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. 



Logarithms, then, being not abfolute but relative quanti- 

 ties, we may affume 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. 



Hence it follows, that there may be different fyflems, 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 r for the common ratio of the geometrical feries, 

 x for any number or term of the feries, 

 b 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 b. 

 Then we have x zz r y , and b zz r m , and becaufe by the nature of 



logarithms log x : log b\ : y : m, therefore log x = 2L x log b -, 



m 



but by hypothefis log bzzi f therefore log * =: — . 



52. Lit 



