M (2/ 1 1 -- n n 3 n 9 + 27 81 243 3n3 1 1 2n2 3n3 1 1 1 + 4 8 2n3 3n3 n1 n5 16 32 1 "4n1 16 4n4 9 27 81 2n2 3n3 4n1 &c.) (4) &c.) (B) + — &c.) (C) log.(n-3)—log.n=M( Combine these equations thus: B-3C-D; we shall find log. (n+2)-3 log. (n+1)+3 log. n-log. (n-1)= 3 6 10 + + &c.) no 4. 1 5n5 32 5n5 + &c. 243 5n5 &c.) (D) &c.) (E) — &c.) (F) Make the combination B-4C-4D-E; then log. (n+2)-4 log. (n+1)+6 log. n-4 log. (n-1)+log. (n-2)= -M 6 20 63 204 + + + + &c.) (69) Make the combination A-5 B+10 C+5 D-E; then log. (n+3)-5 log. (n+2)+10 log. (n+1)-10 log. n + 5 log. (n-1)-log. (n-2)= M 24 60 240 630 (68) Make the combination A-6 B+15 C+15 D-6 E+F; then log. (n+3)—6 log. (n + 2) +15 log. (n + 1)—20 log. n + 15 log. (n-1)—6 log. (n − 2) + log. (n − 3) = 120 1260 10584 + + &c.) n8 210 (70) (71) Such formulæ may be multiplied at pleasure, as the law of their formation is easily recognized. They express the relation between a number (m+1) of logarithms of consecutive numbers; the terms in the first members are alternately positive and negative; the coefficients are those of the (mth) power of the binomial, and the denominator of the first term of the series is nm. Formula (70) is perhaps the most useful, as it is very easy to multiply by the numbers 5 and 10; but the series in (69) and (71) are more convergent than the series in (70). 111. Logarithms may also be found by the following method. Instead of the logarithm of the number itself, we seek that of any root of the number which is less than 2. This logarithm will be found by (34); we then multiply it by the index of the root to obtain the logarithm of the given number, (Arts. 62 and 63). log.p=nM n log.p= log.p-M p-1- m In this formula, as n is arbitrary, we may assume it so great that the nth root of the number p will be less than 2; then pr-1 will be less than 1, and its powers will decrease. This transformation is the same as that given in Art. 92, in finding the modulus; and if we substitute in (72) the value of M, given by (42), we shall have 1 [ p=_1-¦ (p=_1)2 + } (p = −1) '—&c.] (72) (p = -1)*—&c.] 1 am-1 (73) which is the formula given by LAGRANGE. It expresses the logarithm of any number in terms of that number and the base of the 1 system, n and m being arbitrary numbers, but such that pa and am are less than 2. 1 This method is similar to that actually employed by Briggs in the construction of his tables; but the tedious operation of extracting roots which it requires, has been rendered unnecessary by the more expeditious formula which we have already given. APPLICATION OF THE FORMULA. 112. In the actual construction of a table, it is necessary to employ the preceding formulæ only in finding the logarithms of prime numbers; for the logarithms of composite numbers will be found by adding together the logarithms of their factors, (Art. 60). We shall have to find by the formulæ the logs. of the prime numbers 2, 3, 5, 7, 11, 13, 17, &c., but those of the composite numbers will be derived from these as follows: = log. 4= log. 23-2 log. 2, log. 8 log. 23 3 log. 2, log. 10= log. 2+ log. 5, = = log. 6=log. 2 + log. 3, 113. If we wish to find naperian logarithms we make M=1 in all the formulæ, which then take the simplest form for calculation. But in finding common logarithms we make M=.4342, &c., (Art. 93). This value of M may be found directly as in Art. 93, or more conveniently, as in Art. 99, from the naperian logarithm of 10. L. 2=2 The most natural method of finding the naperian logarithm of 10 is to find that of 2, then that of 5, and then that of 10=L. 2+L. 5. For this purpose, in (64) let M=1 and p=1; then L. p=0, and we have 1 1 1 (1/3+ + + and by actual computation, taking ten terms of the series, we have L. 2=0.69314,71805,6. + &c. Then L. 4-2 L. 2=1.38629,43611,2. Making p=4 in (64), we have 1 1 1 L. 5=L. 4+2 (+ + + + &c.) 3.93 5.95 7.97 (taking six terms of the series) =1.38629,43611,2+.22314,25513,1, =1.60943,79124,3. Hence L. 10=0.69314,71805,6+1.60943,79124,3, =2.30258,50929,9. From this we find, as in Art. 99, Briggs' modulus = reciprocal of L. 10 .43429,44819,03. 114. The preceding example sufficiently illustrates the mode of using the formulæ. We have only to substitute numbers for letters, and compute the series by adding a number of terms, greater or less according to the degree of accuracy required. It will be observed that, generally, the most convergent formulæ are those which require the greatest number of logarithms to be previously known. Thus, (59), which requires no logarithms to be given, is not so convergent as (64), in which one logarithm is supposed to be known; formulæ (65....71) are still more convergent, requiring three, four, five, or more logarithms to be given. To find the common logarithms of 2 and 5, we have only to make M=.434, &c. in the formulæ and compute the series as above; or we may multiply their naperian logarithms already obtained, by .434, &c., according to Art. 98. We may then proceed to find the logs. of the succeeding prime numbers as follows: We might thus compute the entire table with this formula only; but the labour may be much abridged by various devices, some of which we shall here explain. log. 11=1+.434 &c. 115. In commencing the construction of a table, we know only the logarithms of the integral powers of the base of the system; from these we may find the logs. of all numbers greater or less than these powers by unity, by the formulæ of Art. 106. Thus, in Briggs' system we know the logs. of 10, 100, 1000, &c. to be 1, 2, 3, &c. From these we may find those of 9, 11, 99, 101, 999, 1001, &c. by making successively n=10, 100, &c. in (60) and (61); we thus find 1 1 -(10+ 2.100 +3.1000 &c. 14,47648,27 8685,89 62,04 48 .04357,50878,58 &c. The similarity of these series enables us to compute them all at the same time, by observing that the significant figures of the first, second, third, &c. terms respectively, are necessarily the same in all the series, but are preceded by a greater or less number of ciphers. We subjoin the computation of the preceding examples, designating the terms by A, B, C, &c. To find log. 11 and log. 9, A.04342,94481,90 F= H= 1 4.10000 K 1 4.10000 = 1 4.100002 + &c. + + &c.) 1+ &c.) 1 4.10000o B = .00217,14724,10 D = ;+ &c.) 1,08573,62 5,43 04 .00218,24027,01 |