PORISM, in Geometry, is a name given by the ancient geometers to two classes of mathematical propositions. Euclid gives this name to propositions which are involved in others which he is professedly investigating, and which, although not his principal object, are yet obtained along with it, as is expressed by their name porismata, "acquisitions." Such propositions are now called
called corollaries. But he gives the same name, by way of eminence, to a particular class of propositions which he collected in the course of his researches, and selected from among many others on account of their great subservience to the business of geometrical investigation in general. These propositions were so named by him, either from the way in which he discovered them, while he was investigating something else, by which means they might be considered as gains or acquisitions, or from their utility in acquiring farther knowledge as steps in the investigation. In this sense they are porismata; for porism signifies both to investigate and to acquire by investigation. These propositions formed a collection, which was familiarly known to the ancient geometers by the name of Euclid's porisms; and Pappus of Alexandria says, that it was a most ingenious collection of many things conducive to the analysis or solution of the most difficult problems, and which afforded great delight to those who were able to understand and to investigate them.
Unfortunately for mathematical science, this valuable collection is now lost, and it still remains a doubtful question in what manner the ancients conducted their researches upon this curious subject. We have, however, reason to believe that their method was excellent both in principle and extent; for their analysis led them to many profound discoveries, and was restricted by the severest logic. The only account we have of this class of geometrical propositions, is in a fragment of Pappus, in which he attempts a general description of them as a set of mathematical propositions, distinguishable in kind from all others; but of this description nothing remains, except a criticism on a definition of them given by some geometers, and with which he finds fault, as defining them only by an accidental circumstance, "A Porism is that which is deficient in hypothesis from a local theorem."
Pappus then proceeds to give an account of Euclid's porisms; but the enunciations are so extremely defective, at the same time that they refer to a figure now lost, that Dr Halley confesses the fragment in question to be beyond his comprehension.
The high encomiums given by Pappus to these propositions have excited the curiosity of the greatest geometers of modern times, who have attempted to discover their nature and manner of investigation. M. Fermat, a French mathematician of the 17th century, attaching himself to the definition which Pappus criticises, published an introduction (for this is its modest title) to this subject, which many others tried to elucidate in vain. At length Dr Simson, Professor of Mathematics in the University of Glasgow, was so fortunate as to succeed in restoring the Porisms of Euclid. The account he gives of his progress and the obstacles he encountered will always be interesting to mathematicians. In the preface to his treatise De Porismatibus, he says, "Postquam vero apud Pappum legeram Porismata Euclidis Collectionem fuisse artificiosissimam multarum rerum, que spectant ad analysin difficiliorum et generalium problematum, magno desiderio tenebar, aliquid de iis cognoscendi; quare sæpius et multis variisque viis tum Pappi præpositionem generalem, mancam et imperfectam, tum primum lib. 1. porisma, quod, ut dictum fuit, 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 temporis contumperint, atque tandem molestæ admodum evaserint, firmiter animum induxi nunquam in posterum investigare; præsertim cum optimus Geometra Halleyus spem omnem de iis intelligendis abjecisset. Unde quacies mentis occurrerant, toties eas arcebam. Postea tamen accidit ut improvidum et propositi immemorem invaserint, meque detinuerint donec tandem lux quædam effulserit quæ spem mihi faciebat invenendi saltem Pappi præpositionem generalem, quam quidem multa investigatione tandem restitui. Hæc autem paulo post una cum Porismate primo lib. 1. impressa est inter Transactiones Philosophicas anni 1723, No 177."
Dr Simson's Restoration has all the appearance of being just; it precisely corresponds to Pappus's description of them. All the lemmas which Pappus has given for the better understanding of Euclid's propositions are equally applicable to those of Dr Simson, which are found to differ from local theorems precisely as Pappus affirms those of Euclid to have done. They require a particular mode of analysis, and are of immense service in geometrical investigation; on which account they may justly claim our attention.
While Dr Simson was employed in this inquiry, he carried on a correspondence upon the subject with the late Dr M. Stewart, professor of mathematics in the university of Edinburgh; who, besides entering into Dr Simson's views, and communicating to him many curious porisms, pursued the same subject in a new and very different direction. He published the result of his inquiries in 1746, under the title of General Theorems, not wishing to give them any other name, lest he might appear to anticipate the labours of his friend and former preceptor. The greater part of the propositions contained in that work are porisms, but without demonstrations; therefore, whoever wishes to investigate one of the most curious subjects in geometry, will there find abundance of materials, and an ample field for discussion.
Dr Simson defines a porism to be "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 shown to take place."
This definition is not a little obscure, but will be plainer if expressed thus: "A porism is a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions." This definition agrees with Pappus's idea of these propositions, so far at least as they can be understood from the fragment already mentioned; for the propositions here defined, like those which he describes, 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 resolved, 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 problems.
We shall endeavour to make our readers understand this
Porism. this subject distinctly, by considering them in the way in which it is probable they occurred to the ancient geometers in the course of their researches: this will at the same time show the nature of the analysis peculiar to them, and their great use in the solution of problems.
It appears to be certain, that it has been the solution of problems which, in all states of the mathematical sciences, has led to the discovery of geometrical truths: the first mathematical inquiries, in particular, must have occurred in the form of questions, where something was given, and something required to be done; and by the reasoning necessary to answer these questions, or to discover the relation between the things given and those to be found, many truths were suggested, which came afterwards to be the subject of separate demonstrations.
The number of these was the greater, because the ancient geometers always undertook the solution of problems, with a scrupulous and minute attention, inasmuch that they would scarcely suffer any of the collateral truths to escape their observation.
Now, as this cautious manner of proceeding gave an opportunity of laying hold of every collateral truth connected with the main object of inquiry, these geometers soon perceived, that there were many problems which in certain cases would admit of no solution whatever, in consequence of a particular relation taking place among the quantities which were given. Such problems were said to become impossible; and it was soon perceived, that this always happened when one of the conditions of the problem was inconsistent with the rest. Thus, when it was required to divide a line, so that the rectangle contained by its segments might be equal to a given space, it was found that this was possible only when the given space was less than the square of half the line; for when it was otherwise, the two conditions defining, the one the magnitude of the line, and the other the rectangle of its segments, were inconsistent with each other. Such cases would occur in the solution of the most simple problems; but if they were more complicated, it must have been remarked, that the constructions would sometimes fail, for a reason directly contrary to that just now assigned. Cases would occur, where the lines, which by their intersection were to determine the thing sought, instead of intersecting each other as they did commonly, or of not meeting at all, as in the above-mentioned case of impossibility, would coincide with one another entirely, and of course leave the problem unresolved. It would appear to geometers upon a little reflection, that since, in the case of determinate problems, the thing required was determined by the intersection of the two lines already mentioned, that is, by the points common to both; so in the case of their coincidence, as all their parts were in common, every one of these points must give a solution, or, in other words, the solutions must be indefinite in number.
Upon inquiry, it would be found that this proceeded from some condition of the problem having been involved in another, so that, in fact, the two formed but one, and thus there was not a sufficient number of independent conditions to limit the problem to a single or to any determinate number of solutions. It would soon be perceived, that these cases formed very curious propositions
of an intermediate nature between problems and theorems; and that they admitted of being enunciated in a manner peculiarly elegant and concise. It was to such propositions that the ancients gave the name of porisms. This deduction requires to be illustrated by an example: suppose, therefore, that it were required to resolve the following problem.
A circle ABC (fig. 1.), a straight line DE, and a point F, being given in position, to find a point G in the straight line DE such, that GF, the line drawn from it to the given point, shall be equal to GB, the line drawn from it touching the given circle.
Suppose G to be found, and GB to be drawn touching the given circle ABC in B, let H be its centre, join HB, and let HD be perpendicular to DE. From D draw DL, touching the circle ABC in L, and join HL; also from the centre G, with the distance GB or GF, describe the circle BKF, meeting HD in the points K and K'. It is evident that HD and DL are given in position and magnitude: also because GB touches the circle ABC, HBG is a right angle; and since G is the centre of the circle BKF, HB touches that circle, and consequently or ; but because is bisected in D, , therefore . But , therefore and . But DL is given in magnitude, therefore DK is given in magnitude, and consequently K is a given point. For the same reason K' is a given point, therefore the point F being given in position, the circle KFK' is given in position. The point G, which is its centre, is therefore given in position, which was to be found. Hence this construction:
Having drawn HD perpendicular to DE, and DL touching the circle ABC, make DK and DK' each equal to DL, and find G the centre of the circle described through the points KFK'; that is, let FK' be joined and bisected at right angles by MN, which meets DE in G, G will be the point required; or it will be such a point, that if GB be drawn touching the circle ABC, and GF to the given point, GB is equal to GF.
The synthetical demonstration is easily derived from the preceding analysis; but it must be remarked, that in some cases this construction fails. For, first, if F fall anywhere in DH, as at F', the line MN becomes parallel to DE, and the point G is nowhere to be found; or, in other words, it is at an infinite distance from D. This is true in general; but if the given point F coincide with K, then MN evidently coincides with DE; so that, agreeable to a remark already made, every point of the line DE may be taken for G, and will satisfy the conditions of the problem; that is to say, GB will be equal to GK, wherever the point G is taken in the line DE: the same is true if F coincide with K. Thus we have an instance of a problem, and that too a very simple one, which, in general, admits but of one solution; but which, in one particular case, when a certain relation takes place among the things given, becomes indefinite, and admits of innumerable solutions. The proposition which results from this case of the problem is a porism, and may be thus enunciated:
"A circle ABC being given by position, and also a straight line DE, which does not cut the circle, a point K may be found, such, that if G be any point whatever
in DE, the straight line drawn from G to the point K shall be equal to the straight line drawn from G touching the given circle ABC."
The problem which follows appears to have led to the discovery of many porisms.
A circle ABC (fig. 2.) and two points D, E, in a diameter of it being given, to find a point F in the circumference of the given circle; from which, if straight lines be drawn to the given points E, D, these straight lines shall have to one another the given ratio of to , which is supposed to be that of a greater to a less.—Suppose the problem resolved, and that F is found, so that FE has to ED the given ratio of to ; produce EF towards B, bisect the angle EFD by FL, and DFB by FM: therefore , that is in a given ratio, and since ED is given, each of the segments EL, LD, is given, and the point L is also given; again, because DFB is bisected by FM, , that is, in a given ratio, and therefore M is given. Since DFL is half of DFE, and DFM half of DFB, therefore LFM is half of (DFE + DFB), that is, the half of two right angles, therefore LFM is a right angle; and since the points L, M, are given, the point F is in the circumference of a circle described upon LM as a diameter, and therefore given in position. Now the point F is also in the circumference of the given circle ABC, therefore it is in the intersection of the two given circumferences, and therefore is found. Hence this construction: Divide ED in L, so that EL may be to LD in the given ratio of to , and produce ED also to M, so that EM may be to MD in the same given ratio of to ; bisect LM in N, and from the centre N, with the distance NL, describe the semicircle LFM; and the point F, in which it intersects the circle ABC, is the point required.
The synthetical demonstration is easily derived from the preceding analysis. It must, however, be remarked, that the construction fails when the circle LFM falls either wholly within or wholly without the circle ABC, so that the circumferences do not intersect; and in these cases the problem cannot be solved. It is also obvious that the construction will fail in another case, viz. when the two circumferences LFM, ABC, entirely coincide. In this case, it is farther evident, that every point in the circumference ABC will answer the conditions of the problem, which is therefore capable of numberless solutions, and may, as in the former instances, be converted into a porism. We are now to inquire, therefore, in what circumstances the point L will coincide with A, and also the point M with C, and of consequence the circumference LFM with ABC. If we suppose that they coincide, , and , or by conversion, —, O being the centre of the circle ABC; therefore, also, , and by composition, , therefore . Hence, if the given points E and D (fig. 3.) be so situated that , and at the same time, , the problem admits of numberless solutions; and if either of the points D or E be given, the other point, and also the ratio which will render the problem indeterminate, may be found. Hence we have this porism:
"A circle ABC, and also a point D being given, another point E may be found, such that the two lines
inflected from these points to any point in the circumference ABC, shall have to each other a given ratio, which ratio is also to be found." Hence also we have an example of the derivation of porisms from one another; for the circle ABC, and the points D and E remaining as before (fig. 3.), if, through D we draw any line whatever HDB, meeting the circle in B and H; and if the lines EB, EH be also drawn, these will cut off equal circumferences BF, HG. Let EC be drawn, and it is plain from the foregoing analysis, that the angles DFC, CFB, are equal; therefore if OG, OB, be drawn, the angles BOC, COG, are also equal; and consequently the angles DOB, DOG. In the same manner, by joining AB, the angle DBE being bisected by BA, it is evident that the angle AOF is equal to AOH, and therefore the angle FOH to HOG; hence the arch FB is equal to the arch HG. It is evident that if the circle ABC, and either of the points DE were given, the other point might be found. Therefore we have this porism, which appears to have been the last but one of the third book of Euclid's porisms. "A point being given, either within or without a circle given by position. If there be drawn, anyhow through that point, a line cutting the circle into two points; another point may be found, such, that if two lines be drawn from it to the points in which the line already drawn cuts the circle, these two lines will cut off from the circle equal circumferences."
The proposition from which we have deduced these two porisms also affords an illustration of the remark, that the conditions of a problem are involved more another in the porismatic or indefinite case; for several independent conditions are laid down, by the help of which the problem is to be resolved. Two points D and E are given, from which two lines are to be inflected, and a circumference ABC, in which these lines are to meet, as also a ratio which these lines are to have to each other. Now these conditions are all independent of one another, so that any one may be changed without any change whatever in the rest. This is true in general; but yet in one case, viz. when the points are related to another that the rectangle under their distances from the centre is equal to the square of the radius of the circle; it follows, from the preceding analysis, that the ratio of the inflected lines is no longer a matter of choice, but a necessary consequence of this disposition of the points.
From what has been already said, we may trace the imperfect definition of a porism which Pappus ascribes to the later geometers, viz. that it differs from a local theorem, by wanting the hypothesis assumed in that theorem.—Now, to understand this, it must be observed, that if we take one of the propositions called loci, and make the construction of the figure a part of the hypothesis, we get what was called by the ancient geometers, a local theorem. If, again, in the enunciation of the theorem, that part of the hypothesis which contains the construction be suppressed, the proposition thence arising will be a porism, for it will enunciate a truth, and will require to the full understanding and investigation of that truth, that something should be found, viz. the circumstances in the construction supposed to be omitted.
Thus, when we say, if from two given points D, E (fig. 3.) two straight lines EF, FD, are inflected to a third point F, so as to be to one another in a given ratio,
Porism. tio, the point F is in the circumference of a given circle, we have a locus. But when conversely it is said, if a circle ABC, of which the centre is O, be given by position, as also a point E; and if D be taken in the line EO, so that ; and if from E and D the lines EF, DF be inflected to any point of the circumference ABC, the ratio of EF to DF will be given, viz. the same with that of EA to AD, we have a local theorem.
Lastly, when it is said, if a circle ABC be given by position, and also a point E, a point D may be found, such that if EF, FD be inflected from E and D to any point F in the circumference ABC, these lines shall have a given ratio to one another, the proposition becomes a porism, and is the same that has just now been investigated.
Hence it is evident, that the local theorem is changed into a porism, by leaving out what relates to the determination of D, and of the given ratio. But though all propositions formed in this way from the conversion of loci, are porisms, yet all porisms are not formed from the conversion of loci; the first, for instance, of the preceding cannot by conversion be changed into a locus; therefore Fermat's idea of porisms, founded upon this circumstance, could not fail to be imperfect.
To confirm the truth of the preceding theory, it may be added, that Professor Dugald Stewart, in a paper read a considerable time ago before the Philosophical Society of Edinburgh, defines a porism to be "A proposition affirming the possibility of finding one or more conditions of an indeterminate theorem;" where, by an indeterminate theorem, he means one which expresses a relation between certain quantities that are determinate and certain others that are indeterminate; a definition which evidently agrees with the explanation which has been here given.
If the idea which we have given of these propositions be just, it follows, that they are to be discovered by considering those 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 problem required, happening to coincide with one another. A porism may therefore be deduced from the problem to which it belongs, just as propositions concerning the maxima and minima of quantities are deduced from the problems of which they form limitations; and such is the most natural and obvious analysis of which this class of propositions admits.
The following porism is the first of Euclid's, and the first also which was restored. It is given here to exemplify the advantage which, in investigations of this kind, may be derived from employing the law of continuity in its utmost extent, and pursuing porisms to those extreme cases where the indeterminate magnitudes increase ad infinitum.
This porism may be considered as having occurred in the solution of the following problem: Two points A, B, (fig. 4.) and also three straight lines DE, FK, KL, being given in position, together with two points H and M in two of these lines, to inflect from A and B to a point in the third line, two lines that shall cut off from KF and KL two segments, adjacent to the given points H and M, having to one another the given ratio of to . Now, to find whether a porism be connected with this
problem, suppose that there is, and that the following proposition is true. Two points A and B, and two straight lines DE, FK, being given in position, and also a point H in one of them, a line LK may be found, and also a point in it M, both given in position, such that AE and BE inflected from the points A and B to any point whatever of the line DE, shall cut off from the other lines FK and LK segments HG and MN adjacent to the given points H and M, having to one another the given ratio of to .
First, let AE', BE', be inflected to the point E', so that AE' may be parallel to FK, then shall E'B be parallel to KL, the line to be found; for if it be not parallel to KL, the point of their intersection must be at a finite distance from the point M, and therefore making as to , so this distance to a fourth proportional, the distance from H at which AE' intersects FK, will be equal to that fourth proportional. But AE' does not intersect FK, for they are parallel by construction; therefore BE' cannot intersect KL, which is therefore parallel to BE', a line given in position. Again, let AE'', BE'', be inflected to E'', so that AE'' may pass through the given point H: then it is plain that BE'' must pass through the point to be found M; for if not, it may be demonstrated just as above, that AE'' does not pass through H, contrary to the supposition. The point to be found is therefore in the line E'B, which is given in position. Now if from E there be drawn EP parallel to AE', and ES parallel to BE', BS : SE :: BL
; therefore the ratio of FG to LN is compounded of the ratios of AF to BL, PE to ES, and BS to AP; but PE : SE :: AE' : BE', and BS : AP :: DB : DA, for DB : BS :: DE' : E'E :: DA : AP; therefore the ratio of FG to LN is compounded of the ratios of AF to BL, AE' to BE', and DB to DA. In like manner, because E'' is a point in the line DE and AE'', BE'' are inflected to it, the ratio of FH to LM is compounded of the same ratios of AF to BL, AE' to BE', and DB to DA; therefore FH : LM :: FG : NL (and consequently) :: HG : MN; but the ratio of HG to MN is given, being by supposition the same as that of to ; the ratio of FH to LM is therefore also given, and FH being given, LM is given in magnitude. Now LM is parallel to BE', a line given in position; therefore M is in a line QM, parallel to AB, and given in position; therefore the point M, and also the line KLM, drawn through it parallel to BE', are given in position, which were to be found. Hence this construction: From A draw AE' parallel to FK, so as to meet DE in E'; join BE', and take in it BQ, so that , and through Q draw QM parallel to AB. Let HA be drawn, and produced till it meet DE in E'', and draw BE'', meeting QM in M; through M draw KML parallel to BE', then is KML the line and M the point which were to be found. There are two lines which will answer the conditions of this porism; for if in QB, produced on the other side of B, there be taken Bq = BQ, and if qm be drawn parallel to AB, cutting MB in m; and if m be drawn parallel to BQ, the part mn, cut
Porism. off by EB produced, will be equal to MN, and have to HG the ratio required. It is plain, that whatever be the ratio of to , and whatever be the magnitude of FH, if the other things given remain the same, the lines found will be all parallel to BE. But if the ratio of to remain the same likewise, and if only the point H vary, the position of KL will remain the same, and the point M will vary.
Another general remark which may be made on the analysis of porisms is, that it often happens, as in the last example, that the magnitudes required may all, or a part of them, be found by considering the extreme cases; but for the discovery of the relation between them, and the indefinite magnitudes, we must have recourse to the hypothesis of the porism in its most general or indefinite form; and must endeavour so to conduct the reasoning, that the indefinite magnitudes may at length totally disappear, and leave a proposition asserting the relation between determinate magnitudes only.
For this purpose Dr Simson frequently employs two statements of the general hypothesis, which he compares together. As for instance, in his analysis of the last porism, he assumes not only E, any point in the line DE, but also another point O, anywhere in the same line, to both of which he supposes lines to be inflected from the points A, B. This double statement, however, cannot be made without rendering the investigation long and complicated; nor is it even necessary, for it may be avoided by having recourse to simple porisms, or to loci, or to propositions of the data. The following porism is given as an example where this is done with some difficulty, but with considerable advantage both with regard to the simplicity and shortness of the demonstration. It will be proper to premise the following lemma. Let AB (fig. 7.) be a straight line, and D, L any two points in it, one of which D is between A and B; also let CL be any straight line. Then shall
For place CL perpendicular to AB, and through the points A, C, B describe a circle, and let CL meet the circle again in E, and join AE, BE. Also draw DG parallel to CE, meeting AE and BE in H and G, and draw EK parallel to AB. Then, from the elements of geometry,
therefore, (Geom. Sect. III. Theor. 8.)
From the three equations which we have deduced from
the first, second, and fifth of these propositions, it is manifest that
Again, because
And because
these two last propositions we have
but , and , and therefore . Now it has been proved, that
was to be demonstrated.