Porisms (Gr. , from , to supply or deduce), a class of geometric propositions treated by the ancient Greek geometers, the precise nature of which is a matter of dispute. The only original authorities we have upon the subject are the seventh book of the " Mathematical Collections " of Pappus and the commentary on Euclid's " Elements " by Pro-cms. In both authors the language is so vague and the text so corrupt that they have served rather to stimulate the curiosity and exercise the ingenuity of scholars than to afford any real insight into the subject. Euclid is said to have written three books of porisms, but our information in regard to them is substantially confined to the imperfect account of Pappus above mentioned. According to this, a porism is a proposition intermediate between a theorem and a problem. "A theorem," says Pappus, "is a proposition requiring demonstration, a problem one requiring construction, a porism one requiring investigation." This is too vague to afford much assistance in the restoration of this class of propositions. The first important step in this direction was made by Robert Simson in a work published in 1776, after his death.
His definition was substantially the same as that afterward given by Playfair in a paper contained in the " Transactions of the Royal Society of Edinburgh," vol. i., which we quote: "A porism is a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions." The most important recent work on the subject is by M. Chasles, Les trois linres de porismes d'Euolide retablis (Paris, 1860). According to him, "a porism is an incomplete theorem expressing certain relations between things varying according to a common law indicated in the enunciation. The theorem would be complete if the magnitude and position of certain things which result from the hypothesis were determined, but which the enunciation of the porism does not explain." Other views have been presented by mathematicians of great ability, and the subject must still be considered as involved in obscurity.