Porism - Pappus On Euclid's Porism

Pappus On Euclid's Porism

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 the following: 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 these 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 + 1, of which n can turn about as many points fixed on the (n + 1)th. These n straight lines cut, two and two, in 1/2n(n − 1) points, 1/2n(n − 1) being a triangular number whose side is n − 1. If, then, they are made to turn about the n fixed points so that any n − 1 of their 1/2n(n − 1) points of intersection, chosen subject to a certain limitation, lie on n − 1 given fixed straight lines, then each of the remaining points of intersection, 1/2n(n − 1)(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 expressed 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 enunciations given by Pappus are incomplete, and he merely says that he gives thirty-eight lemmas for the three books of porisms; and these include 171 theorems. The lemmas which Pappus gives in connexion with the porisms are interesting historically, because he gives:

1. the fundamental theorem that the cross or an harmonic 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 quadrilateral;
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 this subject seems to have had great fascination for mathematicians, and many geometers have attempted to restore the lost porisms. Thus Albert Girard says in his Traité de trigonometrie (1626) that he hopes to publish a restoration. About the same time Pierre de Fermat wrote a short work under the title Porismatum euclidaeorum renovata doctrina et sub forma isagoges recentioribus geometeis exhibita (see Œuvres de Fermat, i., Paris, 1891); but two at least of the five examples of porisms which he gives do not fall within the classes indicated by Pappus.