' ANT'ECEDEN'TAL CALCULUS, 79 



<hird the fame with that of A to -^^^Sr. JIr^.__- and fo 



«n. 



The general expreflion, (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 demonflrated in the fame manner as above, that if 

 A+M and G-f N have refpedlively to A and C ratios nearer to 

 that of equahty than any given or affigned ratio, or than by 

 any given or affigned magnitude, this expreffion alfo has to 



__::: L_ a ratio nearer to that of equahty than any given ra- 

 tio, or than by any given magnitude. And the demonflration 

 is exacflly 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- 

 preffion thence arifmg, 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 p== . But it is readily demon- 

 flrated, as above, that if A-j-M and C+N have refpedlively to A 

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

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



^ ,^ 1 CD.M — AD.N . , r 



preliion alio has to q% a ratio nearer to that 01 equa- 

 lity than any given ratio, or than by any given or affigned 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 affigned, and ratios 

 Clearer to that of equality than any that may be given or affign- 

 ed. 



