PORISMS are a species of proposition in geometry, much employed by the ancients : they appear to have been highly valued, and to have assisted them greatly in their geometrical researches. The name is almost all that re mains to us of their labours, and although, from the singu larly curious nature of the subject, and the facilities they afford in the cultivation of every branch of geometrical science, it is improbable that the subject should have oc cupied the attention of only a single author, yet we are acquainted with but one geometer among the ancients who composed a work expressly on the subject, and, un fortunately for the science, the whole of that work is lost, if we except a small fragment preserved by I'appus, in such an imperfect manner, as almost to have rendered hopeless any divination of its meaning. The three books on Porisms, by the author of the Elements of Geometry, form the only work expressly devoted to this subject, whose title has been handed down to us; and even the meaning of the word Porism was involved in considerable obscurity. Papus Alexandrinus, through whom alone we derive any information on this subject, has given two defi nitions of the word Porism ; the first he blames as insuf ficient, because it might include some loci, and the se cond, which he adopts, is so general and indefinite, as to convey to us no precise knowledge of their nature.
In the abstract he has given of the labours of those who preceded him, he has mentioned the three books on Po risms by Euclid, and has given thirty-eight geometrical propositions, of no very considerable difficulty, as use ful for the comprehension of the work itself. These, to gether with the imperfect definition, and an example of a porism which refers to a figure that is lost, and which is so remarkably confused as almost to render its reconstruction impossible, are all the data that remained. From these materials modern geometers have attempted to restore the work of Euclid ; and although the task is one of extreme difficulty, yet when executed, a very probable estimate may be formed of its resemblance to the original.
Albert Girard, in a work on trigonometry, printed at the Hague in 1629, mentions the lost books of porisms, and says that he had made a restoration of them : and again, in an edition of the works of Stevinus, f he declares that he had reinvented the porisms of Euclid, and intended shortly to publish them. Unfortunately he died before this intention was accomplished. As the works of Stevinus were printed by his widow after his death, possibly the manuscript may still exist in some of the libraries in Hol land. From the subsequent discoveries of Dr. Simson, however, it appears that the idea which Girard had of the species of propositions to which he annexed the name of porisms, was by no means the same as that which the former writer has so ably proved to have been attached to it by the ancient geometers.
After Girard, the next attempt to explain the nature of porisms was made by Bullialdus ; but this seems to have been derived from a communication with Fermat, to which distinguished mathematician we must now advert.
Amongst his posthumous works § is a short paper, entitled, Porismatum Euclidmeorum renovata doctrina," from which it appears that he had approached nearer than any of his predecessors to the true meaning of this class of propositions ; and, in fact, several of those with which he illustrates his view of the subject are in reality porisms : but he did not arrive at any definition which should clearly separate porisms from local theorems, nor did he even conjecture that there existed some peculiar mode of analy sis by which such propositions might be discovered, nor attempt to restore any of those of Euclid ; his promised restoration of the whole of the three books never having been published.
Dr. Halley, who possessed an extensive and profound acquaintance with the ancient geometry, made some at tempts to decypher the enunciation of the porism given by Pappus. He had successfully restored the 8th book of the conics of Appollonius, and the two books of the same author, De Sectione Spatii ; and had achieved a still more difficult labour, that of translating from the Arabic (a language with which he was unacquainted) the work of Apollonius De Sectione Rationis ; yet he was baffled by the obscurity which pervaded the mutilated description of Pappus, and observes, " Hactenus Porismatum descrip tio nec mihi intellecta nec lectori profutura." The failure of all who preceded in elucidating this ob scure subject, as well as the high rank which Pappus as signed to these propositions, seems to have stimulated the curiosity of one whose unabated perseverance has been rewarded by complete success. Dr. Robert Simson has described the progress he made in this subject, in a way which cannot fail to interest the attention of those who have devoted even a small portion of their time to geome trical inquiries. " Postquam vero spud Pappum legeram porismata Euclidis collectionem fuisse artiliciosissimam multarum rerum, qux spectant ad analysin difficiliorum et generalium problematum, magno desiderio tenebar, aliquid de its cognoscendi ; quare sxpius et rnultis variis que viis tum Pappi propositionem generalem mancam et imperfectam, turn primum, lib. 1. Porisma quod solum ex omnibus in tribus libris integrum adhuc manet, intel ligere et restituere conabar ; frustra tamen, nihil enim proficiebam. Cumque cogitationes de hac re multum mi hi temporis consumpserint, atque molestx admodum eva serint, firmiter animum induxi haec nunquam imposterum investigare ; prxsertim cum optimus geometra Hallcius spem omnem de its intelligendis abjecisset. Uncle quo ties mente occurrebant, toties eas arcebam. Postea (amen accidit, ut improvidum et prxpositi immemorem invase riot, meque detinuerint donee tandem lux quxdam effulse rit, qux spem mihi faciebat inveniendi saltem Pappi pro positionem generalem ; quam quidem multa investigatione tandem restitui.