SIMSON. 151 



we ever have arrived at a knowledge of the hidden 

 value ;* and he shows how, in his fifth proposition, the 

 porism flows from a locus, or rather he confounds po- 

 risms with loci, saying porisms generally are loci, and 

 so he treats his own fifth proposition as a locus; and 

 yet the locus to a circle which he states as that from 

 which his proposition flows has no connexion with it, 

 according to Dr. Simson's just remark ('Opera Reliqua,' 

 p. 345). That the definition on which he relies is truly 

 imperfect, appears from this : there could be no alge- 

 braical porism, were every porism connected with a 

 local theorem. But an abundant variety of geometri- 

 cal porisms can be referred to, which have no possible 

 connexion with loci. Thus, it has never been denied 

 that most of the Propositions in the Higher Geometry, 

 which I investigated in 1 797, were porisms, yet many 

 of them were wholly unconnected with loci ; as that 

 affirming the possibility of describing an hyperbola 

 which should cut in a given ratio all the areas of the 

 parabolas lying between given straight lines.t Here 

 the locus has nothing to do with the solution, as if the 

 proposition were a kind of a local theorem : it is only 

 the line dividing the curvilineal areas, and it divides 

 innumerable such areas. Professor Playfair, who had 

 thoroughly investigated the whole subject, never in 

 considering this proposition doubted for a moment its 

 being most strictly a porism. 



Therefore, although Fermat must be allowed to 

 have made a considerable step, he was unacquainted 

 with the true nature of the porism; and instead of 

 making good his boast that he could restore the lost 

 books, he never even attempted to restore the investi- 

 gation of the first proposition, the only one that re- 

 mains entire. A better proof can hardly be given tf 

 the difficulty of the whole subject, t 



Var. Op. p. 118. 



f Phil. Trans. 1798, p 111. 



j The respect due to the great name of Fermat, a venerable magistrate 



