ANTECEDENTAL CALCULUS. 67 



ceeds the geometrical expreflion, 



aforefaid geometrical expreflion. 



It is almoft unneceflary to obferve, that the two expreflionSj 

 which have refpedlively to B ratios, having to the ratios of 

 A+N to B, and A— N to B the ratio of R to Q^ give us 



R R R_Q R R-Q, R-2Q^^, , , 



A — +— .A-— -->'.N + — . — --^A— — -^N^+ +&C. 



— -^:^= — -^^= ^^^^ ^^Q^ -^^^ for the g€0- 



metrical magnitude, which has to B a ratio, having to the ratio 

 of A±N to B the ratio of R to Q^ But as this expreflion mufl; 

 vary indefinitely with the endlefs variations in the quantity of 

 the magnitude B, its geometrical flandard of comparifon, fo, 

 when we fuppofe it to become numerical, we get an indefinite 

 number of arithmetical formulae, referring to difl^erent flandards 

 of comparifon. For B may be then reprefented by 



I, 2, 3,4, 5, &c. 



I, '/2, /3, v'4,V5» &c. 



-i. -1 — — — &c 



or &c. &cc.ftne limit e» 



And in that particular cafe, when it is reprefented by i or 

 unit, this geometrical formula gives the arithmetical one, (put- 

 ting r and q for R and Q»^) 



A^ + -L. A^==L . N + -il . JTX. Ai:=^.N^ + Jl . ITIL . Ii:!£. aI=3^^ 



which 



