INVESTIGATION of PORISMS. 183 



of the fquares of D E and D F to the fquare of D G, which 

 is the fame with that of A B to N. 



Hence this conftruclion : Divide A B in L, fo that A L may 

 be to LB as the fquare of A H to the fquare of B K, and 

 make as the fquare of A H to the fquare of A B, fo A L to N ; 

 and, laftly, having drawn from L upon A C and C B the per- 

 pendiculars LO and L M, make LG perpendicular to AB, and 

 fuch. that as A B to N, fo the fum of the fquares of LO and 

 L M to the fquare of LG; G will be the point required, and 

 the given ratio, which the fquares on D F and D E have to the 

 fquare on D G, will be that of A B to N. 



This is the conftruclion which follows mod directly from 

 the analyfis ; but it may be rendered more fimple. For fince 

 A H 2 : A B 2 :: A L : N, and BK : : AB 2 ::BL:N, therefore A H 2 

 -f B K 2 : A B 2 :: A B : N. Likewife. if A G, B G be joined, 

 AB:N:: AH 2 : AG 2 , and A B : N :: B K 2 : B G' ; wherefore 

 A B : N :: A H 2 -f B K. 2 : A G 2 -f B G 2 , that is, A H 2 -f B K 2 : 

 A B 2 :: A H 2 + B K 2 : A G 2 + BG\ and AG'-j-GB 1 -3 

 A B s . The angle A G B is therefore a right angle, and 

 A L : L G : L B. if therefore AB be divided in L, aa in the 

 preceding conftrueYion ; and if L G, a mean proportional be- 

 tween A L and LB, be placed at right angles to AB,. G will be 

 the point required. 



Cor. It is evident from the conftrucKon, that if at the points 

 A and B we fuppofe weights to be placed that are as the fquares 

 of the fines of the angles C A B, C B A, L will be the centre of 

 gravity of thefe weights. For A L is to LB as A C 2 to C B 2 , 

 or inverfely as the fquares of the fines of the angles at A and B. 



25. Now, the ftep in this analyfis by which a fecond intro- 

 duction of the general hypothefis is avoided, is that in which 

 the angle G L D is concluded to be a right angle. This con- 

 clufion follows from the excefs of the fquare of D G above 

 the fquare of G L, having a given ratio to the fquare of L D, 

 at the lame time that L D is of no determinate magnitude. For, 



if 



