3 595727 4-91 .90 632401 5-8i 367699 41 14 596021 4.90 963271 .90 63275o 5-8i 367260 46 i5 5963i5 4.90 4. § 9 963217 963163 .90 633o 9 8 5-8o 366902 45 16 596609 59690J • 90 633447 5-8o 366553 44 \l 4.89 963io8 ■91 63379; 5-8o 366205 43 597196 4-89 963o54 ■ -91 634143 5.79 365857 42 '9 597490 4-88 962999 962945 •91 634490 5.79 3655 10 41 20 597763 g. 59OO75 4-88 •91 634838 t]l 365i62 40- 21 4-87 9.962890 9628J6 •91 9-635i85 io-3648i5 ll 22 5 9 8368 4-8 7 :9' 635532 6-78 364468 33 598660 4-87 4-86 962781 .91 635879 5.78 3641 2 1 ll 24 598952 962727 •91 636226 5-77 363774 25 599244 4-86 962672 •91 6365 7 2 5-77 363428 35 26 599536 4-85 962617 ■91 636919 637265 5-77 363o8i 34 11 599827 600118 4-85 962662 ■91 5-77 362735 33 4-85 962508 •91 .637611 5-76 362389 32 2 9 600409 4.84 962453 •91 637966 638302 5-76 362044 3t 3o ■600700 4.84 962398 •92 5-76 361698 io-36i353 3o 3i 9.600990 4-84 9-962343 •92 9-638647 5.75 29 32 601280 4-83 962288 •92 638992 5- 7 5 36ioo8 28 33 601670 4-83 962233 •92 639337 5- 7 5 36o663 27 34 601860 4-82 962178 •92 639682 5-74 36o3i8 26 35 602 1 5o 4-82 962123 •92 640027 5-74 359973 25 36 602439 4-82 962067 •92 640371 5-74 359629 24 u 602728 4-Si 962012 •92 640716 5- 7 3 350284 23 603017 4-8i 961957 ■92 641060 5- 7 3 358940 358596 22 39 6o33o5 4-8i 961002 96 1 846 •92 641404 5- 7 3 31 40 6o35o4 4- 80 •92 641747 5-72 358253 20 41 9.603882 4-8o 9.961791 9617J5 •92 9-642091 642434 5-72 10-357909 357666 !8 4r 604170 4-79 ■92 5-7? 43 604457 4-79 961680 •92 642777 5-72 357223 '7 44 604745 4-79 961624 ■ 9 3 643120 5-71 356880 16 45 6o5o32 4-78 96.569 96i5i3 • 9 3 643463 5- 7 i 356537 15 46 6o53i9 4-78 • 9 3 6438o6 5-71 356194 355852 14 % 6o56oo 4-78 961468 ■ • 9 3 644148 5-70 i3 605892 4-77 961402 • 9 3 644490 644832 5.70 3555io 12 49 606179 4-77 961346 • 9 3 5.70 355i68 11 5o 606465 4-76 961290 9.961235 • 9 3 645174 5-69 354826 10 5i 9.606761 4-76 • 9 3 9-6455i6 5-69 10-354484 52 607036 4-76 961170 961123 ■ 9 3 646857 5-69 354U3 53 607322 4-75 • 9 3 646199 5-69 3538oi 54 607607 4.75 961067 • 9 3 646540 5-68 35346o 55 607892 608177 4-74 961011 • 9 3 646881 5-68 353u 9 352778 5 56 4-74 960955 • 9 3 647222 5-68 I U 608461 4-74 960899 960843 • 9 3 647562 5-67 352438 3 608745 4-73 •94 647903 5-67 352097 a 59 609029 609313 4-73 960786 •94 648243 5-67 5-66 351757 1 60 4-73 960730 •94 648583 351417 M. It Cosine D. Sine 06 c Cotoug. D. Tang. 42 (2- t DEGREES.) A TABLE OF LOGARITHMIC Sine 9-60931; 1 1> ' Cosine D. Tang. 1 D - Cotting. io-35i4i- 60 1 4-73 9 • 96073c • gi 9-64858; 5-66' i 60959" 609880 4-72 96067* •9< 64892; 5-66 35107- 89 58 2 4-72 9606 1 £ •94 64926; 5-66 3507J' 350398 35ooS8 3 610164 4-72 960561 •94 64960: 5-66 57 4 610447 4-71 96o5o5 .94 64994: 5-65 56 5 610729 4-71 960446 ■94 65o28i 5-65 349719 55 6 61101: 4-70 96039: .94 65o62c 5-65 34938o 54 I 61 129: 4-70 96o335 • 94 650985 5-64 349041 348703 53 611576 611808 4-70 960279 ■ 94 65i29 ; 65i636 5-64 52 9 4-69 96022: .94 5-64 348364 5i 10 612140 4-69 9601 65 .94 651974 9-652JU 5-63 348026 5o 11 9-612421 4.69 9-960109 .98 5-63 10.347688 40 12 612702 4-68 96005: .95 65265c 5-63 34i35o 48 i3 612983 4-68 939995 959038 .93 652o88 653326 5-63 347012 % i4 6i3264 4-67 .95 5-62 346674 346337 i5 6i3545 4-67 939882 .95 653663 5-12 45 16 6i3825 4-67 4-66 95982; .95 654000 5-62 346000 44 »7 614100 959768 .95 65433- 5-6! 345663 43 18 6U385 4-66 90971 1 .95 65467,/ 5-6i 345326 42 <9 6i4665 4-66 95965*: ■95 655on 5-6i 344989 41 •20 614944 4-65 959596 9 • 959539 .95 655348 5-6i 344652 40 21 9-6id223 4-65 .,5 9 '655684 5-6o ic-3443i6 IS 22 61 ?5o2 4-65 959482 ■9? 656020 5- 6a 343980 23 615781 4-64 959425 .95 656356 5- 6a 343644 ll 24 616060 4-64 939368 •95 656692 5-5 9 3433o8 25 6i6338 4-64 959310 .96 657028 5-5 9 342972 35 26 616616 4-63 959253 .96 657364 5-5 9 342636 34 20 616894 4-63 939195 959138 .96 657699 5-5 9 5-58 3423oi 33 617172 4-62 .96 658o34 341966 34i63i 32 ^ 617400 4-62 959081 ■ 56 65836 9 5-58 3i Jo 617727 -4-62 959023 .96 658704 5-58 341296 3o 3i 9-618004 4-6: 9-958963 .96 9' 659039 65937J 659708 5-58 10-340961 29 32 6182S1 4-6i 958908 .96 5-57 340627 28 33 6i8558 4-6i 95885o .96 5-57 340292 339958 27 34 618834 4-6o 958792 9587J4 .96 660042 5-5 7 26 35 619110 4-6o .96 660376 5-5 7 339624 25 36 619386 4.60 958677 .96 660710 5-56 339290 338907 24 n 619662 4- 5 9 958619 .96 661043 5-56 23 619938 4-5g 958561 .96 661377 5-56 338623 22 3 9 620213 . 4-59 4-58 9585o3 ■97 661710 5-55 338290 337937 21 40 620488 958445 •97 6*62043 5-55 20 41 9-620763 4-58 9-958387 ■97 9-662376 5-55 10-337624 \i 43 62io38 4-57 958329 •97 662709 5-54 337291 336908 43 62i3i3 4-57 958271! ■97 663o42 5-54 n 44 621587 4-5t 4-56 9582i3 -cj7 663375 5.54 336625 16 45 621861 958 1 54 •97 663707 5-54 336293 i5 46 622135 4-56 958096 958o38 •97 664039 5-53 335ooi 335629 14 % 622409 4-56 •97 664371 5-53 i3 622682 4-55 957979 •97 664703 5-53 335297 12 p 622956 4-55 957921 •97 665o35 5-53 334965 11 5o 623229 4-55 95 7 863 ■97 665366 5-52 334634 10 5i 9-6235o2 4-54 9 '957804 •97 9-6656 9 7 5-52 !0'3343o3 8 52 • 623774 4-54 957746 .98 666029 5-52 333971 53 624047 4-54 957687 957628 .98 66636o 5-5i 333640 I 54 624319 4-53 .98 666691 5-5i 333309 55 624591 4-53 957570 .08 667021 5-5i 332979 332648 5 56 624863 4-53 957511 .98 667352 5-5i 4 u 625i35 4-52 957452 .98 667682 668oi3 5-5o 3323i8 3 625406 4-52 1 957393 957335 .98 5-5o 331987 . 33i65 7 33i3a8 2 5, 625677 625948 4-52 .98 668343 5-5o ■ 60 4-5l 957276 ■Is 668672 5-5o Cosine | D- ! Sine |65° Cotanff. D. Tang. J M. BINES AND TANGENTS. (25 DEGREES. 1 43 M. o Si no D. Cosine D. T-.mg. D, Cotang. io-l3i32-7 330998 60 9 '62 5g48 4-5i 9,957276 98 9.668673 5 So - 1 626219 4 5l 957217 957108 98 669002 5 49 5o 2 626490 4 5i 98 669332 5 49 33o668 58 3 626760 4 5o 957099 9 8 669661 5 49 33o33 9 57 4 627030 4 5o 957040 98 669991 5 48 330009 56 5 627300 •4 5o 956981 98 670320 5 48 329680 55 6 627570 4 49 956021 956862 99 670649 5 48 329351 54 7 627840 628109 4 49 99 670977 671306 5 48 329023 53 8 4 49 9568o3 99 5 47 32S694 52 9 628378 4 48 956744 99 671634 5 47 328366 5i 10 628647 4 48 906684 99 671963 5 47 328o3 7 5o II 9-628916 4 47 9.906625 99 9.672291 5 47 ,0.327709 49 12 629185 4 47 956566 99 672619 5 46 327381 48 l3 629453 4 47 9565o6 99 672947 5 46 327053 47 14 629721 4 46 956447 99 673274 5 46 326726 3263 9 8 46 i5 629989 4 46 9 5638 7 99 673602 5 46 45 16 630257 4 46 95632T 956268 99 673929 5 45 326071 44 \l 63o524 4 46 99 674257 5 45 325743 43 630792 4 45 956208 00 674084 5 45 325416 42 l 9 ■ 63 1 039 4 45 956148 00 674910 5 44 325090 41 20 63i326 4 45 956089 00 675237 5 44 324763 40 21 9-63i593 4 44 g-956029Jl 00 n. 675564 5 44 io . 324436 u 22 63i85q 632I2D 4 44 955969 I 00 V 6 7 58 9 5 44 324110 23 4 44 955909 00 676216 5 43 323784 37 24 632392 632658 4 43 955849 00 676543 5 43 323457 36 25 4 43 955789 00 676869 5 43 323i3i 35 26 632923 4 43 955729 00 677194 5 43 322806 34 3 633i8g 633454 4 4 42 42 955669 955600 00 00 677520 677846 678171 5 5 42 42 322480 322i54 33 32 29 633719 4 42 955548 00 5 42 321829 3i 3o 633984 4 41 955488 00 678496 5 42 32i5o4 3o 3i 9-634249 4 41 9-955428 01 9.678821 5 41 I0-32U79 32o854 11 32 6345 1 4 4 40 9 55368 1 01 679U6 5 41 33 634778 4 4o 955307 01 67947I 5 41 32o52o 320200 27 34 635o42 4 40 955247 01 679795 6SOI20 5 41 26 35 6353o6 4 3 9 955i86 01 5 40 319880 25 36 635570 4 3 9 955126 01 680444 5 40 319556 24 u 635834 4 3 955o65 01 680768 5 4o 319232 318908 3 1 8584 23 636097 4 955oo5 01 681092 5 40 22 39 63636o 4 38 954944 954883 01 68l4l6 5 39 21 40 636623 4 38 01 681740 5 39 318260 20 41 9-636886 4 37 9-954823 01 q. 682063 5 39 ,0.317937 \l 42 637148 4 37 954762 01 682387 5 ll 317613 43 63741 1 4 37 9 547^ 954640 91 682710 5 317290 >7 44 63 7 6 7 3 4 37 01 683o33 5 38 316967 16 45 637935 4 36 954579 c: 683356 5 38 316644 i5 46 638197 4 36 954518 02 683679 5 38 3i632i U % 638458 4 36 954457 02 684001 5 37 315999 i3 638720 4 35 954396 954335 02 684324 5 37 3i5676 12 49 638981 4 35 02 684646 5 37 3i 5354 11 56 639242 4 35 954274 02 684968 5 37 3i5o32 10 5i 9* 6395o3 4 34 9-954213 02 9.685290 5 36 io-3i47io 9 52 639764 4 34 954152 02 685612 5 36 3U388 8 53 640024 4 34 954090 02 685 9 34 5 36 3i4o66 54 640284 4 33 954020 953968 02 686255 5 36 3 13745 55 64o544 4 33 02 # 686577 5 35 3 1 3423 5 56 640804 4 33 953906 953845 02 686898 5 35 3i3io2 4 u 641064 4 32 02 687219 5 35 312781 3 1 641324 4 32 953783 02 687540 5 35 312460 2 59 641 584 4 32 953722 o3 687861 688182 5 34 3i2i3g 3n8i8 I 60 641842 4-3i 953660 i-o3 5 34 Cosine D. Sine 84° Cotanpr. D. Tang. 11. 13 44 (20 DEGREES.) A TABLE OF LOGARITHMIC M. Sine jj. Cosine D. ^o3 Tang. 9-688182 D. CotfUlg. "" 9-641842 4>3i 9.953660 5-34 10-3ii8i8 60 i 642101 4 3i 953599 •o3 688502 5 34 311498 59 2 642360 4 3i 953537 ■o3 688823 5 34 311177 58 3 642618 4 3o 953473 -o3 689143 5 33 3 10857 57 4 642877 4 3o 9334i3 -o3 68 9 463 5 33 3 io537 56 5 643l 33 4 3o 933352 • o3 689783 5 33 310217 55 6 643393 4 3o 933290 ■ o3 • 690103 5 33 309897 54 I 643600 4 29 033228 • o3 690423 5 33 309577 53 643908 4 29 033 166 • o3 690742 5 32 3o 9 258 52 9 644163 4 29 . 9.33104 ■ o3 691062 5 32 3o8o38 308619 5i 10 644423 4 28 953042 •oj 691381 5 32 5o n 9-644680 4 28 9-912980 •04 9-691700 5 3i io-3o83oo 49 12 644936 4 28 932918 • 04 692019 5 3i 307981 48 i3 645193 4 27 952855 •04 692338 5 3i 307602 47 U 643430 4 2 7 , . 952793 •04 692656 5 3i 307344 46 i5 643706 4 27 952731 •04 692975 5 3i 307025 306707 45 16 640962 4 26 952669 ■ 04 693293 5 3o 44 \l 646218 4 26 932606 •04 693612 5 3o 3o6388 43 646474 4 26 952544 -04 693930 5 3o 306070 3o5732 42 »9 646729 4 25 952481 ■ 04 694248 5 3o 41 20 646984 4 25 952419 .04 6g4566 5 29 3o5434 40 21 9-647240 4 25 9.932356 • 04 9-694883 5 29 io-3o5ii7 3 9 22 647494 4 24 932294 ■04 695201 5 29 304799 3o4402 38 23 647749 4' 24 952231 • 04 6g55i8 5 29 37 24 648004 4 24 952168 • o5 6 9 5836 5 ll 304164 36 25 648258 4 24 952106 -o5 696153 5 3o3847 35 26 648512 4 23 952043 -o5 696470 696787 5 23 3o353o 34 2 7 648766 4 23 951980 -o5 5 28 3o32i3 33 28 649020 4 23 951917 -o5 697 1 o3 5 28 302897 3o23HO 32 29 649274 4 22 95i854 ■ o5 697420 5 27 3i 3o 649527 4 22 951791 ■o5 697736 5 27 302264 3o 3i 9-649781 4 22 9.931128 931666 -o5 g. 698053 5 27 10.301947 ll 32 65oo34 4 22 • o5 6o836 9 698685 5 20- 3oi63i 33 630287 4 21 951602 -o5 5 3oi3i5 ll 34 65o539 4 21 95i539 • o5 699001 5 26 300999 3oo684 35 650792 4 21 951476 • o5 699316 5 26 25 36 631044 4 20 95i4i2 -o5 699632 5 26 3oo368 24 ll 651297 4 20 931349 .06 699947 700263 5 26 3ooo53 23 65 1 549 4 20 931286 -06 5 25 299737 22 3 9 65i8oo 4 '9 951222 • 06 700578 5 25 299422 21 40 632052 4 '9 931159 .06 700S93 5 25 299107 20 41 9-6523o4 4 '9 9.931096 .06 9.701208 5 24 10-298792 \i 42 632555 4 18 95io32 .06 701 523 5 24 298477 2 9 8i63 43 632806 4 l8 950968 ■ 06 • 701837 5 24 \i 44 653o5t 4 18 950905 •06 •J02I52 5 24 297848 45 6533o8 4 18 950841 • 06 702466 5 24 2^7534 i5 46 653558 4 '7 950778 .06 702780 5 23 297220 14 % 6538o8 4 17 950714 •06 703095 5 23 296905 296391 i3 634039 4 \l 95o65o •06 703400 703723 5 23 12 49 654309 4 9 5o586 •06 5 23 296277 11 5o 634558 4 16 95o522J • 07 704036 5 22 295964 10 5i 9-654808 4 16 9-95o458| •07 9-704350 5 22 10-295650 8 52 655o58 4 16 93o3o4 95o33o! •07 704663 5 22 295337 53 655307 4 13 •°7 704977 5 22 293023 I 54 655556 4 |5 950266J •07 705290 5 22 294710 294397 294084 55 655So5 4 13 950202 •07 7o56o3 5 21 5 56 656o54 4 ■4 95oi 38! •07 703916 5 21 4 u 656302 4 14 950074! ■07 706228 5 21 293772 3 65655i 4 14 95ooio| •07 706541 5 21 293459 3 59 656799 4 i3 949945] 949881 1 ■07 706854 5 21 293146 1 60 657047 4>i3 •07 707166 5-20 292834 : Cosine D. Sine G3° Cotang. D. . .T** . M._ SINES JtND TANGENTS. . (27 DEGREES.') 45 M. Sine I I> " Cosine D. Tang. D. Cotang. 9-657047 657293 4-i3 9-949881 1-07 9-707166 5-20 10-292834! 60 i 4-i3 949816 1 -07 707478 5-20 2 9 252S 5 9 2 657542 4-12 949752 1-07 707790 708102 5-20 29221c 58 3 657790 4-12 949688 1-08 5-20 291896 291 586 57 4 658o37 4-12 94962; i-o8 7084 H 5-19 56 5 658284 4-12 949558 1-08 708726 5-. 9 291 27* 55 6 65853i 4-n 949494 1-08 700037 709349 5-19 29096c 54 I 658 77 8 4-u 949429 1-08 5-19 290651 53 659025 4-n 949364 1-08 709660 5-19 5-i8 290340 52 9 659271 4-io 949300 1-08 709971 710282 29002c 2&97ii 5i 10 659517 4-io 949235 i-o8 5-i8 5o ii 9 ■ 659763 4-io 9-949I70 1-08 9-710593 • 5-i8 10-28940- % 12 66000c 66o255 4-09 949105 i-oS 710904 5-i8 289096 288785 |3 4-09 949040 1-08 711215 5-i8 % U 66o5oi 4-og 948975 1-08 7ii525 5-i 7 288475 288164 15 660746 4-09 4-o8 948910 948845 1-08 711836 5-i 7 45 16 660991 1-08 712146 5-i 7 287804 44 \l 66i236 4-o8 948780 1-09 712456 5-i 7 5-10 287544 43 661481 4-o8 948715 1-09 712766 287234 42 >9 661726 4-07 94865o 1-09 713076 713386 5-i6 286924 41 20 661970 4-07 948584 1 -09 5-i'6 286614 40 21 9-662214 4-07 9-948519 1-09 9-713696 5- 16 io-2863o/| IS 22 66245o 662703 4-07 4-oo 948454 1-09 714005 5-i6 285 99 5 23 948388 1-09 714314 5i5 •285686 u 24 662946 4-o6 948323 1-09 714624 5-i5 285376 25 663190 663433 663677 4-o6 948257 1-09 7U933 5-i5 285067 35 26 4-o5 948192 1-09 715242 5i5 284758 34 3 4o5 948126 1 -09 7i555i 5-i4 28444 9 33 663920 4-o5 948060 1 -09 7i5S6o 5-U 284140 32 ? 9 664163 4-o5 947995 I -10 716168 5-U 283832 3i 3o 664406 4-o4 947929 I-IO 716477 9-716785 5-14 283523 3o 3i 9-664648 4-o4 9-947863 I-IO 5-i4 io-2832i5 3 32 664891 665 1 33 4-o4 947797 94773 1 I-IO 717093 5-i3 282 9 07 33 4o3 I-IO 717401 5-i3 282S99 'I 34 6653 7 5 4-o3 947665 I-IO 717709 718017 5-i3 282291 281983 26 35 665617 4-o3 947600 I-IO 5-i3 25 36 66585g 4-02 947533 I-IO 718325 5-i3 281670 281367 24 3 666100 4-02 947467 I-IO 7i8633 5-12 23 666342 4-02 947401 I-IO 718940 5-12 281060 22 3 9 666583 4-02 947335 I-IO 719248 5-12 280752 21 4o 666824 4-oi 947269 9-947203 I-IO 719555 5-12 280445 20 4i 9-667065 4-oi I-IO 9-719862 5-12 io-28oi38 ;g 42 667305 4-oi 947136 I -II 720169 5-u 27 9 83i 43 667546 4-oi 947070 I'll 720476 720783 5-n 27 9 524 ■7 44 667786 668027 » 4-oo 947004 946937 I'll 5-n 279217 27891 1 16 45 4-oo I-II 721089 5- 11 i5 46 668267 4-oo 946871 I • 1 1 721396 5-u 278604 14 % 6685o6 3-g 9 946804 I-II 721702 5-io 278298 i3 668746 3- 99 946738 I-II 722009 7223i5 5-io 277991 12 4 9 66S986 3- g9 946671 I-II 5-io 277685 11 5o 669225 S 9 466o4 I-II 722621 5-io 277379 10-277073 276768 10 5i 9. 669464 9 -946538 I-II 9.722927 5-io I 52 669703 3- 9 8 946471 I-II 723232 5-09 53 669942 3- 9 8 946404 I-II 723538 5-09 276462 I 54 670181 X'97 946337 I-II 723844 5-o 9 276156 55 670419 670658 •T97 946270 I-I2 724149 5-OQ 275851 5 56 3-97 946203 I - 12 724454 5-o 9 275546 4 n 670896 671134 3-97 946136 1-12 724759 5-o8 275241 3 3- 9 6 946069 I - 1 2 725o65 5.o8 274935 2 &r 671372 3- 9 6 946002 I - 12 725369 5-o8 27463i I 671609 3- 9 6 945935 I -12 725674 5-o8 274326 1 Cosine D. Sine 62° Cotang. D. Tang. ML 46 (28 DEGREES.) A TABLE OF LOGARITHMIC IM. Sine 1 »■ Cosine | D. Tang. 1 D. Cotang. ) 60 9-671601 , 3.96 0-945935:i. 15 ' 945868 i-i: 9.725674] 5-o8 10 -27432* I 67184- 3.95 725979 5-o8 27402 59 2 67208; , 3.93 945800 1 • 1 ; 726284 5-07 273716 58 3 67232 3-95 i-i; 729929 73023; 5-o5 270071 46 i5 3.92 94492: 1-13 5-o5 26976- 45 16 675390 3-91 94485^ i-i3 73o535 5-o5 269465 44 \l 675624 3.91 944786 1 . i3 73o838 5-o4 269162 43 675859 3-91 9447i8.i-i3 73n4i 5-04 268859 42 19 67609,; 3.91 ■o4465o 1. 1 3 7 3i444 5-o4 268556 41 20 676328 3.90 944582li.i4 731746 5-o4 268254 40 21 9*676562 3.90 9-9445i4Ji-i4 9-732048 5-o4 10-267952 3 9 . 22 676796 6770J0 3.90 944446 1 -14 73235i 5-o3 267649 38 23 3.90 3-89 94437711.14 732653 5-o3 267347 37 24 677264 944309 1 -14 732955 5-o3 267045 36 25 677498 67773 1 3-89 944241 1 -14 733257 733558 5-o3 266743 35 26 3-8o 944172 1 -14 5-o3 266442 34 11 677964 .678197 678430 3-88 944104 1 -14 73386o 5-02 266140 33 3-88 944o36 1..4 734162 5-02 265838 32 ?9 3-88 943967 943099 9'94383o 1-14 734463 5-02 265537 3i 3o 678663 3-88 i-U 734764 5-02 265236 3o 3i 9-678895 H 7 1 -14 9- 735o66 5-02 10-264934 3 32 679128 3-8 7 943761 1 -U 735367 735668 5-02 264633 33 679360 3-87 943693 i-i5 5-oi 264332 2 7 34 679592 3-8 7 3-86 943624 i-i5 735969 5-oi 26403 1 26 35. 679824 ■ 68oo56 943555 i-i5 736269 5-oi 263 7 3i 25 36 3-86 943486 i-i5 736570 5-oi 26343o 24 u 680288 3-86 943417 943348 i-i5 736871 5-oi 263 1 29 23 680519 3-85 i-i5 737HI 5-oo 262829 22 39 680750 3-85 943279 i-i5 737471 5-oo 262529 21 40 680982 3-85 943210 i-i5 73777' 5-oo 262229 20 41 9-68i2i3 3-85 9-943i4i i-i5 9-738071 5-oo 10-261929 IO 42 681443 3-84 943072 i-i5 738371 5-oo 261629 l8 43 681674 3-84 943oo3 1 -i5 738671 4.99 261329 \l 44 68i 9 o5 3-84 9429341 - 15 738 97 i 4-99 261029 45 682135 3-84 942864 i-i 5 739271 4.99 260729 i5 46 682365 3-83 942795 1-16 739570 4.99 260430 14 % 682595 3-83 942726 1-16 739870 4.99 26oi3o 13 682825 3-83 942656 1-16 740160 740468 4.99 25 9 83i 12 i 9 683o55 3-83 94258 7 1-16 4.98 ' 259532 11 5o 683284 3-82 9425i7 1-16 740767 9.741066 4.98 250233 10 5i 9-6835i4 3-82 9-942448 I- 16 4.98 10-258934 I 52 683743 3-82 942378 1-16 741365 4-98 258635 53 683972 3-82 9423o8 I- 16 741664 4-98 258336 I 54 684201 3-8i 942239 I- 16 741962 4-97 258o38 55 68443o 3-8i 942169 I- 16 742261 4-97 257739 5 56 684658 3-8i 942099 1 - 16 742559 4-97 257441 4 n 684887 3-8o 942029(1- 16 742858 4-97 257142 3 685 1 1 5 3-8o 9419591- 16 743 1 56 4-97 256844 2 $> 685343 3-8o 941889 1 -17 743454 4-97 4-9° 256546 ■ 60 685571 3-8o 941819 1-17 743752 256248 m7 Cosine D. Sine 81° Cotang. D. Tans;. BINES AND TANGENTS. (29 degrees. ) 47 M. Sine B. Cosine | D. Tang. D. Ootang. 9-685571 3-8o 9-941819 1-17 9-743752 4.96 10-256248 60 i 685799 3-79 941749 1-17 •744o5o 4-96 255g5o 59 3 686027 3-79 941679 1-17 744348 4.96 255652 58 3 686254 3-79 941609 1-17 744645 4.96 255355 57 4 686482 l]l 941539 1-17 744943 4.96 255o57 56 5 686709 941469 1-17 745240 4.96 254760 55 6 686 9 36 3-78 941398 1-17 745538 4-95 254462 54 I 687163 3- 7 8 94i328 1-17 745835 4-95 254165 53 687389 3- 7 8 941258 1-17 746i32 4-95 253868 52 9 687616 3.77 941187 1-17 746429 4-95 253571 5i 10 687843 3.77 94U17 1-17 746726 4-95 253274 5o ii 9-688069 3-77 9-941046 1-18 9-747023 4-94 10-252977 49 12 688293 ?■" 940975 1-18 747319 4.94 25268i 48 i3 688521 3-76 940005 1-18 747616 4-94 252384 47 14 688747 3-76 940834 1-18 747913 748209 7485o5 4-94 252087 46 15 688972 3- 7 6 040763 i-i8 4-94 251791 45 15 . 689198 3- 7 6 940693 1-18 4-93 25U95 44 •7 6S9423 3- 7 5 940622 i-i8 748801 4-93 25i 199 250900 43 18 6896/18 3- 7 5 94o55 1 1-18 749097 4-93 42 19 68 9 8 7 3 3- 7 5 940480 1-18 7493o3 4-93 2 5o6o7 41 20 690098 3- 7 5 940409 1-18 749689 9-749985 4-93 25o3ii 40 21 9-690323 3-74 9-94o338 i-i8 4-93 io-25ooi5 3 9 22 690548 3.74 940267 1-18 750281 4-92 249719 38 23 690772 3-74 940196 i-i8 750576 4-92 249424 3 7 . 24 690996 3-74 940125 1-19 750872 4-92 249128 248833 36 25 691220 3-73 940054 1-19 751167 4-9 2 35 26 691444 3-73 939982 1-19 751462 4-92 248538 34 27 691668 3-73 939911 1-19 751757 4-92 248243 33 28 691892 3-73 939840 1 -19 752052 4.91 247948 32 29 692115 3.72 939768 939697 1-19 752347 4-91 247653 3l 3o 692339 3-72 1-19 /52642 4-91 247358 3o 3i 9-692562 3-72 9-939625 1 -19 9-752937 4-91 10-247063 29 32 692785 3-71 '939554 1-19 75j23i 4-91 246769 28 33 693008 3-.7I 939482 1-19 753526 4-91 246474 246180 11 34 693231 3-71 939410 1-19 753820 4-90 35 693453 3.71 939339 1-19 7541 i5 4.90 245885 25 36 693676 3-70 939267 1-20 754409 75470J 4.90 245591 24 -3 7 693898 3-70 939195 I -20 4-9° 245297 23 38 694120 3-70 939 1 23 1-20 754997 4.90 245oo3 22 3 9 694342 3-70 93oo52 938980 1-20 755291 4-90 4-89 244709 244415 2t 4o 694564 3-6 9 1-20 755585 20 41 9-694786 3-6 9 9-938908 9 38836 1-20 9-755S78 4.89 10-244122 \l 42 695007 3-6 9 1-20 756172 4-89 243828 43 695229 3-6 9 9 38 7 63 1-20 756465 4.89 243535 '7 44 695450 3-63 938691 1-20 756759 4.89 243241 16 45 695671 3-63 9.38619. 1-20 757052 4-8o 242948 i5 46 695892 3-63 938547 1-20 757345 4-88 242655 14 % 6961 i3 3-63 938475 1-20 757638 4-88 242362 i3 6 9 6334 3-67 938402 1-21 757931 4-88 242069 12 49 696554 3-67 938330 I - 2 I 758224 4-88 241776 11 5o 696775 3-67 9 3825S 1-21 7585i 7 4-88 241483 10 5i 0-696995 3-67 9- 9 38i85 1 -21 9-7588io 4-88 10-241190 52 69721 5 3-66 93Sn 3 1-21 759102 4-87 240898 8 53 607435 3-66 938040 1-21 759395 759687 4-87 24o6o5 7> 54 697654 3-66 937967 1-21 4-87 24o3 1 3 6 55 697874 . 698094 3-66 937895 1-21 759979 4-87 24002 1 5 56 3-65 937822 1-21 760272 4-87 239728 4 57 C 9 K3i3 3-65 937749 I-2I 7&o564 4-87 239436 3 58 (,98532 3-65 937676 r-2i 76o856 4-86 239144 238852 2 '■><) 69815; 3-65 937604 I-2I 761 148 4-86 1 6o 698970 Cosine 3-64 937531 I - 21 . 761439 4-86 238561 Taiitf. w _ D. Sine BOO Co tang. D. 18 (30 DEGREES.) A rABLE OF LOGARITHMIC M. Sine I). Cosine | D. Tang. Ii. Cotang. ' 9-698970 3-64 9-93753I 1-21 9-761439 4-86 io- 23856i 60 1 699189 3 64 937458 1 22 761731 4-86 238269 u 3 699407 3 64 9 37385 1 22 762023 4-86 237977 3 699626 699844 3 64 937312 1 22 762314 4-86 237686 57 4 3 63 937238 1 22 762606 4-85 237394 56 5 700062 3 63 937 i65^i 22 762897 763188 4-85 237103 55 6 700280 3 63 937092,1 22 4-85 236813 54 I 700498 3 63 9 3 7 0, 9| I 22 7634T9 4-85 236521 53 700716 3 63 9360461 22 763770 4-85 23623o 52 9 700933 3 62 936872 1 22 764061 4-85 235939 235648 5i 10 70IIDI 3 62 936799J 1 9- 9 36725ji 22 764352 4-84 5o ii 9-7oi368 3 62 22 9-764643 4-84 io-235357 % 12 701585 3 62 9 36652 1 23 764933 4-84 235067 i3 701802 3 61 936578 23 765224 4-84 234776 47 U 702019 3 61 9365o5 23 7655i4 4-84 234486 46 i5 702236 3 61 93643i 23 7658o5 4-84 234195 45 16 702452 3 61 936357 23 766095 4-84 233905 44 \l 702669 702885 3 60 936284 1 23 766385 4-83 2336i5 43 3 60 936210 1 23 766675 4-83 233325 42 >9 7o3ioi 3 60 936i36;i 23 766965 4-83 233o35 41 20 703317 3 60 936062 1 23 767255 4.- 83 232745 40 21 9-703 533 3 5 9 9-935988 1 23 9-767545 4-83 ■0-232455 18 22 703749 3 5 9 935914 1 23 767834 768124 4-83 232166 23 703964 3 5 9 935840 1 23 4-82 231876 37 24 704179 704396 3 5 9 935766 24 768413 4-82 23 1 587 36 25 3 5 9 935692 24 768703 4-82 231297 35 26 704610 3 58 9356i8 24 768992 4-82 23 1008 34 3- 704825 3 58 935543 24 769281 4-8z 230719 33 7o5o4o 3 58 935469 935390 24 769570 4-82 23o43o 32 ? 9 705254 3 58 24 769860 4-8i 23oi4o 3i 3o 7052469 9 -705683 3 57 935320 24 770148 4-8i 22qS52 3o 3i 3 57 9-935246 24 9.770437 4-8i 10-229563 11 32 . 705898 3 5 7 9 35i 7 i 24 770726 4-8i 229274 228985 33 7061 12 3 57 935097 24 771015 4-8i 27 34 706326 3 56 935o22 24 77i3o3 4*81 223697 228408 26 35 7o653o 706753 3 56 934948 934873 24 77 'i-y 771880 4-8i 25 36 3 56 24 4-8o 228120 24 32 706967 3 56 934798 25 772168 4-8o ■227832 23 707180 3 55 934723 25 772457 772745 4-So 227543 22 39 707393 3 55 934649 25 4-So 227255 21 4o 707606 3 55 934574 25 773o33 4-8o 226967 20 41 9-7°7Si9 3 55 9-934499 25 9-773321 4-8o 10-226679 ;s 42 708032 3 54 934424 25 773608 4-79 226392 43 708245 3 54 934349 25 773896 4-79 226104 \i 44 708458 3 54 934274 25 774184 4-79 225Si6 45 708670 708882 3 54 934199 25 774471 4-79 225529 i5 46 3 53 934123 1 25 774759 4-79 225241 14 47 709094 3 53 934048 25 77 ^ 4 5 4-79 224954 i3 48 49 709306 709518 3 3 53 53 $$ 25 26 775333 7 7 562i 4-79 4-78 224667 224379 12 11 5o 709730 3 53 933822 26 775908 4-78 224092 10 5i 9-70994I 3 52 9-933747 26 9.776195 776482 4-78 >o-2238o5 § 52 7ioi53 3 52 933671 26 4-78 2235i8 53 710364 3 52 933596 26 776769 777055 4-78 223a3i 7 54 710575 710786 3 52 933520 26 4-78 222945 6 55 3 5! 933445 26 777342 4 78 222658 5 56 710997 3 5i 933369 93329J 26 777628 4-77 22J372 4 ii 71 1208 3 5i 26 7779> 5 4-77 222085 3 711419 3 5i 933217 26 778201 4-77 221799 2 5, 711629 3 5o 933141 26 778487 4-77 22l5l2 1 6o 711839 3-5o 933o66 26 778774 4-77 221226 Cosine I). Sine 69° Cotang. D. Tan«. mT BINES AND TANGENTS. (31 DEGREES. 49 o Sine 9 • 7 1 1 83q D. Cosine ] D. Tang. r>. Cotang. T 1 1 _ 3 5o 9*933o66li -26 9-770774 4-77 I0-22122( 1 60 I 712030 3 • 5o 932990! i- 2- 779060 4-77 220941 59 3 712263 3 •5o 932914 1 -2-; 779346 4-76 22065/1 58 3 712469 3 49 932838[i-2- 779632 4-76 22o36E 57 4 712679 3 49 932762 1-27 779918 4-76 22008: 56 5 712889 713098 3 49 932685 1-27 780203 4-76 21979" 55 6 3 ■49 932609 932533 1-27 780489 780773 4-76 t 2193Il 54 o 7i33o8 3 % 1-27 4-76 2lg22£ 2 1 894c 53 7i35i7 3 932457 1-27 781060 4-76 52 9 713726 3 48 93238o 1-27 781346 4-75 218654 5i lO 713935 3 48 9323o4 1-27 78i63i 4-75 218369 5o 11 9-714144 3 48 9-932228 1-27 9-781916 4-75 10-21808/ 49 48 12 714352 3 47 932i5i 1-27 782201 4-75 "7799 i3 I456i 3 47 932075JI-2S 782486 4-75 217514 47 ;4 714760 3 47 . 93i998 | i-28 782771 4-75 217229 46 i5 714978 3 47 931921 1-28 783006 4-75 21694/ 45 16 7t5i86 3 47 93i845'i-28 783341 4-75 216659 44 17 715394 3 46 93176811-28 783626 4-74 21637/ 43 iS 715602 3 46 931691 1-28 783910 4-74 216090 42 19 715809 3 46 931614 1-28 784195 4-74 2i58o5 41 20 716017 3 46 g3i537 1-28 784479 4-74 215521 40 21 9-716224 3 45 9-931460 1-28 9-784764 4-74 io-2i5236 39 22 716432 \ 3 45 9 3i383 1-28 785048 4-74 214952 38 23 716639 716846 3 45 93i3o6 1-28 785332 4-73 214668 37 24 3 45 931229 1-29 783616 4-73 214384 36 25 717053 L 3 45 93n52 1-29 783900 4-73 214100 -35 26 717259 3 44 93i07D ( i -29 7S6184 4-73 2i38i6 34 11 717466 3 44 930998,1-29 736468 4-73 2i3532 33 717673 3 44 933921 1-29 933843^-29 736752 4-73 2i3248 32 29 T.7S79 718080 3 44 737036 4-73 2 1 2964 3l 3o 3 43 9307661 1 -29 7"?3io 9-787603 4-72 212681 3o 3i 9-718291 3 43 9-93o68S,i -29 '4-72 10-212397 29 32 718497 3 43 93o6 1 1 1-29 787S86 4-72 2121)4 28 33 718703 3 43 93i533|i-29 788170 4-72 2 i i 83o 27 34 718909' 3 43 93o456! 1 -29 788453 4-72 2 1 1 547 26 35 719ml 3 42 933378,1 -29 738 7 36 4-72 21 I264 25 36 719320 3 42 93o3oo 1 -3o 789019 4-72 210981 24 I 1 719525 3 42 93o223,i -3o 789302 4-71 210698 23 38 719730 3 42 .93oi45 1 -3o 78 9 585 471 210415 22 3 9 71993* 3 41 930067 [ 1 • 3o 789868 4-71 210132 21 < 4o 720140 3 41 929980 1. 3o 7901 5i 471 209849 20 4! 9-72o345 3 41 9-929911 i-3o 929833 j 1 -3o 9-790433 4-7! 10-209367 \l 42 ' 720549 3 41 790716 4-71 209284 43 720754 3 4o 92975511-30 790999 4-71 209001 17 44 720958 3 40 929677 1 -3o • 791281 4-71 208719 16 45 72116a 3 40 92959911 -3o 79 1 563 4-70 208437 i5 46 72i366 3 40 929521 i-3o 791846 4-70 208154 14 % 721570 3 40 929442 i-3o 792128 4-70 207872 13 721774 3 3 9 929364 i-3i 792410 4-70 207590 12 49 721978 3 39 929286 i-3i 792692 4-70 207308! 11 5o 722 181 3 ^ 929207 i-3i 79 2 974 4-70 207026 10 5i 9-722385 3 3 9 9.929129 i-3i 9-793256 4-70 10-206714 % 52 722588 3 3o 92oo5o 928972 i-3i 7g3538 4-6 9 206462 53 722 7911 3 38 1 -3i 793819 4-69 206181 7 54 722994 3 33 92889311-31 794101 4.69 205899 6 55 723197, 3 38 9 288i5!i-3i 794383 4-69 , 205617 5 56 723400, 3 38 928736'! -3i 794664 4-6 9 205336 4 57 7236o3; 3 37 928657 1-3 1 794945 4-69 2o5o55 3 58 7238o5 3- 37 92S578 i-3i 795227 .4-6o 4-68 2047731 2 59. 724007 3 37 928499 1 -3i 795308 204492 j 1 60 724210 Cosine 3 3 7 928420! i-3i 58° 795789 4-68 20421 11 " I>- Si,,,- I Cotiunr. r>. Tarig* | M. ^ 50 (32 DEGREES.) A TABLE OF LOGARITHMIC It. Sine IX Cosine | D. Tang. D. Cotang. 9.724210 3-37 9.92842011-32 9-796785 4-68 io- 20421 1 60 i 724412 3 ■37 928342a -32 79607c 4-68 203930 5o 5§ 2 724614 3 ■36 92826311 -32 796351 4-68 203649 3 724816 3 .'36 ' 928183 1-32 796632 4-68 2o336( 57 4 725oi- 3 •36 928104I1 -32 79691; 4-68 203087 56 5 725219 3 •36 928o25|i-32 797 '94 4-68 202806 55 6 725420 725622 3 •35 92794611-32 927867 1-32 79747^ 4.68 ' 20252!! 54 I 3 •35 797755 4-68 202245 53 725823 3 35' 92778tIi-32 798036 4-67 201964 52 9 726024 3 35 92770811-32 92762911 -32 7 9 83 16 4-67 201684 5i 10 726225 3 35 798596 4-67 201404 5o ii 9.726426 3 34 9-927549! I -32 9-798877 4-67 IO-20I 123 49 12 726626 3 34 927470 1-33 799157 4-67 200843 48 i3 726.827 3 34 927390 1-33 799437 4-67 2oo563 47 14 727027 727228 3 34 927310 1-33 7997 '7 4-67 200283 46 i5 3 34 927231 1-33 799997 4-66 2O0003 45 16 727428 3 33 927151 1-33 000277 800557 4-66 199723 44 \l 727628 3 33 927071 1-33 4-66 199443 43 727828 3 33 926991 1-33 0oo836 4-66 i9or64 1 98884 42 ■9 728027 3 33 92691 1 1-33 801 1 16 4-66 41 20 728227 3 33 926831 1-33 801396 4-66 198604 40 21 9-728427 3 32 9-926751 1-33 9-801675 4-66 10.198325 3 9 22 723626 3 32 926671 1-33 801955 4-66 198045 38 23 728825 3 32 926591 1-33 802234 4-65 197766 ll 24 729024 3 32 926DII 1-34 8o 2 5i3 4-65 197487 25 72922? 3 3i 926431 1-34 802792 4-65 197208 35 26 729422 3 3i 926351 i-34 803072 4-65 196928 34 27 729621 3 3i • 926270 1-34 8o335i 4-65 196649 33 2 34 742652 3 '7 920772' 1 40 821S80 4 57 178120 26 35 .742842 3 17 920688! 1 40 8221.54 4 57 177846 2:") 36 743o33 3 '7 920604 1 40 822429 822703 4 57 177571 24 37 743223 3 17 920520jl 40 4 57 177297 23 38 7434 t 3 3 16 920436 I 40 822977 4 56 177023 22 3 9 7436o2 3 16 92o352 : i 40 82J25o 4 56 1767,50 21 40 743792 3 16 920268II 40 823524 4 56 176476 20 41 9-743982 3 16 9-920i84[i 40 9-823798 4 56 10-1762021 19 42 744171 3 16 920099J 1 40 82-1072 4 56 175928I 18 43 744361 3 i5 920015.1 40 S243/.5 4 56 175655) 17 44 744550 3 i5 919931 |i 41 82461O 824893 4 56 I7538il 16 45 744739 744928 3 i5 919846 1 41 4 56 175107] i5 46 3 1 5 919762 1 41 825l66 4 56 174834! 14 2 745117 3 |5 9i9 6 77 1 41 825439 82571 3 4 55 I7456i' 1 3 7453o6 3 14 919393 1 41 4 55 174287! 12 49 7 15494 3 14 919508 1 41 82-986 826259 4 55 I74qi4 11 5o 745683 3 14 919424 1 41 4 55 1737.11: 10 5i 9-74587I 3 14 9-919339 1 41 9-826532 4 55 io- 1 73468 1 9 1731951 8 32 746o5o 746248 3 14 919254 1 41 826So5 . 4 55 53 3 i3 919169 1 919085 1 41 827078 4 55 172922I 7 54 746436 4 i3 41 82 7 33i 4 55 1726491 6 55 746624 3 ■ 3 9 1 0000' I 41 827624 4 55 172376; 5 56 746812 3 i3 918915.1 42 827897 4 54 1721031 ( 57 746999 747187 3 i3 9i883o ! i 42 828170 4 54 i7i83oi 3 58 3 12 918745,1 42 828442 4 54 171558 3 39 .747374 3 13 918659I1 42 828715 4 54 17128s 1 66 74756a 3 12 918574I1 42 828987 4 54 171013 O Cosine D. _ '-'■ine 5 8° Cotane. I Tang. »-, 62 (34 DEGREES ) A TABLE OF LOGARITHMIC M. Sine I>. Cosine 1 D. Tang. D. Cotang. | 9-747562 3-12 9-918574 1-42 9-828987 4 54 10-171013 60 i 747749 3 12 918489 1-42 829260 4 54 170740 38 2 747936 3 12 918404 1-42 829532 4 54 170468 3 748123 3 11 9 r83i8 1-42 829805 4 54 170195 5 7 4 7483 10 3 11 918233 1-42 830077 4 54 169923 56 5 748497 3 11 918147 1-42 83o349 4 53 1 6965 1 55 6 748683 3 11 918062 1-42 83o62i 4 53 169379 54 I 748870 3 11 • 917976 1-43 83o8 9 3 4 53 169107 53 749056 3 10. 917891 1-43 83 1 1 65 4 53 108335 52 9 749243 3 10 917805 1-43 83U37 4 53 ■68563 5i 10 749429 9-749615 3 10 917719 1-43 83 1 709 4 53 168291 5o II 3 10 9-917634 1-43 9-831981 4 53 lo- 168019 % 12 749801 3 10 917548 ■ •43 832253 4 53 '67747 i3 749987 3 09 917462 i-43 83 2 5 2 5 4 53 . 167476 47 14 750172 3 09 917376 1-43 832796 4 53 167204 46 i5 75o358 3 09 917290 1-43 833o68 4 52 166932 45 16 75o543 3 09 91720411-43 83333g 4 52 1 6666 1 44 \l 750729 3 00 917118 1.44 8336i 1 4 52 i6638o 43 750914 3 917032 1-44 833882 4 52 166118 42 '9 751099 751284 3 08 916946 1-44 834154 4 52 165846 41 20 3 08 916859 9-916773 i-44 834425 4 52 165575 40 21 9-751469 3 08 i-44 9-834696 4 52 io- i653o4 3 9 22 75 1 654 3 08 916687 1.44 804967 4 52 i65o33 38 23 75i83o 752023 3 08 916600 1-44 835238 4 52 164762 37 24 3 07 9 1 65 1 4 i-44 8355o9 4 52 1 6449 1 36' 23 732208 3 07 916427 i-44 833780 4 6( 164220 35 '26 752392 3 07 916341 i-44 836o5i 4 31 163949 34 27 752576 3 °7 916244. J--44 836322 4 5i 163678 33 28 752760 3 07 916167 i-45 8365 9 3 4 5i 163407 32 ?° 752944 3 06 916081 1-45 836364 4 5i 1 63 1 36 3i 3o 753128 3 06 915994 1-45 837134 4 5i 162866 3o 3i 9-7533i2 3 06 9-915907 915S20 1-45 9-8374o5 4 5i 10-162395 29 32 753495 3 06 1-45 837675 4 31 162325 28 3-3 753679 3 06 915733 1-45 83 79 46 4 5i 162054 27 34 733362 3 o5 915646 1-45 838216 4 5i 161784 26 33 754046 3 o5 915559 1-45 83848 7 4 5o i6^5i3 25 36 754229 3 o5 915472 1-45 83875 7 4 5o 161243 24 37 754412 3 o5 9i5385 1-45 839027 4 5o 160973 23 33 754595 3 o5 916297 1-45 839297 4 5o 160703 22 3 9 754778 3 04 915210 1-45 83 9 568 4 5o 160432 21 40 754960 3 04 915123 1-46 839838 4 5o 160162 20 -41 o-755i43 3 04 9*9i5o35 i- 4ft 9-840108 4 5o 10-159892 19 41 755326 3 04 914948 914860 1-46 84037S 4 5o 159622 18 43 7555o8 3 04 1-46 840647 4 5o i5 9 353 17 44 755690 3 04 9U773 1-46 840917 4 49 i5 9 o83 16 45 755872 3 o3 914685 1-46 841 187 4 49 i588i3 i5 46 ■J56o54 3 o3 914598 1-46 841467, 4 49 158543 14 47 756236 3 o3 914510 1.46 841726 4 49 168274 i3 48 756418 3 o3 914422 1-46 841996 4 49 1 38004 12 49 736600 3 o3 9I4334 1-46 842266 4 49 1 5 77 34 11 so 756782 3 02 914246 1-47 842535 4 49 157465 10 5i 9-756963 3 02 9-914158 1.47 9-842805 4 49 io- 157193 9 52 757144 3 02 . 914070 913982 913894 ■■47 843074 4 49 156926 8 53 757326 3 02 1-47 843343 4 49 156657 7 54 757507 3 02 1-47 843612 4 % 156388 6 55 757688 3 01 913806 1-47 843882 4 i56n8 5 56 737869 3 01 913718 1-47 844 1 5 1 4 48 155849 4 n 7:58o5o 3 01 9i363o 1-47 844420 4 48 1 555Ko 3 738230 3 01 9i354i 1-47 844689 4 48 i553u 1 5 9 75841 1 3 01 9i3453 1 -47 844968 4 48 i55o42 1 60 758591 3 01 913365 1-47 846227 4-48 164773 Cosine r >7~ _ Sine 53° Cotanjr. IX Tang. M. SINES AND TANGENTS. (35 DEGREES. ) 53 to. Sico D. Cosine D. Tang. D. Cotana;. 9-758591 3 01 9-91.3365 1 ■47 9-840227 4-48 io- 154773 60 1 708772 758962 3 00 913276 1 :% 845496 4.48 1 545o;j 5 9 2 3 00 913187 1 845764 4.48 154236 58 3 73913a 3 00 913099 1 •48 846o33 4-48 1 53 9 67 5 7 4 759312 3 00 9i3oio 1 .48 846302 4.48 1 536 9 8 1 5343o 56 5 759492 3 00 912922 1 912833 1 .48 846570 4-47 55 6 739672 2 99 ■48 846839 4-47 i53i6i 54 I 759852 2 99 912744 1 ■ 48 847107 847376 "4-47 i528 9 3 53 76003 1 2 29 912655 1 • 48 4-47 152624 52 9 7602 1 1 2 99 912566 1 ■ 48 847644 4-47 152356 5i 10 760390 2 99 912477 ' •48 847913 4-47 152087 5o li 9 ■760560 760748 2 9^ 9-912388 1 ■ 48 9-848181 4-47 I0'i5i8i 9 49 12 2 % 912299 1 .49 848449 4-47 i5i55i 48 i3 760927 2 ^ 912210 1 .49 838717 4-47 i5i283 47 a 761106 2 ^ 912I2I 1 •49 848986 4-47 i5ioi4 46 i5 761285 2 '2 9120.3 1 I •49 849254 4-47 1 50746 45 16 761464 2 9 3 911942 I •49| 849522 4-47 1 50478 44 n 761642 2 97 9 n853 1 ■491 849790 4-46 l5o2IO 43' 18 761821 2 97 91 1763 1 •49 85oo58 4-46 U9942 42 19 761999 2 97 91167411 9115841 .49 85o325 4-46 1 4967 5 41 20 762177 2 97 .49 85o5 9 3 4.46 149407 4o 21 9 '762356 2 97 9-911495 1 .49 9-85o86i 4.46 10-149139 3 9 22 762534 2 9 6 91 i4o5|i .49 831129 85i3 9 6 4-46 148871 38 23 762712 762889 2 96 91 i3i5: 1 • 5o 4.46 148604 37 24 2 96 911226 1 ■ 5o ' 85 1 664 4-46 148336 36 25 763067 763245 2 9 6 9iu36i • 5o 831931 4-46 148069 35 26 2 9 6 9110461 • 5o 852199 852466 4-46 147801 34 =7 763422 2 9 5 9109561 •5o 4.46 147534 33 2S 763600 2 95 910866 1 ■5o 852733 4-45 . U7267 32 29 763777 763954 2 9 5 9-0776; 1 910680?! • 5o 853ooi 4-45 146999 146732 3i 3o 2 95 • - 5o 853268 4-45 3o 3i 9-764131 2 95 9-9io5 9 6,i •5o 9-853535 4-45 10-146465 29 32 764308 2 9 5 9 io5o6 1 • 5o 8538o2 4-45 146198 i45o3i 145664 28 33 7&44S5 2 94 910415 1 • 5o 854069 4-45 27 3< 764662 2 94 910325 1 • 5i 854336 4-45 26 35 764838 2 94 910235 1 • 5i 8546o3 4-45 145397 i45i3o 25 36 ■;65oi5 2 94 910144 1 • 5i 854870 4-45 24 37 765191 2 94 910054 1 • 5i 855i3 7 4-45 144863 23 3S 765367 2 94 909963 1 ■5i 855404 4-45 144596 22 3 9 765544 2 93 909873 1 909782,1 ■ 5i 8556 7 i 4-44 144329 21 4o 765720 9*765896 2 9 3 ■ • 5i 855 9 38 4-44 144062 20 4i 2 93 9.909691 1 • 5i 9-856204 4.44 10-143796 \l 42 766072 2 93 909601 1 ■ 5i 856471 856 7 3 7 4-44 U3529 U3263 43 766247 2 93 909510:1 • 5i 4-44 17 44 766423 2 93 . 9°94i9:i • 5i 857004 4-44 142096 142730 16 45 7665 9 8 2 9 2 909328 1 •52 857270 4.44 i5 46 766774 2 92 909237 1 ■52 85 7 537 4.44 U2463 14 47 766949 2 92 909 1 46 J 1 •52 8578o3 4-44 142197 i4io3i 141064 13 48, 767124 2 92 9ogo55 i ■52 83S069 4.44 12 49 767300 2 9 2 908964 1 908873 1 •52 858336 4-44 11 5o 767475 2 91 •52 858602 4-43 !4i3o8 1 • 1 _, 1 1 ! 2 10 5i 9.767649 2 9 1 9 •908781 1 •52 9-858868 4'43 % 52 767824 2 9> 908690 1 •52 859134 4-43 140866 53 7&IJ999 2 91 908599 1 .53 85g4oo 4.43 140600 7 54 768173 2 9> 908507 1 ■52 85 9 666 4-43 !4o334 6 55 768348 2 9° 90S4161 ■53 8599.3a 4-43 140068 5 55 768322 2 go 9 o8324'i • 53 860198 4-43 139802 4 u 768697 2 9° 90S233I1 • 53 860464 4-43 i3g536 3 768871 2 9° 908141 1 •53 860739 4-43 139270 2 59 769045 2 00 908049 1 •53 86o 99 4-43 139005 1 60 769219 2-90 907958 1 bine' C •53 1° 861261 Cotung 4-43 138739 M." Cosine D. D. Tang. CI (3G DEGREES.) A TABLE OF LOGARITHMIC r ^ Sine D. Cosine | D. Tang. D. Cotang. 60 9-769219 769393 2-90 9 -907958! 1 -53 9-861261 4.43 10-138739 138473 I 2 89 9078661 1 53 86 1 527 4-43 3 1 769566 2 89 9°7774l 1 907682 1 53 861792 862008 4-42 138208 3 769740 2 89 53 4-42 137942 & 4 769913 2 89 907590 53 862323 4-42 137677 56 5 770087 2 89 907498 53 862589 4-42 137411 55 6 770260 2 88 907406 1 53 862854 4-42 137146 54 I 770433 2 88 907.3141 54 863119 863385 4.42 13688 1 53 770606 2 88 907222; 1 54 4-42 i366i5 52 9 770779 2 88 907129 1 54 86365o 4-42 i3635o 5i 10 770952 2 88 907037 1 54 863g 1 5 4-42 i36o85 5o ]i 9-771125 2 88 9-906945 1 906852 1 54 9-864180 4-42 10-135820 %■ 12 771298 2 87 54 S64445 4-42 135555 i3 77'47° 2 87 906760 1 5'. 864710 4-42 135290 47, U 771643 2 87 906667, 1 54 864975 4-4i i35o25 46 15 77ifii5 2 87 906570 1 54 865240 4-41 1 34760 45 16 77'9^7 2 87 906482 1 54 8655o5 4-41 1.34495 44 )l 772159 2 ll 906389 1 55 865770 4-41 1 3423o 43 772331 2 906296 1 55 866o35 4-41 133965 42 '9 7725o3 2 86 906204 1 55 8663oo 4-41 133700 41 20 772675 2 86 9061 1 1 J 1 55 866564 4-41 133436 40 21 9-772847 773oi8 2 S6 9-906018 1 55 9-866829 4-41 io- 133171 3o 22 2 86 905925] 1 55 867094 4-4i 132906 38 23 773190 2 86 9o5S32 1 55 867358 4-41 O2642 37 24 77336i 2 Si 905739I1 55 867623 4-41 i323 77 36 25 773533 2 85 905645 1 55 867887 4-41 i32ii3 35 26 773704 2 85 9o5552'i 55 868 1 52 4.40 131848 34 27 7738 7 5 2 85 9o545g : i 55 868416 4-40 i3i584 33 28 774046 2 85 9o5366!i 56 86S680 4-40 i3i32o 32 29 774217 2 85 905272 1 56 C63 9 45 4-40 i3io55 3i 3o 774308 2 84 906179 1 56 E69209 9-869473 4.40 1 30794 3o 3i 9-774538 2 9-90 5oS5 56 4.40 io- i3o527 3 32 774729 2 84 904092 904898 56 C69737 4.40 1 3026.3 33 7748<>j 2 84 56 870001 4-40 1 29999 27 34 77.1070 2 84 904804 56 870265 4-40 129735 26 35 775240 2 84 904711 56 870529 870793 871007 4-40 1 2947 c 25 36 775410 2 83 904617 9o/,523 56 4-40 129207 24 ll 775:3o 2 83 56 4-4o .289*3 2.3 775750 2 83 9o / i42o 904335 5 7 871321 4-4o .28679 22 *0 775920 2 83 57 8 7 i5S5 4-4o 128415 21 4o 776000 2 83 904241 57 871849 ' 4-39 128i5i 20 4' 9-776269 2 S3 9-904147 57 9-872112 4-39 10-127888 \t 42 776429 776598 2 ?J 90405.3 57 872376 4-39 127624 43 2 82 903959 903864 57 872640 4-39 127360 '7 4.1 775-68 2 82 57 872903 4-39 1 27097 16 45 776937 2 82 903770 57 873167 4-39 1 26833 i5 46 7-7106 2 82 90.3676 57 87343o 4-3 9 1 26570 14 tl 777275 2 81 9035SI 1 ! 57 873694 873907 4-39 1 263o6 |3 777444 2 81 903487; 1 57 4- 3 9 1 26043 12 49 • 7776i3 2 81 90339211 58 874220 4-39 125786 1 1 5o 777731 2 81 9032981 58 874484 4-39 I255i6 10 5( 9-777 9 5o 778119 2 81 9-9o32o3 1 58 9-874747 4-3 9 10 125253 I il 2 81 903108I1 58 871010 4- 3t> 1 24990 ti 778287 2 80 90.3014J1 58 875273 4- 38 124727- 7 '•4 778455 2 80 902919 1 902824! 1 58 875536 4-38 1 24464 6 55 778624 2 80 58 875800 4-38 124200 5 ?6 778792 2 80 902729I1 58 S76063 4-38 1 23 9 3 7 4 57 778960 2 8b 902634! 1 58 876326 4-38 i?36 7 4 3 58 77912H 2 80 » 902539)1 5 9 8 7 658 9 4-38 r>3',n 2 5g 779295 2 79 902444 59 876851 4-38 12.3149 1 6o 779463 2-79 902349 5 9 8771U Ontane;. 4-38 ~7T 122886 ST' 1 Cosine 1>. Sine 58° SINES AND TANGENTS, (37 DEGREE?. ) 58 M. o 1 Bine 1 "• Cosine | D. T-Mlg. D. Cot; ti£. 9-779463 2-79 9-90234911.59 9-877114 4-38 10-122886 60 i 719631 2-79 902253 1 -5c 877377 4-38 1 2262; 5o 3 77979^ 2-79 902 1 58 1 -5c 877640 4-38 I2236o 58 3 779966 2-79 902063 1 - 5c 877903 4-38 12209' I2i835 57 4 78oi33 2-79 2-78 90io67li-5c 901872U -5c 878165 4-38 56 5 78o3oo 878428 4-38 121572 55 6 780467 2-78 901776:1-59 878691 1 4-38 I2i3og 54 I 780634 2-78 901681 1-59 878953 4-37 12104- 53 780801 2-78 90 1 585 1 -59 879216 4-37 12078/1 5a 9 780968 2-78 901490 1 -5c 879478 4-37 120522 5i 10 781 i34 2-78 901394 1 -6o 879-41 4-37 12025c 5o ii 9-78i3oi 2-77 9-901298 1 -6o g-88ajo2 4-3 7 IO • 1 19991 % 12 781468 2-77 901202 i-6o 880265 4-37 119735 'i3 i8i634 2-77 901 106 1 -6o 8So528 4-37 1 19472 47 U 781800 2.77 901010 i-6o (380790 8Sio52 4-3 7 IIL2I0 46 l5 781966 2-77 900914 i-6o 4-3 7 1 1 8948 45 J6 782132 2-77 90081 8j 1-60 88i3i4 4-37 1 18686 44 '7 782298 2-76 90072211-60 88i5 7 6 4-37 1 1 8424 43 18 782464 2-76 900626 i- 60 881839 4-37 Il8l6l 42 <9 782630 2-76 900529 i- 6c 882101 4-3 7 I I7809 1 17637 41 20 782796 2-76 900433 i-6i 882363 4-36 40 21 9-782961 2-76 9-900337 i-Oi 9-882625 4-36 IO-II7375 1% 22 783127 2-76 900240 1-61 882887 883 1 48 ' 4-36 1 171 13 23 783292 2-75 900144 1 -61 4-36 II6S52 n 24 733458 2-75 900047 89995! i-6i 883410 4-36 1 16590 25 783623 2- 7 5 1-61 8836 7 2 4-36 116328 35 26 783788 2-73 899854 1-61 883g34 4-36 1 16066 34 u 783 9 53 2- 7 5 899707 i-6i 884196 884437 4-36 n58o4 33 7841 1 8 2- 7 5 899660 i-6i 4-36 1 15543 32 ?9 784282 2-74 899564 1-61 884719 4-36 ii528j 3i 3o 784447 2-74 899467 1-62 884980 4-36 Il5o20 3o 3i 9-784612 »-7i 9-899370 1 62 9-885242 4-36 10-114758 3 32 784776 2-74 899273 1-62 8855o3 4-36 1 14497 33 784941 2-74 899176 1-62 8S5 7 65 4-36 1 14235 27 34 786103 2-74 899078 898081 1-62 886»«6 4-36 1 13974 26 35 ' 785269 785433 2- 7 3 1-62 886288 4-36 113712 25 36 2- 7 3 898884 1-62 886549 4-35 ' ii345i 24 ll 785597 2.73 898787 1 62 886810 4-35 1 1 3 190 23 785 7 6i 2 -?3 898689 1-62 887072 4-35 1 1 2928 22 39 785925 2- 7 3 898592 1-62 887333 4-35 1 1 2667 21 4o 786089 2.73 898494 1-63 887594 9.887855 8881 16 4-35 1 1 2406 20 41 9-786252 2-72 9-898397 1-63 4-35 10-112145 \l 42 786416 2-72 898299 1-63 4-35 111884 43 786379 2-72 898202 1-63 888377 4-35 1 1 1623 \l 44 786742 2-72 898104 1-63 888639 4-85 iii36i 45 786906 2-72 898006 1-63 888900 4-35 1 moo i5 46 78x069 2-72 897908 897810 1-63 889160 4-35 1 10840 U 47 787232 2-71 1-63 889421 4-35 1 10579 i3 48 787395 787557 2-71 897712 1-63 889682 4-35 iio3i8 12 49 2-71 897614 1-63 889943 4-35 1 10057 11 5o 787720 2-71 897516 1-63 890204 4-34 109796 io- 109535 10 5i 9-787883 2-71 9-897418 1-64 9-890465 4-34 8 52 788045 2-71 897320 1-64 890725 4-34 109275 53 788208 2.71 897222 1-64 890986 4-34 109014 7 54 ' 788370 2-70 897123 1-64 891247 4-34 108753 6 55 788532 2-70 897025 1-64 891507 4-34 108493 108232 5 56 788694 788856 2-70 896926 1-64 891768 4-34 4 u 2-70 896828 1-64 892028 4-34 107972 3 789018 2-70 896729 1-64 892289 4-34 107711 3 59 789180 2-70 8 9 663 1 1-64 892549 4-34 107451 I 6o 789342 2-69 896532 1-64 892810 4-34 107190 Cosire I D. Sine 52°! Cotnncr. D. Tantf. M. 66 (3fc DEGREES.) A TABLE OP LOGARITHMIC M. Sine D. CoBine D. -64 Tang. 9-892810 D. Cctimg. 9-789342 3 .69 9-896532 4 34 10.107190 60' i 789504 2 .69 896433 •65 893070 4 34 106930 u 2 789665 2 .69 8 9 6335 •65 893331 4 34 106669 3 789827 2 .69 ' 896236 •65 893591 8 9 385i 4 34 106409 57 4 789988 2 .69 896137 8 9 6o38 •65 4 34 106149 56 5 790149 2 69 •65 8941 11 4 34 105889 55 6 79o3io 2 68 8 9 5o39 •65 894371 4 34 io562o 54 I 790471 2 68 895840 •65 894632 4 33 io5?68 53 790632 2 68 895741 •65 894892 895 1 52 4 33 io5io8 5a 9 790793 790954 2 68 895641 ■ 65 4 33 104848 5i 10 2 68 895542 r-65 895412 4 33 104588 5o ii 9-791 1 i5 2 68 9-895443 •66 9-895672 4 33 10-104328 % 12 791275 2 67 895343 • 66 8g5 9 32 4 33 104068 i3 79U36 2 67 895244 • 66 896 1 92 8964D2 4 33 io38o8 % 14 79i5q6 791757 2 67 895145 -66 4 33 103548 i5 2 67 8g5o45 -66 8967 1 2 4 33 103288 45 16 791917 2 ?? 894945 894846 •66 896971 4 33 103029 44 \l 792077 2 to • 66 897231 4 33 102769 43 792237 2 894746 • 66 897491 897761 4 33 102609 42 19 792397 792557 2 66 894646 • 66 4 33 102249 41 20 2 66 894546 • 66 898010 4 33 101990 10-101730 40 21 9-792716 2 66 9-894446 •67 9-898270 4 33 is 22 792876 2 66 894346 -6 7 898530 4 33 101470 23 793o35 2 66 894246 •67 898789 4 33 181211 37 24 793 1 n5 793354 7935i4 2 65 894146 ■67 899049 4 32 1 0095 1 36 25 26 2 2 65 65 » t 899308 899568 4 4 32 32 100692 100432 35 34 ll 793673 2 65 893846 .67 899827 4 32 100173 33 7 9 3S32 2 65 893745 .67 900086 4 32 099914 32 ?9 793991 2 65 893645 ■67 900346 4 32 . 099654 3i 3o 7941 DO 2 64 893544 • 6 7 900605 4 32 099395 10-099136 3o 3i 9 -7943o8 2 64 9-893444 • 68 9-900864 4 32 3 32 794467 2 64 893343 • 68 901124 4 32 098876 33 794626 2 64 893243 • 68 90i383 4 32 098617 ll 34 7947S4 2 64 893142 -68 901642 4 32 098358 35 794942 2 64 893041 • 68 901901 4 32 098099 25 36 795101 2 64 892940 ■ 68 902160 4 32 097840 24 ll 795259 2 63 892839 •68 902419 4 32 097581 23 795417 2 63 892739 •68 902679 4 32 097321 22 3 9 795575 795733 9-795891 2 63 89263 8 •68 902938 4 32 097062 21 4o 2 63 8 9 2536 1 •68 903197 9-903455 4 3i 096803 20 41 2 63 9-892435 .69 4 3i 10-096545 !o 42 796049 2 63 892334 i .69 903714 4 3i 096286 43 796206 2 63 892233 1 .69 903973 4 3i 096027 \l 44 796364 2 62 892132 1 .69 904232 4 3i 095768 45 796521 2 62 892030 1 .69 90449' 4 3i 095609 i5 46 796679 2 62 891929 1 .69 904750 4 3l 095a5o 14 % 796836 2 62 891827 1 .69 905008 4 3i 094992 094733 i3 796993 797 1 5o 2 62 891726 1 .69 905267 905526 4 3i 12 $ 9 2 61 891624 1 .69 4 3i 094474 II 5o 797307 2 61 8gi523 1 • 70 905784 4 3i 094216 10 5i 9-797464 2 61 9-891421 1 • 70 9-906043 4 3i 10-093957 3 52 797621 2 61 891319 1 • 70 906302 4 3i •093698 53 797777 7979 3 4 2 6i 891217 1 89I I ID 1 • 70 906560 4 3i 093440 I 54 2 61 • 70 906810 4 3i 093 181 55 . 798091 2 6i 89IOI3 I ■ 70 907071 907336 4 3i 092923 5 56 798247 2 61 8900 1 1 1 •70 4 3i 092664 4 u 798403 2 60 890809 I •70 907594 9078S 2 9081 1 1 4 3i 092406 3 798360 2 60 89O707 I •70 4 3i 092148 2 5, 798716 t 60 890605 I •70 4 3o 091889 1 6o 798872 2-6o 8go5o3 1 • 70 908369 4-3o 091631 m7 Cosine D. Sine 1 »o Cotan£. B. Tang. SINUS AND TANGENTS. (39 DEGREES.' 51 to. Sine D. Cosine D. Tang. I). Cotang. ,' 0-798872 3 60 9 -890503 1 70 9 -908369 4-3o io-ogi63i 60 I 799028 2 60 890400 71 908628 4 3o 091372 5o 3 799184 2 60 890298 71 908886 4 3o 091 1 14 58 3 79 9 33o 799495 79g65i 2 ^ 890195 7 1 909144 4 3o 090856 57 4 2 5 9 890093 7' 909402 4 3o 090598 56 5 2 5 9 889990 889888 71 909660 4 3o ogo34o 55 6 799806 2 5 9 7' 909918 4 3o .090082 54 2 799962 800117 2 5 9 889785 71 910177 4 3o 089823 53 2 5 9 889682 V 910435 4 3o o8g565 52 9 800272 2 58 889579 i 7' 910693 910951 4 3o 089307 5i IO 800427 2 53 889477 ' 7' 4 3o 089049 5o 11 9- 8oo582 2 58 9-889374 72 9-911209 4 3o 10-088791 o88533 % 12 800737 800892 2 58 889271 7 2 91 1467 4 3o i3 2 58 889168 72 911724 4 3o 088276 47 14 801047 2 58 889064 72 9H9S2 4 3o 0880; 8 46 15 801 201 2 58 888961 72 912240 4 3o 087760 45 ■6 8oi356 2 57 888858 72 912498 912756 4 3o 087502 44 n 8oi5n 2 57 888755 72 4 3o 087244 43 18 801 665 2 57 88865i 72 9i3oi4 4 29 086986 42 '9 801819 80197J 2 57 888548 1 72 913271 4 29 086729 41 20 2 57 888444 73 913529 4 29 086471 40 21 9-802128 2 57 9-888341 73 9-913787 4 29 10-086213 18 22 802282 2 56 888237 73 914044 4 29 o85g56 23 802436 2 56 888134 73 914302 4 29 085698 ll 24 802589 802743 2 56 888o3o 73 914560 4 29 oS544o 25 2 56 887926 887822 73 914817 4 29 o85i83 35 26 802897 2 56 73 915075 4 29 084925 34 3 8o3o5o 2 56 887718 73 9i5332 4 29 084668 33 8o3204 2 56 887614 ■3 915590 4 29 084410 32 ?9 8o3357 2 55 887510 73 9' 5847 4 29 0841 53 3l 3o 8o35u 2 55 887406 74 916104 4 29 083896 io-o83638 3o 3i 9-8o3664 2 55 9-887302 74 9-916362 4 29 2I 32 8o38i 7 2 55 887198 74 916619 4 29 o8338i 33 803970 2 55 887093 886989 886885 74 916877 4 29 o83i23 27 34 804123 2 55 74 9'7>34 4 29 082866 26 35 804276 2 54 74 917391 4 29 082609 25 36 804428 2 54 886780 886676 74 917648 4 29 o82352 24 32 8o458i 2 54 74 917905 4 3 082095 081837 23 804734 2 54 886571 74 918163 4 22 3 9 804886 2 54 886466 74 918420 4 28 081 58o 21 4o 8o5o39 2 54 886362 75 918677 4 28 o8i323 20 41 9-805191 2 54 g. 886257 75 9-918934 4 28 10-081066 18 42 8o5343 2 53 8861 52 75 919191 4 28 080809 43 8o5495 2 53 886047 75 919448 4 28 o8o552 \l 44 8o5647 2 53 885 9 42 75 919705 4 28 080295 o8oo38 45 805799 805931 2 53 885837 75 919962 4 28 ■ 5 46 2 53 885 7 32 75 920219 4 28 079781 14 47 8o6io3 2 53 885627 75 920476 4 28 079524 i3 48 806254 2 53 885522 75 920733 4 28 079267 12 49 806406 2 52 8854i"6 75 920990 4 28 079010 078753 11 5o 8o655 7 2 52 8853ii 76 921247 4 28 10 5i 9-806709 806860 2 52 9-885205 76 9-92i5o3 4 28 10-078497 I 52 2 52 885ioo 76 921760 4 28 078240 53 80701 1 2 52 884994 76 922017 4 28 077983 I 54 807163 2 52 884880 884783 76 922274 4 28 077726 55 807314 2 52 76 92253o 4 28 077470 5 56 807465 2 5i 884677 76 922787, 4 28 077213 4 32 807615 2 5i 884572 76 923044 4 28 076956 3 807766 2 5i 884466 76 9233oo 4 28 076700 2 59 807917 808067 2 5i 884360 76 923557 4 27 076443 1 6o 2 5i 884254 77 9238r3 4-27 076187 Tang. HT Cosine D. Sine 5p° Cotang. I 3. B8 (40 DEGREES.) A TABLE OP LOGARITHMIC M. Sine D. Cosine D. Tung. D. Cotang. 9.808067 2-5l 9-884254 '■77 9-9238i3 4 27 IC076187 60 i 808218 2 5l 884148 77 924070 4 27 075930 ll 2 8o8368 2 5l 884042 77 924327 4 27 075073 3 808519 2 5o 883 9 36 883829 883 7 23 77 924583 4 27 075417 n 4 80S669 2 5o 77 924840 4 27 075160 5 808819 2 5o 77 925096 925352 4 27 074904 55 6 808969 2 5o 883617 77 4 27 074648 54 I 8091 19 2 5o 8835io 77 925609 925865 4 27 074391 074135 53 809269 2 5o 883404 11 4 27 52 9 809419 2 49 883297 926122 4 27 073878 5i 10 809569 9-809718 2 49 883191 9-883o84 7« 926378 4 27 073622 5o 1 1 2 49 78 9-926634 4 27 10-073366 % 12 809868 2 49 882977 78 926890 4 27 073110 ■ 3 810017 2 49 882871 78 927147 4 27 072853 % 14 810167 8io3i6 2 % 882764 78 927403 4 27 072597 i5 2 882657 78 927639 927915 928171 4 27 072341 45 ■ 6 8io465 2 48 88255o 78 4 27 072085 44 \l 810614 2 43 882443 78 4 27 071829 07 1 573 43 810763 2 48 882336 79 928427 4 27 43 •9 810912 2 48 8S2229 79 928683 4 27 071317 41 20 811061 2 48 882121 79 928940 4 27 071060 40 21 9-811210 2 48 9-882014 79 9.929196 929452 4 27 10-070804 38 22 8u358 2 47 881907 79 4 27 070548 S3 8n5o7 2 47 881799 79 929708 4 27 070292 070036 ll 24 8n655 2 47 881692 79 929964 4 26 25 81 1804 2 47 88 1 584 79 930220 4 26 069780 069525 35 26 81 1952 2 47 881477 79 93o475 A 26 34 27 812100 2 47 88 1 36 9 79 93o73i 4 26 069269 06901J 068757 068001 33 28 812248 2 % 881261 80 930987 4 26 32 *9 812396 2 88n53 80 93i243 4 26 3i 3o 812544 2 46 881046 80 931499 9.931755 4 26 3o 3i 9-812692 2 46 9-880938 88o83o 80 4 26 10-068245 2I 32 812840 2 46 80 932010 4 25 067990 067734 33 812988 2 46 880722 80 932266 4 26 ll 34 8i3i35 2 46 88o6i3 80 932522 4 26 067478 35 8i3a83 2 46 88o5o5 80 932778 933033 4 26 067222 25 36 8i343o 2 45 880397 880289 80 4 26 066967 24 32 8i35 7 8 2 45 81 933289 933545 4 26 0667 1 1 23 8i3725 8i38 7 2 2 45- 880180 81 4 26 066455 22 3 9 2 45 880072 879963 81 933800 4 26 066200 21 4o 814019 2 45 81 934056 4 26 o65o44 10.065689 065433 20 4i 9-814166 2 45 9.879855 81 9-93431 1 4 26 I? 42 8i43i3 2 45 879746 81 934567 4 26 43 814460 2 44 879637 81 934823 4 26 065177 "7 44 814607 2 44 879529 81 935078 935333 4 26 064922 16 45 814753 2 44 879420 81 4 26 064667 .5 46 814900 2 44 8793 1 1 81 935589 4 26 06441 1 14 % 8i5o46 2 44 879202 82 935844 4 26 0641 56 i3 8i5iq3 8 1 533o 8 1 5485 2 44 879093 878984 878875 9-878766 878656 82 936100 4 26 063900 12 49 2 44 82 9 36355 4 26 063645 11 5o 2 43 82 936610 4 26 o633oo io-o63i34 10 5i 9-8i563i 2 43 82 9-936866 4 25 3 52 815778 2 43 82 937121 4 25 062879 53 815924 2 43 878547 878438 82 937376 937632 4 25 062624 I 54 816069 816210 2 43 82 4 25 062368 55 2 43 8 7 8328 82 937887 4 25 0621 i3 5 56 8i636i 2 43 878219 83 938142 4 25 o6i858 4 3z 816507 2 42 878109 83 9383 9 8 4 25 061602 3 8i6652 2 42 877999 83 9 38653 4 25 061347 2 59 816798 2 42 877890 83 938908 4 25 061092 060837 1 6o 816943 2 42 877780 i-83 939163 4-25 Cosine D. Sine 40° Cotang. D. Twig. M. SINES ATJD TANGENTS. (41 DKGBEES. 59 M. Sine D. Cosine 1 D. Tang. D. Cotang. c 9-816943 2 42 9-877780 1-83 9-939163 4-25 10-060837 60 i 817088 2 42 877670 1-83 9394 i£ 4-23 o6o582 is 2 817233 2 42 877560 1-83 939673 ' 4-25 ' o6o32'; 3 8i 7 3 79 2 42 877450 1-83 939926 4-25 060075 57 4 817524 2 41 877340 1 83 940 j 83 4-23 03981'; 56 5 817668 2 41 877230 1-84 94o43£ 4-25 05956s 55 6 817813 2 41 877120 1 84 940694 4-:5 059306 54 I 817958 2 41 877010 1.84 940941; 4-25 o5go5i 058796 53 8i8io3 2 41 876899 876789 1-84 94-204 4-25 52 9 818247 2 41 1.84 94 458 4-25 o58542 5i 10 8i83o2 o-8i8536 2 41 876678 g- 876568 1-84 941 '14 4-25 058286 5o II 2 40 1.84 9-04196? 4-25 io.o58o32 % 12 8 1 8681 2 40 876457 1-84 94 * 2*3 4-25 057777 i3 818825 2 40 876347 1-84 942478 4-25 057523 47 i4 818969 819113 2 40 876236 1-85 94-^33 4-25 057267 46 l5 2 40 876125 1-85 9/ -;88 4-25 057012 45 16 819257 2 40 876014 1-85 94-243 4-25 056757 44 17 819401 2 40 875904 1-85 943498 943733 4-25 o565o2 43 ■ 8 819545 2 39 875793 1-85 4-25 056248 4» 19 819689 2 39 875682 1-85 044007 4-25 055993 41 20 819832 2 39 875571 1-85 944262 4-a5 o55 7 38 40 21 9-819976 2 39 9.875459 • 875348 1-85 9-944517 4-25 10-055483 3 9 22 820120 2 39 1-85 944771 4-24 055229 38 23 820263 2 39 875237 1-85 945026 4-24 054974 37 24 820406 2 ll 875126 1-86 945281 4-24 054719 05446b 36 25 82o55o 2 875014 1-86 945535 4-24 35 26 820693 820836 2 38 874903 1-86 945790 4-24 054210 34 27 2 38 874791 874680 1-86 946045 4-24 053955 33 28 820979 2 38 1-86 946299 946554 4-24 053701 32 29 821122 2 38 874568 1-86 4-24 053446 3i 3o 821265 2 38 874456 1-86 946808 4-24 o53ig2 3o 3i 9-821407 2 38 9.874344 1-86 9.947063 4-24 10-052937 2 32 82i55o 2 38 874232 1-87 947318 4-24 052682 33 821693 2 37 874121 1 -87 947572 4-24 052428 27 34 821835 2 37 874009 .-87 947826 4-24 052174 26 35 821977 2 37 873896 8 7 3 7 84 ..87 948081 4-24 031919 25 36 822120 2 37 1-87 948336 4-24 051664 24 37 822262 2 37 873672 1-87 948590 4-24 o5i4io 23 38 822404 2 37 873560 1.87 948844 4-24 o5n56 22 3 9 822546 2 37 873448 1 -87 949099 949333 4-24 050901 21 40 822688 2 36 873335 1.87 4-24 050647 20 41 9-822830 2 36 9-873223 1.87 9-949607 4-24 io-o5o393 \l 42 822972 2 36 873 1 IO 1-88 949862 4-24 o5oi38 43 823i 14 2 36 872998 1-88 950116 4-24 049884 :z 44 823255 2 36 872885 1-88 950370 4-24 049630 45 823397 823539 2 36 872772 1-88 950625 4-24 049375 i5 46 2 36 872639 1-88 950879 g5 1 [33 4-24 049121 ■4 2 82368o 2 35 872547 1-88 4-24 048867 i3 823821 2 35 872434 1-88 9 5i388 4-24 048612 13 49 823963 2 35 872321 1-88 951642 4-24 048358 11 5o 824104 2 35 872208 1-88 951896 9-952100 4-24 048104 10 5t 9-824245 2 35 9-872093 871981 1-89 4-24 io-04 7 85o I 52 824386 2 35 1-89 932405 4-24 ^ 047395 53 824527 2 35 871868 1-89 932659 95291J 4 24 047341 7 54 824668 2 34 871755 1-89 4-24 047087 5 55 824808 2 34 871641 1.89 953167 4-23 046833 5 56" 824949 2 34 87I528 1-89 953421 4-23 046579 046325 4 u 825090 82523o 2 34 87UU ..8? 953675 4-23 3 2 34 871301 1.89 953929 934183 4-23 046071 2 59 825371 2 34 871187 1-89 4-23 045817 1 60 8255ii 2-34 871073 1-90 954437 4-23 045563 Cosine 1). Sine 48° Ootang. D. _Tang. 14 60 (42 DEGREES.) A TABLE 07 LOGARITHMIC M. Sine D. Cosine ) D. J Tang. D. Cotang. 9-8255n 2-34 9.871073 •9 C 1 9-954437 4-23 10-045563 60 i 82565i 2 33 870960 870846 • 9 c 954691 4-23 o453oq o45o55 11 2 825791 8259J1 2 33 •9 C 954945 4-23 3 2 -33 870732 •9 C 955200 4-23 044800 57 4 826071 2 33 870618 .90 955454 4-23 o44546 56 5 82621 1 2 33 870504 •90 955707 4-23 044293 044039 043785 55 6 82635i 2 -33 870390 • 90 955961 4-23 54 I 826491 82663i 2 33 870276 • 9 r 9562i5 4-23 53 2_ •33 870161 ■90 95646c 956723 4-23 o4353i 52 9 826770 2~ "32 870047 •91 4-23 043277 o43o23 5i 10 826910 2 32 86 9 o33 9-869818" ■QI 9 56 977 4-23 5o n 9.827049 2 32 ■91 9.957231 4-23 10-042769 042015 % 12 827189 827328 2 32 869704 •9> 9 5 7 485 4-23 i3 2 32 869589 • 91 957739 957993 4-23 042261 47 U 827467 2 32 869474 ■91 4-23 042007 46 i5 827606 2 32 869360 .91 958246 4-23 041754 45 16 827745 2 32 869245 .91 9585oo 4-23 o4i5oo 44 \l 827884 2 3i 869130 • 91 958754 4-23 041246 43- 828023 2 3i 86901 5 868900 .92 95900$: 4-23 040992 040738 42 ' >9 828162 2 3i .92 959262 4-23 41 20 8283oi 2 3i 868786 • 92 959516 4-23 040484 40 21 9-828439 828578 2 3i 9-868670 ■ 95 9 -959769 96002.J 4-23 10 -04023 1 ll 22 2 3i 868555 • 92 4-23 039977 23 828716 828855 2 3i 868440 .92 960277 4-23 039723I 37 24 2 3o 868324 •92 96o53i 4-23 039469 36 2D ■ 828993 829131 2 3o 868209 868o 9 3 • 92 960784 4-23 o3o2i6 35 26 2 3o .92 96103S 4-23 038962 34 ll 829269 2 3o 867978 ■93 961291 4-23 038709 33 829407 2 3o 867862 - 9 3 961545 4-23 038455 32 2 9 829545 2 3o 867747 - 9 3 961799 962052 4-23 o382oi 3i 3o 829683 2 3o 86 7 63i - 9 3 4-23 037948 3o 3i 9.829821 2 29 9-8675i5 - 9 3 9.962306 4-23 10-037694 3 32 829959 2 29 867390 867283 • 9 3 962560 4-23 037440 33 830097 830234 2 29 • 9 3 962813 4-23 037187 27 34 2 29 867167 ■ 9 3 963067 , 4-23 036933 26 35 83o372 2 29 867051 ■ 9 3 963320 4-23 o3668o 25 36 83o5o9 2 29 866n35 •94 •963574 4-23 036426 24 ll 83o646 2 29 866819 866703 •94 963827 4-23 036173 23 830784 2 29 ■94 964081 4-23 035919 22 3 9 830921 2 28 866586 ■94 96/,335 4-23 035665 21 40 83io58 2 28 866470 ■94 964588 4-22 o354t2 20 41 9-831195 83i332 2 28 9-866353 •94 9 ■ 964842 4-22 io-o35i58 10 42 2 28 866237 •94 965095 4-22 034905 l8 43 831469 2 28 866120 •94 965349 4-22 03465 1 \l 44 83 1 606 2 28 866004 - 9 5 965602 4-22 034398 45 831742 83i8 79 832oi5 2 28 865887 ] - 9 5 9 65855 4-22 o34U5 i5 46 2 28 865770 1 865653 1 - 9 5 966105 4-22- 033891 033638 14 47 2 27 - 9 5 966362 4-22 i3 4S 832i52 2 27 865536 1 - 9 5 966616 4-22 033384 12 49 832 288 2 2 7 865419 r - 9 5 966869 967123 4-22 o33i3i 11 5o 832425 2 2 7 8653o2 1 - 9 5 4-22 032877 10 5i 9-83256i 2 2 7 9-865i85 1 ■95 9-967376 4-22 10-032624 52 832697 2 27 865o68 1 • 9 5 967629 967883 9 68i36 4-22 032371 8 53 832833 2 27 864950 1 864833 1 -95 4.22 032117 7 54 832969 2 26 ■96 4-22 o3i864 6 55 833 io5 . 2 26 S64716 1 .96 968389 968643 4-22 o3i6ii 5 56 833241 2 26 8645o8 1 •96 4.22 o3i357 4 ll 833377 2 26 R64481 1 ■96 968896 4-22 o3uo4 3 8335i2 2 26 864363 1 .96 969149 96940J 4.22 o3o85i 2 ' 2? 833648 2 26 864245 1 .96 4-22 030597 i i 60 833783 2 26 864127 1 .96 969656 4-22 1 o3o344 -5- I— Cosine D. Pine 4 17° Cot" finer. D. 1 Time- 1L J 6IWEf .AND TANOENTS. (43 DEGREES. 1 61 to. Sine D. Cosine | D. Tan?. D. Cotanff. ~6o~ 9-833783 2-26 9-864I27I1-Q6 9-969656 4-22 io-o3o344 I 833919 2 23 864010 1 •96 969909 4-22 o3oooi 59 2 834054 2 2D 8638 9 2' 1 •97 970162 4-22 02 9 838 58 3 834189 834325 2 25 863774! 1 •97 970416 4-22 029584 57 4 2 25 863656 1 •97 97066^ 4-22 029331 56 5 834460 2 25 863538; 1 ■97 970922 4-22 029078 028825 55 6 8345o5 83473o 2 25 863419 1 ■97 971175 4-22 54 7 2 25 8633oi ! 1 ■97 971429 4-22 028571 53 8 834865 2 25 863i83 1 •97 971682 4-22 0283i8 52 9 834999 835i34| 2 24 863o64i ■97 971935 4-22 028065 5i 10 2 24 862946 1 9-862827 1 -98 972188 4-22 02781 21 5o II 0.835269 8354o3 2 24 .98 9-972441 4-2 2 10-027559 49 0273061 48 12 2 24 86270911 .98 972694 4-22 i3 835538 2 24 862590 1 - 9 3 972948 4-22 027052! 47 14 835672 2 24 862471 1 .98 973201 4-22 026799 46 ■ 5 835807 2 24 862353 1 .98 973454 4-22 026546! 45 ■6 835941 2 24 862234 1 •9? 973707 4-22 026293 44 \l 836o 7 5 2 23 862u5|i .98 973960 4-22 026040 43 836209 836343 2 23 8619961 .98 974213 4-22 025787 42 •9 2 23 861877 1 .98 974466 4-22 025534 41 20 836477 2 23 861758 1 ■ 99 974719 9-974973 4-22 025281 40 21 9-8366i 1 2 23 9-86i638:i ■ 99 4-22 io- 025027 3 9 22 836745 23 86i5ig 1 99 975226 v 4-22 024774 38 23 836878 2 23 861400 1 •99 975479 4-22 024521 37 24 837012 2 22 86i28o'i • 99 975732 4-22 024268 36 25 837146 2 22 861161I1 •99 975985 4-22 024015 35 26 837279 2 22 861041 1 •99 976238 4-22 023762 34 27 837412 2 22 860922 11 ' 860802; 1 99 976491 4-22 o235og 33 28 83 7 546 2 22 99 976744 4-22 023256 32 2 9 837679 2 22 860682 2 • 00 976997 977250 4-22 o23oo3 3i 3o 837812 2 22 860562I2 ■00 4-22 022750 3o 3i 9 .83 79 45 2 22 9-860442 2 •00 9-9775o3 4-22 10*022497 29 32 838078 2 21 86o322 2 •00 977756 4-22 022244: 28 33 83821 1 2 21 86020*2 2 00 978009 4-22 021991 20 34 838344 2 21 860081! <2 00 978262 4-22 021738 35 838477 2 21 859962-2 00 9785 1 5 4-22 021485 25 36 8386io 2 21 859842:2 00 978768 4-22 021232 24 37 838742 2 21 85972IJ2 01 979021 4-22 020979 23 38 838875 2 21 859601,2 01 979274 4-22 020726J 22 3 9 839007 2 21 85 9 48ol2 01 979527 4-22 020473 1 21 40 839140 2 20 . 85g36o2 01 979780 4-22 020220! 20 41 9.839272 2 20 9"S59239 I 2 01 9-9Soo33 4-22 10-019967] 19 019714! 18 42 839404 2 20 SSoug^ 858998.3 858877,2 01 980286 4-22 43 83 9 536 2 20 01 9 8o538 4-22 019462! 17 44 839668 2 20 01 980791 4-21 019209' 16 45 839800 2 20 8587562 02 981044 4-21 018956' i5 46 839932 2 20 858635,2 02 981297 98i55o 4-21 oi87o3j 14 47 840064 2 '9 8585i4'2 02 4-21 oi845o ; i3 48 840196 2 19 858393 2 02 981803 4-21 018197I 12 49 840328 2 ■9 8582722 02 9S2056 4-21 017944 i> 5o 840459 2 •9 858*5- a 02 982309 4-21 017691! 10 5i 9.840591 2 '9 9-8380292 8379082 02 9-982562 4-21 10-017438 9 017186 8 52 840722 2 •9 02 982814 4-21 53 840854 2 '9 837786,2 02 983067 4-21 016933; 7 0166801 6 54 840985 2 ;g 85 7 665,2 o3 983320 4-21 55 841 1 16 2 857543 2 o3 983573 421 016427 5 56 841247 2 18 8574222 o3 933826 4-21 016174 4 5-t 841378 2 18 8573oo' 2 o3 984079 4-21 015921 3 58 841 509 2 18 85717812 o3 9 3433i 4-21 01 566o| 2 oi54i6| 1 59 841640 2 18 85 7 o56l2 o3 984584 4-21 60 84H7' 2 18 856934|2 o3 084837 4-21 oi?i63 il Cosine D. Sin J 148° Cotang. D. 1 Tan*. 62 SINES AND TANGENTS. M. SiiiG D. Cosine D. Tang. D. Cotang. o 9-84I77I 2 18 0.856934 2-03 9.984837 4 21 io-oi5i63 60 I 841902 2 18 856812 2 o3 985090 4 21 014910 a J 842033 2 18 8566go 2 04 985343 4 21 014657 3 842163 2 17 856568 2 04 985596 4 21 014404 n 4 842294 2 '7 ' 856446 2 04 985848 4 21 0i4i52 5 842424 2 17 856323 2 04 986101 4 21 013899 55 6 842555 2 17 856201 2 04 986354 4 21 013646 54 2 842685 2 17 856078 2 04 986607 4 21 013393 53 8428i5 2 17 855 9 56 855833 2 04 986860 4 21 oi3i4o 52 9 842946 2 17 2 04 987112 4 21 012888 5i 10 843076 2 \l 855 7 n 2 o5 987365 4 21 012635 5o ii 9-843206 2 9-855588 2 o5 9-987618 4 21 10-012382 3 12 843336 2 16 855465 2 o5 988123 4 21 012129 i3 843466 2 16 855342 2 o5 4 21 01 1 877 % U 843595 2 16 855219 2 o5 988376 4 21 01 1624 i5 843725 2 16 855oo6 2 o5 988629 4 21 011371 45 16 843855 2 16 854o73 85485o 2 o5 988882 4 21 011118 44 \l 843984 2 16 2 o5 989134 4 21 010866 43 844i 14 2 i5 854727 2 06 989387 4 21 oio6i3 42 >9 844243 2 i5 8546o3 2 06 989640 4 21 oio36o 41 20 844372 2 i5 85448o 2 06 989893 4 21 010107 40 21 9-844502 2 i5 9-854356 2 06 9.990145 4 21 10-009855 18 22 84463i 2 i5 854233 2 06 990398 990651 4 21 009602 23 844760 844889 845oi8 2 i5 854109 2 06 4 21 009349 ll 24 2 ■1 5 853 9 86 2 06 990903 4 21 009097 25 2 i5 853862 2 06 991156 4 21 008844 35 26 845 147 2 i5 853 7 38 8536i4 2 06 991409 4 21 008591 34 s 845276 2 14 2 07 991662 4 21 oo8338 33 845405 2 14 853490 2 °7 991914 4 21 008086 32 ?9 845533 2 14 853366 2 07 992167 4 21 007833 3i 3o 845662 2 14 853242 2 07 992420 4 21 007580 3o 3i 9-845790 2 14 9-853n8 2 0.7 9-992672 4 21 10-007328 3 32 845919 2 14 852994 852869 852745 852620 2 07 992925 4 21 007075 33 846047 2 i4 2 07 993178 4 21 006822 ll 34 846175 2 14 2 °7 993430 4 21 006570 35 846304 2 14 2 3 993683 4 21 006317 25 36 846432 2 i3 852496 2 993g36 4 21 006064 24 32 84656o 2 i3 8523 7 i 2 08 994189 4 21 oo58n 23 846688 2 i3 852247 2 08 994441 4 21 oo555o oo53o6 22 3 9 846816 2 i3 852122 2 08 994694 4 21 21 4o 846944 2 i3 85i99 7 9.851872 2 08 994947 4 21 oo5o53 20 4i 9-847071 2 i3 2 08 9-995199 995452 4 21 10-004801 ;s 42 847199 2 i3 85 1 747 85I022 2 08 4 21 004548 43 847327 2 i3 2 08 995705 4 21 004295 \i 44 847454 2 12 85 1 497 2 09 995957 4 21 004043 45 847582 2 12 85i372 2 09 996210 4 21 003790 oo3537 i5 46 847709 847836 2 12 85i246 2 09 996463 4 21 14 % 2 12 85ii2i 2 09 996715 4 21 oo3285 ■ 3 847964 848091 2 12 85099612 8508702 09 996968 4 21 oo3o32 12 49 2 12 09 997221 4 21 002779 11 5o 848218 2 12 850745 2 09 997473 4 21 002527 10 5i 9-848345 2 12 9-85o6io 800493 2 09 9-997726 4 21 10-002274 8 52 848472 2 II 2 10 997979 4 21 002021 53 848599 2 11 85o368 2 10 998231 4 21 001769 I 54 848726 2 II 850242 2 10 998484 4 21 ooi5i6 55 848852 2 II 85ou6 2 10 998737 4 21 001263 5 56 848979 2 11 849990 2 10 998989 4 21 001011 * ll 849 1 06 2 II 849864 2 10 999242 4 21 000758 3 849232 2 11 849738 8496 11 2 10 999495 4 21 ooo5o5 a 59 849359 849485 2 II 2 10 999748 4 21 000253 1 Co 2 II 849485 2 10 10-000000 4-21 10-000000 M. Cosine D. Sine i 50 Cotan£. ]). Tang.