184 On the O.RIGIN and 



if poflible, let G LD be obtufe, (fig. 7.) and let the perpendicular 

 from G upon A B meet A B in V, which point V is therefore 

 given. And fince the excefs of the fquare of G D above the 

 fquare of L G is equal to the fquare of L D, togetlter with 

 twice the rectangle D L V, therefore by the fuppofition, the 

 fquare of L D, together with twice the rectangle DLV, mud 

 have a given ratio to the fquare of LD; the ratio of the rec- 

 tangle DLVto the fquare of L D, that is, of VL to L D, i» 

 therefore given, fo that V L being given in magnitude, L D is 

 alfo given. But this is contrary to the fuppofition, for L D is 

 indefinite by hypothefis ; and therefore G L D cannot be ob- 

 tufe, nor any other than a right angle. 



The fame conclulion that is here drawn immediately from 

 the indetermination of L D, would be deduced, according to 

 Dr Sim son's method, by affuming another point D', any how, 

 and from the fuppofition, that the excefs of G D'* above G L* 

 was to LD' 2 in the fame ratio that the excefs of GD 1 above 

 G L 2 is to L D 2 , it would follow without much difficulty, that 

 G L D muft 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 fpirit 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 

 muft fubfift between the magnitudes that are given, and thofe 

 that are to be found. 



26. This 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 



