A Compendious Course of Mathematics, Theoretical and Practical

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1855 edition. Excerpt: ... circumstances which must be always borne in mind. 1. That whatever the base a may be, the logarithm of that base, in the system to which it belongs, is always 1; for o'=a, .'. l=log a. 2. The logarithms of the same number, in any two systems, are to one another as the moduli of those systems; for the right-hand member of (A), provided n remain unaltered, would differ from its present value if the system were changed from that in which the modulus is M, to another in which the modulus is M', only in having Mreplaced by M', 'M so that the ratio of the two expressions for log (1 + n) would be-. Let now e stand for the value of a, which satisfies the equation A = 1; then, by the exponential theorem, so that making x=1, we readily find for the Napierian base e, the value e= 1 + 1 +1 +-5 + 5--7+...=2-718281828459 On account of the rapidity with which this series 2 converges, a comparatively small number of the leading 3 terms will suffice to determine e to several places of 4 decimals; thus, taking only ten of the terms 2 + 5 + y; +...., and computing as in the margin, we get Jj e true to seven decimals. 8 Resuming now the general series for log(l +n), and 9 leaving the base of the system, and consequently the 10 modulus-j, or M, open to any hypothesis, as to value, that may be hereafter introduced, we have, by taking n 2 '7182818 first positive and then negative, log(l + )=j)/(_i + 53-i + &c.)....(l) A 0 log(l-n) = M(-n-ln'-l?-n-&o.)....(2) m O 4 1 + % Subtracting, log(l + n)--log(l--n), that is, log = 1--n 2Mn + ln3 + ns + &, c)....(3) o o 59. These series are of but little direct importance in computing a table of logarithms, on account of their slow convergency, except when...