/4ZA*r7\ ASTRONOMY DEPT. NEWCOMB'S MATHEMATICAL COURSE. /. SCHOOL COURSE. Algebra for Schools. Key to Algebra for Schools. Plane Geometry and Trigonometry, with Tables. The Essentials of Trigonometry. II. COLLEGE COURSE. Algebra for Colleges. Key to Algebra for Colleges. Elements of Geometry. Plane and Spherical Trigonometry, with Tables. Trigonometry (separate). Tables (separate). Elements of Analytic Geometry. Elements of the Differential and Integral Calculus. Astronomy, Advanced Course, by Newcomb and Holden. Astronomy, Briefer Course, by Newcomb and Holden. HENRY HOLT & CO., Publishers, New York. NE WC OMB' 8 MATHEMATICAL COURSE LOGARITHMIC AND OTHER MATHEMATICAL TABLES WITH EXAMPLES OF THEIR USE AND HINTS ON THE ART OK COMPUTATION BY SIMON NEWCOMB f Mathematics, in the Johns Hopkins University. NEW YORK HENRY HOLT AND COMPANY LC I ? I ASTRONOMY DEPT COPYRIGHT, 1883, BY HENRY HOLT & COl PBEFACE. IK the present work an attempt is made to present to computers and students a set of logarithmic and trigonometric tables which shall have all the conveniences familiar to those who use German tables. The five-figure tables of F. G. GAUSS, of which fifteen edi- tions have been issued, have, after long experience with them, been taken as the basis of the present ones, but modifications have been introduced wherever any improvement could be made. Five places of decimals have been adopted as an advantageous mean. The results obtained by them, being nearly always reliable to the 10,000th part, are amply accurate for most computations, while the time of the student who uses them is not wasted in unnecessary calculation. The Introduction is intended to serve not only as an explanation of the tables, but as a little treatise on the art of computation, and the methods by which the labor of computation may be abridged. To avoid fostering the growing evil of nearsightedness among students, the author and publishers have spared neither pains nor expense in securing clearness of typography. M192073 CONTENTS. INTRODUCTION TO TABLES. TABLE I. LOGARITHMS OP NUMBERS. KAMI Introductory Definitions 3 The Use of Logarithms 4 Arrangement of the Table of Logarithms 6 Characteristics of Logarithms 8 Interpolation of Logarithms 10 Labor-Saving Devices ' 11 Number Corresponding to Given Logarithm 13 Adjustment of Last Decimal 14 The Arithmetical Complement 16 Practical Hints on the Art of Computation 18 Imperfections of Logarithmic Calculation 20 Applications to Compound Interest and Annuities 25 Accumulation of an Annuity 28 TABLE II. MATHEMATICAL CONSTANTS. Explanation 31 TABLES III. AND IV. LOGARITHMS OP TRIGONOMETRIC FUNCTIONS. Angles less than 45 32 Angles between 45 and 90 33 Angles greater than 90 35 Methods of Writing the Algebraic Signs 36 Angle Corresponding to a Given Function 37 Cases when the Function is very Small or Great 38 TABLE V. NATURAL SINES AND COSINES. Explanation 42 CONTENTS. TABLE VI. ADDITION AND SUBTRACTION LOGARITHMS. PAGE Use in Addition ........................................................ 43 Use in Subtraction .................. t .................................. 44 Special Cases ........................................................... 45 TABLE SQUARES OP NUMBERS. Explanation. .............. .................. .......................... 49 TABLE VIII. HOURS, MINUTES, AND SECONDS INTO DECIMALS OP A DAY. Explanation ........................................... . ................ 51 TABLE IX. To CONVERT TIME INTO ARC. Explanation ............................................................ 53 TABLE X. MEAN AND SIDEREAL TIME. Explanation ............................................................ 55 OF DIFFERENCES AND INTERPOLATION. General Principles ........ . ...... .' ...................................... 56 Fundamental Formulae .................................................. 61 Transformations of the Formulae ......................................... 62 Formulas of Stirling and Bessel ..................................... ..... 68 Special Cases of Interpolation Interpolation to Halves .................... 64 Interpolation to Thirds ................................................ 66 Interpolation to Fifths ........................... , ..................... 70 FORMULAE FOR THE SOLUTION OF PLANE AND SPHERICIAL TRIANGLES. Remarks ............................................................... 74 Formulae ............................................................... 75 TABLES I. TO X. TABLE I. LOGARITHMS OF NUMBERS. 1. Introductory Definitions. Natural numbers are numbers used to represent quantities. The numbers used in arithmetic and in the daily transactions of life are natural numbers. To every natural number may be assigned a certain other number, called its logarithm. The logarithm of a natural number is the exponent of the power to which some assumed number must be raised to produce the first number. The assumed number is called the base. E.g. f the logarithm of 100 with the base 10 is 2, because 10 2 = 100; with the base 2, the logarithm of 64 would be 6, because 2 fl = 64. A system of logarithms means the logarithms of all POSN tive numbers to a given base. Although there may be any number of systems of logarithms, only two are used in practice, namely: 1. The natural or Napierian system, base = e = 2. 718 282. 2. The common system, base = 10. The natural system is used for purely algebraic purposes. The common system is used to facilitate numerical calculations and is the only one employed in this book. If the natural number is represented by n, its logarithm is called log n. A logarithm usually consists of an integer number and a decimal part. The integer is called the characteristic of the logarithm. The decimal part is called the mantissa of the logarithm. A table of logarithms is a table by which the logarithm of any given number, or the number corresponding to any given loga- rithm, may be found. 4 LOGARITHMIC TABLES. The moot simple form of table is that on the first page of Table I., which gives the logarithms of all entire numbers from 1 to 150; each logarithm being found alongside its number. The student may begin his exercises with this table. Mathematical tables in general enable us, when one of two related quantities is given, to find the other. In such tables the quantity supposed to be given is called the argument. The argument is usually printed on the top, bottom, or side of the table. The quantities to be found are called functions of the argu- ment, and are found in the same columns or lines as the argument, feut in the body of the table. In a table of logarithms the natural number is the argument, and the logarithm is the function. 2. The Use of Logarithms. The following properties of logarithms are demonstrated in treatises on algebra. I. The logarithm of a product is equal to the sum of the loga- rithms of its factors. II. The logarithm of a quotient is found by subtracting the loga- rithm of the divisor from that of the dividend. III. The logarithm of any power of a number is equal to the loga- rithm of the number multiplied by the exponent of the power. IV. The logarithm of the root of a number is equal to the loga- rithm of the number divided by the index of the root. "We thus derive the following rules: To find the product of several factors by logarithms. EULE. Add the logarithms of the several factors. Enter the table with the sum as a new logarithm, and find the number corres- ponding to it. This number is the product required. Example 1. To multiply 7x8. We find from the first page of Table I. log? = 0.84510 " 8 = 0.90309 Sum of logs = 1.748 19 = log of product. Having added the logarithms, we look in column log for a num- THE USE OF LOGARITHMS. 5 ber corresponding to 1.788 19 and find it to be 56, which is the pro- duct required. Ex. 2 To find the continued product 2x6x8. log 2, 0.30103 " 6, 0.77815 " 8, 0.90309 Sum of logs, 1.982 27 = log product. The number corresponding to this logarithm is found to be 96, which is the product required. Ex. 3. To find the quotient of 147 -f- 21. log 147, 2.16732 " 21,1.32222 Difference, 0.845 10 We find this difference to be the logarithm of 7, which is tha required quotient. Ex. 4. To find the quotient arising from dividing the continued /roduct 98 X 102 X 148 by the continued product 21 X 37 X 68. log 21, 1.322 22 log 98, 1.991 23 " 37, 1.56820 " 102, 2.00860 " 68, 1.83251 " 148, 2.17026 Sum = log divisor, 4.722 93 Sum = log dividend, 6.170 09 log divisor, 4.72293 Difference = log quotient, 1.447 16 Looking into the table, we find the number corresponding to this logarithm to be 28, which is the required quotient. NOTE. The student will notice that we have found this quotient without actually determining either the divisor or dividend, having used only their loga- rithms. If he will solve the problem arithmetically, he will see how much shorter is the logarithmic process. Ex. 5. To find the seventh power of 2. We have log 2 = 0.301 03 7 2.10721 = log 128 Hence 128 is the required power. Ex. 6. To find the cube root of 125. 3 | 2.09691 0.69897 6 LOGARITHMIC TABLES. The index of the root being 3, we divide the logarithm of 125 by it. Looking in the tables, we find the number to be 5, which is the root required. EXEKCISES. Compute the following products, quotients, powers, and roots by logarithms. OO C 1. 11 . 13. Ans. 143. 5. ~~. Ans. 128. 2. 12'. Ans. 144. 6. 51 ' 98 ^ 81 . Ans. 31. O4: . DO 8. -^-. Ans. 48. 7. ' . Ans. 144. D D 2. 9'. 91. 78 , 54.48 4 13'. 21. 3 * AnS ' 108 - 8 --8T9-' Ans ' 36 ' 3. Arrangement of the Table of Logarithms. A table giving every logarithm alongside its number, as on the first page of Table I., would be of inconvenient bulk. For numbers larger than 150 the succeeding parts of Table I. are therefore used. Here the first three figures of the natural number are given in the left-hand column of the table. The first figure must be understood where it is not printed. The fourth figure is to be sought in the horizontal line at the top or bottom. The mantissa of the logarithm is then found in the same line with the first three digits, and in the column having the fourth digit at the top. To save space the logarithm is not given in the column, but only its last three figures. The first two figures are found in the first column, and are commonly the same for all the logarithms in any one line. Example 1. To find the logarithm of 2090. . We find the number 209, the figure 2 being omitted in printing, in the left-hand column of the table, and look in the column having the fourth figure, 0, at its top or bottom. In this column we find 320 15, which is the mantissa of the logarithm required. Ex. 2. To find the logarithm of 2092. Entering the table with 209 in the left-hand column, and choos- ing the column with 2 at the top, we find the figures 056. Te these we prefix the figures 32 in column 0, making the total logarithm 320 56. Therefore Mantissa of log 2092 = .32056. ARRANGEMENT OF THE TABLE. 7 EXERCISES. Find in the same way the mantissas of the logarithms of the fol- lowing numbers: 2240; 5133; 2242; 5256; 2249; 5504; 2895; 8925; 3644; 9557; 4688; 9780. When the first two figures of the mantissa are not found in the same line in which the number is sought, they are to be found in the first line above which contains them. Example. The first two figures of log 6250 are 79, which be- longs to all the logarithms below as far as 6309. Therefore mantissa of log 6250 = .795 88. EXEECISES. Find the mantissae of the logarithms of 6300; answer, .79934. 6309; " .79996. 6434; 6653; 6755; 6918; 7868. Exception. There are some cases in which the first two figures change in the course of the line. In this case the first two figures are to be sought in the line above before the change and in the line next below after the change. Example. The mantissa of log 6760 is .82995. But the man- tissa of log 6761 is .83001. In this case the figures 83 are to be found in the next line below. To apprise the computer of these cases, each of the logarithms in which the two first figures are found in the line below is indicated by an asterisk. EXERCISES, Find the mantissa of log 1022; answer, .009 45. log 1024; " .01030. 8 LOGARITHMIC TABLES. 1231; 1999; 1387; 3988; 1419; 4675; 1621; 4798; 1622; 5377; 1862; 8512; 1863; 1009. 4. Characteristics of Logarithms. The part of the table here described gives only the mantissa oj 033 pr. pt. for .9, 25.2 pr. pt. for .07, 1.9b " " " .03, 0.8 log, 5.19031 log, 5.19031 * By the complement or arithmetical complement of a decimal fraction is here meant the remainder found by subtracting it from unity or from a unit of the next order higher than itself. Thus : co. .723 = .277 co. .1796 = .8204 co. .9932 = .0068. NUMBER CORRESPONDING TO A GIVEN LOGARITHM. 13 7. To find the Number corresponding to a given Logarithm. The reverse process of finding the number corresponding to a given logarithm will be seen by the following example: To find the number of which the logarithm is 2.027 90. Entering the table, we find that this logarithm does not exactly occur in the table. We therefore take the next smaller logarithm, which we find to be as follows: log 1066 = 2.02776. Subtracting this from the given logarithm we find the latter to be greater by 14, while the difference between the two logarithms of the table is 40. We therefore state the proportion 40 : 14 : : 1 to the required fraction. The result is obtained by dividing 14 by 40, giving a quotient .35. The required number is therefore 106.635. It will be remarked that we take no account of the characteristic and position of the decimal until we write down the final result, when we place the decimal in the proper position. The table of proportional parts is used to find the fifth and sixth figures of the number by the following rule: If the given logarithm is not found in the table, note the ex- cess of the given logarithm above the next smaller one in the table, which call A. Take the difference of the two tabular logarithms, and fjrui it among the large figures which head the proportional parts. That proportional part next smaller than A will be the fifth figure of the required number. Take the excess of A above this proportional part; imagine its decimal point removed one place to the right, and find the nearest number of the table. This number will be the sixth figure of the required number. Example. To find the number of which the logarithm is 2.193 59. Entering the table, we find the next smaller logarithm to be .193 40. Therefore A = 19. Also its tabular difference = 28. Entering the table of proportional parts under 28, we find 16.8 opposite 6 to be the number next smaller than 19 the value of A. Therefore the fifth figure of the number is 6. The excess of 19 above 16.8 is 2.2. Looking in the same tablt for the number 22, we find the nearest to be opposite 8. 14 LOGARITHMIC TABLES. Therefore the fifth and sixth figures of the required number are 68. Now looking at the log .193 40 and taking the corresponding number, we find the whole required number to be 156 168. The characteristic being 2, the number should have three figures before the decimal point. Therefore we insert the decimal point at the proper place, giving as the final result 156.168. 8. Number of Decimals necessary. In the preceding examples we have shown how with these tables the numbers may be taken out to six figures. In reality, however, it will seldom be worth while to write down more than five figures. That is, we may be satisfied by adding only one figure to the four found from the table. In this case, when we enter the table of proportional parts, we take only the number corresponding to the nearest proportional part. To return to the last preceding example, where we find the num- be/ corresponding to 2. 193 59. We find under the difference 28 that th^ number nearest 19 is 19.6, which is opposite 7. Therefore the number to be written down would be 156.17. In the following exercises it would be well for the student to 4n ite six figures when the number is found on one of the first two pa^es of the table and only five when on one of the following page* Tl -e reason of this will be shown subsequently. EXAMPLES AND EXERCISES. \. To find the square root of f . We have log 3, 0.477 12 " 2, 0.30103 log |, 0.17609 -f- 2, log V$, 0.08804 Here we have a case in which the half of an odd number is required. We might have written the last logarithm 0.088045, but we should then have had six decimals, whereas, as our tables only give five decimals, we drop the sixth. If we write 4 for the fifth figure it will be too small by half a unit, and if we write 5 it will be too large by half a unit. It is therefore indifferent which figure we write, so far as mere accuracy is concerned. NUMBER OF DECIMALS NECESSARY. 15 A good rule to adopt in such a case is to write the nearest EVEN" number. For example, for the half of .261 81 we write .130 90; " " .26183 " .13092; " " .26185 " .13092; " " .26187 " .13094; " " .26189 " .13094; " " .26197 " .13098; " " .26199 " .13100. Returning to our example, we find, by taking the number corre- sponding to 0.088 04, Vt = 1.224 72. 2. To find the square root of f . log 2, 0.301 03 " 3, 0.47712 logf, 9.82391 - 10 | log |, 4.911 96 - 5 = log |/f. The last logarithm is the same as 9.911 96 - 10, which is the form in which it is to be written in order to apply the rule of characteristics. The corresponding number is 0.816 50. We have here a case in which, had we neglected considering the surplus 10 as we habitually do, the characteristic of the answer would have been 4 instead of 9 or 1. The easiest way to treat such cases is this: When we have to divide a logarithm in order to extract a root, instead of increasing the characteristic by 10, increase it by 10 X index of root. Thus we write log ^= 19.823 91 - 20. Dividing by 2, log i/f = 9.911 96 - 10, which is in the usual form. , > 3. To find the cube root of |. logl, 0.00000 " 2, 0.30103 log|, 9.69897 -10, which we write in the form log | = 29.69897-30. Dividing this by 3, J log 1 = log VT = 9-899 66 - 10. 16 LOGARITHMIC TABLES. This logarithm is in the usual form, and gives V?= 0.793 70. The affix 30, or 10 x divisor, can be left to be understood in these cases as in others. All that is necessary to attend to is that instead of supposing the characteristic to be one or more units less than 10, as in the usual run of cases, we suppose it to be one or more nnits less than 10 X divisor. Find: 4. The square root of -J; 5. The cube root of 2; 6. The fourth root of f ; 7. The fifth root of 20; 8. The tenth root of 10; 9. The tenth root of . 9. The Arithmetical Complement. When a logarithm is subtracted from zero, the remainder is called its arithmetical complement. If L be any logarithm, its arithmetical complement will be L. Hence if L = log n, then arith. comp. = L = log -; that is, The arithmetical complement of a given logarithm is the logarithm tfthe reciprocal of the number corresponding to the given logarithm. Notation. The arithmetical complement of a logarithm is writ- ten co-log. It is therefore defined by the form co-log n = log . Finding the arithmetical complement. To find the arithmetical Complement of log 2 = 0.301 03, we may proceed thus: 0.00000 log 2, 0,30103 co-log 2, 9.69897-10. We subtract from zero in the usual way; but when we come to the characteristic, we subtract it from 10. This makes the re* mainder too large by 10, so we write 10 after it, thus getting a quantity which we see to be log 0.5. We may leave the 10 to be understood, as already explained. THE ARITHMETICAL COMPLEMENT. 17 The arithmetical complement may be formed by the following rule: Subtract, each figure of the logarithm from 9, except the last sig- nificant one, which subtract from 10. The remainders will form the arithmetical complement. For example, having, as above, the logarithm 0.301 03, we form, mentally, 9-0 = 9; 9-3 = 6; 9-0 = 9; 9-1 = 8; 9-0 = 9; 10 3 = 7; and so write 9.698 97 as the arithmetical complement. To form the arithmetical complement of 3.284 00 we have 9 3 = 6; 9 2 = 7; 9 8 = 1; 10 4 = 6. The complement is therefore 6.71600. The computer should be able to form and write down the arith- metical complement without first writing the tabular logarithm, the subtraction of each figure being performed mentally. Use of the arithmetical complement. The co-log is used to substi- tute addition for subtraction in certain cases, on the principle: To add the co-logarithm is the same as to subtract the logarithm. Example. We may form the logarithm of J in this way by ad- dition: log 3, 0.47712 co-log 2, 9.69897 logj, 0.17609 Here there is really no advantage in using the co-log. But there is an advantage in the following example: 97fi3 N/ J.1Q 9J. To find the value of P = ;L We ad <* to the loga- yy rithms of the numerator the co-log of the denominator, thus: log 2763, 3.44138 log 419.24, 2.62246 co-log 99, 8.00436 logP, 4.06820 '.P = 11700. The use of the arithmetical complement is most convenient when ^o divisor is a little less than some power of 10. J8 LOQAEITHMIC TABLES. EXERCISES. Form by arithmetical complements the values of: 109 X 216.26 1. 2. 3. 0.99316 8263 X 9162.7 92 X 99.618 4 X 6 X 8219 9X992 1C. Practical Hints on the Art of Computation. The student who desires to be really expert in computation should learn to reduce his written work to the lowest limit, and to perform as many of the operations as possible mentally. We have already described the process of taking a logarithm from the table without written computation, and now present some exercises which will facilitate this process. 1. Adding and subtracting from left to right. If one has but two numbers to add it will be found, after practice, more easj and natural to write the sum from the left than from the right. The method is as follows: In adding each figure, notice, before writing the sum, whether the sum of the figures following is less or greater than 9, or equal to it. If the sum is less than 9, write down the sum found, or its last figure without change. If greater than 9, increase the figure by 1 before writing it down. If equal to 9, the increase should be made or not made accord- ing as the first sum following which differs from 9 is greater or less than 9. If the first sum which differs from 9 exceeds it, not only must we increase the number by 1, but must write zeros under all the places where the 9's occur. If the first sun different from 9 is less than 9, write down the 9's without change. The following example illustrates the process: 7502768357858892837 8239171645041102598 15741940002899995435 Here 7 and 8 are 15. 5 + 2 being less than 9, we write 15 without change. 3 + being less than 9, we write 7 without change. 9 + 2 being greater than 9, we increase the sum 3 + by 1 and write down 4. 7 + 1 being PRACTICAL HINTS ON THE ART OF COMPUTATION. 19 less than 9, we write the last figure of 9 -|- 2, or 1, without change. 6 + 7 being greater than 9, we increase 7 + 1 by 1 and write down 9. Under 6 + 7 we write down 3 or 4. To find which, 8+ 1 = 9; 3 + 6 = 9; 5+4= 9; 7 + 5 = 12. This first sum which is different from 9 being greater than 9, we write 4 under 6 + 7, and O's in the three following places where the sums are 9. 7+5 = 12. Since 8 + < 9, we write down 2. Before deciding whether to put 8 or 9 under 8 + 0, we add 5 + 4 = 9; 8 + 1 = 9; 8 + 1 =9; 9 + = 9; 2 + 2 = 4 This being less than 9, we write 8 under 8 + 0, and 9's in the four following places. Since 5 + 8 = 13 > 9, we write 5 under 2 + 2. Since 9+ 3 = 12 > 9, we write 4 under 5 + 8. Since 8 + 7 = 15 > 9, we write 3 under 9 + 3. Finally, under 8 + 7 we write 5. This process cannot be advantageously applied when more than two numbers are to be added. EXERCISES. Let the student practise adding each consecutive pair of the fol- lowing lines, which are spaced so that he can place the upper margin of a sheet of paper under the lines he is adding and write the sum upon it. 250917285316981208 251235964692184368 791615832316646891 208532164379102909 868588964342944825 987654321012345674 Subtracting. We subtract each figure of the subtrahend from the corresponding one of the minuend (the latter increased by 10 if necessary), as in arithmetic. Before writing down the difference, we note whether the follow- ing figure of the subtrahend is greater, less, or equal to the corre- 'gponding figure of the minuend. If greater, we diminish the remainder by 1 and write it down.* If less, we write the remainder without change. If equal, we note whether the subtrahend is greater or less than the minuend in the first following figure in which they differ. If greater, we diminish the remainder by 1, as before, and write 9's under the equal figures. * If the student is accustomed to carrying 1 to the figures of the minuend when he has increased the figure of his subtrahend by 10, he may find it easier to defer each subtraction until he sees whether the remainder is or is not to be diminished by 1, and, in the latter case, to increase the minuend by 1 before subtracting. 20 LOGARITHMIC TABLES. If less, write the remainder unchanged, putting O's under the equal figures. Example. 72293516214394 24268518014198 48024998200196 Here 7 2 =5; because 4 > 2, we write 4 12 4 = 8; because 2 = 2 and 6 < 9, we write 8; and write in the following place. 96 = 3; be- cause 8 > 3, we write 2. 13 8 = 5; 5 = 5; 1 = 1; 8>6; so under 13 8 we write 4, with 9's in the two next places. 16 8 = 8; because < 2, we write 8. 2 = 2; 1 = 1; 4 = 4; 1 < 3; so under 2 we write 2, followed by O's. 3 1 = 2; because 9 = 9, 8 > 4, we write 1, with 9 in the next place. 14 8 = 6, which we write as the last figure. EXERCISES. The preceding exercises in addition will serve as exercises in sub- traction by subtracting each line from that above or below it. The student should be able to subtract with equal facility whether the minuend is written above or below the subtrahend. Mental addition and subtraction. When an expert computer has to add or subtract two logarithms, as in forming a product or quo- tient of two quantities, he does not necessarily write both of them, but prefers to write the first and, taking the other mentally, add (or subtract) each figure in order from left to right, and write down the sum (or difference). He thus saves the time spent in writing one number, and, sometimes, the inconvenience of writing it where there is not sufficient room for it. This process of inverted addition is most useful in adding the proportional part in taking a logarithm from the table. It is then absolutely necessary to save the computer the trouble of copying both logarithm and proportional part. Expert computers can add seven-figure logarithms in this way without trouble. But with those who do not desire to become ex- perts it will be sufficient to learn to add two or three figures, so as to be able to take a five-figure or seven -figure logarithm from the table without writing anything but the result. 11. Imperfections of Logarithmic Calculations. Nearly all practical computations with logarithms are affected by certain sources of error, arising from the omission of deci- mals. It is important that these errors should be understood in IMPERFECTIONS OF LOGARITHMIC CALCULATIONS. 21 order not only to avoid them so far as possible, but to avoid spend- ing labor in aiming at a degree of accuracy beyond that of which the numbers admit. Mathematical results may in general be divided into two classes: (1) those which are absolutely exact, and (2) those which are only to a greater or less degree approximate. As an example of the former case, we have all operations upon entire numbers which involve only multiplication and division. For example, the equations 16 s = 256 j^_16 6 a ~ 9 are absolutely exact. But if we express the fraction | as a decimal fraction, we have ^ = .142857. ., etc., ad infinitum. Hence the representation of \ as a decimal fraction can never be absolutely exact. The amount of the error will depend upon how many decimals we include. If we use only two decimals we shall certainly be within one hundredth; if three, within one thou- sandth, etc. Hence the degree of accuracy to which we attain de- pends upon the number of decimals employed. By increasing the number of decimals we can attain to any degree of accuracy. As an example, it is shown in geometry that if the ratio of the circumfer- ence of a circle to its diameter be written to 35 places of decimals, the result will give the whole circumference of the visible universe without an error as great as the minutest length visible in the most powerful microscope. There are no numbers, except the entire powers of 10, of which the logarithms can be exactly expressed in decimals. We must therefore omit all figures of the decimal beyond a certain limit. The number of decimals to be used in any case depends upon the degree of accuracy which is required. The large tables of logarithms con- tain seven decimal places, and therefore give results correct to the ten-millionth part of the unit. This is sufficiently near the truth in nearly all the applications of logarithms. With five places of decimals our numbers will be correct to the hundred-thousandth part of a unit. This is sufficiently near for most practical applications. Accumulation of errors. When a long computation is to be made, the small errors are liable to accumulate so that we cannot rely upon this degree of accuracy in the final result. The manner 22 LOGARITHMIC TABLES. in which the tables are arranged so as to reduce the error to a mini- mum may be shown as follows: We have to seven places of decimals log 17 = 1.230 448 9 " 18 = 1.2552725 When the tables give only five places of decimals the two last figures must be omitted. If the tables gave log 17=. 230 44, the logarithm would be too small by 89 units in the seventh place. It is therefore increased by a unit in the fifth place, and given .23045. This quantity is then too large by 11, and is therefore nearer the truth than the other. The nearest number being always given, we have the result: Every logarithm in the table differs from the truth ~by not more than one half a unit of the last place of decimals. Since the error may range anywhere from zero to half a unit, and is as likely to have one value as another between those limits, we -conclude: The average error of the logarithms in the tables is one fourth of Common ratio = ^ ; Number of terms = n. By College Algebra, 212, the sum of this progression will be / 1 \n , '-(iT-J (1 + <)-! If the first payment is to be made immediately, instead of at the end of a year, the last or th payment will be due in n 1 years, and the progression will be p + r+p + (i + />)* + ' ' * + (i+P)' 1 - 1 * We find the sum of the geometric progression to be EXERCISES. 1. What is the present value of 15 annual payments of $85 each, of which the first is due in one year, the rate being 5 per cent? We find by substitution Present value = 85 __!05^-1__ J5_ 1.05- 1 1.05 16 - 1.05 16 1.05" ' .05 _1700(1.05 16 -1) (1.05) 16 ' log 1.05, 0.021189 1.05", 2.07895 _ 15 1.05" -1, 1.07895 log 1.05", 0.31784 log, 0.03300 co-log 1.05 15 , 9.68216 log 1700, 3.23045 Value, $882.28 log value, 2L945 6? 30 LOQAR1THM10 TABLES. 2. The same thing being supposed, what would be the present value if the rate of interest were 4 per cent ? Ans. $945.80 3. What is the present value of 25 annual payments of $1000 each, the first due immediately, if the rate of interest is 3 per cent ? Ans. $17,935 4. A debtor owing $10,000 wishes to pay it in 10 equal annual instalments, the first being payable immediately. If the rate of interest is 6 per cent, how much should each payment be? Ans. $1281.76. NOTE. This problem is the reverse of the given one, since, in the equation (2), we have given 2 a = 10000, p = 0.06, and n = 10, to find p. 5. The same thing being supposed, what should be the annual payment in case the payments should begin in a year? Ans. $1358.69. Perpetual annuities. If the rate of interest were zero, the present value of an infinity of future payments would be infinite. But with any rate of interest, however small, it will be finite. For if, in the first equation (1), we suppose n infinite, f J will converge toward zero, and we shall have This result admits of being put into a concise form, thus: Since 2 is the present value of the perpetual annuity p, the annual interest on this value will be p*S. But the equation (3) gives p2=p. Hence: The present value of a perpetual annuity is the sum of which the annuity is the annual interest. Example. If the rate of interest were 3| per cent, the present value of a perpetual annuity of $70 would be $2000. EXERCISES. 1. A government owing a perpetual annuity of $1000 wishes to pay it off by 10 equal annual payments. If the rate of interest is 4 per cent, what should be the amount of each payment? Ans. $3082.30. 2. A government bond of $100 is due in 15 years with interest at 6 per cent. The market rate of interest having meanwhile fallen to 3| per cent, what should be the value of the bond? NOTE. We find, separately, the present value of the 15 annual instalments of interest, and of the principal. TABLE II. MATHEMATICAL CONSTANTS. 14. In this table is given a collection of constant quantities which frequently occur in computation, with their logarithms. The logarithms are given to more than five decimals, in order to be useful when greater accuracy is required. When used in five- place computations, the figures following the fifth decimal are to be dropped, and the fifth decimal is to be increased by unity in case the figure next following is 5 or any greater one. TABLES III. AND IT. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 15. By means of these tables the logarithms of the six trigono- metric functions of any angle may be found. The logarithm of the function instead of the function itself is given, because the latter is nearly always used as a factor. We begin by explaining Table IV., because Table III. is used only in some special cases where Table IV. is not convenient. I. Angles less than 45. If the angle of which a function is soiight is less than 45, we seek the number of degrees at the top of the table and the minutes in the left-hand column. Then in the line opposite these minutes we find successively the sine, the tan- gent, the cotangent, and the cosine of the angle, as given at the heading of the page. Example. log sin 31 27' = 9.717 47; log tan 31 27' = 9.78647; log cotan 31 27' = 0.213 53; cos 31 27' = 9.93100. The sine, tangent, and cosine of this angle being all less than unity, the true mantissas of the logarithm are negative; they are therefore increased by 10, on the system already explained. If the secant or cosecant of an angle is required, it can be found by taking the arithmetical complement of the cosine or sine. It is shown in trigonometry that secant = : , cosine and cosecant = -; . sine Therefore log secant = log cosine = co-log cosine; log cosec = log sine = co-log sine. We thus find log sec 31 27' = 0.069 00; log cosec 31 27' = 0.282 53. After each column, upon intermediate lines, is given the differ- LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 33 ence between every two consecutive logarithms, in order to facilitate interpolation. In the case of tangent and cotangent, only one column of differ- ences is necessary for both functions. If we use no fractional parts of minutes, no interpolation is necessary; but if decimals of a minute are employed, we can inter- polate precisely as in taking out the logarithms of numbers. "Where the differences are very small they are sometimes omitted. Tables of proportional parts are given in the margin, the use of which is similar to those given with the logarithms of numbers. Example 1. To find the log sin of 31 27'.7. We have from the tables, log sin 31 27' = 9.717 47 Under diff. 20, P.P. for 7, 14 log sin 31 27'. 7 = 9.71761 Ex. 2. To find log cot 15 44'. 34. The tables give log cot 15 44' = 0.550 19 Under diff. 48, opposite 0.3, P.P., - 14.4 " " 0.4 -7- 10, - 1.9 log cot 15 44'. 34, 0.55003 Since the tabular quantity diminishes as the angle increases, the proportional parts are subtractive. EXEECISES. Find from the tables : 1. log cot 43 29'. 3; 2. log tan 43 29'. 3; 3. log cos 27 10'. 6; 4. log sin 27 10'. 6; 5. log tan 12 9'. 43; 6. log cot 12 9'. 43. In the case of sines and tangents of small angles the differences vary so rapidly that in most cases the exact difference will not be found in the table of proportional parts. In this case, if the pro- portional parts are made use of, a double interpolation will generally be necessary to find the fraction of a minute corresponding to a given sine or tangent. If only tenths of minutes are used, an expert com- puter will find it as easy to multiply or divide mentally as to refer to ihe table. II. Angles between 45 and 90. It is shown in trigonometry that if we compute the values of the trigonometric functions for the 34 LOGARITHMIC TABLES. first 45, we have those for the whole circle by properly exchanging them in the different parts of the circle. First, if we have a + ft = 90, then a and ft are complementary functions, and |. sin ft = cos a\ tan ft = cotan a. Therefore if our angle is between 45 and 90, we may find its complement. Entering the table with this complement, the com- plementary function will then be the required function of the angle. Example. To find the sine of 67 23', we may enter the table with 22 37' (= 90- 67 23') and take out the cosine of 22 37', whichfis the required sine of 67 23. To save the trouble of doing this, the complementary angles and the complementary denominations of the functions are given at the bottom of the page. The minutes corresponding to the degrees at the bottom are given on the right hand. Therefore: To find the trigonometric functions corresponding to an angle between 45 an d 90, we take the degrees at the bottom, of the page and the minutes in the right-hand column. The values of the four func- tions log sine, log tangent, log cotangent, and log cosine, as read at the bottom of the page, are then found in the same line as the minutes. Example 1. For 52 59' we find log sin = 9.90225; log tan = 0.12262; log cot = 9.877 38; log cos = 9.77963. Ex. 2. To find the trigonometric functions of 77 17'. 28. 77 17' 9. sin. tan. cot. cos. 98921 0.64653 9.35347 9.34268 P.P. for 0.2 + 0.6 + 11.8 - 11.8 - 11.2 " 0.08 + 0.2 + 4.7 - 4.7 - 4.5 9. 989 22 0. 646 70 9. 353 30 9. 342 52 Then log sec = co-log cos = 0.657 48; log cosec = co-log sin = 0.010 78. EXERCISES. Find the logarithms of the six functions of the following angles: 1. 45 50'. 74; 3. 74 0'.68; 2. 4849'.37; 4. 83 59'.62. LOGARITHMS OF TRIGONOMETRIC III. When the angle exceeds 90. EULE. Subtract from the angle the greatest multiple of 90 which it contains. If this multiple is 180, enter the table with the excess of the angle over 180 and take out the functions required, as if this excess were itself the angle. If the multiple is 90 or 270, take out the complementary func- tion to that required. By then assigning the proper algebraic sign, as shown in trigo- nometry, the complete values of the function will be obtained. The computer should be able to assign the proper algebraic sign according to the quadrant, without burdening his memory with the special rules necessary in each + case. This he can do by carrying Sine positive in his mind's eye the following scheme. He should have at com- mand the arrangement of the four quadrants as usually represented in trigonometry, so as to know, when an angle is stated, where it will fall relatively to the horizontal and ver- tical lines through the centre of the circle. Then, in the case of Sine or cosecant. If the angle Sine negative is above the horizontal line (which it is between and ISO ), the sine is positive; if below, negative. Cosine or secant. If the angle is to the right of the vertical central line (as it is in the first and fourth quadrants), the cosine and secant are positive; if to the left (as in the second and third quad- rants), negative. Tangent or cotangent. Through the opposite first and third quad- rants, positive; through the opposite second and fourth quadrants, negative. Example 1. To find the tangent and cosine of 122 44'. Sub- tracting 90, we enter the table with 32 44' and find log cot 32 44' = 0.19192; log sin 32 44' = 9.73298. T nerefore, writing the algebraic sign before the logarithm, we hare log tan 122 44' = - 0.191 92; log cos 122 44' = 9.732 98. 86 LOGARITHMIC TABLES. Ex. 2. To find the sine and cotangent of 322 58'. Entering the table with 52 58' = 322 58' 270, and taking out the complementary functions, we find log sin 322 58' = - 9.779 80; log cot 322 58' = 0.122 36. Ex. 3. To find the sine and tangent of 253 5'. Entering with 73 5', we take out the sine and tangent, finding log sin 253 5' = 9.890 79; log tan 253 5' = -f 0.516 93. Ex. 4. To find the six trigonometric functions of 152 38'. We have log sin 152 38' = log cos 62 38' pos. = + 9.662 46; log cos 152 38' = log sin 62 38' neg. = 9.948 45; log tan 152 38' = log cot 62 38' neg. = 9.71401; log cot 152 38' = log tan 62 38' neg. = 0.285 99; log sec = co-log cos = 0.05155; log cosec = co-log sin = + 0.337 54. EXERCISES. Find the six trigonometric functions of the following angles: 276 29'.3; 66 0'.5; 96 59'.8; 252 20'.3; 318 10'. 7; - 25 22'.2; -155 30'. 7. 16. Method of Writing the Algebraic Signs. As logarithms are used in computation, they may always be con- sidered positive. It is true that the logarithms of numbers less than unity are in reality negative, but, for convenience in calculation, we increase them by 10, so as to make them positive. The number corresponding to a given logarithm may, in compu- tation, be positive or negative. There are two ways of distinguishing the algebraic sign of the number, between which the computer may choose for himself. I. Write the algebraic sign of the number before the logarithm. As usually interpreted, the algebraic sign written thus would apply to the logarithm, which it does not. It is therefore necessary for the ANGLE CORRESPONDING TO A GIVEN FUNCTION. 37 computer to bear in mind that the sign belongs, not to the loga- rithm, as written, but to the number. II. Write the letter n after the logarithm when the number i& negative. This plan is theoretically the best, but, should the com- puter accidentally omit the letter, the number will be treated as positive, and a mistake will be made. It therefore requires vigilance on his part. An improvement would be to write a letter not likely to be mistaken for n, s for instance, after all positive logarithms. 17. To Find the Angle Corresponding to a Given Trigonometric Function. Disregarding algebraic signs, there will always be four angles corresponding to each function, one in each quadrant. These angles, will be: The smallest angle, as found in the table; This angle increased by 180; The complementary angle increased by 90; The complementary angle increased by 270. For instance, for the angle of which log tan is 0.611 92, we find 76 16'. But we should get this same tangent for 103 44', 256 16', and 283 44'. Of the four functions corresponding to the four angles, two will always be positive and two negative; so that, in reality, there will only be two angles corresponding to a function of which both the- sign and the absolute value are given. These values are found by selecting from the four possible ones the two for which the functions have the given algebraic sign. After selecting them, they may be checked by the following theorems, which are easily deduced from: the relations between the values of each function as given in trigo- nometry: The sum of the two angles corresponding to the same sine is ISO ' vr 540. The sum of the two angles corresponding to the same cosine is 360. The difference of the two angles corresponding to the same tangenk is 180. Which of the two possible angles is to be chosen depends upon the conditions of the problem or the nature of the figure to which the angle belongs. If neither the conditions nor the figure decida the question, the problem is essentially ambiguous, and either ^~ Doth angles are to be taken. 38 LOGARITHMIC TABLES. EXEECISES. Find tne pairs of values of the angle a from the following values of the trigonometric functions: 1. log sin a = + 9.902 43; 12. log sec a = -{- 0.221 06; 2. log sin a = - 9.902 43; 13. log sec a = - 0.221 06; 3. log cos a = + 9.902 43; 14. log sec a - - 0.099 20; 4. log cos a = 9.902 43; 15. log sec a + 0.123 46; 5. log tan a = + 0.143 16; 16. log sin a = -f 8.990 30; 6. log tan a = 0.143 16; 17. log sin a ~ 8.990 30; 7. log cot a = -f 0.143 16; 18. log cos a = + 9.218 67; 8. log cot a 0.143 16; 19. log cos a = 9.218 67; 9. log tan a = - 9.024 81; 20. log tan a = - 9.136 90; 10. log tan a = - 0.975 19; 21. log tan a = -f- 9.136 90; 11. log tan a = + 0.975 19; 22. log cot a = + 9.136 90. 18. Cases when the Function is very Small or Great. When the angle of which we are to find the functions approaches to zero, the logarithms of the sine, tangent, and cotangent vary so .rapidly that their values to five figures cannot be readily interpolated. The same remark applies to the cosine, cotangent, and tangent of angles near 90 or 270. The mode of proceeding in these cases will depend upon circumstances. In the use of five-place logarithms, there is little advantage in carrying the computations beyond tenths of minutes, though the hundredths may be found when the tangent or cotangent is given. Where greater accuracy than this is required, six- or seven-place tables must be used. If the angles are only carried to tenths of minutes, there is no necessity for taking out the sine, tangent, or cotangent to more than four decimals when the angle is less than 3, and three decimal? suffice for angles less than 30'. The reason is that this number of decimals then suffice to distinguish each tenth of minute. When the decimals are thus curtailed, an expert computer will be able to perform the multiplication and division for the tenths o? minutes mentally. If, however, this is inconvenient, the following rule may be applied. To find the log sine or log tangent of an angle less than 2 to four places of decimals: EULE. Enter the table of logarithms of numbers with the valu* WHEN THE FUNCTION IS VERY SMALL OR GREAT. 39 of the angle expressed in minutes and tenths, and take out the loga- rithm. To this logarithm add the quantity 6.4637. The sum will be the log sine, and the log tangent may be assumed to have the same value. Example 1. To find log sin 1 2$'. 6. 1 22'. 6 = 82'.6 log 82'.6 = 1.9170 constant, 6.4637 log sin 1 22'. 6, 8.3807 This rule is founded on the theorem that the sines and tangents of very small arcs may be regarded as equal to the arcs themselves. Since, in using the trigonometric functions, the radius of the circle is taken as unity, an arc must be expressed in terms of the unit radius when it is to be used in place of its sine or tangent. Now, it is shown in trigonometry that the unit radius is equal to 57. 2958 or 3437'. 747 or 206 264". 8. Hence we must divide the number of angular units in the angle by the corresponding one of these coef- ficients to obtain the length of the corresponding arcs in unit radius. Now, log 3437. 747 = 3.5363 co-log 6.4637 which may be added instead of subtracting the logarithm. To find the cosine of an angle very near 90, we find the sine of its complement, which will then be a very small angle, positive or negative. EXERCISES. Find to four places of decimals: 1. log sin 22'. 73; 2. log sin 1 1M2; 3. log cos 90 0'.78; 4. log tan 88 59'. 35; 5. log cot 90 28'. 76; 6. log cos 89 22'.23; 7. log sin 0'.25. If an angle corresponding to a given sine or tangent is required, the rule is: From the given log sine or tangent subtract 6.4637 or add 3.5363. The result is the logarithm of the number of minutes. Of course this rule applies only to angles less than 2, in the value of which only tenths of minutes are required. 40 LOGARITHMIC TABLES. EXERCISES. Find a from: 1. log sin a = 7.2243; 3. log tan a = - 3.8816; 2. log cot a = 2.8816; 4. log cos a = 6.9218. When the small angle is given in seconds. Although the com, puter may take out his angles to tenths of minutes, cases often arise in which a small angle is given in seconds, or degrees, minutes, and .seconds, and in which the trigonometric function is required to five decimals. In this case the preceding method may not always give = accurate results, because the arc and its sine or tangent may differ by .a greater amount than the error we can admit in the computation. Table III. is framed to meet this case. The following are the quantities given: In the second column : The argument, in degrees and minutes, as already explained for Table IV. In the first column : This argument reduced to seconds. From this column the number of seconds in an arc of less than 2, given in degrees, minutes, and seconds, may be found at sight. Example. How many seconds in 1 28' 39' ? In the table, before 1 28', we find 5280*, which being increased by 39" gives 5319", the number required. Col. 3. The logarithm of the sine of the angle. This is the same as in Table IV. Col. 4. The value of log sine minus log arc; that is, the difference between the logarithm of the sine and the logarithm of the number of seconds in the angle. Col. 5. The same quantity for the tangent. Cols. 6 and 7. The complements of the preceding logarithms, dis- tinguished by accents. The use of the tables is as follows. To find the sine or tangent of an angle less than 2: Express the angle in seconds ~by the first two columns of the table. Write down the logarithm in column 8 or column T, according as the sine or a tangent is required. Find from Table I. the logarithm of the number of seconds. Adding this logarithm to S or T, the sum will be the log sine or log tangent. Example. Find log sin 1 2' 47'.9. 8, 4.68555 1 2' 47*. 9 = 3767*.9; log, 3.576 10 log sin 1 2' 47*.9, 8.261 65 WHEN THE FUNCTION IS VERY SMALL OR GREAT. 4J To find the arc corresponding to a given sine or tangent: Find in the column L. sin. the quantity next greater or next smaller than the given logarithm. Take the corresponding value of S' or T' according as the given function is a sine or tangent, and add it to the given function. The sum is the logarithm of the number of seconds in the required angle. Example. Given log tan x = 8.401 25, to find x. log tan x, 8.401 25 T', 5.31433 logz, 3.71558 x = 5194'. 9 = 1 26' 34'. 9, from col. 2. EXERCISES. Find: 1. log sin 20' 20'.25; 2. log tan 0' 1'.2273; 3. log sin 1 59' 22'. 7; 4. log tan 1 0'59'.7. Find x from: 1. log tans = 8.42796; 2. log tan x = 7.42796; 3. log tan x = 6.427 96; 4. log sin x = 5.35435; 5. log sin x = 4.226 19; 6. log sin x = 8.540 78. When the cosine or cotangent of an angle near 90 or 270 is re- quired, we take its difference from 90 or 270, and find the comple- mentary function by the above rules. Remark. The use of the logarithms of the trigonometric func- tions is so much more extensive than that of the functions themselves that the prefix "log" is generally omitted before the designation of the logarithmic function, where no ambiguity will result from the omission. TABLE V. NATURAL SINES AND COSINES. 19. This table gives the actual numerical values of the sine and cosine for each minute of the quadrant. To find the sine or cosine corresponding to a given angle less than 45, we find the degrees at the top of a pair of columns and the minutes on the left. In the two columns under the degrees and in the line of minutes we find first the sine and then the cosine, as shown at the head of the column. A decimal point precedes the first printed figure in all cases, ex- cept where the printed value of the function is unity. If the given angle is between 45 and 90, find the degrees at the bottom and the minutes at the right. Of the two numbers above the degrees, the right-hand one is the sine and the left-hand one the cosine. For angles greater than 90 the functions are to be found ac- cording to the precepts given in the case of the logarithms of the sines and tangents. TABLE VL ADDITION AND SUBTRACTION LOGARITHMS. 20. Addition and subtraction logarithms are used to solve the problem: Having given the logarithms of two numbers, to find the logarithm of the sum or difference of the numbers. The problem can of course be solved by finding the numbers corresponding to the logarithms, adding or subtracting them, and taking out the logarithm of their sum or difference. The table under consideration enables the result to be obtained by an abbrevi- ated process. I. Use in addition. The principle on which the table is con- structed may be seen by the following reasonings. Let us put 8=*a + b, a and b being two numbers of which the logarithms are given. "We shall have putting, for clearness, x = -. We then have log S = log a + log (1 + x). Since log a and log b are both given, we can find log x from the equation log x = log b log a y which is therefore a known quantity. Now, for every value of log x there will be one definite value of each of the quantities x, 1 + #> and log (I-{- x). Therefore a table may be constructed showing, for every value of log x, the correspond- ing value of log (1 -f- x). Such a table is Table VI. The argument, in column A, being log x, the quantity B in the table is log (1 -j- x). Example, log 0.25 = 9. 397 94. Entering the table with A = 9.397 94, we find J3 = 0.096 91, which is the logarithm of 1.25. 44 LOGARITHMIC TABLES. Therefore, entering the table with log x as the argument, we take out log (1 -}- x), which added to log a will give log S. We have therefore the following precept for using the table in addition: Take the difference of the two given logarithms. Enter the table with this difference as the argument A, and take out the quantity B. Adding B to the subtracted logarithm, the sum will be the required logarithm of the sum. It is indifferent which logarithm is subtracted, but convenience in interpolating will be gained by subtracting the greater logarithm from the lesser increased by 10. The number B will then be added to the greater logarithm. Example. Given log m = 1.62974, log n = 2.203 86 ; find, log (m + n). The required logarithm is found in either of the following two ways: Jog m, 1.629 74 (1) log , 0.676 76 (4) log 'n, 2.203 86 (2) log m, 1.629 74 (1) B, 0.102 64 (4) log , 2.203 86 (2) A = log w -r- n, 9.425 88 (3) log n -=- ra, 0.574 12 (3) log ( m + n ), 2.306 50 (5) log (m + ), 2.306 50 (5) The figures in parentheses show the order in which the numbers are written. EXERCISES. Log a and log b having the following values, find log (a -f- b). 1. log a = 1.700 37; log b = 0.921 69. 2. log a = 0.624 60; log b = 9.881 26. 3. log a = 9.791 86; log b = 9.322 09. 4. log a = 1.601 62; log I = 1.306 06. 5. log a = 0.792 90; log b = 9.221 27. 6. log a = 0.601 32; log b = 9.001 68. 7. log a = 4.796 43; log b = 3.981 86. II. Use in subtraction. The problem is, having given log a and log I, to find the logarithm of D - a - b. Wehave ADDITION AND SUBTRACTION LOGARITHMS. 45 Since log -r is found by subtracting log b from log a, if we can find log IT- l) from log j-, the problem will be solved. From the construction of the table already explained, if we have we must have We now have the following precept for subtraction: Subtract the lesser of the given logarithms from the greater. Enter the table so as to find the difference of the logarithms in the numbers B of the table. Add the corresponding value of A to the lesser of the given loga- rithms. The sum will be the logarithm of the difference. . Example. Find log (n m) in the example of the preceding section. log n, 2.203 86 (1) log m, 1.62974 (2) ^,0.43945 (4) log = B, 0.57412 (3) m log (n - m), 2.069 19 (5) EXERCISES. Find the logarithms of the differences of the quantities a and b in the preceding section. Remark. In the use of addition and subtraction logarithms, the precepts apply to numerical sums and differences, without respect to the algebraic signs of the quantities. For example, the algebraic difference between -f- 1473 and 29 462 is to be found by addition, and the algebraic sum of a positive and negative quantity by subtraction. Case where the quotient is large. Near the end of the table, A and B become nearly equal; the structure of the table is therefore changed so as to simplify its use. It is evident that if b is very small compared with a, the logarithms of a -f- b and a b will not differ much from the logarithm of a itself. Hence, in this case, we shall have smaller numbers to use if we can find the quantity which must be added to log a to give log (a -f- b), or subtracted from 46 LOGARITHMIC TABLES. log a to give log (a 5). Now, the equations already written give, when a > b, log a = log ft -{- A, log (a + 1) = log b + ; whence, by subtraction, log (a + b) log a = B A, or log (a + b) = log a + - A. (with Arg. ^) We find in the same way, log (a - b) = log a - (B - A), (with Arg. B) Now, whenever log a log ~b is greater than 1.65, we shall find it more convenient to take out B A from the table than either A or B. We notice that the last two figures of B in this part of the table vary slowly, and we need only attend to them in interpolate ing. For instance, in the horizontal line corresponding to A = 1.66 we find: for A = 1.660 00; B = 1.669 40; B - A = .009 40; .66100; .67038; .00938; .66200; .67136; .00936; .66300; .67233; ,00933; .66400; .67331; .00931; .66500; .67429; .00929; etc. etc. etc. The interpolation of B A is now very easy whether the quan- tity given is A or B. We note that B A has but three significant figures, of which the first is found in column zero, and the other two are the last two figures of B as printed. As an example, let us find log (a + b) from logfl = 2.79163 log = 1.12819 A = 1.66344 Entering the table with this value of A, we find by column that B A falls between .009 40 and .009 19. Following the hori- zontal line A = 1.66 to column 3 and interpolating the last two figures between 33 and 31 for .44, with the difference 2, we find B - A = .00932 Then log a = 2. 791 63 log(0 + b) = 2.80095 Next, if log (a b) is required, we have to find the difference 1.663 44 in the part B of the table. We find in the table: for B = 1.662 55; B - A = .009 55; for B = 1.663 53; B - A = .009 53. ADDITION AND SUBTRACTION LOGARITHMS. 47 Therefore for B = 1.663 44; B - A = .009 53. Subtracting this from log a, we have log (a - b) = 2.78210. EXEECISES. ^'nd log (a + ) and log (a ) from: 8. log a = 0.367 02; log b = 8.462 83. 9. log a = 0.001 26; log b = 8.329 07. 10. log a = 2.069 23; log b = 0.110 85. 11. log a = 5.807 35; log b = 3.83809. For values of A and B greater than 2.00, the table is so arranged that no interpolation at all is necessary. The computer has only to find what value of A or B given in the table comes nearest his value of log a log b and take the corresponding value of B A. He must remember that column A is to be entered for addition, and B for subtraction. In this part of the table A and B are given to fewer than five decimals; because five decimals are not necessary to give B A with accuracy. The nearer the end of the table is approached, the fewer the decimals necessary in taking the difference. Example. Find log (a -f b) and log (a b) from log a = 1.265 32 log b = 9.22230 log a log, 2.04302 Entering column A with this difference, we find the nearest tabu- lar value of A to be 2.0425, to which corresponds B A = .003 92. Hence log (a + b) = 1.265 32 -f .003 92 = 1.269 24. Entering column B with the same difference, we find B A SB ' .00395; whence log (a - b) = 1.265 32 - .003 95 = 1.261 37. EXERCISES. Find log (a -f- b) and log (a b) from: 1. log a = 4.069 05; log b = 2.001 32. 2. log a = 3.926 93; log b = 1.201 59. 3. log a = 3.061 64; log b = 0.126 15. 4. log a = 1.22C 68; log b = 7.321 56. 5. log a = 0.693 17; log 5 = 6.010 23. 6. log a = 2.306 20; log b = 7.023 01. 48 LOGARITHMIC TABLES. Case of nearly equal numbers. Near the beginning of the table the reverse is true: it is not possible to find A with accuracy to five places of decimals. But here the value of A taken from the tables, though it be found to only two, three, or four places of decimals, will give as accurate a result as the computation of a and b to five places will admit of. Let us suppose, for example, that we have to find log (a b) from log a = 9.883 15 log b = 9.88296 B = 0.00019 We find ^4 = 6.64-10; whence log (a b) = 6.52 10. We note that the value of A may be 6.63 or 6.65 as well as 6.64, so that the result cannot be carried beyond two decimals. To show that these two are as accurate as the work admits of* we find the natural numbers a and b from Table I. a= 0.76410 b = 0.76377 a-b = 0.00033 Since a b has but two significant figures, and the first of these is less than 5, two figures in the logarithm are all that can be accurate. TABLE YIL SQUARES OF NUMBERS. 21. By means of this table the square of any number less than 1000 may be found at sight, and that of any number less than 10 000 by a simple and easy interpolation. The first page gives the squares of the first 100 numbers, which it is often convenient to have by themselves. On the second and third pages (98 and 99) the hundreds of the number to be squared are found at the tops of the several columns, and the tens and units in the left-hand column. The first three or four figures of the square are in the column under the hundreds, and opposite the tens and units, and the last two figures on the right of the page after the column 9 + + Examples. The square of 634 is 401 956; " " 329 " 108241; " " 265 " 70225; " " 153 " 23409; 999 " 998001. The same table may be used for any number of three significant figures by attention to the position of the decimal-point. Thus: 51100 9 = 2611210000; 511 s = 261121; 51.1' = 2611.21; 5.11' = 26.1121; 0.511 9 = 0.261121. When there are four significant figures, an interpolation may be executed in several ways. If n be the nearest number the square of ^hich is found in the table, and h the excess of the given number over this, so that n -{ his the number whose square is required, we shall have where N = n + h 9 the given number. 50 LOGARITHMIC TABLES. We may therefore find the square of 257.4 in the following way: 257 s = 66 049 514.4 X .4 = 205.76 (257. 4) 2 = 66254.76 To find the square of 9037 we proceed thus: 9037 9030 a = 81 540 900 18067 X 7 = 126469 9037 a = 81667369 In many cases only one more figure will be required in the square than in the given number. The square can then be interpolated with all required accuracy by the differences, the last two figures of which are found in the last column of the table, while the remaining figures are found by taking the difference between two consecutive numbers in the principal column. To return to the last example, we find the difference between 257 a and 258 3 to be 515, the first figure being the difference between 660 and 665, and the last two, 15, in the last column. Then 257 3 = 66 049 515 X 0.4 = 206 (257. 4) a = 66255 which is correct to the nearest unit. It will be remarked that the two methods are substantially the same when only five figures are sought in the result. The substantial identity rests upon the general theorem that The difference of the squares of two consecutive numbers is equal to the sum of the numbers. We prove this theorem thus: (n + l) a - n* = 2n + 1 = n + (n + 1). When the tabular difference is taken in the way already described, it will often happen that the difference between the numbers in the columns of hundreds is to be diminished by unity. Thus, although 4173 4160 = 13, the difference between 645 s and 646 2 is not 1391, but 1291. These cases are noted by the asterisk after the number in the last column. The squares of numbers of more than four figures may be found in the same way, but in such cases it will generally be easier to use logarithms than the table of squares. TABLE VIII. TO CONVERT HOURS, MINUTES, AND SECONDS INTO DECIMALS OF A DAY, AND VICE VEKSA. . The familiar method of solving this problem is to convert the seconds into decimals of a minute, and the minutes into decimals of an hour, by dividing by 60, and then the hours into decimals of a day by dividing by 24. The reverse problem is solved by multiply- ing by 24, 60,- and 60. Table VIII. enables us to perform these operations without divi- sion. Column D gives each hundredth of a day, but its numbers may also be regarded as ten thousandths or millionths of a day, according to which of the following three columns is used. In column H.M.S. are found the hours, minutes, and seconds corresponding to these hundredths. In the next column is one hundredth of column H. M. S. 9 or the minutes and seconds in the number of ten thousandths of a day in column D. Finally, column ' ' shows the number of iuu seconds in the number of millionths of a day found in column D. Example. To convert O d .532 946 into hours, minutes, and seconds. O d .53 = 12 h 43 m 12 s .002 9 = 4 m 10 8 .56 .000046= 3 8 .97 O d .532 946 = 12 h 47 m 26 8 .53 It will be seen that we divide the figures of the given decimal of a day into pairs, and enter the three columns of time with these three pairs in succession. If seven decimals are given, we may interpolate the last number, as in taking out a logarithm. Example. Convert O d .050 762 7. O d .05 = l h 12 m O 8 .000 7 = l m s . 48 .000062 = 5 s . 36 .0000007 = .7X.08= 0-.06 l h 13 m 5 8 .90 52 LOGARITHMIC TABLED In practice the computer will perform the interpolation mentally, adding .7 X .08 = .06 to the number 5.36 of the table in his head, and writing down 5 s . 42 as the last quantity to be added. EXERCISES. Convert into hours, minutes, and seconds: 1. O d .2030792; 2. O d . 783 605 8; 3. O d .0102034; 4. O d . 990 990 9. To use the table for the reverse operation, we proceed as in the following example: It is required to convert 17 h 29 m 30 s . 93 into decimals of a day. Looking in the table, we find that the required decimal is between 0.72 and 0.73. Hence the first two figures are 0.72, the equivalent of 17 h 16 m 48 s . Subtracting the lat- 1711 ggm 30*. 93 ter from the given number, we 0.72 = 17 h 16 m 48 s have a remainder 12 m 42 8 .93, to be 12 m 42 s . 93 ,,. . . H.M.S. _. -0088 = 12 m 40 s . 32 sought for in column m ~. This >000 03 2 = " ~2^61 gives 88 as the next two figures. Subtracting the equivalent of .0088 or 12 m 40 8 .32, we have left 2 s . 61, which we are to seek in TT -r nr column ' ' '. We find the corresponding number of column D to 100 be 302. Hence 17 h 29 m 30 8 .93 = O d . 728 830 2. In solving this problem the computer should be able, after a little practice, to perform the subtractions and carry the remainders men- tally, thus saving himself the trouble of writing down the numbers. EXERCISES. Take the answers obtained from the four preceding exercises, subtract each result from 24 h O m B , change the remainder to deci- mals of a day, and see if when added to the decimals of the preceding exercises the sum is l d . 000 000 0, as it should be. TABLE IX. TO CONVERT TIME INTO ARC, AND VICE VERSA. 23. In astronomy the right ascensions of the heavenly bodies are commonly given in hours, minutes, and seconds, the circumfer- ence being divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds. Since 360 r = one circumference, ve have l h = 15; l m = 15'; 1 s = 15'; the signs h , m , and 8 indicating hours, minutes, and seconds of time. Hence we may change time into arc by multiplying by 15, and arc into time by dividing by 15, the denominations being changed in each case. Table IX. enables us to do this by simple addition and subtraction by a process similar to that employed in changing hours, minutes, and seconds into decimals of a day. To turn time into arc, we find in the table the whole number of degrees contained in the time denomination next smaller than the given one, and subtract the former time denomination from the latter. Next we find the minutes of arc corresponding to the given time next smaller than the remainder, and again subtract. Lastly we interpolate the seconds corresponding to the second remainder. Example. Change 15 h 29 m 46 8 .24 to arc. Given time, 15 h 29 m 46 8 .24 The table gives 232 = 15 h 28 m Remainder, l m 46 8 .24 The table gives 26' = l m 44 8 Remainder, 2 s . 24 = 33'. 6 Hence 15 h 29 m 46 8 .24 = 232 26' 33'. 6. 54 LOGARITHMIC TABLES. The computer should be able to go through this operation with- out writing down anything but the result. The operation of changing arc into time is too simple to require description, but it is more necessary to write down the work. EXERCISES. Change the following times to arc, and then check the results by changing the arcs into time and seeing whether the original times are reproduced: 1. 7 h 29 m 17 8 .86; 2. O h 4 m 8 .25; 3. 12 h 4 m s . 25; 4. 13 h 48 m 16 9 .40; 5. 19 h 7 m 59 8 .92. TABLE X. TO CONVERT MEAN TIME INTO SIDEREAL TIME, AND SIDEREAL INTO MEAN TIME. 24. Since 365 solar days = 366^- sidereal days (very nearly),, any period expressed in mean time may be changed to sidereal time- by increasing it by its - part, and an interval of sidereal time- uDO./oO may be changed to mean time by diminishing it by its - part.. ODD. -CO The first part of the table gives, for each 10 minutes of the argu- ment, its Q part, by which it is to be increased. The second: part of the table gives the O^FITH P ar ^ f the argument. The small table in the margin shows the change for periods of less than 10 minutes. Example 1. To change 17 h 48 m 36 s . 7 of mean time to sidereai time. Given mean time, 17 h 48 m 36 s . 70 Corr. for 17 h 40 m , 2 m 54M3 Corr. for 8 m 37', 1 8 .41 Sidereal time, 17 h 51 m 32 s . 24 Ex. 2. To change this interval of sidereal time back to mean time. Corr. for 17 h 50 m , - 2 m 55". 29 Corr. for l m 32% - 8 .25 2 m 55 8 .54 Sidereal time, 17 h 51 m 32 8 .24 Mean time, 17 h 48 m 36 8 .70 EXERCISES. Change to sidereal time: 1. 3 h 42 m 36". 5 m. t.; 3. 22 h 3 m 5 .61 m. t* 2. 18 h 46 m 29 8 .82 " 4. O h l m 12 B .55 " Change to mean time: 5. O h 7 m 16 8 .3 sidereal time; 6. 22 h 17 m 29 8 .65 " 56 OF INTERPOLATION. OF DIFFERENCES AND INTERPOLATION.* 25. General Principles. "We call to mind that the object of a mathematical table is to enable one to find the value of a function corresponding to any value whatever of the variable argument. Since it is impossible to tabulate the function for all values of the argument, we have to construct the table for certain special values only, which values are generally equi- distant. For example, in the tables of sines and cosines in the present work the values of the functions are given for values of the argument differing from each other by one minute. The process of finding the values of functions corresponding to values of the argument intermediate between those given is called interpolation. We have already had numerous examples of interpolation in its' most simple form; we have now to consider the subject in a more general and extended way. In the first place, we remark that, in strictness, no process of interpolation can be applicable to all cases whatever. From the mere facts that To the number 2 corresponds the logarithm 0.301 03, " " " 3 " " " 0.477 12, we are not justified in drawing any conclusion whatever respecting "the logarithms of numbers between 2 and 3. Hence some one or more hypotheses are always necessary as the base of any system of interpolation. The hypotheses always adopted are these two: 1. That, supposing the argument to vary uniformly, the function varies according to some regular law. 2. That this law may be learned from the values of the function given in the table. These hypotheses are applied in the process of differencing, the * The study of this subject will be facilitated by first mastering so much of it as is contained in the author's College Algebra, 299-302. It is also recommended to the beginner in the subject that, before going over the algebraic developments, he practise the methods of computation according to the rules and formulae, so as to have a clear practical understand ing of the notation. He can then more readily work out the developments. GENERAL PRINCIPLES. 57 nature of which will be seen by the following example, where it is applied to the logarithms of the numbers from 30 to 37: Function. A' 4" A"< A" log 30. 1.47712 " 31. 1.491 36 ~ - 45 , " 32. 1.505 15 T JoS - 43 J * + 2 " 33. 1.518 51 J i9Q7 - 39 J * - 8 34. 1.531 48 J J*JJ _ 38 + + 1 35. 1.544 07 + }f* _ 36 + * + 1 " 36. 1.556 30 "tifS - 33 + ' "37. 1.568 20 ~* The column A' gives each difference between two consecutive values of the function, formed by subtracting each number from that next following. These differences are called first differences. The column A" gives the difference between each two consecu- tive first differences. These are called second differences. In like manner the numbers in the succeeding columns, when written, are called third differences, fourth differences, etc. Now if, in continuing the successive orders of differences, we find them to continually become smaller and smaller, or to converge to- ward zero, this fact shows that the values of the functions follow a regular law, and the first hypothesis is therefore applicable. In order to apply interpolation we must then assume that the intermediate values of the function follow the same law. The truth of this assumption must be established in some way before we can interpolate with mathematical rigor, but in practice we may suppose it true in the absence of any reason to the contrary. 26. Effect of errors in the values of the functions. In the pre- ceding example it will be noticed that if we continue the orders of differences beyond the fourth, they will begin to increase and become irregular. This arises from the imperfections of the logarithms, owing to the omission of decimals beyond the fifth, already described in 11. When we find the differences to become thus irregular, we must be able to judge whether this irregularity arises from actual errors in the original numbers, which ought to be corrected, or from the small errors necessarily arising from the omission of decimals* The great advantage of differencing is that any error, however small, in the quantities differenced, unless it follows a regular law, will be detected by the differences. To show the reason of this, we investigate what effect errors in the given functions will have upon the successive orders of differences. 68 OF INTERPOLATION. THEOREM. The differences of the sum of two quantities are equal to the sums of their differences. General proof. Let / /, /, etc., be one set of functions; //, /,',/,', etc., another set. fi +//> /a + //> /s + /'> etc., wil1 then be their sums. In the first of the following columns we place the first differences of/, in the second those of/', and in the third those of / + /', each formed according to the rule : etc. etc. etc. It will be seen that the quantities in the third column are the sums of those in the first two. NUMERICAL EXAMPLE. / A f A' f+f A lt+n l + i ll + n *n ~r H c + 3 t-f. + 14 _ 5 ?- 51 10 + 4 9~ 47 We see that the third set of values of A r follow the theorem. Because the second differences are the differences of the first, the third the differences of the second, etc., it follows that the theorem is true for differences of any order. Now when we write a series of functions in which the decimals ex- ceeding a certain order are omitted, we may conceive each written num- ber to be composed of the algebraic sum of two quantities, namely: 1. The true mathematical value of the function. 2. The negative of the omitted decimals. Example. In the preceding collection of logarithms, since the true value of log 30 is 1.477 121 3 . . . , we may conceive the quantity written to be 1.477 12 = log 30 - .000 001 3 ____ Hence the differences actually written are the differences of the true logarithms minus the differences of the errors. Now suppose the errors to be alternately + 0.5 and 0.5 = the point marking off the last decimal. Their differences will then be as follows: /' J' A" 4'" - 0.5 , x + 2 __ 4 + 0.5 i-2 - 0.5 ~ \ + 2 ' + 0.5 " L - 2 " etc. etc. etc. etc. GENERAL PRINCIPLES. 59 It is evident that the wth order of differences of the errors are equal to 2"- 1 . Hence, in this case, if the nth order of differences of the true values of the function were zero, still, in consequence of the omission of decimals, the actual differences of the nth order would be2- 1 . This, however, is a very extreme case, since it is beyond all proba- bility that the errors should alternate in this way. A more probable average example will be obtained by supposing a single number to have an error of 0.5, while the others are correct. We shall then have: f A 1 4" d'" ^ 1T ^ T A + 0-5 o __ o ' + 0.5 In this case the maximum value of the difference of the nth order is 1.5 in the differences of the third order, 3 in those of the fourth, 5 in those of the fifth, etc. Its general expression is 1 n (n - 1) (n - 2) ____ (n - s -f 1) 2 1.2. 3.... s where n is the order of differences, and n n -1 * = 2 or 2 according as n is even or odd. Thus: A' =1 . 2 > ' - * L 2 ' 1 = i; 1 3 I/ ^'" = g- " 1 = .< & - 1 4. 3 3. 2 * 1. 2 y 1 5. 4 5. 2 ' 1. 2 "~ , etc. etc. This being about the average case, in actual practice the differ- ences may be two or three times as great without necessarily imply- ing an error greater than 0.5 in the numbers written. We have now the following general rule for judging whether a series of numbers do really follow a uniform law: Difference the series until we reach an order of differences in which the 4* and signs either alternate or follow each other irregularly. 60 OF INTERPOLATION. If none of the differences of this order expressed in units of the last place of decimals exceed the limit n (n 1) . . . _. (n s + 1) 1. 2. 3 .... s that is, the value of the largest binomial coefficient of the nth order the given numbers may be assumed to follow a regular law, and therefore to be correct to a unit in the last figure. If some differences exceed this limit, their quotient by the above binomial coefficient may be considered to show the maximum error with which the number opposite it is probably affected. We can thus detect an isolated error in a series of numbers with great certainty. Suppose, for example, an error of 2 in some number of the series. Differencing the series 0, 0, 0, 2, 0, 0, 0, we shall find the four largest differences of the fifth order to be 10, -j- 20, 20, -[- 10, which would enable us to hit at once upon the erro- neous number and judge of the magnitude of its error. An error near the beginning and end of the series of numbers of which the differences are taken cannot be detected by the differences unless it is considerable. If, for instance, the first or last number is in error by 1, the error of each order of differences will only be 1, as we may easily see by the following example: /' A' A" A'" ~ I + I - 1 etc. It is only in those differences which are on or near the same line as the numbers which are magnified in the way we have shown. But at the beginning and end of the series we cannot determine theso differences. Examining the various tables of differences, we see that n numbers have n 1 first differences, n 2 second differences, and so on, the number diminishing by 1 with each succeeding order. Hence, unless the number of given functions exceeds the index expressing the order of differences which we have to form, no certain conclusion can be drawn. What is here said of the correctness of the numbers when the differences run properly must be understood as applicable to isolated errors only. If all the numbers were subject to an error following a regular law, this error would not be detected by the differences be- cause, from the nature of the case, the latter only show deviations from some regular law. FUNDAMENTAL FORMULA. 61 27. Fundamental Formulae of Interpolation. We suppose a series of numbers to be differenced in the way already shown, and the various differences to be designated as in the follow- ing scheme, which is supposed to be a selection from a series preceding and- folio wing it. Function. 1st Diff. 2d Diff. 3d Diff. 4th Diff. 5tn Diff. a /f' 2 /f"' 3 yfv A -I A ,, A _, J-.j ' " , 3 A'", etc. etc. etc. etc. etc. etc. It will be seen that the lower indices are chosen so as to on which line a difference of any order falls. Thus all quantities with index 2 are on one horizontal line, those with index |- = 2 are half a line below, etc. This notation is a little different from that used in algebra, but the change need not cause any confusion. , It is shown in algebra that if n be any index, we have the notation being changed as in the preceding scheme. Now the fundamental hypothesis of interpolation is that this iormula, which can be demonstrated only for integral values of ^>k true also for fractional values; that is, for values of the function u between those given in the table or in the above scheme. We there- fore suppose this formula to express the value of the function u for any value of n between and 1. r . , . For values between -f 1 and -f 2 we have only to increase the indices, of u and its differences by unity, thus: . + etc,, and by supposing n to increase from to 1 in this formula we shall have values of u from u l to w a . 62 OF INTERPOLATION. Increasing the indices again that is, applying our general foi mulae to a row of quantities one line lower we shall have etc. , The equation (a) is known as Newton's formula of interpolation. 28. Transformations of the Formula of Interpolation. In the equation (a) and those following it, the formula of inter- polation is not in its most convenient form. We shall therefore transform it so that the differences employed shall be symmetrical with respect to the functions between which the interpolation is to be made. In working these transformations we shall suppose the sixth and following orders of differences to be so small as not to affect the result. These differences being supposed zero, any two consecutive fifth differences may be supposed equal. First transformation. Let us first find what the original formula (a) will become when, instead of using the series of differences J'*, J" lf ^'"j, A^\, etc., we use J'l, J" , J'"i, J* f etc. To effect the transformation we must find the values of the first series of differences in terms of the second, and substitute them in the formula (a). We find, by the mode of forming the differences, for which, because we suppose the values of J T to be equal, we may put #\ = *\ + 8J; /}', = A\. Making these substitutions in (a), we have =. + J' t + * (* ~ l) (A>\ + J'",) (-!) (-*) ,, 1.2.3.4.5 ** TRANSFORMATIONS OF FORMULAE. 63 Reducing by collecting the coefficients of equal differences, we find - = //'* + n ( n ~ l l A" + ( + l) (-*) A ,n. **n **o !l> t i 12 oi 123 ( + l)n(*-l)(-2) 1.2.3. 4 . v . 1.2.3.4.5 ** Second transformation. Next, instead of the series of this last formula, (J), J't, J"., J'" b ^ , etc., let us use /J'_ t , J"., J"'_ 4 , J\, etc. To effect this transformation we substitute in (S) for d\, 4"i, etc., The series (b) then changes into *-^^* - . i 1.2.3.4 1.2.3.4.5 '-*' tV'-!) n(it--l)(n'-4)k.H-^ . ^ ^ 1.2.3.4 ^ 1.2.3.4.5 2 T 0.| W which is known as Stirling 9 s formula of interpolation. It will be seen that we have put n* - I for (n + 1) (n - 1), w 9 - 4 for (n + 2) (w - 2), etc. etc. Fourth transformation. In the equation (5), instead of the series of differences J'h J"., J'",, J"., etc., let us use A' k , A', 4"', & etc. 64 OF INTERPOLATION. -. To effect this we put J" = A'\ - J'" i5 JiV o = Jl Vi _ JV^ Making these substitutions in (), it becomes * - *J' 11 .f" ^ -- , iv 1.2.3.4" ' ' - . v 1.2.3.4.5 transformation. BesseVs formula. Let us take half the sum of the equations (e) and (b). We then have 1.2.3.4.5 which is commonly known as BesseVs formula of interpolation, and which is the one most convenient to use in practice. , In applying this formula to find a value of the function inter- mediate between two given values, we must always suppose ,^ the index to apply to the given value next preceding that to be found, and the index 1 to apply to that next following. The quantity n will then be a positive proper fraction. 29. Example of interpolation to halves. If we increase the loga- rithms of 30, 31, etc., already given, by unity, we shall have the logarithms of 300, 310, 320, etc. It is required to find, by interpola- tion, the logarithms of the numbers half way between the given ones (omitting the first and last), namely, the logarithms of 315, 325, 335, etc. Here, the required quantities depending upon arguments half way between the given ones, we have n = -J, and the values of the Bessel- ian coefficient, so far as wanted, are n (n - 1) _ !_ 2 " 8 J log (a, - 5) = log a, - - TRANSFORMATIONS OF FORMULA. 65 The subsequent terms are neglected, being insensible. So, if we put a and a l for any consecutive two of the numbers 300, 310, etc., we have (*) where we put A for that first difference between a and a lt These two formulae are two expressions for the same quantitj because a -f- 5 = a l 5. They are both used in such a way as to provide a check upon the accuracy of the work. For this purpose we compute the two quantities log (a. + 5) - log a = -^A\ - - 1, 1 1 A" A" f W log a, - log (..+ 5) = -J'i + - 1 1. J The most convenient and expeditious way of doing the work is shown in the accompanying table, where we give every figure which it is necessary to write, besides those found on p. 57. The following is the plan of computation: Ko. Log. Difl. ^', *'" + '"'. ^+X 310 2.49136 315 .498 31 RS4. + 689.5 - 5.5 -44 320 .50515 UOTt 325 .51188 aao 668.0 - 5.1 - 41 330 .51851 DOO r* Ef o 335 .52504 bOo /? A 4 648.5 -4.8 - 38 340 .53148 b44 AQ/f 345 .53782 OO4: 629.5 - 4.6 - 37 350 .54407 /1-f /> 355 360 .55023 2.55630 616 607 + 611.5 - 4.3 -34 We compute the right-hand column by the formula using the values of A given in the scheme, p. 57. This mode of computing the half sum of two numbers which are nearly equal is easier than adding and dividing by 2. In the next two columns to the left, the sixth place of decimals 66 OF INTERPOLATION. is added in order that the errors may not accumulate by the addition of several quantities. This precaution should always be taken when the interpolated quantities are required to be as accurate as the given ones. The fourth column from the right is formed by adding and sub- tracting the numbers of the second and third columns according to the formula (k). The additional figure is now dropped, because no longer necessary for accuracy. The numbers thus formed are the first differences of the series of logarithms found by inserting the interpolated logarithms between the given ones, as will be seen by equation (&). We write the first logarithm of the series, namely, log 310 = 2.49136, and then form the subsequent ones by continual addition of the dif- ferences, thus: log 315 = log 310 + 695; log 320 = log 315 + 684; log 325 = log 320 + 673; etc. etc. etc. If the work is correct, the alternate logarithms will agree with the given ones in the former table. The continuance of the above process for a few more numbers, say up to 450, is recommended to the student as an exercise. 3O. Interpolation to thirds. Let us suppose the value of a quantity to be given for every third day, and the value for every day to be required. By putting n = -j- and applying formula (/) to each successive given quantity, we shall have the value for each day following one of those given, and by putting n = } we shall have values for the second day following, which will complete the series . But the interpolation can be executed by a much more expeditious process, which consists in computing the middle difference of the interpolated quantities and finding the intermediate differences by a secondary interpolation. Let us put / / / f , etc., the given series of quantities; / /u f*> /> /4> etc -> tne required interpolated series; A' t A", etc., the first differences, second differences, etc., of the given series; $', ", etc., the first differences, second differences, etc., of the interpolated series. TRANSFORMATIONS OF FORMULAS. 67 We may then put /,/, = ^'* (in the given series); /,-/.-*') /,/! = ^ 'f > (in the interpolated series). /.-/.= *Y) We shall then have <*'* + <*', + f| = 4'*. The value of /i / = b\ is given by putting n = $ in the Bes- selian formula (/). Thus we find ,, 1 1 J".+ J". , 1 ., ** = 3 J *~9 2 + i62 J ^^ + ^_J_ r 243 2 1458 Putting w = |, we have the value of/, ~/ , that is, of Thus we find 5 ^ t +^. 1 r 2 r !458 Subtracting these expressions, we have ^ = 3 L ^-^'"* which is most easily computed in the form We see that the computation of tf'f, the middle difference of the interpolated quantities, is much simpler than that of = 2 J + L j.. 2 > ^ 2 '* For the third term, j'" f _ A'" k = J 1 ^. For the fourth term, dropping the terms in d* as too small iu practice, we may put j". + aj". + ^'% = g The difference of the fifth terms may also be dropped, because they contain only sixth differences. Making these substitutions in the value of #" 3 , we find OF INTERPOLATION. 69> By this formula we may compute every third value of #", and then interpolate the intermediate values. By means of these values, we find by addition the intermediate values of d', of which every third value has been computed by formula (m). Then, by continu- ally adding the values of 6', we find those of the function/. As an example of the work, we give the following values of the. sun's declination for every third day of part of July, 1886, for Green* wich mean noon: Date. Q'sDec. A' A" A'" 1886 o / // in n it 6.. . ...22 41 9 > 16 28. 3 212. 4 9.. . ...22 8. 5 20 0. 7 207. 9 + 4.5 12.. . ...21 57 39 9 23 28. 6 203. 4 -f 4.5 15.. . ...21 30 47, 9 26 52. 197. 7 + 5.7 18.. 21 38. 2 30 9. 7 The values of J iv are too small to hav-e any influence. The whole work of interpolation is shown in the following table> fhere, as before, the right-hand column is that first computed, and gives the value of A' -faA'" according to formula (m) : Date. o'sDec. we have the following value of the interpolated first differences im- mediately following a given value of the function: 2 50 6.5.25 2 24 Again, putting n = -fr, and changing the signs, we find for the first difference next preceding a given function 50 6.5.25 24 6.5.20 - The difference of these quantities gives the required second dif- ference, which we find to be 72 LOGARITHMIC TABLES. As an example and exercise we show the interpolation of loga* rithms when every fifth logarithm is given: Number. Logarithm. 6' 3" A p A" 1000 3.0000000 + 21 661 1005 1006 3.002 166 1 .0025980 4319.2 A 0-1 A Q -4.32 - 4.31 - 108 1007 1008 1009 1010 .003 029 5 .0034606 .0038912 3.0043214 'ioi^b. y 4310.6 4306.3 4302.0 A CIA W tV - 4.30 - 4.30 - 4.29 - 4.28 + 21 553 - 107 1011 .0047512 4297.7 A Cir\f*i f - 4.27 1012 .1013 1014 1015 .0051805 .0056094 .006 037 9 3.0064660 4293.5 4289.2 4285.0 4280.8 - 4.26 - 4.23 - 4.20 -4.16 + 21 446 + 21 342 ~" 104 1020 3.0086002 1025 3.0107239 1030 3.0128372 1035 3.0149403 1040 3.0170333 FORMULAE OE THE SOLUTION OF PLANE AND SPHERICAL TRIANGLES. REMARKS. 1. It is better to determine an angle by its tangent than by its sine or cosine, because a small angle or an angle near 180 cannot be accurately determined by its cosine, nor one near either 90 or 270 by its sine, Sometimes, however, the data of the problem are such that the angle can. be determined only through its sine or cosine. Any un- certainty which may then arise from the source pointed out is then inherent in the problem; e.g., if the hypothenuse and one side of a right triangle are 0.39808 and 0.39806 respectively (sixth and follow- ing decimals being omitted), the value of the included angle may be anywhere between 25' and 42', no matter what method of com- putation be adopted. 2. If the sine and cosine can be independently computed, their agreement as to the angle will generally serve as a check on the accuracy of the computation. If they agree, their quotient will give the tangent. 3. It is desirable, when possible, to have a check upon the accu- racy of the computation; that is, to make a computation which must give a certain result if the work is right. But no check can give a positive assurance of accuracy: all it can do is to make it more or less improbable that a mistake exceeding a certain limit exists. 4. In the following list several formulae are sometimes given as applicable to the same problem. In such cases, the most convenient for the special purpose must be chosen. PLANE TRIAXGLE8. Notation, a, b, and c are the three sides. A, B, and C are the opposite angles. PLANE TRIANGLES. Given. Required. s a+ a, b, c, ^, the three one angle. tin A- -4 ~ \/ sides. o s(s a) A, B, C, TT A/( S ~ a ) ( S 0)(S C) m all the s ' angles. tan^ = -*- tan 45- H s V tan -i G tCUl T ^ s j; angle oppo- parts. __ sin C' sin C site one of sin 5 " sin ^4 " them. 76 RIGHT SPHERICAL TRIANGLES. Given. Required. a, A, B, a, c, o, C = 180 - (A + 5); one side the re- , _ a sin ^ and any maining sin J. ' two angles. parts. a sin (7 a sin (^4 -f- J9) sin A sin ^4 RIGHT SPHERICAL TRIANGLES. c is the hypothenuse. , b, A, B, or c. cot A = cot a sin #; the sides cot B = cot J sin a\ containing cos c = cos a cos 5; the right sin # CJl f^i / angle. bill t/ . - Bin -4 A and c. sin c sin A = sin a; sin c cos ^4 = cos a sin 5; cos c = cos a cos 5* sin c sin ^ = sin J; B and c sin c cos B = sin a cos 5. a, c, A, B } or b. . sin G^ sin A . -, one side sin c and the hy- cos .Z? = tan a cot C| pothenuse. ^ cos c C/wb C/ ~ cos a a, A, b, c, or B. sin = tan a cot ^4; one side sin a and the sin c = rj sin -4 opposite . cos A ' r*i >i A. angle. bill X> *,&, b, c, or A. tan b = sin tan B; one side tan a and the tan c = ^; cos 2r adjacent cos ^4 = cos a sin .#. angle. c and A. sin A sin c = sin a; sin ^4 cos c = cos a cos 5; cos A = cos a sin B. QUADRANTAL SPHERICAL TRIANGLES. 77 i*iven. Required. 0, B. b and A. sin .4 sin b = sin a sin B; sin -4 cos # = cos B. c,A, a, b, or B. sin a = sin c sin ^4; the hypo- tan b = tan c cos A; thenuse cot j5 = cos c tan A and one angle. a and B. cos a sin j5 = cos A; cos a cos J? = sin A cos c; sin a = sin -4 sin c. a and b. cos a sin b = cos -4 sin c\ cos a cos # = cos c. A,B, a, b, or c. cos .4 the two sin ^ J angles. , cos jB cos b = -. -. ; sm ^1 cos c = cot ^4 cot B. QUADEANTAL SPHERICAL TEIANGLES. a, 5, the two sides. a, (7, one side and the angle oppo- site the right side. A, B, or C, either angle. A, B, or b. A and b. A and B. cos A = c is the omitted side equal to 90. C is the angle opposite this side. cos a t sin b' cos cos B = sin a cos C = cot cot sin -4 = sin a sin (7; tan B = cos a tan (7; cot # = tan a cos (7. cos ^4 sin Z = cos a; cos A cos b = sin a cos C. sin -4. = sin a sin & cos A sin ^ = cos a sin (7; cos ^4 cos B = cos C. 78 QUADRANTAL SPHERICAL TRIANGLES. Given. one angle and the adjacent side. 0, A, one side and the opposite angle. one angle and the angle oppo- site the right side. A,B, two angles. Required. a, B, or (7. a and B. a and C. b, B, or C. a, b, or B. a, b 9 or C. a and C. I and a cos a = cos A sin 5; tan B = sin A tan b; cot C = cot A cos . sin a sin B = sin J. sin J; sin a cos .Z? = cos &; cos a = cos A sin . sin a sin (7 = sin A \ sin a cos C = cos J. cos 5 = cos a cos A' sin .# = cot a tan .4; . ~ sin A sm # = . sin a sin A sm a%= - ^ sm C" cos b = tan ^4 cot C"; D cos (7 cosjS = 3. cot a = cot A sin B; cot # = sin A cot .5; cos (7 = cos A cos #. sin 7 sin a sin ^4; sin (7 cos a = cos ^4 sin 5; cos (7 = cos A cos 5. sin C sin 5 = sin B\ sin (7 cos b = sin ^4 cos B. SPHERICAL TRIANGLES IN GENERAL. SPHERICAL TRIANGLES IN GENERAL. Given. Required. a, b, c, A, B, G, - \ ), the three sides. the three angles. rr A/ Sm ( S ~~ a ) SI* 1 ( S ~#) Sm ( 5 C ) sins tan 4 A sin (s a) fin 47? tail TT -O : 7T> sm (5 b)' tnr 4 ^ sm (s ) p, , sin a sin 5 sin c sin J[ sin B sin (7* a, 5, C 9 A and ) = sinc cosj(^4 B); all the sin tfcos I (a -f Z>) = cos | c cos J (^4 + B); remaining cos^(7sinj( b) = sinc sin %(A B); parts. cos -J- C'cos J ( b) = cos ^ c sin (^4 + ). A, B, a, two angles &, c, C, all the . .. sin a sin B en T| /) /-f-TTT/^ iTrtli-i/%ci r\-r A\* Dill U : 1 LWU YdtlUUo \JL U}* sin A and an opposite remaining parts. ^*- _ cos 1 (^ + B) tan i (a + b) m wall -g- (/ ' " 1 / /f " 7?\ ' side. cos -|- (^ ^) cot -|- (^4 -j- ^) COS % ( a + ^) A, B, (7, a, I, c, #=(^ + +(7); the three the three P-4/ -cos-S' angles. sides. K cos(^-^) cos (#-.#) cos (#-(7)' tan Ja = P cos ($ ^4); tan %b = P cos ($ ^)^ tan^c = Pcos (S C). TABLES. TABLE I. COMMON LOGARITHMS OF NUMBERS. X. Log. N. Log. N. Log. N. Log. N. Log. 1 2 3 Infinity. 30 31 32 33 I.477I2 60 61 62 63 1.77815 90 91 92 93 1.95424 120 121 122 123 2.07 918 O.OO OOO 0.30 103 0.47 712 1.49 136 I.505I5 1.51 851 1.78533 1.79239 1.79934 .95904 .96 379 .96 848 2.08 2>9 2.08636 2.08 991 4 5 6 0.60 206 0.69 897 0.77815 34 35 36 I.53I43 1.54407 1.55630 64 65 66 i. 80618 1.81 291 1.81 954 94 95 96 97 313 .97 772 .98 227 124 125 126 2.09342 2.09691 2.10037 7 8 9 10 11 12 13 0.84 510 .0.90 309 0.95 424 37 38 39 40 41 42 43 1 . 56 820 1.57978 1.59 106 67 68 69 70 71 72 73 1.82607 1.83251 1.83 885 97 98 99 100 101 102 103 .98 677 .99 123 99 564 127 128 129 130 131 132 133 2.10 380 2.IO 721 2. 1 1 059 I.OOOOO 1. 60 206 1.84 510 2.00000 2. ii 394 .04 139 .07 918 .11 394 1.61 278 1.62323 1.63347 1.85 126 1.85733 1.86332 2.00432 2.00 860 2.01 284 2. 1 1 727 2.12057 2.12385 14 15 16 .14613 .17609 .20412 44 45 46 1.64345 1.65321 1.66 276 74 75 76 1.86923 1.87 506 i. 88081 104 105 106 2.01 703 2.02 119 2.02 531 134 135 136 2.12 710 2,13033 2.13354 17 18 19 20 21 22 23 .23043 .25 527 .27 875 47 48 49 50 51 52 53 1.67 210 1.68 124 1.69 020 77 78 79 80 81 82 83 1.88649 1.89 209 1^89763 107 108 109 110 111 112 113 2.02 938 2.03 342 2.03743 137 138 139 140 141 142 143 2.13672 2.13988 2.I430I .30 103 1.69897 1.90309 2.04139 2.I46I3 .32 222 .34242 .36 173 1.70757 1.71 600 1.72428 i .90 849 1.91 381 1.91 908 2.04532 2.04922 2.05 308 2.14 922 2.15229 2.15534 24 25 26 .38021 39794 41 497 54 55 66 1.73239 1.74036 1.74819 84 85 86 i .92 428 1.92942 1.93450 114 115 116 2.O5 600 2.06070 2.06446 144 145 146 2.15836 2.16 137 2.16435 27 28 29 30 .43 136 .44716 .46 240 57 58 59 60 1.75587 I 76 343 1.77085 87 88 89 90 1.93952 1.94448 1-94939 117 118 119 120 2.06 819 2.07 188 2.07 555 147 148 149 150 2.16732 2.17 026 2.17 319 1.477" 1.77815 1.95424 2.07918 2.17 609 TABLE I. N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 100 01 02 03 04 05 06 07 08 09 110 11 12 13 14 15 16 17 18 19 120 21 22 23 24 25 26 27 28 29 130 31 32 33 34 35 36 37 38 39 140 41 42 43 44 45 46 47 48 49 150 oo ooo 043 087 130 173 217 260 303 346 389 i 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 44 4-4 8.8 13 a 17-6 22.0 26-4 30.8 35-2 39-6 4 4' 8.2 12.3 16.4 20.5 24.6 28.7 32.8 36-9 38 3-8 7.6 ix. 4 15-2 19.0 22.8 26.6 3 4 34 .3 35 3-5 7-o 10.5 14.0 17 5 21. 4 5 28.0 31-5 32 3-2 6-4 9-6 12.8 16.0 19.2 22-4 25-6 28.8 43 43 8.6 12.9 17.2 21. S 2 5 .8 30.1 34 4 38.7 40 4-o 8.0 12.0 16.0 2O- 24.0 28.0 32.0 36.0 37 3-7 7-4 IX. I 14-8 18-5 22.2 25.9 2 9 .6 33-3 .34 3-4 6-8 10 2 I 3 -6 17.0 20.4 23.8 27.2 30.6 31 31 6.2 93 12.4 IS 5 18.6 21.7 24-8 27.9 49 4 8. 12. 16. 21. 25 29. 33-6 37-8 39 39 7-8 11.7 156 '9 5 23.4 27 3 31-2 35 36 3.6 7-a 10.8 14 4 18 o 21 6 25.3 28.8 32 4 33 33 6.6 99 13 3 I6 '5 19-8 23. x- 26.4 29 7 30 3-o 6.0 9-o 12. 15-0 18.0 21. 24.0 37-0 432 860 oi 284 703 02 119 531 938 03 342 743 475 903 326 7 2 5 1 60 572 979 383 782 518 945 368 787 202 612 *oi9 423 822 561 988 410 828 243 653 *o6o 463 862 604 *030 452 870 284 694 *IOO 503 902 647 *072 494 912 325 735 *I 4 I 543 941 689 *ii5 536 953 366 776 *i8i 583 981 732 *I57 578 995 407 816 *222 623 *02I 775 *i99 620 *O36 449 857 *262 663 *o6o 817 *242 662 *078 490 898 * 3 Q2 703 *IOO 04 139 179 218 258 297 336 376 415 454 493 532 922 05 308 690 06 070 446 819 07 1 88 555 5 2' 9 6i 346 729 1 08 483 856 225 591 610 999 385 767 H5 521 893 262 628 650 *038 423 805 183 558 930 298 664 689 *077 461 843 221 595 967 335 700 727 *ii S 500 881 258 633 *oo4 372 737 766 *I54 538 918 296 670 *04i 408 773 805 ="192 576 956 333 707 *078 445 809 844 *23I 614 994 371 744 *ii5 482 846 883 *269 652 *032 408 781 *I 5 I 518 882 918 954 990 *027 *o6 3 *099 *I35 *i 7 i *2O7 *243 08 279 636 991 09 342 691 10 037 380 721 n 059 3H 672 *026 377 726 072 415 755 093 350 707 *o6i 412 760 1 06 449 789 126 386 743 *096 447 795 140 483 823 1 60 422 778 *I 3 2 482 830 175 517 857 193 458 814 *i67 517 864 209 551 890 227 493 849 *202 552 8 99 243 585 924 261 529 884 *2 3 7 587 934 278 619 958 294 565 920 * 27 2 621 968 312 653 992 327 600 955 *3<>7 656 *oo3 346 687 *025 36 1- 394 428 461 494 528 561 594 628 661 694 727 12 057 385 710 13033 354 672 988 14 301 760 090 418 743 066 386 704 *oi9 333 793 123 450 775 098 418 735 *o5i 364 826 483 808 130 450 767 *082 395 860 189 516 840 162 481 799 *ii4 426 893 222 548 8 7 2 194 513 830 *I45 457 926 254 581 3 545 862 *i76 489 959 287 613 937 258 577 893 *208 520 992 320 646 969 290 609 925 *239 55i *O24 $ *OOI 322 640 956 *27O 582 613 644 675 706 737 768 799 829 860 891 922 15 229 534 836 16 137 435 732 17 026 319 953 259 564 866 167 465 761 056 348 983 290 594 897 197 495 791 085 377 *oi4 320 625 927 227 524 820 114 406 *045 351 655 957 256 554 850 H3 435 *076 38i 685 987 286 584 879 f ? 464 *io6 412 715 *oi7 316 613 909 202 493 *i37 442 746 *047 346 643 938 231 522 *i68 473 776 *077 376 673 967 260 55i *i98 503 806 *io7 406 702 289 580 609 O 638 667 2 696 3 725 4 754 MIBMMMM 5 782 - 6 811 7 840 8 869 9 N. 1 Prop. Pts. LOGARITHMS OF NUMBERS. N. O 1 2 3 4 5 6 r 8 9 Prop. Pts. 150 17 609 638 667 696 725 754 782 811 840 869 51 898 926 955 984 *oi3 "041 *o7o *O99 *I2 7 *i$6 29 28 52 18 184 213 241 270 298 327 355 384 412 441 53 469 498 526 554 583 611 639 667 696 724 1 2 2.9 5.8 2.8 5.6 54 752 780 808 837 865 893 921 949 977 *oos 3 8.7 8.4 55 19033 061 089 117 145 173 201 229 257 285 4 11.6 11.2 56 312 340 368 396 424 45 l 479 507 535 562 5 14.5 14.0 57 590 618 645 673 700 ^728 * 756 783 811 838 6 17.4 16.8 58 866 893 921 948 976 *o 5 8 *o85 *II2 7 20.3 19.6 59 20 140 167 194 222 249 276 303 330 358 385 8 23.2 22.4 OK 160 412 439 466 493 520 548 575 602 629 656 5 *030 2 5.0 74 24 055 080 105 130 155 180 204 229 254 279 3 7.5 75 304 329 353 378 403 428 452 477 502 527 4 10.0 76 576 601 625 650 674 699 724 748 773 5 12.5 77 797 822 846 871 895 920 944 969 993 *oi8 6 15.0 7 17.5 78 25 042 066 091 115 139 164 1 88 212 237 261 8 20.0 79 285 310 334 358 382 406 431 455 479 503 180 527 551 573 600 624 648 672 696 720 744 81 768 792 816 840 864 888 912 935 959 983 34 H 82 26 007 031 055 079 102 126 150 174 198 221 1 2.4 2.3 83 245 269 293 3 I6 340 364 387 411 435 458 2 4.8 4.6 84 482 55 529 553 576 600 623 647 670 694 3 7.2 6.9 85 717 764 788 8n 834 858 881 905 928 4 9.G 9.2 86 951 975 998 *02I *045 *o68 "091 "114 *i 3 8 *i6i 5 6 12.0 14.4 11.5 13.8 87 27 184 207 231 254 277 300 323 346 370 393 7 16.8 16.1 88 89 416 646 439 669 g 485 715 508 738 531 761 554 784 577 807 600 830 623 852 8 9 19.2 21.6 18.4 20.7 190 875 898 921 944 967 989 *OI2 *Q35 *os8 *o8i 99 91 91 28 103 126 149 171 194 217 240 262 285 307 mm 21 92 330 353 398 421 443 466 488 533 1 2.2 2.1 93 556 578 601 623 646 668 691 713 735 758 2 4.4 4.2 i 94 780 803 825 847 870 892 914 937 959 981 3 4 6.6 80 6.3 Q 4 , 95 96 29 003 226 026 248 048 270 070 292 092 336 137 358 58 ill 403 203 425 rr 5 6 * O 11.0 13.2 O . rr 10.5 12.6 97 447 469 491 513 535 557 579 601 623 645 7 15.4 14.7 98 667 688 710 732 754 776 798 820 842 863 8 17.6 16.3 99 885 907 929 951 973 994 *oi6 *o 3 8 *o6o *o8i 9 19.818.9 200 30 103 125 146 168 190 211 233 255 276 298 N. O 1 2 3 4 5 6 7 8 1 9 Prop. Pts. TABLE I. N. O 1 9 3 4 5 6 7 8 O Prop. Pts. 1 200 01 02 03 04 05 06 07 08 09 210 11 12 13 14 15 16 17 >18 19 220 21 22 23 24 25 26 27 28 29 280 31 32 33 34 35 36 37 38 39 240 41 42 43 44 45 46 47 48 49 250 MM N. 30 103 123 146 168 190 211 233 255 276 298 2 1 2 2 4 3 6 4 8 511 613 7 15 817 9 19 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 ~Pr 2 21 .2 2.1 .4 4.2 .6 6.3 .8 8.4 .0 10.5 .2 12.6 .4 14.7 ..6 16.8 .818.9 20 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 19 1.9 3.8 5.7 7.6 9.5 11.4 13.3 15.2 17.1 18 1.8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 17 1.7 3.4 5.1 6.8 8.5 10.2 11.9 13.6 15.3 mmm*mwiBmmmmv op. Pts. 320 535 750 963 3i 175 387 597 32 015 34i 557 771 984 197 408 618 827 035 363 578 792 *oo6 218 429 sis 040 056 & 814 *027 239 450 660 869 077 406 621 835 *048 260 471 681 890 098 428 643 8 5 6 *o6 9 281 492 702 9ii 118 449 664 878 *09i 302 513 723 93i 139 471 685 899 *II2 323 534 744 952 1 60 492 707 920 *I33 345 555 765 973 181 5H 728 942 *I54 366 576 785 994 20 1 222 243 263 284 305 325 346 366 387 408 428 634 838 33 041 244 445 646 846 34 044 449 654 858 062 264 465 666 866 064 469 6/5 879 082 284 486 686 885 084 490 695 899 102 304 5 06 7 06 90S 104 510 715 919 122 325 526 726 925 124 53i 736 940 H3 345 546 746 945 U3 552 756 960 163 3 S 566 766 965 163 572 777 980 183 385 586 786 985 183 593 797 *OOI 203 405 606 806 *oo5 203 613 818 *02I 224 425 626 826 *O25 223 242 262 282 301 321 34i 361 380 400 420 439 635 830 35 025 218 411 603 793 984 459 655 850 044 238 430 622 813 *oc>3 479 674 869 064 257 449 641 832 *02I 498 083 2 7 6 468 660 * 85 ' *O4O 5 l8 713 908 I O2 679 870 *059 537 733 928 122 315 507 698 889 *078 557 753 947 141 334 526 717 908 *097 577 772 967 160 353 545 736 927 *u6 596 792 986 1 80 372 564 755 946 *i35 616 Su *oo5 199 392 583 774 965 *I 5 4 36 173 192 211 229 248 267 286 305 324 342 361 549 736 922 37 107 291 S 840 380 568 754 940 125 310 493 676 858 773 959 144 328 694 876 418 605 791 977 162 346 530 712 894 436 624 810 996 181 365 548 73i 912 i 55 642 829 *oi4 199 383 566 749 93i 474 661 847 *0 33 218 401 %* 767 949 493 680 866 *o5i 236 420 603 785 967 511 *070 254 438 621 803 985 530 717 903 *o88 273 457 639 822 *oo3 38 021 039 057 075 093 112 130 148 166 184 2O2 382 561 739 917 39<>94 270 445 620 220 399 578 757 934 in 287 463 637 238 417 596 775 952 129 305 480 655 256 I 35 614 792 970 146 322 498 672 274 453 632 Sio 987 164 340 I 15 690 292 471 650 828 *oo5 182 358 533 707 310 846 *023 199 375 550 724 328 507 686 863 *04i 217 393 568 742 346 525 703 881 *os8 235 410 585 759 364 543 721 899 *076 252 428 602 777 794 MMMMBBMHI Sn |T 829 2 846 ^ : 3 863 4 881 M 5 898 6 915 mammmmm 7 933 8 950 O LOGARITHMS OF NUMBERS. N. O 1 2 3 4 5 6 7 8 9 Pr< >p. PK 250 39 794 811 829 846 863 88 1 898 915 933 950 MMMMHM MBMMMMMM 51 967 985 *002 *io6 *I2 3 18 52 53 40 140 312 157 329 I7 I 346 192 364 209 226 398 243 415 261 432 278 449 295 466 1 2 1.8 3 6 54 483 500 5 I8 535 552 569 586 603 620 637 3 5.4 55 654 671 688 705 722 739 756 773 790 8^7 4 7.2 56 824 841 858 875 892 909 926 943 960 976 5 9.0 57 993 *OIO *027 *044 *o6i * 07 g *095 *m *I28 *i45 6 10.8 58 m 41 162 179 196 212 229 246 263 280 296 3^3 7 12.6 59 330 347 363 380 397 414 430 447 464 481 8 14.4 260 497 5H 53 1 547 564 581 597 614 631 647 ' 61 664 68 1 697 7H 731 747 764 780 797 814 17 62 830 847 863 880 896 913 * 92 2 946 963 979 i 1 7 63 996 *OI2 *029 *045 *062 *o 7 8 *m "127 *I44 J. 2 -l.i 3.4 64 65 42 160 325 177 341 193 357 210 374 226 390 406 259 423 275 439 292 455 308 472 3 4 5.1 6.8 66 488 504 521 537 553 570 586 602 619 635 5 8.5 67 651 66 7 684 700 716 732 749 765 781 797 6 7 10.2 UQ 68 8i3 830 846 862 878 * 894 * 9 " 927 * 943 959 t . y 1 Q ft 69 975 991 *oo8 *034 *040 *o88 *I20 9 lo . O 15.3 270 43 136 152 169 185 201 217 233 249 265 281 71 297 313 329 345 361 377 393 409 425 441 16 72 457 473 489 505 521 537 553 569 584 600 1 1.8 73 616 632 648 664 680 696 712 727 743 759 2 74 775 791 807 823 838 854 870 886 902 * 9 ' 7 3 4^8 75 933 949 965 981 996 *OI2 *028 *44 *o 59 4 6.4 76 44091 107 122 138 154 170 185 20 1 217 232 5 Q 8.0 (i 77 248 264 279 295 3" 326 342 358 373 389 7 / vl 11.2 78 79 404 560 420 576 436 592 45i 607 467 623 483 638 498 654 669 S2Q 685 545 700 8 9 12.8 14.4 280 716 73 i 747 762 778 793 809 824 840 855 81 871 886 902 917 932 948 963 979 994 *OIO 15 82 45 025 040 056 071 086 1 02 117 133 148 163 1 1.5 83 179 194 209 225 240 255 271 286 301 317 2 3.0 84 332 347 362 378 393 408 423 439 454 469 3 4" 4.5 6f\ 85 484 500 515 530 545 561 576 606 621 .0 7e 86 637 652 667 682 697 712 728 743 758 773 6 .5 9.0 87 788 803 818 834 849 864 879 894 909 924 7 10.5 88 939 954 969 984 *000 *oi5 *O3O *45 *o6o "075 8 12.0 89 46 090 105 1 20 135 150 165 1 80 195 210 225 9 13.5 200 240 255 270 285 300 315 330 345 359 374 91 389 404 419 434 449 464 479 494 509 5 2 3 14 92 93 g 553 702 568 716 583 746 613 761 627 776 642 790 657 805 820 1 2 1.4 2.8 94 835 850 864 879 ^894 * 9 9 * 923 * 938 953 967 3 A 4.2 6 a 95 982 997 *OI2 *026 *IOO *n 4 TE ft U 7 O 96 47 129 144 159 *73 1 88 202 217 232 246 261 *J 6 1 Vf 8.4 97 276 290 305 319 334 349 363 378 392 407 7 9.8 98 422 436 451 465 480 494 509 '524 538 553 8 11.2 99 567 582 59 6 6n 625 640 654 669 683 698 9 12.6 800 712 727 741 756 770 784 799 813 828 842 N. O 1 2 3 4 5 6 7 8 9 Pr p. Pts. TABLE I. " N. 1 2 3 4 5 6 7 8 Prop. Pts. 800 47 712 727 741 756 770 784 799 813 828 842 01 8 5 7 871 885 900 914 929 943 958 972 986 02 48 ooi 015 029 044 058 073 087 101 116 130 03 144 159 173 187 202 216 230 244 259 273 15 04 287 302 316 330 344 359 373 387 401 416 1 1.5 05 06 430 572 AAA M r*r 586 458 601 6if 487 629 643 515 657 530 671 in 558 700 2 3 3.0 4.5 07 714 728 742 756 770 78J 799 813 827 841 4 6.0 08 855 869 883 J*& 911 926 940 954 968 982 5 7.5 09 996 *OIO *O24 *o66 *o8o *io8 *I22 6 7" 9.0 1ft K 810 49 136 150 164 178 192 206 220 234 248 262 8 1U.O 12.0 11 276 290 304 318 332 346 360 374 388 402 9 13.5 12 13 554 429 568 443 582 457 596 471 610 485 624 513 651 679 14 693 707 721 734 748 762 776 790 803 8l 7 ' 15 831 845 859 872 886 900 * 9 * 4 927 941 * 9 " 14 16 969 982 996 *OIO *024 "037 "065 1 1.4 17 50 106 120 133 H7 161 174 1 88 202 215 229 2 2.8 18 243 2 5 6 270 284 297 325 338 352 365 3 4.2 19 379 393 406 420 433 447 461 474 488 SOI 4 5.6 7/v 820 515 529 542 556 569 583 596 610 623 637 6 .0 8 4 21 22 23 X 920 664 799 934 678 813 947 691 826 961 70S 840 974 718 987 732 866 *OOI 745 880 759 893 *028 772 % 907 7 8 9 9.8 11.2 12.6 24 51 055 068 08 1 095 1 08 121 135 148 162 175 25 26 188 322 202 335 348 228 362 242 375 388 402 282 415 2 95 308 441 27 455 468 481 495 508 521 534 548 561 574 18 28 587 601 614 627 640 654 667 680 693 706 1 1.3 29 720 733 746 759 772 7 86 799 812 825 838 2 2.6 880 851 865 878 891 904 917 930 943 957 970 3 3.9 31 983 996 "009 *022 *Q35 *048 *o6i *075 *o88 *IOI 4 5.2 6K 32 52 114 127 140 153 1 66 179 192 205 218 231 O 7Q 33 244 257 270 284 297 310 323 336 349 362 7 .0 9.1 34 375 388 401 414 427 440 453 466 479 49* 8 10.4 35 36 504 634 I 17 647 543 673 556 686 569 699 582 711 595 724 608 737 621 750 9 11.7 37 763 776 789 802 815 827 840 853 866 * 879 38 892 905 917 930 943 956 969 982 994 39 53 020 033 046 058 071 084 097 no 122 135 12 840 148 161 173 1 86 199 212 224 237 250 263 1 1.2 41 275 288 301 3H 326 339 352 364 377 390 2 2.4 3/> 42 403 415 428 441 466 479 491 504 .6 43 529 542 555 567 580 593 605 618 631 643 4 5 4.8 6 44 45 656 782 668 794 681 807 694 820 706 832 719 845 732 857 744 870 882 769 895 6 7 7.2 8.4 46 908 920 933 945 958 970 983 995 *oo8 *020 8 9.6 47 54033 045 058 070 083 095 108 120 133 145 9 10.8 48 158 170 183 195 208 220 233 245 258 270 49 283 295 307 320 332 345 357 370 382 394 850 407 419 432 444 456 469 481 4941 506 518 N. 1 2 3 4 5 6 7 1 8 9 Prop. Pts. LOGARITHMS OF NUMBERS. N. BMMMMM 350 51 52 53 54 , 55 56 57 58 59 360 61 62 63 64 65 66 67 68 69 370 71 72 73 74 75 76 77 78 79 880 81 82 83 84 85 86 87 88 89 390 91 92 93 94 95 96 97 98 99 400 N. O MHMMMMMMM 54 407 lj_ 419 2 432 3 444 4 456 5 469 6 481 7 494 8 ~ 9 ~ Prc p. Pts. 13 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 12 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 11 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 10 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 p. Pts. 53i 654 777 900 55 023 145 267 388 509 630 75i 871 991 56 1 10 467 585 703 543 667 790 913 035 157 279 400 522 555 679 802 925 047 169 291 4i3 534 568 691 814 937 060 182 303 425 546 580 704 827 949 072 194 315 437 558 593 716 839 962 084 206 328 449 570 605 728 851 974 096 218 340 461 582 617 741 864 986 1 08 230 352 473 594 630 753 876 998 121 242 364 485 606 642 765 888 *OII 133 255 376 497 618 642 654 666 678 691 703 715 727 739 763 883 *OO3 122 241 360 478 597 7H 775 895 *oi5 134 253 372 490 608 726 787 907 *027 146 265 384 502 620 738 799 919 "038 158 277 396 5H 632 750 811 93i *o5o 170 289 407 526 644 761 823 * 9 J 3 *062 182 301 419 1% 656 773 835 955 "074 194 312 43i 549 667 785 847 967 *o86 205 324 443 561 679 797 859 979 *098 217 336 453 573 691 808 820 832 844 855 867 879 891 902 914 926 937 57 054 171 287 403 519 634 749 864 978 9 i? 066 183 299 415 530 646 7 6i 875 961 078 194 310 426 542 657 772 887 972 089 206 322 438 553 669 784 898 984 101 217 334 449 565 680 795 910 996 "3 229 345 461 576 692 807 921 *oo8 124 241 357 473 588 88 933 *oi9 136 252 368 484 600 715 830 944 "031 148 264 380 496 61-1 726 841 955 *043 159 276 392 507 623 738 8f 967 990 *OOI *OI 3 *O24 *Q35 *047 "058 *O7O *o8i 58 092 206 320 433 546 659 771 883 995 104 218 331 444 557 670 782 894 *oo6 "5 229 343 456 569 681 794 906 *OI 7 127 240 354 467 e 805 917 *028 138 252 365, 478 591 704 816 928 *O4o 149 263 377 490 602 715 827 939 *o5i 161 274 388 501 614 726 838 172 286 399 512 625 737 850 961 *073 184 297 410 524 636 749 861 973 *o84 195 309 422 I 3 * 647 760 872 984 *o 9 s 59 106 118 129 140 151 162 173 184 195 207 218 329 439 770 879 60 097 229 340 450 I 61 671 780 890 240 461 III 791 901 *OIO 119 2 I' 362 472 a 802 912 *02I 130 262 373 483 594 704 813 923 "032 141 273 384 494 605 715 824 934 "043 152 284 395 506 616 726 835 945 *Q54 163 295 406 517 627 737 846 956 *o6ij 173 306 417 528 638 748 857 966 "076 184 3i8 428 539 649 759 868 977 *o86 195 206 O 217 mfmm^mmmm 1 228 2 239 3 249 4 260 5 271 6 282 7 293 8 304 9 TABLE I. 1 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 400 60 206 217 228 239 249 260 271 282 293 304 01 3H 325 336 347 358 369 379 390 401 412 02 423 433 444 455 466 477 487 498 509 520 03 531 541 552 563 574 584 595 606 617 627 04 638 649 660 670 68 1 692 703 713 724 735 05 746 756 767 778 788 799 810 821 83' 842 06 853 863 874 885 895 906 917 927 938 949 11 07 959 970 981 991 *O02 "013 *023 *034 *045 *055 1 1.1 08 61 066 077 087 098 I0 9 119 130 140 151 162 2 2.2 09 172 183 194 204 215 225 236 247 257 268 3 A 3.3 A A 410 278 289 300 310 321 33i 342 352 363 374 I 5 tb.4 5.5 11 384 395 405 416 426 437 448 458 469 479 6 6.6 12 13 490 595 500 606 5" 616 SJ 532 637 542 648 553 658 g I 74 679 584 690 7 8 7.7 8.8 14 15 700 805 711 815 721 826 731 836 742 847 752 857 763 868 III 784 888 794 899 9 9.9 16 909 920 930 941 951 962 972 982 993 *c3 17 62 014 024 034 045 055 066 076 086 097 107 18 118 128 138 149 159 170 1 80 190 201 211 19 221 232 242 252 263 273 284 294 304 315 420 325 335 346 356 366 377 387 397 408 4 l8 21 428 439 449 459 469 480 490 500 511 521 IV 22 23 531 634 542 644 IS 562 665 572 675 583 685 603 706 716 624 726 1 2 1.0 2.0 24 737 747 757 767 778 788 798 808 818 829 3 3.0 4 A 25 26 839 941 849 95i s? 870 972 880 982 890 992 900 *002 910 *OI2 921 *022 931 *033 4 5 6 .0 5.0 6.0 27 63 043 053 063 073 083 094 IO4 114 124 134 7 7.0 28 144 155 165 175 185 195 205 215 225 236 8 8.0 29 246 256 266 276 286 296 306 317 327 337 9 9.0 430 347 357 367 377 387 397 407 417 428 438 31 448 458 468 478 488 498 5 08 5 l8 528 538 32 548 558 568 579 589 599 60 9 619 629 639 33 649 659 669 679 689 699 709 719 729 739 34 35 849 759 859 769 869 779 879 789 889 899 80 9 909 819 919 829 929 839 939 36 949 959 969 979 988 998 *oo8 *oi8 *028 *o 3 8 9 37 64 048 058 068 078 088 098 108 118 128 137 1 0.9 38 147 157 167 177 187 197 207 217 227 237 2 1.8 39 246 256 266 276 286 296 306 316 326 335 3 2.7 440 345 355 365 375 385 395 404 414 424 434 4 3.6 4C 41 444 454 464 473 483 493 503 513 523 532 6 .0 5.4 42 542 552 562 572 582 59i 601 611 621 631 7 ft a 43 640 650 660 670 680 689 699 709 719 729 1 8 u %} 7.2 44 738 748 758 768 777 787 797 807 816 826 9 8.1 45 836 846 856 865 875 885 904 914 924 46 933 943 953 963 972 982 992 *O02 *OII *O2I 47 65 031 040 050 060 070 079 089 099 108 118 48 128 137 H7 157 167 176 1 86 I 9 6 205 215 ^ \ 49 225 234 244 254 263 273 283 292 302 312 450 321 33i 341 350 360 369 379 389 398 408 N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS. N. 450 51 52 53 54 55 56 57 58 59 460 61 62 63 64 65 66 67 68 69 470 71 72 73 74 75 76 77 78 79 480 81 82 83 84 85 86 87 88 89 490 91 92 93 94 95 90 97 93 99 500 O 65 321 1 2 MM^M 341 3 350 4 5 ~ 6 - 379 7 m*mmm*mmm 389 8 "398" 9 408 Pro] i 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 IMMHHMB Pro ). PtS. MMMi^M 10 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 9 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 8 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 p. Pts. 418 5H 610 706 801 896 992 66 087 181 276 370 464 558 652 745 839 932 67 025 117 427 523 619 715 811 906 *OOI 096 191 437 533 629 725 820 916 *OII 106 200 447 543 639 734 830 925 *O2O 2IO 456 III 744 839 935 124 219 466 562 658 753 849 944 134 229 475 57i 667 763 858 954 238 485 |8l 677 772 868 963 153 247 495 686 782 877 973 *o68 162 257 504 600 696 792 887 982 172 266 285 295 304 3 T 4 323 332 342 35 1 361 380 474 567 661 755 848 941 034 127 389 483 577 671 764 857 950 043 136 398 492 5 86 680 773 867 960 052 MS 408 502 596 689 783 876 969 062 154 417 33 699 792 885 978 071 164 427 521 614 708 801 894 987 080 173 43 6 530 624 717 811 904 997 089 182 445 539 633 727 820 913 *oo6 099 191 455 549 642 736 829 922 20 1 210 219 228 237 247 256 265 274 284 293 302 394 486 H 8 669 761 852 *o 943 68 034 3" 403 495 587 679 770 861 952 043 321 504 596 688 779 870 961 052 330 422 605 788 879 970 06 1 339 523 614 706 797 888 979 070 348 440 532 624 897 988 079 357 449 541 633 724 906 997 088 367 459 550 642 733 825 916 *oo6 097 376 468 560 651 742 834 925 *oi5 106 385 477 569 660 752 843 934 "024 124 133 142 I 5 I 160 169 178 187 196 205 215 305 395 485 574 664 753 842 224 3H 404 494 583 673 762 851 940 233 323 413 502 592 68 1 771 860 949 242 332 422 601 690 780 869 958 251 431 520 610 699 789 878 966 260 350 440 529 619 708 975 269 359 449 538 628 717 806 895 984 278 368 458 547 637 726 815 904 993 287 377 467 S A 646 735 824 913 *OO2 $ st 744 833 922 *OII 69 020 028 037 046 055 064 073 082 090 099 108 197 285 3 P 461 548 636 810 897 ^^ MHHHMHBBBMC O 117 205 294 469 557 644 732 819 126 214 302 390 478 566 653 740 827 135 223 399 487 574 662 749 836 144 232 320 408 496 583 671 758 845 152 241 329 417 504 592 679 767 854 161 249 338 425 513 601 688 & 170 2 58 346 434 522 609 6 97 784 871 179 267 355 443 705 793 880 1 88 276 364 452 539 627 7H 801 888 906 914 923 3 932 4 940 5 949 6 958 7 966 975 N. 1 2 8 9 10 TABLE I. N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 500 01 02 03 04 05 06 07 08 09 510 11 12 13 14 15 16 17 18 19 520 21 22 23 24 25 26 27 28 29 580 31 32 33 34 35 36 37 38 39 540 41 42 43 44 45 46 47 48 49 550 N. 69897 984 70 070 157 243 329 415 501 586 672 906 914 923 932 940 949 958 966 975 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 ^M^BM^ Pro 9 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 8 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 7 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6 3 P. Pts. 992 079 165 252 338 424 509 595 680 *OOI 088 174 260 346 432 li 8 603 689 *OIO 096 183 269 355 441 526 612 697 *oi8 105 191 278 364 449 I 35 621 706 *02 7 114 2OO 286 372 458 544 629 7H *036 122 .209 295 381 467 III 723 *044 131 217 303 389 475 561 646 73i *Q53 140 226 312 398 484 569 655 740 *062 148 234 32 1 406 492 578 663 749 757 842 927 71 012 096 181 265 ( 349 433 517 766 851 935 020 105 189 273 357 441 525 774 783 ; 791 800 808 817 825 834 859 944 029 113 366 450 533 868 952 037 122 206 290 374 458 542 876 961 046 130 214 299 383 466 550 885 969 054 139 223 307 391 475 559 893 978 06 3 147 231 315 399 483 567 902 986 071 155 240 324 408 492 575 910 995 079 164 248 332 416 500 584 919 *oo3 088 172 257 341 s 592 600 609 617 625 634 642 650 659 667 675 684 767 850 933 72 016 099 181 "^28" 692 III 941 024 107 189 272 354 700 784 867 950 032 "5 362 709 792 875 958 041 123 206 288 370 717 800 883 966 049 132 214 296 378 725 809 892 975 057 140 222 304 387 734 817 900 983 148 230 313 395 742 8 991 074 156 239 321 403 750 834 917 163 247 329 411 759 842 925 *oo8 090 173 255 337 419 436 444 452 460 469 477 485 493 501 509 59i 673 754 835 916 997 73 078 159 518 599 681 762 843 925 *oo6 086 167 526 607 689 770 852 933 *oi4 094 175 I 3 ! 616 697 779 860 941 *022 102 I8 3 542 624 705 787 868 949 *03o in 191 1 S 632 713 795 876 957 *038 119 199 558 640 722 803 884 965 *046 127 207 567 648 730 811 892 973 *054 135 215 575 656 738 819 900 981 *062 143 223 583 665 746 827 908 989 *O7O 151 231 239 247 255 263 272 280 288 296 304 312 320 400 480 560 640 719 1% 957 74 036 - O 328 408 488 568 648 727 807 886 965 336 416 496 i^ 656 735 815 894 973 344 424 504 584 664 743 823 902 981 352 432 512 592 672 751 830 910 989 360 440 520 600 679 759 838 918 997 368 448 528 608 687 767 846 926 *oo3 376 456 536 616 695 775 854 933 *oi3 384 464 544 624 703 783 862 * 941 *O2O 392 472 552 632 711 791 870 949 *028 044 MMMM 1 052 2 060 3 068 4 076 ^MiMB 5 084 6 092 7 099 8 107 9 LOGARITHMS OF NUMBERS. 1 1 fmmmmv^ N. 550 5.1 52 53 54 55 56 57 58 59 660 61 62 63 64 65 66 67 68 69 570 71 72 73 74 75 76 77 78 79 580 81 82 83 84 85 86 87 88 89 590 91 92 93 94 95 96 97 98 99 600 N. mmm^mttmrmm 74 036 1 044 2 052 3 060 4 ^M^^M 068 5 ^^MHMK O76 6 084 r 092 8 099 9 107 Pro] ^M^H 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 "Pr7 >.Pte. 8 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 7 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 p.Pts. US 194 273 351 429 507 586 663 74i 123 202 280 359 437 515 593 671 749 131 210 288 367 445 523 601 679 757 139 218 296 374 453 531 609 687 764 147 225 304 382 46l 539 617 695 772 155 233 312 390 468 547 624 702 780 162 241 320 398 476 554 632 710 788 170 249 327 406 484 562 640 718 796 178 257 335 414 492 570 648 726 803 1 86 265 343 421 500 578 656 733 SH 819 827 834 842 850 858 865 873 881 889 896 974 75051 128 205 282 358 435 5" 904 981 059 136 213 289 366 442 519 912 989 066 143 220 297 374 450 526 920 997 074 I5i 228 305 38i 458 534 927 *ool 082 '59 236 312 389 465 542 935 *OI2 089 166 243 320 397 473 549 943 *020 097 174 251 328 404 481 557 950 *028 105 182 259 335 412 488 565 958 *035 "3 189 266 343 420 496 572 966 *043 120 197 274 351 427 504 580 587 595 603 610 618 626 633 641 648 656 664 740 815 891 967 76 042 118 193 268 343 671 747 823 899 974 050 125 200 275 679 755 83' 906 982 057 133 208 283 686 762 -838 914 989 o6J 140 215 290 694 770 846 921 997 072 148 223 298 702 778 853 929 *oos 080 155 230 305 709 785 861 937 *OI2 087 163 238 313 717 793 868 944 *020 095 170 245 320 724 800 876 952 *O27 103 178 253 328 732 808 884 959 *<>35 no ' 260 335 350 358 365 373 380 388 395 403 410 486" I 59 634 708 782 856 930 *oo4 078 418 492 567 641 716 790 77 012 425 500 574 649 723 797 871 945 019 433 507 582 656 730 805 879 953 026 440 515 589 664 738 812 886 960 034 448 522 597 671 745 819 ^ 967 041 455 530 604 678 753 827 901 975 462 537 612 686 760 834 908 982 056 470 545 619 693 768 842 916 989 063 477 g % 849 923 997 070 085 093 IOO 107 H5 122 129 137 144 151 & 159- 232 305 379 452 525 597 670 743 166 240 313 386 459 532 605 677 750 173 247 320 3 ?I 466 539 612 685 757 181 254 327 401 474 546 619 692 764 188 262 335 408 481 554 627 699 772 195 269 342 415 488 5 6l 634 706 779 203 276 349 422 495 568 641 7H 786 210 28 3 357 430 5>3 576 648 721 793 217 291 364 437 5io 583 656 728 801 225 371 444 517 590 663 7 8ol 815 ^I^HH^^B^B 822 1 830 3 837 3 844 4 851 5 859 6 866 7 873 8 880 9 12 TABLE I. N. O 1 2 3 4 5 6 7 8 9 Pro] E>.Pts. 600 77 815 822 830 837 844 851 859 866 873 880 01 02 03 04 05 06 07 08 09 887 960 78 032 104 176 247 319 390 462 8 9 5 967 039 in 183 254 326 398 469 902 974 046 118 190 262 333 405 476 909 981 053 125 197 269 340 412 483 916 988 061 132 204 276 347 419 490 924 996 068 140 211 283 355 426 497 * 931 *oo3 075 147 219 290 362 433 504 * 938 *OIO 082 154 226 297 369 440 512 945 *oi7 089 161 233 305 376 447 519 952. *025 097 1 68 240 312 383 455 526 1 2 3 8 0.8 1.6 2.4 39 610 533 540 547 554 561 569 576 583 590 597 5 4.0 11 12 13 14 15 16 17 18 19 604 675 746 817 888 958 79 029 099 169 611 682 753 824 895 965 036 1 06 176 618 689 760 831 902 972 043 H3 183 625 696 767 838 909 979 050 1 20 190 633 704 774 845 916 986 057 127 197 640 711 78i 852 923 993 064 134 204 647 718 789 859 930 *000 071 141 211 654 725 796 866 937 *oo7 078 148 218 661 732 803 873 * 944 *oi4 085 155 225 668 739 810 880 95i *02I 092 162 232 6 7 8 9 4.8 5.6 6.4 7.2 620 239 246 253 260 267 274 28l 288 295 302 21 22 23 24 25 26 27 28 29 309 379 449 518 588 657 727 796 865 316 386 456 525 595 664 734 803 872 323 393 463 532 602 671 741 810 879 330 400 470 539 678 748 817 886 337 407 477 546 616 685 754 824 893 344 414 484 553 623 692 761 831 900 351 421 491 S 60^ 630 699 768 837 906 358 428 498 567 637 706 775 844 913 365 435 505 I 74 644 713 782 851 920 372 442 5ii 58i 650 720 789 858 927 1 2 3 4 5 C 7 8 9 7 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 630 934 941 948 955 962 969 975 982 989 996 - 31 32 33 34 1 35 36 37 38 39 80 003 072 140 209 277 346 414 482 550 OIO 079 H7 216 284 353 421 489 557 017 085 154 223 291 359 428 496 564 024 092 161 229 298 366 434 502 570 030 099 1 68 236 305 373 441 509 577 037 106 175 243 312 380 448 516 584 044 H3 182 250 3i8 387 455 523 59i 051 120 188 257 325 393 462 530 598 058 127 195 264 332 400 468 536 604 065 134 202 271 339 407 475 543 611 1 2 3 6 0.6 1.2 1.8 640 618 625 632 638 645 652 659 665 672 679 4 2.4 41 42 43 44 45 46 47 48 49 686 754 821 889. 956 8 1 023 090 158 224 693 760 828 895 963 030 097 164 231 699 767 835 902 969 037 104 171 238 706 774 841 909 976 043 in 178 245 713 781 848 916 983 050 117 184 251 720 787 855 922 990 057 124 191 258 726 794 862 929 996 064 I3J 198 265 733 801 868 93<3 *oo"j 070 137 204 271 740 808 875 * 943 *OIO 077 144 211 2 7 8 747 814 882 949 *oi7 084 151 218 285 5 6 7 8 9 3.0 3.6 4.2 4.8 5.4 650 N. 291 O 298 1 305 2 3ii 3 318 4 325 5 33i 6 338 KMWMM 7 345 8 351 9 Pro p Pts. LOGARITHMS OF NUMBERS. N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 650 81 291 298 305 311 3i8 325 33i 338 345 35i 51 358 365 371 378 385 391 398 405 411 418 52 425 438 445 458 465 47i 478 485 53 491 498 505 5i8 525 538 544 551 54 55 558 624 564 631 637 578 644 584 651 657 Li 604 671 611 677 617 684 56 690 697 704 710 717 723 730 737 743 750 57 757 763 770 776 783 790 796 803 809 816 58 823 829 836 842 849 856 862 869 875 882 59 889 895 902 908 915 921 928 935 941 948 660 954 961 968 974 981 987 994 *ooo *oo7 *oi4 61 82 020 027 033 040 046 053 060 066 073 079 62 086 092 099 105 112 "9 125 132 138 143 1 0.7 63 151 158 164 171 I 7 8 184 191 197 204 210 2 1.4 64 65 217 282 223 289 230 295 236 302 308 249 315 256 321 263 328 269 334 2 7 6 341 3 4 2.1 2.8 3e 66 347 354 36o 367 373 380 387 393 400 406 6 O 4.2 67 413 419 426 432 439 445 452 458 465 471 7 4.9 68 478 484 491 497 504 510 523 530 53 6 8 5.6 69 543 549 556 562 569 575 582 588 595 601 9 6.3 670 607 614 620 627 633 640 646 653 659 666 71 672 679 685 692 698 705 711 718 724 730 72 73 737 802 808 750 814 756 821 763 827 769 834 776 840 782 847 789 853 795 860 74 866 872 879 885 892 898 905 911 918 924 75 930 937 943 * 95 956 963 * 969 975 982 988 76 995 *OOI *oo8 *O2O *027 *046 "052 77 83 059 065 072 078 085 091 097 104 no 117 78 W 123 129 136 142 149 155 161 168 174 181 79 187 193 200 206 213 219 225 232 238 241 680 251 257 264 270 2 7 6 283 289 296 302 308 81 315 321 327 334 340 347 353 359 366 372 6 82 83 378 442 448 391 455 398 461 404 467 410 474 480 423 487 429 493 436 499 1 2 0.6 1.2 84 506 512 518 525 531 537 544 55 556 563 3 1.8 2 A 85 86 569 632 575 639 645 588 651 594 658 664 607 670 613 677 620 683 626. 689 5 6 .4 3.0 3.6 87 696 702 708 715 721 727 734 740 746 753 7 4.2 88 759 765 771 778 784 790 797 809 816 8 4.8 89 822 828 835 841 847 853 860 866 872 879 5.4 690 885 891 897 904 910 916 923 929 935 942 91 948 954 960 967 973 979 985 992 998 *oo4 92 84 on 017 023 029 036 042 048 055 061 067 93 073 080 086 092 098 105 in 117 123 130 94 95 136 142 205 148 211 155 217 161 223 167 230 III 180 242 186 248 192 251 96 261 267 273 280 286 292 2 9 8 305 3" 317 97 98 $ 330 392 336 398 342 404 348 410 354 417 3 6i 423 367 429 373 435 379 442 99 448 454 460 466 473 479 485 491 497 504 700 510 516 522 528 535 54i 547 553 559 566 N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. TABLE I. N. i 2 3 4 5 6 7 8 Prop. Pts. 700 01 1 02 03 1 04 05 OG i 07 1 08 09 710 11 12 13 14 15 16 17 18 19 720 21 22 23 24 25 26 27 28 29 780 31 32 33 34 35 36 37 38 39 740 41 42 43 44 45 46 47 48 49 750 N. 84 510 516 522 528 535 541 547 553 559 566 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 MI^M Pro] 7 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 ).PtS. 572 634 696 757 819 880 942 85 003 065 126 I 78 640 702 763 825 887 948 009 071 584 646 708 770 831 893 954 016 077 590 652 7H 776 837 899 960 022 083 597 658 720 782 844 905 967 028 089 603 665 726 788 850 911 973 034 095 609 671 733 794 856 917 979 040 101 6i5 677 739 800 862 924 107 621 683 743 807 868 93<> 991 052 114 628 689 75i 8i3 874 936 997 058 1 20 132 138 144 150 156 163 169 175 181 187 248 309 370 43i 491 552 612 673 193 254 315 376 437 497 558 618 679 199 260 321 382 443 503 564 625 685 205 266 327 388 449 509 570 631 691 211 272 333 394 455 516 576 6 37 697 217 278 339 400 461 522 582 643 703 224 285 345 406 467 528 588 649 709 230 291 352 412 473 534 594 655 715 236 297 358 418 479 540 600 661 721 242 303 364 425 485 546 606 667 727 733 739 745 751 757 763 769 775 78i 788 794 854 914 ^ 974 86 034 094 153 213 273 800 860 920 980 040 IOO 159 219 279 338" 806 866 926 986 046 106 165 225 285 812 872 932 992 052 112 171 231 291 818 878 938 998 058 118 177 237 297 824 884 944 *(X>4 064 124 183 243 303 830 890 950 *OIO 070 130 189 249 308 836 896 956 *oi6 076 136 195 255 3H 842 902 962 *O22 082 141 201 26l 320 848 908 968 *028 088 147 207 267 326 332 344 404 463 522 I 81 641 700 759 817 876 350 356 362 368 374 380 386 392 45' 510 570 629 688 747 806 864 39o 457 516 576 635 694 753 812 870 410 469 528 587 646 705 764 823 882 415 475 534 593 652 711 770 829 888 421 481 540 599 658 717 776 835 894 427 487 546 605 664 723 782 841 goo 433 493 552 611 670 729 788 847 906 439 499 558 617 676 735 794 853 911 445 504 564 623 682 741 800 859 917 923 929 935 941 947 953 958 964 970 976 982 87 040 099 I5 2 216 274 332 390 448 988 046 105 163 221 280 338 396 454 994 052 in 169 227 286 344 402 460 999 058 116 '75 233 291 349 408 466 *00 5 064 122 181 239 297 355 4i3 47i *on 070 128 186 245 303 361 419 477 *oi7 075 134 192 251 309 367 425 483 *023 08 1 140 198 256 315 373 489 *O29 087 146 204 262 320 379 437 495 *35 093 151 210 268 326 384 442 500 558 9 5 o6 o * 518 2 523 3 529 4 535 5 54i 6 547 7 552 8 LOGARITHMS OF NUMBERS. N. >*m~~ 750 51 52 53 54 55 56 57 58 59 7 CO 61 62 63 64 65 66 67 68 69 770 71 72 73 74 75 76 77 78 79 780 81 82 83 84 85 86 87 88 89 790 91 92 93 94 95 96 97 98 99 800 O 87 506 1 512 2 ~ 3 523 4 529 5 ^HMMMM 535 6 MHMMM* 541 7 MMMM^ 547 8 i^mmmtmm 552 9 Iss" Pro MMMM 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 p. Pts. -^ O.G 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 622 679 737 795 852 910 967 88 024 570 628 685 743 800 858 915 973 030 576 633 691 749 806 864 921 978 036 |8i 639 697 754 812 869 927 984 041 587 645 703 760 818 875 933 990 047 593 65 i 708 766 823 881 938 996 053 599 656 714 772 829 887 944 *OOI 058 604 662 720 777 835 892 950 *oo7 064 610 668 726 783 841 898 * 955 *oi3 070 616 674 73i 789 846 904 961 *oi8 076 081 087 093 098 104 1 10 116 121 127 133 138 195 252 309 366 423 480 536 593 144 20 1 258 315 372 429 485 542 598 150 207 264 321 377 434 491 % 156 213 270 326 383 440 497 553 610 161 218 275 332 389 446 502 559 615 167 224 281 338 395 45i 508 564 621 173 230 287 343 400 457 513 570 627 173 235 292 349 406 463 519 576 632 184 241 298 355 412 468 525 581 638 190 247 3>4 360 417 474 530 I 87 643 649 655 660 666 672 677 683 689 694 700 70S III 874 89042 098 154 711 767 824 880 936 992 048 104 159 717 773 829 885 941 997 053 109 165 722 779 835 891 947 *oo3 059 H5 170 728 784- 840 897 953 *oc9 064 120 176 734 79 846 902 958 *oi4 070 126 182 739 795 852 8 964 *020 076 I 3 I 745 801 857 9 I 3 969 *025 08 1 137 193 750 807 863 919 * 975 *03i 087 $ 756 812 868 925 981 *037 092 148 204 209 215 221 226 232 237 243 248 254 260 265 % 542 597 $ 271 326 382 437 492 548 603 658 713 2 7 6 $ $ 553 609 664 719 282 337 393 448 504 559 614 669 724 287 $ 454 509 564 620 675 730 $ 404 459 5i5 570 625 680 735 298 354 409 465 520 575 686 741 304 360 415 470 526 581 636 691 746 310 365 421 476 53i 5 b<3 642 697 752 807 315 Z7 \ 426 481 537 592 647 702 757 763 768 774 779 785 790 796 801 812 818 873 927 982 90037 091 146 200 253 823 878 933 988 042 097 '*! 206 260 829 883 938 993 048 1 02 '.57 211 266 $ 944 998 33 162 217 271 840 894 949 *004 059 "3 168 222 2 7 6 845 900 955 *oo9 064 119 173 227 282 851 905 960 *oi"5 069 124 179 233 287 856 911 966 *O2O 075 129 I8 4 2 3 8 293 862 916 971 *026 080 135 189 244 298 867 922 977 *03i 086 140 195 249 304 309 314 320 325 331 336 342 347 352 358 N. i 2 3 4 5 <* 7 8 Prop. Pts. 1 6 TABLE I. N. 1 2 3 4 5 7 8 9 ~ Prop. Pts. 800 90 309 3H 320 325 331 ~ 342 347 352 01 363 369 374 380 385 390 396 401 407 412 02 417 423 428 434 439 445 450 455 461 466 03 472 477 482 488 493 499 504 509 515 520 04 526 53I 536 542 547 553 558 563 569 574 05 580 58? 590 596 601 607 612 617 623 628 00 634 639 644 650 655 660 666 671 677 682 07 687 693 698 703 709 714 720 725 730 736 08 74 1 747 752 757 763 768 773 779 784 789 09 795 800 806 Sii 816 822 827 832 838 843 810 849. 854 ~8~59~ 865 870 875 88 1 886 891 897 11 12 902 956 907 961 966 918 972 924 977 929 982 934 988 940 993 998 950 *oo4 1 0.6 13 91 009 014 020 025 030 036 041 046 052 057 2 1.2 14 062 068 073 078 084 089 094 IOO 105 no 3 1.8 2 A 15 116 121 126 132 137 142 148 153 158 164 ,*t 3 A 16 169 174 180 185 190 196 20 1 206 212 217 6 u 3.6 17 222 228 233 238 243 249 254 259 265 270 7 4.2 18 275 281 286 291 297 302 307 312 318 323 8 4.8 19 328 334 339 344 350 355 360 365 371 376 9 5.4 820 381 387 392 397 403 408 413 418 424 429 21 434 440 445 450 455 461 466 47 i 477 482 22 487 492 498 503 508 5H 519 524 529 535 23 540 545 556 561 566 572 577 582 587 24 593 598 603 609 614 619 624 630 635 640 25 645 651 656 661 666 672 677 682 687 693 26 f 698 703 709 7H 719 724 730 735 740 745 27 75 1 756 761 766 772 777 782 787 793 798 28 803 808 814 819 824 829 834 840 845 850 29 855 861 866 871 876 882 887 892 897 93 830 908 913 918 924 929 934 939 944 950 955 31 960 965 971 976 981 986 991 997 *OO2 *oo7 32 02 OI2 018 023 028 033 038 044 049 054 059 1 0.5 33 065 070 075 080 085 091 096 101 106 in 2 1.0 34 117 122 127 132 137 14^ 148 153 158 163 3 1.5 2i\ 35 36 I6 9 221 174 226 179 231 184 236 189 241 19? 247 200 252 205 257 2IO 262 215 267 5 6 .V 2.5 3.0 37 273 : 278 283 288 293 298 304 309 314 319 7 3.5 ' 38 324 330 335 340 345 350 355 361 3 66 371 8 4.0 1 39 376 381 3^7 392 397 402 407 412 4l8 423 4.5 840 428 433 438 443 449 454 459 464 469 474 41 480 485 490 495 500 505 5 11 516 521 526 42 531 536 542 547 557 562 567 572 578 43 588 593 598 603 609 614 619 624 629 44 634 639 645 650 655 660 665 670 675 68 1 45 46 686 737 691 742 747 701 752 706 758 711 763 716 768 722 773 727 778 732 783 47 48 788 840 793 845 799 850 804 85? 809 860 814 865 819 870 824 875 829 881 834 8S6 49 891 896 901 906 911 916 921 927 932 937 850 942 947 952 957 962 967 973 978 983 988 N. O 1 MBMMIM 3 MMM 4 5 6 7 8 9 Prop.Pts. LOGARITHMS OF NUMBERS. N. 1 2 3 4 5 6 7 8 9 Pro] ). PtS. 850 92 942 947 952 957 962 967 973 978 983 988 51 52 53 54 i 55 50 57 58 59 993 93 044 095 146 197 247 298 349 399 998 049 IOO 151 202 252 303 354 404 *oo3 054 105 156 207 258 308 359 409 *oo8 059 no 161 212 263 3 < 3 364 414 *OI 3 064 "5 166 217 268 3 i 8 369 420 *oi8 069 120 171 222 273 323 374 425 *024 075 125 176 227 278 328 379 430 *029 080 131 181 232 283 334 384 435 *Q34 085 136 1 86 237 288 339 389 440 *Q39 090 141 192 242 293 544 394 445 1 2 3 0.6 1.2 1.8 24. 860 450 455 460 465 470 475 480 485 490 495 5 3.0 61 62 63 64 65 66 67 68 60 500 551 601 651 702 752 802 852 902 505 656 707 757 807 857 907 S x 561 611 661 712 762 812 862 912 & 616 666 717 767 817 867 917 520 57i 621 671 722 772 822 872 922 52 ^ 576 626 676 727 777 827 877 927 531 |8i 631 682 732 782 832 882 932 536 586 636 687 737 787 837 887 937 541 59i 641 692 742 792 842 892 942 546 596 646 697 747 797 847 897 947 6 7 8 9 3.6 4.2 4.8 5.4 870 952 957 962 967 972 977 982 987 992 997 6' 71 72 73 74 75 76 77 78 79 94 002 052 101 151 201 250 300 349 399 007 5 2 106 156 206 255 305 354 404 012 062 III 161 211 260 310 359 409 017 067 116 166 216 265 315 364 4H 022 072 121 171 221 270 3 20 369 419 027 077 126 176 226 275 325 374 424 032 082 I3i 181 280 330 379 429 037 086 136 1 86 236 285 335 384 433 042 091 141 191 240 290 340 389 438 047 096 146 196 245 295 345 394 443 1 2 3 4 5 6 7 8 9 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 880 448 453 458 463 468 473 478 483 488 493 81 82 83 84 85 86 87 88 89 498 547 596 645 694 743 792 841 890 503 I 52 601 650 699 748 797 846 895 507 557 606 655 704 753 802 851 900 512 562 660 709 758 807 856 905 517 567 616 665 7H 763 812 86 1 910 522 571 621 670 719 768 817 866 915 52 2 576 626 675 724 773 822 871 919 532 630 680 729 778 827 876 924 537 586 635 685 734 783 832 880 929 542 I 91 640 689 738 787 836 885 934 1 2 3 4 0.4 0.8 1.2 890 939 944 949 954 959 963 968 973 978 983 4 5 1.6 2 91 92 93 94 95 96 97 98 99 988 95 036 085 134 182 231 279 328 376 993 041 090 139 187 236 284 332 38i 998 046 095 143 192 240 289 & *002 051 IOO 148 197 245 294 342 390 *7 056 105 153 202 250 209 347 395 *OI2 061 109 158 207 255 303 352 400 *oi7 066 114 163 211 260 308 357 405 *022 071 119 1 68 216 265 3 < 3 361 410 *027 075 124 173 221 270 318 366 415 *O32 080 129 177 226 274 323 371 419 6 7 8 9 2.4 2.8 3.2 3.6 900 424 429 434 439 444 448 453 458 463 468 N. 1 2 3 4 5 6 7 8 9 Pro p. Pts. i8 TABLE I. N. O 1 2 3 4 5 6 7 8 9 Prop. Ptg. 900 95 424 429 434 439 444 448 453 "458" "463" "468" 01 472 477 482 487 492 497 501 506 5" 516 02 03 521 569 525 574 530 578 535 540 588 545 593 550 598 602 559 607 564 612 04 617 622 626 631 636 641 646 650 655 660 05 665 670 674 679 684 689 694 698 703 708 06 713 718 722 727 732 737 742 746 756 07 761 766 770 775 780 785 789 794 799 804 08 809 813 818 823 828 o 837 842 847 852 09 856 861 866 871 875 880 885 890 895 899 910 904 909 914 918 923 928 933 938 942 947 11 952 957 * 961 966 * 971 976 980 985 990 995 12 999 *oo 4 *oi4 *023 *028 *O33 *o 3 8 *042 1 0.5 13 96047 052 057 061 066 071 076 080 085 090 2 1.0 14 15 095 142 099 147 104 152 109 156 114 161 118 166 123 128 175 III 137 185 3 4 p. 1.5 2.0 9 f\ 16 190 194 199 204 209 213 218 223 227 232 o 6 .O 3.0 17 18 as 242 289 246 294 251 298 256 303 261 308 265 313 270 317 275 322 280 327 7 8 3.5 4.0 19 332 336 341 346 350 355 360 365 369 374 9 4.5 920 379 384 388 393 398 402 407 412 417 421 21 426 431 435 440 445 450 454 459 464 468 22 473 478 483 487 492 497 501 506 5" 515 23 520 525 530 534 539 544 548 553 558 562 24 25 4 572 619 577 624 581 628 586 633 591 638 I 95 642 600 647 605 652 609 656 26 661 666 670 675 680 685 689 694 699 703 27 28 708 713 759 717 764 722 769 727 774 778 736 783 788 745 792 750 29 802 806 811 816 820 825 830 834 839 844 930 848 853 858 862 867 872 876 881 886 890 31 32 895 942 900 946 904 909 956 914 #9 6o 918 965 923 970 928 974 932 979 984 1 0.4 33 988 993 997 *OO2 *OII *oi6 *02I *O2S *O3O 2 0.8 34 97 035 039 044 049 053 058 063 067 072 077 3 1.2 10 35 08 1 086 1 090 095 100 104 109 114 118 123 .0 2(1 36 128 132 1 137 142 146 155 1 60 165 169 6 .V 2.4 37 174 179 183 188 192 197 202 206 211 216 7 2.8 38 220 225 230 234 239 243 248 253 257 262 8 3.2 39 267 271 276 280 285 290 294 299 304 308 9 3.6 940 313 317 322 327 331 336 340 345 350 354 41 359 364 368 373 377 382 387 391 396 400 42 405 410 414 419 424 428 433 437 442 447 43 456 460 465 470 474 479 483 488 493 44 497 502 ! 506 5" 516 520 525 529 534 539 45 543 548 552 557 562 566 575 580 585 46 589 594 598 603 607 612 617 621 626 630 47 633 640 644 649 653 658 663 667 672 676 48 68 1 685 690 695 699 704 708 713 717 722 49 727 736 740 745 749 754 759 763 768 950 772 777 782 786 791 795 800 804 809 813 N. O 1 | 2 3 4 5 6 r 8 9 Prop. Pte. LOGARITHMS OF NUMBERS. N. O 1 2 3 4 5 6 7 8 9 Prop.Pte.|| 950 97 772 777 782 786 791 795 800 804 809 813 51 818 823 827 832 836 841 845 850 855 859 52 864 868 873 877 882 886 891 896 900 90S 53 909 914 918 923 928 932 937 941 946 950 54 955 959 964 968 973 978 982 987 991 996 55 98 ooo 00$ 009 014 019 023 028 032 037 041 56 046 050 055 059 064 068 073 078 082 087 ' 57 58 091 137 096 141 100 146 105 150 !SS 114 159 118 164 III 127 173 132 177 59 182 186 191 195 200 204 209 214 218 223 960 227 232 236 241 245 250 254 259 263 268 61 272 277 281 286 290 295 2 99 304 308 313 6 62 318 322 327 331 336 340 345 349 354 358 1 0.5 63 363 367 372 376 381 385 390 394 399 403 2 1.0 64 65 408 412 457 4J7 462 421 466 426 47i 430 475 480 439 484 444 489 448 493 3 4 1.5 2.0 2 ft 66 498 502 507 5" 516 520 525 529 534 538 6 3.0 67 68 69 F 632 547 592 637 552 597 641 I* 6 601 646 I? 605 650 565 610 655 570 614 659 574 619 664 579 623 668 673 7 8 9 3.& 4.0 4.6 970 677 682 686 691 695 700 704 709 713 717 71 722 726 731 735 740 744 749 753 758 762 1 72 767 771 776 780 784 789 793 798 802 807 1 73 fell 816 820 825 829 834 838 843 847 851 1 74 856 860 865 869 874 878 883 887 892 896 | 75 900 9^5 909 914 918 923 927 932 936 941 I 76 945 949 954 958 963 967 972 976 981 985 1 77 989 994 098 "003 *oo7 *OI2 *oi6 *O2I *025 *029 73 79 99034 078 038 083 043 087 047 092 052 096 056 100 061 105 06 5 109 069 114 074 118 I 980 123 127 131 136 140 145 149 154 158 162 81 167 171 176 1 80 185 189 193 I 9 8 202 207 II 82 211 216 220 224 229 233 238 242 247 251 1 0.4 83 255 260 264 269 273 277 282 286 291 295 2 0.8 84 300 304 308 313 317 322 326 330 335 339 3 1.2 1 ct 85 344 348 352 357 361 366 370 374 379 383 9 n 86 388 392 396 401 405 410 414 419 423 427 6 24 87 432 436 441 445 449 454 458 463 467 471 7 2.8 88 476 480 484 489 493 498 502 506 5 11 515 8 3.2 89 520 524 528 533 537 542 546 550 555 559 9 3.6 990 564 568 572 577 581 585 590 594 599 603 91 607 612 616 621 62$ 629 634 638 642 647 1 92 651 656 660 664 669 673 677 -682 686 691 93 695 699 704 708 712 717 721 726 730 734 94 739 743 747 752 756 760 765 769 774 778 95 782 787 791 795 800 804 808 813 8i7 822 1 96 826 830 835 839 843 848 852 856 861 865 1 97 870 874 878 883 887 801 896 000 904 909 1 98 99 913 957 ^61 922 965 926 970 930 974 935 978 939 983 944 987 94** 991 952 996 1000 00 000 004 009 013 017 022 026 030 035 039 9 N. O 1 2 3 4 5 6 7 8 9 Prop.Pta.ll |l 2O TABLE I. N. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 1000 ooo ooo 043 087 130 174 217 260 304 347 391 1001 434 477 521 564 608 651 694 738 78i 824 1002 868 911 954 998 *O4i *o84 *I28 *I 7 I *2I 4 *2 5 8 1003 ooi 301 344 388 43i 474 517 561 604 647 690 44 1004 734 777 820 863 907 950 993 *036 *o8o *I2 3 1 4.4 1005 002 1 66 209 252 296 339 425 468 512 555 2 8.8 1006 598 641 684 727 771 814 857 900 943 986 3 13.2 1007 003 029 073 116 159 202 245 288 331 374 417 4 17.6 1008 461 504 547 590 633 676 719 762 805 848 5 22.0 1009 891 934 977 *020 *o6 3 *io6 *i 49 *I92 *235 *278 6 IT 26.4 OA Q 1010 004 321 364 407 450 493 536 579 622 665 708 f 8 oU. o 35.2 1011 75i 794 837 880 923 966 *oo9 "052 *095 *I38 9 39.6 1012 005 i 80 223 266 309 352 395 438 481 524 567 1013 609 652 695 738 781 824 867 909 952 995 1014 006 038 08 1 124 166 209 252 295 338 380 423 1015 466 509 552 594 637 680 723 765 808 851 43 1016 894 936 979 *022 *o6s *io7 *i5o *I93 '236 *278 1 4.3 1017 007 321 364 406 449 492 534 577 620 -662 705 2 8.6 1018 748 790 833 876 918 961 *oo4 *046 *o89 *I 3 2 3 12.9 1019 008 174 217 259 302 345 387 430 472 515 558 4 17.2 O1 K 1020 600 643 _6_8s_ 728 770 813 856 898 941 983 6 21. o 25.8 1021 009 026 068 in 153 196 238 281 323 366 408 7 30.1 1022 45i 493 536 578 621 f\f\^ 706 748 791 833 8 34.4 1023 876 918 961 *oo3 *45 *o88 *I30 *I73 *2I 5 *2 5 8 9 38.7 1024 oio 300 342 385 427 470 512 554 597 639 681 1025 724 766 809 851 893 936 978 *020 *o6 3 *ios 1026 on 147 190 232 274 317 359 401 444 486 528 1027 570 613 655 697 740 782 824 866 909 95i 42 1028 993 *035 *o 7 8 *I20 *I62 *2O4 *247 *289 *33i *373 1 4/2 1029 012 415 458 500 542 584 626 669 711 753 795 2 8.4 1030 837 879 922 964 *oo6 *048 *O9O *I 3 2 *I74 *2I 7 8 12.6 1031 013 259 301 343 385 427 469 5ii 553 596 638 4 16.8 91 A 1032 680 722 764 806 848 890 932 974 *oi6 ="058 XI . v OK O 1033 014 100 142 184 226 268 310 352 395 437 479 7 4t). i 29.4 1034 521 563 605 647 689 730 772 814 856 898 8 33.6 1035 940 982 *024 *o66 *io8 *i5o *I 9 2 *234 *276 * 3 i8 9 37.8 1036 015 360 402 444 485 527 569 611 653 695 737 1037 779 821 863 904 946 988 *O3O *O72 *ii4 *I56 1038 016 197 239 281 323 365 407 448 490 532 574 1039 616 657 699 74i 783 824 866 908 950 992 41 1040 017 033 075 117 159 200 242 284 326 367 409 1 4.1 1041 45i 492 534 576 618 659 701 743 784 826 2 8.2 10i2 868 909 95i 993 *034 *076 *n8 *I 59 *20I *243 3 12.3 1043 018 284 326 368 409 451 492 534 576 6I 7 659 4 5 16.4 20.5 1044 700 742 784 825 867 908 950 992 *033 *075 6 24.6 1045 019 116 158 199 241 282 324 366 407 449 490 7 28.7 1046 532 573 615 656 698 739 781 822 864 905 8 32.8 1047 947 988 *030 *07I *ii3 *I54 *I 95 *237 *2 7 8 *320 9 36.9 1048 1049 020 361 775 403 817 444 858 486 900 527 94i 568 982 610 *O24 651 "065 693 *io7 734 *I48 1050 021 189 231 272 313 355 396 437 479 520 561 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS. 21 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 1050 1051 1052 1053 1054 1055 ]056 1057 1058 1059 10CO 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 021 189 231 272 3i3 355 396 437 479 520 561 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 "FT 42 4.2 8.4 12.6 16.8 21.0 25.2 29.4 33.6 37.8 41 4.1 8.2 12.3 16.4 20.5 24.6 28.7 32.8 36.9 40 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 89 3.9 7.8 11.7 15.6 19.5 23.4 27.3 31.2 35.1 M^M^MMMV op. Pts. 603 022 Ol6 4 2S 841 023 252 664 024 075 486 896 644 057 470 882 294 705 116 527 937 685 5" 923 335 746 157 568 978 727 140 552 964 376 787 198 609 *oi9 768 181 593 *oos 417 828 239 650 *o6o 809 222 635 *047 458 870 280 691 *IOI 851 263 676 *oS8 499 911 321 732 *I42 892 3^5 717 *I2 9 541 952 363 773 *i8 3 933 346 758 *I70 582 993 404 814 *22 4 974 387 799 *2II 623 *034 445 855 *26 5 025 306 347 388 429 470 SH 552 593 634 674 , 715 026 125 533 942 027 350 757 028 164 57i 978 029 384 756 165 574 982 390 798 205 612 *oi8 797 20<$ 615 *023 431 839 246 653 *059 838 247 656 "064 472 879 287 693 *IOO 879 288 697 *io5 5i3 920 327 734 *I4O 920 329 737 "146 II? 368 775 *i8i 961 370 778 *i86 594 *002 409 * 8 ' 5 S^T) j *002 411 819 *227 635 *O42 449 856 *262 *043 452 860 *268 676 *o83 490 896 *3Q3 708 *o84 492 901 * 309 716 *I2 4 53i 937 *343 424 465 506 546 587 627 668 749 789 030 195 600 031 004 408 812 032 216 619 033 021 830 * 35 640 045 449 853 256 659 062 871 276 681 085 489 893 296 699 102 9l \ 316 721 126 530 933 337 740 142 952 357 762 166 570 974 377 780 182 77 992 397 802 206 610 *oi4 417 820 223 *Q33 438 843 247 651 *054 458 860 263 665 *073 478 883 287 691 *95 498 901 303 705" *H4 519 923 328 732 *i 35 538 941 343 *I54 559 964 368 772 *i75 ' 5 Z 8 981 384 424 464 504 544 625 745 785 826 034 227 628 035 029 43 830 036 230 629 037 028 866 267 669 069 470 870 269 906 308 709 109 510 9IO 309 709 108 946 348 749 149 550 950 349 749 148 986 388 789 190 590 990 389 789 187 *O27 428 829 230 630 *O3O 227 *o67 468 869 270 670 *070 469 868 267 *io7 508 909 310 710 *IIO 509 908 307 *I 47 548 949 350 750 *i5o 549 948 347 "187 588 989 390 790 *i9o 9 387 426 466 506 546 586 984 382 779 176 573 969 365 761 *i$6 626 665 705 745 *H3 54i 938 335 73i *I2 7 523 919 *3U 708 8 785 825 038 223 620 039 017 414 811 040 207 602 998 041 393 B^-M^ ^B O 865 262 660 057 454 850 246 642 *Q37 904 302 700 097 493 890 286 68 1 *077 914 342 739 136 533 929 325 721 *ii6 *O24 421 819 216 612 *oo9 405 800 *'95 "064 461 859 ? 55 652 *o 4 8 444 840 I 2 3i 630 6 *io3 501 295 484 879 *2 7 4 *i83 580 978 374 771 *i67 563 958 *353 432 MMMMW 1 472 2 5H 3 55i 4 590 5 669 7 748 9 TABLE II. TABLE II. CONSTANTS WITH THEIR LOGARITHMS. Number. Logarithm. Ratio of circumference to diameter, TV, 3.14159265 0.49714 99 .. n\ 9.86960440 0.99429 97 . 27T, 6.28318531 0.79817 99 .. ^, I-77245385 0.2485749 Number of degrees in circumference, 360 2-5563025 minutes 21600' 4-33445 38 seconds 1296000" 6.11260 50 Degrees in arc equal to radius, 57- 2957795 1.75812 26 Minutes .. .. 3437'. 74677 3-53627 39 Seconds 2o6264 // .8o6 5.31442 51 Length of arc of i degree, .01745329 8.24187 7410 ..i minute, .00029089 6.46372 61 10 .. ..i second, .000004848 4.68557 4910 Number of hours in i day, 24 1.38021 12 minutes 1440 3.15836 25 . seconds 86400 4-9365 37 Number of days in Julian year, 365.25 2.5625902 Naperian base, 2.718281828 0.43429 45 Modulus of common logarithms, 0.434294482 9.637784310 Hours in which earth revolves through arc equal to radius, 3.8197186 0.58203 14 Minutes of time .. .. .. 229.18312 2.36018 26 Seconds of time '3750-987 4-1383339 III SINES AND TANGENTS OF SMALL ANGLES. 25 TABLE III. FOR SINES AND TANGENTS OF SMALL ANGLES. TO FIND THE SINE OB TANGENT I Log sin a = log a (in seconds) + & Log tan a = log a (in seconds) + T. TO FIND A SHALL ANGLE FROM ITS SINE OB TANGENTs Log a (in seconds) = log sin a + ?' Log a (in seconds) as log tan a + 2*. 26 TABLE III. 99 9 L. Sin. 8 T S' T' 60 1 2O 1 80 240 I 2 3 4 6.4^73 76476 7.06579 4-68557 .68557 .68557 .68557 .68557 4.68557 .68557 .68557 .68557 .68558 S-3I443 .31443 .31443 .31443 .31443 5-31443 .31443 .31443 .31443 .31442 300 360 420 4 80 540 1 I 9 7.16270 .24188 .30882 .36682 .41797 4.68557 .68557 .68557 .68557 .68557 4.68558 .68558 .68558 .68558 .68558 5-31443 31443 .31443 31443 .31443 5-31442 3 I 442 31442 .31442 .31442 600 660 720 780 840 10 ii 12 13 H 7.46373 .50512" 54291 n -4:68557 .68557 68557 .68557 68557 4-68558 .68558 .68558 .68558 .68558 S.3I443 .31443 .31443 31443 .31443 5-3I442 .31442 .31442 3 I 442 . 3*442 s 960 I02O 1080 1140 !! [I 19 7 :S& .69417 .71900 .74248 4 .6557 .68557 .68557 .68557 .68557 4.68558 .68558 .68558 .68558 .68558 5-31443 .31443 .31443 .31443 .31443 5-31442 .31442 .3H42 .3*442 .3*442 1200 1260 1320 1380 1440 20 21 22 23 24 7.76475 .78594 .80615 .82545 .84393 4.68557 .68557 .68557 .68557 .68557 4-68558 .68558 .68558 .68558 .68558 5-3I443 .31443 .31443 .31443 .31443 5-31442 3 I 442 .31442 ..31442 .31442 1500 1560 1620 1680 1740 3 % 29 7.86166 .87870 .89509 .91088 .92612 4.68557 .68557 .68557 .68557 .68557 4-68558 .68558 .68558 .68558 .68559 5-31443 .31443 .31443 .31443 31443 5.31442 .31442 .31442 3 I 442 .3*44* 1800 1860 1920 1980 2040 30 3i 32 33 34 7.94084 .95508 .96887 .98223 99520 4-68557 .68557 .68557 .68557 .68557 4-68559 .68559 .68559 .68559 .68559 5-31443 .31443 .31443 3 I 443 3 I 443 5.3I44I 3 I 44i 3*441 .3*441 3*44* SHOO 2l6o 2220 2280 2340 8 8 39 8.00779 .02002 .03192 04350 05478 4.68557 68557 .68557 .68557 .68557 4-68559 .68559 .68559 .68559 .68559 5.3I443 .31443 .31443 .31443 .31443 5.3*44* 3*441 .3*44* .31441 .31441 2400 2460 2520 2580 2640 40 4i 42 43 44 8.06578 .07650 .09718 .10717 4 'S 55 1 .68556 .68556 .68556 .68556 4-68559 .68560 68560 .68560 .68560 5-31443 3 I 444 3 I 444 .31444 .31444 5-3*44* .3*440 .31440 .3*440 3*440 2700 2760 2820 2880 2940 9 47 48 49 8.11693 .12647 .13581 .14495 .15391 4.68556 .68556 .68556 .68556 .68556 4.68560 .68560 .68560 .68560 .68560 5-3 I 444 3 I 444 .31444 .3!444 . 3*444 5.3I440 .3*440 .31440 3 I 440 .3*440 3000 3060 3120 3180 3240 50 51 52 S3 54 8.16268 .17128 -I797I .18798 . 19610 4-68556 .68556 .68556 .68556 .68556 4.68561 .68561 .68561 .68561 .68561 5-31444 3 I 444 3M44 .51444 3 I 444 5.31439 .31439 .3H39 .31439 .3H39 3300 3360 3420 3480 3540 P P 59 8.20407 .21189 .21958 .22713 23456 4-68556 .68556 .68555 .68555 .68555 4.68561 .68561 .68561 .68562 .68562 5-31444 3 I 444 3 I 445 .31445 31445 5-3I439 .3H39 .31439 .3H38 .31438 3600 60 8.24186 4-68555 4.68562 5-3I445 5 3H38 SINES AND TANGENTS OF SMALL ANGLES. 1 M t L.Sin. 8 T 8' r 3600 3660 3720 3780 3840 o I 2 3 4 8.24186 .24907 .25609 .26304 .26988 4-68555 .68555 .68555 .68555 .68555 4.68562 .68562 .68562 .68562 .68563 5-3I445 .31445 31445 .31445 .31445 5.31438 31438 .31435 .31438 3H37 i 3900 3960 4020 4080 4140 i I 9 8.27661 .28324 .28977 .29621 .30255 4-68555 .68555 .68555 .68555 .68555 4.68563 .68563 .68563 .68563 .68563 5-31445 .31445 .31445 31445 .31445 5-3H37 1 .3H37 .3H37 .3H37 31437 4200 4260 4320 4440 10 ii 12 13 H 8.30879 .3H95 .32103 .32702 .33292 4-68554 .68554 -68554 .68554 .68554 4-68563 .68564 .68564 .68564 .68564 5-31446 .31446 .31446 .31446 .31446 5.3H37 .31436 .31436 4620 4680 4740 % 19 8.33875 34450 .35018 .35578 .36131 4-68554 .68554 .68554 .68554 .68554 4.68564 .68565 .68565 .68565 .68565 5-31446 .31446 .31446 .31446 .31446 5.3I436 .3H3S .31435 .31435 .31435 4800 4860 4920 498o 5040 20 21 22 23 24 8.36678 .37217 '38276 .38796 4.68554 .6S553 .68553 .68553 .68553 4.68565 .68566 .68566 .68566 .68566 5- 3M46 .31447 .31447 .31447 3 H47 5.31435 .31434 .31434 .3H34 .31434 5100 5160 5220 5280 5340 11 11 29 '40320 .40816 4-68553 .68553 .68553 .68553 .68553 4.68566 .68567 .68567 .68567 .68567 5-3I447 .31447 .31447 .31447 31447 5-3I434 ^433 .31433 .31433 5400 5520 558o 5640 30 31 32 33 34 8 41792 .42272 42746 .43216 .43680 4-68553 .68552 .68552 .68552 .68552 4-68567 .6856$ .68568 .68568 .68568 5-31447 .31448 .31448 .31448 31448 5-3H33 .3H32 '3H32 5700 5760 5820 5880 CO CO CO CO CO 8.44139 .44594 .45044 .45489 .45930 4-68552 .68552 .68552 .68552 .68551 4-68569 .68569 .68569 .68569 .68569 5-3I448 .31448 .31448 .31448 .31449 ^31431 3H3I .3H3I 3I43I 6060 6120 6180 6240 40 41 42 43 44 8.46366 .46799 .47226 .4p5o .48069 4.68551 .68551 .68551 .68551 .68551 4.68570 .68570 .68570 .68570 .68571 5.31449 .31449 3 '449 .31449 .31449 5-3I430 .31430 .3M30 '31429 6300 6360 6420 6480 6540 49 '49304 .49708 .50108 4.68551 .68551 .68550 .68550 .68550 4.68571 .68571 .68572 .68572 .68572 5-3I449 31449 '3H50 5-3I429 .31429 .31428 .31428 .31428 6600 6660 6720 6780 6840 50 51 52 53 54 8.50504 '51673 .52055 4.68550 .68550 .68550 .68550 .68550 4.68572 .68573 .68573 .68573 .68573 5.3I450 .3H50 .3H50 .3H50 5.31428 .31427 31427 .31427 .31427 6900 6960 7020 7080 7I4P 11 11 59 8.52434 .52810 .53183 53552 .53919 4.68549 .68549 .68549 .68549 .68549 4.68574 .68574 .68574 .68575 68575 5.3I45I .3H5I .31451 .3H5I .31451 5.3I426 .31426 .31426 .31425 31425 7200 60 8.54282 4.68549 4-68575 53H5I 5 31425 TABLE IV. LOGARITHMS, ETC. 29 TABLE IV. LOGARITHMS OF THE SINE, COSINE, TANGENT AND COTANGENT FOR EACH MINUTE OF THE QUADRANT. TABLE IV. 9 L. Sin. d. L. Tang. c. d. L. Cotg. L. Cos. Proj ). Pt S. I 2 3 4 6.46373 6 . 76 476 6.94085 7.06 579 30x03 17609 12494 9601 6.46373 o . 76 476 6.94085 7.06579 30103 17609 12494 0601 3.53627 3.23524 3 05915 2.93421 o.ooooo o.ooooo o.ooooo o.ooooo 0.00000 00 3 3 .1 .2 3 3476 348 695 1043 3218 322 644 965 2997 300 599 899 i I 9 7.16 270 7 24188 7.30882 7.36682 7.41 797 7918 6694 5800 55 4576 7.16270 7.24 188 7.30882 7.36682 7.41 797 7918 6694 5800 S5 2.83730 2.75812 2.69 118 2.63318 2.58203 o.ooooo o.ooooo o.ooooo 0.00000 o.ooooo 55 54 53 52 5i 4 5 .x 1390 1738 2&M 280 X28 7 1609 2633 263 1199 1498 2483 248 10 ii 12 13 14 7.46373 7.50512 7-54291 7-57 767 7.60985 4139 3779 3476 32x8 7-46373 7.50512 7.5429I 7.57767 7.60986 4139 3779 3476 3219 2.53627 2.49488 2.45 709 2.42233 2.39014 o.oo ooo o.ooooo o.ooooo o.ooooo o.ooooo 50 4 2 4 8 47 46 .2 3 4 5 500 84X ZI2X S401 2227 527 790 1053 1316 497 745 993 1242 1848 ii Is 7 19 7.63 982 7.66 784 7.69417 7.71900 7.74248 2802 2633 2483 2348 7.63982 7.66 ?8| 7.69418 7.71900 7.74248 2803 2633 2482 2348 2228 2.36018 2.33215 2.30582 2.28 100 2.25 752 0.00000 o.ooooo 9-99999 9-99999 9 99999 45 44 43 42 41 .x .2 3 4 .5 22 3 445 668 Sgx 11x3 202 404 606 808 XOIO 370 554 739 924 20 21 22 23 24 7.76475 7- 78*>4 7.80615 7-82545 7-84393 2119 2O2I X930 x8 4 8 7.76476 7.78595 7.80615 7-82546 7 84394 2119 2O2O 1931 1848 2.23524 2.21 405 2.19385 2.17454 2.15 606 9 99999 9-99999 9-99999 9-99999 9-99999 40 37 36 .1 .2 3 1704 170 5" 1579 158 3x6 474 I47 47 294 442 11 11 29 7.86 166 7.87870 7.89 509 7 91088 7.92 612 1704 x6 39 579 524 7.86 167 7-87871 7.89510 7.91 089 7.92613 1704 1639 579 1524 2.13833 2.12 129 2 . 10 490 2.08911 2.07387 9.99999 9-99999 9 99999 9-99999 9.99998 35 34 33 31 4 5 .1 682 852 1379 138 632 789 1297 130 589 736 1223 122 30 32 33 34 7.94084 7-95 58 7.96 887 7.98223 7-99 520 1472 1424 379 X336 1297 7.94086 7-95 5io 7.96889 7-98225 7.99522 *473 1424 '379 1336 "97 2.05914 2.04490 2.03 III 2.01 77.Pt S. 89 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 31 1 9 L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. Prop. Pts. I 2 3 4 8.24 186 8.24903 8.25609 8.26304 8.26988 717 7 o6 695 68 4 6 73 66 3 653 6 44 634 624 616 608 599 590 583 575 568 560 553 547 539 533 526 520 5H 508 502 496 49 1 485 480 474 470 464 459 455 450 445 441 436 433 427 424 419 416 411 408 404 400 396 393 39 386 382, 379 376 373 369 367 363 8.24 192 8.24910 8.25616 8.26312 8.26996 7 o6 696 68 4 673 66 3 654 643 634 625 617 607 599 584 575 568 553 546 54<> 533 527 520 509 502 496 49 1 486 480 475 470 464 460 455 450 446 437 432 428 424 420 416 413 408 4. I 2 3 4 8.54282 8 . 54 642 8-54999 8-55354 8-55 705 360 357 355 35i 349 8.54308 8.54669 8.55027 8.55382 8-55734 361 358 355 352 349 1.45692 I.4533I 44973 .44618 .44266 9-99974 9-99973 9-99973 9.99972 9.99972 GO 59 58 11 .1 .2 3 360 36 72 108 350 35 70 105 340 3* a 102 I 9 8.56054 8 . 56 400 8.56743 8.57084 8.57421 346 343 341 337 336 8.56083 8.56429 8.56773 8.57114 8-57452 346 344 34i 338 006 .43917 43 571 43 227 .42886 42 548 9.99971 9.99971 9.99970 9.99970 9.99969 55 54 53 52 5i 4 5 .6 7 .8 144 180 216 252 288 140 75 210 245 280 I 3 6 170 204 2 3 8 2 7 2 10 it 12 13 14 8-57757 8 . 58 089 8.58419 8.58747 8.59072 333 330 328 325 323 8.57 788 8.58121 8 58451 8.58779 8.59105 333 330 3?8 y 32* .42 212 .41 879 41 549 .41 221 .40895 9.99969 9.99968 9-99968 9.99967 9-99967 50 8 % 9 .x .3 3 324 330 33 66 99 315 320 32 6 4 9 6 306 310 3* 6a 93 15 10 17 18 19 8-59395 8.59715 8.60033 8.60349 8.60662 320 3i3 316 313 311 8.59428 8 59749 8.60068 8 . 60 384 8.60698 321 319 316 314 .40572 .40251 -39932 .396l6 .39302 9-99967 9.99966 9.99966 9-99965 9.99964 45 44 43 42 41 *4 5 .6 7 .8 132 165 108 231 264 128 160 192 224 256 124 1 55 186 . 217 248 . 20 21 22 2 3 24 8.60973 8.61 282 8.61 589 8.61 894 8.62 196 309 307 305 302 8.61 009 8.61 319 8.61 626 8.61 931 8.62234 310 307 305 303 .38991 .38681 .3f374 .38069 37 766 9.99964 9.99963 9.99963 9.99962 9.99962 40 39 38 y y .X .a 3 297 300 30 60 90 288 290 29 58 87 27$ 285 28.- 57-< 85-5 s 3 29 8.62497 8.62 79! 8.63091 8.63385 8.63678 298 296 294 293 8-62 535 8.62834 8.63 131 8.63426 8.63 718 299 297 295 292 .37465 .37 166 36869 .36574 .36282 9.99961 9.99961 9.99960 9.99960 9-99959 35 34 33 32 3* 4 .5 .6 7 .8 120 ISO 180 210 240 116 145 174 203 232 114.0 142.5 171.0 199.5 228.0 BO 3i 32 33 34 8.63968 8.64256 8.64543 8.64827 8.65 no 288 287 284 283 281 8.64009 8.64298 8.64585 8.64870 8.65 154 289 ,.87 "85 284 281 35 99i 35 702 35415 35 130 .34846 9-99959 9.99958 9-99958 9-99957 9.99956 80 3 11 9 .x .2 3 270 280 28.0 S 6.0 84.0 275 27.5 55-0 82.5 256.5 270 27.0 54-0 81.0 35 36 * 1 39 8-65391 8.65670 8.65947 8.66223 8.66497 279 277 276 274 8.65435 8.65 715 8.65993 8.66269 8.66543 280 278 276 274 .34565 34285 .34007 33 73i 33457 9.99956 9-99955 9-99955 9-99954 9-99954 25 24 23 22 21 4 5 .6 7 .8 112. I40.O 168.0 196.0 224.0 IIO.O 137.5 165.0 192.5 22O.O 108.0 135-0 162.0 189 o 2ld 40 4i 42 43 44 8.66769 8.67039 8.67308 8.67575 8.67841 270 269 267 266 267 8.66816 8.67087 8.67356 8.67624 8.67890 271 269 268 266 _g. 33 184 32913 .32644 .32376 .32 no 9-99953 9.99952 9-99952 9-99951 9-99951 20 JQ !2 .1 .2 3 265 .26.5 53-0 79-5 260 .26.O .52.0 . 7 8.0 255 25.5 .51.0 .76.5 9 % 49 8.68 104 8.68367 8.68627 8.68886 8.69 144 263 260 259 258 2-5 8.68 154 8.68417 8.68678 8.68938 8.69 196 263 261 260 258 .31 846 31 583 .31 322 .31 062 30804 9-99950 9-99949 9.99949 9.99948 9.99948 15 H 13 12 II -4 5 .6 7 .8 132-5 159.0 185-5 212.0 2^8 <; I04.O 130.0 156.0 182.0 208.0 127.5 153.0 178.5 204.0 50 5i 52 S3 54 8.69400 8.69654 8.69907 8.70159 8.70409 254 253 253 250 8.69453 8.69708 8.69962 8.70214 8.70465 25S 254 252 251 .30547 .30292 .30038 .29786 29535 9 99947 9.99946 9.99946 9-99945 9-99944 10 1 I .1 .a 3 250 .25.0 .50.0 .75-0 345 .24.5 .49.0 73-5 240 .24.0 .48.0 .72.0 56 Ji 59 8.70658 8.70905 8.71 151 8.71395 8.71 638 949 247 246 344 243 8.70714 8.70962 8.71 208 71453 8.71697 249 248 246 245 244 .29286 .29038 .28 792 28 547 .28 303 9-99944 9-99943 9.99942 9.99942 9.9994: 5 4 3 i .4 5 .6 7 125.0 150.0 175.0 200.0 295.O 122.5 X47.0 X7I-5 Z96.O 22O.5 I2O.O 144.0 168.0 193.0 2x6.0 GO 8.71 880 8.71940 1.28060 9.99940 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. t Pro P. Pfc i. 1 87 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 33 3 t L. Sin. d. L. Tang. e.d. L. Cotg. L. Cos. P0p. PtS. I 8.71880 8.72 120 340 8.71940 8.72 181 241 1.28060 1.27819 9.99940 9.99940 GO 338 334 339 2 8.72359 339 2-0 8 . 72 420 239 1.27580 9-99939 58 .1 33.8 83-4 33.9 3 4 8.72$97 8.72834 330 837 8.72659 8.72896 239 237 2T.6 1.27341 1.27 104 9-99938 9-99938 y .3 3 47.6 46.8 70.2 45-8 68.7 I 8.73069 8.73303 234 8.73132 8.73366 1.26868 1.26634 9 99937 9.99936 55 54 -4 5 95-2 119.0 93-6 117.0 91.6 "4-5 I 8-73535 8.73767 232 232 8.73600 8.738J2 234 232 i . 26 400 1.26 1 68 9-999J6 9-99935 53 52 .6 7 142.8 166.6 140.4 163.8 137-4 160.3 if 8-73997 229 228 8.74063 231 229 1-25937 9 99934 51 .8 9 190.4 314.2 187.2 2X0.6 183.3 206. x 8.74226 8.74292 1.25 708 9 99934 50 ii 12 8-74454 8.74680 226 226 8.74521 8.74748 229 227 __ 1-25479 1.25 252 9 99933 9-99932 4-Q ., 22.5 22O 32.0 310 3X.6 13 8.74906 8.74974 1.25 026 9.99932 47 .3 45-0 44.0 43-a H 8.75130 224 8-75 199 225 i . 24 801 9-99931 46 3 67-5 66.0 64.8 16 8-75353 8-75575 222 8.75423 8.75645 222 1-24577 1-24355 9.99930 9 99 9-9 45 44 4 5 90.0 112.5 88.0 XIO.O 86.4 108.0 ii 19 8-75795 8.76015 8.76234 220 8X9 217 8.75867 8.76087 8.76306 220 2X9 1.24133 1.23913 1.23694 9 99929 9.99928 9.99927 43 42 41 .6 7 .8 135-0 157.5 180.0 133.0 154.0 176.0 _-0 n 139.6 151.2 173.8 20 21 22 8.76451 8. 76667 8.76883 216 3l6 8.76525 8.76742 8.76958 3J7 2x6 1-23475 1.23258 1.23 042 9-99926 9.99926 9.99925 40 39 38 9 .1 313 21.2 198.0 308 20.8 304 20.4 23 8.77097 8X4 8.77173 8X5 1.22 827 9.99924 37 .3 42.4 41.6 40.8 24 8.77310 213 8.77387 8X4 I.226I3 9.99923 36 3 6 3 .6 63.4 61.3 25 26 8.77522 8-77 733 311 8.77600 8.77811 813 211 I . 22 400 1.22 189 9.99923 9.99922 35 S4 4 5 84.8 1 06.0 83.2 104.0 81.6 103.0 % 29 8-77943 8.78152 8.78360 309 208 208 8 . 78 022 8.78232 8.78441 211 2IO 209 208 I. 21 978 1. 21 768 I-2I559 9.99921 9 99 920 9.99920 33 32 w .6 7 .8 9 127.2 148.4 169.6 190.8 124.8 145.6 166.4 187.2 122.4 142.8 163.2 183.6 30 8.78508 8.78649 I- 21 351 9.99919 S 2 8.78774 8.78979 205 8.78855 8.79061 206 206 I 21 145 1.20939 9.99918 9.99917 28 .1 301 2O. I 197 19.7 193 19.3 33 34 8.79183 8.79386 204 203 2O2 8 . 79 266 8.79470 205 204 1.20734 1 . 20 530 9 99917 9.99916 z .3 3 40.2 60.3 39-4 59- 1 38.6 57-9 35 36 9 39 10" 8.79588 8.79789 8.79990 8.80 189 8.80388 201 201 199 199 197 197 8.79673 8.79875 8 80076 8.80277 8.80476 203 202 201 201 199 I 9 8 x 9 8 1.20327 1.20 125 1 . 19 924 1 . 19 723 1 . 19 524 9-999I5 9.99914 9 999'3 9-999I3 9.99912 25 24 23 22 21 20" 19 4 5 .6 7 .8 9 100.5 120.6 140.7 160.8 180.9 189 98.5 1x8.2 137-9 157.6 185 772 96-5 1x5.8 I35.I 154-4 173-7 181 8.80585 8.80782 8 80674 8.80872 1 . 19 326 I.I9 128 9.99911 9.99910 42 8.80978 196 8.81068 196 1.18932 9.99909 18 i 18.9 18.5 18.1 43 8.81 173 X 95 8.81 264 196 I.I8736 9.99909 17 .2 37. 37.0 36.3 44 8.81 367 194 8.81459 195 1.18 541 9.99 908 16 3 56.7 __ f 55-5 54-3 45 46 49 8.81 560 8.81 752 8.81 944 8.82 134 8.82324 192 192 190 190 1 80 8.81 653 8 81 846 8.82038 8.82 230 8.82420 194 193 192 192 190 1.18347 1.18154 1.17962 1.17 770 1.17 580 9.99907 9.99906 9-99905 9.99904 9.99904 15 13 12 II 4 5 .6 7 .8 75.0 94-5 132.3 151.9 170.1 74.0 92.5 XXX. 129.5 148.0 166.5 73.4 90.5 108.6 126.7 144-8 162.0 50 8.82513 8.82 701 188 8.82610 8.82 799 190 x89 1.17390 I.I7 201 9.99903 9.99 902 10 4 3 a x 52 53 8.82888 8.83075 8.83261 187 x8 7 186 8.82 987 8.83 175 8.83361 1 88 188 1 86 -QJC I.I70I3 i . 16 825 i . 16 639 9.99901 9.99 900 9.99899 I .1 .3 3 0.4 0.8 .2 0.3 O.2 O.X 0.6 0.4 o.a 0.9 0.6 0.3 59 8.83446 8.83630 8.83813 8.83996 8.84177 184 183 183 181 x8x 8-83547 8-83732 8.83916 8 . 84 100 8.84282 185 184 x8 4 182 182 i 16 453 1 . 16 268 i . 16 084 1.15900 1.15 718 9-99898 9.99898 9.99897 9.99896 9-99895 5 4 3 2 I 4 5 .6 7 .8 9 .O 4 .8 .2 3.6 1.5 .0 0.5 x.8 .2 0.6 a. i .4 0.7 3.4 .608 9.7 .809 60 8.84358 8 . 84 464 I-I5536 9.99894 o L. Cos. d. L. Cotg. c.d L. Tang. L. Sin. f Prop. Pts. 86 TABLE IV. 4 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. Prop. Pts. I 2 3 4 8.84358 8.84539 8.84 718 8.84897 8.85075 181 179 179 178 177 177 176 US 175 173 173 173 171 171 171 169 169 169 167 168 166 1 66 165 164 164 163 163 162 162 160 161 59 J 59 159 158 157 157 156 155 155 55 154 153 *52 152 152 151 15* 150 150 149 149 148 147 M7 147 146 146 4S 145 8.84464 8.84646 8.84826 8.85006 8.85 185 182 180 180 179 178 15536 15354 15 *74 .14994 .14815 9.99894 9-99893 9.09892 9.99891 9.99891 60 11 H .x .2 3 .4 -5 .6 7 .8 9 .1 .2 3 4 5 .6 7 .8 9 .t .2 3 4 5 .6 7 .8 9 ") .i .2 3 4 5 .6 7 .8 9 .1 .2 3 4 5 .6 7 .8 9 .2 3 4 5 .6 7 .8 9 z8z 18.1 36.2 54-3 72.4 90.5 108.6 126.7 144-8 162.9 175 i7-5 35-0 53-5 70.0 87-5 105.0 122.5 140.0 157-5 168 .16.8 33-6 50.4 779 17.9 35-8 53-7 71.6 89-5 107.4 125-3 143.2 161.1 173 17-3 34-6 51-9 69.2 86.5 103.8 121. 1 138.4 155-7 166 16.6 /33-2 49.8 66.4 83.0 99.0 116.2 132.8 149.4 159 15.9 31-8 47-7 63.6 79-5 95-4 111.3 127.2 i43- 153 15-3 30.6 45-9 61.2 76.5 91.8 107.1 122.4 "37-7 M7 14.7 29.4 44.1 58.8 73-5 88.2 102.9 117.6 132.3 177 17.7 35-4 53-1 70.8 88.5 106.9 123.9 141.6 *59-3 171 17.1 34- Si-3 68.4 85.5 102.6 119.7 136.8 153-9 164 16.4 32.8 49.8 65.6 820 98-4 114.8 131-3 147.6 157 15-7 31-4 47- 62.8 78-5 94-3 109.9 125.6 MI- 3 ' IS* 15.1 30.2 45-3 60.4 75-5 90.6 105.7 120.8 135.9 z O.I O.S 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I I 9 8.85 252 8.85429 8.85 605 8.85 780 8.8595! 8-85 363 8.85 540 8.85 717 8.85893 8.86069 177 177 176 176 174 174 174 172 172 171 171 170 169 169 168 167 167 1 66 165 165 165 163 163 163 161 162 160 1 60 1 60 !59 158 158 157 157 156 155 155 155 153 154 152 152 152 151 151 150 'So 149 148 149 147 H7 147 146 146 H 637 .14460 . 14 283 .14107 I393 1 9.99890 9.99 889 9.99888 9-99887 9.99886 55 54 53 S 2 5i 10 ii 12 13 14 8.86 128 8.86301 8.86474 8.86645 8-86816 8.86243 8.86417 8.86591 8.86 763 8.86935 13757 !3 5 8 3 13 409 .13237 .13065 9.99885 9.99884 9.99883 9.99882 9.99881 50 49 48 47 46 15 1 6 \l I I9 8.86987 8.87 156 8.87325 8.87494 8.87661 8.87 106 8.87277 8.87447 8.87616 8.87785 .12894 . 12 723 .12553 . 12 384 .12 215 9.99880 9.99879 9.99879 9.99878 9.09877 45 4. 43 42 4i 20 21 22 23 2 4 8.87829 8.87995 8.88 ?6i 8.88326 8.88490 8-87953 8.88 126 8.88287 8.88453 8.88618 . 12 047 .11 880 .11 713 II547 .11 382 9-99876 9 99875 9.99874 9.99873 9.99872 40 i H 2 II 31 32 33 34 8.88654 8.88817 8.88980 8.89 142 8.89304 8.88 783 8.88948 8.89 in 8.89274 8.89437 .11 217 . 1 1 052 . 10 889 . 10 726 . 10 563 9.99871 9.99870 9 . 99 869 9.99868 9.99867 35 34 33 32 3i 67.5 84.0 100.8 117.6 134-4 8.89464 8.89625 8.89 784 8.89943 8.90 102 8.89598 8.89 760 8.89920 8.90080 8.90240 . 10 402 . 10 240 . 10 080 .09920 .09760 9.99866 9.99865 9.99864 9.99863 9 . 99 862 30 2 9 28 % 151.2 ; 162 16.2 33-4 4 8.6 6 4 .8 81.0 97.2 "3-4 129.6 145-8 155 iS-5 31-0 46.5 62.0 77-5 93-o 108.5 124.0 139-5 149 14.9 29.8 44-7 59-6 74-5 89.4 104.3 119.2 I34-J it 3 39 8.9O260 8.90417 8.90 574 8.90 730 8.90885 8.90399 8.90557 8.90 715 8.90872 8.91 029 .09601 .09443 .09285 .09 128 .08 971 9.99861 9.99860 9-99859 9.99858 .9 99857 25 24 23 22 21 "20" ii 'to i 4i i 4* 43 44 8.91 040 8.91 195 8.91 349 8.91 502 8.91 655 8.91 185 8.91 340 8.91 495 8.91 650 8.91 803 .08815 .08 660 .08 505 .08350 .08 197 9.99856 9.99855 9-99854 9.99853 9.99852 9 9 49 8.91 807 8.91 959 8.92 no 8.92261 8.92411 8-91 957 8.92 no 8 . 92 262 8 92 414 8.92565 8.92 716 .8.92866 8.93016 8.9.3 165 8-93 313 .08043 .07890 .07 738 .07 586 07435 9.99851 9.99850 9-99848 9.99847 9.99846 15 14 13 12 II 50 Si 52 53 54 8.92 561 8.92 710 8.92859 8.93007 8-93 154 .07 284 07 134 .06 984 .06 835 .06 687 9-99845 9.99844 9-99843 9.99842 9.99841 10 1 I 55 56 R 59 8.93301 8.93448 8-93594 8.93 740 8.93 885 8.93462 8 9^609 8-9.3756 8.93903 8.94049 .06 538 .06 391 .06244 .06 097 05951 9.99840 9-99839 9.99838 9.99837 9.99836 5 4 3 2 I 60 8.94030 8.94195 1.05805 9-99834 L. Cos. (1. L. Cotg. 1C. d. L. Tang. L. Sin. f Prop. Pts. 85 . | LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 35 5 9 L. Sin. d. L. Tang-. c. d. L. Cots. L. Cos. Pro p. Pfe 1. I 2 3 4 8.94030 8-94 174 8.94317 8.94461 8.94603 144 MS 144 142 8-94 195 8.94340 8.94485 8.94630 8-94 773 145 145 145 43 144 05805 .05 660 05 5i5 05 370 .05 227 9-99834 9-99833 9.99832 9.99831 9.99830 60 3 3 .X .3 3 145 14.5 ay o 43-5 143 4-3 38.6 43.9 14X X4.x 28.9 42-3 \ I 9 8.94746 8.94887 8.95 029 8.95 170 8-95 3io 4 142 4 140 8.94917 8.95060 8.95 202 8.95 344 8.95 486 143 142 142 143 .05 083 .04 940 .04 798 .04 656 04 5H 9.99829 9.99828 9-99827 9.99825 9-99824 55 54 53 52 5i 4 5 .6 7 .8 58.0 73.5 87.0 101.5 116.0 57-2 71.5 85.8 1 00. 1 1x4.4 56.4 70.5 84.6 98.7 XI3.8 10 ii 12 13 14 8-95 450 8-95 589 8.95 728 8.95 867 8.96005 139 139 139 138 178 8.95 627 8.95 767 8.95908 8.96047 8.96 187 140 141 139 140 x?8 04 373 04 233 .04092 03 953 .03 813 9.99823 9.99822 9.99821 9.99820 9.99819 50 2 3 9 .X .9 3 I30-5 139 13-9 37.8 41-7 128.7 138 13.8 ?. 4-4 126.9 136 13.6 07.9 40.8 !i ii 19 8.96143 8.96280 8.96417 8.96 553 8.96689 37 37 136 136 ,,6 8-96325 8.96464 8.96602 8.96 739 8.96877 139 138 37 138 TQ6 03 675 03 536 03 398 .03 261 .03 123 9-99817 9.99816 9.99815 9-99814 9.99813 45 44 43 42 41 4 5 .6 7 .8 55-6 69.S 83-4 97-3 XII. 3 55-2 69.0 82.8 96.6 1x0.4 54-4 68.0 81.6 95. a 1 08. 8 20 21 22 23 24 8.96825 8.96960 8.97095 8.97229 8.97363 35 135 134 134 8.97013 8.97 ijo 8.97285 8.97421 8.97556 37 135 136 J35 .02 987 .02 8^0 .02 7lS .02 579 .02444 9.99812 9.99 810 9.99809 9.99808 9.99807 40 P II 9 .x .9 3 125.1 135 3-5 37.0 40.5 124.2 133 3-3 26.6 39-9 122.4 131 X3.x 26.9 39-3 3 3 29 8.97496 8.97629 8.97 762 8 . 97 894 133 133 132 132 8.97691 8.97825 8-97959 8.98092 8.98225 134 134 133 133 .02309 02 175 .02 041 .01 908 oi 775 9.99806 9-99804 9.99803 9.99802 9.99801 35 34 33 32 3i 4 5 .6 7 .8 S4-o 67.5 81.0 94-5 108.0 53-2 66.5 79-8 93-i ro6.4 Sa-4 6S.S 78.6 9X.7 104.8 ao 31 32 33 34 8.98157 8.98288 8.98419 8.98 549 8.98679 131 131 130 130 8.98358 8.98490 8.98622 8.98 753 8.98884 132 132 131 131 .01 642 .01 510 .01 378 .01 247 .01 116 9.99800 9 99 798 9-99 797 9.99796 9-99795 30 % Vw .9 3 129 13.9 25.8 38.7 .19.7 128 X3.8 25.6 38.4 117.9 126 X3.6 25.9 37-8 1 39 8.98808 8.98937 8.99066 8-99 194 8.99322 129 129 128 128 128 8.99015 8.99145 8.99275 8.99405 8-99534 130 130 130 129 128 .00985 .00855 .00 725 00595 .00466 9-99793 9 99 792 9-99 79i 9.99 790 9.99 788 25 24 23 22 21 4 5 .6 7 .8 51.6 64.5 77-4 90.3 103.2 116 x 51.2 64.0 76.8 89.6 :03.4 50.4 63.0 75-6 88.3 100.8 40 4i 42 43 44 8.99450 8-99577 8-99 704 8.99830 8.99956 127 127 126 126 126 8.99662 8.99 791 8.99919 9.00046 9.00 174 129 128 127 128 .00338 .00209 .00081 0.99954 0.99826 9-99 787 9.99 786 9.99785 9-99 783 9.99 782 20 !l .x .3 3 MS 12.5 35.0 37-5 123 13.3 34.6 36.9 123 12.2 24-4 36.6 .0 3 4^ 1 49 9.00082 9.00207 9.00332 9.00456 9.00 581 125 125 124 125 9.00301 9.00427 9-00553 9.00679 9.00 805 126 126 126 126 0.99699 0-99 573 0.99447 0.99321 0.99 195 9.99 781 9-99 78o 9-99 778 9-99 777 9-99 776 15 H 13 12 II 4 5 .6 7 .8 50.0 62.5 75-o 87.5 100. "13-5 49.2 61.5 73-8 86.1 98.4 *IO.7 61.0 73- 85-4 97.6 IOQ.8 50 Si 52 53 54 9.00 704 9.00828 9.00951 9.01 374 9.01 196 124 123 123 122 9.00 930 9-01 055 9.01 179 9-Oi 303 9.01 427 125 124 124 124 0.99070 0.98 945 0.98821 0.98697 0.98573 9-99 775 9-99773 9-99 772 9-99 77i 9-99 769 10 1 I .x .3 3 121 13. 1 24.2 36.3 .0 . 120 12.0 *4.o 36.0 .0 _ X O.I O.3 0.3 11 11 59 9.01 318 9.01 440 9.01 561 9.01 682 9.01 803 122 121 I2X 121 9-Oi 550 9.01 673 9.01 796 9.01 918 9 . 02 040 123 123 122 122 0.98 450 0.98 327 0.98 204 o 98 082 0.97 960 9.99768 9.99 767 9 99 765 9-99764 9.99763 5 4 3 2 I 4 5 .6 7 .8 48.4 60.5 72.6 84.7 96.8 108.9 60.0 72.0 8 4 .a 96.0 108.0 0-5 0.6 f 0.7 08 o.o GO 9.01 923 9.02 162 0.97838 9-99 76i L. Cos. d. L. Cotgf. c.d. L. Tang. L. Sin. f Pro P. Pfc L 84 6 TABLJ2, IV. 6 I L. Sin. (1. L. Tang. c.d. L.Cotg. L. Cos. Prop. Pts. I 2 3 4 I I 9 lo- ii 12 13 14 9-oi 923 9.02 043 9.02 163 9.02283 9.02402 120 1 2O 120 119 118 119 118 117 118 117 117 116 116 "6 116 "5 "5 114 "5 "3 114 114 "3 112 XI3 112 112 ZI2 XII III III 110 1 10 110 110 109 109 109 108 109 108 107 108 107 107 106 107 106 105 106 105 105 105 105 104 104 104 103 103 103 9.02 162 9.02 283 9 . 02 404 9.02525 9.02 645 121 121 I2X 120 X2X "9 1 2O 119 118 1x9 118 118 117 1x8 116 117 1x6 116 1x6 US "5 "5 "5 114 114 "3 114 "3 1X2 1x3 1X2 112 XI2 III III III no III 1X0 109 no 109 109 108 109 108 108 107 108 107 106 107 106 106 106 106 105 105 105 104 0.97838 0.97717 0.97596 0-97471 0-97355 9 99 76i 9-99 76o 9-99 759 9 99757 9-99 756 GO 59 58 I .1 .2 3 4 5 .6 7 .8 9 .1 .a 3 4 5 .6 7 .8 9 .3 3 4 5 .6 7 .8 9 .3 3 4 5 .6 7 .8 9 .i .a 3 4 5 .6 7 .8 S .3 3 4 5 .6 7 .8 9 tax 12. 1 34.2 36.3 48.4 60.5 73.6 84.7 96.8 108.9 8 xx.8 a 3 .6 35-4 47.3 59-0 70.8 82.6 94-4 106.3 115 11.5 23.0 34-5 46.0 57-5 69.0 80.5 92.0 103.5 iia II. 2 23.4 33-6 44-8 56.0 67.2 78.4 89.6 100.8 109 10.9 31.8 32-7 43.6 54 5 65-4 76-3 87.2 98.1 106 10.6 21.2 31-8 43-4 53-c 6 3 .< 74.3 84.8 95-4 zoo 12.0 24.0 3 e o 48.0 60.0 72.0 84.0 9 6.o 108.0 "7 "7 33.4 35-1 46.8 58.5 70.2 81.9 93-6 105.3 "4 11.4 22.8 34.3 45-6 57-0 68.4 79.8 91.2 IO2.6 III II. I 32.2 33-3 44-4 55-5 66.6 77-7 83.8 99-9 zo8 10.8 21.6 3-4 43-2 54-0 64.8 75-6 86.4 97.2 zos 10.5 21. 3-S 42.0 52 S 63. c 73-5 84. c 94-5 "9 IS. J *3-8 : ,35 7 47 6 I 59 ! 7i-4 83-3 95 -a 107.1 116 li. 6 ' 93.3 34-8 1 46.4 , 58.0 69.6 81.3 92.8 104.4 "3 11.3 22 6 33 9 45-- 56.5 67.8 79-1 90.4 101.7 zzo II. 22.0 33 o 44-0 55-0 66.0 77.0 88.0 99.0 107 10.7 21.4 32.1 42.8 53-5 6 4 .a 74-9 ' 85-6 I 96.3 Z04 10.4 20.8 3*. 2 41.6 52.0 62.4 72.8 8 3 .a 93.6 9.02 520 9.02639 9.02 757 9.02874 9.02992 9.02 766 9.02885 9.03005 9.03 124 9.03242 0.97234 0.97 115 0.96995 0.96 8/6 0.96 758 9-99755 9-99 753 9-99 75 2 9-99751 9-99 749 55 54 53 52 5i 9.03 109 9.03 226 9.03342 9 03458 9-03 574 9.03361 9-03479 9-03 597 9.03714 9.03832 0.96 639 0.96 521 o . 96 403 0.96 286 0.96 1 68 9-99 748 9-99 747 9-99745 9-99 744 9-99 742 50 3 i II 12 19 9.03690 9.03805 9 . 03 920 9.04034 9.04 149 9.03 948 9.04065 9.04 181 9.04297 9.04413 0.96052 0-95 935 0.95819 0-95 73 o 95 587 9-99 74i 9-99 740 9-99 738 9-99 737 9-99 736 45 44 43 42 4i ~w 39 38 3 20 21 22 23 24 9 . 04 262 9.04376 9.04490 9.04603 9-047I5 9.04528 9-04643 9.04758 9.04873 9.04987 0.95472 0-95 357 0.95 242 0.95 127 0.95013 9-99 734 9-99 733 9-99731 9-99730 9 99 728 g 8 29 9.04828 9.04940 9.05052 9.05 164 9.05275 9.05 101 9.05 214 9.05328 9.05441 9-05553 0.94899 0.94 786 0.94672 0-94559 0.94447 9 99 727 9-99 726 9-99 724 9-99 723 9 99 72i 35 34 33 32 3i B 31 32 33 34 9.05386 9-05497 9.05607 9.05 717 9.05 827 9.05 666 9.05 778 9.05 890 9.06002 9.06 113 0-94334 . 94 222 0.94 no o . 93 998 0.93887 9-99 720 9.99718 9.99717 9.99716 9-99 7H 30 29 28 27 26 9 11 39 9-5 937 9.06 046 9-o6 155 9.06264 9.06372 9.06 224 9-06335 9.06445 9.06556 9.06666 0.93 776 0.93 665 0-93555 0.93444 0-93334 9-99 713 9 99 7" 9.99710 9-99 7o8 9.99707 25 24 23 22 21 40 4i 42 43 44 9.06481 9.06 589 9.06 696 9 . 06 804 9.06911 9.06 775 9.06885 9.06994 9.07 103 9.O7 211 0.93225 0.93H5 0.93 006 0.92 897 0.92 789 9 99 705 9-99 704 9.99702 9 99 7oi 9.99699 20 1 9 18 17 16 * 49 9.07 018 9.07 124 9.07231 9-07337 9 07442 9.07320 9.07 428 9 07536 9.07643 9.07 751 0.92 680 0.92572 0.92 464 0.92357 0.92 249 9.99698 9.99696 9-99695 9.99693 9.99692 15 14 13 12 II 50 5i 52 53 54 9.07548 9-07653 9.07758 9 07863 9.07968 9-07858 9.07964 9.08071 9.08 177 9.08283 0.92 142 0.92 036 0.91 929 0.91 823 0.91 717 9.99 690 9.99 689 9.99687 9.99686 9.99684 10 I 9 12 59 60" 9.08 072 9.08 176 9.08280 9.08383 9.08486 9.08389 9.08495 9.08 600 9.08 705 9.08810 0.91 611 o 9i 505 0.91 400 o 91 295 o 91 190 9.99683 9.99681 9.99 680 9.99678 9 99677 5 4 3 2 9-08589 9.08 914 o 91 086 9.99675 L. Cos. (1. L. Cotg. c.d. L. Tang. L. Sin. / Prop. Pts. 83 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 37 7 / L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. BO" P H Prop. Pts. 6 2 3 4 | I 9 9.08589 9.08 692 9.08795 9.08897 9.08999 103 103 102 102 102 lot 102 lot xox 100 tot TOO TOO 99 100 99 99 98 99 98 98 98 98 97 97 97 97 96 97 96 96 95 96 95 95 95 94 95 94 94 93 94 93 93 93 93 93 92 93 93 93 9* 93 9* 9* 9 9 90 9' 9 9.08914 9.09019 9 09 123 9.09227 9.09330 105 104 104 103 104 103 03 103 103 102 XO3 toi X02 IOX IOI XOI XOX 100 100 too too 99 99 99 99 99 98 98 98 98 97 98 97 97 96 97 96 96 96 96 95 95 95 95 95 94 95 94 94 93 94 93 93 93 93 93 93 93 9' 93 0.91 086 0.90981 0.90877 0.90 773 0.90670 9-99675 9.99674 9.99672 9.99670 9.99669 .2 3 4 :i i -9 .1 .2 3 .4 :l :1 9 .1 .2 3 4 i 9 .2 3 -4 i 9 .1 .2 3 4 :i :l 9 105 io-5 21.0 31-5 42.0 52-5 63.0 III 94-5 103 IO.2 20.4 30.6 40.8 i!:S E:t 9 i.8 i .1 < .2 K . 3 2( 4 3< 5 4< : JZ :5 97 9-7 19.4 29.1 38.8 48-5 58.2 67.9 77.6 873 94 ,11 28.2 37-6 47 o 56.4 65.8 III 9* 9 i 18.2 27-3 36-4 45-5 546 11 81^9 104 10.4 20.8 31.2 41.6 52.0 62.4 72.8 83.2 93-6 zox 10. 1 20.2 30-3 40.4 Si Si 90.9 39 ! >-9 < >-8 ic )-7 2< ) 6 3< >-5 4; 1-3 % ?-2 7* >.i SI 96 9.6 19.2 28.8 38-4 48.0 III 76 .8 93 93 18.6 27.9 37-2 46-5 558 65 i 74-4 83-7 90 ,I:S 35 45-o 54.0 63-0 E:S 103 10.3 20.6 30.9 41.2 ill 2: 92.7 100 IO.O 2O. O 30.0 40.0 50.0 6O.O 70.0 80.0 90.0 > >.8 >.6 M ).a |.o 58 5.6 u 95 95 19.0 28.5 38.0 47-5 570 66.5 76.0 85-5 9* 92 18.4 27.6 36.8 46 O 55 2 64-4 73 ^ 82.8 a O.2 1 0.6 0.8 I.O 1.2 :i 1.8 9.09 101 9.09 202 9.09304 9.09405 9.09506 9 09434 9 09537 9.09 640 9.09742 9.09845 0.90566 0.90463 0.90360 0.90 258 0.90155 9.99667 9.99666 9.99664 9.99663 9.99661 55 54 53 52 5i 10 ii 12 '3 14 9.09606 9.09 707 9.09807 9.09907 9.IO006 9-09947 9.10049 9.10 150 9.10252 9.10353 0.90053 0.89951 0.89850 0.89 748 0.89 647 9 99659 9.99658 9.99656 9 99655 9 99653 50 49 48 47 46 ii !1 J9 9.10 106 9.1020f 9-10304 9 . 10 402 9.10501 9.10454 9-10555 9.10650 9.10756 9.10856 o . 89 546 0.89445 0.89344 0.89244 o 89 144 9 99651 9 99 650 9.99648 9.99647 9 99645 45 44 43 42 41 40" i 20 21 22 23 24 9.10599 9-10697 9.10795 9 10893 9.10990 9.10956 9.11 056 9 ii 155 9 ii 254 9-ii 353 o 89044 o 88944 0.88845 o 88 746 0.88647 9 99643 9.99642 9.99640 9.99638 9 99637 11 11 29 9.11 087 9.11 184 9.11 281 9.11377 9.11474 9.11452 9 55i 9.11 649 9 ii 747 9- ii 845 o 88548 0.88449 0.88351 0.88253 o 88155 9 99635 9 99633 9.99632 9.99630 9 99629 35 34 33 32 3i w 3 11 BO 3i 32 33 34 9 "570 9.11 666 9.11 761 9 ii857 9.11 952 9 ii 943 9.12040 9 .12 138 9.1223? 9.12332 0.88057 0.87960 0.87862 o 87765 o 87 668 9 99627 9.99625 9.99624 9 99622 9 . 99 620 35 36 H 39 9-12047 9.12 142 9.12 2^6 9.12331 9.12425 9.12428 9 12 525 9.12621 9.12717 9.12813 o 87 572 o 87475 o 87379 o 87283 0.87 187 9.99618 9.99617 9.99615 9.99613 9.99612 25 24 23 22 21 20" !1 \l 40 4i 42 ' 43 1 44 9.12519 9.12 612 9 12 706 9. 12 799 9 12 892 9 12909 9.13004 9 13099 9 13 194 9.13289 o 87091 0.86996 0.86901 o 86 806 o 86 711 9.99 610 9 99608 9 99607 9 99605 9 99603 8 ? 49 50 51 52 53 54 9.12.985 9 13078 9 U i/i 9 13263 9 U355 9 i33 8 4 9 13478 9 13573 9.13667 9.13 761 o 86616 o 86 522 o 86 427 086333 o 86 239 9 99 601 9 99 600 9 99 598 9 99 596 9 99595 15 14 13 12 II 9 13447 9 13539 9.13630 9.13722 9 13813 9 13854 9.13948 9.14041 9-I4I34 9.14227 0.86 146 0.86052 0.85 959 0.85 866 0-85 773 9 99593 9-99591 9-99589 9-99 588 9.99586 10 1 I P y 59 9.13904 9-13994 9.14085 9-I4I75 9.14266 9.14320 9.14412 9.14504 9-14597 9.14688 0.85 680 0.85 588 0.85 496 o 85403 0.85 312 9.99584 9.99582 9.99581 9-99 579 9-99577 5 4 3 2 I "0" loo 9.I4356 9.14780 0.85 220 9-99575 L. Cos. d. L. Cotg. c. d. L. Tang. L. Sin. f Prop. Pts. 82 8 TABLE I\ r 8 9 L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. F ro P.; Pts b 2 3 4 9-I4356 9-14445 9-14535 9.14624 9.14714 89 90 8 9 90 80 9.14780 9.14872 9.14963 9-15054 9-i5 i45 99 9* 9* 9 0.85 220 0.85 128 0.85 037 o . 84 946 0.84855 9 99575 9-99574 9-99572 9-99570 9.99568 (JO 5 si 1 9 i 9 .2 18 .3 27 a .2 1 i .1 27 x .1 .2 ^ go 9.0 27.0 i 1 9 9.14803 9.14891 9.14980 9.15069 9-15 157 88 89 89 88 88 9.15236 9-15327 9-15417 9.15508 9 15 598 9 1 90 9 90 o . 84 764 0.84673 0.84583 o . 84 492 0.84402 9.99566 9.99565 9 99563 9.99561 9-99559 55 54 53 52 5 1 .436 5 46 6 55 .7 64 18 I 3 .8 .0 .2 4 .6 36 45 8 P 4 i i 36.0 45-Q 54.0 63.0 P' 10 II 12 '3 H 9.15245 9-15333 9.15421 9-I5508 9-I5596 88 88 87 83 87 9.15688 9-15777 9.15867 9.I5956 9.16046 89 90 89 90 80 0.84312 0.84223 0.84133 0.84044 0.83954 9-99557 9-99556 9-99554 9-99 SS 2 9 99550 50 8 S 9 82 .2 3 .8 ! i 2( 81 *9 5-9 7-8 37 9 i i 81.0 M $.8 a hi jS' 19 9-15683 .9.15 770 9-I5857 9-15944 9.16030 87 87 87 86 86 9-16135 9.16224 9.16312 9.16401 9.16489 89 88 89 88 88 0.83865 0.83 776 0.83688 0.83 599 0.83 511 9-99548 9.99546 9-99545 9 99543 9-99541 45 44 43 42 41 -4 1 .:i 3. * i 5-0 1-5 5-4 z-3 1.2 3. 4^ r 7< ) 2 :i 1.6 >-4 20 21 22 23 2 4 9.16 116 9 16203 9.16289 9-16374 9 16 460 87 86 85 86 8e 9-I6577 9.16665 9-16753 9.16841 9.16928 88 88 88 87 88 0.83423 0-83335 0.83247 0.83159 0.83072 9-99539 9-99537 9 )9 535 9i 99 533 9 99532 40 3. 11 9 .1 .2 3 a ck 1 } I J. 1 17 5-7 7-4 ^ ( J i 2 ^.2 36 ?.6 tl '2 2 29 9-16545 9.16631 9.16 716 9.56801 9.16886 86 85 85 85 BA. 9.17016 9.17 103 9.17 190 9.17277 9 17363 87 87 87 86 8? 0.82984 o 82897 0.82 810 0.82 723 o 82 637 9 99530 9.99528 9-99526 9 99524 9-99 522 35 34 33 32 3i 3 is 3< 4. I 6( -< ^& J-5 Z.2 3.9 ?.6 ? _ 3< 4. 1 6J \4. J-o [.6 5.2 $.8 80 31 S 34 9.16970 9-I7055 9.17139 9-17223 9.17307 85 84 84 84 4 9.I7450 9.17536 9.17 622 9.17708 9.17794 86 86 86 86 86 0.82 550 0.82464 0.82378 0.82 292 0.82 206 9.99520 9.99518 9-995I7 9 99515 9 99513 30 1 9 .1 .2 3 7 J i i 2. >-o !5 5-5 7-0 >-5 7 J i< 2 7-4 H ^:S J -? i 39 9.17391 9-17474 9.17558 9.17641 9.17724 83 84 8 3 8 3 8l 9.17880 9.17965 9.18051 9.18 136 9.l8 221 85 86 85 85 a- O.82 I2O 0.82035 0.81 949 0.81 864 0.81 779 9.99511 9 99509 9 99507 9 99505 9 99503 25 24 23 22 21 :! i # * 5 I 7( ^0 2-5 [.0 1:1 3r 3. 4^ ! 7 J-6 s.o ^ 7-2 ;6 40 4? 42 43 44 9.17807 9.17890 9-17973 9.18055 9.18137 8 3 8 3 82 82 8? 9.18306 9.I839I 9-I8475 9.18560 9.18644 85 84 85 84 g. 0.81 694 0.81 609 0.81 525 0.81 440 0.81 356 9.99501 9.99499 9 99497 9 99495 9 99494 20 18 \l .1 .2 3 / l 1 \ K 2*. J 3 1:2 t 9 /. 1 i K 2i w to $.2 il 45 46 49 9.18220 9.18302 9-18383 9-18465 9-18547 82 81 82 82 81 9.18728 9.I88I2 9.18896 9.18979 9.19063 84 84 8 3 84 Q, 0.81 272 0.81 188 0.81 104 0.81 021 0.80937 9.99492 9.99490 9.99488 9.99486 9.99484 15 14 13 12 II '4 3. 4 I 7 i- 2 [ -5 ).8 J.I > 4 (7 *>< 4 4< i 7' S.o [.O )-2 ^ }g SO 5' 52 53 54 9:18628 9.18709 9.18790 9.18871 9-18952 8x Si 81 81 81 9.I9I46 9.19229 9.I93I2 9-19395 9.19478 8 3 83 8 3 83 83 0- 0.80854 0.80771 0.80688 0.80605 0.80522 9.99482 9.99480 9.99478 9.99476 9 99474 10 I I 1 .1 8 .2 16 3 24 I .1 .2 3 / a S 16 24 /. .0 .O .0 0.2 04 0.6 A X 11 H r59 9 1933 9.19113 9.19 193 9 19273 9 19353 80 80 80 80 flrt 9.I956I 9.19643 9.19725 9 19807 9.19889 *3 82 82 82 82 0.80439 0.80357 0.80275 0.80 193 0.80 in 9.99472 9.99470 9.99468 9.99466 9.99464 5 4 3 2 I 4 3 2 5 4Q .6 48 7 56 .8 64 .0 72 4 i . y 3 2 ? 7? .0 .0 :.o .0 o 1.0 1.2 M 1.8 1 j60 9 19433 9.19971 0.80029 9.99462 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. f p ro p.] Pfe . 81 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 39 9 t L. Sin. d. L. Tangr. c.d. L. Cotg. L. Cos. I to P.: Pfe b I 2 3 4 9-19433 9-i95i3 9.19592 9 19672 9 I975I 80 79 80 79 70 9.19971 9.20053 9 20 134 9.20216 9.20297 8a 81 82 81 8x 0.80029 0.79 947 0.79 866 0.79784 0.79703 9.99462 9.99460 9-99458 9-99456 9-99454 00 59 58 11 8 .1 8 2 16 3 24 2 2 4 .6 ! 8 16 24 I .1 .2 3 80 8.0 16.0 24.0 i t 9 9.19830 9.19909 9.19988 9.20067 9.20145 79 79 79 78 .0 9.20378 9.20459 9.20540 9.20621 9.20 701 8x 81 81: 80 RT 0.79622 0.79541 0.79460 0.79379 0.79299 9-99452 9-99450 9-99448 9-99446 9-99444 55 54 53 52 5i 4 32 5 41 -6 49 ig .8 .0 .2 .4 .6 32 4 c 4* 4 . ^ o i 32.0 40.0 48.0 56.0 64.0 10 ii 12 '3 14 9.20223 9.20302 9.20380 9 . 20 458 9 20535 79 78 78 77 .0 9.20 782 9.20862 9.20942 9.21 022 9.21 102 80 80 80 80 0- 0.79218 0.79138 0.79058 0.78978 0.78898 9.99442 9.99440 9-99438 9 99436 9-99434 50 3 3 9 73 .1 .2 3 .8 , t I 2; yu 79 -1 J-7 y i i 2: 72.0 rt 3 J-4 II! !1 19 9.20613 9.20691 9 . 20 768 9.20845 9 . 20 922 78 77 77 77 9-21 l82 9.21 26l 9 21 341 9.21 420 9 21 499 79 80 79 79 0.78818 0.78739 0.78 659 0.78580 0.78501 9-99432 9.99429 9.99427 9 99425 9 99423 45 44 43 42 41 4 i 1 3 3 ( 4 I: [.to ?-5 7-4 5-3 5-2 3 3< 1 t.2 !:| J.6 20 21 22 23 24 9.20999 9 21 076 9 21 153 9 21 229 9 21 306 77 77 76 77 _g 9 21 578 9 21 657 9 21 736 9 21 814 9 21 893 79 79 79 78 79 _0 0.78422 0.78343 0.78 264 0.78 186 0.78 107 9.99421 9.99419 9;994i7 9 99415 9.994I3 40 I 9 .1 .2 3 7 - i 2. 77 r-7 5-4 J -J 7 ( | i i. 2: ? T-6 Ii 2 5 26 3 29 9 21 382 9 21 45 8 9 21 534 9 21 610 9.21 685 76 76 76 75 76 9 21 971 9.22049 9.22 127 9 22 2O5 9 22 283 78 78 78 78 _Q 0.78029 0.77951 0.77873 0-77795 o 77 717 9 99411 9.99409 9.99407 9.99404 9 99402 35 34 33 32 3i 4 i i I 4< i 6 3.8 !:i 59 [.6 5 4. i; fi< d ;.6 !:! ? A so 31 32 33 34 9 21 761 9 21 836 9 21 912 9.21 987 9 . 22 O62 7 75 76 75 75 9.22361 9 22438 9.22 516 9-22593 9.22 670 77 78 77 77 o 77 639 o 77562 0.77484 o 77407 0.77330 9 99400 9 99398 9.99396 9 99394 9.99392 ao 29 28 n .2 3 I. 2: 3 75 7-S >.o 2-5 I I I; 2: J -4 74 [1 1.2 P ii 39 9 22137 9 22 21 1 9.22286 9 22361 9 22435 75 74 75 75 74 9 22 747 9.22824 9.22901 9.22977 9 23054 77 77 77 76 77 o 77253 o 77176 0.77099 o 77023 o 76946 9 99390 9.99388 9 99385 9 99383 9 99381 25 24 23 22 21 4 ii :I 3< 3 4. g 6 J.O 7-S 5-0 2-5 ).0 2C 3' * 5 1! ) 6 7.0 f:| i 6 40 4i 42 1 43 i 44 9 22509 9 22583 9 22657 9 22731 9.22 805 74 74 74 74 74 9 23 130 9.23 206 9 23283 9 23359 9-23435 76 76 77 7 76 0.76870 0.76794 0.76717 o 76 641 0.76565 9 99379 9 99377 9 99375 9 99372 9 99370 20 19 18 17 16 V .1 .2 3 D 1. 2 5 73 I'l '9 i i4 2 .< ^a 7.2 50 11 ;i 49 9.22878 9.22952 9 23025 9.23098 9.23 171 73 74 73 73 73 9.23510 9 23586 9.23661 9 23737 9.23812 75 76 75 76 75 o 76 490 o 76414 o 76339 0.76263 o 76188 9 99368 9 99366 9 99 364 9 99 3 6 2 9 99 359 5 14 13 12 11 4 i II 2< 3< 4. 1 H j'8 [.i 5-4 7 2c 3< 4: 5< & .5 ).0 5-2 ;i , 8 50 5i 52 53 54 9.23244 9 23317 9.23390 9.23462 9 23535 73 73 73 72 73 9-23887 9-23962 9.24037 9.24 112 9.24 186 75 75 75 75 74 0.76 113 o . 76 038 0-75 963 0.75888 0.75814 9 99357 9 99355 9-99353 9 99351 9 99348 10 I ? i 7 2 14 3 21 . -0 ". i .1 2 -3 C c C 3 '3 6 9 O.2 4 0.6 r 8 it 12 59 9.23607 9.23679 9.23752 9.23823 923895 72 72 73 7i 7* 9.24261 9-24335 9.24410 9-24484 9.24558 75 74 75 74 74 0-75 739 0.75665 0.75590 o.755i6 0.75442 9 99346 9 99344 9 99342 9-9934C 9 99337 5 4 3 2 I .4 2% :i2 :I3 96? 4 I I n I I 2 2 1 I i 4 7 I.O 1.2 Ji 1.8 60 9.23967 72 9.24632 74 0.75368 9-99335 L. Cos. d. L. Cot?. c.d. L. Tang. L. Sin. f r ro P Pfc u 80 4 TABLE IV. 10 I / L. Sin. d. L. Tang. c.d. L. Cot?. L.Cos. Prop. Pts. | i 2 6 4 9-23967 9-24039 9.24 no 9.24 181 9 24253 72 7* 7* 72 7 7 7 70 7* 70 7 70 70 70 70 70 70 69 ' 7<> 69 69 69 69 69 69 69 63 69 68 68 68 68 68 68 68 67 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 65 66 66 65 65 66 65 65 "dT 9-24632 9.24706 9-24779 9-24853 9 . 24 926 74 73 74 73 74 73 73 73 73 73 72 73 72 73 7* 7 72 72 72 7 7* 7* 73 7* 7 7* 7 70 7* 7 1 70 7<> 7* 70 70 70 70 69 70 69 70 69 69 69 69 69 69 69 63 69 68 69 68 68 68 68 67 68 68 67 0.75368 0.75294 0.75 221 0-75 H7 0.75074 9-99335 9-99333 9-99331 9.99328 9.99326 >0 P 11 .1 .2 3 .4 :! 9 .1 .2 .3 A e '.I :l .9 .1 ./ .1 .( !s .9 .1 .2 3 4 :1 :i 74 il'i 22.2 29.6 37-o Si 11:1 7 7-2 14.4 21.6 28.8 36.0 43 2 50.4 III 70 7-0 14.0 21.0 28.0 35-0 42.0 490 56.0 63.0 68 6.8 13-6 20.4 27.2 34-0 40.8 47-6 54-4 61.2 66 6.6 13.2 19.8 26.4 33-c 39-6 46.2 52.8 59-4 3 -; 0.6 0-9 1.2 1.5 I.I 2.1 2-4 2.7 73 ,1:1 21.9 29.2 36.5 43-8 Si.i 58.4 65-7 71 7-i 14.2 21.3 28.4 35-5 42.6 49-7 56.6 63-9 69 6.9 13-8 ) 20.7 27.6 34-5 81 ts z, 13.4 20.1 26.8 33-5 40.2 46.9 53-6 60.3 6s 6-5 13.0 195 26.0 32-5 39 o 45-5 52.0 58-5 a O.2 0-4 0.6 c 8 I.O 1.2 11 1.8 I I & ii 12 13 14 9.24324 9-2439? 9.24466 9-24536 9.24607 9.25 ooo 9-25073 9.25 146 9-25219 9-25292 0.75 ooo 0.74927 0.74854 0.74781 0.74708 9 99324 9-99322 9-993I9 9-993I7 9-9931? 55 54 53 52 5i 9.24677 9.24748 9.24818 9.24888 9.24958 9-25365 9-25437 9.25 5io 9-25 582 9-25655 0.74635 0.74563 0.74490 0.74418 0.74345 9 993*3 ; 9-99 3'o 9-99308 9.99306 9-99304 50 49 48 47 46 :* \i 19 20" 21 22 23 24 9 . 25 028 9-25098 9.25 1 68 9 25237 9 25307 9.25 727 9-25 799 9.25871 9-25 943 9.26 015 0.74273 0.74201 0.74129 0.74057 0.73985 9.99301 9.99299 9-99297 9.99294 9-99292 45 44 43 42 41 16" P 35 34 33 32 3i 9-25 376 9-25445 9.25514 9-25583 9 25652 9.26086 9.26 158 9.26229 9.26301 9.26372 0.73914 0.73842 0.73771 o 73699 0.73628 9-99290 9.99 288 9.99285 9.99283 9.99281 3 3 29 9.25 721 9-25 790 9-25858 9-25927 9-25995 9.26443 9.26514 9-26585 9.26655 9.26 726 0.73557 0.73486 0.73415 0.73345 o 73 274 9.99278 9 99276 9-99274 9-99271 9.99269 80 3i 32 33 34 9.26063 9.26 131 9.26199 9.26 267 9 26335 9-26 797 9.26867 9.26937 9.27008 9.27078 0.73203 0.73133 0.73063 0.72992 0.72 922 9.99267 9-99264 9.99 262 9.99260 9 99257 BO 3 'd 3 ? * 41 42 43 44 9 . 26 403 9.26470 9 26538 9.26605 9.26672 9.27 148 9.27218 9.27288 9-27357 9.27427 0.72 852 0.72 782 0.72 712 0.72643 0.72573 9.9925? 9.99252 9.99250 9.99248 9 99245 25 24 23 22 21 9 . 26 806 9.26873 9 . 26 940 9.27007 9-27496 9-27566 9-27635 9.27704 9-27 773 0.72504 0.72434 0.72365 0.72 296 0.72 227 9 99243 9.99241 9.99238 9.99236 9 99233 20 !! \l .1 .2 ,J -A c .6 :l .9 .1 4 i .3 :I .0 A l 46 47 48 49 9.27073 9.27 140 9.27 206 9.27273 9 27339 9.27842 9.27911 9.27980 9.28049 9.28 117 0.72 158 0.72 089 0.72 020 0.71 951 0.71 883 9.99231 9-99229 9.99226 9 99 224 9 99 221 15 14 13 12 II 50 5i 52 53 J.L * 9 59 9.27405 9.27471 9-27537 9.27 602 9.27668 9.28 186 9-28254 9.28323 9-28391 9.28459 0.71 814 0.71 746 0.71677 0.71 609 0.7I54I 9.99219 9.992I7 9-99214 9.99212 9-99209 10 I I 9-27734 9-27799 9.27864 9.27930 9-27995 9 -*So6o 9-28527 9.2859! 9.28662 9.28730 9.28 798 0.7M73 0.71405 0.71338 0.71 270 0.71 202 9.99207 9.99204 9.99202 9.99200 9-99 197 5 4 3 2 I 60 9.28865 0-71 135 9-99 195 L* Cos. L. Cotg. c. d. L. Tang. L. Sin. t Prop. Pts. 79 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 41 i 11 f i iu MIL d. L. Tang. c.d. L. Cotg. L. Cos. Prop. Pts. 1 9.28060 i 1 9.28 125 2 1 9.28 190 3 9-28254 4 9-28319 65 6 4 65 65 64 6 S 64 64 64 63 64 6 3 63 64 63 63 63 63 63 63 62 63 62 62 63 6a 62 61 6a 62 61 62 61 6a 61 61 61 61 61 61 60 61 60 61 60 61 60 60 60 60 59 60 60 59 60 9.28865 9-28933 9.29000 9.29067 9.29 134 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 66 66 66 65 65 65 65 65 65 64 65 64 65 64 65 64 64 64 64 63 64 63 64 6 3 64 63 63 63 63 63 63 63 62 62 63 62 62 62 0.71 i35 0.71 067 0.71 ooo 0.70933 0.70 866 9-99 195 9-99 192 9-99 190 9-99 187 9-99 185 GO 59 58 g .2 3 -4 :I i 9 .1 .2 3 4 9 .1 .2 .3 4 . c 6 k .9 .1 A \l .9 .1 ,4 1 1 .6 ;j .i .1 .6 t :l .9 68 6.8 20.4 27.2 34-0 40.8 47-6 54-4 61.2 66 6.6 13.2 10.8 26.4 33-o 39-6 46.2 52.8 59-4 6.4 12.8 19.2 32.0 38.4 44-8 57^6 6a 6.2 12 ./. 18.6 24.8 31.0 37-2 43-4 49-6 55-8 60 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54-0 3 0.9 1.2 1:1 2.1 2.4 2.7 6? 6.7 I3.4 20.1 26.8 33-5 40.2 46.9 53-6 60.3 65 6-5 13-0 195 26.0 32.5 39-o 45-5 52.0 58.5 63 6-3 12.6 18.9 25.2 44.1 50.4 56.7 6s 6.1 12.2 18-3 24.4 36i 42.7 48.8 54-9 59 5-9 11. 8 17.7 23-6 29-5 35-4 41-3 47.2 53-1 a 0.2 o!s I.O 1.2 i.B 7 9 "tin 12 13 14 9-28384 9.28448 9.28512 9.28577 9 . 28 641 9 . 28 705 9.28769 9-28833 9.28896 9.28960 9.29 201 9.29268 9-29335 9.29402 9 . 29 468 0.70799 0.70 732 0.70665 0.70598 0.70532 9.99 182 9.99 180 9-99 177 9-99 175 9-99 172 55 54 53 52 _51_ 3 47 46 9-29535 9.29 601 9.29668 9-29 734 9.29 800 0.70465 0.70399 0.70332 0.70 266 0.70 200 9-99 170 9.99 167 9-99 165 9.99 162 9.99 160 15 1 9.29024 16 I 9.29087 17 9.29 150 18 1 9.29214 19 1 9.29277 9.29866 9.29932 9-29998 9.30064 9.30 130 0.70 134 0.70068 0.70002 0.69936 0.69 870 9-99 157 9-99 155 9-99 152 9.99150 9 99 H7 45 44 43 42 41 1T P i 20 21 22 2 3 24 9.29340 9.29403 9.29466 9 29 529 9.29591 9-30I95 9.30261 9.30326 9-30391 9-30457 0.69 805 0-69 739 0.69674 0.69609 0.69 543 9 99H5 9-99 '42 9-99 HO 9-99 137 9-99 135 2 7 29 30 32 P 9-29654 9.29716 9.29 779 9.29841? 9-29903 9 . 29 966 9 . 30 028 9.30090 9 30151 9 30213 9.30275 9-30336 9-30398 9.30521 9-30522 9-30587 9.30652 9.30717 9.30782 0.69478 0.69413 0.69348 0.69 283 0.69 218 9-99 132 9 99 13 9 99 127 9-99 124 9.99122 35 34 33 32 9.30846 9.30911 9-30975 9.31 040 9.31 104 0.69 154 0.69 089 o . 69 025 0.68960 0.68896 9.99119 9.99117 9 99 "4 9.99 112 9 99 109 30 11 11 f s 44 9.31 168 9 -31 233 9.31 297 9.31 361 0.68832 0.68 767 0.68 703 0.68639 0.68575 9.99 106 9-99 104 9.99 101 9.99099 9.99096 25 24 23 22 21 9.30582 9-30643 9.30 704 9-30765 9 . 30 826 9.31489 9-31 55| 9.31 616 9.31 679 9-3i 743 0.68511 0.68448 0.68384 0.68321 0.68257 9.99093 9.99091 9.99088 9.99086 9.99083 20 5 |45 46 47 | 4 8 i 49 9-30887 9-30947 9.31 008 9.31 068 9.31 129 9.31806 9.31 870 9-31 933 9.31 996 9-32059 0.68 194 0.6*8 130 0.68067 0.68004 0.67941 9 . 99 080 9.99078 9-99075 9-99072 9.99070 i5 13 12 II 150 52 53 54 9 II ! 59_ 9.31 189 9 31 250 9.31310 9-31 370 9-3 1 430 9.32 122 9-32 185 9-32248 9-32311 9-32373 0.67878 0.67815 0.67 752 0.67 689 0.67 627 9.99067 9.99064 9 . 99 062 9.99059 9-99056 10 I I 9.31 490 9-31 549 9.31 609 9.31 669 9.31 728 9-32436 9.32498 9.32561 9.32623 9.32685 0.67 564 0.67 502 0.67439 0.67377 0.67315 9-99054 9.99051 9.99048 9-99046 9.99043 5 4 3 2 I 9.31 788 9-3 2 747 0.67253 9.99040 L. Cos. d. L. Cotg. c. d. L. Tans. L. Sin. t Prop. Pts. 78 TABLE IV. 12 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. p rop. ] Pts. I 2 3 4 9.31 788 9-3 1 847 9.31 907 9.31 966 9.32025 59 60 59 59 rg 9-32747 9.32810 9.32872 9.32933 9-32995 63 62 61 62 62 0.67253 0.67 190 0.67 128 0.67067 0.67 005 9.99040 9.99038 9-99035 9.99032 9.99030 (JO 3 11 .1 .2 3 63 6.3 12.6 18.9 62 6.2 12 4 18.6 I I 9 9.32084 9-32 H3 9.32202 9.32261 9-32 3*9 59 59 59 58 9-33057 9-33 "9 9-33 * 80 9-33242 9.33303 62 61 62 61 62 o . 66 943 0.66881 0.66 820 0.66758 0.66697 9.99027 9.99024 9 . 99 022 9.99019 9.99 016 55 54 53 S 2 5i 4 :f I 25.2 3J:i 44.1 50.4 24.8 31.0 37-2 43-4 49-6 10 ii 12 13 14 9-32378 9.32437 9-32495 9-32553 9.32 612 59 58 58 59 eg 9.33365 9.33426 9.33487 9-33548 9-33609 61 61 61 61 61 0.66635 0.66574 0.66 513 0.66452 0.66391 9.99013 9.99011 9.99008 9.99005 9.99002 50 3 s 9 .2 3 i> 6 -'; 61 6.1 12.2 18-3 55- 8 60 6.0 12.0 18.0 II ii 19 9-32670 9.32 728 9-32 786 9-32844 9.32902 58 58 58 5 t-8 9.33670 9-33 73 1 9-33 792 9.33853 9.33913 61 61 61 60 61 0.66 330 0.66 269 0.66 208 0.66 147 0.66087 9.99 ooo 9.98997 9.98994 9.98991 9 . 98 989 45 44 43 42 41 4 ii 24.4 30-5 3 6.6 42-7 48.8 24.0 30.0 36.0 42.0 48.0 20 21 22 \ 23 24 9.32960 9.33018 9-33075 9-33 133 9-33 190 58 57 58 57 eg 9-33974 9-34034 9-34095 9.34155 9.34215 60 61 60 60 6z 0.66 026 0.65 966 0.65 905 0.65 845 0.65 785 9.98 986 9.98983 9.98980 9.98978 9-98975 40 3 11 9 54-9 5 .i 5 .2 II 3 IJ 54- 9 'I 2 5 26 3 29 9-33248 9-33305 9-33362 9-33420 9-33477 57 57 58 57 9.34276 9.34336 9.34396 9.34456 9-345I6 60 60 60 60 60 0.65 724 0.65 664 0.65 604 0.65 544 0.65 484 9-98972 9.98969 9.98967 9-98964 9.98961 35 34 33 32 31 4 *J .5 29 6 35 .7 4i .8 45 Q H^ .6 5 4 -3 .2 IT 30 3i 32 33 34 9-33 534 9-33591 9-33647 9-33 704 9-3376I 57 56 57 57 9.34576 9.34635 9.34695 9-34755 9.34814 59 60 60 59 60 0.65 424 0.65 365 0.65 305 0.65 245 0.65 186 9.98958 9.98955 9.98953 9.98950 9.98947 30 3 Z .1 .2 3 y jj 58 5-8 n. 6 17-4 57 5-7 11.4 17.1 it O 35 36 % 39 9.338i8 9-33 874 9-33931 9-33987 9-34043 57 56 57 56 56 9-34874 9-34933 9-34992 9-3505I 9-35IH 59 59 59 60 0.65 126 0.65 067 0.65 008 0.64 949 0.64889 9.98944 9.98941 9.98938 9.98936 9-98933 25 24 23 22 21 ii ii .0 23.2 29.0 34-8 40.6 46.4 tJ2.2 28.5 34-2 39-9 45-6 51.3 40 4i 42 43 44 9-34 loo 9.34I5 6 9.34212 9.34268 9-34324 56 5 56 56 efi 9-35 170 9-35 229 9-35 288 9-35347 9-35 405 59 59 59 58 o . 64 830 0.64 771 0.64 712 0.64653 0.64595 9.98930 9.98927 9.98924 9.98 921 9.98919 20 ii 11 .1 .2 3 56 5-6 II. 2 16.8 122 4 55 5-5 II. 16.5 22 O $ 9 49 9-3438o 9-34436 9-34491 9-34547 9.34602 5 56 55 56 55 9.35464 9.35 5J3 9-35 58i 9-35 640 9-35698 59 58 59 58 0.64536 0.64477 0.64419 0.64 360 0.64 302 9.98 916 9.98913 9.98910 9.98907 9.98904 15 14 13 12 II ii .9 28.0 33-6 39-2 44-8 50.4 27-5 33-0 38-5 44-0 49-5 50 5i 52 53 54 9-34658 9-347I3 9-34769 9-34824 9-34879 5 55 56 55 55 9-35 757 9-35815 9-35 873 9-35931 9-35 989 58 58 58 58 eg 0.64243 0.64 185 0.64 127 0.64069 0.64 on 9.98 901 9.98898 9.98896 9.98893 9.98890 10 I I .1 .2 4 3 -3 0.6 0.9 1.2 a 10.2 0.4 0.6 0.8 9 9 1 59 9-34934 9.34989 9-35044 9.35099 9-35 154 55 55 55 55 55 9.36047 9.36 105 9-36 163 9.36221 9.36279 58 58 58 58 0.63953 0.63 895 0.63 837 0.63 779 0.63 721 9.98887 9.98884 9.98881 9.98 878 9.98875 5 4 3 2 I I I i 4 ii 2.1 2.4 2.7 I.O 1.2 H 1.8 60 9-35209 55 9.36336 0.63 664 9.98 872 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. t I >rop. Pts. 77 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 43 1 3 I I , L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. Prop. Pts. 9-35209 9-36336 eg 0.63 664 9.98872 60 i 9-35 263 9.36394 eg o 63 606 9.98869 59 58 57 2 9-35 3 l8 55 9-36452 0.63 548 9.98867 S8 .1 S-8 5-7 3 9-35373 55 9-36509 0.63491 9.98864 57 .2 11.6 II.4 4 9-35427 54 9.36566 58 0.63434 9.98861 56 3 17.4 I7.I I 9-3548i 9-35 S3 6 55 9.36624 9.36681 57 0.63 376 0.63319 9.98858 9-98855 55 S4 4 23.2 29.0 22.8 28.5 7 9-35590 54 9-36738 57 0.63 262 9.98852 53 .6 34-8 34-2 8 9-35 644 54 9.36 795 57 0.63 205 9.98849 S2 7 40.6 39-9 9 9-35698 54 54 9.36852 57 57 0.63 148 9.98846 51 .8 46.4 45-6 10 9-35 752 9-36909 0.63091 9-98843 50 9 52.2 5^3 ii 9.358o6 54 9.36966 57 0.63 034 9.98 840 49 56 55 12 13 H 9-35 860 9-359H 54 54 9.37023 9.37080 9-37 137 57 57 eg 0.62977 0.62 920 0.62863 9-98837 9.98834 9-98831 48 47 46 .2 -3 5-6 II. 2 16.8 5.5 II. 16.5 11 17 9.36 022 9-36075 9 3 6 129 53 54 9-37 193 9-37250 9.37306 57 56 0.62 807 0.62 750 0.62694 9.98828 9.98825 9.98822 45 44 43 4 22.4 28.0 33-6 22.0 27-5 33-o 18 9.36 182 53 9-373 6 3 57 0.62637 9.98 ^19 42 I 39-2 38-5 19 9-36236 54 53 9.37419 5 0.62 581 9.98816 .8 44.8 44-0 20 9.36289 9-37476 0.62524 9-988i 3 40 9 5'4 49-5 21 9-36342 53 9-37532 e 0.62468 9.98 810 39 54 22 23 24 9-36395 9.36449 9-36502 53 54 53 CO 9.37588 9-37644 9-37 700 5 56 56 0.62412 0.62 356 0.62 300 9.93 807 9.98804 9.98801 38 11 .1 5-4 .2 10.8 3 16.2 % 9.36555 9.36608 53 9.37756 9.37812 56 0.62244 0.62 188 9.98 798 9 98795 35 34 .4 21.6 .5 27.0 27 9.36660 5~ 9.37868 50 0.62 132 9.98 792 33 .6 32.4 29 9.36713 9.36766 53 53 51 9-37924 9.37980 5 56 0.62076 o . 62 020 9.98 789 9.98786 32 'I 37-8 .8 43-2 9..Q 6 30 9.36 819 9-38035 0.61 965 9-98783 80 40.0 9.36871 52 9.38091 56 0.61 909 9.98780 29 53 S* 32 33 34 9.36924 9.36976 9.37028 53 52 52 9.38 147 9.38202 9-38257 50 55 55 0.61 853 0.61 798 0.61 743 9.98 777 9.98 774 9.98771 28 11 .1 .2 3 5-3 10.6 15-9 5-2 10.4 15.6 1 39 9-37o8i 9-37I33 9.371S5 9 37237 9-37289 52 52 52 52 9-383I3 9.38368 9-38423 9-38479 9.38534 55 55 56 55 0.61 687 0.61 632 0.61 577 o.Ci 521 0.61 466 9.98768 9.98 765 9.98 762 9-98 759 9.98 756 25 24 23 22 21 4 7 21 .2 26.5 3? 8 37-i 42.4 20. 26.0 3 1.7 36.4 41.6 46 8 40 9-37341 9-37393 52 9-38589 9.38644 55 55 0.61 411 0.61 356 9.98 753 9.98 750 20 19 . 51 4 42 9-37445 5 2 9.38699 55 o.Ci 301 9.98 746 18 .1 b- 1 0.4 43 44 9-37497 9-37549 S 2 52 9! 38 808 55 54 0.61 246 0.61 192 9-98 743 9.98740 \l .2 3 10.2 15-3 1.2 45 46 9.37600 52 9.38863 9.38918 55 55 0.61 137 0.61 082 9-98 737 9-98 734 15 14 4 25-5 2.0 11 9-37703 9-37755 Si 52 9-38972 9.39027 54 55 0.61 028 0.60973 9.98 73i 9.98 728 13 12 :J Hi 49 9.37806 5 1 9.39082 55 0.60918 9-98 725 II O AC Q \ 6 50 9.37858 9-39 136 54 0.60864 9.98 722 10 51 9-37909 5 X 9.39 190 54 0.60810 9.98 719 9 3 52 53 54 9.37960 9.38011 9.38062 5* 5 9.39245 9.39299 9-39353 55 54 54 0.60 755 0.60 701 0.60647 9.98 715 9.98 712 9.98 709 I .1 .2 -3 0.3 0.6 0.9 I* O.2 O.O o 8 P 9.38 "3 9.38 164 S 9.39407 9.39461 54 54 0.60593 0.60539 9.98 706 9.98 703 5 4 1:1 1.0 1.2 9-38215 9.38266 5 X 5 9.39515 54 54 0.6048? 0.60431 9.98700 9.98697 3 i 2.1 2 A 59 9-38317 5 9-39623 54 0.60377 9.98694 i .9 if .4 2.7 1.8 60 9-38368 9-39677 54 0.60323 9 . 98 690 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. t Prop. Pts. 76 44 TABLE IV. 14 t L. Sin. d. L. Tang. .d. L. Coter. L. Cos. d. T] rop. ] >ts. 1 I 2 3 4 9-38368 9.38418 9.38469 9-38519 9-38570 so 51 50 51 5 9.39677 9-39 73i 9.39785 9-39838 9-39892 54 54 53 54 53 0.60323 o . 60 269 0.60 215 0.60 162 0.60 108 9.98690 9.98687 9.98684 9.98681 9.98678 3 3 3 3 00 59 58 y .1 2 54 5-4 10 8 53 IO O 1 I 9 9.38620 9.38670 9-38721 9.38 771 9.38821 So Si So 5 SO 9-39945 9-39999 9.40052 9.40 106 9-40 159 54 53 54 53 53 0.60055 0.60001 0.59948 0.59894 0.59841 9-98675 9.98671 9.98668 9.98665 9.98662 4 3 3 3 55 54 53 52 5i 3 -4 .7 16.2 21.6 27.0 SI 15-9 21.2 26.q 318 37-1 10 ii 12 13 H 9.38871 9.38921 9.38971 9.39021 9.39071 So So So So 9.40212 9.40266 9.40319 9.40372 9.40425 54 53 53 53 0.59788 0-59734 0.59681 0.59 628 0-59575 9.98659 9.98 656 9-98652 9.98649 9 . 98 646 3 4 3 .3 60 3 % .8 9 &6 5* 42-4 47-7 5 \l 11 19 9-39 121 9 39 170 9.39220 9.39270 9.393I9 49 50 50 49 9.40478 9-40531 9.40584 9.40 636 9.40689 53 53 52 53 53 0.59522 0.59469 0.59416 0.59364 0-593" 9-98643 9.98640 9.98636 9-98633 9.98630 3 4 3 3 45 44 43 42 41 .1 .2 3 4 5-2 10.4 15.6 20.8 26.0 5-x IO.2 15.3 20.4 25-5 20 21 22 23 24 9-39369 9.39418 9-39467 9.395I7 9.39566 49 49 So 49 9.40 742 9-40 795 9.40847 9.40900 9.40952 53 53 53 53 0.59258 0.59205 0.59153 0.59 100 0.59048 9.98627 9.98 623 9 98 620 9.98617 9.98 614 4 3 3 3 40 39 38 11 i 9 31.2 36.4 41.6 46.8 30.0 35-7 40.6 45-9 > 8 3 29 9-396i5 9.39664 9-397I3 9.39762 9.39811 49 49 49 49 9-41 005 9-4i 057 9.41 109 9.41 161 9.41 214 53 53 53 53 52 0.58995 0.58943 0.58891 0.58839 0.58786 9.98 610 9.98607 9.98604 9.98601 9-98597 3 3 3 4 35 34 33 32 3i .1 2 3 A 50 5-0 10. 15.0 20 o 49 4-9 9.1 14.7 IQ.6 80 3i 32 33 34 9.39860 9.39909 9-3995? 9.40006 9-40055 49 49 48 49 48 9.41 266 9.41 318 9.41 370 9.41 422 9.41 474 53 53 53 53 5 a 0.58 734 0.58682 0.58630 0.58578 0.58 526 9.98594 9.98591 9-98588 9.98584 9.98581 3 3 4 3 80 3 2 :! .9 25.0 30.0 35.0 40.0 45.0 24.5 29.4 34-3 39.2 44.1 P H 39 9-40 103 9-40 152 9.40200 9.40249 9.40297 49 48 49 48 9-41 526 9.41 578 9.41 629 9.41 68 i 9-41 733 5 5 5 53 0.58474 0.58422 0.58371 0.58319 0.58267 9.98578 9.98574 9-9857I 9-98568 9-98565 4 3 3 3 25 24 23 22 21 .1 .2 48 4-8 9.6 47 4-7 9.4 40 4i 42 43 44 9.40346 9 40394 9-40442 9.40490 9-40538 48 48 48 48 A a 9.41 784 9.41 836 9.41 887 9.4I939 9.41 990 5* Si 53 S 0.58 216 0.58 164 0.58 113 0.58061 0.58010 9.98 561 9.98558 9.98555 9.98551 9-98548 3 3 4 3 20 18 II 3 4 14.4 19.2 24.0 28.8 3,V6 ls',8 S:ll 32.9 1 3 s 49 9.40586 9.40634 9.40682 9.4073 9.40 778 48 48 48 48 9.42041 9.42093 9.42 144 9.42 195 9.42 246 53 51 Si 5* 0-57959 0.57907 0.57856 0.57805 0-57754 9-98545 9.98541 9.98538 9.98535 9-98531 4 3 3 4 15 14 13 12 II 9 38.4 43 2 4 37.6 : sr ! 5I 1 C2 1 53 54 9.40825 9.40873 9.40921 9 40 968 9.41 016 47 48 48 47 48 9.42297 9-42348 9.42399 9-42450 9.42501 5i 5 Si 5 0-57 703 0.57652 0.57601 0.57550 0.57499 9.98528 998525 9 98521 9-98518 9 985*5 3 3 4 3 3 10 1 .1 .2 3 4 0.4 o.S 1.2 1.6 2.C 0.3 0.6 0.9 1.2 ;-j 5 5^ ? 59 9.41063 9.41 in 9.41 158 9.41 205 9 41 252 48 47 47 47 .0 9.42552 9-42603 9-42653 9-42 704 9-42 755 5* 50 S Si 0.57448 0-57397 0-57347 0.57296 0.57245 9.98511 9-98508 9-98505 9.98501 9.98498 4 3 3 4 3 5 4 3 2 I :l 9 2.4 2.8 3:! 1.8 2.1 2-4 2-7 it jo 9.41 300 9.42805 0-57 195 9.98494 L. Cos. d. L. Cotg. c.d L. Tang L. Sin. d. t 1 *rop. Pts. 75 LOGARITHMS OF SINE, CO.Bl^, TANGENT AND COTANGENT, ETC. 45 15 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. 9.41 300 9.42805 O-57 J 95 9.98494 00 I 9-4i 347 9-42856 0.57 144 9.98491 S9 2 9-41 394 9.42906 0.57094 9.98488 3 S8 SI 5 3 4 9.41441 9.41 488 47 47 9-42957 9.43007 50 50 0.57043 0-56993 9.98484 9.98481 3 4 H .1 .2 IO.2 5-o IO.O i I 9 9-4i 535 9.41 582 9.41 628 9-4i 675 9.41 722 47 46 47 47 46 9-43057 9-43 108 9-43 158 9.43208 9-43258 50 50 50 5 0-56943 0.56 892 0.56 842 0.56 792 0.56 742 9.98477 9.98474 9.98471 9.98467 9.98464 3 3 4 3 55 54 53 52 .3 4 15-3 20.4 25-5 30.6 3S-7 15-0 20. o 25-0 ; 30.0 35-o 10 9.41 768 9-43308 0.56692 9 . 98 460 50 .8 40.8 40.0 12 9.41815 9.41 861 47 46 9-43358 9.43408 5 50 0.56642 0.56592 9-98457 9-98453 3 4 49 48 9 45-9 45-0 13 14 9.41 908 9-41 954 47 46 9-43458 9-43508 5 50 5 0.56542 0.56492 9.98450 9.98447 3- 3 47 46 49 48 ii 9.42001 9.42047 9.42093 9.42 140 46 46 47 9.43558 9.43607 9.43657 9-43 707 49 50 50 0.56442 0.56393 0.56343 0.56293 9-98443 9-98440 9.98436 9-98433 3 4 3 45 44 43 42 .1 .2 3 4 49 9-8 14.7 19.6 4-8 9-6 14.4 19.2 19 9.42 186 40 46 9-43756 49 5 0.56244 9.98429 4 41 5 24-5 21 22 23 9.42232 9.42278 9.42324 9.42370 46 46 46 .f. 9-43855 9-43954 49 50 49 0.56 194 0.56 145 0.56095 0.56046 9.98 426 9.98422 9.98419 9.98415 4 3 4 40 39 38 37 9 29.4 34.3 39-2 44.1 pU 43-2 1 24 9.42416 AC 9.44004 40 0.55996 9.98412 3 36 f ? 29 9.42461 9.42507 9-42553 9.42599 9.42644 4 6 4 6 46 45 46 9-44053 9-44 102 9.44I5I 9.44201 9.44250 49 49 So 49 0-55947 0.55898 0.55849 0-55 799 0.55750 9.98409 9.98405 9.98402 9.98398 9-98395 4 3 4 3 35 34 33 32 31 .1 .2 3 A 47 4-7 9-4 14.1 18 8 4.6 9-2 n.8 18.4 30 9.42690 9.42735 45 .e. 9.44299 49 0.55 701 0-55652 9.98391 9.98388 3 29 23.5 28.2 23.0 27.6 32 33 34 9.42 781 9.42 826 9.42872 40 45 46 45 9-44397 9.44446 9.44495 49 49 49 0.55603 0-55554 0.55505 9-98384 9.98381 9.98377 4 3 4 28 11 .9 32.9 37-6 32.2 36.8 41.4 P 9.42917 9.42962 45 A f. 9-44544 9-44592 48 0.55456 0.55408 9-98373 9.98370 3 25 24 ii 39 9.43008 9.43053 9.43098 4 45 45 9.44641 9-44690 9.44738 49 49 48 0-55359 0.55310 0.55 262 9-98366 9-98363 9.98359 4 3 4 23 22 21 2 45 4-5 9O 44 it 40 9-43 143 9.44787 49 0-55213 9.98356 3 20 .3 41 9-43 l8 8 45 9.44836 49 0-55 164 9.98 352 4 19 4 18.0 17 6 42 43 44 9.43233 9.43278 9.43323 45 45 45 9-44884 9-44933 9.44981 48 49 48 A 9 0.55 116 0.55067 0.55019 9-98349 9-98345 9.98342 3 4 3 \l 22.5 27.0 v-s 22. 26.4 30.8 ? 9.43367 9.43412 9-43457 45 45 9-45029 9-45078 9-45 126 49 48 0-54971 0.54922 0.54874 9-98338 9-98334 9-9833I 4 4 3 15 H 13 9 36.0 40.5 35-2 39-6 48 9-43502 45 9-45 174 4 8 0.54826 9-98327 4 12 49 9-43546 44 9-45222 48 0.54778 9.98324 3 II 4 3 50 9-43591 9-45 271 49 0.54729 9.98320 4 10 .1 04 03 5i 52 9-43635 9.43680 44 45 9-45 319 9-45 367 4 8 48 0.54681 0.54633 9.98317 9-983I3 3 4 1 .2 3 0.8 1.2 0.6 0.9 53 9-43 724 44 9.45 4i5 48 0-54585 9.98309 4 7 4 1.6 1.2 9-43 769 45 9-45463 48 .0 0-54537 9.98306 3 6 2.O 1-j 55 9-43813 9-455" 4 0.54489 9.98302 4 5 .b 2.4 1.8 5<> 9-43857 44 9-45559 48 0.54441 9.98299 3 4 7 2.8 2.1 57 9.43901 44 9-45 606 47 0-54394 9.98295 4 3 .8 3.2 2.4 58 9-43 946 45 9-45 654 48 0.54346 9.98291 4 2 9 3-6 2.7 59 9-43990 44 9.4^702 48 0.54298 9.98288 3 I GO 9-44034 9-45 750 4 o 54 250 9.98284 4 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. , Prop. P^. 74 TABLE IV. 16 / L. Sin. d. L. Tan?. c.d. L. Cot?. L, Cos. d. p rop. 1 Pis. I 2 3 4 9.44034 9.44078 9.44 122 9.44 166 9.44210 44 44 44 44 43 9-45 750 9-45 797 9.4584? 9-45892 9-45 940 47 48 47 48 47 0.54250 0.54203 0.54155 0.54 1 08 0.54060 9 . 98 284 9.98281 9.98277 9.98273 9.98270 3 4 4 3 00 9 9 .1 48 4-8 9f. 47 4-7 I 9 9-44253 9.44297 9-44341 9.44385 9.44428 44 44 44 43 44 9-45 987 9-46035 9.46082 9.46 130 9.46 177 48 47 48 47 0.54013 0.53965 o.539i8 0.53870 0.53823 9.98266 9.98 262 9.98259 9.98255 9.98251 4 3 4 4 55 54 53 52 |l 3 4 7 14.4 19.2 24.0 28.8 33 6 9-4 14.1 18.8 III 72 Q 110 ii 12 13 14 9-44472 9 44 5 l6 9-44559 9.44602 9.44646 44 43 43 44 43 9.46224 9.46271 9-463I9 9-46366 9.46413 47 48 47 47 47 0.53776 0.53729 0.53681 0.53634 0.53587 9.98248 9.98244 9.98240 9.98237 9-98233 3 4 4 3 4 60" % % 8 9 P-4 43- 2 46 37-6 42.3 45 15 16 \l 19 9.44689 9-44 733 9-44 776 9-448I9 9.44862 44 43 43 43 9.46460 9.46 507 9.46554 9.46601 9.46648 47 47 47 47 40 4 40 4> 40 4' 40 40 41 40 40 4<> 40 4 4<> 40 4<> 40 4<> 40 39 40 40 39 4<> 40 39 40 39 40 39 39 39 40 39 39 39 39 39 39 39 39 39 9-48534 9.48579 9.48 624 9.48669 9.48 714 45 45 45 45 45 45 45 45 45 45 45 44 45 45 44 45 44 45 44 45 44 45 44 44 45 44 44 44 44 44 44 44 44 44 44 44 44 43 44 44 44 43 44 43 44 43 44 43 44 43 43 44 43 43 43 43 43 44 43 43 0.51 466 0.51 421 0.51 376 0.51331 0.51 286 9 . 98 060 9.98056 9.98052 9 . 98 048 9.98044 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 4 4 4 4 4 4 4 4 4 4 00 f* y .1 .2 3 4 :i .7 .8 9 .1 .2 3 4 j 9 I .2 3 4 ;2 9 .1 .2 3 4 .b :l 9 .2 3 4 j -9 45 4-5 9.0 !3i 22.5 27.0 3J-S 36.0 40-5 43 tt 12.9 17.2 21-5 258 30.1 34-< 38-7 V 41 8^2 12.3 16.4 20.5 24.6 28.7 32.8 36-9 39 ?:l 11.7 15.6 iQ-5 23-4 27-3 31.2 35-i 4 0.4 0.8 -1.2 1.6 2.0 2-4 2.8 1-6 44 si o.o '3-2 , 17 6 22 26.4 30.8 35-2 39-6 42 4-2 8.4 12.6 16.8 21. 25-2 29.4 336 37-8 t 40 4.0 8.0 12.0 16.0 20. o 24.0 28.0 32.0 36.0 5 0.5 1.0 15 2.0 2-5 30 35 4-0 4-5 3 -3 | 0.6 0.9 1.2 !:i 2.1 2.4 2.7 I I 9 10 ii 12 13 14 9 . 46 800 9.46841 9.46882 9.46923 9.46964 9-48 759 9.48804 9.48849 9-48894 9-48939 0.51 241 0.51 196 0.51 151 0.51 106 0.51 061 9 . 98 040 9 . 98 036 9.98032 9.98029 9.98025 55 54 53 52 5i 9.47005 9-47045 9.47086 9-47 127 9.47168 9.48984 9.49029 9-49073 9.49 118 9-49 103 0.51 016 0.50971 0.50927 O.5O 8\>2 0.50837 9.98021 9.98017 9.98 013 9 98 oc i 9 . 98 005 50 3 8 15 16 17 18 19 9.47209 9.47249 9.47290 9-47330 9-47371 9-49 207 9-49252 9.49296 9-49341 9.49385 0.50793 0.50 748 o . 50 704 0.50659 0.50 615 9.98001 9-97997 9-97993 9.97989 9.97986 45 44 43 42 41 20 21 22 23 24 9.47411 9.47452 9-47492 9-47533 9-47573 9-49430 9-49474 9.49519 9-49 563 9.49607 0.50570 0.50526 0.50481 0.50437 0.50393 9.97982 9.97978 9-97974 9.97970 9.97966 40 3 11 II % 29 9-47613 9.47654 9.47694 9 47 734 9-47 774 9.49652 9.49696 9-49 740 9.49784 9.49828 0.50 348 0.50304 0.50 260 0.50216 o . 50 1 72 9.97962 9-97958 9-97954 9-97950 9.97946 35 34 33 32 31 mT ?8 11 30 3i 32 33 34 9.47814 9-47854 9.47894 9-47934 9 47974 9.49872 9.49916 9.49960 9 . 50 004 9 . 50 048 0.50 128 o . 50 084 o . 50 040 0.49996 0.49952 9.97942 9-97938 9-97934 9.97930 9.97926 9 % 39 9.48014 9-48054 9.48094 9 48 133 9-48 173 9.50092 9.50 136 9.50 180 9.50223 9.50267 0.49 908 0.49 864 0.49820 0.49777 0-49 733 9.97922 9.97918 9.97914 9.97910 9.97906 25 24 23 22 21 40 4i 42 43 44 9.48213 9-48252 9.48292 9-48332 9-4837I 9-503" 9.50355 9-50398 9.50442 9-50485 0.49 689 0.49645 0.49 602 0.49558 OA951S 9.97902 9-97898 9.97894 9.97890 9.97886 20 : \i 45 46 47 48 49 9.48411 9.48450 9.48490 9-48529 9.48568 9-50529 9.50572 9.50616 9-50659 9-50703 0.49471 0.49428 0.49 384 0.49341 0.49 297 9.97882 9.97878 9.97874 9.97870 9.97866 15 14 13 12 II ~w I 5 4 3 2 I ~0" 50 5i 52 53 54 9.48 607 9.48647 9.48686 9-48725 9.48 764 9.50746 9-50789 9-50833 9.50876 9.50919 0.49 254 0.49 211 0.49 167 0.49 124 0.49081 9.97861 9-97857 9-97853 9-97849 9-97845 P 57 58 59 w 9-48803 9.48842 9.48881 9.48920 9-48959 9.50962 9-5IOOS 9-51048 9-51092 9-51 135 0.49038 0.48995 o 48 952 0.48908 0.48865 9.97841 9-97837 9-97833 9.97829 9-97825 9-48998 9-51 178 0.48822 9.97821 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. 1 Prop. Pts. 72 TABLE IV. 18 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. Y I 2 3 4 9.48998 9.49037 9.49076 9-49 H5 9 49 153 39 39 39 38 39 39 38 39 39 33 39 38 28 39 38 38 39 38 38 38 38 38 38 38 38 38 38 38 38 38 37 38 38 37 38 38 37 38 37 37 38 37 37 38 9-5i 178 9.51 221 9.51 264 9.51 306 9-51 349 43 43 42 43 43 43 43 43 43 42 43 43 42 43 42 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 4>I 42 42 41 4* 42 41 42 41 42 4* 41 4 1 0.48822 0.48 779 0.48 736 0.48 694 0.48 651 9.97821 9.97817 9.97812 9.97808 9.97804 4 5 4 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 00 59 58 JL 55 54 53 52 .1 .2 3 4 ^9 .1 .2 3 4 9 .1 .2 3 4 '9 .1 .2 3 4 ii 9 4 8 12 17 21 25 30 3 4 : i 9 l 2 / ii i< ic 2< 23 3' 3f : ^ \ i ii \\ 22 2 2( 3; ( t t i 3 i 9 .2 i 4 7 < \ 12 ie 2C 2^ 2* I 9 9 3 >-5 4 '3 .2 -I 17 1 7 r-4 .1 1:1 1.2 >-9 )-6 J-3 5 ).f I.G z.o 2-5 J-o 5-5 l- 1-5 43 4-2 8-4 12.6 16.8 21.0 25.2 29.4 33 6 37-8 x I f 2 3 '4 >.5 5 >_9 38 3-8 7-6 ii. 4 15-2 19.0 22.8 26.6 30.4 34-2 36 3 .6 10.8 18 o 21 6 ; 25.2 28.8 32-4 4 ; 3-4 i 0.8 1.2 1.6 2.O 2.4 2.8 I I 9 12 14 9.49 192 9.49231 9.49269 9.49308 9-49 347 9.51392 9.5I435 9-5 1 478 9-51 520 9 5 1 563 0.48608 0.48 565 0.48 522 z 48480 ? 48 437 9.97800 9.97796 9.97792 9-97 788 9.97784 9.49424 9.49462 9.49500 9 49539 9.51 606 9.51 648 9.51691 9-5 1 734 9.51 776 0.48394 0.48352 0.48309 0.48 266 0.48 224 9-97779 9-97775 9.97771 9-97767 9-97 763 50 49 48 15 16 \l 19 "20" 21 22 23 24 9-49577 9.49615 9 49654 9-49692 9-49730 9.51819 9.51 861 9-5 1 903 9.51946 9-51988 0.48 181 0.48 139 0.48097 0.48 054 0.48012 9-97 759 9-97754 9-97750 9-97746 9-97 742 45 44 43 42 41 9.49768 9.49 806 9.49844 9.49882 9.49920 9.52031 9-52073 9-52 157 9.52 200 0.47969 0.47927 0.47 885 0-47843 0.47 800 9.97738 9-97734 9.97725 9-97 72i 4 5 4 4 4 4 5 4 4 4 5 4 4 4 5 4 4 4 5 4 4 4 5 4 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 40 9 9 ? 29 31 32 33 f 9 39 9 49958 9.49996 9.50034 9.50072 9.50 no 9.52242 9.52284 9.52326 9-52368 9.52410 0-47 758 0.47716 0.47674 0.47632 0.47590 9.97717 9-97713 9-97 7oS 9.97704 9.97700 35 34 33 32 W 2Q 28 9.50 148 9.50185 9-50223 9.50261 9.50298 9.52494 9.52536 9.52 620 0.47548 0.47 56 0.47464 0.47422 0.47380 9.97696 9.97691 9-97687 9.97683 9.97679 9-50336 9.50374 9.50411 9 50449 9.50486 9.52 661 9-52 703 9.52745 9-52 787 9.52829 0-47 339 0.47297 0-47255 0.47213 0.47 171 9.97670 9.97666 9.97662 9-97657 25 24 23 22 21 40 42 43 1 44 9-50523 9 50 561 9.50598 9 50 635 9-50673 9.52870 9 52912 9 52953 9 52 995 9-53037 0.47 130 o 47 088 o 47 047 o 47005 0.46963 9-97653 9.97649 9-97645 9-97640 9-97636 20 19 17 16 45 46 9 49 9.50710 9.50747 9.50784 9.50 821 9.50858 37 37 37- 37 38 37 37 37 36 9-53078 9-53 120 9 53 161 9.53202 0.46 922 0.46880 0.46 839 0.46 798 0.46 756 9-97632 9.97628 9-97623 9.97619 9-976I5 IS 14 13 12 II 50 52 53 54 9 . 50 896 9.50933 9.50970 9.51 007 9.51 043 9-53285 9.53327 9.53368 9-53409 9.53450 0.46 715 0.46673 0.46 632 0.46591 0.46 550 9.97 610 9.97606 9.97602 9 97597 9-97593 10 I 59 9.51 080 9 5i "7 9-51 154 9.51 191 9.51 227 37 37 37 36 9-53492 9-53533 9-53574 9-53615 9-53656 0.46 508 0.46 467 0.46 426 0.46 385 0.46344 9-97589 9-97584 9.97580 9-97576 9-97571 5 4 3 2 I GO 9.51 264 37 9-53697 0.46303 9-97567 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. t Prop. Pts. ! 71 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 49 19 ! f L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. | I 2 3 4 9.51 264 9-5i3oi 9-5I338 9-5 1 374 9-5 1 4" 37 37 36 37 36 37 36 37 36 36 37 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 35 36 35 36 35 36 35 36 35 35 36 35 35 35 35 35 35 35 36 34 35 35 35 35 35 35 34 35 9-53697 9-53 738 9-53779 9-53820 9-5386I 41 41 41 41 4 4< 4< 4* 40 4i 4 40 41 41 4<> 4 4<> 4^ 40 41 0.46303 0.46 262 0.46 221 0.46 180 0.46 139 9-97567 9-97563 9.97558 9-97554 9-97550 4 5 4 4 5 4 5 4 4 5 4 4 5 4 5 4 5 4 4 5 4 5 4 5 4 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 (JO 59 58 H .1 .2 .3 .4 i 9 .1 .2 3 4 :S :i 9 .1 .2 3 4 :i :l 9 .1 .2 3 4 :1 9 41 4-1 8.2 "3 16.4 20.5 24.6 28.7 32.8 36.9 3 .' 3 .2 7 3 " 4 15 5 19 .6 23 .7 27 8 3i 9 35 37 3-7 7-4 ii. i 14.8 18.5 22.2 25.9 29.6 33-3 35 3-5 7-o 10.5 14.0 17-5 21.0 24-5 28.0 31.5 5 o-5 I.O 15 2.0 2-5 3-o 3-5 4.0 4-5 40 4.0 8.0 12. 16.0 20.0 24.0 28.0 32.0 36.0 9 1 iiii -5 4 3 .2 .1 36 3-6 10.8 14.4 18.0 21.6 25.2 28.8 32.4 34 i: 4 8 10.2 136 17.0 20.4 23-8 2 7 .2 30.6 4 0.4 0.8 1.2 1.6 2.0 :i i:l I I 9 'To ii :2 13 14 9-5 1 447 9.51 484 9-5 1 5 2 o 9.51557 9-5i 593 9-51629 9.51 666 9.51 702 9 5' 738 9 5i 774 9-53902 9-53943 9-53984 9-54025 9.54065 0.46 098 0.46057 o 46016 0-4597 0.45 93 9-97545 9-97541 9-97536 9-97532 9.97528 55 54 53 52 5i 9.54106 9-54147 9-54 187 9.54228 9.54269 0.45894 0-45 853 0.45813 0.45 772 0-45 73 1 9-97523 9 975*9 9-975I5 9.97510 9.97506 50 49 48 47 46 IS \l ft 21 22 23 24 9.51 811 9-5I847 9.51883 9-5i 9i9 9-5I955 9-54309 9-54350 9 54390 9 5443 1 9-54471 0.45 691 0.45 650 0.45 610 0-45 569 0-45 5 2 9 9.97501 9-97497 9.97492 9.97488 9.97484 45 44 43 42 41 40~ 39 38 P 9 5 1 99i 9.52027 9.52063 9.52099 9-5 2 135 9 54512 9 54552 9 54593 9 54633 9 54673 40 41 40 40 4 40 40 4 1 40 4> 40 4<> 40 4 4<> 40 4<> 40 4 4 4<> 4<> 39 4 4> 4 39 40 4<> 39 40 39 40 39 o . 45 488 0.45 448 0.45407 0-45 367 0.45327 9-97479 9-97475 9.97470 9.97466 9.97461 111 3 29 9.52 171 9.52207 9.52242 9.52278 9 5 2 3H 9 547H 9-54754 9-54794 9.54835 9 54875 0.45 286 0.45 246 0.45 206 0.45 165 0.45 125 9 97457 9 97453 9.97448 9-97444 9-97439 35 34 33 32 3i 80 3i 32 33 _34_ 9 3 39 9-52350 9-52385 9.52421 9-52456 9-52492 9-54915 9-54955 9-54995 9-55035 9.55075 0.45 08 0.4504 0.4500; 0.4496; 0.4492 9 97435 9 9743 9.97426 9-97421 9.97417 30 2 9 28 3 9-52527 9-52563 9-52598 9-52634 9.52669 9-55 "5 9-55 155 9-55 195 9.55235 9 55275 0.44885 0.44845 0.44805 0.44 765 0-44 725 9 97412 9.97408 9-97403 9 97399 9 97394 25 24 23 22 21 20" 19 18 !| 40 4i 42 43 44 9 52 705 9.52740 9.52775 9.52 811 9.52 846 9 553J5 9-55355 9-55395 9-55434 9-55474 0.44685 0.44645 0.44605 0.44566 0.44526 9.97390 9 97385 9.97381 9.97376 9.97372 11 % 49 Iso' 1 5 1 52 53 54 9.52881 ,9.52916 9.52951 9-52986 9 53021 9-555H 9-55554 9-55593 9.55633 o . 44 486 0.44446 0.44407 0.44367 0.44327 9 97367 9 97363 9-97358 9 97353 9 97349 15 H 13 12 II 9-53056 9-53092 9-53 126 9-53i6i 9-53I96 9-55712 9 55752 9-55 79i 9-55831 9 55870 0.44288 0.44248 0.44209 0.44 169 0.44 130 9-97344 9-97340 9-97335 9-97331 9.97326 10 6 P P 59 9-53231 9-53266 9-53301 9.53336 9-53370 9-55910 9-55949 9.55989 9.56028 9.56067 39 4<> 39 39 4<> 0.44090 0.44051 0.44011 0.43972 0-43933 9-97322 9-973I7 9.97312 9.97308 9 97303 5 4 3 2 I 60 9 534^5 9-56107 0.43893 9.97299 L. Cos. (1. L. Cots. c.d. L. Tang. L. Sin. d. ' f Prop. Pts. 70 TABLE IV. 20 L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. p rop. Pts. I 2 3 4 9.53405 9-53440 9-53475 9.53509 9-53544 35 35 34 35 34 9.56 107 9.56 146 9-56185 9.56224 9 . 56 264 39 39 39 40 39 0.43893 0.43 854 0.43 815 0.43 776 0.43 736 9.97299 9-97294 9.97 289 9.97285 9.97280 5 5 4 5 00 59 58 i .1 2 40 \l 39 3 -i 7 I I 9 9.53578 9.53$i3 9.53647 9-53682 9-53 7i6 35 34 35 34 35 9-56303 9.56342 9-56381 9.56420 9.56459 39 39 39 39 39 0.43 697 0.43658 0.43619 0.43 580 0.43 54i 9.97276 9.97271 9.97266 9.97262 9-97257 5 5 4 5 55 54 53 52 Si 3 :! .7 12.0 16.0 20.0 7.0 ii. 7 15.6 19-5 23 4 27.7 10 ii 12 13 14 9-53 751 9-53 75 9-53819 9.53854 9.53888 34 34 35 34 34 9.56498 9.56537 9.56576 9.56615 9-56654 39 39 39 39 39 0.43 502 0.43463 0.43 424 0-43 385 0-43 346 9-97252 9.97248 9-97243 9-97238 9-97234 4 5 5 4 w 8 % .8 9 32.0 36.0 38 31-2 35-1 37 15 16 !i 19 9-53922 9-53957 9-53991 9-54025 9 54059 35 34 34 34 34 9.56693 9.56732 9.56771 9.56810 9.56849 39 39 39 39 38 0-43 307 0.43 268 0.43 229 0.43 190 0.43 I5i 9.97229 9-97224 9.97220 9-97215 9.97210 5 4 5 5 45 44 43 42 41 .2 3 4 3 ^ 7-6 11.4 15-2 19.0 3-7 7-4 ii. i 14.8 18.5 20 21 22 23 24 9.54093 9-54I27 9.54i6i 9-54I95 9-54229 34 34 34 34 34 9-56887 9.56926 9-56965 9.57004 9.57042 39 39 39 38 39 0-43 "3 0.43074 0.43035 0.42 996 0.42958 9.97206 9.97201 9-97 196 9.97192 9-97 187 5 5 4 5 40 3 11 i 9 22.8 26.6 30.4 34-2 22.2 25-9 29.6 33-3 > 3 2 29 9-54263 9-54297 9-54331 9-5436? 9-54399 34 34 34 34 34 9.57081 9.57 120 9.57158 9-57197 9.57235 39 38 39 38 39 0.42919 0.42880 0.42 842 0.42 803 (^42 765 9.97 182 9.97178 9-97 173 9.97 168 9-97 163 4 5 5 5 35 34 33 32 3i .1 .2 3 i< A \. 35 J-S 7-0 >-5 1 O 80 3i 32 33 34 9-54433 9.54466 9-54500 9-54534 9.54567 33 34 34 33 OJ 9.57274 9.57312 9-57351 9.57389 9.57428 38 39 38 39 08 0.42 726 0.42688 0.42 649 0.42 611 0.42 572 9-97 159 9-97 154 9-97 149 9-97 H5 9-97 HO 5 5 4 5 30 1 5 * .6 2 .7 2, .8 2! 9 3 7-3 I.O ti [.5 9 !? 39 9-54601 9-54635 9.54668 9.54702 9-54735 34 33 34 33 34 9.57466 9.57504 9-57543 9.57581 9-57619 38 39 38 38 30 0.42534 0.42496 0.42457 0.42419 0.42 381 9 97135 9-97 130 9.97 126 9.97121 9.97 116 5 5 4 5 5 25 24 23 22 21 .1 2 34 n 33 3 6' 3 6 40 4i 42 43 44 9-54769 9.54802 9-54836 9-54869 9-54903 33 34 33 34 33 9.57658 9.57696 9-57734 9-57772 9.57810 38 38 38 38 30 0.42342 0.42304 0.42 266 0.42 228 0.42 190 9.97 in 9-97 107 9.97 102 9.97097 9.97092 5 4 5 5 5 20 J 9 II 3 4 .7 10.2 13-6 17.0 20.4 23.8 9-9 13.2 i6.g 19.8 23-1 s 47 48 49 9-54936 9.54969 9-55003 9-55036 9-55069 33 34 33 33 9.57849 9 57887 9.57925 9-57963 9.58 ooi 38 38 38 38 ^s 0.42 151 0.42 113 0.42075 0.42037 0.41 999 9.97087 9-97083 9.97078 9.97073 9.97068 5 4 5 5 5 15 14 13 12 II .8 9 2 7 .2 3O.6 5 26.4 29.7 4 50 5i 5 2 53 54 9-55 102 9-55 136 9-55 169 9-55202 9.55235 34 33 33 33 q-a 9-58039 9.58077 9-58 n5 9.58153 9-58 191 38 38 38 38 38 0.41 961 0.41 923 0.41 885 0.41 847 0.41 809 9.97063 9-97059 9-97054 9.97049 9-97044 5 4 5 5 5 10 I .1 .2 3 4 5 o-5 I.O i-5 2.0 2-5 0.4 0.8 1.2 1.6 2.0 55 56 ii 59 9-55268 9-55301 9-55334 9.55367 9-55400 33 33 33 33 9-58229 9-58267 9-58304 9-58342 9-58380 38 37 38 38 ,Q 0.41 771 0.41 733 0.41 696 0.41 658 0.41 620 9-97039 9.97035 9.97030 9.97025 9.97020 5 4 5 5 5 5 4 3 2 I .b :l 9 3-o 3-5 4-0 4-5 111 5.8 60 9-55433 9.58418 0.41 582 9.97015 5 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. t F rop. ] Pte. 69 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 5 , 21 r L. Sill. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop.Pte. I 2 3 4 9-55433 9.55466 9-55499 9-55 532 9.55564 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 3* 3 33 33 3 3 33 33 33 33 33 3' 33 33 33 3' 33 3* 33 33 3* 33 3' 3i 33 3 33 3* 3 33 9-58418 9-58455 9.58493 9-58531 9-58569 37 38 38 38 37 38 37 38 38 37 38 37 38 37 37 38 37 38 37 37 37 38 37 37 37 37 38 37 37 37 37 37 37 37 37 37 37 36 37 37 37 37 36 37 37 37 36 37 37 36 37 36 37 36 37 36 37 36 37 36 0.41 582 0.41 545 0.41 507 0.41 469 0.41 431 9.97015 9.97010 9.97005 9.97001 9.96996 5 5 4 5 5 5 5 5 5 5 4 5 5 5 5 5 5 5 5 5 5 5 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 5 5 5 5 5 5 5 5 5 5 5 60 P g .1 .2 3 4 :i i 9 .1 .2 3 :! is 9 .1 .2 3 4 :i 9 .1 .2 3 4 i 9 38 3-8 7-6 11.4 15-2 19.0 22.8 26.6 304 34.2 3 6 3-6 10.8 14.4 18.0 21.6 25.2 28.8 32.4 J" .2 < 3 < .4 i: 5 I( .6 K 7 * .8 l .9 2\ 3> 6.2 93 12.4 21.7 24.8 27-9 5 -5 I.O 1-5 2.0 2.5 3-o 3-5 4.0 4-5 37 3-7 74 II. 1 14.8 18.5 22.2 25.9 29.6 33-3 33 i:i 99 15.2 i&.J 19.8 23.1 26.4 29.7 J J- a >-4 )-6 2.8 3.0 ?-2 |1 3.8 | o'.6 1.2 I 8 2.4 3-o 36 4-2 48 54 4 0.4 j 0.8 1.2 1.6 2.O ;i 3:8 \ 9 \w II 12 13 H 9-55 597 9-55630 9-55663 9-55695 9.55728 9.58606 9.58644 9.58681 9-58719 9-58757 0.41 394 0.41 356 0.41 319 0.41 281 0.41 243 9.96991 9.96986 9.96981 9-96976 9.96971 55 54 53 52 5i 9-55 76i 9-55 793 9.55 826 9.55858 9-5589I 9.58794 9-58832 9-58869 9-58907 9-58944 0.41 206 0.41 168 0.41 131 0.41 093 0.41 056 9.96966 9.96962 9-96957 9-96952 9.96947 60 8 8 3 :; 19 9.55923 9^5956 9.56021 9-56053 9.58981 9.59019 9-59056 9-59094 9-59131 0.41 019 0.40981 0.40 944 0.40 906 0.40 869 9.96942 9.96937 9.96932 9.96927 9.96922 45 44 43 42 4i IT 1 20 21 22 23 24 9.56085 9.56 na 9-56 150 9.56 182 9-56215 9-59 1 68 9-59205 9-59243 9.59280 9.59317 0.40832 0.40 795 0.40 757 0.40 720 0.40 683 9.96917 9.96912 9.96907 9 . 96 903 9.96898 2 29 9.56247 9.56279 9-56311 9-56343 9.56375 9-59354 9.59391 9.59429 9.59466 9-59503 0.40 646 0.40609 0.40571 0.40 534 0.40497 9 . 96 893 9.96888 9-96883 9.96878 9.96873 35 34 33 32 3i 30 3i 32 33 34 9.56408 9.56440 9.56472 9-56504 9-56536 9-59540 9-59577 9-59614 9-5965I 9.59688 9.59725 9.59762 9 59799 9.59835 9.59872 0.40 460 0.40423 0.40386 0.40349 0.40312 9.96868 9-96863 9-96858 9-96853 9.96848 30 11 11 25 24 23 22 21 20- 19 18 g 35 36 9 39 9.56568 9-56599 9-56631 9.56663 9.56695 0.40275 0.40 238 0.40 2OI O.40 165 O.40 125 9 96 843 9-96838 9-96833 9.96828 9-96823 j40 4i 42 43 44 9-56727 9-56759 9-56790 9 56 822 9.56854 9-59909 9^9946 9.59983 9.60019 9.60 056 0.40091 0.40054 0.40017 0.39981 0-39944 9.96818 9-96813 9.96808 9.96803 9.96 798 45 46 47 48 49 50 5i 52 53 54 9.56886 9.56917 9 56949 9 . 56 980 9.57012 9.60093 9.60 130 9.60 1 66 9.60203 9 . 60 240 0.39907 0.39870 o 39834 0-39 797 0.39760 9-96 793 9.96 788 9-96 783 9.96 778 9.96772 15 H 13 12 II "10" 6 9 57044 9-57075 9.57107 9.57138 9.57169 9.60276 9.60313 9.60349 9.60386 9.60422 0.39 724 0.39687 0.39651 0.39614 0-39 578 9.96767 9.96 762 9.96757 9.96 752 9.96 747 55 56 ? ft 9.57201 9-57232 9.57264 9.57295 9-57326 9.60459 9.60495 9.60532 9.60 568 9.60605 o 39 54i 0.39505 0.39468 0.3943 2 0-39395 9.96742 9.96 737 9.96 732 9.96727 9.96 722 5 4 3 2 I 9-57358 9.60641 0-39359 9.96717 L. Cos. d. L. Cots. o. d. L. Tang. L. Sin. d. / Prop. Pts. 68 TABLE IV. 22 I L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. I 9.57358 9-57389 31 9.60641 9.60677 36 0-39359 0.39323 9.96717 9.96711 6 60 >9 2 9.57420 9.60714 ,6 0.39 286 9.96 706 5 S8 37 1 36 3 4 9 57451 9.57482 3 2 9.60750 9.60 786 36 37 0.39250 0.39214 9.96 701 9.96696 5 5 .1 2 3-7 7 4 3-6 7.2 i 9.57514 9-5754? 9.60823 9 . 60 859 36 ,6 0.39177 o-39 HI 9.96691 9.96686 5 55 S4 3 .4 II. I 14.8 14.4 I 9 9-57576 9-57607 9-57638 3 3* 9.60895 9.60931 9-60967 36 36 37 0.39 105 0.39069 0.39033 9.96681 9.96676 9.96670 5 5 6 53 52 1 18-5 22.2 25.9 18.0 21.6 2 5- 2 10 9.57669 9.61 004 ,6 0.38996 9.96665 50 .8 29.6 28.8 ii 9.57700 9.61 040 og 0.38960 9.96 660 5 49 9 33-3 32-4 12 13 14 9-57731 9.57762 9-57793 $ 9.61 076 9.61 112 9.6l 148 36 36 36 0.38924 0.38888 0.38852 9-96655 9 . 96 650 9.96645 5 5 48 8 35 \l 9.57824 9.57855 9.57885 30 9.6l 184 9.61 22O 9.61 256 36 36 ,A 0.38816 0.38 780 0.38 744 9 . 96 640 9.96634 9 . 96 629 6 5 45 44 43 .2 7-0 3 lo-S 18 9.57916 * 9.61 292 ,jt 0.38 708 9.96624 5 42 .4 I4.O 19 9-57947 31 9.61 328 3 36 0.38672 9.96619 5 41 20 21 9.57978 9.58008 30 9-6l 364 9.6l 400 36 0.38636 0.38600 9.96614 9.96608 6 *? .7 24.5 X 28 O 22 23 24 9-58039 9.58070 9.58 101 9.6l 436 9.6l 472 9.6l 508 3 36 36 36 0.38 564 0.38528 0.38492 9.96603 9.96598 9-96593 5 5 5 11 9 3'-5 11 9-58131 9.58 162 9.61 544 9.61 579 35 -e. 0.38456 0.38421 9.96588 9.96582 6 35 34 3 31 3 | % 29 9.58192 9.58223 9-58253 30 30 31 9.61 615 9.61 651 9.61 687 3 36 36 35 0-38385 0.38349 0.38313 9-96577 9.96572 9.96567 5 5 5 33 32 31 .2 3 A T? 8 6.2 9.3 12.4 30 9 S 8 284 9.61 722 ofi 0.38278 9 . 96 562 30 16 o 31 9-583I4 3 9.61 758 30 ,6 0.38242 9 96556 29 .6 19.2 18.6 34, 9-58345 9.58406 30 9.61 794 9.61 830 9,61 865 36 35 16 0.38 206 0.38 170 0.38135 9 96551 9 96546 9.96541 5 5 6 11 9 22.4 25.6 28.8 21.7 24.8 27.9 9-58436 9.58467 3' 9.61 901 9.61 936 35 0.38099 0.38064 9 96535 9 96530 5 25 24 39 9-58497 9-58527 9 58557 30 30 30 9.61972 9.62 008 9.62043 36 36 35 ,6 0.38028 0.37992 0-37957 9-9652? 9.96520 9.96514 5 5 6 23 22 21 .1 2 30 ft 29 2.9 (0 9-58588 3 9 . 62 079 0.37921 9.96509 20 .3 9 8 7 41 9.58618 3 9.62 114 35 0.37886 9-96504 5 19 4 12.0 ii. 6 42 9.58648 3 9.62 150 30 0.37850 9.96498 18 15 14-5 43 9.58678 3 9.62 185 35 -f. 0-37815 9.96493 5 17 .6 18.0 17-4 44 9.58 709 3 1 9.62 221 3 0.37 779 9.9648$ 5 ib .7 21.0 20.3 | 9-58739 9.58769 30 9.62 256 9 . 62 292 36 0-37744 0.37708 9.96483 9.96477 6 15 14 9 24.0 27.0 23.2 26.1 9.58799 9 - 58 829 30 30 9.62327 9.62 362 35 35 0.37673 0.37638 9.96472 9.96467 5 5 13 12 49 9-58859 3 9.62 398 0.37602 9.96461 II 6 5 50 9.58889 9.58919 30 9.62433 9.62468 35 0.37567 0.37532 9.96451 5 10 .1 .2 0.6 1.2 0-5 I.O 52 9-58949 3 9.62 504 36 0.37496 9-96445 6 8 3 1.8 15 53 9-58979 30 9.62 539 35 0.37461 9.96440 5 7 4 2.4 2 54 9.59009 30 9.62574 35 0.37426 5 5 6 3-o 2-5 55 9.59039 9.62609 0.37391 9.96429 5 b 3-6 3-0 56 9.59069 30 9 . 62 645 36 0-37355 9.96424 5 4 'I 4-2 3-5 57 9.59098 29 9.62680 35 0.37320 9.96419 5 3 .8 4-8 4.0 58 59 9-59 128 9-59 IS 8 3 30 9-62 715 9.62 750 35 35 0.37285 0.37250 9-96413 9.96408 6 5 2 I 9 5-4 4-5 60 9.59188 9.62 785 0.37215 9.96403 5 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. t Prop. Pts. 67 | LOGARITHMS OF SINE, COSINE, TAff -tfT AND COTANGENT, ETC. i 23 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. I 2 3 4 9.59 188 9.59218 9-59247 9-59277 9- 5937 30 29 30 30 29 30 30 29 30 29 30 29 30 29 30 29 39 30 29 9 30 29 29 29 *9 30 9 29 29 29 29 29 29 29 9 29 29 29 29 28 29 29 29 28 29 29 29 28 29 28 29 38 29 28 39 28 29 28 28 9-62785 9 . 62 820 9.62 855 9 . 62 890 9.62 926 35 35 35 36 35 35 35 35 35 34 35 35 35 35 35 35 34 35 35 35 35 34 35 35 34 35 34 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 0.37215 0.37 180 0-37 145 0.37 no 0.37074 9.96403 9.96397 9-96392 9.96387 9.96381 6 5 5 6 5 6 5 5 6 5 6 5 5 6 5 6 5 6 5 6 5 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 6 5 6 5 6 5 6 6 5 6 5 6 6 5 6 5 6 00 CQ i .1 4 3 4 i 9 .1 .2 3 :l i 9 .1 .2 3 -4 i i 9 3 3 7 ID 11 21 25 28 32 .1 .2 3 4 :l .1 9 2 1 s 12 IE ll 21 2<: 2; .1 .2 3 .4 i :l .9 ( : i i i 6 .6 2 .8 4- .6 .2 .8 4 I \ 1C ': i; 2C 2; 2; 3< .0 .0 >.o .0 :S .0 ^o r.o i i i 2 2 6 >.6 [.2 [.8 i-4 M 1:5 ;-4 35 3-5 7.0 1 10.5 14.0 17-5 21.0 24-5 28.0 31-5 14 S;i >.2 t-6 r.O ;i r.2 >.6 39 2.9 l l II. 6 14-5 17-4 20.3 2:? 8 2.8 I' 6 1.4 1.2 4..0 5.8 9.6 2.4 52 5 0.5 1.0 '5 20 2-5 30 35 40 4-5 i I 9 lo- ii 12 13 H 9.59336 9-59366 9-59396 9-59425 9-59455 9.62 961 9.62 996 9-63031 9 . 63 066 9.63 101 0.37039 0.37004 0.36969 0.36934 o . 36 899 9-96376 9.96370 9-96365 9-96360 9-96354 55 54 53 52 5i 9-59484 9-595H 9-59543 9 59573 9.59602 9-63 135 9-63 170 9.63205 9.63240 9 63275 0.36865 0.36830 0.36 795 0.36 760 0.36 725 9-96349 996343 9 96338 9 96333 9.96327 60 * % 15 10 11 19 9-59632 9 59661 9.59690 9.59720 9-59749 9.63310 9.63345 9.63379 9.63414 9-63449 0.36690 0.36655 0.36621 0.36586 0.36551 9-96322 9.96316 9.96311 9-96305 9.96300 45 44 43 42 4i 40" 39 38 8 20 21 22 23 24 9-59778 9-59808 9 59837 9.59866 9 59895 9.63484 9.63519 9.63 553 9-63588 9-63623 0.36516 0.36481 0.36447 0.36412 0.36377 9.96294 9.96 289 9.96284 9.96278 9.96273 3 2 29 9-59924 9-59954 9-59983 9.60012 9.60041 9-63657 9.63692 9.63726 9-63 76i 9-63796 0.36343 0.36308 0.36274 0.36239 0.36204 9.96267 9.96262 9 96256 9 96251 9.96245 35 34 33 32 _1L 80 3 27 26 30 3i 32 33 34 9.60070 9.60099 9.60 128 9-60157 9.60186 9 63 830 9-63865 9.63899 9.63934 9.63968 0.36 170 0.36 135 0.36 101 0.36066 0.36032 9 . 96 240 9-96234 9 96 229 9-96223 9.96 218 9 12 i- 41 42 43 44 9.60215 9.60244 9.60273 9.60302 9 60331 9 64003 9.64037 9.64072 9.64 106 9.64 140 0-35 997 0-35 963 0.35 928 0.35894 0.35 860 9.96 212 9.96207 9.96 201 9.96 196 9.96 190 25 24 23 22 21 ~w 19 i! 9-60359 9.60388 9.60417 9.60446 9-60474 9.64I75 9.64209 9-64243 9.64278 9.64312 0-35825 0-35 79i o.35 757 0.35 722 0.35688 9.96 185 9.96 179 9.96174 9.96 168 9.96 162 i* iti 49 9.60503 9.60532 9.60 561 9.60589 9.60618 9 64 346 9-64381 9.64415 9.64449 9.64483 0-35654 0.35619 0-35 585 o.3555i 0-35 5i7 9.96 157 9.96151 9.96146 9.96 140 9-96I35 15 14 13 12 II lo" I 60 5i 52 53 54 9.60646 9-60675 9.60 704 9.60 733 9.60 76. 9-64517 9 6^552 9-64586 9.64620 9.64654 0-35 483, 0.35448 0.354J4 0.35380 0-35 346 9.96129 9.96 123 9.96 1 18 9.96 112 9.96 107 11 % 59 9.60 789 9.60818 9.60846 9.60875 9.60903 9.64688 9.64722 9.64756 9.64790 9.64824 0.35312 0.35 278 0.35 244 0.35210 0.35 176 9.96 ioi 9.96095 9.96090 9 . 96 084 9.96079 5 4 3 2 I ~TT 00 9-60931 9.64858 0-35 142 9.96073 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. t Prop. Pts. 66 j TABLE IV. 24 t L. Sin. d. L. Tang. c.d. L. Cots. L. Cos. d. Proi >. Pts. 2 3 4 9.60931 9.60960 9.60988 9.61 016 9.61045 29 28 28 29 28 9.64858 9 . 64 892 9 . 64 926 9 . 64 960 9.64994 34 34 34 34 34 0-35 J 42 0.35 108 0-35 074 0.35 040 0.35006 9.96073 9.96067 9 . 96 062 9 . 96 056 9 . 96 050 6 5 6 6 60 CQ I 3 .1 3 2 6 4 * S3 i:I I I 9 9.61073 9.61 101 9.61 129 9.61 158 9.61 186 28 28 29 28 28 9 . 65 028 9 . 65 062 9 . 65 096 9.65 130 9.65 164 34 34 34 34 33 0.34972 0.34938 0.34904 0.34870 0.34836 9-96045 9-96039 9.96034 9 . 96 028 9 . 96 022 6 5 6 6 55 54 53 52 5i .3 10 4 13 5 17 .6 20 7 23 2 .6 .0 :i 99 13.2 16 5 19.8 23.1 10 ii 12 3 '4 9.61 214 9.61 242 9.61 270 9.61 298 9.61 326 28 26 28 28 28 9.65 197 9.65231 9.65 265 9.65299 9-65 333 34 34 34 34 0.34803 0-34 769 0-34735 0.34701 0.34667 9.96017 9.96011 9.96005 9.96000 9-95994 6 6 5 6 5 50 8 4 J .8 27 9 30 .2 .6 i 26.4 29.7 >9 II 11 9 9-6i 354 9.61 382 9.61 411 9-61 438 9.61 466 28 29 27 28 28 9.65 400 9.65434 9.65 467 9.65 501 34 34 33 34 0.34634 0.34600 0.34566 0-34533 0-34499 9.95988 9.95982 9-95977 9-95971 9-95965 6 S 6 6 45 44 43 42 41 .1 .2 3 4 t I ii i; 2 8 l l .6 1-5 20 21 22 23 24 9.61 494 9.61 522 9-6i 550 9-6i 578 9.61 606 28 28 28 28 28 9-65 535 9.65 568 9.65 602 9.65 636 9.65 669 33 34 34 33 0.34465 0.34432 0.34398 0.34364 0-34331 9.95960 9 95954 9-95948 9 95942 9 95937 6 6 6 5 6 40 39 38 8 j 9 1 2C 2; 7-4 >-3 11 3 3 29 9-61634 9.61 662 9.61 689 9.61 717 9-6i 74? 28 27 28 28 28 9-65 703 9-65 736 9.65 770 9-65 803 9-65837 33 34 33 34 0.34297 0.34264 0.34230 0-34 197 0.34 163 9-95 93 1 9.95925 9.95920 9.959H 9.95908 6 5 6 6 6 35 34 33 32 3i .1 .2 3 4 T 18 2.8 ii 1 .2 30 3* 32 33 34 9.61 773 9.61 800 9.61 828 9.61 856 9.61 883 27 38 28 27 28 9.65 870 9.65 904 9.65937 9.65971 9.66004 34 33 34 33 0.34 130 0.34096 0.34063 0.34029 0.33996 9.95902 9.95897 9.95 891 9.95885 9.95879 5 6 6 6 5 30 27 26 1 .9 I I I 2 2 *-o 5.8 ? .6 2.4 5-2 8 8 1 39 9.61 911 9.61 939 9.61 966 9.61 994 9.62 O2I 28 27 28 27 38 9.66038 9.66071 9.66 104 9.66 138 9.66 171 33 33 34 33 0-33 962 0.33929 0.33 896 0.33862 0.33829 9-95873 9.95868 9.95 862 9.95856 9-95 850 5 6 6 6 g 25 24 23 22 21 .1 .2 7 2.7 r 4 ,40 4i 42 43 44 9.62 049 9.62076 9.62 IO4 9.62 131 9.62 159 27 28 27 28 9.66204 9.66238 9.66 271 9.66304 9.66337 34 33 33 33 0.33 796 0-33 762 0.33729 0.33696 0.33663 9-95 844 9-95 839 9.95833 9.95827 9.95821 5 6 6 6 g 20 19 18 11 3 4 '-7 I I I n 0.8 Ii 8-9 9 ;i 49 9.62 186 9.62 214 9.62 241 9.62268 9 . 62 296 28 27 27 28 9.66371 9 . 66 404 9.66437 9.66470 9.66503 33 33 33 33 0.33 629 0.33 596 0-33 563 0-33 530 0-33497 9 958i5 9-95 810 9.95 804 9 95798 9 95 792 5 6 6 6 6 15 14 13 12 II .8 9 2 2 6 1.6 4- 3 5 150 5i 52 53 54 9.62323 9.62350 9.62377 9-62405 9.62432 27 27 28 9-66537 9.66570 9.66603 9.66636 9 . 66 669 33 33 33 33 0-33463 0.33430 0-33 397 0.33 364 0-33331 9.95 786 9-95 78o 9-95 775 9-95 769 9-95 763 10 I .1 C .2 1 3 1 4 J 5 : >.6 .2 .8 '-4 ;o 0-5 1 1.0 ; 15 2.0 25 P 11 59 9-62459 9.62486 9-62513 9 62 541 9.62568 37 98 9.66702 9-66735 9.66 768 9.66801 9-66834 33 33 33 33 0.33298 0.33265 0-33 232 0-33 199 0.33 166 9-95 757 9-95 75i 9 95 745 9 95 739 9 95 733 6 6 6 6 5 4 3 2 o : i ; 9 I ; -t> |:S 1 4 30 35 4.0 45 60 9 62595 7 9.66867 o 33 133 9 95 728 L. Cos. d. L. Cotsr. c.d. L. Tang. L. Sin. d. f Pro P- Pts. 65 u JUAKl 1 1 MS L * blJNJL, I .AJMJ NJi, lAJNlj r,JN 1 AJN 1 ) l^U 1AJN< jUJMl, C 1 rc - 5 25 t L.Sin. d. L. Tang. c. d. L. Cotg. L. Cos. d. Proj >.] Pts. i 2 3 4 9-62595 9.62 622 9.62 649 9.62 676 9.62 703 27 27 27 27 27 9.66867 9 . 66 900 9.66933 9.66966 9.66999 33 33 33 33 33 0-33 133 0.33 ioo 0.33067 0.33034 0.33001 9-95 728 9-95 722 9-95 7i6 9-95 7io 9-95 704 6 6 6 6 6 60 59 58 H 3 i 3 2 6 3 33 3-2 6.4 i *<> I 9 9.62 730 9.62 757 9.62 784 9.62 81 i 9.62 838 27 27 27 27 27 9.67032 9 . 67 065 9.67098 9.67131 9.67 163 33 33 33 33 33 0.32968 0.32935 0.32 902 0.32869 0.32837 9.95698 9.95692 9-95^36 9.95 680 9-95674 6 6 6 6 6 55 54 53 52 5i 3 9 4 13 5 l6 .6 19 7 23 9 .2 . I 1 12.8 16.0 19.2 22.4 10 ii 12 13 14 9.62865 9.62 892 9.62 918 9-62945 9.62972 27 26 27 27 27 9.67196 9.67229 9.67262 9.67295 9.67327 33 33 33 32 0.32804 0.32 77i 0.32 738 0-32 705 0.32673 9.95 668 9-95 663 9.95657 9-9565I 9 95645 S 6 6 6 6 50 49 48 47 46 .8 26 .9 29 4 7 2C.6 2.8 7 15 1 6 Is 7 19 9-62999 9.63026 9.63052 9.63079 9.63 106 87 26 27 27 27 9-67360 9.67393 9.67426 9.67458 9.67491 33 33 33 33 0.32 640 0.32 607 0.32574 0.32 542 0.32509 9-95639 9.95633 9-95627 9.95621 9-956I5 6 6 6 6 6 45 44 43 42 41 .1 .2 3 4 3 : t 1C i: 5.7 11 >.8 SI ' 20 21 22 23 24 9-63 133 9-63 159 9.63 186 9.63213 9 63239 26 27 27 26 27 9.67524 9.67556 9.67589 9.67 622 9.67654 32 33 33 33 0.32476 0.32444 0.324" 0.32378 0.32 346 9.95609 9-95603 9-95 597 9-95591 9 95585 6 6 6 6 6 40 i :1 9 i* 2 2^ l'| [.6 L-3 3 3 29 9.63266 9.63292 9.63319 9.63345 9.63372 26 27 26 27 26 9.67687 9.67719 9.67752 9-67785 9.67817 3 33 33 32 0.32313 0.32 281 0.32248 0.32 215 0.32 183 9-95579 9 95 573 9 95567 9 95 56i 9 95 555 6 6 6 6 g 35 34 33 32 3i .1 .2 3 A I T( * 2.6 \l 5 A 80 3i 32 33 34 9-63398 9-63425 9-63451 9.63478 9-63504 27 26 27 26 l%% 9.67915 9.67947 9.67980 33 33 33 33 0.32 150 0.32 118 0.32085 0.32053 0.32020 9 95 549 9-95 543 9-95 537 9-9553' 9-95 525 6 6 6 6 6 30 11 2 i .9 i; i 2( 2, 3-. ii D.8 5-4 P 9 39 9-63 S3 1 9.63557 9-63583 9.63610 9-63 636 26 26 27 2 26 9.68012 9 . 68 044 9.68 077 9 68 109 9.68 142 32 33 32 33 0.31 988 0.31 956 0.31 923 0.31 891 0.31 858 9.95519 9-95 513 9 95 507 9-95500 9-95494 6 6 7 6 5 25 24 23 22 21 .1 .2 < 7 3.7 1-4 40 4i 42 43 44 9.63 662 9.63 689 9-63 715 9.63 741 9.63 767 27 26 26 26 9.68174 9.68206 9 . 68 239 9.68271 9.68303 32 33 33 33 0.31 826 0.31 794 0.31 761 0.31 729 0.31 697 9.95488 9.95482 9.95476 9-95 470 9.95464 6 6 6 6 g 20 11 \l 3 :! 7 t t Z.I 2.8 5-5 \.2 t-9 9 9 49 9-63 794 9.63820 9-63 846 9.63872 9.63898 26 26 96 26 26 9-68336 9.68368 9.68400 9.68432 9.68465 33 32 32 33 0.31 664 0.31 632 0.31 600 0.31 568 0.31 535 9.95458 9-95452 9-95 446 9-95 440 9-95434 6 6 6 6 15 14 13 12 II .8 9 5 I 6 &3 5 50 5i 52 53 54 9.63924 9.63950 9.63976 9.64002 9.64028 26 26 26 26 26 9-68497 9-68529 9.68561 9.68 593 9.68626 33 33 33 33 0.31 503 0.31471 0.31 439 0.31 407 0.31 374 9.95427 9.95421 9.95415 9-95 409 9.95403 6 6 10 1 I .1 C .2 1 3 i 4 a i 3 >.6 .2 .8 4 .0 0-5 t.o i-5 2.0 25 55 56 P 59 9-64054 9.64080 9.64 106 9 64 132 9.64 158 26 26 26 26 26 9.68658 9.68690 9.68 722 9.68 754 9.68 786 33 33 32 32 0.31 342 0.31 310 0.31 278 0.31 246 0.31 214 9-95 397 9-95 39i 9-95384 9.95378 9 95372 5 4 3 2 6 3 i : 9 5 6 .2 .8 4 30 35 4.0 4 5 60 9.64 184 9.68818 0.31 182 9-95 366 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. 9 Proi ). Pts. 64 6 TABLE IV. 26 ll t L. Sin. d. L. Tang. .d. L. Cotg. L. Cos. d. 60- 9 JL 55 54 53 52 5i Prop. Pts. i 2 3 4 9.64 184 9.64 2IO 9.64236 9.64 262 9.64288 26 26 26 26 25 26 86 26 26 25 26 36 25 26 26 25 26 25 26 25 26 25 26 25 26 S 26 5 25 26 25 25 36 25 25 25 26 25 25 25 3 25 26 25 25 25 25 25 25 25 25 25 25 25 24 25 25 25 25 25 9.68818 9.68850 9.68882 9.68 914 9.68946 32 32 32 32 32 32 32 32 32 0.31 182 0.31 150 0.31 118 0.31 086 0.31 054 9-95366 9-9536o 9-95354 9-95 348 9-95341 6 6 6 6 6 6 6 7 6 6 6 6 7 6 6 6 7 6 6 6 7 6 6 6 7 6 6 7 6 6 6 7 6 6 7 6 6 7 6 6 7 6 7 6 6 7 6 6 7 6 7 6 6 7 6 7 6 6 7 & :i I 3 9 .4 12 I l6 .6 19 .7 22 [9 28 .1 .2 3 4 :i i 9 .1 .2 3 4 i :I 9 .1 .2 3 4 :i :I -9 .1 ( .2 3 ' 4 - i ; i : 9 < i j 31 2 3-1 4 6.2 o 9-3 .8 12.4 o 1^.5 .2 18.6 .4 21.7 11$ * \i S:| 15-6 18.2 20.8 ,3,1! as 2-5 5-0 7-5 10. 12. 5 15.0 17.5 20. o 22.5 N :i ll 12.0 !t.i 19.2 21.6 7 )-7 0.6 1.4 1.2 z.i 1.8 B.8 2. 4 J-S 3-0 J-2 3-6 ^9 4-2 j-6 4-8 >-3 5-4 1 I 9 IT ii 12 13 H 11 17 18 19 21 22 23 24 3 3 29 9-643I3 9-64339 9.64365 9.64391 9.64417 9.68978 9.69010 9 . 69 042 9.69074 9.69 106 0.31 022 0.30990 0.30958 0.30926 0.30894 9-95335 9-953 2 9 9-95 323 9-953I7 9-95310 9.64442 9.64468 9.64494 9.64519 9.64545 9-69 138 9.69 170 9.69202 9-69234 9.69 266 32 32 32 32 0.30862 0.30830 0.30 798 0.30 766 0.30734 9-95304 9-95 298 9.95 292 9 95286 9 95279 60 11 8 9.6457I 9.64596 9.64622 9.64647 9.64673 9.69298 9.69329 9.69361 9.69393 9.69425 3 3* 32 32 32 3i 32 32 32 3i 32 32 3 32 32 3* 32 3 32 32 3 33 3 32 3 32 3 32 3* 32 3 3i 32 3 32 3i 3* 32 3 3 32 3 3* 3 32 O.30 702 0.30671 0.30639 0.30607 0.30575 9-95273 9-95 267 9.95261 9-95254 9.95248 45 44 43 42 41 9.64698 9.64 724 9.64749 9.64775 9.64 800 9-69457 9.69488 9.69520 9.69552 9.69584 0.30543 0.30512 0.30480 o . 30 448 0.30416 9.95242 9 95236 9.95229 9 95223 9.95217 40 P ll 9.64826* 9.64851 9.64877. 9.64902 9.64927 9.69615 9.69647 9-69679 0.69 710 9.69742 0.30385 0.30353 0.30321 0.30290 0.30258 9-952II 9.95204 9.95 198 9-95 192 9 95 185 35 34 33 32 i 1 25 24 23 22 21 w 10 ii \l 30 3i 32 33 34 9.64953 9.64978 9.65003 9.65 O29 9-65054 9.69774 9-69805 9.69837 9.69868 9.69900 0.30 226 0.30 195 0.30 163 0.30 132 0.30 100 9 95 179 9-95 173 9 95 167 9-95 160 9-95 '54 5 9 39 9.65079 9.65 104 9.65 130 9.65 155 9.65 180 9.69932 9.69963 9.69995 9.70026 9.70058 0.30068 0.30037 0.30005 0.29974 0.29942 9.95 148 9-95 HI 9-95 135 9 95 129 9 95 122 40 4 42 43 44 9-65205 9-65230 9.65 255 9.65 281 9.65306 9.70089 9.70 121 9.70152 9.70184 9.70215 0.29 911 0.29879 0.29848 0.29 816 0.29 785 9-95 "6 9 95 "0 9 95 103 9.95097 9.95090 9 .a 49 9-6533I 9-65356 9- 653*i 9.65406 9.65431 9.70247 9.70278 9.70309 9.70341 9.70372 0.29 753 0.29 722 0.29 691 0.29 659 o . 29 628 9.95084 9.95078 9.95071 9.95065 9-95059 15 14 13 12 II 60 51 5 2 53 54 9.65456 9.65 481 9.65506 9 65531 9 65 556 9.70404 9-70435 9 . 70 466 9.70498 9-70529 0.29596 0.29565 0.29 534 0.29502 0.29471 9-95052 9.95046 9-95039 9-95033 9.95027 10 I i? 12 59 E 9.65580 9 65605 9 65 630 9-65655 9.65 680 9.70560 9.70592 9.70623 9.70654 9 70 685 0.29440 o . 29 408 0.29377 0.29 346 0.29315 9 95 020 9 95014 9 95007 9.95001 9 94 995 s 4 3 2 I ~cr 9.65 705 9.70717 o . 29 283 9.94988 L. Cos. d. L. Cotg. c.d L. Tang. L. Sin. d. t Prop. Pts. 63 | LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 57 27 9 L. Sin. d. L. Tang. .d. L. Cotgr. L. Cos. d. p rop ,1 >ts. I 2 3 4 9.65 705 9.65 729 9-65754 9-65 779 9.65804 24 25 25 25 24 9.70717 9.70748 9.70779 9.70 810 9.70841 3 3 3 3 o . 29 283 0.29252 0.29 221 O.29 190 0.29 159 9.94988 9.94982 9-94975 9.94969 9.94962 6 7 6 7 6 60 5? 58 Ii .1 .2 32 I 2 /) 31 ii 1 I 9 9.65828 9-65853 9-65878 9.65902 9.65927 25 25 24 25 25 9.70873 9 . 70 904 9.70935 9.70966 9.70997 3i 3 31 3 31 0.29 127 0.29096 0.29065 0.29034 0.29003 9.94956 9.94949 9-94943 9.94936 9-94930 7 6 7 6 55 54 53 52 5i 3 4 :! 7 9 12 16 19 22 6 8 2 4 9-3 12.4 21.7 10 ii 12 13 14 9-65952 9.65976 9 . 66 ooi 9.66025 9.66050 24 23 24 25 25 9.71 028 9.71059 9.71090 9.71 121 9-71 153 3 31 31 32 31 0.28972 0.28941 0.28910 0.28879 0.28 847 9.94923 9.94917 9.94911 9.94904 9.94898 6 6 7 6 50 3 t? .8 9 3 6 8 3 24.8 27.9 15 10 \l 19 9-66075 9.66099 9.66 124 9.66 148 9-66173 24 25 24 5 2 4 9.71 184 9.71 2l| 9.71 246 9.71277 9.71308 3* 3i 3* 3* 0.288l6 0.28 785 0.28 754 0.28 723 0.28 692 9.94891 9.9488; 9.94878 9.94871 9.94865 6 7 7 6 45 44 43 42 41 i .2 3 4 i I $ 12 ;s .0 .O .0 .0 .0 ! o 20 21 22 23 24 9.66 197 9.66 221 9.66246 9.66270 9.66293 24 25 24 25 9.71339 9.71370 9.7I40I 9.7I43I 9.71 462 3 3 30 3 0.28 661 o . 28 630 0.28599 o 28 569 0.28 538 9.94858 9.94852 9.94845 9-94839 9.94832 6 7 6 7 6 40 fs 11 i 9 21 24 2; .O ' [0 r.o 25 20 3 29 9-66319 ris IM 24 25 24 4 9$ 9.7I493 9-71 5 2 4 9.7I555 9.71586 9.7I6I7 3i 3 3 3* 0.28 507 0.28476 0.28445 0.28414 0.28383 9 . 94 826 9.94819 9.94813 9.94806 9-94799 7 6 7 7 6 35 34 33 32 3i .1 .2 3 A a 2 5 7 10 5 5 .0 5 o 4 2 4 4.8 & 30 3i 32 33 34 9.66441 9.6646? 9.66489 9-66513 9-66537 24 24 24 24 25 9 71 648 9.71679 9.71 709 9.71 740 9.71 771 3* 30 3i 3i 0.28352 0.28 321 0.28 291 0.28 260 0.28 229 9-94793 9.94786 9.94780 9-94773 9.94767 7 6 7 6 so 29 28 27 26 ''I 9 12 15 17 20 22 5 .0 -5 .0 -S 12. 14 4 16.8 192 21.6 II 9 39 9.66 562 9.66586 9.66610 9.66634 9.66658 24 24 24 24 9.71 802 9-71833 9.71863 9.71894 9.71 925 3i 30 3* 3 0.28 198 0.28 167 0.28 137 0.28 1 06 0.28075 9.94760 9 94753 9-94 747 9.94740 9 94 734 7 6 7 6 25 24 23 22 21 .1 ? 23 a 40 41 42 43 44 9.66 682 9 . 66 706 9.66731 9-66755 9.66779 24 25 2 4 2 4 9.7I955 9.71 986 9.72017 9 . 72 048 9.72078 3i 3 3i 30 o . 28 045 0.28 014 0.27983 0.27952 0.27922 9.94727 9.94720 9.94714 9-94 707 9.94700 7 6 7 7 | 20 19 il 3 4 :i 7 , i 5, 9 9.2 il 3 4 J 49 9.66803 9.66827 9.66851 9 66875 9.66 899 24 24 24 24 9.72 109 9.72 140 9.72170 9 . 72 201 9.72231 3i 30 3i 30 o 27891 0.27 860 0.27 830 0.27 799 0.27 769 9.94694 9.94687 9 . 94 680 9.94674 9.94667 7 7 6 7 15 14 13 12 II .8 9 i 2 7 8.4 0.7 6 50 5i 52 53 54 9.66 922 9 . 66 946 9.66970 9.66994 9.67018 24 24 24 24 9 . 72 262 9.72293 9-72323 9.72354 9.72384 3i 30 3 30 0.27738 0.27 707 0.27677 0.27646 0.27616 9 . 94 660 9-94654 9.94647 9.94640 9-94634 7 6 7 7 6 10 I .1 .2 3 4 c i 2 2 3 -7 4 . i .S 5 0.6 1.2 1.8 2.4 3-0 I s 59 9.67042 9.67066 9.67090 9.67113 ,9.67137 24 24 23 24 9-72415 9.72445 9.72476 9.72506 9.72537 30 3i 30 3i 0-27585 0-27555 0.27524 .0.27494 0.27463 9.94627 9 . 94 620 9.94614 9.94607 9.94600 7 6 7 7 5 4 3 2 I :1 9 4 4 1 .2 1 3 3-6 4.2 4-8 5-4 60 9.67 161 9.72 567 0.27433 9-94593 L. Cos. d. L. Cotg. c.d L. Tang L. Sin. d. t ] VO] > Pts. 62 g TABLE IV. 28 1 I. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. 9.67 161 9.72567 0.27433 9-94593 00 I 2 3 4 9.67 185 9.67208 9.67232 9.67256 23 24 24 24 9.72598 9.72 628 9.72659 9.72 689 3 1 3 31 0.27402 0.27372 0.27341 0.27311 9-94587 9-9458o 9-94573 9-94567 7 7 6 .1 2 31 3- 1 6 2 30 I 9.67280 9.67303 23 24 9.72 720 9.72750 30 0.27 280 0.27 250 9-9456o 9-94553 7 55 54 3 -4 9-3 12.4 9.0 12.0 I 9.67327 9.67350 23 9.72 780 9.72811 3i 0.27 220 0.27 189 9.94546 9.94540 6 53 S2 i 5 15.0 18.0 9 9-67374 24 9.72841 31 0.27159 9-94533 7 7 21 .7 21 .0 10 9.67398 9.72872 0.27 128 9.94 526 50 .8 24 .8 24.0 ii 12 9.67421 9-67445 9.67468 24 23 9.72902 9-72932 9-72963 3 30 3* 0.27098 0.27068 0.27037 9-94 5 J 9 9-94513 9.94506 7 6 7 49 48 9 27 9 27.0 *4 9.67492 23 9-72993 30 0.27007 9-94499 7 46 39 !<> 9.675I5 9.73023 0.26977 9.94492 45 .1 2.9 16 9-67539 9-73054 3 1 0.26 946 9-94485 7 44 .2 5-8 17 18 9.67562 9.67 586 24 9.73084 9-73ii4 30 0.26916 0.26886 9-94479 9. 94 472 , 7 43 42 3 4 & 19 9.67609 23 24 9-73 144 3 0.26856 9-94465 7 4i 5 14-5 20 9.67633 9 73 175 0.26 825 9-94458 40 7 17-4 20 7 21 9.67656 9-73205 0.26 795 9-94451 39 'I 22 9.67680 9.73235 30 0.26 765 9-94445 38 Q 23.2 26 i 23 24 9.67703 9.67726 23 24 9-73265 9.73295 30 3 0-26735 0.26 705 9-94438 9-9443 1 7 7 y 25 9.67750 9.73326 0.26 674 9-94424 35 34 26 29 9.67773 9-67796 9.67820 9.67843 23 24 23 23 9.73356 9.73386 9.73416 9-73446 3 30 30 30 0.26 644 0.26 614 0.26 584 0.26554 9.94417 9.94410 9.94404 9-94397 7 7 6 7 34 33 32 .1 .2 3 A 7-2 o 6 V 6 6.9 92 30 9.67866 9-73476 0.26 524 9-94390 30 12.0 II. j 32 33 34 9.67890 9.67913 9.67936 9 67959 23 23 23 23 9-73507 9-73537 9-73567 9-73597 3 1 30 30 30 0.26 493 o . 26 463 0.26433 0.26403 9-94383 9-94376 9-94369 9-94362 7 7 7 7 1 i .9 14.4 16.8 19.2 21.6 20.7 % 9.67982 9.68006 24 9.73627 30 0.26373 0.26343 9-94355 9 94 349 6 25 24 B 9.68029 9.68052 23 23 9-73687 9-73 717 30 30 0.26313 0.26 283 9-94342 9-94335 7 7 23 22 33 2 2 39 9.68075 23 9-73 747 30 0.26253 9-94328 7 21 .2 4-4 40 9.68098 9-73777 o . 26 223 9-94321 20 -3 6.6 41 9.68 121 9.73807 3 0.26 193 9-943H 7 19 .4 8.8 42 43 9.68 144 9.68 167 23 23 9.73867 3 30 0.26 163 0.26 133 9-94307 9.94300 7 7 18 17 i II. 13.2 44 9.68 190 23 9.73897 3 0.26 103 9-94293 7 16 .7 9 9.68 213 24 9-73927 9-73957 30 0.26073 0.26043 9.94286 9.94279 7 15 9 17.6 19.8 47 9.68260 9- 73 987 3 0.26013 9-94273 13 48 9.68283 9.74017 30 0.25983 9 . 94 266 7 12 49 9.68305 9.74047 30 0.25 953 9-94259 7 II 7 6 50 9.68328 9.74077 0.25923 9-94252 10 .1 0.7 0.6 5' 9-6835I 23 9-74 107 3 0.25893 9-94245 7 9 .2 ] [-4 1.2 52 53 9-68374 9.68397 23 23 9.74137 9.74166 30 29 0.25863 0.25 834 9.94238 9.94231 7 7 7 -3 4 2.1 2.8 1.8 2.4 54 9.68420 23 23 9.74196 30 0.25 804 9.94224 7 6 5 3-5 8-8 B ii 9.68443 9.68466 9.68489 9-68 512 23 23 23 9.74226 9.74256 9 . 74 286 9.74316 30 30 30 0.25 774 0.25 744 0.25 714 0.-25 684 9.94217 9.94 2IO 9.94203 9.94196 7 7 7 5 4 3 2 .b i 9 4.2 4-9 I' 6 6-3 3- 6 5-4 59 9-68534 23 9-74345 29 0-25655 9-94 189 7 I 00 9.68557 9-74375 0.25 625 9.94 182 L. Cos. d. L. Cots. c.d. L. Tancr. L. Sin. d. t Prop. Pts. I 61 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. r 29 i 9 I 2 3 4 L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. 9-68557 9.68580 9 68603 9.68625 9.68648 23 23 22 23 3 23 22 23 23 22 23 22 23 23 23 23 29 23 22 23 32 23 22 23 22 22 23 22 23 22 22 23 22 22 22 23 33 93 33 22 3 32 33 33 33 33 32 22 22 23 32 22 22 22 22 22 23 22 22 23 9-74375 9-74405 9-74435 9.74465 9-74494 3 3 3 29 30 30 29 30 30 30 29 30 30 29 30 30 29 3<> 29 30 29 30 30 29 30 29 30 29 30 29 30 29 30 29 29 30 29 30 29 29 30 29 30 29 29 30 29 29 29 30 29 29 29 30 29 29 29 30 29 29 o . 25 625 25 595 0-25565 0-25 535 0.25 506 9.94 182 9.94175 9.94 168 9.94161 9 94154 7 7 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 8 7 7 7 7 7 8 7 7 7 7 8 7 7 7 8 7 7 7 7 8 7 7 8 7 7 7 8 7 00 3 11 .1 .2 3 4 i 9 .1 .2 3 :I :| 9 .1 .2 3 4 :! i 9 .1 .2 3 4 :i :l 9 .1 C .2 1 .3 ' .4 : i : :l \ .9 ; 30 3 6.0 9.0 12.0 I 5- 18.0 21.0 24.0 27.0 9 2.9 H n. 6 14.5 17-4 20.3 2J.2 20.1 93 i 1 6. 9 9-2 "5 '3-8 IO.I 18.4 20.7 99 2.2 to 8.8 II. O 13-2 15-4 17.6 |I 9 .8 8 7 ).8 0.7 .6 1.4 (.4 2.1 5.2 2.8 [.o 3.5 t.8 4.2 ;.6 4-9 -4 5-6 r.2 6.3 1 9 9.68 671 9 . 68 694 9.68 716 9.68739 9.68 762 9.74524 9^74554 9-74583 9.74613 9.74643 0.25 476 0.25 446 0.25417 0.25387 0.25357 9.94147 9.94 140 9-94I33 9.94 126 9.94 119 55 54 53 52 Si 10 ii 12 3 14 9.68 784 9.68807 9.68829 9.68852 9.68875 9.74673 9.74702 9-74732 9.74762 9-74791 0.25327 0.25 298 0.25 268 0.25 238 0.25 209 9.94 112 9.94105 9.94098 9.94090 9.94083 50 4 2 4 8 47 46 45 44 43 42 41 !2 !2 19 9.68897 9.68920 9.68942 9.68965 9.68987 9.74821 9-74851 9.74880 9.74910 9-74939 0.25 179 0.25 149 O.25 120 O.25 09O 0.25 06 1 9.94076 9.94069 9.94062 9-94055 9.94048 20 21 22 23 24 9.69010 9.69032 9 69055 9.69077 9.69 100 9.74969 9.74998 9.75028 9.75058 9.75087 0.25 031 O.25 002 0.24972 0.24942 0.24913 9.94041 9-94034 9.94027 9.94020 9.94012 40 39 38 1 35 34 33 32 3i % 11 29 9.69 122 9.69144 9.69167 9.69 189 9.69212 9-75 "7 9-75 H6 9-75 176 9-75205 9-75 235 0.24883 0.24854 0.24824 0.24795 0.24765 9.94005 9-93998 9-93991 9-93984 9-93977 n 31 32 33 34 9.69234 9.69256 9.69279 9.69301 9.69323 9-75 264 9-75294 9.75323 9-75353 9.75382 0-24736 o . 24 706 0.24677 0.24647 0.24618 9.93970 9.93963 9-93955 9-93948 9-93941 30 27 26 i 39 IT 41 42 43 44 9-69345 9.69368 9.69390 9.69412 9-69434 9-754" 9-75441 9-75470 9-75500 9.75529 0.24589 0.24559 0.24530 0.24500 0.24471 9-93934 9.93927 9.93920 9.93912 9-93905 25 24 23 22 21 "20" ! 9 8 \l 9-69456 9-69479 9.69501 9-69523 9.69545 9-75558 9-75588 9-756I7 9-75647 9.75676 0.24442 0.24412 0.24383 0.24353 0.24324 9-93898 9.93891 9.93 884 9.93 876 9.93869 9 8 49 9.69567 9.69589 9.69 611 9-69633 9-69655 9-75705 9-75735 9-75 764 9-75793 9-75822 0.24295 0.24 265 0.24236 0.24207 0.24 178 9.93862 9.93855 9.93847 9.93840 9-93833 15 H 13 12 II 50 5i 52 53 54 9.69677 9.69699 9.69721 9 69743 9 69765 9-75852 9.75881 9.75910 9-75939 9.75969 0.24 148 0.24 119 0.24090 0.24061 0.24031 9.93826 9.93819 9.93811 9.93804 9-93 797 10 1 I 55 56 9 59 1ST 9 69787 9.69809 9.69831 9 69853 9 69875 9-75998 9.76027 9.76056 9.76086 9.76 115 0.24002 c 23973 0.23944 0.23 914 0.23885 9-93 789 9-93 782 9-93 775 9.93 768 9 93 76o 5 4 3 2 I "0" 9 69 897 9.76 144 o 23 856 9-93753 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. t Prop. Pts. 60 TABLE IV* i 80 1 t L. Sin. d. L. Tang. .d. L. Cotg. L. Cos. d. Prop. Pis. i ' \ 9.69897 9.69919 9.69941 9 69963 9.69984 23 33 33 21 23 9.76 144 9.76 173 9 . 76 202 9.76231 9.76 26l 39 29 29 30 29 39 29 29 9 29 39 29 29 39 29 29 30 29 39 28 39 39 29 29 39 29 29 29 29 39 29 29 28 29 29 29 29 29 28 29 29 29 29 28 29 29 29 28 29 29 28 29 29 29 28 29 28 29 29 28 0.23 856 0.23 827 o 23 798 0.23 769 o 23 739 9 93 753 9 93 746 9 93 738 9-9373" 9 93 724 7 8 7 7 7 8 7 7 8 7 7 8 7 8 7 7 8 7 7 8 7 8 7 7 8 7 8 7 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 8 7 8 7 8 7 00 3 11 3< :i I 3 9 .4 12 :i !I its .9 27 .1 .2 3 :I i 9 .1 .2 3 .4 i :J 9 .1 .2 3 4 :i i 9 .1 ( .2 3 ' 4 , 5 * .6 - :2 . 9 > o 2.9 o j.8 .0 8.7 1 .0 ii. 6 .0 14.5 .0 17.4 .0 20.3 .0 23.2 .0 20.1 38 2.8 s- 6 1-4 II. 2 14 o 16.8 19.6 22.4 25.2 sa 2.2 ft 8.8 II. 13 2 "5-4 17.6 I 9 .8 at 2.1 4 2 63 8.4 10 5 12.6 2 18.9 8 7 5.8 0.7 [.6 1.4 2.4 2.1 J.2 2.8 *o 3.5 ^.8 4-2 C.6 49 3-4 5-6 7.2 6.3 i 2 9 10 ii 12 13 H \l 11 19 21 22 23 24 9 . 70 006 9 . 70 028 9 . 70 050 9.70072 9.70093 33 32 , 22 21 23 33 33 91 32 33 31 22 81 23 32 21 23 21 22 21 22 31 33 21 33 21 33 81 33 31 31 23 21 31 33 31 31 21 22 21 21 21 22 21 21 21 21 21 23 21 XX 21 21 21 21 9.76290 9.76319 9.76348 9.76377 9.76406 0.23 710 0.23 68 1 0.23652 0.23623 o 23 594 9 93717 9 93 709 9.93702 9-93695 9-93687 9.93680 9 93673 9 93 66 J 9-93658 9 93 650 55 54 53 52 5" 60" 3 2 9.70115 9.70137 9.70159 9.70 180 9 . 70 202 9.76435 9.76464 9.76493 9.76522 9-7655I 0.23565 0.23536 0.23 507 0.23478 0.23449 9.70224 9.70245 9.70 267 9.70288 9.70310 9.76580 9.76609 9.76639 9.76668 9.76697 o . 23 420 0.23391 0.23361 0.23332 0.23303 9-93643 9 93636 9 93 628 9-9362I 9.93614 45 44 43 42 4" 9-70332 9.70352 9.70375 9.70396 9.70418 9.76725 9-76 754 9-76783 9.76 812 9.76841 0.23275 0.23 246 0.23217 0.23 188 0.23 159 9.93606 9-93599 9-9359" 9 93 584 9 93577 40 3 1 35 34 33 32 i I s 2 29 31 32 33 Jl_ $ 12 39 4i 42 43 1 44 9.70439 9.70461 9.70482 9.70504 9-70525 9.76870 9.76899 9.76928 9.76957 9.76986 0.23 130 0.23 101 0.23072 0.23043 0.23 014 9-93569 9 93562 9-93554 9-93547 9 93539 9-70547 9-70568 9.70590 9.70611 9-70633 9.77015 9.77044 9.77073 9.77 101 9.77130 0.22985 0.22 956 0.22 927 . 22 899 0.22 870 9-93532 9-935 2 5 9-93 5"7 9-93510 9-93 502 9.70654 9.70675 9.70697 9.70718 9-70739 9.77159 9.77188 9.77217 9.77246 9.77274 0.22 841 0.22 8l2 0.22 783 0.22 754 0.22 726 9-93495 9 93487 9-9348o 9-93472 9 93465 25 24 23 22 21 w 19 ii 9.70761 9.70782 9.70803 9.70824 9 . 70 846 9.77303 9-77332 9.77361 9-77390 9.77418 0.22 697 . 22 668 O.22 639 O.22 6lO 0.22 582 9 93457 9 93450 9 93442 9 93435 9 93427 4 46 % 49 10" 5i 52 53 J 9 5 5 I I 9.70 867 9.70888 9.70909 9.70931 9.70952 9-77447 9.77476 9- 7755 9-77533 9.77562 0.22 553 0.22 524 0.22495 0.22 467 0.22438 9.93420 9 93412 9 93405 9 93397 9 93390 15 14 "3 12 II lo~ 1 5 4 3 2 I ~0" 9-70973 9.70994 9.71 015 9.71036 9.71058 9-77591 9.77619 9-77648 9.77677 9.77706 . 22 409 22 381 0.22352 0.22 323 22 294 9 93382 9 93 375 9 93 367 9-93360 9 93 352 9.71 079 9.71 loo 9.71 121 9.71 142 9 7i I 6 3 9-77734 9-77 763 9.77791 9.77820 9.77849 0.22 266 0.22237 O.22 2O9 0.22 l80 0.22 151 9-93344 9 93337 9 933 2 9 9 93322 9 93314 9 71 184 9.77877 0.22 123 9 93 37 L. Cos. d. L. Cots c.d, L. Tang L. Sin. d. / Prop. Pts. 59 1 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 6Y 31 / L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. W CQ 57 56 Prop. Pts. I 2 3 4 9.71 184 9.71 205 9.71 226 9.71247 9.71 268 21 21 21 21 21 21 21 21 21 20 21 21 21 21 21 21 20 21 21 21 2O 21 si 91 2O 81 21 20 21 21 20 21 20 21 20 21 20 21 21 20 20 21 20 21 20 21 2O 2O 21 20 20 21 20 20 21 30 2O M 90 30 9.77877 9.77906 9-7793? 9.77963 9-77992 29 29 28 29 28 0.22 123 0.22094 0.22065 0.22037 0.22008 9-93307 9.93299 9-93 291 9.93284 9.93276 8 8 7 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 8 8 7 8 8 8 7 8 g 8 ' .1 .2 3 4 ii 9 .1 .2 3 :! :J 9 .1 .2 3 4 ii 9 .1 .2 .3 4 ii ii 9 .1 < .2 ] 3 - .4 : :i ; :i i 9 ; 39 S* ,!:i H-5 17 4 20.3 li.l 38 2.8 5.6 8.4 II. 2 14-0 16 8 19.6 22.4 25.2 ax 2.1 X 8-4 S:i Z 18.9 20 2.O 4-0 6.0 8.0 IO.O 12. 14-0 16.0 180 8 7 ).8 0.7 1.6 1.4 J.4 2.1 J.2 2.8 ^o 3-5 k8 4-2 ;.6 4.0 )-4 5- 6 r 2 63 1 I 9 9.71289 9.71310 9-7i33i 9-71352 9 7i 373 9 . 78 020 9.78049 9.78077 9.78 106 9.78135 29 28 29 29 28 0.21 980 0.21 951 0.21 923 0.21 894 0.21 865 9.93269 9.93261 9-93253 9-93246 9-93238 55 54 53 5 2 51 "50" 49 48 8 10 ii 12 '3 14 9.71393 9.71414 9-7i 435 9-71 45 6 9.71 477 9-78163 9.78192 9 . 78 220 9.78249 9.78277 29 28 29 28 29 28 29 28 28 29 38 29 98 29 38 28 29 28 29 28 28 29 28 28 29 28 28 29 28 28 28 29 28 28 28 29 28 28 28 28 29 28 28 28 28 28 39 28 28 28 0.21 837 0.21 808 0.21 780 0.21 751 0.21 723 9.93230 9.93223 9-932I5 9-93207 9-93200 11 II 19 9.71 498 9.7i 5*9 9-71 539 9.71 560 9.71 581 9.78306 9-78334 9-78363 9.78391 9.78419 0.21 694 0.21 666 0.21 637 O.2I 609 0.21 581 9-93 192 9-93 l8 4 9-93 177 9-93 169 9.93 161 45 44 43 42 * 9 11 20 21 22 23 24 9.71 602 9.71 622 9-71 643 9.71664 9.71 685 9.78448 9-78476 9-78505 9.78533 9-78562 0.21 552 O.2I 524 0.21 495 0.21 467 0.21 438 9 93 154 9-93 H6 9-93 138 9 93 131 9 93 123 3 27 28 3 3i 32 33 34 9.71 705 9.71 726 9.71 747 9.71 767 9.71 788 9.78590 9.78618 9.78647 9.78675 9.78 704 0.21 410 O.2I 382 0-21353 0.21325 0.21 296 9-93 "5 9.93 1 08 9.93 loo 9.93092 9.93084 35 34 33 32 * % 2 25 24 23 22 21 ~w 19 18 \l 9.71 809 9.71 829 9-71850 9.71 870 9.71 891 9-78732 9.78760 9-78789 9-73817 9-78845 0.21 268 0.21 240 O.2I 211 O.2I 183 0.21 155 9.93077 9.93069 9.93061 9.93053 9.93046 9 II 39 9.71 911 9.71932 9.71952 9-7i 973 9.71994 9-78874 9.78902 9.78930 9-78959 9-78987 0.21 126 0.21 098 0.21 07O O.2I 041 0.21 013 9-93038 9.93030 9.93022 9.93014 9.93007 40 4i 42 43 44 9.72014 9.72034 9.72055 9.72075 9.72096 9.79015 9-79043 9.79072 9.79 loo 9.79 128 0.20985 0.20957 0.20928 O.2O 9OO 0.2072 9.92999 9.92991 9.92983 9-92976 9 . 92 968 S S 49 9.72 116 9.72 137 9.72 157 9.72177 9.72 198 9-79I56 9-79 185 9-792I3 9.79241 9.79269 o . 20 844 0.20815 0.20 787 0.20759 o 20 731 9 . 92 960 9.92952 9.92944 9.92936 9.92929 15 14 J 3 12 II ICT z 50 5i 52 53 54 9.72 218 9.72238 9.72259 9.72279 9-72299 9-79297 9.79326 9-79354 9.79382 9.79410 0.20 703 O.2O 674 . 2O 646 O.206l8 0.20 590 9.92921 9.92913 9.92905 9.92 897 9.92 889 55 56 9 59 9.72320 9.72340 9.72360 9.72381 9.72401 9-79438 9-79466 9-79495 9.795 2 3 9-79551 0.20 562 0.20534 0.20 505 0.20477 0.20449 9.92881 9.92874 9.92866 9-92858 9.92 850 5 4 3 2 I ~0" 60 9.72421 9-79579 O.2O 421 9.92842 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. / Prop. Pts. 58 TABLE IV. 32 9 L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. p roi ). ] Pts. I 2 3 4 9.72421 9.72441 9.72461 9.72482 9.72502 20 20 21 2O 20 9-79579 9.79607 9.79635 9.79663 9.79691 28 28 28 28 98 0.20421 0.20393 0.20 365 0.20337 0.20309 9.92842 9.92834 9.92 826 9.92818 9.92 810 8 8 8 8 GO 59 58 % .1 2 z 2 9 38 2.8 <: 6 i I 9 9.72522 9-72542 9-72562 9.72582 9 . 72 602 20 20 2O 2O 2O 9-79 719 9-79 747 9-79 776 9.79804 9-79832 28 29 28 28 28 O.2O 28l 0.20253 0.20224 0.20 196 0.20 168 9.92 803 9-92 795 9.92 787 9-92 779 9-92 77i 8 8 8 8 g 55 54 53 52 3 4 7 14 17 2C 5 -4 .T, :> - u 8.4 II. 2 14-0 16.8 19.6 12 13 H 9.72622 9.72643 9-72663 9.72683 9-72703 21 2O 20 20 2O 9 . 79 860 9.79888 9.79916 9-79944 9.79972 28 28 28 28 28 0.20 140 0.20 112 0.20084 0.20056 0.20028 9-92 763 9.92755 9.92747 9-92 739 9.92 731 8 8 8 8 3 1 % .8 9 % .2 a 22.4 25.2 7 19 9.72723 9-7? 743 9.72763 9.72783 9.72803 20 20 20 20 2O 9.80000 9.80028 9 . 80 056 9.80084 9.80 112 28 28 28 28 28 0.20000 0.19972 0.19944 O.I9 916 0.19888 9.92 723 9.92 715 9 92 707 9.92699 9.92691 8 8 8 8 3 45 44 43 42 .1 .2 3 4 1 2 i 1C 7 .4 iis 20 21 22 23 24 9-72823 9-72843 9.72863 9-72883 9.72902 2O 20 20 19 9.80 140 9.80168 9-80 195 9.80223 9.80251 28 27 28 28 28 0.19 860 0.19832 0.19805 0.19777 o. 19 749 9.92 683 9.92675 9.92667 9.92659 9.92651 8 8 8 8 3 40 1 1 9 I* 2] 2; *-9 1.6 t-3 27 28 29 9.72922 9-72942 9.72962 o . 72 982 9.73002 20 20 2O 2O 9.80279 9.80307 9.80335 9.80363 9.80391 28 28 28 28 28 0.19 721 0.19 693 0.19 665 0.19637 0.19 609 9-92643 9-92635 9.92627 9.92 619 9.92 611 8 8 8 8 3 35 34 33 32 .1 .2 3 4 -. 2 4 ( J II !.I I 2 H 30 2.0 4.0 6.0 8.0 30 32 33 34 9.73022 9.73041 9.73061 9.73081 9.73101 20 20 20 9.80419 9.80447 9.80474 9 . 80 502 9.80530 28 27 28 28 28 0.19581 0.19553 0.19 526 0.19498 0.19470 9.92603 9-92 595 9.92587 9-92579 9.92571 8 8 8 8 3 30 29 28 % 9 1C i: i, K I. 11 1-7 3.8 5-9 IO.O 12. 14.0 16.0 18.0 9 9 39 9.73121 9.73140 9-73 160 9.73180 9.73200 J 9 20 20 20 10 9.80558 9.80586 9.80614 9 . 80 642 9.80669 28 28 28 27 28 0.19 442 0.19414 0.19 386 0.19358 0.19331 9-92563 9-92555 9.92 546 9-92 538 9-92530 8 9 8 8 3 25 24 23 22 21 .2 [ . C 9 40 41 42 43 44 9.73219 9.73239 9-73259 9.73278 9.73298 20 20 20 9.80697 9-80725 ,9.80753 9.80 781 9.80808 28 28 28 27 28 0.19303 0.19275 0.19247 0.19 219 0.19 192 9.92522 9.92514 9-92506 9.92498 9-92490 8 8 8 8 3 20 19 18 17 16 3 7 i i \l ?-5 i.4 3-3 1:1 4-5 54 6-3 49 9-733I8 9-73337 9-73357 9-73377 9 73396 19 20 20 '9 9.80836 9.80864 9.80892 9.80919 9.80947 28 28 27 28 28 o. 19 164 0.19 136 0.19 108 0.19081 0.19053 9.92482 9-92473 9.92465 9-92457 9.92449 9 8 8 8 3 15 14 13 12 II .8 9 i i 5-2 7-1 8 21 7 W 53 54 9-734i6 9-73435 9-73455 9-73474 9-73494 20 20 9.80975 9.81 003 9.81 030 9.81 058 9.81 086 28 27 28 28 0.19025 0.18997 o. 18 970 0.18 942 0.18 914 9.92441 9-92433 9.92425 9.92416 9 . 92 408 8 8 9 8 3 10 I .1 .2 3 4 < t D.8 1.6 2.4 3-2 *-s o-7 2.1 2.8 3-5 59 9 73513 9-73533 9 73552 9 73572 9 73591 2O 20 19 9.81 113 9.81 141 9.81 169 9.81 196 9!8i 224 28 28 27 28 28 0.18887 0.18859 0.18831 0.18804 o. 18 776 9.92 400 9.92 392 9.92384 9-92367 8 8 8 9 3 s 4 3 2 I 9 t ( \.t> \\ 7-2 4.2 4-9 1.1 !()() 9 73 6n 9.81 252 0.18 748 9-92359 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. f 1 To 1>. Pte. 57 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 33 / L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pte. I 2 3 4 9.73611 9.73630 9-73 6 50 9.73 669 9.73689 19 20 *9 30 19 '9 20 19 J 9 20 19 19 2O 19 *9 20 9 J 9 19 19 20 19 *9 J 9 J 9 20 9 9 >9 9 19 19 J 9 *9 X 9 19 19 9 i9 19 *9 *9 19 i9 *9 19 9 19 18 19 i> 19 19 19 18 19 19 19 18 *9 9.81 252 9.81 279 9.81 307 9-8i 335 9.81 362 7 28 28 27 28 28 27 28 27 28 0.18 748 0.18 721 0.18 693 0.18665 0.18638 9 92359 9-9235 1 9-92343 9-92335 9.92326 8 8 8 9 8 8 8 9 8 8 8 9 8 8 9 . 8 8 8 9 8 & 9 8 8 9 8 8 9 8 8 9 8 8 9 8 9 8 8 9 8 9 8 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 60 3 11 a .1 2 I I .4 II 5 H .6 16 7 19 .8 22 9 25 .1 .2 3 :! i 9 .1 .i .3 .4 ;i i 9 .1 2 3 :1 i 9 .1 C .2 1 3 2 4 2 :l 1 7 * :5 ^ 8 37 .8 2.7 .6 c.4 .4 8.1 .2 10.8 o J 3.5 .8 16.2 .6 18.9 .4 21.6 .2 24.3 90 2.O 1 6.0 8.0 IO.O 12. 14-0 IO.O 18.0 9 5:1 M 95 H.4 J3.3 15.2 17.1 it 1.8 36 5-4 7.2 9.o 10.8 12 6 14.4 16.2 9 >-9 0.8 .8 1.6 .7 2. 4 .6 3.2 5 4.o 4 4.8 3 5-6 .2 6.4 .1 72 I I 9 9.73708 9.73727 9-73747 9.73766 9-73785 9-8i 390 9.81418 9.81445 9 8i473 9.81 500 0.18 610 0.18 582 0.18555 0.18 527 0.18 500 9.92318 9.92310 9.92302 9.92293 9-92285 55 54 53 52 5i 10 ii 12 13 14 9.73805 9.73824 9.73843 9.73863 9.73882 9.81 528 9 81 556 9.81 583 9.81 611 9.81 638 28 27 28 27 28 0.18472 0.18444 0.18417 0.18389 0.18 362 9.92277 9.92269 9.92 260 9 92252 9.92244 50 3 3 15 10 \l 19 ,20 21 22 23 24 9.73901 9.73921 9-73940 9 73959 9.73978 9.81 666 9.81 693 9.81 721 9.81 748 9.81 776 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 27 28 27 28 27 27 28 27 28 27 27 28 27 27 28 27 27 28 27 27 27 28 27 27 27 28 0.18334 0.18307 0.18 279 0.18252 0.18 224 9-92235 9.92227 9.92 219 9.92 211 9 . 92 202 45 44 43 42 41 40" i 9-73997 9.74017 9.74036 9-74055 9.74074 9.81 803 9.81 831 9.81858 9.81 886 9.81913 0.18 197 0.18 169 0.18 142 0.18 114 0.18087 9.92 I 9 4 9.92 186 9-92 177 9.92 169 9.92 161 3 i 27 28 29 9-74093 9.74"3 9-74132 9-74 IS 1 9.74170 9.81941 9.81 968 9.81 996 9.82023 9.82051 0.18 059 0.18032 0.18004 0.17977 0.17949 9-92 152 9.92 144 9.92 136 9.92 127 9.92119 35 34 33 32 3 1 30 29 28 27 26 30 3 1 32 33 34 9.74189 9.74208 9.74227 9.74246 9.74265 9.82078 9.82 106 9.82 133 9.82 161 9.82 1 88 0.17922 0.17894 0.17867 0.17839 0.17 812 9.92 in 9.92 102 9 . 92 094 9 . 92 086 9.92077 9 i? 39 9.74284 9 -7433 9.74322 9-74341 9.74360 9.82 215 9.82243 9.82 270 9.82 298 9.82325 0.17785 0.17757 0.17730 0.17 702 0.17675 9.92069 9.92060 9-92052 9.92044 9-92035 25 24 23 22 21 W il 40 4i 42 43 44 9-74379 9-74398 9.744I7 9.74436 9-74455 9.82352 9.82380 9.82407 9-82435 9 . 82 462 0.17 648 0.17 620 0.17593 0.17565 0.17538 9.92027 9.92 018 9.92010 9 . 92 002 9-91 993 3 s 49 9-74474 9-74493 9-74512 9-74531 9-74549 9.82489 9.82517 9.82544 9.82571 9.82599 0.17511 0.17483 0.17456 0.17429 0.17401 9.91 985 9.91 976 9.91 968 9-9i 959 9-9i 95i 15 H 13 12 II 50 5i 52 53 54 9.74568 9-74587 9.74606 9.74625 9.74644 9.82 626 9.82653 9.82681 9.82 708 9-82 735 0.17374 0.17347 0.17319 0.17 292 0.17265 9.91 942 9-9i 934 9.91 925 9.91 917 9.91 908 10 I I 5 4 3 i 55 56 H 59 9.74662 9.74681 9.74700 9.74719 9-74737 9.82 762 9.82 790 9.82 817 9.82 844 9.82871 0.17238 0.17 210 0.17 183 O.I7I56 O.I7 129 9.91 900 9.91 891 9-91 f3 9.91 874 9.91 866 00 9-74756 9.82899 0.17 ioi 9-91 857 L. Cos. d. L. Cotg. c. d. L. Tang. L. Sin. d. / Prop. Pts. 56 TABLE IV. 34 / L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Pro] 0. Pte. I 2 3 4 9-74756 9-74775 9-74794 9.74812 9-74831 X 9 X 9 18 X 9 9.82899 9.82 926 9-82953 9.82980 9.83008 27 27 27 28 27 0.17 101 0.17074 0.17047 0.17 020 0.16992 9.91 857 9.91 849 9.91 840 9.91 832 9.91 823 8 9 8 9 8 58 11 g .1 2 2 e 8 .8 6 27 2.7 r A i I 9 9.74850 J 9.74868 9-74887 9.74906 9.74924 18 X 9 X 9 18 to 9-83035 9.83062 9-83089 9-83117 9-83 144 27 27 28 27 27 0.16 965 0.16938 0.16911 0.16883 0.16856 9.91815 9.91 806 9.91 798 9.91 789 9.91 781 9 8 9 8 55 54 53 5i 3 8 .4 ii 5 14 .6 16 .7 19 4 .2 .0 .8 .6 11 10.8 13-5 16.2 18.9 10 ii 12 13 9-74943 9.74961 9.74980 9-74999 9.75017 18 X 9 x8 X 9 9-83 171 9.83 198 9-83225 9-83252 9.83280 27 27 27 28 27 0.16 829 0.16 802 0.16775 0.16 748 0.16 720 9.91 772 9-91 763 9-9i 755 9.91 746 9.91 738 9 9 8 9 8 50 8 3 .8 22 9 25 4 .2 i 21.6 24.3 (6 19 9-75036 9-75054 9.75073 9.75091 9 75 " 18 X 9 18 X 9 18 9-83307 9-83334 9-83361 9.83388 9-83415 27 27 27 0.16693 0.16666 0.16639 0.16 612 0.16 585 9.91 729 9.91 720 9.91 712 9.91 703 9.91 695 9 8 9 8 45 44 43 42 41 .1 .2 3 4 1 2 c i i 1C i; >..6 >-4 '6 20 21 22 23 24 9-75 128 9-75 147 9-75 165 9-75 184 9.75202 X 9 18 X 9 18 9.83442 9.83470 9-83497 9-83524 9-83551 28 2? 27 27 0.16 558 o. 16 530 0.16503 0.16476 o 16 449 9.91 686 9.91 677 9.91 669 9.91 660 9.91 651 9 9 8 9 9 40" P H .0 9 ij 2< 2; >- 6 $.2 >.8 M 29 9-75221 9.75239 9-75258 9-75276 9.75294 18 X 9 18 18 9.83578 9.83605 9-83659 9.83686 27 27 27 27 27 0.16422 0.16395 0.16368 0.16341 0.16 314 9.91 643 9.91 634 9.91 625 9.91617 9.91 608 9 9 8 9 35 34 33 32 .1 .2 -3 A ' 1 [9 7 6 80 32 33 34 9-753I3 9-75.331 9-75350 9-75368 9 75386 18 X 9 18 18 ig 9-83 7i3 9.83 740 9-83 768 9-83 795 9.83822 27 28 2 7 27 27 0.16287 0.16260 0.16 232 0.16 205 0.16 178 9-9i 599 9-91 59i 9.91582 9.91573 9 -9i S 6 ? 9 8 9 9 8 30 27 26 9 ( I i; i i >-5 1-4 5-3 5-2 7-1 P 3^ 39 9 75405 9-75423 9 75441 9-75459 9 75478 18 18 18 X 9 18 9-83849 9-83876 9-83903 9-83930 9.83957 27 27 2 7 27 0.16 151 0.16 124 0.16097 0.16 070 o. 16 043 9-91 556 91 547 9.91 530 9.91 521 9 9 8 9 25 24 23 22 21 .1 .2 IS [.8 z.6 40 42 43 44 9-75496 9-755I4 9-75533 9-75551 9-75 569 18 18 18 18 9-83984 9.84 on 9.84038 9.84065 9 . 84 092 27 27 27 27 o. 16 016 0.15989 0.15 962 0.15935 0.15 908 9.91512 9.91 504 9-91 495 9.91 486 9.91 477 9 8 9 9 9 3 20 19 18 17 16 -3 4 i K I 5-4 7-2 ?-o D.8 2.6 47 48 49 9-75587 9-75605 9 75 624 9.75642 9.75660 It 18 18 18 9.84 119 9 . 84 146 9.84173 9 . 84 200 9.84227 7 27 27 27 0.15881 0.15854 0.15 827 0.15 800 0.15 773 9.91 469 9.91 460 9-91 451 9.91 442 9-91433 9 9 9 9 8 15 14 13 12 II 9 9 8 50 5* 52 53 54 9.75696 9-757H 9-75733 9-75751 18 18 X 9 18 18 9.84254 9.84280 9.84307 9.84334 9.84361 27 27 27 0.15 746 o. 15 720 0.15 693 o.'is 666 0.15639 9.91 425 9.91 416 9.91407 9.91 398 9.91 389 9 9 9 9 8 10 I .1 C .2 1 3 '< .4 : 5 4 > 9 .8 5-7 \-6 [5 0.8 i i 6 2 4 3-3 11 56 59 9.75769 9.75787 9.75805 9-75823 9-75841 z8 18 x8 18 18 9.84388 9.84415 9.84442 9.84469 9.84496 27 27 27 0.15 612 0-15585 0.15558 O.I553I 0.15 504 9.91 381 9.91 372 9-91 363 9-91 354 9-91 345 9 9 9 9 5 4 3 2 '.7 * :5 1 -4 >-3 r.2 ;.i 4 .S I 6 6.4 7.2 60 9-75 859 9.84523 0.15477 9-9i S3 6 9 L. Cos. d. L. Cotg. c.d. L. Tancr. L. Sin. d. t Pro > Pis. 55 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 85 t L. Sin. <1. L. Tang. c.d. L. Cotg. L. Cos. d. Proj .1 ?ts. I 2 3 4 9-75859 9.75877 9.75895 9.75913 9-75931 18 It 18 18 18 9-84523 9-8455o 9.84576 9 . 84 603 9.84630 27 26 27 27 27 o.i5477 0.15450 0.15424 0.15397 0.15370 9-91 S3 6 9.91 328 9.91319 9.91 310 9.91 301 8 9 9 9 9 60 5 2 58 H 2 .1 2 .2 5 7 -7 4 26 2.6 C.2 I I 9 9-75949 9.75967 9.75985 9/76003 9.76021 18 18 18 18 18 9-84657 9.84684 9.84711 9.84738 9.84764 27 27 27 26 27 0-15343 0.15 316 0.15 289 0.15 262 0.15 236 9.91 292 9.91 283 9.91 274 9.91 266 9.91 257 9 9 8 9 9 55 54 53 52 5i 3 8 .4 10 : 7 Is .1 .8 5 .2 9 7.8 10.4 I 3 .0 i.6 18.2 10 ii 12 13 14 9.76039 9.76057 9.76075 9.76093 9 . 76 1 1 1 18 18 18 18 18 9.84791 9.84818 9.84845 9 . 84 872 9.84899 27 27 27 27 26 0.15 209 0.15 182 0-15 155 0.15 128 0.15 101 9.91 248 9.91 239 9.91 230 9.91 221 9.91 212 9 9 9 9 50 49 48 47 46 .8 21 9 24 .6 3 i 20.8 23 4 8 11 \l 19 9.76129 9.76146 9.76164 9.76 182 9 . 76 200 7 18 18 18 18 9.84925 9.84952 9.84979 9.85 006 9-85033 27 27 27 27 26 0-15075 0.15 048 0.15 021 0.14994 0.14967 9.91 203 9.91 194 9.91 185 9.91 176 9.91 167 9 9 9 9 45 44 43 42 4i .1 .2 3 4 \ 1 t < .8 1-6 5-4 r-2 ) o )O 20 21 22 23 24 9.76 218 9.76236 9-76253 9.76271 9.76289 18 7 18 18 18 9-85059 9.85086 9-85 "3 9.85 140 9.85 166 27 27 27 26 0.14941 O.I49I4 0.14887 0.14860 0.14834 9.91 158 9.91 149 9.91 141 9.91 I 3 2 9.91 123 9 8 9 9 40 li II i 9 i: i. K .5 1.6 11 11 27 28 29 9.76307 9.76324 9.76342 9.76360 9.76378 7 18 18 18 9-85 193 9.85 220 9-85 247 9.85273 9.85300 27 27 26 27 0.14807 O.I4 780 0-14753 0.14727 0.14 700 9.91 114 9.91 IO5 9 91 096 9.91087 9.91 078 9 9 9 9 35 34 33 32 3i .1 .2 3 A t7 t-7 J-4 Ij 30 3i 32 33 34 9-76395 9.76413 9.76431 9.76448 9 . 76 466 18 18 17 18 18 9-85327 9-85354 9-85380 9.85407 9-85434 27 26 27 27 26 O.I4 673 0.14 646 0.14620 0.14593 0.14566 9.91069 9.91 060 9.91051 9.91042 9 91 033 9 9 9 9 30 2 2 | 11 1 i 9 i i i i *-5 D.2 1 1 36 5-3 9 9 39 9.76484 9.76501 9 76519 9.76537 9.76554 17 18 18 17 18 9.85460 9-85487 9-855I4 9.85540 9-85 567 27 27 26 27 0.14540 O.I45I3 0.14486 0.14460 0-14433 9.91023 9.91014 9.91 005 9.90996 9.90987 9 9 9 9 25 24 23 22 21 .1 .2 zo I.O 2.O 40 41 42 43 44 9.76572 9.76590 9.76607 9-76625 9 . 76 642 18 7 18 *7 x8 9 85594 9.85 620 9.85647 9.85674 9.85 700 26 27 27 26 0.14406 0.14380 0-14353 o. 14 326 0.14300 9.90978 9.90969 9.90960 9.90951 9.90942 9 9 9 9 20 19 ii 3 -4 :i .7 30 4.0 |-o 5.0 7.0 9 9 49 9.76 660 9.76677 9-76695 9.76712 9.76730 17 18 7 18 9.85 727 9-85 754 9.85 780 9-85807 9-85834 27 27 26 27 27 ofi 0.14273 0.14246 0.14220 0.14193 0.14 166 9 90933 9.90924 9.90915 9.90906 9 . 90 896 9 9 9 9 xo 15 14 13 12 II .8 9 9 8.0 30 8 50 5i 52 53 _5!_ 9.76747 9.76765 9.76782 9.76800 9.76817 18 17 18 7 18 9.85860 9.85887 9-859I3 9.85940 9.85967 27 26 2 7 2 7 0.14 140 0.14113 0.14 087 0.14060 0.14033 9.90887 9.90878 9 . 90 869 9 . 90 860 9 . 90 85 1 9 9 9 9 9 10 I .1 C .2 1 3 2 4 2 '} A >-9 'I r-5 0.8 1.6 2.4 32 4 'S 55 56 12 59 9-76835 9-76852 9.76870 9.76887 9.76904 17 18 7 17 18 9 85993 9 . 86 020 9.86046 9.86073 9.86 loo 27 26 27 27 0.14007 0.13980 0-13954 0.13927 o 13 900 9.90842 9.90832 9.90823 9.90814 9 90805 9 xo 9 9 9 5 4 3 2 I :? i .8 I . 9 i 4 3 .2 .1 4-8 56 6.4 7 2 GO 9.76922 9.86 126 o 13874 9 90796 9 L. Cos. d. L. Cotgr. c.d. L. Tang. L. Sin. d. t Pro] > Pfs. 54 66 TABLE IV. 36 1 L. Sin. (1. L. Tang. C.<1. L. Cot!?. L. Cos. <1. Pro] >. Pts. I 2 3 4 9.76922 9 76939 9-76957 9.76974 9.76991 17 18 J7 7 18 9.86 126 9-86 153 9.86179 9.86206 9.86232 27 26 27 26 27 0.13874 0.13847 0.13 821 0.13794 0.13 768 9.90 796 9.90787 9.90777 9.90768 9.90 759 9 10 9 9 GO 3 11 3 I 2 2 C 7 7 | 26 2.6 IT 2 I I 9 9.77009 9.77026 9-77043 9.77061 9.77078 17 17 18 7 17 9.86 259 9.86285 9.86312 9.86338 9.86365 26 2 7 96 27 27 o. 13 741 0.13 7i5 0.13688 0.13 662 0.13635 9.90750 9.90 741 9.90731 9.90 722 9.90 713 9 9 x> 9 9 55 54 53 52 5i 3 * .4 ic * \l 7 iS .1 .8 5 .2 9 fl 10.4 ill 10 ii 12 13 '4 9-77095 9.772 9.77130 9-77 H7 9.77164 7 18 7 *7 *7 9 86392 9.86418 9.86445 9.86471 9.86498 26 27 26 27 26 0.13608 0.13582 0.13555 0.13529 0.13 502 9.90704 9-90694 9.90685 9.90 676 9.90667 9 10 9 9 9 50 3 % .8 21 .9 24 .6 3 20.8 23.4 18 III 17 18 19 9.77181 9.77199 9.77216 9.77233 9-77250 18 17 7 17 18 9-86524 9.86551 9.86577 9.86603 9.86630 27 26 26 27 2<> 0.13476 0.13449 0.13423 0.13397 0.13370 9.90657 9.90648 9-90639 9.90630 9 . 90 620 9 9 9 10 45 44 43 42 41 .1 .2 3 4 I I >, i c .8 1-6 !-4 r.2 ).o 20 21 22 23 24 9.77268 9.77285 9.77302 9-773I9 9-77336 17 7 17 7 17 9.86656 9.86683 9.86709 9-86 736 9.86 762 27 26 27 26 0.13344 0.13317 0.13291 0.13264 0.13 238 9.90611 9.90602 9.90592 9-90583 9-90574 9 9 xo 9 9 ~w fs 11 .0 :l 9 K Ii 1^ I( ).o j.6 M ) 2 2 3 29 9-77353 9-77370 9-77387 9.77405 9.77422 7 17 18 17 17 9.86789 9.86815 9.86842 9.86868 9.86894 26 27 26 26 0.13 211 0.13 185 0.13 158 O.I3I32 0.13 106 9-90565 9.90555 9.90546 9-90537 9.90527 9 10 9 9 xo 35 34 33 32 3i .1 .2 3 ] 17 [-7 5-4 ij 30 3i 32 ! 33 1 34 9-77439 9-77456 9-77473 9.77490 9-77507 7 7 7 '7 17 9.86921 9.86947 9.86974 9.87000 9.87027 26 27 26 2 7 26 0.13079 0.13053 0.13026 0.13000 0.12973 9.90518 9.90509 9-90499 9.90490 9.90480 9 9 10 9 xo 80 1 :! .9 ! 1C i ; i J-5 5.2 'i j.6 53 !$ 12 39 9-77524 9-77541 9-77558 9-77575 9-77592 '7 7 7 17 17 9-87053 9.87079 9.87 106 9-87 132 9.87 158 26 27 26 26 0.12947 0.12 921 0.12894 0.12868 O.I2 842 9.90471 9.90462 9.90452 9-90443 9-90434 9 9 xo 9 9 25 24 23 22 21 .1 .2 16 [.6 i. 2 40 4i 42 43 44 9 . 77 609 9.77626 9-77643 9 77660 9.77677 17 17 i? 17 17 9-87 185 9.87211 9.87238 9.87264 9.87290 26 2 7 26 26 0.12 815 O.I2 789 O.I2 762 0.12 736 0.12 710 9.90424 9.90415 9.90405 9.90396 9.90386 9 xo 9 xo 20 19 il 3 4 :I .7 * i ( f 3 ;:o- >.6 1.2 45 46 4 4 I 49 9.77694 9.77711 9.77728 9-77 744 9.77761 17 17 16 17 9.87317 9-87343 9.87369 9.87396 9.87422 26 26 2 7 26 26 0.12 683 0.12 657 0.12 631 0.12 604 0.12 578 9.90377 9.90368 9-90358 9-90349 9-90339 9 9 xo 9 xo 15 14 13 12 II .8 9 i u i< 2.8 J-4 9 50 5i 5 2 53 54 9.77778 9-77795 9.77812 9-77829 9.77846 J7 7 17 '7 16 9.87448 9.87475 9-87501 9.87527 9.87554 27 26 26 27 26 0.12552 O.J2525 0.12499 0.12473 0.12446 9-90330 9.90320 9.90311 9.90301 9.90292 9 xo 9 xo 9 10 I .1 i .2 2 3 3 4 A '? f .0 .0 .0 .0 .0 ?:2 ai 4-5 11 11 59 9.77862 9.77879 9.77896 9.779I3 9.77930 '7 17 '7 '7 16 9.87580 9.87606 9-87633 9-87659 9.87685 26 2? 26 26 26 O. 12 42O 0.12394 0.12367 O.I234I O.I23I5 9.90 282 9.90273 9 . 90 263 9.90254 9.90244 9 xo 9 xo 5 4 3 2 I .6 c :l I 9 9 .0 .0 .0 .0 1.1 C 00 9.77946 9.87711 0.12 289 9-90235 9 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. 9 Pro] ). . Pts. 53 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 67 1 37 t 1 Sin. d. L. Tang 1 . c.d. L. Cotg-. L. Cos. d. 6JT Prop. Pts. I 2 3 4 9.77946 9.77963 9.77980 9-77997 9.78013 17 17 16 '7 16 17 7 16 7 17 7 16 16 7 17 16 7 16 17 16 17 16 16 17 16 17 16 16 7 16 7 16 i7 16 16 16 17 16 16 16 16 16 7 16 16 16 16 17 16 16 16 9.87711 9-87738 9-87764 9.87 790 9-87817 27 26 26 27 26 26 26 27 26 26 0.12 289 0.12 262 0.12 236 0.12 210 0.12 183 9-90235 9.90225 9.90 216 9.90206 9.90197 xo 9 xo 9 xo 9 xo 9 xo xo 9 xo 9 xo xo 9 xo 9 xo xo 9 xo xo 9 10 xo 9 xo xo 9 xo xo 9 xo 10 xo .9 10 xo xo 9 xo 10 xo 9 xo IO xo xo 9 xo 10 xo xo xo 9 xo xo 10 xo .1 .2 3 4 i :l -9 .1 .2 3 4 4 .1 .2 3 4 :I 9 .2 3 4 '.S 9 .1 l .2 i 3 : 4 4 V 91 * 7 2.7 H 10.8 '3-5 16.2 18.9 21.6 24.3 2.6 5-2 7.8 10.4 13.0 15.6 18.2 20.8 23 4 17 1-7 3-4 5.1 6.8 8-5 10.2 "I 13-6 15-3 16 1.6 |! 8.0 9-6 II. 2 12.8 14.4 to g .0 0.9 5.0 1.8 5-0 2.7 [.o 3.6 >.o 4-5 >-o 5.41 r.o 6.3 5.0. 7.2 i.o[ 8.1 I I 9 10 12 13 9.78030 9.78047 9-78063 9.78080 9.78097 9.87843 9.87869 9.87895 9.87922 9.87948 O.I2I57 0.12 131 0.12 ID? 0.12078 0.12052 9.90187 9.90 178 9.90 168 9-90 159 9-90 149 55 54 53 52 9.78113 9-78130 9.78147 9-78163 9.78 180 9.87974 9.88000 9.88027 9-88053 9.88079 26 27 26 26 0.12026 0.12000 O.II973 O.II947 o.ii 921 9.90139 9-90 13 9.90 120 9.90 III 9.90 ioi 50 4 2 4 8 ~iT 44 43 42 41 !i \l 19 "20" 21 22 23 24 9.78197 9-78213 9.78230 9.78246 9.78263 9.88 105 9 .88 131 9.88 158 9.88 184 9.88210 26 8 7 26 26 26 26 27 26 26 26 26 27 26 26 26 26 26 27 26 26 26 26 26 26 26 27 26 26 26 26 26 26 26 26 26 26 27 26 26 26 26 26 26 26 o.ii 895 O.II 869 o.ii 842 o.ii 816 o.ii 790 9.90091 9.90 082 9.90072 9-90063 9-90053 9.78280 9.78296 9-78313 9.78329 9.78346 9.88236 9.88262 9.88289 9.88315 9.88341 o.ii 764 o.ii 738 o.ii 711 o.ii 685 o.ii 659 9.90043 9.90034 9 . 90 024 9.90014 9.90005 1 27 28 29 9.78362 9.78379 9-78395 9.78412 9.78428 9.88367 9.88393 9.88420 9.88446 9.88472 o.ii 633 o.ii 607 o.ii 580 o.ii 554 o.ii 528 9-89995 9.89 985 9.89976 9.89966 9.89956 35 34 33 32 3' 30 32 33 34 9.78445 9.78461 9.78478 9.78494 9.78510 9.88498 9.88524 9-88550 9.88577 9.88603 o.ii 502 o.ii 476 o.ii 450 o.ii 423 0.11397 9.89947 9-89937 9 '.899i8 9.89908 30 29 28 % I 39 41 42 43 44 9.78527 9.78543 9-78560 9-78592 9.88629 9-88655 9.88681 9.88707 9-88 733 o.ii 371 o.ii 345 o.ii 319 o.ii 293 o.ii 267 9.89898 9.89888 9.89879 9.89869 9-89859 25 24 23 22 21 "20" |2 9.78609 9-78625 9 . 78 642 9.78658 9-78674 9-88 759 9.88 786 9.88812 9.88838 9.88864 o.ii 241 o.ii 214 o.ii 188 o.ii 162 o.ii 136 9.89849 9.89840 9.89830 9 . 89 820 9.89810 47 48 49 9.78691 9.78 707 9.78723 9.88.890 9.88916 9.88942 9.88968 9.88994 O.II IIO o.ii 084 o.ii 058 o.ii 032 O.II 006 9.89801 9.89 791 9.89 781 9.89 771 9.89 761 15 14 13 12 II lo~ I I 50 52 53 54 9.78772 9.78788 9-78805 9.78821 9.78837 9.89020 9.89046 9.89073 9.89099 9-89 125 0.10980 0.10954 o.io 927 0.10901 0.10875 9.89 752 9.89 742 9.89732 9.89 722 9.89 712 1 58 59 9-78853 9.78869 9.78886 9.78902 9.78918 9.89 151 9.89 177 9.89203 9 . 89 229 9-8925? 0.10849 0.10823 0.10797 o.io 771 0.10745 9.89 702 9.89693 9.89683 9.89673 9-89663 5 4 3 2 I nr GO 9-78934 9.89281 o.io 719 9 89653 L. Cos. d. L. Cotfir. c.d. L. Tang-. L. Sin. d. t Prop. Pts. 52 68 TABLE IV. 38 1 L. Sin. d. L. Tanar. c.d. L. Cotg. L. Cos. d. r ro] ). Pts. 2 3 4 9-7*934 9.78950 9.78967 9.78983 9.78999 16 17 16 16 16 9.89281 9-89307 9.89333 9-89359 9-89385 26 26 26 26 26 o.io 719 0.10693 0.10667 0.10641 0.10615 9-89653 9.89643 9-89633 9 . 89 624 9.89 614 xo xo xo xc xo 00 9 11 .1 2 2 2 e 6 .6 2 25 2-5 e o 1 I i 9 9.79015 9.79031 9.79047 9.79063 9.79079 16 *6 16 16 16 9.89411 9.89437 9.89463 9.89489 9-89515 76 26 26 26 26 0.10589 o.io 563 0.10537 o.io 511 0.10485 9.89604 9.89594 9-89584 9-89574 9.89564 xo xo xo xo 55 54 53 52 51 .3 :S .7 7 10 13 ;i .8 -4 .0 .6 .2 75 10. 12.5 15.0 ^S i 10 ii 12 13 H 9-7995 9.79111 9.79128 9.79 144 9.79160 16 17 16 16 16 9.89541 9.89567 9.89593 9.89619 9.89645 26 26 26 26 26 0.10459 0.10433 0.10407 0.10381 0.10355 9-89554 9.89544 9.89534 9.89524 9.89514 xo xo xo xo 50 3 8 .8 9 20 23 .8 4 i 20. o 22.5 7 it \l 19 9.79176 9.79192 9 . 79 208 9.79224 9.79240 16 16 16 16 16 9.89671 9.89697 9.89723 9.89749 9.89775 26 26 26 26 26 0.10329 0.10303 0.10277 0.10251 0.10225 9.89504 9-89495 9.89485 9-89475 9.89465 9 xo xo xo 45 44 43 42 41 .1 .2 3 4 5 i i ! -7 1-4 si .s "20 21 22 23 24 9.79256 9.79272 9.79288 9-79304 9.79319 16 x6 x6 IS 16 9.89801 9.89827 9.89853 9-89879 9.89905 26 26 26 26 26 o.io 199 o.io 173 o.io 147 O.IO 121 0.10095 9-89455 9-89445 9-89435 9.89425 9.89415 xo xo xo xo 40 It 9 :l -9 I i; i. [ -9 *-6 5-3 3 3 29 9-79335 9-79351 9.79367 9-79383 9-79399 16 16 16 16 x6 9.89931 9.89957 9,89983 9.90009 9-90035 26 26 26 26 26 0.10069 0.10043 0.10017 0.09991 0.09965 9.89405 9.89395 9-89385 9.89375 9.89364 IO xo xo XI 35 34 33 32 31 .1 .2 3 .4 i i i ^ f 6 .6 3 > 4 15 '5 3-o *j 30 3i 32 33 34 9-794I5 9-79431 9-79447 9-79463 9.79478 16 16 16 15 16 9.90061 9 . 90 086 9.90 112 9.90138 9.90 164 25 26 26 26 26 0.09 939 0.09 914 0.09 888 0.09 862 0.09 836 9.89354 9.89344 9.89334 9.89324 9.89314 xo 10 xo IO 30 2! 11 1 i .9 i c ii 12 It 1.0 ).6 .2 5.8 I--4 7-5 9.0 10.5 12. '3 5 9 9 39 9-79494 9.79510 9.79526 9-79542 9-79558 16 16 16 16 9.90190 9.90216 9.90242 9.90268 9.90294 26 26 26 26 26 0.09810 0.09 784 0.09 758 0.09 732 0.09 706 9.89304 9.89294 9.89284 9.89274 9.89 264 xo xo 10 xo 25 24 23 22 21 .1 .2 ii [.i 2.2 40 4i 42 43 44 9-79573 9-79589 9-79605 9.79 621 9.79636 x6 16 16 IS 16 9.90320 9.90346 9.90371 9.90397 9.90423 26 25 26 26 26 0.09 680 0.09 654 0.09 629 o . 09 603 0.09577 9.89254 9.89244 9-89233 9.89223 9.89213 xo XX xo xo 20 ii 3 4 7 i ( 3-3 [>4 7-7 2 ;* 7 49 9.79652 9 79668 9.79684 9.79699 9-797I5 16 16 IS 16 16 9.90449 9-90475 9-90501 9.90527 9-90553 26 26 26 26 0.09551 0.09525 0.09499 0.09473 0.09447 9.89 203 9-89193 9.89 183 9-89 173 9.89 W2 zo xo xo XX 15 14 13 12 II .8 9 3 < i 3.8 )-9 9 50 5i 52 53 54 9-79 73 1 9.79746 9-79 762 9.79778 9 79 793 IS 16 16 IS 16 9.90578 9.90604 9.90630 9.90656 9 . 90 682 26 26 26 26 26 0.09 422 0.09 396 0.09370 0.09344 0.09318 9 .8 9 I52 9.89142 9.89 132 9.89 122 9.89 112 xo xo xo xo 10 I I .1 .2 3 4 ] 2 3 A I .0 .0 -o .0 .0 0.9 ii 4-5 55 56 fi 59 9.79809 9-79825 9.79840 9.79856 9.79872 x6 15 x6 16 9 . 90 708 9-90734 9-90759 9.90 785 9.90 811 26 25 26 26 26 0.09 292 0.09 266 0.09 241 0.09 215 0.09 189 9.89 101 9.89 091 9.89081 9.89071 9 . 89 060 xo 10 xo IX 5 4 3 2 I :1 .9 I 9 .0 .0 .0 n K 60 9.79887 9.90837 0.09 163 9 . 89 050 L. os. d. L. Cotg. c.d. L. Tang. L. Sin. d. / p roj ). PtSr 51 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 69 39 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Proj .1 ts. I 2 3 4 9.79887 9-79903 9.79918 9-79934 9-79950 16 15 16 16 15 9.90837 9.90863 9.90889 9.90914 9-90940 26 26 as 26 26 0.09 163 0.09 137 0.09 in 0.09 086 0.09060 9 . 89 050 9 . 89 040 9.89030 9.89020 9.89009 xo IO 10 II IO 00 ii 11 .1 .2 21 2 5 5 .6 .2 \ I 9 9.79981 9.79996 9.80012 9.80027 16 IS 16 15 16 9.90966 9.90992 9.91 018 9.91 043 9.91069 26 26 as 26 26 0.09034 0.09 008 0.08982 0.08957 0.08931 9.88999 9.88989 9.88978 9.88968 9.88958 10 XI xo xo xo 55 54 53 52 51 3 4 '7 7 10 13 !i .8 4 .0 .6 .2 10" ii 12 '3 14 9.80043 9.80058 9 . 80 074 9.80089 9.80 105 IS 16 IS 16 15 9.91095 9.91 121 9.91 147 9.91 172 9.91 198 26 26 26 26 0.08905 0.08879 0.08853 0.08828 0.08802 9.88948 9-88937 9.88927 9.88917 9.88906 XX xo IO XX xo 50 4 i 48 .8 9 20 23 a .8 4 5 19 9.80 120 9.80 136 9.80151 9.80 166 9.80 182 16 IS IS 16 15 9.91 22 4 9 9i 250 9.91 276 9-91 301 9.91 327 26 26 as 26 26 0.08 776 0.08 750 0.08 724 0.08 699 0.08673 9 . 88 896 9.88886 9.88875 9.88865 9.88855 IO XX xo xo jj 45 44 43 42 41 .1 .2 3 4 2 5 7 10 12 5 .0 5 .0 5 20 21 22 23 24 9.80 197 9.80213 9.80228 9.80244 9-80259 16 IS 16 9 9i 353 9-91 379 9.91404 9.91430 9.91 456 26 85 26 26 26 0.08 647 0.08 621 0.08 596 0.08 570 0.08544 9.88844 9-88834 9.88824 9.88813 9.88803 xo IO XX IO 40 3 11 9 J 5 I? 2C 22 5 .0 -5 29 9.80274 9 . 80 290 9-80305 9.80320 9-86336 x6 15 IS x6 9.91 482 9.91 507 9 9i 533 9-91 559 9 9i 585 as 26 26 26 0.08518 0.08493 0.08467 0.08441 0.08 415 9.88793 9.88782 9.88772 9.88 761 9.88751 XX IO IX xo 35 34 33 32 31 .1 .2 3 A 1 1 3 i 6 .6 1:5 30 32 33 34 980351 9.80366 9.80382 9.80397 9.80412 IS 16 IS IS 16 9.91 610 9.91 636 9.91 662 9.91 688 9.91 713 26 26 26 as 26 0.08 390 0.08 364 0.08 338 0.08 312 0.08287 9.88741 9 88730 9.88 720 9.88709 9.88699 IX IO IX IO 30 i I .9 J < i] i: K ! - 4 \l .2 1.8 L-4 39 9.80428 9.80443 9-80458 9.80473 9.80489 IS 15 IS 16 9-9i 739 9.91 765 9 9i 791 9.91 816 9 91 842 26 26 as 26 26 0.08 261 0.08235 0.08 209 0.08 184 0.08 158 9.88688 9.88678 9.88668 9.88657 9.88647 xo xo II IO 25 24 23 22 21 .1 .2 ] <5 5 4i 42 43 44 9 . 80 504 9 80519 980534 9 80550 9 80565 IS 16 IS 9 91 868 9 91 893 9.91 919 9-9 945 9.91971 as 26 26 26 0.08 132 0.08 107 0.08081 0.08055 0.08 029 9.88636 9 . 88 626 9 88615 9.88605 9.88594 IO II 10 II 20 il 3 4 7 i ( ( 1C kS ) O r.5 ) o >-5 s 49 9 . 80 580 9 80 595 9 80 610 9 80 625 9 80 641 15 '5 16 9.91 996 9 92 022 9 92 048 9 92073 9.92099 26 26 25 26 26 0.08004 o 07 978 o 07952 0.07927 0.07 901 9 88 584 988573 9.88563 9 88552 9-88542 II IO II IO 15 14 13 12 II .8 9 i: i; d s.o J 5 xo 50 52 53 54 9 80 656 9.80 671 9.80686 9.80 701 9.80 716 IS 15 IS 15 9.92 125 9-92150 9.92176 9 . 92 202 9.92227 85 26 26 as 06 0.07875 0.07 850 0.07 824 0.07 798 0.07 773 9.88531 9.88521 9.88510 9-88499 9.88489 xo II II 10 10 I I .1 1 .2 : .3 : .4 / 5 l t.i 1.2 5-3 t-4 i-j I.O 2.O 3-o 4.0 i- 55 56 59 9 80731 9 80 746 9 80 762 9.80777 9 . 80 792 15 16 IS 9.92279 9.92304 9.92330 9.92356 26 85 26 26 0.07 747 0.07 721 0.07696 0.07 670 0.07 644 Q 88 478 9 . 88 468 9.88457 9.88447 9.88436 10 XI 10 XX 5 4 3 2 I .6 ( :! i 9 S >.t> 7 -7 5.8 >-9 6.0 7.0 8.0 9.0 (iO 9.80807 9.92381 2 S 0.07 619 9.88425 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. t Pro] P.: Pts. 50 TABLE IV. 40 t L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pte. 9.80807 9.92381 26 0.07619 9.88425 60 i 9.80822 9.92407 0.07 593 9.88415 59 2 9.80837 is 9-92433 0.07567 9.88404 58 36 3 9.80852 15 9.92458 25 26 0.07 542 9.88394 10 57 j 2 6 4 9.80867 9.92484 26 0.07 516 9.88383 M 56 .2 e.2 1 9.80882 9.80897 9.80912 9.80927 15 15 15 9.92 510 9-92535 9-92561 9.92587 25 26 26 0.07490 0.07465 0.07439 0.07413 9.88372 9.88362 9-88351 9.88340 10 II II 55 54 53 S2 3 4 7-8 10.4 130 15.6 9 9.80942 15 9.92 612 25 26 0.07388 9.88330 II 7 lS.2 ii 12 9.80957 9.80972 9.80987 15 IS 9.92 638 9.92663 9.92 689 25 26 0.07362 0.07337 0.07311 9.88319 9.88308 9.88298 II IO 50 49 48 .8 9 20.8 23-4 13 9.8l 002 15 9.92 715 20 0.07285 9.88287 47 14 9.8l 017 15 15 9-92 740 25 26 0.07260 9.88276 IO 46 5 !i 9.8l 032 9.8l 047 IS 9.92 766 9.92 792 26 0.07 234 0.07 208 9.88266 9.88255 XX 45 44 .1 .2 2.5 5-0 IS 9.81 061 9.81 076 14 IS 9.92817 9.92 843 25 26 0.07 183 0.07 157 9.88244 9-88234 to 43 42 3 4 7-5 IO.O 19 9.81 091 IS 15 9.92868 25 26 0.07 132 9.88223 I] 41 5 12.5 20 21 9.81 106 9.81 121 IS 9.92894 9.92 920 26 0.07 106 0.07080 9.88212 9.88201 XI *9 i 17-5 20 o 22 9.81 136 *5 9 92945 25 ?f> 0.07055 9.88 191 38 o 22. C 23 24 9.81 151 9.81 166 IS 9.92971 9.02 996 25 26 0.07029 o . 07 004 9.88 180 9.88 169 II 11 11 9.81 180 9.81 195 9.93022 9.93048 26 0.06 978 0.06952 9.88 158 9 . 88 148 IO 35 34 * f 11 9.8l 210 9.81 225 15 IS 9-93073 9.93099 25 26 0.06927 0.06901 9.88137 9.88126 xr 33 32 .2 J 3-0 29 9.8l 240 15 9 93 124 25 26 0.06876 9.88 115 '. .4 60 30 3' 32 9.8l 254 9.8l 269 9.8l 284 15 IS 9-93 ISO 9 93 175 9.93 2OI 25 26 0.06 850 0.06 825 0.06 799 9.88105 9.88094 9.88083 ii IX 30 3 i 7-5 9.0 10.5 33 34 9.8l 299 9.8l 314 15 . 15 9-93227 9.93252 26 25 26 0.06 773 0.06 748 9.88072 9.88061 IX 10 2 9 12.0 135 P 8 9.81328 9-8i 343 9-8i 358 9.81 372 15 15 14 9.93278 9-93303 9-93329 9-93354 25 26 25 0.06 722 0.06 697 0.06 671 0.06 646 9.88051 9.88040 9 88 029 9.88018 XI II II 25 24 23 22 N I A. 39 9.81387 15 9.9338o 26 0.06 620 9.88007 21 .2 I'i 40 9.81 402 9.93406 0.06 594 9.87996 20 3 4-2 41 9.81417 15 9-93431 25 0.06 569 9.87985 19 .4 5-6 42 9.81 431 14 9 93457 26 0.06 543 9 87975 18 7.0 43 44 9.81446 9.81 461 15 15 9.93482 9-93 5o8 25 26 0.06 518 0.06492 9.87964 9.87953 II 11 7 8.4 9.8 45 9 81475 9-93533 25 0.06467 9.87942 IS .8 II. 2 46 9.81 490 9-8i 505 IS 15 9-93 559 9-93 584 26 25 0.06441 0.06416 9.87931 9.87920 II 14 13 9 12.6 48 9.81 519 14 9.93610 26 0.06390 9.87909 12 49 9 81 534 IS 9.93636 26 0.06 364 9.87898 II IX 10 50 5 2 9.81 549 9-81563 9.81 578 14 15 9.93661 9-93687 9 93 7 12 25 26 25 0.06339 0.06313 0.06 288 9.87887 9.87877 9.87866 xo II 10 .1 I.I I.O .2 2.2 2.0 3 33 3-0 53 9.81 592 14 9-93 73 s 26 0.06 262 9-87855 IX 7 .4 4.4 4.0 54 9.81 607 15 9-93 763 25 0.06237 9.87844 6 -I n I'i 1 9.81 622 9.81 636 9.81 651 9.81 665 14 IS 14 9 93 789 9-93 814 9.93840 25 26 25 0.06211 0.06 186 0.06 160 0.06 135 9-87833 9.87822 9.87811 9.87800 XX XX XX 5 4 3 2 .6 6.6 6.0 1 11 I:S 9 9.9 9.0 59 9.81 680 15 9.93891 26 0.06 109 9.87789 XX I GO 9 81 694 9.93916 25 0.06084 9.87778 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. t Prop. Pts. 49 | LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 7, 1 41 , L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. ' d. Proj ). Pts. ho" 2 * 9.81 694 9.81 709 9.81 723 9.81 738 9.81 752 s 14 15 14 15 9.93916 9-93942 9-93967 9-93 993 9.94018 26 25 26 25 26 0.06084 0.06058 0.06 033 0.06007 0.05 982 9.87778 9.87 767 9.87 756 9-87 745 9-87 734 IX zz ZI IZ ZI 60 9 9 .1 .2 a6 2.6 C.2 * 9 9 81 767 9.81 781 9 81 796 9.81 810 9.81825 14 15 14 15 14 9.94044 9.94069 9-94095 9.94 120 9-94 H6 25 26 25 26 25 0.05 956 0.05 931 0.05 905 0.05 880 0.05 854 9.87723 9 87 712 9.87 701 9.87690 9.87679 zz zz zz zz JI 55 54 53 52 51 3 :! .7 7-8 10.4 13.0 15.6 18.2 10" ii 1 I2 *3 14 9.81 839 9-81854 9.81868 9.81 882 9.81897 IS 14 14 15 14 9.94171 9-94 197 9.94222 9.94248 9-94273 26 25 26 25 26 0.05 829 0.05 803 0.05 778 0.05 752 0.05 727 9.87 668 9.87657 9.87646 9.f7635 9.87624 zz zz zz zz zz 50 9. 9. .8 9 20.8 23.4 m 15 10 17 18 19 9.81 911 9.81 926 9.81 940 9 81955 9.81 969 S H 15 14 14 9.94299 9-94324 9-94350 9 94375 9,94401 S 26 25 26 25 0.05 701 0.05 676 0.05 650 0.05 625 0.05 599 9-87613 9.87601 9.87590 9 87579 ,9-87568 12 zz zz ZI II 45 44 43 42 41 .2 3 4 'I 2-5 5-o 7-5 IO.O 12 5 1C O 20 21 22 1 23 24 9 81 983 9.81 998 9.82 OI2 9 . 82 026 9.82 041 '5 M 14 S 14 9 94 426 9 94452 9-94477 9 94503 9.94528 26 25 26 25 26 0.05 574 0.05 548 0.05 523 0.05497 0.05472 9.87557 9.87546 9 87535 9.87524 9 87513 ZI zz zz ZI ia 40 37 36 i 9 5- 17-5 20. o 22.5 hs~ 26 h \ 2 9 9.82055 9 82 069 9.82084 9.82098 9.82 112 14 IS 14 14 9-94 554 9-94579 9.94604 9 94630 9 94655 25 25 26 25 26 0.05 446 0.05421 0.05 396 0.05 370 0-05 345 9.87501 9.87490 9.87479 9.87468 9 87 457 tz II II ZI ZI 35 34 33 32 31 i .2 3 .4, 5 "5 30 n 30 31 I 32 33 34 9.82 126 9.82 141 9-82 155 9.82 169 9.82 184 IS 14 4 S 9.94681 9.94706 9-94732 9-94757 9-94 783 25 26 25 26 25 0.05319 0.05 294 0.05 268 0.05 243 0.05 217 9 87446 9 87434 9.87423 9 87412 9.87401 IS II ZI II zz 30 2! 11 ii :1 9 7.5 9 .o lo-S 12. '3-5 P 9 39 9.82 198 9 82 212 9 82 226 9 . 82 240 9 82255 14 M 4 IS 9.94808 9-94 834 9.948$9 9.94884 9.94910 26 25 S 26 o 05 192 0.05 166 o 05 141 o 05 116 o 05 090 9 87390 9.87378 9 87 367 9 87356 9 87345 13 zz zz ZI ZI 25 24 23 22 21 .1 .2 14 ;i 40 41 42 43 44 9 82 269 9 82283 9 82297 9.82 311 9.82326 14 14 14 IS 9 94935 9.94961 9 94986 9.95012 9 95037 26 25 26 25 0.05 065 005039 0.05 014 o 04988 0.04963 9 87334 9.87322 9 87311 9.87300 9.87288 12 II II 12 IZ 20 19 il 3 4 i 7 g i;: 9.8 9 9- 49 9 82 340 9 82354 9.82 368 9.82382 9 82396 4 14 14 14 9.95062 995088 9 95 3 9 95 '39 9-95 164 26 25 26 25 26 o 04938 0.04912 0.04 887 0.04 861 o 04 836 9 87277 9 87266 9 87255 9 87 243 9.87232 zz IZ Z2 ZI 15 14 13 12 II .8 9 i II. 2 12.6 a II 150 5 1 52 53 54 9.82 410 9 82424 9.82439 9 82453 9.82467 14 s 14 4 9-95 !90 9-952I5 9-95 240 9-95 266 9-95 291 25 as 26 5 26 0.04 810 0.04 785 0.04 760 0.04 734 0.04 709 9.87 221 9.87209 9.87 I 9 8 9.87 187 9 87175 la zz zz za ff 10 6 .1 i .2 s 3 c 4 4 .2 I.I (.4 2.2 [6 3.3 ^8 4-4 >0 ii 56 19 59 9.82481 9.82495 9-82509 9-82523 9 82537 4 >4 14 4 14 9-953I7 9-95342 9-95 368 9-95393 9.95418 as 26 as 5 96 0.04683 0.04 658 0.04 632 o . 04 607 0.04 582 9.87 164 9-87153 9.87 141 9-87 130 9.87 119 XI 12 xz 11 5 4 3 2 I Ii 9 r.2 6.6 U 7-7 > 6 8.8 >.8 9.9 9-82551 9-95444 0.04 556 9.87 107 1 L. Cos. d. L. Cotgr. c.d. L. Tang. L. Sin. d. / Pro] p. Pts. 4:8 2 LAKLZ, IV 42 / L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Pro] ). ] Pts. I 2 3 4 9.82551 9.82565 9.82579 982593 9.82 607 14 14 14 14 9-95444 9 95469 9-95495 9-95 520 9-95 545 25 26 25 25 26 0.04556 0.04531 0.04505 0.04480 0.04455 9.87 107 9.87096 9-87085 9.87073 9.87062 ii IX 12 II 60 5 2 58 8 2 a 2 i .6 2 1 1 9 9.82 621 9-82635 9 . 82 649 9.82663 9.82677 4 14 14 4 14 9-95571 9-95 596 9-95 622 9-95 647 9-95 672 25 26 25 25 26 o 04 429 o 04 404 0.04 378 0.04353 o 04328 9.87 050 9.87039 9.87028 9.87 016 9 - 87 005 II IX 12 II 12 55 54 53 52 5i 3 .4 :I .7 7 1C 13 !! .8 4 .0 .6 .2 10 ii 12 13 H 9.82 691 9.82 705 9.82 719 9-82 733 9-82747 H 14 J4 4 14 9.95698 9-95 723 9-95 748 9-95 774 9-95 799 25 25 26 25 36 o . 04 302 0.04277 0.04252 0.04 226 0.04 201 9.86993 9 . 86 982 9.86 970 9.86959 9.86947 II l IX 12 50 3 11 .8 9 2C 23 3 .8 4 5 II \l 19 9.82 761 9-82775 9.82 788 9.82802 9.82816 4 13 14 14 1^ 9.95 825 9-95850 9-95875 9-95 901 9 95 926 25 25 26 25 26 0.04 175 0.04 150 0.04 125 0.04099 0.04074 9 . 86 936 9 . 86 924 9 86 913 9.86902 9.86890 12 XI II is ii 45 44 43 42 41 .1 .2 3 4 J 2 5 5 1C 12 5 .0 '5 >.o 5 20 21 22 23 24 9.82830 9.82844 9.82858 9 . 82 872 9.82885 >4 4 14 3 14 9-95 952 9-95 977 9.96002 9 . 96 028 9-96053 25 95 6 *5 o 04 048 0.04023 0.03998 0.03 972 0.03947 9.86879 9.86867 9.86855 9.86844 9 . 86 832 13 12 XX 19 40 P H :i 9 i; 2C 22 O r-5 5 O '5 2 5 26 3 29 9.82899 9.82913 9.82927 9.82941 9 82955 4 14 14 H 13 9.96078 9.96 104 9.96 129 9.96 155 9.96 180 26 25 26 25 0.03 922 0.03 896 0.03871 0.03 845 o 03 820 9.86821 9.86 809 9.86 798 9.86786 9-86775 12 IX 12 XX 35 34 33 32 3i .1 .2 3 A 1 1 t 14 1:1 H 80 3i 32 33 34 9.82968 9.82982 9.82996 9.83 oio 9.83023 14 14 14 3 14 9.96205 9.96231 9.96256 9.96 281 9.96 307 26 25 25 26 0.03 795 0.03 769 0.03 744 0.03 719 0.03693 9.86763 9.86752 9.86 740 9.86 728 9.86717 XX 12 12 XX 12 30 3 27 26 :i :I .9 I ( I i: ii: >.i [.2 2.6 9 9 39 9-83037 9.83051 9.8300? 9.83078 9.83092 4 14 13 14 14 9.96332 9 96357 9-96383 9 . 96 408 9-96433 25 26 25 25 26 0.03 668 0.03 643 0.03 617 0.03592 c 03 567 9.86705 9 . 86 694 9.86682 9.86670 9 86 659 XX 12 13 II 25 24 23 22 21 .1 .2 >3 1-3 2.6 40 41 42 43 44 9.83 106 9.83 I2O 9 83 133 9 83 147 9.83 161 14 13 14 4 *3 9-96459 9.96484 9-96510 9.96535 9.96560 25 26 25 25 26 0.03541 0.03 516 0.03490 0.03 465 0.03440 9 86647 9 86635 9 86 624 9.86612 9.86600 12 XI 12 12 20 19 ii 3 4 :! 7 3 9 1 M 4 S <6 47 48 49 9-83 174 9.83 188 9 . 83 202 9 83215 9.83229 4 14 13 M 9.96586 9.96 61 1 9 . 96 636 9 . 96 662 9.96687 25 25 26 25 0.03 414 0.03389 0.03 364 0.03338 0.03313 9 86 589 9 86577 9 .86 565 9-86554 9.86542 12 12 II 12 15 14 13 12 II .8 9 3 I i 2 3-4 1.7 XI 60 5i 52 53 54 9.83 242 9 83256 9-83270 9.83283 9.83297 14 14 3 4 9.96 712 9.96738 9-96763 9.96 788 9.96 814 2 5 26 25 25 26 0.03 288 0.03 262 0.03 237 0.03 212 0.03 186 9.86530 9.86 518 9.86 507 9.86495 9.86483 12 II 12 12 10 I .1 1 .2 2 -3 I \ \ .2 4 .6 ^s >.o I.I 2.2 3-3 4-4 5-5 fi ? 59 9.83310 9 83324 983338 9-8335I 983365 14 14 13 14 9.96839 9.96864 9.96890 9.96915 9 9 6 940 25 25 26 25 25 0.03 161 o 03 136 0.03 no 0.03 085 o . 03 060 9.86472 9 . 86 460 9.86448 9.86436 9.86425 12 12 12 II 5 4 3 2 * t .8 c, 9 ic .2 -4 .6 >.S o.o Ii 9-9 60 9.83378 9.96 966 0.03034 9.86413 L. Cos. d. L. Cotg. c.d. L. Tang. L. Sin. d. / Pro] > Pts. 47 LOGARITHMS OF SINE, COSINE, TANGENT AND COTANGENT, ETC. 73 1 43 , L. Sin. d. L. Tang. c.d. L. Cotg. L. Cos. d. Prop. Pts. i 2 3 4 9 83378 9-83392 9-83405 9.83419 9-83432 14 13 13 14 13 14 13 13 13 14 13 13 14 13 13 14 3 13 13 13 3 13 4 3 U 14 13 J3 13 13 13 13 '3 13 14 13 13 '3 13 13 9.96966 9.96991 9.97016 9.97042 9.97067 25 25 26 25 25 0.03034 0.03 009 O.02 984 O.O2 958 0.02 933 9-86413 9.86 401 9-86389 9-86377 9.86366 12 12 12 II 12 S3 S3 13 12 II S3 12 13 12 13 12 S3 IS S3 GO ii i 55 54 53 52 .1 .2 3 4 .1 :i -9 .1 .2 3 4 :l 9 .1 .2 3 4 :I 9 .1 .2 3 4 i 9 .1 .2 1 -3 : .4 < ;! i .9 ic aff 2.6 w 10.4 111 18.2 20.8 23 4 25 5-0 7.5 10. 12.5 17^5 20. o 22.5 14 1:2 U 9-8 II. 2 12.6 13 39 7$ 9-1 10.4 11.7 13 II [.2 I.I Z.4 2.2 J-6 3-3 1-8 4-4 >- 5-5 7.2 6.6 !:* Ii ).8 9.9 I 8 9 9.83446 9 83459 9.83473 9.83486 9.83500 9.97092 9-97 "8 9-97 143 9.97 168 9-97 193 26 25 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 26 25 5 25 26 o.o2"9o8 0.02 882 0.02 857 O.02 832 O.O2 807 9-86354 9-86342 9.86330 9.86318 9.86306 10 ii 12 13 H 9.83513 9.83540 9.83554 9-83567 9.97219 9-97244 9-97269 9.97320 0.02 781 O.02 756 0.02 731 0.02 705 0.02 680 9.86295 9.86283 9.86271 9-86259 9.86247 50 4 2 4 8 ii 1 L. 21 22 23 24 9.83581 9-83594 9.83608 9.83 621 9-83634 9-97345 9-97371 9.97396 9-97421 9-97447 0.02 655 O.02 629 O.O2 604 0.02 579 0.02 553 9.86235 9.86223 9.86211 9.86 200 9.86 188 45 44 43 42 9-83648 9.83661 9.83674 9.83688 9.83 701 9-97472 9-97497 9-97548 9-97573 O.O2 528 0.02 503 0.02477 0.02452 O.O2 427 9.86176 9.86 164 9.86 152 9.86 146 9.86 128 S3 S3 S3 S3 S3 13 13 S3 S3 S3 S3 S3 S3 12 S3 S3 13 S3 12 S3 S3 S3 12 12 12 13 13 12 S3 S3 S3 S3 S3 13 S3 S3 S3 S3 S3 40 39 38 37 Jl 35 34 33 32 ? 2 9 9.83715 9-83728 9.83741 9.83755 9-83 768 9-97 598 9-97624 9.97649 9.97674 9.97 700 O.O2 4O2 0.02376 0.02351 0.02 326 0.02 300 9.86 116 9.86 104 9.86092 9.86080 9.86068 3 32 i 33 34 9.83781 9 83 795 9.83808 9.83821 9-83834 9-97 725 9=97750 9-97 776 9.97801 9.97826 O.O2 275 0.02 2JJO 0.02 224 O.O2 199 0.02 174 9.86056 9.86044 9.86032 9.86020 9.86008 30 3 11 35 3 6 39 9.83848 983861 9.83874 9-83887 9.83901 9-97851 9.97877 9.97902 9.97927 9-97953 O.O2 149 0.02 123 0.02098 0.02073 0.02 047 9.85996 9.85984 9.85972 9.85960 9.85948 25 24 23 22 21 w 19 i! 40 42 43 44 45 46 i 48 1 49 9.83914 9.83927 9.83940 9 83954 9.83967 9-97978 9.98003 9.98029 9.98054 9-98079 O.O2 O22 o.oi 997 o.oi 971 o.oi 946 o.oi 921 9-85936 9.85924 9.85912 9.85 900 9.85888 9.83980 9.f3993 9 . 84 006 9 . 84 020 9 84 033 9.98 104 9-98 130 9-98155 9.98 180 9.98 206 o.oi 896 o.oi 870 o.oi 845 o.oi 820 o.oi 794 9.85876 9.85 864 9-85851 9-85839 9.85827 15 13 12 II 50 i 5 ' 1 53 1 54 9 84 046 9 84059 9 84072 9-8408$ 9 . 84 098 9-98231 9-98256 9.98281 9.98307 9-98332 o.oi 769 o.oi 744 o.oi 719 o.oi 693 o.oi 668 9.85815 9-85803 9.85 791 9-85 779 9-85 766 10 S 5 4 3 i 55 5 6 58 59 9.84 112 9.84 125 9.84138 9.84151 9.84 164 9 '.98 p| 9.98408 9.98458 o.oi 643 o.oi 617 o.oi 592 o.oi 567 o.oi 542 9-85 754 9-85742 9-85 730 9.85 718 9-85 7o6 9.84 177 9.98484 o.oi 516 9-85693 L. Cos. d. L. Cotsr. c.d. L. Tang. L. Sin. d. f Prop. Pts. 46 TABLE IV. 44 t L. Sin. (!. L. Tang. e.d. L. Cot. L. Cos. d. Proi ). Pts. I 2 3 4 9.84 177 9.84 190 9-84203 9.84 210 9.84229 3 13 *3 3 9.98484 9.98509 9-98534 9.98 560 9-98585 25 25 26 25 o.oi 516 o.oi 491 O.OI 466 o.oi 440 o.oi 415 9-85693 9.85 68 1 9-85669 9-85657 9-85645 12 12 xa 12 (>0 9 11 ,i 26 2.6 1 2 I I 9 9 . 84 242 9 84255 9 . 84 269 9 84282 9.84295 *3 4 >3 *3 9.98 610 9-98635 9.98661 9.98686 9.98711 25 26 25 25 26 o.oi 390 o.oi 365 o.oi 339 o.oi 314 o.oi 289 9.85632 9.85 620 9.85608 9.85596 9.85 583 12 12 12 3 55 54 53 52 5i 3 4 :I .7 f: 10.4 13.0 15.6 18.2 10 ii 12 *3 H 9.84308 9.84321 9- ^4334 9.84347 9.84360 3 3 13 3 9.98737 9.98 762 9-98 787 9.98812 9.96838 25 25 s 26 o.oi 263 o.oi 238 o.ci 213 o.oi 188 o.oi 162 9.85571 9.85559 9-85 547 9.85534 9-85 522 12 12 13 12 50 49 48 47 46 .8 9 20.8 23.4 25 3- !1 19 9.84373 9.84385 9.84398 9.84411 9.84424 12 3 3 13 9.98863 9.98808 9.98913 9 9*939 9.98964 25 25 26 25 o.oi 137 O.OI 112 o.oi 087 o.oi obi o.oi 036 9.85 510 9.85497 9.85485 9-85473 9.85460 3 12 12 3 45 44 43 42 41 .1 .2 3 4 I 2.5 5-0 7-5 10. 12.5 20 21 22 23 24 9-84437 9.84450 9.84463 9.84476 9.84489 3 3 J3 13 9.98989 9.99015 9.99040 9.99065 9.99090 26 25 25 25 26 O.OI Oil 0.00985 o.oo 960 0.00935 0.00910 9-85448 9-85436 9-85423 9.85411 9.85399 12 '3 12 12 40 I .0 :1 9 I S 17-5 20.0 22.5 S 27 28 29 9-8450^ 9.84515 9-84528 9.84540 9.84553 13 *3 12 13 9.99 116 9-99 HI 9.99 166 9-99 191 9.99217 25 25 85 26 0.00884 0.00859 o.oo 834 0.00809 o.oo 783 9-85386 9-85374 9-85361 9 85349 9-85337 19 <3 12 12 35 34 33 32 3i .1 .2 3 4 M 5:1 4.2 5-6 80 3 32 33 34 9-84566 9 84579 9 84592 9.84605 9.84618 3 3 3 *3 9-99242 9.99267 9-99293 9.99318 9-99343 25 26 25 25 o.oo 758 o.oo 733 o.oo 707 0.00682 0.00657 9-85 324 9-85312 9-85 299 9-85287 9.85274 12 3 12 3 12 80 1 :i i 9 7-0 8-4 9.8 II. 2 12.6 9 9 39 9 84 630 9.84 643 9 . 64 656 9.84669 9.84082 *3 3 3 3 9.99368 9-99394 9.99419 9.99444 9.99469 26 25 25 25 26 0.00632 0.00606 o.oo 581 0.00556 o.oo 531 9 . 85 262 9-85250 9-85237 9-85 225 9.85 212 12 3 12 >3 12 25 24 23 22 21 .1 .2 3 11 40 41 42 43 44 9-84694 9-84707 9.84720 9 84733 9-4 745 3 3 13 ' 12 9-99495 9-99 520 9-99545 9-99570 9-99 59& 25 25 25 26 o.oo 505 o.oo 480 o.oo 455 0.00430 0.00404 9.85 200 9.85 187 9-85 175 9.85 162 9-85 15 3 12 3 12 i>0 ;i 3 3 !i 7 39 l: i 7-8 9-i 3 J2 49 9-84758 9.84 771 9.84 784 9.84 796 9 . 84 809 X 3 13 13 12 3 9.99621 9.99646 9.99672 9.99697 9.99722 8 S 25 26 25 25 0.00379 0.00354 0.00328 0.00303 0.00278 9-85 137 9-85 125 9.85 112 9.85 ioo 9.85087 12 *3 12 13 15 14 13 12 II .8 9 10.4 11.7 xa 50 5i 52 53 54 9.84822 9.8483$ 9-84847 9.84860 9-84873 X 3 3 12 3 3 9-99747 9-99 773 9.99798 9-9982;? 9.99848 25 26 25 25 25 0.00253 0.00227 O.OO 2O2 o.oo 177 o.oo 152 9.85074 9.85062 9.85049 9.85037 9.85024 18 3 12 >3 10 1 I .1 .2 3 4 1 1.2 'I tt P 12 59 9-84885 9.8489$ 9.849" 9.84923 9.84936 3 3 19 3 9.99874 9-99899 9-99924 9.99949 9-99975 25 25 25 96 o.oo 126 0.00 101 0.00076 o.oo 051 0.00025 9.85012 9.84999 9.84986 9.84974 9.84961 3 3 19 3 ft 5 4 3 i .0 i 9 *1 ,2:1 00 9 84949 *3 o.ooooo 25 o.oo ooo 9.84949 L. Cos. d. L. Cotg. c.d. L. Tang:. L. Sin. d. i Pro] [>. PtS. 45 TABLE V NATURAL SINES AND COSINES. 75 TABLE V. NATURAL SINES AND COSINES ) 1 Alil^r* V 1 2 3 40 40 60 59 58 H 55 54 53 52 5i 50 2 t tf. sine N. cos. N. sine S T. cos. N. sine N. cos. N. sine ^. cos. N. sine|N. cos. O I 2 3 4 I 'I 9 1C ii 12 ~^3~ 14 II 1 19 20 21 22 23 24 .00000 .00029 .00058 .00087 .00116 .00145 .00175 .00000 .00000 .00000 .00000 .00000 .00000 .00000 01745 .01774 .01803 .01832 .01862 .01891 .01920 .99985 .99984 .99984 .99983 .99983 .99982 .99982 .03490 03519 03548 -03577 .03606 03635 .03664 99939 .99938 99937 9993 6 99935 99934 99933 35234 .05263 .05292 05321 05350 05379 .05408 .99863 99861 .99860 .99858 99857 99855 99854 .06976 .07005 .07034 .07063 .07092 .07121 .07150 99756 99754 99752 99750 99748 99746 99744 .00204 .00233 .00262 .00291 .00320 .00349 .00000 .00000 .00000 .00000 99999 99999 .01949 .01978 .02007 .02036 .02065 .02094 .99981 .99980 .99980 99979 99979 99978 .03693 .03723 03752 .03781 .03810 03839 99932 99931 .99930 .99929 99927 .99926 5437 .05466 05495 05524 05553 .05582 .99852 .99851 99849 .99847 .99846 .99844 .07179 .07208 .07237 .07266 .07295 07324 .99742 .99740 99738 99736 99734 99731 .00378 .00407 .00436 .00465 .00495 .00524 99999 99999 99999 99999 99999 99999 .02123 .02152 .02181 .02211 .O224O .02269 99977 99977 .99976 99976 99975 99974 .03868 .03897 .03926 03955 .03984 .04013 99925 .99924 .99923 .99922 .99921 .99919 .05611 .05640 .05669 .05698 05727 05756 .99842 .99841 99839 .99838 99836 99834 07353 07382 .07411 .07440 .07469 .07498 .99729 .99727 99725 99723 99721 .99719 9 45 44 43 42 005*3 .00582 .00611 .00640 .00669 .00698 99998 .99998 .99998 99998 .99998 .99998 .02298 .02327 02356 02385 .02414 .02443 99974 99973 99972 .99972 .99971 .99970 .04042 .04071 .04100 .04129 .04159 .04188 .99918 .99917 .99916 99915 99913 .99912 05785 05814 .05844 05873 .05902 05931 99833 .99831 .99829 .99827 .99826 .99824 .07527 07556 07585 .07614 07643 .07672 .99716 .99714 .99712 .99710 .99708 99705 41 40 39 38 37 36 2 3 29 jp_ 3i 32 33 34 35 _36_ 11 39 40 4i 42 43 44 45 46 47 48 .00727 .00756 .00785 .00814 .00844 .00873 99997 99997 99997 99997 .99996 .99996 .02472 .02501 .02530 .02560 .02589 .02618 99969 .99967 .99966 .99966 .04217 .04246 .04275 .04304 04333 .04362 .99911 .99910 .99909 .99907 .99906 .99905 .05960 05989 .06018 .06047 .06076 .06105 .99822 .99821 .99819 99817 .99815 .99813 .07701 .07730 07759 .07788 .07817 .07846 99703 99701 .99699 .99696 99694 .99692 35 34 33 32 3i 30 .00902 .00931 .00960 .00989 .01018 .01047 .99996 .99996 99995 99995 99995 99995 .02647 .02676 .02705 .02734 .02763 .02792 99965 .99964 99963 .99963 .99962 .99961 .04391 .04420 04449 .04478 04507 04536 99904 .99902 .99901 99897 .06134 .06163 .06192 .06221 .06250 .06279 1.99812 .99810 .99808 .99806 .99804 99803 07875 .07904 07933 .07962 .07991 .08020 .99689 .99687 .99685 9968 3 .99680 .99678 % 11 25 24 .01076 .01105 .01134 .01164 .01193 .01222 .99994 99994 99994 99993 99993 99993 .02821 .02850 .02879 .02902 .02938 .02967 .99960 99959 99959 .99958 99957 99956 04565 .04594 .04623 04653 .04682 .04711 .99896 .99894 99893 .99892 .06308 .06337 .06366 06395 .06424 06453 .99801 99799 99797 99795 99793 .99792 .08049 .08078 .08107 .08136 .08165 .0819^ .99676 99673 .99671 .99668 .99666 .99664 23 22 21 2O *9 .01251 .01280 .01309 01338 .01367 .01396 99992 99992 .99991 .99991 .99991 .99990 .02996 .03025 03054 .03083 .O3II2 .03141 99955 99954 99953 .99952 99952 99951 .04740 .04769 .04798 .04827 .04856 .04885 .99888 .99886 .99885 .99883 .99882 .99881 .06482 .06511 .06540 .06569 .06598 .06627 .99790 .99788 99786 .99784 99782 .99780 .08223 .08252 .08281 .08310 .0836* .99661 .99659 99657 .99654 .99652 99649 II 15 H 13 12 II 10 I 49 50 5i ! 52 , 53 5 * .01425 .01454 .01483 .01513 .01542 .01571 99990 99989 99989 .99989 .99988 .99988 .03170 .03199 .03228 03257 .03286 .03316 .99950 99949 99948 99947 99946 99945 .04914 .04943 .04972 .05001 05030 05059 .99879 .99878 .99876 99875 99873 99872 .06656 .06685 .0671^ 06743 .06773 .06802 .99778 99776 99774 99772 .99770 .99768 .08397 .08426 .08455 .08484 S&3 99647 .99644 99642 99639 .99637 99635 1 55 56 57 5 8 .OI6OO .01629 .01658 .01687 .01716 01745 N. cos. .99987 .99987 .99986 .99986 .99985 .99985 N. sine 03345 03374 03403 03432 .03461 .03490 N. cos. 99944 99943 .99942 .99941 .99940 99939 N. sine .05088 .05117 .05146 05175 .05205 05234 N. cos. .99870 .99869 .99867 .99866 .99864 99863 N. sine .06831 06860 .00889 .06918 .06947 .06976 N. cos. .99766 .99764 .99762 .99760 99758 99756 N. sine S .08629 .08658 .08687 .08716 N. cos. 1.99632 .99630 .90627 99625 .99622 .99619 N. sine 5 4 3 2 I f 89' 88 87 86 85 NATURAL SINES AND COSINES. 77 5 6 7 8 9 / o I 2 3 4 1 N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. .08716 .08745 .08774 .08803 .08831 .08860 .08889 .99619 .99617 .99614 .99612 .99609 .99607 .99604 10453 . 10482 .10511 10540 .10569 .10597 . 10626 .99452 99449 99446 99443 .99440 99437 99434 .12187 .12216 .12245 .12274 .12302 12331 .12360 99255 .99251 .99248 99244 .99240 .99237 99233 I39I7 .13946 13975 .14004 14033 .14061 . 14090 .99027 .99023 .99019 .99015 .99011 .99006 .99002 15643 .15672 .15701 15730 15758 15787 .15816 .98769 .98764 .98760 98/55 9875' .98746 .98741 60 9 9 55 ' 54 , I 9 10 ii 12 .08918 .08947 .08976 .09005 .09034 .09063 .99602 99599 .99596 99594 99591 .99588 ! 10684 .10713 .10742 .10771 .10800 9943 1 .99428 .99424 .99421 .99418 99415 .12389 .12418 .12447 .12476 .12504 12533 99230 .99226 .99222 .99219 99215 .99211 .14119 .14148 .14177 .14205 .14234 .14263 .98998 98994 .98990 .98986 .98982 .98978 15845 15873 .15902 I593I 15959 .15988 98737 98732 .98728 .98723 .98718 .98714 53 i 52 Si 50 49 48 13 14 15 11 .09092 .09121 .09150 .09179 .09208 .09237 .99586 99583 .99580 99578 99575 99572 . 10829 . 10858 . 10887 .10916 .10945 .10973 .99412 .99409 .99406 .99402 99399 99396 .12562 .12591 .12620 .12649 .12678 .12706 .99208 .99204 .99200 .99197 99193 .99189 .14292 .14320 14349 14378 .I44CJ .14436 98973 .98969 .98965 .98961 98957 98953 .16017 .16046 .16074 .16103 .16132 .16160 .98709 .98704 .98700 .98695 .98690 .98686 47 46 45 44 43 42 19 2O 21 22 23 24 .09266 .09295 .09324 09353 .09382 .09411 9957 99567 .99564 .99562 99559 99556 .IIOO2 .11031 .11060 .11089 .IIIlS .11147 99393 .99390 .99386 99383 99380 99377 12735 .12764 .12793 .12822 .12851 .12880 .99186 .99182 .99178 99175 .99171 .99167 .14464 .14493 .14522 I455I .14580 .14608 .98948 .98944 .98940 .98936 .98931 .98927 .16189 .16218 .16246 .16275 .16304 16333 .98681 .98676 .98671 .98667 .98662 .98657 41 40 39 38 37 36 2 3 29 30 .09440 .09469 .09498 .09527 09556 09585 99553 99551 .99548 99545 99542 .99540 . 1 1 1 76 .11205 .11234 .11263 .11291 .II32O 99374 99370 99367 99364 9936o 99357 .12908 12937 .12966 .12995 .13024 13053 .99163 .99160 .99156 .99152 .99148 .99144 14637 .14666 .14695 H723 14752 .14781 .98923 .98919 .98914 .98910 .98906 .98902 .16361 .16390 .16419 .16447 .16476 16505 .98652 .98648 98643 .9863$ 98633 .98629 35 34 33 32 31 30 3 1 32 33 34 I .09614 .09642 .09671 .09700 .09729 .09758 99537 99534 99531 .99528 99526 99523 II349 .11378 .11407 .11436 .11465 .11494 99354 99351 99347 99344 99341 99337 .13081 .13110 I3 J 39 .13168 I3I97 .13226 .99141 99137 99133 .99129 .99125 .99122 .14810 14838 .14867 .14896 .14925 14954 .98897 ! 98884 .98880 .98876 16533 .16562 'SB! .16620 .16648 .16677 .98624 .98619 .98614 .98609 .98604 .98600 % 11 25 24 9 39 40 4i 42 .09787 .09816 .09845 .09874 .09903 .09932 .99520 99517 99514 995" .99508 .99506 H523 "552 .11580 .11609 .11638 .11667 99334 9933 1 99327 99324 .99320 99317 13254 13283 I33I2 I334I 13370 13399 .99118 .99114 .99110 .99106 .99192 .99098 .14982 .15011 .15040 .15069 .15097 .15126 .98871 .98867 .98863 .98858 .98854 .98849 .16706 16734 .16763 .16792 .16820 .16849 98595 98585 .98580 98575 .98570 23 22 21 20 43 44 9 3 .09961 .09990 .10019 .10048 .10077 .10106 99503 .99500 99497 99494 .99491 .99488 .11696 .11725 .11754 .11783 .11812 .11840 993M .99310 99307 99303 .99300 99297 13427 13456 13485 I35I4 13543 13572 99094 .99083 99079 99075 I5I55 .15184 .15212 .15241 .15270 .15299 98845 .98841 .98836 .98832 .98827 .98823 .16878 .16906 '6935 .1696^ .16992 .17021 98565 .98561 98556 98551 .98546 .98541 \l 15 H 13 12 49 50 51 52 53 54 IOI 35 .10164 .10192 .10221 .10250 .10279 .99485 .99482 99479 99476 99473 .99470 .11869 .11898 .11927 .11956 .11985 .12014 99293 .99290 .99286 99283 99279 -99276 .13600 .13629 13658 13687 .13716 13744 .99071 .99067 .99063 .99059 99055 .99051 15327 15356 15385 I54H .15442 I547I .98818 .98814 .98809 .98805 .98800 .98796 .17050 .17078 .17107 .17136 .17164 17193 98536 98531 98526 .98521 98516 .98511 11 10 I 55 56 9 S . 10308 10337 .10366 10395 .10424 10453 .99467 .99464 .99461 .99458 99455 9945 2 .12013 .12071 .12100 .12129 .12158 .12187 .99272 .99269 99265 .99262 .99258 99255 13773 .13802 :& 13889 13917 .99047 99043 .99039 99035 .99031 .99027 .15500 15529 15557 .15586 .15615 15643 .98791 .98787 .98782 98778 98773 .98769 . i 7222 17250 .17279 .17308 17336 17365 .98506 .98501 .98496 .98491 .98486 .98481 5 4 3 2 I O ( N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine 9 84 83 82 81 80 7 8 TABLE V. 10 11 % 12 13 14 / N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. o i 2 3 1 17365 17393 .17422 I745I 17479 .17508 17537 .98481 .98476 .98471 .98466 1.98461 .9f455 9845 .19081 .19109 .19138 .19167 .19195 .19224 .19252 98163 98i57 .98152 .98146 .98140 98135 .98129 .20791 .20820 .20848 .20877 .20905 .20933 .20962 978i5 .97809 .97803 97797 .97791 97784 97778 .22495 .22523 22552 .22580 .22637 .22665 97437 9743 97424 97417 .97411 97404 97398 .24192 .24220 .24249 .24277 24305 24333 .24362 .97030 .97023 97015 .97008 .97001 60 59 58 !? 55 54 I 9 10 ii 12 17565 17594 .17623 :$ .17708 98445 .98440 98435 .98430 .98425 .98420 .19281 .19309 I 933 8 .19366 19395 .19423 .98124 .98118 .98112 .98107 .98101 .98096 .20990 .21019 .21047 .21076 .21104 .21132 .97772 .97766 .97760 97754 97748 .97742 .22693 .22722 .22750 .22778 .22807 22835 97391 97384 97378 97371 97365 97358 .24390 .24418 .24446 24474 24503 24531 .96980 .96973 .96966 .96959 .96952 96945 53 52 5i 50 1 '3 H \l II :!$6 7 17794 '17852 ! i 7880 .98414 .98409 .98404 .98399 .98394 .98389 .19452 .19481 .19509 .I953f .19566 19595 .98079 98073 .98067 .98061 .21161 .21189 .21218 .21246 .21275 .21303 97735 97729 97723 .97717 .97711 97705 .22863 .22892 .22920 .22948 .22977 .23005 97351 97345 9733 s 97331 97325 97318 24559 .24587 .24615 .24644 .24672 .24700 .96937 .96930 .96923 .96916 .96909 .96902 9 45 44 43 42 19 20 21 22 23 24 3 2 29 30 .17909 17937 .17966 17995 .18023 .18052 3 '& .98362 98357 .19623 .19652 .19680 .19709 .19737 .19766 .98056 .98050 .98044 .98039 98033 .98027 21331 .21360 .21388 .21417 21445 .21474 .97698 .97692 .97680 97673 .97667 23033 .23062 .23090 .23118 .23146 23175 973H 97304 .97298 97291 .97284 97278 .24728 24756 24784 .24813 .24841 .24869 .96894 .96887 .96880 96873 .96866 .96858 41 40 P 11 .18081 .18109 .18138 .18166 18195 . 18224 98352 9f347 .98341 98336 98331 98325 .19794 .19823 .19851 .19880 .19908 19937 .98021 .98016 .98010 .98004 97998 97992 .21502 21530 21559 .21587 .21616 .21644 .97661 2$ 3& .97630 .23203 .23231 .23260 .23288 .23316 23345 .97271 .97264 .97257 97251 97244 97237 .24897 24925 24954 .24982 .25010 .25038 .96851 .96844 .96837 .96829 .96822 96815 35 34 33 32 31 30 31 32 33 34 $ .18252 .18281 .18309 18338 18367 .18395 .98320 98315 .98310 .98304 .98299 .98294 .19965 .19994 .20022 .20051 .20079 .20108 .97987 .97981 97975 97969 .97963 97958 .21672 .21701 .21729 .21758 .21786 .21814 .97623 .97617 .97611 .97604 97598 97592 .23373 .23401 .23429 23458 .23486 23514 .97230 97223 .97217 .97210 .97203 .97196 .25066 .25094 .25122 .25151 25179 .25207 .96807 .96800 .96793 .96786 .96778 .96771 1 25 24 S 39 40 4i 42 .18424 .18452 .18481 .18509 18538 18567 .98288 .98283 .98277 .98272 .98267 .98261 .20136 .20165 .20193 -2O222 .20250 .20279 97952 .97946 97940 97934 .97928 .97922 .21843 .21871 .21899 .21928 .21956 .21985 97585 97579 97573 97566 97560 97553 .23542 .23571 23599 .23627 .23656 .23684 .97189 .97182 .97176 .97169 .97162 .97155 25235 .25263 25291 25320 25348 25376 .96764 .967.56 .96749 .96742 96734 96727 23 22 21 2O 43 44 i 45 46 2 18595 .18624 .18652 .18681 .18710 .18.738 .98256 .98250 98245 .98240 .98234 .98229 .20307 .20336 20364 20393 .20421 .20450 .97916 .97910 97905 97899 97893 97887 .22013 .22041 .22070 .22098 .22126 22155 97547 97541 97534 97528 97521 97515 .23712 .23740 .23769 .23797 .23825 23853 .97148 .97141 97134 .97127 .97120 97"3 25404 25432 .25460 .25488 .25516 25545 .96719 .96712 .96705 .96697 \l 15 H 13 12 49 50 5i 52 53 54 .18767 i8795 .18824 .18852 .18881 .18910 .98223 .98218 .98212 .98207 .98201 .98196 .20478 .20507 20535 .20563 .20592 .2O62O .97881 97875 .97869 .97863 97857 97851 .22183 .22212 .22240 .22268 .22297 22325 .97508 .97502 .97496 97489 97483 97476 .23882 .23910 20 23995 .24023 .97106 .97100 97093 .97086 .97079 .97072 25573 .25601 .25629 25657 .25685 25713 96075 .96667 .96660 .96653 .96645 .96638 II 10 I I 55 56 H 8 .18938 .18967 .18995 .19024 .19052 .19081 N. cos. .98190 .98185 .98179 .98174 .98168 98163 N. sine .20649 .20677 .20706 20734 20763 .20791 N. cos. 97845 97839 97833 97827 .97821 978i5 N. sine 22353 .22382 .22410 .22438 .22467 22495 N. cos. 97470 97463 97457 .97450 97444 97437 N. sine .24051 .24079 .24108 .24136 .24164 .24192 N. cos. .97065 .97058 .97051 .97044 .97037 97030 N. sine 25741 .25769 .25826 -25854 .25882 N. cos. .96630 '12 [96600 96593 M. sine 5 4 3 2 O / 79 7 77 76 75 NATURAL SINES AND COSINES. 79 15 16 17 18 19 t N". sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. o I 2 3 4 .25882 .25910 25938 .25966 25994 .26022 .26050 .96593 .96585 .96578 .96570 .96562 96555 96547 27564 .27592 .27620 .27648 .27676 .27704 .27731 .96126 .96118 .96110 .96102 ^96078 29237 .29265 29293 29321 .29348 .29376 .29404 95630 .95622 95613 95605 955? 6 .95588 95579 .30902 .30929 3957 30985 .31012 .31040 .31068 .95106 95097 .95088 95079 95070 95061 95052 32557 32584 .32612 .32639 .32667 32694 .32722 94552 94542 94533 94523 94514 94504 94495 60 59 58 11 55 54 9 10 ii 12 .26079 .26107 26135 .26163 .26191 .26219 .96540 96532 .96524 96517 .96509 96502 27759 .27787 .27815 .27843 .27871 .27899 .96070 .96062 .96054 .96046 96037 .96029 .29432 .29460 .29487 29515 29543 29571 95571 95562 95554 95545 95536 95528 31095 3 II2 3 .31151 .31178 .31206 3^233 9543 95033 .95024 95015 .95006 94997 32749 .32777 .32804 .32832 32859 32887 .94485 .94476 .94466 94457 94447 .94438 53 52 5i 50 4 J 13 14 ii || .26247 .26275 .26303 -26331 26359 .26387 .96494 .96486 .96479 .96471 .96463 .96456 .27927 27955 .27983 .28011 .28039 .28067 .96021 .96013 .96005 95997 .95989 .95981 2Q5C9 .29626 .29654 .29682 .29710 29737 95519 955" 95502 95493 95485 95476 .31261 .31289 31316 31344 31372 31399 .94988 94979 94970 .94961 94952 94943 .32914 32942 .32969 32997 33024 33051 .94428 .94418 94409 94399 .94390 .94380 9 45 44 43 42 19 20 21 22 3 24 26415 .26443 .26471 .26500 .26528 .26556 .96448 .96440 96433 96425 .96417 .96410 .28095 .28123 .28150 .28178 .28206 .28234 95972 95964 95956 .95948 .95940 95931 29765 29793 .29821 .29849 .29876 .29904 95467 95459 9545 95441 95433 .95424 31427 3H54 .31482 3i5io 31537 31565 94933 94924 94915 .94906 .94897 .94888 33079 .33106 33!34 33161 33189 .33216 94370 .94361 94351 94342 94332 94322 41 40 9 H % 27 28 29 3 .26584 .26012 '.2.6696 .26724 .96402 .96394 .96386 96379 .96371 96363 .28262 .28290 .28318 28346 28374 .28402 95923 95915 95907 .95898 .95890 .95882 .29932 .29960 .29987 30015 30043 .30071 95415 95407 95398 95589 95380 95372 31593 .31620 .31648 3 l6 75 31703 31730 .94878 .94869 .94860 .94851 .94842 .94832 33JM4 332/1 3329? 33326 33353 33381 94313 94303 94293 .94284 .94274 .94264 35 34 33 32 3i 30 3i 32 33 34 i .26752 .26780 .26808 .26836 .26864 .26892 96355 96347 .96340 .96332 .96324 .96316 .20429 28457 .28485 2C 5 I 3 .28541 .28569 95*74 .95865 95857 .95849 .95841 95832 .30098 .30126 3 OI 54 .30182 .30209 .30237 95363 95354 95345 95337 95328 95319 31758 31786 31813 .31841 .31868 .31896 .94823 .94814 .94805 94795 .94786 91-777 33408 33436 33463 33490 33518 33545 94254 94245 94235 94225 94215 .94206 11 3 25 24 9 39 40 41 1 42 .26920 .26948 .26976 .27004 .27032 .27060 .96308 .96301 96293 96285 .96277 .96269 .28597 .28625 .28652 .28680 .28708 .28736 .95824 .95816 95807 95799 95791 95782 30265 .30292 .30320 30348 .30376 .30403 95310 95301 95293 .95284 95275 .95266 31923 3J95 1 31979 .32006 32034 .32061 .94768 94758 94749 94740 94730 94721 33573 33600 .33627 33655 33682 33710 .94196 .94186 .94176 .94167 94157 .94147 23 22 21 20 11 43 44 45 i 46 i:i .27068 .27116 .27144 .27172 .27200 .27228 .96261 96253 .96246 96238 .96230 .96222 .287^4 .28792 28820 .2^847 28875 .20905 95774 95766 -95757 95749 95740 95732 30431 :$ 30514 30542 30570 95257 .95248 95240 95231 .95222 95213 .32089 .32116 .32144 .32171 32199 .32227 .94712 .94702 .94603 .9461)4 .94674 .CJ46CS 33737 33764 33792 33819 33846 33874 94137 .94127 .94118 .94108 .94098 .94088 !! 15 14 13 12 49 50 51 52 53 54 .27256 .27284 .27312 .27340 27368 27396 .96214 .96206 .96198 .96190 .96182 96174 28931 .28959 .28987 29015 .29042 .29070 95724 95715 .95698 .95690 95681 30597 .30625 30653 .30680 .30708 30736 .95204 95*95 .95186 95177 .95168 95159 32254 .32282 32309 32337 32364 32392 94656 .94646 94637 .94627 .94618 .94609 33901 33929 33956 33983 .34011 34038 .94078 .94068 .94058 .94049 94039 .94029 II 10 I i g .27424 .27452 .27480 .27508 27536 27564 .96166 .96158 .96150 .96142 .96134 .96126 .29098 .29126 .29154 .29182 .29209 .29237 .95673 .95664 95656 95647 95639 95630 N. sine 30763 .30791 30819 .30846 .30874 .30902 N. cos. 95150 95142 -95*33 95124 95^5 .95106 .32419 32447 32474 32502 .32529 32557 94599 .94590 .94580 94571 9456i 94552 34065 34093 34120 34H7 34175 .34202 .94019 .94009 93999 .93989 93979 .93969 5 4 3 2 I O N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine 9 74 73 72 71 70 8o TABLE V. 2O 21 22 23 24 / N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. 60 59 58 y 1 55 54 o I 2 3 i .34202 .34229 .34257 .34284 543" 34339 34366 93969 93959 93949 93939 93929 939 J 9 93909 35837 .35864 35891 35918 35945 35973 .36000 93358 93348 93337 93327 93316 93306 93295 .37461 37488 37515 37542 .37569 37595 .37622 37649 .37676 37703 37730 37757 37784 .92718 .92707 .92697 .92686 .92675 .92664 92653 39073 .39100 .39127 39153 .39180 .39207 39234 .92050 .92039 .92028 .92016 .92005 .91994 .91982 .40674 .40700 .40727 40753 .40780 .40806 40833 .40860 .40886 .40913 .40939 .40992 91355 91343 91331 913*9 91307 .91295 .91283 i 9 10 ii 12 34393 .34421 34448 34475 3453 34530 .93899 .93889 .93879 93869 93859 .93849 .36027 36054 .36081 .36108 36135 .36162 93285 93274 .93264 93253 93243 9323 2 .92642 .92631 .92620 .92609 92598 92587 .39260 .39287 39314 39341 39367 39394 .91971 9*959 .91948 .91936 .91925 .91914 .91272 .91260 .91248 .91236 .91224 .91212 53 52 5i 50 49 48 13 H !i [I 34557 34584 .34612 34639 .34666 34694 93839 .93829 93819 .93809 93799 .36190 .36217 36244 .36271 .36298 36325 .93222 .93211 .93201 .93190 .93180 .93169 .37811 37838 37865 37892 37919 37946 .92576 92565 92554 92543 92532 .92521 .39421 .39448 39474 39501 39528 39555 .91902 .91879 .91868 .91856 .91845 .41019 .41045 .41072 .41098 .41125 .41151 .91200 .91188 .91176 .97164 .91152 .91140 47 46 45 44 43 42 19 20 21 22 23 24 34721 34748 34803 34830 34857 93779 93769 93759 93748 93738 .93728 36352 3637? .36406 36434- .36461 .36488 93159 .93148 93137 93127 .93116 93106 37973 37999 .38026 38*053 .38080 .38107 .92510 92499 .92488 92477 .92466 92455 3958i .39608 19688 39715 91833 .91822 .91810 .91799 .91787 9 J 775 .41178 .41204 .41231 .41257 .41284 .41310 .91128 .91116 .91104 .91092 .91080 .91068 41 40 fs 11 2 2 29 30 .34884 .34912 34939 .34966 34993 35021 93718 93708 .93698 93688 .93677 93667 36515 36542 36569 .36596 36623 .36650 93095 .93084 93074 93063 93052 93042 38134 .38161 .38188 .38215 .38241 .38268 92444 92432 .92421 .92410 .92399 .92388 39741 39768 39795 39822 .39848 39875 .91764 91752 .91741 .91729 .91718 .91706 41337 41363 .41390 .41416 41443 .41469 .91056 .91044 .91032 .91020 .91008 .90996 35 34 33 32 3i 30 3i 32 33 34 8 35048 35075 35102 35130 35157 35184 93657 93647 .93637 .93626 .93616 .93606 .36677 36704 .36731 .36758 36785 .36812 93031 .93020 .93010 .92978 .38295 38322 .3f349 38376 38403 38430 .92377 .92366 92355 92343 92332 92321 .39902 .39928 39955 .39982 .40008 40035 .91694 .91683 .91671 .91660 .91648 91636 .41496 .41522 41549 .41575 .41602 .41628 .90984 .90972 .90960 .90948 .90936 .90924 27 26 25 24 23" 22 21 2O 19 P 39 40 4i 42 35 211 .35239 .35266 .35293 35320 35347 93596 93585 93575 93565 93555 93544 36839 .36867 .36894 36921 .36948 36975 .92967 92956 92945 92935 .92924 .92913 38456 38483 .38510 .38537 .38564 38591 .92310 .92299 .92287 .92276 .92265 92254 .40062 .40088 .40115 .40141 .40168 .40195 .91625 .91613 .91601 .91590 91578 .91566 41655 .41681 .41707 41734 .41760 .41787 !9o875 .90863 90851 43 44 g 48 35375 35402 35429 35456 35484 355" 93534 93524 935'4 93503 93493 93483 .37002 .37029 37056 37083 .37110 -37137 .92902 .92892 .92881 .92870 .92859 .92849 .38617 .38644 .38671 38698 .38725 38752 92243 .92231 .92220 .92209 .92198 .92186 .40221 .40248 40275 40301 .40328 .40355 91555 91543 9i53i 9I5J9 .91508 .91496 .41813 .41840 .41866 .41892 .41919 41945 90839 .90826 .90814 .90802 .90790 .90778 !Z 15 H 13 12 ~II~~ 10 I 49 50 51 52 53 S4 35538 35565 35592 356i9 .35647 35674 93472 93462 93452 93441 93431 .93420 37164 37I9I .37218 .37245 .37272 .37299 .92838 .92827 .92816 .92805 .92794 .92784 .38778 .38805 38832 38859 . 3 8856 .38912 92175 .92164 .92152 .92141 .92130 .92119 .40381 .40408 .40434 .40461 .40488 .40514 .91484 .91472 .91461 .91449 9H37 .91425 .41972 .41998 .42024 .42051 .42077 .42104 .90766 90753 .90741 .90729 .90717 .90704 ft 5 58 7 S 35701 .35728 35755 .35782 35810 35837 .93410 .93400 93389 93379 93368 93358 .37326 37353 37380 37407 37434 .37461 92773 .92762 92751 .92740 .92729 .92718 38939 .38966 38993 .39020 39046 39073 .92107 .92096 .92085 .92073 .92062 .92050 .40541 .40567 .40594 .40621 .40647 .40674 .91414 .91402 .91390 91378 .91366 91355 .42130 .42156 .42183 .42205 42235 .42262 ! 90668 90655 90643 .90631 5 4 3 I N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine f 69 6' 67 66 65 NATURAL SINES AND COSINES. 8l Vw 2, 5 2< J 2' r 21 * 21 > / ^. sine \. cos. X. sine N. cos. \. sine \. cos. N T . sine ^. cos. N. sine N. cos. o I 2 3 4 I .42262 .42288 42315 .42341 .42367 .42394 .42420 .90631 .90618 .90606 90594 .90582 .90569 90557 43837 43863 43889 .43916 43942 .43968 43994 .89879 .89867 .89854 .89841 .89828 .89816 .89803 45399 45425 45451 45477 45503 45529 45554 .89101 .89087 .89074 .89061 .89048 89035 .89021 .46947 46973 46999 .47024 .47050 .47076 .47101 .88295 .88281 .88267 .88254 .88240 .88226 .88213 .48481 .48506 48532 48557 48583 .48608 .48634 .87462 .87448 87434 .87420 .87406 .87391 .87377 60 59 5* 57 5 55 54 I 9 10 ii 12 .42446 42473 .42499 42525 42552 .42578 90545 90532 .90520 .90507 .90495 .90483 .44020 .44046 .44072 .44098 .44124 44151 .89790 89777 .89764 8975 2 89739 .89726 .4558o .45606 45632 45658 .45684 45710 ^88968 88955 .88942 .47127 47153 .47178 .47204 .47229 47255 .88199 .88185 .88172 .88158 .88144 .88130 48659 .48684 .48710 .48735 .48761 .48786 87363 87349 87335 .87321 .87306 .87292 53 52 5i 50 49 48 13 H !i jl .42604 .42631 .42657 42683 .42709 42736 .90470 .90458 .90446 90433 .90421 .90408 44177 .44203 .44229 44255 .44281 44307 89713 .89700 .89687 .89674 .89662 .89649 45736 .45762 45787 45813 45839 45865 .88928 88915 .88902 .88888 88875 .88862 .47281 47306 47332 47358 4733 .47409 .88117 .88103 .88089 .88075 .88062 .88048 .48811 48837 .48862 .48888 .48913 48938 .87278 .87264 .87250 87235 .87221 .87207 47 46 45 44 43 42 19 20 21 22 23 24 .42762 .42788 42815 .42841 .42867 .42894 .90396 90383 .90371 90358 .90346 90334 44333 44359 44385 .44411 44437 44464 .89636 .89623 .89610 89597 .89584 89571 .45891 45917 .45942 .45968 45994 .46020 .88848 .88835 .88822 .88808 .88795 .88782 47434 .47460 .47486 475 47537 .47562 .88034 .88020 .88006 87993 87979 .87965 .48964 .48989 .49014 .49040 .49065 .49090 87193 .87178 .87164 87150 .87136 .87121 41 40 P M 2 3 29 30 .42920 .42946 42972 .42999 43025 43051 .90321 .90309 .90296 .90284 .90271 .90259 .44490 .44516 .44542 .44568 44594 .44620 .89558 89545 89532 89519 .89506 .89493 .46046 .46072 .46097 .46123 .46149 46i75 .88768 88755 .88741 .88728 88715 .88701 47588 476i4 .47639 47665 .47690 .47716 8795! 87937 .87923 .87909 .87896 .87882 .49116 .49141 .49166 .49192 .49217 .49242 .87107 .87093 .87079 .87064 .87050 .87036 35 , 34 33 32 3i 30 31 32 33 34 35 36 4377 .43104 43130 43156 .43182 .43209 .90246 .90233 .90221 .90208 .90196 .90183 .44646 .44672 .44698 44724 44750 .44776 .89480 .89467 .89454 .89441 .89428 .89415 .46201 .46226 .46252 .46278 .46304 46330 .88688 .88674 .88661 .88647 .88634 .88620 47741 47767 47793 .47818 47844 .47869 .87868 .87854 .87840 .87826 .87812 .87798 .49268 .49293 .49318 49344 49369 49394 .87021 .87007 .86993 .86978 .86964 .86949 i 27 26 25 24 11 39 40 4i 42 43235 .43261 .43287 43313 43340 43366 .90171 .90158 .90146 .90133 .90120 .90108 .44802 .44828 .44854 .44880 .44906 .44932 .89402 .89389 89376 89363 89350 89337 46355 .46381 .46407 .46433 46458 .46484 .88607 .88593 .88580 .88566 88553 88539 47895 .47920 .47946 47971 47997 .48022 .87784 .87770 .87756 87743 .87729 87715 .49419 49445 .49470 49495 .49521 .49546 86935 .86921 !86878 .86863 23 22 21 20 19 43 44 45 46 47 1 48 43392 .43418 43445 43471 43497 43523 90095 .90082 .90070 .90057 .90045 .90032 .44958 .44984 45010 45036 .45062 .45088 89324 .89311 .89298 .89285 .89272 .89259 .46510 46536 .46561 .46587 .46613 46639 .88526 .88512 .88499 .88485 .88472 .88458 .48048 .48073 .48099 .48124 .48150 48i75 .87701 .87687 87673 .87659 .87645 .87631 49571 .49596 .49622 .49647 .49672 49697 .86849 .86834 .86820 .86805 .86791 86777 || *5 14 *3 12 49 50 51 52 53 54 43549 43575 .43002 43628 43654 .43680 .90019 .90007 .89994 .89981 .89968 .89956 45 "4 .45140 .45166 .45192 .45218 45243 89245 .89232 .89219 .89206 .89193 .89180 .46664 .46690 .46716 .46742 .46767 46793 .88445 .88431 .88417 .88404 .88390 88377 .48201 .48226 .48252 .48277 48303 .48328 .87617 .87603 .87589 87575 .87561 .87546 49723 49748 49773 .49798 .49824 .49849 .86762 .86748 86733 .86719 .86704 .86690 II 10 7 58 60 .43706 43733 43759 43785 438" .43837 .89943 .89930 .89918 89879 .45269 .45295 45321 45347 45373 45399 .89167 .89153 .89140 .89127 .89114 .89101 .46819 .46844 .46870 .46896 .46921 46947 .88363 .88349 88336 .88322 .88308 .88295 .48354 .48379 .48405 .48430 .48456 .48481 87532 .87518 87504 .87490 .87476 .87462 .49874 .49899 .49924 .49950 49975 .50000 .86675 .86661 .86646 .86632 .86617 .86603 5 4 3 2 I O N. cos N. sine N. cos N. sine N. cos. N. sine N. cos N. sine N. cos. N.sine / 1 4 6 3 6 2 e 1 TABLE V. 30 31 32 33 34 f N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. I 2 3 4 | .50000 50025 .50050 .50076 .50101 .50126 .50151 .86603 .86588 86573 86559 .86544 .86530 .86515 51504 51529 51554 51579 .51604 .51628 5 l6 53 .85717 .85702 .85687 .85672 85657 .85642 -85627 .52992 53017 53041 .53066 53091 53H5 53 HO .84805 .84789 .84774 84759 84743 .84728 .84712 54464 .54488 54513 54537 .54561 54586 .54610 .83867 83851 83835 83819 .83804 83788 .83772 .55919 55943 55968 55992 .56016 .56040 .56064 ^82871 .82855 .82839 .82822 .82806 60 II H 55 54 2 9 10 ii 12 50176 .50201 .50227 .50252 .50277 50302 .86501 .86486 .86471 .86457 .86442 .86427 .51678 STO .51728 51753 .85612 85597 85582 85567 85551 85536 53164 53189 53214 53238 53263 5328S .84697 .84681 .84666 .84650 84635 .84619 54635 54659 54683 .54708 54732 54756 .83750 .83740 .83724 .83708 .83692 .83676 .56088 .56112 56136 .56160 .56184 .56208 .82790 82773 .82757 .82741 .82724 .82708 53 52 5i 50 49 48 13 14 is [I 50327 50352 50377 50403 .50428 5453 .86413 .86398 .86384 .86369 86354 .86340 .51828 .51852 51877 .51902 51927 5!952 85521 .85506 .85491 .85476 .85461 .85446 533 12 53337 5336i 53386 -534" 53435 .84604 .84588 84573 84557 84542 .84526 5478i 54805 54829 54854 .54878 .54902 .83660 83645 .83629 83613 83597 .83581 .56232 .56256 .56280 56305 56329 .56353 .82692 .82675 .82659 .82643 .82626 .82610 47 46 45 44 43 42 19 20 21 22 23 24 .50478 50503 50528 50553 .50578 .50603 .86325 .86310 ! 86266 .86251 51977 .52002 .52026 52051 .52076 .52101 85431 85416 .85401 85385 85370 85355 53460 .53484 53509 53534 53558 53583 .84511 84495 .84480 .84464 .84448 84433 54927 54951 54975 54999 55024 55048 83565 83549 83533 83517 83501 83485 56377 .56401 56425 56449 56473 56497 82593 82577 .82561 82544 .82528 .82511 41 40 1 II 3 29 30 .50628 50654 .50679 .50704 .50729 .50754 86237 .86222 .86207 .86192 .86178 .86163 .52126 52151 52175 .52200 .52225 52250 85340 85325 .85310 .85294 .85279 .85264 53607 53632 53656 .53681 53705 5373 .84417 .84402 .84386 .84370 .84355 84339 55072 55097 55I2I 55145 55169 55194 .83469 83453 83437 .83421 83405 83389 56521 56545 56569 56593 56617 .56641 .82495 .82478 .82462 .82446 .82429 .82413 35 34 33 32 31 30 '11 11 25 24 3i 32 33 34 ii .50779 .50804 .50829 50854 .50879 .50904 .86148 86133 .86119 .86104 .86089 .86074 52275 52299 52324 52349 52374 52399 .85249 85234 .85218 .85203 .85188 85173 53754 53779 53804 .53828 53853 53877 .84324 .84308 .84292 .84277 .84261 84245 .55218 55242 55266 55291 55315 55339 83373 83356 .83340 83324 .83308 .83292 .56665 .56689 56736 .56760 .56784 .82396 .82380 82363 82347 82330 82314 11 39 40 4i 42 .50929 50954 50979 .51004 .51029 51054 .86059 .86045 .86030 .86015 .86000 85985 5 2 423 .52448 52473 .52498 .52522 52547 .85157 85142 .85127 .85112 .85096 .85081 53902 53926 53951 53975 .54000 .54024 84230 .84214 .84198 .84182 .84167 .84151 :P 55412 55436 55460 55484 83276 .83260 .83244 .83228 .83212 83195 .56808 .56832 '56904 .56928 .82297 .82281 .82264 .82248 .82231 .82214 23 22 21 2O 43 44 1 2 51079 .51104 .51129 5H54 5H79 .51204 .85970 85956 .85941 .85926 .85911 .85896 5 2 572 52597 52621 .52646 .52671 .52696 .85066 85051 85035 .85020 .85005 .84989 .54049 54073 54097 .54122 .54146 .54171 84135 .84120 .84104 .84088 .84072 .84057 55509 55533 55557 5558i 55605 55630 83179 83163 83147 83131 .83115 .83098 .56952 .56976 .57000 57024 57047 57071 .82198 .82181 .82165 .82148 .82132 .82115 \l 15 H '3 12 49 50 51 52 53 54 .51229 51254 51279 5 I 34 5 I 3 2 9 :5i354 .85881 .85866 .85851 .85836 .85821 .85806 .52720 52745 52770 .52794 .52819 52844 84974 .84959 .84943 .84928 .84913 .84897 54195 .54220 54244 .54269 54293 54317 .84041 .84025 .84009 83994 .83978 .83962 55654 55678 55702 55726 55750 55775 .83082 .83066 .83050 83034 83017 .83001 57095 57"9 57143 .57167 .57191 57215 .82098 .82082 .82065 .82048 .82032 .82015 II 10 | P R e 5 I 379 .51404 .51429 51454 S479 51504 .85792 85777 .85762 85747 85732 85717 .52869 52893 .52918 52943 52967 .52992 .84882 .84866 .84851 .84836 .84820 .84805 54342 .54366 54391 .54415 .54440 .54464 .83946 83930 83915 .83899 .83883 83867 55799 55823 .55847 55871 55895 55919 .82985 .82969 82953 .82936 .82920 .82904 5723* .57262 .57286 57310 .57334 57358 .81999 .81982 81965 .81949 .81932 81915 5 4 3 2 I O N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine f 59 . 68 57 56 55 NATURAL SINES AND COSINES. 8 35 36 37 38 39 1 f N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. i 2 3 4 I 57358 573^1 57405 57429 57453 57477 57501 81915 .81899 .81882 .81865 .81848 .81832 .81815 .58779 .58802 .58826 .58849 58873 .58896 .58920 .80902 .80885 .80867 .80850 .80833 .80816 .80799 .60182 .60205 .60228 .60251 .60274 .60298 .60321 .79864 .79846 .79829 .79811 79793 79776 79758 .61566 61589 .61612 61635 .61658 .61681 .61704 .78801 78783 78765 .78747 .78729 .78711 .78694 .62932 62955 .62977 .63000 .63022 63045 .63068 .77715 .77696 77678 .77660 .77641 .77623 .77605 60 * H 55 ' 54 I I 9 1C ii 12 13 14 3 17 18 57524 57548 57572 57596 57619 57643 .81798 .81782 .81765 .81748 .81731 .81714 58943 58967 .58990 .59014 59037 .59061 .80782 .80765 .80748 .80730 .80713 .80696 .60344 .60367 .60390 .60414 .60437 .60460 79741 79723 .79706 .79688 .79671 79653 .61726 6i749 .61772 .61795 !6i84i .78676 .78658 .78640 .78622 .78604 .78586 .63090 63113 I 3 ! 3 ! .63180 63203 77586 77568 77550 77531 77513 77494 53 52 5i So 49 48 .57667 57691 57715 57738 57762 57786 .81698 .81681 .81664 .81647 .81631 .81614 .59084 .59108 59I3I 59154 .59178 .59201 .80679 .80662 .80644 .80627 .80610 80593 .60483 .60506 60529 60553 .60576 .60599 .79635 .79618 .79600 .79583 79565 79547 .61864 .61887 .61909 .61932 .61955 .61978 . 78568 78550 78532 .785H .78496 .78478 63225 .63248 .63271 63293 63316 63338 .77476 77458 77439 .77421 .77402 77384 47 46 45 44 43 42 19 20 21 22 23 2 4 57810 57833 57857 .57881 57904 .57928 igg .81563 .81546 .81530 81513 59225 .59248 .59272 59295 59318 59342 .80576 .80558 .80541 .80524 .80507 .80489 .60622 .60645 .60668 .60691 .60714 60738 79530 79512 79494 79477 79459 .79441 .62001 .62024 .62046 .62069 .62092 .62115 .78460 .78442 .78424 .78405 78387 .78369 .63361 * 338 2 .63406 .63428 63451 63473 .77366 77347 77329 .77.310 .77292 .77273 41 40 I 3 2 29 30 57952 .57976 57999 -58023 .58047 .58070 .81496 .81479 .81462 .81445 .81428 .81412 59365 59389 .59412 .59436 -59459 .59482 .80472 80*455 80438 .80420 .80403 .80386 .60761 .60784 .60807 .60830 .60853 .60876 .79424 .79406 .79388 79371 79353 79335 .62138 .62160 .62183 .62206 .62229 .62251 78351 78333 78315 .78297 .78279 .78261 .63496 .63518 63540 63563 63585 .63608 77255 77236 .77218 .77199 .77181 .77162 35 34 33 32 3i 30 31 32 33 34 g .58094 .58118 .58141 .58165 .58189 .58212 .81395 81378 .81361 .81344 81327 .81310 59506 -59529 59552 59576 59599 .59622 .80368 .80351 80334 .80316 .80299 .80282 .60899 .60922 .60945 .60968 .60991 .61015 793 l8 .79300 .79282 .79264 .79247 .79229 .62274 .62297 .62320 .62342 62365 .62388 .78243 .78225 .78206 .78188 .78170 78152 .63630 .63653 63675 .63698 .63720 63742 77144 77125 .77107 .77088 .77070 77051 2 25 24 S 39 40 4i 42 58236 .58260 .58283 58307 58330 58354 81293 .81276 .81259 .81242 .81225 .81208 59646 .59669 59693 59716 59739 59/63 .80264 .80247 .80230 .80212 .80195 .80178 .61038 .61061 .61084 .61107 .61130 6"53 .79211 .79193 .79176 79158 .79140 .79122 .62411 62433 .62456 .62479 .62502 .62524 78i34 .78116 .78098 .78079 .78061 .78043 63765 .63787 .63810 .63832 .63854 .63877 77033 .77014 .76996 .76977 76959 .76940 23 22 21 2O "il 15 H 13 12 43 44 45 46 47 48 58378 .58401 58425 .58449 .58472 .58496 .81191 .81174 8n57 .81140 .81123 .81106 59786 .59809 59832 59856 598/9 .59902 .80160 .80143 .80125 .80108 .80091 .80073 .61176 .61199 .61222 61245 .61268 .61291 79105 79087 .79069 79051 79033 .79016 .62547 .62570 .62592 .62615 .62638 .62660 .78025 . 78007 77988 779/0 77952 77934 .63899 .63922 .63944 .63966 .63989 .64011 -.76921 .76903 .76884 .76866 .76847 .76828 49 50 51 52 53 54 .58519 58543 58567 58590 .58614 58637 .81089 .81072 81055 .81038 .81021 .81004 .59926 59949 59972 59995 .60019 .60042 .80056 .80038 .80021 .80003 .79986 .79968 .61314 :$ ^ .61406 .61429 .78998 .78980 .78962 .78944 .78926 .78908 .62683 .62706 .62728 .62751 62774 .62796 .77916 77897 .77879 .77861 77843 .77824 64033 .64056 .64078 .64100 .64123 .64145 .76810 .76791 .76772 76754 76735 .76717 II IO I 55 56 8 .58661 .58684 58708 58731 .58755 .58779 .80987 .80970 80953 .80936 .80919 .80902 .60065 .60089 .60112 60135 .60158 .60182 79951 79934 .79916 .79899 .79881 .79864 .61451 61474 .61497 .61520 61543 .61566 .78891 78873 78855 78837 .78819 .78801 .62819 .62842 .62864 .62887 .62909 .62932 .77806 .77788 .77769 77751 77733 77715 .64167 .64190 .64212 .64234 .64256 .64279 .76698 76679 .76661 .76642 .76623 .76604 N. sine S 4 3 2 I O N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. i 54 53 52 51 50 TABLE V. , 40 41 42 43 44 9 N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. 60 % H l 55 . 54 O I 2 3 4 I .64279 .64301 64323 .64346 .64368 .64390 .64412 . 76604 76586 7 6 567 .76548 76530 .76511 .76492 .65606 .65628 .65650 .65672 .65694 65716 65738 75471 75452 75433 75414 75395 75375 75356 66913 66935 .66956 .66978 .66999 .67021 .67043 743H 74295 74276 .74256 74237 .74217 .74198 .68200 .68221 .68242 .68264 .68285 .68306 .68327 73135 .73116 73096 .73076 73056 73036 .73016 .69466 .69487 .69508 .69529 .69549 .69570 .69591 71934 .71914 .71894 71873 71853 71833 .71813 i 9 10 ii 12 64435 64457 64479 .64501 .64524 .64546 76473 76455 76436 . 7641 7 . 76398 76380 65759 .65781 65803 .65825 -65847 .65869 75337 753i8 75299 .75280 75261 75241 .67064 .67086 .67107 .67129 .67151 .67172 .74178 .74159 74139 .74120 .74100 .74080 .68349 68370 68391 .68412 .68434 68455 .72996 .72976 72957 72937 .72917 72897 .69612 69633 .69654 '%& .69696 .69717 .71792 .71772 .71752 .71732 .71711 .71691 53 52 5i 50 49 48 13 H :i 17 18 .64568 .64590 .64612 64635 64657 .64679 .76361 .76342 .76323 76304 .76286 .76267 .65891 65913 65935 .65956 .65978 .66000 .75222 75203 75184 75165 75H6 .75126 .67194 67215 .67237 .67258 .67280 .67301 .74061 .74041 .74022 .74002 73983 73963 .68476 .68497 .68518 .68539 .68561 .68582 72877 72857 72837 .72817 .72797 72777 .69737 .69758 .69779 .69800 .69821 .69842 .71671 .71650 .71630 .71610 71590 71569 47 46 45 44 43 42 19 20 21 22 23 24 .64701 .64723 .64746 .64768 .64790 .64812 .76248 .76229 .76210 .76192 76i73 76i54 .66022 .66044 .66066 .66088 .66109 .66131 75107 .75088 .75069 75050 75030 .75011 67323 %m 67387 .67409 6743 73944 73924 73904 73885 73865 .73846 .68603 .68624 .68645 .68666 .68688 .68709 72757 .72737 .72717 .72697 .72677 .72657 .69862 .69883 .69904 .69925 .69946 .69966 71549 71529 .71508 .71488 .71468 71447 41 40 11 H II 2 29 30 .64834 .64856 .64878 .64901 .64923 .64945 76135 .76116 .76097 76078 76059 .76041 66153 I7S .66197 .66218 .66240 .66262 .74992 74973 74953 74934 74915 .74896 .67452 67473 67495 .67516 67538 67559 73826 .73806 73787 .73767 73747 73728 .68730 .68751 .68772 68793 .68814 68835 .72637 .72617 72597 72577 72557 72537 .69987 .70008 .70029 .70049 .70070 .70091 .71427 .71407 71386 .71366 71345 71325 35 34 33 32 31 30 3* 32 33 34 P .64967 .64989 .65011 65033 65055 .65077 . 76022 .76003 75984 75965 75946 75927 .66284 .66306 66327 .66349 66371 .66393 .74876 74857 .74838 .74818 74799 .74780 95 .67623 .67645 .67666 .67688 73708 .73688 .73669 73649 73629 .73610 .68857 .68878 .68899 .68920 .68941 .68962 72517 72497 72477 72457 72437 .72417 .70112 .70132 70153 .70174 70195 .70215 71305 .71284 .71264 71243 .71223 .71203 3 11 25 24 P 39 40 4i 42 .65100 .65122 3% .65188 .65210 75870 .75851 75832 758i3 .66414 .66436 .66458 .66480 .66501 66523 .74760 74741 .74722 74703 .74683 .74664 .67709 .67730 .67752 67773 67795 .67816 73590 73570 73551 73531 735" 73491 .68983 .69004 .69025 .69046 .69067 .69088 72397 .72377 72357 72337 .72317 .72297 .70236 70257 .70277 .70298 .70319 70339 .71182 .71162 .71141 .71121 .71100 .71080 23 22 21 2O lo 43 44 $ i 65232 8 .65298 .65320 65342 75794 75775 75756 75738 75719 .75700 66545 .66566 66588 .66610 .66632 .66653 74644 .74625 .74606 74586 74567 74548 .67837 67859 .67880 .67901 67923 .67944 73472 73452 -73432 73413 73393 73373 .69109 .69130 69151 .69172 .69193 .69214 .72277 72257 .72236 .72216 .72196 .72176 .70360 .70381 .70401 .70422 70443 70463 71059 .71039 .71019 .70998 .70978 70957 11 15 H 13 12 49 50 51 52 53 54 65364 65386 .65408 65430 .65452 65474 .75680 .75661 .75642 75623 .75604 75585 .66675 .66697 .66718 .66740 .66762 .66783 .74528 74509 .74489 .74470 74451 74431 .67965 .68029 .68051 .68072 73353 73333 733!4 73294 73274 73254 69235 .69256 .69277 .69298 .69319 69340 72156 .72136 .72116 .72095 72075 72055 .70484 70505 70525 70546 70567 70587 .70937 .70916 .70896 70875 70855 .70834 II 10 1 6 P II g .65496 .65518 65540 .65562 l& .75566 75547 75528 75509 75490 75471 .66805 .66827 .66848 .66870 .66891 66913 .74412 74392 74373 74353 74334 743H .68093 .68115 .68136 .68157 .68179 .68200 73234 .73215 73*95 73*75 73155 73135 69361 .69382 69403 .69424 69445 .69466 72035 72015 .71995 .71974 .71954 .71934 .70608 .70628 .70649 .70670 .70690 .70711 70813 70793 .70772 .70752 -7073 1 .70711 5 4 3 2 I N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine N. cos. N. sine 9 49 48 47 46 45 TABLE VI. ADDITION AND SUBTRACTION LOGARITHMS. 8 TABLE VI. ; ADDITION AND SUBTRACTION LOGARITHMS. PRECEPTS. I. Wfan difference of given logarithms is less than 2.OO. ADDITION. Enter table with difference between logarithms as Arg. A, and take out B. Add B to subtracted logarithm. SUBTRACTION. Subtract lesser from greater logarithm; enter with the difference as B, and take out A. Add A to the subtracted logarithm. II. When difference of given logarithms exceeds 2.OO. Subtract lesser from greater. ADDITION. Enter table with difference as Arg. A t take out BA and add it to the greater logarithm. SUBTRACTION. Enter column B with difference of logarithms ; take out BA, and subtract it from greater logarithm. A. B. 1 2 3 4 5 6 7 8 9 Prop. Pts. 5- o.oo ooo 001 OOI OOI OOI OOI 002 002 "003 "ooj m^mmm G.O 004 004 005 005 005 005 005 005 005 005 6.1 005 006 006 006 006 006 006 006 007 007 3 4 5 6 6.2 007 007 007 007 008 008 008 008 008 008 i 0.3 o 4 0.3 0.6 6.3 009 009 009 009 OIO OIO OIO OIO OIO Oil 2 0.6 08 I.O 1.3 3 0.9 Z 2 1.5 1.8 6.4 on on Oil OI2 OI2 012 013 013 013 013 4 1.3 i 6 3.O a 4 6.5 014 014 014 015 015 015 016 016 017 017 5 i. 5 3 9 *5 3.0 6.6 017 018 018 019 019 019 020 020 02 1 02 1 6 1.8 a 4 3-0 3-6 6.7 022 022 023 023 024 024 025 026 026 027 7 8 3.1 8.4 3 8 3 2 3-5 4.0 4-a 4.8 68 027 028 029 O2Q O3O O3I Oil O32 o^* 014 6.9 034 035 036 X 037 038 039 J 040 *J O4I ijj 041 JT^ 042 7.0 043 044 045 047 048 049 050 051 052 053 7-1 055 056 057 059 060 061 063 064 066 06 7 j 7 O 7 0.8 9 TO 7-2 069 070 072 074 075 077 079 08 1 083 085 2 w.y 1.4 1.6 0.9 1.8 2.O 7-3 087 089 091 093 095 097 099 102 104 106 3 2.1 3-4 2.7 3-o 7-4 109 in ii/ 117 119 122 125 128 131 134 4 2.8 3-2 3-6 4.0 7-5 137 140 144 147 150 154 157 161 165 169 5 6 3-5 4- 2 4.0 4.8 4-5 5 A 5-o 6.0 7.6 173 177 181 185 I8 9 194 198 203 207 212 7 4-9 5-6 '^ 6-3 7.0 7-7 217 222 227 233 2 3 8 244 249 255 261 267 8 5-6 6-4 7.2 8.0 7-8 273 280 286 293 299 3 06 313 321 328 336 9 3 7.3 8.1 9.0 79 344 352 360 3 68 377 385 394 403 413 422 1 8.0 432 442 452 463 474 485 496 507 519 531 1 A> B. 1 2 a . 4 5 6 7 8 9 Prop. Pts. 86 TABLE VL ADD i lo S<* ~ lo S* = A > Qrm / lo S* ~ lo S<* = & Am Mlo g (0 + ) = log0 + ^. SU3 'Uog(0~) = log + ^. A. rsloo 8.01 8.02 8.03 8.04 ; 8.05 8.06 8.07 8.08 8.09 8.10 8. ii 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 1 8-23' 8.24 8.25 8.26 8.27 8.28 8.29 8.30 .3i 8.32 I- Sip l :5 8.39 8.40 Itt 8.43 8.44 I' 4 ! 8.46 8.47 8.48 8.49 8.50 11. o.oo 432 1 433 i 434 a 435 4 ~ 5 ~ M 438 7 m^m^mm 439 8 440 9 44i "451 462 473 483 495 506 518 530 542 Proi ^^^w I 2 3 i i 9 i 2 3 4 i I 9 i 2 3 4 1 I 9 ).Pt8. 1 ' * 0.8 J:5 \i :\ 3 0.6 ?:I 1:1 * '.' *1 0.8 \ 1.2 1.6 2.0 :i !:J 442 452 463 474 485 496 507 519 53i 443 453 464 475 486 497 508 520 532 444 454 465 476 487 498 510 521 533 445 456 466 477 488 499 5" 523 535 446 457 467 478 489 500 512 524 536 447 458 468 479 490 502 513 525 537 448 459 469 480 49 i 503 5H 526 538 449 460 470 481 492 504 515 527 540 450 461 47i 482 494 505 517 529 541 543 5|6 569 582 595 609 623 638 652 667 683 545 546 547 548 550 55i 552 553 555 557 570 583 597 611 625 639 654 669 558 57i 585 598 612 626 641 655 671 560 $ 599 613 628 642 657 672 561 574 587 601 6i5 629 644 658 674 562 575 589 602 616 630 645 660 675 564 577 590 604 618 632 646 661 677 565 578 59i 605 619 633 648 663 678 566 579 593 606 620 635 649 664 680 567 58i 594 608 622 636 & 681 684 686 688 689 691 692 694 696 697 699 715 731 748 766 783 801 820 839 858 "8^8 898 919 940 962 984 o.oi 006 030 Q53 077 700 716 733 750 767 785 803 822 841 702 718 735 752 769 787 805 823 842 703 720 736 753 771 789 807 825 844 705 721 738 755 773 790 809 827 846 707 723 740 757 774 792 810 829 848 708 725 741 759 776 794 812 831 850 710 726 743 760 778 796 814 833 852 712 728 745 762 780 798 816 835 854 713 730 747 764 78i 799 818 837 856 860 862 864 866 868 870 872 874 876 880 900 921 942 964 986 009 032 056 882 902 923 944 966 988 on 034 058 884 904 925 946 968 99 o 013 037 060 886 906 927 948 970 993 016 039 063 888 908 929 95i 973 995 018 041 065 890 910 93i 953 975 997 020 044 068 892 912 933 955 977 999 022 046 070 894 915 936 957 979 *002 O23 048 073 896 917 938 P? *4 027 51 075 080 082 085 087 090 092 095 097 IOO 1 02 128 153 1 80 207 23 263 292 322 352 105 130 156 183 2IC 23* 266 295 32] 107 133 159 185 21; 240 269 298 328 HO 135 161 1 88 215 243 272 30 33 112 138 164 I 9 I 218 246 275 304 334 H5 140 167 193 221 249 2 7 * 307 337 117 143 169 196 22; 252 280 310 340 120 146 172 199 226 25" 283 313 343 122 148 175 2O2 229 257 286 316 346 125 151 177 204 232 260 289 319 349 35! 358 36 364 36* 37 374 377 380 II A - B. 1 2 8 4 5 6 7 8 9 Prop. Pts. o ADDITION AND SUBTRACTION LOGARITHMS. g Ann J lQ % b "" lo % a ^' STTB J lo S<* lo ^ = A ' I log ( + *) = log* -f A bIJ i log (a - J) = log* + A. A. ~&60 8.5i 8.52 8-53 8.54 8-55 8.56 8.57 8.58 8-59 8.60 8.61 8.62 8.63 8.64 8.65 8.66 8.67 8.68 8.69 8.70 8.7i 8.72 8.73 8.74 8.75 8.76 *.77 8.78 8.79 8.80 8.81 8.82 8.83 8.84 8.85 8.86 8.87 8.88 8.89 8.90 8.91 8.92 8-93 8.94 8.95 8.96 8.97 8.98 8.99 9.00 B. o.oi 352 383 415 447 480 5H 549 584 621 658 695 1 "355 2 IP a ~i 4 364 5 368 6 37i 7 374 8 377 9 380 P I 2 3 4 i I 9 I 2 3 i i 9 i 2 3 I I 9 i 2 3 i I 9 rop. MHM^ s 0.3 0.6 0.9 1.2 *:I 2.1 2.4 2.7 5 0-5 I.O -5 2.O 2-5 3-o 3-5 4-0 4-5 7 0.7 1-4 2.1 2.8 3-5 4.2 4-9 I' 6 0-3 9 !:! 11 4-5 *1 *; Pts. mmm^^m^mmmm 4 0.4 0.8 1.2 1.6 2.O 2:3 & 6 0.6 1.2 1.8 2.4 3-o 3-6 4.2 4.8 54 8 0.8 1.6 2.4 3-2 4.0 4.8 I' 6 6.4 7.2 so I.O 2.O 30 4-0 5-0 6.0 7.0 8.0 9.0 386 418 450 484 518 552 588 624 661 389 421 454 487 521 556 I 9 l 628 665 "703 393 424 457 490 5^5 559 595 632 669 39 6 428 460 494 5 2 8 563 599 635 673 399 43i 464 497 531 566 602 639 676 402 434 467 501 535 570 606 643 680 405 437 470 504 538 574 610 646 684 408 441 474 507 542 577 613 650 688 412 444 477 5" 545 581 617 654 692 699 707 711 715 719 722 726 730 734 774 814 856 898 941 985 O.O2 030 077 124 738 778 8.'8 860 902 945 990 035 08 1 742 782 822 864 906 950 994 040 086 746 786 827 868 911 954 999 044 091 750 790 831 872 9i5 959 *oo3 049 095 754 794 835 877 ?i 9 03 *oo8 053 100 758 798 839 881 924 967 *OI2 058 105 762 802 843 885 928 972 *oi7 063 no 766 806 847 889 932 976 *O2I 06 7 114 770 8ro 851 894 937 981 *026 072 119 129 133 138 143 148 153 158 162 167 172 221 272 323 376 430 485 54i 599 177 226 277 329 38i 435 490 547 604 182 231 282 334 387 441 496 I 52 610 187 236 287 339 392 446 502 558 616 192 241 292 344 397 452 507 564 622 197 246 297 350 403 457 513 570 628 2O2 252 303 355 408 463 518 I 75 634 207 257 308 360 414 468 524 581 639 211 262 3'3 365 419 474 % 645 216 267 3i8 37i 424 479 535 593 651 657 717 779 841 905 971 0.03 037 106 175 247 663 669 675 68 1 687 693 699 705 711 723 785 848 912 977 044 "3 183 254 729 791 854 918 984 051 120 100 26l 735 797 860 925 991 058 126 197 268 742 803 867 931 22 065 133 204 276 748 810 873 938 *oo4 071 140 211 283 754 816 879 944 *on 078 147 218 290 760 822 886 95i "017 085 '54 3 766 829 892 957 *024 092 161 232 305 772 835 899 964 *031 099 168 240 312 320 327 334 342 349 357 364 37i 379 386 394 470 548 627 708 790 5* 961 0.04 049 401 478 555 635 7i6 799 883 970 058 409 485 563 643 724 807 892 979 067 417 493 57i 651 73 2 816 901 987 076 424 501 579 659 74i 824 909 996 085 432 509 587 667 749 832 918 *00 5 094 439 516 595 675 757 841 926 *oi4 103 447 524 603 683 765 849 935 *023 112 455 532 611 691 774 858 944 *0 3 2 121 462 540 619 700 782 866 953 *o4o 130 139 148 '57 167 176 i*5 194 203 213 222 A. B. 2 a 4 6 tf 7 8 9 Prop. Pts. ! 88 TABLE VI. ADD f Io 8* lo % a = A * SUB i lo % a "* lo S^ = A x i log( + J) = log* + ^. bU i log( - *) = log + ^. A. B. 1 2 a 4 5 6 7 8 9 Prop. Pts. 0.00 9.01 9.02 9.03 9.04 9.05 9.06 9.07 9.08 9.09 9.10 9.11 9.12 9-13 9-H 9.15 9.16 9-17 9.18 9.19 9.20 9.21 9.22 9-23 9-24 9.25 9.26 9-27 9.28 9.29 9.30 9.31 9-32 9-33 9-34 9-35 9.36 9-37 9-38 9-39 9.40 9.41 9.42 9-43 9-44 9-45 9.46 9-47 9.48 9-49 9.50 0.04 139 148 157 167 176 185 194 203 213 306 401 499 598 700 803 909 017 127 222 z 3 3 4 5 6 7 8 9 z 3 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 X 2 3 4 5 6 7 8 9 z 3 3 4 5 6 7 8 9 z 2 3 4 5 6 7 8 9 9 0.9 1.8 2.7 3-6 4-5 5-4 6-3 7.2 8.z xa Z.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 '5 -5 3-0 4-5 6.0 7-5 9.0 10.5 13.0 3-5 18 z.8 3-6 S-4 7.2 9.0 10.8 13.6 H.4 16.2 at zo z.o 2.0 3.c 4.0 5.0 6.0 7.0 8.0 9.0 '3 1-3 2.6 3-9 5-2 6-5 7.8 9.1 10.4 zz.7 16 z.6 3.2 4.8 6.4 8.0 9.6 IX. 2 13.8 x 4-4 9 1.9 3.8 5-7 7.6 9-5 II. 4 T 3-3 15.2 17. x 23 XI z.z 3.il 3-3 44 5-5 6.6 7-7 8.8 9-9 M *-4 3.8 4. 5-6 7.0 8.4 9.8 ii. a 12.6 17 ; f 3-4 5-1 6.8 8-5 10. a 11.9 13.6 '5-3 2O a.o 4-0 6.0 8.0 IO.O 12.0 14.0 16.0- 18.0 as 231 325 421 5 ! 9 618 720 824 93i 0.05 039 240 334 43 528 628 73i 835 941 050 250 344 440 I 38 639 74i 845 952 061 259 353 450 548 649 75i 856 963 072 268 363 460 I 58 659 762 867 974 083 278 373 469 568 669 772 877 985 094 287 382 479 I 78 679 782 888 995 105 297 392 489 588 689 793 898 006 116 315 411 509 608 710 814 92O 028 139 150 161 172 183 195 206 217 229 240 251 263 378 496 616 738 863 991 0.06 121 254 274 390 508 628 $ 004 134 267 286 401 519 640 763 889 *oi7 147 281 297 4i3 53i 652 775 901 *030 161 294 308 425 543 664 788 914 1*043 174 308 320 436 555 677 800 927 *o 5 6 187 321 332 448 567 689 813 939 *o69 200 335 343 460 579 701 825 952 *082 214 348 355 472 59i 7H 838 965 *095 227 362 366 484 604 726 851 978 *io8 240 376 389 403 417 430 444 458 472 486 500 513 13 812 959 0.07 108 261 416 575 736 54i 683 827 973 123 276 432 59i 753 I 55 697 841 988 138 291 448 607 769 569 711 856 *oc>3 154 307 463 623 785 583 725 870 *oi8 169 322 479 639 802 597 740 885 *033 184 338 495 655 818 612 754 900 *048 199 354 I 11 671 835 626 769 914 *o63 215 369 527 687 851 640 783 929 *o 7 8 230 385 543 704 868 654 798 944 *093 245 400 559 720 884 901 918 934 95i 968 985 *OOI *oi8 *<>35 *0$2 0.08 069 240 415 592 774 958 0.09 146 338 533 086 257 432 610 792 977 165 357 553 I0j 275 450 628 810 996 184 377 573 120 292 468 646 829 *oi4 204 396 593 137 309 485 664 847 *o 33 223 416 612 154 327 503 683 865 *0 5 2 242 I 35 632 171 344 521 701 884 *o7i 261 455 652 188 362 539 719 206 379 557 737 921 *io8 299 494 692 223 397 574 755 940 *I27 319 5H 712 902 *090 280 474 672 4-2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 24 3.4 4.8 7.2 9.6 12.0 14.4 z6.8 19.2 21.6 4-4 6.6 8.8 II. 13-2 iS-4 17.6 19.8 5 2-5 5.0 7-5 10. 12.5 15.0 17-5 20.0 22. 5 4-6 6.9 9.3 XZ.J 13.8 i6.z 18.4 20.7 26 2.6 5-2 7.8 10.4 13-0 15.6 lS.2 20.8 23-4 732 935 o.io 141 35i 565 783 o.n 005 23 46 69- 933 752 773 793 813 833 853 874 894 914 955 162 373 587 805 028 % 715 976 183 394 609 827 050 277 507 742 996 204 415 630 849 073 3 oc 53i 766 *oi7 225 437 652 872 095 323 554 790 * 03 8 246 458 674 894 118 345 *os8 267 479 696 916 140 368 60 1 837 *0 79 288 501 718 938 163 392 62; 86 *IOO 309 522 739 960 1 86 415 648 885 *I2O 330 544 761 983 208 438 671 909 957 98 *oc5 *030 *054 *o 7 8 *I02 *I27 *i 5 i A. B. 1 2 8 | 6 6 7 8 9 Prop. Pts. ADDITION AND SUBTRACTION LOGARITHMS. 89 A ( log loga = A. 9 i loga log = B. \ x \ log (a + t) = loga + B. bUB * i log(a - b] = log^ + A A. B. 1 2 a 4 5 6 7 *I02 8 9 Prop. Pts. 9.50 9.51 9.52 9-53 9-54 9-55 9.56 9-57 9.58 9-59 9.60 9.61 9.62 9-63 9.64 9.65 9.66 9.67 9.68 9.69 9.70 9.71 9-72 9-73 9-74 9-75 9.76 9-77 9.78 9-79 9.80 9.81 9.82 9-83 9.84 9.85 9.86 9-8'/ 9.88 9.89 9.90 9.91 9.92 9-93 9-94 9-95 9.96 9-97 9.98 9-99 0.00 o." 933 0.12 175 422 673 928 0.13 188 452 721 994 0.14 272 ~554 841 0.15 133 43 , 73i o.io 037 349 665 986 o.i7_3i2 643 980 0.18 322 668 0.19 020 378 740 0.20 108 481 860 957 981 *oo5 *O3O *54 *078 *I2 7 "151 i 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 i 5 3 4 5 e 7 8 9 i 2 3 4 6 7 8 9 37 2.7 5-4 8.1 10.8 13-5 16.2 18.9 21.6 24-3 31 3-i 6.2 9-3 12.4 15-5 18.6 21.7 24.8 27.9 35 3-5 7.0 10.5 14.0 17-5 21.0 24-5 28.0 31-5 39 3-9 7.8 11.7 15-6 19-5 23-4 27-3 31-2 35- 1 43 4-3 8.6 12.9 17.2 21.5 25.8 30-1 34-4 33.7 47 4-7 9-4 14.1 18.8 23-5 28.2 32.9 37-6 43.3 38 2.8 5-6 8.4 II. 2 I 4 .0 16.8 19.6 22.4 25.2 32 3-2 6.4 9-6 12.8 16.0 19.2 22.4 25.6 28.8 36 3-6 7-2 10.8 14.4 18.0 21.6 25.2 28.8 32-4 40 29 2-9 5-8 8.7 ix. 6 14-5 17.4 20.3 23.2 26.1 33 3-3 6.6 9-9 13.2 16.5 19.8 23.1 26.4 29.7 37 3-7 7-4 u. i 14.8 18.5 22.2 25-9 29.6 33-3 41 30 3-o 6.0 9 a 12 ' 15 o 18.0 21.0 24.0 27.0 34 3-4 6.8 10.2 I 3 .6 17.0 20.4 2 3 .8 2 7 .a 30.6 38 3-8 7.6 11.4 15.2 19.0 22.3 26.6 30.4 34-a 43 200 447 698 954 214 479 748 *O2I 300 224 472 724 980 240 505 775 *049 328 249 497 749 *oo6 267 532 802 *077 356 274 522 775 *0 3 2 293 559 829 *I04 384 298 547 800 *o58 3 J 9 586 85* *I32 412 323 572 826 *o84 346 613 884 *i6o 441 348 597 851 *IIO 372 640 911 *i88 469 372 622 877 *I36 399 667 939 *2l6 497 397 648 903 *l62 425 694 966 *244 526 583 611 640 668 697 726 755 783 812 870 162 460 7 6l 068 380 697 *oi8 345 899 192 489 792 099 411 729 *o5i 378 928 221 520 822 I 3 443 761 *o83 411 957 251 550 8 I 3 161 474 793 *ii6 444 986 281 580 884 192 506 825 *I48 477 *oi6 310 610 914 224 538 857 *i8i 510 *045 340 640 945 255 569 889 *2I 4 544 *074 370 670 976 286 601 921 *2 4 7 577 *IO4 400 701 *oo 7 317 633 954 *279 610 677 710 744 777 811 *i5o 494 844 198 558 923 294 670 ^052 845 *i84 529 879 234 9 6o 331 708 *O90 878 912 946 *oi4 356 703 056 414 777 145 519 898 *048 390 738 091 450 813 182 557 937 *082 425 773 127 486 850 220 594 975 *ii6 460 808 163 522 887 257 * 632 *oi3 *2l8 564 914 270 631 997 369 746 *I28 *2 53 599 949 306 667 *Q34 406 784 *i67 *287 633 985 342 704 *o7i 444 822 *206 8.0 12.0 16.0 20. o 24.0 28.0 32.0 36.0 44 4-4 8.8 13-2 17.6 22.0 26. 4 30.8 35-2 39.6 48 4.8 9.6 14.4 19.2 24.0 28.8 33-6 38.4 43-2 8.2 12.3 16.4 20.5 24-6 28.7 32.8 36-9 45 4-5 9.0 13-5 18.0 22.5 27.0 3i-5 36.0 40-5 49 4.9 9.8 14.7 19.6 24-5 29.4 34-3 39-2 44.1 8.4 12.6 16.8 21.0 25.2 29.4 33-6 37-8 46 4.6 9.2 13-8 18.4 23.0 27.6 32.2 36.8 41.4 50 5-0 IO.O iS-o. 20. o 25.0 30.0 35-0 40.0 45-0 0.21 244 634 O.22 O29 430 8 3 6 0.23 247 665 0.24 088 5 I6 950 283 322 361 399 438 477 516 556 595 989 3S9 795 *206 623 *Q45 473 907 *346 673 069 470 877 289 707 130 559 994 712 109 510 918 330 749 173 603 *o 3 8 752 149 55i 959 372 791 216 646 *082 791 189 59i *000 414 833 258 689 *I26 831 229 632 *04I 455 875 301 733 *I 7 870 269 673 *082 497 918 344 776 *2I 4 910 309 713 *I2 3 539 960 387 819 *258 949 349 754 *i65 581 *oo3 430 863 *302 0.25 390 434 479 523 568 612 657 701 746 791 836 0.26 287 744 0.27 207 675 0.28 149 629 0.29 115 606 881 332 790 253 722 197 677 163 655 926 378 836 300 769 245 726 212 705 970 423 882 346 817 292 7 2 4 261 754 *oi6 469 928 393 864 340 822 310 804 *o6i 515 974 440 911 388 871 359 854 *io6 560 *02I 487 959 436 920 409 903 *i 5 i 606 "067 534 *oo6 484 968 458 953 *i 9 6 652 *U4 581 *Q54 532 *oi7 507 *oo3 *2 4 2 698 *i6o 628 *IOI 581 *o66 556 *053 0.30 103 153 2O3 253 303 354 404 454 505 555 1 A. B. 1 2 a 4 5 6 7 8 9 Prop. Pts. ADD { ^ga-logd = A. g ( loga - log = B. ' \ log (a + &) = log + -#. | log (a ) = log + A. A. B. 1 2 3 4 6 7 8 9 Prop. Pts. 0.00 O.OI O.O2 0.03 0.04 O.O5 0.06 0.07 0.08 O.O9 0.10 O.II O.I2 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 O.2J O.26 0.27 0.29 0.30 0.31 0.32 0.33 0-34 0.35 0.36 0.37 0.38 0.39 0.40 0.4 0.42 0.43 0.4 0.4 0.46 0.4 0.4 0.4 0.5 0.30 103 606 0.31 115 629 0.32 149 675 0.33 207 744 0.34 287 836 153 203 253 303 354 404 454 505 555 5 5-0 10.0 15.0 20.0 25.0 30.0 35-o 40.0 > 5-o 54 5-4 ! 10.8 JI6.2 J2I.6 527.0 532.4 737-8 3 43-2 ?4 8.6 58 ' 5-8 2 II. 6 3 17-4 4 23.2 5 29.0 634.8 740.6 846.4 952.2 62 I 6.2 2 12.4 3 18.6 424.8 53i-o 637.2 743-4 849-6 955.8 66 i 6.6 213.2 3 19-8 426.4 533-c 639-6 746.2 852.8 959-4 70 2 14. c 32I.C 4 28. c 535-c 6 42. c 749 < 8 5 6.c 963.* 51 5-i tO. 2 '5-3 20.4 25-5 30.6 J5-7 to. 8 15-9 55 5-5 II. '6.5 22.0 27.5 33-o 38.5 44-o 49-5 59 5-9 ii. 8 17.7 23-6 29-5 35-4 4i-3 47-2 S3-i 63 6-3 12.6 18.9 25.2 37-8 44.1 5-4 56-7 67 6-7 13-4 20.1 26.8 33-5 40.2 46-9 53-6 60.3 71 14.2 21.3 28.4 35.5 42. e 49-3 56.8 6 3 -<3 5-2 to. 4 20.8 26.0 ;i.2 56.4 56 5-6 [1.2 [6.8 22.4 28.0 33-6 39-2 44-8 50.4 60 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54-0 6.4 12.8 19.2 25.6 32.0 38.4 44.8 51.2 57-6 68 6.8 13.6 20.4 27.2 34-c 40.8 47-C 54-4 61.2 73 7.2 14.4 21. e 28. 43-s 50.4 57-< 64.* 53 5-3 0.6 5-9 1.2 6-5 1.8 7-J 2-4 7-7 57 5-7 M 7-i 22.8 28.5 34-a 39-9 45-6 Si-3 6x 6.x 12.2 I8. 3 24.4 30-5 36.6 42.7 4 8.8 54-9 65 6-5 13-0 19.5 26.0 32-5 39-0 45.5 52.0 58.5 69 6.9 13-8 20.7 27.6 34-5 41.4 48.3 55-2 62.1 73 7-3 14.6 21.9 29.9 36.5 43-8 51.1 58.4 65-7 656 1 66 68 1 20 1 728 260 798 342 891 707 217 732 254 852 396 946 758 268 784 306 834 367 906 OOI 809 320 836 359 887 421 960 * 5 6 859 37i 888 411 940 474 '5 561 112 910 422 940 464 993 528 069 616 168 961 474 992 5I 2 046 582 123 670 223 012 526 045 569 100 636 178 726 279 063 577 097 622 153 690 232 78i 334 0.35 390 446 502 558 614 6 7 726 782 838 894 * 95 ? 0.36 516 0.37 088 665 0.38 247 836 0.39 430 0.40 029 634 0.41 244 007 573 723 306 895 489 089 695 *o6 3 630 203 363 954 549 149 756 119 687 260 839 423 *oi3 609 2IO 816 "428 669 297 570 214 864 ?J! 744 897 482 *073 669 270 877 *2 33 80 1 375 955 *I 3 2 729 33 o 938 +289 858 433 *oi4 600 *I9I 789 999 * 34 6 916 491 *072 / 59 849 452 *o6i 403 973 549 718 *3io 909 512 *I22 *459 *O3O 607 777 *370 969 573 *i8 3 306 367 490 552 613 1 487 *I22 763 4 08 *o6o 716 "377 675 737 798 860 0.42 481 0.43 108 740 o.44 378 0.45 020 668 0.46 322 980 922 544 171 804 442 085 387 O/1 ( 984 606 234 867 506 149 799 453 *II2 *io8 360 995 634 279 929 584 *245 *i70 794 423 *o 5 8 698 344 994 650 * 3 n 920 550 *i86 827 473 782 *444 *357 982 613 89? 538 *IGO 8 4 8 * 5 IO *045 677 *3H 956 603 *2 5 6 914 *577 Q.47 643 0.48 312 986 0.49 665 0.50 349 0.51 037 0.52 430 o.53 133 84 o.54 554 710 777 844 910 977 O4^ *iii *I 7 8 *245 918 *597 *280 968 661 *36o *062 770 *483 * 379 73: 417 107 801 500 204 912 447 *I2I Soi 486 176 870 570 274 983 869 555 245 940 640 345 *55 *257 938 624 *OIO 710 416 *I26 * 64 - *oo6 692 384 *o8o 78 486 *i97 716 *393 *o 7 4 761 453 "150 85 *M 7 8 783 * 4 6i *I43 830 522 *220 92 628 *34o 851 *529 *2II 899 592 *289 992 699 *4i 626 697 769 841 912 984 *os6 *I28 *2OO 0.55 272 , 994 0.56 72 o.57 45 0.58 18 92 0.59 67 0.60 42 0.61 17 344 *o66 794 262 748 497 251 416 *i 39 86 59 3 82 57 32 488 *2II 940 672 4ic *i 5 i 897 648 402 *28 746 484 *226 972 723 478 632 *357 *o86 819 558 * 3 oo 79 554 704 *?59 893 63 *37~ *I2 874 6 3 c 777 * 5 02 *2 3 2 967 706 *449 *i 9 94 70 * 4 6 849 *57" * 3 o *0 4 * 78C *27 78 4 * 92 ' *379 *ii 4 854 * S9 8 *347 *IOO 857 *ooc *o8 *i6 *237 *3M * 39 c *54 *6i9 B. 1 2 3 4 5 1 8 9 Prop. Pts. ADDITION AND SUBTRACTION LOGARITHMS. A f loga log b= A. q ( loga log = B. ADD ' i log(a + b) = log* + B. 3 ' i log(0 - b) = log + ^. A. B. 1 2 a 4 5 G 7 8 9 Prop. Pts. 0.50 0.51 0.52 o.53 0.54 o.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 o.73 0.74 0.75 0.76 0.77 0.78 o.79 0.80 0.81 10.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 0.61 933 *cx>9 +085 *l6i *237 * 3 i 4 *39o *466 *542 *6i 9 X 2 3 4 5 6 7 8 9 X 3 3 4 5 6 7 8 9 t a 3 4 5 6 7 8 9 X 2 3 4 5 6 7 8 9 X 3 3 4 5 6 7 8 9 X a 3 4 5 6 7 8 o 74 7-4 14-8 22.2 2 9 .6 37.0 44-4 51.8 59-2 66.6 77 7-7 15-4 30.8 38.5 46.2 53-9 61.6 69.3 80 8.0 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 83 8-3 16.6 24.9 33-2 41.5 49-8 58.1 66.4 74-7 86 8.6 17.2 25.8 34-4 51.6 60.2 68.8 77-4 89 89 2^7 35. 6 44-5 53-4 62.3 7.a 80.1 75 7-5 15-0 22.5 30.0 37-5 45-0 52-5 60.0 67-5 78 7.8 15.6 23-4 31.2 39-0 46.8 54-6 63.4 70.2 81 8.x 16.2 24.3 32.4 40.5 48.6 56.7 64.8 72.9 84 8.4 16.8 25.2 33-6 42.0 50.4 58.8 67.2 75-6 87 8-7 17-4 26.1 34-8 43-5 52.2 60.9 69.6 78.3 90 9.0 18.0 27.0 36.0 45 .e 54-c 63.0 72.0 8x.c 7 6 7.6 15-2 22.8 I 30-4 38.0 | 45-6 53-2 60.8 68.4 79 7-9 15.8 23.7 31.6 39-5 47-4 55-3 63.2 71.1 83 8.2 16.4 24.6 32.8 41.0 49.2 57-4 ! 65.6 73-8 85 8.5 17.0 25.5 34-o 42.5 51-0 59-5 68.0 76.5 88 8.8 17.6 26.4 35-a 44.0 52.8 61.6 70.4 9* 9.1 18.2 27.3 36.4 45-5 54-6 63.7 72.8 81.9 0.62 695 0.63 461 0.64 231 0.65 005 783 0.66 565 0.67 351 0.68 141 935 771 538 308 083 861 644 430 220 *oi4 848 615 386 160 939 722 509 * 3 924 692 463 2 3 8 *oi8 Soi 588 379 *I74 *OOI 768 540 "096 879 667 458 845 618 394 *i 74 958 746 538 *333 *I54 923 695 472 *2 5 2 617 * 4 I 3 *2 3 I *ooo 773 549 *33 *ii5 904 696 *493 *37 "077 850 * 627 *I94 983 776 *573 OOOO W4^vjvO-iU> ononONVJOOO tOOnOO to on U U> ^>J on G04- 4- 0.69 732 812 892 972 *0$2 *I32 *2I2 *293 *373 *453 0.70 533 0.71 338 0.72 146 958 0.73 774 o.74 592 o.75 415 0.76 240 0.77 069 _9oi 0.78 736 o.79 575 0.80 416 0.81 261 0.82 108 959 0.83 812 0.84 668 0.85 527 614 419 227 855 674 497 323 152 694 499 308 *I2I 937 757 579 406 235 774 580 390 *202 839 662 488 318 *IOI 921 744 57i 401 935 742 552 * 3 6 5 *i8 3 *00 3 827 654 485 *oi6 823 633 *447 *o8s 909 737 568 *o 9 6 904 7H * 34 6 *i68 992 820 651 *I77 984 796 *6io *428 *o 7 5 903 734 *257 877 ^692 -"157 818 984 820 659 500 345 193 *044 898 754 613 *o68 *I 5 I *235 *3i8 * 4 02 *48$ *56 9 *653 904 743 585 430 278 *I2 9 983 840 700 987 827 66 9 515 363 *2I4 926 786 911 754 599 448 * 3 oo *I54 *OI2 8 7 2 995 838 684 * 533 *2 4 958 * 239 922 769 618 * 4 7o *i63 *oo7 854 703 * 55 6 *4H 938 788 *497 *355 *2I7 * 49 i *?76 *02 3 873 * 7 2 7 *44i 0.86 389 476 562 648 735 821 908 994 *o8i "167 0.87 254 0.88 121 991 0.89 863 0.90 738 0.91 616 0.92 49 6 0.93 378 0.94 263 340 20 951 826 704 584 466 427 295 914 7 9 I 6 7 2 555 440 382 *2 5 2 *I25 *OOI 879 760 643 529 600 469 *339 *2I 3 967 848 732 617 687 * 556 !3 *I77 *55 936 820 706 774 * 643 *5H *'4: 08 795 861 730 *6oi *475 *352 *2 3 I *ii3 997 883 947 *68 7 * 5 6 3 t 440 *20I *o86 972 904 *776 *4o8 *I74 *o6i 950 Q.95 '50 0.96 039 0.97 82^ 0.98 720 0.99 618 i. oo 519 i.oi 421 1.02 325 1.03 231 239 128 *020 914 810 708 609 5" 322 327 416 505 594 683 772 861 217 *i09 *00 3 900 798 699 601 506 413 *i8 789 692 597 503 395 *288 *l82 879 782 687 594 485 *377 *2 7 2 *i69 *o68 969 873 778 685 *362 *259 *I 5 8 *o6o 963 868 776 663 *45i *349 959 867 *645 * 54 i *44 957 841 *735 *63i *528 *428 "140 1.04 139 230 321 412 503 594 685 776 867 958 1 A. B. 1 2 8 4 6 G 7 8 9 Prop. Pts. 2 TABLE VI. ADD ) l Z a ~~ 10g b = A ' QTTR f lo g * ~ lo * = # * t log (a + b) == log ^+ A i log( - J) = log^ 4- A. ; A. B. 1 2 a 4 5 6 7 8 9 Prop. Pts. 1.00 01 02 03 04 l 06 :>7 08 09 10 ii 12 13 14 15 16 17 18 19 .20 21 22 .23 .24 .25 .26 !28 .29 .30 31 3 2 < -33 : -34 i -35 36 37 38 39 1.04 139 230 321 412 503 594 685 776 867 958 9 i 9 2 18 3 27 4 36 5 45 6 54 9 Si I 2 3 4 i I 9 i 2 3 4 1 9 ! i c 2 I< 32 i 4 3^ 5 4 6 \ 7 6( 8 7 9 8 i 2 3 4 .1 I 9 i .2 3 4 9 93 27 1 65 g 9 ,i 28 37 1 1 )5 )-5 ).o 1:1 J-S 7-c It 5-5 ? 5 IS 2C 1 6; 7' 8; 9* 9.2 18.4 27.6 36.8 46.0 55-2 64.4 g;l I q 2 I I 4 7 \ i .2 .6 .0 i .2 .6 96 96 19.2 28.8 38.4 48 o 57-6 67.2 76.8 86.4 >7 >-7 >-4 !i ii ',i r-3 1.05 049 961 i. 06 875 1.07 790 i. 08 708 1.09 627 i. 10 548 i. i i 470 1. 12 394 140 053 966 882 800 719 640 562 486 232 144 058 974 891 811 732 655 579 323 235 149 065 983 903 824 747 671 414 * 3 26 *2 4 I I 157 *075 995 916 839 764 505 418 332 249 167 087 009 932 857 596 *5o 9 *424 34i 259 179 101 024 949 687 "601 * 5 i6 *432 *35i *27 1 ^193 *ii7 ^042 779 692 607 524 443 363 285 209 134 870 783 6 99 616 535 455 378 301 227 1.13 320 412 503 598 690 783 876 968 06 1 i54 i 14 247 I.I5 175 1.16 106 1.17 J7 97i 1.18 905 1.19 841 1.20 779 1. 21 717 1.22 657 1.23 599 1.24 54i 1.25 485 1.26 430 1.27 376 1.28 323 1.29 272 1.30 221 1.31 J 72 340 268 199 131 064 999 935 872 811 432 361 292 224 157 092 029 966 905 525 454 385 317 251 1 86 122 060 999 618 547 473 411 *344 *27 9 *2l6 *I 54 *0 93 I 11 640 57i 504 * 43 8 *373 * 3 ip *248 *i8 7 804 733 665 597 * S ? ! *467 * 4 03 *342 *28l 897 826 758 691 I 6 ? *56o *497 *435 *375 990 9 2O 8 5 I 784 718 654 591 5 2 9 46 9 083 013 944 877 812 748 685 623 563 751 845 939 *Q34 *I28 *222 * 3 i6 410 504 693 635 579 524 47i 418 367 316 267 787 730 674 619 565 5 J 3 462 411 362 88 1 824 768 714 660 608 557 507 458 975 918 863 808 755 703 652 602 553 *070 *oi3 957 903 850 797 746 697 648 *l64 *I07 *052 997 944 892 841 792 743 *2 5 8 *2O2 *I 4 6 *092 *d^9 987 936 887 838 352 2 9 6 *24I *i8 7 *i 34 *082 $ 933 447 390 *335 *28l *22 9 *I77 *I26 *077 *02 9 1.32 124 219 3H 410 505 600 695 791 886 9 8l 1.33 77 1.34 3 985 1.35 94i 1.36 898 1.37 856 1.38 814 1.39 774 1.40 734 172 126 *o8i *Q37 994 95i 910 870 830 267 221 *i76 *I 3 2 *o89 *O47 *oo6 966 926 363 3i7 *272 *228 *i85 *i43 *IO2 *062 *022 458 * 4 ^ 2 * 3 6 7 * 3 24 *28l *2 39 *i98 *I58 *u 9 55 ^ 508 * 4 6 3 *4i 9 *377 *335 *2 94 * 254 *2I5 649 603 *559 *5i5 *472 * 43 I * 39 o * 3 5o * 3 ii 744 699 *6 54 *6n * 5 68 *527 ^486 *446 * 4 o 7 840 794 *750 *7o6 *664 622 * 5 82 *542 * 5 o 3 935 890 *845 *802 *76o * 7 i8 *678 *6 3 8 *599 .40 4 .42 44 .4 .46 .47 4 .4 1.50 i 4 1 6 95 792 888 984 *oSo *I 7 6 *273 * 3 6 9 * 4 6 5 * 4 28 *39 * 35 6 *32 * 2 87 *2 5 4 *22 *i8 *i5 *56i 1.42 658 1.43 6 2 J 1.44 584 1.45 549 I.465H 1.47 480 1.48 447 1.49 4i 1.50 383 754 717 681 645 61 577 544 % 850 813 777 742 707 674 64 608 577 946 9IC 874 8 3 8 804 770 737 705 674 *043 *oo6 970 935 901 867 834 802 771 *I 39 *I02 *o66 *o 3 997 964 93 895 868 *235 *I 99 *i63 *I2S *o 9 4 *o6c *028 99 6 964 *332 *29 *2 59 *22~ *I 9 C *i 5 *I24 *09 *o6 *524 *488 *452 * 4 i8 * 3 8 4 * 350 *3i8 *286 *255 1.51 35 449 546 64: 740 837 934 *o 3 *I2 *225 A. B. 1 2 3 4 5 1 8 t) Prop. Pts. ADDITION AND SUBTRACTION LOGARITHMS. 93 A ( log a - log b = A. Q ( log a log b = B. x \ log (a + J) = log<* + ^. bu B ' i log( - ) = log* + A. A. 1.50 51 52 53 54 :ii ? 59 .GO .61 .62 63 .64 65 .66 .67 .68 .69 .70 71 .72 73 74 :3 77 .78 79 .80 .81 .82 .8 3 .84 .85 .86 .87 .88 .89 1.90 91 .92 93 94 95 .96 97 .98 99 2.00 B. 1 2 546 3 643 4 5 7 8 9 Prop. Pts. 1-51 352 1.52 322 1.53 292 i . 54 263 1-55 235 1.56 207 1.57 180 1-58 153 1.59 128 I. 60 102 1.61 077 449 740 837 934 *0 3 I *I28 *225 i 2 3 1 I 9 i 2 3 4 i I 9 i 2 3 4 1 9 97 9-7 19.4 38.8 485 58.2 67.0 77.6 87.3 98 9-8 19.6 29.4 39-2 49.0 68^6 78.4 88.2 99 9> i 19.8 29.7 39 6 49.5 1 g;! 419 389 360 332 304 277 251 225 200 516 486 457 429 402 375 348 322 297 613 583 555 526 499 472 446 420 395 710 680 652 624 596 569 543 517 492 807 778 749 721 693 667 640 6i5 590 904 875 846 818 791 764 738 712 687 *OOI 972 943 861 835 810 785 *o 9 8 *o6 9 *O4O *OI 3 985 959 933 907 882 ::n *I 3 8 *IIO *o8 3 ^056 *O 3 O ^005 9 8o 175 273 370 468 565 663 760 858 956 1.62 053 1.63 o 3 o i 64 006 984 1.05 962 i . 66 940 1.67 919 1.68 898 1.69 878 151 127 104 *o8i *059 *o 3 8 *oi7 996 976 248 225 202 *I79 *I57 *I 3 6 *ii5 *094 *074 346 322 299 *277 *255 *233 *2I2 *I 9 2 *I72 444 420 397 *375 *353 *33' * 3 IO *290 *270 54i 518 495 *473 *45i *429 *4o8 * 3 88 * 3 68 * 34 8 639 616 593 *57o * 54 8 * 5 2 7 * 5 o6 *486 *466 737 713 6 9 o *668 *646 *62 5 *6o4 *584 *5 6 4 834 811 788 * 7 66 *744 *723 *702 *682 *662 932 886 *86 4 *842 *82I *8oo * 7 8o * 7 6o 1.70 858 956 *054 *I52 *250 *446 *544 *642 *74i 1.71 859 1.72 820 1.73 801 1-74 783 1.75 766 1.76 748 1-77 73i 1.78 715 1-79 699 937 918 899 881 864 847 8 3 o 813 797 *o 3 5 *oi6 998 980 962 945 928 f *i33 *ii4 *o 9 6 *078 *o6o *Q4 3 *026 *OIO 994 *23I *2I2 *I 94 *I 7 6 *I 59 *i 4 i *I25 *io8 *0 9 2 * 3 2 9 * 3 IO *292 *274 *257 *24O *22 3 *2O7 *i9i *427 *49 * 39 o *373 *355 * 33 8 * 3 2I * 3 os *28 9 * 5 25 *507 * 4 8 9 * 47 i *453 * 43 6 * 4 20 * 4 o 3 * 3 88 *62 3 *6o 5 * 5 8 7 * 5 6 9 * 55 2 *535 * 5 i8 *502 * 4 86 *722 * 703 *685 *667 *65O *6 33 *6i6 *6oo *584 i. 80 68 3 781 880 978 *077 *I75 *274 * 3 72 * 47 i * 5 6 9 1.81 667 1.82 652 i.8 3 6 3 8 1.84 625 1.85 609 1.86 595 1.87 582 1.88 569 1.89 556 766 75 i 736 722 708 694 681 667 655 864 849 835 820 806 793 779 766 753 963 948 933 919 878 865 852 *o6i *046 *0 3 2 *oi8 *oo4 990 977 964 95i *i6o *i45 *I 3 *u6 *I02 *o89 *075 *062 *o5o *258 *244 *22 9 *2I5 *20I *i87 *I74 *i6i *I 4 8 *357 *342 * 3 28 * 3I3 *286 *2 73 *260 *2 4 7 *455 *44i * 4 26 *4I2 * 39 8 * 3 8s *37i *358 * 34 6 *554 *539 *525 *5ii *497 * 4 8 3 *47o *457 *445 i 9Q 543 1.91 53i 1.92 519 1-93 507 1.94 496 1.95 485 1.96 474 1-97 463 1.98 452 1.99442 2.00 432 642 74i 840 938 *0 37 *i 3 6 *2 3 5 *333 *432 630 618 606 PI 573 562 55i 54i 729 717 705 69^: 682 671 66 1 650 640 827 815 804 770 760 749 739 926 914 903 891 880 869 859 848 838 *025 *OI 3 *002 990 979 968 958 947 937 *I2 4 *II2 *IOO *o8 9 *078 *o6 7 *o 57 *o 4 6 *o 3 6 *22 3 *2II *I99 *i88 *i77 *i66 *I 5 6 *I 4 5 *i35 * 3 2I * 3 IO *2 9 8 *28 7 *2 7 6 *26s *254 *244 *2 34 *420 *4o8 *397 * 3 86 ! 3 ^ * 3 64 *353 *343 *333 53i 630 729 828 927 *026 *I25 *22 4 *323 B. 1 2 8 4 5 7 8 9 Prop. Pts. 94 TABLE VI. log a log b = log( + *) = = A. log a log b = B. log* + (B - A). log (a - b) = log a - (B - A) A. B. B-A. A. B. B-A. A. B. B-A. 1.9823 .9833 .9842 .9852 .9862 1.9868 .9878 .9887 .9897 .9907 .00450 449 448 447 446 2.0337 .0348 .0359 .0370 .0381 2.0377 .0388 0399 .0410 .0421 .00400 399 398 397 396 2.0920 .0932 .0945 .0957 .0970 2.0955 .0967 .0980 .0992 .1005 .00350 | 349 348 347 346 1.9872 .9882 .9891 .9901 .9911 1.9917 .9926 9935 9945 9955 .00445 444 443 442 441 2.0392 .0403 .0414 .0425 0437 2.0432 .0443 .0454 .0465 .0476 .00395 394 393 392 39i 2.0982 .0995 .1008 .1020 1033 2.1017 .1029 .1042 .1054 .1067 .00345 344 343 342 34i 1.9921 9931 .9941 .9951 .9961 1.9965 9975 .9985 9995 2.0005 .00440 439 438 437 436 2.0448 .0459 .0470 .0481 .0493 2.0487 .0498 .0509 .0520 .0532 .00390 389 388 387 386 2.1046 .1059 .1072 .1085 .1098 2.1080 .1093 .1106 .1119 .1132 .00340 339 338 337 336 1.9971 .9981 .9991 2.0001 .OOI I 2.0015 .0024 .0034 .0044 .0054 .00435 434 433 432 431 2.0504 .0515 .0527 .0538 .0550 2.0543 0553 .0565 .0576 .0588 .00385 384 383 382 38i 2. mi .1124 1137 .1150 .1163 2.1144 .1157 .1170 .1183 .1196 00335 334 333 332 33i 2.OO2I .OO32 .OO42 .0052 .0062 2.0065 .0075 .0085 .0095 .0105 .00430 429 428 427 426 2.0561 .0573 .0584 .0596 .0607 2.0600 .0611 .0622 .0634 .0645 .00380 379 378 377 376 2.1176 .1190 .1203 .1216 .1229 2.1209 .1223 .1236 .1249 .1262 00330 329 328 327 326 2.0073 .0083 .0093 .OIO4 .0114 2.0115 .0125 .0135 .0146 .0156 .00425 424 423 422 421 2.0619 .0630 .0642 .0654 .0666 2.0656 .0667 .0679 .0691 .0703 .00375 374 373 372 37i 2.1243 .1256 .1270 .1283 .1297 2.1275 .1288 .1302 .1315 .1329 .00325 324 323 322 321 2.0124 .0135 .0145 .0156 .0166 2.0166 .oi7/ .0187 .0198 .0208 .00420 419 418 417 416 2.0677 .0689 .0701 .0713 .0725 2.0714 .0726 .0738 .0750 .0762 .00370 369 368 367 366 2.1310 .1324 .1338 .1351 .1365 2.1342 .1356 .1370 .1383 1397 .00320 319 3i8 317 316 1 2.0177 .0187 .0198 .0208 .0219 2.0218 .0228 .0239 .0249 .0260 .00415 414 413 412 411 2.0737 .0749 .0761 .0773 .0785 2.0773 .0785 .0797 .0809 .0821 .00365 3 6 4 363 362 361 2.1379 .1393 .1407 .1421 H35 2.1410 .1424 .1438 .1452 .1466 .00315 3'4 313 312 3" 2.0229 .0240 .0251 .O26l .O272 2.0270 .0281 .0292 .0302 .0313 .00410 409 408 407 406 2.0797 .0809 .0821 0833 .0845 2.0833 .0845 .0857 .0869 .0881 .00360 359 358 35 2 356 2.1449 .1463 .1477 .1491 .1505 2.1480 .1494 .1508 .1522 1536 .00310 309 308 307 306 2.O283 .0294 .0305 OS'S .0326 2.0324 .0334 .0345 3 II .0366 .00405 404 403 402 401 2.0858 .0870 .0882 .0895 .0907 2.0893 .0905 .0917 .0930 .0942 .00355 354 353 352 35i 2.1520 .1534 .1548 .1563 .'577 2.1550 .1564 .1578 .1593 . 1607 .00305 304 303 302 30i 2.0337 2.0337 .00400 2.0920 2.0955 .00350 2.1592 2 1622 .00300 A. B. B-A. A. B. B-A. A. B. B-A. ADDITION AJND 5UJJTKAUT1U.N -LUUAK.1T.HM5. g Iog0 log as A. loga log = B. log(0 + V] = loga + (B A). log(a V) as loga (B A). A. B. B-A. A. B. B-A. A. B. B A. 2.1592 .1606 .1621 .1635 .1650 2.1622 .1636 .1651 .1665 .1680 .00300 299 298 297 296 2.2386 .2403 .2421 .2439 .2456 2.2411 .2428 .2446 .2464 .2481 .00250 249 248 247 246 2.3358 3379 .3401 .3423 .3446 2.3378 3399 .3421 3443 .3466 .00200 199 198 197 196 2.1665 .1680 .1694 .1710 .1724 2.1694 .1709 .1723 .1739 .1753 .00295 294 293 292 291 2.2474 .2492 .2510 .2528 .2546 2.2498 .2516 .2534 .2552 .2570 .00245 244 243 242 241 2.3468 .3490 .3513 3535 .3558 2.3487 .3509 3532 3554 3577 .00195 194 193 192 191 2.1739 .1754 .1770 .1785 .1800 2.1768 .1783 .1799 .1814 .1829 .00290 289 288 287 286 2.2564 .2582 .2600 .2618 .2637 2.2588 .2606 .2624 .2642 .2661 .00240 239 238 237 236 2.3581 .3604 .3627 .3650 .3673 2.3600 .3646 .3669 .3692 .00190 187 1 86 2.1815 .1830 .1846 .1861 .1877 2.1844 .1858 .1874 .1889 .1905 to to to to to 00 00 0000 OO 1-4 to CO 4^> Cn 2.2656 .2674 .2693 .2711 .2730 2.2679 .2697 .2716 .2734 2753 .00235 234 233 232 231 2.3697 .3720 3744 .3768 .3792 (0 CO CO CO CO OJ <-> OO ONCO i-i O ON tO OOCn .00185 184 183 182 181 2.1892 .1908 .1923 .1939 .1955 2.1920 .1936 .1951 .1967 .1983 .00280 279 278 277 276 2.2749 .2768 .2787 .2806 .2825 2.2772 .2791 .2810 .2829 .2848 .00230 229 228 227 226 2.3816 .3840 .3865 .3889 .39H .3883 .3907 .3932 .00180 179 178 177 176 2.1971 .1987 .2002 .2019 .2035 2.1998 .2014 .2029 .2046 .2062 .00275 274 273 272 271 2.2845 .2864 .2884 .2903 .2923 2.2867 .2886 .2906 .2925 .2945 .00225 224 223 222 221 2-3939 .3964 .3989 .4014 .4039 2.3956 .3981 .4006 .4031 .4056 .00175 174 173 172 171 2 . 205 1 .2067 .2083 .2099 .2116 2 . 2078 .2094 .2110 .2126 .2143 .00270 269 268 267 266 2.2943 .2962 .2982 .3002 .3022 2.2965 .2984 .3004 .3024 .3044 .OO22O 219 218 217 216 2.4065 .4090 .4116 .4142 .4168 2.4082 .4107 .4133 .4159 .4185 .00170 \6 7 1 66 2.2132 .2149 .2165 .2182 .2198 2.2159 .2175 .2191 .2208 .2224 00265 264 263 262 261 2.3043 .3063 .3083 .3104 .3124 2.3064 .3084 .3104 .3125 .3H5 .00215 214 213 212 211 2.4195 .4221 .4248 .4275 .4302 2.4211 .4237 .4264 .4291 .4318 .00165 164 'g 162 161 2.2215 .2232 .2249 .2266 .2283 2.2241 .2258 .2275 .2292 .2309 .00260 259 258 256 .3166 .3187 .3208 .3229 2.3166 .3187 .3208 .3229 .3250 20g 208 207 206 2.4329 .4356 .4383 .4411 4439 2.4345 .4372 4399 .4427 4455 .00160 159 158 157 156 2.2300 .2317 .2334 .2369 2.2325 .2342 .2359 .2376 .2394 .00255 254 253 252 251 to CO CO CO CO CO CO CO tO tO tO CO >-i\O Vjen O\4^co - O 2.3271 .3291 .3313 3334 .3356 .OO2O5 204 20 3 202 201 2.4467 4495 .4523 4552 .4581 2.4482 .4510 .4538 .4567 .4596 .00155 154 153 152 2.2386 2.2411 .00250 2.3358 2.3378 .O0200 2.4609 2 4624 .00150 A. B. B-A. A. B. B-A. A. B. B-A. TABLE VI. 1 Iog0 log b = A. log a log b = B. log (a + ) = log* + (^ ^). log (a ) = log a (^ ^4). A. B. B-A. A. B. B-A. A. B. B-A. 2.4609 .4638 .4668 .4697 .4727 2.4624 .4653 .4683 .4712 .4742 .00150 149 148 H7 146 2.6373 .6416 .6461 .6505 .6550 2.6383 .6426 .6471 5 S i 5 .6560 .OOIOO .00099 98 9 96 2.9385 9474 .9563 .9655 .9748 2.9390 9479 .9568 .9660 9753 .00050 J3 % 2-4757 .4787 .4817 4848 4878 2.4772 .4801 .4831 .4862 .4892 .00145 144 143 142 141 2.6596 .6642 .6688 .6735 .6783 2.6606 .6651 .6697 .6744 .6792 .00095 94 93 92 9i 2.9844 2.9941 3.0041 .0143 .0248 2.9848 2.9945 3-0045 .0147 .0252 .00045 44 43 42 41 4910 4941 4972 5004 5036 2.4924 4955 .4986 .5018 .5050 .00140 139 138 137 136 2.6831 .6880 .6928 .6978 .7028 2 . 6840 .6889 .6937 .6987 .7037 .00090 89 88 87 86 3 ' 3 ^ .0466 .0578 .0694 .0813 3-0360 .0470 .0582 .0698 .0817 .00040 P ! 5068 5100 5 J 33 5165 5199 2.5081 .5H| .5146 .5178 .5212 .00135 134 133 132 131 2.7079 7131 .7183 .7236 .7289 2.7088 .7139 .7191 .7244 .7297 .00085 84 83 82 81 3-0935 .1061 .1191 .1324 .1463 3-0939 .1064 .1194 .1327 .1466 .00035 34 33 32 3i 5232 5266 5299 5333 5368 2.5245 .5279 .5312 .5346 .538i .00130 129 128 127 126 2.7343 .7398 7453 .7509 .7566 2.7351 .7406 .7461 .7517 7574 .00080 79 78 8 3.1606 .1753 .1905 .2063 .2226 3.1609 .1756 .1908 .2066 .2229 .00030 29 28 27 26 5402 5437 5472 5508 5544 2.5415 5449 .5484 .5520 .5556 .00125 124 123 122 121 2.7623 .7682 .7741 .7801 .7862 2.7631 .7689 .7748 .7808 .7869 .00075 74 73 72 7i 3-2396 .2575 .2760 .2952 .3154 3-2399 .2577 .2762 .2954 .3156 .00025 24 23 22 21 5580 5616 5653 5690 5727 2.5592 .5628 .5665 .5702 5739 .OOI20 119 118 117 116 2.7923 .7985 .8050 .8114 .8180 2.7930 .7992 .8057 .8121 .8187 .00070 69 68 67 66 3.3366 3590 .3825 .4072 4335 3.3368 3592 .3827 .4074 4337 .OO020 19 I/ 16 2.5765 5803 5841 5880 .5919 2.5776 .5814 5852 .5891 .5930 .00115 114 H3 112 III 2.8245 .8313 .8381 .8451 .8521 2.8252 .8319 .8387 .8457 .8527 .00065 64 3 62 61 3-4617 49 J 7 .5237 .5587 .5964 3.46i9 .4918 .5238 .5588 .5965 .00015 14 13 12 II 2.5958 5998 .6038 .6079 .6120 2.5969 .6009 .6049 .6090 .6131 .OOIIO 107 106 2.8593 .8666 .8741 .8816 .8893 2.8599 .8672 .8747 .8822 .8899 .00060 59 58 i 3.6377 .6835 7345 .7925 .8595 3.6378 .6836 .7346 .7926 .8596 .OOOIO 09 08 07 06 2.6161 .6202 .6244 .6287 .6329 2.6172 .6212 .6254 6297 .0339 .00105 104 103 102 101 2.8971 .9051 .9132 .9215 .9300 2.8977 .9056 9'37 .9220 .9305 .00055 54 53 52 5i 3.9390 4.0355 4.1600 4.3375 4.6367 3.9391 4.0355 4.1600 4.3375 4.6367 .00005 04 03 02 01 2.6373 2.6383 .OOIOO 2.9385 2.9390 .00050 oo 00 .00000 A. B. B-A. A. B. B-A. A. B. B-A. TABLE VII. SQUARES OF NUMBERS. 97 TABLE VTL> SQUARES OF NUMBERS. No. Square. No. Square. No. Square. No. Square. No. Square. I 2 3 20 21 22 23 400 40 4i 42 43 1600 60 61 62 63 3600 80 Si 82 83 6400 I 4 9 441 484 529 1681 1764 1849 3721 3844 3969 6561 6724 6889 4 5 6 16 25 36 24 25 26 576 625 676 44 45 46 1936 2025 2116 64 65 66 4096 4225 4356 84 85 86 7056 7225 7396 7 8 9 10 ii 12 13 49 64 81 27 28 29 80 3i 32 33 729 784 841 47 48 49 50 5i 52 53 2209 2304 2401 67 68 69 70 7i 72 73 4489 4624 4761 87 88 89 90 9i 92 93 7569 7744 7921 100 900 2500 4900 8100 121 144 I6 9 961 1024 1089 2601 2704 2809 5041 5184 5329 8281 8464 8649 14 15 16 J96 225 2 5 6 34 35 36 1156 1225 1296 54 55 56 2916 3025 3136 74 75 76 5476 5625 5776 94 95 96 8836 9025 9216 17 18 19 20 289 324 3 6l 37 38 39 40 1369 1444 7521 57 58 59 60 3249 3364 348i 77 78 79 80 5929 6084 6241 97 98 99 100 9409 9604 9801 400 1600 3600 6400 10000 o8 SQUARES OF NUMBERS FROM 100 TO 1000. 1<* 9~ 3*+ 4** 5~ 6~ 1~ $ 94* Diff. 00 100 40O 900 1600 2500 3600 4900 6400 8100 00 X 01 102 404 906 1608 2510 3612 4914 6416 8118 01 O2 03 104 106 408 412 912 918 1616 1624 252O 2530 & 4928 4942 6432 6448 8136 8154 04 09 3 5 7 04 108 416 924 1632 2540 3648 495 6 6464 8172 16 05 no 420 930 1640 2550 3660 4970 6480 8190 2 5 j j 06 112 424 936 1648 2560 3672 4984 6496 8208 36 07 114 428 942 1656 2570 3684 4998 6512 8226 49 08 116 432 948 1664 2580 3696 5012 6528 8244 64 * 09 118 436 954 1672 2590 3708 5026 6544 8262 81 19* 10 121 441 961 1681 2601 3721 5041 6561 8281 oo 21 II 123 445 967 1689 26ll 3733 5055 6577 8299 21 12 125 449 973 1697 2621 3745 5069 6593 8317 44 25 13 127 453 979 1705 2631 3757 5083 8335 69 27 14 129 457 985 1713 2641 3769 5097 6625 8353 96 20* " 15 I 3 2 462 992 1722 2652 3782 5112 6642 8372 2 5 OT 16 134 466 998 1730 2662 3794 5126 6658 8390 5 6 J* 33 17 136 470 1004 1738 2672 3806 5140 6674 8408 89 # 18 139 475 IOII 1747 2683 3819 5*55 6691 8427 24 19 141 479 1017 1755 2693 3831 5i 6 9 6707 8445 61 39* 20 144 484 1024 1764 2704 3844 5184 6724 8464 00 4 21 22 I 4 6 145 488 492 1030 1036 1780 2714 2724 3856 3868 5198 5212 6740 6756 8482 8500 84 43 AS* 23 IS' 497 1043 1789 2735 3881 5227 6773 8519 29 45 47 24 3 $ I 5 8 501 506 1049 1056 IO02 1797 1806 1814 2745 2756 2766 3893 3906 5241 5256 5270 6789 6806 6822 P 8574 76 49* 5* 53* 27 28 161 515 519 1069 1075 1823 1831 2777 2787 3931 3943 5285 5299 6839 6855 ?? 29 84 55 29 166 524 1082 1840 2798 3956 53H 6872 8630 4i 59* 30 169 529 1089 1849 2809 3969 5329 6889 8649 00 61 3 i 171 533 1095 1857 2819 3981 5343 6905 8667 61 6,* 32 33 :n 542 1102 1108 1866 1874 2830 2840 3994 4006 5358 5372 6922 6938 8686 8704 n "j 65 | 67* 34 9 ig 547 55J 556 1115 1 122 1128 1883 1892 1900 2851 2862 2872 4019 4032 4044 5387 5402 6972 6988 8723 8742 8760 56 96 69* 7 / 73* 8 1*0 566 "35 1142 1909 1918 2883 2894 4057 4070 5431 5446 7005 7022 8779 69 44 75 * 39 J93 57i "49 1927 2905 4083 7039 88? 7 21 79* 40 196 576 1156 1936 29l6 4096 5476 7056 8836 00 81 41 198 580 1162 1944 2926 4108 5490 7072 8854 81 8 * 42 2OI 585 1169 '953 2937 4121 5505 7o8 9 8873 64 8 * 43 204 590 1176 1962 2948 4134 5520 7106 8892 49 87* 44 207 595 1183 1971 2959 4M7 5535 7123 8911 36 89" $ 210 213 600 605 1190 1197 1980 1989 2970 298l 4160 4173 5550 5565 7140 7i57 8930 8949 3 9t" 93* 47 216 610 1204 1998 2992 4186 558o 7174 8968 09 OS* 48 219 615 I2II 2007 4199 5595 7191 8987 04 95 07* 49 222 620 1218 2016 3 OI 4 4212 5610 7208 9006 01 97 99* 50 225 625 1225 2025 3025 4225 5625 7225 9025 oo SQUARES OF NUMBERS FROM 100 TO 1000 (Continued). 99 1~ 2~ 3~ 4** 5~ 6~ r+ 8~ 0*^ Diff. 60 225 625 1225 2025 3025 4225 5625 7225 9025 00 i 5 i 228 630 1232 2034 3036 4238 5640 7242 9044 01 52 53 231 234 635 640 1239 1246 2043 2052 3047 3058 4251 4264 5655 5670 7259 7276 9063 9082 04 09 3 5 7 54 237 645 1253 2061 3069 4277 5685 7293 9101 16 55 240 650 1260 2070 3080 4290 5700 7310 9120 25 56 243 655 1267 2079 3091 4303 5715 7327 9U9 36 131 ' 57 246 660 1274 2088 3102 43l6 5730 7344 9158 49 58 249 665 1281 2097 31*3 4329 5745 9177 64 59 252 670 1288 2106 3124 4342 5760 7378 9196 Si 19* 60 256 676 1296 2116 3136 4356 5776 7396 9216 00 I 61 259 681 1303 2125 3U7 4369 5791 7413 9235 21 62 63 265 686 691 1310 1317 2134 2143 3158 3169 4382 4395 5806 5821 7430 7447 9254 9273 % 35 27 64 268 696 1324 2152 3l80 4408 5836 7464 9292 96 29* 272 702 1332 2162 3192 4422 5852 7482 9312 25 66 275 707 '339 2171 3203 4435 5867 7499 9331 56 33 % 278 712 718 1346 1354 2180 2190 32H 3226 4448 4462 5882 5898 7534 9350 9370 8 9 24 * 69 285 723 1361 2199 3237 4475 59^3 755' 9389 61 39* 70 289 729 1369 2209 3249 4489 5929 7569 9409 00 41 72 292 295 734 739 1376 1383 2218 2227 3260 3271 4502 4515 5944 5959 7586 7603 9428 9447 41 84 43 73 299 745 I39i 2237 3283 45 2 9 5975 7621 9467 29 47 74 9 302 306 309 750 1398 1406 1413 2246 2256 .2265 3294 3306 4542 4556 4569 5990 6006 6021 7638 7656 7673 9486 9506 9525 76 49* s- 78 313 316 767 772 1421 1428 2275 2284 3329 3340 4583 4596 6037 6052 7691 7708 9545 95 6 4 29 84 55 79 320 778 1436 2294 3352 4610 6068 7726 9584 41 57* 59* 80 324 784 1444 2304 33 6 4 4624 6084 7744 9604 00 61 81 327 789 I45I 2313 3375 4637 6099 7761 9623 61 fia* 82 331 795 U59 2323 3387 4651 6115 7779 9643 24 03 6. 83 334 800 1466 2332 3398 4664 6130 7796 9662 89 J 67* f 4 ii 338 342 345 806 812 817 1474 1482 1489 2342 2352 2361 3422 3433 4678 4692 4705 6146 6162 6177 78H 7832 7849 9682 9702 9721 56 69* 71 73* 87 349 823 1497 2371 3445 4719 6193 7867 9741 69 88 353 829 1505 2381 3457 4733 6209 7885 9761 44 75 89 357 835 '513 2391 3469 4747 6225 793 9781 21 77* 79* 90 361 841 1521 2401 348i 4761 6241 7921 9801 00 81 9 1 364 846 1528 2410 3492 4774 6256 793 9820 81 92 93 368 372 852 858 1536 1544 2420 2430 3504 3516 4788 4802 6272 6288 7956 ^Q74 9840 0860 64 49 8 3* 85' 87* 94 384 864 870 876 1560 1568 2440 2450 2460 3528 3540 3552 4816 4830 4844 6304 6320 6336 79^* 8010 8028 9880 9900 9920 36 s 89* 9** 93* 11 388 392 882 888 1576 2470 2480 35 6 4 3576 4858 4872 6352 636* 8046 8064 9940 9960 09 04 95* 99 396 894 1592 2490 4886 6384 $082 9980 01 97* 99* 100 400 900 1600 2500 3600 4900 6400 8100 1 0000 oo TABLE VIII. DECIMALS OF DAY INTO HOURS, ETC. 101 H. M. S. H.M.S. H. M. S. H.M.S. D. H. M. S. ~"~ o D. H. M. S. IOO IOO IOO IOO* d. h. m. s. M. S. s. d. h. m. s. m, s. *. O.OI o 14 24 o 8.64 0.09 0.51 12 14 24 7 20.64 4.41 0.02 o 28 48 o 17.28 0.17 0.52 12 28 48 7 29.28 4.49 0.03 o 43 12 o 25.92 0.26 0.53 12 43 12 7 37.92 4.58 O.O4 o 57 36 o 34.56 0.35 0-54 12 57 36 7 46.56 4.67 O.O5 I 12 o 43.20 o.43 0.55 13 12 7 55-20 4.75 O.O6 I 26 24 o 51.84 0.52 0.56 13 26 24 8 3.84 4.84 0.07 I 40 48 0.48 0.60 0.57 13 40 48 8 12.48 4.92 0.08 I 55 12 9.12 0.69 0.53 13 55 12 8 21.12 5.01 O.O9 0.10 2 9 36 2 24 17.76 26.40 0.78 0.86 0-59 0.60 14 9 36 14 24 o 8 29.76 8 38.40 5.10 5.18 O.II 2 38 24 35.04 0.95 0.61 14 38 24 8 47.04 5.27 0.12 2 52 48 43-68 .04 0.62 14 52 48 8 55.68 5.36 0.13 3 7 12 52.32 .12 0.63 15 7 12 9 4.32 5.44 0.14 0.15 3 21 36 3 36 o 2 0.96 2 9.60 .21 .30 0.64 0.65 15 21 36 15 36 o 9 12.96 9 21.60 III 0.16 0.17 3 50 24 4 4 48 2 18.24 2 26.88 .38 47 0.66 0.67 15 50 24 16 448 9 30-24 9 38.88 5.70 5.79 0.18 4 19 12 2 35.52 .56 0.68 16 19 12 9 47.52 5.88 0.19 4 33 36 2 44.16 .64 0.69 16 33 36 9 56.16 5.96 O.2O 448 o 2 52.80 73 0.70 16 48 o 10 4.80 6.05 0.21 5 2 24 3 L44 .81 0.71 17 2 24 10 13.44 6.13 0.22 5 16 48 3 10.08 .90 0.72 17 16 48 10 22.08 6.22 0.23 5 3 1 I2 3 18.72 99 0.73 17 31 12 10 30.72 6-31 0.2l 0.2$ 5 45 36 600 3 27.36 3 36.00 2.07 2.16 0.74 o.75 17 45 36 1800 10 39.36 10 48.00 6.39 6.48 O.26 6 14 24 3 44.64 2.25 0.76 18 14 24 10 56.64 6-57 0.27 6 28 48 3 53.28 2-33 o.77 18 28 48 II 5.28 6.6 5 0.28 6 43 12 4 1.92 2.42 0.78 18 43 12 II 13.92 - 6.74 0.29 6 57 36 4 10.56 2.51 0.79 18 57 36 II 22.56 6.83 0.30 7 12 4 19.20 2.59 0.80 19 12 O II 31.20 6.91 0.31 7 26 24 4 27.84 2.68 0.81 19 26 24 II 39.84 7.00 0.32 7 40 48 4 36.48 2.76 0.82 19 4O 48 II 48.48 7.08 0-33 7 55 12 4 45.12 2.85 0.83 19 55 12 II 57.12 7-17 8 9 36 2.94 0.84 20 9 36 12 5.76 7.26 0.35 8 24 o 5 2.40 3.02 0.85 20 24 o 12 14.40 7.34 0.36 8 38 24 5 ".04 3 TI 0.86 20 38 24 12 23.04 7-43 0.37 8 52 48 5 19.68 3-20 0.87 20 52 48 12 31.68 7.52 0.38 9 7 12 5 28.32 3.28 0.88 21 7 12 12 40.32 7.6o o-39 9 21 36 5 36.96 3-37 0.89 21 21 36 12 48.96 7.6 9 0.40 9 36 o 5 45-60 3-46 0.90 21 36 12 57.6O 7.78 0.41 9 50 24 5 54-24 3-54 0.91 21 50 24 13 6.?A 7.86 0.42 10 4 48 6 2.88 3.63 0.92 22 4 48 13 14.88 7.95 o.43 IO 19 12 6 11.52 3-72 0.93 22 19 12 I? 13.52 8.04 0.44 10 33 36 6 20.16 0.94 22 33 36 15, jz.ib 8.12 0.45 10 48 o 6 28.80 3-89 0.95 22 48 15 40.80 8.21 0.46 0.47 II 2 24 ii 16 48 6 37-44 6 46.08 3-97 4.06 0.96 o.97 23 2 24 23 16 48 13 49-44 13 58.08 8.29 8.38 0.48 II 31 12 6 54.72 4.15 0.98 23 31 12 14 6.72 8.47 0.49 ii 45 36 7 3.36 4.23 0-99 23 45 36 14 15.36 8.55 0.50 12 7 12.00 4-32 1. 00 24 o o 14 24.00 8.64 i 102 TABLE IX. ARC INTO TIME AND VICE VERSA. o h, m. o /i. m. h. m. h. m. o h. m. h. m. i m. s. j. o O O 60 4 o 1 20 8 180 12 O 240 16 o 300 20 o 0.000 I o 4 61 4 4 121 8 4 181 12 4 241 16 4 301 20 4 i o 4 i 0.066 2 o 8 62 4 8 122 8 8 182 12 8 242 16 8 302 20 8 2 o 8 2 *33 3 12 63 4 12 123 8 12 183 12 12 243 16 12 33 2O 12 3 12 3 0.200 4 o 16 64 4 16 I2 4 8 16 184 12 16 244 16 16 304 20 16 4 o 16 4 0.266 I o 20 o 24 66 420 I2 5 126 8 20 824 185 186 12 20 12 24 III 16 20 16 24 305 306 20 20 20 24 I O 20 o 24 \ o 333 0.400 i 7 028 67 4 28 127 8 28 187 12 28 247 16 28 20 28 7 28 7 0.466 8 032 68 4 32 128 832 188 12 3 2 248 16 32 308 2o 32 8 o 32 8 o-533 9 o 36 69 436 129 836 189 12 36 249 1636 309 20 36 9 o 36 9 0.600 10 o 40 70 4 40 130 8 40 190 40 250 16 40 310 2O 40 IO o 40 10 0.666 n o 44 4 44 844 191 12 44 251 16 44 3" 20 44 n o 44 n 0-733 12 o 48 72 448 132 8 48 192 12 48 252 1648 312 2048 12 o 48 12 0.800 13 052 73 452 133 852 193 12 52 253 16 52 313 20 52 3 o 52 13 0.866 i '4 o 56 74 456 134 8 56 194 12 56 254 16 56 20 56 M o 56 M 0-933 11 o 4 76 c o 5 4 135 136 Q O 9 4 '95 196 13 o 13 4 255 256 17 o 17 4 3'5 316 21 O 21 4 15 16 o 4 15 1 6 1. 000 1.066 11 8 12 78 5 8 5 '2 137 138 9 8 9 12 197 198 13 8 13 12 257 258 17 8 17 12 317 21 8 21 12 \l 8 12 M I 133 1.200 *9 16 79 516 139 9 16 199 13 16 259 17 16 319 21 16 19 16 19 1.266 20 20 80 5 20 140 9 20 200 13 20 260 17 20 320 21 20 20 20 20 '333 21 22 24 28 Si 82 528 141 142 9 24 9 28 201 202 13 24 n 28 261 262 17 24 17 28 321 322 21 24 21 28 21 22 3 21 22 i 400 1.466 23 32 83 5 3? 143 9 32 203 '3 32 263 17 32 323 21 32 23 32 23 ' 533 2 4 36 84 536 144 9 36 204 13 36 264 17 36 324 21 36 24 36 24 i. 600 11 40 44 II 5 40 5 44 MS 146 9 40 9 44 205 206 3 40 3 44 266 1740 '7 44 1 21 40 21 44 3 40 44 11 1.666 1-733 11 48 52 11 548 5 52 H8 948 9 5 2 207 208 13 48 13 52 267 268 1748 17 52 21 48 21 52 11 48 52 11 i. 806 1.866 29 56 89 5 56 149 9 S^ 20 9 *3 5 6 269 17 56 329 21 56 29 56 29 * 933 30 2 O 90 6 o ISO 1C O 210 14 o 270 18 o 330 22 O 30 2 O 30 2 OOO 31 2 4 6 3 10 4 211 14 4 271 18 4 331 22 4 2 4 31 2.066 32 2 8 92 6 8 152 ic 8 212 14 8 272 18 8 332 22 8 32 2 8 32 2-133 33 2 12 93 6 12 153 10 12 213 14 12 273 18 12 333 22 12 33 2 12 33 2.200 34 2 16 94 6 16 154 ic 16 2I 4 14 16 274 18 16 334 22 16 34 2 16 34 2.266 35 36 2 20 2 24 95 96 6 20 624 155 IO 20 ic 24 215 216 14 20 14 24 III 18 20 18 24 335 336 122 20 22 24 35 3 6 2 20 2 24 1 2-333 2.4OO 37 2 28 6 28 *57 10 28 217 14 28 277 18 28 337 22 28 2 28 2.466 38 2 3 2 98 632 158 10 32 218 M 3 2 278 1832 338 22 32 38 2 32 38 2-533 39 2 3 6 99 6 36 159 10 36 219 M 36 279 18 36 339 22 36 39 2 36 39 2.600 40 240 IOO 640 160 10 40 22O M 40 280 18 40 340 22 40 40 2 40 40 2.666 4' 2 44 101 644 161 10 44 221 M 44 281 18 44 22 44 2 44 2 733 4? 248 102 6 48 162 10 48 222 1448 282 18 48 342 22 48 42 2 4 8 42 2.800 43 2 52 103 652 163 10 52 223 14 52 283 18 52 343 22 S2 43 2 52 43 2.866 44 2 56 104 r ;6 164 10 56 224 14 56 284 18 56 344 22 5 6 44 2 5 6 44 2 933 45 46 3 o 3 4 I0 5 1 06 7 4 II ii 4 III 15 o 15 4 III 19 o 19 4 345 346 23 o 23 4 46 3 o 3 4 46 3.000 3.066 47 3 8 107 7 8 167 n 8 227 15 8 287 19 8 347 23 8 47 3 8 47 3 133 48 3 I2 108 7 12 1 68 II 12 228 15 12 288 19 12 348 23 12 48 3 I2 4^ 3.200 49 316 109 716 169 ii 16 229 15 16 289 19 16 349 23 16 49 316 49 3.266 50 320 no 7 20 170 II 20 230 15 20 290 19 20 350 2 3 20 50 3 20 50 3-333 51 3 24 in 724 171 II 24 2 3 I 15 24 291 19 24 351 23 24 51 51 3.400 52 3 28 112 728 172 II 28 232 15 28 292 19 28 352 23 28 52 3 28 52 3.466 53 3 32 "3 732 173 II 32 233 15 32 293 19 32 353 23 32 53 3 32 53 3 533 54 336 114 736 174 II 36 234 15 36 294 19 36 354 2336 54 336 54 3-600 55 3 40 115 740 175 II 40 235 '5 40 295 19 40 355 23 40 55 3 40 55 3.666 56 3 44 no 7 44 176 ii 44 236 15 44 296 19 44 356 23 44 56 3 44 56 3 733 348 117 748 II 48 237 15 48 297 1948 357 2348 348 57 3.800 58 3 52 118 7 52 178 II 52 238 15 52 298 19 52 358 23 52 58 3 52 58 3-866 .59 356 119 756 179 II 56 239 15 56 299 19 5 6 359 2356 59 356 59 3 933 TABLE Xa. TO CONVERT MEAN INTO SIDEREAL TIME. 103 Mean T. h. m. Correction. m. s. Mean T. h. m. Correction. /. s. Mean T h. m. Correction. m. s. Corr. for min. and sec. m. s. s. o o 0.00 8 o i 18.85 16 o 2 37.70 10 0.03 10 1.64 10 20.50 10 39.35 20 O.Oj 20 3.29 20 22.14 20 40.99 30 0.08 30 4-93 30 23.78 30 42.63 40 O.II 40 6.57 40 25-42 40 44.28 5 0.14 50 8.21 50 27.07 50 45.92 I 0.16 IO 0.19 I o 9.86 9 o i 28.71 17 o 2 47.56 20 3 0.22 C.25 IO 11.50 IO 30.35 IO 49.20 40 0.27 20 13.14 20 31-99 20 50.85 0.30 30 14.78 30 33.64 30 52.49 2 O 0.33 40 50 16.43 16.07 40 50 35.28 36.92 40 50 54.13 55-77 XO 20 0.36 0. 3 3 30 0.41 2 10 o 19.71 21.36 10 10 i 38.56 40.21 18 o 10 2 57.42 59.06 40 50 0.44 0.47 20 23.00 20 41.85 20 3 0.70 3 o o 49 30 c 40 So 24.64 26.28 27.93 30 40 50 43-49 45.14 46.78 30 40 50 2.34 3.99 5.63 IO 20 30 40 0.52 o 55 o 60 50 0.63 3 o o 29.57 II i 48.42 19 o 3 7.27 4 0.66 10 31.21 10 50.06 10 8.92 10 0.68 20 32.86 20 51.71 20 10.56 20 0.71 30 34.50 3 53-35 30 12. 2O 30 0.74 40 36.14 40 54-99 40 13.84 40 0.77 50 37.78 56.64 50 15.49 5 0.79 c o 0.82 4 o o 39-43 12 I 58.28 20 o 3 I7.I3 IO o 85 n RS 10 20 30 41.07 42.71 44.35 10 20 30 59-92 2 1.56 10 20 30 18.77 20.42 22.06 20 30 40 5 o.oo 0.90 093 0.96 40 46.00 40 4*85 40 23.70 50 47.64 50 6.49 50 25.34 6 o 10 o 99 .01 20 04 5 o o 49.28 13 o 2 8.13 21 3 26.99 30 .07 10 50.92 10 9.78 10 28.63 40 IO 20 52 57 20 11.42 20 30.27 50 .12 30 54.21 30 13.06 30 31.91 7 O j- 40 55.85 40 14.70 40 33.56 / 10 '18 50 57.50 50 16.35 50 35-20 20 .21 30 23 6 o 10 o 59.H i 0.78 14 o 10 2 17.99 19.63 22 10 3 36.84 38.48 40 50 .29 20 2.42 20 21.28 20 40.13 8 o .31 30 4.07 30 22.92 30 4L77 IO 34 40 5.71 40 24.56 40 43-41 20 37 50 7-35 50 26.20 50 45.o6 3> 40 .40 .42 So 45 7 o 10 I 9.00 10.64 15 o 10 2 27.85 29.49 23 o 10 3 46.70 48.34 9 o 48 rf\ 20 12.28 20 20 49.98 20 5 30 13.92 30 32.77 30 51-63 3 ?o 40 15.57 40 34.42 40 53.27 40 Pg 50 17.21 50 36.06 50 50 .6, TABLE X. TO CONVERT SIDEREAL INTO MEAN TIME. Sid. T. k. m. Correction. m. s. Sid. T. A. m. Correction. m. s. Sid. T. k. m. Correction, w. s. Corr. for min. and sec. m. s. s. o o.oo 8 o I 18.64 16 o 2 37-27 O IO 0.03 10 20 1.64 3.28 10 20 20.28 21 .91 IO 20 38.91 40.55 20 30 0.05 0.08 30 4.92 30 23.55 30 42.19 40 O.1I 40 6-55 40 25.19 40 43.83 5 0.14 50 8 19 50 26.83 50 45.46 I o. 16 10 o. 19 t o 9.83 9 o I 28.47 17 o 2 47.10 20 3 0.22 0.25 IO 11.47 IO 30.10 10 48.74 40 O.27 20 13." 20 3L74 20 50.38 50 0.30 30 H.74 30 33.38 30 52.02 2 O o ? 3 40 16.38 40 35-02 40 53-66 10 0.35 50 18.02 50 36.66 50 55-29 20 0.38 30 0.41 t IO o 19.66 21.30 10 10 i 38.30 39-93 18 o IO 2 56.93 58.57 40 50 0.44 0.47 20 22.94 20 4L57 20 3 0.21 3 O 0.49 30 40 24.57 26.21 30 40 43-21 44-85 30 40 1.85 3.48 IO 90 0.52 o-55 50 27.85 50 46.49 50 5.12 3 40 0.57 0.60 50 0.63 ^ o o 29.49 II i 48.12 19 o 3 6.76 4- o 0.66 10 31.13 10 49.76 10 8.40 10 0.68 20 32.76 20 51.40 20 10.04 20 0.71 30 34.40 30 53.04 30 11.68 30 0.74 40 36.04 40 54.68 40 13.32 40 0.76 50 37.68 50 56.32 50 J4-95 5 0.79 5 0.82 4 o 10 20 o 39.32 40.96 42.60 12 IO 2O i 57.96 59-59 2 1.23 20 o 10 20 3 16.59 18.23 19.87 10 20 30 4 0.85 0.87 0.90 0.93 30 44.23 30 2.87 30 21.51 5 0.96 40 45.87 40 4-5' 40 23.14 30 47.51 50 6.15 50 24.78 6 o 10 0.98 i .01 20 i .04 C o o 49.15 13 o 2 7.78 21 3 26.42 30 i. 06 10 50.79 IO 9.42 IO 28.06 40 1.09 20 52 42 20 II. 06 20 29.70 5 I . 12 30 54.06 30 12.70 3 31.34 7 .15 40 55-70 40 14-34 40 32.97 IO 1 7 50 57-34 50 15.98 50 34.6i 20 .20 30 2 3 6 o o 58.98 14 o 2 I7.6l 22 3 36.25 40 fr\ .26 10 I 0.62 10 19.25 10 37.89 5 l2 20 2.25 20 20.89 20 39-53 8 o 31 - 3 3.89 30 22.53 30 41.16 10 34 40 5-53 40 24.17 40 42.80 20 37 50 7.17 50 25.80 50 44-44 3 40 39 .42 5 45 7 o i 8.81 15 o 2 27.44 23 o 3 46.08 4*7 IO 10.44 lo 29.08 10 47.72 IO *t/ 20 12.08 20 30.72 20 49.36 20 C? 30 13.72 30 32.36 30 51.00 3 56 40 15.36 40 34-00 40 52.63 40 .58 5 17.00 50 35.64 50 54.27 50 .61 i UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. ASTRONOMY LIBRARY flffT 2 5 11 G' LD 21-100m-ll,'49(B7146sl6)476 YC i 02258 M19307: AsH THE UNIVERSITY OF CALIFORNIA LIBRARY