log. 11=1+0.04357,50878,58—.00218,24027,01)= 1.04139,26852, log. 9=14.04357,50878,58+.00218,24027,01)= 0.95424,25094. To find log. 101 and log. 99, we have from the preceding calculation, simply by changing the position of the decimal point, log. 101=2+0.00434,30895,93—.00002,17158,10)= 2.00432,13738, log. 99=240.00434,30895,93+.00002,17158,10)= 1.99563,51946. We employ twelve places of decimals in the computation, but retain only ten in the logarithms, which may therefore be relied upon as correct in the last place. In the same manner we should find log. 1001 = 3.00043,40775 log. 999 = 2.99956,54882 log. 10001 = 4.00004,34273 log. 9999 = 3.99995,65684 116. These logarithms prepared, we may proceed as follows. We have 1024=210, and log. 1024=10 log. 2; therefore in (53) d let n=1000, d=24, and =.024; then n 10 log. 2= log. 1000+ M[.024—10.024)+ }(.024)3 — &c.] whence, by taking six terms of the series, log. 2= (3 + .01029,99566,4)=0.30102,99956,64, &c.) 1013 To find log. 7, in (64) let p=9800=79.2.10%, p+1=980111%.34; then we have 2 log. 11 + 4 log. 3—2 log. 7— log. 2- 2 log. 10 = 1 2 M [ +(19601)*+ &c. 19601 whence we have 1 log. 7= log. 11+2 log. 3— { log.2—1— M( + &c. 19601 1 Here the second term of the series is less than ; the first term will therefore suffice in finding the logarithm to eleven or twelve places. The same degree of accuracy is obtained with the use of only one term of the series in the following examples. In (64) let p+1=6656=2o.13, p=6655=5.11; then we have log. (p+1)=9 log. 2 + log. 13, log. p= log. 5+3 log. 11, and the formula gives 2M 13311 2M + &c.) Let p+1=5776=192.24, p=5775=3.7.5%.11 ; then M log. 19=} (log. 5775 - log. 16)+: + &c. 11551 Let p+1=8281=7*.13, p=8280=23.34.5.23; then 2M - ( + &c.) 16561 Let p+1=13225= 59.23%, p=13224=23.3.19.29; then 2M + &c. .) log. 29= log. 13225 — log. 456—(26449 Let p+1=8464=232.24, p=8463= 3.7.13.31; then 2M Let p+1-5625=54.3%, p=5424=28.19.37; then 2M log. 37= log. 5625 — log. 152 + &c.) 11249 Let p+1=6561=38, p=6560=25.5.41; then 2M 13121 -( 13121 117. To continue the computation for primes above 41, we may employ Borda's formula, (65), without any transformations such as the above; and the first term will suffice in finding the logarithms to twelve places. Thus, to find log. 43 we make p=41, and find 2M 34399 In like manner 2M log. 47=2(log. 46 — log. 44)+log. 43+ + &c. 45495 2M log. 53=2(log. 52 - log. 50) +log. 49+ + &c. 66249 and with this formula we may compute the logarithms of all the prime numbers under 100. Above 100 it will be more convenient to use (67), which requires only two logarithms to be known, and gives the logarithm correctly to twelve places with the use of only one term of the series. For example, when q=103, we have M log. 103=i(log. 102 +log. 104)+ + &c. 21217 Log. 104 is known before log. 103, since 104 is a composite number =8.13. In general, q being a prime number, 2-1, and q+1 must be composite, and their logarithms are found before that of q. * These transformations may be obtained, by inspection, from a table of composite numbers. ; the 118. Above 500 the formulæ of Art. 110 may be employed, and the series omitted entirely except in computing very extended tables. 24 1 Thus, in (70), if n=500 it is evident that is less than n5 1012 series therefore will not affect the eleventh or twelfth place. Omitting the series, the formula may be put in the following form: log. (n+3) = log. (n— 2) + 5[log. (n+2) — log. (n-1)]10[log. (n + 1)— log. n]. (74) For example, the common logarithm of 509 may be found from the five logarithms immediately preceding it in the table, by a computation like the following, where we put D= log, (n+2)— log. (n-1), D'=log. (n+1) - log.n, and the characteristics are omitted. n 5D= N. Logarithms. Calculation. n - 2150470243,05364,45 70243,05364,45 -1505 70329,13781,19 1286,16708,25 n 506 70415,05168,40 71529,22072,70 n +1 507 70500,79593,33 857,44249,30 n+ 2 508 70586,37122,84 log. (n+3) = 70671,77823,40 10 D' = When n=10000 in (68), we may omit the series, which will not affect the eleventh place of decimals; and we shall have log. (n+2) = log. (n-1)-3[log. (n+1) - log. n.] (75) Or we may omit the series in (65), and we shall have log. (P+2)= log. (p—2)+2[log. (p+1)— log. (p-1.)] (76) With these formulæ we may continue the computation as far as log. 100,000, which is the usual limit of the tables. 119. Above 100,000 we may use (67) without the series, if we require only ten places; we shall then compute by the formula log.q=i[log. (9-1) + log. (9+1)], . 12 (77) from which it appears that the logarithm of any number above 100,000 is an arithmetical mean between the log. immediately preceding and that immediately succeeding it—at least as far as the tenth place of decimals. With this formula, then, we may extend the table at pleasure. 120. By retaining the series in the formulæ, and using one, two, or more terms, we may compute logarithms to twenty or more places, as has been done by Briggs and others. The variety of the formulæ here given will enable the computist to test the correctness of his work by occasionally computing the same logarithm by two or more methods; he will thus detect the amount of the error occasioned by the neglected terms of the series, and may operate in such a manner that this error shall not affect the last figure retained. ANTI-LOGARITHMS. 121. Formula (41) of Art. 91, enables us to compute the number corresponding to any given logarithm. We may thus obtain the approximate numbers corresponding to exact logarithms. These numbers have been called anti-logarithms, and tables of them are found convenient in some astronomical computations. In these tables the usual arrangement is reversed; the exact logarithms are placed first, increasing regularly by 1 from 1 to 10000, (or, as in Dodson's Anti-Logarithmic Canon, to 100,000,) and the corresponding nearest numbers in the columns opposite, with their differences and proportional parts. m n mn 1+ D 122. In (41) let b= and log. b= log. m-log. n=D; the formula becomes [ +)*+z73G)*+ &c. (78) In applying this to naperian logarithms we make M=1; we then have D3 + &c.) (79 |