' ANTECEDENTAL CALCULUS. 79 



third the fame with that of A to ~^^ .— — . — , and fo 



on. 



The general expreffion, (p. 5. Antecedental Calculus)^ gives 



— ^— — '- '■ — for the excefs of the magnitude, which has to 



B the ratio, that is produced by compounding the ratio of C+N 

 to D with that of A+M to B, above the magnitude, which has 

 to B the ratio compounded of the ratios of A to B and C to D. 

 But it is demonftrated in the fame manner as above, that if 

 A+M and C+N have refpedlively to A and C ratios nearer to 

 that of equality than any given or afligned ratio, or than by 

 any given or afligned magnitude, tliis expreflion alfo has to 



A.N + CM . •> c ^■ ^ 



a ratio nearer to that or equality than any given ra- 

 tio, or than by any given magnitiide. And the demonftration 

 is exadlly the fame, when any number of ratios are compounded. 

 In hke manner, if the ratio of C+N to D be decompounded 

 with that of A+M to B, we get the difference between the ex- 

 preflion thence arifing, and the magnitude which has to B the 

 ratio produced by decompounding the ratio of C to D with that 



CD.M — AD.N . . 



of A to B, equal to c~^ • ^^^ ^^ ^® readily demon- 

 ftrated, as above, that if A+M and C+N have refpedlively to A 

 and C ratios nearer to that of equality than any given or aflign- 

 ed ratio, or than by any given or afligned magnitude, this ex- 



^ ,^ , CD.M — AD.N . , r 



preliion alio has to qj a ratio nearer to that of equa- 

 lity than any given ratio, or than by any given or a,fllgned mag- 

 nitude. 



It is manifeft then, that in this calculus no indefinitely fmall 

 or infinitely little magnitudes are fuppofed, but only magni- 

 tudes lefs than any that may be given or afligned, and ratios 

 Jiearer to that of equahty than any that may be given or aflign- 



' ed. 



