A = 1.000000 0.0000000 B = 10.000000 1.0000000 C = [root](AB) = 3.162277 0.5000000 D = [root](BC) = 5.623413 0.7500000 E = [root](CD) = 4.216964 0.6250000 F = [root](DE) = 4.869674 0.6875000 G = [root](DF) = 5.232991 0.7187500 H = [root](FG) = 5.048065 0.7031250 I = [root](FH) = 4.958069 0.6953125 K = [root](HI) = 5.002865 0.6992187 L = [root](IK) = 4.980416 0.6972656 M = [root](KL) = 4.991627 0.6982421 N = [root](KM) = 4.997242 0.6987304 O = [root](KN) = 5.000052 0.6989745 P = [root](NO) = 4.998647 0.6988525 Q = [root](OP) = 4.999350 0.6989135 R = [root](OQ) = 4.999701 0.6989440 S = [root](OR) = 4.999876 0.6989592 T = [root](OS) = 4.999963 0.6989668 V = [root](OT) = 5.000008 0.6989707 W = [root](TV) = 4.999984 0.6989687 X = [root](WV) = 4.999997 0.6989697 Y = [root](VX) = 5.000003 0.6989702 Z = [root](XY) = 5.000000 0.6989700 Great attention was devoted to the methods of calculating logarithms during the 17th and 18th centuries. The earlier methods proposed were, like those of Briggs, purely arithmetical, and for a long time logarithms were regarded from the point of view indicated by their name, that is to say, as depending on the theory of compounded ratios. The introduction of infinite series into mathematics effected a great change in the modes of calculation and the treatment of the subject. Besides Napier and Briggs, special reference should be made to Kepler (_Chilias_, 1624) and Mercator (_Logarithmotechnia_, 1668), whose methods were arithmetical, and to Newton, Gregory, Halley and Cotes, who employed series. A full and valuable account of these methods is given in Hutton's "Construction of Logarithms," which occurs in the introduction to the early editions of his _Mathematical Tables_, and also forms tract 21 of his _Mathematical Tracts_ (vol. i., 1812). Many of the early works on logarithms were reprinted in the _Scriptores logarithmici_ of Baron Maseres already referred to. In the following account only those formulae and methods will be referred to which would now be used in the calculation of logarithms. Since log(e)(1 + x) = x - ½x² + (1/3)x³ - ¼x^4 + &c., we have, by changing the sign of x, log(e)(1 - x) = -x - ½x² - (1/3)x³ - ¼x^4 - &c.; whence 1 + x log(e) ----- = 2(x + (1/3)x³ + (1/5)x^5 + &c.), 1 - x p - q and, therefore, replacing x by -----, p + q _ _ p | p - q /p - q\³ /p - q\^5 | log(e) --- = 2 | ----- + (1/3)( ----- ) + (1/5)( ----- ) + &c. |, q |_ p + q \p + q/ \p + q/ _|