520 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 con course 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 Trigonometric a hope that he will be able to restore them. About the same time Fermat wrote a short work under the title Porismatum Evdidxorum 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 jwrismatibus 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, De 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 follow ing : " Porisma est propositio in qua proponitur demon- strare rem aliquam vel plures datas esse, cui vel quibus, ut et cuilibet ex rebus innume ris 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. 8oc. 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 be tween theorems and problems, and were called "porisms." Play fair accordingly defined a porism thus : "A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of in numerable 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." l This definition of a porism appears to be most generally accepted, at least in England. How ever, in Liouville s Journal de mathematiques pures et appliquees (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 Mathematics^ 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 Tr6pur/j.a 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 por- ismes 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 uncer tain whether these lost Porisms formed part of the Arith metics or were an independent treatise. Diophantus refers to them in the Arithmetics in three places, introducing a proposition assumed as known with the words e ^o/zev ev 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 vewrepot 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 pro pounded theories on the subject of porisms. It must, however, suffice merely tO-mention the chief among the rest of the contribu tions to the subject. These are, besides the papers of Vincent and 1 Tliis view of porisms is known exclusively by the name of Play- fair, though, as he himself says, Dugald Stewart had several years before defined a porism to be "a proposition affirming the possibility
of finding one or more of the conditions of an indeterminate theorem.