74 On the PRINCIPLES of the 



than by any given, or affigned magnitude, and.R and Q^are two 

 given magnitudes of the fame kind. 



PROPOSITION I. 



In this cafe, the firft term in each of thefe general expref- 

 fions has to tw^ice the fecond, the fecond to thrice the thifd, the 

 third to four times the fourth, the fourth to five times the fifth, 

 and fo on, a ratio greater than. any given ratio. 



For, if this be denied, let C and D be two given homogene- 

 ous magnitudes, and let the ratio of C to D be greater. 



In each, the ratio of the firft term to twice the fecond, is that 



of A to 5^=^ N, and its inverfe ^^^'^, or (N + N. -^5-\ ^ 



to A, is the ratio compounded of the ratios of R — Qj:o Q^ and 

 N to A, (For. I. Theor. i. Univerfal Comparifoii). Now, the 

 -ratio compounded of this ratio, and that of C to D, is a ratio 

 compounded of the three ratios C to D, R — Q^to Q^ and N to 

 A. But, fince R and Q^are given magnitudes, R — Q^is a gi- 

 ven magnitude, (4. Euc. Data), and the ratio of R — Q^to Q^a 

 given ratio, (i. Data). Wherefore the ratio compounded of the 

 ratios of C to D and R — Qjo Q^is alfo given, (67. Data). This 

 ratio, however, compounded with that of N to A, is the fame 



with the ratio compounded of C to D, and — -^ N to A. But 



fince that of A to N is by the hypothefis greater than any given 

 ratio, the ratio compounded of C to D and R — Qjio Q^ com- 

 pounded with that of N to A, produces a ratio of lefs inequa- 



lity, (Lemma 2.). Confeqxiently^ the ratio of A to .N is 



greater than any given ratio C to D. Wherefore, the fuppofi- 



tion. 



