I2S DEMONSTRATIONS of 



P, Q^ R, S. Let Xa, Xb, Xc, be three flraight lines, fuch, 

 that 3(EP 2 +EQ 2 +ER 2 -|-ES 2 ) = 4 (E« 2 +E£ 2 -|-Ec 2 ), (Cafe 2. 

 of this). Defcribe a triangle def (Fig. 14. No. 2.) having the 

 angle def— aXb, and the angle dfe ~ bXc. Let£- be a point in 

 that triangle, fuch, as to be the centre of gravity of the three 

 points h, k, /, where perpendiculars drawn from it meet the fides, 

 (Theor. 13.). Defcribe a fquare = i(XL 2 +XM 2 +XN 2 + 

 HO 2 ), and divide it into three fquares whofe fides Xm, Xn, 

 Xo, fhall have the mutual ratios of gh. gk, gl. Through X 

 draw Xm, Xn, Xo, perpendicular to Xa, Xb, Xc, and through 

 tn, », 0, draw mp, nq, qp, perpendicular to Xm, Xn, Xo, and 

 meeting JLa, ¥J), Er, in x, y, z. We have, by Theor. 13. 

 3(EF 2 -fEG 2 +EH'+EK 2 ) - 3 (EP 3 +EQ^+ER 2 +ES 2 )+ 

 3(XL 2 +XM 2 +XN*+X0 2 ), and alfo 



4 (Ex 2 +Ej 2 +Es 2 ) rr 4 (Ea 2 +E£ 2 +E<- 2 )+ 4 (Xw J +X« 2 +X* a ). 

 But by conftruction, 



3(EP 2 +EQH~ER 2 +ES J ) = 4(Ea t +-EJ a +Er 9 ), and by Cafe 2. 

 of this, 3 (XL 2 +XM 2 -fX0 2 +XN 2 ) = 4 (X^ 2 +X« 2 +Xo 2 ). 

 Therefore, 



3(EF 2 +EG 2 +EH 1 +EK 2 ) = 4 (E^ i +Ej/ t +Ejs 2 ). Therefore 

 mp, nq, qp, are the lines required to be found. 



The three lines found in this Theorem are determined, in their 

 pofition, only relatively to one another, and not abfolutely j 

 becaufe, in the conftruclion of each of the cafes, an arbitrary 

 fuppofition is unavoidably introduced, and of confequence there 

 are innumerable fets of lines, within certain limits however, 

 that all equally anfwer the conditions required in the propor- 

 tion. When one of thefe is affumed as given in pofition, the 

 other two are necelfarily determined. 



The four propofitions which follow in Dr Stewart's book 

 are extenfions of four that have already been demonftrated 

 here, viz. the 10th, 12th, 13th and i 4 th \ and are related to 

 them juft as the 7th of the preceding is to the fixth, or the 9th 

 to the 8th. The purpofe of them is to apply what has been 



demonftrated 



