LOGARITHMS. 
number. And this was the method Mr. 
Briggs took to make the logarithms. But 
if they are to be made to only seven places 
of figures which are enough for common use, 
they had only occasion to find 25 mean 
proportionals, or, which is the same thing, 
to extract the root of 10. Now 
having the logarithms of 3, 5, and 7, they 
easily got those of 2, 4, 6, 8, and 9 ; for 
since ^ = 2, the logarithm of 2 will be the 
difference of the logarithms of 10 and 5 ; 
the logarithm of 4 will be two times the loga- 
rithm of 2 ; the logarithm of 6 will be the 
sum of the logarithm of 2 and 3 ; and the 
logarithm of 9 double the logaritlim of S. 
So, also having found the logarithms of 13, 
17, and 19, and also of 23 and 29, they did 
easily get those of all the numbers between 
0 and 30, by addition and subtraction 
only j 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 numbers 
is so intolerably laborious and troublesome, 
the more skilful mathematicians that came 
after the first inventors, employing their 
thoughts about abbreviating this method, 
had a vastly more easy and short way offer- 
ed to them from the contemplation and 
mensuration of hyperbolic spaces contained 
between the portions of an asymptote, 
right lines perpendicular to it, and the 
curve of the hyperbola ; for if ECN (Plate 
IX. fig. 6 ) be an hyperbola, and A D, 
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, 
Q N, &r. be drawn from the several points 
B, P, Q, &c. to the curve, then will the 
rjuadrilinear spaces E C M P, P M N Q, &c. 
niz. their numerical measures be the loga- 
rithms 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 propor- 
tionals, the said spaces are equal, as is demon- 
strated by several writers concerning conic 
sections. 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 
metliod how tq measure these spaces, and 
consequently of the construction of loga- 
rithms. 
Let C A (fig. 6)= A F be = 1, and A B 
= A6=x; then will- — — be=:BD, and 
1 I 
= 6 d 5 and putting these expressions 
into series, it will be — ^ — = 1 . — x 4 - — 
1 X ' 
!■' — x^, &c. and 
1 -j- X 
-}- x’ -f- + ‘r^ &c. and ~ ^ — 
X X X — X x‘' X ■ — A- &c. and 
~ ^ = x-\-xx4[-x^x-\-x'^x-\-x*x-\- 
x’’ X, &c. and taking the fluents, we shall 
have the area AFDB = x — — -L-— — 
2^3 
-f- &c. and the area A F d i, = x -4- 
X X x"^ 
— + r + H- j &c. and the sum 6d D B 
= 2x + lx” + 1x^ + 1 &c. 
Nowif AB, or ab, be =x, Cb being = 
0.9, and C B = 1 . 1 , by putting this value 
of X in the equations above, we shall have 
the area 6 d D B = 0.2006706954621511 
for the terms of the series will stand as you 
see in this table. 
Term of the series. 
0.2000000000000000 = first 
6666666666666 = second 
40000000000 = third 
285714286 = fourth 
2222222 = fifth 
18182 = sixth 
154 =; seventh 
1 = eighth 
0.2006706954621511 
If the parts A d and A D of this area he 
added separately, and the lesser DA be 
taken from the greater d A, we shall have 
Ad-AD = x^-f|+'i+l,&c.= 
■C O 4: 
0.0100503358535014, for the terms reduced 
to decimals will stand thus : 
Term of the series. 
0.0100000000000000 = first 
500000000000 = second 
3333333333 = third 
25000000 = fourth 
200000 = fifth 
1667 =:sixlli 
14 = seventh 
0.0100.503358535014 
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, i-iz. 
