University of California Berkeley 
 
 THE THEODORE P. HILL COLLECTION 
 
 of 
 EARLY AMERICAN MATHEMATICS BOOKS 
 

 , ^ 
 
 . 
 
 


 3, 
 
Oat} anb ftljomson's Series. 
 
 A 
 
 TREATISE 
 
 OP 
 
 PLANE TRIGONOMETRY, 
 
 AND THE 
 
 MENSURATION OP HEIGHTS AND DISTANCES. 
 
 TO WHICH 13 PREFIXED 
 
 A SUMMARY VIEW OP THE NATURE AND USB 
 OP 
 
 LOGARITHMS, 
 
 ADAPTED TO 
 
 FHB METHOD OF INSTRUCTION IN SCHOOLS AND ACADEMIES 
 
 BY JEREMIAH DAY, D.D. LL.D. 
 
 LATX PRESIDENT OF TALB COLLEGE. 
 
 NEW YORK: 
 IVISON & PHINNEY, 178 FULTON STREET; 
 
 (SUCCESSORS or NEWMAN fc IVISON, AND MARK H. NEWMAN * co.) 
 CHICAGO: S. C. GRIGGS A CO., Ill LAKE ST. 
 
 BUFFALO: PHINNEY t CO., 188 MAIN STREET 
 
 AUBURN: J. c. IVISON * co. DETROIT : A. M'FARREN. 
 
 CINCINNATI: MOORE, ANDERSON * co. 
 
 1855. 
 
ENTERED, according to Act of Congress, in the year 1848, by 
 JEREMIAH DAY, 
 
 In tla Clerk's Office of the District Court of the United States far the 
 Southern District of New York. 
 
 THOMAS B. SMITH, STERKOTYPER. J. D. BEDFORD, PRINTER, 
 
 216 WILLIAM STRKKT, N. Y- 138 FULTON STREET. 
 
PLANE TRIGONOMETRY. 
 
 SCARCELY any department of Mathematics is more impor- 
 tant, or more extensive in its applications, than Trigonometry.. 
 By it the mariner traces his path on the ocean ; the geogra- 
 pher determines the latitude and longitude of places, the di- 
 mensions and positions of countries, the altitude of mountains, 
 the courses of rivers, &c., and the astronomer calculates tha 
 distances and magnitudes of the heavenly bodies, predicts the 
 eclipses of the sun and moon, and measures the progress of 
 light from the stars. 
 
 The section on right angled triangles in this treatise, may 
 perhaps be considered as needlessly minute. The solutions 
 might, in all cases, be effected by the theorems which are 
 given for oblique angled triangles. But the applications of 
 rectangular trigonometry are so numerous, in navigation, sur- 
 veying, astronomy, &c., that it was deemed important, to ren- 
 der familiar the various methods of stating the relations of the 
 sides and angles ; and especially to bring distinctly into view 
 the principle on which most trigonometrical calculations are 
 founded, the proportion between the parts of the given tri- 
 angle, and a similar one formed from the sines, tangents, &c., 
 in the tables. 
 
 As this treatise is intended to form a part of Day and 
 Thomson's Course of Mathematics for the use of Schools and 
 Academies, the references to Algebra are made to Thomson's 
 Abridgment ; and the references to Geometry, to Thomson's 
 Legendre, as well as to Euclid's Elements. 
 
CONTENTS. 
 
 LOGARITHMS. 
 
 Ftp 
 
 SECTION I. Nature of Logarithms, 7 
 
 II. Directions for taking Logarithms and their Num- 
 bers from the Tables, 14 
 
 m. Methods of calculating by Logarithms, .... 
 
 Multiplication, 22 
 
 Division, 25 
 
 Involution, 27 
 
 Evolution, 29 
 
 Proportion, 32 
 
 Arithmetical Complement, 33 
 
 Compound Proportion, 35 
 
 Compound Interest, 37 
 
 Increase of Population, ........ 40 
 
 Exponential Equations, 45 
 
 TRIGONOMETRY. 
 
 SECTION I. Sines, Tangents, Secants, &c., 47 
 
 n. Explanation of the Trigonometrical Tables, . . 59 
 HI. Solutions of Right angled Triangles, .... 67 
 IV. Solutions of Oblique angled Triangles, ... 85 
 V. Geometrical Construction of Triangles, ... 99 
 VI. Description and use of Gunter's Scale, ... 107 
 
 VII. Trigonometrical Analysis, 116 
 
 Application of Trigonometry to the mensuration of heights and 
 
 distances, 130 
 
 Notes, 148 
 

LOGARITHMS, 
 
 SECTION I. 
 
 NATURE OF LOGARITHMS. 
 
 ART. 1. The operations of Multiplication and Division, 
 when they are to be often repeated, become so laborious, 
 that it is an object of importance to substitute, in their 
 stead, more simple methods of calculation, such as Addition 
 and Subtraction. If these can be made to perform, in an 
 expeditious manner, the office of multiplication and division, 
 a great portion of the time and labor which the latter pro- 
 cesses require, may be saved. 
 
 Now it has been shown, (Algebra, 189, 193,) that powers 
 may be multiplied by adding their exponents, and divided, 
 by subtracting their exponents. In the same manner, roots 
 may be multiplied and divided, by adding and subtracting 
 their fractional exponents. (Alg., 232, 239.) When these 
 exponents are arranged in tables, and applied to the general 
 purposes of calculation, they are called Logarithms. 
 
 2. LOGARITHMS, THEN, ARE THE EXPONENTS OF A 
 
 SERIES OF POWERS AND ROOTS. 
 
 In forming a system of logarithms, some particular num- 
 ber is fixed upon, as the base, radix, or first power, whose 
 logarithm is always 1. From this a series of powers is 
 raised, and the exponents of these are arranged in tables for 
 use. To explain this, let the number which is chosen for the 
 
8 If ATTTRE OF LOGARITHMS. 
 
 first power be represented by a. Then taking a series of 
 powers, both direct and reciprocal, as in Alg. 163 ; 
 
 a 4 , a', a 9 , a 1 , a, d~\ a" 1 , a" 9 , cT 4 , &c. 
 The logarithm of a* is 3, and the logarithm of a~ l is 1, 
 
 of a 1 is 1, of cf is 2, 
 
 of a is 0, of of is 3, <fec. 
 
 Universally, the logarithm of a* is x. 
 
 3. In the system of logarithms in common use, called 
 Briggs's logarithms, the number which is taken for the radix 
 or base is 10. The above series, then, by substituting 10 
 for a, becomes 
 
 10 4 , 10 s , 10 9 , 10 1 , 10, 10~', 10-*, 10-*, &c. 
 Or 10000, 1000, 100, 10, 1, -^, rhr, rsW, &c. 
 
 Whose logarithms are 
 4, 3, 2, 1, 0, 1, 2, 3, <fec. 
 
 4. The fractional exponents of roots, and of powers of 
 roots, are converted into decimals, before they are inserted 
 in the logarithmic tables. See Alg. 208. 
 
 The logarithm of a*, or a ' 3 8 8 , is 0.3333, 
 of eft, or a - 68 ' 6 , is 0.6666, 
 of a*, or a ' 4986 , is 0.4285, 
 of a^ora 8 - 86 ' 6 , is 3.6666, <fec. 
 
 These decimals are carried to a greater or less number of 
 places, according to the degree of accuracy required. 
 
 5. In forming a system of logarithms, it is necessary to 
 obtain the logarithm of each of the numbers in the natural 
 series 1, 2, 3, 4, 5, <fec. ; so that the logarithm of any num- 
 ber may be found in the tables. For this purpose, the radix 
 of the system must first be determined upon ; and then every 
 other number may be considered as some power or root of 
 
NATURE 07 LOGARITHMS. ft 
 
 this. If the radix is 10, as in the common system, every 
 other number is to be considered as some power of 10. 
 
 If the exponent is a fraction, and the numerator be in- 
 creased, the power will be increased ; but if the denominator 
 be increased, the power will be diminished. 
 
 6. To obtain then the logarithm of any number, according 
 to Briggs's system, we have to find a -power or root of 10 
 which shall be equal to the proposed number. The exponent 
 of that power or root is the logarithm required. Thus 
 
 . 84ft 1 
 
 7=10 
 20=10 
 30=10 
 400=10 9 - 6080 
 
 1.8010 
 
 . 4 T7 1 
 
 of 7 is 0.8451 
 therefore the I of 20 is 1.3010 
 logarithm ] of 30 is 1.4771 
 
 .of 400 is 2.6020, &c. 
 
 7. A logarithm generally consists of two parts, an integer 
 and a decimal. Thus the logarithm 2.60206, or, as it is 
 sometimes written, 2 +.60206, consists of the integer 2, and 
 the decimal .60206. The integral part is called the charac- 
 teristic or index* of the logarithm ; and is frequently omitted, 
 in the common tables, because it can be easily supplied, 
 whenever the logarithm is to be used in calculation. 
 
 By art. 3d, the logarithms of 
 
 10000, 1000, 100, 10, 1, .1, .01, .001, &c. 
 are 4, 3, 2, 1, 0, -1, 2, 3, &c. 
 
 As the logarithms of 1 and of 10 are and 1, it is evident, 
 that, if any given number be between 1 and 10, its logarithm 
 will be between and 1, that is, it will be greater than 0, 
 but less than 1. It will therefore have for its index, with 
 a decimal annexed. 
 
 Thus, the logarithm of 5 is 0.69897. 
 
 * The term index, as it is used here, may possibly lead to some con- 
 fusion in the mind of the learner. For the logarithm itself is the index 
 or exponent of a power. The characteristic, therefore, is the index of 
 an index. 
 
JO NATURE OF LOGARITHMS* 
 
 For the same reason, if the given number be between 
 
 10 and 100, \ the log. ( 1 and 2, i.e. 1+the dec. part, 
 
 100 and 1000, > will be ) 2 and 3, 2+the dec. part. 
 
 1000 and 10000, ) between ( 3 and 4, 3-f-the dec. part. 
 
 We have, therefore, when the logarithm of an integer or 
 mixed number is to be found, this general rule : 
 
 8. The index of the logarithm is always one less, than the 
 number of integral figures, in the natural number whose logo- 
 rithm is sought : or, the index shows how far the first figure 
 of the natural number is removed from the place of units. 
 
 Thus, the logarithm of 37 is 1.56820. 
 
 Here, the number of figures being two, the index of the 
 logarithm is 1. 
 
 The logarithm of 253 is 2.40312. 
 
 Here the proposed number 253 consists of three figures, 
 the first of which is in the second place from the unit figure. 
 The index of the logarithm is therefore 2. 
 
 The logarithm of 62.8 is 1.79796. 
 
 Here it is evident that the mixed number 62.8 is between 
 10 and 100. The index of its logarithm must, therefore, 
 bel. 
 
 9. As the logarithm of 1 is 0, the logarithm of a number 
 less than 1, that is, of any proper frac tion, must be negative. 
 
 Thus, by art. 3d, 
 
 The logarithm of iV or .1 is 1, 
 of To7 or .01 is 2, 
 of rsVo or .001 is 3, &c. 
 
 10. If the proposed number is between TT and io\ , its 
 logarithm must be between 2 and 3. To obtain the 
 logarithm, therefore, we must either subtract a certain frac- 
 tional part from 2, or add a fractional part to 3 ; that 
 
NATURE OF LOGARITHMS. 11 
 
 is, we must either annex a negative decimal to 2, or a pos- 
 itive one to 3. 
 
 Thus, the logarithm 
 
 of .008 is either 2 .09691, or 3 + 90309.* 
 
 The latter is generally most convenient in practice, and is 
 more commonly written 3.90309. The line over the index 
 denotes, that that is negative, while the decimal part of th<* 
 logarithm is positive, 
 
 f of 0.3, is T47712, 
 
 The logarithm ) of 0.06, is J2777815, 
 
 ( of 0.009, is 3.95424, 
 
 And universally, 
 
 11. The negative index of a logarithm shows how far the 
 first significant figure of the natural number, is removed from, 
 the place of units, on the right ; in the same manner as a pos- 
 itive index shows how far the first figure of the natural num- 
 ber is removed from the place of units on the left. (Art. 8.) 
 Thus, in the examples in the last article, 
 
 The decimal 3 is in the first place from that of units, 
 6 is in the second place, 
 9 is in the third place ; 
 
 And the indices of the logarithms are 1, 2, and 3. 
 
 12. It is often more convenient, however to make the in- 
 dex of the logarithm positive, as well as the decimal part. 
 This is done by adding 10 to the index. 
 
 Thus, for 1, 9 is written, for 2, 8, &c. 
 Because 1 + 10=9, 2+10=8, <fec. 
 
 * That these two expressions are of the same value will be evident, 
 if we subtract the same quantity, +.90309 from each. The remainders 
 will be equal, and therefore the quantities from which the subtraction is 
 made must be equal. 
 
12 NATURE OF LOGARITHMS. 
 
 With this alteration, 
 
 i T.90309 \ f 9.90309, 
 
 The logarithm 5 .90309 > becomes j 8.90309, 
 
 ( .90309 ) ( 7.90309, &c. 
 
 This is making the index of the logarithm 10 too great. 
 But with proper caution, it will lead to no error in practice. 
 
 13. The sum of the logarithms of two numbers, is the 
 logarithm of the product of those numbers ; and the differ- 
 ence of the logarithms of two numbers, is the logarithm of 
 the quotient of one of the numbers divided by the other. 
 (Art. 2.) In Briggs's system, the logarithm of 10 is 1. (Art. 
 3.) If therefore any number be multiplied or divided by 10, 
 its logarithm will be increased or diminished by 1 : and as 
 this is an integer, it will only change the index of the loga- 
 rithm, without affecting the decimal part. 
 
 Thus, the logarithm of 4730 is 3.67486 
 And the logarithm of 10 is 1. 
 The logarithm of the product 47300 is 4.67486 
 And the logarithm of the quotient 473 is 2.67486 
 
 Here the index only is altered, while the decimal part re- 
 mains the same. We have then this important property, 
 
 14. The DECIMAL PART of the logarithm of any number is 
 the same, as that of the number multiplied or divided by 10, 
 100, 1000, &c. 
 
 Thus the log. of 45670, is 4.65963, 
 
 4567, 3.65963, 
 
 456.7, 2.65963, 
 
 45.67, 1.65963, 
 
 4.567, a65963, 
 
 .4567, T.65963, or 9.65963, 
 
 .04567, 65963, 8.65963, 
 
 .004567, 3765963, 7.65963 
 
 property, which is peculiar to Briggs's system, is ot 
 
NATURE OF LOGARITHMS. 13 
 
 great use in abridging the logarithmic tables. For when 
 we have the logarithm of any number, we have only to 
 change the index, to obtain the logarithm of every other 
 number, whether integral, fractional, or mixed, consisting of 
 the same significant figures. The decimal part of the loga- 
 rithm of a fraction found in this way, is always positive. For 
 it is the same as the decimal part of the logarithm of a 
 whole number. 
 
 IV. If a series of numbers be in GEOMETRICAL progression, 
 their logarithms will be in ARITHMETICAL progression. For, 
 in a geometrical series ascending, the quantities increase by 
 a common multiplier ; (Alg. 359.) That is, each succeeding 
 term is the product of the preceding term into the ratio. 
 But the logarithm of this product is t) e sum of the logarithms 
 of the preceding term and the ratio ; that is, the logarithms 
 increase by a common addition, and are, therefore, in arith- 
 metical progression. (Alg. 326.) In a geometrical progres- 
 sion descending, the terms decrease by a common divisor, and 
 their logarithms, by a common difference* 
 
 Thus, the numbers 1, 10, 100, 1000, 10000, &c., are in 
 geometrical progression. 
 
 And their logarithms 0, 1, 2, 3, 4, &c., are in arithmetical 
 progression. 
 
 See Note A. 
 2 
 
14 THE LOGARITHMIC TABLES. 
 
 SECTION II. 
 
 DIRECTIONS FOR TAKING LOGARITHMS AND THEIR NUM- 
 BERS FROM THE TABLES.* 
 
 ART. 24. The purpose which logarithms are intended to 
 answer, is to enable us to perform arithmetical operations 
 with greater expedition, than by the common methods. Be- 
 fore any one can avail himself of this advantage, he must 
 become so familiar with the tables, that he can readily find 
 the logarithm of any number ; and, on the other hand, the 
 number to which any logarithm belongs. 
 
 In the common tables, the indices to the logarithms of the 
 first 100 numbers are inserted. But, for all other numbers, 
 the decimal part only of the logarithm is given ; while the 
 index is left to be supplied, according to the principles in 
 Arts. 8 and 11. 
 
 25. To find the logarithm of any number between 1 and 
 100: 
 
 Look for the proposed number, on the left ; and against it, 
 in the next column, will be the logarithm, with its index 
 Thus, 
 
 The log. of 18 is 1.25527. The log. of 73 is 1.86332. 
 
 26. To find the, logarithm of any number between 100 and 
 1000 ; or of any number consisting of not more than three 
 significant figures, with ciphers annexed. 
 
 * The best English Tables are Button's in 8vo. and Taylor's in 4to. 
 In these, the logarithms are carried to seven places of decimals, and 
 proportional parts are placed in the margin. The smaller tables are nu- 
 merous ; and, when accurately printed, are sufficient for common cal- 
 culations. 
 
THE LOGARITHMIC TABLES. 16 
 
 In the smaller tables, the three first figures of each num- 
 ber, are generally placed in the left hand column ; and the 
 fourth figure is placed at the head of the other columns. 
 
 Any number, therefore, between 100 and 1000, may be 
 found on the left hand ; and directly opposite, in the next 
 column, is the decimal part of its logarithm. To this the 
 index must be prefixed, according to the rule in Art. 8. 
 
 The log. of 458 is 2.66087, The log. of 935 is 2.97081, 
 of 796 2.90091, of 386 2.58659. 
 
 If there are ciphers annexed to the significant figures, the 
 logarithm may be found in a similar manner. For, by Art. 
 14, the decimal part of the logarithm of any number is the 
 same, as that of the number multiplied into 10, 100, &c. 
 All the difference will be in the index ; and this may be sup- 
 plied by the same general rule. 
 
 The log. of 4580 is 3.66087, The log. of 326000 is 5.51322, 
 of 79600 4.90091, of 8010000 6.90363. 
 
 27. To find the logarithm of any number consisting of 
 FOUR figures, either with, or without, ciphers annexed. 
 
 Look for the three first figures, on the left hand, and for 
 the fourth figure, at the head of one of the columns. The 
 logarithm will be found, opposite the three first figures, and 
 in the column which, at the head, is marked with the fourth 
 figure.* 
 
 The log. of 6234 is 3.79477, The log. of 788400 is 5.89398, 
 of 5231 3.71858, of 6281000 6.79803. 
 
 28. To find tlie logarithm of a number containing MORE 
 than FOUR significant figures. 
 
 By turning to the tables, it will be seep, that if the differ- 
 ences between several numbers be small, in comparison with 
 the numbers themselves ; the differences of the logarithms 
 
 * In Taylor's, Hutton's, and other tables, four figures are placed in 
 the left hand column, and the fifth at the top of the page. 
 
10 THE LOGARITHMIC TABLES. 
 
 will be nearly proportioned to the differences of the numbers. 
 Thus, 
 
 The log. of 1000 is 3.00000, Here the differences in the 
 of 1001 3.00043, numbers are, 1, 2, 3, 4, &c., 
 of 1002 3.00087, and the corresponding dif- 
 of 1003 3.00130, ferences in the logarithms, 
 of 1004 3.00173, <fec. are 43, 87, 130, 173, &c. 
 
 Now 43 is nearly half of 87, one-third of 130, one-fourth 
 of 173, &c 
 
 Upon this principle, we may find the logarithm of a num- 
 ber which is between two other numbers whose logarithms 
 are given by the tables. Thus, the logarithm of 21716 is 
 not to be found in those tables which give the numbers to 
 four places of figures only. 
 
 But by the table, the log. of 21720 is 4.33686 
 and the log. of 21710 is 4.33666 
 
 The difference of the two numbers is 10 ; and that of the 
 logarithms 20. 
 
 Also, the difference between 21710, and the proposed num- 
 ber 21716, is 6. 
 
 If, then, a difference of 10 in the numbers 
 make a difference of 20 in the logarithms : 
 
 A difference of 6 in the numbers will 
 make a difference of 12 in the logarithms. 
 
 That is, 10 : 20 : : 6 : 12. 
 
 If, therefore, 12 be added to 4.33666, the log. of 21710 ; 
 The sum will be 4.33678, the log. of 21716. 
 
 We have, then, this 
 
 RULE. 
 
 To find the logarithm of a number consisting of more than 
 four figures : 
 
THE LOGARITHMIC TABLES. 17 
 
 Take out the logarithm of two numbers, one greater, and 
 the other less, than the number proposed : Find the differ- 
 ence of the two numbers, and the difference of their loga- 
 rithms : Take also the difference between the least of the 
 two numbers, and the proposed number. Then say, 
 As the difference of the two numbers, 
 To the difference of their logarithms ; 
 So is the difference between the least of the two nunv 
 
 bers, and the proposed number, 
 To the proportional part to be added to 
 
 the least of the two logarithms. 
 
 It will generally be expedient to make \hefirstfourfigurc8, 
 in the least of the two numbers, the same as in the proposed 
 number, substituting ciphers, for the remaining figures ; and 
 to make the greater number the same as the less, with the 
 addition of a unit to the last significant figure. Thus, 
 
 For 36843, take 36840, and 36850, 
 For 792674, 792600, 792700, 
 
 For 6537825, 6537000, 6538000, &c. 
 
 The first term of the proportion will then be 10, or 100, 
 or 1000, &c. 
 
 Ex. 1. Required the logarithm of 362572. 
 
 The logarithm of 362600 is 5.55943 
 of 362500 5.55931 
 The differences are 100, and 12. 
 
 Then 100 : 12 : : 72 : 8.64, or 9 nearly. 
 And the log. 5.55931+9=5.55940, the log. required. 
 
 Ex. 2. The log. of 78264 is 4.89356 
 
 3. The log. of 143542 is 5.15698 
 
 4. The log. of 1129535 is 6.05290. 
 
 By a little practice, such a facility in abridging these cal- 
 culations may be acquired, that the logarithms may be taken 
 
 2* 
 
18 THE LOGARITHMIC TABLES. 
 
 out, in a very short time. When great accuracy is not re- 
 quired, it will be easy to make an allowance sufficiently near, 
 without formally stating a proportion. In the larger tables, 
 the proportional parts which are to be added to the loga- 
 rithms, are already prepared, and placed in the margin. 
 
 29. To find the logarithm of a DECIMAL FRACTION. 
 
 The logarithm of a decimal is the same as that of a whole 
 number, excepting the index. (Art. 14.) To find then the 
 logarithm of a decimal, take out that of a whole number 
 consisting of the same figures ; observing to make the neg- 
 ative index equal to the distance of the first significant figure 
 of the fraction from the place of units. (Art. 11.) 
 
 The log. of 0.07643, is "^88326, or 8.88326, (Art. 12.) 
 of 0.00259, |[41330, or 7.41330, 
 of 0.0006278, 4/79782, or 6.79782. 
 
 30. To find the logarithm of a MIXED decimal number. 
 
 Find the logarithm, in the same manner as if all the fig- 
 ures were integers ; and then prefix the index which belongs 
 to the integral part, according to Art. 8. 
 
 The logarithm of 26.34 is 1.42062. 
 
 The index here is 1, because 1 is the index of the loga- 
 rithm of every number greater than 10, and less than 100. 
 (Art. 7.) 
 
 The log. of 2.36 is 0.37291, The log. of 364.2 is 2.56134, 
 of 27.8 1.44404, of 69.42 1.84148. 
 
 31. To find the logarithm of a VULGAR FRACTION. 
 
 From the nature of a vulgar fraction, the numerator may 
 be considered as a dividend, and the denominator as a divi- 
 sor ; in other words, the value of the fraction is equal to 
 the quotient of the numerator divided by the denominator. 
 (Alg. 110.) But in logarithms, division is performed by sub- 
 traction ; that is, the difference of the logarithms of two num- 
 bers, is the logarithm of the quotient of those numbers. 
 
THE LOGARITHMIC TABLES. 19 
 
 (Art. 1 .) To find then the logarithm of a vulgar fraction, 
 subtract the logarithm of the denominator from that of the nu- 
 merator. The difference will be the logarithm of the frac- 
 tion. Or the logarithm may be found, by first reducing the 
 vulgar fraction to a decimal. If the numerator is less than 
 the denominator, the index of the logarithm must be neg- 
 ative, because the value of the fraction is less than a unit. 
 (Art. 9.) 
 
 Required the logarithm of ff . 
 
 The log. of the numerator is 1.53148 
 of the denominator 1.93952 
 
 of the fraction J..591 96, or 9.59196. 
 
 The logarithm of VVW is 2^66362, or 8.66362. 
 
 of T^TS 1T04376, or 7.04376. 
 
 32. If the logarithm of a mixed number is required, re- 
 duce it to an improper fraction, and then proceed as before. 
 
 The logarithm of 3$=^ is 0.57724. 
 
 33. To find the NATURAL NUMBER belonging to any loga- 
 rithm. 
 
 In computing by logarithms, it is necessary, in the first 
 place, to take from the tables the logarithms of the numbers 
 which enter into the calculation ; and, on the other hand, at 
 the close of the operation, to find the number belonging to 
 the logarithm obtained in the result. This is evidently done 
 by reversing the methods in the preceding articles. 
 
 Where great accuracy is not required, look in the tables 
 for the logarithm which is nearest to the given one ; and di- 
 rectly opposite on the left hand, will be found the three first 
 figures, and at the top, over the logarithm, the fourth figure 
 of the number required. This number, by pointing off dec- 
 
20 THE LOGARITHMIC TABLES. 
 
 imals, or by adding ciphers, if necessary, must be made to 
 correspond with the index of the given logarithm, according 
 to Arts. 8 and 11. 
 
 The natural number belonging 
 
 to 3.86493 is 7327, to 1.62572 is 42.24, 
 to 2.90141 796.9, toT.89115 0.07783. 
 
 In the last example, the index requires that the first signi- 
 ficant figure should be in the second place from units, and 
 therefore a cipher must be prefixed. In other instances, it 
 is necessary to annex ciphers on the right, so as to make 
 the number of figures exceed the index by 1. 
 
 The natural number belonging 
 
 to 6.71567 is 5196000, to ^65677 is 0.004537, 
 to 4.67062 46840, to 4~59802 0.0003963. 
 
 34. When great accuracy is required, and the given loga- 
 rithm is not exactly, or very nearly, found in the tables, it 
 will be necessary to reverse the rule in Art. 28. 
 
 Take from the tables two logarithms, one the next greater, 
 the other the next less than the given logarithm. Find 
 the difference of the two logarithms, and the difference of 
 their natural numbers ; also the difference between the least 
 of the two logarithms, and the given logarithm. Then say, 
 As the difference of the two logarithms, 
 To the difference of their numbers ; 
 So is the difference between the given 
 
 logarithm and the least of the other two, 
 To the proportional part to be added to 
 the least of the two numbers. 
 
 Required the number belonging to the logarithm 2.67325. 
 Next great. log.2.67330. Its numb. 471.3. Given log. 2. 67325. 
 Next less 2.67321. Its numb. 471.2. Next less 2.67321. 
 Differences T 0.1 ~~4 
 
THE LOGARITHMIC TABLES. 21 
 
 Then, 9 : 0.1 : : 4 : 0.044, which is to be added 
 
 to the number 471.2 
 The number required is 471.244. 
 
 The natural number belonging 
 
 to 4.37627 is 23783.45, to JL73698 is 54.57357, 
 to 3.69479 4952.08, to T09214 0.123635. 
 
 35. Correction of the Tables. The tables of logarithms 
 have been so carefully and so repeatedly calculated, by the 
 ablest computers, that there is no room left to question their 
 general correctness. They are not, however, exempt from 
 the common imperfections of the press. But an error of this 
 kind is easily corrected, by comparing the logarithm with 
 any two others to whose sum or difference it ought to be 
 equal. (Art. 1.) 
 
 Thus 48=24X2=16X3=12X4=8X6. Therefore, the 
 logarithm of 48 is equal to the sum of the logarithms of 24 
 and 2, of 16 and 3, &c. 
 
 And, 3=f =Y=- L =-V l =-y-, &c. Therefore, the loga- 
 rithm of 3 is equal to the diference of the logarithms of 6 
 and 2, of 12 and 4, &c. 
 
22 MULTIPLICATION BY LOGARITHMS. 
 
 SECTION III. 
 
 RETHODS OF CALCULATING BY LOGARITHMS. 
 
 ART. 36. The arithmetical operations for which logarithms 
 were originally contrived, and on which their great utility 
 depends, are chiefly multiplication, division, involution, evo- 
 lution, and finding the term required in single and compound 
 proportion. The principle on which all these calculations 
 are conducted, is this : 
 
 If the logarithms of two numbers be added, the SUM will be 
 the logarithm of the PRODUCT of the numbers ; and, 
 
 If the logarithm of one number be subtracted from that of 
 another, the DIFFERENCE will be the logarithm of the QUOTIENT 
 of one of the numbers divided by the other. 
 
 In proof of this, we have only to call to mind, that loga- 
 rithms are the EXPONENTS of a series of powers and roots. 
 (Arts. 2, 5.) And it has been shown, that powers and roots 
 are multiplied by adding their exponents ; and divided, bv 
 subtracting their exponents. (Alg. 189, 193, 232, 239.) 
 
 MULTIPLICATION BY LOGARITHMS. 
 
 37. ADD THE LOGARITHMS OF THE FACTORS '. THE SUM 
 WILL BE THE LOGARITHM OF THE PRODUCT. 
 
 In making the addition, 1 is to be carried for every 10, 
 from the decimal part of the logarithm, to the index. (Art. 7.) 
 
 Numbers. Logarithms. Numbers. Logarithms. 
 
 Mult. 36.2 (Art. 30.) 1.55871 Mult 640 2.80618 
 
 Into 7.84 0.89432 Into 2.316 0.36474 
 
 Prod. 283.8 "2745303 Prod. 1482 3.17092 
 
 The logarithms of the two factors are taken from the 
 
MULTIPLICATION BY LOGARITHMS. 23 
 
 tables. The product is obtained, by finding, in the tables, 
 the natural number belonging to the sum. (Art. 33.) 
 
 Mult. 89.24 1.95056 Mult. 134. 2.12710 
 
 Into 3.687 0.56667 Into 25.6 1.40824 
 
 Prod. 329. 2.51723 Prod. 3430 3.53534 
 
 38. When any or all of the indices of the logarithms are 
 negative, they are to be added according to the rules for the 
 addition of positive and negative quantities in algebra. But 
 it must be kept in mind, that the decimal part of the loga- 
 rithm is positive. (Art. 10.) Therefore, that which is car- 
 ried from the decimal part to the index, must be considered 
 positive^also. 
 
 Mult. 62.84 J..79824 Mult. 0.0294 .46835 
 
 Into 0.682 T.83378 Into 0.8372 T.92283 
 
 Prod. 42.86 1.63202 Prod. 0.0246 ^39118 
 
 In each of these examples, +1 is to be carried from the 
 decimal part of the logarithm. This, added to 1, the 
 lower index, makes it ; so that there is nothing to be 
 added to the upper index. 
 
 If any perplexity is occasioned, by the addition of positive 
 and negative quantities, it may be avoided, by borrowing 10 
 to the index. (Art. 12.) 
 
 Mult. .62.84 1.79824 Mult. 0.0294 8.46835 
 
 Into 0.682 9.83378 Into 0.8372 9.92283 
 
 Prod. 42.86 1.63202 Prod. 0.0246 8.39118 
 
 Here 10 is added to the negative indices, and afterwards 
 rejected from the index of the sum of the logarithms. 
 
 Multiply 26.83 1.42862 1.42862 
 
 Into 0.00069 4~83885 or 6.83885 
 
 Product 0.01851 726747 8.26747 
 
24 MULTIPLICATION BY LOGARITHMS. 
 
 Here +1 carried to 4 makes it 3, which added to the 
 upper index +1, gives 2 for the index of the sum. 
 
 Multiply .00845 1T92686 or 7.92686 
 Into 1068. 3.02857 3.02857 
 
 Product 9.0246 0.95543 0.95543 
 
 The product of 0.0362 into 25.38 is 0.9188 
 of 0.00467 into 348.1 is 1.626 
 of 0.0861 into 0.00843 is 0.0007258 
 
 39. Any number of factors may be multiplied together, by 
 adding their logarithms. If there are several positive, and 
 several negative indices, these are to be reduced to one, as in 
 algebra, by taking the difference between the sum of those 
 which are negative, and the sum of those which are positive, 
 increased by what is carried from the decimal part of the 
 logarithms. (Alg. 53.) 
 
 Multiply 6832 
 Into 0.00863 
 
 3.83455 
 3.93601 
 
 3.83455 
 or 7.93601 
 
 And 0.651 
 
 1.81358 
 
 9.81358 
 
 And 0.0231 
 
 1.36361 
 
 or 8.36361 
 
 And 62.87 
 
 1.79844 
 
 1.79844 
 
 Prod. 55.74 
 
 1.74619 
 
 1.74619 
 
 Ex. 2. The prod, of 36.4x7.82X68.91X0.3846 is 7544. 
 
 3. The prod, of 0.00629X2.647X0.082X278.8X0.00063 
 is 0.0002398. 
 
 40. Negative quantities are multiplied, by means of loga- 
 rithms, in the same manner as those which are positive. (Art. 
 16.) But, after the operation is ended, the proper sign must 
 be applied to the natural number expressing the product, 
 according to the rules for the multiplication of positive and 
 negative quantities in algebra. The negative index of a loga- 
 rithm, must not be confounded with the sign which denotes 
 that the natural number is negative. That which the index 
 
DIVISION BY LOGARITHMS. 25 
 
 of the logarithm is intended to show, is not whether the 
 natural number is positive or negative, but whether it is 
 greater or less than a unit. (Art. 16.) 
 
 Mult. +36.42 1.56134 Mult. 2.681 0.42830 
 Into 67.31 1.82808 Into +37.24 1.57101 
 Prod. 2451 3.38942 Prod. 99.84 1.99931 
 
 In these examples, the logarithms are taken from the 
 tables, and added, in the same manner, as if both factors 
 were positive. But after the product is found, the negative 
 sign is prefixed to it, because + is multiplied into . (Alg. 
 82.) 
 
 Mult. 0.263 T.41996 Mult. 0.065 2^81291 
 
 Into 0.00894 T.95134 Into 0.693 T84073 
 Prod. 0.002351 lf.37130 Prod. 0.04504 1^65364 
 
 Here the indices of the logarithms are negative, but the 
 product is positive, because the factors are both positive. 
 
 Mult. 62.59 _L79650 Mult 68.3 1.83442 
 Into 0.00863 ".93601 Into 0.0096 ~3.98227 
 Prod. +0.5402 T.73251 Prod. +0.6557 T.81669 
 
 DIVISION BY LOGARITHMS. 
 
 41. FROM THE LOGARITHM OF THE DIVIDEND, SUB- 
 TRACT THE LOGARITHM OF THE DIVISOR; THE DIF- 
 FERENCE WILL BE THE LOGARITHM OF THE QUO- 
 
 1'IENT. (Art. 36.) 
 
 Logarithms. 
 2.95245 
 0.99330 
 1.95915 
 
 42. The decimal part of the logarithm may be subtracted 
 
 8 
 
 Numbers. 
 
 Divide 6238 
 
 Logarithms. 
 
 3.79505 
 
 Numbers. 
 
 Divide 896.3 
 
 By 2982 
 
 3.47451 
 
 By 9.847 
 
 Cluot. 2.092 
 
 0.32054 
 
 Quot. 91.02 
 
26 
 
 DIVISION BY LOGARITHMS. 
 
 as in common arithmetic. But for the indices, -when either 
 <of them is negative, or the lower one is greater than the 
 upper one, it will be necessary to make use of the general 
 rule for subtraction in algebra ; that is, to change the signs of 
 the subtrahend, and then proceed as in addition. (Alg. 60.) 
 When 1 is carried from the decimal part, this is to be con- 
 sidered affirmative, and applied to the index, before the sign 
 is changed. 
 
 Divide 0.8697 T.93937 or 9.93937 
 By 98.65 1.99410 1.99410 
 
 Quot. 0.008816 3^94527 7.94527 
 
 In this example, the upper logarithm being less than the 
 lower one, it is necessary to borrow 10, as in other cases of 
 subtraction ; and therefore to carry one to the lower indeT, 
 which then becomes +2. This changed to 2, and added 
 to 1 above it, makes the index of the difference of the 
 logarithms 3. 
 
 Divide 29.76 1.47363 1.47363 
 
 By 6254 3.79616 3.79616 
 
 Quot. 0.00476 3~67747 or 7.67747 
 
 Here, 1 carried to the lower index, makes it +4. This 
 changed to 4, and added to 1 above it, gives 3 for the 
 index of the difference of the logarithms. 
 
 Divide 6.832 - _0.8S455 Divide 0.00634 "80209 
 By .0362 12.55871 By 62.18 1.79365 
 
 Quot. 188.73 2.27584 Quot. 0.000102 Z00844 
 
 The quotient of 0.0985 divided by 0.007241, is 13.6 
 The quotient of 0.0621 divided by 3.68, is 0.01687 
 
 43. To divide negative quantities, proceed in the same 
 manner as if they were positive, (Art. 40.) and prefix to the 
 
INVOLUTION BY LOGARITHMS. 27 
 
 quotient, the sign which is required by the rules for division 
 in algebra. 
 
 Divide +3642 3.56134 Divide 0.657 T81757 
 By 23.68 1.37438 By +0.0793 2.89927 
 Quot. 153.8 2.18696 Quot. 8.285 0.91830 
 
 In these examples, the sign of the divisor being different 
 from that of the dividend, the sign of the quotient must be 
 negative. (Alg. 100.) 
 
 Divide 0.364 T.56110 Divide 68.5 1.83569 
 By 2.56 0.40824 By +0.094 .97313 
 Quot. +0.1422 Tl5286 Quot. 728.7 2.86256 
 
 INVOLUTION BY LOGARITHMS. 
 
 44. Involving a quantity is multiplying it into itself. By 
 means of logarithms, multiplication is performed by addition. 
 If, then, the logarithm of any quantity be added to itself, the 
 logarithm of a power of that quantity will be obtained. But 
 adding a logarithm, or any other quantity, to itself, is mul- 
 tiplication. The involution of quantities, by means of loga- 
 rithms, is therefore performed, by multiplying the logarithms. 
 
 Thus the logarithm 
 
 of 100 is 2 
 
 of 100X100, that is, of 100* is 2+2 =2X2 
 
 of 100X100X100, To0 8 is2+2+2 =2X3 
 
 of 100X100X100X100,100* is 2+2+2+2 =2X4 
 
 On the same principle, the logarithm of 100 n is 
 And the logarithm of x n , is (log. x)Xn. Hence, 
 
 45. To involve a quantity by logarithms, MULTIPLY 
 
 THE LOGARITHM OF THE QUANTITY, BY THE INDEX OF THE 
 POWER REQUIRED. 
 
28 INVOLUTION BY LOGARITHMS. 
 
 The reason of the rule is also evident, from the considera- 
 tion, that logarithms are the exponents of powers and roots, 
 and a power or root is involved, by multiplying its index 
 into the index of the power required. (Alg. 170, 242.) 
 
 Ex. 1. What is the cube of 6.296 ? 
 
 Root 6.296, its log. 0.79906 
 
 Index of the power 3 
 
 Power 249.6 2.39718 
 
 2. Required the 4th power of 21.32 
 Root 21.32 log. 1.32879 
 
 Index 4 
 
 Power 206614 5.31516 
 
 3. Required the 6th power of 1.689 
 Root 1.689 log. 0.22763 
 
 Index 6 
 
 Power 23.215 1.36578 
 
 4. Required the 144th power of 1.003 
 Root 1.003 log. 0.00130 
 
 Index 144 
 
 Power 1.539 0.18720 
 
 46. It must be observed, as in the case of multiplication, 
 (Art. 38.) that what is carried from the decimal part of the 
 logarithm is positive, whether the index itself is positive or 
 negative. Or, if 10 be added to a negative index, to render 
 it positive, (Art, 12.) this will be multiplied, as well as the 
 other figures, so that the logarithm of the square, will be 20 
 too great ; of the cube, 30 too great, &c. 
 
 Ex. 1. Required the cube of 0.0649 
 Root 0.0649 log. 2^81224 or 8.81224 
 
 Index 3 3 
 
 Power 0.0002733 4~43672 6.43672 
 
EVOLUTION BY LOGARITHMS. 29 
 
 2. Required the 4th power of 0.1234 
 
 Root 0.1234 log. T09132 or 9.09132 
 
 Index 4 4 
 
 Power 0.0002319 T.36528 6.36528 
 
 8. Required the 6th power of 0.9977 
 
 Root 0.9977 log. T.99900 or 9.99900 
 
 Index 6 6 
 
 Power 0.9863 T.99400 9.99400 
 
 4. Required the cube of 0.08762 
 
 Root 0.08762 log. 1F.94260 or 8.94260 
 
 Index 8 3 
 
 Power 0.0006727 T82780 6.82780 
 
 5. The 7th power of 0.9061 is 0.5015. 
 
 6. The 5th power of 0.9344 is 0.7123. 
 
 EVOLUTION BT LOGARITHMS. 
 
 
 
 47. Evolution is the opposite of involution. Therefore, 
 as quantities are involved, by the multiplication of logarithms, 
 roots are extracted by the division of logarithms ; that is, 
 
 To extract the root of a quantity by logarithms, DIVIDE 
 
 THE LOGARITHM OF THE QUANTITY, BY THE NUMBER EX- 
 PRESSING THE ROOT REQUIRED. 
 
 The reason of the rule is evident also, from the fact, that 
 logarithms are the exponents of powers and roots, and evo- 
 lution is performed, by dividing the exponent, by the number 
 expressing the root required. (Alg. 210.) 
 
 1. Required the square root of 648.3 
 
 Numbers. Logarithms. 
 
 Power 648.3 2)2.81178 
 
 Root 25.46 1.40589 
 
 3* 
 
30 EVOLUTION BY LOGARITHMS. 
 
 2. Required the cube root of 897.1 
 
 Power 897.1 3)2.95284 
 
 Root 9.645 0.98428 
 
 In the first of these examples, the logarithm of the give* 
 number is divided by 2 ; in the other, by 3. 
 
 8. Required the 10th root of 6948. 
 
 Power 6948 10)3.84186 
 
 Root 2.422 0.38418 
 
 4. Required the lOOdth root of 983. 
 
 Power 983 100)2.99255 
 Root 1.071 0.02992 
 
 The division is performed here, as in other cases of deci- 
 mals, by removing the decimal point to the left. 
 
 5. What is the ten thousandth root of 49680000 ? 
 
 Power 49680000 10000)7.69618 
 
 Root 1.00179 0.00077 
 
 We have, here, an example of the great rapidity with 
 which arithmetical operations are performed by logarithms. 
 
 48. If the index of the logarithm is negative, and is not 
 divisible by the given divisor, without a remainder, a diffi- 
 culty will occur, unless the index be altered. 
 
 Suppose the cube root of 0.0000892 is required. The 
 logarithm of this is 5.95036. If we divide the index by 3, 
 the quotient will be 1, with 2 remainder. This remain- 
 der, if it were positive, might, as in other cases of division, 
 be prefixed to the next figure. But the remainder is nega- 
 tive, while the decimal part of the logarithm is positive ; so 
 that, when the former is prefixed to the latter, it will make 
 neither +2.9 nor 2.9, but 2 + .9. This embarrassing 
 intermixture of positives and negatives may be avoided, by 
 adding to the index another negative number, to make it ex- 
 
EVOLUTION BY LOGARITHMS. 31 
 
 actly divisible by the divisor. Thus, if to the index 5 
 there be added 1, the sum 6 will be divisible by 3. But 
 this addition of a negative number must be compensated, by 
 the addition of an equal positive number, which may be pre- 
 fixed to the decimal part of the logarithm. The division 
 may then be continued, without difficulty, through the 
 whole. 
 
 Thus, if the logarithm ~5~.95036 be altered to ~6~+ 1.95036 
 it may be divided by 3, and the quotient will be 2.65012. 
 We have then this rule, 
 
 49. Add to the index, if necessary, such a negative num- 
 b& a& mil make it exactly divisible by the divisor, and prefix 
 an equal positive number to the decimal part of the logarithm. 
 
 1. Required the 5th root of 0.009642 
 
 Power 0.009642 log. 3^98417 
 
 or "6+2.98417 
 
 Root 0.3952 1.59683 
 
 2. Required the 7th root of 0.0004935. 
 
 Power 0.0004935 log. 1L69329 
 
 or 7)7+3.69329 
 
 Root 0.337 T.52761 
 
 50. If, for the sake of performing the division conven- 
 iently, the negative index be rendered positive, it will be ex- 
 pedient to borrow as many tens, as there are units in the 
 number denoting the root. 
 
 What is the fourth root of 0.03698 ? 
 
 Power 0.03698 4)^56797 or 4)38.56797 
 Root 0.4385 T.64199 9.64199 
 
 Here the index, by borrowing, is made 40 too great, that 
 is, +38 instead of 2. When, therefore, it is divided by 4, 
 it is still 10 too great, +9 instead of 1. 
 
32 PROPORTION BY LOGARITHMS. 
 
 What is the 5th root of 0.008926 ? 
 
 Power 0.008926 5)3^95066 or 5)47.95066 
 Hoot 0.38916 T.59013 9.59013 
 
 61. A power of a root may be found by first multiplying 
 the logarithm of the given quantity into the index of the 
 power, (Art. 45.) and then dividing the product by the 
 number expressing the root. (Art. 47.) 
 
 1. What is the value of (53)^, that is, the 6th power of 
 the 7th root of 53 ? 
 
 Given number 53 log. 1.72428 
 Multiplying by 6 
 
 Dividing by 7)10.34568 
 
 Power required 30.06 1.47795 
 
 2. What is the 8th power of the 9th root of 654 ? 
 
 PROPORTION BY LOGARITHMS. 
 
 52. In a proportion, when three terms are given, the 
 fourth is found in common arithmetic, by multiplying to- 
 gether the second and third, and dividing by the first. But 
 when logarithms are used, addition takes the place of mul- 
 tiplication, and subtraction, of division. 
 
 To find, then, by logarithms, the fourth term in a propor- 
 tion, ADD THE LOGARITHMS OF THE SECOND AND THIRD 
 
 TERMS, AND from the sum SUBTRACT THE LOGARITHM OF 
 THE FIRST TERM. The remainder will be the logarithm of 
 the term required. 
 
 Ex. 1. Find a fourth proportional to 7964, 378, and 27960. 
 
 Numbers. Logarithms. 
 
 Second term 378 2.57749 
 
 Third term 27960 4.44654 
 
 7.02403 
 
 First term 7964 3.90113 
 
 Fourth term 1327 3.12290 
 
ARITHMETICAL COMPLEMENT. $3 
 
 2. Find a 4th proportional to 768, 381, and 9780. 
 Second term 381 2.58092 
 
 Third term 9780 3.99034 
 
 6.57126 
 First term 768 2.88536 
 
 Fourth term 4852 3.68590 
 
 ARITHMETICAL COMPLEMENT. 
 
 53. When one number is to be subtracted from another, 
 it is often convenient, first to subtract it from 10, then to 
 add the difference to the other number, and afterwards to re- 
 ject the 10. 
 
 Thus, instead of a b, we may put 10 b-\-a 10. 
 
 In the first of these expressions, b is subtracted from a. 
 In the other, b is subtracted from 10, the difference is added 
 to a, and 10 is afterwards taken from the sum. The two ex- 
 pressions are equivalent, because they consist of the same 
 terms, with the addition, in one of them, of 10 10=0. The 
 alteration is, in fact, nothing more than borrowing 10, for the 
 sake of convenience, and then rejecting it in the result. 
 
 Instead of 10, we may borrow, as occasion requires, 100, 
 1000, &c. 
 
 Thus, a 6=100 6+a 100=1000 b+a 1000, &c. 
 
 54. The DIFFERENCE between a given number and 10, or 
 100, or 1000, &c. y is called the ARITHMETICAL COMPLEMENT 
 of that number. 
 
 The arithmetical complement of a number consisting of 
 one integral figure, either with or without decimals, is found, 
 by subtracting the number from 10. If there are two in- 
 tegral figures, they are subtracted from 100 ; if three, from 
 1000, &c. 
 
 Thus, the arithmetical compl't of 3.46 is 103.46=6.54 
 
 of 34.6 is 10034.6=65.4 
 of 346. is 1000 346.=654. &c. 
 
84 ARITHMETICAL COMPLEMENT. 
 
 According to the rule for subtraction in arithmetic, any 
 number is subtracted from 10, 100, 1000, &c. by beginning 
 on the right hand, and taking each figure from 10, after in- 
 creasing all except the first, by carrying 1. 
 
 Thus, if from 10.00000 
 
 We subtract 7.63125 
 
 The difference, or arith'l compl't is 2.36875, which is 
 obtained by taking 5 from 10, 3 from 10, 2 from 10, 4 from 
 10, 7 from 10, and 8 from 10. But, instead of taking each 
 figure, increased by 1 from 10 ; we may take it without 
 being increased, from 9. 
 
 Thus, 2 from 9 is the same as 3 from 10, 
 
 3 from 9 the same as 4 from 10, <fec. Hence, 
 
 55. To obtain the ARITHMETICAL COMPLEMENT of a num- 
 ber, subtract the right hand significant figure from 10, and 
 each of the other figures from 9. If, however, there are 
 ciphers on the right hand of all the significant figures, they 
 are to be set down without alteration. 
 
 In taking the arithmetical complement of a logarithm, if 
 the index is negative, it must be added to 9 ; for adding a 
 negative quantity is the same as subtracting a positive one. 
 (Alg. 81.) The difference between 3 and +9, is not 6, 
 but 12. 
 
 The arithmetical complement 
 
 of 6.24897 is 3.75103 of "2^70649 is 11.29351 
 of 2.98643 7.01357 of 3.64200 6.35800 
 of 0.62430 9.37570 of 9.35001 0.64999 
 
 66. The principal use of the arithmetical complement, is 
 in working proportions by logarithms ; where some of the 
 terms are to be added, and one or more to be subtracted. 
 In the Rule of Three or simple proportion, two terms are to 
 be added, and from the sum, the first term is to be sub- 
 tracted. But if, instead of the logarithm of the first term, 
 
COMPOUND PROPORTION. 35 
 
 we substitute its arithmetical complement, this may be added 
 to the sum of the other two, or more simply all three may- 
 be added together, by one operation. After the index is 
 diminished by 10, the result will be the same as by the com- 
 mon method. For subtracting a number is the same, as adding 
 its arithmetical complement, and then rejecting 10, 100, or 
 1000, from the sum. (Art. 53.) 
 
 It will generally be expedient, to place the terms in the 
 same order, hi which they are arranged in the statement of 
 the proportion, 
 
 1. As 6273 a. c. 6,20252 2. As 253 a. c. 7.59688 
 Is to 769.4 2.88615 Is to 672.5 2.82769 
 
 So is 37*61 1.57530 So is 497 2.69636 
 
 To 4.613 0.66397 * To 1321.1 3.12093 
 
 3. As 46.34 a. c. 8.33404 4. As 9.85 a. c. 9.00656 
 
 Is to 892.1 2.95041 Is to 643 2.80821 
 
 So is 7.638 0.88298 So is 76.3 1.88252 
 
 To 147 2.16743 To 4981 3.69729 
 
 COMPOUND PROPORTION, 
 
 57. In compound, as in single proportion, the term re- 
 quired may be found by logarithms, if we substitute additioD 
 for multiplication, and subtraction for division. 
 
 Ex. 1. If the interest of $365, for 3 years and 9 months, 
 be $82.13 ; what will be the interest of $8940, for 2 years 
 and 6 months ? 
 
 In common arithmetic, the statement of the question is 
 made in this manner. 
 
 365 dollars (940 dollars > . . 
 
 ) . 
 
 f 
 
 3.75 years f ( 2.5 years 
 
06 COMPOUND PROPORTION. 
 
 And the method of calculation is, to divide the product of 
 the third, fourth, and fifth terms, by the product of the first 
 two.* This, if logarithms are used, will be to subtract the 
 sum of the logarithms of the first two terms, from the sum 
 of the logarithms of the other three. 
 
 ( 365 log. 2.56229 
 First two terms j ^ . 57403 
 
 Sum of the logarithms 3.13632 
 
 8940 3.95134 
 
 Third and fourth terms 
 
 Fifth term 82.13 1.91450 
 
 Sum of the logs, of the 3rd, 4th, and 5th, 6.26378 
 
 Do. 1st and 2nd, 3.13632 
 
 Term required 1341 3.12746 
 
 58. The calculation will be more simple, if, instead of 
 subtracting the logarithms of the first two terms, we odd 
 their arithmetical complements. But, it must be observed, 
 that each arithmetical complement increases the index of the 
 logarithm by 10. If the arithmetical complement be intro- 
 duced into two of the terms, the index of the sum of the 
 logarithms will be 20 too great ; if it be in three terms, the 
 index will be 30 too great, &c. 
 
 _. A A I 365 a. c. 7.43771 
 
 First two terms j 3 . 75 a . c . 9<42597 
 
 t 8940 3.95134 
 Third and fourth terms ^ Q 
 
 Fifth term 82.13 1.91450 
 
 Term required 1341 23.12746 
 
 The result is the same as before, except that the index of 
 the logarithm is 20 too great. 
 
 > See Arithmetic. 
 
COMPOUND INTEREST. 3 
 
 Ex. 2. If the wages of 53 men for 42 days be 2200 dol- 
 lars ; what will be the wages of 87 men for 34 days ? 
 
 53 men ) ( 87 men ) 
 
 42days | : j 34 days { : : 22 ' 
 
 ( 53. a. c. 8.27572 
 First two terms j 4 2. a. c. 8.37675 
 
 . (87 1.93952 
 
 Third and fourth terms i . 153148 
 
 Fifth term 2200 3.34242 
 
 Term required 2923.5 3.46589 
 
 59. In the same manner, if the product of any number of 
 quantities, is to be divided, by the product of several others ; 
 we may add together the logarithms of the quantities to be 
 divided, and the arithmetical complements of the logarithms 
 of the divisors. 
 
 Ex. If 29.67X346.2 be divided by 69.24x7.862X497 ; 
 what will be the quotient ? 
 
 ( 29.67 1.47232 
 
 Numbers to be divided j 346 2 2 53933 
 
 ( 69.24 a.c. 8.15964 
 
 Divisors ! 7.862 a. c. 9.10447 
 
 ( 497 a.c. 7.30364 
 
 Quotient 0.03797 8.5794 
 
 In this way, the calculations in Conjoined Proportion may- 
 be expeditiously performed. 
 
 COMPOUND INTEREST. 
 
 60. In calculating compound interest, the amount for the 
 first year, is made the principal for the second year ; the 
 amount for the second year, the principal for the third 
 
 4 
 
38 COMPOUND INTEREST. 
 
 year, &c. Now the amount at the end of each year, must be 
 proportioned to the principal at the beginning of the year. If 
 the principal for the first year be 1 dollar, and if the amount 
 of 1 dollar for 1 year=a; then, (Alg. 341.) 
 
 ', : a a =the amount for the 2d year, or the prin- 
 cipal for the 3d ; 
 
 ; 9 : a 9 =the amount for the third year, or the 
 principal for the 4th ; 
 
 , 3 : a*=the amount for the 4th year, or the prin- 
 cipal for the 5th. 
 
 That is, the amount of 1 dollar for any number of years 
 is obtained by finding the amount for 1 year, and involving 
 this to a power whose index is equal to the number of years. 
 And the amount of any other principal, for the given time, 
 is found by multiplying the amount of 1 dollar, into the num- 
 ber of dollars, or the fractional part of a dollar. 
 
 If logarithms are used, the multiplication required here 
 may be performed by addition ; and the involution by mul- 
 tiplication. (Art. 45.) Hence, 
 
 61. To calculate Compound Interest, Find the amount of 
 1 dollar for I year; multiply its logarithm by the number of 
 years; and to the product, add the logarithm of the principal. 
 The sum will be the logarithm of the amount for the given 
 time. From the amount subtract the principal, and the re- 
 mainder will be the interest. 
 
 If the interest becomes due 'half yearly or quarterly ; find 
 the amount of one dollar, for the half year or quarter, and 
 multiply the logarithm by the number of half years or quar- 
 ters in the given time. 
 
 If P=the principal, 
 
 a=the amount of 1 dollar for 1 year, 
 w=any number of years, and 
 A=the amount of the given principal for n years ; then. 
 
COMPOUND INTEREST. 39 
 
 Taking the logarithms of both sides of the equation, and 
 reducing it, so as to give the value of each of the four quan- 
 tities, in terms of the others, we have 
 
 1. Log. A=nX log. a+ log. P. 
 
 2. Log. P=log. A nX log. a. 
 
 log. A log. P. 
 
 3. Log. a 
 
 4. n= 
 
 n 
 
 log. A log. P. 
 log a. 
 
 Any three of these quantities being given, the fourth may 
 be found. 
 
 Ex. 1. What is the amount of 20 dollars, at 6 percent, 
 compound interest, for 100 years ? 
 
 Amount of 1 dollar for 1 year 1.06 log. 0.0253059 
 Multiplying by 100 
 
 2.53059 
 
 Given principal 20 1.30103 
 
 Amount required $6786 3.83162 
 
 2. What is the amount of 1 cent at 6 per cent, compound 
 interest, in 500 years ? 
 
 Amount of 1 dollar for 1 year 1.06 log. 0.0253059 
 Multiplying by 500 
 
 12.65295 
 Given principal 0.01 -2.00000 
 
 Amount $44,973,000,000 10.65295 
 
 More exact answers may be obtained, by using logarithms 
 of a greater number of decimal places. 
 
 3. What is the amount of 1000 dollars, at 6 per cent, 
 compound interest, for 10 years? Ans, 1790.80. 
 
40 INCREASE OF POPULATION. 
 
 4. What principal, at 4 per cent, interest, will amount to 
 1643 dollars in 21 years? Ans. 721. 
 
 5. What principal, at 6 per cent., will amount to 202 dol- 
 lars in 4 years ? Ans. 160. 
 
 6. At what rate of interest, will 400 dollars amount to 
 669i, in 9 years ? Ans. 4 per cent. 
 
 7. In how many years will 500 dollars amount to 900, at 
 5 per cent, compound interest? Ans. 12 years. 
 
 8. In what time will 10,000 dollars amount to 16,288, at 
 5 per cent compound interest ? Ans. 10 years. 
 
 9. At what rate of interest, will 11,106 dollars amount to 
 20.000 in 15 years ? Ans. 4 per cent. 
 
 10. What principal, at 6 per cent, compound interest, will 
 amount to 3188 dollars in 8 years? Ans. $2000. 
 
 11. What will be the amount of 1200 dollars, at 6 per 
 cent compound interest, in 10 years, if the interest is con- 
 verted into principal every half year? Ans. 2167.3 dolls. 
 
 12. In what time will a sum of money double, at 6 per 
 cent compound interest ? Ans. 11.9 years. 
 
 13. What is the amount of 5000 dollars, at 6 per cent, 
 compound interest, for 28^- years ? Ans. 25.942 dollars. 
 
 INCREASE OF POPULATION. 
 
 62. The natural increase of population in a country, may 
 be calculated in the same manner as compound interest ; on 
 the supposition, that the yearly rate of increase is regularly 
 proportioned to the actual number of inhabitants. From 
 the population at the beginning of the year, the rate of in- 
 crease being given, may be computed the whole increase 
 during the year. This, added to the number at the begin- 
 ning, will give the amount, on which the increase of the 
 second year is to be calculated, in the same manner as the 
 
INCREASE OF POPULATION. 41 
 
 first year's interest on a sum of money, added to the sum 
 itself, gives the amount on which the interest for the second 
 year is to be calculated. 
 
 If P=the population at the beginning of the year, 
 
 a=l+the fraction which expresses the rate of increase, 
 =any number of years ; and 
 A=the amount of the population at the end of n ye 
 then, as in the preceding article, 
 
 A=a*xP, and 
 
 1. Log. A=rcXlog. a+log. P. 
 
 2. Log. P=log. A wXlog. a. 
 
 log. A log. P. 
 
 3. Log. a 
 
 4. n= 
 
 n 
 log. A log. P. 
 
 log. a. 
 
 Ex. 1. The population of the United States in 1840 was 
 (in round numbers) 1Y,OYO,000.* Supposing the yearly rate 
 
 * For some very interesting views of the progress of population, &c., 
 in the United States, see Prof. George Tucker's elaborate essays, first 
 published in the Merchant's Magazine, 1842 3, and subsequently in a 
 separate volume. 
 
 The following tables show the official census of the United States 
 from 1790 to 1840 with the decennial rate of increase. 
 
 POPULATION. 
 
 1790. | 1800. | 1810. | 1820. | 1830. | 1840. 
 3,929,827 | 5,305,925 | 7,239,814 | 9,638,131 | 12,866,020 | 17,069,453 
 
 DECENNIAL INCREASE. 
 
 1800. 
 
 | 1810. 
 
 1820. 
 
 1830. 
 
 | 1840. 
 
 35.02 
 
 | 36.45 
 
 | 33.35 
 
 | 33.26 
 
 | 33.67 
 
42 INCREASE OF POPULATION. 
 
 of increase to be -& part of the whole, what will be the 
 population in 1850 ? 
 
 Here P==l 7,070,000. n=lO. a=I-{~ 3 \^*. 
 
 And log. A=10Xlog. ff+log. (17,070,000,) 
 Therefore, A=22,810,000, the population in 1850. 
 
 2. If the number of inhabitants in a country be five mil- 
 lions at the beginning of a century ; and if the yearly rate 
 of increase be -V J what will be the number at the end of 50 
 years ? and what at the end of the century ? 
 
 Ans. 25,763,000, and 132,750,000. 
 
 3. If the population of a country, at the end of a century, 
 is found to be 45,860,000 ; and if the yearly rate of increase 
 has been -riir ; what was the population at the commence- 
 ment of the century ? Ans. 20 millions. 
 
 4. The population of the United States in 1810 was 
 7,240,000 ; in 1820, 9,625,000. What was the annual rate 
 of increase between these two periods, supposing the in- 
 crease each year to be proportioned to the population at the 
 beginning of the year ? 
 
 log. 9,625,000 log. 7,240,000 
 Here log. o= - 
 
 Therefore, o=1.029 ; and TlHhr> or 2.9 per cent, is the 
 rate of increase. 
 
 5. The population of the United States on the 1st August, 
 1820, was 9,638,000 in 1830, the time of taking the census 
 was changed to the 1st June, and at that time the popula- 
 tion was 12,866,000. What was the annual rate of increase ? 
 And what would have been the amount of population to be 
 added for the subsequent two months ? 
 
 6. In how many years, will the population of a country 
 advance from two millions to five millions ; supposing the 
 yearly rate of increase to be -3^ ? Ans. 47-J- years. 
 
INCREASE OF POPULATION. 43 
 
 7. If the population of a country, at a given time, be 
 seven millions ; and if the yearly rate of increase be ^th ; 
 what will be the population at the end of 35 years ? 
 
 8. The population of the United States in 1800 was 
 5,306,000. What was it in 1780, supposing the yearly rate 
 of increase to be -^ ? 
 
 9. In what time will the population of a country advance, 
 from four millions to seven millions, if the ratio of increase 
 
 10. What must be the rate of increase, that the population 
 of a place may change from nine thousand to fifteen thou- 
 sand, in 12 years? 
 
 If the population of a country is not affected by immi- 
 gration or emigration, the rate of increase will be equal to 
 the difference between the ratio of the birtfis, and the ratio 
 of the deaths, when compared with the whole population. 
 
 Ex. 11. If the population of a country, at any given time, 
 be ten millions ; and the ratio of the annual number of births 
 to the whole population be -jV, and the ratio of deaths -jV 
 what will be the number of inhabitants, at the end of 60 
 years? 
 
 Here the yearly rate of increase==^ l 4 3V=dhr. 
 
 And the population, at the end of 60 years=3 1,750,000. 
 
 The rate of increase or decrease from immigration or emi- 
 gration, will be equal to the difference between the ratio of 
 immigration and the ratio of emigration ; and if this differ- 
 be added to, or subtracted from, the difference between the 
 ratio of the births and that of the deaths, the whole rate 
 of increase will be obtained. 
 
 Ex. 12. If in a country, the ratio of births be -gV> 
 the ratio of deaths ^, 
 
 the ratio of immigration -5^, 
 the ratio of emigration eV, 
 
44 INCREASE OF POPULATION". 
 
 and if the population this year be 10 millions, what will it 
 be 20 years hence ? 
 
 The rate of the natural increase =- L fa 120 J 
 That of increase from immigration =5*0 Jo sio ; 
 The sum of the two is ^fo ; 
 
 -And the population at the end of 20 years, is 12,611,000. 
 
 13. If the ratio of the births be *V, 
 
 of the deaths -gV, 
 
 of immigration -jV, 
 
 of emigration -V, 
 
 in what time will three millions increase to four and a half 
 millions ? 
 
 If the period in which the population will double be given ; 
 the numbers for several successive periods, will evidently be 
 in a geometrical progression, of which the ratio is 2 ; and as 
 the number of periods will be one less than the number of 
 terms ; 
 
 If P=the first term, 
 
 A=the last term, 
 
 ?i=the number of periods ; 
 Then will A=Px2 n , (Alg. 439.) 
 Or log. A=log. P+wXlog. 2. 
 
 Ex 1 . If the descendants of a single pair double once in 
 25 years, what will be their number at the end of one thou- 
 sand years ? 
 
 The number of periods here is 40. 
 And A=2X2 40 =2,199,200,000,000. 
 
 2. If the descendants of Noah, beginning with his three 
 sons and their wives, doubled once in 20 years for 300 years, 
 what was their number, at the end of this time ? 
 
 Ans. 196,608. 
 
 8. The population of the United States in 1820 being 
 
EXPONENTIAL EQUATIONS. 45 
 
 9,638,000 ; what must it be in the year 2020, supposing it 
 to double once in 25 years ? Ans. 2,467,333,000. 
 
 4. Supposing the descendants of the first human pair to 
 double once in 50 years, for 1650 years, to the time of the 
 deluge, what was the population of the world, at that time ? 
 
 EXPONENTIAL EQUATIONS. 
 
 62. An EXPONENTIAL equation is one in which the letter 
 expressing the unknown quantity is an exponent. 
 
 Thus a x =b, and x x =bc, are exponential equations. These 
 are most easily solved by logarithms. As the two members 
 of an equation are equal, their logarithms must also be equal. 
 If the logarithm of each side be taken, the equation may 
 then be reduced, by the rules given in algebra. 
 
 Ex. What is the value of x in the equation 3*=243 ? 
 
 Taking the logarithms of both sides, log. 3 ar ==log. 243. 
 But the logarithm of a power is equal to the logarithm of 
 the root, multiplied into the index of the power. (Art. 45.) 
 
 Therefore (log. 3)x#=log. 243 ; and dividing by log. 3. 
 
 log. 243 2.38561 
 
 x= - = - =5. So that 3 6 =243. 
 log 3. 0.47712 
 
 64. The exponent of a power may be itself a power, as in 
 the equation 
 
 where x is the exponent of the power m*, which is the ex- 
 
 X 
 
 ponent of the power a m . 
 
 X 
 
 Ex. 4. Find the value of x, in the equation 9 8 =1000. 
 
 log. 1000. 
 3*X(log. 9) log, 1000. Therefore, 3*-^ Io & 
 
46 EXPONENTIAL EQUATIONS. 
 
 Then, as 3^=3.14. a-(log. 3)=log. 3.14 
 
 log. 3.14 
 
 Therefore, x= =^ffff=i.o4. 
 
 log. 3 
 
 In cases like this, where the factors, divisors, &c. are loga- 
 rithms, the calculation may be facilitated, by taking the 
 logarithms of the logarithms. Thus the value of the fraction 
 *?7?in is most easily found, by subtracting the logarithm 
 of the logarithm which constitutes the denominator, from 
 the logarithm of that which forms the numerator. 
 
 ba x +d 
 5. Find the value of x t in the equation ==m 
 
 log. (cm d) log. 6. 
 
 log. o. 
 
TRIGONOMETRY, 
 
 SECTION I. 
 
 SINES, TANGENTS, SECANTS, &C. 
 
 AKT. Vl. TRIGONOMETRY treats of the relations of the 
 sides and angles of TRIANGLES. Its first object is to deter- 
 mine the length of the sides, and the quantity of the angles. 
 In addition to this, from its principles are derived many in- 
 teresting methods of investigation in the higher branches of 
 analysis, particularly in physical astronomy. 
 
 72. Trigonometry is either plane or spherical. The for- 
 mer treats of triangles bounded by right lines ; the latter, 
 of triangles bounded by arcs of circles. 
 
 Divisions of the Circle. 
 
 73. In a triangle there are two classes of quantities which 
 are the subjects of inquiry, the sides and the angles. For 
 the purpose of measuring the latter, a circle is introduced. 
 
 The periphery of every circle, whether great or small, is 
 supposed to be divided into 360 equal parts called degrees, 
 each degree into 60 minutes, each minute into 60 seconds, 
 each second into 60 thirds, &c., marked with the characters 
 , ', ", "', &c. Thus, 32 24' 13" 22'" is 32 degrees, 24 min- 
 utes, 13 seconds, 22 thirds. 
 
 A degree, then, is not a magnitude of a given length; but 
 
48 
 
 TRIGONOMETRY. 
 
 a certain portion of the whole circumference of any circle. 
 It is evident that the 360th part of a large circle is greater 
 than the same part of a small one. On the other hand, the 
 number of degrees in a small circle, is the same as in a large 
 one. 
 
 The fourth part of a circle is called a quadrant, and con- 
 tains 90 degrees. 
 
 74. To measure an angle, a circle is so described that its 
 center shall be the angular point, and its periphery shall cut 
 the two lines which include the angle. The arc between the 
 two lines is considered a measure of 
 
 the angle, because, by Euc. 33. 6, 
 angles at the center of a given circle, 
 have the same ratio to each other, as the 
 arcs on which they stand. Thus the 
 arc AB, is a measure of the angle 
 ACB. 
 
 It is immaterial what is 
 the size of the circle, pro- 
 vided it cuts the lines 
 which include the angle. 
 Thus, the angle ACD is 
 measured by either of 
 the arcs AG, ag. For 
 ACD is to ACH, as AG 
 to AH, or as ag to ah. (Euc. 33. 6.) 
 
 75. In the circle ADGH, let the 
 two diameters AG and DH be perpen- 
 dicular to each other. The angles 
 ACD, DCG, GCH, and HCA, will 
 be right angles; and the periphery 
 of the circle will be divided into four 
 equal parts, each containing 90 
 
 degrees. As a right angle is subtended by an arc of 
 90, the angle itself is said to contain 90. Hence, in two 
 
SINES, TANGENTS, <feC. 49 
 
 right angles, there are 180; in four right angles, 360; 
 and in any other angle, as many degrees, as in the arc by 
 which it is subtended. 
 
 76. The sum of the three angles of any triangle being 
 equal to two right angles, (Euc. 32. 1.*) is equal to 180. 
 Hence, there can never be more than one obtuse angle in a 
 triangle. For the sum of two obtuse angles is more than 
 180. 
 
 77. The COMPLEMENT of an arc or an a-rigle, is the differ- 
 mce between the arc or angle and 90 degrees. 
 
 The complement of the arc AB is DB ; and title comple- 
 ment of the angle ACB is DOB. The complement of the 
 rc BDG is also DB, 
 
 The complement of 10 is 80, of 60 is 30, 
 of 20 is 70, of 120 is 30, 
 of 50 is 40, of 170 is 80, &c, 
 
 Hence, an acute angle and its com- 
 plement are always equal to 90. 
 The angles ACB and DCB are to- 
 gether equal to a right angle. The 
 two acute angles of a right angled 
 triangle are equal to 90 : therefore 
 each is the complement of the other. 
 
 78. The SUPPLEMENT of an arc or an angle is the differ- 
 ence between the arc or angle and 180 degrees. 
 
 The supplement of the arc BDG is AB ; and the supple- 
 ment of the angle BCG is BOA. 
 
 The supplement of 10 is 170, of 120 is 60, 
 
 of 80 is 100, of 150 is 30, &c. 
 
 Hence an angle and its supplement are always equal to 
 
 * Thomson's Geometry, 38, 1. 
 5 
 
50 TRIGONOMETRY. . 
 
 180o. The angles BOA and BOG are together equal to two 
 right angles. 
 
 79. Cor. As the three angles of a plane triangle are equal 
 to two right angles, that is, to 180 (Euc. 32. 1.) the sum 
 of any two of them is the supplement of the other. So 
 that the third angle may be found, by subtracting the sum 
 of the other two from 1CO. Or the sum of any two may 
 be found, by subtracting the third from 180. 
 
 80. A straight line drawn from the centre of a circle to 
 any part of the periphery, is called a radius of the circle, 
 In many calculations, it is convenient to consider the radius, 
 whatever be its length, as a unit. (Alg. 510.) To this must 
 be referred the numbers expressing the lengths of other 
 lines. Thus, 20 will be twenty times the radius, and 0.75, 
 three-fourths of the radius. 
 
 Definitions of Sines, Tangents, Secants, <&c. 
 
 81. To facilitate the calculations in Trigonometry, there 
 are drawn, within and about the circle, a number of straight 
 lines, called Sines, Tangents, Secants, <6c. With these the 
 learner should make himself perfectly familiar. The direct 
 and proper measure of an angle is an arc of a circle. (Art. 74.) 
 But trigonometrical solutions are commonly made with the 
 aid of certain straight lines, which have known relations to 
 the arcs to which they belong. 
 
 82. The SINE of an arc is a straight line drawn from 
 one end of the arc, perpendicular to a diameter which passes 
 through the other end. 
 
 Thus, BG is the sine of the arc AG. For BG is a line 
 drawn from the end G of the arc, perpendicular to the diam- 
 eter AM which passes through the other end A of the arc. 
 
 Cor. The sine is half the chord of double the arc. The 
 sine BG is half PG, which ia the chord of the arc PAG, 
 double the arc AG. 
 
SINES, TANGENTS, AC. 
 
 51 
 
 83. The VEBSED SINE 
 of an arc is that part of 
 the diameter which is 
 between the sine and the 
 arc. 
 
 Thus, BA is the versed 
 sine of the arc AG. 
 
 84. The TANGENT of 
 an arc, is a straight line 
 drawn perpendicularly 
 from the extremity of the 
 diameter which passes 
 
 through one end of the arc, and extended till it meets a line 
 drawn from the centre through the other end. 
 Thus, AD is the tangent of the arc AG. 
 
 85. The SECANT of an arc is a straight line draim from 
 the centre, through one end of the arc, and extended to the 
 tangent which is drawn from the other end. 
 
 Thus CD is the secant of the arc AG. 
 
 86. In Trigonometry, the terms tangent and secant have 
 a more limited meaning, than in Geometry. In both, indeed, 
 the tangent touches the circle, and the secant cuts it. But 
 in Geometry, these lines are of no determinate length; 
 whereas, in Trigonometry, they extend from the diameter to 
 the point in which they intersect each other. 
 
 87. The lines just defined are sines, tangents, and se- 
 cants of arcs. BG is the sine of the arc AG. But this 
 arc subtends the^ngle GCA. BG is then the sine of the 
 arc which subtends the angle GCA. This is more con- 
 cisely expressed, by saying that BG is the sine of the angle 
 GCA. And universally, the sine, tangent, and secant of an 
 arc, are said to be the sine, tangent, and secant of the angle 
 which stands at the centre of the circle, and is subtended by 
 the arc. Whenever, therefore, the sine, tangent, or secant 
 of an angle is spoken of ; we are to suppose a circle to be 
 
52 TRIGOMOMETRY. 
 
 drawn whose centre is the angular point ; and that the lines 
 mentioned belong to that arc of the periphery which sub- 
 tends the angle. 
 
 88. The sine and tangent of an acute angle, are opposite 
 to the angle. But the secant is one of the lines which in- 
 clude the angle. Thus, the sine BG, and the tangent AD, 
 are opposite to the angle DC A. But the secant CD is one 
 of the lines which include the angle. 
 
 89. The sine complement or COSINE of an angle, is the sine 
 of the COMPLEMENT of that angle. Thus, if the diameter 
 HO be perpendicular to MA, the angle HCG is the com- 
 plement of ACG ; (Art. 77.) and LG, or its equal CB, 
 is the sine of HCG. (Art. 82.) It is, therefore, the cosine of 
 GCA. On the other hand, GB is the sine of GCA, and the 
 cosine of GCH. 
 
 So also the cotangent of an angle is the tangent of the 
 complement of the angle. Thus, HF is the cotangent of 
 GCA. And the cosecant of an angle is the secant of the 
 complement of the angle. Thus, CF is the cosecant of GCA. 
 
 Hence, as in a right angled triangle, one of the acute 
 angles is the complement of the other; (Art. 77.) the sine, 
 tangent, and secant of one of these angles, are the cosine, 
 cotangent, and cosecant of the other. 
 
 90. The sine, tangent, and secant of the supplement of an 
 angle, are each equal to the sine, tangent, and secant of the 
 angle itself. It will be seen, by applying the definition. (Art 
 82.) to the figure,, that the sine of the obtuse angle GCM is 
 BG, which is also the sine of the acute angle GCA. It 
 should be observed, however, tiat the sine of an acute angle 
 is opposite to it ; while the sine of an obtuse angle falls 
 without the angle, and is opposite to its supplement. Thus 
 BG, the sine of the angle MCG, is not opposite to MCG, 
 but to its supplement ACG. 
 
 The tangent of the obtuse angle MCG is MT, or its equal 
 
SINES, TANGENTS, <feC. 53 
 
 AD, which is also the tangent of ACG. And the secant of 
 MCG is CD, which is also the secant of ACG. 
 
 91. But the versed+sine of an angle is not the same as that 
 of its supplement. The versed sine of an acute angle is 
 equal to the difference between the cosine and radius. But 
 the versed sine of an obtuse angle is equal to the sum of 
 the cosine and radius. Thus, the versed sine of ACG is 
 AB=AC BC. (Art. 83.) But the versed sine of MCG is 
 MB=MC+BC. 
 
 Relations of Sines, Tangents, Secants, &c., to each other. 
 
 92. The relations of the sine, tangent, secant, cosine, &c., 
 to each other, are easily derived from the proportions of the 
 sides of similar triangles. (Euc. 4. 6.*) In the quadrant 
 ACH, these lines form 
 
 three similar triangles, 
 viz. ACD,BCGorLCG, 
 and HCF. For, in each 
 of these, there is one 
 right angle, because the 
 sines and tangents are, 
 by definition, perpen- 
 dicular to AC ; as the 
 cosine and cotangent are 
 to CH. The lines CH, 
 BG, and AD, are paral- 
 lel, because C A makes a right angle with each. (Euc. 27. l.f) 
 For the same reason, CA, LG, and HF, are parallel. The 
 alternate angles GCL, BGC, and the opposite angle CDA, 
 are equal; (Euc. 29. l.f) as are also the angles GCB, LGC, 
 and HFC. The triangles ACD, BCG, and &CF, are there- 
 fore similar. 
 
 * Thomson, 18. 4. 
 
 t Ibid. 18. 1. 
 5* 
 
 J Ibid. 24. 1. 
 
TRIGONOMETRY. 
 
 It should also be observed, that the line BC, between the 
 sine and the centre of the circle, is parallel and equal to the 
 cosine ; and that LC, between the cosine and centre, is par- 
 allel and equal to the sine ; (Euc. 34. 1.*) so that one may be 
 taken for the other in any calculation. 
 
 93. From these sim- 
 ilar triangles, are de- 
 rived the following pro- 
 portions ; in which R is 
 put for radius, 
 
 sin for sine, 
 cos for cosine, 
 tan for tangent, 
 cot for cotangent, 
 sec for secant, 
 cosec for cosecant. 
 
 By comparing the triangles CBG and CAD, 
 
 1. AC BC : : AD BG, that is, R : cos : : tan : sin. 
 
 2. CG CD : : BG AD R : sec : : sin : tan. 
 8. CB CA : : CG CD cos : R : : R : see. 
 
 Therefore R*=cosXsec. 
 
 By comparing the triangles CLG and CHF, 
 
 4. Cfl CL : : HF LG, that is, R : sin :: cot : cos. 
 
 5. CG CF : : LG HF, R : cosec : : cos : cot. 
 
 6. CL CH : : CG CF sin : R : : R : cosec. 
 
 Therefore R a =sin X cosec. 
 By comparing the triangles CAD and CHF, 
 
 7. CH 
 
 AD 
 
 ::CF 
 
 CD, that is, R : 
 
 tan : : cosec : sec. 
 
 8. CA 
 9. AD 
 
 HF 
 Afc 
 
 : : CD 
 ::CH 
 
 CF R : 
 HF tan 
 
 cot : : sec ; cosec. 
 I R : : R : cot. 
 
 
 
 
 Therefore R 8 = 
 
 =tanXcot. 
 
 * Thomson, 26. 1. 
 
IFB8, TANGENTS, AC. 
 
 It will not be necessary for the learner to commit these 
 proportions to memory. But he ought to make himself so 
 familiar with the manner of stating them from the figure, as 
 to be able to explain them, whenever they are referred to. 
 
 94. Other relations of the sine, tangent, &c., may be de- 
 rived from the proposition, that the square of the hypothe- 
 nuse is equal to the sum of the squares of the perpendicular 
 sides. (Euc. 47. 1. Thomson 11. 4.) 
 
 In the right angled triangles CBG, CAD, and CHF, 
 
 1. CG 2 =CB" a -f-BG :a , that is, R 2 =cos 2 +sin 2 ,* 
 
 2. CD 3 =CA 9 -f AD 8 sec*=R a +tan 1 , 
 
 3. CF a =CH a 4-HF a cosec a =R a +cot 3 , 
 
 And, extracting the root of both sides, (Alg. 296.) 
 
 E>=V cos*+sin*= V sec 2 tan 2 =V cosec 2 cot 2 
 Hence, if R=l, (Alg 385.) 
 
 Sin=V 1 cos 2 Sec=V 1 +tan a 
 
 Cos=V 1 sin 1 Cosec=Vl-fcot* 
 
 95. The sine of 90 C 
 
 The chord of GO 
 
 are, in any circle, each equal 
 
 And the tangent of 45 
 to the radius, and therefore equal to each other . 
 
 Demons tration. 
 
 1. In the quadrant ACH, (figure on the next page,) the 
 arc AH is 90. The sine of this, according to the definition, 
 (Art. 82.) is CH, the radius of the circle. 
 
 * Sin2 is here put for the square of the sine, cos2 for the square of the 
 cosine, &c. 
 
56 TRIGONOMETRY. 
 
 2. Let AS be an arc of 60. 
 Then the angle ACS, being mea- 
 sured by this arc, will also con- 
 tain 60 ; (Art. 75.) and the tri- 
 angle ACS will be equilateral. 
 For the sum of the three angles 
 is equal to 180. (Art. 76.) 
 From this, taking the angle ACS, 
 which is 60, the sum of the re- 
 maining two is 120. But these two are equal, because they 
 are subtended by the equal sides, CA and CS, both radii of 
 the circle. Each, therefore, is equal to half 120, that is, to 
 60. All the angles being equal, the sides are equal, and there- 
 fore AS, the chord of 60, is equal to CS, the radius. 
 
 3. Let AR be an arc of 45. AD will be its tangent, and 
 the angle ACD subtended by the arc, will contain 45. The 
 angle CAD is a right angle, because the tangent is, by defi- 
 nition, perpendicular to the radius AC. (Art. 84.) Sub- 
 tracting ACD, which is 45, from 90, (Art. 77.) the other 
 acute angle ADC will be 45 also. Therefore the two legs 
 of the triangle ACD are equal, because they are subtended 
 by equal angles ; (Euc. 6. 1.) that is, AD the tangent of 45, 
 is equal to AC the radius. 
 
 Cor. The cotangent of 45 is also equal to radius. For 
 the complement of 45 is itself 45. Thus, HD, the cotan- 
 gent of ACD, is equal to AC the radius. 
 
 96. The sine of 30 is equal to half radius. For the 
 sine of 30 is equal to half the chord of 60. (Art. 82. cor.) 
 But by the preceding article, the chord of 60 is equal to 
 radius. Its half, therefore, which is the sine of 30, is equal 
 to half radius. 
 
 Cor. 1. The cosine of 60 is equal to half radius. For 
 the cosine of 60 is the sine of 30. (Art. 89.) 
 
 Cor. 2. The cosine of 30=V3. For 
 
 Cos 2 30=R 2 sin 2 30=1 i=f. 
 
SINES, TANGENTS, AO. 57 
 
 Therefore, 
 
 Cos 30=vi=i-V3. 
 
 1 
 
 96. b. The sine of 45== For 
 
 V2 
 
 R J =l=sin a 45+cos a 45=2 sin 9 45 
 
 1 
 
 Therefore, Sin 45=Vi= 
 V2 
 
 97. The chord of any arc is a mean proportional, between 
 the diameter of the circle, and the versed sine of the arc. 
 
 Let ADB, be an arc, of which 
 AB is the chord, BF the sine, 
 and AF the versed sine. The 
 
 Tl 
 
 angle ABH is a right angle, (Euc. 
 31. 3.*) and the triangles ABH, 
 and ABF, are similar. (Euc. 8. 6.f) 
 Therefore, 
 
 AH : AB : : AB : AF. 
 
 That is, the diameter is to the chord, as the chord to the 
 versed sine. 
 
 Let the arc AD=a, and ADB=2a. Draw BF perpen- 
 dicular to AH. This will divide the right angled triangle 
 ABH into two similar triangles. (Euc. 8. 6.) The angles 
 ACD and AHB are equal. (Euc. 20. 3.J) Therefore the 
 four triangles ACG, AHB, FHB, and FAB are similar ; and 
 the line BH is twice CG, because BH : CG : : HA : CA. 
 
 The sides of the four triangles are, 
 
 AG=sin a, CG=cos a. HF=vers. sup. 2o, 
 AB=2 sin a, BH=2 cos a. AC=the radius, 
 BF=sin 2a, AF=vers 2a, AH=the diameter. 
 
 * Thomson, 13. 2. Cor. 2. f Ibid. 22. 4. J: Ibid. 13. 9. 
 
ONOMETRY. 
 
 A variety of proportions may be stated, between the 
 homologous sides of these triangles : For instance, 
 
 By comparing the triangles ACG and ABF, 
 AC : AG : : AB : AF, that is, R : sin a : : 2 sin a ; vers 2a 
 AC : CG : : AB : BF, R : cosa : : 2 sin a : sin 2a 
 
 AG : CG : : AF : BF, Sin a : cos a : : vers 2a : sin 2a 
 
 Therefore, 
 
 2a=2sin a a 
 2o=2sin aXcos a 
 2a=vers 2aXcos a 
 
 Sin 
 
 By comparing the triangles 
 ACG and BFH, 
 
 AC ; CG : : BH : HF, that is, R : cos a : : 2cos a : vers. sup. 2a 
 AG : CG : : BF : HF, Sin a : cos a : : sin 2a : vers. sup. 2a 
 
 Therefore, 
 
 BXvers. sup. 2a=2 cos a a 
 Sin a X vers. sup. 2a=cos aXsin 2a 
 &c. <fec. 
 
 That is, the product of radius into the versed sine of the 
 supplement of twice a given arc, is equal to twice the square 
 of the cosine of the arc. 
 
 And the product of the sine of an arc, into the versed 
 sine of the supplement of twice the arc, is equal to the pro- 
 duct of the cosine of the arc, into the sine of twice the 
 arc, <fec., <fec. 
 
THE TRIGONOMETRICAL TABLES. 
 
 SECTION II. 
 
 THE TRIGONOMETRICAL TABLES. 
 
 ART. 98. To facilitate the operations in trigonometry, the 
 sine, tangent, secant, &c., have been calculated for every 
 degree and minute, and in some instances, for every second, 
 of a quadrant, and arranged in tables. These constitute 
 what is called the Trigonometrical Canon. It is not neces- 
 sary to extend these tables beyond 90 ; because the sines, 
 tangents, and secants, are of the same magnitude, in one of 
 the quadrants of a circle, as in the others. Thus the sine of 
 30 is equal to that of 150. (Art 90.) 
 
 99. And in any instance, if we have occasion for the sine, 
 tangent, or secant of an obtuse angle, we may obtain it, by 
 looking for its equal, the sine, tangent, or secant of the sup- 
 plementary acute angle^ 
 
 100. The tables are calculated for a circle whose radius is 
 supposed to be a unit. It may be an inch, a yard, a mile, 
 or any other denomination of length. But the sines, tan- 
 gents, &c., must always be understood to be of the same de- 
 nomination as the radius. 
 
 101. All the sines, except that of 90, are less than radius, 
 (Art. 82.) and are expressed in the tables by decimals. 
 
 Thus the sine of 20 is 0.34202, of 60 is 0.86603, 
 
 of 40 is 0.64279, of 89 is 0.99985, &c. 
 
 When the tables are intended to be very exact, the decimal 
 is carried to a greater number of places. 
 
 The tangents of all angles less than 45 are also less than 
 radius. (Art. 95.) But the tangejits of angles greater than 
 45, are greater than radius, and are expressed by a whole 
 
60 THE TRIGONOMETRICAL TABLES. 
 
 number and a decimal. It is evident that all the secants also 
 must be greater than radius, as they extend from the centre, 
 to a point without the circle. 
 
 102. The numbers in the table here spoken of, are called 
 natural sines, tangents, &c. They express the lengths of 
 the several lines which have been defined in Arts. 82, -?, 
 &c. By means of them, the angles and sides of triangles 
 may be accurately determined. But the calculations must 
 be made by the tedious processes of multiplication and 
 division. To avoid this inconvenience, another set of tables 
 has been provided, in which are inserted the logarithms of 
 the natural sines, tangents, c. By the use of these, ad- 
 dition and subtraction are made to perform the office of mul- 
 tiplication and division. On this account, the tables of loga- 
 rithmic, or as they are sometimes called, artificial sines, 
 tangents, &c., are much more valuable, for practical pur- 
 poses, than the natural sines, &c. Still it must be remem- 
 bered that the former are derived from the latter. The arti- 
 ficial sine of an angle, is the logarithm of the natural sine 
 of that angle. The artificial tangent is the logarithm of the 
 natural tangent, <fec. 
 
 103. One circumstance, - however, is to be attended to, in 
 comparing the two sets of tables. The radius to which the 
 natural sines, &c., are calculated, is unity. (Art. 100.) The 
 secants, and a part of the tangents are, therefore, greater 
 than a unit ; while the sines, and another part of the tan- 
 gents, are less than a unit. When the logarithms of these 
 are taken, some of the indices will be positive, and others 
 negative ; (Art. 9.) and the throwing of them together in the 
 same table, if it does not lead to error, will at least be at- 
 tended with inconvenience. To remedy this, 10 is added to 
 each of the indices. (Art. 12.) They are then all positive. 
 Thus the natural sine of 20 is 0.34202. The logarithm of 
 this is "L53405. But the index, by the addition of 10, be- 
 
THE TRIGONOMETRICAL TABLES. 61 
 
 comes 10 1=9. The logarithmic sine in the tables is 
 therefore 9.53405.* 
 
 Directions for taking Sines, Cosines, &c.,from the tables. 
 
 104. The cosine, cotangent, and cosecant of an angle, are 
 the sine, tangent, and secant of the complement of the angle. 
 (Art. 89.) As the complement of an angle is the differ- 
 ence between the angle and 90, and as 45 is the half of 90 ; 
 if any given angle within the quadrant is greater than 45, 
 its complement is less ; and, on the other hand, if the angle 
 is less than 45, its complement is greater. Hence, every 
 cosine, cotangent, and cosecant of an angle greater than 45, 
 has its equal among the sines, tangents, and secants of angles 
 less than 45, and v. v. 
 
 Now, to bring the trigonometrical tables within a small 
 compass, the same column is made to answer for the sines 
 of a number of angles above 45, and for the cosines of an 
 equal number below 45. 
 
 Thus 9.23967 is the log. sine of 10, and the cosifie of 80, 
 9.53405 the sine of 20, and the cosine of 70, &c. 
 
 The tangents and secants are arranged in a similar man- 
 ner. Hence, 
 
 105. To find the Sine, Cosine, Tangent, <&c., of any num- 
 ber of degrees and minutes. 
 
 If the given angle is less than 45, look for the degrees 
 at the top of the table, and the minutes on the left ; then, 
 opposite to the minutes, and under the word sine at the 
 head of the column, will be found the sine ; under the word 
 tangent, will be found the tangent, &c. 
 
 * Or the tables may be supposed to be calculated to the radios 
 10000000000, whose logarithm is 10. 
 
62 THE TRIGONOMETRICAL TABLES. 
 
 Thelog. sin of 43 25' is 9.83715 The tan of 17 20' is 9.49430 
 of 17 20' 9.47411 of 8 46' 9.18812 
 
 The cos of 17 20' 9.97982 The cot of 17 20' 10.50570 
 of 8 46' 9.99490 of 8 46' 10.81188 
 
 The first figure is the index ; and the other figures are the 
 decimal part of the logarithm. 
 
 106. If the given angle is between 45 and 90 ; look for 
 the degrees at the bottom of the table, and the minutes on 
 the right ; then, opposite to the minutes, and over the word 
 sine at the foot of the column, will be found the sine ; over 
 the word tangent, will be found the tangent, &c. 
 
 Particular care must be taken, when the angle is less than 
 45, to look for the title of the column, at the top, and for 
 the minutes on the left ; but when the angle is between 45 
 and 90, to look for the title of the column at the bottom, 
 and for the minutes, on the right. 
 
 The log. sine of 81 21' is 9.99503 
 The cosine of 72 10' 9.48607 
 
 The tangent of 54 40' 10.14941 
 The cotangent of 63 22' 9.70026 
 
 107. If the given angle is greater than 90, look for the 
 sine, tangent, &c., of its supplement. (Art. 98, 99.) 
 
 The log. sine of 96 44' is 9.99699 
 
 The cosine of 171 16' 9.99494 
 
 The tangent of 130 26' 10.06952 
 
 The cotangent of 156 22' 10.35894 
 
 108. To find the sine, cosine, tangent, &c., of any number 
 of degrees, minutes, and SECONDS. 
 
 In the common tables, the sine, tangent, &c., are given 
 only to every minute of a degree.* But they may be found 
 to seconds, by taking proportional parts of the difference of 
 
 * In the very valuable tables of Michael Taylor, the sines and tan- 
 gents are given to every second. 
 
THE TRIGONOMETRICAL TABLES. 63 
 
 the numbers as they stand in the tables. For, -within a sin- 
 gle minute, the variations in the sine, tangent, <fcc., are 
 nearly proportional to the variations in the angle. Hence, 
 
 To find the sine, tangent, &c., to seconds : Take out the 
 number corresponding to the given degree and minute ; and 
 also that corresponding to the next greater minute, and find 
 their difference. Then state this proportion ; 
 
 As 60, to the given number of seconds ; 
 
 So is the difference found, to the correction for the seconds. 
 
 This correction, in the case of sines, tangents, and secants, 
 is to be added to the number answering to the given degree 
 and minute ; but for cosines, cotangents, and cosecants, the 
 correction is to be subtracted ; 
 
 For, as the sines increase, the cosines decrease. 
 
 Ex. 1. What is the logarithmic sine of 14 43' 10" ? 
 
 The sine of 14 43' is 9.40490 
 of 14 44' 9.40538 
 
 Difference 48 
 
 Here it is evident that the sine of the required angle is 
 greater than that of 14 43', but less than that of 14 44'. 
 And as the difference corresponding to a whole minute or 
 60" is 48 ; the difference for 10" must be a proportional 
 part of 48. That is, 
 
 60" : 10" : : 48 : 8 
 the correction to be added to the sine of 14 43'. 
 
 Therefore the sine of 14 43' 10" is 9.40498. 
 
 2. What is the logarithmic cosine of 32 16' 45" ? 
 
 The cosine of 32 16' is 9.92715 
 of 32 17' 9.92707 
 Difference 8 
 
 Then, 60'' ; 45" : : 8 : 6 the correction to be subtracted 
 from the cosine of 32 16'. 
 
64 THE TRIGONOMETRICAL TABLES. 
 
 Therefore the cosine of 32 16' 45" is 9.92709. 
 
 The tangent of 24 15' 18" is 9.65376 
 
 The cotangent of 31 50' 5" is 10.20700 
 
 The sine of 58 14' 32" is 9.92956 
 
 The cosine of 55 10' 26" is 9.75670 
 
 If the given number of seconds be any even part of 60, 
 as i> "3* i> &c., the correction may be found, by taking a like 
 part of the difference of the numbers in the tables, without 
 stating a proportion in form. 
 
 109. To find the degrees and minutes belonging to any 
 given sine, tangent, &c. 
 
 This is reversing the method of finding the sine, tangent, 
 &c,. (Art. 105,*6, 7.) 
 
 Look in the column of the same name, for the sine, tan- 
 gent, &c., which is nearest to the given one ; and if the title 
 be at the head of the column, take the degrees at the top of 
 the table, and the minutes on the left ; but if the title be at 
 the foot of the column, take the degrees at the bottom, and 
 the minutes on the right. 
 
 Ex. 1. What is the number of degrees and minutes be- 
 longing to the logarithmic sine 9.62863? 
 
 The nearest sine in the tables is 9.62865. The title of 
 sine is at the head of the column in which these numbers are 
 found. The degrees at the top of the page are 25, and the 
 minutes on the left are 10. The angle required is, therefore 
 25 10'. 
 
 The angle belonging to 
 
 the sine 9.87993 is 49 20' the cos 9.97351 is 19 48' 
 the tan 9.97955 43 39' the cotan 9.75791 60 12' 
 the sec 10.65396 77 11' the cosec 10.49066 18 51' 
 
 110. To find the degrees, minutes, and SECONDS, belonging 
 to any given sine, tangent, t&c. 
 
THE TRIGONOMETRICAL TABLES. 65 
 
 This is reversing the method of finding the sine, tangent, 
 <fcc., to seconds. (Art. 108.) 
 
 First find the difference between the sine, tangent, &c., 
 next greater than the given one, and that which is next less ; 
 then the difference between this less number and the given 
 one; then 
 
 As the difference first found, is to the other difference ; 
 
 So are 60 seconds, to the number of seconds, which, in the 
 case of sines, tangents, and secants, are to be added to the 
 degrees and minutes belonging to the least of the two num- 
 bers taken from the tables ; but for cosines, cotangents, and 
 cosecants are to be subtracted. 
 
 Ex. 1 . What are the degrees, minutes, and seconds, be- 
 longing to the logarithmic sine 9.40498 ? 
 
 Sine next greater 14 44' 9.40538 Given sine 9.40498 
 Next less 14 43' 9.40490 Next less 9.40490 
 
 Difference 48 Difference 8 
 
 Then, 48 : 8 : : 60" : 10", which added to 14 43', gives 
 14 43' 10" for the answer. 
 
 2. What is the angle belonging to the cosine 9.09773 ? 
 
 Cosine next greater 82 48' 9.09807 Given cosine 9.09773 
 
 Next less 82 49' 9.09707 Next less 9.09707 
 
 Difference 100 Difference 66 
 
 Then, 100 : 66 : : 60" : 40", which subtracted from 82 
 49', gives 82 48' 20" for the answer. 
 
 It must be observed here, as in all other cases, that of the 
 two angles, the less has the greater cosine. 
 
 The angle belonging to 
 
 the sin 9.20621 is 9 15' 6'' the tan 10.43434 is 69 48' 16" 
 the cos 9.98157 16 34' 30" the cot 10.33554 24 47' 16" 
 
 6* 
 
00 THE TRIGONOMETRICAL TABLES. 
 
 Method of Supplying the Secants and Cosecants. 
 
 111. In some trigonometrical tables, the secants and cose- 
 cants are not inserted. But they may be easily obtained 
 from the sines and cosines. For, by Art. 93, proportion 3d, 
 
 cosXsec=R 2 . 
 
 That is, the product of the cosine and secant, is equal to 
 the square of radius. But, in logarithms, addition takes the 
 place of multiplication; and, in the tables of logarithmic 
 sines, tangents, <fec., the radius is 10. (Art. 103.) There- 
 fore, in these tables, 
 
 cos-f sec=20. Or sec=20 cos. 
 Again, by Art 93, proportion 6, 
 
 sinXcosec=R a . 
 Therefore, in the tables, 
 sin+cosec=20. Or, cosec=20 sin. Hence, 
 
 112. To obtain the secant, subtract the cosine from 20; 
 and to obtain the cosecant, subtract the sine from 20. 
 
 These subtractions are most easily performed, by taking 
 the right hand figure from 10, and the others from 9, as in 
 finding the arithmetical complement of a logarithm ; (Art. 
 55.) observing, however, to add 10 to the index of the secant 
 or cosecant. In fact the secant is the arithmetical comple- 
 ment of the cosine, with 10 added to the index. 
 
 For the secant =20 cos. 
 
 And the arith. comp. of cos =10 cos. (Art. 54.) 
 
 So also the cosecant is the arithmetical complement of the 
 sine, with 10 added to the index. The tables of secants and 
 cosecants are, therefore, of use, in furnishing the arithmetical 
 complement of the sine and cosine, in the following simple 
 manner : 
 
RIGHT ANGLED TRIANGLES. 07 
 
 113. For the arithmetical complement of the sine, sub- 
 tract 10 from the index of the cosecant; and for the arith- 
 metical complement of the cosine, subtract 10 from the index 
 of the secant. 
 
 By this, we may save the trouble of taking each of the 
 figures from 9. 
 
 SECTION III. 
 
 SOLUTIONS OF RIGHT ANGLED TRIANGLES. 
 
 ART. 114. In a triangle there are six parts, three sides, 
 and three angles. In every trigonometrical calculation, it is 
 necessary that some of these should be known, to enable us 
 to find the others. TJie number of parts which must be 
 given, is THREE, one of which must be a SIDE. 
 
 If only two parts be given, they will be either two sides, 
 a side and an angle, or two angles ; Cither of which will 
 limit the triangle to a particular form and size. 
 
 If two sides only be given, they may make any angle with 
 each other ; and may, there- 
 fore, be the sides of a thousand 
 different triangles. Thus, the 
 two lines a and b may belong 
 either to the triangle ABC, or 
 ABC', or ABC". So that it 
 
 will be impossible, from know- * 
 
 ing two of the sides of a trian- 
 gle, to determine the other parts. 
 
68 BIGHT ANGLED TRIANGLES. 
 
 Or, if a side and an angle 
 only be given, the triangle 
 will be indeterminate. Thus, 
 if the side AB and the an- 
 gle at A be given; they 
 may be parts either of the 
 triangle ABC, or ABC', or 
 ABC". 
 
 Lastly, if two angles, or even if all the angles be given, 
 they will not determine the length of the sides. "For the tri- 
 angles ABC, A'B'C', A"B"C", 
 and a hundred others which 
 might be drawn, with sides par- 
 allel to these, will all have the 
 same angles. So that one of 
 the parts given must always be 
 a side. If this and any other 
 two parts, either sides or angles, be known, the other three 
 may be found, as will be shown, in this and the following 
 section. 
 
 115. Triangles are either right angled or oblique angled. 
 The calculations of the former are the most simple, and those 
 which we have the most frequent occasion to make. A great 
 portion of the problems in the mensuration of heights and 
 distances, in surveying, navigation and astronomy, are solved 
 by rectangular trigonometry. Any triangle whatever may 
 be divided into two right angled triangles, by drawing a per- 
 pendicular from one of the angles to the opposite side. 
 
 116. One of the six parts in aright angled triangle, is 
 always given, viz. the right angle. This is a constant quan- 
 tity ; while the other angles and the sides are variable. It 
 is also to be observed, that, if one of the acute angles is 
 given, the other is known of course. For one is the com- 
 plement of the other. (Art. 76, 77.) So that, in a right angled 
 triangle, subtracting one of the acute angles from 90 gives the 
 
RIGHT ANGLED TRIANGLES. 
 
 69 
 
 other. There remain, then, only four parts, one of the acute 
 angles, and the three sides, to be sought by calculation. If 
 any two of these be given, with the right angle, the others 
 may be found. 
 
 117. To illustrate the method of 
 calculation, let a case be supposed in 
 which a right angled triangle CAD, 
 has one of its sides equal to the 
 radius to which the trigonometrical 
 tables are adapted. 
 
 In the first place, let the base of the p . 
 triangle be equal to the tabular radius. Then, if a circle be de- 
 scribed, with this radius, about the angle C as a centre, DA 
 will be the tangent, and DC the secant of that angle. (Art. 
 84, 85.) So that the radius, the tangent, and the secant of 
 the angle at C, constitute the three sides of the triangle. 
 The tangent, taken from the tables of natural sines, tangents, 
 <fec., will be the length of the perpendicular ; and the secant 
 will be the length of the hypothenuse. If the tables used be 
 logarithmic, they will give the logarithms of the lengths of 
 the two sides. 
 
 In the same manner, any 
 right angled triangle whatever, 
 whose base is equal to the 
 radius of the tables, will have 
 its other two sides found 
 among the tangents and se- 
 cants. Thus, if the quadrant 
 AH, be divided into portions 
 of 15 each ; then, in the 
 triangle 
 
 CAD, AD will be the tan, and CD the sec of 15, 
 In CAD/, AD' will be the tan, and CD 7 the sec of 30, 
 In CAD", AD" will be the tan, and CD" the sec of 45,<fec. 
 
70 
 
 RIGHT ANGLED TRIANGLES. 
 
 118. In the next place, let 
 the hypothenuse of a right angled 
 triangle CBF, be equal to the 
 radius of the tables. Then, if a 
 circle be described, with the given 
 radius, and about the angle C as 
 a centre; BF will be the sine, 
 and BC the cosine of that angle. 
 (Art. 82, 89.) Therefore the 
 
 sine of the angle at C, taken from the tables, will be the 
 length of the perpendicular, and the cosine will be the length 
 of the base. 
 
 And any right angled triangle 
 whatever, whose hypothenuse 
 is equal to the tabular radius, 
 will have its other two sides 
 found among the sines and co- 
 sines. Thus, if the quadrant 
 AH, be divided into portions of 
 15 each in the points F, F', F", 
 &c. ; then, in the triangle, 
 
 CBF, FB will be the sin, and CB the cos, of 15, 
 In CB'F', F'B' will be the sin, and CB' the cos, of 30, 
 In CB"F", F"B" will be the sin, and CB" the cos, of 45, &c, 
 
 119. By merely turning to the tables, then, we may find 
 the parts of any right angled triangle which has one of its 
 sides equal to the radius of the tables. But for determin- 
 ing the parts of triangles which have not any of their sides 
 equal to the tabular radius, the following proportion is used ; 
 
 As the radius of one circle, 
 To the radius of any other ; 
 So is a sine, tangent, or secant, in one, 
 To the sine, tangent, or secant, of the same number 
 of degrees, in the other. 
 
RIGHT ANGLED TRIANGLES. 
 
 71 
 
 In the two concentric 
 circles AHM, ahm, the 
 arcs AG and ag, contain 
 the same number of de- 
 grees. (Art. 74.) The 
 sines of these arcs are 
 BG and bg, the tangents 
 AD and ad, and the se- 
 cants CD and Cd. The four triangles, CAD, CBG, Cad, 
 and Cbg, are similar. For each of them, from the nature 
 of sines and tangents, contains one right angle ; the angle at 
 C is common to them all ; and the other acute angle in each 
 is the complement of that at C. (Art. 77.) We have, then, 
 the following proportions. (Euc. 4. 6.*) 
 
 1. CG : Cg : : BG : bg. 
 
 That is, one radius is to the other, as one sine to the other. 
 
 2. CA : Ca : : DA : da. 
 
 That is, one radius is to the other, as one tangent to the other. 
 
 3. CA t Ca : : CD ; Cd. 
 
 That is, one radius is to the other, as one secant to the other. 
 Cor. BG : bg : : DA : da : : CD : Cd. 
 
 That is, as the sine in one circle, to the sine in the other ; 
 so is the tangent in one, to the tangent in the other ; and so 
 is the secant in one, to the secant in the other. 
 
 This is a general principle, which may be applied to most 
 trigonometrical calculations. If one of the sides of the pro- 
 posed triangle be made radius, each of the other sides will 
 be the sine, tangent, or secant, of an arc described by this 
 radius. Proportions are then stated, between these lines, 
 and the tabular radius, sine, tangent, <fec. 
 
 * Thomson 18. 4. 
 
RIGHT ANGLED TRIANGLES. 
 
 120. A line is said to be made radius, when a circle is 
 described, or supposed to be described, whose semi-diameter 
 is equal to the line, and whose centre is at one end of it. 
 
 121. In any right angled triangle, if the HYPOTHENUSE be 
 made radius, one of the leys will be a SINE of its opposite 
 angle, and the other leg a COSINE of the same angle. 
 
 Thus, if to the triangle ABC 
 a circle be applied whose radius 
 is AC, and whose centre is A, 
 then BC will be the sine, and 
 BA the cosine, of the angle at 
 A. (Art. 82, 89.) 
 
 If, while the same line is 
 radius, the other end C be made 
 
 the centre, then BA will be the sine, and BC the cosine, of 
 the angle at C. 
 
 122. If either of the LEGS be made radius, the other leg will 
 be a TANGENT of its opposite angle, and the hypothenuse will 
 be a SECANT of the same angle ; that is, of the angle between 
 the secant and the radius. 
 
 Rod 
 
 Thus, if the base AB (Fig. 15.) be made radius, the centre 
 being at A, BC will be the tangent, and AC the secant, of 
 the angle at A. (Art. 84. 85.) 
 
 But, if the perpendicular BC, (Fig. 16.) be made radius, 
 with the centre at C, then AB will be the tangent, and AC 
 the secant, of the angle at C. 
 
 123. As the side which is the sine, tangent, or secant 
 of one of the acute angles, is the cosine, cotangent, or cose- 
 
RIGHT ANGLED TRIANGLES. 73 
 
 cant of the other ; (Art. 89,) the perpendicular BC (Fig. 14.) 
 is the sine of the angle A, and the cosine of the angle C ; 
 while the base AB, is the sine of the angle C, and the cosine 
 of the angle A. 
 
 If the base is made radius, as in Fig 15, the perpendicular 
 BC is the tangent of the angle A, and the cotangent of the 
 angle C ; while the hypothenuse is the secant of the angle 
 A, and the cosecant of the angle C. 
 
 If the perpendicular is made radius, as in Fig. 16, the 
 base AB is the tangent of the angle C, and the cotangent of 
 the angle A ; while the hypothenuse is the secant of the angle 
 C, and the cosecant of the angle A. 
 
 124. Whenever a right angled triangle is proposed, whose 
 sides or angles are required ; a similar triangle may be 
 formed, from the sines, tangents, <fec., of the tables. (Art. 
 117, 118.) The parts required are then found, by stating 
 proportions between the similar 
 sides of the two triangles. If 
 the triangle proposed be ABC, 
 (Fig. 17.) another abc may be 
 formed, having the same angles 
 with the first, but differing from 
 it in the length of its sides, so as 
 to correspond with the numbers in the tables. If similar sides 
 be made radius in both, the remaining similar sides will be 
 lines of the same name ; that is, if the perpendicular in one 
 of the triangles be a sine, the perpendicular in the other will 
 be a sine ; if the base in one be a cosine, the base in the 
 other will be a cosine, &c. 
 
 If the hypothenuse in each triangle be made radius, as in 
 Fig. 14, the perpendicular be, will be the tabular sine of the 
 angle at a ; and the perpendicular BC, will be a sine of 
 the equal angle A, in a circle of which AC is radius. 
 
 If the base in each triangle be made radius, as in Fig. 15, 
 then the perpendicular be, will be the tabular tangent of the 
 
 7 
 
74 RIGHT ANGLED TRIANGLES. 
 
 angle at a ; and BC -will be a tangent of the equal angle A, 
 in a circle of which AB, is radius, &e. 
 
 125. From the relations of the similar sides of these tri- 
 angles, are derived the two following theorems, which are 
 sufficient for calculating the parts of any right angled tri- 
 angle whatever, when the requisite data are furnished. One 
 is used, when a side is to be found ; the other, when an 
 angle is to be found. 
 
 THEOREM I. 
 126. When a side is required ; 
 
 As THE TABULAR SINE, TANGKNT, &<X, OF THE 
 SAME NAME WITH THE GIVEN SIDE, 
 
 To THE GIVEN SIDE; 
 
 SO IS THE TABULAR SINE, TANGENT, <feC., OF THE 
 
 SAME NAME WITH THE REQUIRED SIDE, 
 To THE REQUIRED SIDE. 
 
 It will be readily seen, that this is nothing more than a 
 statement in general terms, of the proportions between the 
 similar sides of two triangles, one proposed for solution, and 
 the other formed from the numbers in the tables. 
 
 Thus, if the hypothenuse be 
 given, and the base or perpen- 
 dicular be required ; then in 
 Fig. 14, where ac is the tabular 
 radius, be the tabular sine of a, 
 or its equal A, and ab the tab- 
 ular sine of C; (Art. 124.) 
 
 ac : AC : : be : BC, that is, B : AC : : sin A : BC. 
 c ; AC : : o& : AB, B : AC : : sin C ; AB. 
 
RIGHT ANGLED TRIANGLES. 
 
 15 
 
 In Fig. 15, where ab is the tabular radius, ac the tabular 
 secant of A, and be the tabular tangent of A ; 
 ac : AC : : be : BC, that is, sec A : AC : : tan A : BC. 
 ac : AC : : ab : AB, sec A : AC : : R ; AB. 
 
 In Fig. 16, where be is the tabular radius, ac the tabular 
 secant of C, and ab the tabular tangent of C ; 
 ac : AC : : be : BC, that is, sec C : AC : : R : BC. 
 ac : AC : : ab : AB, sec C : AC : : tan C ; AB. 
 
 THEOREM II. 
 127. When an angle is required ; 
 
 As THE GIVEN SIDE MADE RADIUS, 
 To THE TABULAR RADIUS J 
 So IS ANOTHER GIVEN SIDE, 
 
 TO THE TABULAR SINE, TANGENT, AC., OF THE 
 SAME NAME. 
 
 Thus, if the side made radius, and one other side be given, 
 then, in Fig. 14, 
 
 AC : ac : : BC : be, that is, AC 
 AC : ac : : AB : ab, AC 
 
 In Fig. 15, 
 
 AB : ab : : BC : be, that is, AB 
 AB : ab : : AC : ac, AB 
 
 In Fig. 16, 
 AB : ab, that is, BC 
 
 BC : be 
 BC ; be 
 
 : AC : ac, 
 
 BO ; 
 
 R: 
 R: 
 
 BC 
 AB 
 
 sin A. 
 sin C. 
 
 R: 
 R: 
 
 BC 
 AC 
 
 tan A. 
 sec A. 
 
 R: 
 
 AB 
 AC 
 
 tanC. 
 sec C. 
 
76 RIGHT ANGLED TRIANGLES. 
 
 It will be observed that in these theorems, angles are not 
 introduced, though they are among the quantities which are 
 either given or required, in the calculation of triangles. But 
 the tabular sines, tangents, &c., may be considered the repre- 
 sentatives of angles, as one may be found from the other, by 
 merely turning to the tables. 
 
 128. In the theorem for finding a side, the first term of 
 the proportion is a tabular number. But, in the theorem foi 
 finding an angle, the first term is a side. Hence, in apply- 
 ing the proportions to particular cases, this rule is to be ob- 
 served ; 
 
 To find a SIDE, begin with a tabular number, 
 To find an ANGLE, begin with a side. 
 
 Radius is to be reckoned among the tabular numbers. 
 
 129. In the theorem for finding an angle, the first term is 
 a side made radius. As in every proportion, the three first 
 terms must be given to enable us to find the fourth, it is evi- 
 dent, that where this theorem is applied, the side made 
 radius must be a given one. But, in the theorem for finding 
 a side, it is not necessary that either of the terms should be 
 radius. Hence, 
 
 130. To find a SIDE, ANY side may be made radius. 
 
 To find an ANGLE, a GIVEN side must be made radius. 
 
 It will generally be expedient, in both cases, to make 
 radius one of the terms in the proportion ; because, in the 
 tables of natural sines, tangents, &c., radius is 1, and in the 
 logarithmic tables it is 10. (Art. 103.) 
 
 131. The proportions in Trigonometry are of the same 
 nature as other simple proportions. The fourth term is found, 
 therefore, as in the Rule of Three in Arithmetic, by multi- 
 plying together the second and third terms, and dividing their 
 product by the first term. This is the mode of calculation, 
 when the tables of natural sines, tangents, <fec., are used. 
 But the operation by logarithms is so much more expeditious, 
 
RIGHT ANGLED TRIANGLES. 77 
 
 that it has almost entirely superseded the other method. In 
 logarithmic calculations, addition takes the place of multipli- 
 cation ; and subtraction the place of division. 
 
 The logarithms expressing the lengths of the sides of a 
 triangle, are to be taken from the tables of common loga- 
 rithms. The logarithms of the sines, tangents, &c. t are found 
 in the tables of artificial sines, &c. The calculation is then 
 made by adding the second and third terms, and subtracting 
 the first. (Art. 52.) 
 
 132. The logarithmic radius 10, or, as it is written in the 
 tables, 10.00000, is so easily added and subtracted, that the 
 three terms of which it is one, may be considered as, in 
 effect, reduced to two. Thus, if the tabular radius is in the 
 first term, we have only to add the other two terms, and 
 then take 10 from the index ; for this is subtracting the first 
 term. If radius occurs in the second term, the first is to be 
 subtracted from the third, after its index is increased by 10. 
 In the same manner, if radius is in the third term, the first is 
 to be subtracted from the second. 
 
 133. Every species of right angled triangles may be 
 solved upon the principle, that the sides of similar triangles 
 are proportional, according to the two theorems mentioned 
 above. There will be some advantages, however, in giving 
 the examples in distinct classes. 
 
 There must be given, in a right angled triangle, two of the 
 parts, besides the right angle. (Art. 116.) These may be; 
 
 1. The hypothenuse and an angle; or 
 
 2. The hypothenuse and a leg ; or 
 
 3. A leg and an angle ; or 
 
 4. The two legs. 
 
 CASK I. 
 
 thenuse, 
 angle, } " "" j Perpendicular. 
 
 134. Given i Th hypothenuse, ) tofind ( The base and 
 ( And an angle, ) ( 
 
78 
 
 RIGHT ANGLED TRIANGLES. 
 
 \C 
 
 \ 
 
 Ex. 1. If the hypothenuse AC,* 
 be 45 miles, and the angle at A 
 32 20', what is the length of the 
 base AB, and the perpendicular 
 BC? 
 
 In this case, as sides only are 
 required, any side may be made 
 radius. (Art. 130.) 
 
 If the hypothenuse be made 
 radius, BC will be the sine of 
 A, and AB the sine of C, or the 
 cosine of A. (Art. 121.) And 
 if abc be a similar triangle, 
 whose hypothenuse is equal to 
 the tabular radius, be will be the 
 tabular sine of A, and db the 
 tabular sine of C. (Art. 124.) 
 
 To find the perpendicular, then, by Theorem I, we have 
 this proportion ; 
 
 ac : AC : : be : BC. 
 Or R : AC : : Sin A : BC. 
 
 Whenever the terms Radius, Sine, Tangent, &c., occur in a 
 proportion like this, the tabular Radius, &c., is to be under- 
 stood, as in Arts. 126, 127. 
 
 The numerical calculation, to find the length of BC, may 
 be made, either by natural sines, or by logarithms. See 
 Art 131. 
 
 By natural Sines. 
 1 t 45 : : 0.53484 : 24.068=BC. 
 
 * The parts which are given are distinguished by a mark across the 
 line, or at the opening of the angle, and the parts required by a cipher. 
 
RIGHT ANGLED TRIANGLES. 
 
 Computation by Logarithms, 
 As radius 10.00000 
 
 To the hypothenuse 45 1.65321 
 
 So is the Sine of A 32 20' 9.72823 
 To the perpendicular 24.068 1.38144 
 
 Here the logarithms of the second and third terms are 
 added, and from the sum, the first term 10 is subtracted, 
 (Art 132.) The remainder is the logarithm of 24.068=BC. 
 Subtracting the angle at A from 90, we have the angle at 
 C=^57 40'. (Art. 116.) Then to find the base AB; 
 
 ac : AC : : ab : AB 
 Or R ; AC : : Sin C : AB=38.023. 
 
 Both the sides required are now found, by making the 
 hypothenuse radius. The results here obtained may be veri- 
 fied, by making either of the other sides radius. 
 
 If the base be made radius, as in Fig. 15, the perpen- 
 dicular will be the tangent, and the hypothenuse the secant 
 of the angle at A. (Art 122.) Then, 
 
 Sec A : AC : : R : AB 
 
 R I AB : : Tan A : BC 
 
 By making the arithmetical calculations, in these two pro- 
 portions, the values of AB and BC, will be found the same 
 as before. 
 
 If the perpendicular be made radius, as in Fig. 16, AB 
 will be the tangent, and AC the secant of the angle at C. 
 Then, 
 
80 
 
 RIGHT ANGLED TRIANGLES. 
 
 Sec C : AC : : R : BC 
 R ; BC : : Tan C : AB 
 
 Ex. 2. If the hypothennse of a right angled triangle be 
 250 rods, and the angle at the base 46 30' ; what is the 
 length of the base and perpendicular ? 
 
 Ans. The base is 172.1 rods, and the perpendic. 181.35. 
 
 CASE II. 
 
 135. Given \ The h yP<*enuse, j to find j The angles and 
 ( And one leg. ) ( The other leg. 
 
 18 
 
 one leg. ) ( The other leg. 
 
 Ex. 1. If the hypothenuse be 35 
 leagues, and the base 26 ; what is 
 the length of the perpendicular* 
 and the quantity of each of the 
 acute angles ? 
 
 To find the angles it is necessary 
 that one of the given sides be made 
 radius. (Art. 130.) 
 
 If the hypothenuse be radius, the base and perpendicular 
 will be sines of their opposite angles. Then, 
 
 R : : AB : Sin C=47 58i' * 
 
 AC 
 
 And to find the perpendicular by Theorem I ; 
 R : AC : : Sin A : BC=23.43 
 
 If the base be radius, the perpendicular will be tangent, 
 and the hypothenuse secant of the angle at A. Then, 
 
 AB : R : : AC : Sec A 
 R : AB : : Tan A : BC 
 
 In this example, where the hypothenuse and base are 
 given, the angles cannot be found by making the perpen- 
 dicular radius. For to find an angle, a given side must be 
 made radius. (Art. 130.) 
 
RIGHT ANGLED TRIANGLES. 
 
 81 
 
 136, Ex. 2. If the hypothenuse be 54 
 miles, and the perpendicular 48 miles, 
 what are the angles, and the base ? 
 
 Making the hypothenuse radius. 
 
 AC : R : : BC : Sin A 
 R I AC : : Sin C : AB 
 
 The numerical calculation will give A=62 44' and AB 
 -=24.74. 
 
 Making the perpendicular radius. 
 
 BC : R : : AC : Sec C 
 R : BC : : Tan C : AB 
 
 The angles cannot be found by making the base radius, 
 when its length is not given. 
 
 CASE III. 
 
 137. Given $ T K an S les > 1 to find j The h 7P tlien e, 
 ( and one leg. ) ( and the other leer. 
 
 Ex. 1. If the base be 60, and the 
 angle at the base 47 12', what is the 
 length of the hypothenuse and the per- 
 pendicular ? 
 
 In this case, as sides only are re- 
 quired, any side may be radius. A 
 
 Making the hypothenuse radius. 
 
 Sin C : AB : : R : AC=88.31 
 R t AC : : Sin A : BC=64.8 
 
82 RIGHT ANGLED TRIANGLES, 
 
 Making the base radius, (Fig. 20.) 
 R : AB : : Sec A : AC 
 R : AB : : Tan A : BO 
 
 Making the perpendicular radius. 
 Tan C : AB : : R : BO 
 R : BC : : Sec G : AC 
 
 138. Ex. 2. If the perpen- 
 dicular be 74, and the angle 
 C 61 27', what is the length 
 of the base and the hypothe- 
 nuse ? 
 
 Making the hypothenuse radius. 
 Sin A : BC : : R : AC 
 R t AC : : sin C : AB 
 
 Making the base radius. 
 
 Tan A : BC : : R : AB 
 R t AB : : sec A : AC 
 
 Making the perpendicular radius. 
 R t BC : : sec C : AC 
 R t BC : : tan C : AB 
 The hypothenuse is 154.83 and the base 136. 
 
 CASE IV. 
 
 139. Given } e ba * e > "* { to find j e hypothenuse, 
 ( Perpendicular, ) ( And the angles. 
 
 Ex. 1. If the base be 284, and 
 the perpendicular 192, what 
 are the angles, and the hypothe- 
 nuse? 
 
 In this case, one of the legs 
 
RIGHT ANGLED TRIANGLES. 83 
 
 must be made radius, to find an angle ; because the hypothe- 
 nuse is not given. 
 
 Making the base radius. 
 
 AB ; R : : BC : tan A=34 4 7 
 R ; AB : : sec A : AC=342.84 
 
 Making the perpendicular radius. 
 
 BC : R : : AB : tan 
 R : BC : : sec C : AC 
 
 Ex. 2. If the base be 640, and the perpendicular 480, 
 what are the angles and hypothenuse ? 
 
 Ans. The hypothenuse is 800, and the angle at the base 
 36 52' 12". 
 
 Examples for Practice. 
 
 1. Given the hypothenuse 68, and the angle at the base 
 
 89 17' ; to find the base and perpendicular. 
 
 2. Given the hypothenuse 850, and the base 594, to find 
 
 the angles, and the perpendicular. 
 
 3. Given the hypothenuse 78, and perpendicular 57, to 
 
 find the base, and the angles. 
 
 4. Given the base 723, and the angle at the base 64 18', 
 
 to find the hypothenuse and perpendicular. 
 
 5. Given the perpendicular 632, and the angle at the base 
 
 81 36', to find the hypothenuse and the base. 
 
 6. Given the base 32, and the perpendicular 24, to find 
 
 the hypothenuse, and the angles, 
 
 140. The preceding solutions are all effected, by means 
 of the tabular sines, tangents, and secants. But, when any 
 two sides of a right angled triangle are given, the third side 
 may be found, without the aid of the trigonometrical tables, 
 by the proposition, that the square of the hypothenuse is equal 
 
84 RIGHT ANGLED TRIANGLES. 
 
 to the sum of the squares of the two perpendicular sides. 
 (Euo. 47. 1.) 
 
 If the legs be given, extracting the square root of the sum 
 of their squares, will give the hypothenuse. Or, if the hypo- 
 thenuse and one leg be given, extracting the square root of 
 the difference of the squares, will give the other leg. 
 
 Let A=the hypothenuse \ 
 
 j=the perpendicular > of a right angled triangle. 
 6==the base ) 
 
 Then #=& 2 -fy, or (Alg. 248.) 
 
 By trans. 6 2 =A 3 p*, or 6=vA 2 ^ 
 
 And p*=h* 6 2 , or 
 
 Ex. 1. If the base is 32, and the perpendicular 24, what 
 is the hypothenuse ? Ans. 40. 
 
 2. If the hypothenuse is 100, and the base 80, what is 
 the perpendicular ? Ans. 60. 
 
 3. If the hypothenuse is 300, and the perpendicular 220, 
 what is the base ? 
 
 Ans. 300 2 220 2 =4160, the root of which is 204 nearly. 
 
 141. It is generally most convenient to find the difference 
 of the squares by logarithms. But this is not to be done by 
 subtraction. For subtraction, in logarithms, performs the 
 office of division. (Art. 41.) If we subtract the logarithm 
 of 6 a from the logarithm of A 2 , we shall have the logarithm, 
 not of the difference of the squares, but of their quotient. 
 There is, however, an indirect, though very simple method, 
 by which the difference of the squares may be obtained by 
 logarithms. It depends on the principle, that the difference 
 of the squares of two quantities is equal to the product of the 
 sum and difference of the quantities. (Alg. 191.) Thus, 
 
OBLIQUE ANGLED TRIANGLES. 85 
 
 as will be seen at once, by performing the multiplication. The 
 two factors may be multiplied by adding their logarithms. 
 Hence, 
 
 142. To obtain the difference of the squares of two quanti~ 
 ties, add the logarithm of the sum of the quantities to the 
 logarithm of their difference. After the logarithm of the 
 difference of the squares is found ; the square root of this 
 difference is obtained, by dividing the logarithm by 2. 
 (Art. 47.) 
 
 Ex. 1. If the hypothenuse be 75 inches, and the base 45, 
 what is the length of the perpendicular ? 
 
 Sum of the given sides 120 log. 2.07918 
 
 Difference of do. 30 1.47712 
 
 Dividing by 2)3.55630 
 
 Side required 60 1.77815 
 
 2. If the hypothenuse is 135, and the perpendicular 108, 
 what is the length of the base ? Ans. 81. 
 
 SECTION IV. 
 
 SOLUTIONS OF OBLiaUE ANGLED TRIANGLES. 
 
 ART. 143. The sides and angles of oblique angled trian- 
 gles may be calculated by the following theorems. 
 
 THEOREM I. 
 In any plane triangle, THE SINES or THE ANGLES ARE AS 
 
 THEIR OPPOSITE SIDES. 
 
 8 
 
86 OBLIQUE ANGLED TRIANGLES. 
 
 Let the angles be denoted by the letters A, B, C, and their 
 opposite sides by a, 6, c, as in Fig. 23 and 24. From one 
 
 of the angles, let the line p be drawn perpendicular to the 
 opposite side. This will fall either within or without the 
 triangle. 
 
 1. Let it fall within as in Fig. 23. Then, in the right 
 angled triangles ACD, and BCD, according to Art. 126, 
 
 R : b : : sin A : p 
 R t a : : sin B ; p 
 
 Here, the two extremes are the same in both proportions. 
 The other four terms are, therefore, reciprocally proportion- 
 al :* that is, 
 
 a : b : : sin A t sin B. 
 
 2. Let the perpendicular^? fall without the triangle, as in 
 Fig. 24. Then in the right angled triangles ACD and 
 BCD; 
 
 E : 5 : : sin A : p 
 R t : * sin B t p 
 Therefore, as before, 
 
 a \ b : : sin A ; sin B. 
 
 Sin A is here put both for the sine of DAC, and for that 
 of BAG. For, as one of these angles is the supplement of 
 the other, they have the same sine. (Art. 90.) 
 i The sines which are mentioned here, and which are used 
 
 * Euclid, 23. 6, 
 
OBLIQUE ANGLED TRIANGLES. 67 
 
 in calculation are tabular sines. But the proportion will be 
 the same, if the sines be adapted to any other radius. (Art. 
 119.) 
 
 THEOREM II. 
 144. In a plane triangle, 
 
 As THE SUM OP ANY TWO OF THE SIDES, 
 
 To THEIR DIFFERENCE ; 
 
 So IS THE TANGENT OF HALF THE SUM OF THE 
 
 OPPOSITE ANGLES J 
 To THE TANGENT OF HALF THEIR DIFFERENCE. 
 
 Thus, the sum 
 
 of AB and AC, A c 
 
 is to their differ- 
 ence ; as the tan- 
 gent of half the 
 sum of the an- 
 gles ACB and 
 ABC, to the tan- 
 gent of half their difference. 
 
 Demonstration. 
 
 Extend CA to G, making' AG equal to AB; then CG is 
 the sum of the two sides AB and AC. On AB, set off AD, 
 equal to AC ; then BD is the difference of the sides AB and 
 AC. 
 
 The sum of the two angles ACB and ABC, is equal to 
 the sum of ACD and ADC ; because each of these sums is 
 the supplement of CAD. (Art. 79.) But as AC==AD by 
 construction, the angle ADC=ACD (Euc. 5 1.*) There- 
 fore ACD is half the sum of ACB and ABC. As AB=AG, 
 the angle AGB=ABG, or DBE. Also, GCE, or ACD=- 
 ADC=BDE. (Euc. 15. l.f) Therefore in the triangles 
 
 * Thomson's Legendre, 11. 1. f rbid - * * 
 
88 
 
 OBLIQUE ANGLED TRIANGLES. 
 
 GCE, and DBE, 
 the two remain- 
 ing angles DEB, 
 and CEG, are 
 equal; (Art. 79.) 
 So that CE is 
 perpendicular to 
 BG. (Euc. Def. 
 7. 1.*) If then CE is made radius, GE is the tangent 
 of GCE, (Art. 84.) that is, the tangent of half the sum of 
 the angles opposite to AB and AC. 
 
 If from the greater of the two angles ACB and ABC, 
 there be taken ACD their half sum; the remaining angle 
 ECB will be their half difference. The tangent of this an- 
 gle, CE being radius, is EB, that is, the tangent of half the 
 difference of the angles opposite to AB and AC. We have then, 
 
 CG=the sum of the sides AB and AC ; 
 
 DB=their difference ; 
 
 GE=the tangent of half the sum of the opposite angles ; 
 
 EB=the tangent of half their difference. 
 
 But by similar triangles, 
 CG : DB : : GE : EB. Q. E. D. 
 
 THEOREM III. 
 
 145. If upon the longest side of a triangle, a perpendicular 
 be drawn from the opposite angle ; 
 
 As THE LONGEST SIDE, 
 
 To THE SUM OF THE TWO OTHERS J 
 
 So 18 THE DIFFERENCE OF THE 
 LATTER, 
 
 To THE DIFFERENCE OF THE SEG- 
 MENTS MADE BY THE PERPEN- 
 DICULAR. 
 
 Thomfon'i Legendrt, Def 12, 1. 
 
OBLIQUE ANGLED TRIANGLES. 89 
 
 In the triangle ABC, if a perpendicular be drawn from 
 C upon AB; 
 
 AB ; CB+CA : : CB CA : BP PA.* 
 
 Demonstration. 
 
 Describe a circle on the centre C, and with the radius BC. 
 Through A and C, draw the diameter LD, and extend BA 
 to H. Then by (Euc. 35. 3. f) 
 
 ABxAH=ALxAD 
 Therefore, 
 
 AB : AD : : AL : AH 
 
 But AD=CD+CA=CB+CA 
 And AL=CL CA=CB CA 
 And AH=HP PA=BP PA (Euc. 3. 3. Thorn. 6. 2.) 
 
 If, then, for the three last terms hi the proportion, we sub- 
 stitute their equals, we have, 
 
 AB : CB+CA : : CB CA : PB PA. 
 
 146. It is to be observed, that the greater segment is next 
 the greater side. If BC is greater than AC, PB is greater 
 than AP. With the radius AC, describe the arc AN. The 
 segment NP=AP. (Euc. 3.3.) But BP is greater than NP. 
 
 14 7. The two segments are to each other, as the tangents 
 of the opposite angles, or the cotangents of the adjacent an- 
 gles. For, in the right angled triangles AGP, and BCP, if 
 CP be made radius, (Art. 126.) 
 
 R t PC : : Tan ACP : AP 
 R : PC : : Tan BCP : BP 
 
 Therefore, by equality of ratios, (Alg. 346.J) 
 Tan ACP : AP : : Tan BCP : BP 
 
 * See note B. f Thomson's Legendre, 28. 4. Cor. J Eue. 11. & 
 
 8* 
 
90 OBLIQUE ANGLED TRIANGLES. 
 
 That is, the segments are as the tangents of the opposite 
 angles. And the tangents of these are the cotangents of the 
 adjacent angles A and B. (Art. 89.) 
 
 Cor. The greater segment is opposite to the greater angle. 
 And of the angles at the base, the less is next the greater 
 side. If BP is greater than AP, the angle BOP is greater 
 than AGP ; and B is less than A. (Art. 77.) 
 
 148. To enable us to find the sides and angles of an 
 oblique angled triangle, three of them must be given. (Art. 
 114.) 
 
 These may be, either 
 
 1. Two angles and a side, or 
 
 2. Two sides and an angle opposite one of them, or 
 
 3. Two sides and the included angle, or 
 
 4. The three sides. 
 
 The two first of these cases are solved by Theorem I, (Art 
 143.) the third by Theorem II, (Art. 144.) and the fourth by 
 Theorem III. (Art. 145.) 
 
 149 In making the calculations, it must be kept in mind, 
 that the greater side is always opposite to the greater angle, 
 (Euc 18, 19. 1.*) that there can be only one obtuse angle hi 
 a triangle, (Art. 76.) and therefore, that the angles opposite 
 to the two least sides must be acute. 
 
 CASE I. 
 
 150. Given, 
 
 Two angles, and ) ( The remaining angle, and 
 
 A side, ) i ( The other tw 
 
 * Thomson's Legendre, 13. 1. 
 
OBLIQUE ANGLED TRIANGLES. 
 
 The third angle is found by merely subtracting the sum 
 of the two which are given from 180. (Art. 79.) 
 
 The sides are found, by stating, according to Theorem I, 
 the following proportion ; 
 
 As the sine of the angle opposite the given side, 
 To the length of the given side ; 
 So is the sine of the angle opposite the required side 
 To the length of the required side. 
 
 As a side is to be found, it is necessary to begin with a 
 tabular number. 
 
 Ex. 1. In the triangle ABC, the 
 side b is given 32 rods, the angle A 
 56 20', and the angle C 49 10', to 
 find the angle B, and the sides a 
 and c. 
 
 The sum of the two given angles 
 56 20'+49 10'=105 30' ; which 
 subtracted from 180, leaves 74 30' 
 the angle B. Then, 
 
 Sin A ! a 
 
 Calculation by logarithms. 
 
 As the sine of B 
 To the side b 
 So is the sine of A 
 To the side a 
 
 As the sine of B 
 To the side b 
 So is the sine of C 
 To the side c 
 
 74 30 ; a. c. 
 
 0.01609 
 
 32 
 
 1.50515 
 
 56 20' 
 
 9.92027 
 
 27.64 
 
 1.44151 
 
 74 30' a. c. 
 
 0.01609 
 
 32 
 
 1.50515 
 
 49 1(V 
 
 9.87887 
 
 25.13 
 
 1.40011 
 
 The arithmetical complement used in the first term here, 
 
02 OBLIQUE ANGLED TRIANGLES. 
 
 may be found in the usual way, or by taking out the cose- 
 cant of the given angle, and rejecting 10 from the index. 
 (Art. 113.) 
 
 Ex. 2. Given the side b 71, the angle A 107 6', and the 
 angle C 27 40' ; to find the angle B, and the sides a and c. 
 The angle B is 45 14'. Then, 
 
 Sin A : 0=95.58 
 Sin B . b . 
 
 When one of the given angles is obtuse, as in this exam- 
 ple, the sine of its supplement is to be taken from the tables. 
 (Art. 99.) 
 
 CASE H. 
 
 151. Given, 
 
 Two sides, and ) ( The remaining side and 
 
 An opposite angle, ) ( The other two angles. 
 
 One of the required angles is found, by beginning with a 
 side, and, according to Theorem I, stating the proportion, 
 
 As the side opposite the given angle, 
 To the sine of that angle ; 
 So is the side opposite the required angle, 
 To the sine of that angle. 
 
 The third angle is found, by subtracting the sum of the 
 other two from 180 ; and the remaining side is found, by 
 the proportion in the preceding article. 
 
 152. In this second case, if the side opposite to the given 
 angle be shorter than the other given side the solution will 
 be ambiguous. Two different triangles may be formed, each 
 of which will satisfy the conditions of the problem. 
 
OBLIQUE ANGLED TRIANGLES. 
 
 Let the side b, the angle 
 A, and the length of the 
 side opposite this angle be 
 given. With the latter for 
 radius, (if it be shorter 
 than 6,)describe an arc, cut- 
 ting the line AH in the 
 
 points B and B'. The lines BC and B'C, will be equal. So 
 that, with the same data, there may be formed two different 
 triangles, ABC and AB'C. 
 
 There will be the same ambiguity in the numerical calcu- 
 lation. The answer found by the proportion will be the sine 
 of an angle. But this may be the sine either of the acute 
 angle AB'C, or of the obtuse angle ABC. For, BC being 
 equal to B'C, the angle CB'B is equal to CBB'. Therefore 
 ABC, which is the supplement of CBB', is also the supple- 
 ment of CB'B. But the sine of an angle is the same, as the 
 sine of its supplement. (Art. 90.) The result of the calcu- 
 lation will, therefore, be ambiguous. In practice, however, 
 there will generally be some circumstances which will deter- 
 mine whether the angle required is acute or obtuse. 
 
 If the side opposite the given angle be longer than the 
 other given side, the angle which is subtended by the latter, 
 will necessarily be acute. For there can be but one obtuse 
 angle in a triangle, and this is always subtended by the long- 
 est side. (Art. 149.) 
 
 If the given angle be obtuse, the other two will, of course, 
 be acute. There can, therefore, be no ambiguity in the 
 solution. 
 
 Ex. 1. Given the angle A, 35 20', the opposite side a 50, 
 and the side b 70 ; te find the remaining side, and the other 
 two angles. 
 
 To find the angle opposite to 6, (Art. 151.) 
 a ; sin A : ; b ; sin B 
 
04 OBLIQUE ANGLED TRIANGLES. 
 
 The calculation here gives the acute angle AB'C 54 3 
 50", and the obtuse angle ABC 125 56' 10". If the latter 
 be added to the angle at A 35 20', the sum will be 161 16 ; 
 10", the supplement of which, 18 43' 50", is the angle 
 ACB. Then in the triangle ABC, to find the side c=AB, 
 
 Sin A : a : : sin C ; c=27.76 
 
 If the acute angle AB'C 54 3' 50" be added to the angle 
 at A 35 20', the sum will be 89 23' 50", the supplement 
 of which, 90 36' 10", is the angle ACB'. Then, in the tri- 
 angle AB'C, 
 
 Sin A ! CB' : : sin C : AB'=86.45. 
 
 Ex. 2. Given the angle at A, 
 63 35', the side b 64, and the 
 side a 72 ; to find the side c, and 
 the angles B and C. 
 
 A B 
 
 a : sin A : : b : sin B=52 45 7 25" 
 Sin A : a : : sin C : c=72.05 
 
 The sum of the angles A and B, is 116 20' 25", the sup- 
 plement of which, 63 39' 35", is the angle C. 
 
 In this example the solution is not ambiguous, because the 
 side opposite the given angle is longer than the other given 
 side. 
 
 Ex. 3. In a triangle of which the angles are A, B, and C, 
 and the opposite sides a, b, and c, as before ; if the angle 
 A be 121 40', the opposite side a 68 rods, and the side b 47 
 rods ; whart are the angles B and C, and what is the length 
 of the side c ? Ans. B is 36 2' 4", C 22 17' 56", and c 
 30.3. 
 
 In this example also, the solution is not ambiguous, be- 
 cause the given angle is obtuse. 
 
OBLIQUE ANGLED TRIANGLES. 05 
 
 CASE III. 
 153. Given, 
 
 Two sides, and ) ( The remaining side, and 
 
 The included angle, ) l ( The other two angles. 
 
 In this case, the angles are found by Theorem II. (Art. 
 144.) The required side may be found by Theorem I. 
 
 In making the solutions, it will be necessary to observe, 
 that by subtracting the given angle from 180, the sum of 
 the other two angles is found ; (Art. 79.) and, that adding 
 half the difference of two quantities to their half sum gives the 
 greater quantity, and subtracting the half difference from the 
 half sum gives the less. The latter proposition may be geo- 
 metrically demonstrated thus ; 
 Let AE, be the 
 
 greater of two 
 
 magnitudes, and 
 
 BE the less. Bisect AB in D, and make AC equal to BE. 
 
 Then, 
 
 AB is the sum of the two magnitudes ; 
 CE their difference ; 
 DA or DB half their sum ; 
 DE or DC half their difference ; 
 But DA-f-DE=AE the greater magnitude ; 
 And DE DE=BE the less. 
 Ex. 1. In the triangle 
 ABC, the angle A is given 26 
 14', the side b 39, and the 
 side c 53 ; to find the angles B 
 and C, and the side a. 
 
 The sum of the sides b and c is 63+39=92 
 And their difference 53 39=14 
 
 The sum of the angles B and C=180 26 14'153 46' 
 And half the sum of B and C is 76 65? 
 
96 OBLIfcUE ANGLED TRIANGLES. 
 
 Then, by Theorem II, (Fig. 30.) 
 (b+c) : (bc) : : tan i(B+C) : tan (B~C) 
 
 To and from the half sum 76 53' 
 
 Adding and subtracting the half difference 33 8 50 
 
 We have the greater angle 110 1 50 
 
 And the less angle 43 44 10 
 
 As the greater of the two given sides is c, the greater 
 angle is C, and the less angle B. (Art. 149.) 
 
 To find the side a, by Theorem I. 
 
 Sin B : b : : sin A : a=24.94. 
 
 Ex. 2. Given the angle A 101 30', the side b 76, and the 
 side c 109 ; to find the angles B and C, and the side a. 
 B is 30 57', C 47 32-J', and a 144.8 
 
 CASE IV. 
 
 154. Given the three sides, to find the angles. 
 
 In this case, the solutions may be made, by drawing a per- 
 pendicular to the longest side, from the opposite angle. This 
 will divide the given triangle into two ri-ght angled triangles. 
 The two segments may be found by Theorem III. (Art. 145.) 
 
 There will then be given, in each of the right angled tri- 
 angles, the hypothenuse and one of the legs, from which the 
 angles may be determined, by rectangular trigonometry. 
 (Art. 135.) 
 
 Ex. 1. In the triangle ABC, 
 the side AB is 39, AC 35, and 
 BC 27. What are the angles ? 
 
 Let a perpendicular be drawn 
 from C, dividing the longest side 
 AB into the two segments AP 
 and BP. Then by Theorem III, 
 
 AB : AC+BC : : AC BC . AP BP. 
 
OBLIQUE ANGLED TRIANGLES. $7 
 
 As the longest side 39 a. c. 8.40894 
 
 To the sum of the two others 62 1.79239 
 
 So is the difference of the latter 8 0.90309 
 
 To the difference of the segments 12.72 1.10442 
 
 The greater of the two segments is AP, because it is next 
 the side AC, which is greater than BC. (Art. 146.) 
 
 To and from half the sum of the segments 19.5 
 
 Adding and subtracting half their difference, (Art. 153.) 6.36 
 We have the greater segment AP 25.86 
 
 And the less BP 13.14 
 
 Then, in each of the right angled triangles APC and BPC, 
 we have given the hypothenuse and base ; and by Art, 135. 
 
 AC : R : : AP : cos A=42 21' 57" 
 BC : R : : BP ; cos B=60 52' 42" 
 
 And subtracting the sum of the angles A and B from 
 180, we have the remaining angle ACB=76 45' 21". 
 
 Ex. 2. If the three sides of a triangle are 78, 96, and 
 104 ; what are the angles ? 
 
 Ans. 45 41' 48", 61 43' 27", and 72 34' 45". 
 
 Examples for Practice. 
 
 1. Given the angle A 54 30', the angle B 63 10', and the 
 side a 164 rods; to find the angle C, and the sides 6 
 and c. 
 
 4. Given the angle A 45 6', the opposite side a 93, and the 
 side b 108 ; to find the angles B and C, and the side c. 
 
 3. Given the angle A 67 24', the opposite side a 62, and 
 
 the side b 46 ; to find the angles B and C, and the side c. 
 
 4. Given the angle A 127 42', the opposite side a 381, and 
 
 the side b 184 ; to find the angles B and C, and the 
 side c, 
 
 9 
 
OBLIQUE ANGLED TRIANGLES. 
 
 5. Given the side b 58, the side c 67, and the included angl* 
 
 A=36 ; to find the angles B and C, and the side a. 
 
 6. Given the three sides, 631, 268, and 546; to find the 
 
 angles. 
 
 155. The three theorems demonstrated in this section, 
 have been here applied to oblique 
 angled triangles only. But they 
 are equally applicable to right 
 angled triangles. 
 
 Thus, in the triangle ABC, ac- 
 cording to Theorem I, (Art. 143.) 
 
 i> 
 
 Sin B : AC : : sin A : BC 
 
 This is the same proportion as one stated in Art 134, ex- 
 cept that, in the first term here, the sine of B is substituted 
 for radius. But, as B is a right angle, its sine is equal to 
 radius. (Art. 95.) 
 
 Again, in the triangle ABC, 
 by the same theorem ; 
 
 Sin A : BC : : sin C : AB 
 
 This is also one of the pro- 
 portions in rectangular trigo- 
 nometry, when the hypothe- 
 nuse is made radius. 
 
 The other two theorems might be applied to the solution of 
 right angled triangles. But, when one of the angles is known 
 to be a right angle, the methods explained in the preceding 
 section, are much more simple in practice. 
 
GEOMETRICAL CONSTRUCTION OF TRIAN&LES. 99 
 
 SECTION V. 
 
 GEOMETRICAL CONSTRUCTION OF TRIANGLES, B TUB 
 PLANE SCALE. 
 
 ART. 156. To facilitate the construction of geometrical 
 figures, a number of graduated lines are put upon the com- 
 mon two feet scale ; one side of which is called the Plane 
 Scale, and the other side, Gunter's Scale. The most im- 
 portant of these are the scales of equal parts, and the line 
 of chords. In forming a given triangle, or any other right 
 lined figure, the parts which must be made to agree with the 
 conditions proposed, are the lines and the angles. For the 
 former, a scale of equal parts is used ; for the latter, a line 
 of chords. 
 
 157. The line on the upper side of the plane scale, is 
 divided into inches and tenths of an inch. Beneath this, on 
 the left hand, are two diagonal scales of equal parts,* divided 
 into inches and half inches, by perpendicular lines. On the 
 larger scale, one of the inches is divided into tenths, by lines 
 which pass obliquely across, so as to intersect the parallel 
 lines which run from right to left. The use of the oblique 
 lines is to measure hundredths of an inch, by inclining more 
 and more to the right, as they cross each of the parallels. 
 
 To take off, for instance, an extent of 3 inches, 4 tenths, 
 and 6 hundredths ; 
 
 Place one foot of the dividers at the intersection of the 
 perpendicular line marked 3 with the parallel line marked 6, 
 
 * These lines are not represented by a figure, as the learner is rap- 
 posed to have the scale before him, 
 
100 GEOMETRICAL CONSTRUCTION OF TRIANGLES. 
 
 and the other foot at the intersection of the latter with the 
 oblique line marked 4. 
 
 The other diagonal scale is of the same nature. The 
 divisions are smaller, and are numbered from left to right. 
 
 158. In geometrical constructions, what is often required, 
 is to make a figure, not equal to a given one, but only sim- 
 ilar. Now figures are similar which have equal angles, and 
 the sides about the equal angles proportional. (Euc. Def. 
 1. 6.*) Thus a land surveyor, in plotting a field, makes the 
 several lines in his plan to have the same proportion to each 
 other, as the sides of the field. For this purpose a scale of 
 equal parts may be used, of any dimensions whatever. If 
 the sides of the field are 2, 5, 7, and 10 rods, and the lines 
 in the plan are 2, 6, 7, and 10 inches, and if the angles are 
 the same in each, the figures are similar. One is a copy of 
 the other, upon a smaller scale. 
 
 So any two right lined figures are similar, if the angles are 
 the same in both, and if the number of smaller parts in each 
 side of one, is equal to the number of larger parts in the cor- 
 responding sides of the other. The several divisions on the 
 scale of equal parts may, therefore, be considered as repre- 
 senting any measures of length, as feet, rods, miles, &c. All 
 that is necessary is, that the scale be not changed, in the 
 construction of the same figure ; and that the several divi- 
 sions and subdivisions be properly proportioned to each 
 other. If the larger divisions, on the diagonal scale, are 
 units, the smaller ones are tenths and hundredths. If the 
 larger are tens, the smaller are units and tenths. 
 
 159. In laying down an angle, of a given number of de- 
 grees, it is necessary to measure it. Now the proper meas- 
 ure of an angle is an arc of a circle. (Art. 74.) And the 
 measure of an arc, where the radius is given, is its chord. 
 For the chord is the distance, in a straight line, from one 
 
 STT""" 1 
 
 * Thomson's Legendre, Def. B. 4. 
 
GEOMETRICAL CONSTRUCTION OF TRIANGLES. 101 
 
 end of the arc to the other. 
 
 Thus the chord AB, is a F \ 
 
 measure of the arc ADB, 
 
 and of the angle ACB. 
 
 To form the line of chords, 
 a circle is described, and the 
 length of its chords deter- 
 
 mined for every degree of the quadrant. These measures 
 are put on the plane scale, on the line marked CHO. 
 
 160. The chord of 60 is equal to radius. (Art. 95.) In 
 laying down or measuring an angle, therefore, an arc must 
 be drawn, with a radius which is equal to the extent from 
 to 60 on the line of chords. There are generally on the 
 scale, two lines of chords. Either of these may be used ; 
 but the angle must be measured by the same line from 
 which the radius is taken. 
 
 161. To make an angle, then, of a given number of de- 
 grees ; from one end of a straight line as a centre, and with 
 a radius equal to the chord of 60 on the line of chords, de- 
 scribe an arc of a circle cutting a straight line. From the 
 point of intersection, extend the chord of the given number 
 of degrees, applying the other extremity to the arc ; and 
 through the place of meeting, draw the other line from the 
 angular point. 
 
 If the given angle is obtuse, take from the scale the chord 
 of half the number of degrees, and apply it twice to the arc. 
 Or make use of the chords of any two arcs whose sum is 
 equal to the given number of degrees. 
 
 A right angle may be constructed, by drawing a perpen- 
 dicular without using the line of chords. 
 
 Ex. 1. To make an angle of 32 degrees. With the point 
 C, in the line CH, for a centre, and with the chord of 60 
 for radius, describe the arc ADF. Extend the chord of 32 
 from A to B ; and through B, draw the line BC. Then is 
 ACB an angle of 32 degrees. 
 
 9* 
 
102 GEOMETRICAL CONSTRUCTION OF TRIANGLES. 
 
 2. To make an angle of 140 degrees. On the line CH, 
 with the chord of 60, describe the arc ADF ; and extend 
 
 the chord of 70 from A to D, and from D to B. The arc 
 ADB=70X2=140 . 
 
 On the other hand : 
 
 162. To measure an angle ; On the angular point as a 
 centre, and with the chord of 60 for radius, describe an arc 
 to cut the two lines which include the angle, The distance 
 between the points of intersection, applied to the line of 
 chords, will give the measure of the angle in degrees. If the 
 angle be obtuse, divide the arc into two parts. 
 
 Ex. 1. To measure the angle ACB. (Fig. 33, page 101.) 
 Describe the arc ADF, cutting the lines CH and CB. The 
 distance AB, will extend 32 on the line of chords. 
 
 2. To measure the angle ACB. (Fig. 34.) Divide the 
 arc ADB into two parts, either equal or unequal, and meas- 
 ure each part, by applying its chord to the scale. The sum 
 of the two will be 140. 
 
 163. Besides the lines of chords, and of equal parts, on 
 the plane scale ; there are also lines of natural sines, tangents, 
 and secants, marked Sin., Tan., and Sec. ; of semitangents, 
 marked S. T. ; of longitude, marked Lon. or M. L. ; of 
 rhumbs, marked Rhu. or Rum., &c. These are not neces- 
 sary in trigonometrical construction. Some of them are used 
 in Navigation ; and some of them in the projections of the 
 Sphere. 
 
GEOMETRICAL CONSTRUCTION OF TRIANGLES. 103 
 
 164. In Navigation, the quadrant, instead of being grad- 
 uated in the usual manner, is divided into eight portions, 
 called Rhumbs. The Rhumh lime, on the scale, is a line of 
 chords, divided into rhumbs and quarter-rhumbs, instead of 
 degrees. 
 
 165. The line of Longitude is intended to show the num- 
 ber of geographical miles in a degree of longitude, at differ- 
 ent distances from the equator. It is placed over the line 
 of chords, with the numbers in an inverted order: so that 
 the figure above shows the length of a degree of longitude, 
 in any latitude denoted by the figure below.* Thus, at the 
 equator, where the latitude is 0, a degree of longitude is 60 
 geographical miles. In latitude 40, it is 46 miles ; in lati- 
 tude 60, 30 miles, &c. 
 
 166. The graduation on the line of secants begins where 
 the line of sines ends. For the greater sine is only equal to 
 radius ; but the secant of the least arc is greater than 
 radius. 
 
 167. The semitangents are the tangents of half the given 
 arcs. Thus, the semitangent of 20 is the tangent of 10. 
 The line of semitangents is used in one of the projections of 
 the sphere. 
 
 168. In the construction of triangles, the sides and angles 
 which are given, are laid down according to the directions in 
 Arts. 158, 161. The parts required are then .measured, ac- 
 cording to Arts. 158, 162. The following problems corres- 
 pond with the four cases of oblique angled triangles ; (Art. 
 148.) but are equally adapted to right angled triangles. 
 
 169. PROB. I. The angles and one side of a triangle being 
 given ; to find, by construction, the other two sides. 
 
 Draw the given side. From the ends of it, lay off two 
 
 * Sometimes the line of longitude is placed under the line of chords. 
 
104 
 
 GEOMETRICAL CONSTRUCTION OF TRIANGLES. 
 
 of the given angles. Extend the other sides till they inter- 
 sect; and then measure their lengths on a scale of equal 
 parts. 
 
 Ex. 1. Given the side b 32 rods, 
 the angle A 56 20', and the angle 
 C 49 W ; to construct the trian- 
 gle, and find the lengths of the sides 
 a and c. 
 
 Their lengths will be 25 and 
 27*. 
 
 2. In a right angled triangle, 
 given the hypothenuse 90, and the 
 angle A 32 20', to find the base 
 and perpendicular. 
 
 The length of AB will be 76, and 
 of BC 48. 
 
 3. Given the side AC 68, the an- A B 
 gle A 124, and the angle C 37 : 
 
 to construct the triangle. 
 
 ' 170. PROB. II. Two sides and an opposite angle being 
 given, to find the remaining side, and the other two angles. 
 
 Draw one of the given sides ; from one end of it, lay off 
 the given angle ; and extend a line indefinitely for the re- 
 quired side. From the other end of the first side, with the 
 remaining given side for radius, describe an are cutting the 
 indefinite line. The point of intersection will be the end 
 of the required side. 
 
 If the side opposite the given angle be less than the othei 
 given side, the case will be ambig- 
 uous. (Art. 152.) 
 
 Ex. 1. Given the angle A 63 
 35', the side 632, and the side a 36. 
 
 The side AB will be 36 nearly, 
 the angle B 52 45i', and C 63 
 
GEOMETRICAL CONSTRUCTION OF TRIANGLES. 105 
 
 2. Given the. angle A 
 35 20', the opposite side 
 a 25, and the side b 35. 
 
 Draw the side b 35, 
 make the angle A 35 20', 
 
 and extend AH indefinite- _ _ 
 
 ly. From C with radius 
 
 25, describe an arc cutting AH in B and B'. Draw CB and 
 CB', and two triangles will be formed, ABC and AB'C, each 
 corresponding with the conditions of the problem. 
 
 3. Given the angle A 116, the opposite side a 38, and 
 the side b 26 ; to construct the triangle. 
 
 171. PROS. III. Two sides and the included angle being 
 given ; to find the other side and angles. 
 
 Draw one of the given sides. From one end of it lay off 
 the given angle, and draw the other given side. Then con- 
 nect the extremities of this and the first line. 
 
 Ex. 1. Given the angle A 
 26 14', the side b 78, and 
 the side c 106 ; to find B, C, 
 and a. 
 
 The side a will be 50, the 
 angle B 43 44', and C 
 110 2'. 
 
 2. Given A 86, b 65, and c 83 ; to find B, C, and a. 
 
 172. PROB. IV. The three sides being given; to find the 
 angles. 
 
 Draw one of the sides, and from one end of it, with an 
 extent equal to the second side, describe an arc. From the 
 other end, with an extent equal to the third side, describe a 
 second arc cutting the first ; and from the point of intersec- 
 tion draw the two sides. (Euc. 22. 1.) 
 
10'6 
 
 GEOMETRICAL CONSTRUCTION OF TRIAKQL^S. 
 
 Ex. 1. Given AB 78, AC 70, 
 and BC 54, to find the angles. 
 
 The angles will be A 42 22', 
 B 60 52|' and C 76 45i'. 
 
 2. Given the three sides 58, 
 39, and 46 ; to find the angles. 
 
 173. Any right lined figure 
 
 whatever, whose sides and angles are given, may be con- 
 structed, by laying down the sides from a scale of equal 
 parts, and the angles from a line of chords. 
 
 Ex. Given the sides AB= 
 20> BC=*22, CD=30, DE= 
 12 ; and the angles B=102, B f 
 C=130, D=108, to con- 
 struct the figure. 
 
 frraw the side AB=20, 
 make the angle B=102, draw 
 BC=22, make C=130, draw CD=30, make D=108, 
 draw DE=12, and connect E and A. 
 
 The last line, EA, may be measured on the scale of equal 
 parts j and the angles E and. A, by a line of chords. 
 
GUNTER'S SCALE. 107 
 
 SECTION VI. 
 
 IRT. 174, An expeditious method of solving the problems 
 in /igonometry, and making other logarithmic calculations, 
 m <u mechanical way, has been contrived by Mr. Edmund 
 irunter. The logarithms of numbers, of sines, tangents, 
 <fec., are represented by lines. By means of these, multipli- 
 cation, division, the rule of three, involution, evolution, &c., 
 may be p^rfoimed much more rapidly, than in the usual 
 method by figures. 
 
 The logaritharic lines are generally placed on one side 
 only of the scale in common use. They are, 
 
 A line of artificial Sines divided into ^Rhumbs, and 
 
 marked, S. R. 
 
 A line of artificial Tangent*, do T . R. 
 
 A line of the logarithms of Numbers, Num. 
 
 A line of artificial Sines, to every degree, SIN. 
 
 A line of artificial Tangents, do TAN. 
 
 A line of Versed Sims. V. S. 
 
 To these are added a line of equal parts, and a line of 
 Meridional Parts, which are not logarithmic. The latter is 
 used in Navigation. 
 
 The Line of Numbers. 
 
 175. Portions of the line of Numbers, are intended to 
 represent the logarithms of the natural series of numbers 
 2, 3, 4, 5, <fec. 
 
 The logarithms of 10, 100, 1000, &c., are 1, 2, 3, <kc. 
 (Art. 3.) 
 
108 CfI7NTER r S SCALE, 
 
 If, then, the log. of 10 be represented by a line of 1 foot ; 
 the log. of 100 will be repres'd by one of 2 feet; 
 the log. of 1000 by one of 3 feet ; 
 
 the lengths of the several lines being proportional to the cor- 
 responding logarithms in the tables. Portions of a foot will 
 represent the logarithms of numbers between 1 and 10 ; and 
 portions of a line 2 feet long, the logarithms of numbers be- 
 tween 1 and 100. 
 
 On Gunter's scale, the line of the logarithms of numbers 
 begins at a brass pin on the left, and the divisions are num- 
 bered 1, 2, 3, <fec., to another pin near the middle. From 
 this the numbers are repeated, 2, 3, 4, &c., which may be 
 read 20, 30, 40, &c. The logarithms of numbers between 
 1 and 10, are represented by portions of the first half of the 
 line; and the logarithms of numbers between 10 and 100, 
 by portions greater than half the line, and less than the 
 whole. 
 
 176. The logarithm of 1, which is 0, is denoted, not by 
 any extent of line, but by a point under 1, at the commence- 
 ment of the scale. The distances from this point to differ- 
 ent parts of the line, represent other logarithms, of which 
 the figures placed over the several divisions are the natural 
 numbers. For the intervening logarithms, the intervals be- 
 tween the figures, are divided into tenths, and sometimes 
 into smaller portions. On the right hand half of the scale, 
 as the divisions which are numbered are tens, the subdivisions 
 are units. 
 
 Ex. 1. To take from the scale the logarithm of 3.6; set 
 one foot of the dividers under 1 at the beginning of the 
 scale, and extend the other to the 6th division after the first 
 figure 3. 
 
 2. For the logarithm of 47 ; extend from 1 at the be- 
 ginning, to the 7th subdivision after the second figure 4*. 
 
 * If the dividers will not reach the distance required ; first open them so 
 as to take off half, or any part of the distance, and then the remaining part. 
 
GtNTER's SCALE. 109 
 
 177. It will be observed, that the divisions and sub- 
 divisions decrease, from left to right ; as in the tables of logo,' 
 rithms, the differences decrease. The difference between the 
 logarithms of 10 and 100, is no greater, than the difference 
 between the logarithms of 1 and 10. 
 
 178. The line of numbers, as it has been here explained, 
 furnishes the logarithms of all numbers between 1 and 100. 
 
 And if the indices of the logarithms be neglected, the 
 same scale may answer for all numbers whatever. For the 
 decimal part of the logarithm of any number is the same, 
 as that of the number multiplied or divided by 10, 100, 
 &c. (Art. 14.) In logarithmic calculations, the use of the 
 indices is to determine the distance of the several figures 
 of the natural numbers from the place of units. (Art. 11.) 
 But in those cases in which the logarithmic line is com- 
 monly used, it will not generally be difficult to determine the 
 local value of the figures in the result. 
 
 179. We may, therefore, consider the point under 1 at the 
 left hand, as representing the logarithm of 1, or 10, or 100; 
 or iV, or T-tar, &c., for the decimal part of the logarithm of 
 each of these is 0. But if the first 1 is reckoned 10, all the 
 succeeding numbers must also be increased in a tenfold 
 ratio ; so as to read, on the first half of the line, 20, 30, 40, 
 &c., and on the other half, 200, 300, &c. 
 
 The whole extent of the logarithmic line, 
 
 is from 1 to 100, or from 0.1 to 10, 
 
 or from 10 to 1000, or from 0.01 to 1, 
 
 or from 100 to 10000, &c. or from 0.001 to 0.1, &c. 
 
 Different values may, on different occasions, be assigned 
 to the several numbers and subdivisions marked on this line. 
 But for any one calculation, the value must remain the 
 same. 
 
 Ex. Take from the scale 365. 
 
 As this number is between 10 and 1000, let the 1 at the 
 10 
 
beginning of the scale, be reckoned 10. Then, from this 
 point to the second 3 is 300 ; to the 6th dividing stroke is 
 60 ; and half way from this to the next stroke is 5. 
 
 180. Multiplication, division, &c., are performed by the 
 line of numbers, on the same principle, as by common loga- 
 rithms. Thus, 
 
 To multiply by this line, add the logarithms of the two 
 factors; (Art. 37.) that is, take off, with the dividers, that 
 length of line which represents the logarithm of one of the 
 factors, and apply this so as to extend forward from the end 
 of that which represents the logarithm of the other factor. 
 The sum of the two will reach to the end of the line rep- 
 resenting the logarithm of the product. 
 
 Ex. Multiply 9 into 8. The extent from 1 to 8, added to 
 that from 1 to 9, will be equal to the extent from 1 to 72, the 
 product. 
 
 181. To divide by the logarithmic line, subtract the loga- 
 rithm of the divisor from that of the dividend ; (Art. 41.) 
 that is, take off the logarithm of the divisor, and this extent 
 set back from the end of the logarithm of the dividend, will 
 reach to the logarithm of the quotient! 
 
 Ex. Divide 42 by 7. The extent from 1 to 7, set back 
 from 42, will reach to 6, the quotient. 
 
 182. Involution is performed in logarithms, by multiply- 
 ing the logarithm of the quantity into the index of the 
 power ; (Art. 45.) that is, by repeating the logarithms as 
 many times as there are units in the index. To involve a 
 quantity on the scale, then, take in the dividers the linear 
 logarithm, and double it, treble it, &c., according to the index 
 of the proposed power. 
 
 Ex. 1. Required the square of 9. Extend the dividers 
 from 1 to 9. Twice this extent will reach to 81, the square. 
 
 2. Required the cube of 4. The extent from 1 to 4 re- 
 peated three times, will reach to 64 the cube of 4. 
 
 183. On the other hand, to perform evolution on the scale ; 
 
111 
 
 take half, one-third, <kc., of the logarithm of the quantity, ac- 
 cording to the index of the proposed root. 
 
 Ex. 1. Required the square root of 49. Half the extent 
 from 1 to 49, will reach from 1 to 7, the root. 
 
 2. Required the cube root of 27. One third the distance 
 from 1 to 27, will extend from 1 to 3, the root. 
 
 184. The Rule of Three may be performed on the scale, 
 in the same manner as in logarithms, by adding the two 
 middle terms, and from the sum, subtracting the first term 
 (Art. 52.) But it is more convenient in practice to begin by 
 subtracting the first term from one of the others. If four 
 quantities are proportional, the quotient of the first divided 
 by the second, is equal to the quotient of the third divided 
 by the fourth. (Alg. 315.) 
 
 Thus, if a ; 6 : : c : d, theni=-, and^LJ-. (Alg. 344.) 
 6 d c d 
 
 But in logarithms, subtraction takes the place of division ; 
 so that, 
 
 log. a log. 6=log. c log. d. Or, log. a log. c=log. b 
 log. d. 
 
 Hence, 
 
 185. On the scale, the difference between the first and 
 second terms of a proportion, is equal to the difference between 
 the third and fourth. Or, the difference between the first 
 and third terms, is equal to the difference between the 
 second and fourth. 
 
 The difference between the two terms is taken, by ex- 
 tending the dividers from one to the other. If the 
 second term be greater than the first ; the fourth must be 
 greater than the third; if less, less.* Therefore, if the 
 dividers extend forward from left to right, that is, from a 
 less number to a greater, from the first term to the second ; 
 
 * Euclid, 14,5. 
 
112 GUNTER'S SCALE. 
 
 they must also extend forward from the third to the fourth. 
 But if they extend backward, from the first term to the 
 second ; they must extend the same way, from the third to 
 the fourth. 
 
 Ex. 1. In the proportion 3 : 8 t : 12 : 32, the extent 
 from 3 to 8, will reach from 12 to 32 ; Or, the extent from 
 8 to 12, will reach from 8 to 32. 
 
 2. If 54 yards of cloth cost 48 dollars, what will 18 yds. 
 cost? 
 
 54 : 48 : : 18 : 16 
 
 The extent from 64 to 48, will reach backwards from 18 
 to 16, 
 
 3. If 63 gallons of wine cost 81 dollars, what will 35 gal- 
 lons cost ? 
 
 63 : 81 : : 35 : 45 
 The extent from 63 to 81, will reach from 35 to 45. 
 
 The Line of Sines. 
 
 186. The line on Gunter's scale marked SIN. is a line of 
 logarithmic sines, made to correspond with the line of num- 
 bers. The whole extent of the line of numbers, (Art, 179.) 
 
 is from 1 to 100, whose logs, are 0.00000 and 2.00000, 
 or from 10 to 1000, whose logs, are 1.00000 and 3.00000, 
 or from 100 to 10000, whose logs, are 2.00000 4.00000, 
 
 the difference of the indices of the two extreme logarithms 
 being in each case 2. 
 
 Now the logarithmic sine of 34' 22" 41"' is 8.00000 
 And the sine of 90 (Art. 95.) is 10.00000 
 
 Here also the difference of the indices is 2. If then the 
 point directly beneath one extremity of the line of numbers, 
 be marked for the sine of 34' 22" 41"' ; and the point 
 
GUNTER'S SCALE. 113 
 
 beneath the other extremity, for the sine of 90 ; the interval 
 may furnish the intermediate sine ; the divisions on it being 
 made to correspond with the decimal part of the logarithmic 
 sines in the tables.* 
 
 The first dividing stroke in the line of Sines is generally 
 at 40', a little farther to the right than the beginning of 
 the line of numbers. The next division is at 50' ; then 
 begins the numbering of the degrees, 1, 2, 3, 4, &c., from 
 left to right. 
 
 The Line of Tangents. 
 
 187. The first 45 degrees on this line are numbered from 
 left to right, nearly in the same manner as on the line of 
 Sines. 
 
 The logarithmic tangent of 34' 22" 35'" is 8.00000 
 And the tangent of 45, (Art. 95.) is 10.00000 
 
 The difference of the indices being 2, 45 degrees will 
 reach to the end of the line. For those above 45 the scale 
 ought to be continued much farther to the right. But as 
 this would be inconvenient, the numbering of the degrees, 
 after reaching 45, is carried back from right to left. The 
 same dividing stroke answers for an arc and its complement, 
 one above and the other below 45. For, (Art. 93. Pro- 
 por. 9.) 
 
 tan : R : : R : cot 
 In logarithms, therefore, (Art. 184.) 
 tan R=R cot. 
 
 That is, the difference between the tangent and radius, is 
 equal to the difference between radius and the cotangent : in 
 
 * To represent the sines less than 34' 22" 41'", the scale must be ex- 
 tended on the left indefinitely. For, as the sine of an arc approaches 
 to 0, its logarithm, which is negative, increases without limit. (Art. 15.) 
 
114 
 
 other words, one is as much greater than the tangent of 45, 
 as the other is less. In taking, then, the tangent of an arc 
 greater than 45, we are to suppose the distance between 
 45 and the division marked with a given number of degrees, 
 to be added to the whole line, in the same manner as if the 
 line were continued out. In working proportions, extending 
 the dividers lack, from a less number to a greater, must 
 be considered the same as carrying them forward in other 
 cases. See Art. 185. 
 
 Trigonometrical Proportions on the Scale. 
 
 188. In working proportions in trigonometry by the scale; 
 the extent from the first term to the middle term of the same 
 name, will reach from the other middle term to the fourth 
 term. (Art. 185.) 
 
 In a trigonometrical proportion, two of the terms are the 
 lengths of sides of the given triangle ; and the other two are 
 tabular sines, tangents, &c. The former are to be taken from 
 the line of numbers ; the latter, from the lines of logarith- 
 mic sines and tangents. If one of the terms is a secant, the 
 calculation cannot be made on the scale, which has com- 
 monly no line of secants. It must be kept in mind that 
 radius is equal to the sine of 90, or to the tangent of 45. 
 (Art. 95.) Therefore, whenever radius is a term in the pro- 
 portion, one foot of the dividers must be set on the end of 
 the line of sines or of tangents. 
 
 189. The following examples are taken from the propor- 
 tions which have already been solved by numerical calcula- 
 tion. 
 
 Ex. 1. In Case I, of right angled triangles, (Art. 134. 
 ex. 1.) 
 
 R I 45 : : sin 32 20' : 24 
 
 Here the third term is a sine ; the first term radius is, 
 therefore, to be considered as the sine of 90. Then the 
 
GUNTER'S SCALE. 115 
 
 extent from 90 to 32 20' on the line of sines, will reach 
 from 45 to 24 on the line of numbers. As the dividers 
 are set back from 90 to 32 20' ; they must also be set back 
 from 45. (Art. 185.) 
 
 2. In the same case, if the base be made radius, (page 60.) 
 
 R : 38 : : tan 32 20' ; 24 
 
 Here, as the third term is a tangent, the first term radius 
 is to be considered the tangent of 45. Then the extent 
 from 45 to 32 20' on the line of tangents, will reach from 
 38 to 24 on the line of numbers. 
 
 3. If the perpendicular be made radius, (page 62.) 
 
 R t 24 : : tan 57 40 ; : 38 
 
 The extent from 45 to 57 40' on the line of tangents, 
 will reach from 24 to 38 on the line of numbers. For the 
 tangent of 5*7 40' on the scale, look for its complement 32 
 20'. (Art. 187.) In this example, although the dividers 
 extend back from 45 to 57 40' ; yet, as this is from a less 
 number to a greater, they must extend for ward on the line 
 of numbers. (Arts. 185, 187.) 
 
 4. In Art. 135, 
 
 35 : R : : 26 : sin 48 
 The extent from 35 to 26 will reach from 90 to 48. 
 
 5. In Art. 136, 
 
 R I 48 : : tan 27-J- : 24| 
 The extent from 45 to 2^|, will reach from 48 to 24f. 
 
 6. In Art. 150, ex. 1. 
 
 Sin 74 30' : 32 : : sin 56 20' : 27. 
 
 For other examples, see the several cases in Sections IEL 
 and IV. 
 
 190. Though the solutions in trigonometry may be ef- 
 
116 TRIGONOMETRICAL ANALYSIS. 
 
 fected by the logarithmic scale, or by geometrical construc- 
 tion, as well as by arithmetical computation ; yet the latter 
 method is by far the most accurate. The first is valuable 
 principally for the expedition with which the calculations are 
 made by it. The second is of use, in presenting the/orm 
 of the triangle to the eye. But the accuracy which attends 
 arithmetical operations, is not to be expected, in taking lines 
 from a scale with a pair of dividers.* 
 
 SECTION VII. 
 
 THE FIRST PRINCIPLES OF TRIGONOMETRICAL ANALYSIS. 
 
 ART. 191. In the preceding sections, sines, tangents, and 
 secants have been employed in calculating the sides and an- 
 gles of triangles. But the use of these lines is not confined 
 to this object. Important assistance is derived from them, 
 in conducting many of the investigations in the higher 
 branches of analysis, particularly in physical astronomy. It 
 does not belong to an elementary treatise of trigonometry, 
 to prosecute these inquiries to any considerable extent. But 
 this is the proper place for preparing the formula, the appli- 
 cations of which are to be made elsewhere. 
 
 Positive and Negative SIGNS in Trigonometry. 
 
 192. Before entering on a particular consideration of the 
 algebraic expressions which are produced by combinations 
 of the several trigonometrical lines, it will be necessary to 
 attend to the positive and negative signs in the different 
 
 * See note 0. 
 
TRIGONOMETRICAL ANALYSIS. 
 
 quarters of the circle. The sines, tangents, <kc., in the 
 tables, are calculated for a single quadrant only. But these 
 are made to answer for the whole circle. For they are of the 
 same length in each of the four quadrants. (Art. 90.) Some 
 of them however, are positive; while others are negative. 
 In algebraic processes, this distinction must not be neg- 
 lected. 
 
 193. For the purpose of tracing the changes of the signs, 
 in different parts of the circle, let it be supposed that a 
 straight line CT is 
 fixed at one end C, 
 while the other end 
 is carried round, like 
 a rod moving on a 
 pivot ; so that the 
 point S shall describe 
 
 the circle ABDH. If 
 the two diameters 
 AD and BH, be per- 
 pendicular to each 
 other, they will di- 
 vide the circle into 
 quadrants. 
 
 194. In the first quadrant AB, the sine, cosine, tangent, 
 &c., are considered all positive. In the second quadrant BD, 
 the sine P'S' continues positive ; because it is still on the 
 upper side of the diameter AD, from which it is measured. 
 But the cosine, which is measured from BH, becomes nega- 
 tive, as soon as it changes from the right to the left of this 
 line. (Alg. 382.) In the third quadrant the sine becomes 
 negative, by changing from the upper side to the under side 
 of DA. The cosine continues negative, being still on the 
 left of BH. In the fourth quadrant, the sine continues neg- 
 ative. But the cosine becomes positive, by passing to the 
 right of BH. 
 
118 TRIGONOMETRICAL ANALYSIS. 
 
 195. The signs of the tangents and secants may be de- 
 rived from those of the sines and cosines. The relations of 
 these several lines to each other must be such, that a uniform 
 method of calculation may extend through the different 
 quadrants. 
 
 In the first quadrant, (Art. 93. Propor. 1.) 
 
 K : cos t : tan ; sin, that is, Tan Rxsin . 
 
 cos 
 
 The sign of the quotient is determined from the signs of 
 the divisor and dividend. (Alg. 100.) The radius is con- 
 sidered as always positive. If then the sine and cosine be 
 both positive or both negative, the tangent will be positive, 
 But if one of these be positive, while the other is negative, 
 the tangent will be negative. 
 
 Now by the preceding article, 
 
 In the 2d quadrant, the sine is positive, and the cosine 
 negative. 
 
 The tangent must therefore be negative. 
 
 In the 3d quadrant, the sine and cosine are both negative. 
 The tangent must therefore be positive. 
 
 In the 4th quadrant, the sine is negative, and the cosine 
 positive. 
 
 The tangent must therefore be negative. 
 
 196. By the 9th, 3d, and 6th proportions in Art. 93. 
 
 R a 
 
 1. Tan : R : : R : cot, that is Cot= 
 
 tan 
 
 Therefore, as radius is uniformly positive, the cotangent 
 must have the same sign as the tangent. 
 
 R a 
 
 2. Cos : R : : R : sec," that is, Sec*- 
 
 cos 
 
TRIGONOMETRICAL ANALYSIS. 
 
 119 
 
 The secant, therefore, must have the same sign as the 
 cosine. 
 
 3. Sin : B : : R : cosec, that is, 
 
 sn 
 
 The cosecant, therefore, must have the same sign as the 
 sine. 
 
 The versed sine, as it is measured from A, in one direction 
 only, is invariably positive. 
 
 197. The tangent 
 AT increases, as the T 
 arc extends from A 
 towards B. (See also 
 Fig 11. p. 69.) Near 
 B the increase is very 
 rapid ; and when the 
 difference between 
 the arc and 90, is 
 less than any assign- 
 able quantity, the 
 tangent is greater 
 than any assignable 
 quantity, and is said 
 
 to be infinite. (Alg. 447.) If the arc is exactly 90 degrees, 
 it has, strictly speaking, no tangent. For a tangent is a line 
 drawn perpendicular to the diameter which passes through 
 one end of the arc, and extended till it meets a line proceed- 
 ing from the centre through the other end. (Art. 84.) But 
 if the arc is 90 degrees, as AB, the angle ACB is a right 
 angle, and therefore AT is parallel to CB ; so that, if these 
 lines be extended ever so far, they never can meet. Still, as 
 an arc infinitely near to 90 has a tangent infinitely great, it 
 is frequently said, in concise terms, that the tangent of 90 
 is infinite. 
 
 In the second quadrant, the tangent is, at first, infinitely 
 
120 
 
 TRIGONOMETRICAL ANALYSIS. 
 
 great, and gradually 
 diminishes, till at D 
 it is reduced to no- 
 thing. In the third 
 quadrant, it increases 
 again, becomes infi- 
 nite near H, and is 
 reduced to nothing 
 at A. 
 
 The cotangent is in- 
 versely as the tan- 
 gent. It is there- 
 fore nothing at B 
 and H, and infinite 
 near A and D. 
 
 198. The secant increases with the tangent, through the 
 first quadrant, and becomes infinite near B ; it then dimin- 
 ishes, in the second quadrant, till at D it is equal to the 
 radius CD. In the third quadrant it increases again, becomes 
 infinite near H, after which it diminishes, till it becomes 
 equal to radius. 
 
 The cosecant decreases, as the secant increases, and v. v. 
 It is therefore equal to radius at B and H, and infinite near 
 A and D. 
 
 199. The sine increases through the first quadrant, till at 
 B it is equal to radius. (See also Fig. 13. page 70.) It then 
 diminishes, and is reduced to nothing at D. In the third 
 quadrant, it increases again, becomes equal to radius at H, 
 and is reduced to nothing at A. 
 
 The cosine decreases through the first quadrant, and is re- 
 duced to nothing at B. In the second quadrant, it increases, 
 till it becomes equal to radius at D. It then diminishes 
 again, is reduced to nothing at H, and afterwards increases 
 till it becomes equal to radius at A. 
 
TRIGONOMETRICAL ANALYSIS. 121 
 
 In all these cases, the arc is supposed to begin at A, and 
 to extend round in the direction of BDH, 
 
 200. The sine and cosine vary from nothing to radius, 
 which they never exceed. The secant and cosecant are never 
 less than radius, but may be greater than any given length. 
 
 The tangent and cotangent have every value from nothing 
 to infinity. Each of these lines, after reaching its greatest 
 limit, begins to decrease ; and as soon as it arrives at its least 
 limit, begins to increase. Thus, the sine begins to decrease, 
 after becoming equal to radius, which is its greatest limit. 
 But the secant begins to increase after becoming equal to 
 radius, which is its least limit. 
 
 201. The substance of several of the preceding articles is 
 comprised in the following tables. The first shows the signs 
 of the trigonometrical lines, in each of the quadrants of the 
 circle. The other gives the values of these lines, at the ex- 
 tremity of each quadrant. 
 
 Quadrant 1st 2d 3d 4th 
 
 Sine and cosecant -h + 
 
 Cosine and secant + + 
 
 Tangent and cotangent + + 
 
 90 180 270 360 
 
 Sine r r 
 
 Cosine r r r 
 
 Tangent oc oc 
 
 Cotangent oc oc oc 
 
 Secant r oc r cc r 
 
 Cosecant oc r oc r oc 
 
 Here r is put for radius, and oc for infinite, 
 
 202. By comparing these two tables, it will be seen, that 
 each of the trigonometrical lines changes from positive to 
 negative, or from negative to positive, in that part of the 
 circle in which the line is either nothing or infinite. Thus, 
 
 11 
 
122 TRIGONOMETRICAL ANALYSIS. 
 
 the tangent changes from positive to negative, in passing 
 from the first quadrant to the second, through the place 
 where it is infinite. It becomes positive again, in passing 
 from the second quadrant to the third, through the point in 
 which it is nothing. 
 
 203. There can be no more than 360 degrees in any circle. 
 But a body may have a number of successive revolutions in 
 the same circle ; as the earth moves round the sun, nearly in 
 the same orbit, year after year. In astronomical calculations, 
 it is frequently necessary to add together parts of different 
 revolutions. The sum may be more than 360. But a body 
 which has made more than a complete revolution in a circle, 
 is only brought back to a point which it had passed over 
 before. So the sine, tangent, &c., of an arc greater than 
 360, is the same as the sine, tangent, &c., of some arc less 
 than 360. If an entire circumference, or a number of cir- 
 cumferences, be added to any arc, it will terminate in the 
 same point as before. So that, if be put for a whole cir- 
 cumference, or 360, and x be any arc whatever ; 
 
 sin z=sin (C+z)=sin (2C+z)=sin (3 C+x), &c. 
 tan z=tan (C+x)==tan (2 C-f#>=tan 
 
 204. It is evident also, that, in a number of successive 
 revolutions, in the same circle ; 
 
 The first quadrant must coincide with the 5th, 9th, 13th, lYth, 
 The second, with the 6th, 10th, Uth, 18th, &c. 
 
 The third, with the 7th, Ilth, 15th, 19th, <fec. 
 
 The fourth, with the 8th, 12th, 16th, 20th, &c. 
 
 205. If an arc, extending in a certain direction from a 
 given point, be considered positive ; an arc extending from 
 the same point, in an opposite direction, is to be considered 
 
TRIGONOMETRICAL ANALYSIS. 
 
 128 
 
 negative. (Alg. 382.) 
 Thus, if the arc ex- 
 tending from A to S, 
 be positive; an arc 
 extending from A to 
 S ;// will be negative. 
 The latter will not 
 terminate in the same 
 quadrant as the other 
 and the signs of 
 the tabular lines 
 must be accommo- 
 dated to this circum- 
 stance. Thus, the 
 
 sine of AS will be positive, while that of AS'" will be nega- 
 tive. (Art. 194.) When a greater arc is subtracted from a 
 less, if the latter be positive, the remainder must be nega- 
 tive. (Alg. 40.) 
 
 TRIGONOMETRICAL FORMULAE. 
 
 206. From the view which has been here taken of the 
 changes in the trigonometrical lines, it will be easy to see, 
 in what parts of the circle each of them increases or de- 
 creases. But this does not determine their exact values, ex- 
 cept at the extremities of the several quadrants. In the 
 analytical investigations which are carried on by means of 
 these lines, it is necessary to calculate the changes produced 
 in them, by a given increase or diminution of the arcs to 
 which they belong. In this there would be no difficulty, if 
 the sines, tangents, <fec., were proportioned to their arcs. But 
 this is far from being the case. If an arc is doubled, its 
 sine is not exactly doubled. Neither is its tangent or se- 
 cant. We have to inquire, then, in what manner the sine, 
 
124 
 
 TRIGONOMETRICAL ANALYSIS. 
 
 tangent, &c., of one arc may be obtained, from those of other 
 arcs already known. 
 
 The problem on which almost the whole of this branch 
 of analysis depends, consists in deriving, from the sines and 
 cosines of two given arcs, expressions for the sine and cosine 
 of their sum and difference. For, by addition and subtrac- 
 tion, a few arcs may be so combined and varied, as to pro- 
 duce others of almost every dimension. And the expres- 
 sions for the tangents and secants may be deduced from those 
 of the sines and cosines. 
 
 Expressions for the SINE and COSINE of the SUM and DIFFER- 
 ENCE of arcs. 
 
 207. Let o=AH, 
 the greater of the 
 given arcs, 
 
 And &=HL=HD, 
 the less. 
 
 Then a+6=AH+ 
 HL=AL, the sum of 
 the two arcs, 
 
 And a 6=AH 
 HD=AD, their differ- 
 ence. 
 
 Draw the chord DL, and the radius CH, which may be 
 represented by R. As DH is, by construction, equal to 
 HL ; DQ is equal to QL, and therefore DL is perpendicular 
 to CH. (Euc. 3. 3.) Draw DO, HN, QP, and LM, each 
 perpendicular to AC ; and DS and QB parallel to AC. 
 
 From the definitions of the sine and cosine, (Arts. 82, 9.) 
 it is evident, that 
 
TRIGONOMETRICAL ANALYSIS. 
 
 125 
 
 of AH, that is, sin o=HN, 
 
 The sine 
 
 of HL, 
 of AL, 
 of AD, 
 
 sin 6=QL, 
 sin (a+5)=LM, 
 sin (a 6)=DO, 
 
 f of AH, that is, cos o=CN, 
 
 The cosine ofHL > cos 6=CQ, 
 
 of AL, cos (a+&)=CM, 
 
 I of AD, cos (a 6)=CO. 
 
 The triangle CHN is obviously similar to CQP ; and it is 
 also similar to BLQ, because the sides of the one are per- 
 pendicular to those of the other, each to each. We have, 
 then, 
 
 1. CH 
 
 CQ 
 
 ::HN 
 
 QP, that 
 
 is, R 
 
 cos b : 
 
 : sin a ; QP, 
 
 2. CH 
 
 QL 
 
 :: GST 
 
 BL, 
 
 R 
 
 sin 6 : 
 
 : cos a : BL, 
 
 3. CH 
 
 CQ 
 
 ::CN 
 
 CP 
 
 R 
 
 cos b : 
 
 : cos a : CP, 
 
 4. CH 
 
 QL 
 
 ::HN 
 
 QB, 
 
 R 
 
 sin b : 
 
 : sin a : QB, 
 
 Converting each of these proportions into an equation ; 
 
 1. QP= 
 
 sin a cos b* 
 
 2. BL 
 
 sin 6 cos a 
 
 R 
 
 4. Q] 
 
 cos a cos b 
 
 ~~TT~ 
 
 sin a sin b 
 
 R 
 
 Then adding the first and second, 
 
 sm a cos ft+sin b cos a 
 
 QP-L.PT, 
 
 R 
 
 Subtracting the second from the first, 
 
 sin a cos b sin b cos a 
 
 QP BL 
 
 R 
 
 * In these formulae, the sign of multiplication is omitted ; sin a cos b 
 ang put for sin aXcos b, that is, the product of the sine of a into the 
 
 11* 
 
TRIGONOMETRICAL ANALYSIS. 
 
 Subtracting the fourth from the third, 
 
 cos a cos b sin a sin 
 
 CP QB=- 
 
 R 
 
 Adding the third and fourth, 
 a COS & " 
 
 CP -f QB 
 
 R 
 
 But it Will be seen, from the figure, that 
 
 QP-f-BL=BM+BL==LM=sin (a-f-o) 
 QP BL=QP QS=DO=sm (a 6) 
 CP QB=CP PM=CM=*=cos (a+0) 
 CP+QB=CP+SD=CO=cos (a b) 
 
 208. It then, for the first member of each of the four 
 equations above, we substitute its value, we shall have, 
 
 sm a cos &+ sm 6 cos a 
 
 I. sin (a+b-_ 
 ILsii 
 
 - - 
 R 
 
 sin a cos & sin b cos a 
 
 IH, cos (a+6)= 
 
 R 
 
 cos a cos 6 sin a sin 
 
 TV ( K\ cos a cos ^"^" s ^ n a sm 
 
 Or multiplying both sides by R, 
 
 R sin (a-f&)=sin a cos 6-f-sin b cos a 
 R sin (a 6)=sm a cos b sin b cos a 
 R cos (a+o)= : cos a cos 6 sin a sin 6 
 R cos (a o)=cos a cos 0+ sin a sin o 
 
 That is, tfee product of radius and the sine of the sum of 
 two arcs, is equal to the product of the sine of the first c 
 
TRIGONOMETRICAL ANALYSIS. 127 
 
 into the cosine of the second -f- the product of the sine of 
 the second into the cosine of the first. 
 
 The product of radius and the sine of the difference of 
 two arcs, is equal to the product of the sine of the first arc 
 into the cosine of the second the product of the sine of 
 the second into the cosine of the first. 
 
 The product of radius and the cosine of the sum of tw< 
 arcs, is equal to the product of the cosines of the arcs 
 the product of their sines. 
 
 The product of radius and the cosine of the diference of 
 two arcs, is equal to the product of the cosines of the arcs 
 4- the product of their sines. 
 
 These four equations may be considered as fundamental 
 propositions, in what is called the Arithmetic of Sines and 
 Cosines, or Trigonometrical Analysis. 
 
 Expression for the area of a triangle, in terms of the sides. 
 
 209. Let the sides of the triangle 
 ABC be expressed by <L, 6, and c, the 
 perpendicular CD by p, the seg- 
 ment AD by d, and the area by S. 
 
 Then ='+<:* 2J, (Euc. 13. 2.) 
 
 Transposing and dividing by 2c ; 
 
 2c 
 By Euc. 47, 1, 
 
 Reducing the fraction, (Alg. 120.) and extracting the root 
 of both sides, 
 
128 TRIGONOMETRICAL ANALYSIS. 
 
 V46V (6 2 +c a 
 
 This gives the length of the perpendicular in terms of the 
 sides of the triangle. But the area is equal to the product 
 of the base into half the perpendicular height. (Alg. 393.) 
 That is, 
 
 2 -f-c 2 a 2 ) 2 
 
 Here we have an expression for the area, in terms of the 
 sides. But this may be reduced to a form much better 
 adapted to arithmetical computation. It will be seen, that 
 the quantities 46V 2 , and (6 2 +c 2 a 2 ) a are both squares ; and 
 that the whole expression under the radical sign is the differ- 
 ence of these squares. But the difference of two squares is 
 equal to the product of the sum and difference of their roots. 
 (Alg. 191.) Therefore, 46V (6 2 -fc 2 a 9 ) 2 may be resolved 
 into the two factors, 
 
 26c+(6 2 -hc 2 a 2 ) which is equal to (6-f c) 2 a a 
 26c (6 a +c a -of) which is equal to a 2 (6 c) a 
 
 Each of these also, as will be seen in the expressions on 
 the right, is the difference of two squares ; and may, on the 
 same principle, be resolved into factors, so that, 
 
 { (b+ c ya 2 =(b+c+a)X(b+ca) 
 (a+b c)x(a b+c) 
 
 * The expression for the perpendicular is the same, when one of th 
 angles is obtuse, as in Fig. 24. page 8fr. Let A.D=d. 
 
 (Euc. 12, 2.) And d- 
 
 Therefore, ,fc =Ug. 1C9.) 
 
 v * ' 
 
TRIGONOMETRICAL ANALYSIS. 129 
 
 Substituting, then, these four factors, in the place of the 
 quantity which has been resolved into them, we have, 
 
 Here it will be observed, that all the three sides, a, 6, and 
 c, are in each of these factors. 
 
 Let h=$(a+b+c) half the sum of the sides. Then 
 
 X (h b) X 
 
 210. For finding the area of a triangle, then, when the 
 three sides are given, we have this general rule ; 
 
 From half the sum of the sides, subtract each side severally ; 
 multiply together the half sum and the three remainders ; and 
 extract the square root of the product. 
 
APPLICATION OF TRIGONOMETRY 
 
 TO THE 
 
 i^, MENSURATION ^ 
 
 OF 
 
 HEIGHTS AND DISTANCES. 
 
 ART. 1. The most direct and obvious method of deter- 
 mining the distance or height of any object, is to apply to 
 it some known measure of length, as a foot, a yard, or a rod. 
 In this manner, the height of a room is found, by a joiner's 
 rule ; or the side of a field by a surveyor's chain. But in 
 many instances, the object, or a part, at least, of the line 
 which is to be measured, is inaccessible. We may wish to 
 determine the breadth of a river, the height of a cloud, or 
 the distances of the heavenly bodies. In such cases it is ne- 
 cessary to measure some other line ; from which the required 
 line may be obtained, by geometrical construction, or more 
 exactly, by trigonometrical calculation. The line first meas- 
 ured is frequently called a base line. 
 
 2. In measuring angles, some instrument is used which 
 contains a portion of a graduated circle divided into degrees 
 and minutes. For the proper measure of an angle is an arc 
 of a circle, whose centre is the angular point. (Trig. 74.) 
 The instruments used for this purpose are made in different 
 forms, and with various appendages. The essential parts 
 are a graduated circle, and an index with sight-holes, for 
 taking the directions of the lines which include the angles. 
 
 3. Angles of elevation, and of depression are in a plane 
 
MENSURATION OF HEIGHTS AND DISTANCES. 
 
 131 
 
 perpendicular to the horizon, which is called a. vertical plane* 
 
 An angle of elevation is contained between a parallel to 
 
 the horizon, and an ascending 
 
 line, as BAG. An angle of 
 
 depression is contained between 
 
 a parallel to the horizon, and a 
 
 descending line, as DCA. The 
 
 complement of this is the angle 
 
 ACB. 
 
 4. The instrument by which angles of elevation, and of 
 depression, are commonly meas- 
 ured, is called a Quadrant. In 
 
 its most simple form, it is a 
 portion of a circular board ABC, 
 on which is a graduated arc of 
 90 degrees, AB, a plumb line CP, 
 suspended from the central point 
 C, and two sight-holes D and E, 
 for taking the direction of the 
 object. 
 
 To measure an angle of elevation with this, hold the plane 
 of the instrument perpendicular to the horizon, bring the 
 centre C to the angular point, and direct the edge AC in 
 such a manner, that the object G may be seen through the 
 two sight-holes. Then the arc BO measures the angle BCO, 
 which is equal to the angle of elevation FCG. For as the 
 plumb line is perpendicular to the horizon, the angle FCO 
 is a right angle, and therefore equal to BCG. Taking from 
 these the common angle BCF, there will remain the angle 
 BCO=FCG. 
 
 In taking an angle of depression, as HCL, the eye is placed 
 at C, so as to view the object at L, through the sight-holes 
 D and E. 
 
 5. In treating of the mensuration of heights and dis- 
 
132 MENSURATION OF 
 
 tances, no new principles are to be brought into view. We 
 have only to make an application of the rules for the solu- 
 tion of triangles, to the particular circumstances in which 
 the observer may be placed, with respect to the line to be 
 measured. These are so numerous, that the subject may be 
 divided into a great number of distinct cases. But as they 
 are all solved upon the same general principles, it will not 
 be necessary to give examples under each. The following 
 problems may serve as a specimen of those which most fre- 
 quently occur in practice. 
 
 PROBLEM I. 
 
 TO FIND THE PERPENDICULAR HEIGHT OF AN ACCESSIBLE OB- 
 JECT STANDING ON A HORIZONTAL PLANE. 
 
 6. MEASURE FROM THE OBJECT TO A CONVENIENT STATION, 
 AND THERE TAKE THE ANGLE OF ELEVATION SUBTENDED BY 
 THE OBJECT. 
 
 If the distance AB be meas- 
 ured, and the angle of elevation 
 BAG ; there will be given in 
 the right angled triangle ABC, 
 the base and the angles, to find 
 the perpendicular. (Trig. 137.) 
 
 As the instrument by which 
 the angle at A is measured, is commonly raised a few feet 
 above the ground ; a point B must be taken in the object, 
 so that AB shall be parallel to the horizon. The part BP, 
 may afterwards be added to the height BC, found by trig- 
 onometrical calculation. 
 
 Ex. 1. What is the height of a tower BC, if the distance 
 AB, on a horizontal plane, he 93 feet ; and the angle BAG 
 35i degrees ? 
 
 Making the hypothenuse radius (Tr}g 121.) 
 
HEIGHTS AND DISTANCES. 133 
 
 Cos. BAG : AB : : Sin. BAG : BC=69.9 feet. 
 
 For the geometrical construction of the problem, see Trig. 
 169. 
 
 2. What is the height of the perpendicular sheet of water 
 at the falls of Niagara, if it subtends an angle of 40 degrees, 
 at the distance of 163 feet from the bottom, measured on a 
 horizontal plane ? Ans. 136f feet. 
 
 7. If the height of the object be known, its distance may 
 be found by the angle of elevation. In this case the angles, 
 and the perpendicular of the triangle are given, to find the 
 base. 
 
 Ex. A person on shore, taking an observation of a ship's 
 mast, which is known to be 99 feet high, finds the angle of 
 elevation 3 degrees. What is the distance of the ship from 
 the observer ? Ans. 98 rods. 
 
 8. If the observer be sta- 
 tioned at the top of the perpen- 
 dicular BC, whose height is 
 known ; he may find the length 
 of the base line AB, by meas- 
 uring the angle of depression 
 ACD, which is equal to BAG. 
 
 Ex. A seaman at the top of a mast 66 feet high, looking 
 at another ship, finds the angle of depression 10 degrees. 
 What is the distance of the two vessels from each other ? 
 
 Ans. 22f rods. 
 
 We may find the distance between two objects which are 
 in the same vertical plane with the perpendicular, by calcu- 
 lating the distance of each from the perpendicular. Thus 
 AG is equal to the difference between AB and GB. 
 
 12 
 
134 MENSURATION OF 
 
 PROBLEM II. 
 
 TO FIND THE HEIGHT OF AN ACCESSIBLE OBJECT STANDING 
 
 ON AN INCLINED PLANE. 
 
 9. MEASURE THE DISTANCE FROM THE OBJECT TO A CON- 
 VENIENT STATION, AND TAKE THE ANGLES WHICH THIS BASE 
 MAKES WITH LINES DRAWN FROM ITS TWO ENDS TO THE TOP 
 OF THE OBJECT. 
 
 If the base AB be measured 
 and the angles BAG and ABC ; 
 there will be given, in the ob- 
 lique angled triangle ABC, the 
 side AB, and the angles, to find 
 BC. (Trig. 150.) V'" 
 
 Or the height BC may be 
 
 found by measuring the distances BA, AD, and taking the 
 angles, BAG and BDC. There will then be given in the tri- 
 angle ADC, the angles and the side AD, to find AC ; and 
 consequently, in the triangle ABC, the sides AB and AC 
 with the angle BAG, to find BC. 
 
 Ex. If AB be 76 feet, the angle B 101 25', and the angle 
 A 44 b 42' ; what is the height of the tree BC ? 
 
 Sin. C : AB : : Sin. A : : BC=95.9 feet. 
 
 For the geometrical construction of the problem, see Trig. 
 169. 
 
 10. The following are some of the methods by which the 
 height of an object may be found, without measuring the 
 angle of elevation. 
 
 1. By shadows. Let the staff be be parallel to an ob* 
 
HEIGHTS AND DISTANCES. 
 
 Ans. 115 feet. 
 
 ject BC, whose height is required. 
 If the shadow of BC extend to A, and 
 that of be to a; the rays of light CA 
 and ca coming from the sun may be 
 considered parallel ; and therefore the tri- 
 angles ABC and abc are similar ; so that 
 
 ab : be : : AB ; BC. 
 
 Ex. If ab be 3 feet, be 5 feet, and AB 69 feet, what is the 
 height of BC ? 
 
 2. By parallel rods. If two 
 poles am and en be placed 
 parallel to the object BC, and 
 at such distances as to bring 
 the points C, c, a, in a line, 
 and if ab be made parallel to 
 AB ; the triangles ABC, and A 
 abc will be similar; and we 
 shall have 
 
 ab :bc:: AB : BC. 
 
 One pole will be sufficient, if the observer can place his 
 eye at the point A, so as to bring A, a, and C in a line. 
 
 3. By a mirror. Let the smooth surface of a body of 
 water at A, or any plane mirror 
 parallel to the horizon, be so situ- 
 ated, that the eye of the observer 
 at c may view the top of the ob- 
 ject C reflected from the mirror. 
 By a law of Optics, the angle B AC 
 is equal to 5 Ac ; and if be be made 
 parallel to BC, the triangle 6Ac 
 will be similar to BAG ; so that 
 
 Ab : be : : AB : BC. 
 
186 MENSURATION OF 
 
 PROBLEM III. 
 
 TO FIND THE HEIGHT OF AN INACCESSIBLE OBJECT ABOVE 4 
 HORIZONTAL PLANE. 
 
 11. TAKE TWO STATIONS IN A VERTICAL PLANE PASSING 
 THROUGH THE TOP OF THE OBJECT, MEASURE THE DISTANCE 
 FROM ONE STATION TO THE OTHER, AND THE ANGLE OF ELE- 
 VATION AT EACH. 
 
 If the base AB be measured, 
 with the angle GBP and CAB ; 
 as ABC is the supplement of 
 CBP, there will be given, in 
 the oblique angled triangle 
 ABC, the side AB and the an- 
 gles, to find BC ; and then in 
 the right angled triangle BCP, 
 the hypothenuse and the angles, to find the perpendicular 
 CP. 
 
 Ex. 1. If C be the top of a spire, the horizontal base line 
 AB 100 feet, the angle of elevation BAG 40, and the angle 
 PBC 60 ; what is the perpendicular height of the spire ? 
 
 The difference between the angles PBC and BAG is equal 
 to ACB. (Euc. 32. 1.) 
 
 Then Sin ACB : AB : : Sin BAG : BC=187.9 
 And R : BC : : Sin PBC : CP=162f feet. 
 
 2. If two persons 120 rods from each other, are standing 
 on a horizontal plane, and also in a vertical plane passing 
 through a cloud, both being on the same side of the cloud : 
 and if they find the angles of elevation at the two stations 
 to be 68 and 76 ; what is the height of the cloud ? 
 
 Ans. 2 miles 135.7 rods. 
 
 12. The preceding problems are useful in particular cases. 
 
HEIGHTS AND DISTANCES. 13 Y 
 
 But the following is a general rule, which may be used for 
 finding the height of any object whatever, within moderate 
 distances. 
 
 PROBLEM IV. 
 
 TO FIND THE HEIGHT OF ANY OBJECT, BY OBSERVATIONS AT 
 
 TWO STATIONS. 
 
 13. MEASURE THE BASE LINE BETWEEN THE TWO STATIONS, 
 THE ANGLES BETWEEN THIS BASE AND LINES DRAWN FROM 
 
 EACH OF THE STATIONS TO EACH END OF THE OBJECT, AND 
 THE ANGLE SUBTENDED BY THE OBJECT, AT ONE OF THE 
 STATIONS. 
 
 If BC be the object 
 whose height is requir- 
 ed, and if the distance 
 between the stations A 
 and D be measured, with 
 the angles ADC, DAG, 
 ADB, DAB, and BAG ; 
 there will be given in the triangle ADC, the side AD and the 
 angles, to find AC ; in the triangle ADB, the side AD and the 
 angles, to find AB ; and then, in the triangle BAG, the sides 
 AB and AC with the included angle, to find the required 
 height BC. 
 
 If the two stations A and D be in the same plane with BC, 
 the angle BAC will be equal to the difference between BAD 
 and CAD. In this case it will not be necessary to measure 
 BAC. 
 
 Ex. If AD=83 feet, ( ADB=33 
 
 I ADC=51 { DAB=121 
 
 ( DAG =95 BAC=26, 
 
 What is the height of the object BC ? 
 12* 
 
138 
 
 MENSURATION OP 
 
 Sin ACD : AD : : ADC : AC = 115.3 (Fig. 8.) 
 Sin ABD : AD : : ADB ; AB= 103.1. 
 (AC+AB) : (AC AB) : : Tan i (ABC-fACB) : Tan -| 
 
 (ABC ACB)=1338' 
 Sin ACB ; AB : : Sin BAG : BC= 50.57 feet. 
 
 If the object BC be perpendicular to the horizon, its 
 height, after obtaining AB and AC as before, may be found 
 by taking the angles of elevation BAP and CAP. The dif- 
 ference of the perpendiculars in the right angled triangles 
 ABP and ACP, will be the height required. 
 
 PROBLEM V. 
 TO FIND THE DISTANCE OF AN INACCESSIBLE OBJECT. 
 
 14. MEASURE A BASE LINE BETWEEN TWO STATIONS, AND 
 THE ANGLES BETWEEN THIS AND LINES DRAWN FROM EACH 
 OF THE STATIONS TO THE OBJECT. 
 
 If C be the object, and if the 
 distance between the stations A 
 and B be measured, with the an- 
 gles at B and A ; there will be 
 given, in the oblique angled tri- 
 angle ABC, the side AB and the 
 angles, to find AC and BC, the 
 distances of the object from the 
 two stations. 
 
 For the geometrical construction, see Trig. 169. 
 
 Ex. 1. What are the distances of the two stations A and 
 B from the house C, on the opposite side of a river ; if 
 AB be 26.6 rods, B 92 46', and A 38 40' ? 
 
 The angle C=180 (A+B)=48 34'. Then 
 Sin A : BC=22.17 
 
 Sin C : AB : : 
 
 Sin B : AC=35.44. 
 
HEIGHTS AND DISTANCES. 139 
 
 2. Two ships in a harbor, wishing to ascertain how far 
 they are from a fort on shore, find that their mutual distance 
 is 90 rods, and that the angles formed between a line from 
 one to the other, and lines drawn from each to the fort are 
 45 and 56 15'. What are their respective distances from 
 the fort ? Ans. 76.3 and 64.9 rods. 
 
 15. The perpendicular distance of the object from the 
 line joining the two stations may be easily found, after the 
 distance from one of the stations is obtained. The perpen- 
 dicular distance PC is one of the sides of the right angled 
 iriangle BCP. Therefore 
 
 R : BC : : Sin B : PC. 
 
 PROBLEM VI. 
 
 TO FIND THE DISTANCE BETWEEN TWO OBJECTS, WHEN THE 
 PASSAGE FROM ONE TO THE OTHER, IN A STRAIGHT LINE 
 IS OBSTRUCTED. 
 
 16. MEASURE THE RIGHT LINES FROM ONE STATION TO 
 
 EACH OF THE OBJECTS, AND THE ANGLE INCLUDED BETWEEN 
 
 THESE LINES. 
 
 If A and B be the two objects, 
 and if the distances BC and AC be 
 measured, with the angle at C ; there 
 will be given, in the oblique angled 
 triangle ABC, two sides and the in- 
 eluded angle, to find the other two 
 angles, and the remaining side. (Trig. 
 153.) 
 
 Ex. The passage between the two objects A and B being 
 obstructed by a morass, the line BC was measured and found 
 to be 109 rods, the line AC 76 rods, and the angle at 
 101 30'. What is the distance AB ? 
 
 Ans. 144.7 rods. 
 
140 MENSURATION OF 
 
 PROBLEM VII. 
 TO FIND THE DISTANCE BETWEEN TWO INACCESSIBLE OBJECTS. 
 
 17. MEASURE A BASE LINE BETWEEN TWO STATIONS AND 
 THE ANGLES BETWEEN THIS BASE AND LINES DRAWN FROM 
 EACH OF THE STATIONS TO EACH OF THE OBJECTS. 
 
 If A and B be the two 
 objects, and if the distance 
 between the stations C and D 
 be measured, with the angles 
 BDC, BCD, ADC, and ACD ; 
 the lines AC and BC may be 
 found as in Problem V, and 
 then the distance AB as in 
 Problem VI. 
 
 This rule is substantially the same as that in Art. 13. 
 The two stations are supposed to be in the same plane with 
 the objects. If they are not, it will be necessary to meas- 
 ure the angle ACB. 
 
 18. The same process by which we obtain the distance 
 of two objects from each other, will enable us to find the 
 distance between one of these and a third, between that and 
 a fourth, and so on, till a connection is formed between a 
 great number of remote points. This is the plan of the 
 great Trigonometrical Surveys, which have been lately car- 
 ried on, with surprising exactness, particularly in England 
 and France. 
 
 19. In the preceding problems for determining altitudes, 
 the objects are supposed to be at such moderate distances, 
 that the observations are not sensibly affected by the spher- 
 ical figure of the earth. The height of an object is meas- 
 ured from an horizontal plane, passing through the station 
 at which the angle of elevation is taken. But in an extent 
 
HEIGHTS AND DISTANCES. 141 
 
 of several miles, the figure of the earth ought to be taken 
 into account. 
 
 Let AB be a portion of the 
 earth's surface, H an object above 
 it, and AT a tangent at the point 
 A, or a horizontal line passing 
 through A. Then HT, the oblique 
 height of the object above the ho- 
 rizon of A, is only zpart of the 
 height above the surface of the 
 earth, or the level of the ocean. 
 To obtain the true altitude, it is 
 necessary to add BT to the height HT found by obser- 
 vation. The height BT may be calculated, if the diameter 
 of the earth and the distance AT be previously known. Or 
 if the height BT be first determined from observation, witn 
 the distance AT ; the diameter of the earth may be thence 
 deduced. 
 
 PROBLEM VIII. 
 
 TO FIND THE DIAMETER OF THE EARTH, FROM THE KNOWN 
 HEIGHT OF A DISTANT MOUNTAIN, WHOSE SUMMIT IS JUST 
 VISIBLE IN THE HORIZON. 
 
 20. FROM THE SQUARE OF THE DISTANCE DIVIDED BY THE 
 HEIGHT, SUBTRACT THE HEIGHT. 
 
 If BT (above figure) be a mountain whose height is 
 known, with the distance AT ; and if the summit T be just 
 visible in the horizon at A ; then AT is a tangent at the 
 point A. 
 
 Let 2BC=D, the diameter of the earth, 
 AT=c?, the distance of the mountain, 
 BT=, its height. 
 
U2 
 
 MENSURATION OF 
 
 Then considering AT as a straight line, and the earth as a 
 sphere, we have (Euc. 36. 3.)* 
 
 (2BC+BT)xBT=AT a ; that is, (D+A)Xft=*, 
 and reducing the equation, 
 
 -h 
 h 
 
 Ex. The highest point of the Andes is about 4 miles above 
 the level of the ocean. If a straight line from this touch 
 the surface of the water at the distance of 178-j- miles ; what 
 is the diameter of the earth ? Ans. 7940 miles. 
 
 21. If the distance AT be un- 
 known, it may be found by meas- 
 uring with a quadrant the angle 
 ATC. Draw BG perpendicular 
 to BC; and join CG. The trian- 
 gles ACG and BCG are equal, be- 
 cause each has a right angle, the 
 sides AC and BC are equal, and the 
 hypothenuse C G is common. There- 
 fore BG and AG are equal. In 
 
 the right angled triangle BGT, the angle BTG is given, and 
 the perpendicular BT. From these may be found BG and 
 TG, whose sum is equal to AT, the distance required.f 
 
 22. In the common measurement of angles, the light is 
 supposed to come from the object to the eye in a straight 
 .line. But this is not strictly true. The direction of the 
 light is affected by the refraction of the atmosphere. If the 
 object be near, the deviation is very inconsiderable. But in 
 an extent of several miles, and particularly in such nice ob- 
 
 * Thomson's Legendre, 30. 4. 
 
 f Tins method of determining the diameter of the earth is not a ac- 
 curate as that by measuring a degree of Latitude. 
 
HEIGHTS AND DISTANCES. 
 
 148 
 
 serrations as determining the height of distant mountains, 
 and the diameter of the earth, it is necessary to make allow- 
 ance for the refraction. 
 
 PROBLEM IX. 
 
 TO FIND THE GREATEST DISTANCE AT WHICH A GIVEN OBJECT 
 CAN BE SEEN ON THE SURFACE OF THE EARTH. 
 
 23. To THE PRODUCT OF THE HEIGHT OF THE OBJECT INTO 
 THE DIAMETER OF THE EARTH, ADD THE SQUARE OF THE 
 HEIGHT ; AND EXTRACT THE SQUARE ROOT OF THE SUM. 
 
 Let 2BC=D,the diameter of the earth, (Fig. 12.) 
 BT=h, the height of the object, 
 AT=e?, the distance required. 
 
 Then (D +A) X h=d\ And d=Wh+h\ 
 
 Ex. If the diameter of the earth be 7940 miles, and 
 Mount JEtna 2 miles high ; how far can its summit be seen 
 at sea ? Ans. 126 miles. 
 
 The actual distance at which an object can be seen, is in- 
 creased by the refraction of the air. 
 
 24. In this problem the eye is supposed to be placed at 
 the level of the ocean. But 
 
 if the observer be elevated 
 above the surface, as on the 
 deck of a ship, he can see 
 to a greater distance. If 
 BT be the height of the 
 object, and B'T' the height 
 of the eye above the level 
 of the ocean ; the distance 
 
 \j 
 
 at which the object can be 
 
 seen, is evidently equal to the sum of the tangents AT and 
 
 AT'. 
 
144 
 
 MENSURATION OF 
 
 Ex. The top of a ship's mast 132 feet high is just visible 
 in the horizon, to an observer whose eye is 33 feet above the 
 surface of the water. What is the distance of the ship ? 
 
 Ans. 21^- miles. 
 
 25. The distance to which a person can see the smooth 
 surface of the ocean, if no allowance be made for refraction, 
 is equal to a tangent to the earth drawn from his eye, as 
 T'A. (Fig. 13.) 
 
 Ex. If a man standing on the level of the ocean, has his 
 eye raised 5 feet above the water : to what distance can he 
 
 see the surface ? 
 
 26. If the distance AT, with 
 the diameter of the earth be given, 
 and the height BT be required ; 
 the equation in Art. 23 gives 
 
 Ans. 2$ miles. 
 
 27. When the diameter of the earth is ascertained, this 
 may be made a base line for determining the distance of the 
 heavenly bodies. A right angled triangle may be formed, the 
 perpendicular sides of which shall be the distance required, 
 and the semi-diameter of the earth. If then one of the an- 
 gles be found by observation, the required side may be easily 
 calculated. 
 
 Let AC be the 
 semi-diameter of 
 the earth, AH the 
 sensible horizon at 
 A, and CM the ra- 
 tional horizon, par- 
 allel to AH,passing 
 
HEIGHTS AND DISTANCES. 145 
 
 through the moon M. The angle HAM may be found by 
 astronomical observation. This angle, which is called the 
 Horizontal Parallax, is equal to AMC, the angle at the moon 
 subtended by the semi-diameter of the earth. (Euc. 29. 1.) 
 
 PROBLEM X. 
 
 TO FIND THE DISTANCE OF ANY HEAVENLY BODY WHOSE HOR- 
 IZONTAL PARALLAX IS KNOWN. 
 
 28. AS RADIUS, TO THE SEMI-DIAMETER OF THE EARTH J SO 
 IS THE CO-TANGENT OF THE HORIZONTAL PARALLAX, TO THE 
 DISTANCE. 
 
 In the right angled triangle ACM, (Fig. 14.) if AC be 
 made radius ; 
 
 R I AC : : Cot. AMC ; CM. 
 
 Ex. If the horizontal parallax of the moon be 5*1', and 
 the diameter of the earth 7940 miles ; what is the distance 
 of the moon from the centre of the earth ? 
 
 Ans. 239,414 miles. 
 
 29. The fixed stars are too far distant to have any sensi- 
 ble horizontal parallax. But from late observations it would 
 seem, that some of them are near enough, to suffer a small 
 apparent change of place, from the revolution of the earth 
 round the sun. The distance of the sun, then, which is the 
 semi-diameter of the earth's orbit, may be taken as a base 
 line, for finding the distance of the stars. 
 
 We thus proceed by degrees from measuring a line on the 
 surface of the earth, to calculate the distances of the 
 heavenly bodies. From a base line on a plane, is deter- 
 mined the height of a mountain ; from the height of a 
 mountain, the diameter of the earth ; from the diameter of 
 the earth, the distance of the sun, and from the distance of 
 the sun the distance of the stars. 
 
 13 
 
146 MENSURATION OF 
 
 30. After finding the distance of a heavenly body, its mag- 
 nitude is easily ascertained ; if it have an apparent diameter, 
 sufficiently large to be measured by the instruments which 
 are used for taking angles. 
 
 Let AEB be the an- 
 gle which a heavenly 
 body subtends at the 
 eye. Half this angle, 
 if C be the centre of 
 the body, is AEG; the 
 line EA is a tangent 
 to the surface, and therefore EAC is a right angle. Then 
 making the distance EC radius, 
 
 R I EC : : Sin. AEG : AC. 
 
 That is, radius is to the distance, as the sine of half the 
 angle which the body subtends, to its semi-diameter. 
 
 Ex. If the sun subtends an angle of 32' 2' , and if his 
 distance from the earth be 95 million miles ; what is his 
 diameter? Ans. 885 thousand miles. 
 
 PROMISCUOUS EXAMPLES. 
 
 1. On the bank of a river, the angle of elevation of a tree 
 on the opposite side is found to be 46 ; and at another sta- 
 tion 100 feet directly back on the same level, 31. What is 
 the height of the tree ? Ans. 143 feet. 
 
 2. On a horizontal plane, observations were taken of a 
 tower standing on the top of a hill. At one station the an- 
 gle of elevation of the top of the tower was found to be 50 ; 
 that at the bottom 39 ; and at another station 150 feet di- 
 rectly back, the angle of elevation of the top of the tower 
 was 32. What are the heights of the hill and the tower ? 
 
 Ans. The hill is 134 feet high; the tower 63. 
 
HEIGHTS AND DISTANCES. 14T 
 
 3. What is the altitude of the sun, when the shadow of a 
 tree, cast on a horizontal plane, is to the height of the tree as 
 4 to 3? Ans. 36 62' 12". 
 
 4. If a straight line from the top of the White Mountains 
 in New Hampshire touch the ocean at the distance of 103-J- 
 miles ? what is the height of the mountains ? 
 
 Ans. 7100 feet. 
 
 5. From the top of a perpendicular rock 55 yards high, 
 the angle of depression of the nearest bank of a river is 
 found to be 55 54', that of the opposite bank 33 20'. Re- 
 quired the breadth of the river, and the distance of its near- 
 est bank from the bottom of the rock. 
 
 The breadth of the river is 46.4 yards ; 
 Its distance from the rock 37.2. 
 
 6. If the moon subtend an angle of 31' 14", when her dis- 
 tance is 240,000 miles ; what is her diameter? 
 
 Ans. 2180 miles. 
 
 7. Observations are made on the altitude of a balloon, by 
 two persons standing on the same side of the balloon, and in 
 a vertical plane passing through it. The distance of the 
 stations is half a mile. At one, the angle of elevation is 30 
 58', at the other 36 52'. What is the height of the bal- 
 loon above the ground ? Ans. 1-J- miles. 
 
 8. The shadow of the top of a mountain, when the alti- 
 tude of the sun on the meridian is 32, strikes a certain point 
 on a level plain below ; but when the meridian altitude of 
 the sun is 67, the shadow strikes half a mile farther south, 
 on the same plain. What is the height of the mountain 
 above the plain ? Ans. 2245 feet. 
 
NOTES. 
 
 NOTE A. p. 13. 
 
 IT is common to define logarithms to be a series of numbers 
 in arithmetical progression, corresponding with another 
 series in geometrical progression. This is calculated to per- 
 plex the learner, when, upon opening the tables, he finds that 
 the natural numbers, as they stand there, instead of being hi 
 geometrical, are in arithmetical progression ; and that the 
 logarithms are not in arithmetical progression. 
 
 It is true, that a geometrical series may be obtained, by 
 taking out, here and there, a few of the natural numbers ; 
 and that the logarithms of these will form an arithmetical 
 series. But the definition is not applicable to the whole of 
 the numbers and logarithms, as they stand in the tables. 
 
 NOTE B. p. 89. 
 
 If the perpendicular be drawn from the angle opposite 
 the longest side, it will 
 always fall within the tri- 
 angle ; because the other 
 two angles must, of course, 
 be acute. But if one of 
 the angles at the base be 
 obtuse, the perpendicular 
 will fall without the trian- 
 gle, as CP. 
 
 In this casv, the side on 
 which the perpendicular 
 
tfOTES. 149 
 
 falls, is to the sum of the other two ; as the difference of the 
 latter, to the sum of the segments made by the perpendicular. 
 
 The demonstration is the same, as in the other case, ex- 
 cept that AH=BP+PA, instead of BP PA. 
 
 Thus, in the circle BDHL, of which C is the centre, 
 
 ABx AH=ALx AD ; therefore AB : AD : : AL : : AH. 
 
 But AD=CD+CA=CB-{-CA 
 And AL=CL CA=CB CA 
 And AH=HP+PA=BP-f PA 
 
 Therefore, 
 
 AB : CB+CA : : CB CA : BP+PA 
 
 When the three sides are given, it may be known whether 
 one of the angles is obtuse. For any angle of a triangle is 
 obtuse or acute, according as the square of the side sub- 
 tending the angle is greater, or less, than the sum of the 
 squares of the sides containing the angle. (Euc. 12, 13. 2.)* 
 
 NOTE C. p. 000. 
 
 Gunter's Sliding Rule, is constructed upon the same prin- 
 ciple as his scale, with the addition of a slider, which is so 
 contrived as to answer the purpose of a pair of dividers, in 
 working proportions, multiplying, dividing, <fec. The lines 
 on the fixed part are the same as on the scale. The slider 
 contains two lines of numbers, a line of logarithmic sines, 
 and a line of logarithmic tangents. 
 
 To multiply by this, bring 1 on the slider, against one of 
 the factors on the fixed part ; and against the other factor on 
 the slider, will be the product on the fixed part. To divide, 
 bring the divisor on the slider, against the dividend on the 
 fixed part ; and against 1 on the slider, will be the quotient 
 
 * Thomson's Legendre, 12, 13. 4. 
 
NOTES. 
 
 on the fixed part. To work a proportion, bring the first term 
 on the slider, against one of the middle terms on the fixed 
 part ; and -against the other middle term on the slider, will 
 be the fourth term on the fixed part. Or the first term 
 may be taken on the fixed part ; and then the fourth term 
 wall be found on the slider. 
 
 Another instrument frequently used in trigonometrical con- 
 structions, is 
 
 THE SECTOR. 
 
 This consists of two equal scales movable about a point 
 as a centre. The lines which are drawn on it are of two 
 kinds, some being parallel to the sides of the instrument, and 
 others diverging from the central point, like the radii of a cir- 
 cle. The latter are called the double lines, as each is re- 
 peated upon the two scales. The single lines are of the 
 same nature, and have the same use, as those which are put 
 upon the common scale ; as the lines of equal parts, of 
 chords, of latitude, &c., on one face ; and the logarithmic 
 lines of numbers, of sines, and of tangents, on the other. 
 
 The double lines are 
 
 A line of Lines, or equal parts, marked Lin. or L. 
 
 A line of Chords, Cho. or C. 
 
 A line of natural Sines, Sin. or S. 
 
 A line of natural Tangents to 45, Tan. or T. 
 
 A line of tangents, above 45, Tan. or T. 
 
 A line of natural Secants, Sec. or S. 
 
 A line of Polygons, Pol. or P. 
 
 The double lines of chords, of sines, and of tangents to 
 45, are all of the same radius ; beginning at the central 
 point, and terminating near the other extremity of each 
 scale ; the chords at 60, the shies at 90, and the tangents 
 at 45. (See Art. 95.) The line of lines is also of the same 
 length, containing ten equal parts which are numbered, and 
 
NOTES. 151 
 
 which are again subdivided. The radius of the lines of se- 
 cants and of tangents above 45, is about one-fourth of the 
 length of the other lines. From the end of the radius, 
 which for the secants is at 0, and for the tangents at 45, 
 these lines extend to between 70 and 80. The line of 
 polygons is numbered 4, 5, 6, &e., from the extremity of 
 each scale, towards the centre. 
 
 The simple principle on which the utility of these several 
 pairs of lines depends is this, that the sides of similar trian- 
 gles are proportional. (Euc. 4. 6.) So that sines, tangents, 
 &c., are furnished to 
 
 A r* 
 
 any radius, within the 
 extent of the opening 
 of the two scales. Let 
 AC and AC' be any 
 pair of lines on the sec- 
 tor, and AB and AB' 
 equal portions of these 
 
 lines. As AC and AC' are equal, the triangle AGO 7 is 
 isosceles, and similar to ABB'. Therefore, 
 
 AB : AC : : BB' : CC'. 
 
 Distances measured from the centre on either scale, as AB 
 and AC, are called lateral distances. And the distances be- 
 tween corresponding points of the two scales, as BB' and 
 CC', are called transverse distances. 
 
 Let AC and CC' be radii of two circles. Then if AB be 
 the chord, sine, tangent, or secant, of any number of degrees 
 in one ; BB' will be the chord, sine, tangent, or secant of 
 the same number of degrees in the other. (Art. 119.) Thus, 
 to find the chord of 80, to a radius of four inches, open the 
 sector so as to make the transverse distance from 60 to 60, 
 on the lines of chords, four inches ; and the distance from 
 80 to 30, on the same lines, will be the chord required. To 
 find the sine of 28, make the distance from 90 to 90, on the 
 
152 KOTES. 
 
 lines of sines, equal to radius ; and the distance from 28 to 
 28 will be the sine. To find the tangent of 37, make the 
 distance from 45 to 45, on the lines of tangents, equal to 
 radius ; and the distance from 37 to 37 will be the tangent. 
 In finding secants, the distance from to must be made 
 radius. (Art. 201.) 
 
 To lay down an angle of 34, describe a circle, of any 
 convenient radius, open the sector, so that the distance from 
 60 to 60 on the lines of chords shall be equal to this radius, 
 and to the circle apply a chord equal to the distance from 
 34 to 34. (Art. 161.) For an angle above 60, the chord 
 of half the number of degrees may be taken, and applied 
 twice on the arc, as in Art. 161. 
 
 The line of polygons contains the chords of arcs of a cir- 
 cle which is divided into equal portions. Thus, the distances 
 from the centre of the sector to 4, 5, 6, and 7, are the chords 
 f } i i"> an d -f of a circle. The distance 6 is the radius. 
 (Art. 95.) This line is used to make a regular polygon, or 
 to inscribe one in a given circle. Thus, to make a pentagon 
 with the transverse distance from 6 to 6 for radius, describe a 
 circle, and the distance from 5 to 5 will be the length of one 
 of the sides of a pentagon inscribed in that circle. 
 
 The line of lines is used to divide a line into equal or pro- 
 portional parts, to find fourth proportionals, &c. Thus, to 
 divide a line into 7 equal parts, make the length of the given 
 line the transverse distance from 7 to 7, and the distance 
 from 1 to 1 will be one of the parts. To find -f of a line, 
 make the transverse distance from 5 to 5 equal to the given 
 line ; and the distance from 3 to 3 will be ^ of it. 
 
 In working the proportions in trigonometry on the sector, 
 the lengths of the sides of triangles are taken from the line 
 of lines, and the degrees and minutes from the lines of 
 sines, tangents, or secants. Thus, in Art. 135, ex. 1, 
 
 35 : R : : 26 : sin 48. 
 
NOTES. 153 
 
 To find the fourth term of this proportion by the sector, 
 make the lateral distance 35 on the line of lines, a transverse 
 distance from 90 to 90 on the lines of sines ; then the lateral 
 distance 26 on the line of lines, will be the transverse dis- 
 tance from 48 to 48 on the lines of shies. 
 
 For a more particular account of the construction and uses 
 of the Sector, see Stone's edition of Bion on Mathematical 
 Instruments, Button's Dictionary, and Robertson's Treatise 
 on Mathematical Instruments. 
 
anb Thomson's Series. 
 A 
 
 PRACTICAL APPLICATION 
 
 OP 
 
 THE PRINCIPLES OF GEOMETRY 
 
 TO THE 
 
 MENSURATION 
 
 OP 
 
 SUPERFICIES AND SOLIDS. 
 
 ADAPTED TO 
 
 THE METHOD OF INSTRUCTION IN SCHOOLS AND ACADEMIES. 
 
 BY JEREMIAH DAY, D.D. LL.D. 
 
 LATI PRESIDENT OP TALE COLLEGE. 
 
 NEW YORK: 
 
 PUBLISHED BY MARK H. NEWMAN & CO., 
 No. 199 BROADWAY. 
 
 1848. 
 
ENTERED, according to Act of Congress, in the year 1848, bj 
 JEREMIAH DAY, 
 
 In the Clerk's Office of the District Court of the United States for the 
 Southern District of New York. 
 
 THOMAS B. SMITH, STEREOTYPIR, 
 316 WILLIAM STREET, 17. Y. 
 
CONTENTS. 
 
 Page 
 
 SECTION T. Areas of figures bounded by right lines, ... 5 
 
 II. The Quadrature of the Circle and its parts, . 19 
 
 Promiscuous examples of Areas, 34 
 
 III. Solids bounded by plane surfaces, 37 
 
 IV. The Cylinder, Cone, and Sphere, 56 
 
 Promiscuous examples of Solids, . ^ . . . 76 
 
 V. Isoperimetry, . . ^ 78 
 
 APPENDIX. 
 
 Gauging of Casks, 92 
 
 Notes, 99 
 
SECTION I. 
 
 AREAS OP FIGURES BOUNDED BY RIGHT LINES. 
 
 ART. 1. The following definitions, which are nearly the 
 same as in Euclid, are inserted here for the convenience of 
 reference. 
 
 I. Four-sided figures have different names, according to 
 the relative position and length of the sides. A parallelo- 
 gram has its opposite sides equal and parallel, as ABCD. 
 
 (Fig. 2.) A rectangle, or right parallelogram, has its oppo- 
 site sides equal, and all its angles right angles ; as AC. 
 (Fig. 1.) A square has all its sides equal, and all its angles 
 right angles ; as ABGH. (Fig. 3.) A rhombus has all its 
 
 sides equal, and its angles oblique; as ABCD. (Fig. 3.) A 
 rhomboid has its opposite sides equal, and its angles oblique ; 
 as ABCD. (Fig. 2.) A trapezoid has only two of its sides 
 parallel ; as ABCD. (Fig. 4.) Any other four sided figure 
 is called a trapezium. 
 
MENSURATION OF PLANE SURFACES. 
 
 II. A figure which has more than four sides is called a 
 polygon. A regular polygon has all its sides equal, and all 
 its angles equal. 
 
 III. The height of a triangle 
 is the length of a perpen- 
 dicular, drawn from one of the 
 angles to the opposite side ; as 
 CP. The height of a, four sided 
 figure is the perpendicular dis- 
 tance between two of its par- 
 allel sides ; as CP. (Fig. 4.) 
 
 IV. The area or superficial contents of a figure is the 
 space contained within the line or lines by which the figure 
 is bounded. 
 
 2. In calculating areas, some particular portion of surface 
 is fixed upon, as the measuring unit, with which the given 
 figure is to be compared. This is commonly a square ; as a 
 square inch, a square foot, a square rod, <kc. For this rea- 
 son, determining the quantity of surface in a figure is called 
 squaring it, or finding its quadrature; that is, finding a 
 square or number of squares to which it is equal. 
 
 3. The superficial unit has generally the same name, as 
 the linear unit which forms the side of the square. 
 
 The side of a square inch is a linear inch ; 
 of a square foot, a linear foot ; 
 of a square rod, a linear rod, &c. 
 
 There are some superficial measures, however, which have 
 no corresponding denominations of length. The acre, for 
 instance, is not a square which has a line of the same name 
 for its side. 
 
 The following tables contain the linear measures in com- 
 mon use, with their corresponding square measures. 
 
MENSURATION OF PLANE SURFACES. 
 
 Linear Measures. 
 
 12 inches =1 foot. 
 3 feet =1 yard. 
 6 feet =1 fathom. 
 16i feet =1 rod. 
 5 yards =1 rod. 
 4 rods =1 chain. 
 40 rods =1 furlong. 
 
 Square Measures. 
 
 144 inches =1 foot. 
 9 feet =1 yard. 
 36 feet =1 fathom. 
 272-J- feet =1 rod. 
 30-J- yards =1 rod. 
 16 rods =1 chain. 
 1600 rods =1 furlong. 
 
 320 rods =1 mile. 
 
 102400 rods =1 mile. 
 
 An acre contains 160 square rods, or 10 square chains. 
 By reducing the denominations of square measure, it will 
 be seen that 
 
 1 sq. mile=640 acres=102400 rods=27878400 feet=401 4489600 inches. 
 I acre=10 chains=160 rods=43560 feet=6272640 inches. 
 
 The fundamental problem in the mensuration of super- 
 ficies is the very simple one of determining the area of a 
 right parallelogram. The contents of other figures, partic- 
 ularly those which are rectilinear, may be obtained by find- 
 ing parallelograms which are equal to them, according to the 
 principles laid down in Euclid. 
 
 PROBLEM I. 
 
 To find the area of a PARALLELOGRAM, square, rhombus, or 
 rhomboid. 
 
 4. MULTIPLY THE LENGTH BY THE PERPENDICULAR HEIGHT 
 
 OR BREADTH. 
 
 It is is evident that the number of D c 
 
 square inches in the parallelogram AC 
 is equal to the number of linear 
 inches in the length AB, repeated as 
 many times as there are inches in the 
 breadth BC. For more particular 
 illustration of this see Alg. 386 389. 
 
8 
 
 MENSURATION OF PLANE SURFACES. 
 
 The oblique parallelogram or rhomboid ABCD, (Fig. 2.) 
 is equal to the right parallelogram GHCD. (Euc. 36. 1.)* The 
 
 area, therefore, is equal to the length AB multiplied into the 
 perpendicular height HC. And the rhombus ABCD, (Fig. 3.) 
 is equal to the parallelogram ABGH. As the sides of a 
 square are all equal, its area is found, by multiplying one of 
 the sides into itself. 
 
 Ex. 1. How many square feet are there in a floor 23 feet 
 long, and 18 feet broad ? Ans. 23^X18=423. 
 
 2. What are the contents of a piece of ground which is 
 66 feet square ? Ans. 4356 sq. feet=16 sq. rods. 
 
 3. How many square feet are there in the four sides of a 
 room which is 22 feet long, 17 feet broad, and 11 feet high? 
 
 Ans. 858. 
 
 ART. 5. If the sides and angles of a parallelogram are 
 given, the perpendicular height may be easily found by trig- 
 onometry. Thus, CH (Fig. 2.) is the perpendicular of a 
 right angled triangle, of which BC is the hypothenuse. 
 Then, (Trig. 134.) 
 
 R I BC : : sin B ; CH. 
 
 The area is obtained by multiplying CH thus found, into 
 the length AB. 
 
 * Thomson's Legendre, i. 5. 
 
MENSURATION OF PLANE SURFACES. 
 
 Or, to reduce the two operations to one, 
 
 As radius, 
 
 To the sine of any angle of a parallelogram ; 
 
 So is the product of the sides including that angle, 
 
 To the area of the parallelogram. 
 
 For the araz=ABxCH, (Fig. 2.) But CH= BC 
 
 R 
 
 Therefore, 
 
 in B . Or,R : sin B : : ABxBC : thearea. 
 
 Ex. If the side AB be 58 rods, BC 42 rods, and the angle 
 B 63, what is the area of the parallelogram ? 
 
 As radius 10.00000 
 
 To the sine of B 63 9.94988 
 
 ( So is the product of AB 58 1.76343 
 
 ( Into BC (Trig. 39.) 42 1.62325 
 
 To the area 2170.5 sq. rods 3.33656 
 
 2. If the side of a rhombus is 67 feet, and one of the 
 angles 73, what is the area ? Ans. 4292.7 feet. 
 
 6. When the dimensions are given in feet and inches, the 
 multiplication may be conveniently performed by the arith- 
 metical rule of Duodecimals ; in which each inferior denom- 
 ination is one-twelfth of the next higher. Considering a foot 
 as the measuring unit, a prime is the twelfth part of a foot ; 
 a second, the twelfth part of a prime, <fec. It is to be ob- 
 served, that, in measures of length, inches are primes ; but in 
 superficial measure they are seconds. In both, a prime is iV 
 of a foot. But iV of a square foot is a parallelogram, a foot 
 long and an inch broad. The twelfth part of this is a square 
 inch, which is TJT f a square foot. 
 1* 
 
10 MENSURATION OF PLANE SURFACES. 
 
 Ex. 1 What is the surface of a board 9 feet 5 inches, by 
 2 feet 7 inches. 
 
 F 
 
 9 5' 
 2 7 
 
 18 10 
 5 5 11 
 
 24 3 11", or 24 feet 47 inches. 
 
 2. How many feet of glass are there in a window 4 feet 
 11 inches high, and 3 feet 5 inches broad ? 
 
 Ans. 16 F. 9' 7", or 16 feet 115 inches. 
 
 7. If the area and one side of a parallelogram be given, 
 the other side may be found by dividing the area by the 
 given side. And if the area of a square be given, the side 
 may be found by extracting the square root of the area. This 
 is merely reversing the rule in Art. 4. See Alg. 520, 521. 
 
 Ex. 1. What is the breadth of a piece of cloth which is 
 36 yds. long, and which contains 63 square yards. 
 
 Ans. If yds. 
 
 2. What is the side of a square piece of land containing 
 289 square rods ? 
 
 3. How many yards of carpeting If yard wide, will cover 
 a floor 30 feet long and 22 broad ? 
 
 Ans. 30X22^ feet=10x7i=75 yds. And 75-i-H=60. 
 
 4. What is the side of a square which is equal to a par- 
 allelogram 936 feet long and 104 broad ? 
 
 5. How many panes of 8 by 10 glass are there, in a win- 
 dow 5 feet high, and 2 feet 8 inches broad ? 
 
MENSURATION OF PLANE SURFACES. 
 
 11 
 
 PROBLEM II. 
 To find the area of a TRIANGLE. 
 
 8. RULE I. MULTIPLY ONE SIDE BY HALF THE PERPEN- 
 DICULAR FROM THE OPPOSITE ANGLE. Or, multiply half the 
 side by the perpendicular, Or, multiply the whole side by 
 the perpendicular and take half the product. 
 
 The area of the triangle ABC, 
 is equal to % PCxAB, because 
 a parallelogram of the same 
 base and height is equal to PC 
 X AB, (Art. 4.) and by Euc. 41, 
 1,* the triangle is half the par- 
 allelogram. 
 
 Ex. 1. If AB be 65 feet, and PC 31.2, what is the area 
 of the triangle ? Ans. 1014 square feet. 
 
 2. What is the surface of a triangular board, whose base 
 is 3 feet 2 inches, and perpendicular height 2 feet 9 inches ? 
 . Ans. 4 F. 4' 3", or 4 feet 51 inches. 
 
 9. If two sides of a triangle and the included angle, are 
 given, the perpendicular on one of these sides may be easily 
 found by rectangular trigonometry. And the area may be 
 calculated in the same man- D c 
 
 ner as the area of a parallel- 
 ogram in Art. 5. In the tri- 
 angle ABC, 
 
 R-BC::sinB:CH 
 
 And because the triangle is half the parallelogram of the 
 same base and height, 
 
 Thomson's Legendre, 2. 4. 
 
1# .MENSURATION OF PLANE SURFACED 
 
 As radius, 
 
 To the sine of any angle of a triangle ; 
 
 So is the product of the sides including that angle, 
 
 To twice the area of the triangle. (Art. 5.) 
 
 Ex. If AC be 39 feet, AB 
 65 feet, and the angle at A 53 
 7' 48", what is the area of the 
 triangle ? Ans. 1014 square feet. 
 
 9. 6. If one side and the angles 
 are given ; then 
 
 As the product of radius and the sine of the angle oppo- 
 site the given side, 
 
 To the product of the sines of the two other angles ; 
 So is the square of the given side, 
 To twice the area of the triangle. 
 
 If PC be perpendicular to AB. 
 
 R t sin B : : BC : CP 
 sin ACB : sin A : : AB : BC 
 
 Therefore, (Alg. 351, 345.) 
 
 _JK x sin ACB : sin A X sin B : : AB X BC : CP X BC : : 
 AB* : ABxCP= twice the area of the triangle. 
 
 Ex. If one side of a triangle be 57 feet, and the angles at 
 the ends of this side 50 and 60, what is the area ? 
 
 Ans. 1147 sq. feet. 
 
 10. If the sides only of a triangle are given, an angle may 
 be found, by oblique trigonometry, Case IV, and then the 
 perpendicular and the area may be calculated. But the area 
 may be more directly obtained, by the following method. 
 
 RULE II. When the three sides are given, from half their 
 sum subtract each side severally, multiply together the half 
 sum and the three remainders, and extract the square root qf 
 the product. 
 
MENSURATION OF PLANE SURFACES. 
 
 13 
 
 If the sides of the triangle are a, b, and c, and if A=half 
 their sum, then 
 
 The area=Vhx(ha) X (hb) X (hc) 
 
 Ex. 1. In the triangle ABC, 
 given the sides a 52 feet, b 39, 
 and c 65 ; to find the side of a 
 square which has the same area 
 as the triangle. 
 
 h a=26 
 
 h 6=39 
 h c=13 
 
 Then the area=v78X 26X39X13=1014 square feet. 
 By logarithms. 
 
 The half sum 
 First remainder 
 Second do. 
 Third do. 
 
 le area required 
 
 =78 
 = 26 
 = 39 
 = 13 
 
 = 1014 
 
 1.89209 
 1.41497 
 1.59106 
 1.11394 
 
 2)6.01206 
 2)3.00603 
 
 Side of the square =31.843 (Trig. 47.) 1.50301 
 
 2. If the sides of a triangle are 134, 108, and 80 rods, 
 what is the area? Ans. 4319. 
 
 3. What is the area of a triangle whose sides are 371, 264, 
 and 225 feet ? 
 
 PROBLEM III. 
 To find the area of a TRAPEZOID. 
 
 21. MULTIPLY HALF THE SUM OF THE PARALLEL SIDES 
 INTO THEIR PERPENDICULAR DISTANCE. 
 
14 
 
 MENSURATION OF PLANE SURFACES. 
 
 The area of the trapezoid 
 ABCD, is equal to half the 
 sum of the sides AB and CD, 
 multiplied into the perpendic- 
 ular distance PC or AH. For 
 the whole figure is made up of 
 the two triangles ABC and 
 
 ADC ; the area of the first of which is equal to the product 
 of half the base AB into the perpendicular PC, (Art. 8.) 
 and the area of the other is equal to the product of half the 
 base DC into the perpendicular AH or PC. 
 
 Ex. If AB be 46 feet, BC 31, DC 38, and the angle B 
 70, what is the area of the trapezoid ? 
 
 R ; BC : : sin B : PC=29.13. And 42 X 29.13 = 1223*. 
 
 2. What are the contents of a field which has two par- 
 allel sides 65 and 38 rods, distant from each other 27 rods ? 
 
 PROBLEM IV. 
 To find the area of a TRAPEZIUM, or of an irregular POLYGON. 
 
 13. DIVIDE THE WHOLE FIGURE INTO TRIANGLES, BY DRAW- 
 ING DIAGONALS, AND FIND THE SUM OF THE AREAS OF THESE 
 TRIANGLES. (Alg. 394.) 
 
 If the perpendiculars in two triangles fall upon the same 
 diagonal, the area of the trapezium formed of the two trian- 
 gles, is equal to half the 
 product of the diagonal into 
 the sum of the perpendiculars. 
 
 Thus the area of the trape- 
 zium ABCH, is 
 
 iBHxAL-KBHxCM=iBHx(AL-fCM.) 
 Ex. In the irregular polygon ABCDH, 
 
MENSURATION OF PLANE SURFACES. 
 
 15 
 
 == 5.3 
 
 !BH=36 . 
 
 CH ~ 32 ' and the perpendiculars ) CM=9.3 
 
 The area= 
 
 14. If the diagonals of a trapezium are given, the area 
 may be found, nearly in the same manner as the area of a 
 parallelogram in Art. 5, and the area of a triangle in Art. 9. 
 
 In the trapezium ABCD, the sines of the four angles at 
 N, the point of intersec- 
 tion of the diagonals, are 
 all equal. For the two 
 acute angles are supple- 
 ments of the other two, 
 and therefore have the 
 same sine. (Trig. 90.) 
 Putting, then, sin N for 
 
 the sine of each of these angles, the areas of the four tri- 
 angles of which the trapezium is composed, are given by the 
 following proportions; (Art. 9.) 
 
 R : sin 
 
 - 
 
 2 area ABN" 
 2 area BCN 
 2 area CDN 
 2 area ADN 
 
 And by addition, (Alg. 349, Cor. 1.)* 
 
 R : 8 
 
 2 area ABCD. 
 
 The 3d tenn=(AN+CN)x(BN-f-DN)=ACxBD, by 
 the figure. 
 
 Therefore R : sin N : : AC X BD : 2 area ABCD. That is, 
 
 * Euclid, 2, 5. Gor. 
 
10 MENSURATION OF PLANS SURFACES. 
 
 As Radius, 
 
 To the sine of the angle at the intersection of the 
 
 diagonals of a trapezium ; 
 So is the product of the diagonals, 
 To twice the area of the trapezium. 
 
 It is evident that this rule is applicable to a parallelogram, 
 as well as to a trapezium. 
 
 If the diagonals intersect at right angles, the sine of N is 
 equal to radius ; (Trig. 95.) and therefore the product of the 
 diagonals is equal to twice the area. (Alg. 356.)* 
 
 Ex. 1. If the two diagonals of a trapezium are 37 and 62, 
 and if they intersect at an angle of 54, what is the area of 
 the trapezium ? Ans. 928. 
 
 2. If the diagonals are 85 and 93, and the angle of inter- 
 section 74, what is the area of the trapezium ? 
 
 PROBLEM V. 
 To find the area of a REGULAR POLYGON. 
 
 15. MULTIPLY ONE OF ITS SIDES INTO HALF ITS PERPEN- 
 DICULAR DISTANCE FROM THE CENTRE, AND THIS PRODUCT 
 INTO THE NUMBER OF SIDES. 
 
 A regular polygon contains as many equal triangles as the 
 figure has sides. Thus, the hexagon 
 ABDFGH contains six triangles, each 
 equal to ABC. The area of one of 
 them is equal to the product of the 
 side AB, into half the perpendicular 
 CP. (Art. 8.) The area of the whole, 
 therefore, is equal to this product 
 multiplied into the number of sides. 
 
 * Euclid, 14. 5. 
 
MENSURATION OF PLANE SURFACES. 17 
 
 Ex. 1. What is the area of a regular octagon, in which the 
 length of a side is 60, and the perpendicular from the centre 
 72.426? Ans. 17382. 
 
 2. What is the area of a regular decagon whose sides are 
 46 each, and the perpendicular 70.7867 ? 
 
 16. If only the length and number of sides of a regular 
 polygon be given, the perpendicular from the centre may be 
 easily found by trigonometry. The periphery of the circle 
 in which the polygon is inscribed, is divided into as many 
 equal parts as the polygon has sides. (Euc. 16.4. Schol.)* 
 The arc, of which one of the sides is a chord, is therefore 
 known ; and of course, the angle at the centre subtended by 
 this arc. 
 
 Let AB be one side of a regular polygon inscribed in the 
 circle ABDG. The perpendicular CP bisects the line AB, 
 and the angle ACB. (Euc. 3. 3.)f Therefore, BCP is the 
 same part of 360, which BP is of the perimeter of the 
 polygon. Then, in the right angled triangle BCP, if BP be 
 radius, (Trig. 122.) 
 
 R : BP : : cot BCP : CP. That is, 
 
 As Radius, 
 
 To half of one of the sides of the polygon ; 
 So is the cotangent of the opposite angle, 
 To the perpendicular from the centre. 
 
 Ex. 1. If the side of a regular hexagon be 38 inches, what 
 is the area ? 
 
 The angle BCP^ of 360=30. Then, 
 R : 19 : : cot 30 ; 32.909=CP, the perpendicular, 
 And the area=19X32.909X6 = 
 
 * Thomson's Legendre, 2. 5. Schol. f ^id. 6. 2. 
 
18 
 
 MENSURATION OF PLANE SURFACES. 
 
 2. What is the area of a regular decagon whose sides are 
 each 62 feet ? Ans. 29576. 
 
 17. From the proportion in the preceding article, a table 
 of perpendiculars and areas may be easily formed, for a series 
 of polygons, of which each side is a unit. Putting R=l, 
 (Trig. 100.) and w=the number of sides, the proportion be- 
 comes 
 
 Q Af) 
 
 1 ; : : cot ; the perpendicular 
 
 So that, the perp.=% co 
 
 360 
 
 And the area is equal to half the product of the perpen- 
 dicular into the number of sides. (Art. 15.) 
 
 Thus, in the trigon, or equilateral triangle, the perpendic- 
 
 QfiftO 
 
 ular=i cot =j cot 60=0.2886752. 
 
 G 
 
 And the area= 0.4330127. 
 
 In the tetragon, or square, the perpendicular = co 
 
 .360 
 IT 
 
 = cot 45=0.5. And the area=l. 
 
 In this manner, the following table is formed, in which 
 the side of each polygon is supposed to be a unit. 
 
 A TABLE OF REGULAR POLYGONS. 
 
 Names. | Sides, j Angles. 
 
 Perpendiculars. 
 
 Areas. 
 
 Trigon, 
 
 3 
 
 60 
 
 ,0.2886752 
 
 0.4330127 
 
 Tetragron, 
 
 4 
 
 45 
 
 0.5000000 
 
 1.0000000 
 
 Pentagon, 
 
 5 
 
 36 
 
 0.6881910 
 
 1.7204774 
 
 Hexagon, 
 
 6 
 
 30 
 
 0.8660254 
 
 2.5980762 
 
 Heptagon, 
 
 7 
 
 25* 
 
 1.0382601 
 
 3.6339124 
 
 Octagon, 
 
 8 
 
 22 
 
 1.2071069 
 
 4.8284271 
 
 Nonagon, 
 
 9 
 
 20 
 
 1.3737385 
 
 6.1818242 
 
 Decagon, 
 
 10 
 
 18 
 
 1.5388418 
 
 7.6942088 
 
 Undecagon, 
 
 11 
 
 16^r 
 
 1.7028439 
 
 9.3656399 
 
 Dodecagon, 
 
 12 
 
 15 
 
 1.8660252 
 
 11.1961524 
 
MENSURATION OP THE CIRCLE. 19 
 
 By this table may be calculated the area of any other reg- 
 ular polygon, of the same number of sides with one of these. 
 For the areas of similar polygons are as the squares of their 
 homologous sides. (Euc. 20. 6.)* 
 
 To find, then, the area of a regular polygon, multiply the 
 square of one of its sides by the area of a similar polygon of 
 which the side is a unit. 
 
 Ex. 1. What is the area of a regular decagon whose sides 
 are each 102 rods ? Ans. 80050.5 rods. 
 
 2. What is the area of a regular dodecagon whose sides 
 are each 87 feet ? 
 
 SECTION II. 
 
 THE QUADRATURE OF THE CIRCLE AND ITS PARTS. 
 
 ART. 18. Definition I. A circle is a plane bounded by a 
 line which is equally distant in all its parts from a point 
 within called the centre. The bounding line is called the 
 circumference or periphery. An arc is any portion of the 
 circumference. A semi-circle is half, and a quadrant one- 
 fourth of a circle. 
 
 II. A Diameter of a circle is a straight line drawn through 
 the centre, and terminated both ways by the circumference. 
 A Radius is a straight line extending from the centre to the 
 circumference. A Chord is a straight line which joins the 
 two extremities of an arc. 
 
 III. A Circular Sector is a space contained between an 
 arc and the two radii drawn from the extremities of the arc. 
 
 * Thomson's Legendre 1. 5. Cor. 
 
20 
 
 MENSURATION OF THE CIRCLE. 
 
 It may be less than a semi-circle, as 
 ACBO, or greater, as ACBD. 
 
 IV. A Circular Segment is the 
 space contained between an arc and 
 its chord, as ABO or ABD. The chord 
 is sometimes called the base of the 
 segment. The height of a segment 
 is the perpendicular from the middle 
 of the base to the arc, as PO. 
 
 V A Circular Zone is the space 
 between two parallel chords, as 
 AGHB. It is called the middle 
 zone, when the two chords are 
 equal, as GHDE. 
 
 VI. A Circular Ring is the space between the peripheries 
 of two concentric circles, as AA', BB'. (Fig. 13.) 
 
 VII. A Lune or Crescent is the space between two circu- 
 lar arcs which intersect each other, as ACBD. (Fig. 14.) 
 
 19. The Squaring of the Circle is a problem which has 
 exercised the ingenuity of distinguished mathematicians for 
 many centuries. The result of their efforts has been only 
 an approximation to the value of the area. This can be car* 
 ried to a degree of exactness far beyond what is necessary 
 for practical purposes. 
 
MENSURATION OF THE CIRCLE. 21 
 
 20. If the circumference of a circle of given diameter 
 were known, its area could be easily found. For the area is 
 equal to the product of half the circumference into half the 
 diameter. (Sup. Euc. 5, l.*)f But the circumference of a 
 circle has never been exactly determined. The method of 
 approximating to it is by inscribing and circumscribing poly- 
 gons, or by some process of calculation which is, in principle, 
 the same. The perimeters of the polygons can be easily 
 and exactly determined. That which is circumscribed is 
 greater, and that which is inscribed is less, than the peri- 
 phery of the circle ; and by increasing the number of sides, 
 the difference of the two polygons may be made less than 
 any given quantity. (Sup. Euc. 4, 1.) 
 
 21. The side of a liexagon in- 
 scribed in a circle, as AB, is the 
 chord of an arc of 60, and there- 
 fore equal to the radius. (Trig. 95.) 
 The chord of half this arc, as BO, 
 is the side of a polygon of 12 equal 
 sides. By repeatedly bisecting the 
 arc, and finding the chord, we may 
 
 obtain the side of a polygon of an immense number of sides. 
 Or we may calculate the sine, which will be half the chord 
 of double the arc, (Trig. 82, cor.,) and the tangent, which 
 will be half the side of a similar circumscribed polygon. 
 Thus the sine AP, is half of AB, a side of the inscribed 
 hexagon ; and the tangent NO is half of NT, a side of the 
 circumscribed hexagon. The difference between the sine 
 and the arc AO is less than the difference between the sine 
 and the tangent. In the section on the computation of the 
 canon, (Trig. 223.) by 12 successive bisections, beginning 
 with 60 degrees, an arc is obtained which is the irrSTT f 
 the whole circumference. 
 
 * In this manner, the Supplement to Play fair's Euclid is referred t& in 
 this work. t Thomson's Legendre, 11. 5. 
 
22 MENSURATION OF THE CIRCLE. 
 
 The cosine of this, if radius be 1, is found to be .99999996732 
 The sine is .00025566346 
 
 And the tangent sme (Trig. 93.) =.00025566347 
 cosine 
 
 The diff. between the sine and tangent is only .00000000001 
 And the difference between the sine and the arc is still less. 
 
 Taking then, .000255663465 for the length of the arc, 
 multiplying by 24576, and retaining 8 places of decimals, 
 we have 6.28318531 for the whole circumference, the radius 
 being 1. Half of this, 
 
 3.14159265 
 
 is the circumference of a circle whose radius is , and diam- 
 eter 1. 
 
 22. If this be multiplied by 7, the product is 21. 99 -for 
 22 nearly. So that, 
 
 Diam : Circum : : 7 : 22, nearly. 
 
 If 3.14159265 be multiplied by 113, the product is 
 354.9999+, or 355, very nearly. So that, 
 
 Diam ; Circum : : 113 ; 355, very nearly. 
 
 The first of these ratios was demonstrated by Archimedes. 
 
 There are various methods, principally by infinite series 
 and fluxions, by which the labor of carrying on the approx- 
 imation to the periphery of a circle may be very much 
 abridged. The calculation has been extended to nearly 150 
 places of decimals. But four or five places are sufficient 
 for most practical purposes. 
 
 After determining the ratio between the diameter and the 
 circumference of a circle, the following problems are easily 
 solved. 
 
MENSURATION OF THE CIRCLE. 23 
 
 PROBLEM I. 
 To find the CIRCUMFERENCE of a circle from its diameter" t 
 
 23. MULTIPLY THE DIAMETER BY 3*14159.* 
 
 Or, 
 
 Multiply the diameter by 22 and divide the product by 1. Or, 
 multiply the diameter by 355, and divide the product by 
 113. (Art. 22.) 
 
 Ex. 1. If the diameter of the earth be 7930 miles, what 
 is the circumference? Ans. 249128 miles. 
 
 2. How many miles does the earth move, in revolving 
 round the sun ; supposing the orbit to be a circle whose 
 diameter is 190 million miles ? Ans. 596,902,100. 
 
 3. What is the circumference of a circle whose diameter 
 is 769843 rods ? 
 
 PROBLEM II. 
 To find the DIAMETER of a circle from its circumference. 
 
 24. DIVIDE THE CIRCUMFERENCE BY 3. 14 159* 
 
 Or, 
 
 Multiply the circumference by 7, and divide the product by 
 22. Or, multiply the circumference by 113, and divide 
 the product by 355. (Art. 22.) 
 
 Ex. 1. If the circumference of the sun be 2,800,000 miles, 
 what is his diameter? Ans. 891,267. 
 
 2. What is the diameter of a tree which is 5 feet round ? 
 
 25. As multiplication is more easily performed than divis- 
 ion, there will be an advantage in exchanging the divisor 
 
 * In many cases, 3.1416 will be sufficiently accurate. 
 
24 MENSURATION OF THE CIRCLE. 
 
 3.14159 for a multiplier which will give the same result. 
 In the proportion 
 
 3.14159 : 1 : : Circum : Diam. 
 
 to find the fourth term, we may divide the second by the 
 first, and multiply the quotient into the third. Now, 1~ 
 3.14159=0.31831. If, then, the circumference of a circle 
 be multiplied by .31831, the product will be the diameter. 
 
 Ex. 1. If the circumference of the moon be 6850 miles, 
 what is her diameter ? Ans. 2180. 
 
 2. If the whole extent of the orbit of Saturn be 5650 
 million miles, how far is he from the sun ? 
 
 3. If the periphery of a wheel be 4 feet 7 inches, what is 
 its diameter? 
 
 PROBLEM III. 
 To find the length of an ARC of a circle. 
 
 26. As 360, to the number of degrees in the arc ; 
 
 So is the circumference of the circle, to the length of the arc. 
 
 The circumference of a circle being divided into 360, 
 (Trig. 73.) it is evident that the length of an arc of any less 
 number of degrees must be a proportional part of the whole. 
 
 Ex. What is the length of an arc of 16, in a circle whose 
 radius is 50 feet ? 
 
 The circumference of the circle is 314.159 feet. (Art. 23.) 
 Then 360 : 16 : : 314.159 : 13.96 feet. 
 
 2. If we are 95 millions of miles from the sun, and if the 
 earth revolves round it in 365^ days, how far are we carried 
 in 24 hours ? Ans. 1 million 634 thousand miles. 
 
 27. The length of an arc may also be found, by multiply- 
 ing the diameter into the number of degrees in the arc, and 
 
MENSURATION OF THE CIRCLE. 
 
 25 
 
 this product into .0087266, which is the length of one de- 
 gree, in a circle whose diameter is 1. For 3. 141 59 -f- 360= 
 0.00872 66. And in different circles, the circumferences, and 
 of course the degrees, are as the diameters. (Sup. Euc. 8, 1.)* 
 
 Ex. 1. What is the length of an arc of 10 15' in a circle 
 whose radius is 68 rods ? Ans. 12.165 rods. 
 
 2. If the circumference of the earth be 24913 miles, what 
 is the length of a degree at the equator ? 
 
 28. The length of an arc is frequently required, when 
 the number of degrees is not given. But if the radius of the 
 circle, and either the chord or the height of the arc, be 
 known ; the number of degrees 
 may be easily found. 
 
 Let AB be the chord, and PO 
 the height, of the arc AOB, As 
 the angles at P are right angles, and 
 AP is equal to BP; (Art. 18. Def. 
 4.) AO is equal to BO. (Euc. 4, 
 l.)f Then, 
 
 BP is the sine, and CP the cosine, 
 
 OP the versed sine, and BO the chord, 
 
 of half the arc AOB. 
 
 CB : R : : 
 
 And in the right angled triangle CBP, 
 
 BP : sin BCP or BO. 
 CP : cos BCP or BO. 
 
 Ex. 1. If the radius C0=25, and the chord AB=43.3; 
 what is the length of the arc AOB ? 
 
 CB : R : : BP : sin BCP or B0=60 very nearly. 
 The circumference of the circle =3.14159X50=157.08. 
 And 360 : 60 : : 157.08 : 26.1 8=OB. Therefore,AOB=52.36, 
 
 * Thomson's Legendre, 10. 5. 
 
 t Ibid., 5. 1. 
 
20 MENSURATION OF THE CIRCLE. 
 
 2. What is the length of an arc whose chord is 21 6^-, in a 
 circle whose radius is 125 ? Ans. 261.8. 
 
 20. If only the chord and the height of an arc be given, 
 the radius of the circle may be 
 found, and then the length of the 
 arc. 
 
 If BA be the chord, and PO the 
 the height of the arc AOB, then 
 (Euc. 35. 3.)* 
 
 DP 
 
 . And DO=OP+DP==OP+gl. 
 
 That is, the diameter is equal to the height of the arc, + 
 the square of half the chord divided by the height. 
 
 The diameter being found, the length of the arc may be 
 calculated by the two preceding articles. 
 
 Ex. 1. If the chord of an arc be 173.2, and the height 50, 
 what is the length of the arc ? 
 
 The diameter =50 -f^?!=200. The arc contains 120, 
 50 
 
 (Art. 28.) and its length is 209.44. (Art. 26.) 
 
 2. What is the length of an arc whose chord is 120, and 
 height 45 ? Ans. 160.8. 
 
 PROBLEM IV. 
 To find the AREA of a CIRCLE. 
 
 30. MULTIPLY THE SQUARE OF THE DIAMETER BY TUB 
 DECIMALS .7854* 
 
 * Thomson's Legendre, 10. 5. 
 
MENSURATION OF THE CIRCLE. 27 
 
 Or, 
 
 MULTIPLY HALF THE DIAMETER INTO HALF THE CIRCUM- 
 FERENCE. Or, multiply the whole diameter into the whole 
 circumference, and take -J- of the product. 
 
 The area of a circle is equal to the product of half the 
 diameter into half the circumference ; (Sup. Euc. 5, 1.) or, 
 which is the same thing, \ the product of the diameter and 
 circumference. If the diameter be 1, the circumference is 
 3.14159; (Art. 23.) one-fourth of which is 0.7854 nearly. 
 But the areas of different circles are to each other, as the 
 squares of their diameters. (Sup. Euc. 7, 1.)* The area of 
 any circle, therefore, is equal to the product of the square of 
 its diameter into 0.7854, which is the area of a circle whose 
 diameter is 1. 
 
 Ex. 1. What is the area of a circle whose diameter is 623 
 feet? Ans. 304836 square feet. 
 
 2. How many acres are there in a circular island whose 
 diameter is 124 rods. Ans. 75 acres, and 76 rods. 
 
 3. If the diameter of a circle be 113, and the circumfer- 
 ence 355, what is the area? Ans. 10029. 
 
 4. How many square yards are there in a circle whose 
 diameter is 7 feet ? 
 
 31. If the circumference of a circle be given, the area may 
 be obtained, by first finding the diameter ; or, without finding 
 the diameter, by multiplying the square of the circumference 
 by .07958. 
 
 For, if the circumference of a circle be 1, the diameter = 
 1-7-3.14159=0.31831 ; and the product of this into the 
 circumference is .07958 the area. But the areas of different 
 circles, being as the squares of their diameters, are also as 
 the squares of their circumferences. (Sup. Euc. 8, 1.) 
 
 * Thomson's Legendre, 28. 4. Cor. 
 
28 
 
 MENSURATION OF THE CIRCLE, 
 
 Ex. 1. If the circumference of a circle be 136 feet, what 
 is the area? Ans. 1472 feet. 
 
 2. What is the surface of a circular fish-pond, which is 
 10 rods in circumference ? 
 
 32. If the area of a circle be given, the diameter may be 
 found, by dividing the area by .7854, and extracting the 
 square root of the quotient. 
 
 This is reversing the rule in Art. 30. 
 
 Ex. 1. What is the diameter of a circle whose area is 
 380.1336 feet ? 
 
 Ans. 380.1336-r-.7854=484. And V484=22. 
 
 2. What is the diameter of a circle whose area is 19.635 ? 
 
 33. The area of a circle, is to the area of the circumscribed 
 square / as .7854 to 1 ; and to that of the inscribed square 
 as .7854 to |. 
 
 Let ABDF be the inscribed 
 square, and LMNO the circum- 
 scribed square, of the circle ABDF. 
 The area of the circle is equal to 
 AF 2 X-7854. (Art. 30.) But the A 
 area of the circumscribed square 
 
 (Art. 4.) is equal to ON*=ADl 
 And the smaller square is half of 
 the larger one. For the latter con- 
 tains 8 equal triangles, of which the former contains only 4. 
 
 Ex. What is the area of a square inscribed in a circle 
 whose area is 159 ? Ans. .7854 : i : : 159 : 101.22. 
 
 PROBLEM V. 
 To find the area of a SECTOR of a circle. 
 
 34. MULTIPLY THE RADIUS INTO HALF THE LENGTH OF 
 
 THE ARC. 
 
MENSURATION OF THE CIRCLE. 
 
 29 
 
 Or, 
 
 As 360, TO THE NUMBER OF DEGREES IN THE ARC J 
 So IS THE AREA OF THE CIRCLE, TO THE AREA OF THE 
 SECTOR. 
 
 It is evident, that the area of the sector has the same 
 ratio to the area of the circle, which the length of the arc 
 has to the length of the whole circumference ; or which the 
 number of degrees in the arc has to the number of degrees 
 in the circumference. 
 
 Ex. 1. If the arc AOB be 120, 
 and the diameter of the circle 226 ; 
 what is the area of the sector 
 AOBC? 
 
 The area of the whole circle is 
 40115. (Art. 30.) 
 
 And 360 : 120 : : 40115 : 
 133711, the area of the sector. 
 
 2. What is the area of a quadrant whose radius is 621 ? 
 
 3. What is the area of a semi- circle, whose diameter is 
 328? 
 
 4. What is the area of a sector which is less than a semi- 
 circle, if the radius be 15, and the chord of its arc 12 ? 
 
 Half the chord is the sine of 23 34f ' nearly. (Art. 28.) 
 
 The whole arc, then, is 47 9' 
 
 The area of the circle is 706.86 
 
 And 360 : 47 9' : : 706.86 : 92.6 the area of the sector. 
 
 5. If the arc ABB be 240 degrees, and the radius of the 
 circle 113, what is the area of the sector ADBC ? 
 
 PROBLEM VI. 
 
 To find the area of a SEGMENT of a circle. 
 35. FIND THE AREA OF THE SECTOR WHICH HAS THE 
 
80 MENSURATION Of THE CIRCLE. 
 
 SAME ARC, AND ALSO THE AREA OF A TRIANGLE FORMED BT 
 THE CHORD OF THE SEGMENT AND THE RADII OF THE SEC- 
 TOR. 
 
 THEN, IF THE SEGMENT BE LESS THAN A SEMI-CIRCLE; 
 SUBTRACT THE AREA OF THE TRIANGLE FROM THE AREA OF 
 THE SECTOR. BUT, IF THE SEGMENT BE GREATER THAN A 
 
 SEMI-CIRCLE, ADD THE AREA OF THE TRIANGLE TO THE AREA 
 OF THE SECTOR. 
 
 If the triangle ABC, be taken 
 from the sector AOBC, it is evi- 
 dent the difference will be the seg- 
 ment AOBP, less than a semi-cir- 
 cle. And if the same triangle be 
 added to the sector ADBC, the 
 sum will be the segment ADBP, 
 greater than a semi-circle. 
 
 The area of the triangle (Art. 8.) 
 
 is equal to the product of half the chord AB into CP, which 
 is the difference between the radius and PO the height of the 
 segment. Or CP is the cosine of half the arc BOA. If this 
 cosine and the chord of the segment are not given, they 
 may be found from the arc and the radius. 
 
 Ex. 1. If the arc AOB be 120, and the radius of the 
 circle be 113 feet, what is the area of the segment AOBP ? 
 
 In the right angled triangle BCP, 
 R : BC : : sin BCO : BP=97.86, half the chord. (Art. 28.) 
 
 The cosine PC=i CO (Trig. 96, Cor.) =56.5 
 
 The area of the sector AOBC (Art. 34.) =13371.67 
 
 The area of the triangle ABC=BPxPC = 5528.97 
 
 The area of the segment, therefore, = 7842.7 
 
 2. If the base of a segment, less than a semi-circle, be 10 
 
MENSURATION OF THE CIRCLE. 31 
 
 feet, and the radius of the circle 12 feet, what is the area 
 of the segment ? 
 
 The arc of the segment contains 49-}- degrees. (Art. 28.) 
 The area of the sector =61.89 (Art. 34.) 
 
 The area of the triangle =54.54 
 
 And the area of the segment = 7.35 square feet. 
 
 3. What is the area of a circular segment, whose height 
 is 19.2 and base 70 ? Ans. 947.86. 
 
 4. What is the area of the segment ADBP, (Fig. 9.) if 
 the base AB be 195.7, and the height PD 169.5 ? 
 
 Ans. 32272. 
 
 36. The area of any figure which is bounded partly by 
 arcs of circles, and partly by right lines, may be calculated, 
 by finding the areas of the segments under the arcs, and then 
 the area of the rectilinear space between the chords of the 
 arcs and the other right lines. 
 
 Thus, the Gothic arch ACB, 
 contains the two segments 
 ACH, BCD, and the plane tri- 
 angle ABC. 
 
 Ex. If AB be 110, each of 
 the lines AC and BC 100, and 
 the height of each of the seg- 
 ments ACH, BCD 10.435 ; 
 what is the area of the whole figure ? 
 
 The areas of the two segments are 
 The area of the triangle ABC is 
 And the whole figure is 
 
82 MENSURATION OF THE CIRCLE. 
 
 PROBLEM VII. 
 To find the area of a circular ZONE. 
 
 37. FROM THE AREA OF THE WHOLE CIRCLE, SUBTRACT 
 
 THE TWO SEGMENTS. ON THE SIDES OF THE ZONE. 
 
 If from the whole circle there be taken the two segments 
 ABC and DBH, there will remain 
 the zone ACDH. 
 
 Or, the area of the zone may be 
 found by subtracting the segment 
 ABC from the segment HBD : Or, 
 by adding the two small segments 
 GAH and VDC, to the trapezoid 
 ACDH. (See Art. 36.) 
 
 The latter method is rather the 
 
 most expeditious in practice, as the two segments at the end 
 of the zone are equal. 
 
 Ex. 1. What is the area of the zone ACDH, if AC is 
 7.75, DH 6.93, and the diameter of the circle 8 ? 
 
 The area of the whole circle is 50.26 
 
 of the segment ABC 17.32 
 
 of the segment DFH 9.82 
 
 of the zone ACDH 23.12 
 
 2. What is the area of a zone, one side of which is 23.25, 
 and the other side 20.8, in a circle whose diameter is 24 ? 
 
 Ans. 208. 
 
 38. If the diameter of the circle is not given, it may be 
 found from the sides and the breadth of the zone. 
 
 Let the centre of the circle be at O. Draw ON perpen- 
 dicular to AH, NM perpendicular to LB, and HP perpen- 
 dicular to AL. Then, 
 
MENSURATION OF THE CIRCLE. 88 
 
 AN=*AH, (Euc. 3. 3.)* MN=i(LA+RH) 
 LM=iLR, (Euc. 2. 6.)f PA =LA RH. 
 
 The triangles APH and OMN are similar, because the 
 sides of one are perpendicular to those of the other, each to 
 each, Therefore, 
 
 PH : PA : : MN : MO 
 MO being found, we have ML MO=OL. 
 
 And the radius CO=vOL 2 +CL a . (Euc. 47. l.)J 
 
 Ex. If the breadth of the zone ACDH (Fig. 12.) be 6.4, 
 and the sides 6.8 and 6 ; what is the radius of the circle ? 
 
 PA=3.4 3=0.4. And, MN=i(3.4+3)=3.2. 
 Then, 6.4 : 0.4 :: 3.2 : 0.2=MO. And, 3.2 0.2=3=OL 
 
 And the radius CO=V3 2 + (3.4) 2 =4.534. 
 
 PROBLEM VIII. 
 To find the area of a LUNE or crescent. 
 
 39. FIND THE DIFFERENCE OF THE TWO SEGMENTS WHICH 
 ARE BETWEEN THE ARCS OF THE CRESCENT AND ITS CHORD. 
 
 If the segment ABC, be 
 taken from the segment ABD ; 
 there will remain the lune or 
 crescent ACBD. 
 
 Ex. If the chord AB be 88, 
 the height CH 20, and the 
 height DH 40 ; what is the 
 area of the crescent ACBD ? 
 
 The area of the segment ABD is 2698 
 
 of the segment ABC 1220 
 
 of the crescent ACBD 1478 
 
 * Thomson's Legendre, 6. 2. f Ibid. 15. 4. $ Ibid. 11. 4. 
 
 2* 
 
34 MENSURATION OF THE CIRCLE. 
 
 PROBLEM IX. 
 
 To find the area of a RING, included between the periphe- 
 ries of two concentric circles. 
 
 40. FlND THE DIFFERENCE OF THE AREAS OF THE TWO 
 CIRCLES. 
 
 Or, 
 
 Multiply the product of the sum and difference of the two 
 diameters by .7854. 
 
 The area of the ring is evidently equal to the difference 
 between the areas of the two cir- \ B 
 
 cles AB and A'B'. 
 
 But the area of each circle is equal 
 to the square of its diameter multi- 
 plied into .7854. (Art. 30.) And 
 the difference of these squares is equal 
 to the product of the sum and dif- 
 ference of the diameters. (Alg. 191.) Therefore the area of 
 the ring is equal to the product of the sum and difference 
 of the two diameters multiplied by .7854. 
 
 Ex. 1. If AB be 221, and A'B' 106, what is the area of 
 
 the ring? Ans. (221 a X.7854>- <106 9 X.7854)=29535. 
 
 2. If the diameters of Saturn's larger ring be 205,000 
 and 190,000 miles, how many square miles are there on one 
 side of the ring ? 
 
 Ans. 395000X15000X-7854=4,653,495,000. 
 
 PROMISCUOUS EXAMPLES OF AREAS. 
 
 Ex. 1. What is the expense of paving a street 20 rods 
 long and 2 rods wide, at 5 cents for a square foot ? 
 
 Ans. 54 4 dollars. 
 
MENSURATION OF THE CIRCLE. 36 
 
 2. If an equilateral triangle contains as many square feet 
 as there are inches in one of its sides ; what is the area of 
 the triangle ? 
 
 Let x=ihe number of square feet in the area. 
 
 /j* 
 
 Then -= the number of linear feet in one of the sides. 
 1 "2 
 
 And, (Art. 11.) x=i X VB=- 
 
 eHt* 
 
 Reducing the equation, x= =332.55 the area. 
 V3 
 
 3. What is the side of a square whose area is equal to that 
 of a circle 452 feet in diameter ? 
 
 Ans. V(452) a X- 7854=400.574. (Arts. 30 and 7.) 
 
 4. What is the diameter of a circle which is equal to a 
 square whose side is 36 feet ? 
 
 Ans. V(36) a -r 0.7854= 40.62 17. (Arts. 4 and 32.) 
 
 5. What is the area of a square inscribed in a circle whose 
 diameter is 132 feet? 
 
 Ans. 8712 square feet. (Art. 33.) 
 
 6. How much carpeting, a yard wide, will be necessary to 
 cover the floor of a room which is a regular octagon, the 
 sides being eight feet each ? Ans. 34-^ yards. 
 
 7. If the diagonal of a square be 16 feet, what is the 
 area? Ans. 128 feet. (Art. 14.) 
 
 8. If a carriage-wheel four feet in diameter revolve 300 
 times, in going round a circular green ; what is the area of 
 the green ? 
 
 Ans. 4154^ sq. rods, or 25 acres, 3 qrs. and 34-fc rods. 
 
 9. What will be the expense of papering the sides of a 
 room, at 10 cents a square yard ; if the room be 21 feet long, 
 
MENSURATION OF THE CIRCLE. 
 
 18 feet broad, and 12 feet high ; and if there be deducted 3 
 windows, each 5 feet by 3, two doors 8 feet by 4, and one 
 fire-place 6 feet by 4 ? Ans. 8 dollars 80 cents. 
 
 10. If a circular pond of water 10 rods in diameter be 
 surrounded by a gravelled walk 8-J- feet wide ; what is the 
 area of the walk? Ans. 16 sq. rods. (Art. 40.) 
 
 11. If CD, the base of the 
 isosceles triangle VCD, be 60 
 feet, and the area 1200 feet; 
 and if there be cut off, by the 
 line LG parallel to CD, the tri- 
 angle VLG, whose area is 432 
 feet ; what are the sides of the 
 latter triangle ? 
 
 Ans. 30, 30, and 36 feet. 
 
 12. What is the area of an equilateral triangle inscribed 
 in a circle whose diameter is 52 feet ? 
 
 Ans. 878.15 sq. ft. 
 
 13. If a circular piece of land is inclosed by a fence, in 
 which 10 rails make a rod in length ; and if the field con- 
 tains as many square rods, as there are rails in the fence ; 
 what is the value of the land at 120 dollars an acre ? 
 
 Ans. 942.48 dollars. 
 
 14. If the area of the equilat- 
 eral triangle ABD be 219.5375 
 feet ; what is the area of the cir- 
 cle OBDA, in which the triangle 
 is inscribed ? 
 
 The sides of the triangle are each 
 22.5167. (Art. 11.) 
 
 And the area of the circle is 530.93. 
 
MENSURATION OF SOLIDS. 87 
 
 15. If 6 concentric circles are so drawn, that the space 
 between the least or 1st, and the 2d is 21.2058, 
 
 between the 2d and the 3d is 35.343, 
 
 between the 3d and the 4th is - 49.4802, 
 between the 4th and the 5th is 63.6174, 
 
 between the 5th and the 6th is 77.7546 ; 
 
 what are the several diameters, supposing the longest to be 
 
 equal to 6 times the shortest ? 
 
 Ans. 3, 6, 9, 12, 15, and 18. 
 
 16. If the area between two concentric circles be 1202.64 
 square inches, and the diameter of the lesser circle be 19 
 inches, what is the diameter of the other ? 
 
 17. What is the area of a circular segment, whose height 
 is 9, and base 24 ? 
 
 SECTION III. 
 
 SOLIDS BOUNDED BY PLANE SURFACES. 
 
 ART. 41. DEFINITION I. A prism is a solid bounded by 
 plane figures or faces, two of which are parallel, similar, and 
 equal ; and the others are parallelograms. 
 
 II. The parallel planes are sometimes called the bases or 
 ends; and the other figures the sides of the prism. The 
 latter taken together constitute the lateral surface. 
 
 III. A prism is right or oblique, according as the sides are 
 perpendicular or oblique to the bases. 
 
 IV. The height of a prism is the perpendicular distance 
 between the planes of the bases. In a right prism, there- 
 fore, the height is equal to the length of one of the sides. 
 
 V. A Parallelepiped is a prism whose bases are parallelo- 
 grams. 
 
88 MENSURATION OF SOLIDS. 
 
 VI. A Cube is a solid bounded by six equal squares. It 
 is a right prism whose sides and bases are all equal. 
 
 VII. A Pyramid is a solid bounded by a plane figure 
 called the base, and several triangular planes, proceeding from 
 the sides of the base, and all terminating in a single point. 
 These triangles taken together constitute the lateral surface. 
 
 VIII. A pyramid is regular, if its base is a regular poly- 
 gon, and if a line from the centre of the base to the vertex 
 of the pyramid is perpendicular .. to the base. This line is 
 called the axis of the pyramid. 
 
 IX. The height of a pyramid is the perpendicular distance 
 from the summit to the plane of the base. In a regular pyr- 
 amid, it is the length of the axis. 
 
 X. The slant-height of a regular pyramid, is the distance 
 from the summit to the middle of one of the sides of the base. 
 
 XI. A frustum or trunk of a pyramid is a portion of the 
 solid next the base, cut off by a plane parallel to the base. 
 The height of the frustum is the perpendicular distance 
 of the two parallel planes. The slant height of a frus- 
 tum of a regular pyramid, is the distance from the middle of 
 one of the sides of the base, to the middle of the corres- 
 ponding side in the plane above. It is a line passing on 
 the surface of the frustum, through the middle of one of 
 its sides. 
 
 XII. A Wedge is a solid of five sides, viz. a rectangular 
 base, two rhomboidal 
 
 sides meeting in an J& H 
 
 edge, and two tri- 
 angular ends ; as 
 ABHG. The base 
 is ABCD, the sides 
 are ABHG and 
 DCHG, meeting in 
 the edge GH, and 
 the ends are BCH and ADG. The height of the wedge is a 
 
MENSURATION OF SOLIDS. 39 
 
 perpendicular drawn from any point in the edge, to the plane 
 of the base, as GP. 
 
 XIII. A Prismoid is a solid whose ends or bases are par- 
 allel, but not similar, and whose sides are quadrilateral. It 
 differs from a prism or a frustum of a pyramid, in having its 
 ends dissimilar. It is a rectangular prismoid, when its ends 
 are right parallelograms. 
 
 XIV. A linear side or edge of a solid is the line of intersec- 
 tion of two of the planes which form the surface. 
 
 42. The common measuring unit of solids is a cube, whose 
 sides are squares of the same name. The sides of a cubic 
 inch are square inches ; of a cubic foot, square feet, <kc. 
 Finding the capacity, solidity* or solid contents of a body, is 
 finding the number of cubic measures, of some given de- 
 nomination contained in the body. 
 
 In solid measure. 
 
 1728 cubic inches =1 cubic foot, 
 27 cubic feet =1 cubic yard, 
 4492i cubic feet =1 cubic rod, 
 32768000 cubic rods =1 cubic mile, 
 282 cubic inches =1 ale gallon, 
 231 cubic inches =1 wine gallon, 
 2150.42 cubic inches =1 bushel, 
 
 1 cubic foot of pure water weighs 1000 
 avoirdupois ounces, or 62 pounds. 
 
 PROBLEM I. 
 To find the SOLIDITY of a PRISM. 
 
 43. MULTIPLY THE AREA OF THE BASE BY THE HEIGHT. 
 
 This is a general rule, applicable to parallelepipeds 
 whether right or oblique, cubes, triangular prisms, &c. 
 
 * See note A. 
 
40 MENSURATION OF SOLIDS. 
 
 As surfaces are measured, by comparing them with a right 
 parallelogram (Art. 3.) ; so solids are measured, by com- 
 paring them with a right parallelepiped. 
 
 If ABCD be the base of a right D o 
 
 parallelepiped, as a stick of timber 
 standing erect, it is evident that the 
 number of cubic feet contained in one 
 foot of the height, is equal to the 
 number of square feet in the area of 
 the base. And if the solid be of any 
 
 other height, instead of one foot, the contents must have the 
 same ratio. For parallelepipeds of the same base are to 
 each other as their heights. (Sup. Euc. 9. 3.)* The solidity 
 of a right parallelepiped, therefore, is equal to the product 
 of its length, breadth, and thickness. See Alg. 39*7. 
 
 And an oblique parallelepiped being equal to a right one 
 of the same base and altitude, (Sup. Euc. 7. 3)f is equal to 
 the area of the base multiplied into the perpendicular height. 
 This is true also of prisms, whatever be the form of their 
 bases. (Sup. Euc. 2. Cor. to 8, 3. Thomson's Legendre, 12. 7.) 
 
 44. As the sides of a cube are all equal, the solidity is 
 found by cubing one of its edges. On the other hand, if the 
 solid contents be given, the length of the edges may be 
 found, by extracting the cube root. 
 
 45. When solid measure is cast by Duodecimals, it is to 
 be observed that inches are not primes of feet, but thirds. 
 If the unit is a cubic foot, a solid which is an inch thick and 
 a foot square is a prime ; a parallelepiped a foot long, an 
 inch broad, and an inch thick is a second, or the twelfth part 
 of a prime ; and a cubic inch is a third, or the twelfth part 
 of a second. A linear inch is -^ of a foot, a square inch lif 
 of . a foot, and a cubic inch -p^Vs- of a foot. 
 
 * Thomson's Legendre, 9. 7. f Ibid., 7. 7. 
 
MENSURATION OP SOLIDS. 41 
 
 Ex. 1. What are the solid contents of a stick of timber 
 which is 31 feet long, 1 foot 3 inches broad, and 9 inches 
 thick? Ans. 29 feet 9", or 29 feet 108 inches. 
 
 2. What is the solidity of a wall which is 22 feet long, 12 
 feet high, and 2 feet 6 inches thick ? 
 
 Ans. 660 cubic feet. 
 
 3. What is the capacity of a cubical vessel which is 2 feet 
 3 inches deep ? 
 
 Ans. 11 F. 4' 8" 3'", or 11 feet 675 inches. 
 
 4. If the base of a prism be 108 square inches, and the 
 height 36 feet, what are the solid contents ? 
 
 Ans. 27 cubic feet. 
 
 5. If the height of a square prism be 2^ feet, and each 
 side of the base 10-^ feet, what is the solidity ? 
 
 The area of the base = 10iXlOi=106|- sq. feet. 
 And the solid contents =106x 2-^=240^ cubic feet. 
 
 6. If the height of a prism be 23 feet, and its base a reg- 
 ular pentagon, whose perimeter is 18 feet, what is the so- 
 lidity ? Ans. 512.84 cubic feet. 
 
 46. The number of gallons or bushels which a vessel will 
 contain may be found, by calculating the capacity in inches, 
 and then dividing by the number of inches in 1 gallon or 
 bushel. 
 
 The weight of water in a vessel of given dimensions is 
 easily calculated ; as it is found by experiment, that a cubic 
 foot of pure water weighs 1000 ounces avoirdupois. For 
 the weight in ounces, then, multiply the cubic feet by 1000 ; 
 or for the weight in pounds, multiply by 62. 
 
 Ex. 1. How many ale gallons are there in a cistern which 
 is 11 feet 9 inches deep, and whose base is 4 feet 2 inches 
 square ? 
 
MEKSOHATION OF SOLIDS. 
 
 The cistern contains 352500 cubic inches; 
 And 352500-r282 = 
 
 2. How many wine gallons will fill a ditch 3 feet 1 1 inches 
 wide, 3 feet deep, and 462 feet long ? Ans. 40608. 
 
 3. What weight of water can be put into a cubical vessel 
 4 feet deep ? Ans. 4000 Ibs. 
 
 PROBLEM II. 
 To find the LATERAL SURFACE of a RIGHT PRISM. 
 
 47. MULTIPLY THE LENGTH INTO THE PERIMETER OF THE 
 BASE. 
 
 Each of the sides of the prism is a right parallelogram, 
 whose area is the product of its length and breadth. But 
 the breadth is one side of the base ; and therefore, the sum 
 of the breadths is equal to the perimeter of the base. 
 
 Ex. 1. If the base of a right prism be a regular hex- 
 agon whose sides are each 2 feet 3 inches, and if the height 
 be 16 feet, what is the lateral surface ? 
 
 Ans. 216 square feet. 
 
 If the areas of the two ends be added to the lateral sur- 
 face, the sum will be the whole surface of the prism. And 
 the superficies of any solid bounded by planes, is evidently 
 equal to the areas of all its sides. 
 
 2. If the base of a prism be an equilateral triangle 
 whose perimeter is 6 feet, and if the height be 1 7 feet, what 
 is the surface ? 
 
 The area of the triangle is 1.732. {Art. 11.) 
 
 And the whole surface is 105.464. 
 
MENSURATION OP SOLIDS. 4d 
 
 PROBLEM III. 
 
 To find the SOLIDITY of a PYRAMID. 
 48. MULTIPLY THE AREA OF THE BASE INTO OF THE 
 
 HEIGHT. 
 
 The solidity of a prism is equal to the product of the 
 area of the base into the height. (Art. 43.) And a pyramid 
 is f of a prism of the same base and altitude. (Sup. Euc. 
 15, 3. Cor. 1.)* Therefore the solidity of a pyramid 
 whether right or oblique, is equal to the product of the base 
 into of the perpendicular height. 
 
 Ex. 1. What is the solidity of a triangular pyramid, 
 whose height is 60, and each side of whose base is 4 ? 
 
 The area of the base is 6.928 
 And the solidity is 138.56. 
 
 2. Let ABC be one side of an oblique 
 pyramid whose base is 6 feet square ; 
 let BC be 20 feet, and make an angle 
 of 70 degrees with the plane of the 
 base ; and let CP be perpendicular to this 
 plane. What is the solidity of the pyr- 
 amid ? 
 
 In the right angled triangle BCP, (Trig. 134.) 
 R I BC : : sin B : : PC=18.79. 
 And the solidity of the pyramid is 225.48 feet, 
 
 3. What is the solidity of a pyramid whose perpendicular 
 height is 72, and the sides of whose base are 67, 54, and 
 40? Ans. 25920; 
 
 * Thomson's Legendre, 15 and 18. 7. 
 
44 MENSURATION OF SOLIDS. 
 
 PROBLEM IV. 
 To find the LATERAL SURFACE of a REGULAR PYRAMID. 
 
 49. MULTIPLY HALF THE SLANT-HEIGHT INTO THE PERIM- 
 ETER OF THE BASE. 
 
 Let the triangle ABC be one of 
 the sides of a regular pyramid. As 
 the sides AC and BC are equal, the 
 angles A and B are equal. Therefore 
 a line drawn from the vertex C to the 
 middle of AB is perpendicular to AB. 
 The area of the triangle is equal to A 
 the product of half this perpendicular into AB. (Art. 8.) 
 The perimeter of the base is the sum of its sides, each of 
 which is equal to AB. And the areas of all the equal tri- 
 angles which constitute the lateral surface of the pyramid, 
 are together equal to the product of the perimeter into half 
 the slant-height CP. 
 
 The slant-height is the hypothenuse of a right angled tri- 
 angle, whose legs are the axis of the pyramid, and the dis- 
 tance from the centre of the base to the middle of one of the 
 sides. See Def. 10. 
 
 Ex. 1. What is the lateral surface of a regular hexagonal 
 pyramid, whose axis is 20 feet, and the sides of whose base 
 are each 8 feet ? 
 
 The square of the distance from the centre of the base to 
 one of the sides. (Art. 16.)=48. 
 
 The slant-height (Euc. 47. l.)*=V4&H-(20) 9 =21.16 
 
 And the lateral surface=21. 16X4X6 = 507. 84 sq. feet. 
 
 2. What is the whole surface of a regular triangular pyr 
 * Thomson's Legendre, 11.4. 
 
MENSURATION OF SOLIDS. 
 
 45 
 
 amid whose axis is 8, and the sides of whose base are each 
 
 20.78 ? 
 
 The lateral surface is 312 
 
 The area of the base is 187 
 
 And the whole surface is 499 
 
 3. What is the lateral surface of a regular pyramid 
 whose axis is 12 feet, and whose base is 18 feet square ? 
 
 Ans. 540 square feet. 
 
 The lateral surface of an oblique pyramid may be found, 
 by taking the sum of the areas of the unequal triangles 
 which form its sides. 
 
 PROBLEM V. 
 To find the SOLIDITY of a FRUSTUM of a pyramid. 
 
 50. ADD TOGETHER THE AREAS OF THE TWO ENDS, AND 
 THE SQUARE ROOT OF THE PRODUCT OF THESE AREAS J AND 
 MULTIPLY THE SUM BY \ OF THE PERPENDICULAR HEIGHT 
 OF THE SOLID, 
 
 Let CDGL be a vertical 
 section, through the middle 
 of a frustum of a right pyr- 
 amid CDV, whose base is a 
 square. 
 
 Let CD=a, LG=&, RN=A. 
 
 By similar triangles, 
 LG : CD : : RV : NV. 
 
 Subtracting the antecedents, (Alg. 349.) 
 LG : CD LG : : RV : NV RV=RN. 
 
 f v RNxLG hb 
 Therefore RV = ^-^= 
 
46 MENSURATION OF SOLIDS. 
 
 The square of CD is the base v 
 
 of the pyramid CD V ; 
 
 And the square of LG is 
 the base of the small pyr- 
 amid LGV. 
 
 Therefore, the solidity of / 
 
 the larger pyramid (Art. 
 
 48) is c 
 
 \ = _*?! 
 
 a 67 Ba 
 
 36 
 And the solidity of the smaller pyramid is equal to 
 
 LG^XlRV=6'X ~ W 
 3a 36~~3a 36 
 
 If the smaller pyramid be taken from the larger, there 
 will remain the frustum. CDLG, whose solidity is equal to 
 
 (Alg. 194. a.) 
 Or, because Va 2 6 2 =a6, (Alg. 210. a.) 
 
 Here h, the height of the frustum, is multiplied into a 
 and 6*, the areas of the two ends, and into V 2 6 2 the square 
 root of the products of these areas. 
 
 In this demonstration the pyramid is supposed to be 
 square. But the rule is equally applicable to a pyramid of 
 any other form. For the solid contents of pyramids are 
 equal, when they have equal heights and bases, whatever be 
 the figure of their bases. (Sup. Euc. 14. 3.)* And the sec- 
 
 * Thomson's Legendre, 14. 7. 
 
MENSURATION OF SOLIDS. 47 
 
 tions parallel to the bases, and at equal distances, are equal 
 to one another. (Sup. Euc. 12. 3. Cor. 2.)* 
 
 Ex. 1. If one end of the frustum of a pyramid be 9 feet 
 square, the other end 6 feet square, and the height 36 feet, 
 what is the solidity ? 
 
 The areas of the two ends are 81 and 36. 
 The square root of their product is 54. 
 And the solidity of the frustum=(81+36+54)xl2=2052. 
 
 2. If the height of a frustum of a pyramid be 24, and 
 the areas of the two ends 441 and 121 ; what is the solid- 
 ity ? Ans. 6344. 
 
 3. If the height of a frustum of a hexagonal pyramid be 
 48, each side of one end 26, and each side of the other end 
 16 ; what is the solidity ? Ans. 56034*, 
 
 PROBLEM VI. 
 
 To find the LATERAL SURFACE of a FRUSTUM of a regular 
 pyramid. 
 
 51. MULTIPLY HALF THE SLANT-HEIGHT BY THE SUM OF 
 THE PERIMETERS OF THE TWO ENDS. 
 
 Each side of a frustum of a regular pyramid is a trapezoid, 
 as ABCD. The slant-height HP, (Def. 
 11.) though it is oblique to the base of 
 the solid, is perpendicular to the line AB. 
 The area of the trapezoid is equal to the 
 product of half this perpendicular into 
 the sum of the parallel sides AB and DC. 
 (Art. 12.) Therefore the area of all the 
 equal trapezoids which form the lateral 
 surface of the frustum, is equal to the 
 
 * Thomson's Legendre, 13, 7. Cor. 
 
48 
 
 MENSURATION OF SOLIDS. 
 
 product of half the slant-height into the sum of the pen- 
 meters of the ends. 
 
 Ex. If the slant-height of a frustum of a regular octag- 
 onal pyramid be 42 feet, the sides of one end 5 feet each, and 
 the sides of the other end 3 feet each ; what is the lateral 
 surface ? Ans. 1344 square feet. 
 
 52. If the slant-height be not given, it may be obtained 
 from the perpendicular 
 
 height and the dimensions 
 of the two ends. Let GD 
 be the slant-height of the 
 frustum CDGL, RN or GP L/ 
 
 V 
 
 \ 
 
 \ 
 
 17 
 
 G 
 
 the perpendicular height, f 
 ND and RG the radii of the / 
 circles inscribed in the pe- / 
 
 
 \ 
 
 rimeters of the two ends. c 
 Then, PD is the difference of the two radii 
 
 i 
 
 P D 
 
 And the slant-height GD=V(GP a +PD a ). 
 
 Ex. If the perpendicular height 
 of a frustum of a regular hexagonal 
 pyramid be 24, the sides of one end 
 13 each, and the sides of the other 
 end 8 each ; what is the whole sur- 
 face? 
 
 V(BC 9 BP a )=CP, that is, V(13 a 6.5 3 )=11.258 
 And V8 8 4 a = 6.928 
 The difference of the two radii is, therefore 4.33 
 
 The slant-height=V(24 a +4.33 a )=24.3875. 
 The lateral surface is 1536.4 
 
 And the whole surface, 2 1 4 1 . Y5 . 
 
MENSURATION OF SOLIDS. 49 
 
 The height of the whole pyramid may be calculated from 
 the dimensions of the frustum. Let VN (Fig. 17.) be the 
 height of the pyramid, RN or GP the height of the frus- 
 tum, ND and RG the radii of the circles inscribed in the 
 perimeters of the ends of the frustum. 
 
 Then, in the similar triangles GPD and VND, 
 DP : GP : : DN : VK 
 
 The height of the frustum subtracted from VN, gives VR 
 the height of the small pyramid VLG. The solidity and 
 lateral surface of the frustum may then be found, by sub- 
 tracting from the whole pyramid, the part which is above 
 the cutting plane. This method may serve to verify the cal- 
 culations which are made by the rules in Arts. 50 and 51. 
 
 Ex. If one end of the frustum CDGL (Fig. 17.) be 90 feet 
 square, the other end 60 feet square, and the height RN 36 
 feet ; what is the height of the whole pyramid VCD : and 
 what are the solidity and lateral surface of the frustum ? 
 
 DP=DNGR=45 30=15. And, GP=RN=36. 
 
 Then, 15 : 36 : : 45 : 108=VN, the height of the whole 
 pyramid. 
 
 And, 108 36=72=VR, the height of the part VLG." 
 
 The solidity of the large pyramid is 291600 (Art. 48.) 
 of the small pyramid 86400 
 
 of the frustum CDGL 205200 
 
 The lateral surface of the large pyramid is 21060 (Art. 49.) 
 of the small pyramid 9360 
 
 of the frustum 11700 
 
MENSURATION OF SOLIDS. 
 
 PROBLEM VII. 
 To find the SOLIDITY of a WEDGE. 
 
 54. ADD THE LENGTH OF THE EDGE TO TWICE THE 
 LENGTH OF THE BASE, AND MULTIPLY THE SUM BY OF THE 
 
 PRODUCT OF THE HEIGHT OF THE WEDGE AND THE BREADTH 
 OF THE BASE. 
 
 Let L = AB the 
 
 length of the base. 
 Let Z=GH the length 
 
 of the edge. 
 Let 6=BC the breadth 
 
 of the base. 
 Let A=PG the height 
 
 of the wedge. 
 
 Then, L /=AB GH=AM. 
 
 If the length of the base and the edge be equal, as BM 
 and GH, the wedge MBHG is half a parallelepiped of the 
 same base and height. And the solidity (Art. 43.) is equal 
 to half the product of the height, into the length and breadth 
 of the base ; that is \ Ihl. 
 
 If the length of the base be greater than that of the edge, 
 as -ABGH ; let a section be made by the plane GMN, par- 
 allel to HBC. This will divide the whole wedge into two 
 parts MBHG and AMG. The latter is a pyramid, whose 
 solidity (Art. 48.) is i bhx(Ll) 
 
 The solidity of the parts together, is, therefore, 
 
 If the length of the base be less than that of the edge, it 
 is evident that the pyramid is to be subtracted from half the 
 parallelepiped, which is equal in height and breadth to the 
 wedge, and equal in length to the edge. 
 
MENSURATION OF SOLIDS. 
 
 The solidity of the wedge is, therefore, 
 
 51 
 
 Ex. 1. If the base of a wedge be 35 by 15, the edge 55, 
 and the perpendicular height 12.4 ; what is the solidity? 
 
 Ans. 
 
 2. If the base of a wedge be 27 by 8, the edge 36, and 
 the perpendicular height 42 ; what is the solidity ? 
 
 Ans. 5040. 
 
 PROBLEM VIII. 
 To find the SOLIDITY of a rectangular PRISMOID. 
 
 55. To THE AREAS OF THE TWO ENDS, ADD FOUR TIMES 
 THE AREA OF A PARALLEL SECTION EQUALLY DISTANT FROM 
 THE ENDS, AND MULTIPLY THE SUM BY OF THE HEIGHT. 
 
 Let L and B be the length and 
 
 breadth of one end, 
 Let I and b be the length and breadth 
 
 of the other end, 
 Let M and m be the length and 
 
 breadth of the section in the middle. 
 And h be the height of the pris- 
 
 moid. 
 
 The solid may be divided into two wedges whose bases are 
 the ends of the prismofd, and whose edges are L and /. The 
 solidity of the whole, by the preceding article is, 
 
 \ 
 
 As M is equally distant from L and I, 
 
 2M=L+Z, 
 
 [bL+lb. 
 
52 MENSURATION OF SOLIDS. 
 
 Substituting 4 Mm for its value, in the preceding ex- 
 pression for the solidity, we have 
 
 That is, the solidity of the prismoid is equal to of the 
 height, multiplied into the areas of the two ends, and 4 
 times the area of the section in the middle. 
 
 This rule may be applied to prismoids of other forms. 
 For, whatever be the figure of the two ends, there may be 
 drawn in each, such a number of small rectangles, that the 
 sum of them shall differ less, than by any given quantity, 
 from the figure in which they are contained. And the solids 
 between these rectangles will be rectangular prismoids. 
 
 Ex. 1. If one end of a rectangular prismoid be 44 feet by 
 23, the other end 36 by 21, and the perpendicular height 
 72 ; what is the solidity ? 
 
 The area of the larger end =44X23 = 1012 
 of the smaller end =36X21= 756 
 of the middle section =40X22= 880 
 And the solidity=(1012-l-756-f4X880)x 12 = 63456 feet. 
 
 2. What is the solidity of a stick of hewn timber, whose 
 ends are 30 inches by 27, and 24 by 18, and whose length 
 is 48 feet? Ans. 204 feet. 
 
 Other solids not treated of in this section, if they be 
 bounded by plane surfaces, may be measured by supposing 
 them to be divided into prisms, pyramids, and wedges. And, 
 indeed, every such solid may be considered as made up of 
 triangular pyramids. 
 
MENSURATION OF REGULAR SOLIDS. 53 
 
 THE FIVE REGULAR SOLIDS, 
 
 56. A SOLID IS SAID TO BE REGULAR, WHEN ALL ITS 
 BOLID ANGLES ARE EQUAL, AND ALL ITS SIDES ARE EQUAL 
 AND REGULAR POLYGONS. 
 
 The following figures are of this description ; 
 1. The Tetraedron, 
 
 2. The Hexaedron or cube, 
 
 3. The Octaedron, 
 
 4. The Dodecaedron, 
 
 5. The Icosaedron, 
 
 whose 
 sides are 
 
 four triangles ; 
 six squares ; 
 eight triangles ; 
 twelve pentagons; 
 L twenty triangles.* 
 
 Besides these five there can be no other regular solids. 
 The only plane figures which can form such solids, are tri- 
 angles, squares, and pentagons. For the plane angles which 
 contain any solid angle, are together less than four right an- 
 gles or 360. (Sup. Euc. 21, 2.) And the least number 
 which can form a solid angle is three. (Sup. Euc. Def. 8, 2.) 
 If they are angles of equilateral triangles, each is, 60. The 
 sum of three of them is 180, of four 240, of Jive 300, and 
 of six 360. The latter number is too great for a solid angle. 
 
 The angles of squares are 90 each. The sum of three of 
 these is 270, of four 360, and of any other greater number, 
 still more. 
 
 The angles of regular pentagons are 108 each. The sum 
 of three of them is 324 : of four, or any other greater num- 
 ber, more than 360. The angles of all other regular poly- 
 gons are still greater. 
 
 In a regular solid, then, each solid angle must be con- 
 tained by three, four, or five equilateral triangles, by three 
 squares, or by three regular pentagons. 
 
 * For the geometrical construction of these solids, see Legendre's 
 Geometry ; Appendix to Books VI. and VII., or Thomson's Legendre, 
 p. 214. 
 
54 MENSURATION OF REGULAR SOLIDS. 
 
 57. As the sides of a regular solid are similar and equal, 
 and the angles are also alike ; it is evident that the sides 
 are all equally distant from a central point in the solid. If 
 then, planes be supposed to proceed from the several edges 
 to the centre, they will divide the solid into as many equal 
 pyramids, as it has sides. The base of each pyramid will be 
 one of the sides ; their common vertex will be the central 
 point; and their height will be a perpendicular from the 
 centre to one of the sides. 
 
 PROBLEM IX. 
 To find the SURFACE of a REGULAR SOLID. 
 
 58. MULTIPLY THE AREA OF ONE OF THE SIDES BY THE 
 NUMBER OF SIDES. 
 
 Or, 
 
 MULTIPLY THE SQUARE OF ONE OF THE EDGES, BY THE 
 SURFACE OF A SIMILAR SOLID WHOSE EDGES ARE 1. 
 
 As all the sides are equal, it is evident that the area of 
 one of them, multiplied by the number of sides, will give the 
 area of the whole. 
 
 Or, if a table is prepared, containing the surfaces of the 
 several regular solids whose linear edges are unity ; this may 
 be used for other regular solids, upon the principle, that the 
 areas of similar polygons are as the squares of their homolo- 
 gous sides. (Euc. 20. 6.)* Such a table is easily formed, by 
 multiplying the area of one of the sides, as given in Art. 1 7, 
 by the number of sides. Thus, the area of an equilateral 
 triangle whose side is 1, is 0.4330127. Therefore, the sur- 
 face 
 
 * Thomson's Legendre, 27. 4. 
 
MENSURATION OF REGULAR SOLIDS. 55 
 
 Of a regular tetraedron =.4330127x4 =1.7320508. 
 Of a regular octaedron =.4330127X8 =3.4641016. 
 Of a regular icosaedron =.4330127X20=8.6602540. 
 
 See the table in the following article. 
 
 Ex. 1. What is the surface of a regular dodecaedron whose 
 edges are each 25 inches ? 
 
 The area of one of the sides is 1075.3 
 And the surface of the whole solid =1075.3X12=12903.6. 
 
 2. What is the surface of a regular icosaedron whose 
 edges are each 102? Ans. 90101.3. 
 
 PROBLEM X. 
 
 To find the SOLIDITY of a REGULAR SOLID. 
 59. MULTIPLY THE SURFACE BY -J- OF THE PERPENDICULAB 
 
 DISTANCE FROM THE CENTRE TO ONE OF THE SIDES. 
 
 Or, 
 
 MULTIPLY THE CUBE OF ONE OF THE EDGES, BY THE 
 SOLIDITY OF A SIMILAR SOLID WHOSE EDGES ARE 1. 
 
 As the solid is made up of a number of equal pyramids, 
 whose bases are the sides, and whose height is the perpendic- 
 ular distance of the sides from the centre (Art. 57.) ; the 
 solidity of the whole must be equal to the areas of all the 
 sides multiplied into - of this perpendicular. (Art. 48.) 
 
 If the contents of the several regular solids whose edges 
 are 1, be inserted in a table, this may be used to measure 
 other similar solids. For two similar regular solids contain 
 the same number of similar pyramids ; and these are to each 
 other as the cubes of their linear sides or edges. (Sup. Euc. 
 15. 3. Cor. 3.)* 
 
 Thomson's Legendre, 20. 7. 
 
MENSURATION OS 1 THE CYLINDER. 
 
 A TABLE OF REGULAR SOLIDS WHOSE EDGES ARE 1. 
 
 Names. JNo. of sides.) Surfaces. 
 
 Solidities. 
 
 Tetraedron 
 Hexaedron 
 Octaedron 
 Dodecaedron 
 Icosaedron 
 
 4 
 6 
 
 8 
 12 
 20 
 
 1.7320508 
 6.0000000 
 3.4641016 
 
 20.6457288 
 8.6602540 
 
 0.1178513 
 1.0000000 
 0.4714045 
 7.6631189 
 2.1816950 
 
 For the method of calculating the last column of this table, 
 see Button's Mensuration, Part. III. Sec. 2. 
 
 Ex. What is the solidity of a regular octaedron whose 
 edges are each 32 inches ? Ans. 15447 inches. 
 
 SECTION IV. 
 
 THE CYLINDER, CONE, AND SPHERE. 
 
 ART. 61. DEFINITION I. A right cylinder is a solid de- 
 scribed by the revolution of a rectangle about one of its 
 sides. The ends or bases are evidently equal and parallel 
 circles. And the axis, which is a line passing through the 
 middle of the cylinder, is perpendicular to the bases. 
 
 The ends of an oblique cylinder are also equal and paral- 
 lej circles ; but they are not perpendicular to the axis. The 
 height of a cylinder is the perpendicular distance from one 
 base to the plane of the other. In a right cylinder, it is the 
 length of the axis. 
 
 II. A right cone is a solid described by the revolution of 
 a right angled triangle about one of the sides which contain 
 the right angle. The base is a circle, and is perpendicular to 
 
MENSURATION OF THE CYLINDER. 
 
 the axis, which proceeds from the middle of the base to the 
 vertex. 
 
 The base of an oblique cone is also a circle, but is not per- 
 pendicular to the axis. The height of a cone is the perpen- 
 dicular distance from the vertex to the plane of the base. In 
 a right cone, it is the length of the axis. The slant-height 
 of a right cone is the distance from the vertex to the circum- 
 ference of the base. 
 
 III. A. frustum of a cone is a portion cut off by a plane 
 parallel to the base. The height of the 
 
 frustum is the perpendicular distance of 
 the two ends. The slant-height of a 
 frustum of a right cone, is the distance 
 between the peripheries of the two 
 ends, measured on the outside of the 
 solid ; as AD. 
 
 IV. A sphere or globe is a solid which 
 has a centre equally distant from every 
 
 part of the surface. It may be described by the revolution 
 of a semicircle about a diameter. A radius of the sphere is 
 a line drawn from the centre to any part of the surface. A 
 diameter is a line passing through the centre, and terminated 
 at both ends by the surface. The circumference is the same 
 as the circumference of a circle whose plane passes through 
 the centre of the sphere. Such a circle is called a great 
 circle. 
 
 V. A segment of a sphere is a part cut off by any plane. 
 The height of the segment is a per- 
 pendicular from the middle of the 
 
 base to the convex surface, as LB. 
 
 VI. A spherical zone or frustum is 
 a part of the sphere included be- 
 tween two parallel planes. It is 
 called the middle zone, if the planes 
 are equally distant from the centre. 
 
 3* 
 
58 MENSURATION OF THE CYLINDER. 
 
 The height of a zone is the distance of the two planes, as 
 LR* 
 
 VII. A spherical sector is a solid produced by a circular 
 sector, revolving in the same manner 
 
 as the semicircle which describes the 
 whole sphere. Thus a spherical sec- 
 tor is described by the circular sec- 
 tor AGP or GCE revolving on the 
 axis CP. 
 
 VIII. A solid described by the 
 revolution of any figure about a fixed 
 axis, is called a solid of revolution. 
 
 PROBLEM I. 
 
 To find the CONVEX SURFACE of a RIGHT CYLINDER. 
 62. MULTIPLY THE LENGTH INTO THE CIRCUMFERENCE OF 
 
 THE BASE. 
 
 If a right cylinder be covered with a thin substance like 
 paper, which can be spread out into a plane ; it is evident 
 that the plane will be a parallelogram, whose length and 
 breadth will be equal to the length and circumference of the 
 cylinder. The area must, therefore, be equal to the length 
 multiplied into the circumference. (Art. 4.) 
 
 Ex. 1. What is the convex surface of a right cylinder 
 which is 42 feet long, and 15 inches in diameter? 
 
 Ans. 42X1-25X3.14159 = 164.933 sq. feet. 
 
 2. What is the whole surface of a right cylinder, which 
 is 2 feet in diameter and 36 feet long ? 
 
 * According to some writers, a spherical segment is either a solid 
 which is cut off from the sphere by a single plane, or one which is in- 
 cluded between two planes : and a zone is the surface of either of these. 
 In this sense, the term zone is commonly used in geography. 
 
MENSURATION OF THE CYLINDER. 59 
 
 The convex surface is 226.1945 
 
 The area of the two ends (Art. 30.) is 6.2832 
 The whole surface is 232.4777 
 
 3. What is the whole surface of a right cylinder whose 
 axis is 82, and circumference 71 ? Ans. 6624.32. 
 
 63. It will be observed that the rules for the prism and 
 pyramid in the preceding section, are substantially the same, 
 as the rules for the cylinder and cone in this. There may be 
 some advantage, however, in considering the latter by them- 
 selves. 
 
 In the base of a cylinder, there may be inscribed a poly- 
 gon, which shall differ from it less than by any given space. 
 (Sup. Euc. 6. 1. Cor.)* If the polygon be the base of a 
 prism, of the same height as the cylinder, the two solids 
 may differ less than by any given quantity. In the same 
 manner, the base of a pyramid may be a polygon of so many 
 sides, as to differ less than by any given quantity, from the 
 base of a cone in which it is inscribed. A cylinder is there- 
 fore considered, by many writers, as a prism of an infinite 
 number of sides ; and a cone, as a pyramid of an infinite 
 number of sides. (For the meaning of the term " infinite," 
 when used in the mathematical sense, see Alg. Sec. XV.) 
 
 PROBLEM II. 
 To find the SOLIDITY of a CYLINDER. 
 
 64. MULTIPLY THE AREA OF THE BASE BY THE HEIGHT. 
 
 The solidity of a parallelopiped is equal to the product of 
 the base into the perpendicular altitude. (Art. 43.) And a 
 parallelopiped and a cylinder which have equal bases and 
 altitudes are equal to each other. (Sup. Euc. 17. 3.)t 
 
 Thomson's Legendre, 9. 5. 
 
60 
 
 MENSURATION OF THE CYLINDER. 
 
 Ex. 1. What is the solidity of a cylinder, whose height is 
 121, and diameter 45.2 ? 
 
 Ans. 45.2 a X. '7854X121 = 194156.6. 
 
 2. What is the solidity of a cylinder, whose height is 424, 
 and circumference 213 ? Ans. 1530837. 
 
 3. If the side AC of an oblique 
 cylinder be 27, and the area of the base 
 32.61, and if the side make an angle 
 of 62 44' with the base, what is the 
 solidity ? 
 
 E : AC : : sin A : BC=24 the per- 
 pendicular height. 
 
 And the solidity is 782.64. 
 
 4. The Winchester bushel is a hollow cylinder, 18^- inches 
 in diameter, and 8 inches deep. What is its capacity ? 
 The area of the base=(18.5) 2 X- 7853982 = 268.8025. 
 And the capacity is 2150.42 cubic inches. See the 
 table in Art. 42. 
 
 PROBLEM III. 
 To find the CONVEX SURFACE of a RIGHT CONE. 
 
 65. MULTIPLY HALF THE SLANT-HEIGHT INTO THE CIR- 
 CUMFERENCE OF THE BASE. 
 
 If the convex surface of a right cone be spread out into a 
 plane, it will evidently form a sector of a circle whose radius 
 is equal to the slant-height of the cone. But the area of the 
 sector is equal to the product of half the radius into the 
 length of the arc. (Art. 34.) Or if the cone be considered 
 as a pyramid of an infinite number of sides, its lateral sur- 
 
MENSURATION OF THE CONE. 61 
 
 face is equal to the product of half the slant-height into the 
 perimeter of the base. (Art. 49.) 
 
 Ex. 1. If the slant-height of a right cone be 82, and the 
 diameter of the base 24, what is the convex surface ? 
 
 Ans. 41X24X3.14159 = 3091.3 square feet. 
 
 2. If the axis of a right cone be 48, and the diameter of 
 the base 72, what is the whole surface ? 
 
 The slant-height = V(36 Q +48 a )=60. (Euc. 47. 1.) 
 
 The convex surface is 6786 
 
 The area of the base 4071.6 
 
 And the whole surface 10857.6 
 
 3. If the axis of a right cone be 16, and the circumfer- 
 ence of the base 75.4 ; what is the whole surface ? 
 
 Ans. 1206.4. 
 
 PROBLEM IV. 
 To find the SOLIDITY of a CONE. 
 
 66. MULTIPLY THE AREA OF THE BASE INTO OF THE 
 HEIGHT. 
 
 The solidity of a cylinder is equal to the product of the 
 base into the perpendicular height. (Art. 64.) And if a cone 
 and a cylinder have the same base and altitude, the cone is 
 $ of the cylinder. (Sup. Euc. 18. 3.)* Or if a cone be con- 
 sidered as a pyramid of an infinite number of sides, the so- 
 lidity is equal to the product of the base into -J of the height, 
 by Art. 48. 
 
 Ex. 1. What is the solidity of a right cone whose height 
 is 663, and the diameter of whose base is 101 ? 
 
 Ans. loT a X.7854x221 = 
 
 * Thomson's Legendre, 4. 8. Cor. 
 
62 MENSURATION OF THE CONE. 
 
 2. If the axis of an oblique cone be 738, and make an 
 angle of 30 with the plane of the base ; and if the circum- 
 ference of the base be 355, what is the solidity ? 
 
 Ans. 1233536. 
 
 PROBLEM V. 
 
 To find the CONVEX SURFACE of a FRUSTUM of a right cone. 
 67. MULTIPLY HALF THE SLANT-HEIGHT BY THE SUM OP 
 
 THE PERIPHERIES OF THE TWO ENDS. 
 
 This is the rule for a frustum of a pyramid ; (Art. 51.) 
 and is equally applicable to a frustum of a cone, if a cone be 
 considered as a pyramid of an infinite number of sides. 
 (Art. 63.) 
 
 Or thus, 
 
 Let the sector ABV represent the A i 
 convex surface of a right cone, (Art. 
 65.) and DCV the surface of a portion 
 of the cone, cut off by a plane parallel 
 to the base. Then will ABCD be the 
 surface of the frustum. 
 
 Let AB=a, DC =6, VD=c?, AD=h. $ 
 
 Then the area ASV=^ax(h-\-d)=^ah+^ad. (Art. 34.) 
 
 And the area VCV=bd. 
 Subtracting the one from the other, 
 
 The area A.EDC=iah+ad $bd. 
 
 But d : d+h ::b : a. (Sup. Euc. 8. 1.)* Therefore iad 
 The surface of the frustum then, is equal to 
 
 ' . i. .i-n ., ... 
 
 * Thomson's Legendre, 10. 5. Cor. 
 
MENSURATION OF THE CONE. 63 
 
 Cor. The surface of the frustum is equal to the product 
 of the slant-height into the circumference of a circle which 
 is equally distant from the two ends. Thus, the surface 
 ABCD is equal to the product of AD into MN. For MIST is 
 equal to half the sum of AB and DC. 
 
 Ex. 1. What is the convex surface of a frustum of a right 
 cone, if the diameters of the two ends be 44 and 33, and 
 the slant-height 84 ? Ans. 10159.8. 
 
 2. If the perpendicular height of a frustum of a right 
 cone be 24, and the diameters of the two ends 80 and 44, 
 what is the whole surface ? 
 
 Half the difference of the diameters is 18. 
 
 And V 18 a +24 a =30, the slant-height, (Art. 52.) 
 The convex surface of the frustum is 5843 
 The sum of the areas of the two ends is 6547 
 
 And the whole surface is 12390 
 
 PROBLEM VI. 
 To find the SOLIDITY of a FRUSTUM of a cone. 
 
 68. ADD TOGETHER THE AREAS OF THE TWO ENDS, AND 
 THE SQUARE ROOT OF THE PRODUCT OF THESE AREAS ', AND 
 MULTIPLY THE SUM BY OF THE PERPENDICULAR HEIGHT. 
 
 This rule, which was given for the frustum of a pyramid, 
 (Art. 50.) is equally applicable to the frustum of a cone ; be- 
 cause a cone and a pyramid which have equal bases and alti- 
 tudes are equal to each other. 
 
 Ex. 1. What is the solidity of a mast which is 72 feet 
 long, 2 feet in diameter at one end, and 18 inches at the 
 other? Ans. 174.36 cubic feet. 
 
64 
 
 MENSURATION OF THE SPHERE. 
 
 2. What is the capacity of a conical cistern which is 9 
 feet deep, 4 feet in diameter at the bottom, and 3 feet at the 
 top ? Ans. 87.18 cubic feet^652.15 wine gallons. 
 
 3. How many gallons of ale can be put into a vat in the 
 form of a conic frustum, if the larger diameter be 7 feet, the 
 smaller diameter 6 feet, and the depth 8 feet ? 
 
 PROBLEM VII. 
 
 To find the SURFACE of a SPHERE. 
 69. MULTIPLY THE DIAMETER BY THE CIRCUMFERENCE. 
 
 Let a hemisphere be described by the quadrant CPD, 
 revolving on the line CD. Let 
 AB be the side of a regular poly- 
 gon inscribed in the circle of 
 which DBF is an arc. Draw AO 
 and BN perpendicular to CD, 
 and BH perpendicular to AO. 
 Extend AB till it meets CD con- 
 tinued. The triangle AOV, re- 
 volving on 0V as an axis, will 
 describe a right cone. (Defin. 2.) 
 AB will be the slant-height of a 
 frustum of this cone extending 
 from AO to BN. From G the middle of AB, draw GM 
 parallel to AO. The surface of the frustum described by 
 AB. (Art. 67. Cor.) is equal to 
 
 AKXcirc GM.* 
 
 From the centre C draw CG, which will be perpendicular 
 to AB, (Euc. 3. 3.) and the radius of a circle inscribed in 
 
 * By circ GM is meant the circumference of a circle the radius of 
 which is GM. 
 
MENSURATION OF THE SPHERE. 60 
 
 the polygon. The triangles ABH and CGM are similar, be- 
 cause the sides are perpendicular, each to each. Therefore, 
 
 HB or ON : AB : : GM : GO : : circ GM : circ GO. 
 
 So that ON X circ GC=ABx>c GM, that is, the sur- 
 face of the frustum is equal to the product of ON the per- 
 pendicular height, into circ GC, the perpendicular distance 
 from the centre of the polygon to one of the sides. 
 
 In the same manner it may be proved, that the surfaces 
 produced by the revolution of the lines BD and AP about 
 the axis DC, are equal to 
 
 NDxcirc GC, and COx^Vc GC. 
 
 The surface of the whole solid, therefore, (Euc. 1.2.) is equal to 
 
 GDxcirc GC. 
 
 The demonstration is applicable to a solid produced by 
 the revolution of a polygon of any number of sides. But a 
 polygon may be supposed which shall differ less than by 
 any given quantity from the circle in which it is inscribed ; 
 (Sup. Euc. 4. 1.)* and in which the perpendicular GC shall 
 differ less than by any given quantity from the radius of the 
 circle. Therefore, the surface of a hemisphere is equal to 
 the product of its radius into the circumference of its base ; 
 and the surface of a sphere is equal to the product of its 
 diameter into its circumference. 
 
 Cor. 1. From this demonstration it follows, that the sur- 
 face of any segment or zone of a sphere is equal to the 
 product of the height of the segment or zone into the cir- 
 cumference of the sphere. The surface of the zone pro- 
 duced by the revolution of the arc AB about ON, is equal 
 to ON x circ CP. And the surface of the segment pro- 
 
 * Thomson's Legendre, 9. 5. 
 
MENSURATION OP THE SPHERE. 
 
 duced by the revolution of BD about DN is equal to 
 eirc CP. 
 
 Cor. 2. The surface of a sphere is equal to four times the 
 area of a circle of the same diameter ; and therefore, the 
 convex surface of a hemisphere is equal to twice the area of 
 its base. For the area of a circle is equal to the product of 
 half the diameter into half the circumference ; (Art. 30.) 
 that is, to the product of the diameter and circumference. 
 
 Cor. 3. The surface of a sphere, or the convex surface 
 of any spherical segment or zone, 
 is equal to that of the circum- 
 scribing cylinder. A hemis- 
 phere described by the revolu- 
 tion of the arc DBF, is cir- 
 cumscribed by a cylinder pro- 
 duced by the revolution of the 
 parallelogram DdCP. The con- 
 vex surface of the cylinder is 
 equal to its height multiplied 
 by its circumference. (Art. 62.) 
 And this is also the surface of 
 the hemisphere. 
 
 So the surface produced by the revolution of AB is equal 
 to that produced by the revolution of ab. And the surface 
 produced by BD is equal to that produced by bd. 
 
 Ex. 1. Considering the earth as a sphere 7930 miles in 
 diameter, how many square miles are there on its surface ? 
 
 Ans. 197,558,500. 
 
 2. If the circumference of the sun be 2,800,000, what is 
 his surface ? Ans. 2,495,547,600,000 sq. miles. 
 
 3. How many square feet of lead will it require, to cover 
 a hemispherical dome whose base is 13 feet across ? 
 
 Ans. 
 
MENSURATION OF THE SPHERE. 67 
 
 PROBLEM VIII. 
 
 To find the SOLIDITY of a SPHERE. 
 YO. 1. MULTIPLY THE CUBE OF THE DIAMETER BY .5230. 
 
 Or, 
 
 2. MULTIPLY THE SQUARE OF THE DIAMETER BY -J- OF THE 
 
 CIRCUMFERENCE. 
 
 Or, 
 
 3. MULTIPLY THE SURFACE BY % OF THE DIAMETER. 
 
 1. A sphere is two-thirds of its circumscribing cylinder. 
 (Sup. Euc. 21. 3.)* The height and diameter of the cylin- 
 der are each equal to the diameter of the sphere. The solid- 
 ity of the cylinder is equal to its height multiplied into the 
 area of its base, (Art. 64.) that is putting D for the diam- 
 eter, 
 
 DxD 8 X.?854 or D 8 X.V854. 
 
 And the solidity of the sphere, being f of this, is 
 D 8 X.5236. 
 
 2. The base of the circumscribing cylinder is equal to half 
 the circumference multiplied into half the diameter ; (Art. 
 80.) that is, if C be put for the circumference, 
 
 -J-CxD ; and the solidity is -J-CxD 1 . 
 
 Therefore, the solidity of the sphere is 
 
 -f ofiCxD'=D'XiC. 
 8. In the last expression, which is the same as CxDx|D, 
 
 * Thomson's Legendre, 12. 8. 
 
68 MENSURATION OF THE SPHERE. 
 
 we may substitute S, the surface, for C X D. (Art. 69.) We 
 then have the solidity of the sphere equal to 
 
 Or, the sphere may be supposed to be filled with small 
 pyramids, standing on the surface of the sphere, and having 
 their common vertex in the centre. The number of these 
 may be such, that the difference between their sum and the 
 sphere shall be less than any given quantity. The solidity 
 of each pyramid is equal to the product of its base into 
 of its height. (Art. 48.) The solidity of the whole, there- 
 fore, is equal to the product of the surface of the sphere 
 into \ of its radius, or of its diameter. 
 
 71. The numbers 3.14159, .7854, .5236, should be made 
 perfectly familiar. The first expresses the ratio of the 
 circumference of a circle to the diameter ; (Art. 23.) the 
 second, the ratio of the area of a circle to the square of the 
 diameter (Art. 30.) ; and the third, the ratio of the solidity 
 of a sphere to the cube of the diameter. The second is -J- 
 of the first, and the third is of the first. 
 
 As these numbers are frequently occurring in mathemat- 
 ical investigations, it is common to represent the first of them 
 by the Greek letter n. According to this notation, 
 
 7r=3.14159, }" 
 
 If D=the diameter, and R=the radius of any circle 01 
 sphere ; 
 
 Then, D=2R D a =4R 2 D 3 =8R 3 . 
 
 . =the 
 
 Or, 27tR J or nR 3 $ the circ. or f^R' $ 
 
 solidity of the sphere. 
 
 Ex 1. What is the solidity of the earth, if it be a sphere 
 7930 miles in diameter ? 
 
 Ans. 261,107,000,000 cubic miles. 
 
MENSURATION OF THE SPHERE. 69 
 
 2. How many wine gallons will fill a hollow sphere 4 feet 
 in diameter ? 
 
 Ans. The capacity is 33.5104 feet=250f gallons. 
 
 3. If the diameter of the moon be 2180 miles, what is its 
 solidity? Ans. 5,424,600,000 miles. 
 
 72. If the solidity of a sphere be given, the diameter may 
 be found by reversing the first rule in the preceding article ; 
 that is, dividing by .5236 and extracting the cube root of the 
 quotient. 
 
 Ex. 1 . What is the diameter of a sphere whose solidity is 
 65.45 cubic feet ? Ans. 5 feet. 
 
 2. What must be the diameter of a globe to contain 16755 
 pounds of water? Ans. 8 feet. 
 
 PROBLEM IX. 
 
 To find the CONVEX SURFACE of a SEGMENT or ZONE of a 
 sphere. 
 
 73. MULTIPLY THE HEIGHT OF THE SEGMENT OR ZONK 
 INTO THE CIRCUMFERENCE OF THE SPHERE. 
 
 For the demonstration of this rule, see Art. 69. 
 
 Ex. 1. If the earth be considered a perfect sphere 7930 
 miles in diameter, and if the polar circle be 23 28' from the 
 pole, how many square miles are there in one of the frigid 
 zones ? 
 
 If PQOE be a meridian on the 
 earth, ADB one of the polar circles, 
 and P the pole ; then the frigid zone 
 is a spherical segment described by 
 the revolution of the arc APB about 
 PD. The angle ACD subtended by 
 the arc AP is 23 28'. And in the 
 right angled triangle ACD, 
 
70 MENSURATION OF THE SPHERE. 
 
 R : AC : : cos ACD : CD=3637. 
 
 Then, CP CD =3965 3637=328=PD the height of 
 the segment. 
 
 And 328X7930X3.14159=8171400 the surface. 
 
 2. If the diameter of the earth be 7930 miles, what is the 
 surface of the torrid zone, extending 
 
 23 28' on each side of the equator ? 
 
 If EQ be the equator, and GH one 
 of the tropics, then the angle ECG is 
 23 28'. And in the right angled 
 triangle GCM, 
 
 R : CG : : sin ECG : GM=CN=1578.9 the height of 
 half the zone. 
 
 The surface of the whole zone is 78669700. 
 
 3. What is the surface of each of the temperate zones ? 
 
 The height DN=CP CN PD = 2058.1* 
 And the surface of the zone is 51273000. 
 
 The surface of the two temperate zones is 102,546,000 
 
 of the two frigid zones 16,342,800 
 
 of the torrid zone 78,669,700 
 
 of the whole globe 197,558,500 
 
 PROBLEM X. 
 To find the SOLIDITY of a spherical SECTOR. 
 
 74. MULTIPLY THE SPHERICAL SURFACE BY % OF THK 
 EAJDIUS OF THE SPHERE. 
 
 The spherical sector produced by the revolution of ACBD 
 
MENSURATION OF THE SPHEBB. 
 
 71 
 
 about CD, may be supposed to be filled 
 
 with small pyramids, standing on the 
 
 spherical surface ADB, and terminating 
 
 in the point C. Their number may be 
 
 so great, that the height of each shall 
 
 differ less than by any given length 
 
 from the radius CD, and the sum of their 
 
 bases shall differ less than by any given .- 
 
 quantity from the surface ABD. The 
 
 solidity of each is equal to the product of its base into % of 
 
 the radius CD. (Art. 48.) Therefore, the solidity of all of 
 
 them, that is, of the sector ADBC, is equal to the product 
 
 of the spherical surface into $ of the radius. 
 
 Ex. Supposing the earth to be a 
 sphere 7930 miles in diameter, and 
 the polar circle ADB to be 23 28' 
 from the pole ; what is the solidity of 
 the spherical sector ACBP ? 
 
 Ans. 10,799,867,000 miles. 
 
 PROBLEM XL 
 To find the SOLIDITY of a spherical SEGMENT. 
 
 75. MULTIPLY HALF THE HEIGHT OF THE SEGMENT INTO 
 THE AREA OF THE BASE, AND THE CUBE OF THE HEIGHT 
 INTO .5236 ; AND ADD THE TWO PRODUCTS. 
 
 As the circular sector AOBC consists of two parts, the 
 segment AOBP and the triangle 
 ABC ; (Art. 35.) so the sphericul 
 sector produced by the revolution of 
 AOC about OC consists of two parts, 
 the segment produced by the revolu- 
 tion of AOP, and the cone produced 
 by the revolution of AGP. If then 
 
MENSURATION OF THE SPHERE. 
 
 the cone be subtracted from the sec- 
 tor, the remainder will be the seg- 
 ment. 
 
 Let CO=R, the radius of the sphere, 
 PB=r, the radius of the base of 
 
 the segment. 
 
 P0=h, the height of the segment, 
 Then PC=R h, the axis of the cone. 
 
 The sector =27rRxfcXiR( Arts. 71, 73, 74.) = 
 The cone^^xi (R A) (Arts. 71, 66.)=in 
 
 9- 
 
 Subtracting the one from the other, 
 
 The segment =$nhR* %ni*R.+\nki*. 
 
 But DOxPO=B0 3 (Trig. 97.*)=PO~ a +PB 2 (Euc. 47. 1.) 
 That is, 2R=#'+r 3 . So that, R=^!!l 
 
 Substituting then, for R and R 2 , then* values, and multi- 
 plying the factors, 
 
 The 
 
 Which, by uniting the terms, becomes 
 
 The first term here is i&X 7 ^ 8 , half the height of the seg- 
 ment multiplied into the area of the base ; (Art. 71.) and the 
 other A'Xi 7 *, the cube of the height multiplied into .5236. 
 
 * Euclid 31, 3, and 8, 6. Cor. 
 
MENSURATION OF THE SPHERE. 78 
 
 If the segment be greater than a hemisphere, as ABD ; 
 the cone ABC must be added to the sector ACBD. 
 
 Let PD=Athe height of the segment, 
 Then PC=h R the axis of the cone. 
 
 The sector ACBD=f7r7*R a 
 The cone=7rr 2 Xi(& R)=i^r 2 ^ir'R 
 Adding them together, we have as before, 
 The segment =$nhR> %nr*R+$nhr 9 . 
 
 Cor. The solidity of a spherical segment is equal to half a 
 cylinder of the same base and height -f- a sphere whose 
 diameter is the height of the segment. For a cylinder is 
 equal to its height multiplied into the area of its base ; and 
 a sphere is equal to the cube of its diameter multiplied by 
 .5236. 
 
 Thus, if Oy be half Ox, the spher- 
 ical segment produced by the revo- 
 lution, of Oxt is equal to the cylin- 
 der produced by tvyx -f- the sphere 
 produced by Oyxz ; supposing each 
 to revolve on the line Ox. 
 
 Ex. 1. If the height of a spherical segment be 8 feet, and 
 the diameter of its base 25 feet ; what is the solidity ? 
 
 Ans. (25) a X.'7854X4+8 8 X.5236=2231.58 feet. 
 
 2. If the earth be a sphere 7930 miles in diameter, and the 
 polar circle 23 28' from the pole, what is the solidity of 
 one of the frigid zones ? Ans. 1,303,000,000 miles. 
 
 4 
 
MENSURATION OF THE SPHERE. 
 
 PROBLEM XII. 
 To find the SOLIDITY of a spherical ZONE or frustum. 
 
 76. FROM THE SOLIDITY OF THE WHOLE SPHERE, SUB- 
 TRACT THE TWO SEGMENTS ON THE SIDES OF THE ZONE. 
 
 Or, 
 
 ADD TOGETHER THE SQUARES OF THE RADII OF THE TWO 
 ENDS, AND -J- THE SQUARE OF THEIR DISTANCE ; AND MULTIPLY 
 THE SUM BY THREE TIMES THIS DISTANCC, AND -THE PRODUCT 
 BY .5236. 
 
 If from the whole sphere, there 
 be taken the two segments ABP and 
 GHO, there will remain the zone 
 or frustum ABGH. 
 
 Or, the zone ABGH is equal to 
 the difference between the segments 
 GHP and ABP. 
 
 Let NP=H 
 
 TT , 
 
 ~~ J the heights of the two segments, 
 the radii of their bases. 
 
 AD=r 
 
 DN=d=H h the distance of the two bases, or the 
 height of the zone. 
 
 Then the larger segment=7iHR 2 +i-7iH 3 ) / Arfc * 5 ^ 
 And the smaller segment=^7rAr a -|-^-7r A 3 ( 
 
 Therefore the zone ABGH=i-n; (3HR 3 +H 3 3r a h*) 
 By the properties of the circle, (Euc. 35, 3.) 
 ONxH=R 2 . Therefore, (ON+H)xH=R 2 +H a 
 Or, OP=?L+51 
 
MENSURATION OF THE SPHERE. 75 
 
 In the same manner, OP= 21 
 
 h 
 
 Therefore, 3Hx (r 2 +^)=3Ax(R a -f H*.) 
 Or, 3Hr a +3H/i 2 - 3AR a 3MF = 0. (Alg. 178.) 
 
 To reduce the expression for the solidity of the zone to 
 the required form, without altering its value, let these terms 
 be added to it : and it will become 
 
 f7(3HR a +3Hr' 3AR a 
 Which is equal to 
 
 in X 3(H h) X (R 3 +r a -H (H A) 2 ) 
 
 Or, as in equals .5236 (Art. 71.) and H h equals d, 
 
 The zone=.5236X3rfx(R a +r 2 +irf a .) 
 
 Ex. 1. If the diameter of one end of a spherical zone is 
 24 feet, the diameter of the other end 20 feet, and the dis- 
 tance of the two ends, or the height of the zone 4 feet ; 
 what is the solidity? Ans. 1566.6 feet. 
 
 2. If the earth be a sphere 7930 miles in diameter, and 
 the obliquity of the ecliptic 23 28' ; what is the solidity of 
 one of the temperate zones ? 
 
 Ans. 55,390,500,000 miles. 
 
 3. What is the solidity of the torrid zone ? 
 
 Ans. 147,720,000,000 miles. 
 
 The solidity of the two temperate zones is 110,781,000,000 
 of the two frigid zones 2,606,000,000 
 
 of the torrid zone 147,720,000,000 
 
 of the whole globe 261,107,000,000 
 
 4. What is the convex surface of a spherical zone, whose 
 breadth is 4 feet, on a sphere of 25 feet diameter ? 
 
Y6 MENSURATION OF SOLIDS. 
 
 5. What is the solidity of a spherical segment, whose 
 height is 18 feet, and the diameter of its base 40 feet ? . 
 
 PROMISCUOUS EXAMPLES OF SOLIDS. 
 
 Ex. 1. How much water can be put into a cubical vessel 
 three feet deep, which has been previously filled with cannon 
 balls of the same size, 2, 4, 6, or 9 inches in diameter, regu- 
 larly arranged in tiers, one directly above another ? 
 
 Ans. 96 wine gallons. 
 
 2. If a cone or pyramid, whose height is three feet, be 
 divided into three equal portions, by sections parallel to the 
 base ; what will be the heights of the several parts ? 
 
 Ans. 24.961, 6.488, and 4.551 ihches. 
 
 3. What is the solidity of the greatest square prism which 
 can be cut from a cylindrical stick of timber, 2 feet 6 
 inches in diameter and 56 feet long ?* 
 
 Ans. 175 cubic feet. 
 
 4. How many such globes as the earth are equal in bulk 
 to the sun; if the former is 7930 miles in diameter, and the 
 latter 890,000 ? Ans. 1,413,678. 
 
 * The common rule for measuring round limber is to multiply the 
 square of the quarter-girt by the length. The quarter-girt is one-fourth 
 of the circumference. This method does not give the whole solidity. It 
 makes an allowance of about one-fifth, for waste in hewing, bark, &c. 
 The solidity of a cylinder is equal to the product of the height into the 
 area of the base. 
 
 If C=the circumference, and rr=3.14159, then (Art. 31.) 
 
 C2 / C \ 2 / C \ 2 
 The area of the base= -= (_) = (^) 
 
 If then the circumference were divided by 3.545, instead of 4, and the 
 quotient squared, the area of the base would be correctly found. See 
 noteB. 
 
MENSURATION OF SOLIDS. 77 
 
 6. How many cubic feet of wall are there in a conical 
 tower 66 feet high, if the diameter of the base be 20 feet 
 from outside to outside, and the diameter of the top 8 feet ; 
 the thickness of the wall being 4 feet at the bottom, and de- 
 creasing regularly, so as to be only two feet at the top ? 
 
 Ans. 7188. 
 
 6. If a metallic globe filled with wine, which cost as much 
 at 5 dollars a gallon, as the globe itself at 20 cents for every 
 square inch of its surface ; what is the diameter of the globe ? 
 
 Ans. 55.44 inches. 
 
 7. If the circumference of the earth be 25,000 miles, 
 what must be the diameter of a metallic globe, which, when 
 drawn into a wire / - of an inch in diameter, would reach 
 round the earth ? Ans. 15 feet and 1 inch. 
 
 8. If a conical cistern be 3 feet deep, 7, feet in diameter 
 at the bottom, and 5 feet at the top ; what will be the depth 
 of a fluid occupying half its capacity ? 
 
 Ans. 14.535 inches. 
 
 9. If a globe 20 inches in diameter, be perforated by a 
 cylinder 16 inches in diameter, the axis of the latter passing 
 through the centre of the former ; what part of the solidity, 
 and the surface of the globe, will be cut aWay by the cyl- 
 inder? 
 
 Ans. 3284 inches of the solidity, and 502,655 of the surface. 
 
 10. What is the solidity of the greatest cube which can 
 be cut from a sphere three feet in diameter ? 
 
 Ans. 5i feet. 
 
 11. What is the solidity of a conic frustum, the altitude 
 of which is 36 feet, the greater diameter 16, and the lesser 
 diameter 8 ? 
 
 12. What is the solidity of a spherical segment 4 feet 
 high, cut from a sphere 16 feet in diameter ? 
 
78 ISOPERIMETRf. 
 
 SECTION V. 
 
 ISOPERIMETRY. 
 
 ART. 77. It is often necessary to compare a number of 
 different figures or solids, for the purpose of ascertaining 
 which has the greatest area, within a given perimeter, or the 
 greatest capacity under a given surface. We may have oc- 
 casion to determine, for instance, what must be the form of 
 a fort, to contain a given number of troops, with the least 
 extent of wall ; or what the shape of a metallic pipe to con- 
 vey a given portion of water, or of a cistern to hold a given 
 quantity of liquor, with the least expense of materials. 
 
 78. Figures which have equal perimeters are called Iso- 
 perimeters. When a quantity is greater than any other of the 
 same class, it is called a maximum. A multitude of straight 
 lines, of different lengths, may be drawn within a circle. 
 But among them all, the diameter is a maximum. Of all 
 sines of angles, which can be drawn in a circle, the sine of 
 90 is a maximum. 
 
 When a quantity is less than any other of the same class, 
 it is called a minimum. Thus, of all straight lines drawn 
 from a given point to a given straight line, that which is per- 
 pendicular to the given line is a minimum. Of all straight 
 lines drawn from a given point in a circle, to the circumfer- 
 ence, the maximum and the minimum are the two parts of 
 the diameter which pass through that point. (Euc. 7, 3.) 
 
 In isoperimetry, the object is to determine, on the one 
 hand, in what cases the area is a maximum, within a given 
 perimeter ; or the capacity a maximum, within a given sur- 
 face : and .on the other hand, in what cases the perimeter is 
 
ISOPERIMETRY. 79 
 
 a minimum for a given area, or the surface a minimum, for a 
 given capacity. 
 
 PROPOSITION I. 
 
 79. An ISOSCELES TRIANGLE has a greater area than any 
 scalene triangle, of equal base and perimeter. 
 
 If ABC be an isosceles trian- 
 gle whose equal sides are AC and D/ 
 BC ; and if ABC' be a scalene tri- 
 angle on the same base AB, and 
 having AC' + BC' = AC+BC; 
 then the area of ABC is greater 
 than that of ABC'. 
 
 Let perpendiculars be raised 
 from each end of the base, extend 
 AC to D, make C'D' equal to AC', join BD, and draw CH 
 and C'H' parallel to AB. 
 
 As the angle CAB= ABC, (Euc. 5, 1.) and ABD is a right 
 angle, ABC+CBD=CAB+CDB=ABC+CDB. Therefore 
 CBD=CDB, so that CD=CB ; and by construction, C'D'= 
 AC'. The perpendiculars of the equal right angled triangles 
 CHD and CHB are equal ; therefore, BH=BD. In the 
 same manner, AH'=iAD'. The line AD=AC+BC=AC / 
 +BC'=D'C'4-BC'. But D'C'+BC'>BD'. (Euc. 20, 1.) 
 Therefore, AD>BD' ; BD>AD', (Euc. 47, 1.) and * BD> 
 \ AD'. But iBD, or BH, is the height of the isosceles tri- 
 angle ; (Art. 1.) and AD' or AH', the height of the scalene 
 triangle ; and the areas of two triangles which have the same 
 base are as their heights. (Art. 8.) Therefore the area of 
 ABC is greater than that of ABC'. Among all triangles, 
 then, of a given perimeter, and upon a given base, the isos 
 celes triangle is a maximum. 
 
 Cor. The isosceles triangle has a less perimeter than any 
 scalene triangle of the same base and area. The triangle 
 
80 
 
 ISOPERIMETRY. 
 
 ABC' being less than ABC, it is evident the perimeter of the 
 former must be enlarged, to make its area equal to the area 
 of the latter. 
 
 PROPOSITION II. 
 
 80. A triangle in which two given sides mcike a RIGHT 
 ANGLE, has a greater area than any triangle in which the same 
 sides make an oblique angle. 
 
 If BC, BC' and BC" be equal, 
 and if BC be perpendicular to 
 AB ; then the right angled trian- 
 gle ABC, has a greater area than 
 the acute angled triangle ABC', or 
 the oblique angled triangle ABC". 
 
 Let P'C' and PC" be perpen- 
 dicular to AP. Then, as the 
 three triangles have the same base AB, their areas are as 
 their heights ; that is, as the perpendiculars BC, P'C', and 
 PC". But BC is equal to BC', and therefore greater than 
 P'C'. (Euc. 47. 1.) BC is also equal to BC", and therefore 
 greater than PC". 
 
 PROPOSITION III. 
 
 81. If all the sides EXCEPT ONE of a polygon be given, 
 the area will be the greatest, when the given sides are so dis- 
 posed that the figure may be INSCRIBED IN A SEMICIRCLE, of 
 which the undetermined side is the diameter. 
 
 If the sides AB, BC, CD, DE, 
 be given, and if their position 
 be such that the area, included 
 between these and another side 
 whose length is not determined, 
 is a maximum ; the figure may 
 
ISOPERIMETRY. 81 
 
 be inscribed in a semicircle, of which the undetermined side 
 AE is the diameter. 
 
 Draw the lines AD, AC, EB, EC. By varying the angle 
 at D, the triangle ADE may be enlarged or diminished, with- 
 out affecting the area of the other parts of the figure. The 
 whole area, therefore, cannot be a maximum, unless this tri- 
 angle be a maximum, while the sides AD and ED are given. 
 But if the triangle ADE be a maximum, under these con- 
 ditions, the angle ADE is a right angle ; (Art. 80.) and 
 therefore the point D is in the circumference of a circle, of 
 which AE is the diameter. (Euc. 31,3.) In the same man- 
 ner it may be proved, that the angles ACE and ABE are 
 right angles, and therefore that the points C and B are in 
 the circumference of the same circle. 
 
 The term polygon is used in this section to include trian- 
 gles, and four-sided figures, as well as other right-lined 
 figures. 
 
 82. The area of a polygon, inscribed in a semicircle, in 
 the manner stated above, will not be altered by varying the 
 order of the given sides. 
 
 The sides AB, BC, CD, DE, are the chords of so 
 many arcs. The sum of these arcs, in whatever order 
 they are arranged, will evidently be equal to the semicircum- 
 ference. And the segments between the given sides and the 
 arcs will be the same in whatever part of the circle they are 
 situated. But the area of the polygon is equal to the area 
 of the semicircle, diminished by the sum of these segments. 
 
 83. If a polygon, of which all the sides except one are 
 given, be inscribed in a semicircle whose diameter is the un- 
 determined side ; a polygon having the same given sides, 
 cannot be inscribed in any other semicircle which is either 
 greater or less than this, and whose diameter is the undeter- 
 mined side. 
 
 The given sides AB, BC, CD, DE, are the chords of arcs 
 whose sum is 180 degrees. But in a larger circle, each 
 
 4* 
 
82 ISOPERIMETRY. 
 
 would be the chord of a less number of degrees, and there- 
 fore the sum of the arcs would be less than 180 : and in a 
 smaller circle, each would be the chord of a greater number 
 of degrees, and the sum of the arcs would be greater than 
 180. 
 
 PROPOSITION IV. 
 
 84. A polygon INSCRIBED IN A CIRCLE has a greater area, 
 than any polygon of equal perimeter, and the same number of 
 sides, which cannot be inscribed in a circle. 
 
 If in the circle ACHF, (Fig. 30.) there be inscribed a 
 c 
 
 polygon ABCDEFG ; and if another polygon dbcdefg (Fig. 
 31.) be formed of sides which are the same in number and 
 length, but which are so disposed, that the figure cannot be 
 inscribed in a circle; the area of the former polygon is 
 greater than that of the latter. 
 
 Draw the diameter AH, and the chords DH and EH. 
 Upon de make the triangle deh equal and similar to DEH, 
 and join ah. The line ah divides the figure abcdhefg into two 
 parts, of which one at least cannot, by supposition, be in- 
 scribed in a semicircle of which the diameter is AH, nor in 
 any other semicircle of which the diameter is the undeter- 
 mined side. (Art. 83.) It is therefore less than the corres- 
 ponding part of the figure ABCDHEFG. (Art. 81.) And 
 the other part of abcdhefg is not greater than the correspond- 
 
ISOPERIMETRY. 83 
 
 ing part of ABCDHEFG. Therefore, the whole figure 
 ABODHEFG is greater than the whole figure abcdhefg. If 
 from these there be taken the equal triangles DEH and deh, 
 there will remain the polygon ABCDEFG greater than the 
 polygon abcdefg. 
 
 85. A polygon of which all the sides are given in num- 
 ber and length, cannot be inscribed in circles of different 
 diameters. (Art. 83.) And the area of the polygon will not 
 be altered by changing the order of the sides. (Art. 82.) 
 
 PROPOSITION V. 
 
 86. When a polygon has a greater area than any other, of 
 the same number of sides, and of equal perimeter, the sides are 
 EQUAL. 
 
 The polygon ABCDF (Fig. 29.) 
 cannot be a maximum, among all 
 polygons of the same number of 
 sides, and of equal perimeters, un- P 
 less it be equilateral. For if any 
 two of the sides, as CD and FD, 
 are unequal, let CH and FH be 
 equal, and their sum the same as 
 the sum of CD and FD. The 
 
 isosceles triangle CHF is greater than the scalene triangle 
 CDF (Art. 79.); and therefore the polygon ABCHF is 
 greater than the polygon ABCDF ; so that the latter is not 
 a maximum. 
 
 PROPOSITION VI. 
 
 87. A REGULAR POLYGON has a greater area than any 
 other polygon of equal perimeter, and of the same number of 
 itides. 
 
84 ISOPERIMETRY 
 
 For, by the preceding article, the polygon which is a max- 
 imum among others of equal perimeters, and the same num- 
 ber of sides, is equilateral, and by Art. 84, it may be m- 
 scribed in a circle. But if a poly- 
 gon inscribed in a circle is equilat- 
 eral, as ABDFGH, it is also equian- 
 gular. For the sides of the polygon 
 are the bases of so many isosceles 
 triangles, whose common vertex is 
 the centre C. The angles at these 
 bases are all equal ; and two of them, 
 
 as AHC and GHC, are equal to AHG one of the angles of 
 the polygon. The polygon, then, being equiangular, as well 
 as equilateral, is a regular polygon. (Art. 1. Def. 2.) 
 
 Thus an equilateral triangle has a greater area, than any 
 other triangle of equal perimeter. And a square has a 
 greater area than any other four-sided figure of equal pe- 
 rimeter. 
 
 Cor. A regular polygon has a less perimeter than any 
 other polygon of equal area, and the same number of 
 sides. 
 
 For if, with a given perimeter, the regular polygon is 
 greater than one which is not regular ; it is evident the pe- 
 rimeter of the former must be diminished, to make its area 
 equal to that of the latter. 
 
 PROPOSITION VII. 
 
 88. If a polygon be DESCRIBED ABOUT A CIRCLE, the areas 
 of the two figures are as their perimeters. 
 
 Let ST be one of the sides of a polygon, either regular or 
 
ISOPERIMETRT. 85 
 
 not, which is described about the cir- 
 cle LNR. Join OS and OT, and 
 to the point of contact M draw the 
 radius OM, which will be perpen- 
 dicular to ST. (Euc. 18, 3.) The 
 triangle OST is equal to half the 
 base ST multiplied into the radius 
 OM. (Art. 8.) And if lines be 
 drawn, in the same manner, from 
 the centre of the circle, to the extremities of the sev- 
 eral sides of the circumscribed polygon, each of the trian- 
 gles thus formed will be equal to half its base multiplied 
 into the radius of the circle. Therefore the area of the 
 whole polygon is equal to half its perimeter multiplied into 
 the radius : and the area of the circle is equal to half its cir- 
 cumference multiplied into the radius. (Art 30.) So that 
 the two areas aie to each other as their perimeters. 
 
 Cor. 1. If different polygons are described about the 
 same circle, their areas are to each other as their perimeters. 
 For the area of each is equal to half its perimeter, multi- 
 plied into the radius of the inscribed circle. 
 
 Cor. 2. The tangent of an arc is always greater than the 
 arc itself. The triangle OMT is to OMN, as MT to MN. 
 But OMT is greater than OMN, because the former includes 
 the latter. Therefore, the tangent MT is greater than the 
 arc MN. 
 
 PROPOSITION VIII. 
 
 89. A CIRCLE has a greater area than any polygon of equal 
 perimeter. 
 
 If a circle and a regular polygon have the same centre, 
 and equal perimeters ; each of the sides of the polygon 
 must fall partly within the circle. For the area of a drcum- 
 
86 
 
 ISOPERIMETRY. 
 
 scribing polygon is greater than the area of the circle, as 
 the one includes the other : and therefore, by the preceding 
 article, the perimeter of the former is greater than that of 
 the latter. 
 
 Let AD then be one side of a 
 regular polygon, whose perimeter 
 is equal to the circumference of 
 the circle RLN. As this falls 
 partly within the circle, the per- 
 pendicular OP is less than the 
 radius OR. But the area of the 
 polygon is equal to half its pe- 
 rimeter multiplied into this per- 
 pendicular (Art. 15.) ; and the area of the circle is equal to 
 half its circumference multiplied into the radius. (Art. 30.) 
 The circle then is greater than the given regular polygon ; 
 and therefore greater than any other polygon of equal pe- 
 rimeter. (Art. 87.) 
 
 Cor. 1. A circle has a less perimeter, than any polygon of 
 equal area. 
 
 Cor. 2. Among regular polygons of a given perimeter, 
 that which has the greatest number of sides, has also the 
 greatest area. For the greater the number of sides, the 
 more nearly does the perimeter of the polygon approach to 
 a coincidence with the circumference of a circle. 
 
 PROPOSITION IX. 
 
 90. A right PRISM whose bases are REGULAR POLYGONS, has 
 a less surface than any other right prism of the same solidity, 
 the same altitude, and the same number of sides. 
 
 If the altitude of a prism is given, the area of the base is 
 as the solidity (Art. 43.) ; and if the number of sides is 
 
IS OPS RIME TRY". 87 
 
 also given, the perimeter is a minimum when the base is a 
 regular polygon. (Art. 87. Cor.) But the lateral surface is 
 as the perimeter. (Art. 47.) Of two right prisms, then, 
 which have the same altitude, the same solidity, and the 
 same number of sides, that whose bases are regular polygons 
 has the least lateral surface, while the areas of the ends are 
 equal. 
 
 Cor. A right prism whose bases are regular polygons has 
 a greater solidity, than any other right prism of the same 
 surface, the same altitude, and the same number of sides. 
 
 PROPOSITION X. 
 
 91. A right CYLINDER has a less surface than any right 
 prism of the same altitude and solidity. 
 
 For if the prism and cylinder have the same altitude and 
 solidity, the areas of their bases are equal. (Artr. 64.) But 
 the perimeter of the cylinder is less, than that of the prism 
 (Art. 89. Cor. 1.) ; and therefore its lateral surface is less, 
 while the areas of the ends are equal. 
 
 Cor. A right cylinder has a greater solidity, than any right 
 prism of the same altitude and surface. 
 
 PROPOSITION XI. 
 
 92. A CUBE has a less surface than any other right paral- 
 lelopiped of the same solidity. 
 
 A parallelepiped is a prism, any one of whose faces may- 
 be considered a base. (Art. 41. Def. I and V.) If these are 
 not all squares, let one which is not a square be taken for a 
 base. The perimeter of this may be diminished, without 
 altering its area (Art. 87. Cor.) ; and therefore the surface 
 
88 ISOPERIMETRY. 
 
 of the solid may be diminished, without altering its altitude 
 or solidity. (Art. 43, 47.) The same may be proved of 
 each of the other faces which are not squares. The surface 
 is therefore a minimum, when all the faces are squares, that 
 is, when the solid is a cube. 
 
 Cor. A cube has a greater solidity than any other right 
 parallelepiped of the same surface. 
 
 PROPOSITION XII. 
 
 93. A CUBE has a greater solidity than any other right par- 
 allelopiped, the sum of whose length, breadth and depth, is equal 
 to ike sum of the corresponding dimensions of the cube. 
 
 The solidity is equal to the product of the length, breadth, 
 and depth. If the length and breadth are unequal, the 
 solidity may be increased, without altering the sum of the 
 three dimensions. For the product of two factors whose 
 sum is giveji, is the greatest when the factors are equal. 
 (Euc. 27. 6.) In the same manner, if the breadth and 
 depth are unequal, the solidity may be increased, without 
 altering the sum of the three dimensions. Therefore, the 
 solid cannot be a maximum, unless its length, breadth, and 
 depth are equal. 
 
 PROPOSITION XIII. 
 
 94. If O, PRISM BE DESCRIBED ABOUT A CYLINDER, the 
 
 capacities of the two solids are as their surfaces. 
 
 The capacities of the solids are as the areas of their bases, 
 that is, as the perimeters of their bases. (Art. 88.) But the 
 lateral surfaces are also as the perimeters of the bases. 
 Therefore the whole surfaces are as the solidities. 
 
 Cor. The capacities of different prisms, described about 
 the same right cylinder, are to each other as their surfaces. 
 
ISOPERIMETRY. 80 
 
 PROPOSITION XIV. 
 
 95. A right cylinder WHOSE HEIGHT is EQUAL TO THE 
 DIAMETER OF ITS BASE has a greater solidity than any other 
 right cylinder of equal surface. 
 
 Let C be a right cylinder whose height is equal to the di- 
 ameter of its base ; and C' another right cylinder having the 
 same surface, but a different altitude. If a square prism P 
 be described about the former, it will be a cube. But a 
 square prism P' described about the latter will not be a cube. 
 
 Then the surfaces of C and P are as their bases (Art. 47. 
 and 88.) ; which are as the bases of C' and P', (Sup. Euc. 
 7, 1.); so that, 
 
 mrfC : surfP : : base C : base P : : base C' : base P' : : surfC' : 
 surfP'. 
 
 But the surface of C is, by supposition, equal to the sur- 
 face of C'. Therefore, (Alg. 395.) the surface of P is equal 
 to the surface of P'. And by the preceding article, 
 
 wild P : solid C : : surfP :surfC : : surfP' : surfQ' : : solid 
 P' : solid C'. 
 
 But the solidity of P is greater than that of P'. (Art. 92. 
 Cor.) Therefore the solidity of C is greater than that of C'. 
 
 Schol. A right cylinder whose height is equal to the di- 
 ameter of its base, is that which circumscribes a sphere. It 
 is also called Archimedes' cylinder ; as he discovered the 
 ratio of a sphere to its circumscribing cylinder; and these 
 are the figures which were put upon his tomb. 
 
 Cor. Archimedes' cylinder has a less surface, than any 
 other right cylinder of the same capacity. 
 
90 ISOPflRIMETRY. 
 
 PROPOSITION XV. 
 
 96. If a SPHERE BE CIRCUMSCRIBED by a solid bounded by 
 plane surfaces ; the capacities of the two solids are as their 
 surfaces. 
 
 If planes be supposed to be drawn from the centre of the 
 sphere, to each of the edges of the circumscribing solid, 
 they will divide it into as many pyramids as the solid has 
 faces. The base of each pyramid will be one of the faces ; 
 and the height will be the radius of the sphere. The 
 capacity of the pyramid will be equal, therefore, to its base 
 multiplied into ^ of the radius (Art. 48.); and the capacity 
 of the whole circumscribing solid, must be equal to its whole 
 surface multiplied into ^ of the radius. But the capacity of 
 the sphere is also equal to its surface multiplied into of its 
 radius. (Art. 70.) 
 
 Cor. The capacities of different solids circumscribing the 
 same sphere, are as their surfaces. 
 
 PROPOSITION XVI. 
 
 97. A SPHERE has a greater solidity than any regular poly- 
 edron of equal surface. 
 
 If a sphere and a regular polyedron have the same centre, 
 and equal surfaces ; each of the faces of the polyedron must 
 fall partly within the sphere. For the solidity of a circum- 
 scribing solid is greater than the solidity of the sphere, as 
 the one includes the other : and therefore, by the preceding 
 article, the surface of the former is greater than that of the 
 latter. 
 
 But if the faces of the polyedron fall partly within the 
 sphere, their perpendicular distance from the centre must be 
 less than the radius. And therefore, if the surface of the 
 
ISOPERIMETRY. "01 
 
 polyedron be only equal to that of the sphere, its solidity 
 must be less. For the solidity of the polyedron is equal to 
 its surface multiplied into ^ of the distance from the centre. 
 (Art. 59.) And the solidity of the sphere is equal to its 
 surface multiplied into - of the radius. 
 
 Cor. A sphere has a less surface than any regular poly- 
 edron of the same capacity. 
 
APPENDII 
 
 GAUGING OF CASKS. 
 
 ART. 119. GAUGING is a practical art, which does not ad- 
 mit of being treated in a very scientific manner. Casks are 
 not commonly constructed in exact conformity with any reg- 
 ular mathematical figure. By most writers on the subject, 
 however, they are considered as nearly coinciding with one 
 of the following forms : 
 
 1. ) C of a spheroid, 
 
 The m,ddle frustum spind , e 
 
 . _ of a paraboloid, 
 
 4 The equal frustums 
 
 The second of these varieties agrees more nearly than any 
 of the others, with the forms of casks, as they are com- 
 monly made. The first is too much curved, the third too 
 little, and the fourth not at all, from the head to the bung. 
 
 120. Rules have already been given, for finding the capa- 
 city of each of the four varieties of casks. (Arts. 68, 110, 
 112, 118.) As the dimensions are taken in inches, these rules 
 will give the contents in cubic inches. To abridge the com- 
 putation, and adapt it to the particular measures used in 
 gauging, the factor .7854 is divided by 282 or 231 ; and 
 the quotient is used instead of .7854, for finding the capa- 
 city in ale gallons or wine gallons. 
 
GAUGING. 98 
 
 Now' =,002785, or .0028 nearly ; 
 
 282 
 
 And --^=.0034 
 231 
 
 If then .0028 and .0034 be substituted for .7854, in the 
 rules referred to above ; the contents of the cask will be 
 given in ale gallons and wine gallons. These numbers are 
 to each other nearly as 9 to 11. 
 
 PROBLEM L 
 
 To calculate the contents of a cask, in the form of a middle 
 frustum of a SPHEROID. 
 
 121. Add together the square of the head diameter, and 
 twice the square of the bung diameter : multiply the sum by 
 of the length, and the product by ,0028 for ale gallons, or 
 by .0034 for wine gallons, 
 
 If D and d=ihe two diameters, and Z=the length; 
 The capacity in inches=(2D 2 +c? 2 )x^X.7854. (Art. 110.) 
 
 And by substituting .0028 or ,0034 for ,7854, we have 
 the capacity in ale gallons or wine gallons. 
 
 Ex. What is the capacity of a cask of the first form, 
 whose length is 30 inches, its head diameter 18, and its bung 
 diameter 24 ? 
 
 Ans. 41.3 ale gallons, or 50.2 wine gallons. 
 
 PROBLEM II. 
 
 To calculate the contents of a cask, in the form of the mid- 
 dle frustum of a PARABOLIC SPINDLE. 
 
 122. Add together the square of the head diameter, and 
 twice the square of the bung diameter, and from the sum 
 
94 GAUGING. 
 
 subtract -f of the square of the difference of the diameters ; 
 multiply the remainder by i of the length, and the product 
 by .0028 for ale gallons, or .0034 for wine gallons. 
 
 The capacity in inches =(2D 2 +cP | (D </) 2 )Xi^X 
 .7854. (Art. 118.) 
 
 Ex. What is the capacity of a cask of the second form, 
 whose length is 30 inches, its head diameter 18, and its 
 bung diameter 24 ? 
 
 Ans. 40.9 ale gallons, or 49.7 wine gallons. 
 
 PROBLEM III. 
 
 To calculate the contents of a cask, in the form of two equal 
 frustums of a PARABOLOID. 
 
 123. Add together the square of the head diameter, and 
 the square of the bung diameter ; multiply the sum by half 
 tfoe length, and the product by .0028 for ale gallons, or 
 .0034 for wine gallons. 
 
 The capacity in inches =(D 2 -j-6? 2 )xi?X-7854. (Art. 112 
 Cor.) 
 
 Ex. What is the capacity of a cask of the third form, 
 whose dimensions are, as before, 30, 18, and 24 ? 
 
 Ans. 37.8 ale gallons, or 45.9 wine gallons. 
 
 PROBLEM IV. 
 
 To calculate the contents of a cask, in the form of two equal 
 frustums of a CONE. 
 
 124. Add together the square of the head diameter, the 
 square of the bung diameter, and the product of the two 
 diameters ; multiply the sum by of the length, and the 
 product by.0028 for ale gallons, or .0034 for wine gallons. 
 The capacity in inches=(D 9 +G? 2 +Dc?)X^X.7854. (Art. 68.) 
 
GAUGING. 05 
 
 Ex. What is the capacity of a cask of the fourth form, 
 whose length is 30, and its diameters 18 and 24? 
 
 Ans. 37.3 ale gallons, or 45.3 wine gallons. 
 
 125. The preceding rules, though correct in theory, are 
 not very well adapted to practice, as they suppose the form 
 of the cask to be known. The two following rules, taken 
 from Hut ton's Mensuration, may be used for casks of the 
 usual forms. For the first, three dimensions are required, 
 the length, the head diameter, and the bung diameter. It 
 is evident that no allowance is made by this, for different 
 degrees of curvature from the head to the bung. If the 
 cask is more or less curved than usual, the following rule is 
 to be preferred, for which four dimensions are required, the 
 head and bung diameters, and a third diameter taken in the 
 middle between the bung and the head. For the demon- 
 stration of these rules, see Button's Mensuration, Part V. 
 Sec. 2. Ch. 5 and 7. 
 
 PROBLEM V. 
 
 To calculate the contents of any common cask, from THREE 
 dimensions. 
 
 126. Add together 
 
 25 times the square of the head diameter, 
 
 39 times the square of the bung diameter, and 
 
 26 times the product of the two diameters ; 
 Multiply the sum by the length, divide the product by 90, 
 
 and multiply the quotient by .0028 for ale gallons, or .0034 
 for wine gallons. 
 
 The capacity in inches=(39 D a +25cP + 26Drf)X j_x.'7854. 
 
 90 
 
 Ex. What is the capacity of a cask whose length is 30 
 inches, the head diameter 18, and the bung diameter 24? 
 Ans. 39 ale gallons, or 47i wine gallons. 
 
96 GAUGING. 
 
 PROBLEM VI. 
 
 To calculate the contents of a cask from FOUR dimensions, the 
 length, the head and bung diameters, and a diameter taken 
 in the middle between the head and the bung. 
 
 127. Add together the squares of the head diameter, of 
 the bung diameter, and of double the middle diameter ; 
 multiply the sum by of the length, and the product by 
 .0028 for ale gallons, or .0034 for wine gallons. 
 
 If D=the bung diameter, c?==the head diameter, m=the 
 middle diameter, and /=the length ; 
 
 The capacity in inches=(D a 
 
 Ex. What is the capacity of a cask, whose length is 30 
 inches, the head diameter 18, the bung diameter 24, and the 
 middle diameter 22 ? 
 
 Ans. 41 ale gallons, or 49f wine gallons. 
 
 128. In making the calculations in gauging, according to 
 the preceding rules, the multiplications and divisions are fre- 
 quently performed by means of a Sliding Rule, on which 
 are placed a number of logarithmic lines, similar to those on 
 Gunter's Scale. See Trigonometry, Sec. VI., and Note C. 
 p. 149. 
 
 Another instrument commonly used in gauging is the 
 Diagonal Rod. By this, the capacity of a cask is very ex- 
 peditiously found, from a single dimension, the distance from 
 the bung to the intersection of the opposite stave with the 
 head ; but this process is not considered sufficiently accurate 
 for casks of a capacity exceeding 40 gallons. The measure 
 is taken by extending the rod through the cask, from the 
 bung to the most distant part of the head. The number of 
 gallons corresponding to the length of the line thus found, is 
 marked on the rod. The logarithmic lines on the gauging 
 
GAUGING. 97 
 
 rod are to be used in the same manner, as on the sliding 
 rule. 
 
 ULLAGE OF CASKS. 
 
 129. When a cask is partly filled, the whole capacity is 
 divided, by the surface of the liquor, into two portions ; the 
 least of which, whether full or empty, is called the ullage. 
 In finding the ullage, the cask is supposed to be in one of 
 two positions ; either standing, with its axis perpendicular to 
 the horizon ; or lying, with its axis parallel to the horizon. 
 The rules for ullage which are exact, particularly those for 
 lying casks, are too complicated for common use. The fol- 
 lowing are considered as sufficiently near approximations. 
 See Button's Mensuration. 
 
 PROBLEM VII. 
 To calculate the ullage of a STANDING cask. 
 
 130. Add together the squares of the diameter at the sur- 
 face of the liquor, of the diameter of the nearest end, and 
 of double the diameter in the middle between the other two ; 
 multiply the sum by of the distance between the surface 
 and the nearest end, and the product by .0028 for ale gal- 
 lons, or .0034 for wine gallons. 
 
 If D=the diameter of the surface of the liquor, 
 rf=the diameter of the nearest end, 
 m=the middle diameter, and 
 
 Z=the distance between the surface and the nearest end ; 
 The ullage in inches=(D a +c?'+2^ r )X^X.V854. 
 
 Ex. If the diameter at the surface of the liquor, in a stand- 
 ing cask, be 32 inches, the diameter of the nearest end 24, 
 the middle diameter 29, and the distance between the sur- 
 
 5 
 
98 GAUGING. 
 
 face of the liquor and the nearest end 1 2 ; what is the ul- 
 lage? Ans. 27-f- ale gallons, or 33f wine gallons. 
 
 PROBLEM VIII. 
 To calculate the ullage of a LYING cask. 
 
 131. Divide the distance from the bung to the surface of 
 the liquor, by the whole bung diameter, find the quotient in 
 the column of heights or versed sines in a table of circular 
 segments, take out the corresponding segment, and multiply 
 it by the whole capacity of the cask, and the product by 1-J- 
 for the part which is empty. 
 
 If the cask be not half full, divide the depth of the liquor 
 by the whole bung diameter, take out the segment, multiply, 
 &c., for the contents of the part which is full. 
 
 Ex. If the whole capacity of a lying cask be 41 ale gal- 
 lons, or 49f wine gallons, the bung diameter 24 inches and 
 the distance from the bung to the surface of the liquor 6 
 inches ; what is the ullage ? 
 
 Ans. 7f ale gallons, or 9 wine gallons. 
 
NOTES. 
 
 NOTE A. p. 39. 
 
 THE term solidity is used here in the customary sense, to 
 express the magnitude of any geometrical quantity of three 
 dimensions, length, breadth, and thickness ; whether it be a 
 solid body, or a fluid, or even a portion of empty space. This 
 use of the word, however, is not altogether free from objec- 
 tion. The same term is applied to one of the general prop- 
 erties of matter ; and also to that peculiar quality by which 
 certain substances are distinguished from fluids. There 
 seems to be an impropriety in speaking of the solidity of a 
 body of water ) or of a vessel which is empty. Some writers 
 have therefore substituted the word volume for solidity. But 
 the latter term, if it be properly defined, may be retained 
 without danger of leading to mistake. 
 
 NOTE B. p. 76. 
 
 The following simple rule for the solidity of round timber, 
 or of any cylinder, is nearly exact : 
 
 Multiply the length into twice the square of ^ of the circum* 
 ference. 
 
 If C=the circumference of a cylinder; 
 
 The area of the base But 2( )=j- 
 
 It is common to measure hewn timber, by multiplying the 
 length into the square of the quarter-girt. This gives ex- 
 
100 NOTES. 
 
 actly the solidity of a parallelopiped, if the ends are squares. 
 But if the ends are parallelograms, the area of each is less 
 than the square of the quarter-girt. (Euc. 27. 6.) 
 
 Timber which is tapering may be exactly measured by the 
 rule for the frustum of a pyramid or cone (Art. 50, 68.) ; 
 or, if the ends are not similar figures, by the rule for a pris- 
 moid. (Art. 55.) But for common purposes, it will be suf- 
 ficient to multiply the length by the area of a section in the 
 middle between the two ends. 
 
TABLE 
 
 OP 
 
 LOGARITHMS OF NUMBERS 
 
 FROM 
 
 1 TO 10,000. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 1 
 
 0.000000 
 
 26 
 
 .414973 
 
 51 
 
 1.707570 
 
 76 
 
 .880814 
 
 2 
 
 0.301030 
 
 27 
 
 .431364 
 
 52 
 
 1.716003 
 
 77 
 
 .886491 
 
 3 
 
 0.477121 
 
 28 
 
 .447158 
 
 53 
 
 1.724276 
 
 78 
 
 .892095 
 
 4 
 
 0.602060 
 
 29 
 
 .462398 
 
 54 
 
 .732394 
 
 79 
 
 .897627 
 
 5 
 
 0.698970 
 
 30 
 
 .477121 
 
 55 
 
 .740363 
 
 80 
 
 .903090 
 
 6 
 
 0.778151 
 
 31 
 
 .491362 
 
 56 
 
 .748188 
 
 81 
 
 .908485 
 
 7 
 
 0.845098 
 
 32 
 
 .505150 
 
 57 
 
 .755875 
 
 82 
 
 .913814 
 
 8 
 
 0.903090 
 
 33 
 
 .518514 
 
 58 
 
 .763428 
 
 83 
 
 .919078 
 
 9 
 
 .954243 
 
 34 
 
 .531479 
 
 59 
 
 .770852 
 
 84 
 
 .924279 
 
 10 
 
 .000000 
 
 35 
 
 .544068 
 
 GO 
 
 .778151 
 
 85 
 
 .929419 
 
 11 
 
 .041393 
 
 36 
 
 .556303 
 
 61 
 
 .785330 
 
 86 
 
 .934498 
 
 12 
 
 .079181 
 
 37 
 
 .568202 
 
 62 
 
 .792392 
 
 87 
 
 .939519 
 
 J3 
 
 .113943 
 
 38 
 
 .579784 
 
 03 
 
 .799341 
 
 88 
 
 .944483 
 
 14 
 
 .146128 
 
 39 
 
 .591065 
 
 64 
 
 .806180 
 
 89 
 
 .949390 
 
 15 
 
 .176091 
 
 40 
 
 .602060 
 
 ft> 
 
 .812913 
 
 90 
 
 .954243 
 
 16 
 
 .204120 
 
 41 
 
 .612784 
 
 66 
 
 .819544 
 
 91 
 
 .959041 
 
 17 
 
 .230449 
 
 42 
 
 .623249 
 
 67 
 
 .826075 
 
 92 
 
 .963788 
 
 18 
 
 .255273 
 
 43 
 
 .633468 
 
 68 
 
 .832509 
 
 93 
 
 .968483 
 
 19 
 
 .278754 
 
 44 
 
 .643453 
 
 69 
 
 .838849 
 
 94 
 
 .973128 
 
 20 
 
 .301030 
 
 45 
 
 .653213 
 
 70 
 
 .845098 
 
 95 
 
 .977724 
 
 21 
 
 .322219 
 
 46 
 
 .662758 
 
 71 
 
 .851258 
 
 96 
 
 .982271 
 
 22 
 
 .342423 
 
 47 
 
 .672098 
 
 72 
 
 .857333 
 
 97 
 
 .986772 
 
 23 
 
 .361728 
 
 48 
 
 .681241 
 
 73 
 
 .863323 
 
 98 
 
 .991226 
 
 24 
 
 .380211 
 
 49 
 
 .690196 
 
 74 
 
 .869232 
 
 99 
 
 .995635 
 
 25 
 
 .397940 
 
 50 
 
 .698970 
 
 75 
 
 1.875061 
 
 100 
 
 2.000000 
 
 N. B. In the following table, in the last nine columns of each page, 
 where the first or leading figures change from 9's to O's, points or dots 
 are introduced instead of the O's through the rest of the line, to catch 
 the eye, and to indicate that from thence the annexed first two figures 
 of the Logarithm in the second column stand in the next lower line. 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. | 0|1|2|3 4 5 | 6 | 7 | 8 | 9 | D. 
 
 100 
 
 000000 
 
 0434 
 
 0868 
 
 1301 
 
 1734 
 
 2166 
 
 2598 
 
 3029 
 
 3461 
 
 3891 
 
 432 
 
 101 
 
 4321 
 
 4751 
 
 5181 
 
 5609 
 
 6038 
 
 6466 
 
 6894 
 
 7321 
 
 7748 
 
 8174 
 
 428 
 
 102 
 
 8600 
 
 9026 
 
 9451 
 
 9876 
 
 .300 
 
 .724 
 
 1147 
 
 1570 
 
 1993 
 
 2415 
 
 424 
 
 103 
 
 012837 
 
 3259 
 
 3680 
 
 4100 
 
 4521 
 
 4940 
 
 5360 
 
 5779 
 
 6197 
 
 6616 
 
 419 
 
 104 
 
 7033 
 
 7451 
 
 7868 
 
 8284 
 
 8700 
 
 9116 
 
 9532 
 
 9947 
 
 .361 
 
 .775 
 
 416 
 
 105 
 
 021189 
 
 1603 
 
 2016 
 
 2428 
 
 2841 
 
 3252 
 
 3664 
 
 4075 
 
 4486 
 
 4896 
 
 412 
 
 106 
 
 5306 
 
 5715 
 
 6125 
 
 6533 
 
 6942 
 
 7350 
 
 7757 
 
 8164 
 
 8571 
 
 8978 
 
 408 
 
 107 
 
 9384 
 
 9789 
 
 .195 
 
 .600 
 
 1004 
 
 1408 
 
 1812 
 
 2216 
 
 2619 
 
 3021 
 
 404 
 
 103 
 
 033424 
 
 3826 
 
 4227 
 
 4628 
 
 5029 
 
 5430 
 
 5830 
 
 6230 
 
 6629 
 
 7028 
 
 400 
 
 109 
 
 7426 
 
 7825 
 
 8223 
 
 8620 
 
 9017 
 
 9414 
 
 9811 
 
 .207 
 
 .602 
 
 .998 
 
 396 
 
 110 
 
 041393 
 
 1787 
 
 2182 
 
 2576 
 
 2969 
 
 3362 
 
 3755 
 
 4148 
 
 4540 
 
 4932 
 
 393 
 
 111 
 
 5323 
 
 5714 
 
 6105 
 
 6495 
 
 6885 
 
 7275 
 
 7664 
 
 8053 
 
 8442 
 
 8830 
 
 389 
 
 112 
 
 9218 
 
 9606 
 
 9993 
 
 .380 
 
 .766 
 
 1153 
 
 1538 
 
 1924 
 
 2309 
 
 2694 
 
 386 
 
 113 
 
 053078 
 
 3463 
 
 3846 
 
 4230 
 
 4613 
 
 4996 
 
 5378 
 
 5760 
 
 6142 
 
 6524 
 
 382 
 
 114 
 
 6905 
 
 7286 
 
 7666 
 
 8046 
 
 8426 
 
 8805 
 
 9185 
 
 9563 
 
 9942 
 
 .320 
 
 379 
 
 115 
 
 060698 
 
 1075 
 
 1452 
 
 1829 
 
 2206 
 
 2582 
 
 2958 
 
 3333 
 
 3709 
 
 4083 
 
 376 
 
 116 
 
 4458 
 
 4832 
 
 5206 
 
 5580 
 
 5953 
 
 6326 
 
 6699 
 
 7071 
 
 7443 
 
 7815 
 
 372 
 
 117 
 
 8186 
 
 8557 
 
 8928 
 
 9298 
 
 966b 
 
 ..38 
 
 .407 
 
 .776 
 
 1145 
 
 1514 
 
 369 
 
 118 
 
 071882 
 
 2250 
 
 2617 
 
 2985 
 
 3352 
 
 3718 
 
 4085 
 
 4451 
 
 4816 
 
 5182 
 
 366 
 
 119 
 
 5547 
 
 5912 
 
 6276 
 
 6640 
 
 7004 
 
 7368 
 
 7731 
 
 8094 
 
 8457 
 
 8819 
 
 363 
 
 120 
 
 079181 
 
 9543 
 
 9904 
 
 .266 
 
 .626 
 
 .987 
 
 1347 
 
 1707 
 
 2067 
 
 2426 
 
 360 
 
 121 
 
 082785 
 
 3144 
 
 3503 
 
 3861 
 
 4219 
 
 4576 
 
 4934 
 
 5291 
 
 5647 
 
 6004 
 
 357 
 
 122 
 
 6360 
 
 6716 
 
 7071 
 
 7426 
 
 7781 
 
 8136 
 
 8490 
 
 8845 
 
 9198 
 
 9552 
 
 355 
 
 123 
 
 9905 
 
 .258 
 
 .611 
 
 .963 
 
 1315 
 
 1667 
 
 2018 
 
 2370 
 
 2721 
 
 3071 
 
 351 
 
 124 
 
 093422 
 
 3772 
 
 4122 
 
 4471 
 
 '1820 
 
 5169 
 
 5518 
 
 5866 
 
 6215 
 
 6562 
 
 349 
 
 125 
 
 6910 
 
 7257 
 
 7604 
 
 7951 
 
 8-29H 
 
 8644 
 
 8990 
 
 9335 
 
 9681 
 
 ..26 
 
 346 
 
 128 
 
 100371 
 
 0715 
 
 1059 
 
 1403 
 
 1747 
 
 2091 
 
 2434 
 
 2777 
 
 3119 
 
 3462 
 
 343 
 
 127 
 
 3804 
 
 4146 
 
 4487 
 
 4828 
 
 5169 
 
 5510 
 
 5851 
 
 6191 
 
 6531 
 
 6871 
 
 340, 
 
 128 
 
 7210 
 
 7549 
 
 7888 
 
 8227 
 
 8565 
 
 8903 
 
 9241 
 
 9579 
 
 9916 
 
 .253 
 
 338 fc 
 
 129 
 
 110590 
 
 0926 
 
 1263 
 
 1599 
 
 1934 
 
 2270 
 
 2605 
 
 2940 
 
 3275 
 
 3609 
 
 335 
 
 130 
 
 113943 
 
 4277 
 
 4611 
 
 4944 
 
 5278 
 
 5611 
 
 5943 
 
 6276 
 
 6608 
 
 6940 
 
 333 
 
 131 
 
 7271 
 
 7603 
 
 7934 
 
 8265 
 
 8595 
 
 8926 
 
 9256 
 
 9586 
 
 9915 
 
 .245 
 
 330 
 
 132 
 
 120574 
 
 0903 
 
 1231 
 
 1560 
 
 1888 
 
 2216 
 
 2544 
 
 2871 
 
 3198 
 
 3525 
 
 328 
 
 133 
 
 3852 
 
 4178 
 
 4504 
 
 4830 
 
 5156 
 
 5481 
 
 5806 
 
 6131 
 
 6456 
 
 6781 
 
 325 
 
 134 
 
 7105 
 
 7429 
 
 7753 
 
 8076 
 
 8399 
 
 8722 
 
 9045 
 
 9368 
 
 9690 
 
 ..12 
 
 323 
 
 135 
 
 130334 
 
 0655 
 
 0977 
 
 1298 
 
 1619 
 
 1939 
 
 2260 
 
 2580 
 
 2900 
 
 3219 
 
 321 
 
 136 
 
 3539 
 
 3858 
 
 4177 
 
 4496 
 
 4814 
 
 5133 
 
 5451 
 
 5769 
 
 6086 
 
 6403 
 
 318 
 
 137 
 
 6721 
 
 7037 
 
 7354 
 
 7671 
 
 7987 
 
 8303 
 
 8618 
 
 8934 
 
 9249 
 
 9564 
 
 315 
 
 138 
 
 9879 
 
 .194 
 
 .508 
 
 .822 
 
 1136 
 
 1450 
 
 1763 
 
 2076 
 
 2389 
 
 2702 
 
 314 
 
 139 
 
 143015 
 
 3327 
 
 3639 
 
 3951 
 
 4263 
 
 4574 
 
 4885 
 
 5196 
 
 5507 
 
 5818 
 
 311 
 
 140 
 
 146128 
 
 6438 
 
 6748 
 
 7058 
 
 7367 
 
 7676 
 
 7985 
 
 8294 
 
 8603 
 
 8911 
 
 309 
 
 141 
 
 9219 
 
 9527 
 
 9835 
 
 .142 
 
 .449 
 
 .756 
 
 1063 
 
 1370 
 
 1676 
 
 1982 
 
 307 
 
 142 
 
 152288 
 
 2594 
 
 2900 
 
 3205 
 
 3510 
 
 3815 
 
 4120 
 
 4424 
 
 4728 
 
 5032 
 
 305 
 
 143 
 
 5336 
 
 5640 
 
 5943 
 
 6246 
 
 6549 
 
 6852 
 
 7154 
 
 7457 
 
 7759 
 
 8061 
 
 303 
 
 144 
 
 8362 
 
 8664 
 
 8965 
 
 9266 
 
 9567 
 
 9868 
 
 .168 
 
 .469 
 
 .769 
 
 1068 
 
 301 
 
 145 
 
 161368 
 
 1667 
 
 1967 
 
 2266 
 
 2564 
 
 2863 
 
 3161 
 
 3460 
 
 3758 
 
 4055 
 
 299 
 
 146 
 
 4353 
 
 4650 
 
 4947 
 
 5244 
 
 5541 
 
 5838 
 
 6134 
 
 6430 
 
 6726 
 
 7022 
 
 297 
 
 147 
 
 7317 
 
 7613 
 
 7908 
 
 8203 
 
 8497 
 
 8792 
 
 9086 
 
 9380 
 
 9674 
 
 9968 
 
 295 
 
 148 
 
 170262 
 
 0555 
 
 0848 
 
 1141 
 
 1434 
 
 1726 
 
 2019 
 
 2311 
 
 2603 
 
 2895 
 
 293 
 
 149 
 
 3186 
 
 3478 
 
 3769 
 
 4060 
 
 4351 
 
 4641 
 
 4932 
 
 5222 
 
 5512 
 
 5802 
 
 291 
 
 150 
 
 176091 
 
 6381 
 
 6670 
 
 6959 
 
 7248 
 
 7536 
 
 7825 
 
 8113 
 
 8401 
 
 8689 
 
 289 
 
 151 
 
 8977 
 
 9264 
 
 9552 
 
 9839 
 
 .126 
 
 .413 
 
 .699 
 
 .985 
 
 1272 
 
 1558 
 
 287 
 
 152 
 
 181844 
 
 2129 
 
 2415 
 
 2700 
 
 2985 
 
 3270 
 
 3555 
 
 3839 
 
 4123 
 
 4407 
 
 285 
 
 153 
 
 4691 
 
 4975 
 
 52.59 
 
 5542 
 
 5825 
 
 6108 
 
 6391 
 
 6674 
 
 6956 
 
 7239 
 
 283 
 
 154 
 
 7521 
 
 7803 
 
 8084 
 
 8366 
 
 8647 
 
 8928 
 
 9209 
 
 9490 
 
 9771 
 
 ..51 
 
 281 
 
 155 
 
 190332 
 
 0612 
 
 0892 
 
 1171 
 
 1451 
 
 1730 
 
 2010 
 
 2289 
 
 2567 
 
 2846 
 
 279 
 
 156 
 
 3125 
 
 3403 
 
 3681 
 
 3959 
 
 4237 
 
 4514 
 
 4792 
 
 5069 
 
 5346 
 
 5623 
 
 278 
 
 157 
 
 5899 
 
 6176 
 
 6453 
 
 6729 
 
 7005 
 
 7281 
 
 7556 
 
 7832 
 
 8107 
 
 8382 
 
 276 
 
 158 
 
 8657 
 
 8932 
 
 9206 
 
 9481 
 
 9755 
 
 ..29 
 
 .303 
 
 .577 
 
 .850 
 
 1124 
 
 274 
 
 159 
 
 201397 
 
 1670 
 
 1943 
 
 2216 
 
 2488 
 
 2761 
 
 3033 
 
 3305 
 
 3577 
 
 3848 
 
 272 
 
 N. | 1 | 2 3 | 4 5 | 6 | 7 | 8 | 9 | D. 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N.j 
 
 
 
 1. 
 
 2 
 
 3 
 
 4 
 
 s 
 
 i; 
 
 7 
 
 8 
 
 ! 
 
 D. 
 
 !(>() 
 
 204120 
 
 4391 
 
 4663 
 
 4934 
 
 5204 
 
 . r >475 
 
 5746 
 
 6016 
 
 62S6 
 
 6556 
 
 271 
 
 161 
 
 6826 
 
 7096 
 
 7365 
 
 7634 
 
 7904 
 
 8173 
 
 8441 
 
 8710 
 
 8979 
 
 9247 
 
 269 
 
 162 
 
 9515 
 
 9783 
 
 ..51 
 
 .319 
 
 .586 
 
 .853 
 
 1121 
 
 1388 
 
 1654 
 
 1921 
 
 267 
 
 1G3 
 
 212188 
 
 2454 
 
 27-20 
 
 2986 
 
 3252 
 
 3518 
 
 3783 
 
 4049 
 
 4314 
 
 4579 
 
 266 
 
 164 
 
 4844 
 
 5109 
 
 5373 
 
 5638 
 
 5902 
 
 6166 
 
 6430 
 
 6694 
 
 6957 
 
 7221 
 
 264 
 
 105 
 
 7484 
 
 7747 
 
 8010 
 
 8273 
 
 8536 
 
 8798 
 
 9060 
 
 9323 
 
 9585 
 
 9846 
 
 262 
 
 166 
 
 220108 
 
 0370 
 
 0631 
 
 0892 
 
 1153 
 
 1414 
 
 1675 
 
 1936 
 
 21% 
 
 2456 
 
 261 
 
 167 
 
 2716 
 
 2976 
 
 3236 
 
 3496 
 
 3755 
 
 4015 
 
 4274 
 
 4533 
 
 4792 
 
 5051 
 
 259 
 
 168 
 
 5309 
 
 5568 
 
 5826 
 
 6084 
 
 6342 
 
 6600 
 
 6858 
 
 7115 
 
 7372 
 
 7630 
 
 258 
 
 169 
 
 7887 
 
 8144 
 
 8400 
 
 8657 
 
 8913 
 
 9170 
 
 9426 
 
 9682 
 
 9938 
 
 .193 
 
 251 i 
 
 170 
 
 230449 
 
 0704 
 
 0960 
 
 1215 
 
 1470 
 
 1724 
 
 1979 
 
 2234 
 
 2488 
 
 2742 
 
 254 
 
 171 
 
 2996 
 
 3250 
 
 3504 
 
 3757 
 
 4011 
 
 4264 
 
 4517 
 
 4770 
 
 5023 
 
 5276 
 
 253 
 
 172 
 
 5528 
 
 5781 
 
 6033 
 
 6285 
 
 6537 
 
 6789 
 
 7041 
 
 7292 
 
 7544 
 
 7795 
 
 252 
 
 173 
 
 8046 
 
 8297 
 
 8548 
 
 8799 
 
 9049 
 
 9299 
 
 9550 
 
 9800 
 
 ..50 
 
 .300 
 
 250 
 
 174 
 
 240549 
 
 0799 
 
 1048 
 
 1297 
 
 1546 
 
 1795 
 
 2044 
 
 2293 
 
 2541 
 
 2790 
 
 249 
 
 175 
 
 3038 
 
 3286 
 
 3534 
 
 3782 
 
 4030 
 
 4277 
 
 4525 
 
 4772 
 
 5019 
 
 5266 
 
 248 
 
 176 
 
 5513 
 
 5759 
 
 6006 
 
 6252 
 
 6499 
 
 6745 
 
 6991 
 
 7237 
 
 7482 
 
 7728 
 
 246 
 
 177 
 
 7973 
 
 8219 
 
 8464 
 
 8709 
 
 8954 
 
 9198 
 
 9443 
 
 9687 
 
 9932 
 
 .176 
 
 245 
 
 178 
 
 250420 
 
 0664 
 
 0908 
 
 1151 
 
 1395 
 
 1638 
 
 1881 
 
 2125 
 
 2368 
 
 2610 
 
 243 
 
 179 
 
 2853 
 
 30% 
 
 3338 
 
 3580 
 
 3822 
 
 4064 
 
 4306 
 
 4548 
 
 4790 
 
 5031 
 
 242 
 
 180 
 
 255273 
 
 5514 
 
 5755 
 
 5996 
 
 6237 
 
 6477 
 
 6718 
 
 6958 
 
 7198 
 
 7439 
 
 241 
 
 181 
 
 7679 
 
 7918 
 
 8158 
 
 8398 
 
 8637 
 
 8877 
 
 9116 
 
 9355 
 
 9594 
 
 9833 
 
 239 
 
 182 
 
 260071 
 
 0310 
 
 0548 
 
 0787 
 
 1025 
 
 1263 
 
 1501 
 
 1739 
 
 1976 
 
 2214 
 
 238 
 
 183 
 
 2451 
 
 2688 
 
 2925 
 
 3162 
 
 3399 
 
 3636 
 
 3873 
 
 4109 
 
 4346 
 
 4582 
 
 237 
 
 184 
 
 4818 
 
 5054 
 
 5290 
 
 5525 
 
 5761. 
 
 5996 
 
 6232 
 
 6467 
 
 6702 
 
 6937 
 
 235 
 
 185 
 
 7172 
 
 7406 
 
 7641 
 
 7875 
 
 8110 
 
 8344 
 
 8578 
 
 8812 
 
 %46 
 
 9279 
 
 234 
 
 186 
 
 9513 
 
 9746 
 
 9980 
 
 .213 
 
 .446 
 
 .679 
 
 .912 
 
 1144 
 
 1377 
 
 1609 
 
 233 
 
 187 
 
 271842 
 
 2074 
 
 2306 
 
 2538 
 
 2770 
 
 3001 
 
 3233 
 
 3464 
 
 36% 
 
 3927 
 
 232 
 
 188 
 
 4158 
 
 4389 
 
 4620 
 
 4850 
 
 5081 
 
 5311 
 
 5542 
 
 5772 
 
 6002 
 
 6232 
 
 230 
 
 189 
 
 6462 
 
 6692 
 
 6921 
 
 7151 
 
 7380 
 
 7609 
 
 7838 
 
 8067 
 
 82% 
 
 8525 
 
 229 
 
 190 
 
 278754 
 
 8982 
 
 9211 
 
 9439 
 
 9667 
 
 9895 
 
 .123 
 
 .351 
 
 .578 
 
 .806 
 
 223 
 
 191 
 
 281033 
 
 1261 
 
 1488 
 
 1715 
 
 1942 
 
 2169 
 
 2396 
 
 2622 
 
 2849 
 
 3075 
 
 227 
 
 192 
 
 3301 
 
 3527 
 
 3753 
 
 3979 
 
 4205 
 
 4431 
 
 4656 
 
 4882 
 
 5107 
 
 5332 
 
 226 
 
 193 
 
 5557 
 
 5782 
 
 6007 
 
 6232 
 
 6456 
 
 6681 
 
 6905 
 
 7130 
 
 7354 
 
 7578 
 
 225 
 
 194 
 
 7802 
 
 8026 
 
 82-19 
 
 8473 
 
 8696 
 
 8920 
 
 9143 
 
 9366 
 
 9589 
 
 9812 
 
 223 
 
 195 
 
 290035 
 
 0257 
 
 0480 
 
 0702 
 
 0925 
 
 1147 
 
 1369 
 
 1591 
 
 1813 
 
 2034 
 
 222 
 
 196 
 
 2256 
 
 2478 
 
 2699 
 
 2920 
 
 3141 
 
 3363 
 
 3584 
 
 3804 
 
 4025 
 
 4246 
 
 221 
 
 197 
 
 4466 
 
 4687 
 
 4907 
 
 5127 
 
 5347 
 
 5567 
 
 5787 
 
 6007 
 
 6226 
 
 6446 
 
 220 
 
 198 
 
 6665 
 
 6884 
 
 7104 
 
 7323 
 
 7542 
 
 7761 
 
 7979 
 
 8198 
 
 8416 
 
 8635 
 
 219 
 
 199 
 
 8853 
 
 9071 
 
 9289 
 
 9507 
 
 9725 
 
 9943 
 
 .161 
 
 .378 
 
 .595 
 
 .813 
 
 218 
 
 200 
 
 301030 
 
 1247 
 
 1464 
 
 1681 
 
 1898 
 
 2114 
 
 2331 
 
 2547 
 
 2764 
 
 2980 
 
 217 
 
 201 
 
 3196 
 
 3412 
 
 3628 
 
 3844 
 
 4059 
 
 4275 
 
 4491 
 
 4706 
 
 4921 
 
 5136 
 
 216 
 
 202 
 
 5351 
 
 5566 
 
 5781 
 
 5996 
 
 6211 
 
 6425 
 
 6639 
 
 6854 
 
 7068 
 
 7282 
 
 215 
 
 203 
 
 7486 
 
 7710 
 
 7924 
 
 8137 
 
 8351 
 
 8564 
 
 8778 
 
 8991 
 
 9204 
 
 9417 
 
 213 
 
 204 
 
 9630 
 
 9843 
 
 ..56 
 
 .268 
 
 .481 
 
 .693 
 
 .906 
 
 1118 
 
 1330 
 
 1542 
 
 212 
 
 305 
 
 311754 
 
 19(i6 
 
 2177 
 
 2389 
 
 2600 
 
 2812 
 
 3023 
 
 3234 
 
 3445 
 
 3656 
 
 211 
 
 206 
 
 3867 
 
 4078 
 
 4289 
 
 4499 
 
 4710 
 
 4920 
 
 5130 
 
 5340 
 
 5551 
 
 5760 
 
 210 
 
 207 
 
 5970 
 
 6180 
 
 6390 
 
 6599 
 
 6809 
 
 7018 
 
 7227 
 
 7436 
 
 7646 
 
 7854 
 
 209 
 
 208 
 
 8063 
 
 8272 
 
 8481 
 
 8689 
 
 8898 
 
 9100 
 
 9314 
 
 9522 
 
 9730 
 
 9938 
 
 208 
 
 209 
 
 320146 
 
 0354 
 
 0562 
 
 0769 
 
 0977 
 
 1184 
 
 1391 
 
 1598 
 
 1805 
 
 2012 
 
 207 
 
 210 
 
 322210 
 
 2426 
 
 2633 
 
 2839 
 
 3046 
 
 3252 
 
 3458 
 
 3665 
 
 3871 
 
 4077 
 
 206 
 
 211 
 
 4282 
 
 4488 
 
 4694 
 
 4899 
 
 5105 
 
 5310 
 
 5516 
 
 5721 
 
 5926 
 
 6131 
 
 205 
 
 212 
 
 633G 
 
 6541 
 
 6745 
 
 6950 
 
 7155 
 
 7359 
 
 7563 
 
 7767 
 
 7972 
 
 8176 
 
 204 
 
 213 
 
 8380 
 
 8583 
 
 8787 
 
 8991 
 
 9194 
 
 9398 
 
 9601 
 
 9805 
 
 ...8 
 
 .211 
 
 203 
 
 214 
 
 330414 
 
 0617 
 
 0819 
 
 1022 
 
 1225 
 
 1427 
 
 1630 
 
 1832 
 
 2034 
 
 2236 
 
 202 
 
 215 
 
 2438 
 
 2640 
 
 2842 
 
 3044 
 
 3246 
 
 3447 
 
 3649 
 
 3850 
 
 4051 
 
 4253 
 
 202 
 
 216 
 
 4454 
 
 4655 
 
 4856 
 
 5057 
 
 5257 
 
 5458 
 
 5658 
 
 5650 
 
 6059 
 
 6-260 
 
 201 
 
 217 
 
 6460 
 
 6660 
 
 6860 
 
 7060 
 
 7260 
 
 7459 
 
 7659 
 
 7858 
 
 8058 
 
 8257 
 
 200 
 
 218 
 
 8456 
 
 8656 
 
 HHf>.-> 
 
 9054 
 
 9253 
 
 9451 
 
 9650 
 
 9849 
 
 ..47 
 
 .246 
 
 199 
 
 2J9 
 
 340444 
 
 0642 
 
 0841 
 
 1039 
 
 1237 
 
 1435 
 
 1632 
 
 1830 
 
 2028 
 
 2225 
 
 198 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 I 
 
 4 
 
 5 
 
 6 I 
 
 7 
 
 8 I 
 
 9 
 
 D. 
 
A TABLE OF LOGARITHMS FROM 1 TO 10.000. 
 
 N. | | 1 | 2 | 3 4 | 5 | 6 | 7 8 | 9 | D. 
 
 220 
 
 342423 
 
 2620 
 
 2817 
 
 3014 
 
 3212 
 
 3409 
 
 3606 
 
 3802 
 
 3999 
 
 4196 
 
 197 
 
 221 
 
 4392 
 
 4589 
 
 4785 
 
 4981 
 
 5178 
 
 5374 
 
 5570 
 
 5766 
 
 591:2 
 
 6157 
 
 196 
 
 222 
 
 . 6353 
 
 6549 
 
 6744 
 
 6939 
 
 7135 
 
 7330 
 
 7525 
 
 7720 
 
 7915 
 
 8110 
 
 195 
 
 223 
 
 8305 
 
 8500 
 
 8694 
 
 8889 
 
 9083 
 
 9278 
 
 9472 
 
 9666 
 
 9860 
 
 ..54 
 
 194 1 
 
 1 224 
 
 350248 
 
 0442 
 
 0636 
 
 0829 
 
 1023 
 
 1216 
 
 1410 
 
 1603 
 
 1790 
 
 1989 
 
 193 - 
 
 225 
 
 2183 
 
 2375 
 
 2568 
 
 2761 
 
 2954 
 
 3147 
 
 3339 
 
 3532 
 
 3724 
 
 3916 
 
 193 
 
 226 
 
 4108 
 
 4301 
 
 4493 
 
 4685 
 
 4876 
 
 5068 
 
 5260 
 
 5452 
 
 5643 
 
 5834 
 
 192 
 
 227 
 
 6026 
 
 6217 
 
 6408 
 
 6599 
 
 6790 
 
 G981 
 
 '172 
 
 7363 
 
 7554 
 
 7744 
 
 191 
 
 228 
 
 7935 
 
 8125 
 
 8316 
 
 8506 
 
 8696 
 
 8886 
 
 9076 
 
 9266 
 
 9456 
 
 9646 
 
 190 
 
 229 
 
 9835 
 
 ..25 
 
 .215 
 
 .404 
 
 .593 
 
 .783 
 
 .972 
 
 1161 
 
 1350 
 
 1539 
 
 189 
 
 230 
 
 361728 
 
 1917 
 
 2105 
 
 2294 
 
 2482 
 
 2671 
 
 2859 
 
 3048 
 
 3236 
 
 3424 
 
 188 
 
 231 
 
 3612 
 
 3800 
 
 3988 
 
 4176 
 
 4363 
 
 4551 
 
 4739 
 
 4926 
 
 5113 
 
 5301 
 
 188 | 
 
 232 
 
 5488 
 
 5675 
 
 5862 
 
 G049 
 
 6236 
 
 6423 
 
 6610 
 
 6796 
 
 6983 
 
 7169 
 
 187 
 
 238 
 
 7356 
 
 7542 
 
 7729 
 
 7915 
 
 8101 
 
 8287 
 
 8473 
 
 8659 
 
 8845 
 
 9030 
 
 186 
 
 234 
 
 9216 
 
 9401 
 
 9587 
 
 9772 
 
 9958 
 
 .143 
 
 .328 
 
 .513 
 
 .698 
 
 .883 
 
 185 
 
 235 
 
 371068 
 
 1253 
 
 1437 
 
 1622 
 
 1806 
 
 1991 
 
 2175 
 
 2360 
 
 2544 
 
 2728 
 
 184 
 
 236 
 
 2912 
 
 3096 
 
 3280 
 
 3464 
 
 3647 
 
 3831 
 
 4015 
 
 4198 
 
 4382 
 
 4565 
 
 184 
 
 237 
 
 4748 
 
 4932 
 
 5115 
 
 5298 
 
 5481 
 
 5664 
 
 5846 
 
 6029 
 
 6212 
 
 6394 
 
 183 
 
 238 
 
 6577 
 
 6759 
 
 6942 
 
 7124 
 
 7306 
 
 7488 
 
 7670 
 
 7852 
 
 8034 
 
 8216 
 
 182 
 
 239 
 
 8398 
 
 8580 
 
 8761 
 
 8943 
 
 9124 
 
 930G 
 
 9487 
 
 9668 
 
 9849 
 
 ..30 
 
 181 
 
 240 
 
 380211 
 
 0392 
 
 0573 
 
 0754 
 
 0934 
 
 1115 
 
 1296 
 
 1476 
 
 1656 
 
 1837 
 
 181 
 
 241 
 
 2017 
 
 2197 
 
 2377 
 
 2557 
 
 2737 
 
 2917 
 
 3097 
 
 3277 
 
 3456 
 
 3636 
 
 180 
 
 242 
 
 3815 
 
 3995 
 
 4174 
 
 4353 
 
 4533 
 
 4712 
 
 4891 
 
 5070 
 
 5249 
 
 5428 
 
 179 
 
 243 
 
 5606 
 
 5785 
 
 5964 
 
 6142 
 
 6321 
 
 6499 
 
 6677 
 
 6856 
 
 7034 
 
 7212 
 
 178 
 
 244 
 
 7390 
 
 7568 
 
 7746 
 
 7923 
 
 8101 
 
 8279 
 
 8456 
 
 8634 
 
 8811 
 
 8989 
 
 178 
 
 245 
 
 9166 
 
 9343 
 
 9520 
 
 9698 
 
 9875 
 
 ..51 
 
 .228 
 
 .405 
 
 .582 
 
 .759 
 
 177 
 
 246 
 
 390935 
 
 1112 
 
 1288 
 
 1464 
 
 1641 
 
 1817 
 
 1993 
 
 21(59 
 
 2345 
 
 2521 
 
 176 
 
 247 
 
 2697 
 
 2873 
 
 3048 
 
 3224 
 
 3400 
 
 3575 
 
 3751 
 
 3926 
 
 4101 
 
 4277 
 
 176 
 
 248 
 
 4452 
 
 4627 
 
 4802 
 
 4977 
 
 5152 
 
 5326 
 
 5501 
 
 5676 
 
 5850 
 
 G025 
 
 175 
 
 249 
 
 6199 
 
 6374 
 
 6548 
 
 6722 
 
 6896 
 
 7071 
 
 7245 
 
 7419 
 
 7592 
 
 7766 
 
 174 
 
 250 
 
 397940 
 
 8114 
 
 8287 
 
 8461 
 
 8634 
 
 8808 
 
 8981 
 
 9154 
 
 9328 
 
 9501 
 
 173 
 
 251 
 
 9674 
 
 9847 
 
 ..20 
 
 .19-2 
 
 .365 
 
 .538 
 
 .711 
 
 .883 
 
 1056 
 
 1228 
 
 173 
 
 252 
 
 401401 
 
 1573 
 
 1745 
 
 1917 
 
 2089 
 
 2261 
 
 2433 
 
 2605 
 
 2777 
 
 2949 
 
 172 
 
 253 
 
 3121 
 
 3292 
 
 3464 
 
 3(535 
 
 3807 
 
 3978 
 
 4149 
 
 4320 
 
 4492 
 
 4663 
 
 171 
 
 254 
 
 4834 
 
 5005 
 
 5176 
 
 5346 
 
 5517 
 
 5688 
 
 5858 
 
 6029 
 
 6199 
 
 6370 
 
 171 
 
 255 
 
 6540 
 
 6710 
 
 6881 
 
 7051 
 
 7221 
 
 7391 
 
 7561 
 
 7731 
 
 7901 
 
 8070 
 
 170 
 
 256 
 
 8240 
 
 8410 
 
 8579 
 
 8749 
 
 8918 
 
 9087 
 
 9257 
 
 9426 
 
 9595 
 
 9764 
 
 1(59 
 
 257 
 
 9933 
 
 .102 
 
 .271 
 
 .440 
 
 .609 
 
 .777 
 
 .946 
 
 1114 
 
 1283 
 
 1451 
 
 169 
 
 258 
 
 411620 
 
 1788 
 
 1956 
 
 2124 
 
 2293 
 
 2461 
 
 2629 
 
 2796 
 
 2964 
 
 3132 
 
 168 
 
 259 
 
 3300 
 
 3467 
 
 3635 
 
 3803 
 
 3970 
 
 4137 
 
 4305 
 
 4472 
 
 4639 
 
 4806 
 
 167 
 
 260 
 
 414973 
 
 5140 
 
 5307 
 
 5474 
 
 5641 
 
 5808 
 
 5974 
 
 6141 
 
 6308 
 
 6474 
 
 167 
 
 261 
 
 6641 
 
 6807 
 
 6973 
 
 7139 
 
 7306 
 
 7472 
 
 7638 
 
 7804 
 
 7970 
 
 8135 
 
 166 
 
 262 
 
 8301 
 
 8467 
 
 8633 
 
 8798 
 
 8964 
 
 9129 
 
 9295 
 
 9460 
 
 9625 
 
 9791 
 
 165 
 
 263 
 
 9956 
 
 .121 
 
 .286 
 
 .451 
 
 .616 
 
 .781 
 
 .945 
 
 1110 
 
 1275 
 
 1439 
 
 165 
 
 264 
 
 421604 
 
 1768 
 
 ]i)33 
 
 2097 
 
 2261 
 
 2426 
 
 2590 
 
 2754 
 
 2918 
 
 3082 
 
 164 
 
 265 
 
 3246 
 
 3410 
 
 3574 
 
 3737 
 
 3901 
 
 4065 
 
 4228 
 
 4392 
 
 4555 
 
 4718 
 
 164 
 
 266 
 
 4882 
 
 5045 
 
 5208 
 
 5371 
 
 5534 
 
 5697 
 
 5860 
 
 6023 
 
 6186 
 
 6349 
 
 163 
 
 267 
 
 6511 
 
 6674 
 
 6336 
 
 6999 
 
 7161 
 
 7324 
 
 7486 
 
 76-48 
 
 7811 
 
 7973 
 
 162 
 
 268 
 
 8135 
 
 8297 
 
 8459 
 
 8621 
 
 8783 
 
 8944 
 
 9106 
 
 9268 
 
 9429 
 
 9591 
 
 162 
 
 269 
 
 9752 
 
 9914 
 
 ..76 
 
 .236 
 
 .398 
 
 .559 
 
 .720 
 
 .881 
 
 1042 
 
 1203 
 
 161 
 
 270 
 
 431364 
 
 1525 
 
 1685 
 
 1846 
 
 2007 
 
 2167 
 
 2328 
 
 2488 
 
 2649 
 
 2809 
 
 161 
 
 271 
 
 2969 
 
 3130 
 
 3290 
 
 3450 
 
 3610 
 
 3770 
 
 3930 
 
 4090 
 
 4249 
 
 4409 
 
 160 
 
 272 
 
 4569 
 
 4729 
 
 4888 
 
 5048 
 
 5207 
 
 5367 
 
 5526 
 
 5685 
 
 5844 
 
 6004 
 
 159 
 
 273 
 
 6163 
 
 6322 
 
 6481 
 
 6640 
 
 6798 
 
 6957 
 
 7116 
 
 7275 
 
 7433 
 
 7592 
 
 159 
 
 274 
 
 7751 
 
 7909 
 
 8007 
 
 8226 
 
 8384 
 
 8542 
 
 8701 
 
 8859 
 
 9017 
 
 9175 
 
 158 
 
 275 
 
 9333 
 
 9491 
 
 9648 
 
 9806 
 
 9964 
 
 .122 
 
 .279 
 
 .437 
 
 .594 
 
 .752 
 
 158 
 
 276 
 
 440909 
 
 1066 
 
 HM4 
 
 1381 
 
 1.538 
 
 1095 
 
 1852 
 
 2009 
 
 2166 
 
 2323 
 
 157 
 
 277 
 
 2480 
 
 2637 
 
 2793 
 
 29M 
 
 3106 
 
 3263 
 
 3419 
 
 3576 
 
 3732 
 
 3889 
 
 157 
 
 278 
 
 4045 
 
 4-201 
 
 4357 
 
 4513 
 
 4669 
 
 4825 
 
 4981 
 
 5137 
 
 5293 
 
 5449 
 
 156 
 
 279 
 
 5604 
 
 5760 
 
 5915 
 
 6071 
 
 6226 
 
 6382 
 
 6537 
 
 6692 
 
 6848 
 
 7003 
 
 155 
 
 JNT| | 1 | 2 
 
 3|4|5|6|7|8|9|D. | 
 
A TABLE OP LOGARITHMS PROM 1 TO 10,000. 
 
 N. | 0|l|2|3|4|5|6|-7 8 | 9 | D. 
 
 280 
 
 447158 
 
 7313 
 
 7468 
 
 7623 
 
 7778 
 
 7933 
 
 8088 
 
 8242 
 
 8397 
 
 8552 
 
 155 
 
 281 
 
 8706 
 
 8861 
 
 9015 
 
 9170 
 
 9324 
 
 9478 
 
 9633 
 
 9787 
 
 9941 
 
 ..95 
 
 154 
 
 282 
 
 450249 
 
 0403 
 
 0557 
 
 0711 
 
 0865 
 
 1018 
 
 1172 
 
 1326 
 
 1479 
 
 1633 
 
 154 
 
 283 
 
 1786 
 
 1940 
 
 2093 
 
 2247 
 
 2400 
 
 2553 
 
 2706 
 
 2859 
 
 3012 
 
 3165 
 
 153 
 
 284 
 
 3318 
 
 3471 
 
 3624 
 
 3777 
 
 3930 
 
 4082 
 
 4235 
 
 4387 
 
 4540 
 
 4(592 
 
 153 
 
 285 
 
 4845 
 
 4997 
 
 5150 
 
 530-2 
 
 5454 
 
 5606 
 
 5758 
 
 5910 
 
 60(52 
 
 6214 
 
 152 
 
 286 
 
 6366 
 
 6518 
 
 6670 
 
 6821 
 
 G973 
 
 7125 
 
 7276 
 
 7428 
 
 7. r >7'.l 
 
 7731 
 
 152 
 
 287 
 
 7882 
 
 8033 
 
 8184 
 
 8336 
 
 8487 
 
 8638 
 
 8789 
 
 8940 
 
 9091 
 
 9242 
 
 151 
 
 288 
 
 9392 
 
 9543 
 
 9694 
 
 9845 
 
 9995 
 
 .146 
 
 .296 
 
 .447 
 
 .597 
 
 .748 
 
 151 
 
 289 
 
 460898 
 
 1048 
 
 1198 
 
 1348 
 
 1499 
 
 1649 
 
 Ii99 
 
 1948 
 
 2098 
 
 2248 
 
 150 
 
 290 
 
 462398 
 
 2548 
 
 2697 
 
 2847 
 
 2997 
 
 3146 
 
 3296 
 
 3445 
 
 3594 
 
 3744 
 
 150 
 
 201 
 
 3893 
 
 4042 
 
 4191 
 
 4340 
 
 4490 
 
 4639 
 
 4788 
 
 4936 
 
 5085 
 
 5234 
 
 149 
 
 903 
 
 5383 
 
 5532 
 
 5680 
 
 5829 
 
 5977 
 
 6126 
 
 6274 
 
 6423 
 
 6571 
 
 6719 
 
 149 
 
 293 
 
 6868 
 
 7016 
 
 7164 
 
 7312 
 
 7460 
 
 7608 
 
 7756 
 
 7904 
 
 8052 
 
 8200 
 
 148 
 
 294 
 
 8347 
 
 8495 
 
 8643 
 
 8790 
 
 8938 
 
 9085 
 
 9233 
 
 9380 
 
 9527 
 
 9675 
 
 148 
 
 295 
 
 9822 
 
 9969 
 
 .116 
 
 .263 
 
 .410 
 
 .557 
 
 .704 
 
 .851 
 
 .998 
 
 1145 
 
 147 
 
 296 
 
 471292 
 
 1438 
 
 1585 
 
 1732 
 
 1878 
 
 2025 
 
 2171 
 
 2318 
 
 2464 
 
 2610 
 
 146 
 
 297 
 
 2756 
 
 2903 
 
 3049 
 
 3195 
 
 3341 
 
 3487 
 
 3633 
 
 3779 
 
 3925 
 
 4071 
 
 146 
 
 298 
 
 4216 
 
 4362 
 
 4508 
 
 4653 
 
 4799 
 
 4944 
 
 5090 
 
 5235 
 
 5381 
 
 5526 
 
 146 
 
 299 
 
 5671 
 
 5816 
 
 5962 
 
 6107 
 
 6252 
 
 6397 
 
 6542 
 
 6687 
 
 6832 
 
 6976 
 
 145 
 
 300 
 
 477121 
 
 7266 
 
 7411 
 
 7555 
 
 7700 
 
 7844 
 
 7989 
 
 8133 
 
 8278 
 
 8422 
 
 145 
 
 301 
 
 8566 
 
 8711 
 
 8855 
 
 8999 
 
 9143 
 
 9287 
 
 9431 
 
 9575 
 
 9719 
 
 9863 
 
 144 
 
 302 
 
 480007 
 
 0151 
 
 0294 
 
 0438 
 
 0582 
 
 0725 
 
 0869 
 
 1012 
 
 1156 
 
 1299 
 
 144 
 
 303 
 
 1443 
 
 1586 
 
 1729 
 
 1872 
 
 2016 
 
 2159 
 
 2302 
 
 2445 
 
 2588 
 
 2731 
 
 143 
 
 304 
 
 2874 
 
 3016 
 
 3159 
 
 33(1-2 
 
 3445 
 
 3587 
 
 3730 
 
 3872 
 
 4015 
 
 4157 
 
 143 
 
 305 
 
 4300 
 
 4442 
 
 4585 
 
 47-27 
 
 4869 
 
 5011 
 
 5153 
 
 5295 
 
 5437 
 
 5579 
 
 142 
 
 306 
 
 5721 
 
 5863 
 
 5005 
 
 6147 
 
 6289 
 
 6430 
 
 6572 
 
 6714 
 
 6855 
 
 .6997 
 
 142 
 
 307 
 
 7138 
 
 7280 
 
 7421 
 
 7563 
 
 7704 
 
 7845 
 
 7986 
 
 8127 
 
 8269 
 
 8410 
 
 141 
 
 308 
 
 8551 
 
 8692 
 
 8833 
 
 8974 
 
 9114 
 
 9255 
 
 9396 
 
 9537 
 
 9677 
 
 9818 
 
 141 
 
 309 
 
 9958 
 
 ..99 
 
 .239 
 
 .380 
 
 .520 
 
 .661 
 
 .801 
 
 .941 
 
 1081 
 
 1222 
 
 140 
 
 310 
 
 491362 
 
 1502 
 
 1642 
 
 1782 
 
 1922 
 
 2062 
 
 2201 
 
 2341 
 
 2481 
 
 2621 
 
 140 
 
 311 
 
 2760 
 
 2900 
 
 3040 
 
 3179 
 
 3319 
 
 3458 
 
 3597 
 
 3737 
 
 3876 
 
 4015 
 
 139 
 
 312 
 
 4155 
 
 4294 
 
 4433 
 
 4572 
 
 4711 
 
 4850 
 
 4989 
 
 5128 
 
 5267 
 
 5406 
 
 139 
 
 313 
 
 5544 
 
 5683 
 
 5822 
 
 5960 
 
 6099 
 
 6238 
 
 6376 
 
 6515 
 
 6653 
 
 6791 
 
 139 
 
 314 
 
 6930 
 
 7068 
 
 7206 
 
 7344 
 
 7483 
 
 7621 
 
 7759 
 
 7897 
 
 8035 
 
 8173 
 
 138 
 
 315 
 
 8311 
 
 8448 
 
 8586 
 
 8724 
 
 8862 
 
 8999 
 
 9137 
 
 9275 
 
 9412 
 
 9550 
 
 138 
 
 316 
 
 9687 
 
 9824 
 
 9962 
 
 ..99 
 
 .236 
 
 .374 
 
 .511 
 
 .648 
 
 .785 
 
 .922 
 
 137 
 
 317 
 
 501059 
 
 1196 
 
 1333 
 
 1470 
 
 1607 
 
 1744 
 
 1880 
 
 2017 
 
 2154 
 
 2291 
 
 137 
 
 318 
 
 2427 
 
 2564 
 
 2700 
 
 2837 
 
 2973 
 
 3109 
 
 3-246 
 
 3382 
 
 3518 
 
 3655 
 
 136 
 
 319 
 
 3791 
 
 3927 
 
 4063 
 
 4199 
 
 4335 
 
 4471 
 
 4607 
 
 4743 
 
 4878 
 
 5014 
 
 136 
 
 320 
 
 505150 
 
 5286 
 
 5421 
 
 5557 
 
 5693 
 
 5828 
 
 5964 
 
 6099 
 
 6234 
 
 6370 
 
 136 
 
 321 
 
 6505 
 
 6640 
 
 6776 
 
 6911 
 
 7046 
 
 7181 
 
 7316 
 
 7451 
 
 7586 
 
 7721 
 
 135 
 
 322 
 
 7856 
 
 7991 
 
 81& 
 
 8260 
 
 8395 
 
 8530 
 
 8664 
 
 8799 
 
 8934 
 
 9068 
 
 135 
 
 323 
 
 9203 
 
 9337 
 
 9471 
 
 9606 
 
 9740 
 
 9874 
 
 ...9 
 
 .143 
 
 .277 
 
 .411 
 
 134 
 
 324 
 
 510545 
 
 0679 
 
 0813 
 
 0947 
 
 1081 
 
 1215 
 
 1349 
 
 1482 
 
 1616 
 
 1750 
 
 134 
 
 325 
 
 1883 
 
 2017 
 
 2151 
 
 2284 
 
 2418 
 
 2551 
 
 2684 
 
 2818 
 
 2951 
 
 3084 
 
 133 
 
 i 326 
 
 3218 
 
 3351 
 
 3484 
 
 3617 
 
 3750 
 
 3883 
 
 4016 
 
 4149 
 
 4282 
 
 4414 
 
 133 
 
 327 
 
 4548 
 
 4681 
 
 4813 
 
 4946 
 
 5079 
 
 5211 
 
 5344 
 
 5476 
 
 5609 
 
 5741 
 
 133 
 
 32H 
 
 5874 
 
 6001) 
 
 6139 
 
 6271 
 
 6403 
 
 6535 
 
 6668 
 
 6800 
 
 6932 
 
 7064 
 
 132 
 
 329 
 
 7196 
 
 7328 
 
 7460 
 
 7592 
 
 7724 
 
 7855 
 
 7987 
 
 8119 
 
 8251 
 
 8382 
 
 132 
 
 330 
 
 5ia->i4 
 
 8646 
 
 8777 
 
 8909 
 
 9040 
 
 9171 
 
 9303 
 
 9434 
 
 9566 
 
 9697 
 
 131 
 
 331 
 
 9828 
 
 9959 
 
 ..90 
 
 .221 
 
 .353 
 
 .484 
 
 .615 
 
 .745 
 
 .876 
 
 1007 
 
 131 
 
 332 
 
 521138 
 
 1269 
 
 1400 
 
 1530 
 
 1661 
 
 1792 
 
 1922 
 
 2053 
 
 2183 
 
 2314 
 
 131 
 
 333 
 
 2444 
 
 2575 
 
 2705 
 
 2835 
 
 2966 
 
 3096 
 
 3226 
 
 3356 
 
 3486 
 
 3616 
 
 130 
 
 334 
 
 3746 
 
 3876 
 
 4006 
 
 4136 
 
 4266 
 
 4396 
 
 4526 
 
 4656 
 
 4785 
 
 4915 
 
 130 
 
 335 
 
 5045 
 
 5174 
 
 5304 
 
 5434 
 
 5563 
 
 5693 
 
 58S3 
 
 5951 
 
 6081 
 
 6210 
 
 129 
 
 336 
 
 6339 
 
 6469 
 
 6598 
 
 6727 
 
 6856 
 
 6985 
 
 7114 
 
 7243 
 
 737-2 
 
 7501 
 
 129 
 
 337 
 
 7630 
 
 7759 
 
 7888 
 
 8016 
 
 8145 
 
 8274 
 
 8402 
 
 8531 
 
 8660 
 
 8788 
 
 129 
 
 338 
 
 8917 
 
 9045 
 
 9174 
 
 9302 
 
 9430 
 
 9559 
 
 9687 
 
 9815 
 
 9943 
 
 ..72 
 
 128 
 
 339 
 
 530200 
 
 0328 
 
 0456 
 
 0584 
 
 0712 
 
 0840 
 
 0968 
 
 1096 
 
 1223 
 
 1351 
 
 128 
 
 N. | | 1 | 2 | 3 | 4 | 5 |6|7 8 | 9 | D. 
 
A TABLE OF LOGARITHMS FROM 1 TO ] 0,000. 
 
 N. 1 
 
 2|3 4 5 6|7 8|9 I). 
 
 340 
 
 531479 
 
 1607 
 
 1734 
 
 1862 
 
 1990 
 
 2117 
 
 2245 
 
 2372 
 
 2501) 
 
 -J&JV 
 
 128 
 
 341 
 
 2754 
 
 2882 
 
 3009 
 
 3136 
 
 3264 
 
 3391 
 
 3518 
 
 3645 
 
 3772 
 
 3899 
 
 127 
 
 342 
 
 4026 
 
 4153 
 
 4280 
 
 4407 
 
 4534 
 
 4661 
 
 4787 
 
 4914 
 
 5041 
 
 5167 
 
 127 
 
 343 
 
 5294 
 
 5421 
 
 5547 
 
 5674 
 
 5800 
 
 5927 
 
 6053 
 
 6180 
 
 6306 
 
 6432 
 
 126 
 
 344 
 
 6558 
 
 6685 
 
 6811 
 
 6937 
 
 7063 
 
 7189 
 
 7315 
 
 7441 
 
 7567 
 
 7693 
 
 126 
 
 345 
 
 7819 
 
 7945 
 
 8071 
 
 8197 
 
 8322 
 
 8448 
 
 8574 
 
 8699 
 
 8825 
 
 8951 
 
 126 
 
 346 
 
 9076 
 
 9202 
 
 9327 
 
 9452 
 
 9578 
 
 9703 
 
 9829 
 
 9954 
 
 ..79 
 
 .204 
 
 125 
 
 347 
 
 540329 
 
 0455 
 
 0580 
 
 0705 
 
 0830 
 
 0955 
 
 1080 
 
 1205 
 
 1330 
 
 1454 
 
 125 
 
 348 
 
 1579 
 
 1704 
 
 1829 
 
 1953 
 
 2078 
 
 2203 
 
 2327 
 
 2452 
 
 2576 
 
 2701 
 
 125 
 
 340 
 
 2825 
 
 2950 
 
 3074 
 
 3199 
 
 3323 
 
 3447 
 
 3571 
 
 3696 
 
 3820 
 
 3944 
 
 124 
 
 350 
 
 544068 
 
 4192 
 
 4316 
 
 4440 
 
 4564 
 
 4688 
 
 4812 
 
 4936 
 
 5060 
 
 5183 
 
 124 
 
 351 
 
 5307 
 
 5431 
 
 5555 
 
 5678 
 
 5802 
 
 5925 
 
 6049 
 
 6172 
 
 6296 
 
 6419 
 
 124 
 
 352 
 
 6543 
 
 6666 
 
 6789 
 
 6913 
 
 7036 
 
 7159 
 
 7282 
 
 7405 
 
 7529 
 
 7652 
 
 123 
 
 353 
 
 7775 
 
 7898 
 
 8021 
 
 8144 
 
 8267 
 
 8389 
 
 8512 
 
 8635 
 
 8758 
 
 8881 
 
 123 
 
 354 
 
 9003 
 
 9126 
 
 9249 
 
 9371 
 
 9494 
 
 9f>16 
 
 9739 
 
 9861 
 
 9984 
 
 .106 
 
 123 
 
 355 
 
 550228 
 
 0351 
 
 0473 
 
 0595 
 
 0717 
 
 0840 
 
 0962 
 
 1084 
 
 1206 
 
 1328 
 
 122 
 
 356 
 
 1450 
 
 1572 
 
 1694 
 
 1816 
 
 1938 
 
 2060 
 
 2181 
 
 2303 
 
 2425 
 
 2547 
 
 122 
 
 357 
 
 2668 
 
 2790 
 
 2911 
 
 3033 
 
 3155 
 
 3276 
 
 3398 
 
 3519 
 
 3640 
 
 3762 
 
 121 
 
 358 
 
 3883 
 
 4004 
 
 4126 
 
 4247 
 
 4368 
 
 4489 
 
 4610 
 
 4731 
 
 4852 
 
 4973 
 
 121 
 
 359 
 
 5094 
 
 5215 
 
 5336 
 
 5457 
 
 5578 
 
 5699 
 
 5820 
 
 5940 
 
 6061 
 
 6182 
 
 121 
 
 360 
 
 556333 
 
 6423 
 
 6544 
 
 6664 
 
 6785 
 
 6905 
 
 7026 
 
 7146 
 
 7267 
 
 7387 
 
 120 
 
 361 
 
 7507 
 
 7627 
 
 7748 
 
 7868 
 
 7988 
 
 8108 
 
 8228 
 
 8349 
 
 8469 
 
 8589 
 
 120 
 
 362 
 
 8709 
 
 8829 
 
 8948 
 
 9068 
 
 9188 
 
 9308 
 
 9428 
 
 9548 
 
 9(367 
 
 9787 
 
 120 
 
 363 
 
 9907 
 
 ..26 
 
 .146 
 
 .265 
 
 .385 
 
 .504 
 
 .624 
 
 .743 
 
 .863 
 
 .982 
 
 119 
 
 364 
 
 561101 
 
 1221 
 
 1340 
 
 1459 
 
 1578 
 
 1698 
 
 1817 
 
 1936 
 
 2055 
 
 2174 
 
 119 
 
 365 
 
 2293 
 
 2412 
 
 2531 
 
 2650 
 
 2769 
 
 2887 
 
 3006 
 
 3125 
 
 3244 
 
 3362 
 
 119 
 
 356 
 
 3481 
 
 3630 
 
 3718 
 
 3837 
 
 3955 
 
 4074 
 
 4102 
 
 4311 
 
 4429 
 
 4548 
 
 119 
 
 367 
 
 4686 
 
 4784 
 
 4903 
 
 5021 
 
 5139 
 
 5257 
 
 5376 
 
 5494 
 
 5612 
 
 5730 
 
 118 
 
 368 
 
 5848 
 
 5966 
 
 6J84 
 
 6232 
 
 6320 
 
 6437 
 
 6555 
 
 6673 
 
 6791 
 
 0909 
 
 118 
 
 369 
 
 7026 
 
 7144 
 
 7262 
 
 7379 
 
 7497 
 
 7614 
 
 7732 
 
 7849 
 
 7967 
 
 8084 
 
 118 
 
 370 
 
 568202 
 
 8319 
 
 8436 
 
 8554 
 
 8671 
 
 8788 
 
 8905 
 
 9023 
 
 9140 
 
 9257 
 
 117 
 
 371 
 
 9374 
 
 9491 
 
 9608 
 
 9725 
 
 9842 
 
 9959 
 
 -.76 
 
 .193 
 
 309 
 
 .426 
 
 117 
 
 372 
 
 570543 
 
 0660 
 
 0776 
 
 0893 
 
 1010 
 
 1126 
 
 1243 
 
 1359 
 
 1476 
 
 1592 
 
 117 
 
 373 
 
 1709 
 
 1825 
 
 1942 
 
 2358 
 
 2174 
 
 2291 
 
 2407 
 
 2523 
 
 2639 
 
 2755 
 
 116 
 
 374 
 
 2872 
 
 2988 
 
 3104 
 
 3220 
 
 3336 
 
 3452 
 
 3568 
 
 3684 
 
 38.K) 
 
 3915 
 
 116 
 
 375 
 
 4031 
 
 4147 
 
 4263 
 
 4379 
 
 4494 
 
 4610 
 
 4726 
 
 4841 
 
 4957 
 
 5072 
 
 116 
 
 376 
 
 5188 
 
 5303 
 
 5419 
 
 5534 
 
 5650 
 
 5765 
 
 5880 
 
 5996 
 
 6111 
 
 622(3 
 
 115 
 
 377 
 
 6341 
 
 6457 
 
 6572 
 
 6687 
 
 6832 
 
 6917 
 
 7032 
 
 7147 
 
 7262 
 
 7377 
 
 115 
 
 378 
 
 7492 
 
 7607 
 
 7722 
 
 7836 
 
 7951 
 
 8066 
 
 8181 
 
 8295 
 
 8410 
 
 8525 
 
 115 
 
 379 
 
 8639 
 
 8754 
 
 8868 
 
 8983 
 
 9097 
 
 9212 
 
 9326 
 
 9441 
 
 9555 
 
 9669 
 
 114 
 
 380 
 
 579784 
 
 9898 
 
 ..12 
 
 .126 
 
 .241 
 
 .355 
 
 .469 
 
 .583 
 
 .697 
 
 .811 
 
 114 
 
 381 
 
 580925 
 
 1039 
 
 1153 
 
 1267 
 
 1381 
 
 1495 
 
 1608 
 
 1722 
 
 1836 
 
 1950 
 
 114 
 
 382 
 
 2063 
 
 2177 
 
 2291 
 
 2404 
 
 2518 
 
 2631 
 
 2745 
 
 2858 
 
 2972 
 
 3085 
 
 114 
 
 383 
 
 3199 
 
 3312 
 
 3426 
 
 3539 
 
 3652 
 
 3765 
 
 3879 
 
 3992 
 
 4105 
 
 4218 
 
 113 
 
 384 
 
 4331 
 
 4444 
 
 4557 
 
 4670 
 
 4783 
 
 4896 
 
 5039 
 
 5122 
 
 5235 
 
 5348 
 
 113 
 
 385 
 
 5461 
 
 5574 
 
 5686 
 
 5799 
 
 5912 
 
 6024 
 
 6137 
 
 6250 
 
 6362 
 
 6475 
 
 113 
 
 386 
 
 6587 
 
 6700 
 
 6812 
 
 6925 
 
 7037 
 
 7149 
 
 7262 
 
 7374 
 
 7486 
 
 7599 
 
 112 
 
 387 
 
 7711 
 
 7823 
 
 7935 
 
 8047 
 
 8160 
 
 8272 
 
 8384 
 
 8496 
 
 8608 
 
 8720 
 
 112 
 
 388 
 
 8832 
 
 8944 
 
 9056 
 
 9167 
 
 9279 
 
 9391 
 
 9503 
 
 9(315 
 
 9726 
 
 9838 
 
 112 
 
 389 
 
 9950 
 
 ..61 
 
 .173 
 
 .284 
 
 .396 
 
 .507 
 
 .619 
 
 .730 
 
 .842 
 
 .953 
 
 112 
 
 390 
 
 591065 
 
 1178 
 
 1287 
 
 1399 
 
 1510 
 
 1621 
 
 1732 
 
 1843 
 
 1955 
 
 2066 
 
 111 
 
 391 
 
 2177 
 
 2283 
 
 2399 
 
 2510 
 
 2621 
 
 2732 
 
 2843 
 
 2934 
 
 3064 
 
 3175 
 
 111 
 
 392 
 
 3286 
 
 3397 
 
 3508 
 
 3618 
 
 3729 
 
 3840 
 
 3950 
 
 4061 
 
 4171 
 
 4282 
 
 111 
 
 393 
 
 4393 
 
 4503 
 
 4614 
 
 4724 
 
 4834 
 
 '4945 
 
 5055 
 
 5165 
 
 5276 
 
 5386 
 
 110 
 
 394 
 
 5496 
 
 5606 
 
 5717 
 
 5827 
 
 5937 
 
 6047 
 
 6157 
 
 6267 
 
 6377 
 
 6487 
 
 110 
 
 395 
 
 6597 
 
 6707 
 
 6817 
 
 6927 
 
 7037 
 
 7146 
 
 7256 
 
 7366 
 
 7476 
 
 7586 
 
 110 
 
 396 
 
 7695 
 
 7805 
 
 7914 
 
 8024 
 
 8134 
 
 8243 
 
 8353 
 
 8462 
 
 8572 
 
 8681 
 
 110 
 
 397 
 
 8791 
 
 8900 
 
 9309 
 
 9119 
 
 9228 
 
 9337 
 
 9446 
 
 9556 
 
 9565 
 
 9774 
 
 109 
 
 398 
 
 9883 
 
 9992 
 
 .10] 
 
 .210 
 
 .319 
 
 .428 
 
 .537 
 
 .646 
 
 .755 
 
 .854 
 
 109 
 
 399 
 
 600973 
 
 1082 
 
 1191 
 
 1299 
 
 1408 
 
 1517 
 
 1625 
 
 1734 
 
 1843 
 
 1951 
 
 109 
 
 N. 1 | 2 
 
 3 | 4 | 5 6(7 
 
 8 | 9 | D. 
 
A TABLE or LOGARITHMS FROM 1 TO 10,000. 
 
 N. |0|1|2|3|4|5 6 | 7 8 | 9 D. 
 
 400 
 
 6021X50 
 
 2169 
 
 2277 
 
 2386 
 
 2494 
 
 2(503 
 
 2711 
 
 2819 
 
 2tS8 
 
 303(5 
 
 108 
 
 401 
 
 3144 
 
 :{-2:>:i 
 
 33(51 
 
 34159 
 
 3577 
 
 3(5815 
 
 3794 
 
 3902 
 
 4010 
 
 4118 
 
 108 
 
 402 
 
 42-26 
 
 4334 
 
 4442 
 
 4.~).">0 
 
 465H 
 
 4760 
 
 4874 
 
 4982 
 
 5089 
 
 5197 
 
 108 
 
 403 
 
 5305 
 
 5413 
 
 5521 
 
 5'WH 
 
 5736 
 
 5844 
 
 5951 
 
 6059 
 
 6166 
 
 6274 
 
 108 
 
 404 
 
 6381 
 
 6489 
 
 6f>9(5 
 
 6704 
 
 6811 
 
 (UU'.I 
 
 702I5 
 
 7133 
 
 7241 
 
 7348 
 
 107 
 
 405 
 
 7455 
 
 75152 
 
 76(59 
 
 7777 
 
 7884 
 
 T'.I'.M 
 
 8098 
 
 8205 
 
 8312 
 
 8419 
 
 107 
 
 406 
 
 8526 
 
 81533 
 
 8740 
 
 8847 
 
 8954 
 
 9061 
 
 9167 
 
 9274 
 
 9381 
 
 9488 
 
 107 
 
 407 
 
 B594 
 
 'J7.ll 
 
 9808 
 
 1)914 
 
 ..21 
 
 .128 
 
 .234 
 
 .341 
 
 .447 
 
 .554 
 
 107' 
 
 408 
 
 6106(50 
 
 07(57 
 
 0873 
 
 0979 
 
 108(5 
 
 1192 
 
 1298 
 
 1405 
 
 1511 
 
 1617 
 
 106 
 
 41)9 
 
 17-2:5 
 
 1829 
 
 19315 
 
 2042 
 
 2148 
 
 2254 
 
 2360 
 
 2466 
 
 2572 
 
 2678 
 
 106 
 
 410 
 
 612784 
 
 2890 
 
 2998 
 
 3102 
 
 3207 
 
 3313 
 
 3419 
 
 3525 
 
 3630 
 
 3736 
 
 106 
 
 411 
 
 3842 
 
 3047 
 
 4053 
 
 4159 
 
 42(54 
 
 4370 
 
 4475 
 
 4581 
 
 468(5 
 
 4-;<)2 
 
 100 
 
 412 
 
 4897 
 
 5003 
 
 5108 
 
 5213 
 
 5319 
 
 5424 
 
 5529 
 
 5634 
 
 5740 
 
 5845 
 
 105 
 
 413 
 
 5950 
 
 6055 
 
 6160 
 
 6265 
 
 6370 
 
 647(5 
 
 6581 
 
 6686 
 
 6790 
 
 6895 
 
 105 
 
 414 
 
 7000 
 
 7105 
 
 7210 
 
 7315 
 
 7420 
 
 7525 
 
 7629 
 
 7734 
 
 7839 
 
 7943 
 
 105 
 
 415 
 
 8048 
 
 8153 
 
 8257 
 
 8362 
 
 8466 
 
 8571 
 
 8676 
 
 8780 
 
 8884 
 
 8989 
 
 105 
 
 416 
 
 9093 
 
 9198 
 
 !)3()-2 
 
 9406 
 
 9511 
 
 9615 
 
 9719 
 
 9824 
 
 9928 
 
 ..32 
 
 104 
 
 417 
 
 620136 
 
 0240 
 
 0344 
 
 0448 
 
 0552 
 
 0658 
 
 0760 
 
 0864 
 
 0968 
 
 1072 
 
 104 
 
 418 
 
 1176 
 
 1280 
 
 1384 
 
 1488 
 
 1592 
 
 1695 
 
 1799 
 
 1903 
 
 2007 
 
 2110 
 
 104 
 
 419 
 
 2214 
 
 2318 
 
 2421 
 
 2525 
 
 2628 
 
 2732 
 
 2835 
 
 2939 
 
 3042 
 
 3146 
 
 104 
 
 420 
 
 623249 
 
 3353 
 
 3456 
 
 3559 
 
 3663 
 
 3766 
 
 3869 
 
 3973 
 
 4076 
 
 4179 
 
 103 
 
 421 
 
 4282 
 
 4385 
 
 4488 
 
 4591 
 
 4695 
 
 4798 
 
 4901 
 
 5004 
 
 5107 
 
 5210 
 
 103 
 
 422 
 
 5312 
 
 5415 
 
 5518 
 
 5(521 
 
 5724 
 
 5827 
 
 5929 
 
 6032 
 
 6135 
 
 6238 
 
 103 
 
 423 
 
 6340 
 
 6443 
 
 6546 
 
 6648 
 
 6751 
 
 6853 
 
 6956 
 
 7058 
 
 7161 
 
 7263 
 
 103 
 
 4-24 
 
 7:;ii 
 
 7468 
 
 7571 
 
 7673 
 
 7775 
 
 7878 
 
 7980 
 
 8082 
 
 8185 
 
 8287 
 
 102 
 
 425 
 
 8389 
 
 8491 
 
 8593 
 
 8695 
 
 8797 
 
 8900 
 
 9002 
 
 9104 
 
 9206 
 
 9308 
 
 102 
 
 426 
 
 9410 
 
 9512 
 
 9613 
 
 9715 
 
 9817 
 
 9919 
 
 ..21 
 
 .123 
 
 .224 
 
 .326 
 
 102 
 
 427 
 
 630428 
 
 0530 
 
 0631 
 
 0733 
 
 0835 
 
 0936 
 
 1038 
 
 1139 
 
 1241 
 
 J342 
 
 102 
 
 428 
 
 1444 
 
 1545 
 
 1647 
 
 1748 
 
 1849 
 
 1951 
 
 2052 
 
 2153 
 
 2255 
 
 2356 
 
 101 
 
 429 
 
 2457 
 
 2559 
 
 2660 
 
 2761 
 
 2882 
 
 2953 
 
 3064 
 
 3165 
 
 3266 
 
 3367 
 
 101 
 
 430 
 
 633468 
 
 3569 
 
 3670 
 
 3771 
 
 3872 
 
 3973 
 
 4074 
 
 4175 
 
 4276 
 
 4376 
 
 100 
 
 431 
 
 4477 
 
 4578 
 
 4679 
 
 4779 
 
 4880 
 
 4981 
 
 5081 
 
 5182 
 
 5283 
 
 5383 
 
 100 
 
 4.i2 
 
 5484 
 
 5584 
 
 5685 
 
 5785 
 
 5886 
 
 5986 
 
 6087 
 
 6187 
 
 6287 
 
 6388 
 
 100 
 
 433 
 
 6488 
 
 6588 
 
 61T88 
 
 6789 
 
 6889 
 
 (5989 
 
 7089 
 
 7189 
 
 7290 
 
 7390 
 
 100 
 
 434 
 
 7493 
 
 7590 
 
 7690 
 
 7790 
 
 7890 
 
 7990 
 
 8090 
 
 8190 
 
 8291) 
 
 8389 
 
 99 
 
 435 
 
 848J 
 
 8589 
 
 8689 
 
 8789 
 
 8888 
 
 8988 
 
 9088 
 
 9188 
 
 9287 
 
 9387 
 
 99 
 
 436 
 
 948(5 
 
 9586 
 
 9686 
 
 9785 
 
 9885 
 
 9984 
 
 ..84 
 
 .183 
 
 .283 
 
 .382 
 
 99 
 
 437 
 
 640481 
 
 0581 
 
 0680 
 
 0779 
 
 0879 
 
 0978 
 
 1077 
 
 1177 
 
 1276 
 
 1375 
 
 99 
 
 438 
 
 1474 
 
 1573 
 
 1672 
 
 1771 
 
 1871 
 
 1970 
 
 2069 
 
 2168 
 
 2267 
 
 2366 
 
 99 
 
 439 
 
 2465 
 
 2563 
 
 2662 
 
 2761 
 
 2860 
 
 2959 
 
 3058 
 
 3156 
 
 3255 
 
 3354 
 
 99 
 
 440 
 
 r,i:u.-,3 
 
 3551 
 
 3650 
 
 3749 
 
 3847 
 
 3946 
 
 4044 
 
 4143 
 
 4242 
 
 4340 
 
 98 
 
 441 
 
 4439 
 
 4537 
 
 4636 
 
 4734 
 
 4832 
 
 4931 
 
 5029 
 
 5127 
 
 522(5 
 
 .1324 
 
 98 
 
 442 
 
 54-22 
 
 5521 
 
 5619 
 
 5717 
 
 5815 
 
 5913 
 
 6011 
 
 6110 
 
 6208 
 
 6306 
 
 98 
 
 443 
 
 6404 
 
 6502 
 
 0600 
 
 6698 
 
 6796 
 
 6894 
 
 6992 
 
 7089 
 
 7187 
 
 7285 
 
 98 
 
 444 
 
 7383 
 
 7481 
 
 7579 
 
 7676 
 
 7774 
 
 7H72 
 
 7969 
 
 8067 
 
 8165 
 
 8262 
 
 98 
 
 445 
 
 8360 
 
 8458 
 
 8555 
 
 8H53 
 
 8750 
 
 8848 
 
 8945 
 
 9043 
 
 9140 
 
 9237 
 
 97 
 
 4 tii 
 
 9335 
 
 9432 
 
 9530 
 
 9827 
 
 9724 
 
 9821 
 
 9919 
 
 ..16 
 
 .113 
 
 .210 
 
 97 
 
 447 
 
 650308 
 
 0405 
 
 0502 
 
 0599 
 
 0896 
 
 0793 
 
 0890 
 
 0987 
 
 1084 
 
 1181 
 
 97 
 
 448 
 
 1278 
 
 1375 
 
 1472 
 
 1569 
 
 1666 
 
 1762 
 
 1859 
 
 1956 
 
 2053 
 
 2150 
 
 97 
 
 449 
 
 2246 
 
 2343 
 
 2440 
 
 2536 
 
 2633 
 
 2730 
 
 2826 
 
 2923 
 
 3019 
 
 3116 
 
 97 
 
 450 
 
 653213 
 
 3309 
 
 3405 
 
 3502 
 
 3598 
 
 3695 
 
 3791 
 
 3888 
 
 3984 
 
 4080 
 
 96 
 
 451 
 
 4177 
 
 4273 
 
 43(59 
 
 4465 
 
 4562 
 
 4658 
 
 4754 
 
 4850 
 
 4946 
 
 5042 
 
 96 
 
 452 
 
 5138 
 
 5235 
 
 5331 
 
 5427 
 
 5523 
 
 5819 
 
 5715 
 
 5810 
 
 5906 
 
 6002 
 
 96 
 
 453 
 
 BOOS 
 
 6194 
 
 6290 
 
 6386 
 
 6482 
 
 6577 
 
 6673 
 
 6769 
 
 6864 
 
 6960 
 
 96 
 
 454 
 
 7056 
 
 7152 
 
 7247 
 
 7343 
 
 7438 
 
 7534 
 
 7629 
 
 7725 
 
 7820 
 
 7916 
 
 96 
 
 455 
 
 8011 
 
 8107 
 
 8202 
 
 8298 
 
 8393 
 
 8488 
 
 8584 
 
 8679 
 
 8774 
 
 8870 
 
 95 
 
 456 
 
 8955 
 
 9060 
 
 9155 
 
 9250 
 
 9346 
 
 9441 
 
 9536 
 
 9631 
 
 9726 
 
 9821 
 
 95 
 
 457 
 
 9916 
 
 ..11 
 
 .105 
 
 .201 
 
 .295 
 
 .331 
 
 .486 
 
 .581 
 
 .676 
 
 .771 
 
 95 
 
 458 
 
 660865 
 
 0960 
 
 1055 
 
 1150 
 
 1-245 
 
 1339 
 
 1434 
 
 1529 
 
 1623 
 
 1718 
 
 95 
 
 459 
 
 1813 
 
 1907 
 
 2002 
 
 2098 
 
 2191 
 
 2286 
 
 23HO 
 
 2475 
 
 2569 
 
 2663 
 
 95 
 
 N. | | 1 | 2 | 3 j 4 I 5 6 | 7 8 | 9 | D. 
 
A TABLE OF LOGARITHMS FROM 1 TO 10.000. 
 
 N. | 1|2J3|4|5|6|7|8|9|D. 
 
 460 
 
 662758 
 
 2852 
 
 2947 
 
 3041 
 
 3135 
 
 3230 
 
 3324 
 
 3418 
 
 3512 
 
 3607 
 
 94 
 
 461 
 
 3701 
 
 3795 
 
 3889 
 
 3983 
 
 4078 
 
 4172 
 
 4266 
 
 4360 
 
 4454 
 
 4548 
 
 94 
 
 462 
 
 4642 
 
 4736 
 
 4830 
 
 4924 
 
 5018 
 
 5112 
 
 5206 
 
 5299 
 
 5393 
 
 5487 
 
 94 
 
 463 
 
 5581 
 
 5675 
 
 5769 
 
 5862 
 
 5955 
 
 6050 
 
 6143 
 
 6237 
 
 6331 
 
 64-24 
 
 94 
 
 .464 
 
 6518 
 
 6612 
 
 0705 
 
 6799 
 
 6892 
 
 0986 
 
 7079 
 
 7173 
 
 7266 
 
 7360 
 
 94 
 
 465 
 
 7453 
 
 7546 
 
 7640 
 
 7733 
 
 782a 
 
 7920 
 
 8013 
 
 8106 
 
 8199 
 
 8293 
 
 93 
 
 466 
 
 838ti 
 
 8479 
 
 8572 
 
 8665 
 
 8759 
 
 8852 
 
 8945 
 
 9038 
 
 9131 
 
 9224 
 
 93 
 
 407 
 
 9317 
 
 9410 
 
 9503 
 
 9596 
 
 9689 
 
 9782 
 
 9875 
 
 9967 
 
 ..60 
 
 .153 
 
 93 
 
 468 
 
 670246 
 
 0339 
 
 0431 
 
 0524 
 
 0617 
 
 0710 
 
 0802 
 
 0895 
 
 0988 
 
 1080 
 
 93 
 
 469 
 
 1173 
 
 1265 
 
 1358 
 
 1451 
 
 1513 
 
 1636 
 
 1728 
 
 1821 
 
 1913 
 
 2005 
 
 93 
 
 470 
 
 672098 
 
 2190 
 
 2283 
 
 2375 
 
 2467 
 
 2560 
 
 2652 
 
 2744 
 
 2836 
 
 2929 
 
 92 , 
 
 471 
 
 3021 
 
 3113 
 
 3205 
 
 3297 
 
 3390 
 
 3482 
 
 3574 
 
 3666 
 
 3758 
 
 3850 
 
 92 
 
 472 
 
 3942 
 
 4034 
 
 4126 
 
 4218 
 
 4310 
 
 4402 
 
 4494 
 
 4586 
 
 4677 
 
 4769 
 
 92 
 
 473 
 
 4861 
 
 4953 
 
 5045 
 
 5137 
 
 5228 
 
 5320 
 
 5412 
 
 5503 
 
 5595 
 
 5687 
 
 92 
 
 474 
 
 5778 
 
 5870 
 
 5962 
 
 6053 
 
 6145 
 
 6236 
 
 6328 
 
 6419 
 
 6511 
 
 6602 
 
 92 
 
 475 
 
 6694 
 
 6785 
 
 6876 
 
 6968 
 
 7059 
 
 7151 
 
 7242 
 
 7333 
 
 7424 
 
 751b 
 
 91 
 
 476 
 
 7607 
 
 7698 
 
 7789 
 
 7881 
 
 7972 
 
 8063 
 
 8154 
 
 8245 
 
 8336 
 
 8427 
 
 91 
 
 477 
 
 8518 
 
 8609 
 
 8700 
 
 8791 
 
 8882 
 
 8973 
 
 9064 
 
 9155 
 
 9246 
 
 9337 
 
 91 
 
 478 
 
 9428 
 
 9519 
 
 9610 
 
 9700 
 
 9791 
 
 9882 
 
 9D73 
 
 ..63 
 
 .154 
 
 .245 
 
 91 
 
 479 
 
 680336 
 
 0426 
 
 0517 
 
 0607 
 
 0698 
 
 0789 
 
 0879 
 
 0970 
 
 1060 
 
 1151 
 
 91 
 
 480 
 
 681241 
 
 1332 
 
 1422 
 
 1513 
 
 1603 
 
 1693 
 
 1784 
 
 1874 
 
 1964 
 
 2055 
 
 90 
 
 481 
 
 2145 
 
 2235 
 
 2326 
 
 2416 
 
 2506 
 
 2598 
 
 2686 
 
 2777 
 
 2867 
 
 2957 
 
 90 
 
 482 
 
 3047 
 
 3137 
 
 3227 
 
 3317 
 
 3407 
 
 3497 
 
 3587 
 
 3677 
 
 3767 
 
 3857 
 
 90 
 
 483 
 
 3917 
 
 4037 
 
 4127 
 
 4217 
 
 4307 
 
 4396 
 
 4486 
 
 4576 
 
 4666 
 
 4756 
 
 90 
 
 484 
 
 4845 
 
 4935 
 
 5025 
 
 5114 
 
 5204 
 
 5294 
 
 5383 
 
 5473 
 
 5563 
 
 5652 
 
 90 
 
 485 
 
 5742 
 
 5831 
 
 5921 
 
 6010 
 
 6100 
 
 6189 
 
 6279 
 
 6368 
 
 6458 
 
 6547 
 
 89 
 
 486 
 
 6036 
 
 6726 
 
 6815 
 
 6904 
 
 6994 
 
 7083 
 
 7172 
 
 7261 
 
 7351 
 
 7440 
 
 89 
 
 487 
 
 7529 
 
 7618 
 
 7707 
 
 7796 
 
 7886 
 
 7975 
 
 8064 
 
 8153 
 
 8242 
 
 8331 
 
 89 
 
 488 
 
 8420 
 
 8509 
 
 8598 
 
 8ii87 
 
 8776 
 
 88W 
 
 8953 
 
 9042 
 
 9131 
 
 9220 
 
 89 
 
 489 
 
 9309 
 
 9398 
 
 9486 
 
 9575 
 
 9664 
 
 9753 
 
 9841 
 
 9930 
 
 ..19 
 
 .107 
 
 89 
 
 490 
 
 690196 
 
 0285 
 
 0373 
 
 0462 
 
 0550 
 
 0639 
 
 0728 
 
 0816 
 
 0905 
 
 0993 
 
 89 
 
 491 
 
 1081 
 
 1170 
 
 1258 
 
 1347 
 
 1435 
 
 1524 
 
 1612 
 
 1700 
 
 1789 
 
 1877 
 
 88 
 
 492 
 
 1965 
 
 2053 
 
 2142 
 
 2230 
 
 2318 
 
 2406 
 
 2494 
 
 2583 
 
 2671 
 
 2759 
 
 88 
 
 493 
 
 2847 
 
 2935 
 
 3J23 
 
 3111 
 
 3199 
 
 3287 
 
 3375 
 
 3463 
 
 3551 
 
 3(539 
 
 88 
 
 494 
 
 3727 
 
 3815 
 
 3903 
 
 399 1 
 
 4078 
 
 4166 
 
 4254 
 
 4342 
 
 4430 
 
 4517 
 
 88 
 
 495 
 
 4505 
 
 4693 
 
 4781 
 
 4868 
 
 4956 
 
 5044 
 
 5131 
 
 5219 
 
 5307 
 
 5394 
 
 88 
 
 496 
 
 5482 
 
 55;>9 
 
 5657 
 
 5744 
 
 5832 
 
 5919 
 
 6007 
 
 6094 
 
 6182 
 
 6269 
 
 87 
 
 497 
 
 6356 
 
 6444 
 
 6531 
 
 6618 
 
 6706 
 
 6793 
 
 6880 
 
 6968 
 
 7055 
 
 7142 
 
 87 
 
 498 
 
 7229 
 
 7317 
 
 7401 
 
 7491 
 
 7578 
 
 7665 
 
 7752 
 
 7839 
 
 7926 
 
 8014 
 
 87 
 
 499 
 
 8101 
 
 8188 
 
 8275 
 
 8362 
 
 8449 
 
 8535 
 
 8622 
 
 8709 
 
 8796 
 
 8883 
 
 87 
 
 500 
 
 698970 
 
 9057 
 
 9144 
 
 9231 
 
 9317 
 
 9404 
 
 9491 
 
 9578 
 
 9664 
 
 9751 
 
 87 
 
 5.11 
 
 9838 
 
 9924 
 
 ..11 
 
 ..98 
 
 .184 
 
 .271 
 
 .358 
 
 .444 
 
 .531 
 
 .617 
 
 87 
 
 502 
 
 700704 
 
 0790 
 
 0877 
 
 01)63 
 
 1050 
 
 1136 
 
 1222 
 
 1309 
 
 1395 
 
 1482 
 
 86 
 
 503 
 
 1568 
 
 1654 
 
 1741 
 
 1827 
 
 1913 
 
 1999 
 
 2086 
 
 2172 
 
 2258 
 
 2344 
 
 86 
 
 504 
 
 2431 
 
 2517 
 
 2603 
 
 2>i89 
 
 2775 
 
 2861 
 
 2947 
 
 3033 
 
 3119 
 
 3205 
 
 86 
 
 505 
 
 3291 
 
 3377 
 
 3463 
 
 3549 
 
 3635 
 
 3721 
 
 3807 
 
 3895 
 
 3979 
 
 4065 
 
 86 
 
 506 
 
 4151 
 
 4236 
 
 4322 
 
 4408 
 
 4494 
 
 4579 
 
 4665 
 
 4751 
 
 4837 
 
 4922 
 
 86 
 
 507 
 
 5008 
 
 5094 
 
 5179 
 
 5265 
 
 5350 
 
 5436 
 
 5522 
 
 5607 
 
 5693 
 
 5778 
 
 86 
 
 508 
 
 5864 
 
 5949 
 
 6035 
 
 6120 
 
 6206 
 
 6291 
 
 6376 
 
 6462 
 
 6547 
 
 6632 
 
 85 
 
 509 
 
 6718 
 
 680J3 
 
 6888 
 
 6974 
 
 7059 
 
 7144 
 
 7229 
 
 7315 
 
 7400 
 
 7485 
 
 85 
 
 510 
 
 707570 
 
 7655 
 
 7740 
 
 7826 
 
 7911 
 
 7996 
 
 8081 
 
 8166 
 
 8251 
 
 8336 
 
 85 
 
 511 
 
 8421 
 
 8506 
 
 8591 
 
 8676 
 
 8761 
 
 884fi 
 
 8931 
 
 9015 
 
 9100 
 
 9185 
 
 85 
 
 512 
 
 9270 
 
 9355 
 
 9440 
 
 9524 
 
 9609 
 
 9694 
 
 9779 
 
 9863 
 
 9948 
 
 ..33 
 
 85 
 
 513 
 
 710117 
 
 0202 
 
 0287 
 
 0371 
 
 0456 
 
 0540 
 
 0825 
 
 0710 
 
 0794 
 
 0879 
 
 85 
 
 514 
 
 0963 
 
 1048 
 
 1132 
 
 1217 
 
 1301 
 
 1385 
 
 1470 
 
 1554 
 
 1639 
 
 1723 
 
 84 
 
 515 
 
 1807 
 
 1892 
 
 1976 
 
 2060 
 
 2144 
 
 2-220 
 
 2313 
 
 2397 
 
 2481 
 
 2566 
 
 84 
 
 516 
 
 2650 
 
 2734 
 
 2818 
 
 2902 
 
 2986 
 
 3070 
 
 3154 
 
 3238 
 
 3333 
 
 3407 
 
 84 
 
 517 
 
 3491 
 
 3575 
 
 3650 
 
 374-2 
 
 3826 
 
 3910 
 
 3994 
 
 4078 
 
 4162 
 
 4246 
 
 84 
 
 518 
 
 4330 
 
 4414 
 
 4497 
 
 4581 
 
 4665 
 
 4749 
 
 4833 
 
 4916 
 
 5000 
 
 5084 
 
 84 
 
 519 
 
 5167 
 
 5251 
 
 5335 
 
 5418 
 
 5502 
 
 5586 
 
 5669 
 
 5753 
 
 5836 
 
 5920 
 
 84 
 
 IN. | 0|1|2|3|4|5|6 7 8 | 9 | D. 
 
A TABLB OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. | 1 2 | 3 4 5 | 6 7 | 8 9 D. 
 
 520 
 
 716003 
 
 6087 
 
 6170 
 
 6254 
 
 6337 
 
 6421 
 
 r.:.o t 
 
 6588 
 
 6671 
 
 6754 
 
 83 
 
 521 
 
 6838 
 
 69-21 
 
 701)4 
 
 7088 
 
 7171 
 
 7254 
 
 73,'W 
 
 7421 
 
 7504 
 
 7587 
 
 83 
 
 522 
 
 Ttn 
 
 7754 
 
 7837 
 
 7920 
 
 81)03 
 
 8086 
 
 8169 
 
 P2.-.3 
 
 8336 
 
 8419 
 
 83 
 
 523 
 
 6903 
 
 8585 
 
 8668 
 
 8751 
 
 8834 
 
 8917 
 
 90:K) 
 
 SHKJ 
 
 911)5 
 
 9248 
 
 83 
 
 524 
 
 9:01 
 
 9414 
 
 911)7 
 
 9580 
 
 9663 
 
 9745 
 
 9828 
 
 9911 
 
 9991 
 
 ..77 
 
 83 
 
 525 
 
 72.) 159 
 
 0242 
 
 0325 
 
 0407 
 
 0490 
 
 0573 
 
 0655 
 
 0738 
 
 OH21 
 
 0903 
 
 83 
 
 526 
 
 0936 
 
 1068 
 
 1151 
 
 1233 
 
 1316 
 
 1398 
 
 1481 
 
 J.-)!i3 
 
 1(546 
 
 1728 
 
 82 
 
 527 
 
 1811 
 
 1893 
 
 1975 
 
 2058 
 
 2140 
 
 2222 
 
 2305 
 
 23ri7 
 
 2469 
 
 2552 
 
 82 
 
 528 
 
 3634 
 
 2716 
 
 2798 
 
 2881 
 
 2;i;i:5 
 
 3045 
 
 3127 
 
 3209 
 
 '3291 
 
 3374 
 
 82 
 
 529 
 
 3456 
 
 3538 
 
 34BQ 
 
 3702 
 
 3784 
 
 38156 
 
 3943 
 
 4030 
 
 4112 
 
 4194 
 
 82 
 
 530 
 
 724276 
 
 4358 
 
 4440 
 
 4522 
 
 4604 
 
 4685 
 
 4767 
 
 4849 
 
 4931 
 
 5013 
 
 82 
 
 531 
 
 5095 
 
 5176 
 
 5258 
 
 5340 
 
 5422 
 
 5503 
 
 5585 
 
 50(57 
 
 5748 
 
 5830 
 
 82 
 
 538 
 
 5912 
 
 5993 
 
 6075 
 
 6156 
 
 (i23S 
 
 63-20 
 
 (i401 
 
 6483 
 
 6564 
 
 6646 
 
 82 
 
 533 
 
 0727 
 
 6809 
 
 6890 
 
 6972 
 
 7053 
 
 7134 
 
 7216 
 
 7297 
 
 7379 
 
 7460 
 
 81 
 
 534 
 
 7.') 1 1 
 
 7623 
 
 7704 
 
 7785 
 
 7866 
 
 7948 
 
 8029 
 
 8110 
 
 8191 
 
 8273 
 
 81 
 
 535 
 
 8354 
 
 8435 
 
 8516 
 
 8597 
 
 8678 
 
 8759 
 
 8841 
 
 S922 
 
 9003 
 
 9084 
 
 81 
 
 536 
 
 9165 
 
 9246 
 
 9327 
 
 9408 
 
 9489 
 
 9570 
 
 9651 
 
 9732 
 
 9813 
 
 9893 
 
 81 
 
 537 
 
 9974 
 
 ..55 
 
 .136 
 
 .217 
 
 .293 
 
 .378 
 
 .459 
 
 .540 
 
 .621 
 
 .702 
 
 81 
 
 538 
 
 730782 
 
 0863 
 
 Oi)41 
 
 1024 
 
 1105 
 
 1186 
 
 1266 
 
 1347 
 
 1428 
 
 1508 
 
 81 
 
 539 
 
 1539 
 
 1669 
 
 1750 
 
 1830 
 
 1911 
 
 1991 
 
 072 
 
 2152 
 
 2233 
 
 2313 
 
 81 
 
 540 
 
 732394 
 
 2474 
 
 2535 
 
 2635 
 
 2715 
 
 2796 
 
 2876 
 
 2956 
 
 3037 
 
 3117 
 
 80 
 
 541 
 
 3197 
 
 3278 
 
 3353 
 
 3438 
 
 3518 
 
 3598 
 
 3679 
 
 3759 
 
 3839 
 
 3919 
 
 80 
 
 542 
 
 3999 
 
 4079 
 
 4160 
 
 4240 
 
 4320 
 
 4400 
 
 4480 
 
 4560 
 
 4640 
 
 4720 
 
 80 
 
 543 
 
 4800 
 
 4880 
 
 4960 
 
 5040 
 
 5120 
 
 5200 
 
 5279 
 
 5359 
 
 5439 
 
 5519 
 
 80 
 
 544 
 
 5599 
 
 5679 
 
 5759 
 
 5838 
 
 5-) 18 
 
 5998 
 
 6078 
 
 6157 
 
 6237 
 
 6317 
 
 80 
 
 545 
 
 0397 
 
 6476 
 
 655!! 
 
 6635 
 
 6715 
 
 6795 
 
 6874 
 
 6954 
 
 7034 
 
 7113 
 
 80 
 
 54(5 
 
 7193 
 
 7272 
 
 7352 
 
 7431 
 
 7511 
 
 7590 
 
 7670 
 
 7749 
 
 7829 
 
 7908 
 
 79 
 
 547 
 
 7987 
 
 8067 
 
 8146 
 
 8225 
 
 8305 
 
 8384 
 
 8463 
 
 8543 
 
 8622 
 
 8701 
 
 79 
 
 . 548 
 
 8781 
 
 88(50 
 
 8939 
 
 9018 
 
 9097 
 
 9177 
 
 9256 
 
 9335 
 
 9414 
 
 9493 
 
 79 
 
 549 
 
 9572 
 
 9651 
 
 9731 
 
 9810 
 
 9839 
 
 9968 
 
 ..47 
 
 .126 
 
 .205 
 
 .284 
 
 79 
 
 550 
 
 740363 
 
 0412 
 
 0521. 
 
 0600 
 
 0678 
 
 0757 
 
 0836 
 
 0915 
 
 0994 
 
 1073 
 
 79 
 
 551 
 
 1152 
 
 1330 
 
 1309 
 
 1388 
 
 1467 
 
 1546 
 
 1624 
 
 1703 
 
 1782 
 
 1860 
 
 79 
 
 552 
 
 1939 
 
 2018 
 
 2096 
 
 2175 
 
 2254 
 
 2332 
 
 2411 
 
 2489 
 
 2568 
 
 2646 
 
 79 
 
 553 
 
 2725 
 
 2804 
 
 2882 
 
 2961 
 
 3039 
 
 3118 
 
 3196 
 
 3275 
 
 3353 
 
 3431 
 
 78 
 
 554 
 
 3510 
 
 3588 
 
 3667 
 
 3745 
 
 3323 
 
 3902 
 
 3980 
 
 4053 
 
 4136 
 
 4215 
 
 78 
 
 555 
 
 4293 
 
 4371 
 
 4449 
 
 4528 
 
 4606 
 
 4684 
 
 4762 
 
 4840 
 
 4919 
 
 4997 
 
 78 
 
 553 
 
 5075 
 
 5153 
 
 5231 
 
 5309 
 
 5387 
 
 5465 
 
 5543 
 
 5621 
 
 5699 
 
 5777 
 
 78 
 
 557 
 
 5855 
 
 5933 
 
 6011 
 
 6089 
 
 6167 
 
 6245 
 
 6323 
 
 6401 
 
 6479 
 
 6556 
 
 78 
 
 558 
 
 8634 
 
 6712 
 
 6790 
 
 6868 
 
 15945 
 
 7023 
 
 7101 
 
 7179 
 
 725(5 
 
 7334 
 
 78 
 
 559 
 
 7412 
 
 7489 
 
 7567 
 
 7(545 
 
 7722 
 
 7800 
 
 7878 
 
 7955 
 
 8033 
 
 8110 
 
 78 
 
 560 
 
 748188 
 
 8266 
 
 8343 
 
 8421 
 
 8493 
 
 8576 
 
 8(153 
 
 8731 
 
 8808 
 
 8885 
 
 77 
 
 5(51 
 
 89153 
 
 9940 
 
 9118 
 
 9195 
 
 9272 
 
 9350 
 
 9427 
 
 9504 
 
 9582 
 
 9659 
 
 77 
 
 562 
 
 9738 
 
 9814 
 
 9891 
 
 9968 
 
 ..45 
 
 .123 
 
 .200 
 
 .277 
 
 .354 
 
 .431 
 
 77 
 
 563 
 
 750508 
 
 058(5 
 
 0563 
 
 0740 
 
 0817 
 
 0894 
 
 0971 
 
 1048 
 
 1125 
 
 1202 
 
 77 
 
 564 
 
 1279 
 
 1356 
 
 1433 
 
 1510 
 
 1587 
 
 1G64 
 
 1741 
 
 1818 
 
 1895 
 
 1972 
 
 77 
 
 565 
 
 2048 
 
 2125 
 
 2203 
 
 2279 
 
 2356 
 
 2433 
 
 2509 
 
 2586 
 
 2663 
 
 2740 
 
 77 
 
 566 
 
 :2816 
 
 2893 
 
 2970 
 
 3047 
 
 3123 
 
 3200 
 
 3277 
 
 3353 
 
 3430 
 
 3506 
 
 77 
 
 5!>7 
 
 3583 
 
 3660 
 
 3736 
 
 3813 
 
 3889 
 
 3966 
 
 4042 
 
 4119 
 
 4195 
 
 4272 
 
 77 
 
 568 
 
 4348 
 
 4425 
 
 4501 
 
 4578 
 
 4654 
 
 4730 
 
 4807 
 
 4883 
 
 4960 
 
 5036 
 
 76 
 
 569 
 
 5112 
 
 5189 
 
 5265 
 
 5341 
 
 5417 
 
 5494 
 
 5570 
 
 5646 
 
 5722 
 
 5799 
 
 76 
 
 570 
 
 755875 
 
 5951 
 
 6027 
 
 6103 
 
 6180 
 
 6256 
 
 6332 
 
 6408 
 
 6484 
 
 6560 
 
 76 
 
 571 
 
 PQ38 
 
 6712 
 
 6788 
 
 6864 
 
 6940 
 
 7016 
 
 7092 
 
 71(58 
 
 7244 
 
 7320 
 
 76 
 
 572 
 
 7396 
 
 7472 
 
 7548 
 
 7624 
 
 7700 
 
 777.-. 
 
 7351 
 
 7927 
 
 8003 
 
 8079 
 
 76 
 
 573 
 
 8155 
 
 8230 
 
 8306 
 
 8382 
 
 8458 
 
 8533 
 
 8609 
 
 8685 
 
 8761 
 
 8836 
 
 76 
 
 574 
 
 8912 
 
 8988 
 
 9063 
 
 9139 
 
 9214 
 
 9290 
 
 9366 
 
 9441 
 
 9517 
 
 9592 
 
 76 
 
 575 
 
 9668 
 
 9743 
 
 9819 
 
 9894 
 
 9970 
 
 ..45 
 
 .121 
 
 .198 
 
 .272 
 
 .347 
 
 75 
 
 576 
 
 760422 
 
 0498 
 
 0573 
 
 0649 
 
 0724 
 
 0799 
 
 0875 
 
 0950 
 
 1025 
 
 1101 
 
 75 
 
 577 
 
 1176 
 
 1251 
 
 1326 
 
 1402 
 
 1477 
 
 1552 
 
 1627 
 
 1702 
 
 1778 
 
 1853 
 
 75 
 
 578 
 
 1928 
 
 2003 
 
 2078 
 
 2153 
 
 2228 
 
 2303 
 
 2378 
 
 2453 
 
 2529 
 
 2604 
 
 75 
 
 579 
 
 2679 
 
 27.->l 
 
 3898 
 
 2904 
 
 2978 
 
 31).-.:$ 
 
 3128 
 
 3203 
 
 3278 
 
 3353 
 
 75 
 
 N. 1 | 1 2 | 3 j 4 | 5 | 6 7 | 8 | 9 | D. 
 
10 
 
 A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. | | 1 | 2 3|4 5 6 | 7 | 8 | 9 | D. j 
 
 580 
 
 763428 
 
 3503 
 
 3578 
 
 3653 
 
 3727 
 
 3802 3-577 
 
 31)52 
 
 4027 1 4101 
 
 75 
 
 581 
 
 4176 
 
 4251 
 
 4326 
 
 4400 
 
 4475 
 
 4550 
 
 4624 
 
 4699 
 
 4774 
 
 4848 
 
 75 
 
 58-2 
 
 4923 
 
 4998 
 
 5072 
 
 5147 
 
 5221 
 
 5296 
 
 5370 
 
 5445 
 
 5520 
 
 5594 
 
 75 
 
 583 
 
 5669 
 
 5743 
 
 5818 
 
 5892 
 
 591)6 
 
 6041 
 
 6115 
 
 6190 
 
 6264 
 
 6338 
 
 74 
 
 584 
 
 6-113 
 
 6487 
 
 6562 
 
 6636 
 
 67JO 
 
 6785 
 
 6859 
 
 6933 
 
 7007 
 
 7032 
 
 74 
 
 585 
 
 7156 
 
 7230 
 
 7304 
 
 7379 
 
 7453 
 
 7527 
 
 7601 
 
 7675 
 
 7749 
 
 7823 
 
 74 
 
 586 
 
 7898 
 
 7972 
 
 8046 
 
 8120 
 
 8194 
 
 8268 
 
 8342 
 
 8416 
 
 8490 
 
 8564 
 
 74 
 
 587 
 
 8638 
 
 8712 
 
 8786 
 
 8860 
 
 8934 
 
 9008 
 
 9082 
 
 9156 
 
 9230 
 
 9303 
 
 74 
 
 588 
 
 9377 
 
 9451 
 
 9.525 
 
 9599 
 
 9673 
 
 9746 
 
 9820 
 
 9894 
 
 99G8 
 
 ..42 
 
 74 
 
 589 
 
 770115 
 
 0189 
 
 0263 
 
 0336 
 
 0410 
 
 0484 
 
 0557 
 
 0631 
 
 0705 
 
 0778 
 
 74 
 
 590 
 
 770852 
 
 0926 
 
 0999 
 
 1073 
 
 1146 
 
 1220 
 
 1293 
 
 1367 
 
 1440 
 
 1514 
 
 74 
 
 591 
 
 1587 
 
 1661 
 
 1734 
 
 1808 
 
 1881 
 
 1955 
 
 2028 
 
 2102 
 
 2175 
 
 2248 
 
 73 
 
 592 
 
 2322 
 
 2395 
 
 2468 
 
 2542 
 
 2615 
 
 2688 
 
 2762 
 
 2835 
 
 2908 
 
 2981 
 
 73 
 
 593 
 
 3055 
 
 3128 
 
 3201 
 
 3274 
 
 3348 
 
 3421 
 
 3494 
 
 3567 
 
 3640 
 
 3713 
 
 73 
 
 594 
 
 3786 
 
 3860 
 
 3!)33 
 
 4006 
 
 4079 
 
 4152 
 
 4225 
 
 4298 
 
 4371 
 
 4444 
 
 73 
 
 595 
 
 4517 
 
 4590 
 
 4663 
 
 4736 
 
 4809 
 
 4882 
 
 4955 
 
 5028 
 
 5100 
 
 5173 
 
 73 
 
 596 
 
 5246 
 
 5319 
 
 5392 
 
 5465 
 
 5538 
 
 5610 
 
 5683 
 
 5756 
 
 5829 
 
 5902 
 
 73 
 
 597 
 
 5974 
 
 6047 
 
 6120 
 
 6193 
 
 6265 
 
 6338 
 
 6411 
 
 6483 
 
 6556 
 
 6629 
 
 73 
 
 598 
 
 6701 
 
 6774 
 
 C846 
 
 69 U) 
 
 6992 
 
 TOM 
 
 7137 
 
 7209 
 
 7282 
 
 7354 
 
 73 
 
 599 
 
 7427 
 
 7499 
 
 7572 
 
 7644 
 
 7717 
 
 7789 
 
 7862 
 
 7934 
 
 8006 
 
 8079 
 
 72 
 
 600 
 
 778151 
 
 8224 
 
 8296 
 
 8368 
 
 8441 
 
 8."> ] 3 
 
 8585 
 
 8658 
 
 8730 
 
 8802 
 
 72 
 
 601 
 
 8874 
 
 8947 
 
 9019 
 
 9091 
 
 9163 
 
 9236 
 
 9308 
 
 9380 
 
 9452 
 
 9524 
 
 72 
 
 602 
 
 9596 
 
 9669 
 
 9741 
 
 9813 
 
 9885 
 
 9957 
 
 ..29 
 
 .101 
 
 .173 
 
 .245 
 
 72 
 
 603 
 
 780317 
 
 0389 
 
 0461 
 
 0533 
 
 0605 
 
 0677 
 
 0749 
 
 0821 
 
 0893 
 
 0965 
 
 72 
 
 604 
 
 1037 
 
 1109 
 
 1J81 
 
 1253 
 
 1324 
 
 1396 
 
 1468 
 
 1540 
 
 1612 
 
 1684 
 
 72 
 
 605 
 
 1755 
 
 1827 
 
 1899 
 
 1971 
 
 2042 
 
 2114 
 
 2186 
 
 2258 
 
 2329 
 
 2401 
 
 72 
 
 606 
 
 2473 
 
 2544 
 
 2616 
 
 2688 
 
 2759 
 
 2831 
 
 2902 
 
 2974 
 
 3046 
 
 3117 
 
 72 
 
 6')7 
 
 3189 
 
 3260 
 
 3332 
 
 3-103 
 
 3475 
 
 35-16 
 
 3618 
 
 3689 
 
 3761 
 
 3832 
 
 71 
 
 608 
 
 3904 
 
 3975 
 
 4046 
 
 4118 
 
 4189 
 
 4261 
 
 4332 
 
 4403 
 
 4475 
 
 4546 
 
 71 
 
 609 
 
 4617 
 
 4689 
 
 4760 
 
 4831 
 
 4902 
 
 4974 
 
 5045 
 
 5116 
 
 5187 
 
 5259 
 
 71 
 
 610 
 
 785330 
 
 5401 
 
 5472 
 
 5543 
 
 5615 
 
 5686 
 
 5757 
 
 5828 
 
 5899 
 
 5970 
 
 71 
 
 611 
 
 6041 
 
 6112 
 
 6183 
 
 6254 
 
 6325 
 
 6396 
 
 6467 
 
 6538 
 
 6609 
 
 6680 
 
 71 
 
 612 
 
 6751 
 
 6822 
 
 6893 
 
 6964 7035 
 
 7106 
 
 7177 
 
 7248 
 
 7319 
 
 731K) 
 
 71 
 
 613 
 
 7460 
 
 7531 
 
 7602 
 
 7673 
 
 7744 
 
 7815 
 
 7885 
 
 7956 
 
 80-27 
 
 8098 
 
 71 
 
 614 
 
 8168 
 
 8239 
 
 8310 
 
 8381 
 
 8451 
 
 8522 
 
 8593 
 
 8663 
 
 8734 
 
 8804 
 
 71 
 
 615 
 
 8875 
 
 8946 
 
 9016 
 
 9087 
 
 9157 
 
 9228 
 
 9299 
 
 9369 
 
 9440 
 
 9510 
 
 71 
 
 616 
 
 9581 
 
 9851 
 
 9722 
 
 9792 
 
 9863 
 
 9933 
 
 ...4 
 
 ..74 
 
 .144 
 
 .215 
 
 70 
 
 617 
 
 790285 
 
 0356 
 
 0426 
 
 0496 
 
 0567 
 
 0637 
 
 0707 
 
 0778 
 
 0848 
 
 0918 
 
 70 
 
 618 
 
 0988 
 
 1059 
 
 1129 
 
 1199 
 
 1269 
 
 1340 
 
 1410 
 
 14cO 
 
 1550 
 
 1620 
 
 70 
 
 619 
 
 1691 
 
 1761 
 
 1831 
 
 1901 
 
 1971 
 
 2041 
 
 2111 
 
 2181 
 
 2252 
 
 2322 
 
 70 
 
 620 
 
 792392 
 
 2462 
 
 2532 
 
 2602 
 
 2672 
 
 2742 
 
 2812 
 
 2882 
 
 2952 
 
 3022 
 
 70 
 
 621 
 
 3092 
 
 3162 
 
 3231 
 
 3301 
 
 3371 
 
 3441 
 
 3511 
 
 3581 
 
 3651 
 
 3721 
 
 70 
 
 622 
 
 3790 
 
 3860 
 
 3930 
 
 400!) 
 
 4070 
 
 4139 
 
 4209 
 
 4279 
 
 4349 
 
 4418 
 
 70 
 
 623 
 
 4488 
 
 4558 
 
 4627 
 
 4697 
 
 4767 
 
 4836 
 
 4906 
 
 4976 
 
 5045 
 
 5115 
 
 70 
 
 624 
 
 5185 
 
 5254 
 
 5324 
 
 5393 
 
 5463 
 
 5532 
 
 5602 
 
 5672 
 
 5741 
 
 5811 
 
 70 
 
 625 
 
 5880 
 
 5949 
 
 6019 
 
 6088 
 
 6158 
 
 6227 
 
 6297 
 
 6366 
 
 6436 
 
 6505 
 
 69 
 
 626 
 
 6574 
 
 6644 
 
 6713 
 
 6782 
 
 6852 
 
 6921 
 
 6990 
 
 7060 
 
 7129 
 
 7198 
 
 69 
 
 627 
 
 7268 
 
 7337 
 
 7406 
 
 7475 
 
 7545 
 
 7614 
 
 7683 
 
 7752 
 
 7821 
 
 7890 
 
 69 
 
 628 
 
 7960 
 
 8029 
 
 8098 
 
 8167 
 
 8236 
 
 8305 
 
 8374 
 
 8443 
 
 8513 
 
 8582 
 
 69 
 
 629 
 
 8651 
 
 8720 
 
 8789 
 
 8858 
 
 8927 
 
 8996 
 
 9065 
 
 9134 
 
 9203 
 
 9272 
 
 69 
 
 630 
 
 799341 
 
 9409 
 
 9478 
 
 9547 
 
 9616 
 
 9685 
 
 9754 
 
 9823 
 
 9892 
 
 9961 
 
 69 
 
 631 
 
 800029 
 
 0098 
 
 0167 
 
 0236 
 
 0305 
 
 0373 
 
 0442 
 
 0511 
 
 0580 
 
 0648 
 
 69 
 
 632 
 
 0717 
 
 0786 
 
 0854 
 
 0923 
 
 0992 
 
 1061 
 
 1129 
 
 1198 
 
 1266 
 
 1335 
 
 69 
 
 633 
 
 1404 
 
 1472 
 
 1541 
 
 1609 
 
 1678 
 
 1747 
 
 1815 
 
 1884 
 
 1952 
 
 2021 
 
 69 
 
 634 
 
 2089 
 
 2158 
 
 2226 
 
 2295 
 
 2363 
 
 2432 
 
 2500 
 
 2568 
 
 2837 
 
 2705 
 
 69 
 
 635 
 
 2774 
 
 2842 
 
 2910 
 
 2979 
 
 3047 
 
 3116 
 
 3184 
 
 3252 
 
 3321 
 
 3389 
 
 68 
 
 636 
 
 3457 
 
 3525 
 
 3594 
 
 3662 
 
 3730 
 
 3798 
 
 3867 
 
 3935 
 
 4003 
 
 4071 
 
 68 
 
 637 
 
 4139 
 
 4208 
 
 4276 
 
 4344 
 
 4412 
 
 4480 
 
 4548 
 
 4616 
 
 4685 
 
 4753 
 
 (58 
 
 638 
 
 4821 
 
 4889 
 
 4957 
 
 5025 
 
 5093 
 
 5161 
 
 5229 
 
 5297 
 
 5365 
 
 5433 
 
 68 
 
 639 5501 
 
 5569 
 
 5637 
 
 5705 
 
 5773 
 
 5841 
 
 5908 
 
 5976 
 
 6044 
 
 6112 
 
 68 
 
 N. | 0|l|2|3j4|5 
 
 6 | 7 | 8 | 9 D. 
 
A TAET,E OF LOGARITHMS FROM 1 TO 10,000. 
 
 11 
 
 N. | 1 | 2 | 3 4 | 5 L 6 | 7 | 8 | 9 | D. 
 
 640* 
 
 rHIOIsil 
 
 8948 
 
 6316 
 
 5384 
 
 o-ir.i 
 
 0519 
 
 df>87 
 
 i;t;.-,r> 
 
 07-23 
 
 6790 
 
 68 
 
 641 
 
 6858 
 
 osi-20 
 
 0994 
 
 7001 
 
 7128 
 
 7197 
 
 7264 
 
 7332 
 
 7400 
 
 7467 
 
 68 
 
 64-2 
 
 ::>:(.-> 
 
 7003 
 
 7670 
 
 7738 
 
 780fl 
 
 7873 
 
 7941 
 
 8008 
 
 8076 
 
 8143 
 
 68 
 
 643 
 
 8211 
 
 827!) 
 
 8346 
 
 8414 
 
 H48 1 
 
 8549 
 
 8016 
 
 8084 
 
 8751 
 
 8818 
 
 67 
 
 644 
 
 8886 
 
 8953 
 
 9021 
 
 9068 
 
 9156 
 
 9-2-23 
 
 9290 
 
 9358 
 
 9425 
 
 9492 
 
 67 
 
 645 
 
 9.">00 
 
 9027 
 
 9094 
 
 970-2 
 
 9829 
 
 9890 
 
 9904 
 
 ..31 
 
 -.98 
 
 .165 
 
 67 1 
 
 64S 
 
 810233 
 
 0300 
 
 0367 
 
 0434 
 
 0501 
 
 0509 
 
 0030 
 
 0703 
 
 0770 
 
 0837 
 
 67 1 
 
 647 
 
 0904 
 
 0971 
 
 1039 
 
 1106 
 
 1173 
 
 1240 
 
 1307 
 
 1374 
 
 1441 
 
 1508 
 
 67 
 
 648 
 
 1575 
 
 1642 
 
 1709 
 
 1776 
 
 1843 
 
 1910 
 
 1977 
 
 2044 
 
 2111 
 
 2178 
 
 67 
 
 649 
 
 2245 
 
 2312 
 
 2379 
 
 2445 
 
 2512 
 
 2579 
 
 2646 
 
 2713 
 
 2780 
 
 2847 
 
 67 
 
 650 
 
 812913 
 
 2980 
 
 3047 
 
 3114 
 
 3181 
 
 3247 
 
 3314 
 
 3381 
 
 3448 
 
 3514 
 
 67 
 
 651 
 
 3581 
 
 3648 
 
 3714 
 
 3781 
 
 3848 
 
 3914 
 
 3981 
 
 4048 
 
 4114 
 
 4181 
 
 67 
 
 652 
 
 4248 
 
 4314 
 
 4381 
 
 4447 
 
 4514 
 
 4581 
 
 4647 
 
 4714 
 
 4780 
 
 4847 
 
 67 
 
 653 
 
 4913 
 
 4980 
 
 5046 
 
 5113 
 
 5179 
 
 5246 
 
 5312 
 
 5378 
 
 5445 
 
 5511 
 
 66 
 
 654 
 
 5578 
 
 5644 
 
 5711 
 
 5777 
 
 5843 
 
 5910 
 
 5976 
 
 6042 
 
 6109 
 
 6175 
 
 66 
 
 655 
 
 6241 
 
 6308 
 
 6374 
 
 6440 
 
 6506 
 
 6573 
 
 6039 
 
 6705 
 
 6771 
 
 6838 
 
 66 
 
 656 
 
 6904 
 
 6970 
 
 7036 
 
 7102 
 
 7169 
 
 7235 
 
 7301 
 
 7367 
 
 7433 
 
 7499 
 
 66 
 
 657 
 
 7565 
 
 7631 
 
 7698 
 
 7764 
 
 7830 
 
 7896 
 
 7902 
 
 8028 
 
 8094 
 
 8160 
 
 66 
 
 658 
 
 8220 
 
 8292 
 
 8358 
 
 8424 
 
 8490 
 
 8556 
 
 8622 
 
 8688 
 
 8754 
 
 8820 
 
 66 
 
 659 
 
 8885 
 
 8951 
 
 9017 
 
 9083 
 
 9149 
 
 9215 
 
 9281 
 
 9346 
 
 9412 
 
 9478 
 
 66 
 
 660 
 
 819544 
 
 9610 
 
 9676 
 
 9741 
 
 9807 
 
 9873 
 
 9939 
 
 ...4 
 
 ..70 
 
 .136 
 
 66 
 
 P$l 
 
 820201 
 
 0267 
 
 0333 
 
 0399 
 
 0464 
 
 0530 
 
 0595 
 
 0061 
 
 0727 
 
 0792 
 
 66 
 
 b62 
 
 0858 
 
 0924 
 
 0989 
 
 1055 
 
 1120 
 
 1186 
 
 1251 
 
 1317 
 
 1382 
 
 1448 
 
 66 
 
 663 
 
 1514 
 
 1579 
 
 1045 
 
 1710 
 
 1775 
 
 1841 
 
 1906 
 
 1972 
 
 2037 
 
 2103 
 
 65 
 
 1 664 
 
 2168 
 
 2233 
 
 2299 
 
 2364 
 
 2430 
 
 2495 
 
 2560 
 
 2620 
 
 2691 
 
 2756 
 
 65 
 
 665 
 
 2822 
 
 2887 
 
 2952 
 
 3018 
 
 3083 
 
 3148 
 
 3213 
 
 3279 
 
 3344 
 
 3409 
 
 65 
 
 G06 
 
 3474 
 
 3539 
 
 3605 
 
 3670 
 
 3735 
 
 3800 
 
 3865 
 
 3930 
 
 3990 
 
 4001 
 
 65 
 
 667 
 
 4126 
 
 4191 
 
 425(5 
 
 4321 
 
 4386 
 
 4451 
 
 4516 
 
 4581 
 
 4646 
 
 4711 
 
 65 
 
 668 
 
 4776 
 
 4841 
 
 4906 
 
 4971 
 
 5036 
 
 5101 
 
 5166 
 
 5231 
 
 529(5 
 
 5361 
 
 65 
 
 609 
 
 5426 
 
 5491 
 
 5556 
 
 5621 
 
 5686 
 
 5751 
 
 5815 
 
 5880 
 
 5945 
 
 6010 
 
 65 
 
 670 
 
 826075 
 
 OHO 
 
 6204 
 
 6269 
 
 6334 
 
 6399 
 
 6464 
 
 6528 
 
 6593 
 
 6658 
 
 65 
 
 671 
 
 6723 
 
 6787 
 
 6852 
 
 6917 
 
 6981 
 
 7046 
 
 7111 
 
 7175 
 
 7240 
 
 7305 
 
 65 
 
 07-2 
 
 7359 
 
 7434 
 
 7499 
 
 7563 
 
 7628 
 
 7692 
 
 7757 
 
 7821 
 
 7886 
 
 7951 
 
 65 
 
 673 
 
 8015 
 
 8080 
 
 8144 
 
 8209 
 
 8273 
 
 8338 
 
 8402 
 
 8467 
 
 8531 
 
 8595 
 
 64 
 
 674 
 
 8660 
 
 8724 
 
 8789 
 
 8853 
 
 8918 
 
 8982 
 
 9046 
 
 9111 
 
 9175 
 
 9239 
 
 64 
 
 675 
 
 9304 
 
 9368 
 
 9432 
 
 9497 
 
 9561 
 
 9625 
 
 9690 
 
 9754 
 
 9818 
 
 9882 
 
 64 
 
 676 
 
 9947 
 
 ..11 
 
 ..75 
 
 .139 
 
 .204 
 
 .208 
 
 .332 
 
 .390 
 
 .460 
 
 .525 
 
 64 
 
 677 
 
 830589 
 
 0653 
 
 0717 
 
 0781 
 
 0845 
 
 0909 
 
 0973 
 
 1037 
 
 1102 
 
 1166 
 
 64 
 
 678 
 
 1230 
 
 1294 
 
 1358 
 
 1423 
 
 1486 
 
 1550 
 
 1014 
 
 1078 
 
 1742 
 
 1806 
 
 64 
 
 679 
 
 1870 
 
 1934 
 
 1998 
 
 2062 
 
 2126 
 
 2189 
 
 2253 
 
 2317 
 
 2381 
 
 2445 
 
 64 
 
 680 
 
 832509 
 
 2.Y?3 
 
 2637 
 
 2700 
 
 2704 
 
 2828 
 
 2892 
 
 2956 
 
 3020 
 
 3083 
 
 64 
 
 681 
 
 3147 
 
 321] 
 
 3275 
 
 3338 
 
 3402 
 
 3460 
 
 3530 
 
 3593 
 
 3657 
 
 3721 
 
 64 
 
 682 
 
 3784 
 
 3848 
 
 3912 
 
 3975 
 
 4039 
 
 4103 
 
 4106 
 
 4220 
 
 4294 
 
 4357 
 
 64 
 
 683 
 
 4421 
 
 4484 
 
 4548 
 
 4611 
 
 4075 
 
 4739 
 
 4802 
 
 4800 
 
 4tt&' 
 
 4993 
 
 64 
 
 684 
 
 505(5 
 
 5120 
 
 5183 
 
 5247 
 
 5310 
 
 5373 
 
 5437 
 
 5500 
 
 55(54 
 
 5027 
 
 C3 
 
 685 
 
 5091 
 
 5754 
 
 5817 
 
 5881 
 
 5944 
 
 6007 
 
 6071 
 
 6134 
 
 6197 
 
 6261 
 
 63 
 
 686 
 
 6324 
 
 6387 
 
 6451 
 
 6514 
 
 6577 
 
 6641 
 
 6704 
 
 6707 
 
 6830 
 
 6894 
 
 63 
 
 687 
 
 6957 
 
 7020 
 
 7083 
 
 7146 
 
 7210 
 
 7273 
 
 7336 
 
 7399 
 
 7402 
 
 7525 
 
 63 
 
 688 
 
 7588 
 
 7652 
 
 7715 
 
 7778 
 
 7841 
 
 7904 
 
 7967 
 
 8030 
 
 8093 
 
 8156 
 
 63 
 
 689 
 
 8219 
 
 8282 
 
 8345 
 
 8408 
 
 8471 
 
 8534 
 
 8597 
 
 8660 
 
 8723 
 
 8786 
 
 63 
 
 690 
 
 838849 
 
 8912 
 
 8975 
 
 9038 
 
 9101 
 
 9164 
 
 9227 
 
 9289 
 
 9352 
 
 9415 
 
 63 
 
 691 
 
 9478 
 
 9541 
 
 9604 
 
 9067 
 
 9729 
 
 9792 
 
 9855 
 
 9918 
 
 9981 
 
 ..43 
 
 63 
 
 692 
 
 "84-JlOrt 
 
 0169 
 
 0232 
 
 0294 
 
 0357 
 
 0420 
 
 0482 
 
 0545 
 
 0608 
 
 0671 
 
 63 
 
 1 693 
 
 0733 
 
 0796 
 
 0859 
 
 0921 
 
 0984 
 
 1046 
 
 1109 
 
 1172 
 
 1234 
 
 1297 
 
 03 
 
 694 
 
 [399 
 
 1422 
 
 1485 
 
 1547 
 
 1610 
 
 1672 
 
 1735 
 
 1797 
 
 1860 
 
 1922 
 
 63 
 
 695 
 
 toes 
 
 2047 
 
 2110 
 
 2172 
 
 2235 
 
 2297 
 
 2360 
 
 2422 
 
 2484 
 
 2547 
 
 62 
 
 696 
 
 2009 
 
 2672 
 
 2734 
 
 2796 
 
 2859 
 
 2921 
 
 2983 
 
 3046 
 
 3108 
 
 3170 
 
 62 
 
 697 
 
 3233 
 
 3295 
 
 3357 
 
 3420 
 
 3482 
 
 3544 
 
 >0<i 
 
 3669 
 
 3731 
 
 3793 
 
 62 
 
 , 698 
 
 3855 
 
 3918 
 
 3980 
 
 4042 
 
 4104 
 
 4166 
 
 4229 
 
 4291 
 
 4353 
 
 4415 
 
 62 
 
 699 
 
 4477 
 
 4539 
 
 4601 
 
 4664 
 
 4726 
 
 4788 
 
 l-.-)0 
 
 4912 
 
 4974 
 
 5036 
 
 62 
 
 N. | | 1 
 
 2|3J4|5|6|7|8|9 
 
 IT 
 
12 
 
 A TABLE OP LOGARITHMS FROM 1 TO 10,000. 
 
 N. | | 1 | 2 3 | 4 | 5 6 7 8 | 9 D. 
 
 1 700 
 
 845093 
 
 5160 
 
 52-22 
 
 5234 
 
 534S 
 
 5408 
 
 5470 
 
 5532 
 
 5594 
 
 56.56 
 
 62 ! 
 
 701 
 
 5718 
 
 5780 
 
 5842 
 
 5904 
 
 5966 
 
 6023 
 
 601)0 
 
 6151 
 
 6-213 
 
 6275 
 
 62 ! 
 
 702 
 
 6337 
 
 6399 
 
 64f51 
 
 6523 
 
 6585 
 
 6646 
 
 6708 
 
 6770 
 
 6832 
 
 6894 
 
 62 j 
 
 703 
 
 6955 
 
 7017 
 
 7079 
 
 7141 
 
 7202 
 
 7284 
 
 7326 
 
 7383 
 
 7449 
 
 7511 
 
 62 
 
 704 
 
 7573 
 
 7634 
 
 7696 
 
 7758 
 
 7819 
 
 7881 
 
 7943 
 
 8004 
 
 8066 
 
 8128 
 
 62 
 
 705 
 
 8189 
 
 8251 
 
 8312 
 
 8374 
 
 8435 
 
 8497 
 
 8559 
 
 8630 
 
 8682 
 
 8743 
 
 G2 
 
 706 
 
 8805 
 
 8866 
 
 8928 
 
 8989 
 
 9051 
 
 9112 
 
 9174. 
 
 9235 
 
 9237 
 
 9358 
 
 61 
 
 707 
 
 9419 
 
 9481 
 
 9542 
 
 9604 
 
 9665 
 
 9726 
 
 9788 
 
 9849 
 
 9911 
 
 9972 
 
 61 
 
 708 
 
 850033 
 
 0095 
 
 0150 
 
 0217 
 
 0279 
 
 0340 
 
 0401 
 
 0462 
 
 0524 
 
 0585 
 
 61 
 
 709 
 
 0646 
 
 0707 
 
 0769 
 
 0830 
 
 0891 
 
 0952 
 
 1014 
 
 1075 
 
 11J6 
 
 1197 
 
 
 710 
 
 851258 
 
 1320 
 
 1381 
 
 1442 
 
 1503 
 
 1564 
 
 1625 
 
 1686 
 
 1747 
 
 1809 
 
 61 
 
 711 
 
 1870 
 
 1931 
 
 1992 
 
 2053 
 
 2114 
 
 2175 
 
 2236 
 
 2297 
 
 2358 
 
 2419 
 
 61 
 
 712 
 
 2480 
 
 2541 
 
 2602 
 
 2663 
 
 2724 
 
 2785 
 
 2846 
 
 2907 
 
 2968 
 
 3029 
 
 61 
 
 713 
 
 3090 
 
 3150 
 
 3211 
 
 3272 
 
 3333 
 
 3394 
 
 3455 
 
 3516 
 
 3577 
 
 3637 
 
 61 
 
 714 
 
 36d8 
 
 3759 
 
 3820 
 
 3881 
 
 3941 
 
 4002 
 
 4003 
 
 4124 
 
 4185 
 
 4245 
 
 61 
 
 715 
 
 4306 
 
 4367 
 
 4428 
 
 4488 
 
 4549 
 
 4610 
 
 4670 
 
 4731 
 
 4792 
 
 4852 
 
 61 
 
 716 
 
 4913 
 
 4974 
 
 5034 
 
 5095 
 
 5156 
 
 5216 
 
 5277 
 
 5337 
 
 5393 
 
 5459 
 
 61 
 
 717 
 
 5519 
 
 5580 
 
 5640 
 
 5701 
 
 57(51 
 
 5822 
 
 5882 
 
 5943 
 
 6003 
 
 6064 
 
 61 
 
 ; 718 
 
 6124 
 
 6185 
 
 6245 
 
 6306 
 
 6366 
 
 6427 
 
 6487 
 
 6548 
 
 6608 
 
 66(58 
 
 GO 
 
 719 
 
 6729 
 
 6789 
 
 6850 
 
 6910 
 
 6970 
 
 7031 
 
 7091 
 
 7152 
 
 7212 
 
 7272 
 
 60 < 
 
 720 
 
 857332 
 
 7393 
 
 7453 
 
 7513 
 
 7574 
 
 7634 
 
 7694 
 
 7755 
 
 7815 
 
 7875 
 
 GO 
 
 721 
 
 7935 
 
 7935 
 
 8056 
 
 8116 
 
 8176 
 
 8236 
 
 8207 
 
 8357 
 
 8417 
 
 8477 
 
 60 
 
 722 
 
 8537 
 
 8597 
 
 8657 
 
 8718 
 
 8778 
 
 8838 
 
 8898 
 
 8958 
 
 9018 
 
 9078 
 
 60 
 
 723 
 
 9138 
 
 9198 
 
 9253 
 
 9318 
 
 9379 
 
 9439 
 
 9499 
 
 9559 
 
 9619 
 
 9679 
 
 60 
 
 724 
 
 9739 
 
 9799 
 
 9859 
 
 9918 
 
 9978 
 
 -.33 
 
 ..98 
 
 .158 
 
 .218 
 
 .278 
 
 60 
 
 725 
 
 860338 
 
 0393 
 
 0453 
 
 0518 
 
 0578 
 
 0637 
 
 0697 
 
 0757 
 
 0817 
 
 0877 
 
 60 
 
 726 
 
 0937 
 
 0996 
 
 1056 
 
 1116 
 
 1176 
 
 1236 
 
 1295 
 
 1355 
 
 1415 
 
 1475 
 
 60 
 
 727 
 
 1534 
 
 1594 
 
 1654 
 
 1714 
 
 1773 
 
 1833 
 
 1893 
 
 1952 
 
 2012 
 
 2072 
 
 60 
 
 728 
 
 2131 
 
 2191 
 
 2251 
 
 2310 
 
 2370 
 
 2430 
 
 2489 
 
 2549 
 
 2608 
 
 26(58 
 
 ilii 
 
 729 
 
 2728 
 
 2737 
 
 2847 
 
 2900 
 
 2936 
 
 302o 
 
 3085 
 
 3144 
 
 32J4 
 
 3:2! S3 
 
 60 
 
 730 
 
 863323 
 
 333-2 
 
 3442 
 
 3501 
 
 3501 
 
 36-20 
 
 3080 
 
 3739 
 
 3709 
 
 3858 
 
 59 
 
 731 
 
 3917 
 
 3977 
 
 4036 
 
 4098 
 
 4155 
 
 4214 
 
 4274 
 
 4333 
 
 4392 
 
 4452 
 
 59 
 
 732 
 
 4511 
 
 4570 
 
 4630 
 
 4 f i89 
 
 4748 
 
 4608 
 
 4067 
 
 4326 
 
 4935 
 
 5045 
 
 59 
 
 733 
 
 5104 
 
 5163 
 
 5222 
 
 5282 
 
 5341 
 
 5400 
 
 5459 
 
 5519 
 
 5578 
 
 5637 
 
 59 
 
 734 
 
 5696 
 
 5755 
 
 5314 
 
 5374 
 
 5933 
 
 5392 
 
 G051 
 
 6110 
 
 6169 
 
 6223 
 
 59 
 
 735 
 
 6287 
 
 6346 
 
 6405 
 
 6405 
 
 6524 
 
 6583 
 
 6642 
 
 6701 
 
 .67(-0 
 
 6819 
 
 5st - 
 
 736 
 
 6378 
 
 6937 
 
 6'J9'5 
 
 7055 
 
 7114 
 
 7173 
 
 7232 
 
 
 7350 
 
 JZ49T 
 
 '59 
 
 737 
 
 7467 
 
 7526 
 
 7585 
 
 7644 
 
 7703 
 
 77-12 
 
 7821 
 
 7883 
 
 
 7998 
 
 59 
 
 738 
 
 8056 
 
 8115 
 
 8174 
 
 8233 
 
 8292 
 
 8350 
 
 8409 
 
 
 8527 
 
 8586 
 
 59 
 
 739 
 
 8644 
 
 8703 
 
 8762 
 
 8821 
 
 8879 
 
 8938 
 
 
 9056 
 
 9114 
 
 9173 
 
 59 
 
 740 
 
 869232 
 
 9290 
 
 9349 
 
 9408 
 
 9468. 
 
 ^So25^ 
 
 9584 
 
 9042 
 
 9701 
 
 9760 
 
 59 
 
 741 
 
 9818 
 
 9877 
 
 9935 
 
 9994, 
 
 ^53 
 
 111 
 
 .170 
 
 .228 
 
 .287 
 
 .345 
 
 59 
 
 742 
 
 870404 
 
 0462 
 
 0521 
 
 ^9579 
 
 0638 
 
 0696 
 
 0755 
 
 0813 
 
 0872 
 
 0930 
 
 58 
 
 743 
 
 744 
 
 0989 
 15J3- 
 
 104Z- 
 "T631 
 
 -06 
 1690 
 
 1164 
 
 1748 
 
 1223 
 
 1806 
 
 1281 
 1865 
 
 1339 
 1923 
 
 1398 
 1981 
 
 1456 
 2040 
 
 1515 
 
 2098 
 
 58 
 58 
 
 745 
 
 -"12156 
 
 2215 
 
 2273 
 
 2331 
 
 2389 
 
 2448 
 
 2505 
 
 2564 
 
 2022 
 
 2681 
 
 58 
 
 746 
 
 2739 
 
 2797 
 
 2355 
 
 2913 
 
 2972 
 
 3030 
 
 3088 
 
 3146 
 
 3204 
 
 3262 
 
 58 
 
 747 
 
 3321 
 
 3379 
 
 3437 
 
 3495 
 
 3553 
 
 3611 
 
 3669 
 
 3727 
 
 3785 
 
 3844 
 
 58 
 
 748 
 
 3902 
 
 3960 
 
 4018 
 
 4076 
 
 4134 
 
 4192 
 
 4250 
 
 4308 
 
 4366 
 
 4424 
 
 58 
 
 749 
 
 4482 
 
 4540 
 
 4598 
 
 4656 
 
 4714 
 
 4772 
 
 4830 
 
 4888 
 
 4945 
 
 5003 
 
 58 
 
 750 
 
 875061 
 
 5119 
 
 5177 
 
 5235 
 
 5293 
 
 5351 
 
 5409 
 
 5466 
 
 5524 
 
 5582 
 
 58 
 
 751 
 
 5640 
 
 5698 
 
 5756 
 
 5813 
 
 5871 
 
 5929 
 
 5987 
 
 6045 
 
 G102 
 
 6160 
 
 58 
 
 752 
 
 6218 
 
 6276 
 
 6333 
 
 6391 
 
 6449 
 
 6.507 
 
 6564 
 
 6622 
 
 6680 
 
 6737 
 
 58 
 
 753 
 
 6795 
 
 6853 
 
 6910 
 
 6968 
 
 7026 
 
 7083 
 
 7141 
 
 7199 
 
 7256 
 
 7314 
 
 58 
 
 754 
 
 7371 
 
 7429 
 
 7487 
 
 7544 
 
 7602 
 
 7659 
 
 7717 
 
 7774 
 
 7832 
 
 788B 
 
 68 
 
 755 
 
 7947 
 
 8004 
 
 8062 
 
 8119 
 
 8177 
 
 8234 
 
 8292 
 
 8349 
 
 8407 
 
 8464 
 
 57 
 
 756 
 
 8522 
 
 8579 
 
 8637 
 
 8694 
 
 8752 
 
 8809 
 
 8886 
 
 8934 
 
 8981 
 
 9039 
 
 57 
 
 757 
 
 9096 
 
 9153 
 
 9211 
 
 9268 
 
 9325 
 
 9383 
 
 9440 
 
 9497 
 
 9555 
 
 9612 
 
 57 
 
 758 
 
 9669 
 
 9726 
 
 9784 
 
 9841 
 
 9898 
 
 9956 
 
 ..13 
 
 ..70 
 
 .127 
 
 .185 
 
 57 
 
 759 
 
 880242 
 
 0299 
 
 0356 
 
 0413 
 
 0471 
 
 0528 
 
 0585 
 
 0642 
 
 0699 
 
 0756 
 
 57 
 
 N. | | 1 
 
 2)3(4(5 6 | 7 | 8 | 9 | D. 
 
A TABLE OP LOGARITHMS PROM 1 TO 10,000. 
 
 13 
 
 N. | 1 2 3 | 4 | ' 5 6 7 | 8 | 9 D. 
 
 760 
 
 880814 
 
 0871 
 
 0998 
 
 0985 
 
 1042 
 
 1099 
 
 U56 
 
 1213 
 
 1271 
 
 1328 
 
 57 
 
 701 
 
 1385 
 
 1442 
 
 1 HI!) 
 
 1556 
 
 1613 
 
 1670 
 
 1727 
 
 1784 
 
 1841 
 
 was 
 
 57 
 
 762 
 
 I'.I.V, 
 
 201-2 
 
 2069 
 
 2126 
 
 21H3 
 
 2240 
 
 22!)7 
 
 2354 
 
 2411 
 
 2468 
 
 57 
 
 703 
 
 8583 
 
 2581 
 
 2638 
 
 2695 
 
 27.V2 
 
 280!) 
 
 9866 
 
 21123 
 
 2980 
 
 3037 
 
 57 
 
 784 
 
 3093 
 
 3150 
 
 3207 
 
 32H4 
 
 3391 
 
 3377 
 
 3434 
 
 34!) 1 
 
 35 18 
 
 3605 
 
 57 
 
 705 
 
 3661 
 
 3718 
 
 3775 
 
 :W32 
 
 3888 
 
 3945 
 
 4002 
 
 4059 
 
 4115 
 
 4172 
 
 57 
 
 706 
 
 4229 
 
 4985 
 
 4342 
 
 4:!!)!) 
 
 4455 
 
 4512 
 
 4569 
 
 4625 
 
 4682 
 
 4739 
 
 57 
 
 167 
 
 4795 
 
 4859 
 
 4909 
 
 4!)l>5 
 
 5032 
 
 5078 
 
 5135 
 
 5192 
 
 5248 
 
 5305 
 
 57 
 
 7G8 
 
 5361 
 
 5418 
 
 5474 
 
 5531 
 
 5587 
 
 5644 
 
 5700 
 
 5757 
 
 5813 
 
 5870 
 
 57 
 
 769 
 
 5926 
 
 5983 
 
 6039 
 
 6096 
 
 61.5-2 
 
 6209 
 
 6265 
 
 6321 
 
 6378 
 
 6434 
 
 56 
 
 770 
 
 886491 
 
 6547 
 
 6604 
 
 6660 
 
 6716 
 
 6773 
 
 6829 
 
 6885 
 
 6942 
 
 6998 
 
 56 
 
 771 
 
 7054 
 
 7111 
 
 7167 
 
 7223 
 
 7280 
 
 7336 
 
 73i)2 
 
 7449 
 
 7505 
 
 7561 
 
 56 
 
 772 
 
 7617 
 
 7674 
 
 7730 
 
 7786 
 
 7842 
 
 7898 
 
 7955 
 
 8011 
 
 8067 
 
 8123 
 
 56 
 
 773 
 
 8179 
 
 8236 
 
 8292 
 
 8348 
 
 8404 
 
 8460 
 
 8516 
 
 8573 
 
 8629 
 
 8685 
 
 56 
 
 774 
 
 8741 
 
 8797 
 
 8853 
 
 8909 
 
 8<)05 
 
 902J 
 
 9077 
 
 9134 
 
 9190 
 
 9246 
 
 56 
 
 775 
 
 9302 
 
 9358 
 
 9414 
 
 9470 
 
 9520 
 
 9582 
 
 9638 
 
 9(594 
 
 9750 
 
 9806 
 
 56 
 
 776 
 
 9862 
 
 9918 
 
 9974 
 
 ..30 
 
 ..86 
 
 .141 
 
 .197 
 
 .253 
 
 .309 
 
 .365 
 
 56 
 
 777 
 
 890421 
 
 0477 
 
 0533 
 
 0589 
 
 0645 
 
 0700 
 
 075(3 
 
 0812 
 
 0868 
 
 0924 
 
 56 
 
 778 
 
 0980 
 
 1035 
 
 1091 
 
 1147 
 
 1203 
 
 1259 
 
 1314 
 
 1370 
 
 1426 
 
 1482 
 
 56 
 
 779 
 
 1537 
 
 1593 
 
 1649 
 
 1705 
 
 1760 
 
 1816 
 
 1872 
 
 1928 
 
 1983 
 
 2039 
 
 56 
 
 780 
 
 892095 
 
 2150 
 
 2206 
 
 2262 
 
 2317 
 
 2373 
 
 2429 
 
 2484 
 
 2540 
 
 2595 
 
 56 
 
 781 
 
 2651 
 
 2707 
 
 2762 
 
 2818 
 
 2873 
 
 2i)2;) 
 
 2985 
 
 3040 
 
 3096 
 
 3151 
 
 56 
 
 782 
 
 3307 
 
 3969 
 
 3318 
 
 3373 
 
 3 129 
 
 3484 
 
 3540 
 
 3595 
 
 3651 
 
 3706 
 
 56 
 
 783 
 
 3762 
 
 3817 
 
 3873 
 
 3928 
 
 3984 
 
 4039 
 
 4094 
 
 4150 
 
 4205 
 
 4261 
 
 55 
 
 784 
 
 4316 
 
 4371 
 
 4427 
 
 4482 
 
 4S3& 
 
 4593 
 
 4648 
 
 4704 
 
 4759 
 
 4814 
 
 55 
 
 785 
 
 4870 
 
 4!)-.'.-) 
 
 4980 
 
 5030 
 
 5091 
 
 5146 
 
 5-201 
 
 5257 
 
 5312 
 
 5367 
 
 55 
 
 78fT 
 
 51-23 
 
 5478 
 
 5533 
 
 5588 
 
 5644 
 
 5IKK) 
 
 5754 
 
 5809 
 
 5864 
 
 5920 
 
 55 
 
 787 
 
 5975 
 
 6030 
 
 
 6140 
 
 8195 
 
 6251 
 
 6306 
 
 6:561 
 
 6410 
 
 6471 
 
 55 
 
 788 
 
 6636 
 
 6581 
 
 6636 
 
 8693 
 
 6747 
 
 6802 
 
 6857 
 
 6912 
 
 6967 
 
 7022 
 
 55 
 
 789 
 
 7077 
 
 7132 
 
 7187 
 
 7242 
 
 7297 
 
 7352 
 
 7407 
 
 7462 
 
 7517 
 
 7572 
 
 55 
 
 790 
 
 897627 
 
 7682 
 
 7737 
 
 7792 
 
 7847 
 
 7902 
 
 7957 
 
 8012 
 
 8067 
 
 8122 
 
 55 
 
 791 
 
 8176 
 
 8231 
 
 8286 
 
 8341 
 
 8396 
 
 8451 
 
 8506 
 
 8561 
 
 8615 
 
 8670 
 
 55 
 
 792 
 
 8725 
 
 8780 
 
 8835 
 
 8890 
 
 8944 
 
 891)9 
 
 9054 
 
 9109 
 
 9164 
 
 9218 
 
 55 
 
 793 
 
 9273 
 
 9:5-28 
 
 9383 
 
 9437 
 
 9492 
 
 9547 
 
 9602 
 
 9656 
 
 9711 
 
 9766 
 
 55 
 
 794 
 
 9821 
 
 9875 
 
 9930 
 
 9985 
 
 ..39 
 
 ..94 
 
 .149 
 
 .203 
 
 .258 
 
 .312 
 
 55 
 
 795 
 
 900367 
 
 0422 
 
 0471) 
 
 0531 
 
 0586 
 
 0640 
 
 0695 
 
 0749 
 
 0804 
 
 0859 
 
 55 
 
 79S 
 
 0913 
 
 09G6 
 
 1022 
 
 1077 
 
 1131 
 
 1186 
 
 1240 
 
 1295 
 
 1349 
 
 1404 
 
 55 
 
 797 
 
 1458 
 
 1513 
 
 1567 
 
 1622 
 
 1676 
 
 1731 
 
 1785 
 
 1840 
 
 1894 
 
 1948 
 
 54 
 
 793 
 
 2003 
 
 9057 
 
 2112 
 
 3166 
 
 2-2-21 
 
 2275 
 
 2329 
 
 2384 
 
 2438 
 
 2492 
 
 54 
 
 799 
 
 2517 
 
 2601 
 
 2655 
 
 2710 
 
 2764 
 
 2818 
 
 2873 
 
 2927 
 
 2981 
 
 3036 
 
 54. 
 
 800 
 
 903093 
 
 3144 
 
 3199 
 
 3253 
 
 3307 
 
 3361 
 
 3416 
 
 3470 
 
 :;:,_> i 
 
 3578 
 
 54 
 
 801 
 
 3, i33 
 
 3687 
 
 3741 
 
 3795 
 
 3849 
 
 3904 
 
 3958 
 
 4012 
 
 4066 
 
 4120 
 
 54 
 
 802 
 
 4174 
 
 4229 
 
 4283 
 
 4337 
 
 4391 
 
 4445 
 
 4499 
 
 4553 
 
 4607 
 
 4661 
 
 54 
 
 803 
 
 4716 
 
 4770 
 
 4824 
 
 4878 
 
 4!)32 
 
 4986 
 
 5040 
 
 5094 
 
 5148 
 
 5202 
 
 54 
 
 804 
 
 5256 
 
 5310 
 
 5364 
 
 5418 
 
 5479 
 
 5526 
 
 5580 
 
 5634 
 
 5688 
 
 5742 
 
 54 
 
 805 
 
 5796 
 
 5850 
 
 5904 
 
 5958 
 
 6012 
 
 6066 
 
 6119 
 
 6173 
 
 6227 
 
 6281 
 
 54 
 
 806 
 
 6335 
 
 6389 
 
 0443 
 
 6497 
 
 6551 
 
 6604 
 
 6658 
 
 6712 
 
 6766 
 
 6820 
 
 54 
 
 807 
 
 6874 
 
 6927 
 
 6981 
 
 7035 
 
 7089 
 
 7143 
 
 7196 
 
 7250 
 
 7304 
 
 7358 
 
 54 
 
 808 
 
 7411 
 
 7465 
 
 7519 
 
 7573 
 
 7626 
 
 7680 
 
 7734 
 
 7787 
 
 7841 
 
 7895 
 
 54 
 
 809 
 
 7949 
 
 8002 
 
 8056 
 
 8110 
 
 8163 
 
 8217 
 
 8270 
 
 8324 
 
 8378 
 
 8431 
 
 54 
 
 810 
 
 908485 
 
 8539 
 
 8592 
 
 8646 
 
 8699 
 
 8753 
 
 8807 
 
 8360 
 
 8914 
 
 8967 
 
 54 
 
 811 
 
 909 
 
 9074 
 
 9128 
 
 9181 
 
 9235 
 
 9289 
 
 9342 
 
 !!Mi 
 
 9449 
 
 9503 
 
 54 
 
 812 
 
 9556 
 
 9610 
 
 9303 
 
 9716 
 
 9770 
 
 9823 
 
 9877 
 
 9930 
 
 9984 
 
 ..37 
 
 53 
 
 Hi3 
 
 SMOIliM 
 
 0144 
 
 0197 
 
 0251 
 
 0304 
 
 0358 
 
 0411 
 
 0464 
 
 051* 
 
 0571 
 
 53 
 
 814 
 
 0!>24 
 
 0678 
 
 (173! 
 
 0784 
 
 0838 
 
 0891 
 
 0944 
 
 0998 
 
 1051 
 
 1 104 
 
 53 
 
 815 
 
 1158 
 
 1211 
 
 1864 
 
 1317 
 
 1371 
 
 L494 
 
 1477 
 
 1530 
 
 1584 
 
 1637 
 
 53 
 
 , 81(5 
 
 1690 
 
 1743 
 
 1797 
 
 1850 
 
 1903 
 
 1956 
 
 2009 
 
 2063 
 
 2116 
 
 2169 
 
 53 
 
 817 
 
 9898 
 
 2275 
 
 2328 
 
 2381 
 
 2435 
 
 2488 
 
 2541 
 
 2594 
 
 2647 
 
 2700 
 
 53 
 
 818 
 
 2753 
 
 9806 
 
 2859 
 
 2,u:5 
 
 2986 
 
 3019 
 
 3072 
 
 3125 
 
 3178 
 
 3931 
 
 53 
 
 819 
 
 3284 
 
 3337 
 
 3390 
 
 3443 
 
 3496 
 
 3549 
 
 3(302 
 
 3655 
 
 3708 
 
 3761 
 
 53 
 
 N. | 
 
 1|2|3J4|5|G|7|8|9|D. 
 
A TABLE OP LOGARITHMS PROM 1 TO 10,000. 
 
 N. 0|1 2(3 4 5 6 7 | 8 | 9 D. 1 
 
 820 
 
 913814 
 
 3867 
 
 3920 
 
 3973 
 
 4026 
 
 4079 
 
 4132 
 
 4184 
 
 .4237 
 
 4290 
 
 53 j 
 
 8-21 
 
 4343 
 
 4396 
 
 444!) 
 
 4502 
 
 4555 
 
 4608 
 
 4660 
 
 4713 
 
 4766 
 
 4819 
 
 53 
 
 ! 22 
 
 4872 
 
 4925 
 
 4977 
 
 5030 
 
 5083 
 
 5136 
 
 5189 
 
 5241 
 
 5294 
 
 5347 
 
 53 
 
 823 
 
 5400 
 
 5453 
 
 5505 
 
 5558 
 
 5611 
 
 5664 
 
 5716 
 
 5709 
 
 5822 
 
 5875 
 
 53 
 
 824 
 
 5927 
 
 5980 
 
 6033 
 
 6085 
 
 6138 
 
 6191 
 
 6243 
 
 6296 
 
 6349 
 
 6401 
 
 53 
 
 825 
 
 6454 
 
 6507 
 
 6559 
 
 6612 
 
 6664 
 
 6717 
 
 6770 
 
 6822 
 
 S875 
 
 6927 
 
 53 
 
 826 
 
 6980 
 
 7033 
 
 7085 
 
 7138 
 
 7190 
 
 7243 
 
 7295 
 
 7348 
 
 7400 
 
 7453 
 
 53 
 
 827 
 
 7506 
 
 7558 
 
 7611 
 
 7663 
 
 7716 
 
 7768 
 
 7820 
 
 7873 
 
 7825 
 
 7978 
 
 52 
 
 828 
 
 8030 
 
 8083 
 
 8135 
 
 8188 
 
 8240 
 
 8293 
 
 8345 
 
 8397 
 
 8450 
 
 8502 
 
 52 
 
 829 
 
 8555 
 
 8607 
 
 8659 
 
 8712 
 
 8764 
 
 8816 
 
 8869 
 
 8921 
 
 8973 
 
 9026 
 
 52 
 
 830 
 
 919078 
 
 9130 
 
 9183 
 
 9235 
 
 9287 
 
 9340 
 
 9392 
 
 9444 
 
 9496 
 
 9549 
 
 52 
 
 831 
 
 9601 
 
 9653 
 
 9706 
 
 9758 
 
 9810 
 
 9862 
 
 9914 
 
 9967 
 
 ..19 
 
 ..71 
 
 52 
 
 832 
 
 920123 
 
 0176 
 
 0228 
 
 0280 
 
 0332 
 
 0384 
 
 043G 
 
 0489 
 
 0541 
 
 0593 
 
 52 
 
 833 
 
 0645 
 
 0697 
 
 0749 
 
 0801 
 
 0853 
 
 0906 
 
 0958 
 
 1010 
 
 1002 
 
 1114 
 
 52 
 
 834 
 
 1166 
 
 1218 
 
 1270 
 
 1322 
 
 1374 
 
 1420 
 
 1478 
 
 1530 
 
 1582 
 
 1634 
 
 52 
 
 835 
 
 1086 
 
 1738 
 
 1790 
 
 1842 
 
 1894 
 
 1946 
 
 1998 
 
 2050 
 
 2102 
 
 2154 
 
 52 
 
 8:J6 
 
 2206 
 
 2258 
 
 2310 
 
 2362 
 
 2414 
 
 2466 
 
 2518 
 
 2570 
 
 2622 
 
 2674 
 
 52 
 
 837 
 
 2725 
 
 2777 
 
 2829 
 
 2881 
 
 2933 
 
 2985 
 
 3037 
 
 3089 
 
 3140 
 
 3192 
 
 52 
 
 838 
 
 3244 
 
 3296 
 
 3348 
 
 3399 
 
 34 r >l 
 
 3503 
 
 3555 
 
 3607 
 
 3658 
 
 3710 
 
 52 
 
 839 
 
 3762 
 
 3814 
 
 3865 
 
 3917 
 
 3909 
 
 4021 
 
 4072 
 
 4124 
 
 4176 
 
 4228 
 
 52 
 
 840 
 
 924279 
 
 4331 
 
 4383 
 
 4434 
 
 448 
 
 4538 
 
 4589 
 
 4641 
 
 4693 
 
 4744 
 
 52 
 
 ;J 4l 
 
 4796 
 
 4848 
 
 4899 
 
 4951 
 
 5003 
 
 5054 
 
 5106 
 
 5157 
 
 5209 
 
 5201 
 
 52 
 
 I 842 
 
 5312 
 
 5364 
 
 5415 
 
 5467 
 
 5518 
 
 5570 
 
 5621 
 
 5673 
 
 5725 
 
 5776 
 
 52 
 
 843 
 
 5828 
 
 5879 
 
 5931 
 
 5982 
 
 6034 
 
 6085 
 
 6137 
 
 6188 
 
 6240 
 
 6291 
 
 51 
 
 844 
 
 6342 
 
 6394 
 
 6445 
 
 6497 
 
 6548 
 
 6600 
 
 6651 
 
 6702 
 
 6754 
 
 6805 
 
 51 
 
 845 
 
 6857 
 
 6908 
 
 6959 
 
 7011 
 
 7062 
 
 7114 
 
 7165 
 
 7216 
 
 7208 
 
 7319 
 
 51 
 
 846 
 
 7370 
 
 7422 
 
 7473 
 
 7524 
 
 7576 
 
 7627 
 
 7678 
 
 7730 
 
 7781 
 
 7832 
 
 51 
 
 847 
 
 7883 
 
 7935 
 
 798(5 
 
 8037 
 
 8088 
 
 8140 
 
 8191 
 
 8242 
 
 8293 
 
 8345 
 
 51 
 
 848 
 
 8396 
 
 8447 
 
 8498 
 
 8549 
 
 8601 
 
 8652 
 
 8703 
 
 8754 
 
 8805 
 
 8857 
 
 51 
 
 849 
 
 8908 
 
 8959 
 
 9010 
 
 9061 
 
 9112 
 
 9163 
 
 9215 
 
 9206 
 
 9317 
 
 9368 
 
 51 
 
 850' 
 
 929419 
 
 9470 
 
 9521 
 
 9572 
 
 9623 
 
 9674 
 
 9725 
 
 9770 
 
 9827 
 
 9879 
 
 51 
 
 851 
 
 9930 
 
 9981 
 
 ..32 
 
 ..83 
 
 .134 
 
 .185 
 
 .236 
 
 .287 
 
 .338 
 
 .389 
 
 51 
 
 852 
 
 930440 
 
 0491 
 
 0542 
 
 0592 
 
 0643 
 
 0094 
 
 (.745 
 
 0796 
 
 0847 
 
 0898 
 
 51 
 
 853 
 
 0949 
 
 1000 
 
 1051 
 
 1102 
 
 1153 
 
 1204 
 
 1254 
 
 1305 
 
 1350 
 
 1407 
 
 51 
 
 854 
 
 1458 
 
 150!) 
 
 1560 
 
 1610 
 
 16fil 
 
 1712 
 
 1763 
 
 1814 
 
 1865 
 
 1915 
 
 51 
 
 855 
 
 1966 
 
 2017 
 
 2008 
 
 2118 
 
 2169 
 
 2220 
 
 2271 
 
 2322 
 
 2372 
 
 2423 
 
 51 
 
 850 
 
 2474 
 
 2524 
 
 2575 
 
 2626 
 
 2677 
 
 2727 
 
 2778 
 
 2829 
 
 2879 
 
 2930 
 
 51 
 
 857 
 
 2981 
 
 3031 
 
 3082 
 
 3133 
 
 3183 
 
 3234 
 
 3285 
 
 3335 
 
 3386 
 
 3437 
 
 51 
 
 858 
 
 3487 
 
 3538 
 
 3589 
 
 3639 
 
 3690 
 
 3740 
 
 3791 
 
 3841 
 
 3892 
 
 3943 
 
 51 
 
 859 
 
 3993 
 
 4044 
 
 4094 
 
 4145 
 
 4195 
 
 4246 
 
 4296 
 
 4347 
 
 4397 
 
 4448 
 
 51 
 
 860 
 
 934498 
 
 4549 
 
 4599 
 
 4650 
 
 4700 
 
 4751 
 
 4801 
 
 4852 
 
 4902 
 
 4953 
 
 50 
 
 80 1 
 
 5003 
 
 5054 
 
 5104 
 
 5154 
 
 5205 
 
 5255 
 
 5306 
 
 5356 
 
 5406 
 
 5457 
 
 50 
 
 862 
 
 5507 
 
 5558 
 
 5608 
 
 5658 
 
 5709 
 
 5759 
 
 5809 
 
 5860 
 
 5910 
 
 5960 
 
 50 
 
 863 
 
 6011 
 
 6061 
 
 6111 
 
 6162 
 
 62J2 
 
 6262 
 
 6313 
 
 6363 
 
 6413 
 
 6463 
 
 50 
 
 864 
 
 6514 
 
 6564 
 
 6614 
 
 6665 
 
 6715 
 
 6765 
 
 6815 
 
 6865 
 
 6916 
 
 6966 
 
 50 
 
 865 
 
 7016 
 
 7066 
 
 7117 
 
 7167 
 
 7217 
 
 7267 
 
 7317 
 
 7367 
 
 7418 
 
 7468 
 
 50 
 
 866 
 
 7518 
 
 7568 
 
 7618 
 
 7668 
 
 7718 
 
 ?;()> 
 
 7819 
 
 7869 
 
 7919 
 
 7969 
 
 50 
 
 867 
 
 8019 
 
 8069 
 
 8119 
 
 8169 
 
 8219 
 
 8269 
 
 8320 
 
 8370 
 
 8420 
 
 8470 
 
 50 
 
 868 
 
 8520 
 
 8570 
 
 8620 
 
 8670 
 
 8720 
 
 8770 
 
 8820 
 
 8870 
 
 8920 
 
 8970 
 
 50 
 
 869 
 
 9020 
 
 9070 
 
 9120 
 
 9170 
 
 9220 
 
 9270 
 
 9320 
 
 9369 
 
 9419 
 
 9469 
 
 50 
 
 870 
 
 939519 
 
 9569 
 
 9619 
 
 9669 
 
 9719 
 
 9769 
 
 9819 
 
 9869 
 
 9918 
 
 9968 
 
 .50 
 
 871 
 
 940018 
 
 0068 
 
 0118 
 
 0168 
 
 0218 
 
 0267 
 
 0317 
 
 0367 
 
 0417 
 
 0467 
 
 50 
 
 872 
 
 0516 
 
 0566 
 
 0616 
 
 0666 
 
 0716 
 
 0765 
 
 0815 
 
 0865 
 
 0915 
 
 0964 
 
 50 
 
 873 
 
 1014 
 
 1064 
 
 1114 
 
 1163 
 
 1213 
 
 1263 
 
 1313 
 
 1362 
 
 1412 
 
 1462 
 
 50 
 
 874 
 
 1511 
 
 1501 
 
 1611 
 
 1000 
 
 1710 
 
 1760 
 
 1809 
 
 1859 
 
 1909 
 
 1958 
 
 50 
 
 875 
 
 2008 
 
 2058 
 
 2107 
 
 2157 
 
 2207 
 
 2256 
 
 2306 
 
 2355 
 
 2405 
 
 2455 
 
 50 
 
 870 
 
 2504 
 
 2554 
 
 2603 
 
 2653 
 
 2702 
 
 2752 
 
 2801 
 
 2851 
 
 2901 
 
 2950 
 
 50 
 
 877 
 
 3000 
 
 3049 
 
 3099 
 
 3148 
 
 3198 
 
 3247 
 
 3297 
 
 3346 
 
 3396 
 
 3445 
 
 49 
 
 878 
 
 3495 
 
 3544 
 
 3593 
 
 3643 
 
 3(i92 
 
 3742 
 
 3791 
 
 3841 
 
 3890 
 
 3939 
 
 49 
 
 879 
 
 3989 
 
 4038 
 
 4088 
 
 4137 
 
 4186 
 
 4236 
 
 4285 
 
 4335 
 
 4384 
 
 4433 
 
 49 
 
 N. 0|1|2 3j4|5|6 7 | 8 | 9 | D. 
 
A TABF.E OP LOGARITHMS FROM 1 TO 10,000. 
 
 N. | 1 | 2 3 | 4 5|6|7 8 9D. 
 
 880 
 
 944483 
 
 4532 
 
 4581 
 
 4631 
 
 4680 
 
 4729 
 
 4779 
 
 4828 
 
 4877 
 
 4927 
 
 49 
 
 881 
 
 4976 
 
 5025 
 
 51174 
 
 5134 
 
 5173 
 
 5222 
 
 5272 
 
 5321 
 
 5370 
 
 5419 
 
 49 
 
 882 
 
 MBO 
 
 5518 
 
 5567 
 
 5->16 
 
 5(ir>,-> 
 
 5715 
 
 5764 
 
 5813 
 
 5862 
 
 5912 
 
 49 
 
 883 
 
 5901 
 
 (5010 
 
 6059 
 
 6108 
 
 8157 
 
 6207 
 
 6356 
 
 6305 
 
 G354 
 
 6403 
 
 49 
 
 884 
 
 6452 
 
 6501 
 
 6551 
 
 6600 
 
 (5649 
 
 (56HH 
 
 (5747 
 
 6796 
 
 6845 
 
 6894 
 
 49 
 
 885 
 
 6943 
 
 6992 
 
 7041 
 
 7090 
 
 7140 
 
 7189 
 
 7238 
 
 7987 
 
 7336 
 
 7335 
 
 49 
 
 83(5 
 
 7434 
 
 7483 
 
 7532 
 
 7581 
 
 7630 
 
 7679 
 
 77,'H 
 
 7777 
 
 7826 
 
 7875 
 
 49 
 
 887 
 
 7924 
 
 7973 
 
 H022 
 
 8070 
 
 8119 
 
 8168 
 
 8217 
 
 8266 
 
 8315 
 
 8364 
 
 49 
 
 888 
 
 8413 
 
 841)2 
 
 8511 
 
 8580 
 
 8609 
 
 8:i57 
 
 8706 
 
 8755 
 
 8804 
 
 8353 
 
 49 
 
 889 
 
 8932 
 
 8951 
 
 8999 
 
 9048 
 
 9097 
 
 111 Hi 
 
 9195 
 
 9244 
 
 9292 
 
 9341 
 
 49 
 
 800 
 
 94!)39'J 
 
 !)43! 
 
 9488 
 
 9536 
 
 9585 
 
 9634 
 
 9683 
 
 9731 
 
 9780 
 
 9829 
 
 49 
 
 81)1 
 
 9878 
 
 9926 
 
 9975 
 
 ..24 
 
 ..73 
 
 .121 
 
 .170 
 
 .219 
 
 .267 
 
 .316 
 
 49 
 
 893 
 
 JjJMi 
 
 0414 
 
 04(52 
 
 0511 
 
 0560 
 
 0608 
 
 0657 
 
 0706 
 
 0754 
 
 0803 
 
 49 
 
 893 
 
 0851 
 
 0900 
 
 0949 
 
 0997 
 
 1046 
 
 1095 
 
 1143 
 
 1192 
 
 1240 
 
 1289 
 
 49 
 
 8i)4 
 
 1338 
 
 1386 
 
 1435 
 
 1483 
 
 1532 
 
 1580 
 
 1629 
 
 1(577 
 
 1726 
 
 1775 
 
 49 
 
 895 
 
 1823 
 
 1872 
 
 1920 
 
 1969 
 
 2017 
 
 2:ir,ij 
 
 2114 
 
 2163 
 
 2211 
 
 2260 
 
 48 
 
 898 
 
 2308 
 
 2356 
 
 2405 
 
 2453 
 
 2502 
 
 2:,5I) 
 
 2599 
 
 2647 
 
 2696 
 
 2744 
 
 48 
 
 897 
 
 9799 
 
 2841 
 
 288!) 
 
 2938 
 
 29815 
 
 3034 
 
 3083 
 
 3131 
 
 3180 
 
 3228 
 
 48 
 
 898 
 
 327t> 
 
 3325 
 
 3373 
 
 3421 
 
 3470 
 
 3518 
 
 35156 
 
 3615 
 
 3663 
 
 3711 
 
 48 
 
 899 
 
 3760 
 
 3808 
 
 3356 
 
 3905 
 
 3953 
 
 4001 
 
 4049 
 
 4098 
 
 4146 
 
 4194 
 
 48 
 
 900 
 
 954243 
 
 4291 
 
 4339 
 
 4387 
 
 4435 
 
 4484 
 
 4532 
 
 4580 
 
 4628 
 
 4677 
 
 48 
 
 901 
 
 4725 
 
 4773 
 
 4821 
 
 4869 
 
 4918 
 
 4966 
 
 5014 
 
 5062 
 
 5110 
 
 5158 
 
 48 
 
 902 
 
 5207 
 
 5255 
 
 5303 
 
 5351 
 
 5399 
 
 5447 
 
 5495 
 
 5543 
 
 5592 
 
 5640 
 
 48 
 
 933 
 
 5688 
 
 5736 
 
 5784 
 
 5832 
 
 5880 
 
 5928 
 
 5976 
 
 6024 
 
 6072 
 
 6120 
 
 48 
 
 934 
 
 6108 
 
 6216 
 
 6265 
 
 6313 
 
 6361 
 
 6409 
 
 6457 
 
 6505 
 
 6553 
 
 6601 
 
 48 
 
 905 
 
 6649 
 
 6697 
 
 6745 
 
 6793 
 
 6840 
 
 6888 
 
 6936 
 
 6984 
 
 7032 
 
 7080 
 
 48 
 
 906 
 
 71-28 
 
 7176 
 
 7224 
 
 7272 
 
 7320 
 
 7368 
 
 7416 
 
 7464 
 
 7512 
 
 7559 
 
 48 
 
 907 
 
 7607 
 
 7(555 
 
 7703 
 
 7751 
 
 7793 
 
 7847 
 
 7894 
 
 7942 
 
 7990 
 
 8038 
 
 48 
 
 90S 
 
 8086 
 
 8134 
 
 8181 
 
 8229 
 
 8277 
 
 8325 
 
 8373 
 
 8421 
 
 8468 
 
 8516 
 
 48 
 
 909 
 
 85J54 
 
 8612 
 
 8ti59 
 
 8707 
 
 8755 
 
 88J3 
 
 8850 
 
 8898 
 
 8946 
 
 8994 
 
 48 
 
 910 
 
 959041 
 
 9089 
 
 9137 
 
 9185 
 
 9232 
 
 9280 
 
 9328 
 
 9375 
 
 9423 
 
 9471 
 
 48 
 
 911 
 
 9518 
 
 9566 
 
 9(514 
 
 9661 
 
 9709 
 
 9757 
 
 9804 
 
 9852 
 
 9900 
 
 9947 
 
 48 
 
 91-2 
 
 9995 
 
 ..42 
 
 ..90 
 
 .138 
 
 .185 
 
 .233 
 
 .280 
 
 .328 
 
 .376 
 
 .423 
 
 48 
 
 913 
 
 930471 
 
 0518 
 
 0566 
 
 0613 
 
 0861 
 
 0709 
 
 0756 
 
 08:)4 
 
 0851 
 
 0899 
 
 48 
 
 914 
 
 0946 
 
 0994 
 
 1041 
 
 1089 
 
 1136 
 
 1184 
 
 1231 
 
 1279 
 
 1326 
 
 1374 
 
 47 
 
 915 
 
 1421 
 
 1469 
 
 1516 
 
 15(53 
 
 1611 
 
 1658 
 
 1706 
 
 1753 
 
 1801 
 
 1848 
 
 47 
 
 91(5 
 
 1895 
 
 1943 
 
 19!)0 
 
 2038 
 
 2083 
 
 2132 
 
 2180 
 
 2227 
 
 2275 
 
 2322 
 
 47 
 
 917 
 
 2369 
 
 2417 
 
 2464 
 
 2511 
 
 2559 
 
 21506 
 
 2653 
 
 2701 
 
 2748 
 
 2795 
 
 47 
 
 918 
 
 2843 
 
 2890 
 
 2937 
 
 2985 
 
 3032 
 
 3079 
 
 3126 
 
 3174 
 
 3221 
 
 3268 
 
 47 
 
 919 
 
 :wi6 
 
 3363 
 
 3410 
 
 3457 
 
 3504 
 
 3552 
 
 3599 
 
 3646 
 
 3693 
 
 3741 
 
 47 
 
 920 
 
 953788 
 
 3835 
 
 3883 
 
 3929 
 
 3977 
 
 4024 
 
 4071 
 
 4118 
 
 4165 
 
 4212 
 
 47 
 
 921 
 
 4260 
 
 4307 
 
 4354 
 
 4401 
 
 4448 
 
 4495 
 
 4542 
 
 4590 
 
 4637 
 
 4684 
 
 47 
 
 922 
 
 4731 
 
 4778 
 
 4825 
 
 4872 
 
 4919 
 
 4966 
 
 5013 
 
 5061 
 
 5108 
 
 5155 
 
 47 
 
 923 
 
 5202 
 
 5249 
 
 5296 
 
 5343 
 
 5390 
 
 5437 
 
 5484 
 
 5531 
 
 5578 
 
 5625 
 
 47 
 
 924 
 
 5672 
 
 5719 
 
 5766 
 
 5813 
 
 5860 
 
 5907 
 
 59.54 
 
 6001 
 
 6048 
 
 6095 
 
 47 
 
 925 
 
 6142 
 
 6189 
 
 6236 
 
 6283 
 
 6329 
 
 6376 
 
 6423 
 
 6470 
 
 6517 
 
 65(54 
 
 47 
 
 92f> 
 
 6611 
 
 6658 
 
 6705 
 
 6752 
 
 6799 
 
 (5845 
 
 6892 
 
 6939 
 
 6986 
 
 7033 
 
 47 
 
 927 
 
 7080 
 
 7127 
 
 7173 
 
 7220 
 
 7267 
 
 7314 
 
 7361 
 
 7408 
 
 7154 
 
 7501 
 
 47 
 
 928 
 
 7548 
 
 7595 
 
 7642 
 
 7688 
 
 7735 
 
 7782 
 
 7829 
 
 7875 
 
 7922 
 
 7969 
 
 47 
 
 929 
 
 8016 
 
 8062 
 
 8109 
 
 8156 
 
 8203 
 
 8249 
 
 8296 
 
 8343 
 
 8390 
 
 8436 
 
 47 
 
 930 
 
 968483 
 
 8530 
 
 8576 
 
 8623 
 
 8670 
 
 8716 
 
 8763 
 
 8810 
 
 8856 
 
 8903 
 
 47 
 
 931 
 
 8950 
 
 8996 
 
 9043 
 
 9090 
 
 9136 
 
 9183 
 
 9229 
 
 9276 
 
 9323 
 
 9369 
 
 47 
 
 932 
 
 9416 
 
 9463 
 
 9509 
 
 9556 
 
 9602 
 
 91549 
 
 9695 
 
 9742 
 
 9789 
 
 9835 
 
 47 
 
 1 933 
 
 9882 
 
 9928 
 
 9975 
 
 ..21 
 
 ..68 
 
 .114 
 
 .161 
 
 .207 
 
 .254 
 
 .300 
 
 47 
 
 934 
 
 970347 
 
 0393 
 
 0440 
 
 0486 
 
 0533 
 
 0579 
 
 0626 
 
 0672 
 
 0719 
 
 0765 
 
 46 
 
 935 
 
 0812 
 
 (H.-H 
 
 09!)4 
 
 0951 
 
 0997 
 
 1044 
 
 1090 
 
 1137 
 
 1183 
 
 12-29 
 
 46 
 
 936 
 
 1276 
 
 1322 
 
 1369 
 
 1415 
 
 1461 
 
 1508 
 
 1554 
 
 1601 
 
 1647 
 
 1693 
 
 46 
 
 937 
 
 1740 
 
 1786 
 
 1832 
 
 1879 
 
 1925 
 
 11171 
 
 2018 
 
 2064 
 
 2110 
 
 2157 
 
 46 
 
 938 
 
 2203 
 
 2249 
 
 2295 
 
 2342 
 
 2388 
 
 2434 
 
 2481 
 
 2527 
 
 2573 
 
 2619 
 
 46 
 
 939 
 
 2666 
 
 2712 
 
 2758 
 
 2804 
 
 2851 
 
 2897 
 
 2943 
 
 2989 
 
 3035 
 
 3082 
 
 46 
 
 N. | | 1 | 2 | 3 | 4 5 6|7|8 9 | D. J 
 
16 
 
 A TABLE OP LOGARITHMS FROM 1 TO 10.000 
 
 N. | | 1 2 3 I 4 | 5 I 6 7|8|9|D. 
 
 940 
 
 973128 
 
 3174 
 
 3220 
 
 32615 
 
 3313 
 
 3359 I 3405 
 
 3451 
 
 3497 
 
 3543 
 
 46 
 
 941 
 
 3590 
 
 3!536 
 
 3682 
 
 3728 
 
 3774 
 
 3820 
 
 38(56 
 
 3913 
 
 3959 
 
 40J5 
 
 46 
 
 942 
 
 4051 
 
 4097 
 
 4143 
 
 4189 
 
 4235 
 
 4281 
 
 4327 
 
 4374 
 
 4420 
 
 4466 
 
 46 
 
 943 
 
 4512 
 
 4558 
 
 4(504 
 
 4(550 
 
 46% 
 
 4742 
 
 4788 
 
 4S34 
 
 4880 
 
 4926 
 
 46 
 
 944 
 
 4972 
 
 5018 
 
 5064 
 
 5110 
 
 5156 
 
 5202 
 
 5248 
 
 5294 
 
 5340 
 
 5386 
 
 46 
 
 945 
 
 5432 
 
 5478 
 
 5524 
 
 5570 
 
 5616 
 
 5(5(52 
 
 5707 
 
 5753 
 
 5799 
 
 5845 
 
 46 
 
 946 
 
 5891 
 
 5937 
 
 5983 
 
 6029 
 
 6075 
 
 6121 
 
 6167 
 
 6212 
 
 6258 
 
 6304 
 
 46 
 
 917 
 
 6350 
 
 6396 
 
 6442 
 
 6488 
 
 6533 
 
 6579 
 
 66-25 
 
 6671 
 
 6717 
 
 6763 
 
 46 
 
 948 
 
 6808 
 
 6854 
 
 6900 
 
 6346 
 
 6992 
 
 7037 
 
 7083 
 
 7129 
 
 7175 
 
 7220 
 
 46 
 
 949 
 
 7266 
 
 7312 
 
 7358 
 
 7403 
 
 7449 
 
 7495 
 
 7541 
 
 7586 
 
 7632 
 
 7678 
 
 46 
 
 959 
 
 977724 
 
 7769 
 
 7815 
 
 7861 
 
 7906 
 
 7952 
 
 7998 
 
 8043 
 
 8089 
 
 8135 
 
 46 
 
 951 
 
 8181 
 
 8226 
 
 8272 
 
 8317 
 
 8363 
 
 8409 
 
 8454 
 
 8500 
 
 8546 
 
 8591 
 
 46 
 
 952 
 
 8637 
 
 8683 
 
 8728 
 
 8774 
 
 8819 
 
 8865 
 
 8911 
 
 81)56 
 
 9002 
 
 9047 
 
 46 
 
 953 
 
 9093 
 
 9138 
 
 9184 
 
 9230 
 
 9275 
 
 9321 
 
 9366 
 
 9412 
 
 9457 
 
 9503 
 
 46 
 
 954 
 
 9548 
 
 9594 
 
 9639 
 
 9;>85 
 
 9730 
 
 9776 
 
 9821 
 
 9867 
 
 9912 
 
 9958 
 
 46 
 
 955 
 
 980003 
 
 0049 
 
 0094 
 
 0140 
 
 0185 
 
 0231 
 
 0276 
 
 0322 
 
 03(57 
 
 0412 
 
 45 
 
 956 
 
 0458 
 
 0503 
 
 0549 
 
 0594 
 
 0640 
 
 0685 
 
 0730 
 
 0776 
 
 0821 
 
 0867 
 
 45 
 
 957 
 
 0912 
 
 0957 
 
 1003 
 
 1048 
 
 1093 
 
 1139 
 
 1184 
 
 1229 
 
 1275 
 
 1320 
 
 45 
 
 958 
 
 1366 
 
 1411 
 
 1456 
 
 1501 
 
 1547 
 
 1592 
 
 1637 
 
 1683 
 
 1728 
 
 ms 
 
 45 
 
 959 
 
 1819 
 
 1864 
 
 1909 
 
 1954 
 
 2000 
 
 2045 
 
 2090 
 
 2135 
 
 2181 
 
 2226 
 
 45 
 
 960 
 
 982271 
 
 2316 
 
 2362 
 
 2407 
 
 2452 
 
 2497 
 
 2543 
 
 2588 
 
 2633 
 
 2678 
 
 45 
 
 961 
 
 2723 
 
 2769 
 
 2814 
 
 2859 
 
 2904 
 
 2949 
 
 -2994 
 
 3040 
 
 3085 
 
 3130 
 
 45 
 
 962 
 
 3175 
 
 3220 
 
 32(55 
 
 3310 
 
 3356 
 
 3401 
 
 3446 
 
 3401 
 
 3536 
 
 3581 
 
 45 
 
 983 
 
 3626 
 
 3671 
 
 3716 
 
 3762 
 
 3807 
 
 3852 
 
 3897 
 
 3942 
 
 3987 
 
 4032 
 
 45 
 
 964 
 
 4077 
 
 4122 
 
 4167 
 
 4212 
 
 4257 
 
 4302 
 
 4347 
 
 4392 
 
 4437 
 
 4482 
 
 45 
 
 965 
 
 4527 
 
 4572 
 
 4617 
 
 4662 
 
 4707 
 
 4752 
 
 4797 
 
 4842 
 
 4887 
 
 4932 
 
 45 
 
 9C6 
 
 4977 
 
 5022 
 
 5067 
 
 5112 
 
 5157 
 
 5202 
 
 5247 
 
 5292 
 
 5337 
 
 5382 
 
 45 
 
 967 
 
 5426 
 
 5471 
 
 5516 
 
 5561 
 
 5606 
 
 5051 
 
 5696 
 
 5741 
 
 5786 
 
 5830 
 
 45 
 
 968 
 
 5875 
 
 5920 
 
 5965 
 
 6010 
 
 6055 
 
 6100 
 
 6144 
 
 6189 
 
 6234 
 
 6279 
 
 45 
 
 969 
 
 6324 
 
 6369 
 
 6413 
 
 6458 
 
 6503 
 
 6548 
 
 6593 
 
 6637 
 
 6682 
 
 6727 
 
 45 
 
 970 
 
 986772 
 
 6817 
 
 6861 
 
 6906 
 
 6951 
 
 6996 
 
 7040 
 
 7085 
 
 7130 
 
 7175 
 
 45 
 
 971 
 
 7219 
 
 7264 
 
 7309 
 
 7353 
 
 7398 
 
 7443 
 
 7488 
 
 7532 
 
 7577 
 
 7622 
 
 45 
 
 972 
 
 7666 
 
 7711 
 
 7756 
 
 7800 
 
 7845 
 
 7890 
 
 7934 
 
 7979 
 
 8024 
 
 8068 
 
 45 
 
 973 
 
 8113 
 
 8157 
 
 82!)2 
 
 8247 
 
 8291 
 
 8336 
 
 8381 
 
 8425 
 
 8470 
 
 8514 
 
 45 
 
 974 
 
 8559 
 
 8604 
 
 8(548 
 
 8693 
 
 8737 
 
 8782 
 
 8826 
 
 8871 
 
 8916 
 
 8960 
 
 45 
 
 1 975 
 
 9005 
 
 9049 
 
 9094 
 
 9138 
 
 9183 
 
 9227 
 
 9272 
 
 9316 
 
 9361 
 
 9405 
 
 45 
 
 976 
 
 9450 
 
 9494 
 
 9539 
 
 9583 
 
 9G28 
 
 9672 
 
 9717 
 
 9761 
 
 9806 
 
 9850 
 
 44 
 
 , 977 
 
 9895 
 
 9939 
 
 9983 
 
 ..28 
 
 ..72 
 
 .117 
 
 .161 
 
 .206 
 
 .250 
 
 .294 
 
 44 
 
 978 
 
 990339 
 
 0383 
 
 0428 
 
 0472 
 
 0516 
 
 0561 
 
 0605 
 
 0650 
 
 0694 
 
 0738 
 
 44 
 
 979 
 
 0783 
 
 0827 
 
 0871 
 
 0916 
 
 0960 
 
 1004 
 
 1049 
 
 1093 
 
 1137 
 
 1182 
 
 44 
 
 980 
 
 991226 
 
 1270 
 
 1315 
 
 1359 
 
 1403 
 
 1448 
 
 1492 
 
 1536 
 
 1580 
 
 1625 
 
 44 
 
 981 
 
 1669 
 
 1713 
 
 1758 
 
 1802 
 
 1848 
 
 1890 
 
 1935 
 
 1979 
 
 2023 
 
 2067 
 
 44 
 
 982 
 
 2111 
 
 2156 
 
 2200 
 
 2244 
 
 2288 
 
 2333 
 
 2377 
 
 2421 
 
 2465 
 
 2509 
 
 44 
 
 983 
 
 2554 
 
 2598 
 
 2642 
 
 2686 
 
 2730 
 
 2774 
 
 2819 
 
 2863 
 
 2907 
 
 2951 
 
 44 
 
 984 
 
 2995 
 
 3039 
 
 3083 
 
 3127 
 
 3172 
 
 3216 
 
 3260 
 
 3304 
 
 3348 
 
 3392 
 
 44 
 
 985 
 
 3436 
 
 3480 
 
 3524 
 
 3568 
 
 3613 
 
 3657 
 
 3701 
 
 3745 
 
 3789 
 
 3833 
 
 44 
 
 986 
 
 3877 
 
 3921 
 
 39155 
 
 4009 
 
 4053 
 
 4097 
 
 4141 
 
 4185 
 
 4229 
 
 4273 
 
 44 
 
 987 
 
 4317 
 
 4361 
 
 4405 
 
 4449 
 
 4493 
 
 4537 
 
 4581 
 
 4625 
 
 4669 
 
 4713 
 
 44 
 
 988 
 
 4757 
 
 4801 
 
 4845 
 
 4889 
 
 4933 
 
 4977 
 
 5021 
 
 5065 
 
 5108 
 
 -5152 
 
 44 
 
 989 
 
 5196 
 
 5240 
 
 5284 
 
 5328 
 
 5372 
 
 5416 
 
 5460 
 
 5504 
 
 5547 
 
 5591 
 
 44 
 
 990 
 
 995635 
 
 5679 
 
 5723 
 
 5767 
 
 5811 
 
 5854 
 
 5898 
 
 5942 
 
 5986 
 
 6030 
 
 44 
 
 991 
 
 6074 
 
 6117 
 
 6161 
 
 6205 
 
 6249 
 
 6293 
 
 6337 
 
 6330 
 
 6424 
 
 64(58 
 
 44 
 
 992 
 
 6512 
 
 6555 
 
 6599 
 
 6643 
 
 6!587 
 
 6731 
 
 6774 
 
 6818 
 
 f>862 
 
 690(5 
 
 44 
 
 &93 
 
 G919 
 
 6993 
 
 7037 
 
 7080 
 
 7124 
 
 7168 
 
 7212 
 
 7255 
 
 7299 
 
 7343 
 
 44 
 
 994 
 
 7386 
 
 7430 
 
 7474 
 
 7517 
 
 7561 
 
 7605 
 
 7648 
 
 7692 
 
 7736 
 
 7779 
 
 44 
 
 995 
 
 7823 
 
 7867 
 
 7910 
 
 7954 
 
 7>J98 
 
 8041 
 
 8085 
 
 8129 
 
 8172 
 
 8216 
 
 44 
 
 996 
 
 8259 
 
 8303 
 
 8347 
 
 839!) 
 
 8434 
 
 8477 
 
 8521 
 
 8564 
 
 8(508 
 
 8652 
 
 44 
 
 997 
 
 8695 
 
 8739 
 
 8782 
 
 8326 
 
 88(59 
 
 8913 
 
 8956 
 
 9000 
 
 9043 
 
 9087 
 
 44 
 
 998 
 
 9131 
 
 9174 
 
 9213 
 
 2f51 
 
 9305 
 
 9348 
 
 9392 
 
 9435 
 
 9479 
 
 9522 
 
 44 
 
 999 
 
 9565 
 
 9609 
 
 9652 
 
 9596 
 
 973!) 
 
 9783 
 
 9826 
 
 9870 
 
 9913 
 
 9957 
 
 43 
 
 N. | 
 
 1 2 | 3 4 | 5 6 7 | 8 
 
 9 D. 
 
SINES AND TANGENTS, 
 
 FOR EVERY 
 
 DEGREE AND MINUTE 
 
 OF THE QUADRANT, 
 
 N.B. The minutes in the left-hand column of each page, increasing 
 downwards, belong to the degrees at the top ; and those increasing up- 
 wards, in the right-hand column, belong to the degrees below. 
 
18 
 
 (0 Degree.) A TABLE OF LOGARITHMIC 
 
 M. Sine [ D | Cosine. | D. Tang. | D Cotang. ! 
 
 
 
 0.000000 
 
 
 10.000000 
 
 
 0.000000 
 
 
 Infinite. 
 
 60 
 
 1 
 
 6.463726 
 
 501717 
 
 000000 
 
 00 
 
 6.4i;37-2r> 
 
 501717 
 
 13.536274 
 
 59 
 
 2 
 
 764756 
 
 293485 
 
 030000 
 
 00 
 
 764756 
 
 2J3433 
 
 235-244 
 
 58 
 
 3 
 
 940847 
 
 208231 
 
 000000 
 
 00 
 
 940847 
 
 206331 
 
 053153 
 
 57 
 
 4 
 
 7.065786 
 
 161517 
 
 000000 
 
 00 
 
 7.065785 
 
 1-J1517 
 
 12.934214 
 
 5<> 
 
 5 
 
 162695 
 
 1319(58 
 
 000000 
 
 03 
 
 162696 
 
 1319(i9 
 
 837304 
 
 55 
 
 6 
 
 241877 
 
 111575 
 
 9.999993 
 
 01 
 
 241878 
 
 111578 
 
 758122 
 
 54 
 
 7 
 
 :io-^24 
 
 93653 
 
 939393 
 
 01 
 
 308825 
 
 93 ! 553 
 
 601175 
 
 53 
 
 8 
 
 366816 
 
 85254 
 
 909093 
 
 01 
 
 366317 
 
 85254 
 
 633183 
 
 52 
 
 9 
 
 4179S8 
 
 76263 
 
 999399 
 
 01 
 
 417970 
 
 762(53 
 
 532030 
 
 51 
 
 10 
 
 463725 
 
 68988 
 
 999998 
 
 01 
 
 463727 
 
 63933 
 
 5315-273 
 
 50 
 
 11 
 
 7.505118 
 
 62981 
 
 9.993993 
 
 01 
 
 7.505120 
 
 62981 
 
 12.494883 
 
 49 
 
 12 
 
 542906 
 
 57936 
 
 999397 
 
 01 
 
 542909 
 
 57933 
 
 457091 
 
 48 
 
 13 
 
 577668 
 
 53641 
 
 993997 
 
 01 
 
 577672 
 
 53642 
 
 422328 
 
 47 
 
 14 
 
 609853 
 
 49938 
 
 999398 
 
 01 
 
 609857 
 
 49039 
 
 :uni3 
 
 46 
 
 15 
 
 639816 
 
 46714 
 
 999395 
 
 01 
 
 6393-30 
 
 46715 
 
 3:50180 
 
 45 
 
 16 
 
 667845 
 
 43881 
 
 993995 
 
 01 
 
 667849 
 
 43832 
 
 332151 
 
 44 
 
 17 
 
 694173 
 
 41372 
 
 999935 
 
 01 
 
 694] 79 
 
 41373 
 
 305821 
 
 43 
 
 18 
 
 718997 
 
 39135 
 
 999394 
 
 01 
 
 719J03 
 
 39136 
 
 230337 
 
 42 
 
 19 
 
 742477 
 
 37127 
 
 939393 
 
 01 
 
 742484 
 
 37128 
 
 257516 
 
 41 
 
 20 
 
 764754 
 
 35315 
 
 999933 
 
 01 
 
 764761 
 
 35136 
 
 235239 
 
 40 
 
 21 
 
 7.785943 
 
 33672 
 
 9.999992 
 
 01 
 
 7.785951 
 
 33673 
 
 1-2.214049 
 
 39 
 
 22 
 
 806146 
 
 32175 
 
 999991 
 
 01 
 
 806155 
 
 32171) 
 
 193845 
 
 38 
 
 23 
 
 825451 
 
 308J5 
 
 993993 
 
 01 
 
 825460 
 
 30838 
 
 174540 
 
 37 ! 
 
 24 
 
 843934 
 
 29547 
 
 993989 
 
 02 
 
 843344 
 
 29549 
 
 155356 
 
 35 i 
 
 25 
 
 881662 
 
 28388 
 
 999938 
 
 02 
 
 881674 
 
 28393 
 
 138325 
 
 35 
 
 23 
 
 878S95 
 
 27317 
 
 999383 
 
 02 
 
 878703 
 
 27318 
 
 121-2J2 
 
 34 
 
 27 
 
 895085 
 
 26323 
 
 999937 
 
 02 
 
 895093 
 
 20325 
 
 104 9J1 
 
 33 
 
 28 
 
 910879 
 
 25399 
 
 999985 
 
 02 
 
 9J0394 
 
 25401 
 
 083106 
 
 32 
 
 29 
 
 926119 
 
 24538 
 
 999985 
 
 02 
 
 925134 
 
 24540 
 
 073365 
 
 31 
 
 30 
 
 940842 
 
 23733 
 
 999983 
 
 02 
 
 940858 
 
 23735 
 
 053142 
 
 30 
 
 31 
 
 7.955082 
 
 22930 
 
 9.9999.32 
 
 02 
 
 7.955100 
 
 22981 
 
 12.044903 
 
 29 
 
 32 
 
 963870 
 
 22273 
 
 999381 
 
 02 
 
 968889 
 
 22275 
 
 031111 
 
 23 
 
 33 
 
 982233 
 
 21*508 
 
 993930 
 
 02 
 
 932253 
 
 21610 
 
 017747 
 
 27 
 
 34 
 
 995198 
 
 2J331 
 
 999379 
 
 02 
 
 995219 
 
 20933 
 
 004781 
 
 28 
 
 35 
 
 8.037787 
 
 S03UO 
 
 933977 
 
 02 
 
 8.007809 
 
 20392 
 
 11.99-2191 
 
 25 
 
 38 
 
 020021 
 
 1U331 
 
 9J3376 
 
 02 
 
 020045 
 
 19333 
 
 979955 
 
 24 
 
 37 
 
 031919 
 
 1930-2 
 
 939975 
 
 02 
 
 031945 
 
 19305 
 
 9S8J55 
 
 23 
 
 38 
 
 043501 
 
 18801 
 
 999373 
 
 02 
 
 043527 
 
 18333 
 
 956473 
 
 22 
 
 39 054781 
 
 18325 
 
 933372 
 
 02 
 
 054809 
 
 18327 
 
 945191 
 
 21 
 
 40 
 
 OS5776 
 
 17872 
 
 993371 
 
 02 
 
 065806 
 
 17874 
 
 934194 
 
 20 
 
 41 
 
 8.076500 
 
 17441 
 
 9.939969 
 
 03 
 
 8.076531 
 
 17144 
 
 11.9=23469 
 
 19 
 
 42 
 
 088965 
 
 17031 
 
 999918 
 
 02 
 
 086397 
 
 17034 
 
 913303 
 
 18 
 
 43 
 
 097183 
 
 16639 
 
 !).).).) iii 
 
 02 
 
 037217 
 
 16642 
 
 902783 
 
 17 
 
 44 
 
 107167 
 
 16255 
 
 9333 : >4 
 
 03 
 
 107202 
 
 16268 
 
 892797 
 
 16 
 
 45 
 
 116926 
 
 15938 
 
 993963 
 
 03 
 
 11(5933 
 
 15910 
 
 883037 
 
 15 
 
 46 
 
 126471 
 
 15566 
 
 993931 
 
 03 
 
 126510 
 
 15568 
 
 873490 
 
 14 
 
 47 
 
 135810 
 
 15238 
 
 939359 
 
 03 
 
 135851 
 
 15241 
 
 864149 
 
 13 
 
 48 
 
 144953 
 
 14924 
 
 999958 
 
 03 
 
 144996 
 
 149.27 
 
 855004 
 
 12 
 
 49 
 
 153907 
 
 14522 
 
 999356 
 
 03 
 
 153952 
 
 14627 
 
 846048 
 
 11 
 
 50 
 
 162(581 
 
 14333 
 
 999954 
 
 03 
 
 162727 
 
 14336 
 
 837273 
 
 10 
 
 51 
 
 8.171280 
 
 14054 
 
 9.9903.52 
 
 03 
 
 8.171328 
 
 14057 
 
 11.828672 
 
 9 
 
 52 
 
 179713 
 
 13786 
 
 999950 
 
 03 
 
 179763 
 
 13790 
 
 820237 
 
 8 
 
 53 
 
 187985 
 
 13529 
 
 939948 
 
 03 
 
 188036 
 
 13532 
 
 811934 
 
 
 54 
 
 193102 
 
 13280 
 
 993946 
 
 03 
 
 196156 
 
 13284 
 
 833844 
 
 6 
 
 55 
 
 204070 
 
 13041 
 
 939344 
 
 03 
 
 204126 
 
 115044 
 
 795874 
 
 5 
 
 58- 
 
 211895 
 
 12810 
 
 939342 
 
 04 
 
 211953 
 
 12814 
 
 788047 
 
 4 
 
 57 
 
 219581 
 
 12587 
 
 993340 
 
 04 
 
 219541 
 
 12590 
 
 78J359 
 
 3 
 
 58 
 
 227134 
 
 12372 
 
 33338 
 
 04 
 
 227195 
 
 12376 
 
 772805 
 
 2 
 
 59 
 
 234557 
 
 12154 
 
 993936 
 
 04 
 
 234621 
 
 12168 
 
 765379 
 
 1 
 
 60 
 
 241855 
 
 11963 
 
 939334 
 
 04 
 
 241921 
 
 11987 
 
 758079 
 
 
 
 | Cosine | Sine | | Cotang. | Tang. | M. 
 
 89 Degrees. 
 
SINES AND TANGENTS. (1 Degree.) 
 
 19 
 
 M. Sine D. | Cosine | D. | Tang. | IX | Cotang. 
 
 
 
 ri.-J-HH.V) 
 
 119U3 
 
 9.999934 
 
 04 
 
 8.241921 
 
 11967 
 
 11.758079 
 
 60 
 
 1 
 
 2-4 9033 
 
 11768 
 
 999932 
 
 04 
 
 249102 
 
 11772 
 
 750898 
 
 59 
 
 2 
 
 256094 
 
 11. VO 
 
 999929 
 
 04 
 
 25(1165 
 
 11584 
 
 743835 
 
 58 
 
 3 
 
 263042 
 
 11398 
 
 999927 
 
 04 
 
 263115 
 
 11402 
 
 736885 
 
 57 
 
 4 
 
 269881 
 
 11221 
 
 9! Ml: l-.'/i 
 
 04 
 
 2(>99f><; 
 
 11225 
 
 730044 
 
 56 ! 
 
 5 
 
 276614 
 
 11050 
 
 9999-22 
 
 04 
 
 276691 
 
 11054 
 
 723309 
 
 K 
 
 1 (i 
 
 2rf43 
 
 10883 
 
 999920 
 
 04 
 
 283323 
 
 10887 
 
 716677 
 
 54 
 
 7 
 
 289773 
 
 10721 
 
 999918 
 
 04 
 
 289856 
 
 10726 
 
 710144 
 
 53 
 
 8 
 
 296807 
 
 10565 
 
 999915 
 
 04 
 
 296292 
 
 10570 
 
 703708 
 
 52 
 
 9 
 
 302546 
 
 10413 
 
 99H913 
 
 04 
 
 3<Wi34 
 
 10418 
 
 697366 
 
 51 
 
 10 
 
 m-o.n 
 
 10266 
 
 999910 
 
 04 
 
 308884 
 
 10270 
 
 691116 
 
 50 
 
 11 
 
 8.314954 
 
 10122 
 
 9.999907 
 
 04 
 
 8.315046 
 
 10126 
 
 11.684954 
 
 49 
 
 12 
 
 321027 
 
 9982 
 
 999905 
 
 04 
 
 321122 
 
 9987 
 
 678878 
 
 48 
 
 13 
 
 327016 
 
 9847 
 
 999902 
 
 04 
 
 327114 
 
 9851 
 
 672886 
 
 47 
 
 14 
 
 332924 
 
 9714 
 
 999899 
 
 05 
 
 333025 
 
 9719 
 
 666975 
 
 46 
 
 15 
 
 338753 
 
 9586 
 
 999897 
 
 05 
 
 338856 
 
 9590 
 
 661144 
 
 45 
 
 16 
 
 344504 
 
 9460 
 
 999894 
 
 05 
 
 344610 
 
 9465 
 
 655390 
 
 44 
 
 17 
 
 350181 
 
 9338 
 
 999891 
 
 05 
 
 350289 
 
 9343 
 
 649711 
 
 43 
 
 18 
 
 355783 
 
 9219 
 
 999888 
 
 05 
 
 355895 
 
 9224 
 
 644105 
 
 42 
 
 19 
 
 361315 
 
 9103 
 
 999885 
 
 05 
 
 361430 
 
 9108 
 
 638570 
 
 41 
 
 20 
 
 366777 
 
 8990 
 
 999882 
 
 05 
 
 366895 
 
 8995 
 
 633105 
 
 40 
 
 21 
 
 8.372171 
 
 8880 
 
 9.999879 
 
 05 
 
 8.372292 
 
 8885 
 
 11.627708 
 
 39 
 
 22 
 
 377499 
 
 8772 
 
 999876 
 
 05 
 
 377622 
 
 8777 
 
 622378 
 
 38 
 
 23 
 
 382762 
 
 8667 
 
 999873 
 
 05 
 
 382889 
 
 8672 
 
 617111 
 
 37 
 
 24 
 
 387962 
 
 8564 
 
 999870 
 
 05 
 
 388092 
 
 8570 
 
 611908 
 
 36 
 
 25 
 
 393101 
 
 8464 
 
 999867 
 
 05 
 
 393234 
 
 8470 
 
 606766 
 
 35 
 
 26 
 
 398179 
 
 8366 
 
 999864 
 
 05 
 
 398315 
 
 8371 
 
 601685 
 
 34 
 
 27 
 
 403199 
 
 8271 
 
 999861 
 
 05 
 
 403338 
 
 8276 
 
 596662 
 
 33 
 
 28 
 
 408161 
 
 8177 
 
 999858 
 
 05 
 
 408304 
 
 8182 
 
 591696 
 
 32 
 
 29 
 
 413068 
 
 8086 
 
 999854 
 
 05 
 
 413213 
 
 8091 
 
 586787 
 
 31 
 
 30 
 
 417919 
 
 7996 
 
 999851 
 
 06 
 
 418068 
 
 8002 
 
 581932 
 
 30 
 
 31 
 
 8.422717 
 
 7909 
 
 9.999848 
 
 06 
 
 8.422869 
 
 7914 
 
 11.577131 
 
 29 
 
 32 
 
 427462 
 
 7823 
 
 999844 
 
 06 
 
 427618 
 
 7830 
 
 572382 
 
 28 
 
 33 
 
 432156 
 
 7740 
 
 999841 
 
 06 
 
 432315 
 
 7745 
 
 567685 
 
 27 
 
 34 
 
 436800 
 
 7657 
 
 999838 
 
 06 
 
 436962 
 
 7663 
 
 563038 
 
 26 
 
 35 
 
 441394 
 
 7577 
 
 999834 
 
 06 
 
 441560 
 
 7583 
 
 558440 
 
 25 I 
 
 36 
 
 445941 
 
 7499 
 
 999831 
 
 06 
 
 446110 
 
 7505 
 
 553890 
 
 24 
 
 37 
 
 450440 
 
 7422 
 
 999827 
 
 06 
 
 450613 
 
 7428 
 
 549387 
 
 23 
 
 38 
 
 454893 
 
 7346 
 
 999823 
 
 06 
 
 455070 
 
 7352 
 
 544930 
 
 22 
 
 39 
 
 459301 
 
 7273 
 
 999820 
 
 06 
 
 459481 
 
 7279 
 
 540519 
 
 21 
 
 40 
 
 463665 
 
 7200 
 
 999816 
 
 06 
 
 4G3849 
 
 7206 
 
 536151 
 
 20 
 
 41 
 
 8.467985 
 
 7129 
 
 9.999812 
 
 06 
 
 8.468172 
 
 7135 
 
 11.531828 
 
 19 
 
 42 
 
 472263 
 
 7060 
 
 999809 
 
 06 
 
 472454 
 
 7066 
 
 527546 
 
 18 
 
 43 
 
 476498 
 
 6991 
 
 999805 
 
 06 
 
 476693 
 
 6998 
 
 523307 
 
 17 
 
 44 
 
 480693 
 
 6924 
 
 999801 
 
 06 
 
 480892 
 
 6931 
 
 519108 
 
 16 
 
 45 
 
 484848 
 
 6859 
 
 999797 
 
 07 
 
 485050 
 
 6865 
 
 514950 
 
 15 
 
 46 
 
 488963 
 
 6794 
 
 999793 
 
 07 
 
 489170 
 
 6801 
 
 510830 
 
 14 
 
 47 
 
 493040 
 
 6731 
 
 999790 
 
 07 
 
 493250 
 
 6738 
 
 506750 
 
 13 
 
 48 
 
 497078 
 
 6669 
 
 999786 
 
 07 
 
 497293 
 
 6676 
 
 502707 
 
 12 
 
 49 
 
 501080 
 
 6608 
 
 999782 
 
 07 
 
 501298 
 
 6615 
 
 498702 
 
 11 
 
 50 
 
 505045 
 
 65-18 
 
 999778 
 
 07 
 
 505267 
 
 6555 
 
 494733 
 
 10 
 
 51 
 
 8.508974 
 
 6489 
 
 9.999774 
 
 07 
 
 8.509200 
 
 6496 
 
 11.490800 
 
 9 
 
 52 
 
 512867 
 
 6431 
 
 999769 
 
 07 
 
 513098 
 
 6439 
 
 4<36902 
 
 8 
 
 53 
 
 516726 
 
 6375 
 
 999765 
 
 07 
 
 516961 
 
 6382 
 
 483039 
 
 7 
 
 54 
 
 520551 
 
 6319 
 
 999761 
 
 07 
 
 520790 
 
 6326 
 
 479210 
 
 6 
 
 ' 55 
 
 524343 
 
 6264 
 
 999757 
 
 07 
 
 524586 
 
 6272 
 
 475414 
 
 5 
 
 56 
 
 528102 
 
 6211 
 
 999753 
 
 07 
 
 528349 
 
 6218 
 
 471651 
 
 4 
 
 57 
 
 531828 
 
 6158 
 
 999748 
 
 07 
 
 532080 
 
 6165 
 
 467920 
 
 3 
 
 58 
 
 535523 
 
 6106 
 
 999744 
 
 07 
 
 535779 
 
 6113 
 
 464221 
 
 2 
 
 59 
 
 539186 
 
 6055 
 
 999740 
 
 07 
 
 539447 
 
 6062 
 
 460553 
 
 1 
 
 60 
 
 542819 
 
 6004 
 
 999735 
 
 07 
 
 543084 
 
 6012 
 
 456916 
 
 
 
 Cosine | | Sine | | Cotang. | | Tang. | M. 
 
 89 Degrees. 
 
so 
 
 (2 Degrees.) A TABLE OP LOGARITHMIC 
 
 M. Sine | D. | Cosine. 
 
 D. | Tang. D. | Cotang. 
 
 
 
 8.542819 
 
 6004 
 
 9.999735 
 
 07 
 
 8.543084 
 
 6012 
 
 11.456916 
 
 60 
 
 1 
 
 546422 
 
 5955 
 
 999731 
 
 07 
 
 546601 
 
 5962 
 
 453309 
 
 59 
 
 2 
 
 549995 
 
 5906 
 
 999726 
 
 07 
 
 550268 
 
 5914 
 
 449732 
 
 58 
 
 3 
 
 553539 
 
 5858 
 
 999722 
 
 08 
 
 553817 
 
 5866 
 
 446183 
 
 57 
 
 4 
 
 557051 
 
 5811 
 
 999717 
 
 08 
 
 557336 
 
 5819 
 
 442664 
 
 56 
 
 5 
 
 530540 
 
 5765 
 
 999713 
 
 08 
 
 560828 
 
 5773 
 
 439172 
 
 55 
 
 . 6 
 
 563999 
 
 5719 
 
 993708 
 
 08 
 
 564231 
 
 57-27 
 
 435709 
 
 54 
 
 7 
 
 567431 
 
 5674 
 
 999704 
 
 08 
 
 557727 
 
 5682 
 
 432273 
 
 53 ; 
 
 8 
 
 570836 
 
 5630 
 
 999699 
 
 08 
 
 571137 
 
 5638 
 
 428863 
 
 52 
 
 9 
 
 574214 
 
 5587 
 
 999694 
 
 08 
 
 574520 
 
 5595 
 
 425480 
 
 51 
 
 10 
 
 577566 
 
 5544 
 
 999689 
 
 08 
 
 577877 
 
 5552 
 
 422123 
 
 50 
 
 11 
 
 8.580892 
 
 5502 
 
 9.999685 
 
 08 
 
 8.581208 
 
 5510 
 
 11.418792 
 
 49 
 
 12 
 
 584193 
 
 5460 
 
 999G80 
 
 08 
 
 584514 
 
 5468 
 
 415486 
 
 48 
 
 13 
 
 587469 
 
 5419 
 
 999675 
 
 08 
 
 587795 
 
 5427 
 
 412205 
 
 47 
 
 14 
 
 590721 
 
 5379 
 
 999670 
 
 08 
 
 591051 
 
 5387 
 
 408949 
 
 46 
 
 15 
 
 593948 
 
 5339 
 
 999665 
 
 08 
 
 594283 
 
 5347 
 
 405717 
 
 45 
 
 16 
 
 597152 
 
 5300 
 
 999660 
 
 08 
 
 597492 
 
 5308 
 
 402508 
 
 44 
 
 17 
 
 600332 
 
 5261 
 
 999655 
 
 08 
 
 600677 
 
 5270 
 
 399323 
 
 43 
 
 18 
 
 603489 
 
 5223 
 
 999850 
 
 08 
 
 603839 
 
 5232 
 
 396161 
 
 42 
 
 19 
 
 606623 
 
 5186 
 
 999645 
 
 09 
 
 605978 
 
 5194 
 
 393022 
 
 41 
 
 20 
 
 609734 
 
 5149 
 
 999640 
 
 09 
 
 610094 
 
 5158 
 
 389906 
 
 40 
 
 21 
 
 8.612823 
 
 5112 
 
 9.999535 
 
 09 
 
 8.613189 
 
 5121 
 
 11.386811 
 
 39 
 
 22 
 
 615891 
 
 5076 
 
 999629 
 
 09 
 
 616262 
 
 5085 
 
 383738 
 
 38 
 
 23 
 
 618937 
 
 5041 
 
 999624 
 
 09 
 
 619313 
 
 5050 
 
 380687 
 
 37 
 
 24 
 
 621962 
 
 5006 
 
 999619 
 
 09 
 
 622343 
 
 5015 
 
 377657 
 
 36 
 
 25 
 
 624965 
 
 4972 
 
 999614 
 
 09 
 
 625352 
 
 4981 
 
 374648 
 
 35 
 
 26 
 
 627948 
 
 4938 
 
 999608 
 
 09 
 
 628340 
 
 4947 
 
 371660 
 
 34 
 
 27 
 
 630911 
 
 4904 
 
 999603 
 
 09 
 
 631308 
 
 4913 
 
 368692 
 
 33 
 
 28 
 
 633854 
 
 4871 
 
 999597 
 
 09 
 
 634256 
 
 4880 
 
 365744 
 
 32 
 
 29 
 
 636776 
 
 4839 
 
 999592 
 
 03 
 
 637184 
 
 4848 
 
 362816 
 
 31 
 
 30 
 
 639680 
 
 4806 
 
 999586 
 
 09 
 
 640093 
 
 4816 
 
 359907 
 
 30 
 
 31 
 
 8.642563 
 
 4775 
 
 9.999581 
 
 09 
 
 8.642982 
 
 4784 
 
 11.357018 
 
 29 
 
 32 
 
 645423 
 
 4743 
 
 999575 
 
 09 
 
 645853 
 
 4753 
 
 354147 
 
 28 
 
 33 
 
 648274 
 
 4712 
 
 999570 
 
 09 
 
 648704 
 
 4722 
 
 351296 
 
 27 
 
 34 
 
 651102 
 
 4682 
 
 999564 
 
 09 
 
 651537 
 
 4691 
 
 34S463 
 
 28 
 
 35 
 
 653911 
 
 4652 
 
 999558 
 
 10 
 
 654352 
 
 4661 
 
 345648 
 
 25 
 
 36 
 
 656702 
 
 4622 
 
 999553 
 
 10 
 
 657149 
 
 4631 
 
 342851 
 
 24 
 
 37 
 
 659475 
 
 4592 
 
 999547 
 
 10 
 
 659928 
 
 4602 
 
 340072 
 
 23 
 
 38 
 
 662230 
 
 4563 
 
 999541 
 
 10 
 
 662689 
 
 4573 
 
 337311 
 
 22 
 
 39 
 
 664968 
 
 4535 
 
 999535 
 
 10 
 
 665433 
 
 4544 
 
 334567 
 
 21 
 
 40 
 
 667689 
 
 4506 
 
 999529 
 
 10 
 
 668J60 
 
 4526 
 
 331840 
 
 20 
 
 41 
 
 8.670393 
 
 4479 
 
 9.999524 
 
 10 
 
 8.670870 
 
 4488 
 
 11.329130 
 
 19 
 
 42 
 
 673080 
 
 4451 
 
 999518 
 
 10 
 
 673563 
 
 4461 
 
 326437 
 
 18 
 
 43 
 
 675751 
 
 4424 
 
 999512 
 
 10 
 
 676239 
 
 4434 
 
 323761 
 
 17 
 
 44 
 
 678405 
 
 4397 
 
 999506 
 
 10 
 
 678900 
 
 4417 
 
 321100 
 
 16 
 
 45 
 
 681043 
 
 4370 
 
 999500 
 
 10 
 
 681544 
 
 4380 
 
 318456 
 
 15 
 
 46 
 
 683665 
 
 4344 
 
 999493 
 
 10 
 
 684172 
 
 4354 
 
 315828 
 
 14 
 
 47 
 
 686272 
 
 4318 
 
 999487 
 
 10 
 
 686784 
 
 4328 
 
 313216 
 
 13 
 
 48 
 
 688863 
 
 4292 
 
 999481 
 
 10 
 
 689381 
 
 4303 
 
 310619 
 
 12 
 
 49 
 
 691438 
 
 4267 
 
 999475 
 
 10 
 
 691963 
 
 4277 
 
 308037 
 
 11 
 
 50 
 
 693998 
 
 4242 
 
 999469 
 
 10 
 
 694529 
 
 4252 
 
 305471 
 
 10 
 
 51 
 
 8.696543 
 
 4217 
 
 9.999463 
 
 11 
 
 8.697081 
 
 4228 
 
 11.302919 
 
 9 
 
 52 
 
 699073 
 
 4192 
 
 999456 
 
 11 
 
 699617 
 
 4203 
 
 300383 
 
 8 
 
 53 
 
 701589 
 
 4168 
 
 999450 
 
 11 
 
 702139 
 
 4179 
 
 297861 
 
 7 
 
 54 
 
 704090 
 
 4144 
 
 999443 
 
 11 
 
 704646 
 
 4155 
 
 295354 
 
 6 
 
 55 
 
 706577 
 
 4121 
 
 999437 
 
 11 
 
 707140 
 
 4132 
 
 292860 
 
 5 
 
 56 
 
 709049 
 
 4097 
 
 999431 
 
 11 
 
 709618 
 
 4108 
 
 290382 
 
 4 
 
 57 
 
 711507 
 
 4074 
 
 999424 
 
 11 
 
 712083 
 
 4085 
 
 287917 
 
 3 
 
 58 
 
 713952 
 
 4051 
 
 999418 
 
 11 
 
 714534 
 
 4062 
 
 285465 
 
 2 
 
 1 59 
 
 716383 
 
 4029 
 
 999411 
 
 11 
 
 716972 
 
 4040 
 
 283028 
 
 1 
 
 60 
 
 718800 
 
 4006 
 
 999404 
 
 11 
 
 719396 
 
 4017 
 
 280604 
 
 
 
 Cosine | | Sine Cotang. | | Tang. 
 
 M 
 
 87 Degrees. 
 
SINES AND TANGENTS. (3 Degrees.) 
 
 M. | Sine D. | Cosine D. | Tang, j D. | Cotang. 
 
 
 
 8.71f<HUO 
 
 4006 
 
 9.999404 
 
 11 
 
 8.719396 
 
 4017 
 
 ll.-JHIIiltl 
 
 60 
 
 1 
 
 721204 
 
 3984 
 
 999398 
 
 11 
 
 721806 
 
 3995 
 
 278194 
 
 59 
 
 2 
 
 723595 
 
 .'tow 
 
 999391 
 
 11 
 
 734304 
 
 3974 
 
 275796 
 
 58 
 
 3 
 
 725972 
 
 :w4i 
 
 999384 
 
 11 
 
 
 3952 
 
 273412 
 
 57 
 
 4 
 
 728337 
 
 3jn 
 
 999378 
 
 11 
 
 7-Jsiir.it 
 
 3930 
 
 271041 
 
 56 
 
 f> 
 
 730688 
 
 3898 
 
 999371 
 
 11 
 
 731317 
 
 3909 
 
 268683 
 
 55 
 
 6 
 
 733027 
 
 3877 
 
 999364 
 
 12 
 
 7:5:!(ii;:i 
 
 3889 
 
 266337 
 
 54 
 
 7 
 
 735354 
 
 3857 
 
 999357 
 
 12 
 
 73.y.)!M5 
 
 3868 
 
 264004 
 
 53 
 
 8 
 
 7:!7i;<)7 
 
 3836 
 
 999350 
 
 12 
 
 738317 
 
 3848 
 
 261683 
 
 52 
 
 9 
 
 739969 
 
 3816 
 
 999343 
 
 12 
 
 740T.26 
 
 3827 
 
 259374 
 
 51 
 
 10 
 
 74-J-r,!) 
 
 3796 
 
 W8936 
 
 12 
 
 7429-22 
 
 3807 
 
 257078 
 
 50 
 
 11 
 
 8.744536 
 
 3776 
 
 9.999329 
 
 12 
 
 8.745207 
 
 3787 
 
 11.254793 
 
 49 
 
 12 
 
 746802 
 
 37f>li 
 
 999322 
 
 12 
 
 747479 
 
 3768 
 
 252521 
 
 48 
 
 13 
 
 7-1 !K!.-,r> 
 
 37:17 
 
 999315 
 
 12 
 
 749740 
 
 3749 
 
 250260 
 
 47 
 
 14 
 
 751297 
 
 3717 
 
 999308 
 
 12 
 
 751989 
 
 3729 
 
 248011 
 
 46 
 
 15 
 
 753528 
 
 3698 
 
 999301 
 
 12 
 
 754227 
 
 3710 
 
 245773 
 
 45 
 
 16 
 
 755747 
 
 3679 
 
 999294 
 
 12 
 
 756453 
 
 3692 
 
 243547 
 
 44 
 
 17 
 
 757955 
 
 366 1 
 
 999286 
 
 12 
 
 758668 
 
 3673 
 
 241332 
 
 43 
 
 18 
 
 760151 
 
 3642 
 
 999279 
 
 12 
 
 760872 
 
 3655 
 
 239128 
 
 42 
 
 19 
 
 762337 
 
 3624 
 
 999272 
 
 12 
 
 763065 
 
 3636 
 
 236935 
 
 41 
 
 20 
 
 764511 
 
 3606 
 
 999265 
 
 12 
 
 765246 
 
 3618 
 
 234754 
 
 40 
 
 21 
 
 8.766675 
 
 3588 
 
 9.999257 
 
 12 
 
 8.767417 
 
 3600 
 
 11.232583 
 
 39 
 
 22 
 
 768828 
 
 3570 
 
 999250 
 
 13 
 
 769578 
 
 3583 
 
 230422 
 
 38 
 
 23 
 
 770970 
 
 3553 
 
 999242 
 
 13 
 
 771727 
 
 3565 
 
 228273 
 
 37 
 
 24 
 
 773101 
 
 3535 
 
 999235 
 
 13 
 
 773866 
 
 3548 
 
 226134 
 
 36 
 
 25 
 
 775223 
 
 3518 
 
 999227 
 
 13 
 
 775995 
 
 3531 
 
 224005 
 
 35 
 
 26 
 
 777333 
 
 3501 
 
 999220 
 
 13 
 
 778114 
 
 3514 
 
 221886 
 
 34 
 
 27 
 
 779434 
 
 3484 
 
 999212 
 
 13 
 
 i 8{hJ2 
 
 3497 
 
 219778 
 
 33 
 
 28 
 
 781524 
 
 3467 
 
 999205 
 
 13 
 
 782320 
 
 3480 
 
 217680 
 
 32 
 
 29 
 
 783605 
 
 3451 
 
 999197 
 
 13 
 
 784408 
 
 3464 
 
 215592 
 
 31 
 
 30 
 
 785675 
 
 3431 
 
 999189 
 
 13 
 
 786486 
 
 3447 
 
 213514 
 
 30 
 
 31 
 
 8.787736 
 
 3418 
 
 9.999181 
 
 13 
 
 8.788554 
 
 3431 
 
 11.211446 
 
 29 
 
 32 
 
 789787 
 
 3402 
 
 999174 
 
 13 
 
 790613 
 
 3414 
 
 209387 
 
 28 
 
 33 
 
 791828 
 
 3386 
 
 999166 
 
 13 
 
 792662 
 
 3399 
 
 207338 
 
 27 
 
 34 
 
 793859 
 
 3370 
 
 999158 
 
 13 
 
 794701 
 
 3383 
 
 205299 
 
 26 
 
 35 
 
 795881 
 
 3354 
 
 999150 
 
 13 
 
 796731 
 
 3368 
 
 203269 
 
 25 
 
 36 
 
 797894 
 
 3339 
 
 999142 
 
 13 
 
 798752 
 
 3352 
 
 201248 
 
 24 
 
 37 
 
 799897 
 
 33-23 
 
 999134 
 
 13 
 
 800763 
 
 3337 
 
 199237 
 
 23 
 
 38 
 
 801892 
 
 3308 
 
 999126 
 
 13 
 
 802765 
 
 3322 
 
 197235 
 
 22 
 
 39 
 
 803876 
 
 3293 
 
 999118 
 
 13 
 
 804758 
 
 3307 
 
 195242 
 
 21 
 
 40 
 
 805852 
 
 3278 
 
 999110 
 
 13 
 
 806742 
 
 3292 
 
 193258 
 
 20 
 
 41 
 
 8.807819 
 
 3263 
 
 9.999102 
 
 13 
 
 8.808717 
 
 3278 
 
 11.191283 
 
 19 
 
 42 
 
 809777 
 
 3249 
 
 999094 
 
 14 
 
 810683 
 
 3262 
 
 189317 
 
 18 
 
 43 
 
 811726 
 
 3234 
 
 999086 
 
 14 
 
 812641 
 
 3248 
 
 187359 
 
 17 
 
 44 
 
 8136G7 
 
 3219 
 
 9i)9077 
 
 14 
 
 814589 
 
 3233 
 
 185411 
 
 16 
 
 45 
 
 815599 
 
 3205 
 
 999069 
 
 14 
 
 816529 
 
 3219 
 
 183471 
 
 15 
 
 46 
 
 817522 
 
 3191 
 
 9990(51 
 
 14 
 
 818461 
 
 3205 
 
 181539 
 
 14 
 
 47 
 
 819436 
 
 3177 
 
 999053 
 
 14 
 
 820384 
 
 3191 
 
 179616 
 
 13 
 
 48 
 
 821343 
 
 3163 
 
 999044 
 
 14 
 
 822298 
 
 3177 
 
 177702 
 
 12 
 
 49 
 
 823240 
 
 3149 
 
 999036 
 
 14 
 
 824205 
 
 3163 
 
 175795 
 
 11 
 
 50 
 
 825130 
 
 3135 
 
 999027 
 
 14 
 
 826103 
 
 3150 
 
 173897 
 
 10 
 
 51 
 
 8.827011 
 
 3122 
 
 9.999019 
 
 14 
 
 8.827992 
 
 3136 
 
 11.172008 
 
 9 
 
 52 
 
 828884 
 
 3108 
 
 999010 
 
 14 
 
 829874 
 
 3123 
 
 170126 
 
 8 
 
 53 
 
 830749 
 
 3095 
 
 999002 
 
 14 
 
 831748 
 
 3110 
 
 168252 
 
 7 
 
 54 
 
 832607 
 
 3082 
 
 998993 
 
 14 
 
 833613 
 
 3096 
 
 166387 
 
 6 
 
 55 
 
 834456 
 
 3069 
 
 998934 
 
 14 
 
 835471 
 
 3083 
 
 164529 
 
 5 
 
 56 
 
 836297 
 
 3056 
 
 998976 
 
 14 
 
 837321 
 
 3070 
 
 162679 
 
 4 
 
 57 
 
 838130 
 
 3043 
 
 998967 
 
 15 
 
 839163 
 
 3057 
 
 160837 
 
 3 
 
 68 
 
 839956 
 
 3030 
 
 998958 
 
 15 
 
 840998 
 
 3045 
 
 159002 
 
 2 
 
 59 
 
 841774 
 
 3017 
 
 998950 
 
 15 
 
 843835 
 
 3032 
 
 157175 
 
 1 
 
 60 
 
 843585 
 
 3000 
 
 9989-11 
 
 15 
 
 844644 
 
 3019 
 
 155356 
 
 
 
 | Cosine | Sine | | Cotang. | Tang. M | 
 
 86 Degrees. 
 
(4 Degrees.) A TABLE or LOGARITHMIC 
 
 j M. Sine D. Cosine IX Tang. | D Cotang. | 
 
 
 
 8.843585 
 
 3005 
 
 9.998941 
 
 15 
 
 8.844644 
 
 3019 
 
 11. 155;. 6 
 
 tk) 
 
 1 
 
 845387 
 
 2992 
 
 
 15 
 
 846455 
 
 3037 
 
 153545 
 
 59 
 
 2 
 
 847183 
 
 2980 
 
 996923 
 
 15 
 
 848-260 
 
 2995 
 
 1517 10 
 
 ",-: 
 
 3 
 
 848971 
 
 25)07 
 
 998914 
 
 15 
 
 850057 
 
 2982 
 
 149943 
 
 57 
 
 4 
 
 850751 
 
 2955 
 
 998905 
 
 15 
 
 851846 
 
 2370 
 
 148154 
 
 53 
 
 5 
 
 85-25-25 
 
 2943 
 
 998896 
 
 15 
 
 853623 
 
 2958 
 
 14!>:i72 
 
 55 
 
 6 
 
 854231 
 
 2931 
 
 998387 
 
 15 
 
 855403 
 
 2946 
 
 144597 
 
 54 
 
 7 
 
 856049 
 
 2919 
 
 998378 
 
 15 
 
 857171 
 
 2095 
 
 142329 
 
 53 ! 
 
 8 
 
 8578J1 
 
 291)7 
 
 998339 
 
 15 
 
 858932 
 
 3833 
 
 1410(53 
 
 52 1 
 
 9 
 
 859546 
 
 23i)5 
 
 933860 
 
 15 
 
 830683 
 
 2911 
 
 133314 
 
 51 
 
 10 
 
 861283 
 
 2884 
 
 993351 
 
 15 
 
 , 832433 
 
 9M 
 
 137567 
 
 50 
 
 11 
 
 8.81)3014 
 
 2373 
 
 9.993841 
 
 15 
 
 8.864173 
 
 2888 
 
 11,135827 
 
 49 
 
 12 
 
 86473S 
 
 28(>1 
 
 993332 
 
 15 
 
 835906 
 
 2377 
 
 134094 
 
 48 
 
 13 
 
 836455 
 
 2850 
 
 993323 
 
 16 
 
 807632 
 
 2366 
 
 1323(38 
 
 47 
 
 14 
 
 838165 
 
 2339 
 
 993813 
 
 16 
 
 839351 
 
 2354 
 
 130648 
 
 46 
 
 15 
 
 8693158 
 
 2828 
 
 998804 
 
 16 
 
 871064 
 
 2343 
 
 128336 
 
 45 
 
 16 
 
 871565 
 
 2817 
 
 993795 
 
 16 
 
 872770 
 
 2W32 
 
 127230 
 
 44 
 
 17 
 
 873255 
 
 2306 
 
 998785 
 
 16 
 
 874469 
 
 2321 
 
 125531 
 
 43 
 
 18 
 
 874938 
 
 2795 
 
 998776 
 
 16 
 
 876162 
 
 2811 
 
 123333 
 
 42 
 
 19 
 
 876615 
 
 2783 
 
 993766 
 
 16 
 
 877849 
 
 2800 
 
 122151 
 
 41 
 
 20 
 
 878285 
 
 2773 
 
 993757 
 
 16 
 
 879529 
 
 2789 
 
 120471 
 
 40 
 
 21 
 
 8.879949 
 
 2763 
 
 9.998747 
 
 16 
 
 8.831202 
 
 2779 
 
 11.118793 
 
 39 
 
 22 
 
 881607 
 
 2752 
 
 998738 
 
 16 
 
 832369 
 
 2768 
 
 117131 
 
 38 
 
 23 
 
 88-3258 
 
 2742 
 
 998728 
 
 16 
 
 884.530 
 
 2758 
 
 115470 
 
 37 
 
 24 
 
 834903 
 
 2731 
 
 998718 
 
 16 
 
 836185 
 
 2747 
 
 113815 
 
 36 
 
 25 
 
 886542 
 
 2721 
 
 998708 
 
 16 
 
 887833 
 
 2737 
 
 112167 
 
 35 
 
 26 
 
 888174 
 
 2711 
 
 998:593 
 
 16 
 
 889476 
 
 2727 
 
 110524 
 
 34 
 
 27 
 
 889801 
 
 2700 
 
 998689 
 
 16 
 
 891112 
 
 2717 
 
 108888 
 
 33 
 
 28 
 
 891421 
 
 2693 
 
 993679 
 
 16 
 
 83274-2 
 
 2707 
 
 107258 
 
 32 
 
 29 
 
 893035 
 
 2680 
 
 998G69 
 
 17 
 
 894366 
 
 2697 
 
 1056-34 
 
 31 
 
 30 
 
 894643 
 
 2870 
 
 998659 
 
 17 
 
 895984 
 
 2687 
 
 101016 
 
 30 
 
 3] 
 
 8.896246 
 
 2660 
 
 9.993649 
 
 17 
 
 8.897593 
 
 2677 
 
 11.102404 
 
 29 
 
 32 
 
 897842 
 
 2651 
 
 05)3639 
 
 17 
 
 839203 
 
 2667 
 
 100797 
 
 28 
 
 33 
 
 89943-2 
 
 2641 
 
 983029 
 
 17 
 
 90J803 
 
 2658 
 
 099197 
 
 27 
 
 34 
 
 901017 
 
 3f!31 
 
 9931519 
 
 17 
 
 902398 
 
 2648 
 
 097602 
 
 26 
 
 35 
 
 90-259-5 
 
 21522 
 
 9j3 i it!9 
 
 17 
 
 903937 
 
 2338 
 
 093:)13 
 
 25 
 
 36 
 
 904169 
 
 2612 
 
 903593 
 
 17 
 
 905570 
 
 2629 
 
 09-1430 
 
 24 
 
 37 
 
 90573(5 
 
 2(503 
 
 938539 
 
 17 
 
 9J7147 
 
 2620 
 
 032353 
 
 23 
 
 38 
 
 907-297 
 
 2593 
 
 933578 
 
 17 
 
 908719 
 
 2610 
 
 091281 
 
 22 
 
 39 
 
 908353 
 
 2584 
 
 998568 
 
 17 
 
 910235 
 
 2301 
 
 089715 
 
 21 
 
 40 
 
 910104 
 
 2575 
 
 938558 
 
 17 
 
 911846 
 
 2592 
 
 088154 
 
 20 
 
 41 
 
 8.911949 
 
 2566 
 
 9.998548 
 
 17 
 
 8.913401 
 
 2583 
 
 11.086599 
 
 19 
 
 42 
 
 913483 
 
 2556 
 
 993537 
 
 17 
 
 914951 
 
 2574 
 
 085049 
 
 18 
 
 43 
 
 915022 
 
 2547 
 
 933527 
 
 17 
 
 913495 
 
 2565 
 
 083505 
 
 17 
 
 44 
 
 916550 
 
 2538 
 
 998516 
 
 18 
 
 913034 
 
 2556 
 
 081966 
 
 16 
 
 45 
 
 918073 
 
 2529 
 
 9i>8.503 
 
 18 
 
 919563 
 
 2547 
 
 080432 
 
 15 
 
 46 
 
 919591 
 
 2520 
 
 998495 
 
 18 
 
 921096 
 
 2538 
 
 078904 
 
 14 
 
 47 
 
 921103 
 
 2512 
 
 998485 
 
 18 
 
 922619 
 
 2530 
 
 077381 
 
 13 
 
 48 
 
 922610 
 
 2503 
 
 998474 
 
 18 
 
 924136 
 
 2521 
 
 075864 
 
 12 
 
 49 
 
 924112 
 
 2494 
 
 998461 
 
 18 
 
 9-25649 
 
 2512 
 
 074351 
 
 11 
 
 50 
 
 925809 
 
 2486 
 
 998453 
 
 18 
 
 9-27156 
 
 2503 
 
 072344 
 
 10 
 
 51 
 
 8.927100 
 
 2477 
 
 9.998442 
 
 18 
 
 8.928658 
 
 2495 
 
 11.071342 
 
 9 
 
 52 
 
 9-28587 
 
 2469 
 
 993431 
 
 18 
 
 930155 
 
 2486 
 
 069345 
 
 8 
 
 53 
 
 930068 
 
 2460 
 
 998421 
 
 18 
 
 931647 
 
 2478 
 
 068353 
 
 
 54 
 
 931544 
 
 2452 
 
 933410 
 
 18 
 
 933134 
 
 2470 
 
 0(508(56 
 
 6 
 
 55 
 
 933015 
 
 2443 
 
 993399 
 
 18 
 
 934(516 
 
 2461 
 
 0(55384 
 
 5 
 
 56 
 
 934481 
 
 2435 
 
 933333 
 
 18 
 
 936093 
 
 2453 
 
 033907 
 
 4 
 
 57 
 
 93594-2 
 
 2427 
 
 993377 
 
 18 
 
 937565 
 
 2445 
 
 062435 
 
 3 
 
 58 
 
 937398 
 
 2419 
 
 998366 
 
 18 
 
 939032 
 
 2437 
 
 060968 
 
 o 
 
 59 
 
 9383.50 
 
 2411 
 
 998355 
 
 18 
 
 940434 
 
 2430 
 
 059506 
 
 1 
 
 60 
 
 940-295 
 
 2403 
 
 938344 
 
 18 
 
 941952 
 
 2421 
 
 053048 
 
 
 
 Cosine | Sine | 
 
 Cotang. 1 | Tang. M. 
 
 85 Degrees. 
 
SINKS AND TANGENTS. (5 Degrees.) 
 
 M. Sine | D. | Cosine | D. Tang. | D. Cotang. | 
 
 
 
 8.940296 
 
 2403 
 
 9.91K544 
 
 19 
 
 8.94111.12 
 
 2421 
 
 H.t).iS,|.lH 
 
 60 
 
 1 
 
 941738 
 
 2394 
 
 938333 
 
 19 
 
 943404 
 
 2413 
 
 056596 
 
 59 
 
 2 
 
 943174 
 
 21587 
 
 1)118322 
 
 19 
 
 9448.i2 
 
 2405 
 
 0551-18 
 
 58 
 
 3 
 
 944606 
 
 2379 
 
 99831] 
 
 19 
 
 946295 
 
 2397 
 
 053705 
 
 57 
 
 4 
 
 946034 
 
 2371 
 
 998300 
 
 19 
 
 947734 
 
 2390 
 
 (1.122116 
 
 56 
 
 5 
 
 9474,-)ti 
 
 3363 
 
 998289 
 
 1J 
 
 949 Hi8 
 
 2382 
 
 050832 
 
 55 
 
 6 
 
 948874 
 
 23.-).-) 
 
 998277 
 
 19 
 
 950597 
 
 2374 
 
 049-103 
 
 54 
 
 7 
 
 950287 
 
 
 998266 
 
 19 
 
 953031 
 
 2366 
 
 047979 
 
 53 
 
 8 
 
 95169!) 
 
 2340 
 
 9 i J255 
 
 19 
 
 953441 
 
 2360 
 
 046559 
 
 52 
 
 9 
 
 953100 
 
 3332 
 
 '..;> Ji:-. 
 
 19 
 
 954856 
 
 2351 
 
 045144 
 
 51 i 
 
 10 
 
 954499 
 
 HM 
 
 993232 
 
 19 
 
 956267 
 
 2344 
 
 043733 
 
 50 
 
 11 
 
 8.955894 
 
 2317 
 
 9,999830 
 
 19 
 
 8.957674 
 
 2337 
 
 11.042326 
 
 49 
 
 12 
 
 957384 
 
 2310 
 
 998209 
 
 19 
 
 959075 
 
 23tt 
 
 040925 
 
 48 
 
 13 
 
 958670 
 
 23,12 
 
 998197 
 
 19 
 
 960473 
 
 2323 
 
 039527 
 
 47 
 
 14 
 
 960052 
 
 UBS 
 
 998186 
 
 19 
 
 981856 
 
 2314 
 
 038134 
 
 46 
 
 15 
 
 961429 
 
 2288 
 
 998174 
 
 J9 
 
 9;;32.15 
 
 2307 
 
 036745 
 
 45 
 
 16 
 
 962801 
 
 22.80 
 
 998163 
 
 19 
 
 984639 
 
 2300 
 
 035361 
 
 44 
 
 17 
 
 954170 
 
 2273 
 
 998151 
 
 19 
 
 986019 
 
 2293 
 
 033981 
 
 43 
 
 13 
 
 965534 
 
 2266 
 
 91)8139 
 
 20 
 
 967394 
 
 2286 
 
 032605 
 
 42 
 
 19 
 
 966893 
 
 2-2.".: > 
 
 998128 
 
 20 
 
 968766 
 
 2279 
 
 031234 
 
 41 
 
 20 
 
 908349 
 
 2359 
 
 998116 
 
 20 
 
 970133 
 
 2271 
 
 029867 
 
 40 
 
 21 
 
 8.969(500 
 
 2214 
 
 9.998104 
 
 21) 
 
 8.971495 
 
 22G5 
 
 11.028504 
 
 39 
 
 22 
 
 970947 
 
 2238 
 
 998092 
 
 20 
 
 972855 
 
 2257 
 
 027145 
 
 38 
 
 23 
 
 972289 
 
 2231 
 
 998080 
 
 20 
 
 974209 
 
 2251 
 
 025791 
 
 37 
 
 24 
 
 973628 
 
 2224 
 
 993068 
 
 20 
 
 975560 
 
 2244 
 
 024440 
 
 36 
 
 25 
 
 974962 
 
 2217 
 
 'I9S1.H; 
 
 20 
 
 976906 
 
 2237 
 
 023094 
 
 35 
 
 26 
 
 976293 
 
 2210 
 
 998044 
 
 20 
 
 978248 
 
 2230 
 
 021752 
 
 34 
 
 27 
 
 977619 
 
 2303 
 
 998032 
 
 20 
 
 979586 
 
 2223 
 
 020414 
 
 33 
 
 28 
 
 978941 
 
 2197 
 
 998020 
 
 20 
 
 98093] 
 
 2217 
 
 019079 
 
 32 
 
 29 
 
 980259 
 
 2190 
 
 998008 
 
 20 
 
 932251 
 
 2-210 
 
 017749 
 
 31 
 
 30 
 
 981573 
 
 2183 
 
 997998 
 
 20 
 
 983577 
 
 2204 
 
 016423 
 
 30 
 
 31 
 
 8.982883 
 
 2177 
 
 9.997984 
 
 20 
 
 8.984899 
 
 21-97 
 
 11.015101 
 
 29 
 
 32 
 
 934189 
 
 2170 
 
 997972 
 
 2J 
 
 986217 
 
 2191 
 
 013783 
 
 28 
 
 33 
 
 935191 
 
 2163 
 
 997959 
 
 20 
 
 987532 
 
 2184 
 
 012468 
 
 27 
 
 34 
 
 986789 
 
 2157 
 
 997917 
 
 20 
 
 938342 
 
 2178 
 
 011158 
 
 26 
 
 35 
 
 988083 
 
 2150 
 
 997935 
 
 31 
 
 990149 
 
 2171 
 
 009351 
 
 25 
 
 36 
 
 98J374 
 
 2144 
 
 997922 
 
 21 
 
 991451 
 
 2165 
 
 008549 
 
 24 
 
 37 
 
 99.1660 
 
 2138 
 
 997910 
 
 21 
 
 992750 
 
 2158 
 
 OJ7250 
 
 23 
 
 38 
 
 991943 
 
 2131 
 
 9J78U7 
 
 31 
 
 994045 
 
 2152 
 
 005955 
 
 22 
 
 39 
 
 9:13222 
 
 2125 
 
 997885 
 
 21 
 
 995337 
 
 2146 
 
 004(5(53 
 
 21 
 
 40 
 
 994497 
 
 2119 
 
 997872 
 
 21 
 
 9-J6624 
 
 2140 
 
 003376 
 
 20 
 
 41 
 
 8.9D5708 
 
 2112 
 
 9.997850 
 
 21 
 
 8.997908 
 
 2134 
 
 11.002092 
 
 19 
 
 42 
 
 99703fi 
 
 2106 
 
 997847 
 
 21 
 
 999188 
 
 2127 
 
 000812 
 
 18 
 
 43 
 
 998291) 
 
 2100 
 
 997835 
 
 21 
 
 9.000465 
 
 2121 
 
 10.999535 
 
 17 
 
 44 
 
 999560 
 
 2094 
 
 997822 
 
 21 
 
 001733 
 
 2115 
 
 993262 
 
 16 
 
 45 
 
 9.000816 
 
 2087 
 
 997809 
 
 21 
 
 003007 
 
 2109 
 
 996993 
 
 15 
 
 46 
 
 002069 
 
 2082 
 
 997797 
 
 21 
 
 004272 
 
 2103 
 
 995728 
 
 14 
 
 47 
 
 003313 
 
 2076 
 
 997784 
 
 21 
 
 005534 
 
 2097 
 
 994466 
 
 13 
 
 48 
 
 0045(53 
 
 2070 
 
 997771 
 
 21 
 
 006792 
 
 2091 
 
 993208 
 
 12 
 
 49 
 
 005805 
 
 2064 
 
 997758 
 
 21 
 
 008047 
 
 2035 
 
 991953 
 
 11 
 
 50 
 
 007044 
 
 2058 
 
 997745 
 
 21 
 
 009298 
 
 2080 
 
 990702 
 
 10 
 
 51 
 
 9.008278 
 
 2052 
 
 9.997732 
 
 21 
 
 9.010546 
 
 2074 
 
 10.989454 
 
 9 
 
 52 
 
 009510 
 
 2046 
 
 997719 
 
 21 
 
 011790 
 
 2068 
 
 988210 
 
 8 
 
 53 
 
 010737 
 
 2040 
 
 997706 
 
 21 
 
 013031 
 
 2062 
 
 986959 
 
 7 
 
 54 
 
 01 1962 
 
 2034 
 
 997693 
 
 22 
 
 014268 
 
 2056 
 
 985732 
 
 6 
 
 55 
 
 013182 
 
 2029 
 
 997680 
 
 22 
 
 015502 
 
 2051 
 
 984498 
 
 5 
 
 56 
 
 014400 
 
 2023 
 
 9976(57 
 
 22 
 
 (11 (57:52 
 
 2045 
 
 983268 
 
 4 
 
 57 
 
 015613 
 
 2017 
 
 997654 
 
 22 
 
 017959 
 
 2040 
 
 982041 
 
 3 
 
 58 
 
 016824 
 
 2012 
 
 997041 
 
 22 
 
 019183 
 
 2033 
 
 980817 
 
 2 
 
 59 
 
 (50 
 
 018031 
 019235 
 
 2006 
 2000 
 
 997628 
 997614 
 
 22 
 22 
 
 020403 
 021620 
 
 2028 
 2023 
 
 979597 
 978380 
 
 1 
 
 
 | Cosine | | Sine | Cotang. | | Tang. M. | 
 
 84 Degress. 
 
(6 Degrees.) A TABLE OF LOGARITHMIC 
 
 M. \ Sine | D. Cosine | D. | Tang. D. Cotang. 
 
 
 
 9.019235 
 
 2000 
 
 9.997614 
 
 22 
 
 9.021620 
 
 2023 
 
 10.978380 
 
 60 
 
 1 
 
 020435 
 
 1995 
 
 997601 
 
 22 
 
 022834 
 
 2017 
 
 977166 
 
 59 
 
 2 
 
 021632 
 
 1989 
 
 997588 
 
 22 
 
 024044 
 
 2011 
 
 975956 
 
 58 
 
 3 
 
 022825 
 
 1984 
 
 997574 
 
 22 
 
 025-251 
 
 2006 
 
 974749 
 
 57 
 
 4 
 
 024016 
 
 1978 
 
 997561 
 
 23 
 
 026455 
 
 2000 
 
 973545 
 
 56 
 
 5 
 
 . 025203 
 
 1973 
 
 997547 
 
 22 
 
 027655 
 
 1995 
 
 972345 
 
 55 
 
 6 
 
 026386 
 
 1967 
 
 997534 
 
 23 
 
 028852 
 
 1990 
 
 971148 
 
 54 
 
 7 
 
 0275S7 
 
 1962 
 
 997520 
 
 23 
 
 030046 
 
 1985 
 
 969954 
 
 53 
 
 8 
 
 028744 
 
 1957 
 
 997507 
 
 23 
 
 031237 
 
 1979 
 
 968763 
 
 52 
 
 9 
 
 029918 
 
 1951 
 
 997493 
 
 23 
 
 032425 
 
 1974 
 
 967575 
 
 5] , 
 
 M 
 
 031089 
 
 1947 
 
 997480 
 
 23 
 
 033^09 
 
 1969 
 
 966391 
 
 50 
 
 11 
 
 9.032257 
 
 1941 
 
 9.997466 
 
 23 
 
 9.034791 
 
 1964 
 
 10.965209 
 
 49 
 
 12 
 
 033421 
 
 1936 
 
 997452 
 
 23 
 
 035969 
 
 1958 
 
 964031 
 
 48 
 
 13 
 
 034582 
 
 1930 
 
 997439 
 
 23 
 
 037144 
 
 1953 
 
 962856 
 
 47 
 
 14 
 
 035741 
 
 1925 
 
 997425 
 
 23 
 
 038316 
 
 1948 
 
 961684 
 
 46 
 
 15 
 
 036896 
 
 1920 
 
 997411 
 
 23 
 
 039485 
 
 1943 
 
 960515 
 
 45 
 
 16 
 
 038048 
 
 1915 
 
 997397 
 
 23 
 
 040651 
 
 1938 
 
 959349 
 
 44 
 
 17 
 
 039197 
 
 1910 
 
 997383 
 
 23 
 
 041813 
 
 1933 
 
 958187 
 
 43 
 
 18 
 
 040342 
 
 1905 
 
 997369 
 
 23 
 
 042973 
 
 1928 
 
 957027 
 
 42 
 
 19 
 
 041485 
 
 1899 
 
 997355 
 
 23 
 
 044130 
 
 1923 
 
 955870 
 
 41 
 
 20 
 
 042625 
 
 1894 
 
 997341 
 
 23 
 
 045284 
 
 1918 
 
 9547U6 
 
 40 
 
 21 
 
 9.043762 
 
 1889 
 
 9.997327 
 
 24 
 
 9.046434 
 
 1913 
 
 10.953566 
 
 39 
 
 22 
 
 044895 
 
 1884 
 
 997313 
 
 24 
 
 047582 
 
 1908 
 
 952418 
 
 38 
 
 23 
 
 046026 
 
 1879 
 
 997299 
 
 1 
 
 048727 
 
 1903 
 
 951273 
 
 37 
 
 24 
 
 047154 
 
 1875 
 
 997285 
 
 24 
 
 049869 
 
 1898 
 
 950131 
 
 36 
 
 25 
 
 048279 
 
 1870 
 
 997271 
 
 24 
 
 051008 
 
 1893 
 
 948992 
 
 35 
 
 26 
 
 049400 
 
 1865 
 
 997257 
 
 24 
 
 052144 
 
 1889 
 
 947856 
 
 34 
 
 27 
 
 050519 
 
 1860 
 
 997242 
 
 24 
 
 053277 
 
 1884 
 
 946723 
 
 33 
 
 28 
 
 051635 
 
 1855 
 
 997228 
 
 24 
 
 054407 
 
 1879 
 
 945593 
 
 32 
 
 29 
 
 052749 
 
 1850 
 
 997214 
 
 24 
 
 055535 
 
 1874 
 
 944465 
 
 31 
 
 30 
 
 053859 
 
 1845 
 
 997199 
 
 24 
 
 056659 
 
 1870 
 
 943341 
 
 30 
 
 31 
 
 9.054966 
 
 1841 
 
 9.997185 
 
 24 
 
 9.057781 
 
 1865 
 
 10.942219 
 
 29 
 
 32 
 
 056071 
 
 1836 
 
 997170 
 
 24 
 
 058900 
 
 1869 
 
 941100 
 
 28 
 
 33 
 
 057172 
 
 1831 
 
 997156 
 
 24 
 
 060016 
 
 1855 
 
 939984 
 
 27 
 
 34 
 
 058271 
 
 1827 
 
 997141 
 
 24 
 
 061130 
 
 1851 
 
 938870 
 
 26 
 
 35 
 
 059367 
 
 1822 
 
 997127 
 
 24 
 
 062240 
 
 1846 
 
 937760 
 
 25 
 
 36 
 
 0604(50 
 
 1817 
 
 997112 
 
 24 
 
 063348 
 
 1842 
 
 936652 
 
 24 
 
 37 
 
 " 061551 
 
 1813 
 
 997098 
 
 24 
 
 064453 
 
 1837 
 
 935547 
 
 23 
 
 38 
 
 OG2639 
 
 1808 
 
 997083 
 
 25 
 
 OC.5556 
 
 1833 
 
 934444 
 
 22 
 
 39 
 
 063724 
 
 1804 
 
 997068 
 
 25 
 
 066655 
 
 1828 
 
 933345 
 
 al- 
 
 40 
 
 064806 
 
 1799 
 
 997053 
 
 25 
 
 067752 
 
 1824 
 
 932248 
 
 so 
 
 41 
 
 9.065885 
 
 1794 
 
 9.997039 
 
 25 
 
 9.068^41} 
 
 1819 
 
 10.931154 
 
 19 
 
 42 
 
 066902 
 
 1790 
 
 997024 
 
 25 
 
 069938 
 
 1815 
 
 930062 
 
 18 
 
 43 
 
 068036 
 
 1786 
 
 997009 
 
 25 
 
 071027 
 
 1810 
 
 928973 
 
 17 
 
 44 
 
 069107 
 
 1781 
 
 996994 
 
 25 
 
 072113 
 
 1806 
 
 927887 
 
 16 
 
 45 
 
 070176 
 
 1777 
 
 996979 
 
 25 
 
 073197 
 
 1802 
 
 926803 
 
 15 
 
 46 
 
 071242 
 
 1772 
 
 996904 
 
 25 
 
 074278 
 
 1797 
 
 925722 
 
 14 
 
 47 
 
 072306 
 
 1768 
 
 996949 
 
 25 
 
 075356 
 
 1703 
 
 924644 
 
 13 
 
 48 
 
 073366 
 
 1763 
 
 996934 
 
 25 
 
 076432 
 
 1789 
 
 923568 
 
 12 
 
 49 
 
 074424 
 
 1759 
 
 995919 
 
 25 
 
 077505 
 
 1784 
 
 922495 
 
 1] 
 
 50 
 
 075480 
 
 1755 
 
 996904 
 
 25 
 
 078576 
 
 1780 
 
 921424 
 
 10 
 
 51 
 
 9.076533 
 
 1750 
 
 9.996889 
 
 25 
 
 9.079G44 
 
 1776 
 
 10.920356 
 
 9 
 
 52 
 
 077583 
 
 1746 
 
 996874 
 
 25 
 
 080710 
 
 1772 
 
 919290 
 
 8 
 
 53 
 
 078631 
 
 1742 
 
 996858 
 
 25 
 
 081773 
 
 1767 
 
 918227 
 
 7 
 
 54 
 
 879676 
 
 1738 
 
 996843 
 
 25 
 
 082833 
 
 1763 
 
 917167 
 
 6 
 
 55 
 
 080719 
 
 1733 
 
 996828 
 
 25 
 
 083891 
 
 1759 
 
 916109 
 
 5 
 
 56 
 
 081759 
 
 1729 
 
 996812 
 
 26 
 
 084947 
 
 1755 
 
 915053 
 
 4 
 
 57 
 
 082797 
 
 1725 
 
 996797 
 
 26 
 
 086000 
 
 1751 
 
 914000 
 
 3 
 
 58 
 
 0&3832 
 
 1721 
 
 996782 
 
 26 
 
 087050 
 
 1747 
 
 912950 
 
 2 
 
 59 
 
 084804 
 
 1717 
 
 996766 
 
 26 
 
 Q88098 
 
 1743 
 
 911902 
 
 1 
 
 60 
 
 085894 
 
 1713 
 
 996751 
 
 26 
 
 089144 
 
 1738 
 
 910856 
 
 
 
 | Cosine | | Sine | Cotang. | Tang. M. 
 
 83 Degrees. 
 
SINES AND TANGENTS. (7 Degrees.) 
 
 M. | Sine | D. | Cosine | D. Tang. D. | Cotang. | 
 
 
 
 9.085894 
 
 1713 
 
 9.996751 
 
 26 1 9.089144 
 
 1738 
 
 10.910856 
 
 60 
 
 1 
 
 086922 
 
 170!) 
 
 998735 
 
 26 | 090187 
 
 1734 
 
 909813 
 
 59 
 
 2 
 
 067947 
 
 1704 
 
 <<)07-20 
 
 20 
 
 091228 
 
 1730 
 
 908772 
 
 58 
 
 3 
 
 088970 
 
 1700 
 
 998764 
 
 2(5 
 
 092206 
 
 17-27 
 
 907734 
 
 57 
 
 4 
 
 089990 
 
 1696 
 
 990088 
 
 26 
 
 093302 
 
 1722 
 
 906698 
 
 56 
 
 5 
 
 091008 
 
 1692 
 
 996673 
 
 26 
 
 094336 
 
 1719 
 
 905664 
 
 55 
 
 6 
 
 099024 
 
 1688 
 
 !i:tOO:>7 
 
 26 
 
 095367 
 
 1715 
 
 904633 
 
 54 
 
 7 
 
 093037 
 
 1684 
 
 9915041 
 
 26 
 
 090395 
 
 1711 
 
 903605 
 
 53 
 
 8 
 
 09-1047 
 
 1680 
 
 9K0025 
 
 26 
 
 097422 
 
 1707 
 
 902578 
 
 52 
 
 9 
 
 <)'j.->nr>r> 
 
 1676 
 
 998610 
 
 26 
 
 098446 
 
 1703 
 
 901554 
 
 51 
 
 10 
 
 OSMiOTrt 
 
 1673 
 
 996594 
 
 26 
 
 099468 
 
 1699 
 
 900532 
 
 50 
 
 11 
 
 9.097005 
 
 1668 
 
 9.996578 
 
 27 
 
 9.100487 
 
 1695 
 
 10.899513 
 
 49 
 
 K 
 
 008866 
 
 1605 
 
 990562 
 
 27 
 
 101504 
 
 1091 
 
 898496 
 
 48 
 
 13 
 
 099005 
 
 1(561 
 
 99(5546 
 
 27 
 
 102519 
 
 1687 
 
 897481 
 
 47 
 
 14 
 
 100002 
 
 1657 
 
 996530 
 
 27 
 
 103532 
 
 1084 
 
 896468 
 
 46 
 
 i ir> 
 
 10105G 
 
 1653 
 
 996514 
 
 27 
 
 104542 
 
 1680 
 
 895458 
 
 45 
 
 16 
 
 1H2048 
 
 1649 
 
 996498 
 
 27 
 
 105550 
 
 1676 
 
 894450 
 
 44 
 
 17 
 
 103037 
 
 1645 
 
 996482 
 
 2? 
 
 106556 
 
 1672 
 
 893444 
 
 43 
 
 18 
 
 104025 
 
 1641 
 
 996465 
 
 O 1 ? 
 
 107559 
 
 1669 
 
 892441 
 
 42 
 
 19 
 
 iu.-,oio 
 
 1638 
 
 996449 
 
 27 
 
 108560 
 
 1665 
 
 891440 
 
 41 
 
 20 
 
 105992 
 
 1634 
 
 996433 
 
 27 
 
 109559 
 
 1661 
 
 890441 
 
 40 
 
 21 
 
 9.100973 
 
 1630 
 
 9.996417 
 
 27 
 
 9.110556 
 
 1658 
 
 10.889444 
 
 39 
 
 33 
 
 107951 
 
 1627 
 
 996400 
 
 27 
 
 111551 
 
 1654 
 
 888449 
 
 38 
 
 33 
 
 108927 
 
 1623 
 
 996384 
 
 27 
 
 112543 
 
 1650 
 
 887457 
 
 37 
 
 24 
 
 109901 
 
 1619 
 
 996368 
 
 27 
 
 113533 
 
 1646 
 
 886467 
 
 36 
 
 25 
 
 1 10--73 
 
 1616 
 
 996351 
 
 27 
 
 114521 
 
 1643 
 
 885479 
 
 35 
 
 2(5 
 
 111842 
 
 1612 
 
 990335 
 
 27 
 
 115507 
 
 1639 
 
 884493 
 
 34 
 
 27 
 
 112809 
 
 1608 
 
 996318 
 
 27 
 
 110491 
 
 1636 
 
 883509 
 
 33 
 
 28 
 
 113774 
 
 1605 
 
 996302 
 
 28 
 
 117472 
 
 1632 
 
 882528 
 
 32 
 
 29 
 
 114737 
 
 1601 
 
 996385 
 
 28 
 
 118452 
 
 1629 
 
 881548 
 
 31 
 
 10 
 
 115698 
 
 1597 
 
 996269 
 
 28 
 
 119429 
 
 1625 
 
 880571 
 
 30 
 
 31 
 
 9.116650 
 
 1594 
 
 9.998252 
 
 28 
 
 9.120404 
 
 1622 
 
 10.879596 
 
 29 
 
 32 
 
 117613 
 
 1590 
 
 996235 
 
 28 
 
 121377 
 
 1618 
 
 878623 
 
 28 
 
 83 
 
 118567 
 
 1587 
 
 996219 
 
 28 
 
 122348 
 
 1615 
 
 877652 
 
 27 
 
 34 
 
 119519 
 
 1583 
 
 990202 
 
 28 
 
 123317 
 
 1611 
 
 876683 
 
 26 
 
 35 
 
 12040!) 
 
 1580 
 
 996185 
 
 23 
 
 124284 
 
 1607 
 
 875716 
 
 25 
 
 36 
 
 121417 
 
 1570 
 
 996168 
 
 28 
 
 125249 
 
 1604 
 
 874751 
 
 24 
 
 37 
 
 12-2302 
 
 1573 
 
 990151 
 
 38 
 
 126211 
 
 1601 
 
 873789 
 
 23 
 
 38 
 
 1233015 
 
 1569 
 
 9915134 
 
 28 
 
 127172 
 
 1597 
 
 872828 
 
 22 
 
 39 
 
 124248 
 
 1306 
 
 990117 
 
 28 
 
 128130 
 
 1594 
 
 871870 
 
 21 
 
 40 
 
 125187 
 
 1562 
 
 996100 
 
 28 
 
 129087 
 
 1591 
 
 870913 
 
 20 
 
 41 
 
 9.12(5125 
 
 1559 
 
 9.99(5083 
 
 29 
 
 9.130041 
 
 1587 
 
 10.869959 
 
 19 
 
 42 
 
 127060 
 
 1556 
 
 996066 
 
 29 
 
 130994 
 
 1584 
 
 869006 
 
 18 
 
 43 
 
 127993 
 
 1552 
 
 996049 
 
 29 
 
 131944 
 
 1581 
 
 868056 
 
 17 
 
 44 
 
 1*8925 
 
 1549 
 
 99(5032 
 
 29 
 
 13-W93 
 
 1577 
 
 867107 
 
 16 
 
 45 
 
 129854 
 
 1545 
 
 99(5015 
 
 29 
 
 133839 
 
 1574 
 
 800101 
 
 15 
 
 40 
 
 130781 
 
 1542 
 
 995998 
 
 29 
 
 134784 
 
 1571 
 
 865216 
 
 14 
 
 47 
 
 131706 
 
 1539 
 
 995980 
 
 29 
 
 135726 
 
 1567 
 
 864274 
 
 13 
 
 48 
 
 132030 
 
 1535 
 
 995963 
 
 29 
 
 136667 
 
 1564 
 
 863333 
 
 12 
 
 49 
 
 133551 
 
 1532 
 
 995946 
 
 29 
 
 137605 
 
 1561 
 
 862395 
 
 11 
 
 50 
 
 134470 
 
 1529 
 
 995928 
 
 29 
 
 138542 
 
 1558 
 
 861458 
 
 10 
 
 51 
 
 9.135337 
 
 1525 
 
 9.995911 
 
 29 
 
 9.139476 
 
 1555 
 
 10.860524 
 
 9 
 
 52 
 
 138303 
 
 1522 
 
 995894 
 
 29 
 
 140409 
 
 1551 
 
 859591 
 
 8 
 
 53 
 
 137216 
 
 1519 
 
 995876 
 
 29 
 
 141340 
 
 1548 
 
 58660 
 
 7 
 
 54 
 
 138128 
 
 1516 
 
 995859 
 
 29 
 
 142269 
 
 1545 
 
 857731 
 
 6 
 
 55 
 
 139037 
 
 1512 
 
 995841 
 
 29 
 
 143196 
 
 1542 
 
 856804 
 
 5 
 
 50 
 
 139914 
 
 1509 
 
 995823 
 
 29 
 
 144121 
 
 1539 
 
 855879 
 
 4 
 
 57 
 
 140850 
 
 150(5 
 
 995806 
 
 29 
 
 145044 
 
 1535 
 
 854956 
 
 3 
 
 58 
 
 11 17.V, 
 
 1503 
 
 995788 
 
 29 
 
 145966 
 
 1532 
 
 854034 
 
 2 
 
 59 
 
 14-20f>:> 
 
 1500 
 
 91)5771 
 
 29 
 
 14(5885 
 
 1529 
 
 853115 
 
 1 
 
 60 
 
 143555 
 
 1496 
 
 995753 
 
 29 
 
 1 17.-03 
 
 1526 
 
 852197 
 
 
 
 ' | Cosine | | Sine | | Cotang. | | Tang. | M. 1 
 
 82 Degrees. 
 
 2 
 
(8 Degrees.) A TABLE OF LOGARITHMIC 
 
 M. | Sine | D | Cosine | D. | Tang. | D. | Cotang. | 
 
 
 
 9.143555 
 
 1496 
 
 9.995753 
 
 30 
 
 9.147803 
 
 1526 
 
 10.852197 
 
 (iO 
 
 1 
 
 144453 
 
 1493 
 
 995735 
 
 30 
 
 148718 
 
 1523 
 
 851282 
 
 59 
 
 2 
 
 145349 
 
 1490 
 
 995717 
 
 30 
 
 149832 
 
 1520 
 
 8503(38 
 
 58 
 
 3 
 
 146243 
 
 1487 
 
 995699 
 
 30 
 
 150544 
 
 1517 
 
 849456 
 
 57 
 
 4 
 
 147136 
 
 1484 
 
 995681 
 
 30 
 
 151454 
 
 1514 
 
 848546 
 
 50 
 
 5 
 
 148026 
 
 1481 
 
 995664 
 
 30 
 
 152363 
 
 1511 
 
 847637 
 
 55 
 
 6 
 
 148915 
 
 1478 
 
 995646 
 
 30 
 
 153269 
 
 1508 
 
 846731 
 
 54 
 
 7 
 
 149802 
 
 1475 
 
 995628 
 
 30 
 
 154174 
 
 1505 
 
 845826 
 
 53 
 
 8 
 
 150686 
 
 1472 
 
 995610 
 
 30 
 
 155077 
 
 1502 
 
 844923 
 
 52 
 
 9 
 
 151569 
 
 1469 
 
 995591 
 
 30 
 
 155978 
 
 1499 
 
 844022 
 
 51 
 
 10 
 
 152451 
 
 1466 
 
 995573 
 
 30 
 
 156877 
 
 1496 
 
 843123 
 
 50 
 
 11 
 
 9.153330 
 
 1463 
 
 9.995555 
 
 30 
 
 9.157775 
 
 1493 
 
 10.842225 
 
 49 
 
 12 
 
 154208 
 
 1460 
 
 995537 
 
 30 
 
 158671 
 
 1490 
 
 841329 
 
 48 
 
 13 
 
 155083 
 
 1457 
 
 995519 
 
 30 
 
 159565 
 
 1487 
 
 840435 
 
 47 
 
 14 
 
 155957 
 
 1454 
 
 995501 
 
 31 
 
 160457 
 
 1484 
 
 839543 
 
 46 
 
 15 
 
 156830 
 
 1451 
 
 995482 
 
 31 
 
 161347 
 
 1481 
 
 838C53 
 
 45 
 
 16 
 
 157700 
 
 1448 
 
 995464 
 
 31 
 
 162236 
 
 1479 
 
 837764 
 
 44 
 
 17 
 
 158569 
 
 1445 
 
 995446 
 
 31 
 
 163123 
 
 1476 
 
 836877 
 
 43 
 
 18 
 
 159435 
 
 1442 
 
 995427 
 
 31 
 
 164008 
 
 1473 
 
 835992 
 
 42 
 
 19 
 
 160301 
 
 1439 
 
 995409 
 
 31 
 
 164892 
 
 1470 
 
 835108 
 
 41 
 
 20 
 
 161164 
 
 1436 
 
 995390 
 
 31 
 
 165774 
 
 1467 
 
 834226 
 
 40 
 
 21 
 
 9.162025 
 
 1433 
 
 9.995372 
 
 31 
 
 9.166654 
 
 1464 
 
 10.833346 
 
 39 
 
 22 
 
 162885 
 
 1430 
 
 995353 
 
 31 
 
 167532 
 
 1461 
 
 832468 
 
 38 
 
 23 
 
 163743 
 
 1427 
 
 995334 
 
 31 
 
 168409 
 
 1458 
 
 831591 
 
 37 
 
 24 
 
 164(100 
 
 1424 
 
 995316 
 
 31 
 
 169284 
 
 1455 
 
 8307] 6 
 
 36 
 
 25 
 
 165454 
 
 1422 
 
 995297 
 
 31 
 
 170157 
 
 1453 
 
 829843 
 
 35 
 
 26 
 
 166307 
 
 14J9 
 
 995278 
 
 31 
 
 171029 
 
 1450 
 
 828971 
 
 34 
 
 27 
 
 167159 
 
 1416 
 
 9952CO 
 
 31 
 
 .171899 
 
 1447 
 
 828101 
 
 33 
 
 28 
 
 168008 
 
 1413 
 
 995241 
 
 32 
 
 172767 
 
 1444 
 
 827233 
 
 32 
 
 29 
 
 168856 
 
 1410 
 
 995222 
 
 32 
 
 173634 
 
 1442 
 
 826366 
 
 31 
 
 30 
 
 169702 
 
 1407 
 
 995203 
 
 32 
 
 174499 
 
 1439 
 
 825501 
 
 30 
 
 31 
 
 9.170547 
 
 1405 
 
 9.995184 
 
 32 
 
 9.175362 
 
 1436 
 
 10.824638 
 
 29 
 
 32 
 
 171389 
 
 1402 
 
 995165 
 
 32 
 
 176224 
 
 1433 
 
 823776 
 
 28 
 
 33 
 
 172230 
 
 1399 
 
 995146 
 
 32 
 
 177084 
 
 1431 
 
 822916 
 
 27 
 
 34 
 
 173070 
 
 1396 
 
 995127 
 
 32 
 
 177942 
 
 1428 
 
 822058 
 
 26 
 
 35 
 
 173908 
 
 1394 
 
 995108 
 
 32 
 
 178799 
 
 1425 
 
 821201 
 
 25 
 
 36 
 
 174744 
 
 1391 
 
 995089 
 
 32 
 
 179655 
 
 1423 
 
 820345 
 
 24 
 
 37 
 
 175578 
 
 1388 
 
 995070 
 
 32 
 
 180508 
 
 1420 
 
 819492 
 
 23 
 
 38 
 
 176411 
 
 1386 
 
 995051 
 
 32 
 
 181360 
 
 1417 
 
 818640 
 
 22 
 
 39 
 
 177242 
 
 1383 
 
 995032 
 
 32 
 
 182211 
 
 1415 
 
 817789 
 
 21 
 
 40 
 
 178072 
 
 1380 
 
 995013 
 
 32 
 
 183059 
 
 1412 
 
 816941 
 
 20 
 
 41 
 
 9.178900 
 
 1377 
 
 9.994993 
 
 32 
 
 9.183907 
 
 1409 
 
 10.816093 
 
 19 
 
 42 
 
 179726 
 
 1374 
 
 994974 
 
 32 
 
 184752 
 
 1407 
 
 815248 
 
 18 
 
 43 
 
 180551 
 
 1372 
 
 994955 
 
 32 
 
 185597 
 
 1404 
 
 814403 
 
 17 
 
 44 
 
 181374 
 
 1369 
 
 994935 
 
 32 
 
 186439 
 
 1402 
 
 8] 3561 
 
 16 
 
 45 
 
 182196 
 
 1366 
 
 994916 
 
 33 
 
 187280 
 
 1399 
 
 812720 
 
 15 
 
 46 
 
 183016 
 
 1364 
 
 994896 
 
 33 
 
 188120 
 
 1396 
 
 811880 
 
 14 
 
 47 
 
 183834 
 
 1361 
 
 994877 
 
 33 
 
 188958 
 
 13!)3 
 
 811042 
 
 13 
 
 48 
 
 184651 
 
 1359 
 
 994857 
 
 33 
 
 189794 
 
 1391 
 
 810206 
 
 12 
 
 49 
 
 185466 
 
 1356 
 
 994838 
 
 33 
 
 190629 
 
 1389 
 
 809371 
 
 11 
 
 50 
 
 186280 
 
 1353 
 
 994818 
 
 33 
 
 191462 
 
 1386 
 
 808538 
 
 10 
 
 51 
 
 9.187092 
 
 1351 
 
 9.994798 
 
 33 
 
 9.192294 
 
 1384 
 
 10.807706 
 
 9 
 
 52 
 
 187903 
 
 1348 
 
 994779 
 
 33 
 
 193124 
 
 1381 
 
 806876 
 
 8 
 
 53 
 
 188712 
 
 1346 
 
 994759 
 
 33 
 
 193953 
 
 1379 
 
 806047 
 
 7 
 
 54 
 
 189519 
 
 1343 
 
 994739 
 
 33 
 
 194780 
 
 1376 
 
 805220 
 
 6 
 
 55 
 
 190325 
 
 1341 
 
 994719 
 
 33 
 
 195606 
 
 1374 
 
 804394 
 
 5 
 
 56 
 
 191130 
 
 1338 
 
 994700 
 
 33 
 
 196430 
 
 1371 
 
 803570 
 
 4 
 
 57 
 
 191933 
 
 1336 
 
 994680 
 
 33 
 
 197253 
 
 1369 
 
 802747 
 
 3 
 
 58 
 
 192734 
 
 1333 
 
 994660 
 
 33 
 
 198074 
 
 1366 
 
 801926 
 
 2 
 
 59 
 
 193534 
 
 1330 
 
 994640 
 
 33 
 
 198894 
 
 1364 
 
 801106 
 
 1 
 
 60 
 
 194332 
 
 1328 
 
 994020 
 
 33 
 
 199713 
 
 1361 
 
 800287 
 
 
 
 | Cosine | | Sine | Cotang. | | Tang. | M. ) 
 
 81 Degrees. 
 
SINES AND TANGENTS. (9 Degrees.) 
 
 27 
 
 M. | Sine D. Cosine | D. | Tang. D. | Cotang. | 
 
 
 
 9.191332 
 
 1338 
 
 9.1)94620 
 
 33 
 
 9. 1!U?13 
 
 13til 
 
 10.800287 
 
 60 
 
 1 
 
 1951-29 
 
 13-26 
 
 994000 
 
 33 
 
 2005-29 
 
 1359 
 
 799471 
 
 59 
 
 2 
 
 195935 
 
 1333 
 
 994580 
 
 33 
 
 201345 
 
 1356 
 
 798655 
 
 58 
 
 3 
 
 196719 
 
 1321 
 
 !):)4500 
 
 34 
 
 202159 
 
 1354 
 
 797841 
 
 57 
 
 4 
 
 W7511 
 
 1318 
 
 994.-) 10 
 
 34 
 
 202971 
 
 1352 
 
 797038 
 
 56 
 
 5 
 
 198303 
 
 13 10 
 
 994519 
 
 34 
 
 203782 
 
 1349 
 
 796218 
 
 55 
 
 6 
 
 199091 
 
 1313 
 
 994499 
 
 34 
 
 204592 
 
 1347 
 
 795408 
 
 54 
 
 7 
 
 199879 
 
 1311 
 
 994479 
 
 34 
 
 205400 
 
 1345 
 
 794600 
 
 53 
 
 8 
 
 900666 
 
 1308 
 
 991459 
 
 34 
 
 200207 
 
 1342 
 
 793793 
 
 52 
 
 9 
 
 "2()]4r>i 
 
 1306 
 
 994438 
 
 34 
 
 207013 
 
 1340 
 
 792987 
 
 51 
 
 10 
 
 202234 
 
 1304 
 
 994418 
 
 34 
 
 207817 
 
 1338 
 
 792183 
 
 50 
 
 11 
 
 9.203017 
 
 1301 
 
 9.934397 
 
 34 
 
 9.208619 
 
 1335 
 
 10.791381 
 
 49 
 
 19 
 
 203797 
 
 1299 
 
 994377 
 
 34 
 
 209420 
 
 1333 
 
 790.580 
 
 48 
 
 13 
 
 204577 
 
 l-2i)li 
 
 994357 
 
 34 
 
 210220 
 
 1331 
 
 789780 
 
 47 
 
 14 
 
 205354 
 
 1294 
 
 994330 
 
 34 
 
 211018 
 
 1328 
 
 788982 
 
 46 
 
 15 
 
 2l)!>131 
 
 1292 
 
 994316 
 
 34 
 
 211815 
 
 1326 
 
 788185 
 
 45 
 
 16 
 
 2(G906 
 
 1289 
 
 994295 
 
 34 
 
 212611 
 
 1324 
 
 787389 
 
 44 
 
 17 
 
 207679 
 
 1287 
 
 994274 
 
 35 
 
 213405 
 
 1321 
 
 786595 
 
 43 
 
 18 
 
 208452 
 
 1285 
 
 994254 
 
 35 
 
 214198 
 
 1319 
 
 785802 
 
 42 
 
 19 
 
 209222 
 
 1282 
 
 991233 
 
 35 
 
 214989 
 
 1317 
 
 785011 
 
 41 
 
 20 
 
 20D9J2 
 
 1280 
 
 994212 
 
 35 
 
 215780 
 
 1315 
 
 784220 
 
 40 
 
 21 
 
 9.210760 
 
 1278 
 
 9.994191 
 
 35 
 
 9.216568 
 
 1312 
 
 10.783432 
 
 39 
 
 22 
 
 211526 
 
 127fr 
 
 994171 
 
 35 
 
 217356 
 
 1310 
 
 782644 
 
 38 
 
 23 
 
 212291 
 
 1273 
 
 994150 
 
 35 
 
 218142 
 
 1308 
 
 781858 
 
 37 
 
 24 
 
 213055 
 
 1271 
 
 994129 
 
 35 
 
 218926 
 
 1305 
 
 781074 
 
 36 
 
 25 
 
 213818 
 
 1268 
 
 91)4108 
 
 35 
 
 219710 
 
 1303 
 
 780290 
 
 35 
 
 26 
 
 214579 
 
 1286 
 
 991087 
 
 35 
 
 2204P2 
 
 1301 
 
 779508 
 
 34 
 
 27 
 
 215338 
 
 1264 
 
 994086 
 
 35 
 
 221272 
 
 1299 
 
 778728 
 
 33 
 
 28 
 
 216097 
 
 1261 
 
 994045 
 
 35 
 
 222052 
 
 1297 
 
 777948 
 
 32 
 
 29 
 
 216854 
 
 1259 
 
 994024 
 
 35 
 
 222830 
 
 1294 
 
 777170 
 
 31 
 
 30 
 
 217609 
 
 1257 
 
 994003 
 
 35 
 
 223606 
 
 1292 
 
 776394 
 
 30 
 
 31 
 
 9.218333 
 
 1255 
 
 9.993981 
 
 35 
 
 9.224382 
 
 1290 
 
 10.775618 
 
 29 
 
 32 
 
 219116 
 
 1253 
 
 993960 
 
 35 
 
 225156 
 
 1288 
 
 774844 
 
 28 
 
 33 
 
 219868 
 
 1250 
 
 993939 
 
 35 
 
 225929 
 
 1286 
 
 774071 
 
 27 
 
 34 
 
 220618 
 
 1248 
 
 993918 
 
 35 
 
 226700 
 
 1284 
 
 773300 
 
 26 
 
 35 
 
 221367 
 
 1246 
 
 99381)6 
 
 36 
 
 227471 
 
 1281 
 
 772529 
 
 25 
 
 36 
 
 222115 
 
 1244 
 
 993875 
 
 36 
 
 228239 
 
 1279 
 
 771761 
 
 24 
 
 37 
 
 222861 
 
 1242 
 
 993854 
 
 36 
 
 229007 
 
 1277 
 
 770993 
 
 23 
 
 38 
 
 223606 
 
 1239 
 
 993832' 
 
 36 
 
 229773 
 
 1275 
 
 770227 
 
 22 
 
 39 
 
 224349 
 
 1237 
 
 993811 
 
 36 
 
 230539 
 
 1273 
 
 769461 
 
 21 
 
 40 
 
 225092 
 
 1235 
 
 993789 
 
 36 
 
 231302 
 
 1271 
 
 768698 
 
 20 
 
 41 
 
 9.225833 
 
 1233 
 
 9.993768 
 
 36 
 
 9.232065 
 
 1269 
 
 10.767935 
 
 19 
 
 42 
 
 226573 
 
 1231 
 
 993746 
 
 36 
 
 232826 
 
 1267 
 
 767174 
 
 18 
 
 43 
 
 227311 
 
 1228 
 
 993725 
 
 36 
 
 233586 
 
 1265 
 
 701)414 
 
 17 
 
 44 
 
 228048 
 
 12-26 
 
 993703 
 
 36 
 
 234345 
 
 1262 
 
 765655 
 
 16 
 
 45 
 
 223784 
 
 1224 
 
 993681 
 
 36 
 
 235103 
 
 1260 
 
 764897 
 
 15 
 
 46 
 
 229518 
 
 1222 
 
 993660 
 
 36 
 
 235859 
 
 1258 
 
 764141 
 
 14 
 
 47 
 
 230252 
 
 1220 
 
 993638 
 
 36 
 
 236614 
 
 1256 
 
 763380 
 
 13 
 
 48 
 
 230984 
 
 1218 
 
 993616 
 
 36 
 
 237368 
 
 1254 
 
 762632 
 
 12 
 
 49 
 
 231714 
 
 1216 
 
 9J3594 
 
 37 
 
 238120 
 
 1252 
 
 761880 
 
 11 
 
 50 
 
 232444 
 
 1214 
 
 993572 
 
 37 
 
 238872 
 
 1250 
 
 761128 
 
 10 
 
 51 
 
 9.233172 
 
 1212 
 
 9.993550 
 
 37 
 
 9.239022 
 
 1248 
 
 10.760378 
 
 9 
 
 52 
 
 233899 
 
 1209 
 
 993528 
 
 37 
 
 240371 
 
 1246 
 
 759629 
 
 8 
 
 53 
 
 234625 
 
 1207 
 
 993506 
 
 37 
 
 241118 
 
 1244 
 
 758882 
 
 7 
 
 54 
 
 235349 
 
 1205 
 
 993484 
 
 37 
 
 241865 
 
 1242 
 
 758135 
 
 6 
 
 55 
 
 23S073 
 
 12!)3 
 
 993402 
 
 37 
 
 342!) 10 
 
 1240 
 
 757390 
 
 5 
 
 56 
 
 23(5795 
 
 1201 
 
 993440 
 
 37 
 
 243354 
 
 1238 
 
 750646 
 
 4 
 
 57 
 
 237515 
 
 1199 
 
 993418 
 
 37 
 
 244097 
 
 1236 
 
 755903 
 
 3 
 
 58 
 
 238235 
 
 1197 
 
 993390 
 
 37 
 
 244839 
 
 1234 
 
 755161 
 
 2 
 
 59 
 
 238953 
 
 1195 
 
 993374 
 
 37 
 
 245579 
 
 1232 
 
 754421 
 
 1 
 
 60 
 
 239570 
 
 1193 
 
 993351 
 
 37 
 
 246319 
 
 1230 
 
 753681 
 
 
 
 | Cosine | Sine | | Cotang. Tang. | M. 
 
 85 Degrees. 
 
(10 Degrees.) A TABLE OF LOGARITHMIC 
 
 M. j Sine D. Cosine D. | Tang. D. Cotang. 
 
 
 
 9.239670 
 
 1193 
 
 9.993351 
 
 37 
 
 9.246319 
 
 1-230 10.753681 
 
 60 
 
 1 
 
 24033G 
 
 1191 
 
 993329 
 
 37 
 
 217.157 
 
 1228 
 
 752943 
 
 59 
 
 2 
 
 241101 
 
 1189 
 
 993307 
 
 37 
 
 247794 
 
 1226 
 
 752206 
 
 58 
 
 3 
 
 211814 
 
 1187 
 
 9932-55 
 
 37 
 
 248530 
 
 1224 
 
 751470 
 
 57 
 
 4 
 
 342526 
 
 1185 
 
 9932IJ2 
 
 37 
 
 219-254 
 
 1-222 
 
 750736 
 
 56 
 
 5 
 
 2432 57 
 
 1183 
 
 993240 
 
 37 
 
 249998 
 
 122:) 
 
 750002 
 
 55 
 
 6 
 
 243947 
 
 1181 
 
 993-217 
 
 38 
 
 259730 
 
 1218 
 
 74!(270 
 
 54 
 
 7 
 
 244656 
 
 1179 
 
 993195 
 
 38 
 
 251461 
 
 1217 
 
 748539 
 
 53 
 
 8 
 
 2453(53 
 
 1177 
 
 99317-2 
 
 38 
 
 252191 
 
 1215 
 
 747809 
 
 52 
 
 9 
 
 246009 
 
 1175 
 
 993149 
 
 38 
 
 25-29-20 
 
 1213 
 
 747080 
 
 51 
 
 10 
 
 246775 
 
 1173 
 
 993127 
 
 38 
 
 253648 
 
 1211 
 
 74035-2 
 
 50 
 
 11 
 
 9.247478 
 
 1171 
 
 9.993104 
 
 38 
 
 9.254374 
 
 1209 
 
 10.745f>2i) 
 
 49 
 
 12 
 
 248181 
 
 1169 
 
 9930H1 
 
 38 
 
 255100 
 
 1207 
 
 744900 
 
 48 
 
 13 
 
 248883 
 
 1167 
 
 993059 
 
 38 
 
 255824 
 
 1205 
 
 744176 
 
 47 
 
 14 
 
 249583 
 
 1165 
 
 993036 
 
 38 
 
 256547 
 
 1203 
 
 743453 
 
 46 
 
 15 
 
 25028-2 
 
 1163 
 
 993013 
 
 38 
 
 257269 
 
 1201 
 
 742731 
 
 45 
 
 16 
 
 250980 
 
 1161 
 
 992990 
 
 38 
 
 257990 
 
 1200 
 
 742010 
 
 44 
 
 17 
 
 251677 
 
 1159 
 
 99-2957 
 
 38 
 
 258710 
 
 1198 
 
 741290 
 
 43 
 
 18 
 
 252373 
 
 1153 
 
 '.u-jii 11 
 
 38 
 
 2594-29 
 
 1196 
 
 740571 
 
 42 
 
 19 
 
 253067 
 
 1156 
 
 9929-21 
 
 38 
 
 260146 
 
 1194 
 
 73>8.)4 
 
 41 
 
 20 
 
 253761 
 
 1154 
 
 992898 
 
 38 
 
 260863 
 
 1192 
 
 739137 
 
 40 
 
 21 
 
 9.254453 
 
 1152 
 
 9.992375 
 
 38 
 
 9.261578 
 
 1190 
 
 10.738422 
 
 39 
 
 22 
 
 255144 
 
 1150 
 
 992852 
 
 38 
 
 25-229-2 
 
 1189 
 
 737703 
 
 38 
 
 23 
 
 255834 
 
 1148 
 
 992829 
 
 39 
 
 263005 
 
 1187 
 
 736995 
 
 37 
 
 24 
 
 256523 
 
 1146 
 
 992806 
 
 39 
 
 253717 
 
 1185 
 
 736283 
 
 38 
 
 25 
 
 257211 
 
 1144 
 
 992783 
 
 39 
 
 284428 
 
 1183 
 
 735572 
 
 35 
 
 26 
 
 2578D8 
 
 1142 
 
 932759 
 
 39 
 
 265138 
 
 1181 
 
 734862 
 
 34 
 
 27 
 
 258583 
 
 1141 
 
 992736 
 
 39 
 
 265847 
 
 1179 
 
 734153 
 
 33 
 
 28 
 
 259268 
 
 1139 
 
 992713 
 
 39 
 
 266555 
 
 1173 
 
 733445 
 
 32 
 
 29 
 
 259951 
 
 1137 
 
 99-2690 
 
 39 
 
 267261 
 
 1176 
 
 732739 
 
 31 
 
 30 
 
 260633 
 
 1135 
 
 992666 
 
 39 
 
 267937 
 
 1174 
 
 732033 
 
 30 
 
 31 
 
 9.261314 
 
 1133 
 
 9.992043 
 
 39 
 
 9.268671 
 
 1172 
 
 10.731329 
 
 29 
 
 32 
 
 261994 
 
 1131 
 
 992619 
 
 39 
 
 269375 
 
 1170 
 
 730625 
 
 28 
 
 33 
 
 262673 
 
 1130 
 
 992595 
 
 39 
 
 270077 
 
 1109 
 
 729923 
 
 27 
 
 31 
 
 263351 
 
 1128 
 
 992572 
 
 39 
 
 270779 
 
 1167 
 
 729221 
 
 26 
 
 35 
 
 2640-27 
 
 1126 
 
 99-2549 
 
 39 
 
 271479 
 
 1165 
 
 728521 
 
 25 
 
 36 
 
 264703 
 
 1124 
 
 992525 
 
 39 
 
 272178 
 
 1164 
 
 727822 
 
 24 
 
 37 
 
 265377 
 
 1122 
 
 99-2501 
 
 39 
 
 272876 
 
 1162 
 
 727124 
 
 23 
 
 38 
 
 26(5051 
 
 1120 
 
 992478 
 
 40 
 
 273573 
 
 1160 
 
 726127 
 
 22 
 
 39 
 
 2667-23 
 
 1119 
 
 992454 
 
 40 
 
 274259 
 
 1158 
 
 725731 
 
 21 
 
 40 
 
 267395 
 
 1117 
 
 992430 
 
 40 
 
 274964 
 
 1157 
 
 725036 
 
 20 
 
 41 
 
 9.268065 
 
 1115 
 
 9.992406 
 
 40 
 
 9.275658 
 
 1155 
 
 10.724342 
 
 19 
 
 42 
 
 268734 
 
 1113 
 
 992382 
 
 40 
 
 276351 
 
 1153 
 
 723649 
 
 18 
 
 43 
 
 269402 
 
 1111 
 
 992359 
 
 40 
 
 277043 
 
 1151 
 
 722957 
 
 17 
 
 44 
 
 270089 
 
 1110 
 
 99-2335 
 
 40 
 
 277734 
 
 1150 
 
 722266 
 
 16 
 
 45 
 
 270735 
 
 1108 
 
 992311 
 
 40 
 
 2784-24 
 
 1148 
 
 721576 
 
 15 
 
 48 
 
 271400 
 
 1106 
 
 992287 
 
 40 
 
 279113 
 
 1147 
 
 720837 
 
 14 
 
 47 
 
 272064 
 
 1105 
 
 992263 
 
 40 
 
 279801 
 
 1145 
 
 720199 
 
 13 
 
 48 
 
 272726 
 
 1103 
 
 99-2-239 
 
 40 
 
 280488 
 
 1143 
 
 719512 
 
 12 
 
 49 
 
 273388 
 
 1101 
 
 992214 
 
 40 
 
 281174 
 
 1141 
 
 718826 
 
 11 
 
 50 
 
 274049 
 
 1099 
 
 992190 
 
 40 
 
 281858 
 
 1140 
 
 718142 
 
 10 
 
 51 
 
 9.274708 
 
 1098 
 
 9.99216G 
 
 40 
 
 9.282542 
 
 1138 
 
 10.717458 
 
 9 
 
 52 
 
 275367 
 
 1096 
 
 99-2142 
 
 40 
 
 283225 
 
 1136 
 
 716775 
 
 8 
 
 53 
 
 276024 
 
 1094 
 
 992117 
 
 41 
 
 283907 
 
 1135 
 
 716093 
 
 7 
 
 54 
 
 276681 
 
 1092 
 
 692093 
 
 41 
 
 284588 
 
 1133 
 
 715412 
 
 6 
 
 55 
 
 277337 
 
 1091 
 
 992059 
 
 41 
 
 285268 
 
 1131 
 
 714732 
 
 5 
 
 56 
 
 277991 
 
 108:) 
 
 99-2044 
 
 41 
 
 2859 i7 
 
 1130 
 
 714053 
 
 4 
 
 57 
 
 278644 
 
 1087 
 
 992020 
 
 41 
 
 286624 
 
 1128 
 
 713376 
 
 3 
 
 58 
 
 279297 
 
 1086 
 
 99-199!) 
 
 41 
 
 287301 
 
 1126 
 
 712699 
 
 2 
 
 59 
 
 279948 
 
 1084 
 
 991971 
 
 41 
 
 287977 
 
 1125 
 
 712023 
 
 1 
 
 60 
 
 280599 
 
 1082 
 
 991947 
 
 41 
 
 288652 
 
 1123 
 
 711348 
 
 
 
 | Cosine 
 
 | Sine | 
 
 Cotang. | | Tang. M. 
 
SINES AND TANGi'.NTS. (11 DegrCCS.) 
 
 29 
 
 M. Sine D. Cosine | D. | Tang. D. Cotang. 
 
 
 
 9.280599 
 
 1082 
 
 9.991947 
 
 41 
 
 9.288652 
 
 1123 
 
 10.711348 
 
 60 
 
 1 
 
 281248 
 
 1081 
 
 991922 
 
 41 
 
 289326 
 
 1122 
 
 710674 
 
 59 
 
 2 
 
 281897 
 
 1079 
 
 991897 
 
 41 
 
 289999 
 
 1120 
 
 710001 
 
 58 
 
 3 
 
 282544 
 
 1077 
 
 991873 
 
 4] 
 
 290671 
 
 1118 
 
 709329 
 
 57 
 
 4 
 
 283190 
 
 1076 
 
 9918-18 
 
 41 
 
 291342 
 
 1117 
 
 708658 
 
 56 
 
 5 
 
 283836 
 
 1074 
 
 991823 
 
 41 
 
 292013 
 
 1115 
 
 707987 
 
 55 
 
 
 
 284480 
 
 1072 
 
 99T99 
 
 41 
 
 293682 
 
 1114 
 
 707318 
 
 54 
 
 7 
 
 285124 
 
 1071 
 
 991774 
 
 42 
 
 293350 
 
 1112 
 
 706650 
 
 53 
 
 8 
 
 285766 
 
 1069 
 
 991749 
 
 42 
 
 294017 
 
 1111 
 
 705983 
 
 52 
 
 9 
 
 286408 
 
 1067 
 
 991724 
 
 42 
 
 294684 
 
 1109 
 
 705316 
 
 51 
 
 10 
 
 287048 
 
 1066 
 
 991699 
 
 42 
 
 295349 
 
 1107 
 
 704651 
 
 50 
 
 11 
 
 9.287687 
 
 1064 
 
 9.991674 
 
 42 
 
 9.296013 
 
 1106 
 
 10.703987 
 
 49 
 
 12 
 
 288326 
 
 10(53 
 
 991649 
 
 42 
 
 296677 
 
 1104 
 
 703323 
 
 48 
 
 13 
 
 288964 
 
 1061 
 
 991624 
 
 42 
 
 297339 
 
 1103 
 
 702661 
 
 47 
 
 14 
 
 289600 
 
 1059 
 
 991599 
 
 42 
 
 298001 
 
 1101 
 
 701999 
 
 46 
 
 15 
 
 290236 
 
 1058 
 
 991574 
 
 42 
 
 298662 
 
 1100 
 
 701338 
 
 45 
 
 16 
 
 290870 
 
 1056 
 
 991549 
 
 42 
 
 29932-3 
 
 1098 
 
 700678 
 
 44 
 
 17 
 
 291504 
 
 1054 
 
 991524 
 
 42 
 
 299980 
 
 1096 
 
 700020 
 
 43 
 
 18 
 
 292137 
 
 1053 
 
 991498 
 
 42 
 
 300638 
 
 1095 
 
 699362 
 
 42 
 
 11) 
 
 292768 
 
 1051 
 
 991473 
 
 42 
 
 301295 
 
 1093 
 
 698705 
 
 41 
 
 20 
 
 293399 
 
 1050 
 
 991448 
 
 42 
 
 301951 
 
 1092 
 
 698049 
 
 40 
 
 21 
 
 9.294029 
 
 1048 
 
 9.991422 
 
 42 
 
 9.302607 
 
 1090 
 
 10.697393 
 
 39 
 
 22 
 
 294658 
 
 1046 
 
 991397 
 
 42 
 
 303261 
 
 1089 
 
 696739 
 
 38 
 
 23 
 
 295286 
 
 1045 
 
 991372 
 
 43 
 
 303914 
 
 1087 
 
 698086 
 
 37 
 
 24 
 
 295913 
 
 1043 
 
 991346 
 
 43 
 
 304567 
 
 1086 
 
 695433 
 
 36 
 
 25 
 
 296539 
 
 1042 
 
 991321 
 
 43 
 
 305218 
 
 1084 
 
 694782 
 
 35 
 
 26 
 
 297164 
 
 1040 
 
 991295 
 
 43 
 
 305869 
 
 1083 
 
 694131 
 
 34 
 
 27 
 
 297788 
 
 1039 
 
 991270 
 
 43 
 
 30(5519 
 
 1081 
 
 693481 
 
 33 
 
 28 
 
 298412 
 
 1037 
 
 991244 
 
 43 
 
 307168 
 
 1080 
 
 692832 
 
 32 
 
 29 
 
 299034 
 
 1036 
 
 991218 
 
 43 
 
 307815 
 
 1078 
 
 692185 
 
 31 
 
 30 
 
 299655 
 
 1034 
 
 991193 
 
 43 
 
 308463 
 
 1077 
 
 691537 
 
 30 
 
 31 
 
 9.300276 
 
 1032 
 
 9.991167 
 
 43 
 
 9.309109 
 
 1075 
 
 10.690391 
 
 29 
 
 32 
 
 300895 
 
 1031 
 
 991141 
 
 43 
 
 309754 
 
 1074 
 
 690246 
 
 28 
 
 33 
 
 301514 
 
 10-29 
 
 991115 
 
 43 
 
 310398 
 
 1073 
 
 689602 
 
 27 
 
 34 
 
 302132 
 
 1028 
 
 991090 
 
 43 
 
 311042 
 
 1071 
 
 688958 
 
 26 
 
 35 
 
 302748 
 
 1026 
 
 991064 
 
 43 
 
 311685 
 
 1070 
 
 688315 
 
 25 
 
 3(5 
 
 303364 
 
 1025 
 
 991038 
 
 43 
 
 312327 
 
 1068 
 
 687673 
 
 24 
 
 37 
 
 :)397 
 
 1023 
 
 991012 
 
 43 
 
 312967 
 
 1067 
 
 687033 
 
 23 
 
 38 
 
 304593 
 
 1022 
 
 99098(5 
 
 43 
 
 313608 
 
 1065 
 
 686392 
 
 22 
 
 39 
 
 305207 
 
 1020 
 
 990960 
 
 43 
 
 314247 
 
 1064 
 
 685753 
 
 21 
 
 40 
 
 305819 
 
 1019 
 
 990934 
 
 44 
 
 314885 
 
 1062 
 
 685115 
 
 20 
 
 41 
 
 9.306430 
 
 1017 
 
 9.990908 
 
 44 
 
 9.315523 
 
 1061 
 
 10.684477 
 
 19 
 
 42 
 
 307041 
 
 1016 
 
 990882 
 
 44 
 
 316159 
 
 1060 
 
 683841 
 
 18 
 
 43 
 
 307650 
 
 1014 
 
 990655 
 
 44 
 
 316795 
 
 1058 
 
 683205 
 
 17 
 
 44 
 
 308259 
 
 1013 
 
 990829 
 
 44 
 
 317430 
 
 1057 
 
 682570 
 
 16 
 
 45 
 
 308867 
 
 1011 
 
 990803 
 
 44 
 
 318064 
 
 1055 
 
 681936 
 
 15 
 
 46 
 
 309474 
 
 1010 
 
 990777 
 
 44 
 
 318697 
 
 1054 
 
 681303 
 
 14 
 
 47 
 
 310080 
 
 1008 
 
 990750 
 
 44 
 
 319329 
 
 1053 
 
 680671 
 
 13 
 
 48 
 
 310685 
 
 1007 
 
 990724 
 
 44 
 
 319961 
 
 1051 
 
 680039 
 
 12 
 
 49 
 
 311289 
 
 1005 
 
 iMK>97 
 
 44 
 
 320592 
 
 1050 
 
 679408 
 
 11 
 
 50 
 
 311893 
 
 1004 
 
 990671 
 
 44 
 
 321222 
 
 1048 
 
 678778 
 
 10 
 
 51 
 
 9.312495 
 
 1003 
 
 9.990644 
 
 44 
 
 9.321851 
 
 1047 
 
 10.678149 
 
 9 
 
 52 
 
 313097 
 
 1001 
 
 990618 
 
 44 
 
 322479 
 
 1045 
 
 677521 
 
 8 
 
 53 
 
 313698 
 
 1000 
 
 iHK>9I 
 
 44 
 
 323106 
 
 1044 
 
 676894 
 
 7 
 
 54 
 
 314297 
 
 998 
 
 990565 
 
 44 
 
 323733 
 
 1043 
 
 676267 
 
 6 
 
 55 
 
 314897 
 
 997 
 
 990538 
 
 44 
 
 324358 
 
 1041 
 
 675642 
 
 5 
 
 56 
 
 315495 
 
 996 
 
 99051] 
 
 45 
 
 324983 
 
 1040 
 
 675017 
 
 4 
 
 57 
 
 316092 
 
 994 
 
 990485 
 
 45 
 
 325607 
 
 1039 
 
 674393 
 
 3 
 
 58 
 
 316689 
 
 993 
 
 990458 
 
 45 
 
 326231 
 
 1037 
 
 673769 
 
 2 
 
 59 
 
 317284 
 
 991 
 
 990431 
 
 45 
 
 326853 
 
 1036 
 
 673141 
 
 1 
 
 GO 
 
 317879 
 
 990 
 
 <i!H!ini 
 
 45 
 
 327475 
 
 1035 
 
 672525 
 
 
 
 Cosine Sine | I Cotang. | Tang. M. 
 
 78 Degrees. 
 
30 
 
 (12 Degrees.) A TABLE or LOGARITHMIC 
 
 i M Sine D Cosine | D. | Tang. | D. | Cotang. | 
 
 
 
 9.317879 
 
 990 
 
 9.990404 
 
 45 
 
 9.327474 
 
 1035 
 
 10.672526 
 
 60 
 
 1 1 
 
 318473 
 
 988 
 
 99U378 
 
 45 
 
 328095 
 
 1033 
 
 671905 
 
 .V.I 
 
 i 2 
 
 319066 
 
 987 
 
 990351 
 
 45 
 
 328715 
 
 1032 
 
 671285 
 
 58 
 
 3 
 
 319658 
 
 986 
 
 990324 
 
 45 
 
 329334 
 
 1030 
 
 670666 
 
 57 
 
 4 
 
 320249 
 
 984 
 
 990297 
 
 45 
 
 329953 
 
 1G29 
 
 670047 
 
 56 
 
 5 
 
 320840 
 
 983 
 
 990-270 
 
 45 
 
 330570 
 
 1028 
 
 GT9430 
 
 55 
 
 G 
 
 321430 
 
 982 
 
 990243 
 
 45 
 
 331187 
 
 1026 
 
 668813 
 
 54 
 
 7 
 
 322019 
 
 980 
 
 990215 
 
 45 
 
 :mso3 
 
 1025 
 
 668197 
 
 53 
 
 8 
 
 322C07 
 
 979 
 
 990188 
 
 45 
 
 332418 
 
 1024 
 
 667582 
 
 52 
 
 9 
 
 323194 
 
 977 
 
 990161 
 
 45 
 
 333033 
 
 1023 
 
 G6f967 
 
 51 
 
 10 
 
 323780 
 
 976 
 
 990134 
 
 45 
 
 333646 
 
 1021 
 
 666354 
 
 50 
 
 11 
 
 9.324366 
 
 975 
 
 9.990107 
 
 46 
 
 9.334259 
 
 1020 
 
 10.665741 
 
 49 
 
 12 
 
 324950 
 
 973 
 
 990079 
 
 46 
 
 334871 
 
 10J9 
 
 665129 
 
 48 
 
 13 
 
 325534 
 
 972 
 
 99QC52 
 
 46 
 
 335482 
 
 1017 
 
 664518 
 
 47 
 
 14 
 
 326117 
 
 970 
 
 990025 
 
 46 
 
 336093 
 
 1016 
 
 663907 
 
 46 
 
 15 
 
 326700 
 
 969 
 
 989997 
 
 46 
 
 336702 
 
 1015 
 
 663298 
 
 45 
 
 1 16 
 
 327281 
 
 968 
 
 989970 
 
 46 
 
 337311 
 
 1013 
 
 662689 
 
 44 
 
 1 I? 
 
 327862 
 
 966 
 
 989942 
 
 46 
 
 337919 
 
 1012 
 
 662081 
 
 43 
 
 18 
 
 328442 
 
 965 
 
 989915 
 
 46 
 
 338527 
 
 1011 
 
 661473 
 
 42 
 
 19 
 
 329021 
 
 964 
 
 989887 
 
 46 
 
 339133 
 
 1010 
 
 660867 
 
 41 
 
 20 
 
 329599 
 
 962 
 
 989860 
 
 46 
 
 339739 
 
 1008 
 
 600261 
 
 40 
 
 21 
 
 9.330176 
 
 961 
 
 9.989832 
 
 46 
 
 9.340344 
 
 1007 
 
 10.659056 
 
 39 
 
 22 
 
 330753 
 
 960 
 
 989804 
 
 46 
 
 340948 
 
 1006 
 
 (559052 
 
 38 
 
 23 
 
 331329 
 
 958 
 
 989777 
 
 46 
 
 341552 
 
 1004 
 
 658448 
 
 37 
 
 24 
 
 331903 
 
 957 
 
 989749 
 
 47 
 
 342155 
 
 1003 
 
 65? 845 
 
 36 
 
 25 
 
 332478 
 
 956 
 
 989721 
 
 47 
 
 342757 
 
 1002 
 
 657243 
 
 35 
 
 26 
 
 333051 
 
 954 
 
 989693 
 
 47 
 
 343358 
 
 1000 
 
 050042 
 
 34 
 
 27 
 
 333624 
 
 953 
 
 9896C5 
 
 47 
 
 342958 
 
 999 
 
 656042 
 
 33 
 
 28 
 
 334195 
 
 952 
 
 989637 
 
 47 
 
 344558 
 
 998 
 
 655442 
 
 32 
 
 29 
 
 334766 
 
 950 
 
 989609 
 
 47 
 
 345157 
 
 997 
 
 654843 
 
 31 
 
 30 
 
 335337 
 
 949 
 
 989582 
 
 47 
 
 345755 
 
 996 
 
 654245 
 
 30 
 
 31 
 
 9.335906 
 
 948 
 
 9.989553 
 
 47 
 
 9.340353 
 
 994 
 
 10.C53647 
 
 29 
 
 32 
 
 336475 
 
 946 
 
 989525 
 
 47 
 
 34(5949 
 
 993 
 
 653051 
 
 28 
 
 33 
 
 337043 
 
 945 
 
 9894SI7 
 
 47 
 
 347545 
 
 992 
 
 652455 
 
 27 
 
 34 
 
 337610 
 
 944 
 
 989469 
 
 47 
 
 348141 
 
 991 
 
 651859 
 
 26 
 
 35 
 
 338176 
 
 943 
 
 989441 
 
 47 
 
 348735 
 
 990 
 
 651265 
 
 25 
 
 36 
 
 338742 
 
 941 
 
 989413 
 
 47 
 
 349329 
 
 988 
 
 650671 
 
 24 
 
 37 
 
 339306 
 
 940 
 
 989384 
 
 47 
 
 349922 
 
 987 
 
 650078 
 
 23 
 
 38 
 
 339871 
 
 939 
 
 989356 
 
 47 
 
 350514 
 
 986 
 
 649486 
 
 22 
 
 39 
 
 340434 
 
 937 
 
 989328 
 
 47 
 
 351106 
 
 985 
 
 648894 
 
 21 
 
 40 
 
 340996 
 
 936 
 
 989300 
 
 47 
 
 351697 
 
 983 
 
 648303 
 
 20 
 
 41 
 
 9.341558 
 
 935 
 
 9.989271 
 
 47 
 
 9.352287 
 
 982 
 
 10.647713 
 
 19 
 
 42 
 
 342119 
 
 934 
 
 989243 
 
 47 
 
 352876 
 
 981 
 
 647124 
 
 18 
 
 43 
 
 342679 
 
 932 
 
 989214 
 
 47 
 
 353465 
 
 980 
 
 646535 
 
 17 
 
 44 
 
 343239 
 
 931 
 
 989186 
 
 47 
 
 354053 
 
 979 
 
 645947 
 
 16 
 
 45 
 
 343797 
 
 930 
 
 989157 
 
 47 
 
 354640 
 
 977 
 
 645360 
 
 15 
 
 46 
 
 344355 
 
 929 
 
 989128 
 
 48 
 
 355227 
 
 976 
 
 644773 
 
 14 
 
 47 
 
 344912 
 
 927 
 
 989100 
 
 48 
 
 355813 
 
 975 
 
 C44187 
 
 13 
 
 48 
 
 345469 
 
 926 
 
 989071 
 
 48 
 
 356398 
 
 974 
 
 643602 
 
 12 
 
 49 
 
 346024 
 
 925 
 
 989042 
 
 48 
 
 356982 
 
 973 
 
 643018 
 
 11 
 
 50 
 
 346579 
 
 924 
 
 989014 
 
 48 
 
 357566 
 
 971 
 
 642434 
 
 10 
 
 51 
 
 9.347134 
 
 922 
 
 9.988985 
 
 48 
 
 9.358149 
 
 970 
 
 10.641851 
 
 9 
 
 52 
 
 347687 
 
 921 
 
 988956 
 
 48 
 
 358731 
 
 969 
 
 641269 
 
 8 
 
 53 
 
 348240 
 
 Q-20 
 
 988927 
 
 48 
 
 359313 
 
 968 
 
 640C87 
 
 7 
 
 54 
 
 348792 
 
 919 
 
 988898 
 
 48 
 
 359893 
 
 967 
 
 640107 
 
 6 
 
 53 
 
 349343 
 
 917 
 
 988869 
 
 48 
 
 360474 
 
 966 
 
 639526 
 
 5 
 
 56 
 
 349893 
 
 916 
 
 988840 
 
 48 
 
 361053 
 
 965 
 
 638947 
 
 4 
 
 57 
 
 350443 
 
 915 
 
 988811 
 
 49 
 
 361632 
 
 963 
 
 638308 
 
 3 
 
 58 
 
 350992 
 
 914 
 
 988782 
 
 49 
 
 362210 
 
 962 
 
 637790 
 
 2 
 
 59 
 
 351540 
 
 913 
 
 988753 
 
 49 
 
 332787 
 
 961 
 
 637213 
 
 1 
 
 60 
 
 352088 
 
 911 
 
 988724 
 
 49 
 
 363364 
 
 960 
 
 636636 
 
 
 
 Cosine 
 
 | Sine | | Cotang. | | Tang. 
 
 M. 
 
 77 Degrees. 
 
SINES AND TANGENTS. (13 Degrees.) 
 
 M. Sine | D. | Cosine | D. Tang. D. Coking. 
 
 
 
 !.:C,JIIH,S 
 
 911 
 
 9.988724 
 
 49 
 
 J.:n53:i'i i 
 
 960 
 
 10.636(536 
 
 60 
 
 1 
 
 352835 
 
 910 
 
 988895 
 
 49 
 
 383940 
 
 959 
 
 036060 
 
 59 
 
 2 
 
 353181 
 
 909 
 
 988666 
 
 49 
 
 364515 
 
 9.18 
 
 635485 
 
 58 
 
 9 
 
 :i:>:!?-j(> 
 
 908 
 
 ' 988(536 
 
 49 
 
 365090 
 
 957 
 
 634910 
 
 57 
 
 4 
 
 35427J 
 
 go? 
 
 988807 
 
 49 
 
 3(55664 
 
 955 
 
 634336 
 
 56 
 
 5 
 
 354815 
 
 905 
 
 988578 
 
 49 
 
 366237 
 
 9->4 
 
 6337(53 
 
 55 
 
 
 
 355358 
 
 904 
 
 988548 
 
 49 
 
 366810 
 
 953 
 
 633190 
 
 54 
 
 7 
 
 355901 
 
 903 
 
 988519 
 
 49 
 
 367382 
 
 999 
 
 632(518 
 
 53 
 
 8 
 
 356443 
 
 902 
 
 938489 
 
 49 
 
 367953 
 
 951 
 
 632047 
 
 52 
 
 n 
 
 
 901 
 
 988460 
 
 49 
 
 368524 
 
 950 
 
 631476 
 
 51 
 
 10 
 
 357521 
 
 899 
 
 988430 
 
 49 
 
 3li!H)'Jl 
 
 '.(4!) 
 
 630906 
 
 50 
 
 u 
 
 9.3580r>4 
 
 898 
 
 9. 98*401 
 
 43 
 
 9.3696i53 
 
 948 
 
 10.630337 
 
 49 
 
 12 
 
 358803 
 
 897 
 
 988371 
 
 49 
 
 370232 
 
 946 
 
 621)708 
 
 48 
 
 13 
 
 359141 
 
 896 
 
 988342 
 
 49 
 
 370799 
 
 945 
 
 629201 
 
 47 
 
 14 
 
 359678 
 
 895 
 
 988312 
 
 50 
 
 371367 
 
 944 
 
 628633 
 
 46 
 
 15 
 
 360215 
 
 893 
 
 988282 
 
 50 
 
 371933 
 
 943 
 
 628067 
 
 45 
 
 16 
 
 360752 
 
 892 
 
 988252 
 
 50 
 
 372499 
 
 942 
 
 627501 
 
 44 
 
 17 
 
 361287 
 
 891 
 
 988223 
 
 50 
 
 373064 
 
 941 
 
 626936 
 
 43 
 
 18 
 
 361822 
 
 890 
 
 988193 
 
 50 
 
 373629 
 
 940 
 
 626371 
 
 42 
 
 19 
 
 3(12356 
 
 889 
 
 988163 
 
 50 
 
 374193 
 
 939 
 
 625807 
 
 41 
 
 20 
 
 362889 
 
 888 
 
 988133 
 
 50 
 
 374756 
 
 938 
 
 625244 
 
 40 
 
 21 
 
 9.3(53422 
 
 887 
 
 9.988103 
 
 50 
 
 9.375319 
 
 937 
 
 10.624681 
 
 39 
 
 22 
 
 303U54 
 
 885 
 
 988073 
 
 50 
 
 375881 
 
 935 
 
 624119 
 
 38 
 
 23 
 
 364485 
 
 884 
 
 988J43 
 
 50 
 
 37(5442 
 
 934 
 
 623558 
 
 37 
 
 24 
 
 3G5016 
 
 883 
 
 988013 
 
 50 
 
 377003 
 
 933 
 
 622997 
 
 36 
 
 25 
 
 365546 
 
 882 
 
 987983 
 
 50 
 
 377563 
 
 932 
 
 622437 
 
 35 
 
 26 
 
 31)0075 
 
 881 
 
 987953 
 
 50 
 
 378122 
 
 931 
 
 621878 
 
 34 
 
 27 
 
 3IMU504 
 
 880 
 
 987922 
 
 50 
 
 378681 
 
 930 
 
 621319 
 
 33 
 
 28 
 
 3157131 
 
 879 
 
 987892 
 
 50 
 
 379239 
 
 929 
 
 620761 
 
 32 
 
 29 
 
 367659 
 
 877 
 
 987862 
 
 50 
 
 379797 
 
 928 
 
 620203 
 
 31 
 
 30 
 
 368185 
 
 876 
 
 987832 
 
 51 
 
 380354 
 
 927 
 
 619646 
 
 30 
 
 31 
 
 9.3(58711 
 
 875 
 
 9.987801 
 
 51 
 
 9.380910 
 
 926 
 
 10.619090 
 
 29 
 
 32 
 
 3*59236 
 
 874 
 
 987771 
 
 51 
 
 381466 
 
 925 
 
 618534 
 
 28 
 
 -S3 
 
 3697(51 
 
 873 
 
 987740 
 
 51 
 
 382020 
 
 924 
 
 617980 
 
 27 
 
 34 
 
 370285 
 
 872 
 
 987710 
 
 51 
 
 38-2575 
 
 923 
 
 617425 
 
 26 
 
 35 
 
 370808 
 
 871 
 
 987679 
 
 51 
 
 383129 
 
 922 
 
 616871 
 
 25 
 
 36 
 
 371330 
 
 870 
 
 987649 
 
 51 
 
 383(582 
 
 921 
 
 616318 
 
 24 
 
 37 
 
 371852 
 
 869 
 
 987618 
 
 51 
 
 384234 
 
 920 
 
 615766 
 
 23 
 
 38 
 
 372373 
 
 867 
 
 ' 987588 
 
 51 
 
 384786 
 
 919 
 
 615214 
 
 22 
 
 39 
 
 372894 
 
 866 
 
 987557 
 
 51 
 
 385337 
 
 918 
 
 614663 
 
 21 
 
 40 
 
 373414 
 
 865 
 
 987526 
 
 51 
 
 385888 
 
 917 
 
 614112 
 
 20 
 
 41 
 
 9.373933 
 
 864 
 
 9.987496 
 
 51 
 
 9.386438 
 
 915 
 
 10.613562 
 
 19 
 
 42 
 
 374452 
 
 863 
 
 987465 
 
 51 
 
 38(5987 
 
 914 
 
 613013 
 
 18 
 
 43 
 
 374970 
 
 862 
 
 987434 
 
 51 
 
 387536 
 
 913 
 
 612464 
 
 17 
 
 44 
 
 375487 
 
 861 
 
 987403 
 
 52 
 
 388084 
 
 912 
 
 611916 
 
 16 
 
 45 
 
 37(iO()3 
 
 860 
 
 987372 
 
 52 
 
 388631 
 
 911 
 
 611369 
 
 15 
 
 46 
 
 376519 
 
 859 
 
 9^7341 
 
 52 
 
 389178 
 
 910 
 
 610822 
 
 14 
 
 47 
 
 377035 
 
 858 
 
 987310 
 
 52 
 
 389724 
 
 909 
 
 610276 
 
 13 
 
 48 
 
 377549 
 
 857 
 
 987279 
 
 52 
 
 390270 
 
 908 
 
 609730 
 
 12 
 
 49 
 
 378063 
 
 856 
 
 987248 
 
 52 
 
 390815 
 
 907 
 
 609185 
 
 11 
 
 50 
 
 378577 
 
 854 
 
 987217 
 
 52 
 
 391360 
 
 906 
 
 608640 
 
 10 
 
 51 
 
 9.379089 
 
 853 
 
 9.987186 
 
 52 
 
 9.391903 
 
 905 
 
 10.608097 
 
 9 
 
 52 
 
 379601 
 
 852 
 
 987155 
 
 52 
 
 392447 
 
 9i)4 
 
 (Tl 17553 
 
 8 
 
 53 
 
 380113 
 
 851 
 
 987124 
 
 52 
 
 392989 
 
 903 
 
 607011 
 
 7 
 
 54 
 
 380624 
 
 850 
 
 9&7092 
 
 52 
 
 :w:r>:u 
 
 902 
 
 6015469 
 
 6 
 
 55 
 
 381134 
 
 849 
 
 9870(51 
 
 52 
 
 394073 
 
 901 
 
 605U27 
 
 5 
 
 5G 
 
 38*643 
 
 848 
 
 987030 
 
 52 
 
 394614 
 
 900 
 
 605386 
 
 4 
 
 57 
 
 382152 
 
 847 
 
 9855998 
 
 52 
 
 395154 
 
 899 
 
 604846 
 
 3 
 
 58 
 
 3H-J661 
 
 846 
 
 986967 
 
 52 
 
 3!>."i''.94 
 
 898 
 
 604306 
 
 2 
 
 59 
 
 383168 
 
 845 
 
 986936 
 
 52 
 
 396233 
 
 897 
 
 6037(57 
 
 1 
 
 60 
 
 383675 
 
 844 
 
 98^904 
 
 52 
 
 396771 
 
 896 
 
 603229 
 
 
 
 | Cosiiu- | | Sine | | Cotang. | | Tang. M. 
 
 76 Degrees. 
 
(14 Degrees.) A TABLE OF LOGARITHMIC 
 
 M. Sine D j Cosine D. | Tang. | D. 
 
 Cotang. | 
 
 
 
 9.383675 
 
 844 
 
 9.986904 
 
 52 
 
 9.396771 
 
 896 
 
 10.603229 60 
 
 1 
 
 384182 
 
 843 
 
 986873 
 
 53 
 
 397309 
 
 896 
 
 602691 .TO 
 
 2 
 
 384687 
 
 842 
 
 986841 
 
 53 
 
 397846 
 
 895 
 
 602154 
 
 58 
 
 3 
 
 385192 
 
 841 
 
 980809 
 
 53 
 
 398383 
 
 894 
 
 G01617 
 
 57 
 
 4 
 
 385697 
 
 840 
 
 986778 
 
 53 
 
 3i)8<m) 
 
 893 
 
 601081 
 
 56 
 
 5 
 
 38(5201 
 
 839 
 
 986746 
 
 53 
 
 399455 
 
 892 
 
 600545 
 
 55 
 
 6 
 
 386704 
 
 838 
 
 986714 
 
 53 
 
 399990 
 
 891 
 
 600010 
 
 54 
 
 7 
 
 387207 
 
 837 
 
 986683 
 
 53 
 
 400534 
 
 890 
 
 599476 
 
 53 
 
 8 
 
 387709 
 
 836 
 
 986651 
 
 53 
 
 401058 
 
 889 
 
 598942 
 
 52 
 
 9 
 
 388210 
 
 835 
 
 986619 
 
 53 
 
 401591 
 
 888 
 
 598409 
 
 51 
 
 10 
 
 388711 
 
 834 
 
 986587 
 
 53 
 
 402124 
 
 887 
 
 887878 
 
 50 
 
 11 
 
 9.389211 
 
 833 
 
 9.986555 
 
 53 
 
 9.402S55 
 
 886 
 
 10.597344 
 
 49 
 
 12 
 
 389711 
 
 832 
 
 986C--23 
 
 53 
 
 403187 
 
 885 
 
 596813 
 
 48 
 
 13 
 
 390210 
 
 831 
 
 986491 
 
 53 
 
 403718 
 
 884 
 
 596282 
 
 47 
 
 14 
 
 390708 
 
 830 
 
 986459 
 
 53 
 
 404249 
 
 883 
 
 595751 
 
 46 
 
 15 
 
 SSI206 
 
 828 
 
 986427 
 
 53 
 
 404778 
 
 882 
 
 595222 
 
 45 
 
 J*^ 
 
 391703 
 
 827 
 
 986395 
 
 53 
 
 405308 
 
 881 
 
 594692 
 
 44 
 
 17 
 
 392199 
 
 826 
 
 986363 
 
 54 
 
 405836 
 
 880 
 
 594164 
 
 43 
 
 18 
 
 392695 
 
 825 
 
 986331 
 
 54 
 
 406364 
 
 879 
 
 593636 
 
 42 
 
 19 
 
 393191 
 
 824 
 
 986299 
 
 54 
 
 406892 
 
 878 
 
 593108 
 
 41 
 
 20 
 
 393G85 
 
 823 
 
 986266 
 
 54 
 
 407419 
 
 877 
 
 592581 
 
 40 
 
 21 
 
 9.394179 
 
 822 
 
 9.986234 
 
 54 
 
 9.407945 
 
 876 
 
 10.592055 
 
 39 
 
 22 
 
 394673 
 
 821 
 
 986202 
 
 54 
 
 408471 
 
 875 
 
 591529 
 
 38 
 
 23 
 
 395166 
 
 820 
 
 986169 
 
 54 
 
 408997 
 
 874 
 
 591003 
 
 37 
 
 24 
 
 395658 
 
 819 
 
 986137 
 
 54 
 
 409521 
 
 874 
 
 590479 
 
 36 
 
 25 
 
 396150 
 
 818 
 
 986104 
 
 54 
 
 410045 
 
 873 
 
 589955 
 
 35 
 
 26 
 
 396641 
 
 817 
 
 986072 
 
 54 
 
 4105139 
 
 872 
 
 589431 
 
 54 
 
 27 
 
 397132 
 
 817 
 
 986039 
 
 54 
 
 411092 
 
 871 
 
 588908 
 
 33 
 
 28 
 
 397621 
 
 816 
 
 986007 
 
 54 
 
 411615 
 
 870 
 
 588385 
 
 32 
 
 29 
 
 398111 
 
 815 
 
 985974 
 
 54 
 
 412] 37 
 
 869 
 
 587863 
 
 31 
 
 30 
 
 398600 
 
 814 
 
 985942 
 
 54 
 
 412658 
 
 808 
 
 587342 
 
 30 
 
 31 
 
 9.399088 
 
 813 
 
 9.985909 
 
 55 
 
 9.413179 
 
 867 
 
 10.586821 
 
 29 
 
 32 
 
 399575 
 
 812 
 
 985876 
 
 55 
 
 413699 
 
 866 
 
 586301 
 
 28 
 
 33 
 
 400062 
 
 811 
 
 985843 
 
 55 
 
 414219 
 
 865 
 
 585781 
 
 27 
 
 34 
 
 400549 i 810 
 
 985811 
 
 55 
 
 414738 
 
 864 
 
 585262 
 
 2<f 
 
 35 
 
 401035 
 
 809 
 
 985778 
 
 55 
 
 415257 
 
 864 
 
 584743 
 
 25 
 
 36 
 
 401520 
 
 808 
 
 985745 
 
 55 
 
 415775 
 
 863 
 
 584225 
 
 24 
 
 37 
 
 402005 
 
 807 
 
 985712 
 
 55 
 
 416293 
 
 862 
 
 583707 
 
 23 
 
 38 
 
 402-189 
 
 806 
 
 985679 
 
 55 
 
 416810 
 
 861 
 
 583190 
 
 22 
 
 39 
 
 402972 
 
 805 
 
 985646 
 
 55 
 
 417326 
 
 860 
 
 582674 
 
 21 
 
 40 
 
 403455 
 
 804 
 
 985613 
 
 55 
 
 417842 
 
 859 
 
 582158 
 
 20 
 
 41 
 
 9.403938 
 
 803 
 
 9.985580 
 
 55 
 
 9.418358 
 
 858 
 
 10.581642 
 
 19 
 
 42 
 
 404420 
 
 802 
 
 985547 
 
 55 
 
 418873 
 
 857 
 
 581127 
 
 18 
 
 43 
 
 404901 
 
 801 
 
 985514 
 
 55 
 
 419387 
 
 856 
 
 580613 
 
 17 
 
 44 
 
 405382 
 
 800 
 
 985480 
 
 55 
 
 419901 
 
 855 
 
 580099 
 
 16 
 
 45 
 
 405862 
 
 799 
 
 965447 
 
 55 
 
 420415 
 
 855 
 
 579585 
 
 15 
 
 46 
 
 406341 
 
 798 
 
 985414 
 
 56 
 
 420927 
 
 854 
 
 579073 
 
 14 
 
 47 
 
 406820 
 
 797 
 
 985380 
 
 56 
 
 421440 
 
 853 
 
 578560 
 
 13 
 
 48 
 
 407299 
 
 796 
 
 985347 
 
 56 
 
 421952 
 
 852 
 
 578048 
 
 12 
 
 49 
 
 407777 
 
 795 
 
 985314 
 
 56 
 
 422463 
 
 851 
 
 577537 
 
 11 
 
 50 
 
 408254 
 
 794 
 
 985280 
 
 56 
 
 422974 
 
 850 
 
 577026 
 
 10 
 
 51 
 
 9.408731 
 
 794 
 
 9.985247 
 
 56 
 
 9.423484 
 
 849 
 
 10.576516 
 
 9 
 
 52 
 
 409207 
 
 793 
 
 985213 
 
 56 
 
 423993 
 
 848 
 
 576007 
 
 R 
 
 53 
 
 409682 
 
 792 
 
 985180 
 
 56 
 
 424503 
 
 848 
 
 575497 
 
 7 
 
 54 
 
 410157 
 
 791 
 
 985146 
 
 56 
 
 425011 
 
 847 
 
 574989 
 
 6 
 
 55 
 
 410632 
 
 790 
 
 985113 
 
 56 
 
 425519 
 
 846 
 
 574481 
 
 5 
 
 56 
 
 411106 
 
 789 
 
 985079 
 
 56 
 
 426027 
 
 845 
 
 573973 
 
 4 
 
 57 
 
 411579 
 
 788 
 
 985045 
 
 56 
 
 426534 
 
 844 
 
 573466 
 
 3 
 
 58 
 
 412052 
 
 787 
 
 985011 
 
 56 
 
 427041 
 
 843 
 
 572959 
 
 2 
 
 59 
 
 412524 
 
 786 
 
 984978 
 
 56 
 
 427547 
 
 843 
 
 572453 
 
 1 
 
 60 
 
 412996 
 
 785 
 
 984944 
 
 56 
 
 428052 
 
 842 
 
 571948 
 
 
 
 Cosine | Sine | | Cotang. | Taiig. M. 
 
 75 Degrees. 
 
SINES AND TANGENTS. (15 Degrees.) 
 
 33 
 
 M. | Sine | D. | Cosine | D. | Tang. | D. | Cotang. | 1 
 
 
 
 <j.4i--".nm 
 
 785 
 
 
 57 
 
 9.438052 
 
 842 
 
 10.571948 
 
 60 J 
 
 1 
 
 4134157 
 
 784 
 
 984910 
 
 57 
 
 428557 
 
 841 
 
 571443 
 
 59 
 
 2 
 
 413938 
 
 783 
 
 934B70 
 
 57 
 
 42i)062 
 
 840 
 
 570938 
 
 58 
 
 3 
 
 414408 
 
 7,^3 
 
 984842 
 
 57 
 
 429566 
 
 839 
 
 570434 
 
 57 
 
 4 
 
 414878 
 
 782 
 
 984808 
 
 57 
 
 430070 
 
 838 
 
 569930 
 
 50 
 
 5 
 
 415347 
 
 781 
 
 984774 
 
 57 
 
 430573 
 
 838 
 
 509427 
 
 55 
 
 6 
 
 41W15 
 
 780 
 
 984740 
 
 57 
 
 431075 
 
 837 
 
 508925 
 
 51 
 
 7 
 
 416383 
 
 77'.) 
 
 984706 
 
 57 
 
 431577 
 
 836 
 
 508423 
 
 53 
 
 8 
 
 41675] 
 
 778 
 
 984673 
 
 57 
 
 433079 
 
 835 
 
 567921 
 
 52 
 
 9 
 
 417217 
 
 777 
 
 984637 
 
 57 
 
 432580 
 
 834 
 
 567420 
 
 51 
 
 10 
 
 417684 
 
 776 
 
 984603 
 
 57 
 
 433080 
 
 833 
 
 566920 
 
 50 
 
 11 
 
 9.418150 
 
 775 
 
 9-984569 
 
 r>7 
 
 9.433580 
 
 832 
 
 10.50(5420 
 
 49 
 
 12 
 
 418G15 
 
 774 
 
 984535 
 
 57 
 
 434080 
 
 832 
 
 565920 
 
 48 
 
 13 
 
 419079 
 
 773 
 
 984500 
 
 57 
 
 434579 
 
 831 
 
 565421 
 
 47 
 
 14 
 
 419544 
 
 773 
 
 984406 
 
 57 
 
 435078 
 
 830 
 
 564922 
 
 46 
 
 15 
 
 4-201)07 
 
 772 
 
 984433 
 
 58 
 
 435576 
 
 829 
 
 5(54424 
 
 45 
 
 16 
 
 420470 
 
 771 
 
 984397 
 
 58 
 
 ) 36073 
 
 828 
 
 563927 
 
 44 
 
 17 
 
 420933 
 
 770 
 
 984363 
 
 56 
 
 436570 
 
 828 
 
 563430 
 
 43 
 
 18 
 
 421395 
 
 769 
 
 984328 
 
 58 
 
 437067 
 
 827 
 
 562933 
 
 42 
 
 19 
 
 421857 
 
 768 
 
 984294 
 
 58 
 
 437563 
 
 826 
 
 562437 
 
 41 
 
 2U 
 
 422318 
 
 767 
 
 984259 
 
 58 
 
 438059 
 
 825 
 
 561941 
 
 40 
 
 21 
 
 9.422778 
 
 767 
 
 9.984224 
 
 58 
 
 9.438554 
 
 824 
 
 10.561446 
 
 39 
 
 2-2 
 
 423238 
 
 768 
 
 984190 
 
 58 
 
 439048 
 
 823 
 
 560952 
 
 38 
 
 23 
 
 423097 
 
 765 
 
 984155 
 
 58 
 
 439543 
 
 823 
 
 560457 
 
 37 
 
 24 
 
 424156 
 
 764 
 
 984120 
 
 58 
 
 440036 
 
 822 
 
 559964 
 
 36 
 
 25 
 
 424615 
 
 703 
 
 984085 
 
 58 
 
 440529 
 
 821 
 
 559471 
 
 35 
 
 20 
 
 425073 
 
 762 
 
 984050 
 
 58 
 
 411022 
 
 820 
 
 558978 
 
 34 
 
 27 
 
 425530 
 
 701 
 
 9840J5 
 
 53 
 
 441514 
 
 819 
 
 558486 
 
 33 
 
 J- 
 
 425987 
 
 7CO 
 
 
 58 
 
 442006 
 
 819 
 
 557994 
 
 :w 
 
 29 
 
 4*26443 
 
 780 
 
 983946 
 
 58 
 
 442497 
 
 818 
 
 557503 
 
 31 
 
 30 
 
 426899 
 
 759 
 
 983911 
 
 58 
 
 442988 
 
 817 
 
 557012 
 
 30 
 
 31 
 
 9.427354 
 
 758 
 
 9.983875 
 
 58 
 
 9.443479 
 
 816 
 
 10.550521 
 
 29 
 
 :w 
 
 427809 
 
 757 
 
 983840 
 
 59 
 
 443968 
 
 816 
 
 556032 
 
 28 
 
 33 
 
 428963 
 
 756 
 
 983805 
 
 59 
 
 444458 
 
 815 
 
 555542 
 
 27 
 
 34 
 
 428717 
 
 755 
 
 983770 
 
 59 
 
 444947 
 
 814 
 
 555053 
 
 26 
 
 35 
 
 429170 
 
 754 
 
 983735 
 
 59 
 
 445435 
 
 813 
 
 554565 
 
 25 
 
 30 
 
 429623 
 
 753 
 
 983700 
 
 59 
 
 445923 
 
 812 
 
 554077 
 
 24 
 
 37 
 
 430075 
 
 752 
 
 983664 
 
 59 
 
 446411 
 
 812 
 
 553589 
 
 23 
 
 38 
 
 430527 
 
 753 
 
 983629 
 
 59 
 
 446898 
 
 811 
 
 553102 
 
 22 
 
 39 
 
 430978 
 
 7.51 
 
 983594 
 
 59 
 
 447384 
 
 810 
 
 552616 
 
 21 
 
 40 
 
 431429 
 
 750 
 
 983558 
 
 59 
 
 447870 
 
 809 
 
 552130 
 
 20 
 
 41 
 
 9.431879 
 
 749 
 
 9.983523 
 
 59 
 
 9.448356 
 
 809 
 
 10.551(544 
 
 19 
 
 42 
 
 432329 
 
 749 
 
 983487 
 
 59 
 
 448841 
 
 808 
 
 551159 
 
 18 
 
 43 
 
 432778 
 
 748 
 
 983452 
 
 59 
 
 449326 
 
 807 
 
 550674 
 
 17 
 
 44 
 
 433226 
 
 747 
 
 983416 
 
 59 
 
 449810 
 
 806 
 
 ' 550190 
 
 16 
 
 45 
 
 433675 
 
 746 
 
 983381 
 
 59 
 
 450294 
 
 806 
 
 549706 
 
 15 
 
 46 
 
 434122 
 
 745 
 
 983345 
 
 59 
 
 450777 
 
 805 
 
 549223 
 
 14 
 
 47 
 
 434569 
 
 744 
 
 983309 
 
 59 
 
 451260 
 
 804 
 
 548740 
 
 13 
 
 48 
 
 435016 
 
 744 
 
 9&3273 
 
 60 
 
 451743 
 
 803 
 
 548257 
 
 12 
 
 49 
 
 435462 
 
 743 
 
 983238 
 
 60 
 
 452225 
 
 802 
 
 547775 
 
 11 
 
 50 
 
 435908 
 
 742 
 
 983202 
 
 60 
 
 452706 
 
 802 
 
 547294 
 
 10 
 
 51 
 
 9-436353 
 
 741 
 
 9.983166 
 
 60 
 
 9.453187 
 
 801 
 
 10.546813 
 
 9 
 
 52 
 
 436798 
 
 740 
 
 983130 
 
 60 
 
 453668 
 
 800 
 
 546332 
 
 8 
 
 53 
 
 437242 
 
 740 
 
 983094 
 
 60 
 
 454148 
 
 799 
 
 545852 
 
 7 
 
 54 
 
 437686 
 
 739 
 
 983058 
 
 60 
 
 454628 
 
 799 
 
 5-15372 
 
 6 
 
 55 
 
 438129 
 
 738 
 
 983022 
 
 60 
 
 455107 
 
 798 
 
 544893 
 
 5 
 
 56 
 
 438572 
 
 737 
 
 982988 
 
 60 
 
 455586 
 
 797 
 
 544414 
 
 4 
 
 57 
 
 439014 
 
 730 
 
 982950 
 
 60 
 
 456064 
 
 796 
 
 543936 
 
 3 
 
 58 
 
 439456 
 
 738 
 
 982914 
 
 60 
 
 456542 
 
 796 
 
 543458 
 
 2 
 
 59 
 
 439897 
 
 735 
 
 982878 
 
 60 
 
 457019 
 
 795 
 
 542981 
 
 i ! 
 
 00 
 
 440338 
 
 734 
 
 982842 
 
 60 
 
 457496 
 
 794 
 
 5-12504 
 
 
 
 | Cosine 
 
 Sine I | Cotang. | | Tang. | M. 
 
 74 Degrees. 
 
 2* 
 
34 
 
 (16 Degrees.) A TABLE or LOGARITHMIC 
 
 M ! Sine D. | Cosine | D. | Tang. | D Cotang. 
 
 
 
 9.440338 
 
 734 
 
 9.9*-'.-4-2 
 
 1)0 
 
 9.4. r >T4:)li 
 
 794 
 
 10.54-2.104 
 
 60 
 
 1 
 
 440778 
 
 733 
 
 982805 
 
 60 
 
 457973 
 
 793 
 
 542027 
 
 59 
 
 2 
 
 441218 
 
 732 
 
 982769 
 
 61 
 
 458449 
 
 793 
 
 541551 
 
 58 
 
 3 
 
 441658 
 
 731 
 
 982733 
 
 61 
 
 458925 
 
 792 
 
 541075 
 
 57 
 
 4 
 
 442096 
 
 731 
 
 98-2690 
 
 61 
 
 4594:)0 
 
 791 
 
 540600 
 
 56 
 
 5 
 
 442535 
 
 730 
 
 98-2600 
 
 61 
 
 459875 
 
 790 
 
 540125 
 
 55 
 
 6 
 
 442973 
 
 789 
 
 982624 
 
 61 
 
 400349 
 
 790 
 
 539051 
 
 54 
 
 7 
 
 44:5410 
 
 728 
 
 982587 
 
 61 
 
 460823 
 
 789 
 
 539177 
 
 53 
 
 8 
 
 443847 
 
 727 
 
 982551 
 
 61 
 
 461297 
 
 788 
 
 538703 
 
 52 
 
 9 
 
 444284 
 
 727 
 
 <J.~2.-)14 
 
 61 
 
 461770 
 
 788 
 
 538230 
 
 51 
 
 10 
 
 444720 
 
 726 
 
 982477 
 
 61 
 
 462242 
 
 787 
 
 537758 
 
 50 
 
 11 
 
 9.445155 
 
 725 
 
 9.98-2441 
 
 61 
 
 9.402714 
 
 786 
 
 10.537286 
 
 49 
 
 12 
 
 445590 
 
 724 
 
 982404 
 
 61 
 
 403186 
 
 785 
 
 53(5814 
 
 48 
 
 13 
 
 446025 
 
 723 
 
 982367 
 
 61 
 
 463658 
 
 785 
 
 530342 
 
 47 
 
 14 
 
 446459 
 
 723 
 
 982331 
 
 61 
 
 464129 
 
 784 
 
 535871 
 
 46 
 
 15 
 
 446893 
 
 722 
 
 982294 
 
 61 
 
 464599 
 
 783 
 
 535401 
 
 45 
 
 16 
 
 4473-26 
 
 721 
 
 982-257 
 
 61 
 
 465069 
 
 783 
 
 534931 
 
 44 
 
 : 17 
 
 447759 
 
 720 
 
 982220 
 
 62 
 
 465539 
 
 782 
 
 534461 
 
 43 
 
 18 
 
 448191 
 
 720 
 
 982183 
 
 62 
 
 466008 
 
 781 
 
 533992 
 
 42 
 
 19 
 
 448623 
 
 719 
 
 982146 
 
 62 
 
 4G6476 
 
 780 
 
 533524 
 
 41 
 
 20 
 
 449054 
 
 7J8 
 
 98-2109 
 
 62 
 
 466945 
 
 780 
 
 533055 
 
 40 
 
 21 
 
 9.449485 
 
 717 
 
 9.982072 
 
 62 
 
 9.467413 
 
 779 
 
 10.532587 
 
 39 
 
 "ft 
 
 449915 
 
 716 
 
 982035 
 
 62 
 
 467880 
 
 778 
 
 532120 
 
 38 
 
 23 
 
 450345 
 
 716 
 
 981998 
 
 62 
 
 468347 
 
 778 
 
 531653 
 
 37 
 
 24 
 
 450775 
 
 715 
 
 981961 
 
 62 
 
 468814 
 
 777 
 
 531186 
 
 36 
 
 25 
 
 451204 
 
 714 
 
 981924 
 
 62 
 
 469280 
 
 776 
 
 530720 
 
 35 
 
 20 
 
 451632 
 
 713 
 
 981885 
 
 62 
 
 40974G 
 
 775 
 
 530254 
 
 34 
 
 27 
 
 452060 
 
 713 
 
 981849 
 
 62 
 
 4/u211 
 
 775 
 
 529789 
 
 33 
 
 28 
 
 452488 
 
 712 
 
 981812 
 
 62 
 
 470G76 
 
 V74 
 
 529324 
 
 32 
 
 29 
 
 452915 
 
 711 
 
 981774 
 
 62 
 
 471141 
 
 773 
 
 6-23859 31 
 
 30 
 
 453342 
 
 710 
 
 981737 
 
 62 
 
 471605 
 
 773 
 
 528395 
 
 iwi , 
 
 31 
 
 9.453768 
 
 710 
 
 9. 98 1099 
 
 63 
 
 9.472068 
 
 772 
 
 10.527932 
 
 29 
 
 32 
 
 454194 
 
 709 
 
 - 981662 
 
 63 
 
 47-25.32 
 
 771 
 
 527468 
 
 28 
 
 33 
 
 454619 
 
 708 
 
 981625 
 
 63 
 
 472995 
 
 771 
 
 52.005 
 
 27 
 
 34 
 
 455044 
 
 707 
 
 981587 
 
 63 
 
 473457 
 
 770 
 
 526543 
 
 26 
 
 35 
 
 455469 
 
 707 
 
 981549 
 
 63 
 
 473919 
 
 769 
 
 524)081 
 
 25 
 
 36 
 
 455893 
 
 706 
 
 981512 
 
 63 
 
 474381 
 
 769 
 
 525619 
 
 24 
 
 37 
 
 4563] 6 
 
 705 
 
 981474 
 
 63 
 
 474842 
 
 768 
 
 525158 
 
 23 
 
 38 
 
 456739 
 
 704 
 
 981436 
 
 63 
 
 475303 
 
 767 
 
 524697 
 
 22 
 
 39 
 
 457162 
 
 704 
 
 981399 
 
 63 
 
 475763 
 
 707 
 
 524237 
 
 21 
 
 40 
 
 457584 
 
 703 
 
 981361 
 
 63 
 
 476223 
 
 766 
 
 523777 
 
 20 
 
 41 
 
 9.458006 
 
 702 
 
 9.981323 
 
 63 
 
 9.476683 
 
 765 
 
 10.523317 
 
 19 
 
 42 
 
 458427 
 
 701 
 
 981285 
 
 63 
 
 477142 
 
 765 
 
 522858 
 
 18 
 
 43 
 
 458848 
 
 701 
 
 981247 
 
 63 
 
 477601 
 
 764 
 
 522399 
 
 17 
 
 44 
 
 4592<58 
 
 700 
 
 981209 
 
 63 
 
 478059 
 
 763 
 
 521941 
 
 16 
 
 1 4."> 
 
 459688 
 
 699 
 
 981171 
 
 63 
 
 478517 
 
 763 
 
 521483 
 
 15 
 
 46 
 
 460108 
 
 698 
 
 981133 
 
 64 
 
 478975 
 
 762 
 
 521025 
 
 14 
 
 47 
 
 460527 
 
 698 
 
 981095 
 
 64 
 
 479432 
 
 761 
 
 520568 
 
 13 
 
 48 
 
 46094K 
 
 697 
 
 981057 
 
 64 
 
 479889 
 
 701 
 
 520111 
 
 12 
 
 49 
 
 461364 
 
 698 
 
 981019 
 
 64 
 
 480345 
 
 760 
 
 519655 
 
 11 
 
 50 
 
 461782 
 
 695 
 
 980981 
 
 64 
 
 480801 
 
 759 
 
 519199 
 
 10 
 
 51 
 
 9.462199 
 
 695 
 
 9.980942 
 
 64 
 
 9.481257 
 
 759 
 
 10.513743 
 
 9 
 
 52 
 
 462616 
 
 694 
 
 980904 
 
 64 
 
 481712 
 
 758 
 
 518288 
 
 8 
 
 53 
 
 463032 
 
 693 
 
 980866 
 
 64 
 
 482167 
 
 757 
 
 517833 
 
 7 
 
 54 
 
 463448 
 
 693 
 
 980827 
 
 64 
 
 482621 
 
 7f>7 
 
 517379 
 
 6 
 
 55 
 
 463864 
 
 692 
 
 980789 
 
 64 
 
 483075 
 
 756 
 
 516925 
 
 5 
 
 56 
 
 464279 
 
 691 
 
 980750 
 
 64 
 
 483529 
 
 755 
 
 510471 
 
 4 
 
 57 
 
 464694 
 
 690 
 
 980712 
 
 64 
 
 483982 
 
 755 
 
 516018 
 
 3 
 
 58 
 
 465108 
 
 690 
 
 980673 
 
 64 
 
 484435 
 
 754 
 
 515565 
 
 2 
 
 59 
 
 465522 
 
 689 
 
 980635 
 
 64 
 
 484887 
 
 753 
 
 515113 
 
 1 
 
 60 
 
 465935 
 
 688 
 
 980596 
 
 64 
 
 485339 
 
 753 
 
 514661 
 
 
 
 | Cosine | | Sine | | Cotang. | Tang. | M. 1 
 
 73 Degrees. 
 
SINES AND TANGENTS. (17 Degrees.) 
 
 35 
 
 i M. | Sine D. | Cosine D. Tang. D. | Cotanj?. ' 
 
 
 
 9.4I>5:I35 
 
 BBS 
 
 9.980596 
 
 64 
 
 9.485339 
 
 V>5 
 
 10.5l4(i()l 
 
 60 
 
 1 
 
 460318 
 
 688 
 
 080558 
 
 64 
 
 485791 
 
 752 
 
 514209 
 
 59 
 
 o 
 
 466781 
 
 687 
 
 930519 
 
 65 
 
 480242 
 
 751 
 
 513758 
 
 58 
 
 3 
 
 167173 
 
 686 
 
 980480 
 
 65 
 
 4Si)fi!)3 
 
 751 
 
 513307 
 
 57 
 
 4 
 
 467585 
 
 685 
 
 9804452 
 
 65 
 
 487143 
 
 750 
 
 512857 
 
 56 
 
 5 
 
 487996 
 
 685 
 
 060403 
 
 65 
 
 487593 
 
 749 
 
 512407 
 
 55 
 
 6 
 
 468407 
 
 684 
 
 9803(54 
 
 65 
 
 488043 
 
 749 
 
 511957 
 
 54 
 
 1 
 
 468817 
 
 883 
 
 080325 
 
 65 
 
 488492 
 
 748 
 
 511508 
 
 53 
 
 B 
 
 469-227 
 
 683 
 
 989286 
 
 65 
 
 488941 
 
 747" 
 
 511059 
 
 52 
 
 9 
 
 469U37 
 
 682 
 
 980947 
 
 65 
 
 489390 
 
 747 
 
 510510 
 
 51 
 
 10 
 
 470046 
 
 681 
 
 980208 
 
 65 
 
 489838 
 
 746 
 
 510162 
 
 50 
 
 11 
 
 9.470455 
 
 680 
 
 9.980169 
 
 65 
 
 9.490236 
 
 746 
 
 10.509714 
 
 49 
 
 12 
 
 470863 
 
 689 
 
 980130 
 
 63 
 
 490733 
 
 745 
 
 509267 
 
 48 
 
 13 
 
 471271 
 
 679 
 
 980091 
 
 65 
 
 491180 
 
 744 
 
 508820 
 
 47 
 
 14 
 
 471679 
 
 678 
 
 980052 
 
 65 
 
 491627 
 
 744 
 
 508373 
 
 46 
 
 15 
 
 472086 
 
 678 
 
 930012 
 
 65 
 
 492073 
 
 743 
 
 507927 
 
 45 
 
 16 
 
 472492 
 
 677 
 
 979973 
 
 65 
 
 492519 
 
 743 
 
 507481 
 
 44 
 
 17 
 
 47-2898 
 
 676 
 
 979934 
 
 66 
 
 492965 
 
 742 
 
 507035 
 
 43 
 
 18 
 
 473304 
 
 676 
 
 979895 
 
 66 
 
 493410 
 
 741 
 
 508590 
 
 42 
 
 19 
 
 473710 
 
 675 
 
 979855 
 
 66 
 
 493854 
 
 740 
 
 506146 
 
 41 
 
 20 
 
 474115 
 
 674 
 
 979816 
 
 66 
 
 494299 
 
 740 
 
 505701 
 
 40 
 
 21 
 
 9-474519 
 
 674 
 
 9.979776 
 
 66 
 
 9.494743 
 
 740 
 
 10.505257 
 
 39 
 
 33 
 
 474923 
 
 673 
 
 979737 
 
 66 
 
 495186 
 
 739 
 
 504814 
 
 38 
 
 23 
 
 475327 
 
 672 
 
 979G97 
 
 66 
 
 495630 
 
 738 
 
 504370 
 
 37 
 
 24 
 
 475730 
 
 672 
 
 979558 
 
 60 
 
 496073 
 
 737 
 
 503927 
 
 36 
 
 25 
 
 476133 
 
 671 
 
 979618 
 
 m 
 
 496515 
 
 737 
 
 503485 
 
 35 
 
 26 
 
 476536 
 
 670 
 
 979579 
 
 66 
 
 496957 
 
 736 
 
 503043 
 
 34 
 
 27 
 
 476938 
 
 669 
 
 979539 
 
 66 
 
 497399 
 
 736 
 
 502G01 
 
 33 
 
 28 
 
 477340 
 
 669 
 
 979499 
 
 63 
 
 497841 
 
 735 
 
 502159 
 
 32 
 
 29 
 
 477741 
 
 668 
 
 979459 
 
 66 
 
 498282 
 
 T34 
 
 501718 
 
 31 
 
 30 
 
 478142 
 
 667 
 
 979420 
 
 t>6 
 
 498722 
 
 734 
 
 501278 
 
 30 
 
 31 
 
 9.478542 
 
 667 
 
 9.979380 
 
 66 
 
 9.499163 
 
 733 
 
 10.500837 
 
 29 
 
 3-2 
 
 478942 
 
 666 
 
 979340 
 
 66 
 
 499603 
 
 733 
 
 500397 
 
 28 
 
 33 
 
 479342 
 
 665 
 
 979300 
 
 67 
 
 500042 
 
 732 
 
 499958 
 
 27 
 
 34 
 
 479741 
 
 665 
 
 9792(50 
 
 67 
 
 500481 
 
 731 
 
 499519 
 
 26 
 
 35 
 
 480140 
 
 664 
 
 979220 
 
 67 
 
 500920 
 
 731 
 
 499080 
 
 25 
 
 36 
 
 480539 
 
 663 
 
 979180 
 
 67 
 
 501359 
 
 730 
 
 498;i41 
 
 24 
 
 37 
 
 480437 
 
 663 
 
 979140 
 
 67 
 
 501797 
 
 730 
 
 498203 
 
 23 
 
 38 
 
 481334 
 
 662 
 
 979100 
 
 67 
 
 502-235 
 
 729 
 
 497765 
 
 22 
 
 39 
 
 481731 
 
 661 
 
 979059 
 
 67 
 
 50-2672 
 
 728 
 
 497328 
 
 21 
 
 40 
 
 482128 
 
 661 
 
 979019 
 
 67 
 
 503109 
 
 728 
 
 493891 
 
 20 
 
 41 
 
 9.482525 
 
 660 
 
 9.978979 
 
 67 
 
 9.503546 
 
 727 
 
 10.496454 
 
 19 
 
 42 
 
 482921 
 
 659 
 
 978939 
 
 67 
 
 503982 
 
 727 
 
 496018 
 
 18 
 
 43 
 
 483316 
 
 659 
 
 978898 
 
 67 
 
 504418 
 
 726 
 
 495582 
 
 17 
 
 44 
 
 483712 
 
 658 
 
 978858 
 
 67 
 
 504854 
 
 725 
 
 495146 
 
 16 
 
 45 
 
 484107 
 
 657 
 
 978817 
 
 67 
 
 505289 
 
 725 
 
 494711 
 
 15 
 
 46 
 
 484501 
 
 657 
 
 978777 
 
 67 
 
 505724 
 
 724 
 
 494276 
 
 14 
 
 47 
 
 484895 
 
 656 
 
 978736 
 
 67 
 
 506159 
 
 724 
 
 493841 
 
 13 
 
 48 
 
 485289 
 
 655 
 
 978696 
 
 68 
 
 50(5593 
 
 723 
 
 493407 
 
 12 
 
 49 
 
 485682 
 
 655 
 
 978655 
 
 63 
 
 507027 
 
 722 
 
 492973 
 
 11 
 
 50 
 
 486075 
 
 654 
 
 978615 
 
 68 
 
 507460 
 
 722 
 
 492540 
 
 10 
 
 51 
 
 9.486467 
 
 653 
 
 9.978574 
 
 68 
 
 9.507893 
 
 721 
 
 10.492107 
 
 9 
 
 5-1 
 
 488860 
 
 653 
 
 978533 
 
 68 
 
 508326 
 
 721 
 
 491674 
 
 8 
 
 53 
 
 487251 
 
 652 
 
 978493 
 
 68 
 
 508759 
 
 720 
 
 491241 
 
 7 
 
 54 
 
 487643 
 
 651 
 
 978452 
 
 68 
 
 509191 
 
 719 
 
 490809 
 
 6 
 
 55 
 
 488034 
 
 651 
 
 978411 
 
 68 
 
 509622 
 
 719 
 
 490378 
 
 5 
 
 56 
 
 4HH4-24 
 
 650 
 
 978370 
 
 68 
 
 510054 
 
 718 
 
 489946 
 
 4 
 
 57 
 
 4888J4 
 
 650 
 
 978329 
 
 68 
 
 510485 
 
 718 
 
 489515 
 
 3 
 
 58 
 
 489204 
 
 649 
 
 978288 
 
 68 
 
 510916 
 
 717 
 
 489084 
 
 2 
 
 59 
 
 483593 
 
 648 
 
 978247 
 
 68 
 
 511346 
 
 716 
 
 488654 
 
 1 
 
 60 
 
 489932 
 
 648 
 
 978206 
 
 68 
 
 51L776 
 
 716 
 
 4H8-224 
 
 
 
 | Cosine | | Sine | | Cotang. | | Tang. M. 
 
 72 Degrees. 
 
(18 Degrees.) A TABLE or LOGARITHMIC 
 
 M. 
 
 Sine 
 
 D 
 
 Cosine 
 
 D. 
 
 ! Tang. 
 
 ! D. 
 
 | Cotang. 
 
 
 
 
 9 489982 
 
 648 
 
 9.978206 
 
 68 
 
 9.511776 
 
 716 
 
 10.488224 
 
 60 
 
 1 
 
 490371 
 
 648 
 
 978165 
 
 68 
 
 512206 
 
 718 
 
 487794 
 
 59 
 
 2 
 
 490759 
 
 647 
 
 978124 
 
 68 
 
 512(535 
 
 7J5 
 
 487365 
 
 58 
 
 3 
 
 491147 
 
 046 
 
 978083 
 
 69 
 
 513064 
 
 714 
 
 48G936 
 
 57 
 
 4 
 
 491535 
 
 646 
 
 978042 
 
 69 
 
 513493 
 
 714 
 
 486507 
 
 56 
 
 5 
 
 491922 
 
 645 
 
 978001 
 
 69 
 
 513921 
 
 713 
 
 486079 
 
 55 
 
 6 
 
 49230? 
 
 644 
 
 977959 
 
 69 
 
 514349 
 
 713 
 
 485651 
 
 54 
 
 7 
 
 49269J 
 
 644 
 
 977918 
 
 69 
 
 514777 
 
 712 
 
 485223 
 
 53 
 
 8 
 
 493081 
 
 H43 
 
 977877 
 
 69 
 
 515204 
 
 712 
 
 484796 
 
 52 
 
 9 
 
 493466 
 
 642 
 
 977835 
 
 69 
 
 515631 
 
 711 
 
 484369 
 
 51 
 
 10 
 
 493851 
 
 642 
 
 977794 
 
 69 
 
 516057 
 
 710 
 
 483943 
 
 50 
 
 11 
 
 9.494236 
 
 .641 
 
 9.977752 
 
 69 
 
 9.516484 
 
 710 
 
 10.483516 
 
 49 
 
 12 
 
 494621 
 
 641 
 
 977711 
 
 69 
 
 516910 
 
 709 
 
 483090 
 
 48 
 
 13 
 
 495005 
 
 640 
 
 9776G9 
 
 69 
 
 517335 
 
 709 
 
 482665 
 
 47 
 
 14 
 
 495388 
 
 639 
 
 977628 
 
 69 
 
 517761 
 
 708 
 
 482239 
 
 46 
 
 15 
 
 495772 
 
 639 
 
 977586 
 
 69 
 
 518185 
 
 708 
 
 481815 
 
 45 
 
 16 
 
 496154 
 
 638 
 
 977544 
 
 70 
 
 518610 
 
 707 
 
 481390 
 
 44 
 
 17 
 
 496537 
 
 637 
 
 977503 
 
 70 
 
 519034 
 
 706 
 
 480966 
 
 43 
 
 18 
 
 496919 
 
 637 
 
 977461 
 
 70 
 
 519458 
 
 70G 
 
 480542 
 
 42 
 
 19 
 
 497301 
 
 636 
 
 977419 
 
 70 
 
 519882 
 
 705 
 
 480118 
 
 41 
 
 20 
 
 497682 
 
 636 
 
 977377 
 
 70 
 
 520305 
 
 705 
 
 479CD5 
 
 40 
 
 21 
 
 9.498064 
 
 635 
 
 9.977335 
 
 70 
 
 9.520728 
 
 704 
 
 10.479272 
 
 ao 
 
 22 
 
 498444 
 
 634 
 
 977293 
 
 70 
 
 521151 
 
 703 
 
 478849 
 
 38 
 
 23 
 
 498825 
 
 634 
 
 977251 
 
 70 
 
 521573 
 
 703 
 
 478427 
 
 37 
 
 24 
 
 499204 
 
 633 
 
 977209 
 
 70 
 
 521995 
 
 703 
 
 478005 
 
 36 
 
 25 
 
 499584 
 
 632 
 
 977167 
 
 70 
 
 522417 
 
 7G2 
 
 77583 
 
 33 
 
 26 
 
 499963 
 
 632 
 
 9771-25 
 
 70 
 
 52-3838 
 
 702 
 
 77JG2 
 
 ::: 
 
 27 
 
 500342 
 
 631 
 
 977083 
 
 70 
 
 523259 
 
 701 
 
 76741 
 
 33 
 
 28 
 
 500721 
 
 631 
 
 977041 
 
 70 
 
 523(!8 
 
 701 
 
 70320 
 
 32 
 
 29 
 
 501099 
 
 630 
 
 U76999 
 
 70 
 
 524100 
 
 700 
 
 75900 
 
 31 
 
 30 
 
 501476 
 
 629 
 
 97G957 
 
 70 
 
 524520 
 
 699 
 
 75480 
 
 30 
 
 31 
 
 9.501854 
 
 629 
 
 9.976914 
 
 70 
 
 9.524339 
 
 699 
 
 10.475061 
 
 29 
 
 32 
 
 502231 
 
 628 
 
 976872 
 
 71 
 
 525359 
 
 698 
 
 474641 
 
 28 
 
 33 
 
 502607 
 
 628 
 
 976830 
 
 71 
 
 525778 
 
 698 
 
 474222 
 
 27 
 
 34 
 
 502984 
 
 627 
 
 970787 
 
 71 
 
 5-2I.J97 
 
 697 
 
 473803 
 
 26 
 
 35 
 
 50;i3GO 
 
 626 
 
 976745 
 
 71 
 
 526G15 
 
 697 
 
 473385 
 
 25 
 
 36 
 
 503735 
 
 626 
 
 976702 
 
 71 
 
 527033 
 
 696 
 
 472967 
 
 24 
 
 37 
 
 504110 
 
 625 
 
 970000 
 
 71 
 
 527451 
 
 696 
 
 472549 
 
 23 
 
 38 
 
 504485 
 
 625 
 
 976617 
 
 71 
 
 527868 
 
 695 
 
 472132 
 
 22 
 
 39 
 
 504860 
 
 624 
 
 976574 
 
 71 
 
 528285 
 
 695 
 
 471715 
 
 21 
 
 40 
 
 505234 
 
 623 
 
 976532 
 
 71 
 
 528702 
 
 694 
 
 471298 
 
 20 
 
 41 
 
 9.505608 
 
 623 
 
 9.976489 
 
 71 
 
 9.529119 
 
 693 
 
 10.470831 
 
 19 
 
 42 
 
 505981 
 
 622 
 
 976446 
 
 71 
 
 529535 
 
 693 
 
 470405 
 
 18 
 
 43 
 
 506354 
 
 622 
 
 976404 
 
 71 
 
 529950 
 
 693 
 
 470050 
 
 17 
 
 44 
 
 506727 
 
 621 
 
 976361 
 
 71 
 
 530366 
 
 692 
 
 409634 
 
 16 
 
 45 
 
 507099 
 
 620 
 
 976318 
 
 71 
 
 530781 
 
 691 
 
 4(59219 
 
 15 
 
 46 
 
 507471 
 
 620 
 
 976275 
 
 71 
 
 531196 
 
 691 
 
 468804 
 
 14 
 
 47 
 
 507843 
 
 619 
 
 976232 
 
 72 
 
 531611 
 
 690 
 
 468389 
 
 13 
 
 48 
 
 508214 
 
 619 
 
 976189 
 
 72 
 
 532021 
 
 690 
 
 467975 
 
 12 
 
 49 
 
 508585 
 
 618 
 
 976146 
 
 72 
 
 532439 
 
 689 
 
 467561 
 
 11 
 
 50 
 
 508956 
 
 618 
 
 976103 
 
 72 
 
 532853 
 
 689 
 
 467147 
 
 10 
 
 51 
 
 9.509326 
 
 617 
 
 9-976060 
 
 72 
 
 9.533266 
 
 688 
 
 10.466734 
 
 9 
 
 52 
 
 509696 
 
 616 
 
 976017 
 
 72 
 
 533679 
 
 688 
 
 466321 
 
 8 
 
 53 
 
 510065 
 
 616 
 
 975974 
 
 72 
 
 534092 
 
 687 
 
 465908 
 
 7 
 
 54 
 
 510434 
 
 615 
 
 975930 
 
 72 
 
 534504 
 
 687 
 
 465496 
 
 6 
 
 55 
 
 510803 
 
 615 
 
 975887 
 
 72 
 
 534916 
 
 686 
 
 465084 
 
 5 
 
 56 
 
 511172 
 
 614 
 
 975844 
 
 72 
 
 535328 
 
 686 
 
 464672 
 
 4 
 
 57 
 
 511540 
 
 613 
 
 975800 
 
 72 
 
 535739 
 
 685 
 
 464261 
 
 3 
 
 58 
 
 511907 
 
 613 
 
 975757 
 
 72 
 
 536150 
 
 685 
 
 463850 
 
 2 
 
 59 
 
 512275 
 
 612 
 
 975714 
 
 72 
 
 536561 
 
 684 
 
 463439 
 
 1 
 
 60 
 
 512642 
 
 612 
 
 975670 
 
 72 
 
 536972 
 
 684 
 
 463028 
 
 
 
 
 Cosine | 
 
 I 
 
 Sine | 
 
 
 Cotang. | 
 
 1 
 
 Tang. | 
 
 M. 
 
 71 Degrees. 
 
SINES AND TANGENTS. (19 Degrees.) 
 
 37 
 
 M. | Sine | D. | Cosine | D. | Tang. D. 
 
 Cotang. | 
 
 
 
 9.512IU2 
 
 612 
 
 9.97.5670 
 
 73 
 
 9.530972 084 
 
 10. 463028 
 
 eo 
 
 1 
 
 513009 
 
 611 
 
 975627 
 
 73 
 
 537383 
 
 683 
 
 462018 
 
 59 
 
 a 
 
 :> 1:075 
 
 611 
 
 975583 
 
 73 
 
 537792 
 
 683 
 
 462208 
 
 58 
 
 3 
 
 513741 
 
 610 
 
 975539 
 
 73 
 
 538202 
 
 682 
 
 461798 
 
 57 
 
 4 
 
 514107 
 
 609 
 
 975496 
 
 73 
 
 53861] 
 
 682 
 
 401389 
 
 56 
 
 5 
 
 514472 
 
 609 
 
 975452 
 
 73 
 
 539020 
 
 681 
 
 460980 
 
 55 
 
 6 
 
 5 14SU7 
 
 608 
 
 975408 
 
 73 
 
 539429 
 
 681 
 
 460571 
 
 54 
 
 7 
 
 515202 
 
 608 
 
 975365 
 
 73 
 
 539837" 
 
 680 
 
 460163 
 
 53 
 
 8 
 
 5155(56 
 
 607 
 
 975321 
 
 73 
 
 540245 
 
 680 
 
 459755 
 
 52 
 
 9 
 
 515930 
 
 607 
 
 975277 
 
 73 
 
 540653 
 
 679 
 
 459347 
 
 51 
 
 10 
 
 510294 
 
 60(5 
 
 975233 
 
 73 
 
 5-1 1 (Mil 
 
 679 
 
 458939 
 
 50 
 
 11 
 
 9.516657 
 
 605 
 
 9.975189 
 
 73 
 
 9.541468 
 
 678 
 
 10.458532 
 
 49 
 
 12 
 
 517020 
 
 605 
 
 975145 
 
 73 
 
 541875 
 
 678 
 
 458125 
 
 48 
 
 13 
 
 517382 
 
 604 
 
 975101 
 
 73 
 
 542281 
 
 677 
 
 457719 
 
 47 
 
 14 
 
 517745 
 
 604 
 
 975057 
 
 73 
 
 542688 
 
 677 
 
 457312 
 
 46 
 
 15 
 
 518107 
 
 603 
 
 975013 
 
 73 
 
 543094 
 
 676 
 
 456906 
 
 45 
 
 16 
 
 518468 
 
 603 
 
 974909 
 
 74 
 
 543499 
 
 676 
 
 450501 
 
 44 
 
 17 
 
 518829 
 
 602 
 
 974925 
 
 74 
 
 543905 
 
 675 
 
 456095 
 
 43 
 
 18 
 
 519190 
 
 601 
 
 974880 
 
 74 
 
 544310 
 
 675 
 
 455690 
 
 42 
 
 19 
 
 519551 
 
 601 
 
 974836 
 
 74 
 
 544715 
 
 674 
 
 455285 
 
 41 
 
 20 
 
 519911 
 
 600 
 
 974792 
 
 74 
 
 545119 
 
 674 
 
 454881 
 
 40 
 
 21 
 
 9.520271 
 
 600 
 
 9.974748 
 
 74 
 
 9.545524 
 
 673 
 
 10.454476 
 
 39 
 
 22 
 
 520031 
 
 599 
 
 974703 
 
 74 
 
 545928 
 
 673 
 
 454072 
 
 38 
 
 23 
 
 520990 
 
 599 
 
 974659 
 
 74 
 
 546331 
 
 672 
 
 453(569 
 
 37 
 
 24 
 
 521349 
 
 598 
 
 974014 
 
 74 
 
 546735 
 
 672 
 
 453265 
 
 36 
 
 25 
 
 521707 
 
 598 
 
 974570 
 
 74 
 
 547138 
 
 671 
 
 452862 
 
 35 
 
 98 
 
 539066 
 
 597 
 
 974525 
 
 74 
 
 547540 
 
 671 
 
 452460 
 
 34 
 
 27 
 
 522424 
 
 596 
 
 974431 
 
 74 
 
 547943 
 
 670 
 
 452057 
 
 33 
 
 28 
 
 522781 
 
 596 
 
 974436 
 
 74 
 
 548345 
 
 670 
 
 451655 
 
 32 
 
 29 
 
 523138 
 
 595 
 
 974391 
 
 74 
 
 548747 
 
 669 
 
 451253 
 
 31 
 
 30 
 
 523495 
 
 595 
 
 974347 
 
 75 
 
 549149 
 
 669 
 
 450851 
 
 30 
 
 31 
 
 9.523852 
 
 594 
 
 9.974302 
 
 75 
 
 9.549550 
 
 668 
 
 10.450450 
 
 29 
 
 32 
 
 524208 
 
 594 
 
 974-257 
 
 75 
 
 549951 
 
 668 
 
 450049 
 
 28 
 
 33 
 
 524564 
 
 593 
 
 974212 
 
 75 
 
 550352 
 
 667 
 
 449648 
 
 27 
 
 34 
 
 524920 
 
 593 
 
 974167 
 
 75 
 
 550752 
 
 667 
 
 449248 
 
 26 
 
 35 
 
 525275 
 
 592 
 
 974122 
 
 75 
 
 551152 
 
 666 
 
 448848 
 
 25 
 
 36 
 
 525630 
 
 591 
 
 974077 
 
 75 
 
 551552 
 
 666 
 
 448448 
 
 24 
 
 37 
 
 525984 
 
 591 
 
 974032 
 
 75 
 
 55J953 
 
 665 
 
 448048 
 
 23 
 
 38 
 
 536339 
 
 590' 
 
 973987 
 
 75 
 
 552351 
 
 665 
 
 447649 
 
 22 
 
 39 
 
 52H093 
 
 590 
 
 973942 
 
 75 
 
 552750 
 
 665 
 
 447250 
 
 21 
 
 40 
 
 527046 
 
 589 
 
 973897 
 
 75 
 
 553149 
 
 664 
 
 446851 
 
 20 
 
 41 
 
 9.527400 
 
 589 
 
 9.973852 
 
 75 
 
 9.553548 
 
 664 
 
 10.446452 
 
 19 
 
 42 
 
 527753 
 
 588 
 
 973807 
 
 75 
 
 553946 
 
 663 
 
 446054 
 
 18 
 
 43 
 
 528105 
 
 588 
 
 973701 
 
 75 
 
 554344 
 
 663 
 
 445656 
 
 17 
 
 44 
 
 528458 
 
 587 
 
 973716 
 
 76 
 
 554741 
 
 662 
 
 445259 
 
 16 
 
 45 
 
 528810 
 
 587 
 
 973671 
 
 7(5 
 
 555138 
 
 662 
 
 444861 
 
 15 1 
 
 46 
 
 529161 
 
 586 
 
 973625 
 
 76 
 
 555536 
 
 661 
 
 444464 
 
 14 , 
 
 47 
 
 529513 
 
 586 
 
 973580 
 
 76 
 
 555933 
 
 661 
 
 444067 
 
 13 
 
 48 
 
 539864 
 
 585 
 
 973535 
 
 76 
 
 556329 
 
 660 
 
 443671 
 
 12 
 
 49 
 
 530215 
 
 585 
 
 973489 
 
 76 
 
 556725 
 
 660 
 
 443275 
 
 11 
 
 50 
 
 530565 
 
 584 
 
 973444 
 
 76 
 
 557121 
 
 659 
 
 442879 
 
 10 
 
 51 
 
 9.530915 
 
 584 
 
 9.973398 
 
 76 
 
 9.557517 
 
 659 
 
 10.442483 
 
 9 
 
 52 
 
 531265 
 
 583 
 
 973352 
 
 76 
 
 557913 
 
 659 
 
 442087 
 
 8 
 
 53 
 
 531614 
 
 582 
 
 973307 
 
 76 
 
 558308 
 
 658 
 
 441692 
 
 7 
 
 54 
 
 531903 
 
 582 
 
 973261 
 
 76 
 
 558702 
 
 658 
 
 441298 
 
 6 
 
 55 
 
 532312 
 
 581 
 
 973215 
 
 76 
 
 559097 
 
 657 
 
 440903 
 
 5 
 
 5(5 
 
 532661 
 
 581 
 
 973169 
 
 76 
 
 559491 
 
 657 
 
 440509 
 
 4 
 
 57 
 
 KWOOH 
 
 580 
 
 973124 
 
 76 
 
 559885 
 
 656 
 
 440115 
 
 3 
 
 58 
 
 533357 
 
 580 
 
 973078 
 
 76 
 
 560279 
 
 656 
 
 439721 
 
 2 
 
 59 
 
 533704 
 
 579 
 
 973032 
 
 77 
 
 560673 
 
 655 
 
 439327 
 
 1 
 
 60 
 
 534052 
 
 578 
 
 972986 
 
 77 
 
 561066 
 
 655 
 
 438934 
 
 
 
 | Cosine | | Sine | | Cotang. | _ Tang. | M. 
 
 70 Degrees. 
 
Degrees.) A TABLE OF LOGARITHMIC 
 
 M. Sine D. Cosine | D. | Tang. | D. Cotang. | 
 
 
 
 9.534052 
 
 578 
 
 9.97293S 
 
 77 
 
 9.551066 
 
 655 
 
 10.4 89 4 
 
 60 
 
 1 
 
 534399 
 
 577 
 
 972940 
 
 77 
 
 5(51459 
 
 654 
 
 438341 
 
 59 
 
 2 
 
 534745 
 
 577 
 
 972394 
 
 77 
 
 5618.")! 
 
 654 
 
 438149 
 
 58 
 
 3 
 
 535092 
 
 577 
 
 97-2348 
 
 77 
 
 562244 
 
 653 
 
 437756 
 
 57 
 
 4 
 
 535438 
 
 576 
 
 97-2302 
 
 77 
 
 502636 
 
 653 
 
 437364 
 
 56 
 
 5 
 
 535783 
 
 576 
 
 972755 
 
 77 
 
 563028 
 
 653 
 
 ' 43(5972 
 
 55 
 
 6 
 
 533129 
 
 575 
 
 972709 
 
 77 
 
 533419 
 
 652 
 
 43:>531 
 
 54 
 
 7 
 
 53(5474 
 
 574 
 
 972663 
 
 77 
 
 563811 
 
 652 
 
 436189 
 
 53 
 
 8 
 
 536818 
 
 574 
 
 972617 
 
 77 
 
 5(542)2 
 
 651 
 
 435798 
 
 52 
 
 9 
 
 537163 
 
 573 
 
 972570 
 
 77 
 
 564592 
 
 651 
 
 435408 
 
 51 
 
 10 
 
 5375J7 
 
 573 
 
 972524 
 
 77 
 
 5154933 
 
 650 
 
 435017 
 
 50 
 
 11 
 
 9.537851 
 
 572 
 
 9.972478 
 
 77 
 
 9.565373 
 
 650 
 
 10.434627 
 
 49 
 
 1-2 
 
 538194 
 
 572 
 
 972431 
 
 78 
 
 565763 
 
 619 
 
 434237 
 
 48 
 
 13 
 
 838538 
 
 571 
 
 972335 
 
 78 
 
 566153 
 
 649 
 
 433847 
 
 47 
 
 14 
 
 538880 
 
 571 
 
 972338 
 
 73 
 
 556542 
 
 649 
 
 433458 
 
 46 
 
 15 
 
 539223 
 
 570 
 
 9722.) 1 
 
 78 
 
 566932 
 
 648 
 
 433068 
 
 45 
 
 16 
 
 5395155 
 
 570 
 
 972345 
 
 78 
 
 567320 
 
 648 
 
 4326 Si) 
 
 44 
 
 17 
 
 539907 
 
 569 
 
 972198 
 
 78 
 
 567709 
 
 647 
 
 432291 
 
 43 
 
 18 
 
 540-249 
 
 569 
 
 972151 
 
 78 
 
 563093 
 
 647 
 
 431902 
 
 42 
 
 19 
 
 540590 
 
 568 
 
 972105 
 
 78 
 
 563488 
 
 646 
 
 431514 
 
 41 
 
 20 
 
 54J931 
 
 568 
 
 972058 
 
 78 
 
 563373 
 
 646 
 
 431127 
 
 40 
 
 21 
 
 9.541272 
 
 567 
 
 9.972011 
 
 78 
 
 9.5*59261 
 
 645 
 
 10.430739 
 
 39 
 
 22 
 
 541613 
 
 567 
 
 971914 
 
 73 
 
 569(548 
 
 645 
 
 430352 
 
 33 
 
 23 
 
 541953 
 
 566 
 
 971917 
 
 78 
 
 570035 
 
 645 
 
 4299-55 
 
 37 
 
 24 
 
 542293 
 
 566 
 
 971870 
 
 78 
 
 570422 
 
 644 
 
 429578 
 
 36 
 
 25 
 
 542832 
 
 565 
 
 971823 
 
 /8 
 
 570309 
 
 644 
 
 421)191 
 
 35 
 
 26 
 
 542971 
 
 565 
 
 971776 
 
 78 
 
 571195 
 
 643 
 
 4288J5 
 
 34 
 
 27 
 
 543310 
 
 584 
 
 971729 
 
 79 
 
 571.581 
 
 643 
 
 428419 
 
 33 
 
 28 
 
 543!549 
 
 564 
 
 971682 
 
 79 
 
 571967 
 
 642 
 
 428033 
 
 32 
 
 29 
 
 543937 
 
 5!53 
 
 971635 
 
 79 
 
 57-2352 
 
 642 
 
 427648 
 
 31 
 
 30 
 
 514325 
 
 563 
 
 971583 
 
 79 
 
 572733 
 
 642 
 
 427252 
 
 30 
 
 31 
 
 9.544663 
 
 5(52 
 
 9.971540 
 
 79 
 
 8. 573123 
 
 641 
 
 10.426877 
 
 29 
 
 32 
 
 5450JO 
 
 562 
 
 971493 
 
 79 
 
 573507 
 
 641 
 
 42!>493 
 
 28 
 
 33 
 
 545338 
 
 561 
 
 97144!! 
 
 79 
 
 573893 
 
 640 
 
 4-2(5108 
 
 27 
 
 34 
 
 545674 
 
 561 
 
 971398 
 
 79 
 
 574276 
 
 640 
 
 425724 
 
 26 
 
 35 
 
 54(5011 
 
 580 
 
 971351 
 
 79 
 
 574560 
 
 639 
 
 425340 
 
 25 
 
 36 
 
 516347 
 
 5f50 
 
 9713J3 
 
 79 
 
 575044 
 
 639 
 
 424956 
 
 24 
 
 37 
 
 546683 
 
 559 
 
 971256 
 
 79 
 
 575427 
 
 639 
 
 424573 
 
 23 
 
 38 
 
 547019 
 
 559 
 
 97 J 208 
 
 79 
 
 5758JO 
 
 638 
 
 424193 
 
 22 
 
 39 
 
 547354 
 
 553 
 
 9711(51 
 
 79 
 
 576193 
 
 638 
 
 423807 
 
 21 
 
 40 
 
 547689 
 
 553 
 
 971113 
 
 79 
 
 576576 
 
 637 
 
 423424 
 
 20 
 
 41 
 
 9.548024 
 
 557 
 
 9.971066 
 
 80 
 
 9.576958 
 
 637 
 
 10.423041 
 
 19 
 
 42 
 
 548359 
 
 557 
 
 971018 
 
 88 
 
 577341 
 
 636 
 
 422659 
 
 18 
 
 43 
 
 5481593 
 
 555 
 
 970970 
 
 80 
 
 577723 
 
 636 
 
 422277 
 
 17 
 
 44 
 
 549027 
 
 556 
 
 970922 
 
 80 
 
 578104 
 
 636 
 
 421896 
 
 ]6 
 
 45 
 
 549360 
 
 555 
 
 970874 
 
 80 
 
 578486 
 
 635 
 
 421514 
 
 15 
 
 46 
 
 549593 
 
 555 
 
 970827 
 
 8D 
 
 578887 
 
 635 
 
 421133 
 
 14 
 
 47 
 
 550026 
 
 554 
 
 970779 
 
 80 
 
 579248 
 
 634 
 
 420752 
 
 13 
 
 48 
 
 550359 
 
 554 
 
 970731 
 
 80 
 
 579829 
 
 634 
 
 420371 
 
 12 
 
 49 
 
 550692 
 
 553 
 
 970683 
 
 80 
 
 580009 
 
 634 
 
 419991 
 
 11 
 
 50 
 
 551024 
 
 553 
 
 970635 
 
 80 
 
 580389 
 
 633 
 
 419811 
 
 10 
 
 51 
 
 9 551356 
 
 552 
 
 9.970586 
 
 80 
 
 9.580769 
 
 633 
 
 10.419231 
 
 9 
 
 52 
 
 551687 
 
 552 
 
 970538 
 
 80 
 
 581149 
 
 632 
 
 418851 
 
 8 
 
 53 
 
 552018 
 
 552 
 
 970490 
 
 83 
 
 581528 
 
 632 
 
 413472 
 
 7 
 
 54 
 
 552349 
 
 551 
 
 970442 
 
 80 
 
 581907 
 
 632 
 
 418093 
 
 6 
 
 55 
 
 552680 
 
 551 
 
 970394 
 
 80 
 
 582286 
 
 631 
 
 417714 
 
 5 
 
 58 
 
 553010 
 
 550 
 
 970345 
 
 81 
 
 582665 
 
 631 
 
 417335 
 
 4 
 
 57 
 
 553341 
 
 550 
 
 970297 
 
 81 
 
 583043 
 
 630 
 
 416957 
 
 3 
 
 58 
 
 553670 
 
 549 
 
 970249 
 
 81 
 
 583422 
 
 630 
 
 416578 
 
 2 
 
 59 
 
 554000 
 
 549 
 
 970230 
 
 81 
 
 583800 
 
 629 
 
 416200 
 
 1 
 
 CO 
 
 554329 
 
 548 
 
 970152 
 
 81 
 
 584177 
 
 629 
 
 415823 
 
 
 
 Cosine | Sine | | Cotang. | Tang. | M. 
 
 69 Degrees 
 
SINES AND TANGENTS. (21 Degrees.) 
 
 39 
 
 M. Sine D. | Cosine | D. | Tang. D. Cotang. | 
 
 
 
 9.5J43-29 
 
 548 
 
 9.970152 
 
 81 
 
 9.584177 
 
 629 
 
 10.415823 
 
 60 
 
 1 
 
 5541)58 
 
 548 
 
 970103 
 
 81 
 
 f>-> i :,:>.-> 
 
 62J 
 
 415445 
 
 59 
 
 2 
 
 554987 
 
 547 
 
 9700:>3 
 
 81 
 
 584932 
 
 628 
 
 415068 
 
 58 
 
 3 
 
 555315 
 
 547 
 
 970U06 
 
 81 
 
 585309 
 
 628 
 
 414691 
 
 57 
 
 4 
 
 555643 
 
 546 
 
 969957 
 
 81 
 
 585686 
 
 627 
 
 414314 
 
 56 
 
 5 
 
 555971 
 
 546 
 
 959909 
 
 81 
 
 586062 
 
 627 
 
 413938 
 
 55 
 
 6 
 
 556299 
 
 545 
 
 969860 
 
 81 
 
 586439 
 
 627 
 
 413561 
 
 54 
 
 7 
 
 55T.626 
 
 545 
 
 989811 
 
 81 
 
 586815 
 
 626 
 
 413185 
 
 53 
 
 8 
 
 556953 
 
 544 
 
 969762 
 
 81 
 
 587190 
 
 626 
 
 412810 
 
 52 
 
 9 
 
 557-280 
 
 544 
 
 969714 
 
 81 
 
 587566 
 
 625 
 
 412434 
 
 51 
 
 10 
 
 557606 
 
 543 
 
 969665 
 
 81 
 
 587941 
 
 625 
 
 412059 
 
 50 
 
 11 
 
 9.557OT2 
 
 543 
 
 9.969616 
 
 82 
 
 9.588316 
 
 625 
 
 10.411684 
 
 49 
 
 12 
 
 558-258 
 
 543 
 
 969567 
 
 82 
 
 588691 
 
 624 
 
 411309 
 
 48 
 
 13 
 
 558583 
 
 542 
 
 969518 
 
 82 
 
 589086 
 
 624 
 
 410934 
 
 47 
 
 14 
 
 558909 
 
 542 
 
 969469 
 
 82 
 
 589440 
 
 623 
 
 410560 
 
 46 
 
 15 
 
 55<h234 
 
 541 
 
 969420 
 
 82 
 
 589814 
 
 623 
 
 410186 
 
 45 
 
 16 
 
 559558 
 
 541 
 
 969370 
 
 82 
 
 590188 
 
 623 
 
 409812 
 
 44 
 
 17 
 
 559883 
 
 540 
 
 969321 
 
 82 
 
 590562 
 
 622 
 
 409438 
 
 43 
 
 18 
 
 560207 
 
 540 
 
 969272 
 
 82 
 
 591)935 
 
 622 
 
 409065 
 
 42 
 
 19 
 
 560531 
 
 539 
 
 969223 
 
 82 
 
 591308 
 
 622 
 
 408692 
 
 41 
 
 21) 
 
 560855 
 
 539 
 
 969173 
 
 82 
 
 591681 
 
 621 
 
 408319 
 
 40 
 
 21 
 
 9.561178 
 
 538 
 
 9.969124 
 
 82 
 
 9.592054 
 
 621 
 
 10.407946 
 
 39 
 
 22 
 
 561501 
 
 538 
 
 969075 
 
 82 
 
 592426 
 
 620 
 
 407574 
 
 38 
 
 23 
 
 561824 
 
 537 
 
 969025 
 
 82 
 
 592798 
 
 620 
 
 407202 
 
 37 
 
 24 
 
 562146 
 
 537 
 
 968976 
 
 82 
 
 593170 
 
 619 
 
 406829 
 
 36 
 
 25 
 
 562468 
 
 536 
 
 968926 
 
 83 
 
 593542 
 
 619 
 
 406458 
 
 35 
 
 26 
 
 562790 
 
 536 
 
 968877 
 
 83 
 
 593914 
 
 618 
 
 406086 
 
 34 
 
 27 
 
 563112 
 
 536 
 
 968827 
 
 83 
 
 594285 
 
 618 
 
 405715 
 
 33 
 
 28 
 
 563433 
 
 535 
 
 988777 
 
 83 
 
 594656 
 
 618 
 
 405344 
 
 32 
 
 29 
 
 563755 
 
 535 
 
 968728 
 
 83 
 
 595027 
 
 617 
 
 404973 
 
 31 
 
 30 
 
 564075 
 
 534 
 
 968678 
 
 83 
 
 595398 
 
 617 
 
 404602 
 
 30 
 
 31 
 
 9.564396 
 
 534 
 
 9.968628 
 
 83 
 
 9.595768 
 
 617 
 
 10.404232 
 
 29 
 
 32 
 
 564716 
 
 533 
 
 968578 
 
 83 
 
 596138 
 
 616 
 
 403862 
 
 28 
 
 33 
 
 565036 
 
 533 
 
 968528 
 
 83 
 
 596508 
 
 616 
 
 403492 
 
 27 
 
 34 
 
 565356 
 
 532 
 
 968479 
 
 83 
 
 596878 
 
 616 
 
 403122 
 
 26 
 
 35 
 
 565676 
 
 532 
 
 968429 
 
 83 
 
 597247 
 
 615 
 
 402753 
 
 25 
 
 36 
 
 565995 
 
 531 
 
 968379 
 
 83 
 
 597616 
 
 615 
 
 402384 
 
 24 
 
 37 
 
 566314 
 
 531 
 
 9(38329 
 
 83 
 
 597985 
 
 615 
 
 402015 
 
 23 
 
 38 
 
 566632 
 
 531 
 
 068378 
 
 83 
 
 598354 
 
 614 
 
 401646 
 
 22 
 
 39 
 
 566951 
 
 530 
 
 968228 
 
 84 
 
 598722 
 
 614 
 
 401278 
 
 21 
 
 40 
 
 567269 
 
 530 
 
 968178 
 
 84 
 
 599091 
 
 613 
 
 400909 
 
 20 
 
 41 
 
 9.567587 
 
 529 
 
 9.968128 
 
 84 
 
 9.599459 
 
 613 
 
 10.400541 
 
 19 
 
 42 
 
 587904 
 
 529 
 
 968078 
 
 84 
 
 599827 
 
 613 
 
 400173 
 
 18 
 
 43 
 
 568222 
 
 528 
 
 968027 
 
 84 
 
 600194 
 
 612 
 
 399806 
 
 17 
 
 44 
 
 568539 
 
 528 
 
 967977 
 
 84 
 
 600562 
 
 612 
 
 399438 
 
 16 
 
 45 
 
 568856 
 
 528 
 
 967927 
 
 84 
 
 600929 
 
 611 
 
 399071 
 
 15 
 
 46 
 
 569172 
 
 527 
 
 967876 
 
 84 
 
 601296 
 
 611 
 
 398704 
 
 14 
 
 47 
 
 569488 
 
 527 
 
 957826 
 
 84 
 
 601662 
 
 611 
 
 398338 
 
 13 
 
 48 
 
 569804 
 
 526 
 
 967775 
 
 84 
 
 602029 
 
 610 
 
 397971 
 
 12 
 
 49 
 
 570120 
 
 526 
 
 967725 
 
 84 
 
 602395 
 
 610 
 
 397605 
 
 11 
 
 50 
 
 570435 
 
 525 
 
 967674 
 
 84 
 
 602761 
 
 610 
 
 397239 
 
 10 
 
 51 
 
 9.570751 
 
 525 
 
 9.987624 
 
 84 
 
 9.603127 
 
 609 
 
 10.396873 
 
 9 
 
 52 
 
 571066 
 
 524 
 
 957573 
 
 84 
 
 603493 
 
 609 
 
 396507 
 
 8 
 
 53 
 
 571380 
 
 524 
 
 967522 
 
 85 
 
 603858 
 
 609 
 
 396142 
 
 7 
 
 54 
 
 571695 
 
 523 
 
 967471 
 
 85 
 
 604223 
 
 608 
 
 395777 
 
 6 
 
 55 
 
 572009 
 
 523 
 
 967421 
 
 85 
 
 604588 
 
 608 
 
 395412 
 
 5 
 
 56 
 
 5723-23 
 
 523 
 
 9-77370 
 
 85 
 
 604953 
 
 607 
 
 395047 
 
 4 
 
 57 
 
 572636 
 
 522 
 
 967319 
 
 85 
 
 605317 
 
 607 
 
 394683 
 
 3 
 
 53 
 
 572950 
 
 5-22 
 
 967268 
 
 85 
 
 605682 
 
 607 
 
 394318 
 
 I 
 
 59 
 
 573263 
 
 521 
 
 987217 
 
 85 
 
 606046 
 
 606 
 
 393!).>4 
 
 1 
 
 60 
 
 573575 
 
 521 
 
 967168 
 
 85 
 
 61)6410 
 
 606 
 
 393590 
 
 
 
 Cosine | | Sine | | Cotang. | Tang. | M. ) 
 
 Degrees. 
 
40 
 
 Degrees.) A TABI.K OP LOGARITHMIC 
 
 M. Sine | D. | Cosine | D. Tang. | D. Cotang | 
 
 
 
 9.573575 
 
 521 
 
 9.967166 85 9.600410 
 
 606 
 
 10.393590 
 
 60 
 
 1 
 
 573888 
 
 520 
 
 967115 
 
 85 
 
 606773 
 
 606 
 
 393227 
 
 59 
 
 2 
 
 574200 
 
 520 
 
 967064 
 
 85 
 
 607137 
 
 605 
 
 392863 
 
 58 
 
 3 
 
 574512 
 
 519 
 
 967013 
 
 85 
 
 607500 
 
 605 
 
 392500 
 
 57 
 
 4 
 
 57-18-24 
 
 519 
 
 966961 
 
 85 
 
 607863 
 
 604 
 
 392137 
 
 56 
 
 5 
 
 575136 
 
 519 
 
 966910 
 
 85 
 
 608225 
 
 604 
 
 391775 
 
 55 
 
 6 
 
 575447 
 
 518 
 
 966859 
 
 85 
 
 608588 
 
 604 
 
 391412 
 
 54 
 
 7 
 
 575758 
 
 518 
 
 9G68U8 
 
 85 
 
 608950 
 
 603 
 
 391050 
 
 53 
 
 8 
 
 576069 
 
 517 
 
 966756 
 
 86 
 
 609312 
 
 603 
 
 390688 
 
 52 
 
 9 
 
 576379 
 
 517 
 
 966705 
 
 86 
 
 609674 
 
 603 
 
 390326 
 
 51 
 
 10 
 
 576689 
 
 516 
 
 966653 
 
 86 
 
 610036 
 
 602 
 
 389964 
 
 50 
 
 11 
 
 9.576999 
 
 516 
 
 9.966602 
 
 86 
 
 9.61&397 
 
 602 
 
 10.389603 
 
 49 
 
 12 
 
 577309 
 
 516 
 
 966550 
 
 86 
 
 610759 
 
 602 
 
 389241 
 
 48 
 
 13 
 
 577618 
 
 515 
 
 966499 
 
 86 
 
 611120 
 
 601 
 
 388880 
 
 47 
 
 14 
 
 577927 
 
 515 
 
 966447 
 
 86 
 
 61 J 480 
 
 601 
 
 388520 
 
 46 
 
 15 
 
 578236 
 
 514 
 
 966395 
 
 86 
 
 611841 
 
 601 
 
 388 J 59 
 
 45 
 
 16 
 
 578545 
 
 514 
 
 966344 
 
 86 
 
 612201 
 
 600 
 
 387799 
 
 44 
 
 17 
 
 578853 
 
 513 
 
 966292 
 
 86 
 
 612561 
 
 600 
 
 387439 
 
 43 
 
 18 
 
 579162 
 
 513 
 
 966240 
 
 86 
 
 612921 
 
 600 
 
 387079 
 
 42 
 
 19 
 
 579470 
 
 513 
 
 966188 
 
 86 
 
 613281 
 
 599 
 
 386719 
 
 41 
 
 20 
 
 579777 
 
 512 
 
 966136 
 
 86 
 
 613641 
 
 599 
 
 380359 
 
 40 
 
 21 
 
 9.580085 
 
 512 
 
 9.966085 
 
 87 
 
 9.614000 
 
 598 
 
 10.386000 
 
 39 
 
 -22 
 
 580392 
 
 511 
 
 966033 
 
 87 
 
 614359 
 
 598 
 
 385641 
 
 38 
 
 23 
 
 580699 
 
 511 
 
 965981 
 
 87 
 
 614718 
 
 598 
 
 385282 
 
 37 
 
 24 
 
 581005 
 
 511 
 
 965928 
 
 87 
 
 615077 
 
 597 
 
 384923 
 
 36 
 
 25 
 
 581312 
 
 510 
 
 965876 
 
 87 
 
 615435 
 
 597 
 
 384565 
 
 35 
 
 26 
 
 581618 
 
 510 
 
 965824 
 
 87 
 
 615793 
 
 597 
 
 384207 
 
 34 
 
 27 
 
 581924 
 
 509 
 
 965772 
 
 87 
 
 616151 
 
 596 
 
 383849 
 
 33 
 
 28 
 
 5822-29 
 
 509 
 
 965720 
 
 87 
 
 616509 
 
 596 
 
 383491 
 
 32 
 
 29 
 
 582535 
 
 509 
 
 965668 
 
 87 
 
 616867 
 
 596 
 
 383133 
 
 31 
 
 30 
 
 582840 
 
 508 
 
 965615 
 
 87 
 
 617224 
 
 595 
 
 382776 
 
 30 
 
 31 
 
 9.583145 
 
 508 
 
 9.965563 
 
 87 
 
 9.617582 
 
 595 
 
 10.382418 
 
 29 
 
 32 
 
 583449 
 
 507 
 
 965511 
 
 87 
 
 617939 
 
 595 
 
 382061 
 
 28 
 
 33 
 
 583754 
 
 507 
 
 965458 
 
 87 
 
 6J8295 
 
 594 
 
 381705 
 
 27 
 
 34 
 
 584058 
 
 506 
 
 965406 
 
 87 
 
 618652 
 
 594 
 
 381348 
 
 26 
 
 35 
 
 584361 
 
 506 
 
 965353 
 
 88 
 
 619008 
 
 594 
 
 380992 
 
 25 
 
 36 
 
 584665 
 
 506 
 
 965301 
 
 88 
 
 619364 
 
 593 
 
 380636 
 
 24 
 
 37 
 
 584968 
 
 505 
 
 965248 
 
 88 
 
 619721 
 
 593 
 
 380279 
 
 23 
 
 38 
 
 585272 
 
 505 
 
 965195 
 
 88 
 
 620076 
 
 593 
 
 379924 
 
 22 
 
 39 
 
 585574 
 
 504 
 
 965143 
 
 88 
 
 620432 
 
 59 : 2 
 
 379568 
 
 21 
 
 40 
 
 585877 
 
 504 
 
 965090 
 
 88 
 
 620787 
 
 592 
 
 379213 
 
 20 
 
 41 
 
 9.586179 
 
 503 
 
 9.965037 
 
 88 
 
 9.621142 
 
 592 
 
 10.378858 
 
 19 
 
 42 
 
 586482 
 
 503 
 
 964984 
 
 88 
 
 621497 
 
 591 
 
 378503 
 
 18 
 
 43 
 
 586783 
 
 503 
 
 964931 
 
 88 
 
 621852 
 
 591 
 
 378148 
 
 17 
 
 44 
 
 587085 
 
 502 
 
 964879 
 
 88 
 
 62-2-207 
 
 590 
 
 377793 
 
 16 
 
 45 
 
 587386 
 
 502 
 
 964826 
 
 88 
 
 622561 
 
 590 
 
 377439 
 
 15 
 
 46 
 
 587688 
 
 501 
 
 964773 
 
 88 
 
 62-2915 
 
 590 
 
 377085 
 
 14 
 
 47 
 
 587989 
 
 501 
 
 9(54719 
 
 88 
 
 623269 
 
 589 
 
 376731 
 
 13 
 
 48 
 
 588289 
 
 501 
 
 964666 
 
 89 
 
 6-23623 
 
 589 
 
 37(5377 
 
 12 
 
 49 
 
 588590 
 
 500 
 
 964613 
 
 89 
 
 623976 
 
 589 
 
 37f'024 
 
 11 
 
 50 
 
 588890 
 
 500 
 
 964560 
 
 89 
 
 624330 
 
 588 
 
 375670 
 
 10 
 
 51 
 
 9.589190 
 
 499 
 
 9.964507 
 
 89 
 
 9.624C83 
 
 588 
 
 10.375317 
 
 9 
 
 52 
 
 589489 
 
 499 
 
 964454 
 
 89 
 
 625036 
 
 588 
 
 3749(54 
 
 8 
 
 53 
 
 589789 
 
 499 
 
 964400 
 
 89 
 
 625388 
 
 587 
 
 374612 
 
 7 
 
 54 
 
 590088 
 
 498 
 
 964347 
 
 89 
 
 625741 
 
 587 
 
 374259 
 
 6 
 
 55 
 
 590387 
 
 498 
 
 964294 
 
 89 
 
 626093 
 
 587 
 
 373907 
 
 5 
 
 56 
 
 593686 
 
 497 
 
 964940 
 
 89 
 
 62(5445 
 
 586 
 
 373555 
 
 4 
 
 57 
 
 590984 
 
 497 
 
 964187 
 
 89 
 
 tvjt;-;i7 
 
 586 
 
 373203 
 
 3 
 
 58 
 
 591282 
 
 497 
 
 9B4133 
 
 89 
 
 6-27149 
 
 586 
 
 372851 
 
 2 
 
 59 
 
 591580 
 
 496 
 
 964080 
 
 89 
 
 627501 
 
 585 
 
 372499 
 
 1 
 
 60 
 
 59187Q 
 
 496 
 
 9W026 
 
 89 
 
 627852 
 
 585 
 
 372148 
 
 
 
 Coeine | | Sine | Cotang. | Tang, j M. ! 
 
 67 Degrees. 
 
SINES AND TANGENTS. (23 Degrees.) 
 
 4! 
 
 M. | Sine | D. | Cosine D. | Tang. | D. | Cotang. | 
 
 
 
 9.591878 
 
 496 
 
 0.984036 
 
 89 
 
 9.6278:12 
 
 585 
 
 10.372148 
 
 C,u 
 
 1 
 
 59-2170 
 
 495 
 
 9(53972 
 
 89 
 
 628203 
 
 585 
 
 3717!)7 
 
 59 
 
 2 
 
 59-2473 
 
 495 
 
 , 963919 
 
 89 
 
 628554 
 
 585 
 
 371446 
 
 58 
 
 3 
 
 592770 
 
 495 
 
 983865 
 
 90 
 
 628905 
 
 584 
 
 371095 
 
 57 
 
 4 
 
 593067 
 
 494 
 
 9153811 
 
 90 
 
 629255 
 
 584 
 
 370745 
 
 56 
 
 5 
 
 593303 
 
 494 
 
 ;Mi:i7^7 
 
 90 
 
 fi29;50 
 
 583 
 
 370394 
 
 55 
 
 6 
 
 5t3659 
 
 493 
 
 963704 
 
 90 
 
 629956 
 
 583 
 
 370044 
 
 54 
 
 7 
 
 593:>5.1 
 
 493 
 
 963650 
 
 90 
 
 630306 
 
 583 
 
 369694 
 
 53 
 
 8 
 
 59425J 
 
 493 
 
 9(53596 
 
 90 
 
 630656 
 
 583 
 
 369344 
 
 52 
 
 9 
 
 594547 
 
 492 
 
 963542 
 
 90 
 
 631005 
 
 582 
 
 368995 
 
 51 
 
 10 
 
 594842 
 
 492 
 
 963488 
 
 90 
 
 631355 
 
 582 
 
 368645 
 
 50 
 
 11 
 
 9.595137 
 
 491 
 
 9.963434 
 
 90 
 
 9.631704 
 
 582 
 
 10.368296 
 
 49 
 
 12 
 
 595432 
 
 491 
 
 963379 
 
 93 
 
 632053 
 
 581 
 
 367947 
 
 48 
 
 13 
 
 595727 
 
 491 
 
 963325 
 
 90 
 
 632401 
 
 581 
 
 367599 
 
 47 
 
 14 
 
 59602] 
 
 490 
 
 903271 
 
 90 
 
 632750 
 
 581 
 
 367250 
 
 46 
 
 15 
 
 508315 
 
 490 
 
 963217 
 
 90 
 
 633098 
 
 580 
 
 366902 
 
 45 
 
 16 
 
 59:5609 
 
 489 
 
 963163 
 
 9D 
 
 633447 
 
 580 
 
 366553 
 
 44 
 
 17 
 
 590903 
 
 489 
 
 963108 
 
 91 
 
 633795 
 
 580 
 
 366205 
 
 43 
 
 18 
 
 51)7191} 
 
 489 
 
 963054 
 
 91 
 
 634143 
 
 579 
 
 365857 
 
 42 
 
 lit 
 
 597490 
 
 488 
 
 962999 
 
 91 
 
 634490 
 
 579 
 
 365510 
 
 41 
 
 20 
 
 597783 
 
 488 
 
 962945 
 
 91 
 
 634838 
 
 579 
 
 365162 
 
 40 
 
 21 
 
 9.598075 
 
 487 
 
 9.962390 
 
 91 
 
 9-635185 
 
 578 
 
 10.364815 
 
 39 
 
 22 
 
 598368 
 
 487 
 
 962836 
 
 91 
 
 635532 
 
 578 
 
 364468 
 
 38 
 
 23 
 
 598600 
 
 487 
 
 962781 
 
 91 
 
 635879 
 
 578 
 
 364121 
 
 37 
 
 24 
 
 538952 
 
 486 
 
 962727 
 
 91 
 
 636226 
 
 577 
 
 363774 
 
 36 
 
 25 
 
 599244 
 
 486 
 
 962672 
 
 91 
 
 636572 
 
 577 
 
 36342S 
 
 35 
 
 86 
 
 599530 
 
 485 
 
 962617 
 
 91 
 
 636919 
 
 577 
 
 363081 
 
 34 
 
 27 
 
 5998-27 
 
 485 
 
 962562 
 
 91 
 
 637265 
 
 577 
 
 362735 
 
 33 
 
 28 
 
 61)0118 
 
 485 
 
 962508 
 
 91 
 
 637611 
 
 576 
 
 382389 
 
 32 
 
 29 
 
 600409 
 
 484 
 
 962153 
 
 91 
 
 637956 
 
 576 
 
 362044 
 
 31 
 
 30 
 
 61)0700 
 
 484 
 
 9ii23J8 
 
 92 
 
 638302 
 
 576 
 
 361698 
 
 30 
 
 31 
 
 9.600990 
 
 484 
 
 9.962343 
 
 92 
 
 9.638647 
 
 575 
 
 10.351353 
 
 29 
 
 32 
 
 601280 
 
 483 
 
 962288 
 
 92 
 
 638992 
 
 575 
 
 361008 
 
 28' 
 
 33 
 
 601570 
 
 483 
 
 9(52233 
 
 92 
 
 639337 
 
 575 
 
 360663 
 
 27 
 
 34 
 
 601880 
 
 482 
 
 962178 
 
 92 
 
 639682 
 
 574 
 
 360318 
 
 26 
 
 35 
 
 602150 
 
 482 
 
 962123 
 
 92 
 
 640027 
 
 574 
 
 359973 
 
 25 
 
 M 
 
 602439 
 
 482 
 
 983067 
 
 92 
 
 640371 
 
 574 
 
 359629 
 
 24 
 
 37 
 
 602728 
 
 481 
 
 982012 
 
 92 
 
 610716 
 
 573 
 
 359284 
 
 23 
 
 38 
 
 603017 
 
 481 
 
 981957 
 
 92 
 
 641000 
 
 573 
 
 358940 
 
 22 
 
 39 
 
 60331)5 
 
 481 
 
 951902 
 
 92 
 
 641404 
 
 573 
 
 358596 
 
 21 
 
 ; " 
 
 603594 
 
 480 
 
 961846 
 
 93 
 
 641747 
 
 572 
 
 358253 
 
 20 
 
 41 
 
 9.603882 
 
 480 
 
 9.961791 
 
 92 
 
 9-642091 
 
 572 
 
 10.357909 
 
 19 
 
 42 
 
 604170 
 
 479 
 
 9:51735 
 
 92 
 
 642434 
 
 572 
 
 35750(5 
 
 18 
 
 43 
 
 604457 
 
 479 
 
 961680 
 
 92 
 
 642777 
 
 572 
 
 357223 
 
 17 
 
 44 
 
 604745 
 
 479 
 
 901624 
 
 93 
 
 643120 
 
 571 
 
 356880 
 
 16 
 
 45 
 
 605032 
 
 478 
 
 961569 
 
 93 
 
 643463 
 
 571 
 
 356537 
 
 15 
 
 4(5 
 
 605319 
 
 478 
 
 961513 
 
 93 
 
 64381)0 
 
 571 
 
 356194 
 
 14 
 
 v47 
 
 605(503 
 
 478 
 
 961458 
 
 93 
 
 644148 
 
 570 
 
 355852 
 
 13 
 
 IB 
 
 605892 
 
 477 
 
 93] 402 
 
 93 
 
 644490 
 
 570 
 
 355510 
 
 12 
 
 49 
 
 606179 
 
 477 
 
 981346 
 
 93 
 
 644832 
 
 570 
 
 355168 
 
 11 
 
 50 
 
 606465 
 
 476 
 
 981299 
 
 93 
 
 645174 
 
 569 
 
 354826 
 
 10 
 
 51 
 
 9.696751 
 
 476 
 
 9.981235 
 
 93 
 
 9.645516 
 
 569 
 
 10.354484 
 
 9 
 
 -)_' 
 
 607038 
 
 476 
 
 961179 
 
 93 
 
 645857 
 
 569 
 
 3.141 13 
 
 8 
 
 so 
 
 607322 
 
 475 
 
 901123 
 
 93 
 
 640199 
 
 569 
 
 353801 
 
 7 
 
 54 
 
 607607 
 
 475 
 
 981067 
 
 93 
 
 646540 
 
 568 
 
 3534(50 
 
 6 
 
 55 
 
 607892 
 
 474 
 
 981011 
 
 93 
 
 646881 
 
 568 
 
 353119 
 
 5 
 
 5ti 
 
 808177 
 
 474 
 
 900955 
 
 93 
 
 647222 
 
 568 
 
 352778 
 
 4 
 
 57 
 
 608461 
 
 474 
 
 980899 
 
 93 
 
 647562 
 
 567 
 
 352438 
 
 3 
 
 58 
 
 608745 
 
 473 
 
 900813 
 
 94 
 
 647903 
 
 567 
 
 352097 
 
 2 
 
 59 
 
 609029 
 
 473 
 
 960786 
 
 91 
 
 648243 
 
 567 
 
 351757 
 
 1 
 
 60 
 
 609313 
 
 473 
 
 960730 
 
 94 
 
 MKJki 
 
 -( 
 
 351417 
 
 
 
 | Cosine | | Sine | Cotang. | Tang. | M. 1 
 
42 
 
 (24 Degrees.) A TABLE OF LOGARITHMIC 
 
 M. | Sine | D Cosine j D. | Tang | D. | Cotang. | 
 
 
 
 9.609313 
 
 473 
 
 9.960730 
 
 94 
 
 9.648583 
 
 566 
 
 10.351417 
 
 GO 1 
 
 1 
 
 C09597 
 
 472 
 
 960674 
 
 M 
 
 648923 
 
 566 
 
 351077 
 
 59 
 
 2 
 
 609880 
 
 472 
 
 960618 
 
 94 
 
 649263 
 
 566 
 
 350737 
 
 58 
 
 3 
 
 610164 
 
 472 
 
 960561 
 
 94 
 
 649602 
 
 568 
 
 350398 
 
 57 
 
 4 
 
 610447 
 
 471 
 
 960505 
 
 94 
 
 649942 
 
 565 
 
 350058 
 
 56 
 
 5 
 
 610729 
 
 471 
 
 9G0448 
 
 94 
 
 650281 
 
 565 
 
 349719 
 
 55 
 
 6 
 
 611012 
 
 470- 
 
 960332 
 
 94 
 
 650620 
 
 565 
 
 349380 
 
 54 
 
 7 
 
 6J 1294 
 
 470 
 
 960335 
 
 94 
 
 650959 
 
 564 
 
 349041 
 
 53 1 
 
 8 
 
 611576 
 
 470 
 
 960279 
 
 94 
 
 651297 
 
 564 
 
 648703 
 
 52 
 
 9 
 
 611858 
 
 469 
 
 960222 
 
 94 
 
 651636 
 
 564 
 
 348364 
 
 51 
 
 10 
 
 612140 
 
 469 
 
 9G01G5 
 
 94 
 
 G51974 
 
 563 
 
 318026 
 
 50 
 
 11 
 
 9.612421 
 
 469 
 
 9.960109 
 
 95 
 
 9.652312 
 
 563 
 
 10.347688 
 
 49 
 
 12 
 
 612702 
 
 468 
 
 960052 
 
 95 
 
 052650 
 
 563 
 
 347350 
 
 4d 
 
 13 
 
 612983 
 
 468 
 
 959995 
 
 95 
 
 652988 
 
 563 
 
 347012 
 
 47 
 
 14 
 
 613264 
 
 467 
 
 959938 
 
 95 
 
 6533-26 
 
 562 
 
 34R674 
 
 46 
 
 15 
 
 613545 
 
 467 
 
 959882 
 
 95 
 
 653663 
 
 562 
 
 346337 
 
 45 
 
 16 
 
 613825 
 
 467 
 
 959825 
 
 95 
 
 654000 
 
 562 
 
 341.000 
 
 44 
 
 17 
 
 614105 
 
 466 
 
 959768 
 
 95 
 
 654337 
 
 561 
 
 345663 
 
 43 
 
 18 
 
 614385 
 
 466 
 
 9597J1 
 
 95 
 
 654674 
 
 561 
 
 345326 
 
 42 
 
 19 
 
 614C65 
 
 466 
 
 959654 
 
 95 
 
 655011 
 
 561 
 
 344989 
 
 41 
 
 20 
 
 614944 
 
 465 
 
 959506 
 
 95 
 
 655348 
 
 561 
 
 344652 
 
 40 
 
 21 
 
 9.615223 
 
 465 
 
 9.959539 
 
 95 
 
 9.655684 
 
 560 
 
 10.344316 
 
 39 
 
 22 
 
 615502 
 
 465 
 
 959482 
 
 95 
 
 656020 
 
 560 
 
 343980 
 
 38 
 
 23 
 
 615781 
 
 464 
 
 959425 
 
 95 
 
 656356 
 
 560 
 
 343644 
 
 37 
 
 24 
 
 616060 
 
 464 
 
 9593(58 
 
 95 
 
 656692 
 
 559 
 
 343308 
 
 36 | 
 
 25 
 
 616338 
 
 464 
 
 959310 
 
 96 
 
 657028 
 
 559 
 
 342972 
 
 35 
 
 26 
 
 61(1616 
 
 463 
 
 959253 
 
 96 
 
 657364 
 
 559 
 
 342636 
 
 34 
 
 27 
 
 616894 
 
 463 
 
 959195 
 
 96 
 
 6571)99 
 
 559 
 
 342201 
 
 33 
 
 28 
 
 617172 
 
 462 
 
 959138 
 
 96 
 
 658034 
 
 558 
 
 341960 
 
 32 
 
 29 
 
 617450 
 
 462 
 
 959081 
 
 96 
 
 658369 
 
 558 
 
 341631 
 
 31 
 
 30 
 
 617727 
 
 462 
 
 959023 
 
 96 
 
 658704 
 
 558 
 
 341296 
 
 30 
 
 31 
 
 9.618004 
 
 461 
 
 9.958965 
 
 90 
 
 9.659039 
 
 558 
 
 10. 340901 
 
 29 
 
 32 
 
 618381 
 
 461 
 
 9o8908 
 
 96 
 
 659373 
 
 557 
 
 340G-27 
 
 28 
 
 33 
 
 618558 
 
 461 
 
 958850 
 
 96 
 
 659708 
 
 557 
 
 340292 
 
 27 
 
 34 
 
 618834 
 
 460 
 
 958792 
 
 96 
 
 660042 
 
 557 
 
 3399.58 
 
 26 
 
 35 
 
 619110 
 
 460 
 
 958734 
 
 95 
 
 660376 
 
 557 
 
 339624 
 
 25 
 
 36 
 
 619386 
 
 460 
 
 958677 
 
 96 
 
 660710 
 
 556 
 
 339290 
 
 24 
 
 37 
 
 619662 
 
 459 
 
 958G19 
 
 96 
 
 661043 
 
 556 
 
 338957 
 
 23 
 
 38 
 
 619938 
 
 459 
 
 958561 
 
 96 
 
 661377 
 
 556 
 
 338623 
 
 22 
 
 39 
 
 620213 
 
 459 
 
 9.58503 
 
 97 
 
 661710 
 
 555 
 
 338290 
 
 21 
 
 40 
 
 620488 
 
 458 
 
 958445 
 
 97 
 
 662043 
 
 555 
 
 337957 
 
 20 
 
 41 
 
 9.6207G3 
 
 458 
 
 9.958387 
 
 97 
 
 9.662376 
 
 555 
 
 10.337624 
 
 19 
 
 42 
 
 621038 
 
 457 
 
 958329 
 
 97 
 
 662709 
 
 554 
 
 337291 
 
 18 
 
 43 
 
 621313 
 
 457 
 
 958271 
 
 97 
 
 663042 
 
 554 
 
 336958 
 
 17 
 
 44 
 
 621587 
 
 457 
 
 958213 
 
 97 
 
 663375 
 
 554 
 
 336625 
 
 16 
 
 45 
 
 6218G1 
 
 456 
 
 958154 
 
 97 
 
 663707 
 
 554 
 
 336293 
 
 15 
 
 46 
 
 622135 
 
 456 
 
 958096 
 
 97 
 
 664039 
 
 553 
 
 335961 
 
 14 
 
 47 
 
 622409 
 
 456 
 
 958038 
 
 97 
 
 664371 
 
 553 
 
 335629 
 
 13 
 
 48 
 
 622682 
 
 455 
 
 957979 
 
 97 
 
 664703 
 
 553 
 
 335297 
 
 12 
 
 49 
 
 622936 
 
 455 
 
 957921 
 
 97 
 
 665035 
 
 553 
 
 334965 
 
 11 
 
 50 
 
 623229 
 
 455 
 
 957863 
 
 97 
 
 665366 
 
 552 
 
 334634 
 
 10 
 
 51 
 
 9.623502 
 
 454 
 
 9.957804 
 
 97 
 
 9.665697 
 
 552 
 
 10.334303 
 
 9 
 
 52 
 
 623774 
 
 454 
 
 957746 
 
 98 
 
 66G029 
 
 552 
 
 333971 
 
 8 
 
 53 
 
 621047 
 
 454 
 
 957087 
 
 98 
 
 666360 
 
 551 
 
 333640 
 
 7 
 
 54 
 
 624319 
 
 453 
 
 957628 
 
 98 
 
 666691 
 
 551 
 
 333309 
 
 6 
 
 55 
 
 624591 
 
 453 
 
 957570 
 
 98 
 
 667021 
 
 551 
 
 332979 
 
 5 
 
 56 
 
 G248f>3 
 
 453 
 
 957511 
 
 98 
 
 667352 
 
 551 
 
 332648 
 
 4 
 
 57 
 
 623135 
 
 452 
 
 957452 
 
 98 ' 
 
 667682 
 
 550 
 
 332318 
 
 3 
 
 58 
 
 625406 
 
 452 
 
 957393 
 
 98 
 
 6f8:)13 
 
 550 
 
 331987 
 
 
 
 ' 59 
 
 625(777 
 
 452 
 
 957335 
 
 9H 
 
 668343 
 
 550 
 
 331657 
 
 1 
 
 60 
 
 625948 
 
 451 
 
 95727G 
 
 98 
 
 668372 
 
 550 
 
 331328 
 
 
 
 1 Cosine | | Sine | Cotang. | | Tang. | M. 
 
 Go Degrees. 
 
SINES AND TANGENTS. (25 Degrees.) 
 
 j M. Sine D. Cosine D. | Tang. D. CoLing. | 
 
 
 
 9.625948 
 
 451 
 
 9.957276 
 
 98 U. 668673 
 
 550 
 
 10.331327 
 
 6( 
 
 1 
 
 626219 
 
 451 
 
 957217 
 
 98 
 
 669002 
 
 549 
 
 330998 
 
 5 
 
 2 
 
 626490 
 
 451 
 
 957158 
 
 98 
 
 (5f!93:J2 
 
 549 
 
 330668 
 
 5 
 
 3 
 
 6207(50 
 
 450 
 
 957099 
 
 98 
 
 669(561 
 
 549 
 
 330339 
 
 57 
 
 4 
 
 627030 
 
 450 
 
 957040 
 
 98 
 
 669991 
 
 548 
 
 330009 
 
 56 
 
 5 
 
 637300 
 
 450 
 
 956981 
 
 98 
 
 670320 
 
 548 
 
 329080 
 
 55 
 
 6 
 
 637570 
 
 449 
 
 956921 
 
 99 
 
 670049 
 
 548 
 
 329351 
 
 54 
 
 7 
 
 627840 
 
 449 
 
 956843 
 
 99 
 
 070977 
 
 548 
 
 329023 
 
 51 
 
 8 
 
 628109 
 
 449 
 
 956803 
 
 99 
 
 671306 
 
 547 
 
 328694 
 
 5 
 
 9 
 
 628:578 
 
 448 
 
 950744 
 
 99 
 
 671634 
 
 547 
 
 328366 
 
 5] 
 
 10 
 
 628647 
 
 448 
 
 956684 
 
 99 
 
 671963 
 
 547 
 
 328037 
 
 50 
 
 11 
 
 9.628916 
 
 447 
 
 9.956625 
 
 99 
 
 9.672291 
 
 547 
 
 10.327709 
 
 4:1 : 
 
 12 
 
 629185 
 
 447 
 
 956506 
 
 99 
 
 672619 
 
 546 
 
 327381 
 
 48 
 
 13 
 
 629453 
 
 447 
 
 950506 
 
 99 
 
 672947 
 
 546 
 
 327053 
 
 47 
 
 14 
 
 6297-21 
 
 446 
 
 956447 
 
 99 
 
 673274 
 
 546 
 
 326726 
 
 46 
 
 15 
 
 629989 
 
 446 
 
 956387 
 
 99 
 
 673602 
 
 546 
 
 326398 
 
 45 
 
 16 
 
 630257 
 
 446 
 
 956327 
 
 99 
 
 673929 
 
 545 
 
 326071 
 
 44 
 
 17 
 
 630.124 
 
 446 
 
 9562(58 
 
 99 
 
 674257 
 
 545 
 
 325743 
 
 43 
 
 18 
 
 630792 
 
 445 
 
 956208 
 
 100 
 
 674584 
 
 545 
 
 325416 
 
 42 
 
 M 
 
 631051) 
 
 445 
 
 956148 
 
 100 
 
 674910 
 
 544 
 
 325090 
 
 41 
 
 20 
 
 631326 
 
 445 
 
 956089 
 
 100 
 
 675237 
 
 544 
 
 324763 
 
 40 
 
 21 
 
 9.631593 
 
 444 
 
 9.956029 
 
 100 
 
 9.675564 
 
 544 
 
 10.324436 
 
 39 
 
 22 
 
 631859 
 
 444 
 
 955969 
 
 100 
 
 675890 
 
 544 
 
 324110 
 
 38 
 
 23 
 
 632125 
 
 444 
 
 955909 
 
 100 
 
 676216 
 
 543 
 
 323784 
 
 37 
 
 24 
 
 632392 
 
 443 
 
 955849 
 
 100 
 
 676543 
 
 543 
 
 323457 
 
 36 
 
 25 
 
 632658 
 
 443 
 
 955789 
 
 100 
 
 676869 
 
 543 
 
 323131 
 
 35 
 
 26 
 
 632923 
 
 443 
 
 955729 
 
 100 
 
 677194 
 
 543 
 
 3i2806 
 
 34 
 
 27 
 
 633189 
 
 442 
 
 955669 
 
 100 
 
 677520 
 
 542 
 
 322480 
 
 33 
 
 28 
 
 633454 
 
 442 
 
 955609 
 
 100 
 
 677846 
 
 542 
 
 322154 
 
 32 
 
 29 
 
 633719 
 
 442 
 
 955548 
 
 100 
 
 678171 
 
 542 
 
 321829 
 
 31 
 
 30 
 
 633984 
 
 441 
 
 955488 
 
 100 
 
 67iM96 
 
 542 
 
 321504 
 
 30 
 
 31 
 
 9.634249 
 
 441 
 
 9.955428 
 
 101 
 
 9.678821 
 
 541 
 
 10.321179 
 
 29 
 
 32 
 
 634514 
 
 440 
 
 955368 
 
 101 
 
 679146 
 
 541 
 
 320854 
 
 28 
 
 33 
 
 634778 
 
 440 
 
 955307 
 
 101 
 
 679471 
 
 541 
 
 320529 
 
 27 
 
 34 
 
 635042 
 
 440 
 
 955247 
 
 101 
 
 679795 
 
 541 
 
 320205 
 
 26 
 
 35 
 
 635306 
 
 439 
 
 955186 
 
 101 
 
 680120 
 
 540 
 
 319880 
 
 25 
 
 36 
 
 635570 
 
 439 
 
 955126 
 
 101 
 
 680444 
 
 540 
 
 319556 
 
 24 
 
 37 
 
 635834 
 
 439 
 
 955065 
 
 101 
 
 680768 
 
 540 
 
 319232 
 
 23 
 
 38 
 
 636097 
 
 438 
 
 955005 
 
 101 
 
 681092 
 
 540 
 
 318908 
 
 22 
 
 39 
 
 636360 
 
 438 
 
 954944 
 
 101 
 
 681416 
 
 539 
 
 318584 
 
 21 
 
 40 
 
 636623 
 
 438 
 
 954883 
 
 101 
 
 681740 
 
 539 
 
 318260 
 
 20 
 
 41 
 
 9.636886 
 
 437 
 
 9.954823 
 
 101 
 
 9.682063 
 
 539 
 
 10.317937 
 
 19 
 
 42 
 
 637148 
 
 437 
 
 954762 
 
 101 
 
 682387 
 
 539 
 
 317613 
 
 18 
 
 43 
 
 637411 
 
 437 
 
 954701 
 
 101 
 
 682710 
 
 538 
 
 317290 
 
 17 
 
 44 
 
 637673 
 
 437 
 
 954640 
 
 101 
 
 683033 
 
 538 
 
 316967 
 
 16 
 
 45 
 
 637935 
 
 436 
 
 954579 
 
 101 
 
 683356 
 
 538 
 
 316644 
 
 15 
 
 46 
 
 638197 
 
 436 
 
 954518 
 
 102 
 
 683679 
 
 538 
 
 316321 
 
 14 
 
 47 
 
 638458 
 
 436 
 
 954457 
 
 102 
 
 684001 
 
 537 
 
 315999 
 
 13 
 
 48 
 
 638720 
 
 435 
 
 954396 
 
 102 
 
 684324 
 
 537 
 
 315676 
 
 12 
 
 49 
 
 638981 
 
 435 
 
 954335 
 
 102 
 
 684646 
 
 537 
 
 315354 
 
 11 
 
 50 
 
 639242 
 
 435 
 
 954274 
 
 102 
 
 684968 
 
 537 
 
 315032 
 
 10 
 
 51 
 
 9.639503 
 
 434 
 
 9.954213 
 
 102 
 
 9.685290 
 
 536 
 
 10.314710 
 
 9 
 
 52 
 
 639764 
 
 434 
 
 954152 
 
 102 
 
 685672 
 
 536 
 
 314388 
 
 8 
 
 53 
 
 640024 
 
 434 
 
 95-1090 
 
 102 
 
 685934 
 
 536 
 
 314066 
 
 7 
 
 54 
 
 640284 
 
 433 
 
 954029 
 
 102 
 
 686255 
 
 536 
 
 313745 
 
 6 
 
 55 
 
 640544 
 
 433 
 
 953908 
 
 102 
 
 686577 
 
 535 
 
 313423 
 
 5 
 
 56 
 
 640804 
 
 433 
 
 953906 
 
 102 
 
 686898 
 
 535 
 
 313102 
 
 4 
 
 57 
 
 641064 
 
 432 
 
 953845 
 
 102 
 
 087*19 
 
 535 
 
 312781 
 
 3 
 
 58 
 
 641324 
 
 432 
 
 953783 
 
 102 
 
 687540 
 
 535 
 
 312460 
 
 2 
 
 59 
 
 641584 
 
 432 
 
 953722 
 
 103 
 
 687861 
 
 534 
 
 312139 
 
 1 
 
 60 
 
 641842 
 
 431 
 
 953660 
 
 103 
 
 r,Krflr2 
 
 534 
 
 3I1H1H 
 
 o I 
 
 | Cosine | Sine | Cotang. Tang. M. 
 
 64 Degrees. 
 
44 
 
 (26 Degrees.) A TABLE OF LOGARITHMIC 
 
 ' M. | Sine | D. | Cosine | D. | Tang. | D. | Cotang. 
 
 
 
 
 9.641842 
 
 431 
 
 9-953660 
 
 103 
 
 9.688182 
 
 534 
 
 10.311818 
 
 60 I 
 
 1 
 
 642101 
 
 431 
 
 953599 
 
 103 
 
 088502 
 
 534 
 
 311498 
 
 59 
 
 2 
 
 642360 
 
 431 
 
 953537 
 
 103 
 
 688823 
 
 534 
 
 311177 
 
 58 
 
 3 
 
 642618 
 
 430 
 
 953475 
 
 103 
 
 689143 
 
 533 
 
 310857 
 
 57 
 
 4 
 
 642877 
 
 430 
 
 953413 
 
 103 
 
 689463 
 
 533 
 
 310537 
 
 56 
 
 5 
 
 643135 
 
 430 
 
 953352 
 
 103 
 
 689783 
 
 533 
 
 310217 
 
 55 
 
 6 
 
 643393 
 
 430 
 
 953290 
 
 103 
 
 690103 
 
 533 
 
 309897 
 
 54 
 
 7 
 
 643050 
 
 429 
 
 953228 
 
 103 
 
 690423 
 
 533 
 
 309577 
 
 53 
 
 8 
 
 643908 
 
 429 
 
 953166 
 
 103 
 
 690742 
 
 532 
 
 309258 
 
 52 
 
 9 
 
 644165 
 
 429 
 
 953104 
 
 103 
 
 691062 
 
 532 
 
 308938 
 
 51 
 
 10 
 
 644423 
 
 428 
 
 953042 
 
 103 
 
 691381 
 
 532 
 
 308619 
 
 50 
 
 11 
 
 9.644680 
 
 428 
 
 9.952980 
 
 104 
 
 9.691700 
 
 531 
 
 10.308300 
 
 49 
 
 12 
 
 644936 
 
 428 
 
 952918 
 
 104 
 
 692019 
 
 531 
 
 307981 
 
 48 
 
 13 
 
 645193 
 
 427 
 
 952855 
 
 104 
 
 692338 
 
 531 
 
 307662 
 
 47 
 
 14 
 
 645450 
 
 427 
 
 952793 
 
 104 
 
 692656 
 
 531 
 
 307344 
 
 46 
 
 15 
 
 645706 
 
 427 
 
 952731 
 
 104 
 
 692975 
 
 531 
 
 307025 
 
 45 
 
 16 
 
 645962 
 
 426 
 
 952669 
 
 104 
 
 693293 
 
 530 
 
 306707 
 
 44 
 
 17 
 
 646218 
 
 426 
 
 952606 
 
 104 
 
 693612 
 
 530 
 
 308388 
 
 43 
 
 18 
 
 646474 
 
 426 
 
 952544 
 
 104 
 
 693930 
 
 530 
 
 306070 
 
 42 
 
 19 
 
 646729 
 
 425 
 
 952481 
 
 104 
 
 694248 
 
 530 
 
 305752 
 
 41 
 
 20 
 
 646984 
 
 425 
 
 952419 
 
 104 
 
 694566 
 
 529 
 
 305434 
 
 40 
 
 21 
 
 9.647240 
 
 425 
 
 9.952356 
 
 104 
 
 9.694883 
 
 529 
 
 10.305117 
 
 39 
 
 22 
 
 647494 
 
 424 
 
 952294 
 
 104 
 
 695201 
 
 529 
 
 304799 
 
 38 
 
 23 
 
 647749 
 
 424 
 
 952231 
 
 104 
 
 695518 
 
 529 
 
 304482 
 
 37 
 
 24 
 
 648004 
 
 424 
 
 952168 
 
 105 
 
 695836 
 
 529 
 
 304164 
 
 36 
 
 25 
 
 648258 
 
 424 
 
 952106 
 
 105 
 
 696153 
 
 528 
 
 303847 
 
 35 
 
 26 
 
 648512 
 
 423 
 
 952043 
 
 105 
 
 696470 
 
 528 
 
 303530 
 
 34 
 
 27 
 
 648766 
 
 423 
 
 951980 
 
 105 
 
 696787 
 
 528 
 
 303213 
 
 33 
 
 28 
 
 649020 
 
 423 
 
 951917 
 
 105 
 
 697103 
 
 528 
 
 302897 
 
 32 
 
 29 
 
 649274 
 
 422 
 
 951854 
 
 105 
 
 697420 
 
 527 
 
 302580 
 
 31 
 
 30 
 
 649527 
 
 422 
 
 951791 
 
 105 
 
 697736 
 
 527 
 
 302264 
 
 30 
 
 31 
 
 9-649781 
 
 422 
 
 9.951728 
 
 105 
 
 9.693053 
 
 527 
 
 10.301947 
 
 29 
 
 32 
 
 650034 
 
 422 
 
 951665 
 
 105 
 
 693369 
 
 527 
 
 301631 
 
 28 
 
 33 
 
 650287 
 
 421 
 
 951602 
 
 105 
 
 698685 
 
 526 
 
 301315 
 
 27 
 
 34 
 
 650539 
 
 421 
 
 951539 
 
 105 
 
 699001 
 
 526 
 
 300999 
 
 26 
 
 35 
 
 650792 
 
 421 
 
 951476 
 
 105 
 
 699316 
 
 526 
 
 300684 
 
 25 
 
 36 
 
 651044 
 
 420 
 
 951412 
 
 105 
 
 699632 
 
 526 
 
 300368 
 
 24 
 
 37 
 
 651297 
 
 420 
 
 951349 
 
 106 
 
 699947 
 
 526 
 
 3001)53 
 
 23 
 
 38 
 
 651549 
 
 420 
 
 951286 
 
 106 
 
 700263 
 
 525 
 
 299737 
 
 22 
 
 39 
 
 651800 
 
 419 
 
 951222 
 
 106 
 
 700578 
 
 525 
 
 299422 
 
 21 
 
 40 
 
 652052 
 
 419 
 
 951159 
 
 106 
 
 700893 
 
 525 
 
 299107 
 
 20 
 
 41 
 
 9.652304 
 
 419 
 
 9.951096 
 
 106 
 
 9.701208 
 
 524 
 
 10.298792 
 
 19 
 
 42 
 
 652555 
 
 418 
 
 951032 
 
 106 
 
 701523 
 
 524 
 
 298477 
 
 18 
 
 43 
 
 652806 
 
 418 
 
 950968 
 
 106 
 
 701837 
 
 524 
 
 298163 
 
 17 
 
 44 
 45 
 
 653057 
 653308 
 
 418 
 418 
 
 950905 
 950841 
 
 106 
 106 
 
 702152 
 702466 
 
 524 
 524 
 
 297848 
 297534 
 
 16 
 15 
 
 46 
 
 653558 
 
 417 
 
 950778 
 
 106 
 
 702780 
 
 523 
 
 297220 
 
 14 
 
 47 
 
 653808 
 
 417 
 
 950714 
 
 106 
 
 703095 
 
 523 
 
 296905 
 
 13 
 
 48 
 
 654059 
 
 417 
 
 950650 
 
 106 
 
 703409 
 
 523 
 
 296591 
 
 12 
 
 49 
 
 654309 
 
 416 
 
 950586 
 
 106 
 
 703723 
 
 523 
 
 296277 
 
 11 
 
 50 
 
 654558 
 
 416 
 
 950522 
 
 107 
 
 704036 
 
 522 
 
 295964 
 
 10 
 
 51 
 
 9.654808 
 
 416 
 
 9.950458 
 
 107 
 
 9.704350 
 
 522 
 
 10.295650 
 
 9 
 
 52 
 
 655058 
 
 416 
 
 950394 
 
 107 
 
 704663 
 
 522 
 
 295337 
 
 8 
 
 53 
 
 655307 
 
 415 
 
 950330 
 
 107 
 
 704977 
 
 522 
 
 295023 
 
 7 
 
 54 
 
 655556 
 
 415 
 
 950266 
 
 107 
 
 705290 
 
 522 
 
 294710 
 
 6 
 
 55 
 
 655805 
 
 415 
 
 950202 
 
 107 
 
 705603 
 
 521 
 
 294397 
 
 5 
 
 56 
 
 656054 
 
 414 
 
 950138 
 
 107 
 
 705916 
 
 521 
 
 294084 
 
 4 
 
 57 
 
 656302 
 
 414 
 
 IS0074 
 
 107 
 
 706228 
 
 521 
 
 293772 
 
 3 
 
 58 
 
 656551 
 
 414 
 
 950010 
 
 107 
 
 706541 
 
 521 
 
 293459 
 
 2 
 
 59 
 
 656799 
 
 413 
 
 949945 
 
 107 
 
 706854 
 
 521 
 
 293146 
 
 1 
 
 60 
 
 657047 
 
 413 
 
 949881 
 
 107 
 
 707166 
 
 520 
 
 292834 
 
 
 
 | Cosine | | Sine | Cotang. | 
 
 Tang. | M. | 
 
 63 Degrees. 
 
SINES AND TANGENTS. (27 Degrees.) 
 
 -15 
 
 M. | Sine | D. | Cosine | D. Tang. | D. Cotonar. 1 
 
 
 
 9.657047 
 
 413 
 
 9.949881 
 
 107 
 
 9. 707 Kiii 
 
 520 
 
 10.2!>2H34 
 
 60 
 
 1 
 
 657295 
 
 413 
 
 949816 
 
 107 
 
 707478 
 
 520 
 
 292522 
 
 59 
 
 2 
 
 657542 
 
 41-2 
 
 949752 
 
 107 
 
 707790 
 
 520 
 
 292210 
 
 58 
 
 3 
 
 657790 
 
 412 
 
 949688 
 
 108 
 
 708102 
 
 520 
 
 291898 
 
 57 
 
 4 
 
 658037 
 
 412 
 
 949823 
 
 108 
 
 708414 
 
 519 
 
 291586 
 
 56 
 
 5 
 
 658284 
 
 419 
 
 949558 
 
 108 
 
 708726 
 
 519 
 
 291274 
 
 55 
 
 6 
 
 658531 
 
 411 
 
 949494 
 
 108 
 
 709037 
 
 519 
 
 291)963 
 
 54 
 
 7 
 
 6r>>77H 
 
 411 
 
 949439 
 
 108 
 
 701)349 
 
 519 
 
 290651 
 
 53 
 
 8 
 
 659025 
 
 411 
 
 9493154 
 
 108 
 
 709(560 
 
 519 
 
 290340 
 
 52 
 
 9 
 
 (J5927I 
 
 410 
 
 . 949300 
 
 108 
 
 709971 
 
 518 
 
 290029 
 
 51 
 
 10 
 
 659517 
 
 410 
 
 949235 
 
 108 
 
 710883 
 
 518 
 
 289718 
 
 50 
 
 11 
 
 9.659763 
 
 410 
 
 9.949170 
 
 108 
 
 9.710593 
 
 518 
 
 10.289407 
 
 49 
 
 12 
 
 6600:)9 
 
 409 
 
 949105 
 
 108 
 
 710904 
 
 518 
 
 289096 
 
 48 
 
 13 
 
 <i<K)2.5 
 
 409 
 
 949040 
 
 108 
 
 711215 
 
 818 
 
 288785 
 
 47 
 
 14 
 
 660501 
 
 409 
 
 948975 
 
 108 
 
 711525 
 
 517 
 
 288475 
 
 46 
 
 15 
 
 6JHI74G 
 
 409 
 
 948910 
 
 108 
 
 711836 
 
 517 
 
 288164 
 
 45 
 
 16 
 
 660991 
 
 408 
 
 948845 
 
 108 
 
 712146 
 
 517 
 
 287854 
 
 44 
 
 17 
 
 661236 
 
 408 
 
 948780 
 
 109 
 
 712456 
 
 517 
 
 287544 
 
 43 
 
 18 
 
 661481 
 
 408 
 
 948715 
 
 109 
 
 71-27(5(5 
 
 516 
 
 287234 
 
 42 
 
 19 
 
 661726 
 
 407 
 
 948650 
 
 109 
 
 713076 
 
 516 
 
 286924 
 
 41 
 
 20 
 
 661970 
 
 407 
 
 948584 
 
 109 
 
 713386 
 
 516 
 
 286614 
 
 40 
 
 21 
 
 9.662214 
 
 407 
 
 9.948519 
 
 109 
 
 9.71369S 
 
 516 
 
 10.286304 
 
 39 
 
 22 
 
 662459 
 
 407 
 
 948454 
 
 109 
 
 714005 
 
 516 
 
 285995 
 
 38 
 
 23 
 
 (5(52703 
 
 406 
 
 948388 
 
 109 
 
 714314 
 
 515 
 
 285686 
 
 37 
 
 24 
 
 6G2946 
 
 40(5 
 
 948323 
 
 109 
 
 714624 
 
 515 
 
 285376 
 
 36 
 
 25 
 
 663190 
 
 406 
 
 948257 
 
 109 
 
 714933 
 
 515 
 
 285067 
 
 35 
 
 20 
 
 663433 
 
 405 
 
 948192 
 
 109 
 
 715242 
 
 515 
 
 284758 
 
 34 
 
 27 
 
 (563677 
 
 405 
 
 9481215 
 
 109 
 
 715551 
 
 514 
 
 284449 
 
 33 
 
 28 
 
 663920 
 
 405 
 
 948060 
 
 109 
 
 715800 
 
 514 
 
 234140 
 
 32 
 
 29 
 
 664163 
 
 405 
 
 947995 
 
 110 
 
 716168 
 
 514 
 
 283832 
 
 31 
 
 30 
 
 664406 
 
 404 
 
 947929 
 
 110 
 
 716477 
 
 514 
 
 283523 
 
 30 
 
 31 
 
 9.664648 
 
 404 
 
 9.947863 
 
 110 
 
 9.716785 
 
 514 
 
 10.283215 
 
 29 
 
 32 
 
 664891 
 
 404 
 
 947797 
 
 110 
 
 717093 
 
 513 
 
 282907 
 
 28 
 
 33 
 
 665133 
 
 403 
 
 917731 
 
 110 
 
 717401 
 
 513 
 
 282599 
 
 27 
 
 34 
 
 665375 
 
 403 
 
 947665 
 
 110 
 
 717709 
 
 513 
 
 282291 
 
 26 
 
 35 
 
 6(55617 
 
 403 
 
 947C.OO 
 
 110 
 
 718017 
 
 513 
 
 281983 
 
 25 
 
 36 
 
 665859 
 
 402 
 
 947533 
 
 110 
 
 718325 
 
 513 
 
 281675 
 
 24 
 
 37 
 
 6G610i) 
 
 402 
 
 94"; 4H7 
 
 110 
 
 718633 
 
 512 
 
 281367 
 
 23 
 
 38 
 
 666342 
 
 402 
 
 947401 
 
 110 
 
 718940 
 
 512 
 
 281060 
 
 22 
 
 39 
 
 668583 
 
 -102 
 
 947:j:{.-> 
 
 110 
 
 719248 
 
 512 
 
 280752 
 
 21 
 
 40 
 
 666824 
 
 401 
 
 947289 
 
 110 
 
 719555 
 
 512 
 
 280445 
 
 20 
 
 41 
 
 9.667065 
 
 401 
 
 9.947203 
 
 110 
 
 9.719862 
 
 512 
 
 10.280138 
 
 19 
 
 49 
 
 667305 
 
 401 
 
 94713(5 
 
 111 
 
 720169 
 
 511 
 
 279831 
 
 18 
 
 43 
 
 667546 
 
 401 
 
 947070 
 
 111 
 
 720476 
 
 511 
 
 279524 
 
 17 
 
 44 
 
 667786 
 
 400 
 
 947004 
 
 111 
 
 720783 
 
 511 
 
 279217 
 
 16 
 
 45 
 
 668027 
 
 400 
 
 946937 
 
 111 
 
 721089 
 
 511 
 
 278911 
 
 15 
 
 46 
 
 668267 
 
 400 
 
 946871 
 
 111 
 
 721398 
 
 511 
 
 278604 
 
 14 
 
 47 
 
 668508 
 
 399 
 
 946804 
 
 111 
 
 721702 
 
 510 
 
 278298 
 
 13 
 
 48 
 
 668746 
 
 399 
 
 946738 
 
 111 
 
 722009 
 
 510 
 
 277991 
 
 12 
 
 49 
 
 668936 
 
 399 
 
 946671 
 
 111 
 
 722315 
 
 510 
 
 277685 
 
 11 
 
 50 
 
 669225 
 
 399 
 
 946604 
 
 111 
 
 722621 
 
 510 
 
 277379 
 
 10 
 
 51 
 
 9.669464 
 
 398 
 
 9.946538 
 
 111 
 
 9.722927 
 
 510 
 
 10.277073 
 
 9 
 
 B 
 
 669703 
 
 398 
 
 946471 
 
 111 
 
 723232 
 
 509 
 
 276768 
 
 8 
 
 53 
 
 669942 
 
 398 
 
 946404 
 
 111 
 
 723538 
 
 509 
 
 276462 
 
 7 
 
 54 
 
 670181 
 
 397 
 
 946337 
 
 111 
 
 723844 
 
 509 
 
 276156 
 
 6 
 
 55 
 
 610419 
 
 397 
 
 946270 
 
 112 
 
 7-24149 
 
 509 
 
 275851 
 
 5 
 
 56 
 
 670658 
 
 397 
 
 946203 
 
 112 
 
 724454 
 
 509 
 
 275546 
 
 4 
 
 57 
 
 67089S 
 
 397 
 
 94(5136 
 
 112 
 
 724759 
 
 508 
 
 275241 
 
 3 
 
 58 
 
 671134 
 
 396 
 
 946069 
 
 112 
 
 725065 
 
 508 
 
 274935 
 
 2 
 
 59 671372 
 
 396 
 
 946002 
 
 112 
 
 725369 
 
 508 
 
 274(531 
 
 1 
 
 60 1 671609 
 
 396 
 
 945935 
 
 112 
 
 725674 
 
 508 
 
 274326 
 
 
 
 | Cosine 1 Sine j | Cotang. | | Tang. | M. ' 
 
 62 Degrees 
 
Degrees.) A TABLE OT 
 
 M. ) Sine I D. I Cosine f D. j Tang. D. j Cotang. 
 
 
 
 9.671009 
 
 396 
 
 9.1M5U:J5 
 
 112 
 
 9.725674 
 
 508 
 
 10.274326 1 00 
 
 1 
 
 671847 
 
 395 
 
 9458(18 
 
 112 
 
 725979 
 
 508 
 
 27402J 
 
 59 
 
 o 
 
 67-2084 
 
 395 
 
 945800 
 
 112 
 
 7-J62C4 
 
 507 
 
 273718 
 
 58 
 
 3 
 
 672321 
 
 385 
 
 945733 
 
 112 
 
 726583 
 
 507 
 
 273412 
 
 57 
 
 4 
 
 672f>58 
 
 395 
 
 945666 
 
 112 
 
 7-ji ; e.<i-. 
 
 507 
 
 27.1 JOS 
 
 56 
 
 5 
 
 672795 
 
 394 
 
 9455JW 
 
 1J2 
 
 727197 
 
 507 
 
 272803 
 
 55 
 
 6 
 
 673032 
 
 394 
 
 945531 
 
 112 
 
 727501 
 
 507 
 
 272499 
 
 54 
 
 7 
 
 673268 
 
 394 
 
 9454154 
 
 113 
 
 727805 
 
 506 
 
 272195 
 
 53 
 
 8 
 
 673505 
 
 394 
 
 945390 
 
 113 
 
 728109 
 
 506 
 
 271891 
 
 52 
 
 9 
 
 673741 
 
 393 
 
 945328 
 
 113 
 
 728412 
 
 506 
 
 27 J 588 
 
 51 
 
 10 
 
 673977 
 
 3U3 
 
 945261 
 
 113 
 
 728716 
 
 506 
 
 271284 
 
 50 
 
 11 
 
 9.674213 
 
 393 
 
 9-945193 
 
 113 
 
 9.729020 
 
 50G 
 
 10.3705)80 
 
 49 
 
 12 
 
 674448 
 
 392 
 
 9451 -25 
 
 113 
 
 729323 
 
 505 
 
 270677 
 
 48 |. 
 
 13 
 
 674684 
 
 392 
 
 945058 
 
 113 
 
 729(526 
 
 505 
 
 270374 
 
 47 
 
 14 
 
 674919 
 
 392 
 
 944990 
 
 113 
 
 729929 
 
 505 
 
 270071 
 
 4(5 
 
 15 
 
 675155 
 
 392 
 
 9449-22 
 
 113 
 
 730333 
 
 505 
 
 859767 
 
 45 
 
 16 
 
 675390 
 
 391 
 
 944854 
 
 113 
 
 730535 
 
 505 
 
 2694(55 
 
 44 
 
 17 
 
 675624 
 
 391 
 
 944786 
 
 113 
 
 730838 
 
 504 
 
 ^69162 
 
 43 
 
 18 
 
 675859 
 
 391 
 
 944718 
 
 113 
 
 731141 
 
 504 
 
 268859 
 
 42 
 
 19 
 
 676094 
 
 391 
 
 944650 
 
 113 
 
 731444 
 
 504 
 
 2(58556 
 
 41 
 
 20 
 
 676328 
 
 390 
 
 . 944582 
 
 114 
 
 731746 
 
 504 
 
 268254 
 
 40 
 
 21 
 
 9.676562 
 
 390 
 
 9.944514 
 
 114 
 
 9.732048 
 
 504 
 
 10.267952 
 
 39 
 
 22 
 
 676796 
 
 390 
 
 94444(i 
 
 114 
 
 732351 
 
 503 
 
 267049 
 
 38 
 
 23 
 
 677030 
 
 390 
 
 944377 
 
 114 
 
 732653 
 
 503 
 
 267347 
 
 37 
 
 24 
 
 677264 
 
 389 
 
 944309 
 
 114 
 
 732955 
 
 503 
 
 267045 
 
 36 
 
 25 
 
 677498 
 
 389 
 
 944241 
 
 114 
 
 733257 
 
 503 
 
 266743 
 
 35 
 
 26 
 
 677731 
 
 389 
 
 944172 
 
 114 
 
 733558 
 
 503 
 
 266442 
 
 34 
 
 27 
 
 677964 
 
 388 
 
 944104 
 
 114 
 
 733800 
 
 502 
 
 206140 
 
 33 
 
 28 
 
 678197 
 
 388 
 
 944036 
 
 114 
 
 734162 
 
 502 
 
 205838 
 
 32 
 
 29 
 
 678430 
 
 388 
 
 943967 
 
 114 
 
 734463 
 
 502 
 
 265537 
 
 31 
 
 30 
 
 678663 
 
 388 
 
 943899 
 
 114 
 
 7347(54 
 
 502 
 
 265236 
 
 30 
 
 31 
 
 9.678895 
 
 387 
 
 9.943830 
 
 114 
 
 9.735066 
 
 502 
 
 10.2/54934 
 
 29 
 
 32 
 
 679128 
 
 387 
 
 943761 
 
 114 
 
 735367 
 
 502 
 
 84633 
 
 28 
 
 33 
 
 679360 
 
 387 
 
 943<>93 
 
 115 
 
 735(568 
 
 501 
 
 264332 
 
 27 
 
 34 
 
 679592 
 
 387 
 
 943024 
 
 115 
 
 735980 
 
 501 
 
 204031 
 
 26 
 
 35 
 
 679824 
 
 386 
 
 943555 
 
 115 
 
 7302(79 
 
 501 
 
 203731 
 
 25 
 
 36 
 
 680056 
 
 386 
 
 94348G 
 
 115 
 
 736570 
 
 501 
 
 263430 
 
 24 
 
 37 
 
 680288 
 
 386 
 
 943417 
 
 115 
 
 730871 
 
 501 
 
 263129 
 
 23 
 
 38 
 
 6805J9 
 
 385 
 
 943348 
 
 115 
 
 737171 
 
 500 
 
 2G2829 
 
 22 
 
 39 
 
 680750 
 
 385 
 
 943279 
 
 115 
 
 737471 
 
 500 
 
 262529 
 
 21 
 
 40 
 
 680982 
 
 385 
 
 943210 
 
 115 
 
 737771 
 
 500 
 
 2G2229 
 
 20 
 
 41 
 
 9.681213 
 
 385 
 
 9.943141 
 
 115 
 
 9.738071 
 
 500 
 
 10.261929 
 
 19 
 
 42 
 
 681443 
 
 384 
 
 943072 
 
 115 
 
 738371 
 
 500 
 
 261029 
 
 18 
 
 43 
 
 681674 
 
 384 
 
 943003 
 
 115 
 
 738071 
 
 499 
 
 261329 
 
 17 
 
 44 
 
 681905 
 
 384 
 
 942934 
 
 115 
 
 738971 
 
 499 
 
 2GI029 
 
 16 
 
 45 
 
 682135 
 
 384 
 
 942864 
 
 115 
 
 739271 
 
 499 
 
 260729 
 
 15 
 
 46 
 
 682365 
 
 383 
 
 942795 
 
 116 
 
 739570 
 
 499 
 
 200430 
 
 14 
 
 47 
 
 682595 
 
 383 
 
 942726 
 
 116 
 
 739870 
 
 499 
 
 2GOI30 
 
 13 
 
 48 
 
 682825 
 
 383 
 
 942056 
 
 116 
 
 7401 09 
 
 499 
 
 259831 
 
 12 
 
 49 
 
 683055 
 
 383 
 
 942587 
 
 116 
 
 740468 
 
 498 
 
 259532 
 
 11 
 
 50 
 
 683284 
 
 382 
 
 942517 
 
 116 
 
 740767 
 
 498 
 
 259233 
 
 10 
 
 51 
 
 9. 683514 
 
 382 
 
 9.942448 
 
 116 
 
 9 741006 
 
 498 
 
 10.258934 
 
 9 
 
 52 
 
 683743 
 
 382 
 
 942378 
 
 116 
 
 741305 
 
 498 
 
 258635 
 
 8 
 
 53 
 
 683972 
 
 382 
 
 942308 
 
 116 
 
 741664 
 
 498 
 
 258336 
 
 7 
 
 54 
 
 684201 
 
 381 
 
 94223!) 
 
 116 
 
 7419(12 
 
 497 
 
 2.18038 
 
 6 
 
 55 
 
 684430 
 
 381 
 
 942109 
 
 11(5 
 
 74-2261 
 
 497 
 
 257739 
 
 5 
 
 56 
 
 684658 
 
 381 
 
 942099 
 
 116 
 
 742559 
 
 497 
 
 257441 
 
 4 
 
 57 
 
 684887 
 
 380 
 
 942029 
 
 116 
 
 742858 
 
 497 
 
 257142 
 
 3 
 
 58 
 
 685115 
 
 380 
 
 941959 
 
 116 
 
 743156 
 
 497 
 
 256844 
 
 2 
 
 59 
 
 685343 
 
 380 
 
 941889 
 
 117 
 
 743454 
 
 497 
 
 256546 
 
 1 
 
 60 
 
 685571 
 
 380 
 
 9-31819 
 
 117 
 
 743752 
 
 496 
 
 256248 
 
 
 
 | Cosine | | Sine | Cotang. | | Tang. M. 
 
 61 Degrees. 
 
AND TANGENTS. (29 Degrees.) 
 
 47 
 
 M. 
 
 Sine D. Cosine D. | Tang. D. Cotang. | 
 
 
 
 9.685571 
 
 :(-.) 
 
 9.941819 
 
 117 
 
 J. 743752 
 
 496 
 
 10.256248 
 
 60 
 
 
 885799 
 
 379 
 
 941749 
 
 117 
 
 744950 
 
 406 
 
 255950 
 
 59 
 
 
 
 686037 
 
 379 
 
 941679 
 
 117 
 
 741318 
 
 49'5 
 
 35565S 
 
 58 
 
 3 
 
 689254 
 
 379 
 
 941609 
 
 117 
 
 744(545 
 
 49(5 
 
 255355 
 
 57 
 
 4 
 
 686482 
 
 St9 
 
 941539 
 
 117 
 
 744943 
 
 496 
 
 255057 
 
 56 
 
 5 
 
 IN 57 M 
 
 378 
 
 941469 
 
 117 
 
 745340 
 
 49i> 
 
 854760 
 
 55 
 
 
 
 686936 
 
 878 
 
 941398 
 
 117 
 
 745.138 
 
 495 
 
 254462 
 
 54 
 
 7 
 
 687163 
 
 378 
 
 941328 
 
 117 
 
 745835 
 
 495 
 
 254165 
 
 53 
 
 8 
 
 687389 
 
 378 
 
 941258 
 
 117 
 
 74IH3-2 
 
 495 
 
 253868 
 
 52 
 
 9 
 
 687016 
 
 377 
 
 941187 
 
 117 
 
 74ii4-2i) 
 
 495 
 
 253571 
 
 51 
 
 10 
 
 
 377 
 
 941117 
 
 117 
 
 74;572(j 
 
 495 
 
 253274 
 
 50 
 
 11 
 
 9.686069 
 
 377 
 
 9.941046 
 
 118 
 
 9.747023 
 
 494 
 
 10.252377 
 
 49 
 
 12 
 
 688295 
 
 377 
 
 1)10.17.-) 
 
 118 
 
 747319 
 
 494 
 
 252(581 
 
 48 
 
 13 
 
 683521 
 
 376 
 
 040J05 
 
 118 
 
 747(516 
 
 494 
 
 252384 
 
 47 
 
 14 
 
 688747 
 
 373 
 
 940834 
 
 118 
 
 747913 
 
 494 
 
 252037 
 
 46 
 
 15 
 
 
 37(5 
 
 9407(53 
 
 118 
 
 748209 
 
 494 
 
 251791 
 
 45 
 
 1(5 
 
 68.) 198 
 
 376 
 
 940693 
 
 118 
 
 748505 
 
 493 
 
 251495 
 
 44 
 
 17 
 
 689423 
 
 375 
 
 940622 
 
 118 
 
 748301 
 
 493 
 
 251199 
 
 43 
 
 18 
 
 639648 
 
 375 
 
 94U551 
 
 118 
 
 749097 
 
 493 
 
 250903 
 
 42 
 
 19 
 
 689873 
 
 375 
 
 940480 
 
 118 
 
 749393 
 
 493 
 
 250607 
 
 41 
 
 20 
 
 69U098 
 
 375 
 
 940409 
 
 118 
 
 749689 
 
 493 
 
 250311 
 
 40 
 
 21 
 
 9.690323 
 
 374 
 
 9.940338 
 
 118 
 
 9.749985 
 
 493 
 
 10.250015 
 
 39 
 
 22 
 
 69J548 
 
 374 
 
 94 ( )267 
 
 118 
 
 750281 
 
 492 
 
 249719 
 
 38 
 
 23 
 
 690772 
 
 374 
 
 940196 
 
 118 
 
 750576 
 
 492 
 
 249424 
 
 37 
 
 24 
 
 69l)i6 
 
 374 
 
 940125 
 
 119 
 
 750872 
 
 492 
 
 249128 
 
 36 
 
 25 
 
 691230 
 
 373 
 
 94W54 
 
 119 
 
 751167 
 
 492 
 
 248833 
 
 35 
 
 20 
 
 691444 
 
 373 
 
 939932 
 
 119 
 
 7514!52 
 
 492 
 
 248538 
 
 34 
 
 27 
 
 691(568 
 
 373 
 
 939911 
 
 119 
 
 751757 
 
 492 
 
 248243 
 
 33 
 
 28 
 
 691892 
 
 373 
 
 939840 
 
 119 
 
 752052 
 
 491 
 
 247948 
 
 32 
 
 29 
 
 692115 
 
 372 
 
 939768 
 
 119 
 
 752347 
 
 491 
 
 247(553 
 
 31 
 
 30 
 
 093339 
 
 372 
 
 939697 
 
 119 
 
 752(542 
 
 491 
 
 247358 30 
 
 31 
 
 b 6^62 
 
 372 
 
 9.93!K52.-> 
 
 119 
 
 9.732937 
 
 491 
 
 10.247063 
 
 29 
 
 32 
 
 iKKS 
 
 371 - 
 
 939554 
 
 119 
 
 753231 
 
 491 
 
 246769 
 
 28 
 
 33 
 
 .IbM* { 171 
 
 939482 
 
 119 
 
 7535-26 
 
 491 
 
 246474 
 
 27 
 
 34 
 
 6^:3; | VI 
 
 939410 
 
 119 
 
 753820 
 
 490 
 
 246180 
 
 26 
 
 35 
 
 C%>153 -V.l 
 
 939339 
 
 119 
 
 754115 
 
 490 
 
 245885 
 
 25 
 
 36 
 
 ."593S57I5 l 'aro 
 
 93!l2ii7 
 
 120 
 
 754409 
 
 490 
 
 245591 
 
 24 
 
 37 
 
 693898 I i>70 
 
 939195 
 
 '120 
 
 754703 
 
 490 
 
 245297 
 
 23 
 
 38 
 
 694130 
 
 T0 
 
 9391-2:1 
 
 1-20 
 
 754997 
 
 490 
 
 245003 
 
 22 
 
 39 
 
 594342 
 
 ?70 
 
 939052 
 
 120 
 
 755291 
 
 49'J 
 
 244709 
 
 21 
 
 40 
 
 694564 
 
 360 938J8J 
 
 120 
 
 755585 
 
 489 
 
 244415 
 
 20 
 
 41 
 
 C. 694785 
 
 909 
 
 9.938908 
 
 120 
 
 9.755878 
 
 489 
 
 10.244122 
 
 19 
 
 42 
 
 695007 
 
 369 
 
 938336 
 
 120 
 
 756173 
 
 489 
 
 243828 
 
 18 
 
 43 
 
 695229 
 
 3(59 
 
 933763 
 
 120 
 
 751)4(55 
 
 489 
 
 243535 
 
 17 
 
 44 
 
 695450 
 
 368 
 
 938(591 
 
 120 
 
 75(5759 
 
 489 
 
 243241 
 
 16 
 
 45 
 
 695671 
 
 368 
 
 .138619 
 
 120 
 
 757052 
 
 489 
 
 243948 
 
 15 
 
 46 
 
 695892 
 
 368 
 
 638547 
 
 120 
 
 757345 
 
 488 
 
 242(555 
 
 14 
 
 47 
 
 696113 
 
 368 
 
 638475 
 
 120 
 
 757638 
 
 488 
 
 242362 
 
 13 
 
 48 
 
 696334 
 
 367 
 
 933402 
 
 121 
 
 757931 
 
 488 
 
 242069 
 
 12 
 
 49 
 
 696554 
 
 367 
 
 938330 
 
 121 
 
 758224 
 
 488 
 
 241776 
 
 11 
 
 50 
 
 69(3775 
 
 367 
 
 938258 
 
 121 
 
 758517 
 
 488 
 
 241483 
 
 10 
 
 51 
 
 f. 686995 
 
 367 
 
 9.938185 
 
 121 
 
 g. 758810 
 
 488 
 
 10.241190 
 
 9 
 
 52 
 
 697215 
 
 366 
 
 938113 
 
 121 
 
 759102 
 
 487 
 
 240898 
 
 8 
 
 53 
 
 697435 
 
 366 
 
 938040 
 
 I-JI 
 
 759395 
 
 487 
 
 240605 
 
 7 
 
 54 
 
 697654 
 
 366 
 
 937967 
 
 121 
 
 759687 
 
 487 
 
 240313 
 
 6 
 
 55 
 
 697874 
 
 366 
 
 93789S 
 
 121 
 
 759979 
 
 487 
 
 240021 
 
 5 
 
 56 
 
 698094 
 
 365 
 
 937822 
 
 121 
 
 760272 
 
 487 
 
 239728 
 
 4 
 
 57 
 
 898313 
 
 365 
 
 937749 
 
 121 
 
 760564 
 
 487 
 
 239436 
 
 3 I 
 
 58 
 
 698532 
 
 365 
 
 93767(5 
 
 121 
 
 760856 
 
 486 
 
 239144 
 
 2 
 
 59 
 
 698751 
 
 365 
 
 937604 
 
 121 
 
 761148 
 
 486 
 
 238352 
 
 1 
 
 60 
 
 698970 
 
 364 
 
 937531 
 
 121 
 
 761439 
 
 486 
 
 238561 
 
 
 
 
 Cosine | | Sine | Cotang. | | Tang. | M. 
 
 60 Degress. 
 
48 
 
 (30 Degrees.) A TABLE OF LOGARITHMIC 
 
 M. 1 Sine D | Cosine D. Tang. I D. Cotang. | 
 
 
 
 9.6118970 
 
 364 
 
 9.937531 1 121 
 
 9.7151439 
 
 486 
 
 10.238561 
 
 60 
 
 1 
 
 699189 
 
 364 
 
 937 4. 58 122 
 
 761731 
 
 486 
 
 
 59 
 
 2 
 
 6<)9407 
 
 364 
 
 937385 
 
 122 
 
 762033 
 
 486 
 
 237977 
 
 58 
 
 .'{ 
 
 699626 
 
 364 
 
 937312 
 
 122 
 
 762314 
 
 486 
 
 237686 
 
 57 
 
 4 
 
 699844 
 
 363 
 
 937238 
 
 122 
 
 762606 
 
 485 
 
 237394 
 
 56 
 
 5 
 
 700062 
 
 363 
 
 937165 
 
 122 
 
 762897 
 
 485 
 
 237103 
 
 55 
 
 G 
 
 700280 
 
 363 
 
 937092 
 
 122 
 
 763188 
 
 485 
 
 236812 
 
 54 
 
 ; 
 
 700498 
 
 363 
 
 937019 
 
 122 
 
 763479 
 
 485 
 
 231)5-21 
 
 53 
 
 e 
 
 700716 
 
 363 
 
 936946 
 
 122 
 
 703770 
 
 485 
 
 236230 
 
 52 
 
 9 
 
 700933 
 
 362 
 
 936872 
 
 122 
 
 764061 
 
 485 
 
 235939 
 
 51 
 
 10 
 
 701151 
 
 362 
 
 936799 
 
 122 
 
 704352 
 
 484 
 
 235648 
 
 50 
 
 u 
 
 9.701368 
 
 362 
 
 9.936725 
 
 122 
 
 9.764643 
 
 484 
 
 10.235:?.57 
 
 49 
 
 12 
 
 701585 
 
 362 
 
 936652 
 
 123 
 
 7(54933 
 
 484 
 
 SfcVMiT 48 
 
 13 
 
 701802 
 
 361 
 
 936578 
 
 123 
 
 765224 
 
 484 
 
 234776 47 ' 
 
 14 
 
 702019 
 
 361 
 
 936505 
 
 123 
 
 765514 
 
 484 
 
 -234481) 
 
 46 
 
 15 
 
 70-2-236 
 
 361 
 
 936431 
 
 123 
 
 765805 
 
 484 
 
 234 J 95 
 
 45 
 
 16 
 
 70-2452 
 
 361 
 
 936357 
 
 123 
 
 766095 
 
 484 
 
 233905 
 
 44 
 
 17 
 
 70-2009 
 
 360 
 
 936284 
 
 123 
 
 766385 
 
 483 
 
 233615 
 
 43 
 
 18 
 
 702885 
 
 360 
 
 936210 
 
 123 
 
 766675 
 
 483 
 
 233325 
 
 42 
 
 19 
 
 703101 
 
 360 
 
 936136 
 
 1-23 
 
 766965 
 
 483 
 
 233035 
 
 41 
 
 20 
 
 703317 
 
 360 
 
 936062 
 
 1-23 
 
 767255 
 
 483 
 
 232745 
 
 40 
 
 21 
 
 9.703533 
 
 359 
 
 9.935988 
 
 123 
 
 9.767545 
 
 383 
 
 10.232455 
 
 39 
 
 'i 22 
 
 703749 
 
 359 
 
 9359M 
 
 123 
 
 767834 
 
 483 
 
 232166 
 
 38 
 
 23 
 
 703964 
 
 359 
 
 935840 
 
 123 
 
 768124 
 
 482 
 
 231876 
 
 37 
 
 24 
 
 704179 
 
 359 
 
 935766 
 
 124 
 
 768413 
 
 482 
 
 231587 
 
 36 
 
 25 
 
 704395 
 
 359 
 
 935692 
 
 124 
 
 768703 
 
 482 
 
 231-297 
 
 35 
 
 26 
 
 704610 
 
 358 
 
 935618 
 
 1-24 
 
 768992 
 
 482 
 
 231008 
 
 34 
 
 27 
 
 704825 
 
 3.58 
 
 935543 
 
 124 
 
 769281 
 
 482 
 
 230719 
 
 33 
 
 28 
 
 705040 
 
 358 
 
 93.54(59 
 
 124 
 
 769570 
 
 482 : 230430 
 
 32 
 
 29 
 
 705254 
 
 358 
 
 935395 
 
 124 
 
 769860 
 
 481 
 
 230140 
 
 31 
 
 30 
 
 705469 
 
 357 
 
 935320 
 
 124 
 
 770148 
 
 481 
 
 229852 
 
 30 
 
 31 
 
 9.705683 
 
 357 
 
 9.935246 
 
 124 
 
 9.770437 
 
 481 
 
 10.229563 
 
 29 
 
 32 
 
 705898 
 
 357 
 
 935171 
 
 124 
 
 770726 
 
 481 
 
 2-29-274 
 
 28 
 
 33 
 
 706112 
 
 357 
 
 935097 
 
 124 
 
 771015 
 
 481 
 
 228985 
 
 27 
 
 34 
 
 7063-20 
 
 356 
 
 9.'io022 
 
 124 
 
 771303 
 
 481 
 
 228697 
 
 26 
 
 35 
 
 706539 
 
 356 
 
 934948 
 
 124 
 
 771592 
 
 481 
 
 228408 
 
 25 
 
 36 
 
 7UU753 
 
 356 
 
 934873 
 
 124 
 
 771880 
 
 480 
 
 2-28120 
 
 24 
 
 37 
 
 706967 
 
 356 
 
 934798 
 
 125 
 
 772168 
 
 480 
 
 227832 
 
 23 
 
 38 
 
 707180 
 
 355 
 
 9317-23 
 
 1-25 
 
 772457 
 
 480 
 
 227543 
 
 22 
 
 39 
 
 707393 
 
 355 
 
 934649 
 
 1'25 
 
 77-274.5 
 
 480 
 
 227255 
 
 21 
 
 40 
 
 707606 
 
 355 
 
 934574 
 
 125 
 
 773033 
 
 480 
 
 2:26967 
 
 20 
 
 41 
 
 9.707819 
 
 355 
 
 9.934499 
 
 125 
 
 9.773321 
 
 480 
 
 10.226679 
 
 19 
 
 42 
 
 708032 
 
 354 
 
 934424 
 
 125 
 
 773608 
 
 479 
 
 226392 
 
 18 
 
 43 
 
 708245 
 
 354 
 
 934349 
 
 125 
 
 773896 
 
 479 
 
 226104 
 
 17 
 
 44 
 
 708458 
 
 354 
 
 934274 
 
 125 
 
 774184 
 
 479 
 
 225816 
 
 16 
 
 45 
 
 708670 
 
 354 
 
 934199- 
 
 125 
 
 774471 
 
 479 
 
 225529 
 
 15 
 
 46 
 
 708882 
 
 353 
 
 934123 
 
 125 
 
 774759 
 
 479 
 
 225241 
 
 14 
 
 47 
 
 709094 
 
 353 
 
 93-1048 
 
 125 
 
 775046 
 
 479 
 
 22^954 
 
 13 
 
 48 
 
 709306 
 
 353 
 
 933973 
 
 125 
 
 77.5:5:j:{ 
 
 479 
 
 224667 
 
 12 
 
 49 
 
 709518 
 
 3.53 
 
 933898 
 
 126 
 
 77.5! 1-21 
 
 478 
 
 224379 
 
 11 
 
 50 
 
 709730 
 
 353 
 
 933822 
 
 126 
 
 775908 
 
 478 
 
 2-24092 
 
 10 
 
 51 
 
 9.70-J941 
 
 352 
 
 9.933747 
 
 126 
 
 9.776195 
 
 478 
 
 10.223805 
 
 9 
 
 52 
 
 710153 
 
 352 
 
 933671 
 
 126 
 
 776482 
 
 478 
 
 223518 
 
 8 
 
 53 
 
 710364 
 
 3.5-2 
 
 933596 
 
 126 
 
 776769 
 
 478 
 
 223231 
 
 7 
 
 54 
 
 710575 
 
 352 
 
 933520 
 
 126 
 
 777055 
 
 478 
 
 2-22945 
 
 6 
 
 55 
 
 710786 
 
 351 
 
 933445 
 
 126 
 
 777342 
 
 478 
 
 222658 
 
 5 
 
 56 
 
 710997 
 
 351 
 
 933309 
 
 126 
 
 777628 
 
 477 
 
 222372 
 
 4 
 
 57 
 
 711208 
 
 351 
 
 933293 
 
 126 
 
 777915 
 
 477 
 
 222085 
 
 3 
 
 58 
 
 711419 
 
 351 
 
 933217 
 
 126 
 
 778201 
 
 477 
 
 221799 
 
 2 
 
 59 
 
 7116-29 
 
 350 
 
 933141 
 
 
 778487 
 
 477 
 
 221512 
 
 1 
 
 60 
 
 711839 
 
 350 
 
 9330!>6 126 778774 
 
 477 
 
 221226 
 
 
 
 | Cosine | | Sine Cotang. | Tang. | M. 
 
 69 Degrees. 
 
SINKS AND TANGENTS. (31 Degrees.) 
 
 M. | Sine D. | Cosine D. Tang. | D. Cotang. | 
 
 
 
 9.711839 
 
 350 
 
 J.i330!i5 
 
 126 
 
 9.778774 
 
 477 
 
 10.221236 
 
 60 
 
 1 
 
 713050 
 
 390 
 
 9329:M) 
 
 127 
 
 779060 
 
 477 
 
 220940 
 
 59 
 
 2 
 
 713360 
 
 390 
 
 93-2914 
 
 1-27 
 
 771)316 
 
 476 
 
 220654 
 
 58 
 
 3 
 
 71-246!) 
 
 349 
 
 932838 
 
 127 
 
 779633 
 
 470 
 
 220368 
 
 57 
 
 4 
 
 712(579 
 
 319 
 
 932762 
 
 127 
 
 779918 
 
 476 
 
 220082 
 
 56 
 
 5 
 
 713B89 
 
 349 
 
 932685 
 
 1-37 
 
 780203 
 
 476 
 
 219797 
 
 55 
 
 6 
 
 713098 
 
 349 
 
 932609 
 
 1-27 
 
 780489 
 
 47(5 
 
 219511 
 
 54 
 
 7 
 
 713308 
 
 349 
 
 932533 
 
 127 
 
 781)775 
 
 476 
 
 211)2-25 
 
 53 
 
 8 
 
 718517 
 
 348 
 
 933457 
 
 127 
 
 781060 
 
 476 
 
 218940 
 
 52 
 
 9 
 
 713728 
 
 348 
 
 932380 
 
 1-27 
 
 781346 
 
 475 
 
 218654 
 
 51 
 
 1 ]!) 
 
 713935 
 
 348 
 
 932304 
 
 1-27 
 
 781631 
 
 475 
 
 218369 
 
 50 
 
 II 
 
 9.714144 
 
 348 
 
 9.932228 
 
 127 
 
 9.781916 
 
 475 
 
 10.218084 
 
 49 
 
 13 
 
 714352 
 
 347 
 
 932151 
 
 1*27 
 
 78-2201 
 
 475 
 
 217799 
 
 48 
 
 13 
 
 714561 
 
 347 
 
 932075 
 
 128 
 
 782486 
 
 475 
 
 217514 
 
 47 
 
 14 
 
 7147(59 
 
 347 
 
 931998 
 
 128 
 
 782771 
 
 475 
 
 217229 
 
 46 
 
 15 
 
 714978 
 
 347 
 
 931921 
 
 128 
 
 783056 
 
 475 
 
 216944 
 
 45 
 
 1(5 
 
 71518(5 
 
 347 
 
 931845 
 
 128 
 
 783341 
 
 475 
 
 216659 
 
 44 
 
 17 
 
 7 J 5394 
 
 346 
 
 931768 
 
 128 
 
 783626 
 
 474 
 
 918374 
 
 43 
 
 18 
 
 7l.vi()-2 
 
 346 
 
 931691 
 
 128 
 
 783910 
 
 474 
 
 216090 
 
 42 
 
 19 
 
 715809 
 
 346 
 
 931614 
 
 128 
 
 784195 
 
 474 
 
 215805 
 
 41 
 
 20 
 
 710017 
 
 346 
 
 931537 
 
 128 
 
 784479 
 
 474 
 
 215521 
 
 40 
 
 21 
 
 9.716224 
 
 345 
 
 9.931460 
 
 128 
 
 9.784764 
 
 474 
 
 10.215236 
 
 39 
 
 22 
 
 71(143-2 
 
 345 
 
 931383 
 
 128 
 
 785048 
 
 474 
 
 214952 
 
 38 
 
 33 
 
 716(539 
 
 345 
 
 931306 
 
 128 
 
 785332 
 
 473 
 
 214(568 
 
 37 
 
 24 
 
 716846 
 
 345 
 
 931-2-29 
 
 129 
 
 785616 
 
 473 
 
 214384 
 
 36 
 
 25 
 
 717053 
 
 345 
 
 931152 
 
 129 
 
 785900 
 
 473 
 
 214100 
 
 35 
 
 26 
 
 717259 
 
 344 
 
 931075 
 
 129 
 
 786184 
 
 473 
 
 213816 
 
 34 
 
 27 
 
 71746G 
 
 344 
 
 930998 
 
 129 
 
 786468 
 
 473 
 
 213532 
 
 33 
 
 S8 
 
 717673 
 
 344 
 
 930921 
 
 129 
 
 786752 
 
 473 
 
 213248 
 
 32 
 
 29 
 
 717879 
 
 344 
 
 930843 
 
 129 
 
 78703S5 
 
 473 
 
 212964 
 
 31 
 
 30 
 
 718085 
 
 343 
 
 930766 
 
 129 
 
 787319 
 
 472 
 
 212681 
 
 30 
 
 31 
 
 9.718291 
 
 343 
 
 9.930688 
 
 129 
 
 9.787603 
 
 472 
 
 10.212397 
 
 29 
 
 32 
 
 718497 
 
 343 
 
 930611 
 
 129 
 
 787886 
 
 472 
 
 212114 
 
 28 
 
 33 
 
 718703 
 
 343 
 
 930533 
 
 129 
 
 788170 
 
 472 
 
 211830 
 
 27 
 
 34 
 
 718909 
 
 343 
 
 930456 
 
 129 
 
 788453 
 
 472 
 
 211547 
 
 26 
 
 35 
 
 719114 
 
 342 
 
 930378 
 
 129 
 
 783736 
 
 472 
 
 211264 25 ' 
 
 36 
 
 719320 
 
 342 
 
 930301) 
 
 130 
 
 789019 
 
 472 
 
 210981 
 
 24 | 
 
 37 
 
 7195-25 
 
 349 
 
 930323 
 
 130 
 
 789302 
 
 471 
 
 210698 
 
 23 
 
 38 
 
 719730 
 
 342 
 
 930145 
 
 130 
 
 789585 
 
 471 
 
 210415 
 
 22 
 
 39 
 
 7199:55 
 
 341 
 
 930067 
 
 130 
 
 789868 
 
 471 
 
 210132 
 
 21 
 
 40 
 
 T20140 
 
 341 
 
 929989 
 
 130 
 
 790151 
 
 471 
 
 209849 
 
 20 
 
 41 
 
 1). 7-2034.1 
 
 341 
 
 9.929911 
 
 130 
 
 9.790433 
 
 471 
 
 10.209567 
 
 19 
 
 42 
 
 720549 
 
 341 
 
 929833 
 
 130 
 
 790716 
 
 471 
 
 209284 
 
 18 
 
 43 
 
 720754 
 
 340 
 
 929755 
 
 130 
 
 790999 
 
 471 
 
 209001 
 
 17 
 
 44 
 
 7201)58 
 
 340 
 
 929577 
 
 130 
 
 791281 
 
 471 
 
 208719 
 
 16 
 
 45 
 
 7211(52 
 
 340 
 
 929599 
 
 130 
 
 791563 
 
 470 
 
 208437 
 
 15 
 
 46 
 
 7213(56 
 
 340 
 
 929531 
 
 130 
 
 791846 
 
 470 
 
 308154 
 
 14 
 
 47 
 
 7-21570 
 
 340 
 
 !)-2<)44-2 
 
 130 
 
 792128 
 
 470 
 
 307872 
 
 13 
 
 48 
 
 721774 
 
 339 
 
 929364 
 
 131 
 
 792410 
 
 470 
 
 207590 
 
 12 
 
 49 
 
 721978 
 
 339 
 
 929286 
 
 131 
 
 792692 
 
 470 
 
 307308 
 
 11 
 
 50 
 
 722181 
 
 339 
 
 9-29207 
 
 131 
 
 792974 
 
 470 
 
 307026 
 
 10 
 
 51 
 
 9.722385 
 
 339 
 
 9.929129 
 
 131 
 
 9.793-236 
 
 470 
 
 10.206744 
 
 9 
 
 52 
 
 7-2-2.-HH 
 
 339 
 
 929050 
 
 131 
 
 793538 
 
 469 
 
 206462 
 
 8 
 
 53 
 
 722791 
 
 338 
 
 928972 
 
 131 
 
 793819 
 
 469 
 
 306181 
 
 7 
 
 54 
 
 722994 
 
 338 
 
 928893 
 
 131 
 
 794101 
 
 469 
 
 305899 
 
 6 
 
 55 
 
 723197 
 
 339 
 
 928815 
 
 131 
 
 794333 
 
 4(51) 
 
 905617 
 
 5 
 
 56 
 
 733400 
 
 338 
 
 9-28736 
 
 131 
 
 794664 
 
 469 
 
 305336 
 
 4 
 
 57 
 
 
 337 
 
 938657 
 
 13] 
 
 794915 
 
 469 
 
 905055 
 
 3 
 
 58 
 
 733803 
 
 337 
 
 928578 
 
 131 
 
 795227 
 
 469 
 
 204773 
 
 2 
 
 59 
 
 724007 
 
 337 
 
 9-2849J 
 
 i:u 
 
 795508 
 
 468 
 
 20441)2 
 
 1 
 
 60 
 
 724210 
 
 337 
 
 9-2*120 
 
 131 
 
 795789 
 
 468 
 
 201-211 
 
 
 
 | Cosine Sine | | Cotang. | Tang. | M. 
 
 68 Degrees. 
 
 3 
 
GO 
 
 (32 Degrees.) A TABLK OF LOGARITHMIC 
 
 M. Sine | D Cosine j D. | Tang, j D. 
 
 Cotang. | 
 
 
 
 9.724210 
 
 337 
 
 9.938420 
 
 132 
 
 9.7957S9 
 
 468 
 
 10.204-211 
 
 60 
 
 1 
 
 724412 
 
 337 
 
 928342 
 
 132 
 
 796070 
 
 408 
 
 203930 
 
 59 
 
 2 
 
 724614 
 
 336 
 
 928203 
 
 132 
 
 796351 
 
 468 
 
 203649 
 
 58 
 
 3 
 
 724816 
 
 838 
 
 928183 
 
 132 
 
 79(5(532 
 
 408 
 
 2033li8 
 
 57 
 
 4 
 
 725017 
 
 336 
 
 928104 
 
 132 
 
 796913 
 
 468 
 
 203087 
 
 56 
 
 5 
 
 725219 
 
 336 
 
 928025 
 
 132 
 
 797194 
 
 468 
 
 903806 
 
 55 
 
 - 6 
 
 725420 
 
 335 
 
 92794(5 
 
 132 
 
 797475 
 
 468 
 
 202525 
 
 54 
 
 7 
 
 725622 
 
 335 
 
 9278(57 
 
 132 
 
 797755 
 
 468 
 
 202245 
 
 53 
 
 8 
 
 725823 
 
 335 
 
 927787 
 
 132 
 
 798036 
 
 467 
 
 2019(54 
 
 52 
 
 9 
 
 726024 
 
 335 
 
 927708 
 
 132 
 
 798316 
 
 467 
 
 201684 
 
 51 
 
 10 
 
 726225 
 
 335 
 
 927629 
 
 132 
 
 798596 
 
 467 
 
 201404 
 
 50 
 
 11 
 
 9.726426 
 
 334 
 
 9.927549 
 
 132 
 
 9.798877 
 
 467 
 
 10.201123 
 
 49 
 
 12 
 
 726626 
 
 334 
 
 927470 
 
 133 
 
 799157 
 
 467 
 
 200843 
 
 48 
 
 13 
 
 726H27 
 
 334 
 
 927390 
 
 133 
 
 799437 
 
 407 
 
 200563 
 
 47 
 
 14 
 
 727027 
 
 334 
 
 927310 
 
 133 
 
 799717 
 
 467 
 
 200283 
 
 46 
 
 15 
 
 727228 
 
 334 
 
 927231 
 
 133 
 
 799997 
 
 466 
 
 200003 
 
 45 
 
 16 
 
 727428 
 
 333 
 
 927151 
 
 133 
 
 800277 
 
 466 
 
 199723 
 
 44 
 
 17 
 
 727628 
 
 333 
 
 927071 
 
 133 
 
 800557 
 
 466 
 
 199443 
 
 43 
 
 18 
 
 727828 
 
 333 
 
 926991 
 
 133 
 
 800836 
 
 466 
 
 199164 
 
 42 
 
 19 
 
 728027 
 
 333 
 
 926911 
 
 133 
 
 801116 
 
 466 
 
 198884 
 
 41 
 
 20 
 
 728227 
 
 333 
 
 926831 
 
 133 
 
 801396 
 
 466 
 
 11)8004 
 
 40 
 
 21 
 
 9.728427 
 
 332 
 
 9.926751 
 
 133 
 
 9.801675 
 
 466 
 
 10.198325 
 
 39 
 
 22 
 
 728626 
 
 332 
 
 926671 
 
 133 
 
 801955 
 
 466 
 
 198045 
 
 38 
 
 23 
 
 728825 
 
 332 
 
 926591 
 
 133 
 
 802234 
 
 465 
 
 197766 
 
 37 
 
 24 
 
 729024 
 
 332 
 
 926511 
 
 134 
 
 802513 
 
 465 
 
 197487 
 
 36 
 
 25 
 
 729223 
 
 331 
 
 92(5431 
 
 134 
 
 802792 
 
 465 
 
 197208 
 
 35 
 
 20 
 
 729422 
 
 331 
 
 926351 
 
 134 
 
 803072 
 
 465 
 
 196928 
 
 34 
 
 27 
 
 729621 
 
 331 
 
 926270 
 
 134 
 
 803351 
 
 465 
 
 196649 
 
 33 
 
 28 
 
 729820 
 
 331 
 
 926190 
 
 134 
 
 803630 
 
 465 
 
 196370 
 
 32 
 
 29 
 
 730018 
 
 330 
 
 926110 
 
 134 
 
 803908 
 
 465 
 
 196092 
 
 31 
 
 30 
 
 730216 
 
 330 
 
 926029 
 
 134 
 
 804187 
 
 465 
 
 195813 
 
 30 
 
 31 
 
 9.730415 
 
 330 
 
 9.925949 
 
 134 
 
 9.804466 
 
 464 
 
 10.195534 
 
 29 
 
 32 
 
 730613 
 
 330 
 
 925868 
 
 134 
 
 804745 
 
 464 
 
 195255 
 
 28 
 
 33 
 
 730811 
 
 330 
 
 925788 
 
 134 
 
 805023 
 
 M64 
 
 194977 
 
 27 
 
 34 
 
 731009 
 
 329 
 
 925707 
 
 134 
 
 805302 
 
 464 
 
 194698 
 
 26 
 
 35 
 
 731206 
 
 329 
 
 925626 
 
 134 
 
 805580 
 
 464 
 
 194420 
 
 25 
 
 36 
 
 731404 
 
 329 
 
 925545 
 
 135 
 
 805859 
 
 464 
 
 194141 
 
 24 
 
 37 
 
 731002 
 
 329 
 
 925465 
 
 135 
 
 806137 
 
 4'64 
 
 193863 
 
 23 
 
 38 
 
 731799 
 
 329 
 
 925384 
 
 135 
 
 806415 
 
 463 
 
 193585 
 
 22 
 
 39 
 
 731996 
 
 328 
 
 925303 
 
 135 
 
 806693 
 
 463 
 
 193307 
 
 21 
 
 40 
 
 732193 
 
 328 
 
 925222 
 
 135 
 
 806971 
 
 463 
 
 193029 
 
 20 
 
 41 
 
 9.732390 
 
 328 
 
 9.925141 
 
 135 
 
 9.807249 
 
 463 
 
 10.192751 
 
 19 
 
 42 
 
 732587 
 
 328 
 
 925060 
 
 135 
 
 807527 
 
 463 
 
 192473 
 
 18 
 
 43 
 
 732784 
 
 328 
 
 924979 
 
 135 
 
 807805 
 
 463 
 
 192195 
 
 17 
 
 44 
 
 732980 
 
 327 
 
 924897 
 
 135 
 
 808083 
 
 463 
 
 191917 
 
 16 
 
 45 
 
 733177 
 
 327 
 
 92J816 
 
 135 
 
 808361 
 
 463 
 
 191639 
 
 15 
 
 46 
 
 733373 
 
 327 
 
 924735 
 
 136 
 
 808638 
 
 462 
 
 191362 
 
 14 
 
 47 
 
 733569 
 
 327 
 
 924654 
 
 136 
 
 808916 
 
 462 
 
 191084 
 
 13 
 
 48 
 
 733765 
 
 327 
 
 924572 
 
 136 
 
 809193 
 
 4(52 
 
 190807 
 
 12 
 
 49 
 
 733961 
 
 326 
 
 924491 
 
 136 
 
 809471 
 
 462 
 
 1905-29 
 
 11 
 
 50 
 
 734157 
 
 326 
 
 924409 
 
 136 
 
 809748 
 
 462 
 
 190252 
 
 10 
 
 51 
 
 9.734353 
 
 326 
 
 9.924328 
 
 136 
 
 9.810025 
 
 462 
 
 10.189975 
 
 9 
 
 52 
 
 734549 
 
 326 
 
 924246 
 
 136 
 
 810302 
 
 462 
 
 189698 
 
 8 
 
 53 
 
 734744 
 
 325 
 
 924164 
 
 136 
 
 810580 
 
 462 
 
 189420 
 
 7 
 
 54 
 
 734939 
 
 325 
 
 924083 
 
 136 
 
 810857 
 
 462 
 
 189143 
 
 6 
 
 55 
 
 73.1135 
 
 325 
 
 92400 1 
 
 136 
 
 811134 
 
 4G1 
 
 18886(5 
 
 5 
 
 56 
 
 735330 
 
 325 
 
 923919 
 
 136 
 
 811410 
 
 461 
 
 188590 
 
 4 
 
 57 
 
 7355-25 
 
 325 
 
 923837 
 
 136 
 
 811687 
 
 461 
 
 188313 
 
 3 
 
 58 
 
 735719 
 
 324 
 
 &375S 
 
 137 
 
 811964 
 
 461 
 
 188036 
 
 2 
 
 5!) 
 
 735914 
 
 324 
 
 923673 
 
 137 
 
 812241 
 
 461 
 
 187759 
 
 1 
 
 60 
 
 736109 
 
 324 
 
 923591 
 
 137 
 
 812517 
 
 461 
 
 187483 
 
 
 
 | Cosine Sine | Cotang. | Tang. | M. 
 
 57 Degrees. 
 
SINES AND TANGENTS. (33 Degrees.') 
 
 M. j Sine D. Cosine D. Tang. | D Totang 
 
 
 
 9. 731 i 109 
 
 3-24 
 
 9.9-235U1 
 
 137 
 
 9. SI -2:. 17 
 
 401 
 
 10. J 874 82 
 
 00 
 
 1 
 
 730303 
 
 3-24 
 
 933509 
 
 137 
 
 812794 
 
 401 
 
 187206 
 
 59 
 
 2 
 
 7:i4!)S 
 
 3-24 
 
 033437 
 
 137 
 
 813070 
 
 401 
 
 180930 
 
 58 
 
 1 3 
 
 73oo<i-2 
 
 323 
 
 923345 
 
 137 
 
 813347 
 
 4(50 
 
 186(553 
 
 57 
 
 4 
 
 73(W86 
 
 3-23 
 
 933263 
 
 137 
 
 813023 
 
 400 
 
 180377 
 
 56 
 
 5 
 
 737080 
 
 3-23 
 
 923181 
 
 137 
 
 813899 
 
 460 
 
 186101 
 
 55 
 
 6 
 
 737-274 
 
 323 
 
 923098 
 
 137 
 
 814175 
 
 460 
 
 185825 
 
 54 
 
 7 
 
 737407 
 
 323 
 
 923016 
 
 137 
 
 814452 
 
 400 
 
 185548 
 
 53 
 
 8 
 
 737861 
 
 322 
 
 9-2-2933 
 
 137 
 
 814728 
 
 400 
 
 185272 
 
 52 
 
 9 
 
 737855 
 
 322 
 
 U22851 
 
 137 
 
 815004 
 
 460 
 
 184996 
 
 51 
 
 10 
 
 738048 
 
 3-22 
 
 922768 
 
 138 
 
 815279 
 
 460 
 
 184721 
 
 50 
 
 11 
 
 9.738241 
 
 322 
 
 9.922686 
 
 138 
 
 9.815555 
 
 459 
 
 10.18-1445 
 
 49 
 
 12 
 
 738434 
 
 3-2-2 
 
 9S9003 
 
 138 
 
 815831 
 
 459 
 
 184109 
 
 48 
 
 13 
 
 738627 
 
 321 
 
 922520 
 
 138 
 
 816107 
 
 459 
 
 183893 
 
 47 
 
 14 
 
 7388-20 
 
 3-21 
 
 9-2-2-138 
 
 138 
 
 816382 
 
 459 
 
 183018 
 
 40 
 
 15 
 
 739013 
 
 321 
 
 922355 
 
 138 
 
 816658 
 
 459 
 
 183342 
 
 45 
 
 1(5 
 
 739-200 
 
 321 
 
 9-2-2-27-2 
 
 138 
 
 816933 
 
 459 
 
 183007 
 
 44 
 
 17 
 
 739398 
 
 321 
 
 922189 
 
 138 
 
 817209 
 
 459 
 
 182791 
 
 43 
 
 18 
 
 739590 
 
 320 
 
 92210(5 
 
 138 
 
 817484 
 
 459 
 
 182516 
 
 42 
 
 19 
 
 739783 
 
 320 
 
 922023 
 
 138 
 
 817759 
 
 459 
 
 182241 
 
 41 
 
 20 
 
 739975 
 
 320 
 
 9-21940 
 
 138 
 
 818035 
 
 458 
 
 181905 
 
 40 
 
 21 
 
 9-740167 
 
 320 
 
 9.921857 
 
 139 
 
 9.818310 
 
 458 
 
 10.181690 
 
 39 
 
 22 
 
 740359 
 
 320 
 
 921774 
 
 139 
 
 818585 
 
 458 
 
 181415 
 
 38 
 
 23 
 
 740550 
 
 319 
 
 921691 
 
 139 
 
 818860 
 
 458 
 
 181140 
 
 37 
 
 24 
 
 74074-2 
 
 319 
 
 981607 
 
 139 
 
 819135 
 
 458 
 
 180805 
 
 36 
 
 25 
 
 740934 
 
 319 
 
 921524 
 
 139 
 
 819410 
 
 458 
 
 180590 
 
 35 
 
 2(5 
 
 741125 
 
 3J9 
 
 921441 
 
 139 
 
 819(584 
 
 458 
 
 180316 
 
 34 
 
 27 
 
 741316 
 
 319 
 
 921357 
 
 139 
 
 819959 
 
 458 
 
 180041 
 
 33 
 
 28 
 
 741508 
 
 318 
 
 921274 
 
 139 
 
 820234 
 
 458 
 
 17G700 
 
 32 
 
 '29 
 
 74 J 699 
 
 318 
 
 921190 
 
 139 
 
 820508 
 
 457 
 
 179492 
 
 31 
 
 30 
 
 741889 
 
 318 
 
 921107 
 
 139 
 
 820783 
 
 457 
 
 179217 
 
 30 
 
 31 
 
 9.742080 
 
 318 
 
 9.921023 
 
 139 
 
 9.821057 
 
 437 
 
 10.178943 
 
 29 
 
 32 
 
 742271 
 
 318 
 
 920939 
 
 140 
 
 821332 
 
 457 
 
 178(508 
 
 28 
 
 33 
 
 742402 
 
 317 
 
 95*0856 
 
 140 
 
 821000 
 
 457 
 
 178394 
 
 27 
 
 34 
 
 742052 
 
 317 
 
 920772 
 
 140 
 
 821880 
 
 457 
 
 178120 
 
 26 
 
 35 
 
 742842 
 
 317 
 
 920088 
 
 140 
 
 822154 
 
 357 
 
 177846 
 
 25 
 
 36 
 
 743033 
 
 317 
 
 920604 
 
 140 
 
 822429 
 
 457 
 
 177571 
 
 24 
 
 37 
 
 743-2-23 
 
 317 
 
 920520 
 
 140 
 
 822703 
 
 457 
 
 177297 
 
 23 
 
 38 
 
 743413 
 
 316 
 
 920436 
 
 140 
 
 822977 
 
 45*5 
 
 177023 
 
 22 
 
 39 
 
 743002 
 
 316 
 
 990353 
 
 140 
 
 823250 
 
 456 
 
 176750 
 
 21 
 
 40 
 
 743792 
 
 316 
 
 920268 
 
 140 
 
 823524 
 
 456 
 
 176476 
 
 20 
 
 41 
 
 9.743982 
 
 316 
 
 9.920184 
 
 149 
 
 9.823798 
 
 456 
 
 10.176202 
 
 19 
 
 42 
 
 744171 
 
 316 
 
 920099 
 
 140 
 
 824072 
 
 456 
 
 175928 
 
 18 
 
 43 
 
 744301 
 
 315 
 
 9-20015 
 
 140 
 
 824345 
 
 456 
 
 175055 
 
 17 
 
 44 
 
 744550 
 
 315 
 
 919931 
 
 141 
 
 824019 
 
 456 
 
 175381 
 
 16 
 
 45 
 
 744739 
 
 315 
 
 919840 
 
 141 
 
 824893 
 
 456 
 
 175107 
 
 15 
 
 4(5 
 
 744928 
 
 315 
 
 919702 
 
 141 
 
 825166 
 
 456 
 
 174834 
 
 14 
 
 47 
 
 745117 
 
 315 
 
 919077 
 
 141 
 
 825439 
 
 455 
 
 174561 
 
 13 
 
 48 
 
 74530(5 
 
 314 
 
 919593 
 
 141 
 
 825713 
 
 455 
 
 174287 
 
 12 
 
 49 
 
 745494 
 
 314 
 
 919508 
 
 141 
 
 825986 
 
 455 
 
 174014 
 
 11 
 
 50 
 
 745C83 
 
 314 
 
 919424 
 
 141 
 
 826259 
 
 455 
 
 173741 
 
 10 
 
 51 
 
 9.745871 
 
 314 
 
 9.919339 
 
 141 
 
 9.826532 
 
 455 
 
 10.173468 
 
 9 
 
 52 
 
 740059 
 
 314 
 
 919254 
 
 141 
 
 826805 
 
 455 
 
 173195 
 
 8 
 
 53 
 
 740248 
 
 313 
 
 919109 
 
 141 
 
 827078 
 
 455 
 
 172922 
 
 7 
 
 54 
 
 74043(5 
 
 313 
 
 919085 
 
 141 
 
 827351 
 
 455 
 
 172(549 
 
 6 
 
 55 
 
 7401 i24 
 
 313 
 
 919000 
 
 141 
 
 827024 
 
 455 
 
 172376 
 
 5 
 
 50 
 
 740812 
 
 313 
 
 918915 
 
 142 
 
 827897 
 
 454 
 
 172103 
 
 4 
 
 57 
 
 74W.H) 
 
 313 
 
 918830 
 
 142 
 
 828170 
 
 454 
 
 171H30 
 
 3 
 
 58 
 
 74"/]87 
 
 312 
 
 918745 
 
 142 
 
 828442 
 
 454 
 
 171558 
 
 2 
 
 59 
 
 747374 
 
 312 
 
 918059 
 
 142 
 
 828715 
 
 454 
 
 171285 
 
 1 
 
 BO 
 
 747502 
 
 312 
 
 918574 
 
 142 
 
 828987 
 
 454 
 
 171013 
 
 
 
 Cosine | | Sine Cotang. j | Tang. 
 
 M. 1 
 
 66 Degrees. 
 
5-3 
 
 (34 Degree.?.) A TABLE OP LOGARITHMIC 
 
 M. Sine D. | Cosine D. Tang. | D. Cotang. | 
 
 
 
 9.747562 
 
 312 
 
 9.918574 
 
 142 
 
 9.8-28987 
 
 454 
 
 10.171013 
 
 60 , 
 
 1 
 
 747749 
 
 312 
 
 918489 
 
 142 
 
 8-2U-260 
 
 454 
 
 170740 
 
 59 
 
 2 
 
 747936 
 
 312 
 
 918404 
 
 142 
 
 829532 
 
 454 
 
 170468 
 
 58 
 
 3 
 
 7481-23 
 
 311 
 
 918318 
 
 14-2 
 
 829305 
 
 454 
 
 170195 
 
 57 
 
 4 
 
 748310 
 
 311 
 
 918233 
 
 142 
 
 830077 
 
 454 
 
 169923 
 
 56 
 
 5 
 
 748497 
 
 311 
 
 918147 
 
 14-2 
 
 83:1349 
 
 453 
 
 169651 
 
 55 ! 
 
 6 
 
 748683 
 
 311 
 
 918062 
 
 142 
 
 830H-21 
 
 453 
 
 169379 
 
 54 
 
 .7 
 
 748870 
 
 311 
 
 917976 
 
 143 
 
 830393 
 
 4.33 
 
 169107 
 
 53 
 
 8 
 
 74905(5 
 
 310 
 
 917891 
 
 143 
 
 831165 
 
 453 
 
 168835 
 
 5-2 
 
 9 
 
 749243 
 
 310 
 
 917805 
 
 143 
 
 831437 
 
 453 
 
 168563 
 
 51 
 
 10 
 
 749429 
 
 310 
 
 917719 
 
 143 
 
 831709 
 
 453 
 
 168291 
 
 50 
 
 11 
 
 9.749015 
 
 310 
 
 9.917634 
 
 143 
 
 9.831981 
 
 453 
 
 10.168019 
 
 49 
 
 1-2 
 
 7498J1 
 
 310 
 
 917548 
 
 143 
 
 8322.13 
 
 453 
 
 K17747 
 
 48 
 
 13 
 
 749337 
 
 309 
 
 917415-2 
 
 143 
 
 8325-25 
 
 453 
 
 167475 
 
 47 
 
 14 
 
 750172 
 
 309 
 
 917376 
 
 143 
 
 832796 
 
 453 
 
 167204 
 
 46 
 
 15 
 
 750358 
 
 309 
 
 917-290 
 
 143 
 
 833063 
 
 452 
 
 1(56932 
 
 45 
 
 16 
 
 750543 
 
 309 
 
 917204 
 
 143 
 
 833339 
 
 452 
 
 16(5661 
 
 44 
 
 
 7.307-29 
 
 309 
 
 917118 
 
 144 
 
 833611 
 
 452 
 
 166389 
 
 43 
 
 18 
 
 750914 
 
 308 
 
 917032 
 
 144 
 
 833882 
 
 4.32 
 
 166118 
 
 42 
 
 19 
 
 751099 
 
 308 
 
 91G946 
 
 144 
 
 834154 
 
 452 
 
 165846 
 
 41 
 
 20 
 
 751284 
 
 303 
 
 916359 
 
 144 
 
 834425 
 
 452 
 
 165575 
 
 40 
 
 21 
 
 9.751469 
 
 308 
 
 9.916773 
 
 144 
 
 9.834696 
 
 452 
 
 10.165304 
 
 39 
 
 22 
 
 7516.34 
 
 308 
 
 916(587 
 
 144 
 
 834957 
 
 4.32 
 
 165033 
 
 38 
 
 23 
 
 751839 
 
 308 
 
 916600 
 
 144 
 
 835-238 
 
 452 
 
 164762 
 
 37 
 
 24 
 
 752023 
 
 307 
 
 916514 
 
 144 
 
 835509 
 
 452 
 
 164491 
 
 36 
 
 25 
 
 752-208 
 
 307 
 
 916427 
 
 144 
 
 835780 
 
 451 
 
 164220 
 
 35 
 
 26 
 
 752:592 
 
 337 
 
 916341 
 
 144 
 
 836051 
 
 451 
 
 163949 
 
 34 
 
 27 
 
 752576 
 
 307 
 
 916254 
 
 144 
 
 836322 
 
 451 
 
 163678 
 
 33 
 
 28 
 
 752760 
 
 307 
 
 916167 
 
 145 
 
 8315593 
 
 4.31 
 
 163407 
 
 32 
 
 29 
 
 752944 
 
 306 
 
 916081 
 
 145 
 
 830864 
 
 451 
 
 163136 
 
 31 
 
 30 
 
 753128 
 
 306 
 
 915994 
 
 145 
 
 837134 
 
 451 
 
 1628t56 
 
 30 
 
 31 
 
 9.753312 
 
 306 
 
 9.915907 
 
 145 
 
 9.8:57405 
 
 451 
 
 10.162595 
 
 29 
 
 32 
 
 753495 
 
 306 
 
 915320 
 
 115 
 
 837675 
 
 451 
 
 182325 
 
 28 
 
 33 
 
 753679 
 
 306 
 
 915733 
 
 145 
 
 837946 
 
 451 
 
 162054 
 
 27 
 
 34 
 
 75386-2 
 
 sas 
 
 915646 
 
 145 
 
 8:58216 
 
 451 
 
 161784 
 
 26 
 
 35 
 
 754046 
 
 305 
 
 915559 
 
 145 
 
 838487 
 
 450 
 
 161513 
 
 25 
 
 23 
 
 7542251 
 
 305 
 
 915472 
 
 145 
 
 838757, 
 
 450 
 
 161243 
 
 24 
 
 37 
 
 754412 
 
 305 
 
 915385 
 
 145 
 
 839027 
 
 450 
 
 160973 
 
 23 
 
 38 
 
 754595 
 
 305 
 
 915-297 
 
 145 
 
 839297 
 
 450 
 
 160703 
 
 22 
 
 39 
 
 754778 
 
 304 
 
 915210 
 
 145 
 
 839568 
 
 450 
 
 160432 
 
 21 
 
 40 
 
 754960 
 
 304 
 
 915123 
 
 146 
 
 839833 
 
 450 
 
 160162 
 
 20 
 
 41 
 
 9.755143 
 
 304 
 
 9.915035 
 
 146 
 
 9.840108 
 
 450 
 
 10.159892 
 
 19 
 
 42 
 
 755326 
 
 304 
 
 914948 
 
 146 
 
 840378 
 
 450 
 
 159622 
 
 18 
 
 43 
 
 755503 
 
 304 
 
 914860 
 
 146 
 
 840647 
 
 450 
 
 159353 
 
 17 
 
 44 
 
 755090 
 
 304 
 
 914773 
 
 146 
 
 840917 
 
 449 
 
 159083 
 
 16 
 
 45 
 
 755872 
 
 303 
 
 914685 
 
 146 
 
 841187 
 
 449 
 
 158813 
 
 15 
 
 46 
 
 7515054 
 
 303 
 
 914598 
 
 146 
 
 841457 
 
 449 
 
 158543 
 
 14 
 
 47 
 
 1*56236 
 
 303 
 
 914510 
 
 146 
 
 841726 
 
 449 
 
 158274 
 
 13 
 
 48 
 
 756418 
 
 303 
 
 914422 
 
 146 
 
 841996 
 
 449 
 
 158004 
 
 12 
 
 49 
 
 756600 
 
 303 
 
 914334 
 
 146 
 
 84-2266 
 
 449 
 
 157734 
 
 11 
 
 50 
 
 756782 
 
 302 
 
 914216 
 
 147 
 
 84-2535 
 
 449 
 
 157465 
 
 10 
 
 51 
 
 9.756963 
 
 302 
 
 9.914158 
 
 147 
 
 9.842805 
 
 449 
 
 10.157195 
 
 9 
 
 52 
 
 757144 
 
 302 
 
 914070 
 
 147 
 
 843074 
 
 419 
 
 15H9-26 
 
 8 
 
 53 
 
 7573-215 
 
 304 
 
 913932 
 
 147 
 
 843343 
 
 449 
 
 156657 
 
 7 
 
 54 
 
 757507 
 
 302 
 
 913334 
 
 147 
 
 843612 
 
 449 
 
 151J388 
 
 6 
 
 55 
 
 757688 
 
 301 
 
 9138;)6 
 
 147 
 
 843882 
 
 448 
 
 156118 
 
 5 
 
 56 
 
 7573S59 
 
 301 
 
 913718 
 
 147 
 
 844151 
 
 448 
 
 155819 
 
 4 
 
 57 
 
 75805!) 
 
 301 
 
 913630 
 
 147 
 
 844420 
 
 448 
 
 155580 
 
 3 
 
 53 
 
 758230 
 
 301 
 
 913541 
 
 147 
 
 844689 
 
 448 
 
 155311 
 
 2 
 
 59 
 
 758411 
 
 301 
 
 913453 
 
 147 
 
 844SI.38 
 
 448 
 
 155042 
 
 1 
 
 60 
 
 758591 
 
 301 
 
 913305 
 
 147 
 
 845227 
 
 448 
 
 154773 
 
 
 
 Cosine | Sine | Cotang. | Tang. | M. 
 
 55 Degrees. 
 
SINE-S AND TANGKMTS. (35 Decrees.) 
 
 53 
 
 M. Sine | D. | Cosine | D. Tang. D. Cotang. | 
 
 
 
 9.758591 
 
 :!oi 
 
 9.9133(55 
 
 147 
 
 9.845-2-27 
 
 448 
 
 10.154773 
 
 60 
 
 1 
 
 758772 
 
 300 
 
 9n-276 
 
 147 
 
 845496 
 
 448 
 
 154504 
 
 59 
 
 2 
 
 758952 
 
 300 
 
 913187 
 
 148 
 
 845764 
 
 448 
 
 154-236 
 
 58 
 
 3 
 
 7r>!H:?-2 
 
 300 
 
 913099 
 
 148 
 
 8415033 
 
 448 
 
 U3987 
 
 57 
 
 4 
 
 759312 
 
 300 
 
 913010 
 
 148 
 
 846302 
 
 448 
 
 153698 
 
 56 
 
 5 
 
 7.-i!l Kt-2 
 
 300 
 
 913992 
 
 148 
 
 84(5570 
 
 447 
 
 153430 
 
 55 1 
 
 6 
 
 759672 
 
 299 
 
 912833 
 
 148 
 
 846839 
 
 447 
 
 153161 
 
 54 
 
 7 
 
 759852 
 
 299 
 
 91-2744 
 
 118 
 
 847107 
 
 447 
 
 152893 
 
 53 
 
 8 
 
 7601)31 
 
 2i)J 
 
 91-26.1,-) 
 
 148 
 
 847376 
 
 447 
 
 152824 
 
 52 
 
 B 
 
 768211 
 
 299 
 
 9125(56 
 
 148 
 
 817*544 
 
 447 
 
 15-2356 
 
 51 
 
 10 
 
 760391) 
 
 289 
 
 912477 
 
 148 
 
 847913 
 
 447 
 
 152087 
 
 50 
 
 11 
 
 9.760569 
 
 298 
 
 9.912388 
 
 148 
 
 9.848181 
 
 447 
 
 10.151819 
 
 49 
 
 12 
 
 760748 
 
 298 
 
 91 2-299 
 
 149 
 
 848449 
 
 447 
 
 151551 
 
 48 
 
 13 
 
 760927 
 
 298 
 
 91 -2-2 10 
 
 149 
 
 848717 
 
 447 
 
 151283 
 
 47 
 
 14 
 
 761106 
 
 298 
 
 912121 
 
 149 
 
 848986 
 
 447 
 
 151014 
 
 46 
 
 15 
 
 7(51285 
 
 298 
 
 912031 
 
 149 
 
 849254 
 
 447 
 
 150746 
 
 45 
 
 16 
 
 761464 
 
 298 
 
 911942 
 
 149 
 
 84D.V2-2 
 
 447 
 
 150478 
 
 44 
 
 17 
 
 76164-2 
 
 297 
 
 911853 
 
 149 
 
 849790 
 
 446 
 
 150210 
 
 43 
 
 18 
 
 761821 
 
 297 
 
 911763 
 
 149 
 
 850058 
 
 446 
 
 149942 
 
 42 
 
 19 
 
 761999 
 
 297 
 
 911674 
 
 149 
 
 8.VW25 
 
 446 
 
 149075 
 
 41 
 
 20 
 
 762177 
 
 297 
 
 911584 
 
 149 
 
 850593 
 
 446 
 
 149407 
 
 40 
 
 21 
 
 9.762356 
 
 297 
 
 9.911495 
 
 149 
 
 9.850881 
 
 446 
 
 10.149139 
 
 39 
 
 22 
 
 762534 
 
 296 
 
 91140.') 
 
 149 
 
 851129 
 
 446 
 
 148871 
 
 38 
 
 23 
 
 7(5-27 1-2 
 
 296 
 
 911315 
 
 150 
 
 851396 
 
 446 
 
 148604 
 
 37 
 
 24 
 
 762889 
 
 MB 
 
 9112-26 
 
 150 
 
 851664 
 
 446 
 
 148336 
 
 36 
 
 25 
 
 763067 
 
 296 
 
 911136 
 
 150 
 
 851931 
 
 446 
 
 J 430(59 
 
 35 
 
 26 
 
 7f : 3-245 
 
 296 
 
 911046 
 
 150 
 
 852199 
 
 446 
 
 147801 
 
 34 
 
 27 
 
 715:142-2 
 
 296 
 
 <)10<).->6 
 
 150 
 
 85-21(56 
 
 446 
 
 147534 
 
 33 
 
 28 
 
 763r>i>n 
 
 295 
 
 910866 
 
 150 
 
 8.12733 
 
 445 
 
 1472(57 
 
 32 
 
 29 
 
 7153777 
 
 295 
 
 910776 
 
 150 
 
 853001 
 
 445 
 
 146999 
 
 31 
 
 30 
 
 763954 
 
 295 
 
 910686 
 
 150 
 
 8532G8 
 
 445 
 
 146732 
 
 30 
 
 31 
 
 9.764131 
 
 295 
 
 9.910596 
 
 150 
 
 9.853535 
 
 445 
 
 10.146465 
 
 29 
 
 32 
 
 764308 
 
 295 
 
 910506 
 
 150 
 
 853802 
 
 445 
 
 146198 
 
 28 
 
 33 
 
 764485 
 
 294 
 
 910415 
 
 150 
 
 854069 
 
 445 
 
 145931 
 
 27 
 
 34 
 
 7646(52 
 
 294 
 
 9103-25 
 
 151 
 
 &J4336 
 
 445 
 
 145664 
 
 26 
 
 35 
 
 764838 
 
 294 
 
 910235 
 
 151 
 
 854'5U3 
 
 445 
 
 145397 
 
 25 
 
 30 
 
 7fi5:)15 
 
 2i)4 
 
 910144 
 
 151 
 
 854870 
 
 445 
 
 145130 
 
 24 
 
 37 
 
 7(55191 
 
 294 
 
 910054 
 
 151 
 
 855137 
 
 445 
 
 144863 
 
 23 
 
 38 
 
 765367 
 
 294 
 
 <);><>963 
 
 151 
 
 855404 
 
 445 
 
 144596 
 
 22 
 
 39 
 
 765544 
 
 293 
 
 909873 
 
 151 
 
 855671 
 
 444 
 
 144329 
 
 21 
 
 40 
 
 765720 
 
 293 
 
 909782 
 
 151 
 
 855938 
 
 444 
 
 144063 
 
 20 
 
 41 
 
 9.765896 
 
 293 
 
 9.90%91 
 
 151 
 
 9.856-204 
 
 444 
 
 10.1437% 
 
 19 
 
 42 
 
 766072 
 
 293 
 
 909601 
 
 151 
 
 856471 
 
 444 
 
 143.V29 
 
 18 
 
 43 
 
 71H5247 
 
 293 
 
 909510 
 
 151 
 
 856737 
 
 444 
 
 143-2(53 
 
 17 
 
 44 
 
 7(56423 
 
 293 
 
 909419 
 
 151 
 
 857004 
 
 444 
 
 142996 
 
 16 
 
 45 
 
 765598 
 
 292 
 
 909328 
 
 152 
 
 H57270 
 
 444 
 
 142730 
 
 15 
 
 46 
 
 766774 
 
 292 
 
 903237 
 
 152 
 
 H.-)7.-)37 
 
 444 
 
 142463 
 
 14 
 
 47 
 
 786949 
 
 29-2 
 
 909146 
 
 152 
 
 857803 
 
 444 
 
 14-2197 
 
 13 
 
 40 
 
 767124 
 
 292 
 
 90.4055 
 
 152 
 
 858069 
 
 444 
 
 141931 
 
 12 
 
 49 
 
 767300 
 
 292 
 
 903964 
 
 152 
 
 858336 
 
 444 
 
 141664 
 
 11 
 
 50 
 
 767475 
 
 291 
 
 908873 
 
 152 
 
 858602 
 
 443 
 
 141398 
 
 10 
 
 51 
 
 ^. 767649 
 
 291 
 
 9.908781 
 
 152 
 
 9.858868 
 
 443 
 
 10.141132 
 
 9 
 
 52 
 
 767824 
 
 291 
 
 9^81590 
 
 152 
 
 859134 
 
 443 
 
 140866 
 
 8 
 
 53 
 
 767999 
 
 291 
 
 908599 
 
 152 
 
 859400 
 
 443 
 
 14!)600 
 
 7 
 
 54 
 
 768173 
 
 291 
 
 909507 
 
 IM 
 
 8596(56 
 
 443 
 
 140334 
 
 6 
 
 55 
 
 768348 
 
 290 
 
 908416 
 
 153 
 
 859932 
 
 443 
 
 140068 
 
 5 
 
 56 
 
 768522 
 
 290 
 
 908334 
 
 L53 
 
 860153 
 
 443 
 
 
 4 
 
 57 
 
 768697 
 
 290 
 
 90H-233 
 
 153 
 
 800464 
 
 413 
 
 139536 
 
 3 
 
 58 
 
 768371 
 
 290 
 
 908141 
 
 153 
 
 8-10730 
 
 443 
 
 139-270 
 
 2 
 
 59 
 
 769043 
 
 290 
 
 908049 
 
 153 
 
 8f>0995 
 
 443 
 
 i3!>:)ti.-> 
 
 1 
 
 60 
 
 769219 
 
 290 
 
 907958 
 
 153 
 
 861261 
 
 443 
 
 138739 
 
 
 
 1 | Cosine | Sine | | Cotang. | | Tang. | M. 
 
 54 Degrees. 
 
54 
 
 (3G Degrees.) A TABLE or LOGARITHMIC 
 
 M. Sine | D Cosine | D. Tang. D. | Cotang. i 
 
 
 
 9.769219 
 
 290 
 
 9.907953 
 
 153 
 
 9.861-261 
 
 443 
 
 10.138739 
 
 60 
 
 1 
 
 769393 
 
 289 
 
 9J7866 
 
 153 
 
 861527 
 
 443 
 
 138473 
 
 59 
 
 o 
 
 709566 
 
 289 
 
 907774 
 
 153 
 
 8(51792 
 
 442 
 
 138-208 
 
 58 
 
 3 
 
 769740 
 
 289 
 
 907682 
 
 153 
 
 803058 
 
 442 
 
 137942 
 
 57 
 
 4 
 
 769913 
 
 289 
 
 907590 
 
 153 
 
 862323 
 
 442 
 
 137677 
 
 56 
 
 5 
 
 770087 
 
 289 
 
 907498 
 
 153 
 
 8(5-2589 
 
 442 
 
 137411 
 
 55 
 
 6 
 
 770260 
 
 288 
 
 9:7406 
 
 153 
 
 8(52854 
 
 442 
 
 137146 
 
 54 
 
 7 
 
 770433 
 
 288 
 
 907314 
 
 154 
 
 8(53119 
 
 442 
 
 130881 
 
 53 
 
 8 
 
 770606 
 
 288 
 
 907222 
 
 154 
 
 863385 
 
 442 
 
 130615 
 
 52 
 
 9 
 
 770779 
 
 288 
 
 907129 
 
 154 
 
 863f50 
 
 442 
 
 130350 
 
 51 
 
 10 
 
 770952 
 
 288 
 
 907037 
 
 154 
 
 863915 
 
 442 
 
 136085 
 
 50 
 
 11 
 
 9.771125 
 
 288 
 
 9.906945 
 
 154 
 
 9.864180 
 
 442 
 
 10.135820 
 
 49 
 
 12 
 
 771298 
 
 287 
 
 906852 
 
 154 
 
 864445 
 
 442 
 
 135555 
 
 48 
 
 13 
 
 771470 
 
 287 
 
 906760 
 
 154 
 
 8(54710 
 
 442 
 
 135-290 
 
 47 
 
 14 
 
 771643 
 
 287 
 
 906667 
 
 154 
 
 864975 
 
 441 
 
 135025 
 
 4(5 
 
 15 
 
 771815 
 
 287 
 
 906575 
 
 154 
 
 865240 
 
 441 
 
 134700 
 
 45 
 
 16 
 
 771987 
 
 287 
 
 906482 
 
 154 
 
 865505 
 
 441 
 
 134495 
 
 44 
 
 17 
 
 772159 
 
 287 
 
 906389 
 
 ' 155 
 
 8(55770 
 
 441 
 
 134230 
 
 43 
 
 18 
 
 772331 
 
 286 
 
 906296 
 
 155 
 
 8(56035 
 
 441 
 
 1339(55 
 
 42 
 
 19 
 
 772503 
 
 288 
 
 906204 
 
 155 
 
 866300 
 
 441 
 
 133700 
 
 41 
 
 20 
 
 772675 
 
 286 
 
 906111 
 
 155 
 
 866564 
 
 441 
 
 133436 
 
 40 
 
 21 
 
 9.772847 
 
 286 
 
 9.90C018 
 
 155 
 
 9.8668-29 
 
 441 
 
 10.133171 
 
 39 
 
 22 
 
 7730 J 8 
 
 286 
 
 905925 
 
 155 
 
 867094 
 
 441 
 
 132906 
 
 38 
 
 | 23 
 
 773190 
 
 286 
 
 905832 
 
 155 
 
 867358 
 
 441 
 
 132642 
 
 37 
 
 24 
 
 773361 
 
 285 
 
 905739 
 
 155 
 
 867623 
 
 441 
 
 132377 
 
 36 
 
 25 
 
 773533 
 
 285 
 
 905645 
 
 155 
 
 867887 
 
 441 
 
 132113 
 
 35 
 
 26 
 
 773704 
 
 285 
 
 905552 
 
 155 
 
 868152 
 
 440 
 
 131848 
 
 34 
 
 27 
 
 773875 
 
 285 
 
 905459 
 
 155 
 
 868416 
 
 440 
 
 131584 
 
 33 
 
 28 
 
 774046 
 
 285 
 
 905366 
 
 156 
 
 868680 
 
 440 
 
 131320 
 
 32 
 
 29 
 
 774217 
 
 2H5 
 
 905272 
 
 156 
 
 868945 
 
 440 
 
 131055 
 
 31 
 
 30 
 
 774388 
 
 284 
 
 905179 
 
 156 
 
 869209 
 
 440 
 
 130791 
 
 30 
 
 31 
 
 9.774558 
 
 284 
 
 9.905085 
 
 156 
 
 9.869473 
 
 440 
 
 10.130527 
 
 23 
 
 32 
 
 774729 
 
 284 
 
 904992 
 
 156 
 
 869737 
 
 440 
 
 130-263 
 
 28 
 
 33 
 
 77-1899 
 
 284 
 
 904898 
 
 156 
 
 870001 
 
 440 
 
 129999 
 
 27 
 
 34 
 
 775070 
 
 284 
 
 904804 
 
 156 
 
 870-265 
 
 440 
 
 129735 
 
 26 
 
 35 
 
 775240 
 
 284 
 
 9047] 1 
 
 156 
 
 870529 
 
 440 
 
 129471 
 
 25 
 
 36 
 
 775410 
 
 283 
 
 904617 
 
 156 
 
 870793 
 
 440 
 
 129-207 
 
 24 
 
 37 
 
 ' 775580 
 
 283 
 
 904523 
 
 156 
 
 871057 
 
 440 
 
 128943 
 
 23 
 
 36 
 
 775750 
 
 283 
 
 904429 
 
 157 
 
 871321 
 
 440 
 
 128679 
 
 22 
 
 39 
 
 775920 
 
 283 
 
 904335 
 
 157 
 
 871585 
 
 440 
 
 128415 
 
 21 
 
 40 
 
 77609!) 
 
 283 
 
 904241 
 
 157 
 
 871849 
 
 439 
 
 128151 
 
 20 
 
 41 
 
 9.776259 
 
 283 
 
 9.904147 
 
 157 
 
 9.872112 
 
 439 
 
 10.127888 
 
 19 
 
 42 
 
 776429 
 
 282 
 
 9JMO.-.3 
 
 157 
 
 872376 
 
 439 
 
 127624 
 
 18 
 
 43 
 
 776598 
 
 282 
 
 903959 
 
 157 
 
 87-2640 
 
 439 
 
 1273(50 
 
 17 
 
 44 
 
 776768 
 
 282 
 
 903864 
 
 157 
 
 872903 
 
 439 
 
 1-27097 
 
 16 
 
 45 
 
 776937 
 
 282 
 
 903770 
 
 157 
 
 873167 
 
 439 
 
 126833 
 
 15 
 
 46 
 
 777106 
 
 282 
 
 903676 
 
 157 
 
 873430 
 
 439 
 
 1 20570 
 
 14 
 
 47 
 
 777275 
 
 281 
 
 903581 
 
 157 
 
 873694 
 
 439 
 
 126306 
 
 13 
 
 48 
 
 777444 
 
 281 
 
 903487 
 
 157 
 
 873957 
 
 439 
 
 120043 
 
 12 
 
 49 
 
 777613 
 
 281 
 
 903392 
 
 158 
 
 874220 
 
 439 
 
 125780 
 
 11 
 
 50 
 
 777781 
 
 281 
 
 903298 
 
 158 
 
 874484 
 
 439 
 
 125516 
 
 10 
 
 51 
 
 9.777950 
 
 281 
 
 9.903203 
 
 158 
 
 9.874747 
 
 439 
 
 10.125253 
 
 9 
 
 52 
 
 778119 
 
 281 
 
 903108 
 
 158 
 
 875010 
 
 439 
 
 124990 
 
 8 
 
 53 
 
 778287 
 
 280 
 
 903014 
 
 158 
 
 875273 
 
 438 
 
 124727 
 
 7 
 
 54 
 
 778455 
 
 280 
 
 902919 
 
 158 
 
 875536 
 
 438 
 
 124464 
 
 6 
 
 55 
 
 778624 
 
 280 
 
 902824 
 
 158 
 
 875800 
 
 438 
 
 1-24200 
 
 5 
 
 56 
 
 778792 
 
 280 
 
 902729 
 
 158 
 
 876003 
 
 438 
 
 123937 
 
 4 
 
 57 
 
 778900 
 
 280 
 
 90-2634 
 
 158 
 
 876326 
 
 438 
 
 1-23674 
 
 3 
 
 58 
 
 779128 
 
 280 
 
 902539 
 
 159 
 
 876589 
 
 438 
 
 l-2:!4 11 
 
 2 
 
 59 
 
 779295 
 
 279 
 
 91)2444 
 
 159 
 
 876851 
 
 438 
 
 123149 
 
 1 
 
 CO 
 
 779463 
 
 279 
 
 902349 
 
 159 
 
 877114 
 
 438 
 
 12-2886 
 
 
 
 | Cosine | | Sine | | Cotang. | | Tang. M. 1 
 
 53 Degrees. 
 
SINES AND TANGENTS. (37 Degrees.) 
 
 55 
 
 M. | Sine D. Cosine D. Tang. D Cotang 
 
 
 
 
 9.779463 
 
 279 
 
 9.9: 12311) 
 
 I.V.I 
 
 9.B77114 
 
 438 
 
 10.122846 
 
 t50 
 
 J 
 
 779831 
 
 279 
 
 902-253 
 
 159 
 
 877377 
 
 438 
 
 122633 
 
 59 
 
 i 
 
 779798 
 
 379 
 
 90-21. )8 
 
 156 
 
 877(540 
 
 438 
 
 123360 
 
 58 
 
 3 
 
 7799(36 
 
 279 
 
 90-2063 
 
 159 
 
 877903 
 
 438 
 
 12-2097 
 
 57 
 
 4 
 
 780133 
 
 279 
 
 901%7 
 
 158 
 
 8781(55 
 
 438 
 
 121835 
 
 56 
 
 5 
 
 7f<()3l)0 
 
 278 
 
 901872 
 
 159 
 
 8784-28 
 
 438 
 
 121572 
 
 55 
 
 6 
 
 780467 
 
 278 
 
 901776 
 
 159 
 
 878(591 
 
 438 
 
 121309 
 
 54 
 
 7 
 
 7rtli(i34 
 
 278 
 
 DOI6HI 
 
 158 
 
 878953 
 
 437 
 
 121047 
 
 53 
 
 8 
 
 780801 
 
 278 
 
 901535 
 
 159 
 
 879216 
 
 437 
 
 120784 
 
 52 
 
 9 
 
 780J68 
 
 278 
 
 91)1490 
 
 159 
 
 879478 
 
 437 
 
 120522 
 
 51 
 
 10 
 
 781134 
 
 278 
 
 'JO 1394 
 
 160 
 
 879741 
 
 437 
 
 120259 
 
 50 
 
 11 
 
 9.781301 
 
 277 
 
 9.901293 
 
 160 
 
 9.83,)003 
 
 437 
 
 10.119937 
 
 49 
 
 13 
 
 7H14t 
 
 277 
 
 901202 
 
 160 
 
 88,)2iij 
 
 437 
 
 119735 
 
 48 
 
 13 
 
 781634 
 
 277 
 
 901105 
 
 160 
 
 880528 
 
 437 
 
 119472 
 
 47 
 
 14 
 
 781800 
 
 277 
 
 901010 
 
 160 
 
 880790 
 
 437 
 
 119210 
 
 46 
 
 15 
 
 781966 
 
 277 
 
 900914 
 
 160 
 
 881052 
 
 437 
 
 118948 
 
 45 
 
 16 
 
 78-2132 
 
 277 
 
 900S18 
 
 160 
 
 881314 
 
 437 
 
 118686 
 
 44 
 
 17 
 
 782-298 
 
 276 
 
 900722 
 
 160 
 
 881576 
 
 437 
 
 118424 
 
 43 
 
 18 
 
 782464 
 
 276 
 
 9i)0i26 
 
 160 
 
 881839 
 
 437 
 
 118161 
 
 42 
 
 19 
 
 782830 
 
 276 
 
 900529 
 
 160 
 
 882101 
 
 437 
 
 117899 
 
 41 
 
 20 
 
 782796 
 
 276 
 
 901)433 
 
 161 
 
 88-3363 
 
 436 
 
 117637 
 
 40 
 
 21 
 
 9.782961 
 
 276 
 
 9.900337 
 
 161 
 
 9.882625 
 
 436 
 
 10.117375 
 
 39 
 
 22 
 
 783127 
 
 276 
 
 900240 
 
 161 
 
 882887 
 
 436 
 
 117113 
 
 38 
 
 23 
 
 783292 
 
 275 
 
 900144 
 
 161 
 
 883148 
 
 436 
 
 116852 
 
 37 
 
 24 
 
 783458 
 
 275 
 
 900047 
 
 161 
 
 883410 
 
 436 
 
 116590 
 
 36 
 
 25 
 
 783623 
 
 275 
 
 899951 
 
 161 
 
 883672 
 
 436 
 
 116328 
 
 35 
 
 26 
 
 783788 
 
 275 
 
 893854 
 
 161 
 
 8839:14 
 
 436 
 
 116066 
 
 34 
 
 27 
 
 783953 
 
 275 
 
 899757 
 
 161 
 
 884196 
 
 436 
 
 115804 
 
 33 
 
 28 
 
 784118 
 
 275 
 
 8991)60 
 
 161 
 
 884457 
 
 436 
 
 115543 
 
 32 
 
 29 
 
 784282 
 
 274 
 
 899564 
 
 161 
 
 884719 
 
 436 
 
 115281 
 
 31 
 
 30 
 
 784447 
 
 274 
 
 899467 
 
 162 
 
 884980 
 
 436 
 
 115020 
 
 30 
 
 :u 
 
 9.784612 
 
 274 
 
 9.899370 
 
 162 
 
 9.885242 
 
 436 
 
 10.114758 
 
 29 
 
 M 
 
 784776 
 
 274 
 
 899273 
 
 162 
 
 885503 
 
 436 
 
 114497 
 
 28 
 
 33 
 
 784941 
 
 274 
 
 899176 
 
 162 
 
 885765 
 
 436 
 
 114235 
 
 27 
 
 34 
 
 785105 
 
 274 
 
 899078 
 
 162 
 
 88(50-26 
 
 43(5 
 
 113974 
 
 26 
 
 3o 
 
 785369 
 
 273 
 
 898981 
 
 162 
 
 886-288 
 
 436 
 
 113712 
 
 25 
 
 36 
 
 785433 
 
 273 
 
 898884 
 
 162 
 
 881 M 49 
 
 435 
 
 113451 
 
 24 
 
 37 
 
 785507 
 
 273 
 
 898787 
 
 162 
 
 886810 
 
 435 
 
 113190 
 
 23 
 
 38 
 
 785761 
 
 273 
 
 898689 
 
 162 
 
 887072 
 
 435 
 
 112928 
 
 22 
 
 39 
 
 785925 
 
 273 
 
 898592 
 
 162 
 
 887333 
 
 435 
 
 112667 
 
 21 
 
 40 
 
 786089 
 
 273 
 
 898494 
 
 163 
 
 887594 
 
 435 
 
 112406 
 
 20 
 
 41 
 
 9.7H(r252 
 
 272 
 
 9.898397 
 
 163 
 
 9.887855 
 
 435 
 
 10.112145 
 
 19 
 
 42 
 
 786416 
 
 272 
 
 898299 
 
 163 
 
 888116 
 
 435 
 
 111884 
 
 18 
 
 43 
 
 786579 
 
 272 
 
 89820-2 
 
 163 
 
 888377 
 
 435 
 
 111623 
 
 17 
 
 44 
 
 786742 
 
 272 
 
 898104 
 
 163 
 
 888639 
 
 435 
 
 111361 
 
 16 
 
 45 
 
 786906 
 
 272 
 
 898006 
 
 163 
 
 888900 
 
 435 
 
 1]1100 
 
 15 
 
 46 
 
 7871)69 
 
 272 
 
 897908 
 
 163 
 
 8891(50 
 
 435 
 
 110840 
 
 14 
 
 47 
 
 787332 
 
 271 
 
 897810 
 
 163 
 
 889421 
 
 435 
 
 110579 
 
 13 
 
 48 
 
 787395 
 
 271 
 
 897712 
 
 163 
 
 889(582 
 
 435 
 
 110318 
 
 12 
 
 49 
 
 787557 
 
 271 
 
 897614 
 
 163 
 
 889943 
 
 435 
 
 110057 
 
 11 
 
 .50 
 
 787720 
 
 271 
 
 897516 
 
 163 
 
 890204 
 
 434 
 
 109796 
 
 10 
 
 51 
 
 9.787883 
 
 271 
 
 9.897418 
 
 164 
 
 9.890465 
 
 434 
 
 10.109535 
 
 9 
 
 52 
 
 788845 
 
 271 
 
 897320 
 
 164 
 
 890725 
 
 434 
 
 109-275 
 
 8 
 
 53 
 
 788208 
 
 27! 
 
 897222 
 
 164 
 
 890986 
 
 434 
 
 109014 
 
 7 
 
 54 
 
 788370 
 
 270 
 
 897123 
 
 164 
 
 891247 
 
 434 
 
 108753 
 
 6 
 
 55 
 
 783532 
 
 270 
 
 897025 
 
 164 
 
 891507 
 
 434 
 
 108493 
 
 5 
 
 56 
 
 788694 
 
 270 
 
 896926 
 
 164 
 
 8917158 
 
 434 
 
 108232 
 
 4 
 
 57 
 
 788856 
 
 270 
 
 896828 
 
 164 
 
 
 434 
 
 107972 
 
 3 
 
 58 
 
 7891)18 
 
 270 
 
 896729 
 
 164 
 
 892289 
 
 434 
 
 107711 
 
 2 
 
 59 
 
 789180 
 
 270 
 
 89(5631 
 
 164 
 
 892549 
 
 434 
 
 107451 
 
 1 
 
 60 
 
 7*9342 
 
 26J 
 
 896532 
 
 164 
 
 892810 
 
 434 
 
 107190 
 
 
 
 | Cosine | Sine | | Cotang. | Tang. M. j 
 
56 
 
 (38 Degrees.) A TABLE OP LOGARITHMIC 
 
 M. | Sine D I Cosine D, | Tang. D. | Cotang. | 
 
 
 
 9.78934-2 
 
 980 
 
 9.891)532 
 
 Ifi4 
 
 9.892810 
 
 434 
 
 10.107190 I 60 
 
 1 
 
 789504 
 
 369 
 
 896433 
 
 165 
 
 893070 
 
 434 
 
 106930 
 
 59 
 
 2 
 
 789665 
 
 269 
 
 896335 
 
 165 
 
 893331 
 
 434 
 
 106669 
 
 58 
 
 3 
 
 7898-27 
 
 269 
 
 896236 
 
 165 
 
 893591 
 
 434 
 
 106409 
 
 57 
 
 4 
 
 789988 
 
 269 
 
 896137 
 
 lt>5 
 
 893851 
 
 434 
 
 106149 
 
 56 
 
 5 
 
 790149 
 
 269 
 
 896038 
 
 165 
 
 894111 
 
 434 
 
 105889 
 
 55 
 
 6 
 
 790310 
 
 268 
 
 895939 
 
 165 
 
 894371 
 
 434 
 
 105629 
 
 54 
 
 7 
 
 7!K)471 
 
 268 
 
 895840 
 
 165 
 
 8:)4i)32 
 
 433 
 
 105368 
 
 53 
 
 8 
 
 790632 
 
 268 
 
 895741 
 
 1G5 
 
 894892 
 
 433 
 
 105108 
 
 52 
 
 9 
 
 790793 
 
 208 
 
 895641 
 
 165 
 
 895152 
 
 433 
 
 104848 
 
 51 
 
 10 
 
 790954 
 
 268 
 
 895542 
 
 165 
 
 895412 
 
 433 
 
 104588 
 
 50 
 
 11 
 
 9.791115 
 
 268 
 
 9.895443 
 
 166 
 
 9.895672 
 
 433 
 
 10.104328 
 
 49 
 
 12 
 
 791275 
 
 267 
 
 895343 
 
 166 
 
 895932 
 
 433 
 
 104068 
 
 48 
 
 13 
 
 791436 
 
 267 
 
 895244 
 
 166 
 
 896192 
 
 433 
 
 103808 
 
 47 
 
 14 
 
 791596 
 
 267 
 
 895145 
 
 166 
 
 8l>6452 
 
 433 
 
 103548 
 
 46 
 
 15 
 
 791757 
 
 267 
 
 895045 
 
 166 
 
 896712 
 
 433 
 
 103288 
 
 45 
 
 16 
 
 791917 
 
 267 
 
 894945 
 
 166 
 
 896971 
 
 433 
 
 103029 
 
 44 
 
 17 
 
 792077 
 
 367 
 
 894846 
 
 166 
 
 897231 
 
 433 
 
 102769 
 
 43 
 
 18 
 
 792237 
 
 266 
 
 894746 
 
 166 
 
 897491 
 
 433 
 
 102509 
 
 42 
 
 19 
 
 792397 
 
 266 
 
 894646 
 
 166 
 
 897751 
 
 433 
 
 10-2249 
 
 41 
 
 20 
 
 792557 
 
 266 
 
 894546 
 
 166 
 
 898010 
 
 133 
 
 101990 
 
 40 
 
 21 
 
 9.792716 
 
 266 
 
 9.894446 
 
 167 
 
 9.?!i,-}?0 
 
 433 
 
 10.101730 
 
 39 
 
 22 
 
 792876 
 
 266 
 
 894346 
 
 167 
 
 898530 
 
 433 
 
 101470 
 
 38 
 
 23 
 
 793035 
 
 2fi6 
 
 894246 
 
 167 
 
 898789 
 
 433 
 
 101211 
 
 37 
 
 24 
 
 793195 
 
 265 
 
 894146 
 
 167 
 
 899049 
 
 432 
 
 100951 
 
 36 
 
 25 
 
 793354 
 
 265 
 
 894046 
 
 167 
 
 899308 
 
 432 
 
 100692 
 
 35 
 
 26 
 
 793514 
 
 265 
 
 893946 
 
 167 
 
 899568 
 
 432 
 
 100432 
 
 34 
 
 27 
 
 793673 
 
 205 
 
 893846 
 
 167 
 
 899827 
 
 432 
 
 100173 
 
 33 
 
 28 
 
 793332 
 
 265 
 
 893745 
 
 167 
 
 900086 
 
 432 
 
 099914 
 
 32 
 
 29 
 
 793991 
 
 265 
 
 893645 
 
 167 
 
 900346 
 
 432 
 
 099654 
 
 31 
 
 30 
 
 794150 
 
 264 
 
 893544 
 
 167 
 
 900605 
 
 432 
 
 099395 
 
 30 
 
 31 
 
 9.794308 
 
 264 
 
 9.893444 
 
 168 
 
 9.900864 
 
 432 
 
 10.099136 
 
 29 
 
 32 
 
 794467 
 
 264 
 
 893343 
 
 168 
 
 901124 
 
 432 
 
 098876 
 
 28 
 
 33 
 
 794626 
 
 2154 
 
 89*243 
 
 1<!8 
 
 901383 
 
 439 
 
 098617 
 
 27 
 
 34 
 
 794784 
 
 2<i4 
 
 893142 
 
 168 
 
 901642 
 
 432 
 
 098358 
 
 2fi 
 
 35 
 
 794942 
 
 264 
 
 89IJ041 
 
 168 
 
 901901 
 
 432 
 
 098099 
 
 25 
 
 36 
 
 795101 
 
 2G4 
 
 892940 
 
 168 
 
 902160 
 
 432 
 
 097840 
 
 24 
 
 37 
 
 795259 
 
 263 
 
 892839 
 
 168 
 
 902419 
 
 432 
 
 097581 
 
 23 
 
 38 
 
 795417 
 
 263 
 
 892739 
 
 168 
 
 900679 
 
 432 
 
 097321 
 
 22 
 
 39 
 
 795575 
 
 263 
 
 892638 
 
 168 
 
 902938 
 
 432 
 
 097062 
 
 21 
 
 40 
 
 795733 
 
 263 
 
 892536 
 
 168 
 
 903197 
 
 431 
 
 096803 
 
 20 
 
 41 
 
 9.795891 
 
 263 
 
 9.892435 
 
 169 
 
 9.903455 
 
 431 
 
 10.09(5545 
 
 19 
 
 42 
 
 796049 
 
 263 
 
 892334 
 
 169 
 
 9H37 14 
 
 431 
 
 096286 
 
 18 
 
 43 
 
 796206 
 
 263 
 
 892233 
 
 169 
 
 903973 
 
 431 
 
 096027 
 
 17 
 
 44 
 
 796364 
 
 262 
 
 892132 
 
 169 
 
 904232 
 
 431 
 
 095768 
 
 16 
 
 45 
 
 796521 
 
 901 
 
 892030 
 
 169 
 
 904491 
 
 431 
 
 095509 
 
 15 
 
 46 
 
 796679 
 
 262 
 
 891929 
 
 169 
 
 904750 
 
 431 
 
 095250 
 
 14 
 
 47 
 
 796836 
 
 262 
 
 891827 
 
 169 
 
 905008 
 
 431 
 
 094992 
 
 13 
 
 48 
 
 79R993 
 
 262 
 
 891726 
 
 169 
 
 9052(57 
 
 431 
 
 094733 
 
 12 
 
 49 
 
 797150 
 
 261 
 
 . 891624 
 
 1(59 
 
 905526 
 
 431 
 
 094474 
 
 11 
 
 50 
 
 797307 
 
 261 
 
 891523 
 
 170 
 
 905784 
 
 431 
 
 094216 
 
 10 
 
 51 
 
 9.797464 
 
 261 
 
 9-891421 
 
 170 
 
 9.906043 
 
 431 
 
 10.093957 
 
 9 
 
 52 
 
 797621 
 
 261 
 
 891319 
 
 170 
 
 906302 
 
 431 
 
 093698 
 
 8 
 
 53 
 
 797777 
 
 261 
 
 891217 
 
 170 
 
 900560 
 
 431 
 
 093440 
 
 7 
 
 54 
 
 797934 
 
 261 
 
 891115 
 
 170 
 
 906819 
 
 431 
 
 093181 
 
 6 
 
 55 
 
 798091 
 
 261 
 
 891013 
 
 170 
 
 907077 
 
 431 
 
 092923 
 
 5 
 
 56 
 
 798247 
 
 261 
 
 890911 
 
 170 
 
 907336 
 
 431 
 
 002664 
 
 4 
 
 57 
 
 798403 
 
 260 
 
 8W889 
 
 170 
 
 907594 
 
 431 
 
 092406 
 
 3 
 
 58 
 
 798560 
 
 2liO 
 
 890707 
 
 170 
 
 907852 
 
 431 
 
 092148 
 
 2 
 
 59 
 
 798716 
 
 260 
 
 890605 
 
 170 
 
 908111 
 
 430 
 
 091889 
 
 1 
 
 60 1 798872 
 
 260 
 
 890503 
 
 170 
 
 908369 
 
 430 
 
 091631 
 
 
 
 | Cosine | Sine | | Cotang. | | Tang. | M. 
 
SINRS AMI TANGKNTS. (39 Degrees.) 
 
 5? 
 
 M. 
 
 Sine | D. Cosine | D. Tang. D. Cotang | 
 
 
 
 9.798872 
 
 260 
 
 9.890503 
 
 170 
 
 9.908369 
 
 430 
 
 W.i '.lira i 
 
 60 
 
 1 
 
 799028 
 
 260 
 
 890400 
 
 171 
 
 908628 
 
 430 
 
 091372 
 
 59 
 
 o 
 
 798184 
 
 2tiO 
 
 890298 
 
 171 
 
 908886 
 
 430 
 
 091114 
 
 58 
 
 3 
 
 799339 
 
 259 
 
 890195 
 
 171 
 
 909144 
 
 430 
 
 090856 
 
 57 
 
 4 
 
 799495 
 
 259 
 
 890093 
 
 171 
 
 909402 
 
 430 
 
 090598 
 
 56 
 
 5 
 
 799851 
 
 299 
 
 889990 
 
 171 
 
 909660 
 
 430 
 
 090340 
 
 55 
 
 6 
 
 799806 
 
 ..'.V.I 
 
 
 171 
 
 909918 
 
 430 
 
 090082 
 
 54 
 
 7 
 
 799963 
 
 399 
 
 
 171 
 
 '.110177 
 
 4:w 
 
 089823 
 
 53 
 
 8 
 
 800117 
 
 250 
 
 889882 
 
 171 
 
 910439 
 
 430 
 
 089565 
 
 52 
 
 9 
 
 800272 
 
 258 
 
 889579 
 
 17! 
 
 910693 
 
 430 
 
 089307 
 
 51 
 
 10 
 
 800427 
 
 258 
 
 889477 
 
 171 
 
 910051 
 
 430 
 
 089049 
 
 50 
 
 11 
 
 9.800582 
 
 258 
 
 9.89374 
 
 172 
 
 9.911209 
 
 430 
 
 10.088791 
 
 49 
 
 12 
 
 800737 
 
 258 
 
 889271 
 
 172 
 
 911467 
 
 430 
 
 088533 
 
 48 
 
 13 
 
 800899 
 
 258 
 
 889 J 68 
 
 179 
 
 <nn-.M 
 
 430 
 
 088276 
 
 47 
 
 14 
 
 Hill 047 
 
 258 
 
 889064 
 
 172 
 
 911982 
 
 430 
 
 088018 
 
 46 
 
 15 
 
 801301 
 
 258 
 
 888961 
 
 172 
 
 912240 
 
 430 
 
 0877(50 
 
 45 
 
 16 
 
 801356 
 
 257 
 
 886858 
 
 172 
 
 912498 
 
 430 
 
 087502 
 
 44 
 
 17 
 
 801511 
 
 257 
 
 888755 
 
 172 
 
 912756 
 
 430 
 
 087244 
 
 43 
 
 18 
 
 801665 
 
 xT.7 
 
 883*651 
 
 172 
 
 913014 
 
 429 
 
 086986 
 
 42 
 
 19 
 
 801819 
 
 257 
 
 888548 
 
 172 
 
 913271 
 
 429 
 
 086129 
 
 41 
 
 20 
 
 801973 
 
 257 
 
 888444 
 
 173 
 
 913529 
 
 429 
 
 086471 
 
 40 
 
 21 
 
 9.802128 
 
 257 
 
 9-883341 
 
 173 
 
 9.913787 
 
 429 
 
 10.086213 
 
 39 
 
 22 
 
 802282 
 
 256 
 
 888-237 
 
 173 
 
 914044 
 
 429 
 
 085956 
 
 38 
 
 23 
 
 802436 
 
 2.~>6 
 
 888134 
 
 173 
 
 914302 
 
 429 
 
 085698 
 
 37 
 
 24 
 
 802589 
 
 256 
 
 888030 
 
 173 
 
 914560 
 
 429 
 
 085440 
 
 36 
 
 35 
 
 802743 
 
 256 
 
 887926 
 
 173 
 
 914817 
 
 429 
 
 085183 
 
 35 
 
 26 
 
 802897 
 
 256 
 
 887822 
 
 173 
 
 915075 
 
 429 
 
 084925 
 
 34 
 
 27 
 
 803050 
 
 2.Vi 
 
 887718 
 
 173 
 
 915332 
 
 429 
 
 OH468 
 
 33 
 
 28 
 
 803-204 
 
 256 
 
 887614 
 
 173 
 
 915590 
 
 429 
 
 084410 
 
 32 
 
 29 
 
 803357 
 
 255 
 
 887510 
 
 173 
 
 915847 
 
 429 
 
 084153 
 
 31 
 
 30 
 
 803511 
 
 255 
 
 887406 
 
 174 
 
 916104 
 
 429 
 
 083896 
 
 30 
 
 31 
 
 9-803664 
 
 255 
 
 9.887302 
 
 174 
 
 9.916362 
 
 429 
 
 10.083638 
 
 29 
 
 32 
 
 803817 
 
 255 
 
 887198 
 
 174 
 
 916619 
 
 429 
 
 083381 
 
 28 
 
 33 
 
 803970 
 
 . 255 
 
 887093 
 
 174 
 
 916877 
 
 429 
 
 083123 
 
 27 
 
 34 
 
 8(14123 
 
 255 
 
 886989 
 
 lf4 
 
 917134 
 
 429 
 
 082866 
 
 26 
 
 35 
 
 804276 
 
 254 
 
 886885 
 
 174 
 
 917391 
 
 429 
 
 082609 
 
 25 
 
 36 
 
 80443 
 
 254 
 
 886780 
 
 174 
 
 ill 7048 
 
 429 
 
 082352 
 
 24 
 
 37 
 
 804581 
 
 254 
 
 886676 
 
 174 
 
 917905 
 
 429 
 
 082005 
 
 23 
 
 38 
 
 804734 
 
 254 
 
 886571 
 
 175 
 
 918163 
 
 428 
 
 081837 
 
 22 
 
 39 
 
 804886 
 
 254 
 
 886466 
 
 174 
 
 918420 
 
 428 
 
 081580 
 
 21 
 
 40 
 
 805039 
 
 254 
 
 886362 
 
 175 
 
 918677 
 
 428 
 
 081323 
 
 20 
 
 41 
 
 9--805191 
 
 254 
 
 9.886257 
 
 175 
 
 9.918934 
 
 428 
 
 10.081066 
 
 19 
 
 42 
 
 805343 
 
 253 
 
 886152 
 
 175 
 
 919191 
 
 428 
 
 080809 
 
 18 
 
 43 
 
 805495 
 
 253 
 
 886047 
 
 175 
 
 919448 
 
 
 080552 
 
 17 
 
 44 
 
 805647 
 
 253 
 
 885942 
 
 175 
 
 919705 
 
 428 
 
 080295 
 
 16 
 
 45 
 
 805799 
 
 253 
 
 885837 
 
 175 
 
 919962 
 
 428 
 
 080038 
 
 15 1 
 
 46 
 
 805951 
 
 253 
 
 885732 
 
 175 
 
 920219 
 
 428 
 
 079781 
 
 I* . 
 
 47 
 
 806103 
 
 253 
 
 885627 
 
 175 
 
 920476 
 
 428 
 
 079524 
 
 13 
 
 48 
 
 806254 
 
 253 
 
 885522 
 
 175 
 
 920733 
 
 428 
 
 079267 
 
 12 
 
 49 
 
 806406 
 
 252 
 
 885416 
 
 175 
 
 920990 
 
 428 
 
 079010 
 
 11 
 
 50 
 
 806557 
 
 252 
 
 885311 
 
 176 
 
 921247 
 
 428 
 
 078753 
 
 10 
 
 51 
 
 9-806709 
 
 252 
 
 9.885205 
 
 176 
 
 9.921503 
 
 428 
 
 10.078497 
 
 9 
 
 52 
 
 896800 
 
 252 
 
 885100 
 
 176 
 
 921760 
 
 428 
 
 078240 
 
 8 
 
 53 
 
 807011 
 
 252 
 
 884994 
 
 176 
 
 922017 
 
 428 
 
 077983 
 
 7 
 
 54 
 
 807163 
 
 252 
 
 884889 
 
 176 
 
 922274 
 
 428 
 
 077726 
 
 6 
 
 55 
 
 807314 
 
 252 
 
 884783 
 
 176 
 
 922530 
 
 428 
 
 077470 
 
 5 
 
 56 
 
 807465 
 
 251 
 
 884677 
 
 176 
 
 922787 
 
 428 
 
 077213 
 
 4 
 
 57 
 
 807615 
 
 251 
 
 884572 
 
 176 
 
 9-23044 
 
 428 
 
 076956 
 
 3 
 
 58 
 
 807766 
 
 251 
 
 884466 
 
 176 
 
 923300 
 
 428 
 
 076700 
 
 2 
 
 59 
 
 807917 
 
 251 
 
 884360 
 
 176 
 
 923557 
 
 427 
 
 076443 
 
 1 
 
 60 
 
 808067 
 
 251 
 
 884954 
 
 177 
 
 923813 
 
 427 
 
 076187 
 
 
 
 | Cosine | Sine | Cotang. Tang. 
 
 M. , 
 
 60 Degree*. 
 
58 
 
 (40 Degrees.) A TABF.E or LOGARITHMIC 
 
 M. | Sine | D. | Cosine D. Tang. | D. Cotang. | 
 
 
 
 9.808067 
 
 251 
 
 9.884254 
 
 177 
 
 9.923813 
 
 4-27 
 
 10.076187 
 
 60 
 
 1 
 
 8)8218 
 
 251 
 
 884148 
 
 177 
 
 9-2-1070 
 
 427 
 
 075930 
 
 59 
 
 2 
 
 8033: 
 
 251 
 
 884042 
 
 177 
 
 9-213-27 
 
 4-27 
 
 075;i73 
 
 58 . 
 
 3 
 
 808519 
 
 250 
 
 a3393!> 
 
 177 
 
 924583 
 
 427 
 
 075417 
 
 57 
 
 4 
 
 8J8I569 
 
 250 
 
 8833-29 
 
 177 
 
 9-2-1810 
 
 427 
 
 075160 
 
 56 
 
 5 
 
 808819 
 
 250 
 
 883723 
 
 177 
 
 <J-25ii96 
 
 4-27 
 
 074904 
 
 55 
 
 6 
 
 808989 
 
 250 
 
 883(517 
 
 177 
 
 9-2535-2 
 
 427 
 
 074648 
 
 54 
 
 
 809119 
 
 250 
 
 883510 
 
 177 
 
 9-256U9 
 
 427 
 
 0743'Jl 
 
 53 
 
 8 
 
 809-269 
 
 250 
 
 833404 
 
 177 
 
 9258:55 
 
 427 
 
 074135 
 
 52 
 
 9 
 
 809419 
 
 249 
 
 083-2U7 
 
 173 
 
 92151-22 
 
 427 
 
 073878 
 
 51 
 
 10 
 
 809569 
 
 249 
 
 8;31<J1 
 
 178 
 
 'J-2G373 
 
 427 
 
 073622 
 
 50 
 
 11 
 
 9.809718 
 
 249 
 
 9.833084 
 
 178 
 
 9.92G634 
 
 427 
 
 10.07:<366 
 
 49 
 
 12 
 
 809868 
 
 249 
 
 882977 
 
 178 
 
 ( J2(>890 
 
 4-27 
 
 073110 
 
 48 
 
 13 
 
 810017 
 
 249 
 
 832371 
 
 178 
 
 027147 
 
 427 
 
 072353 
 
 47 
 
 14 
 
 810167 
 
 249 
 
 832764 
 
 178 
 
 927403 
 
 427 
 
 072.)97 
 
 46 
 
 15 
 
 810316 
 
 248 
 
 883S57 
 
 178 
 
 927659 
 
 427 
 
 072341 
 
 45 
 
 16 
 
 810465 
 
 248 
 
 8d25.50 
 
 178 
 
 9-27915 
 
 . 427 
 
 07-2085 
 
 44 
 
 17 
 
 810614 
 
 248 
 
 882443 
 
 173 
 
 928171 
 
 427 
 
 071829 
 
 43 
 
 18 
 
 810763 
 
 248 
 
 882336 
 
 179 
 
 923427 
 
 4-27 
 
 071573 
 
 42 
 
 19 
 
 810912 
 
 248 
 
 882-2-29 
 
 179 
 
 928683 
 
 427 
 
 071317 
 
 41 
 
 20 
 
 811061 
 
 248 
 
 882121 
 
 179 
 
 928940 
 
 427 
 
 071060 
 
 40 
 
 21 
 
 9.811210 
 
 248 
 
 9.832014 
 
 179 
 
 9.929196 
 
 427 
 
 10.070804 
 
 39 
 
 22 
 
 811358 
 
 247 
 
 881907 
 
 179 
 
 929452 
 
 427 
 
 070548 
 
 38 
 
 23 
 
 811507 
 
 247 
 
 831799 
 
 179 
 
 9-29708 
 
 427 
 
 070292 
 
 37 
 
 24 
 
 811655 
 
 247 
 
 831692 
 
 179 
 
 929964 
 
 4-26 
 
 070036 
 
 36 
 
 ,25 
 
 811804 
 
 247 
 
 881584 
 
 179 
 
 930-220 
 
 4-26 
 
 069780 
 
 35 
 
 26 
 
 8J 1952 
 
 247 
 
 88 J 477 
 
 179 
 
 930475 
 
 426 
 
 01595-25 
 
 34 
 
 27 
 
 812100 
 
 247 
 
 831369 
 
 179 
 
 9:50731 
 
 426 
 
 0159-269 
 
 33 
 
 28 
 
 812-248 
 
 247 
 
 881261 
 
 180 
 
 93:)987 
 
 4-26 
 
 069013 
 
 32 
 
 29 
 
 812396 
 
 246 
 
 881153 
 
 180 
 
 931243 
 
 426 
 
 068757 
 
 31 
 
 30 
 
 812544 
 
 246 
 
 881046 
 
 180 
 
 931499 
 
 426 
 
 008501 
 
 30 
 
 31 
 
 9.812692 
 
 246 
 
 9.880938 
 
 180 
 
 9.931755 
 
 426 
 
 10.068245 
 
 29 
 
 32 
 
 81-2840 
 
 246 
 
 880830 
 
 180 
 
 93)010 
 
 426 
 
 067990 
 
 28 
 
 33 
 
 812938 
 
 246 
 
 88072-2 
 
 180 
 
 93-2266 
 
 426 
 
 067734 
 
 27 
 
 34 
 
 813135 
 
 246 
 
 880613 
 
 180 
 
 93-J5-2-2 
 
 426 
 
 067478 
 
 26 
 
 35 
 
 81 3-283 
 
 246 
 
 880505 
 
 180 
 
 93-2778 
 
 426 
 
 067-222 
 
 25 
 
 36 
 
 813430 
 
 245 
 
 880397 
 
 180 
 
 933(133 
 
 426 
 
 . 06(59117 
 
 24 
 
 37 
 
 813578 
 
 245 
 
 880239 
 
 181 
 
 933289 
 
 426 
 
 066711 
 
 23 
 
 38 
 
 813725 
 
 245 
 
 880180 
 
 181 
 
 933545 
 
 426 
 
 066455 
 
 22 
 
 i :<'.( 
 
 813872 
 
 245 
 
 880072 
 
 181 
 
 933800 
 
 426 
 
 066-200 
 
 21 
 
 40 
 
 814019 
 
 245 
 
 879963 
 
 181 
 
 934056 
 
 426 
 
 065944 
 
 20 
 
 41 
 
 9.814166 
 
 245 
 
 9.879855 
 
 181 
 
 9.934311 
 
 426 
 
 10.065689 
 
 19 
 
 42 
 
 814313 
 
 245 
 
 879746 
 
 181 
 
 934567 
 
 426 
 
 0(55433 
 
 18 
 
 43 
 
 814460 
 
 244 
 
 879637 
 
 181 
 
 934823 
 
 42f) 
 
 065177 
 
 17 
 
 44 
 
 814607 
 
 244 
 
 879529 
 
 181 
 
 935078 
 
 426 
 
 064922 
 
 16 
 
 45 
 
 814753 
 
 iM4 
 
 879 '20 
 
 181 
 
 935333 
 
 426 
 
 0(54(567 
 
 15 
 
 46 
 
 814900 
 
 244 
 
 879311 
 
 181 
 
 935589 
 
 426 
 
 064411 
 
 14 
 
 47 
 
 815046 
 
 244 
 
 879202 
 
 182 
 
 935844 
 
 426 
 
 064156 
 
 13 
 
 48 
 
 815193 
 
 244 
 
 879: 193 
 
 182 
 
 93(5100 
 
 426 
 
 063900 
 
 12 
 
 49 
 
 815339 
 
 244 
 
 878984 
 
 182 
 
 9:56355 
 
 426 
 
 063(145 
 
 11 
 
 50 
 
 815485 
 
 243 
 
 878875 
 
 182 
 
 936610 
 
 426 
 
 063390 
 
 10 
 
 51 
 
 9.815631 
 
 243 
 
 9.878766 
 
 18-2 
 
 9-936866 
 
 425 
 
 10.063134 
 
 9 
 
 52 
 
 815778 
 
 243 
 
 878656 
 
 182 
 
 937121 
 
 425 
 
 062879 
 
 8 
 
 53 
 
 815924 
 
 243 
 
 878547 
 
 182 
 
 937376 
 
 425 
 
 062624 
 
 7 
 
 54 
 
 816069 
 
 243 
 
 878438 
 
 182 
 
 937632 
 
 425 
 
 0623(58 
 
 6 
 
 55 
 
 816215 
 
 243 
 
 878328 
 
 182 
 
 937887 
 
 425 
 
 015-2113 
 
 5 
 
 56 
 
 816361 
 
 243 
 
 878219 
 
 183 
 
 938142 
 
 425 
 
 061858 
 
 4 
 
 57 
 
 816507 
 
 242 
 
 878109 
 
 183 
 
 938398 
 
 425 
 
 061602 
 
 3 
 
 .V 
 
 816652 
 
 242 
 
 877999 
 
 183 
 
 938653 
 
 425 
 
 061347 
 
 2 
 
 59 
 
 816798 
 
 242 
 
 877890 
 
 183 
 
 938908 
 
 425 
 
 001092 
 
 1 
 
 60 
 
 816943 
 
 242 
 
 877780 | 183 
 
 939163 
 
 425 
 
 060837 
 
 
 
 Cosine Sine | Cotang. | | Tang. | M. 
 
 49 Degrees. 
 
SINES AND TANGENTS. (41 Degrees.) 
 
 M. | Sine | D Cosine D. | Tang. D. Cotang. 1 
 
 8 
 
 9.81(5943 
 
 242 
 
 D.H77780 
 
 183 
 
 9.H39U53 
 
 425 
 
 10.060837 
 
 60 
 
 1 
 
 817088 
 
 242 
 
 877670 
 
 183 
 
 i 13! 141* 
 
 425 
 
 060582 
 
 59 
 
 2 
 
 817233 
 
 242 
 
 877/iW) 
 
 183 
 
 93<W73 
 
 425 
 
 060327 
 
 58 
 
 3 
 
 817379 
 
 242 
 
 877450 
 
 183 
 
 WJil-JH 
 
 425 
 
 060072 
 
 57 
 
 4 
 
 817524 
 
 241 
 
 877340 
 
 183 
 
 940183 
 
 425 
 
 059817 
 
 56 
 
 5 
 
 817668 
 
 241 
 
 877230 
 
 184 
 
 940438 
 
 425 
 
 059562 
 
 55 
 
 G 
 
 817813 
 
 241 
 
 877120 
 
 184 
 
 940694 
 
 425 
 
 059306 
 
 54 
 
 7 
 
 817958 
 
 241 
 
 877010 
 
 184 
 
 940949 
 
 425 
 
 059051 
 
 53 
 
 8 
 
 818103 
 
 241 
 
 87fi8!>9 
 
 184 
 
 941204 
 
 425 
 
 058796 
 
 52 
 
 9 
 
 818247 
 
 241 
 
 876789 
 
 184 
 
 941453 
 
 425 
 
 058542 
 
 51 
 
 10 
 
 818392 
 
 241 
 
 876678 
 
 184 
 
 941714 
 
 425 
 
 058286 
 
 50 
 
 11 
 
 9,818536 
 
 240 
 
 9.876568 
 
 184 
 
 9.941968 
 
 425 
 
 10.058032 
 
 49 
 
 12 
 
 818IJ81 
 
 240 
 
 876457 
 
 184 
 
 942223 
 
 425 
 
 057777 
 
 48 
 
 13 
 
 818825 
 
 240 
 
 876347 
 
 184 
 
 942478 
 
 425 
 
 057522 
 
 47 
 
 14 
 
 818969 
 
 240 
 
 876236 
 
 185 
 
 942733 
 
 425 
 
 057267 
 
 46 
 
 15 
 
 819113 
 
 240 
 
 876125 
 
 185 
 
 942988 
 
 425 
 
 057012 
 
 45 
 
 16 
 
 819257 
 
 240 
 
 876014 
 
 185 
 
 943243 
 
 425 
 
 056757 
 
 44 
 
 17 
 
 819401 
 
 240 
 
 875904 
 
 185 
 
 943498 
 
 425 
 
 056502 
 
 43 
 
 18 
 
 819545 
 
 230 
 
 875793 
 
 185 
 
 943752 
 
 425 
 
 056*48 
 
 42 
 
 19 
 
 819689 
 
 239 
 
 875682 
 
 185 
 
 944007 
 
 425 
 
 055993 
 
 41 
 
 20 
 
 819832 
 
 239 
 
 875571 
 
 185 
 
 944262 
 
 425 
 
 055738 
 
 40 
 
 21 
 
 9-819976 
 
 239 
 
 9.875459 
 
 185 
 
 9,944517 
 
 425 
 
 10.055483 
 
 39 
 
 22 
 
 820120 
 
 239 
 
 875348 
 
 185 
 
 944771 
 
 424 
 
 055229 
 
 38 
 
 23 
 
 820263 
 
 239 
 
 875237 
 
 185 
 
 945026 
 
 424 
 
 054974 
 
 37 
 
 24 
 
 820406 
 
 239 
 
 875126 
 
 186 
 
 945281 
 
 424 
 
 054719 
 
 36 
 
 25 
 
 820550 
 
 238 
 
 875014 
 
 186 
 
 945535 
 
 424 
 
 054465 
 
 35 
 
 26 
 
 820693 
 
 238 
 
 874903 
 
 186 
 
 945790 
 
 424 
 
 054210 
 
 34 
 
 27 
 
 820836 
 
 238 
 
 874791 
 
 186 
 
 946045 
 
 424 
 
 053955 
 
 33 
 
 28 
 
 820979 
 
 238 
 
 874680 
 
 186 
 
 946299 
 
 424 
 
 053701 
 
 32 
 
 29 
 
 821122 
 
 238 
 
 874568 
 
 186 
 
 946554 
 
 424 
 
 053446 
 
 31 
 
 30 
 
 821265 
 
 238 
 
 874456 
 
 186 
 
 946808 
 
 424 
 
 053192 
 
 30 
 
 31 
 
 9.821407 
 
 238 
 
 9.874344 
 
 186 
 
 9.947063 
 
 424 
 
 10.052937 
 
 29 
 
 32 
 
 821550 
 
 238 
 
 874232 
 
 187 
 
 947318 
 
 424 
 
 052682 
 
 28 
 
 33 
 
 821693 
 
 237 
 
 874121 
 
 187 
 
 947572 
 
 424 
 
 052428 
 
 27 
 
 34 
 
 821835 
 
 237 
 
 874009 
 
 187 
 
 947826 
 
 424 
 
 052174 
 
 26 
 
 35 
 
 821977 
 
 237 
 
 873896 
 
 187 
 
 948081 
 
 424 
 
 051919 
 
 25 
 
 36 
 
 822120 
 
 237 
 
 873784 
 
 187 
 
 948336 
 
 424 
 
 05U-64 
 
 24 
 
 37 
 
 822262 
 
 237 
 
 873672 
 
 187 
 
 948590 
 
 424 
 
 051410 
 
 23 
 
 38 
 
 822404 
 
 237 
 
 873560 
 
 187 
 
 948844 
 
 424 
 
 051156 
 
 22 
 
 39 
 
 822546 
 
 237 
 
 873448 
 
 187 
 
 9VJ099 
 
 424 
 
 050901 
 
 21 
 
 40 
 
 SZHffl 
 
 236 
 
 873335 
 
 187 
 
 949353 
 
 424 
 
 050647 
 
 20 
 
 41 
 
 9.822SH) 
 
 236 
 
 9.873223 
 
 187 
 
 9.949607 
 
 424 
 
 10.050393 
 
 19 
 
 42 
 
 822972 
 
 236 
 
 8731 10 
 
 IPS 
 
 949662 
 
 424 
 
 050138 
 
 18 
 
 43 
 
 823114 
 
 236 
 
 872998 
 
 188 
 
 950116 
 
 424 
 
 049884 
 
 17 
 
 44 
 
 823255 
 
 236 
 
 872885 
 
 188 
 
 950370 
 
 424 
 
 049630 
 
 16 
 
 45 
 
 823397 
 
 236 
 
 872772 
 
 188 
 
 950625 
 
 424 
 
 049375 
 
 15 
 
 46 
 
 823.539 
 
 236 
 
 872659 
 
 188 
 
 950879 
 
 424 
 
 049121 
 
 14 
 
 47 
 
 823'680 
 
 235 
 
 872547 
 
 188 
 
 951133 
 
 424 
 
 048867 
 
 13 
 
 48 
 
 
 335 
 
 872434 
 
 188 
 
 951388 
 
 424 
 
 048612 
 
 12 
 
 49 
 
 82.'?%3~ 
 
 235 
 
 872321 
 
 188 
 
 951642 
 
 424 
 
 048358 
 
 11 
 
 50 
 
 824104 
 
 235 
 
 872208 
 
 188 
 
 951896 
 
 424 
 
 048104 
 
 10 
 
 51 
 
 9.824245 
 
 235 
 
 9.872095 
 
 189 
 
 9-952150 
 
 424 
 
 10.047850 
 
 9 
 
 52 
 
 824386 
 
 235 
 
 871981 
 
 189 
 
 952405 
 
 424 
 
 047595 
 
 8 
 
 53 
 
 824527 
 
 235 
 
 871868 
 
 189 
 
 952659 
 
 424 
 
 047341 
 
 7 
 
 54 
 
 824668 
 
 234 
 
 871755 
 
 189 
 
 952913 
 
 424 
 
 047087 
 
 6 
 
 55 
 
 824808 
 
 234 
 
 871641 
 
 189 
 
 953167 
 
 423 
 
 046833 
 
 5 
 
 56 
 
 824949 
 
 234 
 
 871528 
 
 189 
 
 953421 
 
 423 
 
 046579 
 
 4 
 
 57 
 
 825090 
 
 234 
 
 871414 
 
 189 
 
 953675 
 
 423 
 
 046325 
 
 3 
 
 58 
 
 8252:*0 
 
 234 
 
 871301 
 
 189 
 
 953929 
 
 423 
 
 046071 
 
 2 
 
 59 
 
 825371 
 
 234 
 
 871187 
 
 189 
 
 954183 
 
 423 
 
 045817 
 
 1 
 
 60 
 
 825511 
 
 234 
 
 871073 
 
 190 
 
 954437 
 
 423 
 
 045563 
 
 
 
 | Cosine | Sine Cotang. Tang. M. [ 
 
 48 Degrees. 
 

 
 (42 Degrees.) A TABLR or LOGARITHMIC 
 
 M. j Sine D. | Cosine | D. Tang. | D. | Cotang f 
 
 
 
 9.825511 
 
 234 
 
 9,871073 
 
 190 
 
 9.054437 
 
 423 
 
 10.045563 60 
 
 1 
 
 825651 
 
 233 
 
 870960 
 
 190 
 
 954691 
 
 423 
 
 045309 59 ' 
 
 2 
 
 835791 
 
 233 
 
 870P46 
 
 190 
 
 954945 
 
 4-23 
 
 045055 i 58 
 
 3 
 
 825931 
 
 233 
 
 870732 
 
 190 
 
 955200 
 
 423 
 
 044800 57 
 
 4 
 
 826071 
 
 233 
 
 870618 
 
 190 
 
 155454 
 
 4-23 
 
 044541 i 5t> 
 
 5 
 
 826211 
 
 233 
 
 870504 
 
 190 
 
 955707 
 
 423 
 
 044293 i 55 
 
 6 
 
 826351 
 
 233 
 
 870390 
 
 190 
 
 955961 
 
 4-23 
 
 044039 54 
 
 7 
 
 826491 
 
 233 
 
 870276 
 
 190 
 
 956215 
 
 423 
 
 043785 53 
 
 8 
 
 826631 
 
 233 
 
 870161 
 
 190 
 
 956469 
 
 4-23 
 
 013531 52 
 
 9 
 
 821.770 
 
 232 
 
 870047 
 
 191 
 
 958723 
 
 423 
 
 043-277 r,| 
 
 10 
 
 826910 
 
 232 
 
 869933 
 
 191 
 
 956977 
 
 423 
 
 043023 
 
 50 
 
 11 
 
 9.827049 
 
 232 
 
 9.869818 
 
 191 
 
 9.957231 
 
 423 
 
 10.042709 
 
 49 
 
 12 
 
 8-27189 
 
 232 
 
 869704 
 
 191 
 
 957485 
 
 423 
 
 042515 
 
 48 
 
 13 
 
 827328 
 
 232 
 
 869589 
 
 191 
 
 957739 
 
 423 
 
 042261 
 
 47 ! 
 
 14 
 
 827467 
 
 232 
 
 869474 
 
 191 
 
 957993 
 
 423 
 
 042007 
 
 46 
 
 15 
 
 8-27606 
 
 232 
 
 869360 
 
 KB 
 
 958246 
 
 423 
 
 94J754 
 
 45 
 
 16 
 
 8-27745 
 
 232 
 
 8H9245 
 
 191 
 
 958500 
 
 423 
 
 041500 
 
 44 
 
 17 
 
 827884 
 
 231 
 
 869130 
 
 191 
 
 958754 
 
 423 
 
 041246 
 
 43 
 
 18 
 
 828023 
 
 231 
 
 869015 
 
 192' 
 
 959008 
 
 423 
 
 040992 
 
 42 
 
 19 
 
 828162 
 
 231 
 
 868900 
 
 192 
 
 959262 
 
 423 
 
 040738 
 
 41 
 
 20 
 
 828301 
 
 231 
 
 868785 
 
 192 
 
 959516 
 
 423 
 
 040484 
 
 40 
 
 21 
 
 9.828439 
 
 231 
 
 9.8G8670 
 
 192 
 
 9.959769 
 
 423 
 
 10.040231 
 
 39 
 
 22 
 
 828578 
 
 231 
 
 868555 
 
 192 
 
 90-T023 
 
 423 
 
 039977 
 
 3& 
 
 23 
 
 828716 
 
 231 
 
 868440 
 
 192 
 
 960277 
 
 423 
 
 039723 
 
 37 
 
 24 
 
 8-28855 
 
 230 
 
 868324 
 
 192 
 
 960531 
 
 423 
 
 039469 
 
 36 
 
 25- 
 
 828993 
 
 230 
 
 868209 
 
 192 
 
 960784 
 
 423 
 
 03&216 
 
 35 
 
 36 
 
 829131 
 
 230 
 
 868093 
 
 192 
 
 961038 
 
 423 
 
 038962 
 
 34 
 
 27 
 
 829269 
 
 230 
 
 867978 
 
 193 
 
 961291 
 
 423 
 
 038709 
 
 33 
 
 28 
 
 829407 
 
 230 
 
 8(37802 
 
 193 
 
 9.'il545 
 
 423 
 
 038455 
 
 32 
 
 20 
 
 829545 
 
 230 
 
 867747 
 
 193 
 
 961799 
 
 423 
 
 038201 
 
 31 
 
 30 
 
 829683 
 
 230 
 
 867631 
 
 193 
 
 962052 
 
 423 
 
 037948 
 
 30 
 
 31 
 
 9.829821 
 
 229 
 
 9.8SJ7515 
 
 193 
 
 9.9IW306 
 
 423 
 
 10,037694 
 
 29 
 
 32 
 
 829959 
 
 229 
 
 867399 
 
 193 
 
 96-25()0 
 
 423 
 
 037440 
 
 28 
 
 33 
 
 830097 
 
 229 
 
 867283 
 
 193 
 
 962813 
 
 423 
 
 037187 
 
 27 
 
 ! 34 
 
 830234 
 
 229 
 
 867167 
 
 193 
 
 530G7 
 
 423 
 
 036933 
 
 26 
 
 35 
 
 8.30372 
 
 2-29 
 
 867051 
 
 193 
 
 963320 
 
 433 
 
 036680 
 
 25 
 
 36 
 
 830509 
 
 229 
 
 866935 
 
 194 
 
 963574 
 
 423 
 
 036426 
 
 24 
 
 37 
 
 830646 
 
 229 
 
 866819 
 
 194 
 
 963827 
 
 423 
 
 036173 
 
 23 
 
 38 
 
 830784 
 
 229 
 
 866703 
 
 194 
 
 964081 
 
 423 
 
 035919 
 
 22 
 
 39 
 
 830931 
 
 228 
 
 866586 
 
 194 
 
 964335 
 
 423 
 
 035665 
 
 21 
 
 40 
 
 831058 
 
 228 
 
 866470 
 
 194 
 
 964588 
 
 422 
 
 035412 
 
 20 
 
 41 
 
 9.831195 
 
 228 
 
 9-866353 
 
 194 
 
 9.964842 
 
 422 
 
 10.035158 
 
 19 
 
 42 
 
 831332 
 
 228 
 
 866237 
 
 194 
 
 965095 
 
 422 
 
 034905 
 
 18 
 
 43 
 
 831469 
 
 228 
 
 866120 
 
 194 
 
 965349 
 
 422 
 
 034651 
 
 17 
 
 44 
 
 831606 
 
 228 
 
 866004 
 
 195 
 
 965602 
 
 422 
 
 034398 
 
 16 
 
 45 
 
 831742 
 
 228 
 
 865887 
 
 195 
 
 965855 
 
 422 
 
 034145 
 
 15 
 
 46 
 
 831879 
 
 228 
 
 865770 
 
 195 
 
 966109 
 
 422 
 
 033891 
 
 14 
 
 47 
 
 832015 
 
 227 
 
 865653 
 
 195 
 
 966362 
 
 422 
 
 033638 
 
 13 
 
 48 
 
 832152 
 
 227 ' 
 
 865536 
 
 195 
 
 966616 
 
 422 
 
 033384 
 
 12 
 
 49 
 
 832288 
 
 227 
 
 865419 
 
 195 
 
 966869 
 
 422 
 
 033131 
 
 11 
 
 50 
 
 832425 
 
 227 
 
 865302 
 
 195 
 
 967123 
 
 422 
 
 032877 
 
 10 
 
 51 
 
 9.832561 
 
 227 
 
 9-865185 
 
 195 
 
 9.907376 
 
 422 
 
 TO. 032624 
 
 9 
 
 52 
 
 832697 
 
 227 
 
 865068 
 
 195 
 
 967629 
 
 422 
 
 032371 
 
 8 
 
 53 
 
 832833 
 
 227 
 
 864950 
 
 195 
 
 967883 
 
 422 
 
 032117 
 
 7 
 
 54 
 
 832969 
 
 226 
 
 864833 
 
 196 
 
 968136 
 
 4-2-2 
 
 031864 
 
 6 
 
 55 
 
 833105 
 
 226 
 
 864716 
 
 196 
 
 968389 
 
 422 
 
 031611 
 
 5 
 
 56 
 
 833241 
 
 226 
 
 864598 
 
 196 
 
 968643 
 
 4,'-2 
 
 031357 
 
 4 
 
 57 
 
 833377 
 
 226 
 
 864481 
 
 196 
 
 968896 
 
 422 
 
 031104 
 
 3 
 
 58 
 
 833512 
 
 226 
 
 864363 
 
 196 
 
 969149 
 
 422 
 
 030851 
 
 2 
 
 59 
 
 833648 
 
 226 
 
 864245 
 
 196 
 
 969403 
 
 422 
 
 030597 
 
 1 
 
 60 
 
 833783 
 
 226 
 
 864127 
 
 196 
 
 969656 
 
 422 
 
 030344 
 
 
 
 Cosine 
 
 | Sine 
 
 ] Cotang. | [ Tang. | M. 1 
 
 47 Degrees. 
 
SINES AND TANGENTS. (43 Degrees.) 
 
 61 
 
 ' M. | Sine D | Cosine | D. | Tang. D. Cotang. | 
 
 
 
 9.833783 
 
 226 
 
 9.864127 
 
 196 
 
 9.969656 
 
 422 
 
 10.030:544 
 
 60 
 
 1 
 
 833919 
 
 22.5 
 
 864010 
 
 196 
 
 90991)9 
 
 422 
 
 030091 
 
 59 
 
 2 
 
 834054 
 
 225 
 
 863892 
 
 197 
 
 970 162 
 
 422 
 
 0298118 
 
 58 
 
 3 
 
 834189 
 
 225 
 
 863774 
 
 197 
 
 970416 
 
 422 
 
 029584 
 
 57 
 
 4 
 
 834325 
 
 225 
 
 863656 
 
 197 
 
 97000!) 
 
 422 
 
 029331 
 
 56 
 
 5 
 
 834460 
 
 225 
 
 863538 
 
 197 
 
 970922 
 
 422 
 
 029078 
 
 55 
 
 
 
 834595 
 
 223 
 
 8li3419 
 
 197 
 
 971175 
 
 1-22 
 
 028825 
 
 54 
 
 7 
 
 834730 
 
 2-2.-> 
 
 Hii^ioi 
 
 197 
 
 971429 
 
 422 
 
 028571 
 
 53 
 
 8 
 
 834805 
 
 823 
 
 863183 
 
 197 
 
 971682 
 
 422 
 
 028318 
 
 52 
 
 9 
 
 834999 
 
 224 
 
 863064 
 
 197 
 
 97l!):r, 
 
 422 
 
 028065 
 
 51 
 
 10 
 
 835134 
 
 224 
 
 862946 
 
 198 
 
 972188 
 
 422 
 
 027812 
 
 50 
 
 11 
 
 9.835269 
 
 B24 
 
 9.862827 
 
 198 
 
 9.972441 
 
 422 
 
 10.027.-,.-,!) 
 
 49 
 
 1-2 
 
 835403 
 
 224 
 
 802709 
 
 198 
 
 972! 594 
 
 422 
 
 027306 
 
 48 
 
 13 
 
 835538 
 
 834 
 
 862590 
 
 198 
 
 972948 
 
 422 
 
 027052 
 
 47 
 
 14 
 
 835672 
 
 224 
 
 863471 
 
 198 
 
 973201 
 
 422 
 
 020799 
 
 46 
 
 15 
 
 835807 
 
 224 
 
 862353 
 
 198. 
 
 973454 
 
 422 
 
 026546 
 
 45 
 
 16 
 
 835941 
 
 224 
 
 862234 
 
 198 
 
 973707 
 
 422 
 
 020293 
 
 44 
 
 1? 
 
 8:51)075 
 
 223 
 
 862115 
 
 198 
 
 973960 
 
 422 
 
 026040 
 
 43 
 
 18 
 
 836209 
 
 223 
 
 861996 
 
 198 
 
 974213 
 
 422 
 
 025787 
 
 42 
 
 19 
 
 836343 
 
 223 
 
 861877 
 
 198 
 
 974406 
 
 422 
 
 025534 
 
 41 
 
 20 
 
 836477 
 
 223 
 
 861758 
 
 199 
 
 974719 
 
 422 
 
 025281 
 
 40 
 
 21 
 
 9.8361)11 
 
 223 
 
 9-861638 
 
 199 
 
 9.974973 
 
 422 
 
 10.025027 
 
 39 
 
 22 
 
 83(5745 
 
 223 
 
 861519 
 
 199 
 
 975226 
 
 422 
 
 0-24774 
 
 38 
 
 23 
 
 836878 
 
 823 
 
 861400 
 
 199 
 
 975479 
 
 422 
 
 024521 
 
 37 
 
 24 
 
 837012 
 
 222 
 
 861280 
 
 199 
 
 975732 
 
 422 
 
 024268 
 
 36 
 
 25 
 
 837146 
 
 2-22 
 
 Hi; 1 Mil 
 
 199 
 
 975985 
 
 422 
 
 024015 
 
 35 
 
 26 
 
 K5727!) 
 
 222 
 
 861041 
 
 199 
 
 !)702:58 
 
 422 
 
 023762 
 
 34 
 
 27 
 
 837412 
 
 222 
 
 860922 
 
 199 
 
 976491 
 
 422 
 
 023509 
 
 33 
 
 28 
 
 837546 
 
 222 
 
 860802 
 
 199 
 
 976744 
 
 422 
 
 023256 
 
 32 
 
 29 
 
 837679 
 
 222 
 
 860682 
 
 200 
 
 976997 
 
 422 
 
 023003 
 
 31 
 
 30 
 
 837812 
 
 222 
 
 860562 
 
 200 
 
 977250 
 
 422 
 
 022750 
 
 30 
 
 31 
 
 9.8371)45 
 
 2-22 
 
 9-860442 
 
 200 
 
 9.977503 
 
 422 
 
 10.022497 
 
 29 
 
 32 
 
 8:58078 
 
 221 
 
 860322 
 
 200 
 
 977755 
 
 422 
 
 022244 
 
 28 
 
 33 
 
 838211 
 
 221 
 
 860202 
 
 200 
 
 97809 
 
 422 
 
 021991 
 
 27 
 
 34 
 
 838344 
 
 221 
 
 860082 
 
 200 
 
 978262 
 
 422 
 
 021738 
 
 26 
 
 35 
 
 838477 
 
 221 
 
 859962 
 
 2l>0 
 
 978515 
 
 422 
 
 021485 
 
 25 
 
 36 
 
 8:58010 
 
 221 
 
 850842 
 
 200 
 
 978768 
 
 422 
 
 (12 1-2:5-2 
 
 24 
 
 37 
 
 838742 
 
 2-21 
 
 8.19721 
 
 201 
 
 979021 
 
 422 
 
 020979 
 
 23 
 
 38 
 
 838875 
 
 221 
 
 s.iinioi 
 
 201 
 
 979274 
 
 422 
 
 020726 
 
 22 
 
 39 
 
 839007 
 
 221 
 
 859480 
 
 201 
 
 979527 
 
 422 
 
 020473 
 
 21 
 
 40 
 
 839140 
 
 220 
 
 859360 
 
 201 
 
 979780 
 
 422 
 
 0-20-220 
 
 20 
 
 41 
 
 9.839272 
 
 220 
 
 9.859239 
 
 201 
 
 9.930033 
 
 422 
 
 10.019967 
 
 19 
 
 42 
 
 839404 
 
 220 
 
 859119 
 
 201 
 
 980286 
 
 422 
 
 019714 
 
 18 
 
 43 
 
 Kl'.).->:50 
 
 220 
 
 858998 
 
 2J1 
 
 980538 
 
 422 
 
 019462 
 
 17 
 
 44 
 
 839668 
 
 220 
 
 858877 
 
 201 
 
 980791 
 
 421 
 
 019209 
 
 16 
 
 45 
 
 8.J9800 
 
 220 
 
 858756 
 
 202 
 
 981044 
 
 421 
 
 018956 
 
 15 
 
 46 
 
 839932 
 
 220 
 
 858635 
 
 202 
 
 981297 
 
 421 
 
 018703 
 
 14 
 
 47 
 
 840064 
 
 219 
 
 858514 
 
 202 
 
 981550 
 
 421 
 
 018450 
 
 13 
 
 48 
 
 840191) 
 
 219 
 
 858393 
 
 202 
 
 981803 
 
 421 
 
 018197 
 
 12 
 
 49 
 
 840328 
 
 219 
 
 858272 
 
 202 
 
 982056 
 
 421 
 
 0171)44 
 
 11 
 
 50 
 
 840459 
 
 219 
 
 858151 
 
 202 
 
 982M9 
 
 421 
 
 017691 
 
 10 
 
 51 
 
 9.840591 
 
 219 
 
 9.858029 
 
 202 
 
 9.982562 
 
 421 
 
 10.017438 
 
 9 
 
 .V2 
 
 840722 
 
 219 
 
 857908 
 
 202 
 
 982814 
 
 421 
 
 017186 
 
 8 
 
 53 
 
 840854 
 
 219 
 
 857786 
 
 202 
 
 983067 
 
 421 
 
 010933 
 
 7 
 
 54 
 
 840985 
 
 219 
 
 857665 
 
 203 
 
 983320 
 
 421 
 
 016680 
 
 6 
 
 55 
 
 841116 
 
 218 
 
 857543 
 
 203 
 
 983573 
 
 421 
 
 016427 
 
 5 
 
 56 
 
 841247 
 
 218 
 
 857422 
 
 203 
 
 983826 
 
 421 
 
 016174 
 
 4 
 
 57 
 
 841378 
 
 218 
 
 857300 
 
 203 
 
 984079 
 
 421 
 
 015921 
 
 3 
 
 58 
 
 841509 
 
 218 
 
 857178 
 
 203 
 
 984331 
 
 421 
 
 015669 
 
 2 
 
 59 
 
 841640 
 
 218 
 
 857056 
 
 203 
 
 984584 
 
 421 
 
 oi:, no 
 
 1 
 
 60 
 
 841771 
 
 218 
 
 856334 
 
 203 
 
 JW4K17 
 
 421 
 
 015163 
 
 
 
 | Cosine | Sine | ! Cotang. | Tang. | M. 
 
 46 Degrees. 
 
(44 Degrees.) LOG. SINES AND TANGENTS. 
 
 M. Sine D. Cosine D. Tang. D. | Cotang. 
 
 
 
 9.841771 
 
 218 
 
 9.856934 1 203 
 
 9.984837 
 
 421 
 
 10.015163 
 
 60 
 
 1 
 
 841902 
 
 218 
 
 856812 1 203 
 
 985090 
 
 421 
 
 014910 
 
 59 
 
 2 
 
 842033 
 
 218 
 
 856690 
 
 204 
 
 985343 
 
 421 
 
 014657 
 
 58 
 
 3 
 
 842163 
 
 217 
 
 856568 
 
 204 
 
 985596 
 
 4'21 
 
 014404 
 
 57 
 
 4 
 
 842294 
 
 217 
 
 856446 
 
 204 
 
 985848 
 
 421 
 
 014152 
 
 56 
 
 5 
 
 842424 
 
 217 
 
 856323 
 
 204 
 
 98lil01 
 
 421 
 
 013899 
 
 55 
 
 ' 6 
 
 842555 
 
 217 
 
 856201 
 
 204 
 
 98G354 
 
 421 
 
 013646 
 
 54 
 
 7 
 
 842685 
 
 217 
 
 856078 
 
 204 
 
 98G607 
 
 421 
 
 013393 
 
 53 
 
 8 
 
 842815 
 
 217 
 
 855956 
 
 204 
 
 986860 
 
 421 
 
 013140 
 
 52 
 
 9 
 
 812946 
 
 217 
 
 855833 
 
 204 
 
 987112 
 
 421 
 
 012888 
 
 51 
 
 10 
 
 843076 
 
 217 
 
 855711 
 
 205 
 
 987365 
 
 421 
 
 012635 
 
 50 
 
 11 
 
 9.843206 
 
 216 
 
 9.855588 
 
 205 
 
 9.987618 
 
 421 
 
 10.012382 
 
 49 
 
 12 
 
 843336 
 
 216 
 
 855465 
 
 205 
 
 987871 
 
 421 
 
 012129 
 
 48 
 
 13 
 
 843466 
 
 216 
 
 855342 
 
 205 
 
 988123 
 
 421 
 
 011877 
 
 47 
 
 14 
 
 843595 
 
 216 
 
 855219 
 
 205 
 
 988376 
 
 421 
 
 011624 
 
 46 
 
 15 
 
 843725 
 
 216 
 
 855096 
 
 205 
 
 988629 
 
 421 
 
 011371 
 
 45 
 
 16 
 
 843855 
 
 216 
 
 854973 
 
 205 
 
 988882 
 
 421 
 
 011118 
 
 44 
 
 17 
 
 843984 
 
 216 
 
 854850 
 
 205 
 
 989134 
 
 421 
 
 010866 
 
 43 
 
 18 
 
 844114 
 
 215 
 
 854727 
 
 206 
 
 989387 
 
 421 
 
 010613 
 
 42 
 
 19 
 
 844243 
 
 215 
 
 854603 
 
 206 
 
 989640 
 
 421 
 
 010360 
 
 41 
 
 20 
 
 844372 
 
 215 
 
 854480 
 
 206 
 
 989893 
 
 421 
 
 010107 
 
 40 
 
 21 
 
 9.844502 
 
 215 
 
 9-854356 
 
 206 
 
 9-990145 
 
 421 
 
 10.009855 
 
 39 
 
 22 
 
 844631 
 
 215 
 
 854233 
 
 206 
 
 990398 
 
 421 
 
 009602 
 
 38 
 
 23 
 
 844760 
 
 215 
 
 854109 
 
 206 
 
 990651 
 
 421 
 
 009349 
 
 37 
 
 24 
 
 844889 
 
 215 
 
 853986 
 
 206 
 
 990903 
 
 421 
 
 009097 
 
 36 
 
 25 
 
 845018 
 
 215 
 
 853862 
 
 206 
 
 991156 
 
 421 
 
 008844 
 
 35 
 
 26 
 
 845147 
 
 215 
 
 853738 
 
 206 
 
 991409 
 
 421 
 
 008591 
 
 34 
 
 27 
 
 845276 
 
 214 
 
 853614 
 
 207 
 
 39J6R2 
 
 421 
 
 008338 
 
 33 
 
 28 
 
 845405 
 
 214 
 
 853490 
 
 207 
 
 991914 
 
 421 
 
 00808(5 
 
 32 
 
 29 
 
 845533 
 
 214 
 
 853366 
 
 207 
 
 99-2167 
 
 421 
 
 007833 
 
 31 
 
 30 
 
 845662 
 
 214 
 
 853242 
 
 207 
 
 992420 
 
 421 
 
 007580 
 
 30 
 
 31 
 
 9.845790 
 
 214 
 
 9.853118 
 
 207 
 
 9-992672 
 
 421 
 
 10.007328 
 
 29 
 
 32 
 
 845919 
 
 214 
 
 852994 
 
 207 
 
 992925 
 
 421 
 
 007075 
 
 28 
 
 33 
 
 846047 
 
 214 
 
 852869 
 
 207 
 
 993178 
 
 421 
 
 006822 
 
 27 
 
 34 
 
 846175 
 
 214 
 
 852745 
 
 207 
 
 993430 
 
 421 
 
 OOG570 
 
 26 
 
 35 
 
 846304 
 
 214 
 
 852620 
 
 207 
 
 993683 
 
 421 
 
 006317 
 
 25 
 
 36 
 
 846432 
 
 213 
 
 852496 
 
 208 
 
 993936 
 
 421 
 
 000064 
 
 24 
 
 37 
 
 846560 
 
 213 
 
 852371 
 
 208 
 
 994189 
 
 421 
 
 005811 
 
 23 
 
 38 
 
 846688 
 
 213 
 
 852247 
 
 208 
 
 994441 
 
 421 
 
 005559 
 
 22 
 
 39 
 
 846816 
 
 213 
 
 852122 
 
 208 
 
 994694 
 
 421 
 
 005306 
 
 21 
 
 40 
 
 846944 
 
 213 
 
 851997 
 
 208 
 
 994947 
 
 421 
 
 005053 
 
 20 
 
 41 
 
 9-847071 
 
 213 
 
 9-851872 
 
 208 
 
 9-995199 
 
 421 
 
 10.004801 
 
 19 
 
 42 
 
 847199 
 
 213 
 
 851747 
 
 208 
 
 995452 
 
 421 
 
 004548 
 
 18 
 
 43 
 
 847327 
 
 213 
 
 851622 
 
 208 
 
 995705 
 
 421 
 
 004295 
 
 17 
 
 44 
 
 847454 
 
 212 
 
 851497 
 
 209 
 
 995957 
 
 421 
 
 004043 
 
 16 
 
 45 
 
 847582 
 
 212 
 
 851372 
 
 209 
 
 996210 
 
 421 
 
 003790 
 
 15 
 
 46 
 
 847709 
 
 212 
 
 851246 
 
 209 
 
 996463 
 
 421 
 
 003537 
 
 14 
 
 47 
 
 847836 
 
 212 
 
 851121 
 
 209 
 
 996715 
 
 421 
 
 003285 
 
 13 
 
 48 
 
 847964 
 
 212 
 
 850996 
 
 209 
 
 996968 
 
 421 
 
 003032 
 
 12 
 
 49 
 
 848091 
 
 212 
 
 850870 
 
 209 
 
 997221 
 
 421 
 
 002779 
 
 11 
 
 50 
 
 848218 
 
 212 
 
 850745 
 
 209 
 
 997473 
 
 421 
 
 002527 
 
 10 
 
 51 
 
 9-848345 
 
 212 
 
 9.850619 
 
 209 
 
 9.997726 
 
 421 
 
 10.002274 
 
 9 
 
 52 
 
 848-172 
 
 211 
 
 850493 
 
 210 
 
 997979 
 
 421 
 
 002021 
 
 8 
 
 53 
 
 848599 
 
 211 
 
 850368 
 
 210 
 
 998231 
 
 421 
 
 001769 
 
 7 
 
 54 
 
 848726 
 
 211 
 
 850242 
 
 210 
 
 998484 
 
 421 
 
 001516 
 
 6 
 
 55 
 
 848852 
 
 211 
 
 850116 
 
 210 
 
 998737 
 
 421 
 
 001263 
 
 5 
 
 56 
 
 848979 
 
 211 
 
 849990 
 
 210 
 
 998989 
 
 421 
 
 001011 
 
 4 
 
 57 
 
 849106 
 
 211 
 
 849864 
 
 210 
 
 999242 
 
 421 
 
 000758 
 
 3 
 
 58 
 
 849232 
 
 211 
 
 849738 
 
 210 
 
 999495 
 
 421 
 
 000505 
 
 2 
 
 59 
 
 849359 
 
 211 
 
 849611 
 
 210 
 
 999748 
 
 421 
 
 000253 
 
 1 
 
 60 
 
 849485 
 
 211 
 
 849485 
 
 210 
 
 10000000 
 
 421 
 
 000000 
 
 
 
 | Cosine Sine Cotang. | Tang. | M. 
 
 45 Degrees. 
 
TABLE 
 
 OP 
 
 NATURAL SINES AND TANGENTS; 
 
 TO 
 
 EVERY TEN MINUTES OF A DEGREE. 
 
 IF the given angle is less than 45, look for the title of the column, at 
 the top of the page ; and for the degrees and minutes, on the left. But 
 if the angle is between 45 and 90, look for the title of the column, at 
 the bottom; and for the degrees and minutes on the right. 
 
 The Secants and Cosecants, which are not inserted in this table, may 
 be easily supplied. If 1 be divided by the cosine of an arc, the quotient 
 will be the secant of that arc. (Art. 228.) And if 1 be divided by the 
 sine, the quotient will be the cosecant. 
 
NATURAL SINKS. 
 
 
 Deg. 
 
 1 Deg. 
 
 2 Deg. 
 
 3 Deg. 
 
 4 Deg. 
 
 
 
 Nat. 
 
 N.Co 
 
 Nat. iN.Co 
 
 I Nat. 
 
 N.Co 
 
 ~NHtT 
 
 N. Co- 
 
 Nat, 
 
 N. Co- 
 
 
 M 
 
 Sine 
 
 Sine 
 
 Sine. | Sine 
 
 Sine 
 
 Sine 
 
 Sine 
 
 Sine 
 
 Sine 
 
 Sine 
 
 .M 
 
 ~o 
 
 OJOOO 
 
 Unit. 
 
 01745 
 
 991)85 
 
 "03490 
 
 99939 
 
 05234 
 
 998( 3 
 
 00970 
 
 99756 
 
 (50 
 
 1 
 
 00029 
 
 00000 
 
 01774 
 
 99984 
 
 03519 
 
 99938 
 
 05263 
 
 99801 
 
 07005 
 
 99754 
 
 59 
 
 2 
 
 00053 
 
 (10000 
 
 01803 
 
 99984 
 
 03548 
 
 99937 
 
 05292 
 
 99861) 
 
 07034 
 
 99752 
 
 58 
 
 3 
 
 00087 
 
 00000 
 
 01832 
 
 99983 
 
 o:?577 
 
 99930 
 
 05321 
 
 99858 
 
 07003 
 
 99750 
 
 57 
 
 4 
 
 00116 
 
 00000 
 
 018tJ2 
 
 99983 
 
 03606 
 
 99935 
 
 0.1350 
 
 99857 
 
 07092 
 
 H9748 
 
 56 
 
 5 
 
 00145 
 
 00000 
 
 01891 
 
 99982 
 
 03635 
 
 99934 
 
 05379 
 
 99855 
 
 07121 
 
 99746 
 
 55 
 
 6 
 
 00175 
 
 00000 
 
 01920 
 
 99982 
 
 03664 
 
 99933 
 
 05408 
 
 9J854 
 
 07150 
 
 99744 
 
 54 
 
 7 
 
 00204 
 
 00000 
 
 01949 
 
 99981 
 
 03693 
 
 99932 
 
 05437 
 
 99852 
 
 07179 
 
 99742 
 
 53 
 
 8 
 
 00233 
 
 00000 
 
 01978 
 
 99980 
 
 03723 
 
 99931 
 
 05406 
 
 9985J 
 
 07208 
 
 1)9740 
 
 52 
 
 9 
 
 00262 
 
 00000 
 
 02007 
 
 99980 
 
 03752 
 
 99930 
 
 05495 
 
 99849 
 
 07237 
 
 99738 
 
 51 
 
 10 
 
 00291 
 
 00000 
 
 0-2030 
 
 99979 
 
 03781 
 
 1)992! 1 
 
 05124 
 
 99847 
 
 07366 
 
 99730 
 
 50 
 
 1] 
 
 0032!) 
 
 99999 
 
 02065 
 
 99979 
 
 03810 
 
 99927 
 
 05553 
 
 99840 
 
 07295 
 
 99734 
 
 49 
 
 18 
 
 00349 
 
 99999 
 
 02094 
 
 99978 
 
 03839 
 
 99920 
 
 05582 
 
 99844 
 
 07324 
 
 99731 
 
 48 
 
 13 
 
 00378 
 
 99999 
 
 02123 
 
 99977 
 
 03868 
 
 99925 
 
 05611 
 
 99842 
 
 07353 
 
 99729 
 
 47 
 
 14 
 
 00407 
 
 99999 
 
 02152 
 
 99977 
 
 0:^897 
 
 9J924 
 
 05040 
 
 99841 
 
 07382 
 
 99727 
 
 46 
 
 15 
 
 00436 
 
 99999 
 
 02181 
 
 99976 
 
 03926 
 
 99923 
 
 05069 
 
 99839 
 
 07411 
 
 99725 
 
 45 
 
 16 
 
 00405 
 
 99999 
 
 02211 
 
 99976 
 
 03955 
 
 99922 
 
 05098 
 
 99838 
 
 07440 
 
 99723 
 
 44 
 
 17 
 
 00495 
 
 99999 
 
 OJ240 
 
 99975 
 
 03984 
 
 999-21 
 
 05727 
 
 998H6 
 
 07469 
 
 99721 
 
 43 
 
 18 
 
 005-24 
 
 99999 
 
 02209 
 
 99974 
 
 04013 
 
 99919 
 
 05756 
 
 99834 
 
 07498 
 
 99719 
 
 42 
 
 19 
 
 01)553 
 
 99998 
 
 02298 
 
 99974 
 
 04042 
 
 93918 
 
 05785 
 
 99S33 
 
 075-27 
 
 99710 
 
 41 
 
 20 
 
 00582 
 
 99998 
 
 02327 
 
 99973 
 
 04071 
 
 939 17 
 
 05814 
 
 99831 
 
 0755(i 
 
 99714 
 
 40 
 
 21 
 
 00611 
 
 99998 
 
 02356 
 
 99972 
 
 04100 
 
 99:)10 
 
 05844 
 
 9982;) 
 
 07585 
 
 99712 
 
 39 
 
 2-2 
 
 001)40 
 
 99998 
 
 02385 
 
 99972 
 
 04129 
 
 09915 
 
 05873 
 
 99827 
 
 07014 
 
 99710 
 
 38 
 
 23 
 
 00660 
 
 99998 
 
 02414 
 
 99971 
 
 0*138 
 
 99913 
 
 OSMSZ 
 
 99826 
 
 07043 
 
 99708 
 
 37 
 
 24 
 
 00698 
 
 99998 
 
 02443 
 
 99970 
 
 04188 
 
 99912 
 
 05931 
 
 99824 
 
 07672 
 
 5*9705 
 
 36 
 
 25 
 
 007-27 
 
 99997 
 
 02472 
 
 99909 
 
 04217 
 
 99911 
 
 05960 
 
 9:)822 
 
 07701 
 
 99703 
 
 35 
 
 26 
 
 00756 
 
 99997 
 
 02501 
 
 99969 
 
 04246 
 
 99910 
 
 05989 
 
 99821 
 
 07730 
 
 99701 
 
 34 
 
 27 
 
 00785 
 
 99997 
 
 02531) 
 
 99968 
 
 04275 
 
 99909 
 
 06018 
 
 99819 
 
 07759 
 
 J9699 
 
 33 
 
 28 
 
 00814 
 
 99997 
 
 025(50 
 
 99967 
 
 04904 
 
 99907 
 
 00047 
 
 9J8I7 
 
 07788 
 
 99090 
 
 32 
 
 29 
 
 00844 
 
 99996 
 
 02589 
 
 99960 
 
 04333 
 
 99900 
 
 06076 
 
 99815 
 
 07817 
 
 19094 
 
 31 
 
 30 
 
 00873 
 
 99996 
 
 02618 
 
 99966 
 
 04362 
 
 J9905 
 
 06105 
 
 99S13 
 
 07846 
 
 99692 
 
 30 
 
 31 
 
 00902 
 
 99996 
 
 02647 
 
 999R5 
 
 04391 
 
 99904 
 
 06134 
 
 99812 
 
 07875 
 
 99689 
 
 29 
 
 32 
 
 00931 
 
 99990 
 
 02676 
 
 99904 
 
 044-20 
 
 99902 
 
 08M3 
 
 91)814) 
 
 07904 
 
 J9087 
 
 28 
 
 33 
 
 00960 
 
 99995 
 
 02705 
 
 99903 
 
 04449 
 
 99001 
 
 00192 
 
 99803 
 
 07933 
 
 J9085 
 
 27 
 
 34 
 
 00989 
 
 99995 
 
 02734 
 
 99963 
 
 04478 
 
 )990i) 
 
 00-221 
 
 99801) 
 
 07962 
 
 J9083 
 
 20 
 
 35 
 
 01018 
 
 99995 
 
 02763 
 
 99902 
 
 04507 
 
 )9898 
 
 06250 
 
 99804 
 
 07991 
 
 J9080 
 
 25 
 
 36 
 
 01047 
 
 99995 
 
 02792 
 
 99961 
 
 04536 
 
 99897 
 
 06279 
 
 99803 
 
 08020 
 
 99678 
 
 24 
 
 37 
 
 01076 
 
 99994 
 
 02821 
 
 99900 
 
 04505 
 
 .989!i 
 
 06308 
 
 99801 
 
 08049 
 
 99676 
 
 23 
 
 38 
 
 01105 
 
 99994 
 
 0-2850 
 
 99959 
 
 04594 
 
 99894 
 
 06337 
 
 99799 
 
 08078 
 
 91)073 
 
 22 
 
 39 
 
 01134 
 
 99994 
 
 02379 
 
 99959 
 
 04623 
 
 99893 
 
 06306 
 
 99797 
 
 08107 
 
 99671 
 
 21 
 
 40 
 
 01164 
 
 99993 
 
 0-2908 
 
 99958 
 
 04653 
 
 )9892 
 
 OG395 
 
 99795 
 
 03130 
 
 J9fifi8 
 
 20 
 
 41 
 
 01193 
 
 99993 
 
 02938 
 
 99957 
 
 -04682 ' 
 
 )9890 
 
 00424 
 
 99793 
 
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 57 
 
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 58 
 
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 1 
 
 "M" 
 
 N.CS. 
 
 N. S. 
 
 N.08. 
 
 N.S. 
 
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 N.S. 
 
 N.CS. 
 
 N.S. 
 
 \ CS.| N S. 
 
 M 
 
 
 89 Deg. 
 
 88 Deg. 
 
 87 Deg. 
 
 86 Deg. 
 
 85 Deg. 
 
 
NATURAL SINKS. 
 
 
 5 Deg. 
 
 6 Deg. 
 
 7 Deg. 
 
 8 Deg. 9 Deg. 
 
 
 M 
 
 N.a 
 
 N.CS. 
 
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 13917 
 
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 987(9 
 
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 57 
 
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 56 
 
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 10597 
 
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 98740 
 
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 6 
 
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 99604 
 
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 12360 
 
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 98709 
 
 47 
 
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 48 
 
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 39 
 
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 11089 
 
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 38 
 
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 14580 
 
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 98662 
 
 37 
 
 21 
 
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 11147 
 
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 35 
 
 20 
 
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 34 
 
 27 
 
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 33 
 
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 14723 
 
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 1)9(538 
 
 32 
 
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 11291 
 
 9930:1 
 
 131)24 
 
 99148 
 
 14752 
 
 98906 104:0 
 
 98033 
 
 31 
 
 30 
 
 09585 
 
 99540 
 
 11320 
 
 9J357 
 
 13053 
 
 99144 
 
 14781 
 
 98902 
 
 16505 
 
 98629 
 
 30 
 
 31 
 
 09G14 
 
 99537 
 
 11349 
 
 99354 
 
 13081 
 
 1)9141 
 
 14810 
 
 98897 
 
 16533 
 
 98624 
 
 29 
 
 32 
 
 09042 
 
 99534 
 
 11378 
 
 1)!35I 
 
 13110 
 
 9137 
 
 14838 
 
 98893 10502 
 
 98619 
 
 28 
 
 33 
 
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 114(17 
 
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 13139 
 
 99133 
 
 14867 
 
 98889 16591 
 
 98G14 
 
 27 
 
 31 
 
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 11436 
 
 99344 
 
 13168 
 
 99129 
 
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 98884 16020 
 
 98609 
 
 26 
 
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 39 
 
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 40 
 
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 14 
 
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 13 
 
 48 
 
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 6 
 
 55 
 
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 98508 
 
 5 
 
 56 
 
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 12071 
 
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 15529 
 
 18787 
 
 17250 
 
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 4 
 
 57 
 
 10386 
 
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 12100 
 
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 13831 
 
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 172'<1) 
 
 9-41)0 
 
 3 
 
 58 
 
 10395 
 
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 12129 
 
 99262 
 
 13860 
 
 99035 
 
 15586 
 
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 17308 
 
 98491 
 
 2 
 
 59 
 
 10424 
 
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 12158 
 
 99258 
 
 13889 
 
 99031 
 
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 98773 
 
 1733(5 
 
 9848B 
 
 1 
 
 M~ 
 
 N.CU 
 
 N.S. 
 
 N.CS.I N S 
 
 N.CS 
 
 N.S. 
 
 v.cs 
 
 N.S. 
 
 N.CS. 
 
 N.S. 
 
 ~T 
 
 
 84 Deg. 
 
 83 Deg. 
 
 "WDegT 
 
 81 Deg. 
 
 ~80~Deg. 
 
 
66 
 
 NATURAL SINES. 
 
 M 
 
 10 Deg. 
 
 11 Deg. 
 
 12 Deg. 
 
 13 Deg. 14 Deg. 
 
 M 
 
 N.S. 
 
 N.CS 
 
 N.S. 
 
 N.CS 
 
 N.S. 
 
 N.O 
 
 N.S. IN.CS. N.S. 
 
 N. US 
 
 
 
 17305 
 
 98481 
 
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 97815 
 
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 97030 
 
 60 
 
 1 
 
 17393 
 
 98476 
 
 19109 
 
 98157 
 
 20820 
 
 97809 
 
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 59 
 
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 19138 
 
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 20848 
 
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 97424 24241 
 
 97015 
 
 58 
 
 3 
 
 17451 
 
 98466 
 
 19167 
 
 98146 
 
 20877 
 
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 22580 
 
 97417 24277 
 
 97008 
 
 57 
 
 4 
 
 17479 
 
 98461 
 
 19195 
 
 98140 
 
 20905 
 
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 97411 24305 
 
 97001 
 
 56 
 
 e 
 
 17508 
 
 98455 
 
 19224 
 
 98135 
 
 j 20933 
 
 9778- 
 
 22637 
 
 97404 24333 
 
 9C994 
 
 55 
 
 6 
 
 17537 
 
 98450 
 
 19252 
 
 98123 
 
 20962 
 
 97778 
 
 22605 
 
 97398 24302 
 
 90987 
 
 54 
 
 ; 
 
 17565 
 
 98445 
 
 19281 
 
 98124 
 
 2(1! (UO 
 
 97772 
 
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 97391 24390 
 
 90980 
 
 53 
 
 8 
 
 17594 
 
 98440 
 
 19309 
 
 98118 
 
 21019 
 
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 22722 
 
 97384 244 IS 
 
 96973 
 
 52 
 
 g 
 
 17623 
 
 98435 
 
 19338 
 
 98112 
 
 21047 
 
 97700 
 
 22750 
 
 97378 244,46 
 
 90900 
 
 51 
 
 10 
 
 17651 
 
 98430 
 
 19366 
 
 98107 
 
 21076 
 
 97754 
 
 22778 
 
 97371 -"447-. 
 
 90959 
 
 50 
 
 11 
 
 17680 
 
 98425 
 
 19395 
 
 98101 
 
 21104 
 
 97748 
 
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 97305 24503 
 
 915952 
 
 49 
 
 12 
 
 17708 
 
 98420 
 
 19423 
 
 9809P 
 
 21132 
 
 97742 
 
 22835 
 
 97358 24531 
 
 901)45 
 
 48 
 
 13 
 
 17737 
 
 98414 
 
 ; 19452 
 
 98090 
 
 21101 
 
 97735 
 
 22803 
 
 97:i.3l 24559 
 
 96937 
 
 47 
 
 14 
 
 17766 
 
 98409 
 
 19481 
 
 98084 
 
 21189 
 
 977-29 
 
 22892 
 
 97345 24587 
 
 96930 
 
 40 
 
 15 
 
 17794 
 
 98404 
 
 19509 
 
 98079 
 
 21218 
 
 97723 
 
 22920 
 
 '.17338 24015 
 
 96923 
 
 45 
 
 16 
 
 17823 
 
 98399 
 
 19538 
 
 98073 
 
 21246 
 
 97717 
 
 22948 
 
 97331 24044 
 
 96910 
 
 44 
 
 17 
 
 17852 
 
 98394 
 
 19566 
 
 98067 
 
 21275 
 
 9771 J 
 
 22977 
 
 973-25 24072 
 
 90909 
 
 43 
 
 18 
 
 17880 
 
 98389 
 
 19595 
 
 OPflfil 
 
 21303 
 
 97705 
 
 23005 
 
 97318 247 OU 
 
 90902 
 
 42 
 
 19 
 
 17909 
 
 98383 
 
 19(523 
 
 9805f 
 
 21331 
 
 97098 
 
 23033 
 
 97311 247-28 
 
 90894 
 
 41 
 
 20 
 
 17937 
 
 98378 
 
 19652 
 
 98050 
 
 21360 
 
 97092 
 
 '23002 
 
 97304 24750 
 
 96887 
 
 40 
 
 24 
 
 17966 
 
 98373 
 
 19680 
 
 98044 
 
 21388 
 
 9708f 
 
 23090 
 
 97298 24784 
 
 90880 
 
 39 
 
 22 
 
 17995 
 
 98368 
 
 19709 
 
 98039 
 
 21417 
 
 97080 
 
 23118 
 
 97291 24813 
 
 9C873 
 
 38 
 
 23 
 
 18033 
 
 98362 
 
 19737 
 
 98033 
 
 21445 
 
 97673 
 
 23146 
 
 97284 : 24841 
 
 90806 
 
 37 
 
 24 
 
 18052 
 
 98357 
 
 19766 
 
 98027 
 
 21474 
 
 97667 
 
 23175 
 
 97278 24809 
 
 90858 
 
 36 
 
 25 
 
 J8081 
 
 98352 
 
 19794 
 
 98021 
 
 21502 
 
 97601 
 
 23203 
 
 97271 24897 
 
 90851 
 
 35 
 
 26 
 
 18109 
 
 98347 
 
 19823 
 
 98016 
 
 21530 
 
 97655 
 
 23231 
 
 97204 24925 
 
 90844 
 
 34 
 
 27 
 
 18138 
 
 98341 
 
 19851 
 
 98010 
 
 21559 
 
 97648 
 
 23260 
 
 97257 24953 
 
 96837 
 
 33 
 
 28 
 
 18166 
 
 98336 
 
 19880 
 
 98004 
 
 21587 
 
 97642 
 
 23288 
 
 97251 24982 
 
 )(i829 
 
 32 
 
 29 
 
 18195 
 
 98331 
 
 1990S 
 
 97998 
 
 21616 
 
 97636 
 
 23316 
 
 97244 25010 
 
 JC:8-2-2 
 
 31 
 
 30 
 
 18224 
 
 98325 
 
 19937 
 
 97992 
 
 21 044 
 
 97630 
 
 23345 
 
 97237 25038 
 
 J0815 
 
 30 
 
 31 
 
 18252 
 
 98320 
 
 19965 
 
 97987 
 
 21672 
 
 97623 
 
 23373 
 
 97230 25066 
 
 96807 
 
 2s> 
 
 32 
 
 18281 
 
 98315 
 
 19994 
 
 97981 
 
 21701 
 
 97617 
 
 23401 
 
 97223 25094 
 
 96800 
 
 28 
 
 33 
 
 18309 
 
 98310 
 
 200-2-2 
 
 97975 
 
 21729 
 
 97611 
 
 23429 
 
 97217 25122 
 
 96793 
 
 27 
 
 34 
 
 18338 
 
 96304 
 
 20051 
 
 979R9 
 
 21758 
 
 97604 
 
 23458 
 
 97-11) 25151 
 
 JG780 
 
 20 
 
 35 
 
 18367 
 
 98299 
 
 20079 
 
 07903 
 
 21786 
 
 97598 
 
 23486 
 
 97203 | 25179 
 
 96778 
 
 25 
 
 36 
 
 18395 
 
 98294 
 
 20108 
 
 97958 
 
 21814 
 
 97592 
 
 23514 
 
 97196 j 25207 
 
 J6771 
 
 24 
 
 37 
 
 18424 
 
 98288 
 
 20136 
 
 97952 
 
 21843 
 
 97585 
 
 23542 
 
 97189 J25235 
 
 J0764 
 
 23 
 
 38 
 
 18452 
 
 98283 
 
 20165 
 
 97946 
 
 21871 
 
 97579 
 
 23571 
 
 97182 
 
 25263 
 
 J0756 
 
 22 
 
 39 
 
 18481 
 
 98277 
 
 20193 
 
 97910 
 
 21899 
 
 97573 
 
 23599 
 
 97176 
 
 25291 
 
 J0749 
 
 21 
 
 40 
 
 18509 
 
 98272 
 
 20222 
 
 97934 
 
 21928 
 
 97566 
 
 23027 
 
 97169 
 
 25320 
 
 90742 
 
 20 
 
 41 
 
 18538 
 
 98267 
 
 20250 
 
 979-28 
 
 21956 
 
 97500 
 
 23056 
 
 97162 
 
 25348 
 
 90734 
 
 19 
 
 42 
 
 18567 
 
 98261 
 
 20279 
 
 979-22 
 
 21985 
 
 97553 
 
 23684 
 
 97155 
 
 25376 
 
 JG727 
 
 18 
 
 43 
 
 18595 
 
 98256 
 
 20307 
 
 97916 
 
 22013 
 
 97547 
 
 23712 
 
 97148 
 
 25404 
 
 96719 
 
 17 
 
 44 
 
 18024 
 
 98250 
 
 20336 
 
 97910 
 
 22041 
 
 97541 
 
 23740 
 
 97141 
 
 25432 
 
 J6712 
 
 16 
 
 45 
 
 18652 
 
 98245 
 
 20364 
 
 97905 
 
 22070 
 
 97534 
 
 23769 
 
 97134 
 
 25460 
 
 96705 
 
 15 
 
 46 
 
 18681 
 
 98240 
 
 20393 
 
 97899 
 
 22098 
 
 97528 
 
 23797 
 
 97127 
 
 25488 
 
 96697 
 
 14 
 
 47 
 
 18710 
 
 98234 
 
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 79 Deg. 
 
 78 Deg. 
 
 77 Deg. 
 
 76 Deg. 
 
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NATURAL SINES. 
 
 
 15 Deg. 
 
 16 Deg. 
 
 17 Deg. 
 
 18 Deg. 
 
 1 19 Deg. 
 
 
 M 
 
 
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 IT 
 
 
 74 Deg. 
 
 "73 Deg. 
 
 72 Deg. 
 
 71 Deg. 
 
 70 Deg. 
 
 
NATURAL SINES. 
 
 
 20 Deg. 
 
 21 Deg. 
 
 22 Deg. 
 
 23 Deg. 
 
 ' 24 Deg. 
 
 
 M 
 
 N.S. 
 
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 69 Deg. 
 
 68 Deg. 
 
 67 Deg. 
 
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NATURAL SINES. 
 
 
 25 Deg. 
 
 26 Deg. 
 
 27 Deg. 
 
 28 Deg. ! 29 Deg. 
 
 
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 N.CS 
 
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 T 
 
 
 64 Deg. 
 
 63 Deg. 
 
 62 Deg. 
 
 61 Deg. | 
 
 60 Deg. 
 
 
NATURAL SINES. 
 
 
 30 Deg. 
 
 31 Deg. 
 
 32 Deg. 
 
 33 Deg. 
 
 34 Deg. 
 
 
 II 
 
 N.S. 
 
 N.CS 
 
 N.S. (N.CS 
 
 N.S. 
 
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 8-2855 
 
 57 
 
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 55 
 
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 560(54 
 
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 531(34 
 
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 53 
 
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 52 
 
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 54 Deg. 
 
 53 Deg. 
 
 52 Deg. 
 
 51 Deg. , 
 
 50 Deg. 
 
 
NATURAL SINES. 
 
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 40 Deg. 
 
 41 Deg. 
 
 42 Deg. 
 
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 05540 
 
 75528' 06848 
 
 74373 
 
 68130 
 
 73195 
 
 69403 
 
 71995 
 
 -0649 
 
 70772 
 
 3 
 
 58 
 
 6 r >562 
 
 75509 ! 61870 
 
 74353 
 
 68157 
 
 73175 
 
 (9484 
 
 71974 
 
 -0670 
 
 70752 
 
 2 
 
 59 
 
 65584 
 
 75190 60891 
 
 74334 
 
 58179 
 
 73155 
 
 59445 
 
 71954 
 
 -0090 
 
 70731 
 
 1 
 
 60 
 
 5560S 
 
 75471 1. 66913 
 
 74314 
 
 68200 
 
 73135 
 
 094(50 71934 
 
 70711 
 
 70711 
 
 
 IT 
 
 v.cs. 
 
 N.S. j:\.CS. 
 
 N.S. 
 
 N.CS. 
 
 N.S. 
 
 \ T . OS | N. S. 
 
 N.OS.I N.S. 
 
 V! 
 
 
 49 Deg. Jl 48 Deg. 
 
 47 Deg. 
 
 46 Deg. 
 
 45 Deg. 
 
 
NATCRAL TAN'GKNTS. 
 
 
 Degrees. 
 
 1 Degree. 
 
 2 Degrees. 
 
 3 Degrees. 
 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 X. IVi. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan 
 
 N. Cot. 
 
 M 
 
 
 
 00000 
 
 oooo.oo 
 
 ~oT74tT 
 
 57.2900 
 
 03492' 
 
 28.63 i.t 
 
 05241 
 
 79.0811 
 
 "W 
 
 1 
 
 00.12:1 
 
 3437.75 
 
 1)177.1 
 
 56.3506 
 
 0352 1 
 
 28.3991 
 
 05270 ' 
 
 18.9755 
 
 59 
 
 o 
 
 00058 
 
 1718.87 
 
 01804 
 
 55.4415 
 
 03550 
 
 28.16(54 
 
 05299 
 
 18.8711 
 
 58 
 
 3 
 
 00067 
 
 ii4.-i.y-j 
 
 01833 
 
 54.5613 
 
 03579 
 
 27.9372 
 
 053-28 
 
 18.7678 
 
 57 
 
 4 
 
 00116 
 
 859.436 
 
 OM62 
 
 53.7086 
 
 03609 
 
 27.7117 
 
 05357 
 
 18.6656 
 
 56 
 
 5 
 
 00115 
 
 687.54g 
 
 01891 
 
 
 03638 
 
 27.48'MI 
 
 05387 
 
 18.5645 
 
 55 
 
 6 
 
 OD IT.-) 
 
 572.957 
 
 019-20 
 
 52.0801 
 
 03667 
 
 27.2715 
 
 05416 
 
 18.4(545 
 
 54 
 
 
 00204 
 
 491. 106 
 
 01949 
 
 51.303-2 
 
 036!)!i 
 
 27.0566 
 
 05445 
 
 18. 365.") 
 
 53 
 
 8 
 
 00233 
 
 429.718 
 
 01978 
 
 50.5485 
 
 03725 
 
 26.8450 
 
 05474 
 
 18.2677 
 
 52 
 
 9 
 
 00202 
 
 381.971 
 
 0-2007 
 
 49. HI 57 
 
 03754 
 
 26.6367 
 
 05503 
 
 18.1708 
 
 51 
 
 10 
 
 00291 
 
 343.774 
 
 02036 
 
 49. 1039 
 
 03783 
 
 2U.4316 
 
 05533 
 
 18.0750 
 
 5.1 
 
 11 
 
 0032 ) 
 
 312.521 
 
 02066 
 
 48.41-21 
 
 03812 
 
 20.229G 
 
 05562 
 
 L7.98Q] 
 
 49 
 
 1-2 
 
 80349 
 
 
 02095 
 
 47.7395 
 
 03842 
 
 2(5.0307 
 
 05591 
 
 17.8863 
 
 48 
 
 13 
 
 00378 
 
 264.441 
 
 02124 
 
 47.0853 
 
 03871 
 
 25.8348 
 
 05620 
 
 17.7934 
 
 47 
 
 J4 
 
 90407 
 
 24.->. 55-2 
 
 02153 
 
 46. 448 J 
 
 03900 
 
 25.6418 
 
 05649 
 
 17.7015 
 
 46 
 
 15 
 
 004313 
 
 229.182 
 
 02182 
 
 45.8294 
 
 03929 
 
 25.4517 
 
 05678 
 
 17.6106 
 
 45 
 
 10 
 
 00465 
 
 214.858 
 
 0-2211 
 
 45.2261 
 
 03958 
 
 25.2644 
 
 05708 
 
 17.5205 
 
 44 
 
 17 
 
 00495 
 
 202. -2 lit 
 
 (1-224!) 
 
 4. 038!) 
 
 03947 
 
 25.0798 
 
 05737 
 
 17.4314 
 
 43 
 
 18 
 
 00524 
 
 191). 984 
 
 02269 
 
 44.0661 
 
 04016 
 
 24.89-78 
 
 05766 
 
 17.3432 
 
 42 
 
 19 
 
 DO.-).-):? 
 
 180.932 
 
 02298 
 
 43.5081 
 
 0404(i 
 
 24.7185 
 
 05795 
 
 17.2558 
 
 41 
 
 JO 
 
 00582 
 
 171.885 
 
 02328 
 
 42.91)41 
 
 04075 
 
 24.5418 
 
 05824 
 
 17.1693 
 
 40 
 
 21 
 
 O.Kill 
 
 163.700 
 
 02357 
 
 42.4335 
 
 04104 
 
 24.3675 
 
 05854 
 
 17.0837 
 
 39 
 
 22 
 
 00640 
 
 150.251) 
 
 02386 
 
 41.9158 
 
 04133 
 
 24.1957 
 
 05883 
 
 16.9990 
 
 38 
 
 23 
 
 001)61) 
 
 149. 465 
 
 0-2415 
 
 41.4106 
 
 04162 
 
 24.0-2ii:i 
 
 05912 
 
 ] (5. 9 150 
 
 37 
 
 24 
 
 00698 
 
 14 3. -2:57 
 
 02444 
 
 1 1.9174 
 
 04191 
 
 23.8593 
 
 05941 
 
 16.8319 
 
 36 
 
 2.) 
 
 D07-27 
 
 137.507 
 
 0-2473 
 
 40.4358 
 
 042-20 
 
 23.6945 
 
 05970 
 
 16.7496 
 
 35 
 
 2li 
 
 007515 
 
 13-2.219 
 
 02503 
 
 39.9655 
 
 04250 
 
 2::.5:i-2i 
 
 05999 
 
 16.6681 
 
 34 
 
 27 
 
 
 1-27.3-21 
 
 02531 
 
 3; 1.505!) 
 
 04279 
 
 23.3718 
 
 06029 
 
 16.5874 
 
 33 
 
 28 
 
 00814 
 
 1-2-2.774 
 
 02530 
 
 39.0568 
 
 04308 
 
 23.2137 
 
 06058 
 
 16.5075 
 
 32 
 
 29 
 
 0084 i 
 
 118.540 
 
 02589 
 
 38.6177 
 
 04337 
 
 23.0577 
 
 06087 
 
 16.4283 
 
 31 
 
 30 
 
 00873 
 
 114.589 
 
 02619 
 
 38.1885 
 
 04366 
 
 22.9037 
 
 06116 
 
 16.3499 
 
 30 
 
 31 
 
 00903 
 
 110.892 
 
 02648 
 
 37.7686 
 
 04395 
 
 22.7518 
 
 06145 
 
 16.2722 
 
 29 
 
 32 
 
 00931 
 
 107.420 
 
 02677 
 
 37.3579 
 
 04424 
 
 22.6020 
 
 06175 
 
 16.1952 
 
 28 
 
 33 
 
 00960 
 
 104.171 
 
 02706 
 
 36.95(iO 
 
 04454 
 
 22. 4541 
 
 06204 
 
 16.1190 
 
 27 
 
 31 
 
 OOU89 
 
 101.107 
 
 0-27.!.") 
 
 36.5627 
 
 04483 
 
 22.3081 
 
 06233 
 
 16.0435 
 
 26 
 
 35 
 
 01018 
 
 98.2179 
 
 0-2764 
 
 36.1776 
 
 04512 
 
 22.1640 
 
 06262 
 
 15.9687 
 
 25 
 
 36 
 
 01047 
 
 95.4895 
 
 0-2793 
 
 35.8006 
 
 04541 
 
 2-2.0-217 
 
 08291 
 
 15.8345 
 
 24 
 
 37 
 
 01076 
 
 92.0085 
 
 02822 
 
 35.4313 
 
 04570 
 
 21.8813 
 
 06321 
 
 15. 8-2 II 
 
 23 
 
 38 
 
 01105 
 
 !)!. 41533 
 
 02851 
 
 35.0695 
 
 04599 
 
 21.7426 
 
 08350 
 
 15.7483 
 
 22 
 
 39 
 
 01135 
 
 88.1436 
 
 02831 
 
 34.7151 
 
 04(528 
 
 21.6056 
 
 015379 
 
 15. (57(5-2 
 
 21 
 
 ; i 
 
 01164 
 
 B5u939S 
 
 0-21(10 
 
 34.3678 
 
 04658 
 
 21.4704 
 
 06408 
 
 [5.6948 
 
 20 
 
 41 
 
 01193 
 
 83.8435 
 
 02939 
 
 34.0-273 
 
 04687 
 
 21.3369 
 
 00437 
 
 15.5340 
 
 19 
 
 42 
 
 1-22-2 
 
 81.8470 
 
 02968 
 
 33.6935 
 
 04716 
 
 21.2049 
 
 0(5467 
 
 15.4638 
 
 18 
 
 43 
 
 01251 
 
 79.9434 
 
 02997 
 
 33.3662 
 
 04745 
 
 21.0747 
 
 06496 
 
 15.3943 
 
 17 
 
 44 
 
 01283 
 
 7*. 1263 
 
 03026 
 
 33.0452 
 
 01774 
 
 20.9460 
 
 06525 
 
 15.3954 
 
 16 
 
 45 
 
 01309 
 
 76.3900 
 
 03055 
 
 32.7303 
 
 04803 
 
 20.8188 
 
 06554 
 
 15.2571 
 
 15 
 
 40 
 
 01338 
 
 74.7292 
 
 03084 
 
 32.4213 
 
 04832 
 
 20.6932 
 
 06584 
 
 15.1893 
 
 14 
 
 47 
 
 01367 
 
 73. 13 JO 
 
 03114 
 
 :!-2.M8i 
 
 04862 
 
 20.5691 
 
 06613 
 
 L5.1222 
 
 13 
 
 48 
 
 0139(i 
 
 71.6151 
 
 03143 
 
 31.8205 
 
 04891 
 
 20.4465 
 
 0(5642 
 
 15.0557 
 
 12 
 
 49 
 
 01425 
 
 70.1533 
 
 03172 
 
 31.5284 
 
 04920 
 
 20.3253 
 
 06671 
 
 14.9898 
 
 11 
 
 51) 
 
 01455 
 
 68.. 75: > I 
 
 03201 
 
 31. -24 l(i 
 
 04949 
 
 20.205;: 
 
 06700 
 
 L4.9244 
 
 10 
 
 51 
 
 01484 
 
 67.4019 
 
 03230 
 
 30.9599 
 
 04978 
 
 20.087-2 
 
 0(5730 
 
 14.8596 
 
 9 
 
 52 
 
 01513 
 
 66.1055 
 
 03259 
 
 30.6833 
 
 05007 
 
 19.9702 
 
 06759 
 
 14.7954 
 
 8 
 
 53 
 
 01542 
 
 64.8580 
 
 03288 
 
 30.4116 
 
 05037 
 
 
 06788 
 
 14.7317 
 
 7 
 
 54 
 
 01571 
 
 63.6567 
 
 03317 
 
 30.1446 
 
 05066 
 
 19.7403 
 
 0(5817 
 
 14.6685 
 
 6 
 
 55 
 
 01600 
 
 62.4992 
 
 03346 
 
 29.8823 
 
 05095 
 
 19.6273 
 
 06847 
 
 14.6059 
 
 5 
 
 515 
 
 01629 
 
 6l.38/2!t 
 
 03376 
 
 29.6215 
 
 05124 
 
 I9;5156 
 
 06876 
 
 L4.5438 
 
 4 
 
 57 
 
 
 (i:).:!<).->8 
 
 03405 
 
 29.3711 
 
 05153 
 
 19.40)1 
 
 06905 
 
 1 1.4^23 
 
 3 
 
 58 
 
 01687 
 
 59. -2< ;.-,) 
 
 n3i3 
 
 2.(.!'J2!) 
 
 05H-2 ' 19.2959 
 
 06934 
 
 14.4218 
 
 2 
 
 59 
 
 01716 
 
 58.2312 
 
 o:!4 ( >:? 
 
 
 0.7212 
 
 n.H:;) 
 
 05983 
 
 L4.3807 
 
 1 
 
 60 
 
 01746 
 
 57.2900 
 
 
 2-UKJt;3 
 
 05241 
 
 19.0811 
 
 06993 
 
 U.3007 
 
 
 
 M 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. T;iii.; 
 
 N. Cot. 
 
 N. T.tu. 
 
 N. Cot. 
 
 N. Tan. 
 
 M 
 
 
 89 Degrees. 
 
 88 Degrees. 
 
 87 Degrees. 
 
 86 Degrees. 
 
 
NATURAL, TANGENTS. 
 
 
 4 Degrees. 
 
 5 Degrees. 
 
 6 Degrees, if 7 Degrees. 
 
 M 
 
 N.Tan 
 
 N. Cot 
 
 N.Tan 
 
 N. Cot. 
 
 N.Tai 
 
 N. Cot. N.Tan 
 
 N.Cot. M 
 
 
 
 OU993 
 
 14.300 
 
 087-il) 
 
 11.430 
 
 1051U 
 
 9.51430 
 
 1-2273 
 
 8.14435 tiO 
 
 1 
 
 07022 
 
 14.241 
 
 08778 
 
 11.3919 
 
 10540 
 
 9.48781 
 
 12308 
 
 8.1248J 1 59 
 
 2 
 
 07051 
 
 14.18-2 
 
 08807 
 
 11.3541 
 
 10569 
 
 9.46141 
 
 12338 
 
 8.1053( 
 
 ; 58 
 
 3 
 
 07080 
 
 14.123 
 
 08837 
 
 11.3163 
 
 10599 
 
 9.43515 
 
 12367 
 
 8.0860( 
 
 ) 57 
 
 4 
 
 07110 
 
 14.065 
 
 08866 
 
 11.2789 
 
 106-28 
 
 9.40904 
 
 12397 
 
 8.0667- 
 
 I 56 
 
 5 
 
 07139 
 
 14.007 
 
 08895 
 
 11.241- 
 
 10657 
 
 9.38307 
 
 12426 
 
 8 . 0475( 
 
 55 
 
 6 
 
 07168 
 
 13.950 
 
 089-25 
 
 11.2048 
 
 10687 
 
 9.3572 
 
 12456 
 
 8.0284? 
 
 54 
 
 7 
 
 07197 
 
 13.894 
 
 08954 
 
 11.1681 
 
 10716 
 
 9.3315 
 
 12485 
 
 8.0094* 
 
 53 
 
 8 
 
 07227 
 
 13.837 
 
 08983 
 
 11.1316 
 
 10746 
 
 9.3059 
 
 12515 
 
 7 . 99056 
 
 5-2 
 
 
 
 07256 
 
 13.782 
 
 09013 
 
 11.0954 
 
 10775 
 
 9.2805 
 
 12544 
 
 7.97176 51 
 
 10 
 
 07285 
 
 13.726 
 
 09042 
 
 11.0594 
 
 10805 
 
 9.2553 
 
 12574 
 
 7.95302 50 
 
 11 
 
 07314 
 
 13.671 
 
 09071 
 
 11.0237 
 
 10834 
 
 9.2301 
 
 12603 
 
 7.934381 49 
 
 12 
 
 07344 
 
 13.617 
 
 09J01 
 
 10.9881 
 
 10863 
 
 9.2056 
 
 12633 
 
 7.9158x 
 
 48 
 
 13 
 
 07373 
 
 13.563 
 
 09130 
 
 10.9528 
 
 10893 
 
 9.1802 
 
 12662 
 
 7.89734 
 
 47 
 
 14 
 
 07402 
 
 13.5098 
 
 09159 
 
 10.9178 
 
 10922 
 
 9.15554 
 
 12692 
 
 7.87895 
 
 46 
 
 15 
 
 07431 
 
 I3.456b 
 
 09189 
 
 10.8829 
 
 10952 
 
 9.1309 
 
 12722 
 
 7.80064 
 
 45 
 
 16 
 
 07461 
 
 13.4039 
 
 09218 
 
 10.8483 
 
 10981 
 
 9.1064 
 
 12751 
 
 7.84242 
 
 44 
 
 17 
 
 07490 
 
 13.3515 
 
 09247 
 
 10.8139 
 
 11011 
 
 9.0821 
 
 12781 
 
 7.82428 
 
 43 
 
 18 
 
 07519 
 
 13.2996 
 
 09277 
 
 10.7797 
 
 11040 
 
 9.05789 
 
 12810 
 
 7.80622 
 
 42 
 
 19 
 
 07548 
 
 13.2480 
 
 09306 
 
 10.7457 
 
 11070 
 
 9.03379 
 
 12840 
 
 7.78825 
 
 41 
 
 20 
 
 07578 
 
 13.1969 
 
 09335 
 
 10.71J9 
 
 11099 
 
 9.0098: 
 
 12869 
 
 7.77035 
 
 40 
 
 21 
 
 07607 
 
 13.1461 
 
 09365 
 
 10.6783 
 
 11128 
 
 8.98598 
 
 12893 
 
 7.75254 
 
 39 
 
 22 
 
 67636 
 
 13.0958 
 
 09394 
 
 10.6450 
 
 11158 
 
 8.96227 
 
 129-29 
 
 7.73480 
 
 38 
 
 23 
 
 07665 
 
 13.0458 
 
 09423 
 
 10.6118 
 
 11187 
 
 8.9386" 
 
 12958 
 
 7.71715 
 
 37 
 
 24 
 
 07695 
 
 12.9962 
 
 09453 
 
 10.5789 
 
 11217 
 
 8.91520 
 
 12988 
 
 7.69957 
 
 36 
 
 25 
 
 07724 
 
 12.9469 
 
 09482 
 
 10.5462 
 
 11246 
 
 8.89185 
 
 13017 
 
 7.68208 
 
 35 
 
 26 
 
 07753 
 
 12.8981 
 
 09511 
 
 10.5136 
 
 11276 
 
 8.86862 
 
 13047 
 
 7.66466 
 
 34 
 
 27 
 
 07782 
 
 12.8496 
 
 09541 
 
 10.4813 
 
 11305 
 
 8.84551 
 
 13076 
 
 7.64732 
 
 33 
 
 28 
 
 07812 
 
 12.8014 
 
 G9570 
 
 10.4491 
 
 11335 
 
 8.82252 
 
 13106 
 
 7.63005 
 
 32 
 
 29' 
 
 07841 
 
 12.7536 
 
 09600 
 
 10.4172 
 
 11364 
 
 8.79964 
 
 13136 
 
 7.61287 
 
 31 
 
 30 
 
 07870 
 
 12.7062 
 
 09629 
 
 10.3854 
 
 11394 
 
 8.77689 
 
 13165 
 
 7.59575 
 
 30 
 
 31 
 
 07899 
 
 12.6591 
 
 09058 
 
 10.3538 
 
 11423 
 
 8.75425 
 
 13195 
 
 7.57872 
 
 29 
 
 32 
 
 07929 
 
 12-6124 
 
 09688 
 
 10.3224 
 
 11452 
 
 8.73172 
 
 13224 
 
 7.56176 
 
 28 
 
 33 
 
 07958 
 
 12.5660 
 
 09717 
 
 10.2913 
 
 U 482 
 
 8.70931 
 
 13254 
 
 7.54487 
 
 27 
 
 34 
 
 07987 
 
 12.5199 
 
 09746 
 
 10.2602 
 
 H511 
 
 8-68701 
 
 13284 
 
 7.52806 
 
 26 
 
 35 
 
 08017 
 
 12.4742 
 
 09776 
 
 10.2294 
 
 U541 
 
 8.66489 
 
 13313 
 
 7.51132 
 
 25 
 
 36 
 
 08046 
 
 12.4288 
 
 09805 
 
 10.1988 
 
 H570 
 
 8.64275 
 
 13343 
 
 7.49465 
 
 24 
 
 37 
 
 08075 
 
 12.3838 
 
 09834 
 
 10.1683 
 
 11600 
 
 8.02078 
 
 13372 
 
 7.4780fr 
 
 23 
 
 38 
 
 08104 
 
 12.3390 
 
 09864 
 
 10.1381 
 
 U629 
 
 8.59893 
 
 13402 
 
 7.46154 
 
 22 
 
 3'J 
 
 08134 
 
 12.2946 
 
 09893 
 
 10.1080 
 
 11659 
 
 8.57718 
 
 13432 
 
 7.44509 
 
 21 
 
 40 
 
 08163 
 
 12.2505 
 
 09923 
 
 10.0780 
 
 11688 
 
 8.55555 
 
 13461 
 
 7.42871 
 
 20 
 
 41 
 
 08192 
 
 12.2067 
 
 09952 
 
 10.0483 
 
 11718 
 
 8.53402 
 
 13491 
 
 7.41240 
 
 19 
 
 42 
 
 08221 
 
 12.1632 
 
 09981 
 
 10.0187 
 
 11747 
 
 8.51259 
 
 13521 
 
 7.39616 
 
 18 
 
 43 
 
 0825^1 
 
 12.1201 
 
 10011 
 
 9.98930 
 
 11777 
 
 8.49128 
 
 13550 
 
 7.37999 
 
 17 
 
 44 
 
 08280 
 
 12.0772 
 
 10040 
 
 9.96007 
 
 11806 
 
 8.47007 
 
 13580 
 
 7.36389 
 
 16 
 
 45 
 
 08309 
 
 12.0346 
 
 10069 
 
 9.93101 
 
 11836 
 
 8.44896 
 
 13609 
 
 7.34786 
 
 15 
 
 46 
 
 08339 
 
 11.9923 
 
 10099 
 
 9.90211 
 
 11865 
 
 .".4-2795 
 
 13639 
 
 7.33190 
 
 14 
 
 47 
 
 08368 
 
 11.9504 
 
 10128 
 
 9.87338 
 
 11895 
 
 8.40705 
 
 13669 
 
 7.31600 
 
 13 
 
 48 
 
 08397 
 
 11.9087 
 
 10158 
 
 9.84482 
 
 11924 
 
 8.38625 
 
 13698 
 
 7.300J8 
 
 12 
 
 49 
 
 08427 
 
 11.8673 
 
 10187 
 
 9.81641 
 
 11954 
 
 8.36555 
 
 13728 
 
 7.28442 
 
 11 
 
 50 
 
 08456 
 
 11.8262 
 
 10216 
 
 9.78817 
 
 11983 
 
 8. 34496 ! 
 
 13758 
 
 7.26873 
 
 10 
 
 51 
 
 08485 
 
 11.7853 
 
 10246 
 
 9.76009 
 
 12013 
 
 8.32446 
 
 13787 
 
 7.25310 
 
 9 
 
 52 
 
 03514 
 
 11.7448 
 
 10275 
 
 9.73217 
 
 12042 
 
 8.30406 
 
 13817 
 
 7.23754 
 
 8 
 
 53 
 
 08544 
 
 11.7045 
 
 10305 
 
 9.70441! 
 
 13073 
 
 8.28376 
 
 13846 
 
 7.22204 
 
 7 
 
 54 
 
 08573 
 
 11.6645! 
 
 10334 
 
 9. 67680 i 
 
 12101 
 
 8.21)355 
 
 13876 
 
 7-20061! f, 
 
 55 
 
 08602 
 
 11.6248 
 
 10363 
 
 9.64935! 
 
 12131 
 
 8.34345 
 
 13906 
 
 7.191251 5 
 
 56 
 
 08632 
 
 11.5853 
 
 10393 
 
 9.62205! 
 
 121(30 
 
 8.22344 
 
 13935 
 
 7.17594 
 
 4 
 
 57 
 
 08661 
 
 11.5461 
 
 10422 
 
 9. 59490 ' 
 
 12190 
 
 8.20352 
 
 13965 
 
 7.16071 
 
 3 
 
 58 
 
 08690 
 
 11.5072 
 
 10452 
 
 9. 56791 i 
 
 12219 
 
 8.18370 
 
 13995 
 
 7.14553 
 
 2 
 
 59 
 
 08720 
 
 11.4685 
 
 10481 
 
 9.54106 
 
 13349 
 
 8.16398 
 
 14024 
 
 7.13042 
 
 1 
 
 60 
 
 08749 
 
 11.4301 
 
 10510 
 
 9.51436 
 
 12278 
 
 8.14435 
 
 14054 
 
 7.11537 
 
 
 
 M 
 
 N Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cotj N. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 M 
 
 
 85 Degrees. 
 
 84 Degrees. , 
 
 83 Degrees. 
 
 82 Degrees. 
 
 
NATURAL TANGENTS. 
 
 75 
 
 
 8 Degrees. 
 
 9 Degrees. 
 
 10 Degrees. 
 
 11 Degrees. 
 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 N.Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan 
 
 N. Cot. 
 
 M 
 
 
 
 14054 
 
 7.11537 
 
 15838 
 
 6.31375 
 
 17633 
 
 5.6712* 
 
 19438 
 
 5.14455 
 
 (50 
 
 1 
 
 14084 
 
 7.1003H 
 
 15888 
 
 6.30189 
 
 17663 
 
 5. tit 5 1(55 
 
 19486 
 
 5.13658 
 
 59 
 
 2 
 
 14113 
 
 7.08546 
 
 15898 
 
 6.29007 
 
 17693 
 
 5.65205 
 
 19498 
 
 5.12862 
 
 58 
 
 3 
 
 14143 
 
 7.07059 
 
 15928 
 
 6.27H29 
 
 17723 
 
 5.64248 
 
 19529 
 
 5.12069 
 
 57 
 
 4 
 
 14173 
 
 7.05579 
 
 15958 
 
 15.26655 
 
 17753 
 
 5.63-295 
 
 19559 
 
 5.11279 
 
 56 
 
 5 
 
 14202 
 
 7.04105 
 
 15988 
 
 6. 25486 
 
 17783 
 
 5.62344 
 
 19589 
 
 5.10490 
 
 55 
 
 G 
 
 142*2 
 
 7.K2037 
 
 16017 
 
 6.24321 
 
 17813 
 
 5.61397 
 
 19619 
 
 5.0!)704 
 
 54 
 
 1 
 
 14262 
 
 7-01174 
 
 16047 
 
 6. 23 160 
 
 17843 
 
 5.60452 
 
 19649 
 
 5.08921 
 
 53 
 
 8 
 
 14291 
 
 6-99718 
 
 16077 
 
 6.2200:5 
 
 17873 
 
 5.59511 
 
 19680 
 
 5.08139 
 
 52 
 
 9 
 
 j 43-2 1 
 
 6-982(58 
 
 16107 
 
 6.20851 
 
 17903 
 
 5.58573 
 
 19710 
 
 5.073(50 
 
 51 
 
 10 
 
 14351 
 
 6-968-23 
 
 16137 
 
 6.19703 
 
 17933 
 
 5.57638 
 
 19740 
 
 5.06584 
 
 50 
 
 11 
 
 14381 
 
 6.95385 
 
 16167 
 
 6.18559 
 
 17963 
 
 5-56706 
 
 19770 
 
 5.05809 
 
 49 
 
 12 
 
 14410 
 
 6-9395-2 
 
 16196 
 
 6.17419 
 
 17993 
 
 5-55777 
 
 19801 
 
 5.05037 
 
 48 
 
 13 
 
 14440 
 
 H.92.V25 
 
 162-26 
 
 6.16283 
 
 18023 
 
 5.54851 
 
 19831 
 
 5.04267 
 
 47 
 
 14 
 
 14470 
 
 6.91104 
 
 1Q356 
 
 6.15151 
 
 18053 
 
 5-53927 
 
 19861 
 
 5.03499 
 
 46 
 
 15 
 
 14499 
 
 6-89688 
 
 16286 
 
 6.14023 
 
 18083 
 
 5-53007 
 
 19891 
 
 5.02734 
 
 45 
 
 16 
 
 14529 
 
 6.88278 
 
 16316 
 
 6.12899 
 
 18113 
 
 5.52090 
 
 19921 
 
 5.01971 
 
 44 
 
 17 
 
 14559 
 
 6- 86874 
 
 16346 
 
 6.11779 
 
 18143 
 
 5.51176 
 
 19952 
 
 5.01210 
 
 43 
 
 18 
 
 14588 
 
 6-85475 
 
 16376 
 
 6.10664 
 
 18173 
 
 5.50264 
 
 19982 
 
 5.00451 
 
 42 
 
 19 
 
 14618 
 
 6.84082 
 
 16405 
 
 6.09552 
 
 18203 
 
 5.49356 
 
 20012 
 
 4.99695 
 
 41 
 
 20 
 
 14648 
 
 6.82694 
 
 16435 
 
 6.08444 
 
 18233 
 
 5.48451 
 
 20042 
 
 4.98940 
 
 40 
 
 21 
 
 14678 
 
 6-81312 
 
 16465 
 
 6.07340 
 
 18263 
 
 5.47548 
 
 20073 
 
 4.98188 
 
 39 
 
 22 
 
 14707 
 
 6.79936 
 
 16495 
 
 6.06240 
 
 18293 
 
 5.46648 
 
 20103 
 
 4.97438 
 
 38 
 
 23 
 
 14737 
 
 6-78564 
 
 16525 
 
 6.05143 
 
 18323 
 
 5.45751 
 
 20133 
 
 4.96690 
 
 37 
 
 24 
 
 14767 
 
 6.77199 
 
 16555 
 
 6.04051 
 
 18353 
 
 5.44857 
 
 20164 
 
 4.95945 
 
 36 
 
 25 
 
 14796 
 
 6.75838 
 
 16585 
 
 6.0-I62 
 
 18383 
 
 5.43966 
 
 20194 
 
 4.95201 
 
 35 
 
 26 
 
 14826 
 
 6.74483 
 
 16615 
 
 6.01878 
 
 18414 
 
 5.43077 
 
 20224 
 
 4.94460 
 
 34 
 
 27 
 
 14856 
 
 6.73133 
 
 16645 
 
 6.00797 
 
 18444 
 
 5.42192 
 
 20254 
 
 4.93721 
 
 33 
 
 28 
 
 14886 
 
 6.71789 
 
 16674 
 
 5.99720 
 
 18474 
 
 5.41309 
 
 20285 
 
 4.92984 
 
 32 
 
 29 
 
 14915 
 
 6.70450 
 
 16704 
 
 5.93646 
 
 18504 
 
 5.40429 
 
 20315 
 
 4.92249 
 
 31 
 
 30 
 
 14945 
 
 6.69116 
 
 16734 
 
 5.97576 
 
 18534 
 
 5.39552 
 
 20345 
 
 4.91516 
 
 30 
 
 31 
 
 14975 
 
 6.67787 
 
 16764 
 
 5.96510 
 
 18564 
 
 5.38677 
 
 20376 
 
 4.90785 
 
 29 
 
 32 
 
 151)05 
 
 6.6463 
 
 16794 
 
 5.95448 
 
 18594 
 
 5.37805 
 
 20406 
 
 4.90056 
 
 28 
 
 33 
 
 15034 
 
 6.65144 
 
 16824 
 
 5. 943 JO 
 
 18624 
 
 5.36936 
 
 2J436 
 
 4.89330 
 
 27 
 
 34 
 
 15064 
 
 6.63831 
 
 16854 
 
 5.93335 
 
 18654 
 
 5.36070 
 
 20466 
 
 4.88605 
 
 26 
 
 35 
 
 15094 
 
 6.152523 
 
 16884 
 
 5.99383 
 
 18684 
 
 5.3520(5 
 
 20497 
 
 4.87882 
 
 25 
 
 36 
 
 15124 
 
 6.61219 
 
 16914 
 
 5.91235 
 
 18714 
 
 5.34345 
 
 20527 
 
 4.87162 
 
 24 
 
 37 
 
 15153 
 
 (>.5<M-2I 
 
 115914 
 
 5.91)191 
 
 18745 
 
 5.33487 
 
 20557 
 
 4.86444 
 
 23 
 
 38 
 
 15183 
 
 6.58627 
 
 16974 
 
 5.89151 
 
 18775 
 
 5.32631 
 
 20588 
 
 4.R5727 
 
 22 
 
 39 
 
 15-213 
 
 8.57339 
 
 17004 
 
 5.88114 
 
 18805 
 
 5.31778 
 
 20618 
 
 4.85013 
 
 21 
 
 40 
 
 15-243 
 
 6.5(5055 
 
 17033 
 
 5.87080 
 
 18835 
 
 5.30928 
 
 20648 
 
 4.84300 
 
 20 
 
 41 
 
 15-272 
 
 6.54777 
 
 17063 
 
 5.86051 
 
 18865 
 
 5.30080 
 
 20679 
 
 4.83590 
 
 19 
 
 42 
 
 15302 
 
 6.53503 
 
 17093 
 
 5.85024 
 
 18895 
 
 5.29235 
 
 20709 
 
 4.82882 
 
 18 
 
 43 
 
 153:5-2 
 
 6.52-234 
 
 17123 
 
 5.84001 
 
 18925 
 
 5.28393 
 
 20739 
 
 4.82175 
 
 17 
 
 44 
 
 15362 
 
 6.50970 
 
 17153 
 
 5.82982 
 
 18955 
 
 5.27553 
 
 20770 
 
 4.81471 
 
 16 
 
 45- 
 
 15391 
 
 6.49710 
 
 17183 
 
 5.81966 
 
 18986 
 
 5.26715 
 
 20800 
 
 4.80769 
 
 15 
 
 46 
 
 15421 
 
 6.48456 
 
 17213 
 
 5.80953 
 
 19016 
 
 5.25880 
 
 20830 
 
 4.80068 
 
 14 
 
 47 
 
 15451 
 
 6.47206 
 
 17-243 
 
 5.79944 
 
 19046 
 
 5.25048 
 
 20861 
 
 4.79370 
 
 13 
 
 48 
 
 151S1 
 
 6.45961 
 
 17-273 
 
 5.78938 
 
 19076 
 
 5.24218 
 
 20891 
 
 4.78673 
 
 12 
 
 49 
 
 15511 
 
 6.44720 
 
 17303 
 
 5.77936 
 
 19106 
 
 5.23391 
 
 20921 
 
 4.77978 
 
 11 
 
 50 
 
 15540 
 
 6.43484 
 
 17333 
 
 5.76937 
 
 19136 
 
 5. 22566 
 
 20952 
 
 4.77286 
 
 10 
 
 51 
 
 15570 
 
 6.4-2-253 
 
 17363 
 
 5.75941 
 
 19166 
 
 5.21744 
 
 20982 
 
 4.76595 
 
 9 
 
 52 
 
 15600 
 
 6.410-26 
 
 17393 
 
 5.74949 
 
 19197 
 
 5.20925 
 
 21013 
 
 4.75906 
 
 8 
 
 53 
 
 15630 
 
 6.39804 
 
 17423 
 
 5.73960 
 
 19227 
 
 5.20107 
 
 21043 
 
 4.75219 
 
 7 
 
 54 
 
 15 ;i;i) 
 
 6.38587 
 
 17453 
 
 5.72974 
 
 19257 
 
 5.19293 
 
 21073 
 
 4.74534 
 
 6 
 
 55 
 
 15!>89 
 
 6.37374 
 
 17483 
 
 5.71992 
 
 19-287 
 
 5.18480 
 
 21104 
 
 4.73851 
 
 5 
 
 5(5 
 
 15719 
 
 6.36165 
 
 17513 
 
 5.71013 
 
 19317 
 
 5.17671 
 
 21134 
 
 4.73170 
 
 4 
 
 57 
 
 15749 
 
 <>.:M!t;i 
 
 17543 
 
 5.70037 
 
 19347 
 
 5.16863 
 
 21164 
 
 4.72490 
 
 3 
 
 58 
 
 15779 
 
 6.33761 
 
 17573 
 
 5.69064 
 
 19378 
 
 5.10058 
 
 21195 
 
 4.71813 
 
 2 
 
 59 
 
 1580!) 
 
 6.3-2566 
 
 17603 
 
 5.68094 
 
 19408 
 
 5.15256 
 
 21225 
 
 4.71137 
 
 1 
 
 (50 
 
 15833 
 
 6.31375 
 
 17633 
 
 5.671-28 
 
 19438 
 
 5.14455 
 
 21256 
 
 4.70463 
 
 
 
 M 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N.Tan. 
 
 N. Cot.1 N. Tan. 
 
 N. Cot.| N. Tan. 
 
 ~M~ 
 
 
 81 Degrees. 
 
 80 Degrees. 
 
 79 Degrees. 
 
 78 Degrees. 
 
 
NATURAL TANGENTS. 
 
 
 12 Degrees. 
 
 13 Degrees. 
 
 14 Degrees. II 15 Degrees. > 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 N.Tan 
 
 N. Cot. 
 
 N.Tai 
 
 N. Cot. \.T;m 
 
 A . Cot. i M 
 
 
 
 21250 
 
 4.7040: 
 
 23087 
 
 4.33148 
 
 249: S3 
 
 4.01078 
 
 20795 
 
 3.73205 60 
 
 1 
 
 
 4.r<t7i>i 
 
 23117 
 
 4.32573 
 
 24964 
 
 4.00582 
 
 26826 
 
 3.12771 
 
 59 
 
 o 
 
 213 1(3 
 
 4.0;12I 
 
 23148 
 
 4. 320U 1 
 
 24995 
 
 4.(i0086 
 
 26857 
 
 3.7S338 
 
 58 
 
 3 
 
 21347 
 
 4.084.V- 
 
 23179 
 
 4.31430 
 
 25020 
 
 3.99592 
 
 26888 
 
 3.7190- 
 
 57 
 
 4 
 
 21377 
 
 4.67781 
 
 23209 
 
 4.3080* 
 
 25056 
 
 3.99099 
 
 26920 
 
 3.7147( 
 
 50 
 
 5 
 
 21408 
 
 4.071-21 
 
 23240 
 
 4.:>02'Ji 
 
 25087 
 
 3.98007 
 
 20951 
 
 3.71(M( 
 
 55 
 
 
 
 2J438 
 
 4.00458 
 
 23271 
 
 4.29724 
 
 25118 
 
 3.98117 
 
 2i;;is2 
 
 3.700H 
 
 54 
 
 7 
 
 214C9 
 
 4. 05791 
 
 23301 
 
 4.21M5S 
 
 251 J9 
 
 3.97627 
 
 27013 
 
 3.70'188 
 
 53 
 
 8 
 
 21499 
 
 4.05138 
 
 23332 
 
 4.28595 
 
 25180 
 
 3.97139 
 
 27044 
 
 3.097GJ 
 
 52 
 
 9 
 
 21.329 
 
 4.154481 
 
 23303 
 
 4.2803- 
 
 25211 
 
 3. <)f.<v> 1 
 
 27070 
 
 3.69335 
 
 51 
 
 10 
 
 21500 
 
 4.63825 
 
 23393 
 
 4.27471 
 
 252-12 
 
 3.96165 
 
 27107 
 
 3,6890! 
 
 50 
 
 ]JL 
 
 21590 
 
 4.0317 
 
 23424 
 
 4.21)911 
 
 25273 
 
 3.95680 
 
 27138 
 
 3.08-18.1 
 
 4!t 
 
 12 
 
 21(521 
 
 4.02518 
 
 23455 
 
 4.20352 
 
 25304 
 
 3.95196 
 
 27109 
 
 3.08001 
 
 48 
 
 13 
 
 2l(i51 
 
 4.61868 
 
 23484 
 
 4.25795 
 
 25335 
 
 3.94713 
 
 27201 
 
 fctJTflsa 
 
 47 
 
 14 
 
 2ir,s-_> 
 
 4.01-211 
 
 23516 
 
 4.252::9 
 
 25300 
 
 3.94232 
 
 27232 
 
 3.072)7 
 
 40 
 
 15 
 
 21712 
 
 4.00572 
 
 23547 
 
 4.24085 
 
 25397 
 
 3.93751 
 
 27203 
 
 3.66796 
 
 45 
 
 10 
 
 21743 
 
 4.59927 
 
 23578 
 
 4.24132 
 
 25428 
 
 3.93271 
 
 2721)4 
 
 3.66376 
 
 44 
 
 17 
 
 21773 
 
 4.59283 
 
 23008 
 
 4.23580 
 
 25459 
 
 3.92793 
 
 27320 
 
 3-65957 
 
 43 
 
 18 
 
 21804 
 
 4.58041 
 
 23039 
 
 4.23030 
 
 25490 
 
 3.92310 
 
 27357 
 
 3. 05538 
 
 42 
 
 19 
 
 21834 
 
 4.58001 
 
 23070 
 
 4.22-181 
 
 25521 
 
 3.91839 
 
 27388 
 
 3.05121 
 
 41 
 
 20 
 
 21804 
 
 4.57303 
 
 23700 
 
 4.21933 
 
 25552 
 
 3.91304 
 
 27419 
 
 3.64705 
 
 40 
 
 21 
 
 21895 
 
 4.5(57-21 
 
 23731 
 
 4.21387 
 
 25583 
 
 3.90890 
 
 27451 
 
 3.64289 
 
 :-;9 
 
 22 
 
 21925 
 
 4.50091 
 
 231762 
 
 4.20842 
 
 25014 
 
 3.90417 
 
 27482 
 
 3.03874 
 
 38 
 
 23 
 
 21956 
 
 4.55458 
 
 23793 
 
 4.20298 
 
 25645 
 
 3.89945 
 
 27513 
 
 3.0^101 
 
 37 
 
 24 
 
 21980 
 
 4.5482J 
 
 23823 
 
 4.19750 
 
 25676 
 
 3.89474 
 
 27545 
 
 3.63048 
 
 30 
 
 25 
 
 22017 
 
 4.54191 
 
 23854 
 
 4.19215 
 
 25707 
 
 3.89004 
 
 27570 
 
 3.02030 
 
 35 
 
 20 
 
 22047 
 
 4.53508 
 
 23885 
 
 4.18675 
 
 25738 
 
 3.88536 
 
 27607 
 
 3.02224 
 
 34 
 
 27 
 
 22078 
 
 4.52941 
 
 23916 
 
 4.18137 
 
 25709 
 
 3.88008 
 
 27038 
 
 3.01814 
 
 33 
 
 2-8 
 
 2=2108 
 
 4.523K 
 
 23946 
 
 4.17000 
 
 25800 
 
 3.87601 
 
 27(570 
 
 3.01405 
 
 32 
 
 29 
 
 '22139 
 
 4.51093 
 
 23977 
 
 4-17004 
 
 25831 
 
 3.87136 
 
 27701 
 
 3.00990 
 
 31 
 
 30 
 
 2-2169 
 
 4.51071 
 
 24008 
 
 4.1 0530 
 
 25862 
 
 3.86671 
 
 27732 
 
 3.60588 
 
 30 
 
 31 
 
 222GO 
 
 4.50451 
 
 24039 
 
 4.15997 
 
 25893 
 
 3.86208 
 
 27764 
 
 3.00181 
 
 29 
 
 32 
 
 22231 
 
 4.J19832 
 
 24009 
 
 4.15J05 
 
 25924 
 
 3.85745 
 
 27795 
 
 3.59775 
 
 28 
 
 33 
 
 22261 
 
 4.49215 
 
 24100 
 
 4.14934 
 
 25955 
 
 3.85283 
 
 27826 
 
 3.59370 
 
 27 
 
 34 
 
 22392 
 
 4.48DOO 
 
 24131 
 
 4.14405 
 
 25986 
 
 3.84824 
 
 27858 
 
 3.58966 
 
 26 
 
 35 
 
 2-2:5-22 
 
 4.47980 
 
 2410-2 
 
 4.13877 
 
 26017 
 
 3.84364 
 
 27889 
 
 3.58562 
 
 25 
 
 36 
 
 22353 
 
 4.47374 
 
 24193 
 
 4.13350 
 
 26048 
 
 3-83904 
 
 27920 
 
 3.58160 
 
 24 
 
 37 
 
 22383 
 
 4. 407154 
 
 24223 
 
 4.12825 
 
 20079 
 
 3-83449 
 
 27952 
 
 3.57758 
 
 23 
 
 38 
 
 22414 
 
 4.40155 
 
 24254 
 
 4.1230] 
 
 26110 
 
 3-82992 
 
 27983 
 
 3.57357 
 
 22 
 
 39 
 
 22444 
 
 4.45548 
 
 24285 
 
 4.11778 
 
 20141 
 
 3.82537 
 
 28015 
 
 3.50957 
 
 21 
 
 40 
 
 22475 
 
 4.44942 
 
 24316 
 
 4.11250 
 
 20172 
 
 3-82083 
 
 28046 
 
 3.50557 
 
 20 
 
 41 
 
 22505 
 
 4.443:i8 
 
 24347 
 
 4.10736 
 
 20203 
 
 3-8J630 
 
 28077 
 
 3.56159 
 
 19 
 
 42 
 
 22530 
 
 4.43735 
 
 24377 
 
 4.10216 
 
 20235 
 
 3-81177 
 
 28109 
 
 3.55761 
 
 18 
 
 43 
 
 22507 
 
 4.43134 
 
 24408 
 
 4.09099 
 
 20200 
 
 3.80726 
 
 28140 
 
 3.55364 
 
 17 
 
 44 
 
 2-2597 
 
 4.42534 
 
 24439 
 
 4.09182 
 
 20297 
 
 3-80276 
 
 28172 
 
 3.54968 16 
 
 45 
 
 22628 
 
 4.41936 
 
 24470 
 
 4.08006 
 
 26328 
 
 3-79827 
 
 28203 
 
 3.54573J 15 
 
 4G 
 
 2-2658 
 
 4.41340 
 
 24501 
 
 4.08152 
 
 20359 
 
 3.79378 
 
 28234 
 
 3.54179 14 
 
 47 
 
 22689 
 
 4.40745 
 
 24532 
 
 4.07639 
 
 20390 
 
 3.78931 
 
 28266 
 
 3.53785 13 
 
 48 
 
 227 J 9 
 
 4.40152 
 
 24502 
 
 4.07127 
 
 2(5421 
 
 3.78485 
 
 28297 
 
 3.53393 12 
 
 49 
 
 22750 
 
 4.35t;o 
 
 24593 
 
 4.06616 
 
 20452 
 
 3.78040! 
 
 28329 
 
 3.53001 11 
 
 50 
 
 22781 
 
 4-389H9 
 
 24024 
 
 4.06107 
 
 20483 
 
 3.77595 
 
 28360 
 
 3.52609' 10 
 
 51 
 
 22811 
 
 4.38381 
 
 24055 
 
 4.05599 
 
 20515 
 
 3.77152 
 
 28391 
 
 3.52219! 9 
 
 52 
 
 22842 
 
 4.37793 
 
 24086 
 
 4.05092; 
 
 20546 
 
 3.70709! 
 
 28423 
 
 3.51829 8 
 
 53 
 
 22872 
 
 4.37207 
 
 24717 
 
 4.04.VI5 
 
 20577 
 
 3.70268 
 
 28454 
 
 3.51441 7 
 
 5^ 
 
 22903 
 
 4.36023 
 
 24747 
 
 4.04081 
 
 26008 
 
 3.75828 
 
 28486 
 
 3.51053 
 
 55 
 
 2-2934 
 
 4.36040 
 
 24778 
 
 4.03578' 
 
 20639 
 
 3:75388 
 
 28517 
 
 3..W06 5 
 
 5 
 
 22954 
 
 4.35459 
 
 24809 
 
 4.03075! 
 
 26670 
 
 3. 74950 1 
 
 28549 
 
 3.50279 4 
 
 57 
 
 2-2995 
 
 4.34879 
 
 24840 
 
 4.02574 
 
 20701 
 
 3.74512 
 
 28580 
 
 3.49894 3 
 
 58 
 
 23026 
 
 4.343'M> 
 
 24871 
 
 4.02074 
 
 26733 
 
 3.74075; 
 
 28012 
 
 3.49509 2 
 
 59 
 
 23056 
 
 4.33723 
 
 24902 
 
 4.01570 ! 
 
 26764 
 
 3.73644; 
 
 28043 
 
 3.49125 1 
 
 GO 
 
 23087 
 
 4.33148 
 
 24933 
 
 4.01078 
 
 26795 
 
 3.73205 
 
 28675 
 
 3.48741 
 
 M 
 
 \. Cot. 
 
 N. Tan 
 
 N. Cot. 
 
 N.Tan.' 
 
 N. Cot. 
 
 N. T;tn. 
 
 N Cot. 
 
 N. Tem. 
 
 M 
 
 
 77 Degrees. 
 
 76 Degrees. 
 
 75 Degrees. 
 
 74 Degrees. 
 
 
NATURAL TANGENTS. 
 
 77 
 
 M 
 
 16 Degrees. 
 
 17 Degrees. 
 
 18 Degrees. 
 
 19 Degrees. 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 N. Ta i. 
 
 i\. Cot. 
 
 N. T,m. 
 
 iM.Cot. 
 
 N.Tan 
 
 N. Ool. 
 
 ~o~ 
 
 281)75 
 
 3.48741 
 
 30.>73 
 
 3.27U85 
 
 324J2 
 
 "3707708 
 
 34433 
 
 2.9J421 
 
 Ik) 
 
 1 
 
 88706 
 
 3.48359 
 
 30605 
 
 3.2(174.-, 
 
 32524 
 
 3.07I04 
 
 34465 
 
 2.90147 
 
 59 
 
 o 
 
 28738 
 
 3.47977 
 
 3(11137 
 
 3.2.1M,, 
 
 32556 
 
 3.071(10 
 
 34498 
 
 2.89873 
 
 58 
 
 3 
 
 28763 
 
 3.4759 
 
 305(59 
 
 3.28061 
 
 32583 
 
 3.06851 
 
 3 1530 
 
 2.89600 
 
 57 
 
 4 
 
 28800 
 
 3.4721(5 
 
 30700 
 
 3.25729 
 
 32(521 
 
 3.0(5(554 
 
 34583 
 
 2.893-27 
 
 56 
 
 5 
 
 2883-2 
 
 3.4W3< 
 
 30732 
 
 3.25392 
 
 32653 
 
 3.0;;-2.-)2 
 
 34596 
 
 2.89055 
 
 55 
 
 (j 
 
 288 54 
 
 3.4(5458 
 
 307(54 
 
 3.25055 
 
 32685 
 
 3.0595,. 
 
 34(528 
 
 2.88783 
 
 54 
 
 
 23895 
 
 3.4GiHn 
 
 30796 
 
 3.5&4719 
 
 32717 
 
 3.05(149 
 
 31(1(51 
 
 2. 88.") 11 
 
 53 
 
 8 
 
 28927 
 
 3.457U3 
 
 30828 
 
 3.24383 
 
 32749 
 
 3.05349 
 
 341193 
 
 2. ,88240 
 
 52 
 
 9 
 
 28938 
 
 3.45327 
 
 30860 
 
 3.24049 
 
 32782 
 
 3.05049 
 
 34786 
 
 2.87970 
 
 51 
 
 10 
 
 28930 
 
 3.44951 
 
 30891 
 
 3. 237 14 
 
 32814 
 
 3.04749 
 
 34758 
 
 2.877011 
 
 50 
 
 11 
 
 26031 
 
 3.4457(5 
 
 30923 
 
 3.23381 
 
 34846 
 
 3.0445!! 
 
 34791 
 
 2.87430 
 
 49 
 
 12 
 
 29053 
 
 3.44202 
 
 30955 
 
 3.23048 
 
 32878 
 
 3.04152 
 
 31824 
 
 2.87161 
 
 48 
 
 13 
 
 29084 
 
 3.43S29 
 
 30987 
 
 3.u27 15 
 
 329J1 
 
 3.03-51 
 
 34856 
 
 2.86892 
 
 47 
 
 14 
 
 29111) 
 
 3.4345(5 
 
 3 I ill 9 
 
 3.22384 
 
 32943 
 
 3.0355(1 
 
 34889 
 
 2.86624 
 
 46 
 
 15 
 
 29147 
 
 3.430-M 
 
 31051 
 
 3.22053 
 
 32975 
 
 3.03200 
 
 34922 
 
 2.8li35;i 
 
 45 
 
 18 
 
 29179 
 
 3.4-2713 
 
 31083 
 
 3.21722 
 
 33007 
 
 3.02963 
 
 34954 
 
 2.86089 
 
 44 
 
 17 
 
 29210 
 
 3.42313 
 
 31115 
 
 3.2U92 
 
 33040 
 
 3.02661 
 
 34987 
 
 2.85822 
 
 43 
 
 18 
 
 29042 
 
 3.41973 
 
 31 147 
 
 3.21063 
 
 33072 
 
 3.02372 
 
 35019 
 
 2.85555 
 
 42 
 
 19 
 
 29274 
 
 3.41694 
 
 31178 
 
 3.20734 
 
 33104 
 
 3.02977 
 
 35052 
 
 2.85289 
 
 41 
 
 20 
 
 29305 
 
 3.41250 
 
 31210 
 
 3.2040!) 
 
 3313;; 
 
 3.01783 
 
 35085 
 
 2. 85923 
 
 40 
 
 21 
 
 29337 
 
 3.40819 
 
 31242 
 
 3.23079 
 
 33169 
 
 3.01489 
 
 35117 
 
 2.84758 
 
 39 
 
 >-> 
 
 293(5* 
 
 3.4050-J 
 
 31274 
 
 3.I9752 
 
 33201 
 
 3. (11 191- 
 
 35150 
 
 2.84494 
 
 38 
 
 2:1 
 
 29400 
 
 3.4013(5 
 
 31306 
 
 3.19426 
 
 33233 
 
 3.00903 
 
 35183 
 
 2.84229 
 
 37 
 
 24 
 
 29432 
 
 3.39771 
 
 31338 
 
 3.19100 
 
 332 iti 
 
 3.00(511 
 
 35216 
 
 2.83965 
 
 36 
 
 25 
 
 29463 
 
 3.3910.1 
 
 31370 
 
 3.18775 
 
 33-298 
 
 3.00319 
 
 352 18 
 
 2.83702 
 
 35 
 
 26 
 
 29495 
 
 3.39342 
 
 31402 
 
 3.18451 
 
 33330 
 
 3,00028 
 
 35281 
 
 2.83439 
 
 34 
 
 27 
 
 295-2(5 
 
 3.38879 
 
 31434 
 
 3.18127 
 
 33353 
 
 2.9973S 
 
 35314 
 
 2.83176 
 
 33 
 
 28 
 
 23558 
 
 3. 3 S3 17 
 
 31166 
 
 3.17804 
 
 33395 
 
 2.99447 
 
 3534(5 
 
 2.82914 
 
 32 
 
 29 
 
 23593 
 
 :!.37!i:>.-) 
 
 31498 
 
 3.17481 
 
 33427 
 
 2.9915H 
 
 35379 
 
 2.82453 
 
 31 
 
 30 
 
 29,521 
 
 3.37594 
 
 31533 
 
 3.17159 
 
 33460 
 
 2.98868 
 
 35412 
 
 2.82391 
 
 30 
 
 31 
 
 in. 
 
 3.37234 
 
 31562 
 
 3.ir>838 
 
 33492 
 
 2.98580 
 
 35445 
 
 2. 82 139 
 
 23 
 
 3-2 
 
 21 X, 
 
 3.315875 
 
 31594 
 
 3.1()-)I7 
 
 33521 
 
 2.9-29J 
 
 35477 
 
 2.81870 
 
 28 
 
 33 
 
 29716 
 
 3.3(5510 
 
 31(52(5 
 
 3.1(5197 
 
 33557 
 
 2.98004 
 
 35510 
 
 2.81610 
 
 27 
 
 34 
 
 23748 
 
 3.311.-,- 
 
 31(558 
 
 3.13877 
 
 335-i 9 
 
 2.97717 
 
 35543 
 
 2.81350 
 
 26 
 
 35 
 
 29780 
 
 3.3.V30 
 
 31690 
 
 3.] 5.-,58 
 
 3:M21 
 
 2.9743d 
 
 35576 
 
 2.81091 
 
 25 
 
 30 
 
 23811 
 
 3.3.1443 
 
 32722 
 
 3.15210 
 
 33654 
 
 2.97144 
 
 35608 
 
 2.86633 
 
 24 
 
 37 
 
 29- H 
 
 3.:r,iH7 
 
 31754 
 
 3.14.122 
 
 33 i- i 
 
 2. 96858 
 
 35641 
 
 2.80574 
 
 23 
 
 33 
 
 29-75 
 
 3.3473-2 
 
 3178(5 
 
 :i. ii ;:).', 
 
 33718 
 
 2.96573 
 
 351574 
 
 2.80316 
 
 22 
 
 39 
 
 29906 
 
 3.34371 
 
 31818 
 
 3.14288 
 
 33751 
 
 2.9:5288 
 
 35797 
 
 2.80059 
 
 21 
 
 40 
 
 29338 
 
 3.31:1-2:! 
 
 bl850 
 
 3.1397-2 
 
 33783 
 
 2. OlOOl 
 
 35740 
 
 2.79-*02 
 
 20 
 
 41 
 
 29970 
 
 3. 3 Hi 70 
 
 31882 
 
 3.13356 
 
 33816 
 
 2.9572) 
 
 35772 
 
 2.79545 
 
 19 
 
 4-2 
 
 300") 1 
 
 3.33317 
 
 31914 
 
 3. 133 II 
 
 33848 
 
 2.95437 
 
 35805 
 
 2.79289 
 
 18 
 
 4:{ 
 
 :',o m 
 
 3.:i-2i)i:, 
 
 31946 
 
 3. 13)27 
 
 33831 
 
 -2.9515.-, 
 
 
 2.79 133 
 
 17 
 
 44 
 
 311 M15 
 
 3.32(514 
 
 31978 
 
 3.1-2713 
 
 33913 
 
 2.94872 
 
 35871 
 
 2.78778 
 
 16 
 
 45 
 
 30097 
 
 3.3-22I54 
 
 32010 
 
 3.12400 
 
 33945 
 
 2.94593 
 
 35934 
 
 2.78523 
 
 15 
 
 4(5 
 
 39128 
 
 3.31914 
 
 32042 
 
 3.12037 
 
 33978 
 
 2.94309 
 
 35937 
 
 2.78269 
 
 14 
 
 47 
 
 3') l!i() 
 
 3.31585 
 
 32074 
 
 3.11775 
 
 3! 110 
 
 2.94)28 
 
 35999 
 
 2.78014 
 
 13 
 
 48 
 
 3019-2 
 
 3.31-21(5 
 
 3-210(5 
 
 3.114'14 
 
 34 143 
 
 2.93748 
 
 3(1092 
 
 2.777(51 
 
 12 
 
 49 
 
 30224 
 
 3.30868 
 
 32139 
 
 3.11153 
 
 34075 
 
 2.93468 
 
 36035 
 
 2.77507 
 
 11 
 
 50 
 
 30255 
 
 3.30521 
 
 32171 
 
 3.10812 
 
 34108 
 
 2.93UM 
 
 3150(58 
 
 -2.7:2:,! 
 
 10 
 
 51 
 
 30287 
 
 3.30174 
 
 322.13 
 
 3.10532 
 
 34140 
 
 2. 929 In 
 
 3(1101 
 
 2.77002 
 
 9 
 
 5-2 
 
 30319 
 
 3.29829 
 
 32235 
 
 3.19223 
 
 34173 
 
 2.92:132 
 
 3(5134 
 
 2.7(1759 
 
 8 
 
 53 
 
 30351 
 
 3.29483 
 
 322.17 
 
 3.09914 
 
 34205 
 
 2.92351 
 
 36167 
 
 2.7IM9S 
 
 7 
 
 54 
 
 3 i3~'2 
 
 3.29139 
 
 3i299 
 
 '{ 09:50(5 
 
 34238 
 
 2.92076 
 
 33199 
 
 2.76247 
 
 6 
 
 55 
 
 30414 
 
 3.28795 
 
 32331 
 
 3.092.1-' 
 
 34270 
 
 2.91799 
 
 3623-2 
 
 2.75996 
 
 5 
 
 56 
 
 3.1441 
 
 3.-2H1.Y.' 
 
 323i53 
 
 3.08:191 
 
 3 1303 
 
 2. '91 523 
 
 382H5 
 
 2.7574(5 
 
 4 
 
 57 
 
 30478 
 
 3.38109 
 
 33396 
 
 
 34335 
 
 2.91-2II1 
 
 30298 
 
 2. 75 1 9(5 
 
 3 
 
 58 
 
 3350.) 
 
 3,27767 
 
 
 3.08379, 
 
 34368 
 
 2.9.1971 
 
 3033J 
 
 2.7521:5 
 
 2 
 
 59 
 
 30341 
 
 3.274-2!) 
 
 32450 
 
 3.08073 
 
 3 MOO 
 
 2.')f.9' 
 
 3(13 '4 
 
 2.74997 
 
 1 
 
 (ii) 
 
 80573 
 
 3.27.K-, 
 
 32492 
 
 3.07768 
 
 34133 
 
 2.90421 
 
 36397 2.74748 
 
 
 
 if 
 
 N. Cot. 
 
 N. T;i 
 
 
 N. T.'i. i 
 
 X. Cot. 
 
 N.Tan. 
 
 N. Cot] N. Tan. 
 
 ~M~ 
 
 
 73 Degrees. 
 
 7'3 Degrees, i 
 
 71 Degrees. 
 
 70 Degrees. 
 
 
78 
 
 NATURAL TANGENfS. 
 
 
 20 Degrees. 
 
 21 Degrees. 
 
 22 Degrees. 23 Degrees. 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 N.Tan 
 
 N. Cot. 
 
 N.Tan 
 
 N.Cot. 
 
 N.Tan.] N.Cot. j M 
 
 ~0 
 
 36397 
 
 2.74748 
 
 38386 
 
 2.60509 
 
 40403 
 
 2.475U9 
 
 42447 
 
 2.35585 60 
 
 1 
 
 3(5430 
 
 2.744U9 
 
 38420 
 
 2.60283 
 
 40436 
 
 2.47302 
 
 4248-2 
 
 2.35395 59 
 
 2 
 
 36463 
 
 2.74251 
 
 38453 
 
 2.60057 
 
 40470 
 
 2.47095 
 
 4-2516 
 
 2.35-205 
 
 58 
 
 3 
 
 36496 
 
 2.74004 
 
 38487 
 
 2.59831 
 
 40504 
 
 2.46888 
 
 42551 
 
 2.35015 
 
 57 
 
 4 
 
 36529 
 
 2.73756 
 
 385-20 
 
 2.59600 
 
 40538 
 
 2.46682 
 
 42585 
 
 2.34825 
 
 5(5 
 
 5 
 
 36562 
 
 2.73509 
 
 38553 
 
 2.59381 
 
 40572 
 
 2.46476 
 
 42619 
 
 2.3463b 
 
 55 
 
 6 
 
 36595 
 
 2.73263 
 
 38587 
 
 2.59156 
 
 40606 
 
 2.46270 
 
 42054 
 
 2.34447 
 
 54 
 
 7 
 
 361.28 
 
 2.73017 
 
 38620 
 
 2.58932 
 
 40640 
 
 2.46085 
 
 42688 
 
 2.34258 
 
 53 
 
 8 
 
 36fi61 
 
 2.72771 
 
 38654 
 
 2.58708 
 
 40074 
 
 2.45860 
 
 42722 
 
 2.34069 
 
 52 
 
 9 
 
 36694 
 
 2.72526 
 
 38687 
 
 2.58484 
 
 40707 
 
 2.45655 
 
 42757 
 
 2.33881 
 
 51 
 
 10 
 
 36727 
 
 2.72-281 
 
 38721 
 
 2.58261 
 
 40741 
 
 2.45451 
 
 42791 
 
 2.33693 
 
 50 
 
 11 
 
 36760 
 
 2.72036 
 
 38754 
 
 2.58038 
 
 40775 
 
 2.45246 
 
 42826 
 
 2.33505 
 
 49 
 
 12 
 
 36793 
 
 2.71792 
 
 38787 
 
 2. 57815 
 
 40809 
 
 2.45043 
 
 42860 
 
 2.33317 
 
 48 
 
 13 
 
 36826 
 
 2.71548 
 
 38821 
 
 2.57593 
 
 40843 
 
 2.44839 
 
 42894 
 
 2.33130 
 
 47 
 
 14 
 
 36859 
 
 2.71305 
 
 38854 
 
 2.57371 
 
 40877 
 
 2.44636 
 
 42929 
 
 2.32943 
 
 46 
 
 15 
 
 36892 
 
 2.71062 
 
 38888 
 
 2.57150 
 
 40911 
 
 2.44433 
 
 42963 
 
 2.32756 
 
 45 
 
 16 
 
 36925 
 
 2.70819 
 
 38921 
 
 2.56928 
 
 40945 
 
 2.44230 
 
 42998 
 
 2.32570 
 
 44 
 
 17 
 
 36958 
 
 2.70577 
 
 38955 
 
 2.56707 
 
 40979 
 
 2.440-27 
 
 43032 
 
 2.32383 
 
 43 
 
 18 
 
 3699J 
 
 2.70335 
 
 38988 
 
 2.56487 
 
 41013 
 
 2.43825 
 
 43067 
 
 2.32197 
 
 42 
 
 19 
 
 37024 
 
 2.70094 
 
 39022 
 
 2.56266 
 
 41047 
 
 2.43623 
 
 43101 
 
 2.32012 
 
 41 
 
 20 
 
 37057 
 
 2.69853 
 
 39055 
 
 2.56046 
 
 41081 
 
 2.43422 
 
 43136 
 
 2.31826 
 
 40 
 
 21 
 
 37090 
 
 2.69612 
 
 39089 
 
 2.55827 
 
 41115 
 
 2.43220 
 
 43170 
 
 2.31641 
 
 39 
 
 22 
 
 37123 
 
 2.69371 
 
 39122 
 
 2.55608 
 
 41149 
 
 2.43019 
 
 43205 
 
 2.31456 
 
 38 
 
 23 
 
 37157 
 
 2.69131 
 
 39156 
 
 2.55389 
 
 41183 
 
 2.42819 
 
 43239 
 
 2.31271 
 
 37 
 
 24 
 
 37190 
 
 2.68892 
 
 39190 
 
 2.55170 
 
 41217 
 
 2.42618 
 
 43274 
 
 2.31080 
 
 36 
 
 25 
 
 37223 
 
 2.68653 
 
 39223 
 
 2.54952 
 
 41251 
 
 2.42418 
 
 43308 
 
 2.30902 
 
 35 
 
 26 
 
 37256 
 
 2.68414 
 
 39257 
 
 2.54734 
 
 41285 
 
 2.42218 
 
 43343 
 
 2.30718 
 
 34 
 
 27 
 
 37289 
 
 2.68175 
 
 39290 
 
 2.54516 
 
 41319 
 
 2.42019 
 
 43378 
 
 2.30534 
 
 33 
 
 28 
 
 37322 
 
 2.67937 
 
 39324 
 
 2.54299 
 
 41353 
 
 2.41819 
 
 43412 
 
 2.30H5J 
 
 32 
 
 29 
 
 37355 
 
 2.67700 
 
 39357 
 
 2.54082 
 
 41387 
 
 2.41620 
 
 43447 
 
 2.30167 
 
 31 
 
 30 . 
 
 37388 
 
 2.67462 
 
 39391 
 
 2.53865 
 
 41421 
 
 2.41421 
 
 43481 
 
 2.29984 
 
 30 
 
 31 
 
 37422 
 
 2.67225 
 
 39425 
 
 2.53t>48 
 
 41455 
 
 2.41223 
 
 43516 
 
 2.29801 
 
 29 
 
 32 
 
 37455 
 
 2.66989 
 
 39458 
 
 2.53432 
 
 41490 
 
 2.41025 
 
 43550 
 
 2.29619 
 
 28 
 
 33 
 
 37488 
 
 2.66752 
 
 39492 
 
 2.53217 
 
 41524 
 
 2.40827 
 
 43585 
 
 2.29437 
 
 27 
 
 34 
 
 37521 
 
 2.66516 
 
 39526 
 
 2.53001 
 
 41558 
 
 2.40629 
 
 43620 
 
 2.29254 
 
 26 
 
 35 
 
 37554 
 
 2.66281 
 
 39559 
 
 2.52786 
 
 41592 
 
 2.40432 
 
 43(554 
 
 2.29073 
 
 25 
 
 36 
 
 37588 
 
 2.66046 
 
 39593 
 
 2.52571 
 
 41626 
 
 2.40235 
 
 43689 
 
 2.28891 
 
 24 
 
 37 
 
 37621 
 
 2.65811 
 
 39626 
 
 2.52357 
 
 41660 
 
 2.40038 
 
 43724 
 
 2.28710 
 
 23 
 
 38 
 
 37654 
 
 2.C5576 
 
 39660 
 
 2.52142 
 
 41694 
 
 2.39841 
 
 43758 
 
 2.285-28 
 
 22 
 
 39 
 
 37687 
 
 2.65342 
 
 39694 
 
 2.51929 
 
 41728 
 
 2.39045 
 
 43793 
 
 2.28348 
 
 21 
 
 40 
 
 37720 
 
 2.65109 
 
 39727 
 
 2.51715 
 
 41703 
 
 2.39449 
 
 43828 
 
 2.28167 
 
 20 
 
 41 
 
 37754 
 
 2.64875 
 
 39761 
 
 2. 51502 
 
 41797 
 
 2.39253 
 
 438(52 
 
 2.27987 
 
 19 
 
 42 
 
 37787 
 
 2.64642 
 
 39795 
 
 2.51289 
 
 41831 
 
 2.39058 
 
 43897 
 
 2-27806 
 
 18 
 
 43 
 
 37820 
 
 2.61410 
 
 39829 
 
 2.51076 
 
 41865 
 
 2.38862 
 
 43932 
 
 2.27026 
 
 17 
 
 44 
 
 37853 
 
 2.64177 
 
 39862 
 
 2.50864 
 
 41899 
 
 2.38668 
 
 43966 
 
 2.27447 
 
 16 
 
 45 
 
 37887 
 
 2.63945 
 
 30896 
 
 2.50652 
 
 41933 
 
 2.38473 
 
 44001 
 
 2.27267 
 
 15 
 
 4f> 
 
 37920 
 
 2.63714 
 
 39930 
 
 2.50440 
 
 41968 
 
 2.38279 
 
 44036 
 
 2.27088 
 
 14 
 
 47 
 
 37953 
 
 2.63483 
 
 39963 
 
 2.50229 
 
 42002 
 
 2.38084 
 
 44071 
 
 2.20909 
 
 13 
 
 48 
 
 37986 
 
 2.63252 
 
 39997 
 
 2.50018 
 
 42036 
 
 2.37891 
 
 44105 
 
 2.26730 
 
 12 
 
 49 
 
 38020 
 
 2.63021 
 
 40031 
 
 2.49807 
 
 42070 
 
 2.37697 
 
 44140 
 
 2.26552 
 
 11 
 
 50 
 
 38053 
 
 2.62791 
 
 40065 
 
 2.49597 
 
 42105 
 
 2.37504 
 
 44175 
 
 2.26374 
 
 10 
 
 51 
 
 38086 
 
 2.62561 
 
 40098 
 
 2.49386 
 
 42139 
 
 2.37311 
 
 44210 
 
 2.26196 
 
 9 
 
 52 
 
 38120 
 
 2.62332 
 
 40132 
 
 2.49177 
 
 42173 
 
 2.37118 
 
 44244 
 
 2.20018 
 
 8 
 
 53 
 
 38153 
 
 2.62103 
 
 40166 
 
 2.48967 
 
 42207 
 
 2.36925 
 
 44279 
 
 2.25840 
 
 7 
 
 54 
 
 38186 
 
 2.61874 
 
 40200 
 
 2.48758 
 
 4-2242 
 
 2.36733 
 
 44314 
 
 2.25663 
 
 6 
 
 55 
 
 38220 
 
 2.61646 
 
 40234 
 
 2.48549 
 
 42276 
 
 2.36541 
 
 44349 
 
 2.25486 
 
 5 
 
 56 
 
 38253 
 
 2.61418 
 
 40267 
 
 2.48340 
 
 42310 
 
 2.36349 
 
 44384 
 
 2.25309 
 
 4 
 
 57 
 
 38-286 
 
 2.61190 
 
 40301 
 
 2.4813-2 
 
 42345 
 
 2.36158 
 
 44418 
 
 2.25132 
 
 3 
 
 58 
 
 38320 
 
 2. 609(53 
 
 40335 
 
 2.47924 
 
 42379 
 
 2.359(57 
 
 44453 
 
 2.24956 
 
 2 
 
 59 
 
 38353 
 
 2.60736 
 
 40369 
 
 2.47716 
 
 42413 
 
 2.35776 
 
 44488 
 
 2.24780 
 
 1 
 
 60 
 
 38386 
 
 2.60509 
 
 10403 
 
 2.47509! 
 
 42447 
 
 2.35585 
 
 44523 
 
 2.24604 
 
 
 
 M 
 
 N Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 M 
 
 
 69 Degrees. 
 
 68 Degrees. 
 
 67 Degrees. 
 
 66 Degrees. 
 
 
NATURAL TANGENTS. 
 
 79 
 
 
 24 Degrees. 
 
 25 Degrees. 
 
 26 Degrees. 
 
 27 Degrees. 
 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N T;.n. 
 
 L\. Cllt. 
 
 N. Tan 
 
 N. Cot. 
 
 M 
 
 
 
 44523 
 
 2.24lU)4 
 
 48631 
 
 2.14451 
 
 48773 
 
 2.05030 
 
 50953 
 
 .'.H5261 
 
 (50 
 
 1 
 
 44596 
 
 2.244'28 
 
 46666 
 
 2. I428H 
 
 48809 
 
 2.04879 
 
 50989 
 
 .'.Mi 1-20 
 
 59 
 
 
 44593 
 
 2.24252 
 
 46702 
 
 2.141-25 
 
 48845 
 
 2.04728 
 
 51036 
 
 .95979 
 
 58 
 
 3 
 
 44687 
 
 2. -2H>77 
 
 4(i737 
 
 2.139ii3 
 
 48881 
 
 2.04577 
 
 51063 
 
 .95838 
 
 57 
 
 4 
 
 44662 
 
 2. 23902 
 
 46772 
 
 2. 13801 
 
 48917 
 
 2.041-2ii 
 
 51H99 
 
 .951)98 
 
 56 
 
 5 
 
 44(597 
 
 2 . -23727 
 
 46608 
 
 2.13(539 
 
 48953 
 
 -2.04276 
 
 51136 
 
 .95557 
 
 55 
 
 6 
 
 44739 
 
 2.23553 
 
 4(5843 
 
 2.13477 
 
 48989 
 
 2.04125 
 
 51173 
 
 .95417 
 
 54 
 
 7 
 
 44767 
 
 2.2337H 
 
 46879 
 
 a. 133 1 6 
 
 49096 
 
 2.03975 
 
 51909 
 
 .95277 
 
 53 
 
 8 
 
 4480-2 
 
 2. -23204 
 
 46914 
 
 2.13154 
 
 49062 
 
 2.0:1825 
 
 51246 
 
 .95137 
 
 52 
 
 9 
 
 44837 
 
 a. 23030 
 
 46950 
 
 2. |-2993 
 
 49068 
 
 2.03675 
 
 51283 
 
 .94!I97 
 
 51 
 
 10 
 
 44872 
 
 2.-2-2rtr>7 
 
 46985 
 
 2.12832 
 
 49134 
 
 2.03526 
 
 51319 
 
 .94858 
 
 50 
 
 11 
 
 44907 
 
 2. -2-26*3 
 
 47021 
 
 2.12671 
 
 49170 
 
 2.03376 
 
 51356 
 
 .94718 
 
 49 
 
 19 
 
 44942 
 
 2.22510 
 
 47056 
 
 2.12511 
 
 49906 
 
 2.03227 
 
 51393 
 
 -94579 
 
 48 
 
 13 
 
 44!I77 
 
 -2.-22.t37 
 
 47092 
 
 2.J2350 
 
 49242 
 
 2.03078 
 
 51430 
 
 .94440 
 
 47 
 
 14 
 
 45012 
 
 2.22164 
 
 47128 
 
 2.12190 
 
 49278 
 
 2.02929 
 
 51467 
 
 .94301 
 
 46 
 
 15 
 
 45047 
 
 2.21992 
 
 47163 
 
 2.12030 
 
 49315 
 
 2.02780 
 
 51503 
 
 .94162 
 
 45 
 
 Ifi 
 
 45082 
 
 2.21819 
 
 47199 
 
 2.11871 
 
 49351 
 
 2.02631 
 
 51540 
 
 .94023 
 
 44 
 
 17 
 
 45117 
 
 2.21647 
 
 47234 
 
 2.11711 
 
 49387 
 
 2.02483 
 
 51577 
 
 .93885 
 
 43 
 
 18 
 
 45152 
 
 2.21475 
 
 47270 
 
 2.11552 
 
 49423 
 
 2.02335 
 
 51614 
 
 .93746 
 
 42 
 
 19 
 
 45187 
 
 2.21304 
 
 47305 
 
 2.11392 
 
 49459 
 
 2.02187 
 
 51651 
 
 .93608 
 
 41 
 
 20 
 
 45222 
 
 2.21132 
 
 47341 
 
 2.11233 
 
 49495 
 
 2.02039 
 
 51688 
 
 .93470 
 
 40 
 
 21 
 
 45257 
 
 2.209(51 
 
 47377 
 
 2.11075 
 
 49532 
 
 2.01891 
 
 51724 
 
 .93332 
 
 39 
 
 22 
 
 45-292 
 
 2.20790 
 
 47412 
 
 2.10916 
 
 495(58 
 
 2.01743 
 
 51761 
 
 .93197 
 
 38 
 
 33 
 
 453-27 
 
 2.20619 
 
 47448 
 
 2.10758 
 
 496(14 
 
 2.01596 
 
 51798 
 
 .93057 
 
 37 
 
 24 
 
 45362 
 
 2.20449 
 
 47483 
 
 2.10(500 
 
 49640 
 
 -2.01449 
 
 51835 
 
 . 92920 
 
 36 
 
 25 
 
 45397 
 
 2. -20-278 
 
 47519 
 
 -2.10441 
 
 49677 
 
 2.01302 
 
 51872 
 
 .92782 
 
 35 
 
 26 
 
 45432 
 
 2.20108 
 
 47555 
 
 2.10284 
 
 49713 
 
 2.01155 
 
 51909 
 
 .92645 
 
 34 
 
 27 
 
 45467 
 
 2. Hi: 138 
 
 47590 
 
 2.10126 
 
 49749 
 
 2.01008 
 
 51946 
 
 .92508 
 
 33 
 
 28 
 
 45502 
 
 2.19769 
 
 47(526 
 
 2.09969 
 
 49786 
 
 2.00862 
 
 51983 
 
 .92371 
 
 32 
 
 99 
 
 45537 
 
 2. 19599 
 
 47(562 
 
 2.09811 
 
 49822 
 
 2.00715 
 
 52020 
 
 .92235 
 
 31 
 
 30 
 
 45573 
 
 2.19430 
 
 47698 
 
 2.09654 
 
 49856 
 
 2.00569 
 
 52057 
 
 .92098 
 
 30 
 
 31 
 
 45608 
 
 2.19261 
 
 47733 
 
 2.09498 
 
 49894 
 
 2.00423 
 
 52094 
 
 .91962 
 
 29 
 
 3-2 
 
 45643 
 
 2.191192 
 
 47769 
 
 2.09341 
 
 49931 
 
 2.00277 
 
 52131 
 
 .91*25 
 
 28 
 
 33 
 
 45678 
 
 2.18923 
 
 47895 
 
 2.09184 
 
 49967 
 
 2.00131 
 
 - 52168 
 
 .91690 
 
 27 
 
 34 
 
 45713 
 
 2.18755 
 
 47840 
 
 2.09028 
 
 50004 
 
 1.99986 
 
 52205 
 
 .91554 
 
 26 
 
 35 
 
 45748 
 
 2.18587 
 
 47876 
 
 2.08-!72 
 
 50040 
 
 l-'.i'.H41 
 
 59242 
 
 .91418 
 
 25 
 
 3!5 
 
 45784 
 
 2.18419 
 
 47912 
 
 2.0S7I6 
 
 58076 
 
 1.9! 16: (5 
 
 52378 
 
 .91282 
 
 !24 
 
 37 
 
 45819 
 
 2. lr-251 
 
 47948 
 
 2.08560 
 
 50113 
 
 I. < >9550 
 
 59316 
 
 91148 
 
 23 
 
 38 
 
 45854 
 
 2.18084 
 
 47984 
 
 2.08405 
 
 50149 
 
 1. '.19406 
 
 5-2353 
 
 .<)10I7 
 
 22 
 
 39 
 
 45889 
 
 2.17916 
 
 48019 
 
 2.08250 
 
 50185 
 
 1.99261 
 
 52390 
 
 -90876 
 
 21 
 
 40 
 
 45924 
 
 2. 17749 
 
 181)55 
 
 2.<H!)'.M 
 
 50222 
 
 1.99116 
 
 52427 
 
 .90741 
 
 20 
 
 41 
 
 459!50 
 
 3. 17582 
 
 48091 
 
 2.07939 
 
 50258 
 
 1.98972 
 
 52464 
 
 .90607 
 
 19 
 
 42 
 
 45905 
 
 2.17410 
 
 48127 
 
 2.07785 
 
 50295 
 
 1-98828 
 
 52501 
 
 .90472 
 
 18 
 
 43 
 
 46030 
 
 2.17249 
 
 48163 
 
 2.07630 
 
 50331 
 
 1.98684 
 
 52538 
 
 .9)337 
 
 17 
 
 44 
 
 46065 
 
 2.17083 
 
 48198 
 
 2.07476 
 
 503(58 
 
 1.98540 
 
 52575 
 
 .90203 
 
 16 
 
 45 
 
 46101 
 
 2.16917 
 
 48234 
 
 2.07321 
 
 50404 
 
 1.98396 
 
 52613 
 
 .90069 
 
 15 
 
 4(5 
 
 46136 
 
 2.16751 
 
 48270 
 
 2.07167 
 
 50441 
 
 1.98253 
 
 52650 
 
 .89935 
 
 14 
 
 47 
 
 4C.171 
 
 2. 16585 
 
 4830(5 
 
 2.07014 
 
 50477 
 
 1.98110 
 
 5-26S7 
 
 .89801 
 
 13 
 
 48 
 
 46206 
 
 2.16420 
 
 48342 
 
 2. 0^)860 
 
 50514 
 
 1.97'.)6i> 
 
 52724 
 
 .89667 
 
 12 
 
 49 
 
 46242 
 
 2.1(1255 
 
 48378 
 
 2.0(5706 
 
 50550 
 
 1.978-2:1 
 
 52761 
 
 .89533 
 
 11 
 
 50 
 
 40877 
 
 2.16090 
 
 48414 
 
 2.06553 
 
 50587 
 
 1.97680 
 
 5*2798 
 
 .89400 
 
 10 
 
 51 
 
 463U 
 
 2. 15925 
 
 48450 
 
 2.0')400 
 
 5015-23 
 
 1.97538 
 
 52836 
 
 .89-266 
 
 9 
 
 52 
 
 415348 
 
 2. 15760 
 
 48486 
 
 2.06247 
 
 50660 
 
 1.97395 
 
 52873 
 
 .89133 
 
 8 
 
 53 
 
 46383 
 
 2. I559li 
 
 4H.V21 
 
 2.0H094 
 
 50596 
 
 1. 97253 
 
 52910 
 
 .89'IO!> 
 
 7 
 
 54 
 
 46418 
 
 2.15J32 
 
 48557 
 
 2.0591-2 
 
 50733 
 
 1.9711] 
 
 52947 
 
 .88867 
 
 6 
 
 55 
 
 46454 
 
 2.152(58 
 
 48593 
 
 2.05789 
 
 507(59 
 
 1-969(59 
 
 52984 
 
 .88734 
 
 5 
 
 5(5 
 
 46489 
 
 2.15104 
 
 48629 
 
 2.05637 
 
 508*1 
 
 1.96827 
 
 53024 
 
 .88602 
 
 4 
 
 57 
 
 46525 
 
 2.14940 
 
 48(565 
 
 2.05485 
 
 50843 
 
 1.96685 
 
 53059 
 
 .88469 
 
 3 
 
 58 
 
 46540 
 
 2.14777 
 
 48701 
 
 L>.05.->33 
 
 50879 
 
 1.96544 
 
 53096 
 
 .88337 
 
 2 
 
 59 
 
 41 J595 
 
 2.14614 
 
 48737 
 
 2.0518-2 
 
 509115 
 
 1.9(i4D-2 
 
 53134 
 
 .88205 
 
 1 
 
 60 
 
 46631 
 
 2.14451 
 
 48773 
 
 2.05030 
 
 50953 
 
 1.96261 
 
 53171 
 
 .88073 
 
 
 
 M 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. IN. Tan. 
 
 N. Cot. N. Tan. 
 
 M 
 
 
 65 Degrees. 
 
 64 Degrees. 
 
 63 Degrees. 
 
 62 Degrees. 
 
 
80 
 
 NATURAL TANGENTS. 
 
 
 28 Degrees. 
 
 29 Degrees. 
 
 30 Degrees. 31 Degrees. 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 N.Tan 
 
 IS'. Cot. 
 
 N.Tai. j N.Cot. N.Tan 
 
 N. Cot. M i 
 
 
 
 53171 
 
 .88073 
 
 55431 
 
 1.80405 
 
 57735 
 
 .73205 60086 
 
 1.66428 60 
 
 1 
 
 53208 
 
 .87941 
 
 55469 
 
 .80281 
 
 57774 
 
 .73089 60120 
 
 1. 00318 59 
 
 2 
 
 53240 
 
 .87809 
 
 55507 
 
 .80158 
 
 57813 
 
 .72973 G0165 
 
 1.06-201) 
 
 58 
 
 3 
 
 53-283 
 
 .87677 
 
 55545 
 
 .80034 
 
 57851 
 
 .7-2f'57 OU-2U5 
 
 1.60099 
 
 57 
 
 4 
 
 53320 
 
 .87546 
 
 55583 
 
 .79U11 
 
 57890 
 
 .72741 r,f245 
 
 1.65990 
 
 56 
 
 5 
 
 53358 
 
 .87415 
 
 55621 
 
 .79788 
 
 57929 
 
 .7-2ii-,T) : 60284 
 
 1.65881 
 
 55 
 
 6 
 
 53395 
 
 .87283 
 
 55659 
 
 .79665 
 
 57-J68 
 
 .7-2509 1 103-24 
 
 1.65772 
 
 54 
 
 7 
 
 53432 
 
 .87152 
 
 55697 
 
 .79542 
 
 58007 
 
 .72393 60304 
 
 1.65663 
 
 53 
 
 8 
 
 53470 
 
 .87021 
 
 55736 
 
 .79419 
 
 58046 
 
 .72278 00403 
 
 1.65554 
 
 52 
 
 9 
 
 53507 
 
 .86891 
 
 55774 
 
 .79296 
 
 58085 
 
 .7-2103 00443 
 
 1.65443 
 
 51 
 
 10 
 
 53545 
 
 .86760 
 
 55812 
 
 .79174 
 
 58124 
 
 .72047 i 60483 
 
 1.65337 
 
 50 
 
 11 
 
 5358-2 
 
 .86630 
 
 55850 
 
 .79051 
 
 58162 
 
 .71932 60522 
 
 1.65986 
 
 49 
 
 12 
 
 53620 
 
 .86499 
 
 55888 
 
 .78929 
 
 58-201 
 
 .71817 60562 
 
 1-05120 
 
 48 
 
 13 
 
 53657 
 
 .86369 
 
 5592(5 
 
 .78807 
 
 58240 
 
 .71702 6(1002 
 
 1.05011 
 
 47 
 
 14 
 
 53694 
 
 .86239 
 
 56964 
 
 .78685 
 
 58279 
 
 .71588 
 
 60642 
 
 1.04903 
 
 46 
 
 15 
 
 53732 
 
 .86109 
 
 56003 
 
 .78563 
 
 58318 
 
 .71473 
 
 60681 
 
 1.64785 
 
 45 
 
 16 
 
 53769 
 
 .85979 
 
 56041 
 
 .78441 
 
 58357 
 
 .71358 
 
 60721 
 
 1.64687 
 
 44 
 
 17 
 
 53807 
 
 .85850 
 
 56079 
 
 .78319 
 
 58396 
 
 .71244 
 
 60761 
 
 1.64579 
 
 43 
 
 18 
 
 53844 
 
 .85720 
 
 56117 
 
 .78198 
 
 58435 
 
 .71129 OOSlll 
 
 1.64471 
 
 42 
 
 19 
 
 53882 
 
 .85591 
 
 56156 
 
 .78077 
 
 58474 
 
 .71015 
 
 60841 
 
 1.04363 
 
 41 
 
 20 
 
 53920 
 
 .85402 
 
 56194 
 
 .77955 
 
 58513 
 
 .70901 
 
 60881 
 
 1.64256 
 
 40 
 
 21 
 
 53957 
 
 .85333 
 
 56232 
 
 .77834 
 
 58552 
 
 .70787 
 
 60921 
 
 1.64148 
 
 39 
 
 22 
 
 53995 
 
 .85204 
 
 56270 
 
 .77713 
 
 58591 
 
 .70673 
 
 60960 
 
 1.64041 
 
 38 
 
 23 
 
 54032 
 
 .85075 
 
 56309 
 
 .77592 
 
 58631 
 
 .70560 j 61000 
 
 1.03933 
 
 37 
 
 24 
 
 54070 
 
 .84946 
 
 56347 
 
 .77471 
 
 58670 
 
 .70446 | 61040 
 
 1.6382(5 
 
 36 
 
 25 
 
 54107 
 
 .84818 
 
 56385 
 
 .77351 
 
 58709 
 
 .70332 
 
 61080 
 
 1.63719 
 
 35 
 
 26 
 
 54145 
 
 .84689 
 
 564-24 
 
 .77230 
 
 58748 
 
 .70219 
 
 61120 
 
 1.63612 
 
 34 
 
 27 
 
 54183 
 
 84561 
 
 56462 
 
 .77110 
 
 58787 
 
 .70106 
 
 61100 
 
 1.63505 
 
 33 
 
 28 
 
 54220 
 
 .84433 
 
 56500 
 
 .76990 
 
 58826 
 
 .69992 
 
 61200 
 
 1.03398 
 
 32 
 
 29 
 
 54258 
 
 .84305 
 
 56539 
 
 .76869 
 
 58865 
 
 .69879 
 
 61240 
 
 1.63292 
 
 31 
 
 30 
 
 54296 
 
 .84177 
 
 56577 
 
 .7674! 
 
 58904 
 
 .69766 
 
 01 .'80 1.63185 
 
 30 
 
 31 
 
 54333 
 
 .84049 
 
 56616 
 
 .76629 
 
 58944 
 
 .69653 
 
 61320 
 
 1.63079 
 
 29 
 
 3-2 
 
 54371 
 
 .839-22 
 
 56654 
 
 .76510 
 
 58983 
 
 .69541 
 
 61360 
 
 1.03972 
 
 28 
 
 33 
 
 54460 
 
 .83794 
 
 56693 
 
 .703SIO 
 
 59022 
 
 .69428 
 
 61400 
 
 1.62866 
 
 27 
 
 34 
 
 5444!! 
 
 .83667 
 
 56731 
 
 .76271 
 
 59061 
 
 .6i315 
 
 61440 
 
 1.62700 
 
 20 
 
 ! 3.5 
 
 54484 
 
 .83540 
 
 56769 
 
 -7615] 
 
 59101 
 
 .69203 
 
 61480 
 
 1 62054 
 
 25 
 
 36 
 
 54522 
 
 .83413 
 
 56808 
 
 .76032 
 
 59140 
 
 .69091 
 
 61520 
 
 1.62548 
 
 24 
 
 37 
 
 545fiO 
 
 .83286 
 
 56846 
 
 .75913 
 
 59179 
 
 .68979 
 
 61561 
 
 1.62442 
 
 23 
 
 38 
 
 54597 
 
 .83159 
 
 56885 
 
 -75794 
 
 59218 
 
 .68866 
 
 61601 
 
 1.62336 
 
 22 
 
 39 
 
 54635 
 
 .83033 
 
 56923 
 
 .75675 
 
 59258 
 
 .68754 
 
 61641 
 
 1.62230 
 
 21 
 
 40 
 
 54673 
 
 .82906 
 
 57982 
 
 .75556 
 
 59297 
 
 .68643 
 
 61681 
 
 1.62125 
 
 20 
 
 41 
 
 54711 
 
 .82780 
 
 57000 
 
 .75437 
 
 59336 
 
 .68531 
 
 61721 
 
 1.62019 
 
 19 
 
 42 
 
 51748 
 
 .82654 
 
 57039 
 
 .75319 
 
 59376 
 
 .68419 
 
 S1761 
 
 1.61914 
 
 18 
 
 43 
 
 54786 
 
 .82528 
 
 57078 
 
 .75200 
 
 59415 
 
 .68308 
 
 61801 
 
 1.61808 
 
 17 
 
 44 
 
 54824 
 
 .82402 
 
 57116 
 
 .75082 
 
 59454 
 
 .68196 
 
 61842 
 
 1.61703 
 
 16 
 
 45 
 
 54862 
 
 .82276 
 
 57155 
 
 .74964 
 
 59494 
 
 .68085 
 
 61882 
 
 1.61598 
 
 15 
 
 46 
 
 54900 
 
 .82150 
 
 57193 
 
 .74846 
 
 59533 
 
 .07974 
 
 61922 
 
 1.61493 
 
 14 
 
 47 
 
 54933 
 
 .82025 
 
 57232 
 
 .74728 
 
 59573 
 
 .67968 
 
 61962 
 
 1.61388 
 
 13 
 
 48 
 
 54975 
 
 .81899 
 
 57271 
 
 .74610 
 
 59612 
 
 .67752 
 
 6-2003 
 
 1.61283 
 
 12 
 
 49 
 
 55013 
 
 .81774 
 
 57309 
 
 .74492 
 
 59651 
 
 .67641 
 
 62043 
 
 1.61179 
 
 11 
 
 50 
 
 55051 
 
 .81649 
 
 57348 
 
 .74375 
 
 59691 
 
 .67530 
 
 62083 
 
 1.61074 
 
 10 
 
 51 
 
 55088 
 
 .81524 
 
 57386 
 
 .74257; 
 
 59730 
 
 .67419 
 
 62124 
 
 1.60970 
 
 9 
 
 5'2 
 
 55127 
 
 .81399 
 
 57425 
 
 .74140J 
 
 59770 
 
 .67309 
 
 62164 
 
 1.60865 
 
 8 
 
 53 
 
 55165 
 
 .81274 
 
 57464 
 
 .74022 
 
 59809 
 
 .67198 
 
 62204 
 
 1.60761 
 
 7 
 
 54 
 
 55903 
 
 .81150 
 
 57503 
 
 .739(15 
 
 59849 
 
 67088 
 
 69345 
 
 1.60657 
 
 6 
 
 55 
 
 55241 
 
 .81025 
 
 57541 
 
 .73788 
 
 59888 
 
 66978; 
 
 6-2-285 
 
 1.60553 
 
 5 
 
 56 
 
 55279 
 
 .80901 
 
 57580 
 
 .73671 
 
 59928 
 
 668671 
 
 6-2325 
 
 1.60449 
 
 4 
 
 57 
 
 55317 
 
 .80777 
 
 57619 
 
 .73555 
 
 59967 
 
 66757 
 
 62366 
 
 1.60345 
 
 3 
 
 58 
 
 55355 
 
 .80653 
 
 57657 
 
 .73-138 
 
 60007 
 
 66647 
 
 62406 
 
 1.60241 
 
 2 
 
 59 
 
 55393 
 
 .80529 
 
 57696 
 
 .73321 
 
 60046 
 
 .66538 
 
 62446 
 
 1.60137 
 
 1 
 
 60 
 
 55431 
 
 .80405 
 
 57735 
 
 1.73205 
 
 60086 
 
 .66428 
 
 62487 
 
 1.60033 
 
 M 
 
 N Cot. 
 
 N. Tan 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. N. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 M 
 
 
 61 Degrees. 
 
 GO Degrees. i| 59 Degrees. 
 
 58 Degrees. 
 
 
NATURAL TANGKNT-S. 
 
 
 32 Detrn <. 
 
 33 Degrees. 
 
 34 Degrees. 
 
 35 Degrees. 
 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 M.Taa 
 
 A. Cot. 
 
 .VTan 
 
 N. Cot. 
 
 N.Ta ii 
 
 N. Cot. 
 
 M 
 
 
 
 02487 
 
 1. 60D33 
 
 64941 
 
 i. 53981 
 
 674.51 
 
 1.48351 
 
 70021 
 
 1.42815 
 
 60 
 
 1 
 
 6-2527 
 
 1.5'.I93( 
 
 64983 
 
 1.538-^ 
 
 67493 
 
 1.4816: 
 
 70064 
 
 1.437ft) 
 
 59 
 
 2 
 
 62568 
 
 I..V.H-2! 
 
 65033 
 
 L.5379 
 
 67538 
 
 1.48074 
 
 70107 
 
 1.42638 
 
 58 
 
 :i 
 
 63808 
 
 I.5975E 
 
 65065 
 
 1.53693 
 
 67578 
 
 1.47977 
 
 70151 
 
 1 .48554 
 
 57 
 
 4 
 
 6-2649 
 
 L. 59691 
 
 65106 
 
 1 . 535S: 
 
 67620 
 
 [.4788J 
 
 70194 
 
 1. 434ft 
 
 56 
 
 5 
 
 82689 
 
 1.59517 
 
 <;;>) 48 
 
 1.53497 
 
 67883 
 
 1.47793 
 
 70238 
 
 1.4-2374 
 
 55 
 
 i l! 
 
 62730 
 
 1. r,;)l 14 
 
 65189 
 
 l.5340( 
 
 6770.) 
 
 1 ,47699 
 
 70281 
 
 1. 43381 
 
 54 
 
 7 
 
 (i-2770 
 
 i.f93ii 
 
 65231 
 
 1.5330, 
 
 87748 
 
 1.47607 
 
 70325 
 
 1.42198 
 
 53 
 
 8 
 
 62811 
 
 1.59908 
 
 65273 
 
 1.53205 
 
 67790 
 
 1.47514 
 
 70368 
 
 1.42111 
 
 52 
 
 9 
 
 62852 
 
 L.59I05 
 
 65314 
 
 1.53107 
 
 67832 
 
 1.474 '22 
 
 70112 
 
 1.43038 
 
 51 
 
 10 
 
 62892 
 
 1.59002 
 
 6.53.55 
 
 1.530K 
 
 67875 
 
 1.4733(1 
 
 70455 
 
 1.41934 
 
 50 
 
 11 
 
 62933 
 
 1.58901 
 
 65397 
 
 I,fr2'.li: 
 
 671117 
 
 1.47338 
 
 70 MW 
 
 1.41847 
 
 49 
 
 12 
 
 62973 
 
 1.587W 
 
 65438 
 
 1.538K 
 
 67960 
 
 1.47146 
 
 7054-2 
 
 1-41750 
 
 48 
 
 13 
 
 63014 
 
 I.596KS 
 
 65480 
 
 1.53711 
 
 68009 
 
 1.47053 
 
 70586 
 
 1.41673 
 
 47 
 
 11 
 
 153. )f>3 
 
 1.5859: 
 
 65531 
 
 I.52C.2:, 
 
 68045 
 
 1.46982 
 
 70629 
 
 1.41584 
 
 4G 
 
 15 
 
 63095 
 
 i.r>Hioa 
 
 65563 
 
 I.. 5-2.5-2: 
 
 68088 
 
 1.46870 
 
 701573 
 
 1 .41497 
 
 45 
 
 i 1:5 
 
 63134 
 
 1.58388 
 
 (55604 
 
 1.. 5-242! 
 
 68130 
 
 1.46778 
 
 70717 
 
 1.41409 
 
 44 
 
 17 
 
 63177 
 
 1.58386 
 
 65646 
 
 L. 52339 
 
 68173 
 
 1.46686 
 
 70760 
 
 1.41333 
 
 43 
 
 18 
 
 63-217 
 
 1.58184 
 
 65688 
 
 1.52235 
 
 68215 
 
 1.46595 
 
 70804 
 
 1.41235 
 
 42 
 
 19 
 
 63258 
 
 1.581)83 
 
 6572!) 
 
 I.52I3< 
 
 68258 
 
 1 .46503 
 
 70848 
 
 1.41148 
 
 41 
 
 20 
 
 0329;) 
 
 1.57981 
 
 65771 
 
 1.52043 
 
 68301 
 
 1. 4(1411 
 
 70891 
 
 1.41061 
 
 40 
 
 21 
 
 63340 
 
 1.5787!) 
 
 65813 
 
 J. 51946 
 
 68343 
 
 1.46329 
 
 7093.5 
 
 1.40974 
 
 39 
 
 2-2 
 
 63380 
 
 1.57778 
 
 65854 
 
 1.51850 
 
 (58386 
 
 1.46339 
 
 70979 
 
 1.40867 
 
 38 
 
 23 
 
 63421 
 
 1.57676 
 
 65HUO 
 
 1.51754 
 
 68429 
 
 1.46137 
 
 71023 
 
 1.40800 
 
 37 
 
 2t 
 
 63463 
 
 1.57575 
 
 (55938 
 
 1.51658 
 
 68471 
 
 1.4(504(5 
 
 71066 
 
 1.40714 
 
 36 
 
 25 
 
 63503 
 
 1.57474 
 
 65980 
 
 1.51562 
 
 68514 
 
 1.45955 
 
 71110 
 
 1.40(527 
 
 35 
 
 26 
 
 G3544 
 
 1.57372 
 
 86021 
 
 1.51466 
 
 68557 
 
 1.458(54 
 
 71154 
 
 1.40540 
 
 34 
 
 27 
 
 63594 
 
 1.57271 
 
 66063 
 
 1.51370 
 
 68600 
 
 1.45773 
 
 71198 
 
 1.40454 
 
 33 
 
 28 
 
 63635 
 
 1-57170 
 
 66105 
 
 1.51275 
 
 681542 
 
 1.45682 
 
 71242 
 
 1.40368 
 
 32 
 
 29 
 
 63996 
 
 1.57069 
 
 66147 
 
 1.51179 
 
 68685 
 
 1.45593 
 
 71285 
 
 1.40281 
 
 31 
 
 30 
 
 63707 
 
 1.56969 
 
 66189 
 
 1.51084 
 
 68788 
 
 1.45501 
 
 71329 
 
 1.40195 
 
 30 
 
 31 
 
 81748 
 
 1.56863 
 
 66230 
 
 1.50988 
 
 6S771 
 
 1.45410 
 
 71373 
 
 1.40109 
 
 29 
 
 3-2 
 
 63789 
 
 1.56767 
 
 66272 
 
 1.50893 
 
 68814 
 
 1.453-20 
 
 71417 
 
 1.40022 
 
 28 
 
 33 
 
 63830 
 
 1.5(5667 
 
 66314 
 
 1-50797 
 
 68857 
 
 1.4522!) 
 
 71461 
 
 1.39936 
 
 27 
 
 34 
 
 (53871 
 
 1.56566 
 
 66356 
 
 1.50702 
 
 68900 
 
 1.45139 
 
 71505 
 
 1.39850 
 
 26 
 
 35 
 
 6391-2 
 
 I.5ti46)i 
 
 66398 
 
 1.50607 
 
 (58942 
 
 1.45048 
 
 71549 
 
 1.39764 
 
 25 
 
 36 
 
 63953 
 
 1.56366 
 
 66440 
 
 1.50512 
 
 88 185 
 
 1.449.58 
 
 71593 
 
 1.39679 
 
 24 
 
 37 
 
 63994 
 
 l.5->-2f>5 
 
 C.'i IH-2 
 
 1.50417 
 
 69028 
 
 1.44868 
 
 71637 
 
 1.39.593 
 
 23 
 
 38 
 
 64035 
 
 1.56165 
 
 66524 
 
 1.5032-2 
 
 69071 
 
 1.44778 
 
 71681 
 
 1.39507 
 
 22 
 
 39 
 
 6407!> 
 
 1.56085 
 
 66566 
 
 L. 50328 
 
 69114 
 
 1 .44688 
 
 71725 
 
 1.39421 
 
 21 
 
 40 
 
 64117 
 
 l.:>:>%6 
 
 66008 
 
 1.50133 
 
 69157 
 
 1 . 1 1.51H 
 
 7171!!) 
 
 1.39336 
 
 20 
 
 41 
 
 61153 
 
 i.r>.->8!)6 
 
 66550 
 
 1.50038 
 
 69200 
 
 1 . i r>iH 
 
 71813 
 
 1.39350 
 
 19 
 
 4-2 
 
 64199 
 
 i..-37(i6 
 
 f56<>92 
 
 1.49!)44 
 
 69243 
 
 1.44418 
 
 71857 
 
 1.39165 
 
 18 
 
 43 
 
 64240 
 
 1.55666 
 
 66734 
 
 1.49819 
 
 69-286 
 
 1.44339 
 
 71901 
 
 L 39679 
 
 17 
 
 44 
 
 64381 
 
 1.55567 
 
 86778 
 
 1.41)755 
 
 69329 
 
 1 . 1 423SI 
 
 71946 
 
 1.38994 
 
 16 
 
 45 
 
 64322 
 
 1.55467 
 
 66818 
 
 1.49661 
 
 69372 
 
 1.44149 
 
 71990 
 
 1.38900 
 
 15 
 
 415 
 
 64363 
 
 1.55368 
 
 66860 
 
 1.49566 
 
 69416 
 
 1.44060 
 
 7-2034 
 
 i.:;--.-l 
 
 14 
 
 47 
 
 64404 
 
 1.55269 
 
 66902 
 
 1.49472 
 
 69459 
 
 1.43970 
 
 72078 
 
 1.38738 
 
 13 
 
 48 
 
 64446 
 
 1.55170 
 
 66944 
 
 1.49378 
 
 (59502 
 
 1.43881 
 
 72122 
 
 1.38653 
 
 12 
 
 49 
 
 64487 
 
 1.55071 
 
 66986 
 
 1.49284 
 
 69545 
 
 1.43792 
 
 72166 
 
 1.38568 
 
 11 
 
 50 
 
 64528 
 
 1.54972 
 
 67028 
 
 1.4919(1 
 
 69588 
 
 1.43703 1| 72211 
 
 1.38484 
 
 10 
 
 51 
 
 64569 
 
 1.54873 
 
 67071 
 
 1.49097 
 
 69631 
 
 1.43614 73255 
 
 1.38399 
 
 9 
 
 52 
 
 64610 
 
 1.51774 
 
 67113 
 
 1.49003 
 
 69675 
 
 1.43525 
 
 72299 
 
 1.38314 
 
 8 
 
 53 
 
 64652 
 
 1.54675 
 
 67155 
 
 1.48909 
 
 6!)718 
 
 1.43436 
 
 72344 
 
 I.3833U 
 
 7 
 
 54 
 
 64693 
 
 1.54576 
 
 67197 
 
 1.48816 
 
 69761 
 
 1-43347 
 
 72388 
 
 1.38145 
 
 6 
 
 55 
 
 64734 
 
 1.54478 
 
 67239 
 
 1.48722 
 
 69804 
 
 1. 432.58 
 
 72432 
 
 1.3801)0 
 
 5 
 
 56 
 
 64773 
 
 1.51379 
 
 67282 
 
 1.48639 
 
 69847 
 
 1-43169 
 
 72477 
 
 1.371)76 
 
 4 
 
 57 
 
 84817 
 
 1.54381 
 
 67324 
 
 J . 48536 
 
 <i9->!)l 
 
 1.43080 
 
 73591 
 
 1.37891 
 
 3 
 
 58 
 
 
 1.51 183 
 
 67366 
 
 1.48442 
 
 69934 
 
 1.429U2 
 
 72565 
 
 1.37807 
 
 2 
 
 59 
 
 64899 
 
 1.5 1085 j 
 
 67409 
 
 1.4334!) 
 
 69977 
 
 1.42903 
 
 72(510 
 
 1.37722 1 
 
 60 
 
 61941 
 
 1. 5:9S(i 
 
 67451 
 
 1.48256 
 
 70021 
 
 1.42815 
 
 72854 
 
 1.37638 
 
 M 
 
 N Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan 
 
 N. Cot 
 
 N. Tan. 
 
 N. Cot. 
 
 N.Tan. M 
 
 
 57 Degrees. 
 
 56 Degrees. 
 
 55 Degrees. 
 
 54 Degrees. 
 
 4* 
 
8-2 
 
 NATURAL TANGENTS. 
 
 M 
 
 36 Degrees. || 37 Degrees. 
 
 38 Degrees. 
 
 39 Degrees. 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 X. Cot. 
 
 i\ ? . Tun. 
 
 N. Cot. 
 
 N. Tan 
 
 N. Cot. 
 
 
 
 72654 
 
 .37638 
 
 75355 
 
 1.32704 
 
 78129 
 
 .27994 
 
 8o978 
 
 .23489 
 
 60 
 
 1 
 
 72699 
 
 .37554 
 
 75401 
 
 ,3-2(124 
 
 78175 
 
 .27917 
 
 81U27 
 
 1.23411.; 
 
 59 
 
 2 
 
 72743 
 
 .37470 
 
 75447 
 
 .32544 
 
 78832 
 
 .27841 
 
 81075 
 
 .23343 
 
 58 
 
 3 
 
 72788 
 
 .37386 
 
 75492 
 
 .32464 
 
 78269 
 
 .27764 
 
 81123 
 
 .23270 
 
 57 
 
 4 
 
 72832 
 
 .37302 
 
 75538 
 
 .32384 
 
 7*310 
 
 .27688 
 
 81171 
 
 .23196 
 
 56 
 
 5 
 
 72877 
 
 .37218 
 
 75584 
 
 .32304 
 
 78363 
 
 .27011 
 
 81220 
 
 .23123 
 
 55 
 
 6 
 
 72921 
 
 .37134 
 
 75629 
 
 .32224 
 
 78410 
 
 .27.535 
 
 81268 
 
 .23050 
 
 54 
 
 
 72966 
 
 .37050 
 
 7f>07.-> 
 
 .3-2144 
 
 78457 
 
 .27458 
 
 81316 
 
 .2-2977 
 
 53 
 
 8 
 
 73010 
 
 .36967 
 
 75721 
 
 .32064 
 
 78504 
 
 .27382 
 
 81304 
 
 .22904 
 
 52 
 
 9 
 
 73055 
 
 .308H3 
 
 75767 
 
 .31984 
 
 78551 
 
 .27306 
 
 81413 
 
 .22831 
 
 51 
 
 10 
 
 73100 
 
 .36800 
 
 75812 
 
 .31904 
 
 78598 
 
 .27230 
 
 81461 
 
 .2-2758 
 
 50 
 
 11 
 
 73J44 
 
 ,36716 
 
 75858 
 
 .31825 
 
 78645 
 
 .27153 
 
 81510 
 
 .22085 
 
 49 
 
 12 
 
 73189 
 
 .36633 
 
 75904 
 
 .31745 
 
 78692 
 
 .27077 
 
 81558 
 
 .22612 
 
 48 
 
 13 
 
 73234 
 
 .36549 
 
 75950 
 
 .3l(>6(> 
 
 78739 
 
 .27001 
 
 81606 
 
 .22539 
 
 47 
 
 14 
 
 73278 
 
 .36406 
 
 75996 
 
 .31586 
 
 78786 
 
 .26925 
 
 81655 
 
 .22467 
 
 46 
 
 15 
 
 73323 
 
 .36383 
 
 76042 
 
 .31.507 
 
 78834 
 
 .26849 
 
 81703 
 
 .22394 
 
 45 
 
 16 
 
 73368 
 
 .36300 
 
 76088 
 
 .31427 
 
 78881 
 
 .26774 
 
 81752 
 
 .22321 
 
 44 
 
 17 
 
 73413 
 
 .36217 
 
 78134 
 
 .31348 
 
 78928 
 
 .26698 
 
 81800 
 
 .22249 
 
 43 
 
 18 
 
 73457 
 
 .36133 
 
 76180 
 
 .31269 
 
 78975 
 
 .26(522 
 
 81849 
 
 .22176 
 
 42 
 
 19 
 
 73502 
 
 .36051 
 
 76226 
 
 .31190 
 
 79022 
 
 .26540 
 
 81898 
 
 .22104 
 
 41 
 
 20 
 
 73547 
 
 .35968 
 
 76272 
 
 .31110 
 
 79070 
 
 .26471 
 
 8194(5 
 
 .22031 
 
 40 
 
 21 
 
 73592 
 
 .35885 
 
 76318 
 
 .31031 
 
 79117 
 
 .26395 
 
 81995 
 
 .21959 
 
 39 
 
 22 
 
 73637 
 
 .35802 
 
 76364 
 
 .30952 
 
 79164 
 
 .26319 
 
 82044 
 
 .21886 
 
 38 
 
 23 
 
 73681 
 
 .35719 
 
 76410 
 
 .30873 
 
 79212 
 
 .26244 
 
 82092 
 
 .21814 
 
 37 
 
 24 
 
 73726 
 
 .35637 
 
 76456 
 
 .30795 
 
 79-259 
 
 .26169 
 
 82141 
 
 .21742 
 
 36 
 
 25 
 
 73771 
 
 .35554 
 
 76502 
 
 .30716 
 
 79306 
 
 .26093 
 
 82190 
 
 .21670 
 
 35 
 
 26 
 
 73816 
 
 .35472 
 
 76548 
 
 .30637 
 
 79354 
 
 .26018 
 
 82238 
 
 .21598 
 
 34 
 
 27 
 
 73816 
 
 .35389 
 
 76594 
 
 .30558 
 
 79401 
 
 .25943 
 
 82287 
 
 .21526 
 
 33 
 
 28 
 
 7390ti 
 
 .35307 
 
 76640 
 
 .30480 
 
 79449 
 
 .25867 
 
 82336 
 
 .21454 
 
 32 
 
 29 
 
 73951 
 
 .352-24 
 
 76686 
 
 .30401 
 
 79496 
 
 .25792 
 
 82385 
 
 .21382 
 
 31 
 
 30 
 
 73996 
 
 .35142 
 
 76733 
 
 .30323 
 
 79544 
 
 .25717 
 
 82434 
 
 .21310 
 
 30 
 
 31 
 
 74041 
 
 .35060 
 
 76779 
 
 .30-244 
 
 79591 
 
 .25642 
 
 8-2483 
 
 .21238 
 
 29 
 
 32 
 
 74086 
 
 .34978 
 
 76825 
 
 .3016(i 
 
 791)39 
 
 .25567 
 
 82531 
 
 .21166 
 
 28 
 
 33 
 
 74J31 
 
 .34896 
 
 76871 
 
 .30087 
 
 791)80 
 
 .25492 
 
 82580 
 
 .21094 
 
 27 
 
 34 
 
 74176 
 
 .34814 
 
 76918 
 
 .30009 
 
 79734 
 
 .25417 
 
 82629 
 
 .21023 
 
 26 
 
 35 
 
 74221 
 
 .34732 
 
 76964 
 
 .29931 
 
 79781 
 
 .25343 
 
 82678 
 
 .20951 
 
 25 
 
 36 
 
 74267 
 
 .34650 
 
 77010 
 
 .29853 
 
 79829 
 
 .25268 
 
 82727 
 
 .20879 
 
 24 
 
 37 
 
 74312 
 
 .345t<8 
 
 77057 
 
 .29775 
 
 79877 
 
 .25193 
 
 82776 
 
 .20808 
 
 23 
 
 38 
 
 74357 
 
 .34487 
 
 77103 
 
 .39696 
 
 799-24 
 
 .25118 
 
 82825 
 
 .20736 
 
 22 
 
 39 
 
 74402 
 
 .34405 
 
 77149 
 
 .29618 
 
 79972 
 
 .25044 
 
 82874 
 
 .20665 
 
 21 
 
 40 
 
 74447 
 
 .34323 
 
 771 96 
 
 1.29541 
 
 80020 
 
 .24969 
 
 82921} 
 
 .20593 
 
 20 
 
 41 
 
 74492 
 
 .34242 
 
 77242 
 
 .29403 
 
 80067 
 
 -24895 
 
 82972 
 
 .20522 
 
 19 
 
 42 
 
 74538 
 
 .34160 
 
 77289 
 
 .29385 
 
 80115 
 
 .24820 
 
 83022 
 
 .20451 
 
 18 
 
 43 
 
 74583 
 
 .34079 
 
 77335 
 
 .29307 
 
 80163 
 
 .24740 
 
 83071 
 
 .20379 
 
 17 
 
 44 
 
 74628 
 
 .3399,- 
 
 77382 
 
 .29229 
 
 80211 
 
 .24672 
 
 83120 
 
 .20308 
 
 16 
 
 45 
 
 74674 
 
 .33916 
 
 77428 
 
 .2915'2 
 
 80258 
 
 .24597 
 
 83169 
 
 .20237 
 
 15 
 
 46 
 
 74719 
 
 . 33835 
 
 77475 
 
 .29074 
 
 80306 
 
 .24523 
 
 83218 
 
 .20166 
 
 14 
 
 47 
 
 74764 
 
 .33754 
 
 77521 
 
 .28997 
 
 80354 
 
 .24449 
 
 83268 
 
 .20095 
 
 13 
 
 48 
 
 74810 
 
 .33673 
 
 77568 
 
 .28919 
 
 80402 
 
 .24375 
 
 83317 
 
 .20024 
 
 12 
 
 49 
 
 74855 
 
 .33592 
 
 77615 
 
 .28842 
 
 80450 
 
 .24301 
 
 83366 
 
 .19953 
 
 11 
 
 50 
 
 74900 
 
 . .33511 
 
 77661 
 
 .28764 
 
 80498 
 
 .24227 
 
 83415 
 
 -19882 
 
 10 
 
 51 
 
 74946 
 
 .33430 
 
 77703 
 
 .28687 
 
 8054(5 
 
 .24153 
 
 83465 
 
 .19811 
 
 9 
 
 52 
 
 74991 
 
 .33349 
 
 77754 
 
 .28610 
 
 80594 
 
 .24079 
 
 83514 
 
 . 19740 
 
 8 
 
 53 
 
 75037 
 
 .33268 
 
 77801 
 
 .28533 
 
 80642 
 
 .24005 
 
 83564 
 
 .19069 
 
 7 
 
 54 
 
 75082 
 
 .33187 
 
 77H48 
 
 .28456 
 
 80690 
 
 .23931 
 
 83613 
 
 . 19599 
 
 6 
 
 55 
 
 75128 
 
 .33107 
 
 77895 
 
 -28379 
 
 80738 
 
 .23858 
 
 83602 
 
 . 19528 
 
 5 ' 
 
 56 
 
 75174 
 
 .33021) 
 
 77941 
 
 .28302 
 
 80786 
 
 .23784 
 
 83712 
 
 .19457 
 
 4 
 
 57 
 
 75219 
 
 .32946 
 
 77988 
 
 .28225 
 
 80834 
 
 -23710 
 
 83761 
 
 .19387 
 
 3 
 
 58 
 
 75264 
 
 .32865 
 
 77035 
 
 .28148 
 
 80882 
 
 .23637 
 
 83811 
 
 .19316 
 
 2 
 
 59 
 
 75310 
 
 .32785 
 
 77082 
 
 1.28071 
 
 80930 
 
 .23563 
 
 83860 
 
 .19246 
 
 1 
 
 60 
 
 75355 
 
 1.32704 
 
 77129 
 
 1.27994 
 
 80978 
 
 .23490 
 
 83910 
 
 1.19175 
 
 
 
 "M" 
 
 N. Cot. 
 
 N.Tan. 
 
 N. Cot.JN. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 M 
 
 
 53 Degrees. 
 
 52 Degrees. 
 
 51 Degrees. 
 
 50 Degrees. 
 
 
NATtRAi. TANCKNT*. 
 
 
 40 Degrees. 
 
 41 Degrees. 
 
 4'2 Degrees. 
 
 43 Degrees. 
 
 
 M 
 
 N.Tan. 
 
 N. Cot. 
 
 N. Tan. 
 
 N. Cot. 
 
 N.Tan. 
 
 N. Cot. 
 
 N.Tan. 
 
 N. Cot. 
 
 M 
 
 
 
 83910 
 
 1.19175 
 
 86929 
 
 1.15037 
 
 901)40 
 
 .11061 
 
 93252 
 
 .07237 
 
 60 
 
 1 
 
 83960 
 
 1.19105 
 
 86980 
 
 1.14909 
 
 90093 
 
 . lUllllfi 
 
 93306 
 
 .07174 
 
 59 
 
 2 
 
 84009 
 
 1.19035 
 
 87031 
 
 1.14902 
 
 90146 
 
 .10931 
 
 93360 
 
 .07112 
 
 58 
 
 3 
 
 84059 
 
 1.18964 
 
 87082 
 
 1.14^34 
 
 91)199 
 
 . 108li7 
 
 93115 
 
 .07049 
 
 57 
 
 4 
 
 84108 
 
 1.18894 
 
 87133 
 
 1.14767 
 
 90251 
 
 .10802 
 
 9154(19 
 
 .06987 
 
 56 
 
 5 
 
 84158 
 
 1.18824 
 
 87184 
 
 1.14699 
 
 90304 
 
 .10737 
 
 93524 
 
 .()ti!>25 
 
 55 
 
 6 
 
 84208 
 
 1.18754 
 
 87236 
 
 1.14632 
 
 90357 
 
 . 10157-J 
 
 93578 
 
 .06862 
 
 54 
 
 7 
 
 84258 
 
 1.18684 
 
 87287 
 
 1.14565 
 
 90410 
 
 .10607 
 
 93633 
 
 .06800 
 
 53 
 
 8 
 
 84307 
 
 1.18614 
 
 87338 
 
 1.14498 
 
 90463 
 
 . 10543 
 
 93688 
 
 .06738 
 
 52 
 
 9 
 
 84357 
 
 1.18544 
 
 87389 
 
 1.J4430 
 
 90516 
 
 .10478 
 
 93742 
 
 .015670 
 
 51 
 
 10 
 
 84407 
 
 1.18474 
 
 87441 
 
 1.14363 
 
 90569 
 
 .10414 
 
 93797 
 
 .06613 
 
 50 
 
 H 
 
 84457 
 
 1.18404 
 
 87492 
 
 1.14296 
 
 90(121 
 
 . 10349 
 
 93852 
 
 .1111551 
 
 49 
 
 12 
 
 84507 
 
 1.18334 
 
 87543 
 
 1.1422!) 
 
 90674 
 
 . 102*-, 
 
 93906 
 
 .0(5-189 
 
 48 
 
 13 
 
 8455G 
 
 1.18264 
 
 87595 
 
 1.14162 
 
 90727 
 
 .10220 
 
 93961 
 
 .06427 
 
 47 
 
 14 
 
 84606 
 
 1.18194 
 
 87646 
 
 1.14095 
 
 90781 
 
 .10156 
 
 94016 
 
 .0(5365 
 
 46 
 
 15 
 
 84656 
 
 1.18125 
 
 87698 
 
 1.14028 
 
 90834 
 
 .10091 
 
 94071 
 
 .06303 
 
 45 
 
 16 
 
 84706 
 
 1.18055 
 
 87749 
 
 1.13961 
 
 90887 
 
 .10027 
 
 94125 
 
 .06241 
 
 44 
 
 17 
 
 84756 
 
 1.17986 
 
 87801 
 
 1.13894 
 
 9094(1 
 
 .091)63 
 
 94180 
 
 .06179 
 
 43 
 
 18 
 
 84806 
 
 1.17916 
 
 87852 
 
 1.13828 
 
 90993 
 
 .09899 
 
 94235 
 
 .06117 
 
 42 
 
 19 
 
 84856 
 
 1.17846 
 
 87904 
 
 1.13761 
 
 91046 
 
 .09834 
 
 94290 
 
 .oi;o:><; 
 
 41 
 
 20 
 
 84906 
 
 1.17777 
 
 87955 
 
 1.13694 
 
 91099 
 
 .09770 
 
 94345 
 
 .05994 
 
 40 
 
 21 
 
 84956 
 
 1.17708 
 
 88007 
 
 1.13627 
 
 91 153 
 
 .09706 
 
 94400 
 
 .05! 132 
 
 39 
 
 22 
 
 85006 
 
 1-17638 
 
 88059 
 
 1.13561 
 
 91206 
 
 .09642 
 
 94455 
 
 .05870 
 
 38 
 
 23 
 
 85057 
 
 1-17569 
 
 88110 
 
 1.13494 
 
 91259 
 
 .09578 
 
 94510 
 
 .05809 
 
 37 
 
 24 
 
 85107 
 
 1.17500 
 
 88162 
 
 1.13428 
 
 91313 
 
 .09514 
 
 94565 
 
 .05747 
 
 36 
 
 25 
 
 85157 
 
 1.17430 
 
 88214 
 
 1.13361 
 
 91366 
 
 .00450 
 
 94620 
 
 .05685 
 
 35 
 
 26 
 
 85207 
 
 1-17361 
 
 88265 
 
 1.13295 
 
 91419 
 
 .09386 
 
 94(576 
 
 .05(524 
 
 34 
 
 27 
 
 85257 
 
 1.17292 
 
 88317 
 
 1.13228 
 
 91473 
 
 .09322 
 
 94731 
 
 .0.-)5ti2 
 
 33 
 
 28 
 
 85307 
 
 1.17223 
 
 88369 
 
 1.13162 
 
 91526 
 
 .09258 
 
 94786 
 
 .05501 
 
 32 
 
 29 
 
 85358 
 
 1.17154 
 
 88421 
 
 1.13096 
 
 91580 
 
 .09195 
 
 94841 
 
 .05439 
 
 31 
 
 30 
 
 85408 
 
 1.17085 
 
 88473 
 
 1.13029 
 
 91633 
 
 .09131 
 
 94896 
 
 .05378 
 
 30 
 
 31 
 
 85458 
 
 1.17016 
 
 88524 
 
 1.12963 
 
 91687 
 
 .09067 
 
 94952 
 
 .05317 
 
 29 
 
 32 
 
 85509 
 
 1.16947 
 
 88576 
 
 1.12897 
 
 91740 
 
 .09003 
 
 95007 
 
 .05255 
 
 28 
 
 33 
 
 85559 
 
 1.16878 
 
 88628 
 
 1.12831 
 
 91794 
 
 .08940 
 
 95062 
 
 .05194 
 
 27 
 
 34 
 
 85609 
 
 1.16809 
 
 88680 
 
 1.12765 
 
 91847 
 
 .08876 
 
 95118 
 
 .05133 
 
 26 
 
 35 
 
 85660 
 
 1.16741 
 
 88732 
 
 1.12699 
 
 91901 
 
 .08813 
 
 95173 
 
 .05072 
 
 25 
 
 36 
 
 85710 
 
 1.16672 
 
 88784 
 
 1.12633 
 
 91955 
 
 .08749 
 
 9522'J 
 
 .05010 
 
 24 
 
 37 
 
 85761 
 
 1.16603 
 
 88836 
 
 1.12567 
 
 92008 
 
 .08686 
 
 95284 
 
 .0494!) 
 
 23 
 
 38 
 
 85811 
 
 1.16535 
 
 88888 
 
 1.12501 
 
 92062 
 
 .OdfrJ-J 
 
 95340 
 
 .04888 
 
 22 
 
 39 
 
 85862 
 
 1.16466 
 
 88940 
 
 1.12435 
 
 92116 
 
 .08559 
 
 95395 
 
 .04827 
 
 21 
 
 40 
 
 85912 
 
 1.16398 
 
 88992 
 
 1. 12369 
 
 92170 
 
 .0849(5 
 
 95451 
 
 .04766 
 
 20 ! 
 
 41 
 
 85963 
 
 1.16329 
 
 89045 
 
 1.12303 
 
 92223 
 
 .08432 
 
 96506 
 
 .0470;-) 
 
 19 
 
 42 
 
 86014 
 
 1.16261 
 
 89097 
 
 1.12238 
 
 92277 
 
 .08369 
 
 95562 
 
 .04644 
 
 J8 
 
 43 
 
 86064 
 
 1.16192 
 
 89149 
 
 1.12172 
 
 92331 
 
 .08306 
 
 95618 
 
 .04583 
 
 17 
 
 44 
 
 86115 
 
 1.16124 
 
 89201 
 
 1.12106 
 
 92385 
 
 .08243 
 
 95673 
 
 .04522 
 
 1(5 
 
 45 
 
 86166 
 
 1.16056 
 
 89253 
 
 1.12041 
 
 92439 
 
 .08179 
 
 95729 
 
 .04461 
 
 15 
 
 46 
 
 86216 
 
 1.15987 
 
 89306 
 
 1.11975 
 
 92493 
 
 .08116 
 
 95785 
 
 .04401 
 
 14 
 
 47 
 
 86267 
 
 1.15919 
 
 89358 
 
 1.11909 
 
 92547 
 
 .08053 
 
 95841 
 
 .04340 
 
 13 
 
 48 
 
 86318 
 
 1.15851 
 
 89410 
 
 1.11844 
 
 92601 
 
 .07990 
 
 95897 
 
 .0427H 
 
 12 
 
 49 
 
 86368 
 
 1.15783 
 
 89463 
 
 1.11778 
 
 92655 
 
 .07927 
 
 96952 
 
 .04218 
 
 11 
 
 50 
 
 86419 
 
 1.15715 
 
 89515 
 
 1.11713 
 
 92709 
 
 .07864 
 
 96008 
 
 .04158 
 
 10 
 
 51 
 
 86470 
 
 1.15647 
 
 89567 
 
 1.11648 
 
 98783 
 
 .07801 
 
 96064 
 
 .040U7 
 
 !> 
 
 52 
 
 86521 
 
 1.15579 
 
 89620 
 
 1.11582 
 
 92817 
 
 .07738 
 
 99130 
 
 .04036 
 
 8 
 
 53 
 
 86572 
 
 1.15511 
 
 89672 
 
 1.11517 
 
 92872 
 
 .07676 
 
 96176 
 
 .0397(5 
 
 7 
 
 54 
 
 86623 
 
 1.15443 
 
 89725 
 
 1.11452 
 
 92926 
 
 .07613 
 
 W833 
 
 .03915 
 
 (5 
 
 55 
 
 86674 
 
 1.15375 
 
 89777 
 
 1.11387 
 
 92980 
 
 .07550 
 
 96288 
 
 .03855 
 
 5 
 
 56 
 
 86725 
 
 1.15308 
 
 89830 
 
 1.11321 
 
 93034 
 
 .07487 
 
 96344 
 
 .03794 
 
 4 
 
 57 
 
 86776 
 
 1.1 5340 
 
 89883 
 
 i.narifi 
 
 93088 
 
 .07425 
 
 96400 
 
 .0373' 
 
 3 
 
 58 
 
 8G827 
 
 1 . '5J72 
 
 83935 
 
 1.11191 
 
 93143 
 
 .073(5-2 
 
 M457 
 
 .03674 
 
 2 
 
 5i) 
 
 86878 
 
 i. 15104 
 
 89988 
 
 1.11126 
 
 93197 
 
 .07399 
 
 96513 
 
 1.03613 
 
 1 
 
 60 
 
 86929 
 
 1.15037 
 
 90040 
 
 1.11061 
 
 93252 
 
 .07237 
 
 96569 
 
 1.03553 
 
 
 
 M 
 
 N Cot 
 
 N. Tan. 
 
 "N. cot. 
 
 N.Tan. 
 
 ~N. Cot. 
 
 N. Tan. 
 
 NTCoT. 
 
 N.Tan. 
 
 ~M~ 
 
 
 49 Degrees. 
 
 48 Degrees. 
 
 47 Degrees. 
 
 46 Degrees. 
 
 
NATURAL TAN&BNTS. 
 
 
 44 Degrees. 
 
 
 
 44 Degrees. 
 
 
 M 
 
 N. Tan. 
 
 N.Cot. 
 
 M 
 
 M 
 
 N. Tan. 
 
 N. Cot. 
 
 M 
 
 
 
 96569 
 
 .03553 
 
 60 
 
 "sT 
 
 98327 
 
 .01702 
 
 29 
 
 1 
 
 96625 
 
 .03493 
 
 59 
 
 32 
 
 98384 
 
 .01642 
 
 28 
 
 2 
 
 96681 
 
 .03433 
 
 58 
 
 33 
 
 98441 
 
 .01583 
 
 27 
 
 3 
 
 96738 
 
 .03372 
 
 57 
 
 34 
 
 98499 
 
 .01524 
 
 26 
 
 4 
 
 96794 
 
 .03312 
 
 56 
 
 35 
 
 98556 
 
 .01465 
 
 25 
 
 5 
 
 96850 
 
 .03252 
 
 55 
 
 36 
 
 98613 
 
 .01406 
 
 24 
 
 6 
 
 96907 
 
 .03192 
 
 54 
 
 37 
 
 98671 
 
 .01347 
 
 23 
 
 7 
 
 96963 
 
 .03132 
 
 53 
 
 38 
 
 98728 
 
 .01288 
 
 22 
 
 8 
 
 97020 
 
 .03072 
 
 52 
 
 39 
 
 98786 
 
 .01229 
 
 21 
 
 9 
 
 97076 
 
 .03012 
 
 51 
 
 40 
 
 98843 
 
 .01170 
 
 20 
 
 10 
 
 97133 
 
 .02952 
 
 50 
 
 41 
 
 9890J 
 
 .01112 
 
 19 
 
 11 
 
 97189 
 
 .02892 
 
 49 
 
 42 
 
 98958 
 
 .01053 
 
 18 
 
 12 
 
 97246 
 
 .02832 
 
 48 
 
 43 
 
 99016 
 
 .00994 
 
 17 
 
 13 
 
 97302 
 
 .02772 
 
 47 
 
 44 
 
 99073 
 
 .00935 
 
 16 
 
 14 
 
 97359 
 
 .02713 
 
 46 
 
 45 
 
 99131 
 
 .00876 
 
 15 
 
 15 
 
 97416 
 
 .02653 
 
 45 
 
 46 
 
 99189 
 
 .00818 
 
 14 
 
 16 
 
 97472 
 
 .02593 
 
 44 
 
 47 
 
 99247 
 
 .00759 
 
 13 
 
 17 
 
 97529 
 
 .02533 
 
 43 
 
 48 
 
 99304 
 
 .00701 
 
 12 
 
 18 
 
 97586 
 
 .02474 
 
 42 
 
 49 
 
 99362 
 
 .00642 
 
 11 
 
 19 
 
 97643 
 
 .02414 
 
 41 
 
 50 
 
 99420 
 
 .00583 
 
 10 
 
 20 
 
 97700 
 
 .02355 
 
 40 
 
 51 
 
 99478 
 
 .00525 
 
 9 
 
 21 
 
 97756 
 
 .02295 
 
 39 
 
 52 
 
 99536 
 
 .00467 
 
 8 
 
 22 
 
 97813 
 
 .02236 
 
 38 
 
 53 
 
 99594 
 
 .00408 
 
 7 
 
 23 
 
 97870 
 
 .02176 
 
 37 
 
 54 
 
 99652 
 
 .00350 
 
 6 
 
 24 
 
 97927 
 
 .02117 
 
 36 
 
 55 
 
 99710 
 
 .00291 
 
 5 
 
 25 
 
 97984 
 
 .02057 
 
 35 
 
 56 
 
 99768 
 
 .00233 
 
 4 
 
 26 
 
 98041 
 
 .01998 
 
 34 
 
 57 
 
 99826 
 
 .00175 
 
 3 
 
 27 
 
 98098 
 
 .01939 
 
 33 
 
 58 
 
 99884 
 
 1.00116 
 
 2 
 
 28 
 
 98155 
 
 .01879 
 
 32 
 
 59 
 
 99942 
 
 1.00058 
 
 1 
 
 29 
 
 98213 
 
 1.01820 
 
 31 
 
 60 
 
 10000 
 
 1.00000 
 
 
 
 30 
 
 98270 
 
 1.01761 
 
 30 
 
 
 
 
 
 M~ 
 
 N. Cot. 
 
 N. Tan. 
 
 M~ 
 
 ~M~ 
 
 N. Cot. 
 
 N. Tan. 
 
 "M 
 
 
 45 Degrees, 
 
 1 
 
 
 45 Degrees. 
 
 
tl / / x*4^C^> (Let f , v/cWlo 
 
 M . 
 
 **.