THE LANCASHIBE GE0METER8 AND THEIR WHITINGS. 149 



a Problem of this class gained him the prize in that work 

 for the year 1822, and in conjunction with Mr. Butterworth 

 he contemplated a separate publication on the subject of this 

 celebrated Enigma of antiquity, but early death prevented the 

 fulfilment of this design. In common with Mr. Butterworth 

 and Mr. William Simpson, of Bolton-le-Moors, he had formed 

 a clear conception of the real nature of these very general 

 inquiries, and had he lived to complete his intentions, his 

 publication would no doubt have contnbuted to remove some 

 of the difficulties relating to Porisms which have formed an 

 obstacle to Geometers from the time of Pappus down to the 

 present day. 



Dr. Simson and Professor Playfair indeed did much to 

 dispel the cloud of obscurity which enveloped the subject, 

 but, as Professor Davies well remarks, " the great generality 

 of Simson's definition of the Porism has created much diffi- 

 culty in apprehending its meaning. It may reasonably be 

 doubted whether any Geometer ever acquired a distinct and 

 complete idea of the Porism from Simson's definition of it ; — 

 just as it may be doubted whether the Porism itself was 

 originally discovered as an independent form of Proposition. 

 Yet it does happen that when the full idea is once gained,- 

 we feel that Simson's definition expresses it most completely 

 and generally," 



When a Porism is actually stated for solution, several 

 methods of proceeding obviously present themselves. We 

 may, for the purposes of analysis, suppose the Porism to be 

 true, and by making two or more statements of the hypo- 

 thesis, general or particular, as best suits the purpose, thus- 

 get rid of the indeterminates, as is done by Dr. Simson in his 

 Opera Reliqua ; — or by one statement and taking advantage 

 of the law of continuity in extreme cases, as is done by 

 Playfair in Art. 24 of his Essay ; — or by one statement and 

 assuming the things affirmed to be given as actually given 

 and constant, as is done by Noble in the Mathematical 



