in Geometry, is a name given by the ancient geometers to two classes of mathematical propositions. Euclid applies this name to propositions, which are involved in others which he is professedly investigating, and which, although they do not form his principal object, are obtained along with it, as is expressed by their name porismata, or acquisitions. Such propositions are now 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 amongst many others, on account of their great subserviency to the business of geometrical investigation in general. These propositions were so named by him, either from the way in which he discovered them, that is to say, whilst he was investigating something else, so that they might be considered as gains or acquisitions, or from their utility as steps in the investigation. In this sense they are porismata; for ἐπισκοπά (from which, according to Proclus, the term is derived) signifies both to investigate and to acquire by investigation. These propositions formed a collection in three books, 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 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 both comprehensive and excellent; 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 the term given by some geometers, namely, "A Porism is that which is deficient in hypothesis from a local theorem," and which he finds fault with, as defining the porisms only by an accidental circumstance. Pappus also gives 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 pronounced by Pappus on these propositions have excited the curiosity of the greatest geometers of modern times, who have attempted to discover their nature and the manner of investigating them. Fermat gave a few propositions, which have been published in his Opera (Tolosa, 1679), and Bullialdus, in a tract entitled Exercitationes Geometricae Tres (Paris, 1657), attempted the same thing, but with less success. Albert Girard, at a still earlier period, announced that he had restored the whole of the three books of Euclid, but it does not appear that this part of his works was ever published.
At length Dr. Simson, Professor of Mathematics in the University of Glasgow, was so fortunate as to succeed in restoring the porisms of Euclid. In the preface to his treatise *De Porismatibus*, he gives the following account of his progress and of the obstacles he encountered: "Postquam vero apud Pappum legeram Porismata Euclidis Collectionem fuisse artificiosissimam multarum rerum, quae spectant ad analysin difficiliorum et generalium problematum, magno desiderio tenebar, aliquid de eis cognoscendi; quare secundum et multis variisque viis tum Pappi propositionem generalern, mancam et imperfectum, tum primum lib. I. Porisma, quod ut dictum fuit, solum ex omnibus in tribus libris integram adhuc manet, intelligere et restituere conoscar: frustra tamen, nihil enim proficiebam. Cumque cogitationes de hac re multum mihi temporis consumpserint, atque tandem molestae admodum evaserint, firmiter animum induxi hæc numquam in posterum investigare; presertim cum optimus Geometra Halleius spem omnem de eis intelligendis abjectisset. Unde quoties menti occurrirent, toties eas arcebam. Postea tamen accedit ut improvidum et propositi immemorem invaserint, meoque deterrinerunt donec tandem lux quedam effluerit quaem specem mihi faciebat inventiendi saltem Pappi propositionem, generalern, quam quidem multa investigatione tandem restitui."
Dr. Simson's Restoration has every appearance of being just. 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.
Whilst Dr. Simson was employed in this inquiry, he carried on a correspondence upon the subject with the late Dr. Matthew 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 demonstration; and those who wish 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 to be shewn to take place."
This definition is somewhat 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, as 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. 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 condition by which that is brought about, they are of the nature of problems.
In order to give our readers a clear idea of the subject of porisms, we shall consider 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, insomuch that they would scarcely suffer any of the collateral truths to escape their observation.
Now, as this cautious manner of proceeding was not better calculated to avoid error than to lay 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 existing amongst 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 HB² = HK × HK'; but because KK' is bisected in D, KH × HK' + DK² = DH², therefore HK² + DK² = DH². But HK² + LD² = DH², therefore DK² = DL² and DK = DL. But DL 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 K'FK; 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 α to β, 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 FD the given ratio of α to β; produce EF towards G, bisect the angle EFD by FL, and DEG by FM: therefore EL : LD :: ER : FD, 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 DEG is bisected by FM, EM : MD :: EF : FD, that is, in a given ratio, and therefore M is given. Since DFL is half of DFE, and DFM half of DFG, 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 α to β, and produce ED also to M, so that EM may be to MD in the same given ratio of α to β; 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 instance, 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, EA : AD :: α : β :: EC : CD, and EA : EC :: AD : CD, or by conversion, EA : AC :: AD : CD — AD :: AD : 2DO, O being the centre of the circle ABC; therefore, also, EA : AO :: AD : DO, and by composition, EO : AO :: AO : DO, therefore EO × OD = AO². Hence, if the given points E and D (fig. 3), be so situated that EO × OD = AO², and at the same time α : β :: EA : AD :: EC : CD, 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, 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 lines will cut off equal circumferences BF, HG. Let FC 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 FOB 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 in 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 in one another in the porismatic or indefinite case; for here 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 so 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, E, D, two straight lines EF, 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 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 EO × OD = AO²; 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, PD 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.
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.
Another general remark which may be made on the analysis of porisms is, that it often happens 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 accordingly Dr. Simson frequently employs two statements of the general hypothesis, which he compares together. 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 simpler porisms, or to loci, or to propositions of the data. The porism which follows, 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. 4.) 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
\[ \frac{LB}{CL} \cdot AD^2 + \frac{LA}{CL} \cdot BD^2 = LB \cdot AL^2 + \frac{LA}{CL} \cdot BL^2 + \frac{AB}{CL} \cdot DL^2. \]
For place CL perpendicular to AB, and through the points Porism. 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,
\[ \text{CL} : \text{LB} = (\text{LA} : \text{LE}) \cdot \text{LA}^2 : \text{LA} \times \text{LE}, \]
and hence \( \text{LA} \times \text{LE} = \frac{\text{LB}}{\text{CL}} \cdot \text{LA}^2. \)
Also \( \text{CL} : \text{LA} = (\text{LB} : \text{LE}) \cdot \text{LB}^2 : \text{LB} \times \text{LE}, \)
and hence \( \text{LB} \times \text{LE} = \frac{\text{LA}}{\text{CL}} \cdot \text{LB}^2. \)
Now \( \text{CL} : \text{LB} = \text{LA} : \text{LE} : \text{EK or LD : KH}, \)
and \( \text{CL} : \text{LA} = \text{LA} : \text{LE} : \text{EK or LD : KG}, \)
therefore \( \text{CL} : \text{AB} = (\text{LD} : \text{GH}) \cdot \text{LD}^2 : \text{EK} \times \text{GH}, \)
and hence \( \text{EK} \times \text{GH} = \frac{\text{AB}}{\text{CL}} \cdot \text{LD}^2. \)
From the three equations now deduced, there results
\[ \text{LB} \cdot \text{LA}^2 + \text{LA} \cdot \text{LB}^2 + \text{AB} \cdot \text{CL} \cdot \text{LD}^2 = \text{AB} \times \text{LE} + \text{EK} \times \text{GH}. \]
Again, because
\[ \text{CL} : \text{LA} = (\text{LB} : \text{LE} : \text{DB : DG}) \cdot \text{DB}^2 : \text{DB} \times \text{DG}, \]
therefore \( \text{DB} \times \text{DG} = \frac{\text{LA}}{\text{CL}} \cdot \text{DB}^2. \) And because
\[ \text{CL} : \text{LB} = (\text{LA} : \text{LE} : \text{DA : DH}) \cdot \text{DA}^2 : \text{DA} \times \text{DH}, \]
therefore \( \text{DA} \times \text{DH} = \frac{\text{LB}}{\text{CL}} \cdot \text{DA}^2. \) From the result of these two last propositions we have
\[ \frac{\text{LB}}{\text{CL}} \cdot \text{DA}^2 + \frac{\text{LA}}{\text{CL}} \cdot \text{DB}^2 = \text{DA} \times \text{DH} + \text{DB} \times \text{DG}; \]
But \( \text{DA} \times \text{DH} = \text{twice trian. ADH}, \) and \( \text{DB} \times \text{DG} = \text{twice trian. BDG}, \) and therefore \( \text{DA} \times \text{DH} + \text{DB} \times \text{DG} = 2(\text{trian. ADH} + \text{trian. BDG}) = 2(\text{trian. AEB} + \text{trian. HEG}) = \text{AB} \times \text{LE} + \text{EK} \times \text{HG}. \) Now it has been proved,
that \( \text{DA} \times \text{DH} + \text{DB} \times \text{DG} = \frac{\text{LB}}{\text{CL}} \cdot \text{DA}^2 + \frac{\text{LA}}{\text{CL}} \cdot \text{DB}^2; \) and
that \( \text{AB} \times \text{LE} + \text{EK} \times \text{HG} = \frac{\text{LB}}{\text{CL}} \cdot \text{LA}^2 + \frac{\text{LA}}{\text{CL}} \cdot \text{LB}^2 + \frac{\text{AB}}{\text{CL}} \cdot \text{LD}^2; \) therefore \( \frac{\text{LB}}{\text{CL}} \cdot \text{AD}^2 + \frac{\text{LA}}{\text{CL}} \cdot \text{BD}^2 = \frac{\text{LB}}{\text{CL}} \cdot \text{AL}^2 + \frac{\text{LA}}{\text{CL}} \cdot \text{BL}^2 + \frac{\text{AB}}{\text{CL}} \cdot \text{DL}^2, \) as was to be demonstrated.
Porism. Let there be three straight lines AB, AC, CB given in position; (fig. 5) and from any point whatever in one of them, as D, let perpendiculars be drawn to the other two, as DF, DE, a point G may be found, such, that if GD be drawn from it to the point D, the square of that line shall have a given ratio to the sum of the squares of the perpendiculars DF and DE, which ratio is to be found.
Draw AH, BK perpendicular to BC and AC; and in AB take L, so that \( \text{AL} : \text{LB} = \text{AH}^2 : \text{BK}^2 = \text{AC}^2 : \text{CB}^2. \) The point L is therefore given; and if a line N be taken, so as to have to AL the same ratio that \( \text{AB}^2 \) has to \( \text{AH}^2, \) N will be given in magnitude. Also, since \( \text{AH}^2 : \text{BK}^2 = \text{AL} : \text{LB}, \) and \( \text{AB}^2 : \text{AL} : \text{N, ex equo, } \text{BK}^2 : \text{AB}^2 = \text{LB} : \text{N}. \) Draw LO, LM perpendicular to AC, CB; LO, LM are therefore given in magnitude. Now, because \( \text{AB}^2 : \text{BK}^2 = \text{AD}^2 : \text{DF}^2, \text{N} : \text{LB} = \text{AD}^2 : \text{DF}^2, \) and \( \text{DF}^2 = \frac{\text{LB}}{\text{N}} \cdot \text{AD}^2. \) For the same reason \( \text{DE}^2 = \frac{\text{AL}}{\text{N}} \cdot \text{BD}^2. \)
\[ \text{LO}^2 = \frac{\text{LB}}{\text{N}} \cdot \text{AL}^2, \text{and } \text{LM}^2 = \frac{\text{AL}}{\text{N}} \cdot \text{BL}^2. \]
But, by the preceding lemma, \( \frac{\text{LB}}{\text{N}} \cdot \text{AD}^2 + \frac{\text{AL}}{\text{N}} \cdot \text{BD}^2 = \frac{\text{LB}}{\text{N}} \cdot \text{AL}^2 + \frac{\text{AL}}{\text{N}} \cdot \text{BL}^2 + \frac{\text{AB}}{\text{N}} \cdot \text{DL}^2; \) that is, \( \text{DE}^2 + \text{DF}^2 = \text{LO}^2 + \text{LM}^2 + \frac{\text{AB}}{\text{N}} \cdot \text{DL}^2. \) Join LG, then by hypothesis \( \text{LO}^2 + \text{LM}^2 \) has to \( \text{LG}^2, \) the same ratio as \( \text{DF}^2 + \text{DE}^2 \) has to \( \text{DG}^2; \) let it be that of R to N, then \( \text{LO}^2 + \text{LM}^2 = \frac{\text{R}}{\text{N}} \cdot \text{LG}^2; \) and therefore \( \text{DE}^2 + \text{DF}^2 = \frac{\text{R}}{\text{N}} \cdot \text{LG}^2 + \frac{\text{AB}}{\text{N}} \cdot \text{DL}^2; \) but \( \text{DE}^2 + \text{DF}^2 = \frac{\text{R}}{\text{N}} \cdot \text{DG}^2; \) therefore, \( \frac{\text{R}}{\text{N}} \cdot \text{LG}^2 + \frac{\text{BA}}{\text{N}} \cdot \text{DL}^2 = \frac{\text{R}}{\text{N}} \cdot \text{DG}^2, \) and \( \frac{\text{AB}}{\text{N}} \cdot \text{DL}^2 = \frac{\text{R}}{\text{N}} \cdot (\text{DG}^2 - \text{LG}^2); \)
therefore \( \text{DG}^2 - \text{LG}^2 \) has to \( \text{DL}^2 \) a constant ratio, viz. that of AB to R. The angle DLG is therefore a right angle, and the ratio of AB to R that of equality, otherwise LD would be given in magnitude, contrary to the supposition. LG is therefore given in position; and since \( \text{R} : \text{N} :: \text{AB} : \text{N} :: \text{LO}^2 + \text{LM}^2 : \text{LG}^2; \) therefore the square of LG, and consequently LG, is given in magnitude. The point G is therefore given, and also the ratio of \( \text{DE}^2 + \text{DF}^2 \) to \( \text{DG}^2, \) which is the same with that of AB to N.
The construction easily follows from the analysis, but it may be rendered more simple; for since \( \text{AH}^2 : \text{AB}^2 :: \text{AL} : \text{N}, \) and \( \text{BK}^2 : \text{AB}^2 :: \text{BL} : \text{N}; \) therefore \( \text{AH}^2 + \text{BK}^2 : \text{AB}^2 :: \text{AL} : \text{N}. \) Likewise, if AG, BG, be joined, \( \text{AB} : \text{N} :: \text{AH}^2 : \text{AG}^2, \) and \( \text{AB} : \text{N} :: \text{BK}^2 : \text{BG}^2; \) therefore, \( \text{AB} : \text{N} :: \text{AH}^2 + \text{BK}^2 : \text{AG}^2 + \text{BG}^2, \) but it was proved that \( \text{AB} : \text{N} :: \text{AH}^2 + \text{BK}^2 : \text{AB}^2; \) therefore \( \text{AG}^2 + \text{BG}^2 = \text{AB}^2; \) therefore the angle AGB is a right angle, and \( \text{AL} : \text{LG} : \text{LG} : \text{LB}. \) If therefore AB be divided in L, so that \( \text{AL} : \text{LB} :: \text{AH}^2 : \text{BK}^2; \) and if LG, a mean proportional between AL and LB, be placed perpendicular to AB, G will be the point required.
The step in the analysis, by which a second introduction of the general hypothesis is avoided, is that in which the angle GLD is concluded to be a right angle; which follows from \( \text{DG}^2 - \text{GL}^2 \) having a given ratio to \( \text{LD}^2, \) at the same time that LD is of no determinate magnitude. For, if possible, let GLD be obtuse, (fig. 6,) and let the perpendicular from G to AB meet in V, therefore V is given; and since \( \text{GD}^2 - \text{GL}^2 = \text{LD}^2 \times 2 \text{DL} \times \text{LV}; \) therefore, by the supposition, \( \text{LD}^2 + 2 \text{DL} \times \text{LV} \) must have a given ratio to \( \text{LD}^2; \) therefore the ratio of \( \text{LD}^2 \) to \( \text{DL} \times \text{VL}, \) that is, of LD to VL, is given, so that VL being given in magnitude, LD is also given. But this is contrary to the supposition; for LD is indefinite by hypothesis, and therefore GLD cannot be obtuse. nor any other than a right angle. The conclusion that is here drawn immediately from the indetermination of LD would be deduced, according to Dr. Simson's method, by assuming another point D' anyhow, and from the supposition that $GD^2 = GL^2 : LD^2 : GD^2 - GL^2 : LD^2$, it would easily appear that GLD must be a right angle, and the ratio that of equality.
These porisms facilitate the solution of the general problems from which they are derived. For example, let three straight lines AB, AC, BC, (fig. 5), be given in position, and also a point R, to find a point D in one of the given lines, so that DE and DF being drawn perpendicular to BC, AC, and DR, joined; $DE^2 + DF^2$ may have to DR$^2$ a given ratio. It is plain, that having found G, the problem would be nothing more than to find D, such that the ratio of $GD^2$ to DR$^2$, and therefore that of GD to DR, might be given, from which it would follow, that the point D is in the circumference of a given circle, as is well known to geometers.
The same porism also assists in the solution of another problem. For if it were required to find D such that $DE^2 + DF^2$ might be a given space; having found G, $DG^2$ would have to $DE^2 + DF^2$ a given ratio, and DG would therefore be given; whence the solution is obvious.
The connection of this porism with the impossible case of the problem is evident; the point L being that from which, if perpendiculars be drawn to AC and CB, the sum of their squares is the least possible. For since $DF^2 + DE^2 : DG^2 :: LO^2 + LM^2 : LG^2$; and since LG is less than DG, $LO^2 + LM^2$ must be less than $DE^2 + DF^2$.
It is evident from what has now appeared, that in some instances at least there is a close connection between these propositions and the maxima or minima, and of consequence the impossible cases of problems. The nature of this connection requires to be farther investigated, and is the more interesting because the transition from the indefinite to the impossible case seems to be made with wonderful rapidity. Thus in the first proposition, though there be not properly speaking an impossible case, but only one where the point to be found goes off in infinitum, it may be remarked, that if the given point F (fig. 1), be anywhere out of the line HD, the problem of drawing GB equal to GF is always possible, and admits of just one solution; but if F be in DH, the problem admits of no solution at all; the point being then at an infinite distance, and therefore impossible to be assigned. There is, however, this exception, that if the given point be at K in this same line, DH is determined by making DK equal to DL. Then every point in the line DE gives a solution, and may be taken for the point G. Here therefore the case of numberless solutions, and of no solution at all, are as it were conterminal, and so close to one another, that if the given point be at K the problem is indefinite; but if it remove ever so little from K, remaining at the same time in the line DH, the problem cannot be resolved. This affinity might have been determined a priori; for, as we have seen, it is 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 whilst they remain fixed or given, should that third one 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, whilst 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 in some cases be so simple as to arise from the mere coincidence of one condition with another, though in no case whatever can any inconsistency take place between them. There are, however, comparatively few porisms so simple in their origin, or that arise from problems where the conditions are but little complicated; for it usually happens that a problem which can become indefinite may also become impossible; and if so, the connection already explained never fails to take place.
Another species of impossibility may frequently arise from the porismatic case of a problem which will affect in some measure the application of geometry to astronomy, or any of the sciences depending upon experiment or observation. For when a problem is to be resolved by means 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 with which we are provided, yet these data may be so related to one another in the case under consideration, that the problem will become indeterminate, and instead of one solution will admit of an indefinite number. This we have already found to be the case in the foregoing propositions. Such cases may not indeed occur in any of the practical applications of geometry; but there is one of the same kind which has actually occurred in astronomy. Sir Isaac Newton, in his Principia, has considered a small part of the orbit of a comet as a straight line described with an uniform motion. From this hypothesis, by means of four observations made at proper intervals of time, the determination of the path of the comet is reduced to this geometrical problem. Four straight lines being in position, it is required to draw a fifth line across them, so as to be cut by them into three parts, having given ratios to one another. Now this problem had been constructed by Dr. Wallis and Sir Christopher Wren, and also in three different ways by Sir Isaac himself in different parts of his works; yet none of these geometers observed that there was a particular situation of the lines in which the problem admitted of innumerable solutions; and this happens to be the very case in which the problem is applicable to the determination of the comet's path, as was first discovered by Boscovich, who was led to it by finding, that in this way he could never determine the path of a comet with any degree of certainty.
The preceding account of this interesting branch of the ancient geometry, is taken from a very elegant and elaborate paper, On the Origin and Investigation of Porisms, by the late Professor Playfair, published in the third volume of the Transactions of the Royal Society of Edinburgh, and also in his collected Works, (vol. iii. p. 178.) For further particulars on the subject, the reader may consult the original paper of Professor Playfair; Dr. Simson's treatise, De Porismatibus, contained in his Opera Reliqua; a paper in the fourth volume of the Edinburgh Transactions, by Professor Wallace, entitled, Some Geometrical Porisms, with examples of their application to the Solution of Problems; Dr. Traill's Account of the Life and Writings of Robert Simson, M.D.; and Professor Leslie's Geometrical Analysis. At the end of Dr. Stewart's General Theorems, above mentioned, five very remarkable porisms are enunciated, the demonstration of the first of which is given in Leybourn's Mathematical Repository, (vol.i.) and of the remaining four in the fifth volume of the same work published in 1830. The algebraic analysis may frequently be applied with great advantage in the investigation of persons, but this manner of treating the subject does not come within the scope of the present article.