PHILOSOPHICAL TRANSACTIONS. 503 



merator is 1 , are all of them together equal to an unit, or 1 . And so of others, 

 as here below. 



P 

 A R , ^ 



q c qq 



V ^^^^ ■3- "T -if "T -TT ~r "5T- O^C* 



&c. &c. 



A is a rank or series of absolute numbers, or rather of all the fractions whose 

 numerator is 1 ; which is supposed to be continued in infinitum downwards. 

 R is another series of all the roots, whose numerator is 1, of all the powers of 

 such fractions ; supposed likewise to be continued in infinitum, p, all the re- 

 spective powers of such fractions^ viz. squares, cubes, &c. or so many ranks of 

 geometrical proportionals; supposed to be continued in infinitum, both to the 

 right hand, and also downwards. 



Lemma. Each of the said ranks of powers, together with their respective 

 roots, is equal to each of the several numbers under A respectively. 



a c e g f d b 



Demonstration. If from the line a h you take, for instance, ^ part towards a, 

 suppose ac; and also from the other end of the same line ah, you take two such 

 parts, or 4 parts, towards h, suppose hd, viz. a number of parts less by two than 

 the whole line a h was first supposed to be divided into, there will remain the 

 line cc? = cc = -i- of ab. Then again, if from cd you take ^ part thereof to- 

 wards a, suppose ce, and from the other end 4 parts of the same cd, suppose df, 

 there will remain only ef=ce = ^o{cd. And if you still go on without ceas- 

 ing, to take on the side towards a, ^ part of what was taken last before, and 

 twice as much on the other side towards b, there shall be found between the two 

 lines last taken, always remaining 4- part of the line from which they were taken. 

 From which -i- part there may still after the same manner be supposed to be taken 

 two other such lines. But if this be supposed to be done infinite times actually, 

 then there will nothing more remain between, and so the continued division on 

 either side will come exactly to the point g, supposing ag to he -^ of ab and bg 

 = 2 ag. For, because that which was taken away towards b, was always twice 

 as much as that which was taken away towards a, the total sum of all the lines 

 taken away towards b, which altogether make up the line bg, must be twice as 

 much as the line ag, which is the total sum of all the lines taken away towards 



