THE LANCASHIRE GEOMETERS AND THEIR WllITINGS. 



141 



by the Lancashire Geometers to the properties of "Halleys 

 Diagram" which first made its appearance, with fewer 

 lines, in Jones's Synopsis Palmariorum Matheseos, page 244, 

 Lond. 1706. 



This important assemblage of lines subsequently found its 

 way into Burrow's Diary for 1777, where several additional 

 properties are given ; and about the same period, Dr. Henry 

 Clarke> in his Rationale of Circulating Numbers, added a 

 " new proposition" to the number already known, and pointed 

 out its uses in drawing the line A F such " that either the 

 sum, difference, ratio, rectangle, sum of the squares, or differ- 

 ence of the squares of A G and G F, may be of a given 

 magnitude;" the point A being at the intersection of two 

 given circles, and G F the part between the respective cir- 

 cumferences. The utility of these properties in the con- 

 struction of several classes of geometrical problems, led 

 Mr. Lowry to reconsider the subject at considerable length 

 in the original series of Leybourn's Repository, where also 

 another class of properties is elegantly deduced by " Mr. 

 M. A. Harrison," who is generally supposed to represent 

 either Mr. J. H. Swale, of Liverpool, or Mr. Lowry himself. 

 In this state of forwardness the diagram was introduced by 

 Mr. Knowles into the first number of the Student, and under 

 the title of " Modern Geometry," sixty-two regular proposi- 

 tions and their corollaries, most of them original, were fur- 

 nished by his correspondents during the progress of the work. 



With reference to Fig. 2, if we suppose the triangle ABC 

 to be inscribed in the circle A H C B F, R its centre, O the 

 centre of the inscribed circle, O U parallel to A B, and the 

 rest of the lines as in the diagram, Mr. Wolfenden proves 

 that :— 



(1.) CO-OF=OL'HF=2Rr. (Mod,Geom., 



Prop. 85; Davies's Home Geom, Diary, 1835, 



Prop. XI.) 

 (2.) AC-CB = CO(FO + F C). (Mod. Geom., 



Prop. 36.) 



