Dr STEWARTS THEOREMS. 131 



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

 making given angles with AB, AC, &c. will have given ratios 

 to the perpendiculars FJ3, FC, &c. 



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

 as above, and if the figure be conJlruBed as in Theor. 13. (Fig. lo-^, 



EF^ + A EG^+-f EH^+4eK^ &c. = -^^-^-^J^+^- (EY^ + 



EZ')+A^, A'' 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 i8th from the 14th. The i8th is, That if a, 

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

 ppfed as in Theor, 14. (Fig. 14.) three firaight lines mp, nq, qp, 



may be found, fuch, that EF^-f- -^EG^-f -^EH^+ -^ EK% &c. = 



'±*±^(E«'+E^»+Ez^). 



We proceed now to a propofition 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 

 firaight lines may be found, which will be given bv pc/ition, and like- 

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

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

 and ED be joined, then, {makin? hP- zz a given fpace, and B* zz 

 the fourth povoer of a given line^) 



A£*-1-BE*-|-CE* &c. =: mDE*+A^(EY^+EZO+B*. 



r z Let 



