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Porism




PORISM. The subject of porisms is perplexed by the multitude of different views which have been held by famous geometers as to what a porism really was and is. This article must therefore be limited to a short historical account (1) of the principal works of the Greek mathematicians which we know to have been called Porisms, and (2) of some of the principal contributions to the elucidation of these works, and conjectures as to the true signification of the term.

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, namely, defined a theorem as _____, a problem as _____ as ______, and finally a porism as ______.

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 _____.

Proclus gives a definition of a porism which agrees very well with the fact that Euclid used the same word ______ in his Elements for what is now called by the Latin name " corollary." Proclus's definition is _____ (Procl., Comment. Eucl., p. 58; cf. p. 80).

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 inter-section of these lines lie each on a fixed straight line, the remaining point of intersection will also lie on another straight line; or, If the sides of a triangle are made to turn each about one of three fixed points in a straight line, and if two of the vertices are made to move on two fixed straight lines, taken arbitrarily, the third vertex describes a third 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 (re+l)th. These n
.... . . n(n— 1) . n(n—l)
straight lines cut, two and two, in 9— points, ——~—-
being a triangular number whose side is (n — 1). If, then, they are made to turn about the n fixed points so that
any (»- 1) of their -A-^—' points of intersection lie on
(n — 1) given fixed straight lines, then each of the remaining
. (w-l)(»-2)
points of intersection, ^ in number, describes
a straight line. Pappus gives also a complete enuncia-tion 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 A 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 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 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 expresses in his Traite de Trigonométrie a hope that he will be able to restore them. About the same time Fermat wrote a short work under the title Porismatum Euclidxorum renovata doctrina et sub forma isagoges recentioribus geometris exhibita. He seems to have concerned himself only with the character and object of Euclid's work; but, though he seems to assert that he has restored the work, the examples of porisms which he gives have no connexion with those propositions indicated by Pappus. Fermat's idea of a porism was that it is nothing more than a locus. We may next mention Halley, who published the Greek text of the preface to Pappus's seventh book with a Latin translation, but with no comments or elucidations, remarking at the end that he has not been able to understand this description of porisms, which (he maintains) is made unintelligible by corruptions and lacunae in the text. Robert Simson was the first to throw real light on the subject. His first great triumph was the explanation of the only three propositions which Pappus indicates with any completeness. This explana-tion was published in the Philosophical Transactions in 1723; but Simson did not stop there. After his first success he set himself to investigate the subject of porisms generally, and the result appears in a work entitled De porismatibus tractatus; quo doctrinam porismatum satis explicatam, et in posterum ab oblivione tutam fore sperat auctor. This work, however, was not published until after Simson's death; it appeared at Glasgow in 1776 as part of a volume, Roberti Simson, matheseos nuper in academia Glasguensi professoris, opera quxdam reliqua. Simson's treatise, Be porismatibus, begins with definitions of theorem, problem, datum, porism, and locus. Respect-ing 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 demon-strare rem aliquam 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 quandam com-munem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ex quibus 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 illus-tration, and some preliminary lemmas. Playfair's memoir (Trans. Roy. Soc. Edin., vol. iii., 1794) may be said to be a sort of sequel to Simson's treatise, having 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's view was that the careful investigation of all possible particular cases of a proposition led to the observation 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." This definition, he maintained, agreed both with Pappus's account and Simson's definition, the obscurity of which he attempts to remedy by the following translation : " A porism is a proposition in which it is proposed to demonstrate that one or more things are given, between which and every one of innumerable other things not given, but assumed according to a given law, a certain relation, described in the proposition, is to be shown to take place." This definition of a porism appears to be most generally accepted, at least in England. How-ever, in Liouville's Journal de mathématiques pures et appliquées (vol. xx., July, 1855) P. Breton published Recherches nouvelles sur les porismes d'Euclide, in which he propounded a different theory, professedly based on the text of Pappus, as to the essential nature of a porism. This was followed in the same journal 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 Mathematicx exercitationes (1657), in which he gives the name of "porism" to one section. According to Schooten, if we observe the various numerical relations between straight lines in a figure and write them 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 a porism. It must be admitted that, if we are to judge of the meaning by the etymology of the name, this idea of a porism has a great deal to recommend it. We must, however, be on our guard against applying, on this view, the term " porism " to the process of discovery. The Greek word _____ should no doubt strictly signify the result obtained, but the name is still indicative of the process. The porism is the result as obtained by the pro-cess, which is itself the cause of the name. So great an authority as Chasles wrote in 1860 (Les trois livres de porismes d'Euclide) that, in spite of the general assent which Playfair's theory met with, he considered it to be unfounded.

The Porisms of Euclid are not the only representatives of this class of propositions. We know of a treatise of Diophantus which was entitled Porisms. But it is uncertain whether these lost Porisms formed part of the Arithmetics or were an independent treatise. Diophantus refers to them in the Arithmetics in three places, introducing a proposition assumed as known with the words ______. These propositions are not, however, all similar in form, and we cannot by means of them grasp what Diophantus understood to be the nature of a porism. So far as we can judge of his treatise it seems to have been a collection of a number of ordinary propositions in the theory of numbers, some of them being mere algebraical identities. Again, Diophantus should probably be included among the vuoTtpoi who are said to have substituted a new definition for that of the ancients, looking only to accidental not essential characteristics of a true porism. And yet, in so far as Diophantus's Porisms had no connexion with geometry, they do not in the least conform to the second definition of Pappus.

We have by no means exhausted the list of writers who have propounded theories on the subject of porisms. It must, however, suffice merely to mention the chief among the rest of the contributions to the subject. These are, besides the papers of Vincent and Breton, the following books or tracts on the Porisms of Euclid :— Aug. Richter, Porismen nach Simson bearbeitet (Elbing, 1837) ; Ch. Housel, " Les Porismes d'Euclide," in Liouville's Journal de mathématiques pures et appliquées (2d ser., vol. i., 1856) ; M. Cantor, "Ueber die Porismen des Euklid und deren Divinatoren," in Sehlômilch's Zeitsch. f. Math. u. Phy., 1857, and Literaturzeiiung, 1861, p. 3 sq. ; Th. Leidenfrost, Die Porismen des Euklid (Programm der Realschule zu Weimar, 1863) ; Fr. Buchbinder, Euclids Porismen und Data (Programm der kgl- Landesschule Pforta, 1866) (T. L. H.)






The above article was written by: T. L. Heath.



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