on Converging Seriefes, 
equations afford the fame value of the quantity exterminated, 
the proportion is true, otherwife not. 
Cor. From thefe principles can be demon ff rated many pro- 
pofitions given by Pappus and others. 
Ex. Let AD— 2AC — 2.v, DE = a, and EB — If, where AD, 
DE, and EB, are independent quantities; if ABxBE— 
(2 x + a -f b) x b =: CB x BD = (x -f a + b) 
h c ]) E B {a -f b) , then wil 1 CB = x + a + b : BD — 
a + b :: AC x CE —x x (at -f a) : AD x DE 
— 2x x a. From hence can be deduced the two equations 
{b — a) x—a~ + cib and 2 ax + a -f b) — {a + b) x (x -H a) ; re- 
duce thefe two equations to one, fo as to exterminate x, and 
there refults the felf-evident equation - b) x — - ( - a' - ab) 
-f a' + ab — o, and confequently the proportion is true. 
2. If (i) equations involving (/ + r) unknown and inde- 
pendent quantities be predicated of (/) equations involving the 
above-mentioned quantities : reduce the (r) equations and one of 
the above-mentioned (Y) equations to one, fo that (*) unknown 
quantities may be exterminated, and if there refults a felf- 
evident equation, then the above-mentioned equation is juftly 
predicated of the (/) equations ; and in the fame manner wc 
may reafon concerning the remaining (s - 1) equations. 
3. 1. If one equation is juftly predicated of another, and 
in both the unknown quantity exterminated has only one 
dimenfon ; then the latter equation can be predicated of the 
former ; for in this cafe both equations have only one and the 
fame value of the unknown quantity exterminated. 
3. 2. If the quantity exterminated has more dimenfions 
than one in the equations, then the propoftion may not gene- 
L 2 rally 
