^{page 156 note *} Collectiones Math. lib. vii. in init.

^{page 157 note *} De sectione rationis, proem, p. 37.

^{page 157 note †} “Porismatum Euclidæorum renovata doctrina, et sub sorma Isagoges exhibita.” Fermat Opera Varia, p. 116.

^{page 158 note *} Rob. Simson Op. reliqua, p. 319.

^{page 163 note *} This solution of the problem was suggested to me by Professor Robison ; and is more simple than that which I had originally given.

^{page 166 note *} The ratio of α to β is supposed that of a greater to a less.

^{page 169 note *} Simson De Porismatibus, Prop. 53.

^{page 170 note *} The given points, and the centre of the given circle, are understood, throughout, to be in the same straight line.

^{page 172 note *} Simson's Opera Reliqua, p. 323.

^{page 172 note †} The following translation will perhaps be found to remedy some of the obscurity complained of.

“A Porism is a proposition, in which it is proposed to demonstrate, that one or more thing 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.”

It may be proper to remark, that there is an ambiguity in the word given, as used here and on many other occasions, where it denotes indifferently things that are both determinate and known, and things that, though determinat, are unknown, provided they can be found. This holds as to the first application of the term in the above definition; from which however no inconveniency arises, when the reader is apprised of it. In the course of this paper, I have endeavoured, as much as possible, to avoid the like ambiguity.

^{page 174 note *} 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 Ποςισμα, 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.

Imo, One of the significations of ποριζω, is to acquire or obtain; and hence Ποςισμα, the thing obtained or gained. Accordingly, Scapula says, Est vox a geometris desumpta qui theorema aliquid ex demonstrativo syllogismo necessario sequens inferentes, illud qunsi lucrari dicuntur, quod non ex professo quidem theorematis hujus 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 propositions, Porismata. This circumstance creates a presumption, 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 Ποςισμα might be very properly applied, since, in the words of Scapula, already quoted, Non ex professo theorematis hujus instituta sit demonstratio, sed tamen ex demonstratis recte sequatur.

2do, But though this interpretation agrees so well with the supposed origin of Porisms, it is not free from difficulty. The verb ωοςιζω 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, afford great delight, τοις δυναμενοις οςαν ϰΧι ποςιζειν, to those who are able to understand and investigate. Hence comes ποςισμος, the act of finding out, or discovering, and from πορισμος, in this sense, the same author evidently considers Ποςισμα as being derived. His words are, Εφασαν δε (οι αςχαιοι) Πορισμα ειναι το ῶςοτεινομενον εις Πορισμον αυτ8 τ8 ῶςοτεινομεν8, the ancients said, that a Porism is something proposed for the finding 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 ænigmatical form of their enunciation, 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 there, fore have been called Porismata or Investigations, by way of eminence.

^{page 180 note *} Opera Reliqua, de Porismatibus, prop. 25.

^{page 185 note *} This Porism, in the case considered above, vix. when there are three straight lines given in position, was communicated to me several years ago, without any analysis or demonstration, by Dr Trail, Prebendary of Lisburn in Ireland, who told me also, that he had met with it among some of Dr Simson's papers, which had been put into his hands, at the time when the posthumous works of that geometer were preparing for the press. The application of it to the second of the problems, (§ 27.) was also suggested by Dr Trail.

^{page 187 note *} Trans. R. S. Edin. vol. ii. p. 112, &c.

^{page 192 note *} Elements, p. 243. Edit. 3. Simpson's solution is remarkably elegant, but no mention is made in it, of the indeterminate case.

^{page 192 note †} Jos. Boscovich Opera, Bassani. tom. 3. p. 331.

^{page 193 note *} Demonstration.—Through C and E draw CH and EG, both parallel to AB, and let them meet BG, parallel to AE, in H and in G. Let GF and HD be joined; and because AC is to CE, that is, BH to HG as BD to DF, by hypothesis, DH is parallel to GF, and has also a given ratio to it, viz. the ratio of GB to BH, or of EA to AC. Take GK equal to HD, and join EK, and the triangle EGK will be equal to the triangle CHD, and therefore the angle KEG is given, and likewise the angle KEF; and since the ratio of GK to KF is given, if from K there be drawn KL parallel to EG, meeting EF in L, the ratio of EL to LF will be given. But the ratio of EL to LK is given, because the triangle ELK is given in species; .therefore the ratio of FL to LK is given ; and the angle FLK being also given, the triangle FKL is given in species, as also the triangle FGE. The angle FGE being therefore given, the triangle KGE is given in species, and EG has therefore given ratios to EK and EF. But EG is equal to AB, and EK to CD, therefore AB, CD and EF have given ratios to one another Q.E.D.

Hence to find the ratios of AB, CD and EF ; in EF take any part EL, and make as AC is to CE, so EL to LF ; through L draw LK parallel to EG or AB, meeting EK, drawn through E parallel to CD in K ; then if FK be drawn meeting EG in G, the ratios required are the same with the ratios of the lines EG, EK, EF. This is evident from the preceding investigation.

If it be required to find the position of the line AE, drawn through the point A, so as to be cut by CD and EF in a given ratio ; draw Ac, any how, cutting DC in c, and produce Ac to e, so that Ac may be to c e in the ratio which AC is to have to CE ; let eE be drawn parallel to DC, intersecting FE in E, and if AE be joined, it is the line required.

Hence the converse of the lemma is easily demonstrated, vix. that if AE and BF be two lines that are cut proportionally by the three lines AB, CD, EF ; and if AB and EF, the parts of any two of these last, intercepted between AE and BF, be also cut proportionally, any how, in b and f, and if bf be joined, meeting the third line in d, bf will be cut in the same proportion with AE or BF. For if not, let bf′ be drawn from b, meeting CD in d′, and EF in f′, so that bd′:d′f′::AC:CE ; then by the lemma, ab:AB::Ef:EF ; and by supposition, ab:AB::Ef:EF, therefore Ef′ = Ef, which is impossible. Therefore, &c.

^{page 197 note *} Prin. Math. lib. I. lem. 27.

^{page 198 note *} Demonstration.—Complete the parallelogram under AC and AD, viz. AG, and on DG describe the triangle DGH, similar and equal to the triangle ABC. Join FG, BH and HE. Through G also, draw GK, equal and parallel to HE, and join CK; CK will be equal and parallel to BE, and the triangle CGK equal to the triangle BHE. The angle GCK is therefore given, being equal to the given angle HBE; and the angle GCF being given, the angle FCK is also given.

The triangles DHE, DGF are fimilar ; for the angles FDE, GDH being equal, the angles FDG, EDH are likewise equal; and also, by supposition, FD being to DE as GD to DH, FD is to DG as DE to DH. The angle FGD is therefore equal to the angle EHD, and FG is also to EH, or to KG, as FD to DE, or as GD to DH.

But if GL be drawn parallel to HD, the angle KGL will be equal to the angle EHD, that is, to the angle FGD, and therefore the angle KGF to the angle LGD or GDH; and it has been shewn, that FG is to GK as GD to DH; therefore the triangle FGK is similar to the triangle GDH, and is given in species.

Draw GM perpendicular to CF, and GN making the angle MGN equal to the angle FGK or GDH, and let GM be to GN in the given ratio of FG to GK, or of GD to DH. Join CN and NK. Then, because MG:GN::FG:GK, MG:FG::GN:GK ; and the angle MGF being equal NGK, the triangles MGF, NGK are similar, and therefore GNK is a right angle. But since the ratio of MG to GN is given, and also of MG to GC, the triangle CGM being given in species, the ratio of GC to GN is given, and CGN being also a given angle, because each of the angles CGM, MGN is given, the triangle CGN is given in species, and consequently the ratio of CG to CN is given. The angle NCK is therefore given ; and the angle CNK is likewise given, each of the angles CNG, GNK being given, therefore the triangle CNK is also given in species. The ratio of CN to CK is therefore given, and since the ratio of CN to CG is also given, the ratio of CG to CK is given, and the triangle CGK given in species. The angle KGC is therefore given, and the angle KGF being also given, the angle CGF is given, and consequently the ratio of CG to CF. The ratios of the lines CG, CK and CF to one another, that is, of AD, BE and CF to one another, are therefore given. Q. E. D.

Cor. Hence also it appears, how a triangle given in species may be described, having its angles on three straight lines given in position, and one of the angles at a given point in one of the lines. The solution of this problem is therefore taken for granted, in the analysis of the Porism, though, for the sake of brevity, the construction is omitted.

^{page 201 note *} Prin. Math. lib. I. prop. 29.