Dr STEWART'S THEOREMS. 129 



demonftrated of the fquares of the perpendiculars in Prop. 10. 

 &c. to any rectilineal figures whatever, each given in fpecies, 

 defcribed on thofe perpendiculars. 



Their demonftrations are all derived in the fame manner 

 from thofe of their correfponding propofitions, and it will there- 

 fore be fufEcient, at prefent, to give the demonftration of one of 

 them. I have made choice of the 16th, as the 15th is only the 

 fimpleft cafe of the 7th, viz. when all the points given, in that 

 Theorem, are in the fame ftraight line. 



THEOREM XVI. Fig. VIII. 



Let there be any number, 771, of Jlraight lines AB, AC, AD, AE, 

 &c given by po/ition, interfering one another in the point A, and 

 let a, b, c, d, &c. be given magnitudes, as many in number as there 

 are lines given by pofition, two Jlraight lines AX, AY, may be found \ 

 which will be given by pofition, fucb y that if from any point F there 

 be drawn FB, FC, FD, FE, &c. perpendicular to AB, AC, AD, 

 AE, &c. and FX, FY, perpendicular to AX, AY, 



FB*+ - FO+ - FD>+ -FE* &c. = ^^^^c . ( 



'a 'a 'a 2a ' ' 



Let m n 4. Let G be the centre of the circle which pafTes 

 through the points A, B, C, D, E and F ; and let H be the cen- 

 tre of gravity of weights proportional to the magnitudes a, b, 

 c, d, placed at the points B, C, D and E. Join GH ; and let 

 XY, at right angles to GH in H, meet the circumference of 

 the circle ABDF in X and Y : AX, AY, are the lines required 

 to be found.. 



For it may be mown, juft as in Theor. T2. by means of a 

 lemma fimilar to the 3d, that AX and AY make given angles 



Vol. II. r with 



