1 86 On the ORIGIN and 



of the fquares of the perpendiculars drawn from it to AC, and 

 CB, mould be equal to a given fquare, this would be done by- 

 finding G ; and then becaufe the fiim of the fquares of the 

 perpendiculars is given, and has a given ratio to the fquare of 

 DG, DG will be given, and confequently the point D. This, 

 is alfo true, whatever be the number of the lines. 



28. The connection of the Porifms with the impofnble cafes 

 of thefe problems, is abundantly evident ; the point L being 

 that from which, if perpendiculars be drawn to AC and CB, 

 the fum of their fquares is the lean: poffible. For fince (fig. 6.) 

 DF*+DE*:DG*::LOH-LM :LG% and fmce LG is lefs than DG, 

 LO+LM 2 muft be lefs than DF z -fDE*. 



Hence alfo a point Q^may be found, from which, if per- 

 pendiculars be drawn to the fides of the triangle ABC, the fum 

 of the fquares of thefe perpendiculars is lefs than the fum of 

 the fquares of the perpendiculars drawn to the fides of the tri- 

 angle from any other point. 



For if ab (fig. 8.) be any line drawn parallel to AR, and if 

 it be divided in A, fo that aX may be to xb in the duplicate ra- 

 tio of aC to Cb, or of AC to CB, then of all the points 

 in the line ab, A is that from which, if perpendiculars be 

 drawn to the lines AC, CB, the fum of their fquares is the 

 lealt poffible. But fince tf A is to xb as the fquare of AC to the 

 fquare of BC, that is, as AL to LB, therefore the locus of A is 

 the ftraight line LC, joining the given points L and C. The 

 point to be found therefore, or that from which perpendicu- 

 lars being drawn to the (ides of the figure, the fum of their 

 fquares is the leaft poilible, is in the ftraight line LC. 



For let q be any point on either fide of LC, and let the line 

 ab be drawn through q parallel to AB, meeting LC in A, then 

 the fum of the fquares of the perpendiculars from q upon AC, 

 CB, is greater than the fum of the fquares of the perpendicu- 

 lars from A upon the fame lines. Therefore adding the fquare 

 of the perpendicular from q, or A, on AB, to both, the fum of 

 the fquares of the perpendiculars from q, will be greater than 



the 



