Mo = 0.43429 44819 03251 82765 11289 18916 60508 22943 97005 80366 65661 14453 78316 58646 49208 87077 47292 24949 33843 17483 18706 10674 47663 03733 64167 92871 58963 90656 92210 64662 81226 58521 27086 56867 03295 93370 86965 88266 88331 16360 77384 90514 28443 48666 76864 65860 85135 56148 21234 87653 43543 43573 17253 83562 21868 25 which is true certainly to 272, and probably to 273, places (_Proc. Roy. Soc._, 1886, 42, p. 22, where also the values of the other logarithms are given). If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus, log(10) (1 + x) = M (x - ½x² + (1/3)x³ - ¼x^4 + &c.), and so on. As has been stated, Abraham Sharp's table contains 61-decimal Briggian logarithms of primes up to 1100, so that the logarithms of all composite numbers whose greatest prime factor does not exceed this number may be found by simple addition; and Wolfram's table gives 48-decimal hyperbolic logarithms of primes up to 10,009. By means of these tables and of a factor table we may very readily obtain the Briggian logarithm of a number to 61 or a less number of places or of its hyperbolic logarithm to 48 or a less number of places in the following manner. Suppose the hyperbolic logarithm of the prime number 43,867 required. Multiplying by 50, we have 50 × 43,867 = 2,193,350, and on looking in Burckhardt's _Table des diviseurs_ for a number near to this which shall have no prime factor greater than 10,009, it appears that 2,193,349 = 23 × 47 × 2029; thus 43,867 = (1/50)(23 × 47 × 2029 + 1), and therefore log(e) 43,867 = log(e) 23 + log(e) 47 + log(e) 2029 - log(e) 50 1 1 1 + --------- - ½ ------------ + (1/3) ---------- - &c. 2,193,349 (2,193,349)² (193,349)³ The first term of the series in the second line is 0.00000 04559 23795 07319 6286; dividing this by 2 ×2,193,349 we obtain 0.00000 00000 00103 93325 3457, and the third term is 0.00000 00000 00000 00003 1590, so that the series = 0.00000 04559 23691 13997 4419; whence, taking out the logarithms from Wolfram's table, log(e) 43,867 = 10.68891 76079 60568 10191 3661. The principle of the method is to multiply the given prime (supposed to consist of 4, 5 or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by 1 or 2, the prime factors may all be within the range of the logarithmic tables. The logarithm is then obtained by use of the formula