PORISM. Porisms is the title of a lost treatise by Euclid, the author of the Elements, for our knowledge of which we are in debted to the Collection of Pappus of Alexandria, who mentions it and gives a number of lemmas necessary for understanding it. Pappus states that the porisms of Euclid are neither theorems nor problems, but are in some sort intermediate ; and they were regarded accordingly by many geometers, who looked merely at the form of the enunciation, as being actually theorems or prob lems, though the definitions given by the older writers showed that they better understood the distinction between the three classes of propositions. They regarded a theorem as directed to proving what is proposed, a problem as directed to constructing what is proposed and finally a porism as directed to finding what is proposed (etc roptcrOv avroy roi) rporavo0vov). Pappus goes on to say that this last definition was changed by certain later geometers, who defined a porism on the ground of an accidental characteristic as Td XeCroi, Tortica Oaepiip.aros, that which falls short of a locus-theorem by a (or in its) hypothesis.
Proclus points out that the word was used in two senses. One sense is that of corollary, a result unsought, but seen to follow from a theorem. On the porism in the other sense he adds nothing to the definition of the older geometers except to say that the finding of the centre of a circle and the finding of the greatest common measure are porisms (Proclus, ed. Friedlein, p. 301).
Pappus gives a complete enunciation of a porism derived from Euclid, and an extension of it to a more general case. This porism, expressed in modern language, asserts that—given four straight lines of which three turn about the points in which they meet the fourth,if two of the points of intersection of the three lines lie each on a fixed straight line, the remaining point of intersection will also lie on another straight line. The general enunciation applies to any number of straight lines, say (n+ I), of which n can turn about as many points fixed on the (n+ i)th. These n straight lines cut, two and two, in r) points, In(n-- r) being a tri angular number whose side is (n—i). If, then, they are made to turn about the n fixed points so that any (n— 1) of their -in(n— I) points of intersection, chosen subject to a certain limitation, lie on (n— I) given fixed straight lines, then each of the remaining points of intersection, i (n— I) (n-2) in number, describes a straight line. Pappus gives also a complete enunciation of one
porism of the first book of Euclid's treatise. This may be ex pressed thus : If about two fixed points P, Q we make turn two straight lines meeting on a given straight line L, and if one of them cut off a segment AM from a fixed straight line AX, given in position, we can determine another fixed straight line BY, and a point B fixed on it, such that the segment BM' made by the second moving line on this second fixed line measured from B has a given ratio X to the first segment AM. The rest of the enun ciations given by Pappus are incomplete, and he merely says that he gives thirty-eight lemmas for the three books of porisms ; and that these include 171 theorems.
The lemmas which Pappus gives in connection with the porisms are interesting historically, because he gives (I) the fundamental theorem that the cross or anharmonic ratio of a pencil of four straight lines meeting in a point is constant for all transversals; ( 2 ) the proof of the harmonic properties of a complete quadri lateral; (3) the theorem that, if the six vertices of a hexagon lie three and three on two straight lines, the three points of concourse of opposite sides lie on a straight line.
During the last three centuries many geometers have attempted to restore the lost porisms. The geometer P. de Fermat (1601 65) wrote a short work under the title Porismatum euclidae orum renovata doctrina et sub forma isagoges recentioribus geometris exhibita (see Oeuvres de Fermat, i., 1891) ; but two at least of the five examples of porisms which he gives do not fall within the classes indicated by Pappus. Robert Simson was the first to throw real light upon the subject. He first succeeded in explaining the only three propositions which Pappus indicates with any completeness (Phil. Trans., 1723). Later he investigated the subject of porisms generally in a work entitled De porismatibus tractatus ; quo doctrinam porismatum satis explicatam, et in posterum ab oblivione tutam fore sperat auctor, and published after his death in a volume, Roberti Simson opera quaedam reliqua (Glasgow, 1776). Simson's treatise, De porismatibus, begins with definitions of theorem, problem, datum, porism and locus.