182 ‘be the ORIGIN and 
and A H?: AB::: AL:N, ex pradt K?:A Bt: LB:N. From 
L draw LO, LM perpendicular to A C, CB; LO and LM are 
given in magnitude. — 
Now, becaufe A B*: BK?:: AD?: DF, N:LB: AD*:DF, 
: LB 
fothat DF? = Tae AD, and for the fame reafon, DE? = 
Fete LB | 
mee BD*. But (Loci Plani, Append. Lem. 1.) No Dt + 
AL LB AL AB 
“hee 153 Ds tra AL+a— Bi? Ao ae. DL’; that is, 
AB 
DE*+DF = LO‘ LM +5 DLs 
Jorn LG; then by hypothefis, LO>-+ LM? has to L G 
the fame ratio which D F?+ D E* has to D G*; and if this ra- 
R 
tio be that of R to N, LO*+LM**= coh L G*; and therefore 
R AB 
DE+DF = a) GEE De. But DE?-+-DF = 
AB R 
De 5 gs LOe ine = VPs and 
AB 
ar" DL: = — ~(D G*—L G*). The excefs of the fquare of 
DG above the a. of LG, has therefore a conftant ratio to 
the fquare of DL, wz. that of AB to R. The angle DLG 
is therefore a right angle, and the ratio of AB toR, the ratio 
of equality, otherwife LD would be given in widenicaled 
which is contrary to the fuppofition. The line LG is there- 
fore given in pofition ; and fince R is to 'N, that is, AB toN, 
as the fquares of LO and LM to the fquare of LG, therefore 
the fquare of LG, and confequently the line LG, is given in 
magnitude. The point G is therefore given, and alfo the ratio 
of 
