76 



. GHEEK GEOMETRY. 



enunciations are not given in the true porismatic form. Thus, 

 in the most remarkable of them, the fifth, he gives the con- 

 struction as part of the enunciation. So far, however, a con- 

 siderable step was made ; but when he comes to show in what 

 manner he discovered 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 propositions answer 

 the definition ; he adds that it reveals the whole nature of 

 porisms ; he says that by no other account but the one con- 

 tained 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 Eeliqua,' 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 hyperbola which should cut in a given ratio 

 all the areas of the parabolas lying between given straight 

 lines. f 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 innu- 

 merable 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. 



* Var. Op. p. 118. 



t Phil. Trans. 1798, p. 111. Tract I. of this volume. 



