demonstrated by the method of Indeterminates. 7 
“ Again, of in the equation 
A+Bx+B/y+Cx?+C/xy+C’y?4+Dx°+D’x*y+D’ xy? 
+D”y*+&.=0 
A, B, B’, C, C’, C”, D, &e. be definite quantities, and x, y 1nDE- 
FInirEes ; then 
A=0 
Bx+B/y=0 sh y is a function of x. 
Cx?4+C’xy+C”’y?=0 
Sic. = 0 
For, let y = zx, then substituting 
A+x(B+Bz)+x*?(C+C0'z2+C’z?) 
+x*?(D+D24+D"z? + D”z*) + &.=0 
Hence, 
A=0,B+B2=0,C4+C'2z2+C"z? 30, &e. 
and substituting z for z and reducing we get 
A=0, Bx + B’y = 0, &. 
‘“‘In the same manner, if we have an equation involving three or 
more indefinites, it may be shown that the aggregates of the homo- 
geneous terms must each equal zero.” oe 
Some of the more obvious applications of this principle will be 
seen in the following extracts. | 
eres 
“Quantities AND THE Ratios or QuantiTies.—The truth 
of the Lemma does not depend upon the species of quantities, but 
upon their conformity with the following conditions, viz. 
“That they tend continually to equality, and approach nearer to 
each other than by any given difference. 
“Fixire 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 “ Fadlacia 
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. 
