LOGARITHMS. 



g<et those of all the numbers between 10 

 and 30, by addition and substraction only; 

 and so having found the logarithms of 

 other prime numbers, they got those of 

 other numbers compounded of them. 



But since the way above hinted at, for 

 finding the logarithms of the prime num- 

 bers is so intolerably laborious and trou- 

 blesome, the more skilful mathematicians 

 that came after the first inventors, em- 

 ploying their thoughts about abbreviating 

 this method, had a vastly more easy and 

 short way offered to them from the con- 

 templation and mensuration of hyperbo- 

 lic spaces contained between the por- 

 tions of an asymptote, right lines per- 

 pendicular to it, and the curve of the hy- 

 perbola : for if E C N (Plate IX. fig. 5.) 

 be an hyperbola, and AD, A Q, the 

 asymptotes, and A B, A P, A Q, &.c. taken 

 upon one of them, be represented by 

 numbers, and the ordinates B C, P M, 

 QN, See. be drawn from the several 

 points B, P, Q, 8cc. to the curve, thea 

 will the quadrilinear spaces B C M P, 

 P M N Q, &c. viz. their numerical mea- 

 sures, be the logarithms of the quotients 

 of the division of A B by A P, A P by 

 A Q, &c. since when A B, A P, A Q, &c. 

 are continual proportionals, the said 

 spaces are equal, as is demonstrated by 

 several writers concerning conic sec- 

 tions. See HYPERBOLA. 



Having said that these hyperbolic 

 spaces, numerically expressed, may be 

 taken for logarithms, we shall next give 

 a specimen, from the said great Sir Isaac 

 Newton, of the method how to measure 

 these spaces, and consequently of the 

 construction of logarithms. 



Let C A (fig. 6.) = A F be == 1, and 



A B = A b = x ; then will - be = 



B D, and 



1 



b d; and putting these 

 1 



expressons nto seres 



= 1 x + x~ xi -j- x* 



it will be 



&c. and 



&c. and 



I + x 



-f 



X XX -\- X 1 X Xl X 



-f- x* x -f- xTi x -f- x* x -f- x< > , &,c. and 

 taking the fluents, we shall have the area 



AFDB=*_^+?-?-t4 S , fce. 



and the area A F db, = x -f ^-f--f 



OC^" .2?5 



~T"\T &c - an d the sum bdJ) B = 2 x 

 H ~ -j- | xl -f- |. & _j_ i X9) & c . ]S T OW 



if A B, or a b, be J_ =* x, C b being = 

 0.9, and C B = 1.1, by putting this value 

 of or in the equations above, we shall have 

 the area bdDB = 0.2006706954621511 

 for the terms of the series will stand as 

 you see in this table. 



Term of the series. 

 0.20000000000000 M = urst 



6666666666666 = second 

 40UOUOOUUUO = third 

 285714286 = fourth 

 2222222 = fifth 

 18152 = sixth 

 154 = seventh 

 1 = eighth 



0.2006706954621511 



If the parts A d and A D of this area be 

 added separately, and the lesser D A be 

 taken from the greater d A, we shall have 



= 0.0100503358535014, for the terms re 

 duced to decimals will stand thus : 



Term of the series. 



O.OlOOOOOOOOOOOOOu = first 



500000000000 = second 

 3333333333 = third 

 25000000 = fourth 

 200000 = fifth 

 1667 = sixth 

 14 = seventh 



0.0100503358535014 



Now if this difference of the areas be 

 added to, and subtracted from, their sum 

 before found, half the aggregate, viz 

 0.1053605156578263, will be the greater 

 area A d, and half the remainder, viz 

 0.0953101798043249, will be the lesser 

 area A D. 



By the same tables, these areas, A D 

 and A d, will be obtained also when A B 

 -j- A d are supposed to be _ -> _ or C B = 

 1.01, and C b = 0.99, if the numbers are 

 but duly transferred to lower places, as 



