H4 DEMONSTRATIONS of 



Dr Sims on, in his Reftoration of the Loci Plani, has deduced 

 them from a proportion of that book. Vid. hoc. Plan. lib. i. 

 prop. 5. cor. 1. & 3. The fecond and fourth of Dr Stewart's 

 Theorems are particular cafes of this propolition, and are eafily 

 derived from it. 



THEOREM VII. Fig. II. 



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



a, b, c y &c. be given magnitudes , as many in number as there are 



given points, a point X may be found, fuch, that if from A, B, C, &c. 



there be drawn Jlraight lines to any point D, and alfo to X the point 



found, and if DX be joined, 



a. AEM-£.BDM-<-.CD* &c. = a. AX 2 + £.BX 2 -f r.CX'-f (a+b+c)DX\ 

 Let m be — 3. Suppofe the point X found. Join DX ; 

 from the given points A, B, C draw AE, BF, CG perpendicular 

 to DX, and join AX, BX, CX. 



SiKCEa.AT>*+b.BD*+c.CD*=:a.AX*+b.BX*+c.CX*+(a+b+c) 

 DX 2 ; and 



a.AD* — tf.AX 2 +*.DX*— 2a.DX. XE, and 

 J.BD* a b.BX*+b.DX*+2b.DX. XF, and 

 c.CD* s= <:.CX 2 +f.DXM-2<-.DX.XG; r 



a.AD*+b.BD*+c.CD* = a.AX*+b.BX*+c.CX*+(a+b+c)VX*+ 

 2DX ( — tf.XE-f^.XF-ff.XG) ; a.XE muft be equal, and in the 

 oppofite direction to £.XF-f-<\XG. 



This will be effected by the following conftruclion : 

 Join AB, and divide it in H, fo that b.BH =: a. AH ; that is, 

 make AH : BH — b:a, and join HC, and divide it in X, fo 

 that HX : CX = c : a+b \ or (a+b) HX = f.CX. Then X will 

 be the point required. 



From H draw to DX, the perpendicular HK. 



Since 



