THE LANCASHIRE GEOMETERS AND THEIR WRITINGS. 



143 



centre is H and radius II A, wherever the point C is taken 

 in the arc A H B." (Davies's Horcg, Prop. XXVII., Diary, 

 1842.) In Questions 544, 621, 664, and 723, he again recurs 

 to this attractive subject, and by purely geometrical processes 

 demonstrates that : — 



(1.) r, + r, 4- r, = 2 1> + r. (Hone, Diary, 1835, 



Prop. IX.) 

 (2.) RO'+RO;+RO;+RO;= 3D'; a "remarkable 



theorem." (Horce, Diary, 1835, Prop. XIX.) 

 (3.) O. O; + O. O3' + O3 0,» = 4D (r, + r, + r.). 



(Hora, Diary, 1842, Prop. XXVII.) 

 (4.) 00;+ 00/+ 00/ + O. O; + 0,0.'+ 0.0; 



+ R0» + RO.» + RO; + RO; = 15 D'. 



(Horee, Diary, 1835, Prop. XIX. and Cors.) 

 On reviewing the preceding propositions from the Student, 

 and also Mr. Buttervv'orth's demonstrations of these more 

 important properties, the want of a uniform system of notation 

 is at once apparent. This deficiency was in part supplied by 

 one or two of the more analytically-minded correspondents to 

 the Companion, and also by the late Professor Davies, in his 

 paper " On the Symmetrical Properties of Plane Triangles," 

 printed in the Phil. Magazine for July, 1827, shortly after 

 the appearance of the last of Mr. Butterwortb's Theorems ; 

 but the defect was not entirely removed until the happy idea 

 of treating all the sides of the triangle as preceding writers 

 had treated the base only, led him to commence the *^ Horce 

 Geometricce" in the Appendices to the Lady's Diary for 

 1835-6-42, which have since been productive of a series of 

 symmetrical results, no less numerous than el^ant, in the 

 able hands of Weddle and Elliott. Nor is the subject by any 

 means exhausted. The pages of the Lady's and Gentleman s 

 Diary, and other mathematical periodicals, frequently con- 

 tain interesting combinations and extensions of the formulje 

 established by the preceding writers, and but little difficulty 

 would be encountered in adding considerably to their number. 



