AXCIEXT ANALYSIS. PORISMS. 77 



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 investigation of the first proposition, the only one 

 that remains entire. A better proof can hardly be given of 

 the difficulty of the whole subject.* 



Indeed it must be confessed that Pappus's account of it, 

 our only source of knowledge, is exceedingly obscure, all but 

 the panegyric which in a somewhat tantalizing manner, he 

 pronounces upon it. " Collectio," says he, " curiosissima 

 multarum reruin spectantium ad resolutionem difficiliorum et 

 generaliorum problemattim " (lib. vii. Proem). His definition 

 already cited is, as he himself admits, very inaccurate ; 

 because the connexion with a locus is not necessary to the 

 porisniatic nature, although it will very often exist, inasmuch 

 as each point in the curve having the same relation to certain 

 lines, its description will, in most cases, furnish the solution 

 of a problem, whence a porism may be deduced. Nor does 

 Pappus, while admitting the inaccuracy of the definition, give 

 us one of his own. Perhaps we may accurately enough define 

 a porism to be the enunciation of the possibility of finding 

 that case in which a determinate problem becomes indeter- 

 minate, and admits of an infinity of solutions, all of which are 

 given by the statement of the case. 



For it appears essential to the nature of a porism that it 

 should have some connexion with an indeterminate problem 

 and its solution. I apprehend that the poristic case is always 

 one in which the data become such that a transition is made 

 from the determinate to the indeterminate, from the problem 



* The respect clue to the great name of Fermat, a venerable magistrate 

 and most able geometrician, is not to be questioned. He was, indeed, one 

 of the first mathematicians of the age in which he flourished, along with 

 the Kobervals, the Harriots, the Descartes. How near lie approached the 

 differential calculus is well known. His correspondence with Roberval, 

 Gassendi, Pascal, and others, occupies ninety folio pages of his posthumous 

 works, and contains many most ingenious, original, and profound observa- 

 tions on various branches of science. 



