THE LANCASHIBE GEOMETERS AND TBEIB WRITINGS. 



135 



in the same point L,", (Repository, vol. i., p. 170,) he demon- 

 sti'ates that — 



(1.) <CL.A=:<CST=F=<AVT. 



(2.) <NL,T = <CST=br<AVT. 



(3.) The centres M, N, S, R, of these circles, and also 

 the point L, , lie in the same circumference. 



(4.) The feet of perpendiculars from L, upon the sides 



ST, S C, V T, V A, lie in the same straight 



line; besides various other relations of minor 



importance. (Repository, vol. vi., pp. 124 — 



136; 229—234.) 



The elegant property in (4.) has been extended to lines 



making equal angles vvith the sides, for the case of the triangle 



by Mr. Godvvard in Ques. 1731 of the Lady's and Gentleman s 



Diary, and for the quadrilateral by Mr. J. W. Elliott in 



Ques. 1847 of the same work; — Mr. Davies also enunciates, 



but does not prove, the curious properties " that circles 



described upon the three diagonals have the same radical 



axis" and that " their centres range in the same straight 



line." (See Diary for 1854-5. Ques. 1878.) 



Henry Clarke, LL.D., was for many years one of the 

 most distinguished ornaments of the Lancashire School. His 

 versatile talents and extensive acquirements rendered him 

 well adapted to occupy a prominent position in the world 

 of Science, and his numerous publications bear testimony to 

 the unceasing activity of his mind. To the Ladys Diary he 

 communicated many elegant geometrical solutions to ques- 

 tions proposed by himself or his friends, and after the death 

 of Mr. Ainsworth he was the principal supporter of Burrow's 

 Diary under the assumed name of Salfordoniensis. Amongst 

 his separate publications, those on Perspective, Circulating 

 Numbers, and the translation of M. Lorgna on Infinite Series, 

 are the best known ; but a complete list comprises many others 

 on Mathematics, Philosophy, Poetry, History, Languages, and 

 General Literature, His acquaintance with the ancient and 



