PORIS31 (iaptapa). An intermediate class of propositions, between problems and theorems, was, as we are informed by Pappus, distin guished by the ancient geometers under the name of porisms. Unfor tunately, however, the only notices of them by the ancients themselves, which are found in their remaining works, occnr in the Collectiones Mathematics' of Pappus Alexandrinus, and the commentaries of Proclus on the Elements of Euclid, in both places so very imperfectly, that till of late years mathematicians were not agreed on their exact interpretation. The description of porisms by Pappus, which he gives in the preface to the seventh book of his above-mentioned work, in an account of Euclid's work on the subject, is, in all the manuscripts which have been examined, extremely mutilated, and every attempt to restore them, before the masterly hand of Robert Simson took up the subject, had completely failed. The first part of the description, which seems to be entire, is calculated only to excite curiosity, being too general for conveying any precise notion of these propositions, or for giving any effectual assistance for the recovery of them; and the remainder, containing a detail of the contents of Euclid's work, is through the whole so corrupt that all endeavours to explain it were nugatory. Several celebrated geometers indeed flattered themselves that they had obtained possession of the secret ; but even Dr. Halley, with all his acuteness, relinquished the task, and adds, after giving the hactenus porismatuin deseriptio nee mini intellects nec lectori profutum." The definition which Pappus quotes from the ancients is too general to be useful, and perhaps implied more than our acquaintance with the language in which he wrote can enable us to determine. He says that a "theorem is something requiring demonstration, a problem in which something is proposed to be con structed; but a porism, that which requires investigation ; " and though this definition certainly does correspond to the nature of these propositions, yet it is deficient in discrimination, and of itself neither conveys any precise notion of Euclid's porisma, nor gives assistance in the investigation of any individual proposition. Dr. Simson's restored definition is as follows, literally translated :" A porism is a proposition in which it is proposed to demonstrate that some one thing or more things are given, to which, as also to each of innumerable other things not given, but which have the same relation to those which are given, it is to be shown that there belongs some common affection described in the proposition." The following less literal translation may probably be better understood :" A ponsm is a pro position 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, caromed according to a given law, a certain relation described in the proposition is to be shown to take place." Dr. Simson illustrates the propriety and accuracy of this definition by many examples, and it is so framed as to correspond with all the intimations of Pappus respecting porisms, and also with the character of the few individual porisms of Euclid which Dr. Simeon had discovered. It may therefore justly be considered as expressive of the notions on this subject entertained by the ancients, although probably, as iu the cases of theorem and problem, no precise was given of purism. It has been objected to Simson's definition, that it may be inferred from it that a porism partakes more of the nature of a problem than a theorem, and consequently is inconsistent with the "intermediate nature " mentioned by Pappus. In his enunciation it is affirmed that certain things may be found which shall have the relations or properties therein described. Now were it simply proposed to investigate certain things which would have the properties expressed in the porism, it may be regarded as a problem ; but if these things are found by a con struction described in the enunciation, the proposition becomes a theorem affirming the truth of the properties asserted ; and then a demonstration only is required, without any investigation, in the manner which appears to have been practised by the later mathemati cians alluded to by Pappus. The enunciation of a poriam as a problem
is not consistent with the usual character of such propositions. Problems usually, whatever difficulty may attend their solution; are almost immediately recognised, by those having some knowledge of geometry, as either passible in certain circumstances of the data, or as altogether impossible ; and it would be unusual to propose as a problem " to find things with certain properties, respecting the possibility of which no judgment can be formed without an analysis, or such consideration as is equivalent to an analysis." For example,if it had been proposed as a problem in the time of Apollonius, to find in a given parabola a point having the property of the focus, that point being then unknown, such a proposition would not have been considered as a proper problem, but would in reality have been a porism. To take another example : Proclus, in his commentaries on the Elements, mentions the first pro position of the third book, " to find the centre of a circle," as a porism, being in some measure between a problem and a theorem. But Proclus, however distinguished as a philosopher, was no mathematician, and as a circle, from Euclid's of it, must have a centre, the proposition to find that centre seems to be a proper problem. Had the circle been defined from another of its properties, as, for instance, from its being produced by the extremity of a straight line moving at right angles to another straight line, given in magnitude and position, and in the same plane, so that the square of the moving line be always equal to the rectangle by the segments into which it divides the given line ; then the finding of the centre would be a proper porism, and might be enunciated thus :" within a given circle (defined in the manner just mentioned) a point may be found from which all straight lines drawn to the circumference will be equal." Having thus placed before our readers the most probable restoration of the ancient meaning of the term porism, we proceed to notice briefly what modern geometers have given us on the subject. First in im portance stands the admirable paper on porisms by Professor Playfair, in the first volume of the Transactions of the Royal Society of Edinburgh,' which was read before that body in July, 1784. H i e m proves on Simson's definition, and substitutes the following :" A porism is a proposition affirming the possibility of finding such con ditions as will render a certain problem indeterminate, or capable of innumerable solutions." This, it must be confessed, is an important and elegant simplification, and fully conveys every idea contained in the more prolix definition of Simson but at the same time we agree with Dr. Trail in thinking that Dr. Sifnson's is expressed more nearly in the language and manner of the ancient geometers :" Though I admire the ingenuity and fully admit the soundness of this definition, and also the utility of the principle on which it is founded in the discovery of porisms, I must acknowledge my doubt of that particular notion of a porisin having ever been adopted, or even proposed, among the ancient geometricians." (Trail's Life of Simson,' pp. 50, 51.) A paper on porisms, containing some examples in the higher geometry, by Lord Brougham, was inserted in the ' Philosophical Transactions of the Royal Society,' in 1798. Fryer has given a popular history of the discovery of porisms, in the last edition of Simson's Geometry. Lastly, the most complete exposition of them that has yet appeared may be found in the Apereu Historique sur l'Origine et le Ddveloppement des 3I4thodes en G6omdtrie,' 4to, Brims., 1837, by 3L armies, of the French Institute : as well as in his recently published work, ' Les trois Byres de Porismea d'Euclide, ritablies pour In premiere foie; Paris, 1860, 8vo.