ANTECEDENrAL CALCULUS. 77 



tio of the firfl term to N be decompounded with it, or its in- 

 verfe, the ratio of N to the firfl term, be compounded with it, 



there arifes the ratio of A + -~rN to A, which (Cor. 4. 



ProD. 1.) is nearer to a ratio of equaUty than any given ratio. 



In the fecond, the excefs of the fir II term above twice the fe- 

 cond is lefs than the whole exprefiion, and confequently has to 

 N a lefs ratio than the exprefTion itfelf has to N, (8. E. 5.). But 

 if with it the ratio of N to the firft term be compounded, there 



arifes the ratio of A 5 ^° ^' which (Cor. 4. Prop, i.) 



is nearer to a ratio of equality than any given ratio. Q^E. D. 



OTHERWISE: 



If it be denied, that each exprefiion has to N a ratio nearer to 

 the ratio of its firft term to N than any given ratio, let the ra- 

 tio of two given magnitudes C and D be nearer to it, and let 

 the ratio of B to E, compounded with that of the firft term to 



N, or with the given ratio — - . — . ^ to B, be equal to the gi- 



ven ratio C to D. But the magnitude, which has to B the ra- 

 tio compounded of thefe two ratios, is (For. i. Theorem i. 



A^=^ A ^~^ 



Univerfal Comparijon), — _-^ + — .— ^--.— — to B, 



which is greater than the ratio of the firft term to N, and lefs 

 than the ratio of the firft expreffion to N, by the fuppofition, 

 and confequently lefs than the ratio of the firft term with twice 



^he 



