the differences of the logarithms are proportional to the differences of the numbers. Suppose, then, that the logarithm of 14518469 were required. From the tables we find, as before, neglecting for the present the characteristic (see a page of the tables of Callet at the end of this volume), log. 14518 1619068. This is also the logarithm of 14518000, which differs from the logarithm of the next number 14519, or 14519000, viz., 1619367 by 299, while the numbers themselves differ by 1000. But the number 14518000 differs from the given number 14518469 by 469, the last three figures not yet used; hence the proportion which result, added to 1619068, gives 7.1619209 for the logarithm required, 7 being the proper characteristic for the logarithm of a number consisting of eight figures. 2 10+ 9 The proportion is solved by multiplying the difference 469 by 9 Now, by inspecting the last column of the page, this differ ence, 299, will be found ready calculated, and its product as nearly as it can be 1 2 3 expressed in two or three figures by 10' 10' 10' &c., or .1, .2, .3, &c., the multiplier being in the left hand and the product in the right hand of the two small columns of figures under the difference, 299. These multipliers may be regarded as hundredths or thousandths, only giving the products their proper place. With this explanation, the following calculation will be understood: Log. 14518 0.4 0.06 1619068 120 18* 3 215. To find the number corresponding to a given logarithm, say 1619209, look in the column marked 0 for the nearest less logarithm, and take the corresponding number, which is 1451. Run the eye along the horizontal line till the number most nearly approaching 9209, forming the last four figures of the given logarithm, is found. This is 9068, which is found in column 8. Subtract this from 9209, and the difference is 141. Find in the right hand of the two columns of small figures marked dif. et p., or simply dif., at the top of the page, the nearest less number than 141; this is 120, which answers to 4 in the left hand. The difference between 120 and 141 is 21. Multiply 21 by 10, and seek, as before, in the small column, the number nearest 210; this is 209, which answers to 7. The calculation is below. The numbers 4 and 7 thus found may be simply annexed to 14518. * The number in the table is 179; but, as the 9 is rejected, the 7 is increased by 1, since 179 is nearer 180 than 170. If the characteristic of the logarithm had been 6, the number would have been 1451847; 14.51847; 1.451847; This table contains in the first three columns an arrangement for reducing any number of degrees, minutes, and seconds, or hours, minutes, and seconds, to seconds, which is particularly useful in astronomical calculations, where the logarithm of the number of seconds in a given number of degrees, minutes, and seconds is frequently required. EXAMPLE I. Reduce 0° or 0 24' 57" to seconds. In the table (see last page), at the head of the first column, find 0°, and immediately under it 24'; descending this column to 55", near the bottom, and opposite 57", which is understood to be two numbers below, is found 1497, the number of seconds required. If the degrees or hours exceed 3, the proceeding is different. EXAMPLE II. To reduce 4° or 4h 2′ 39′′ to seconds. Find 4° 0' at the head of the second column, and below, in this same column, 2′ 30′′, to which corresponds, in the third column, 1455. Thus, 4° 2′ 30′′=14550′′... 4° 2′ 39′′-14559′′. EXAMPLES OF THE APPLICATION OF LOGARITHMS. (1) To find the value to within 0.01 of the expression log. x= log. 7340+ log. 3549— log. 681.8—log. 593.1. The following is the calculation: 216. The arithmetical complement of a logarithm is what remains after the logarithm is subtracted from 10. Thus, the arithmetical complement of the logarithm 2.7190826 is 10-2.7190826=7.2809174, which is obtained by beginning on the right and subtracting each figure (carrying 1 to all except the first) from 10, or beginning on the left and subtracting each figure of the logarithm from 9, except the last, which is subtracted from 10. 217. The operation of subtraction of logarithms can be replaced by addition, if we use the arithmetic complement; for if, to a given logarithm, log. a, we add the arithmetical complement of another logarithm, such as 10- log. b, we have log. a+10-log. b, from which, rejecting 10, the result is log. a-log. b, the same as would be obtained by simply subtracting the second logarithm from the first. We have then the following rule for operating with arithmetical complements: Add the arithmetical complements of the logarithms of the divisors and the logarithms of the multipliers of a formula together, rejecting 10 from the sum for every arithmetical complement employed. The above example would be wrought by this rule as follows: ar. comp. log. 681.8-7.1663430 ar. comp. log. 593.1-7.2268721 sum rejecting 20=1.8090172=log. x,...x=64.42. We thus obtain the same result as by the other method. The number corresponding need be taken from the tables only to four figures, because, the characteristic being 1, the entire part of the number will contain but two places, which will leave two places for the decimal part, as required, since the value of x was to be obtained to within 0.01. (2) To find the value within 0.00001 of the quotient. log. 146298=4.1321907 ar. comp. log. 988789-5.0040803 sum-10, or log. x=1.1362710 (3) Required by means of logarithms. The division by 11 is performed by adding -10 to the negative part of the logarithm and +10 to the positive. The logarithm to be divided is viewed as if written thus : -11+10.6825796. EXERCISES IN LOGARITHMS. (4) Calculate the logarithm of 8 from the table on page 259. (5) Also of 7, 70, 700, 7000, 70000. (6) Also of 356, 35600, 3560000. (7) From the tables find the logarithms of 314, 3.721, 41.2. (8) Also of 7315, 8416, 91.75, 34760, 1708000. (9) Find the numbers the logarithms of which are 0.13130, 4.56502. (10) Also those the logarithms of which are 3.6520528, 7.4891144. (11) Those the logarithms of which are 4.49010, 0.66200, 5.72403. (12) Find by proportional parts the logarithms of 314761, 440736, 37023400, 2111768. (13) Also of 22.3345, 137.2014, 46.27835. (14) Of .75, .341, .7391, .0347, .000536, .0000083. (15) Of 5 3 6 4 7 7' 8' 11' 13' 40° (16) Find the logarithm of the product of 9.734 and 5.639. (17) Also of 35.98 × 7.433 × 6.543 × 29.78. (18) Also of 22.74 × 31.201 × 0.0067 × 0.9298. (19) Divide 3758000 by 4986 by means of logarithms. 1 (22) Find the logarithm of 133' 114' 4566' 1000' 3946' (26 (33) Find by means of logarithms, using the arithmetical complement, the 218. The common logarithms, or logarithms of Briggs, are applicable only to the operations of multiplication, division, formation of powers, or extraction of roots, and do not apply when the required operation is that of addition or sub traction, indicated in formulas by the quantities to be operated upon being connected by the signs + and A system of logarithms has, however, been invented by Gauss,* designed exclusively for sums and differences. The arrangement of these tables, which contain three columns, marked A, B, C, is founded upon the following simple considerations. We have for the form of a sum p+q, and of a difference p-q, the following identities: and log. (p—q)= log. p— log. (2) The logarithms of the sum p+q and the difference p-q appear, therefore, in these formulas, equal to the sum or difference of two logarithms, the first of which is to be considered as directly given, but the second of which must be found by the Gauss tables. They contain, I. In the column A logarithms of numbers of the form ), increasing from 0.000 to 5.000. II. In column B logarithms of numbers of the form (+), decrensing from 0.30103 to 0.00000. III. In column C logarithms of numbers of the form 0.30103 to 5.00000. Now, therefore, inasmuch as log. (2)= log. p-log. 4, by the tables of common logarithms, the first thing to be done is to take the difference of the common logarithms of P and q, enter with this column A in the Gauss logarithms, and take out the corresponding number from column B. The addition of this number to logarithm p will give, according to (3), the logarithm sought of p+q. In order to find the logarithm of the difference p-q, by means of the logarithms of p and 4, two cases must be considered: 1o. Where P 9 <2... log. p- log. q<0.30103, it is only necessary to enter with this difference column B, and to subtract the adjoining logarithm of column C from logarithm p. For, corresponding to the logarithms of numbers of the form 2>2 2o. If >2... in B, C contains the logarithms of those of the form log. p-log. q>0.30103, and, therefore, is contained in the column C; subtract the corresponding logarithm in column B from loga * They are found in the latest edition of the tables of Vega, and those edited by Köhler. |