72 On the PRINCIPLES of the 



LEMMA IV. 



If from any magnitude there be taken the half, and from the 

 remainder its half, and fo on, the halves fo taken, be their 

 number ever fo great, are together lefs than the magnitude. 



C D E F 



A 1 — : 1 1 1 B 



For let AB be any magnitude, AC the half of AB, CD the 

 half of CB, DE the half of DB, EF the half of EB, and fo 

 on. 



Then it is manifefl, that AC, together with CD and DE and 

 EF, &c. are lefs than AB, from which they are taken. Q^E. D. 



Cor. I. The I'atio of AB to the fucceffive halves AC, CD, 

 DE, EF, &c. taken together, be their number ever fo great, is 

 greater than a ratio of equality ; and the ratio of any one of 

 the terms to all the fucceeding ones taken together, be their 

 number ever fo great, is greater than a ratio of equality. 



Cor. 2. Hence it follows, that of any feries or fucceflion of 

 terms, in which the half of each term has to the immediately 

 fucceeding one a ratio of equality, each term has to all the fuc- 

 ceeding ones, be their number ever fo great, a ratio greater than 

 that of equality. 



CoR. 3. Hence it alfo follows, that if CD, inflead of one half, 

 be one third of AC or CB, DE one fourth of CD, EF one fifth 

 of DE, and fo on, the ratio of any term to all the fucceeding 

 ones taken together, exceeds a ratio of equality more than the 

 ratio it has to the fame number of fucceeding terms, be that 

 number ever fo great, exceeds it, when each term has to the 

 immediately fucceeding one the ratio of two to one. 



Scholium. 



