COMPUTATION OF LOGARITHMS. 149 where we have only to divide successively by 9[=32] and the odd numbers 3, 5, 7, ..., as follows: 3/2.00000000 9 '66666667 166666667 9 00823045 500164609 Having thus obtained the Napierian logarithm of 2, 14= 1(2 • 2) =12+12=212=138629436; and putting m=5, n=4, Suppose, now, it were our object to construct a table extending = from 1 to 10000, we should commence with 100; since, in calculating up to 10000 100 100 we should fall upon 200, 300, 400, &c., = 2. 100, 3. 100, 4. 100, &c., whereby the logarithms of 2, 3, 4, &c., would be determined. Since log. 100= log. (102) 2log. 10 = 2.1 = 2, = log. 101 = log. 100 +0·86858896(201+...) =2+· 0.86858896 After two consecutive logarithms have been obtained, the operation may be shortened; for, putting m = n+1, we have Observe also that 4(2n+3) is the difference of two consecutive divisors, 4(n+1)2 — 1, 4(n + 2)2 — 1. Making n = 100, log. 1022log. 101-log. 100 1.7371779 4. 1012-1 41623)1 7371779(417_ Operation. +4-0172004 - 2.0043214 = ... log. 103 240128373 The student should continue the computation. Before terminating this problem, we will make one other transformation whereby the calculation of logarithms will be rendered far more rapid. COMPUTATION OF LOGARITHMS. 151 and For small numbers, Borda puts* m = (p-1)2(p+2)= p3-3p+2, n = (p+1)2(p-2)=p3-3p-2; m− n = 4, m+n=2p3-6p; therefore, substituting in (275) and making M = 1, we have Making p = 5, 6, 7, 8, successively, we obtain 212-313+17=2[55 ++(35)+...]='036367644171, 12+215-217=2[+ + +(4)+...]='020202707317, -412-15+413=2[11+ (11)3+...]='012422519998, -513+15+217 = 2[++++(x+4)+...]='008196767203; from which eliminating 17, we have 512+215-6/3 = 092937995659, Had the above operation been extended sufficiently far, we should have found (272) which, introduced into the last transformation, gives, for the common logarithm, log. (p+2)=2log.(p + 1) +log.(p − 2) — 2log.(p − 1) 2 2 3 +8685890039 [(p(p2 -3))++ (p(p2 -3)) '+]. (278) ... Employing this form and imitating the above operation for the Napierian logarithms, 12, 13, 15, 17, the student will findt * Francœur, Mathématiques pures. ✦ The number of digits employed in the calculation should exceed, according to the nature of the operation, those which it is intended to retain in the results. log. 11=2log. 10+ log. 7-2log. 8+86... [3+}(331)3 + ...] ; log. 121'07918 12460, log. 131'11394 33523, [log. 12 = log. 3+log. 4.] [p = 11.] log. 14= log. 2+ log. 7, log. 15-log. 3+log. 5, log. 16=2log. 4, log. 17=123044 89214, [p = what?] log. 18=? log. 191627875 36010, log. 20 = ? log. 29146239 79979, log. 30= ? log. 31 log. 35 log. 40 = = = log. 21 = ? log. 22 = ? 1'4913616938, log. 32 ? log. 33 ? log. 34 = ? = ? log. 36=? log. 37 = ? log. 38=? log. 39 = ? = ? log. 50 ? log. 60? log. 100? log. 1000 = ? log. 10000? log. ? log. ? log. 001 = ? =? Oo As p increases, the series (278) increases in convergency very rapidly; so much so, that, when p is no greater than 102, the second term will have its first significant figure in the 18th decimal place; and, confining ourselves to 7 decimals, the last two only will have to be obtained by division. If we confine ourselves to seven digits, after the computation has been made up to 1000, the remaining logarithms may be readily calculated from the table itself. Taking the differences of five consecutive logarithms, as those of 100, 101, 102, 103, 104, and the differences of these differences or the second differences, and the third differences, we find We observe here that the third differences, 8, 9, are nearly equal. We are naturally led to the following problem: PROPOSITION IV. It is required to determine what function y, is of x, when y, depends upon x in such way that, if we attribute to x the particular values x = 0, 1, 2, 3, 4, the third differences of the corresponding values of the function, y = Yo, Y1, y2 y3, Y1, shall be equal to each other. Attributing to the particular values 0, 1, 2, 3, indicating the corresponding particular values of y, by yo, y1, Y2, Yз, taking the differences as above, and denoting the first difference, y1 Yo, of the first differences by D1, the first difference of the second differences by D, the first of the third differences by D3, (which, for the sake of distinction, may be called the first, second, and third differences,) we have, |