PORISM

, Porisma, in Geometry, has by some been desined a general theorem, or canon, deduced from a geometrical locus, and serving for the solution of other general and difficult problems. Proclus derives the word from the Greek porizw, I establish, and conclude from something already done and demonstrated: and accordingly he defines Porisma a theorem drawn occasionally from some other theorem already proved: in which sense it agrees with what is otherwise called corollary.

Pappus says, a Porism is that in which something was proposed to be investigated.

Others derive it from w=o/ros, a passage, and make it of the nature of a lemma, or a proposition necessary for passing to another more important one.

But Dr. Simson, rejecting the crroneous accounts that have been given of a Porism, desines it a proposition, either in the form of a problem or a theorem, in which it is proposed either to investigate, or demonstrate.

Euclid wrote three books of Porisms, being a curious collection of various things relating to the analysis of the more difficult and general problems. Those books however are lost; and nothing remains in the works of the ancient geometricians concerning this subject, besides what Pappus has preserved, in a very imperfect and obscure state, in his Mathematical Collections, viz, in the introduction to the 7th book.

Several attempts have been made to restore these writings in some degree, besides that which Pappus has left upon the subject. Thus, Fermat has given a few propositions of this kind; which are to be found in the collection of his works, in folio, 1679, pa. 116. The like was done by Bullialdus, in his Exercitationes Geometricæ, 4to, 1657. Dr. Robert Simson gave also a specimen, in two propositions, in the Philos. Trans. vol. 32, pa. 330; and besides left behind him a considerable treatise on the subject of Porisms, which has been printed in an edition of his works, at the expence of the earl of Stanhope, in 4to, 1776.

The whole three books of Euclid were also restored by that ingenious mathematician Albert Girard, as appears by two notices that he gave, first in his Trigonometry, printed in French, at the Hague, in 1629, and also in his edition of the works of Stevinus, printed at Leyden in 1634, pa. 459; but whether his intention of publishing them was ever carried into execution, I have not been able to learn.

A learned paper on the subject of Porisms, by the very ingenious Professor Playfair, has just been inserted in the 3d volume of the Transactions of the Royal So- ciety of Edinburgh. As this paper contains a number of curious observations on the geometry of the Ancients in general, as well as forms a complete treatise as it were on Porism in particular, a pretty considerable abstract of it cannot but be deemed in this place very useful and important.

“The restoration of the ancient books of geometry (says the learned professor) would have been impossible, without the coincidence of two circumstances, of which, though the one is purely accidental, the other is essentially connected with the nature of the mathematical sciences. The first of these circumstances is the preservation of a short abstract of those books, drawn up by Pappus Alexandrinus, together with a series of such lemmata, as he judged useful to facilitate the study of them. The second is, the necessary connection that takes place among the objects of every mathematical work, which, by excluding whatever is arbitrary, makes it possible to determine the whole course of an investigation, when only a few points in it are known. From the union of these circumstances, mathematics has enjoyed an advantage of which no other branch of knowledge can partake; and while the critic or the historian has only been able to lament the fate of those books of Livy and Tacitus which are lost, the geometer has had the high satisfaction to behold the works of Euclid and Apollonius reviving under his hands.

“The first restorers of the ancient books were not, however, aware of the full extent of the work which they had undertaken. They thought it sufficient to demonstrate the propositions, which they knew from Pappus, to have been contained in those books; but they did not follow the antient method of investigation, and few of them appear to have had any idea of the elegant and simple analysis by which these propositions were originally discovered, and by which the Greek Geometry was peculiarly distinguished.

“Among these few, Fermat and Halley are to be particularly remarked. The former, one of the greatest mathematicians of the last age, and a man in all respects of superior abilities, had very just notions of the geometrical analysis, and appears often abundantly skilful in the use of it; yet in his restoration of the Loci Plani, it is remarkable, that in the most difficult propositions, he lays aside the analytical method, and contents himself with giving the synthetical demonstration. The latter, among the great number and variety of his literary occupations, found time for a most attentive study of the ancient mathematicians, and was an instance of, what experience shews to be much rarer than might be expected, a man equally well acquainted with the ancient and the modern geometry, and equally disposed to do justice to the merit of both. He restored the books of Apollonius, on the problem De Sectione Spatii, according to the true principles of the ancient analysis.

“These books, however, are but short, so that the first restoration of considerable extent that can be reckoned complete, is that of the Loci Plani by Dr. Simson, published in 1749, which, if it differs at all from the work it is intended to replace, seems to do so only by its greater excellence. This much at least is certain, that the method of the ancient geometers does not appear to greater advantage in the most entire of their| writings, than in the restoration above mentioned; and that Dr. Simson has often sacrificed the elegance to which his own analysis would have led, in order to tread more exactly in what the lemmata of Pappus pointed out to him, as the track which Apollonius had pursued.

“There was another subject, that of Porisms, the most intricate and enigmatical of any thing in the ancient geometry, which was still reserved to exercise the genius of Dr. Simson, and to call forth that enthusiastic admiration of antiquity, and that unwearied perseverance in research, for which he was so peculiarly distinguished. A treatise in three books, which Euclid had composed on Porisms, was lost, and all that remained concerning them was an abstract of that treatise, inserted by Pappus Alexandrinus in his Mathematical Collections, in which, had it been entire, the geometers of later times would doubtless have found wherewithal to console themselves for the loss of the original work. But unfortunately it has suffered so much from the injuries of time, that all which we can immediately learn from it is, that the Ancients put a high value on the propositions which they called Porisms, and regarded them as a very important part of their analysis. The Porisms of Euclid are said to be, “Collectio artificio“sissima multarum rerum quæ spectant ad analysin dif“siciliorum et generalium problematum.” The curiosity, however, which is excited by this encomium is quickly disappointed; for when Pappus proceeds to explain what a Porism is, he lays down two desinitions of it, one of which is rejected by him as imperfect, while the other, which is stated as correct, is too vague and indefinite to convey any useful information.

“These defects might nevertheless have been supplied, if the enumeration which he next gives of Euclid's Propositions had been entire; but on account of the extreme brevity of his enunciations, and their reference to a diagram which is lost, and for the constructing of which no directions are given, they are all, except one, perfectly unintelligible. For these reasons, the fragment in question is so obscure, that even to the learning and penetration of Dr. Halley, it seemed impossible that it could ever be explained; and he therefore concluded, after giving the Greek text with all possible correctness, and adding the Latin translation, “Hactenus Porismatum descriptio nec mihi intellecta, “nec lectori profutura. Neque aliter sieri potuit, tam “ob defectum schematis cujus sit mentio, quam ob “omissa quædam et transposita, vel aliter vitiata in pro“positionis generalis expositione, unde quid sibi velit “Pappus haud mihi datum est conjicere. His adde “dictionis modum nimis contractum, ac in re difficili, “qualis hæc est, minime usurpandum.”

“It is true, however, that before this time, Fermat had attempted to explain the nature of Porisms, and not altogether without success. Guiding his conjectures by the desinition which Pappus censures as imperfect, because it defined Porisms only “ab accidente,” viz. “Porisma est quod desicit hypothesi a Theoremate Lo“cali,” he formed to himself a tolerably just notion of these propositions, and illustrated his general description by examples that are in effect Porisms. But he was able to proceed no farther; and he neither proved, that his notion of a Porism was the same with Euclid's, nor attempted to restore, or explain any one of Euclid'<*> propositions; much less did he suppose, that they were to be investigated by an analysis peculiar to themselves. And so imperfect indeed was this attempt, that the complete restoration of the Porisms was necessary to prove, that Fermat had even approximated to the truth.

“All this did not, however, deter Dr. Simson from turning his thoughts to the same subject, which he appears to have done very early, and long before the publication of the Loci Plani in 1749.

“The account he gives of his progress, and of the obstacles he enconntered, will be always interesting to mathematicians. “Postquam vero apud Pappum le“geram, Porismata Euclidis collectionem fuisse artifi“ciosissimam multarum rerum, quæ spectant ad analysin “difficiliorum et generalium problematum, magno “desiderio tenebar, aliquid de iis cognoscendi; quare “sæpius et multis variisque viis tum Pappi propositio“nem generalem, mancam et imperfectam, tum pri“mum lib. i.

“Porisma, quod solum ex omnibus in tribus libris “integrum adhuc manet, intelligere et restituere “conabar; frustra tamen, nihil enim proficiebam. “Cumque cogitationes de hac re multum mihi tempo“ris consumpserint, atque molestæ admodum evaserint, “firmiter animum induxi hæc nunquam in posterum “investigare; præsertim cum optimus geometra Hal“leius spem omnem de iis intelligendis abjecisset. Un“de quoties menti occurrebant, toties eas arcebam. “Postea tamen accidit, ut improvidum et propositi im“memorem invaserint, meque detinuerint donec tan“dem lux quædam effulserit, quæ spem mihi faciebat “inveniendi saltem Pappi propositionem generalem, “quam quidem multa investigatione tandem restitui. “Hæc autem paulo post una cum Porismate primo “lib. i. impressa est inter Transactiones Phil. anni 1723, “num. 177.”

“The propositions mentioned, as inserted in the Philosophical Transactions for 1723, are all that Dr. Simson published on the subject of Porisms during his life, though he continued his investigations concerning them, and succeeded in restoring a great number of Euclid's propositions, together with their analysis. The propositions thus restored form a part of that valuable edition of the posthumous works of this geometer which the mathematical world owes to the munificence of the late earl Stanhope.

“The subject of Porisms is not, however, exhausted, nor is it yet placed in so clear a light as to need no farther illustration. It yet remains to enquire into the probable origin of these propositions, that is to say, into the steps by which the ancient geometers appear to have been led to the discovery of them.

“It remains also to point out the relations in which they stand to the other classes of geometrical truths; to consider the species of analysis, whether geometrical or algebraical, that belongs to them; and, if possible, to assign the reason why they have so long escaped the notice of modern mathematicians. It is to these points that the following observations are chiefly directed.

“I begin with describing the steps that appear to have led the ancient geometers to the discovery of Porisms; and must here supply the want of express testi-| mony by probable reasonings, such as are necessary, whenever we would trace remote discoveries to their sources, and which have more weight in mathematics than in any other of the sciences.

“It cannot be doubted, that it has been the solution of problems, which, in all states of the mathematical sciences, has led to the discovery of most geometrical truths. The first mathematical enquiries, in particular, must have occurred in the form of questions, where something was given, and something required to be done; and by the reasonings necessary to answer these questions, or to discover the relation between the things that were given, and those that were to be found, many truths were suggested, which came afterwards to be the subjects of separate demonstration. The number of these was the greater, that the ancient geometers always undertook the solution of problems with a scrupulous and minute attention, which would scarcely suffer any of the collateral truths to escape their observation. We know from the examples which they have left us, that they never considered a problem as resolved, till they had distinguished all its varieties, and evolved separately every different case that could occur, carefully remarking whatever change might arise in the construction, from any change that was supposed to take place among the magnitudes which were given.

“Now as this cautious method of proceeding was not better calculated to avoid error, than to lay hold of every truth that was connected with the main object of enquiry, these geometers soon observed, that there were many problems which, in certain circumstances, would admit of no solution whatever, and that the general construction by which they were resolved would fail, in consequence of a particular relation being supposed among the quantities which were given.

“Such problems were then said to become impossible; and it was readily perceived, that this always happened, when one of the conditions prescribed was inconsistent with the rest, so that the supposition of their being united in the same subject, involved a contradiction. Thus, when it was required to divide a given line, so that the rectangle under its segment, should be equal to a given space, it was evident, that if this space was greater than the square of half the given line, the thing required could not possibly be done; the two conditions, the one defining the magnitude of the line, and the other that of the rectangle under its segments, being then inconsistent with one another. Hence an infinity of beautiful propositions concerning the maxima and the minima of quantities, or the limits of the possible relations which quantities may stand in to one another.

“Such cases as these would occur even in the solution of the simplest problems; but when geometers proceeded to the analysis of such as were more complicated, they must have remarked, that their constructions would sometimes fail, for a reason directly contrary to that which has now been assigned. Instances would be found where the lines that, by their intersection, were to determine the thing sought, instead of intersecting one another, as they did in general, or of not meeting at all, as in the above-mentioned case of impossibility, would coincide with one another entirely, and leave the question of consequence unresolved. But though this circumstance must have created considerable embarrassment to the geometers who first observed it, as being perhaps the only instance in which the language of their own science had yet appeared to them ambiguous or obscure, it would not probably be long till they found out the true interpretation to be put on it. After a little reflexion, they would conclude, that since, in the general problem, the magnitude required was determined by the intersection of the two lines above mentioned, that is to say, by the points common to them both; so, in the case of their coincidence, as all their points were in common, every one of these points must afford a solution; which solutions therefore must be infinite in number; and also, though infinite in number, they must all be related to one another, and to the things given, by certain laws, which the position of the two coinciding lines must necessarily determine.

“On enquiring farther into the peculiarity in the state of the data which had produced this unexpected result, it might likewise be remarked, that the whole proceeded from one of the conditions of the problem involving another, or necessarily including it; so that they both together made in fact but one, and did not leave a sufficient number of independent conditions, to confine the problem to a single solution, or to any determinate number of solutions. It was not difficult afterwards to perceive, that these cases of problems formed very curious propositions, of an intermediate nature between problems and theorems, and that they admitted of being enunciated separately, in a manner peculiarly elegant and concise. It was to such propositions, so enunciated, that the ancient geometers gave the name of Porisms.

“This deduction requires to be illustrated by examples.” Mr. Playfair then gives several problems by way of illuftration; one of which, which may here suffice to shew the method, is as follows:

“A triangle ABC being given, and also a point D, to draw through D a straight line DG, such, that, perpendiculars being drawn to it from the three angles of the triangle, viz, AE, BG, CF, the sum of the two perpendiculars on the same side of DG, shall be equal to the remaining perpendicular: or, that AE and BG together, may be equal to CF.

“Suppose it done: Bisect AB in H, join CH, and draw HK perpendicular to DG.

“Because AB is bisected in H, the two perpendioulars AE and BG are together double of HK; and as they are also equal to CF by hypothesis, CF must be double of HK; and CL of LH. Now, GH is given in position, and magnitude; therefore the point L is| given; and the point D being also given, the line DL is given in position, which was to be found.

“The construction was obvious. Bisect AB in H, join CH, and take HL equal to one third of CH; the straight line which joins the points D and L is the line required.

“Now, it is plain, that while the triangle ABC remains the same, the point L also remains the same, wherever the point D may be. The point D may therefore coincide with L; and when this happens, the position of the line to be drawn is left undetermined; that is to say, any line whatever drawn through L will satis<*> fy the conditions of the problem. Here therefore we have another indesinite case of a problem, and of consequence another Porism, which may be thus enunciated: “A triangle being given in position, a point in it may be found, such, that any straight line whatever being drawn through that point, the perpendiculars drawn to this straight line from the two angles of the triangle which are on one side of it, will be together equal to the perpendicular that is drawn to the same line from the angle on the other side of it.

“This Porisin may be made much more general; for if, instead of the angles of a triangle, we suppose ever so many points to be given in a plane, a point may be found such, that any straight line being drawn through it, the sum of all the perpendiculars that fall on that line from the given points on one side of it, is equal to the sum of the perpendiculars that fall on it from all the points on the other side of it.

“Or still more generally, any number of points being given not in the same plane, a point may be found, through which if any plane be supposed to pass, the sum of all the perpendiculars which fall on that plane from the points on one side of it, is equal to the sum of all the perpendiculars that fall on the same plane from the points on the other side of it. It is unnecessary to observe, that the point to be found in these propositions, is no other than the centre of gravity of the given points; and that therefore we have here an example of a Porism very well known to the modern geometers, though not distinguished by them from other theorems.”

After some examples of other Porisms, and remarks upon them, the author then adds,

“From this account of the origin of Porisms, it follows, that a Porism may be defined, A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions.

“To this definition, the different characters which Pappus has given will apply without difficulty. The propositions described in it like those which he mentions, are, strictly speaking, neither theorems nor problems, but of an intermediate nature between both; for they neither simply enunciate a truth to be demonstrated, nor propose a question to be solved: but are affirmations of a truth, in which the determination of an unknown quantity is involved. In as far therefore as they assert, that a certain problem may become indeterminate, they are of the nature of theorems; and in as far as they seek to discover the conditions by which that is brought about, they are of the nature of pro- blems.

“In the preceding definition also, and the instances from which it is deduced, we may trace that imperfect description of Porisms which Pappus ascribes to the later geometers, viz, “Porisma est quod desicit hypothesi a theoremate locali.” Now, to understand this, it must be observed, that if we take the converse of one of the propositions called Loci, and make the construction of the figure a part of the hypothesis, we have what was called by the Ancients a Local Theorem. And again, if, in enunciating this theorem, that part of the hypothesis which contains the construction be suppressed, the proposition arising from thence will be a Porism; for it will enunciate a truth, and will also require, to the full understanding and investigation of that truth, that something should be found, viz, the circumstance in the construction, supposed to be omitted.

“Thus when we say; If from two given points E and D (2d fig. above), two lines EF and FD are inflected to a third point F, so as to be to one another in a given ratio, the point F is in the circumference of a circle given in position: we have a Locus.

“But when conversely it is said; If a circle ABC, of which the centre is O, be given in position, as also a point E, and if D be taken in the line EO, so that the rectangle EO OD be equal to the square of AO, the semidiameter of the circle; and if from E and D, the lines EF and DF be inflected to any point whatever in the circumference ABC; the ratio of EF to DF will be a given ratio, and the same with that of EA to AD: we have a local theorem.

“And, lastly, when it is said; If a circle ABC be given in position, and also a point E, a point D may be found, such, that if the two lines EF and FD be inflected from E and D to any point whatever F, in the circumference, these lines shall have a given ratio to one another: the proposition becomes a Porism.

“Here it is evident, that the local theorem is changed into a Porism, by leaving out what relates to the determination of the point D, and of the given ratio. But though all propositions formed in this way, from the conversion of Loci, be Porisms, yet all Porisms are not formed from the conversion of Loci. The first and second of the precéding, for instance, cannot by conversion be changed into Loci; and therefore the desinition which describes all Porisms as being so convertible, is not sufficiently comprehensive. Fermat's idea of Porisms, as has been already observed, was founded wholly on this definition, and therefore could not fail to be imperfect.

“It appears, therefore, that the definition of Porisms given above agrees with Pappus's idea of these propositions, as far at least as can be collected from the imperfect fragments which contain his general description of them. It agrees also with Dr. Simson's definition, which is this: “Porisma est proposicio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quæ ad ea quæ data sunt eandem habent relationem, convenire ostendendum est affectionem quandam communem in propositione descriptam.

“It cannot be denied, that there is a considerable degree of obscurity in this definition; notwithstanding of which it is certain, that every proposition to which it| applies must contain a problematical part, viz “in qua proponitur demonstrare rem aliquam, vel plures datas esse,” and also a theor<*>tical part, which contains the property, or communis asfectio, affirmed of certain things which have been previously described.

“It is also evident, that the subject of every such proposition, is the relation between magnitudes of three different kinds; determinate magnitudes which are given; determinate magnitudes which are to be found; and indeterminate magnitudes which, though unlimited in number, are connected with the others by some common property. Now, these are exactly the conditions contained in the desinitions that have been given here.

“To confirm the truth of this theory of the origin of Porisms, or at least the justness of the notions founded on it, I must add a quotation from an Essay on the same subject, by a member of this society, the extent and correctness of whose views make every coincidence with his opinions peculiarly flattering. In a paper read several years ago before the Philosophical Society, Professor Dugald Stewart defined a Porism to be “A proposition affirming the possibility of finding one or more of the conditions of an indeterminate theorem.” Where, by an indeterminate theorem, as he had previously explained it, is meant one which expresses a relation between certain quantities that are indeterminate, both in magnitude and in number. The near agreement of this with the definition and explanations which have been given above, is too obvious to require to be pointed out; and I have only to observe, that it was not long after the publication of Simson's posthumous works, when, being both of us occupied in speculations concerning Porisms, we were led separately to the conclusions which I have now stated.

“In an enquiry into the origin of Porisms, the etymology of the term ought not to be forgotten. The question indeed is not about the derivation of the word *porisma, for concerning that there is no doubt; but about the reason why this term was applied to the class of propositions above described. Two opinions may be formed on this subject, and each of them with considerable probability: 1mo. One of the significations of porizw, is to acquire or obtain; and hence *porisma, the thing obtained or gained.

“Accordingly, Scapula says, Es<*> vox a geometris desumpta qui theorema aliquid ex demonstrativo syllogismo necessario sequens inferentes, illud quasi lucrari dicuntur, quod non ex professo quidem theorematis bujus instituta sit demonstratio, sed tamen ex demonstratis recte sequatur. In this sense Euclid uses the word in his Elements of Geometry, where he calls the corollaries of his proposition, Porismata. This circumstance creates a prefumption, that when the word was applied to a particular class of propositions, it was meant, in both cases, to convey nearly the same idea, as it is not at all probable, that so correct a writer as Euclid, and so scrupulous in his use of words, should employ the same term to express two ideas which are perfectly different. May we not therefore conjecture, that these propositions got the name of Porisms, entirely with a reference to their origin. According to the idea explained above, they would in general occur to mathematicians when engaged in the solution of the more difficult problems, and would arise from those particular cases, where one of the conditions of the data involved in it some one of the rest. Thus a particular kind of theorem would be obtained, following as a corollary from the solution of the problem: and to this theorem the term *porisma might be very properly applied, since, in the words of Scapula, already quoted, Non ex professo theorematis bujus instituta sit demonstratio, sed tamen ex demonstratis re<*>e sequatur.

“2do. But-though this interpretation agrees so well with the supposed origin of Porisms, it is not free from difficulty. The verb w=orizw has another signification, to find out, to discover, to devise; and is used in this sense by Pappus, when he says that the propositions called Porisms, asford great delight, tois dunamenois oran kai porizma, to those who are able to understand and inv<*>stigate. Hence comes porismos, the act of finding out or discovering, and from porismos, in this sense, the same author evidently considers *porisma as being derived. His words are, *efasan de (o(i arxaioi) *porisma eina.to w=roteinomanon eis *porismon aut<*> w=rotei<*>oman<*>, the Ancients said, that a Porism is something proposed for the sinding out, or discovering of the very thing proposed. It seems singular, however, that Porisms should have taken their name from a circumstance common to them with so many other geometrical truths; and if this was really the case, it must have been on account of the enigmatical form of their enunciations, which required, that in the analysis of these propositions, a sort of double discovery should be made, not only of the Truth, but also of the Meaning of the very thing which was proposed. They may therefore have been called Porismata, or investigations, by way of eminence.

“We might next proceed to consider the particular Porisms which Dr. Simson has restored, and to shew, that every one of them is the indeterminate case of some problem. But of this it is so easy for any one, who has attended to the preceding remarks, to satisfy himself, by barely examining the enunciations of those propositions, that the detail unto which it would lead seems to be unnecessary. I shall therefore go on to make some observations on that kind of analysis which is particularly adapted to the investigation of Porisms.

“If the idea which we have given of these propositions be just, it follows, that they are always to be discovered by considering the cases in which the construction of a problem fails in consequence of the lines which, by their intersection, or the points which, by their position, were to determine the magnitude required, happening to coincide with one another—a Porism may therefore be deduced from the problem it belongs to, in the same manner that the propositions concerning the ma<*>ima and minima of quantities are deduced from the problems of which they form the limitations; and such no doubt is the most natural and most obvious analysis of which this class of propositions will admit.

“It is not, however, the only one that they will admit of; and there are good reasons for wishing to be provided with another, by means of which, a Porism that is any how suspected to exist, may be found out, independently of the general solution of the problem to which it belongs. Of these reasons, one is, that the Porism may perhaps admit of being investigated more easily than the general problem admits of being resolved:| and another is, that the former, in almost every case, helps to discover the simplest and most elegant solution that can be given of the latter.

“It is desirable to have a method of investigating Porisms, which does not require, that we should have previously resolved the problems they are connected with, and which may always serve to determine, whether to any given problem there be attached a Porism, or not. Dr. Simson's Analysis may be considered as answering to this description; for as that geometer did not regard these propositions at all in the light that is done here, nor in relation to their origin, an independent analysis os this kind, was the only one that could occur to him; and be has accordingly given one which is extremely ingenious, and by no means easy to be invented, but which he uses with great skilfulness and dexterity throughout the whole of his Restoration.

“It is not easy to ascertain whether this be the precise method used by the Ancients. Dr. Simson had here nothing to direct him but his genius, and has the full merit of the sirst inventor. It seems probable, however, that there is at least a great affinity between the methods, since the lemmata given by Pappus as necessary to Euclid's demonstrations, are subservient also to those of our modern geometer.

“It is, as we have seen, a general principle that a problem is converted into a Porism, when one, or when two, of the conditions of it, necessarily involve in them some one of the rest. Suppose then that two of the conditions are exactly in that state which determines the third; then, while they remain fixed or given, should that third one be supposed to vary, or differ, ever so little, from the state required by the other two, a contradiction will ensue. Therefore if, in the hypothesis of a problem, the conditions be so related to one another as to render it indeterminate, a Porism is produced; but if, of the conditions thus related to one another, some one be supposed to vary, while the others continue the same, an absurdity follows, and the problem becomes impossible. Wherever therefore any problem admits both of an indeterminate, and an impossible case, it is certain, that these cases are nearly related to one another, and that some of the conditions by which they are produced, are common to both.

“It is supposed above, that two of the conditions of a problem involve in them a third, and wherever that happens, the conclusion which has been deduced will invariably take place.

“But a Porism may sometimes be so simple, as to arise from the mere coincidence of one condition of a problem with another, though in no case whatever, any inconsistency can take place between them. Thus, in the second of the foregoing propositions, the coincidence of the point given in the problem with another point, viz, the centre of gravity of the given triangle, renders the problem indeterminate; but as there is no relation of distance, or pofition, between these points, that may not exist, so the problem has no impossible case belonging to it. There are, however, comparatively but few Porisms so simple in their origin as this, or that arise from problems in which the conditions are so little complicated; for it usually happens, that a problem which can become indefinite, may also become impossible; and if so, the connection between these cases, which has been already explained, never fails to take place.

“Another species of impossibility may frequently arise from the porifmatic case of a problem, which will very much affect the application of geometry to astronomy, or any of the sciences of experiment or observation. For when a problem is to be resolved by help of data furnished by experiment or observation, the first thing to be considered is, whether the data so obtained, be sufficient for determining the thing sought; and in this a very erroneous judgment may be formed, if we rest satisfied with a general view of the subject: For though the problem may in general be resolved from the data that we are provided with, yet these data may be so related to one another in the case before us, that the problem will become indeterminate, and instead of one solution, will admit of an insinite number.

“Suppose, for instance, that it were required to determine the position of a point F from knowing that it was situated in the circumference of a given circle ABC, and also from knowing the ratio of its distances from two given points E and D; it is certain that in general these data would be sufficient for determining the situation of F. But nevertheless, if E and D should be so situated, that they were in the same straight line with the centre of the given circle; and if the rectangle under their distances from that centre, were also equal to the square of the radius of the circle, then, the position of F could not be determined.

“This particular instance may not indeed occur in any of the practical applications of geometry; but there is one of the fame kind which has actually occurred in astronomy: And as the history of it is not a little singular, affording besides an excellent illustration of the nature of Porisms, I hope to be excused for entering into the following detail concerning it.

“Sir Isaac Newton having demonstrated, that the trajectory of a comet is a parabola, reduced the actual determination of the orbit of any particular comet to the solution of a geometrical problem, depending on the properties of the parabola, but of such considerable difficulty, that it is necessary to take the assistance of a more elementary problem, in order to find, at least nearly, the distance of the comet from the earth, at the times when it was observed. The expedient for this purpose, suggested by Newton himself, was to consider a small part of the comet's path as rectilineal, and described with an uniform motion, so that four observations of the comet being made at moderate intervals of time from one another, four straight lines would be determined, viz, the four lines joining the places of the earth and the comet, at the times of observation, acros<*> which if a straight line were drawn, so as to be cut by them in three parts, in the same ratios with the intervals of time abovementioned; the line so drawn would nearly represent the comet's path, and by its intersection with the given lines, would determine, at least nearly, the distances of the comet from the earth at the time of observation.

“The geometrical problem here employed, of drawing a line to be divided by four other lines given in position, into parts having given ratios to one another, had been already resolved by Dr. Wallis and Sir Chris-| topher Wren, and to their solutions Sir Isaac Newton added three others of his own, in different parts of his works. Yet none of all these geometers observed that peculiarity in the problem which rendered it inapplicable to astronomy. This was first done by M. Boscovich, but not till after many trials, when, on its application to the motion of comets, it had never led to any satisfactory result. The errors it produced in some instances were so considerable, that Zanotti, seeking to determine by it the orbit of the comet of 1739, found, that his construction threw the comet on the side of the sun opposite to that on which he had actually observed it. This gave occasion to Boscovich, some years afterwards, to examine the different cases of the problem, and to remark that, in one of them, it became indeterminate, and that, by a curious coincidence, this happened in the only case which could be supposed applicable to the astronomical problem abovementioned; in other words, he found, that in the state of the data, which must there always take place, innumerable lines might be drawn, that would be all cut in the same ratio, by the four lines given in position. This he demonstrated in a dissertation published at Rome in 1749, and since that time in the third volume of his Opuscula. A demonstration of it, by the same author, is also inserted at the end of Castillon's Commentary on the Arithmetica Universalis, where it is deduced from a construction of the general problem, given by Mr. Thomas Simpson, at the end of his Elements of Geometry. The proposition, in Boscovich's words, is this: Problema quo quæritur recta linea quæ quatuor rectas positione datas<*>ita secet, ut tria ejus segmenta <*>int invicem in ratione data, evadit aliquando indeterminatum, ita ut per quodvis punctum cujusvis ex iis quatuor rectis duci possit recta linea, quæ ei conditioni faciat satis.

“It is needless, I believe, to remark, that the proposition thus enunciated is a Porism, and that it was discovered by Bos<*>ovich, in the same way, in which I have supposed Porisms to have been first discovered by the geometers of antiquity.

“A question nearly connected with the origin of Porisms still remains to be solved, namely, from what cause has it arisen that propositions which are in themselves so important, and that actually occupied so considerable a place in the ancient geometry, have been so little remarked in the modern? It cannot indeed be said, that propositions of this kind were wholly unknown to the Moderns before the restoration of what Euclid had written concerning them; for besides M. Boscovich's proposition, of which so much has been already said, the theorem which asserts, that in every system of points there is a centre of gravity, has been shewn above to be a Porism; and we shall see hereafter, that many of the theorems in the higher geometry belong to the same class of propositions. We may add, that some of the elementary propositions of geometry want only the proper form of enunciation to be perfect Po<*>isms. It is not therefore strictly true, that none of the propositions called Porisms have been known to the Moderns; but it is certain, that they have not met, from them, with the attention they met with from the Ancients, and that they have not been distinguished as a separate class of propositions. The cause of this difference is undoubtedly to be sought for in a comparison of the methods employed for the solution of geometrical problems in ancient and modern times.

“In the solution of such problems, the geometers of antiquity proceeded with the utmost caution, and were careful to remark every particular case, that is to say, every change in the construction, which any change in the state of the data could produce. The different conditions from which the solutions were derived, were supposed to vary one by one, while the others remained the same; and all their possible combinations being thus enumerated, a separate solution was given, whereever any considerable change was observed to have taken place.

“This was so much the case, that the Sectio Rationis, a geometrical problem of no great difficulty, and one of which the solution would be dispatched, according to the methods of the modern geometry, in a single page, was made by Apollonius, the subject of a treatise con- <*>ting of two books. The first book has seven general divisions, and twenty-four cases; the second, fourteen general divisions, and seventy-three cases, each of which cases is separately considered. Nothing, it is evident, that was any way connected with the problem, could escape a geometer, who proceeded with such minuteness of investigation.

“The same scrupulous exactness may be remarked in all the other mathematical researches of the Ancients; and the reason doubtless is, that the geometers of those ages, however expert they were in the use f their analysis, had not sufficient experience in its powers, to trust to the more general applications of it. That principle which we call the law of continuity, and which connects the whole system of mathematical truths by a chain of insensible gradations, was scarcely known to them, and has been unfolded to us, only by a more extensive knowledge of the mathematical sciences, and by that most perfect mode of expressing the relations of quantity, which forms the language of algebra; and it is this principle alone which has taught us, that though in the solution of a problem, it may be impossible to conduct the investigation without assuming the data in a particular state, yet the result may be perfectly general, and will accommodate itself to every case with such wonderful versatility, as is scarcely credible to the most experienced mathematician, and such as often forces h<*>m to stop, in the midst of his calculus, and look back, with a mixture of diffidence and admiration, on the unforeseen harmony of his conclusions. All this was unknown to the Ancients; and therefore they had no resource, but to apply their analysis separately to each particular case, with that extreme caution which has just been deseribed; and in doing so, they were likely to remark many peculiarities, which more extensive views, and more expeditious methods of investigation, might perhaps have induced them to overlook.

“To rest satissied, indeed, with too general results, and not to descend sufficiently into particular details, may be considered as a vice that naturally arises out of the excellence os the modern analysis. The esfect which thi<*> has had, in concealing from us the class of propositions we are now considering, cannot be better illustrated than by the example of the Porism discovered by Boscovich, in the manner related above. Though the problem from which that Porism is derived, was| resolved by several mathematicians of the sir<*> eminence, among whom also was Sir Isaac Newton, yet the Porism which, as it happens is the most important case of it, was not observed by any of them. This is the more remarkable, that Sir Isaac Newton takes notice of the two most simple cases, in which the problem obviously admits of innumerable solutions, viz, when the lines given in position are either all parallel, or all meeting in a point, and these two hypotheses he therefore expressly excepts. Yet he did not remark, that there are other circumstances which may render the solution of the problem indeterminate as well as these; so that the porismatic case considered above, escaped his observation: and if it escaped the observation of one who was accustomed to penetrate so far into matters infinitely more obscure, it was because he satissied himself with a general construction, without pursuing it into its particular cases. Had the solution been conducted after the manner of Euclid or Apollonius, the Porism in question must infallibly have been discovered.”

PORISTIC Method, is that which determines when, by what means, and how many different ways, a problem may be resolved.