Dr STEWARTs THEOREMS. 131 



This corollary is evident, becaufe the lines drawn from F, 

 making given angles with AB, AG, 8cc. will have given ratios 

 to the perpendiculars FB, FC, &c. 



The 17th Theorem is, That if a, b, c, &c. be any magnitudes 

 as above, and if the figure be conjlrucled as in Theor. 13. (Fig. 10. J, 



EP + - EGH- - EH*+ -EK' &c. = *'***"•*'• (EY- + 



a ' a 2a 



EZ 2 )-f-A 2 , A 2 being a given fpace. This is demonftrated from 

 its relation to the 13th, in the fame manner with the preceding, 

 and fo alfo is the 18th from the 14th. The 18th is, That if a, 

 b, e, &c. be any given magnitudes, and if the fame things be fup- 

 pofed as in Theor, 14. (Fig. 14.) three Jiraight lines mp, nq, qp, 



may be found, fuch, that EF*+ -£-EG 2 -f- -^EH*-{- ~ E&% & c- = 



We proceed now to a proportion that relates to the fourth 

 powers of the perpendiculars. 



THEOREM XXVII. Fig. XV. 



Let there be any number, m, of given points A, B, C, &c. two 

 Jiraight lines may be found, which will be given by prfition, and like- 

 wife a point D, fuch, that if from any point E, there be drawn EY, 

 EZ, perpendicular to the two lines found, and if EA, EB, EC, &c. 

 and ED be joined, then, {makin? A* — a given fpace, and B 4 zz 

 the fourth power of a given line,) 



AE 4 -fBE 4 -f-CE* &c. s mOL 4 -f- A 2 (EYH-EZ 2 )-fB 4 . 



r z Let 



