500 SIMSON. 



tion. So far, however, a considerable step was made ; 

 but when he comes to show in what manner he dis- 

 covered the nature of his porisms, and how he defines 

 them, it is plain that he is entirely misled by the 

 erroneous definition justly censured in the passage of 

 Pappus already referred to. He tells us that his pro- 

 positions answer the definition ; he adds that it reveals 

 the whole nature of porisms ; he says that by no other 

 account but the one contained in the definition, could 

 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 

 porisms 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 algebraical porism, were every porism connected 

 with a local theorem. But an abundant variety of 

 geometrical 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 1797, were porisms, 

 yet many of them were wholly unconnected with loci ; 

 as that affirming the possibility of describing an hyper- 

 bola which should cut in a given ratio all the areas 

 of the parabolas lying between given straight lines. f 



* Var. Op., p. 118. 

 | Phil. Trans., 1798, p. 111. 



