Porism
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The subject of porisms is perplexed by the multitude of different views which have been held by geometers as to what a porism really was and is.
The treatise which has given rise to the controversies on this subject is the Porisms of Euclid, the author of the Elements. For as much as we know of this lost treatise we are indebted to the Collection of Pappus of Alexandria, who mentions it along with other geometrical treatises, 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, so that they may be presented either as theorems or as problems; and they were regarded accordingly by many geometers, who looked merely at the form of the enunciation, as being actually theorems or problems, though the definitions given by the older writers showed that they better understood the distinction between the three classes of propositions.
The older geometers 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 (εἰς πορισμὸν αὐτοῦ τοῦ προτεινομένου). 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 τὸ λεῖπον ὑποθέσει τοπικοῦ θεωρήματος, 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," as a result unsought, as it were, 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 (what does not really help) that the finding of the center 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 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:
- 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
- the proof of the harmonic properties of a complete quadrilateral
- the theorem that, if the six vertices of a hexagon lie three and three on two straight lines, the three points of concourse 9f 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.
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. This explanation was published in the Philosophical Transactions in 1723. Later he investigated the subject of porisms generally in a work entitled De porismatibus traclatus; quo doctrinam porisrnatum satis explicatam, et in posterum ab oblivion 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. 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 demonstrate rem aliquam vel plures Batas ease, 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 quandam communem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ex quibus data demonstranda aunt, 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 illustration and some preliminary lemmas.
John Playfair's memoir (Trans. Roy. Soc. Edin., 1794, vol. iii.), a sort of sequel to Simson's treatise, had for its special object the inquiry into the probable origin of porisms, that is, into the steps which led the ancient geometers to the discovery of them. Playfair remarked that the careful investigation of all possible particular cases of a proposition would show that (1) under certain conditions a problem becomes impossible; (2) under certain other conditions, indeterminate 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." Playfair 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 had the support of Michel Chasles. However, in Liouville's Journal de mathematiques pures et appliquées (vol. xx., July, 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 Schooten, put forward in his Mathematicae exercitationes (1657), in which he gives the name of "porism" to one section. According to Frans van 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."
The discussions, however, between Breton and Vincent, in which C. Housel also joined, 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, Paris, 1860) makes full use of all the material found in Pappus. But we may doubt its being a successful reproduction of Euclid's actual work. Thus, in view of the ancillary relation in which Pappus's lemmas generally stand to the works to which they refer, it seems incredible that the first seven out of thirty-eight lemmas should be really equivalent (as Chasles makes them) to Euclid's first seven Porisms. Again, Chasles seems to have been wrong in making the ten cases of the four-line Porism begin the book, instead of the intercept-Porism fully enunciated by Pappus, to which the "lemma to the first Porism" relates intelligibly, being a particular case of it.
An interesting hypothesis as to the Porisms was put forward by H. G. Zeuthen (Die Lehre von den Kegelschnitten im Altertum, 1886, ch. viii.). 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 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. 4547 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 conditions"; they are sufficiently analogous therefore to the geometrical porism as defined in Pappus and Proclus.
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A valuable chapter on porisms (from a philological standpoint) is included in J. L. Heiberg's Litterargeschichtliche Studien über Euklid (Leipzig, 1882) ; and the following books or tracts may also be mentioned:
- August Richter, Porismen nach Simson bearbeitet (Elbing, 1837)
- M. Cantor, "Über die Porismen des Euklid and deren Divinatoren," in Schlomilch's Zeitsch. f. Math. u. Phy. (1857), and Literaturzeitung (1861), p. 3 seq.
- Th. Leidenfrost, Die Porismen des Euklid (Programm der Realschule zu Weimar, 1863)
- Fr. Buch-binder, Euclids Porismen und Data (Programm der kgl. Landesschule Pforta, 1866).
This article incorporates text from the Encyclopædia Britannica Eleventh Edition, a publication now in the public domain.
