THE LANCASHIRE GEOMETERS AND THEIB WRITINGS. 



147 



was proposed for solution by Mr, Butterworth as Ques. 699 of 

 the Mathematical Companion for 18^6. It is there shewn by 

 Professor Gill, Dr. Rutherford, Mr. Wright, and the proposer, 

 that if A B be taken, such " that A Q : Q B : : w* : w% and 

 with B as centre and radius equal to the side of a square 

 whose area 



= BV = ^ AT- + (^-^^ AO - f^- A QL-, 



nr \n* n* J \»* / 



a circle be described, it will be the one required ;" and the 

 general formula is readily adapted to the case of similitude, 

 where m : n is also the ratio of the radii of the given circle 

 and that to be found. If w = «, we have — 



^T. AT*— BV , ,, 



Q L* = pr = 00 , and consequently 



Oie radical axis is really a circle of similitude to an infinite 

 radius. 



In another portion of the Companion^ Mr. James Cunliffe, 

 of Bolton-le-Moors, a Lancashire geometer of ability,* con- 

 sidered the cases of the Loci when: — 



(L) The circles are concentric and the tangents form 

 with each other a given angle. (Clues. 202.) 



* In addition to those previously enumerated, the names of Mr. Jonathan 

 Mabbott, of Oldham, but latterly of Manchester, and Mr. Thomas Bazley, of 

 Bolton, ought to be placed on record as Lancashire mathematicians "of con- 

 siderable eminence. The writings of the former gentleman appear in Whiting's 

 Scientific Receptcicle and Oeomelrical Delights, the Lady's and OeTitleman's Scieniifical 

 Reriository, the Diaries, the Leeds Correspondent, See., and evince his thorough 

 acquaintance with the Doctrine of Series and the higher branches of pure 

 Mathematics. Mr. Bazley 's correspondence occurs in the earlier portions of 

 Leyboum's Mathematical Repository, New Series, the Genilevian's MaOiematiail 

 Companion, &o , where he occasionally assumed the signature of "Helozam 

 Satby," an obvious anagram of his real name. His favourite subjects appear to 

 have been Geometry, Mechanics, and the more abstruse parts of pure and 

 applied Science; — his superiority in these subjects procured him several well- 

 earned prizes in both the periodicals previously mentioned. Mr. W^illiam Smith, 

 of Liverpool, was also another excellent Lancashire geometer. His investigations 

 in the Repository, the Companion, and the Student-, bear testimony to his profound 

 aqaaiatance with the Ancient Geometrical Analysis. 



