demonstrated by the method of Indeterminates. 77 



"Again, if in the equation 

 A+Bx-^-B'y + Cx=^+C'xy-^-C'/y2+Dx3 + D'x2y+D''xy^ 

 + D'''y ^ + &;c. =--0 



A, B, B', C, C, C, D, he. be definite quantities, and x, y inde- 

 finites ; then 



A = 0^ 

 Bx + B'y:=0> when y is a function of x. 

 Cx2 +C'xy-+-C"y ^ =0^ 



&:c. = 

 For, let y = z X, then substituting 



A + x (B + B' z) + X ^' (C + C z + C^' z = ) 

 + X 3 (D + D' z + D^' z - + ^" z 3) + &;c. = 

 Hence, 



A = 0, B + B' z = 0, C + C z + C'' z 2 =0, fee. 



y 

 and substituting - for z and reducing we get 



A = 0, Bx + B'y=:0, &;c. 

 " In the same manner, if we have an equation involving tliree or 

 more indefinites, it may be shown that the aggregates of the homo- 

 geneous terms must each equal zero." 



Some of the more obvious applications of tliis principle will be 

 seen in the following extracts. 



Lemma I. 



"Quantities and the Ratios of Quantities. — The truth 

 of the Lemma does not depend upon the species of quantities, but 

 upon their conformity witli the following conditions, viz. 



" That they tend continually to equality, and approach nearer to 

 each other than by any given difference. 



"Finite Time. — Newton obviously introduces the idea of time in 

 this enunciation, to show illustratively that he supposes the quantities 

 to converge continually to equality, without ever actually reaching or 

 passing that state; and since to fix such an idea, he says, "before 

 the end of that time," it was moreover - necessary to consider the 

 time Finite. Hence our author would avoid the charge of ^'Fallacia 

 Suppositionis,''^ or of ^^ shifting the hypothesis.'''' For it is contended 

 that if you frame certain relations between actual quantities, and af- 

 terwards deduce conclusions from such relations on the supposition 

 of the quantities having vanished, such conclusions are illogically de- 

 duced, and ought no more to subsist than the quantities themselves. 



