INVESTIGATION of PORISMS. 181 



SmuS'ON frequently employs two ftatements of the general hy- 

 pothefis, wiiich he compares together ; as for inftance, in his 

 analyfis of the laft Forifej he affumes, not only E, any point 

 whatfoever in the line D E, bilt alfo another point O, any 

 whatfoever in the fame line, to both of which he fuppofes lines to 

 be inflected from the given points A and B. This double abate- 

 ment, however, cannot be made, without rendering the invefti- 

 gation long and complicated ; and therefore it may be of ufe 

 to remark, that it is never neceflary, but may always be avoided 

 by an appeal to Ampler Porifms, or to Loci, or to the propofi- 

 tions of the data. I fhall give the following Porifm as an ex- 

 ample, where this is done with fome difficulty, but with confi- 

 derable advantage, in regard to the fimplicity and fhortnefs of 

 the inveftigation. 



P R O P. V. PORISM. Fig. 6". 



24. Let there be three ftraight lines A B, A C, C B given in 

 pofition, and from any point whatfoever in one of them 

 as D, let perpendiculars be drawn to the other two, as 

 D F, D E ; a point G may be found, fuch, that if G D be 

 drawn from it to the point D, the fquare of that line fhall: 

 have a given ratio to the fum of the fquares of the per- 

 pendiculars DF and D E, which ratio is to be found. 



Draw from A and B the lines A H, B K at right angles to 

 B C and C A, and divide A B in L, fo that A L may be to L B 

 in the given ratio of the fquare of A H to the fquare of B K 

 or, which is the fame, of the fquare of A C to the fquare of 

 C B. The point L is therefore given ; and if N be taken fo as 

 to have to A L the fame ratio that A B* has to A H 2 , N will 

 be given, in magnitude. Alfo fince A H 2 : B K 2 :: A L : L B, 



and 



