184 On the ORIGIN and 
if poflible, let GL D be obtufe, (fig. 7.) and let the perpendiculay: 
from G upon AB meet AB in V, which point V is therefore 
given. And fince the excefs of the fquare of GD above the 
fquare of IL.G is equal to the fquare of LD, together with 
twice the rectangle DLV, therefore by the fuppofition, the 
fquare of LD, together with twice the rectangle DLV, muft 
have a given ratio to the fquare of LD; the ratio of the rec- 
tangle DL V to the fquare of LD, that is, of VL to LD, is 
therefore given, fo that V L being given in magnitude, LD is 
_ alfo given. But this is contrary to the fuppofition, for L D is 
indefinite by hypothefis ; and therefore GLD cannot be ob- 
tufe, nor any other than a right angle. 
Tue fame conclufion that is here drawn immediately from 
the indetermination of LD, would be deduced, according to 
Dr S1mson’s method, by afluming another point D’, any how, 
and from the fuppofition, that the excefs of G D’* above G L* 
was to I, D’? in the fame ratio that the excefs of GD* above 
GL? is to L D’, it would follow without much difficulty, that 
G LD mutt be a right angle, and the given ratio, a ratio of e- 
quality. The method followed above is fhorter and lefs intri- 
cate than this laft, and has, I think, the advantage of dif- 
covering more plainly the /pivit of the analyfis, and the ef- 
fect which the indefinite nature of the quantities, fuppofed in- 
determinate in the Porifm, has in afcertaining the relation, that 
mutt fubfift between the magnitudes that are given, and thofe 
that are to be found. 
26. Tuis Porifm may be extended to any number of lines 
whatfoever, and may be thus enunciated: “ Let there be any _ 
number of ftraight lines given in pofition, and from any point 
in one of them, let perpendiculars be drawn to all the reft, a 
point may be found, fuch, that the fquare of the line joining 
it, and the point from which the perpendiculars are drawn, 
fhall have to the fum of the fquares of thefe perpendiculars a 
given ratio, which ratio is alfo to be found.” 
: THE 
