QUADRATURE of the CONIC SECTIONS, &c. 319 
one or other of the terms of a geometrical feries whofe firft 
term is unity and common ratio, a number very nearly equal 
to unity, but a little greater; and any quantities proportional 
to the exponents of the terms of the feries, are the logarithms 
of the numbers to which the terms are equal. 
Locaritums, then, being not abfolute but relative quanti- 
ties, we may aflume any number whatever as that whofe loga- 
rithm is unity ; but a particular number being once chofen, the 
logarithms of all other numbers are thereby fixed. 
HeEncE it follows, that there may be different fyftems, ac- 
cording as unity is made the logarithm of one or another num- 
ber; the logarithms of two given numbers, however, will al- 
ways have the fame ratio to each other in every fyftem what- 
ever ; thefe properties which are commonly known, are men- 
tioned here only for the fake of what is to follow, as we have 
already premifed. 
51. TAKING this view of the theory of logarithms as the 
foundation of our inveftigations, 
LET us put 7 for the common ratio of the geometrical feries, 
x for any number or term of the {eries, 
4 for the number whofe logarithm is unity, 
y for the exponent of that power of r which is 
equal to x, 
m for the exponent of the power of r which is 
equal to 4. 
‘Then we have x =7’, and =r”, and becaufe by the nature of 
logarithms log x: log 4::y:m, therefore log x =2 x log 4; 
but by hypothefis log 6 = 1, therefore log « = z 
52. Let 
