142 



MB. T. T. WILKINSON OK 



(3,) C F • I O = ^ <A C — C B)'. (Mod. Geom., 

 Prop. 37.) 



(4.) CF(CV + VB)==CB(CZ + ZF). (Mod. 

 Geom., Prop. 38.) 



(5.) H E : E F : : C B^ w «'. (Mod. Geom,, Prop. 39.) 

 Most of the propositioHs previously to those just enume- 

 rated were collected by Mr. Knowles under the signatures of 

 "Non Sibi" and "N. Selwon;" several of tlie subsequent 

 ones most probably belong to Mr. Hilton, since they bear 

 the initials " W. H." ; Messrs. Nicholson and Wright also 

 furnished a considerable number of properties, and the last 

 number of the work contains Mr. Butterworth's contributions 

 to the common stock. He there proves that: — 



(1.) HO"=FH(HU — EU); CMod.Geom.,Froi>.eO.) 



(2.) IL(AC — CB)=KL(AI — IB); (Mod. Geom,, 

 Prop. 61.) 



(3.) LK'=DH (DU— EU); (Mod. Geom., Prop.Gg). 

 Had the Student been continued for a longer period, there 

 is little doubt that Mr. Butterworth would have added consi- 

 derably to the list of properties of the general diagram, for in 

 later years he gave several Theorems of the same class in the 

 Mathematical Companion, which are by no means unworthy of 

 notice. In dues. 352 of this work, by dropping the perpendi- 

 culars upon the sides from A and B to meet HF in T and W, 

 and making E«? = EF, he shews that AC : BC : : Ww : Tw; 

 and in Ques. 418 he furnishes a demonstration of Lhuilier's 

 property that — 



r r^ r, r^ = A'; where r, r^, r^, r,, are the four radii 

 of contact respectively. (Davies's iloj-^e Geom. Diary, 1835, 

 Prop. X.) 



If we now put O, O^, O^, O,, for the centres of the 

 inscribed and escribed circles, Mr. Butterworth further shews, 

 in Ques. 470, that — 



H O, — H A = H B = H O,; and hence *'the centres 

 O,, O,, are always in the circumference qf a circle whose 



