542 Mr. T. S. Davies on Geometry and Geometers. 



Mr. Swale was not in the habit of putting down the demon- 

 strations of the more simple of his constructions ; and as those 

 which to him appeared simple and capable of easy recal at any 

 moment did not often appear so to others less rapid than him- 

 self^ many of his operations that would have somewhat perplexed 

 a less able geometer to demonstrate are left unproved*. The 

 present, indeed, scarcely deserves to be classed as amongst the 

 difficult ones; but as a demonstration may be satisfactoiy to 

 some readers, 1 annex a sufficiently simple one. 



The line BH being found according to the construction, let C 



be the centre of the circle according to EucHd's definition. Join 

 CB, CH : then it is required to prove that BCH is an equilateral 

 triangle, from which the conclusion follows at once. 



Join AC, PC, AP, PH, BL, PL. 



Then since P is the centre of the circle ABL, we have 



HLB=ALB= ^APB=CPB=CBP; 



and since C is the centre of the circle ABH, we have 

 HLB + HBL=:AHB=APB = 2CPB=2CBP. 



Whence also HBL = CBP = HLB. 



Add PHB + PBH = LBH + PBH, or CBH=PBL. 



But PBL is an angle of an equilateral triangle by construction ; 

 and heuce also CBH is one angle of an equilateral triangle. 

 Moreover, since C is the centre of the circle CB = CH, and hence 

 CHB is another angle of an equilateral triangle. Whence also 

 the third angle HCB, etc. 



In an age like this, when science is so taxed to minister to art, 

 one would look for some improvement in the processes of prac- 

 tical plane geometry. There are two ways in which a work on 

 practical geometry might be composed. One is to give each 

 construction at the place in a systematic course, where its de- 

 monstration would naturally and easily flow fi'om the principles 



* Mr. Swale was much in the habit of writing short poems and scraps of 

 verse. In allusion to this and to the civeumslance noticed in the text, some 

 of his friends were accustomed to rally him upon his " writing so much 

 poetical geometry." 



