Respecting the porism Simson says that Pappus's definition is too general, and therefore he will substitute for it the following: "Porisma est propositio in qua proponitur demonstrare rem ali quam vel plures datas esse, cui vel quibus, ut et cuilibet ex rebus innumeris non quidem datis, sed quae ad ea quae data sunt eandem habent relationem, convenire ostendendum est affectionem quan dam communem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ea quae data demonstranda sunt, invenienda proponantur." A locus (says Simson) is a species of porism. Then follows a Latin translation of Pappus's note on the porisms, and the propositions which form the bulk of the treatise. These are Pappus's thirty-eight lemmas relating to the porisms, ten cases of the proposition concerning four straight lines, twenty-nine porisms, two problems in illustra tion and some preliminary lemmas.
John Playfair's memoir (Trans. Roy. Soc. Edin., 1794), a sort of sequel to Simson's treatise, had for its special object the inquiry into the probable origin of porisms. Playfair remarked that the careful investigation of all possible particular cases of a proposi tion would show that (I) under certain conditions a problem be comes impossible, (2) under certain other conditions, indeter minate or capable of an infinite number of solutions. These cases could be enunciated separately, were in a manner intermediate between theorems and problems and were called "porisms." Play fair accordingly defined a porism thus: "A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions." Though this definition of a porism appears to be most favoured in England, Simson's view has been most generally accepted abroad, and has the support of the great authority of Michel Chasles.
In Liouville's Journal de mathematiques pures et appliquees (1855), P. Breton published Recherches nouvelles sur les porismes d'Euclide, in which he gave a new translation of the text of Pappus, and sought to base thereon a view of the nature of a porism more closely conforming to the definitions in Pappus.
This was followed in the same journal and in La Science by a controversy between Breton and A. J. H. Vincent, who disputed the interpretation given by the former of the text of Pappus, and declared himself in favour of the idea of F. van Schooten, put forward in his Mathematicae exercitationes (1657), in which he gives the name of "porism" to one section. According to Schooten, if the various relations between straight lines in a figure are written down in the form of equations or proportions, then the combination of these equations in all possible ways, and of new equations thus derived from them, leads to the discovery of innumerable new properties of the figure, and here we have porisms. These discussions, however, did not carry forward the work of restoring Euclid's Porisms, which was left for Chasles. His work (Les Trois livres de porismes d'Euclide, 186o) makes full use of all the material found in Pappus, but we may doubt its being a successful reproduction of Euclid's actual work. An interesting hypothesis as to the Porisms was put forward by H. G. Zeuthen (Die Lehre von den Kegelschnitten im Altertum, 1886). Observing, e.g., that the intercept-Porism is still true if the two fixed points are points on a conic, and the straight lines drawn through them intersect on the conic instead of on a fixed straight line, Zeuthen conjectures that the Porisms were a by-product of a fully developed projective geometry of conics. It is a fact that Pappus's Lemma 31 (though it makes no mention of a conic) corresponds exactly to Apollonius's method of determining the foci of a central conic (Conics, iii. with 42).
The three porisms stated by Diophantus in his Arithmetica are propositions in the theory of numbers which can all be enunciated in the form "we can find numbers satisfying such and such condi tions"; they are sufficiently analogous therefore to the geometrical porism as defined in Pappus and Proclus.