a Porentru, PORENTRU, is a town of Swifferland, in Elfgaw, restoration of the porisms of Euclid, which has all the Porism. a good castle, where he resides. It has in it, however, Pappus's description of them. ` All the lemmas which rems precisely as Pappus affirms those of Euclid to This definition is not a little obscure, but will be Unfortunately for mathematical science, however, this plainer if expressed thus : “ A porism is a proposition valuable collection is now loft, and it still remains a doubt. affirming the possibility of finding such conditions as ful question in what manner the ancients conducted their will render a certain problem indeterminate, or capable researches upon this curious fubject. We have, however, of innumerable solutions.” This definition agrees with reason to believe that their method was excellent both Pappus’s idea of these propositions, so far at least as in principle and extent, for their analyfis led them to they can be understood from the fragment already menmany profound discoveries, and was rettricted by the fe- tioned; for the propositions here defined, like those verest logic. The only account we have of this class which he describes, are, ftri&tly speaking, neither theoof geometrical propositions, is in a fragment of Pappus, rems nor problems, but of an intermediate nature bein which he attempts a general definition of them as a tween both; for they neither fimply enunciate a truth fet of mathematical propositions distinguishable in kind to be demonstrated, nor propose a question to be resol. from all others; but of this distinction nothing remains, ved, but are affirmations of a truth in which the deterexcept a criticism on a definition of them given by some mination of an unknown quantity is involved. In as geometers, and with which he finds fault, as defining far, therefore, as they assert that a certain problem may them only by an accidental circumstance, “ Pori/ma become indeterminate, they are of the nature of theoeft quod deficit hypothefi a theoremate locali.” rems; and, in as far as they seek to discover the condi. Pappus then proceeds to give an account of Euclid's tions by which that is brought about, they are of the porisms; but the enunciations are so extremely defec- nature of problems. tive, at the same time that they refer to a figure now We shall endeavour to make our readers understand lost, that Dr Halley confeffes the fragment in question this subject distinctly, by confidering them in the to be beyond his comprehension. way in which it is probable they occurred to the anThe high encomiums given by Pappus to these pro- cient geometers in the course of their researches: this positions have excited the curiosity of the greatest geo- will at the same time show the nature of the analysis pemeters of modern times, who have attempted to dif- culiar to them, and their great use in the solution of cover their nature and manner of investigation. M. problems. Fermat, a French mathematician of the last century, It appears to be certain, that it has been the solution of attaching himself to the definition which Pappus cri- problems which, in all states of the mathematical sciticises, published an introduction (for this is its modest enccs, has led to the discovery of geometrical truths : title) to this subject, which many others tried to eluci- the first mathematical inquiries, in particular, must have date in vain. At length Dr Simson of Glasgow, by occurred in the form of questions, where something was patient inquiry and some lucky thoughts, obtained given, and something required to be done; and by the reasoning a a Porism. reasoning necessary to answer thefe questions, or to dif- ing the given circle ABC in B, let H be its centre, join Porism. cover the relation between the things given and those HB, and let HD be perpendicular to DE. From D describe the circle BKF, meeting HD in the points K HB:= the rectangle K HK ; which rectangle +DK and DK=DL; and since DL is given in mag- Upon inquiry, it would be found that this proceed. is a forism, and may be thus enunciated : de ever in DE, tie straight line drawn from G to the lines shall have to one another the given ratio of to fi, Plate A circle ABC (fig. 1.), a straight line DE, and a which is supposed to be that of a greater to a less.CCCCXII point F, being given in positien, to find a point G in the Suppose the problem resolved, and that F is found, so Itraight line DE such, that GF, the line drawn from that FE has to FD the given ratio of 2 to ng produce DFB by FM: therefore EL.LD::EF: FD, that 3 D 2 ments لب a Porifm. ments EL, LD, is given, and the point L is also given; is equal to AOH, and therefore the angle FOB to Porifin. because DFB is bisected by FM, EM:MD:: EF:FD, HOG, that is, the arch FB to the arch HG. This upon LM as a diameter, and therefore given volved in one another in the porismatic or indefinite case; under their distances from the centre is equal to the therefore EOXOD=A0'. Hence, if the given points (fig. 3.) two straight lines EF, FD, are inflected to a Pla'e E and D (Ag. 3.) be fo ftuated, that EOXOD= third point F, fo as to be to one another in a given raCCCCXII AO', and at the fame time a:ß:: EA: AD::EC: tio, 'the point F is in the circumference of a given CD, the problem admits of numberless folutions; and circle, we have a locus. But when conversely it is from E and D the lines EF, DF be inflected to any therefore : a ; : a Porifnr. therefore Fermat's idea of porisms, founded upon this must pass through the point to be found M; for if not, Porism., circumstance, could not fail to be imperfect. it may be demonstrated just as above, that AE" does To confirm the truth of the preceding theory, it may not pass through H, contrary to the supposition. The be added, that professor Dr Stewart, in a paper read point to be found is therefore in the line E B, which is a confiderable time ago before the Philosophical Society given in pofition. Now if from E there be drawn EP of Edinburgh, defines a porism to be “ A proposition parallel to AE', and ES parallel to BE', BS:SE::BL affirming the poflibility of finding one or more condi SEXBL PEXAF tions of an indeterminate theorem;" where, by an in :LN and AP:PE:: AF: FG= determinate theorem, he meant one which expresses a re BS AP lation between certain quantities that are determinate therefore FG:LN:: PEXAF SEXBL :: PEXAF AP BS LN is compounded of the ratios of AF to BL, PE to DA: AP; therefore the ratio of FG to LN is comproblem fails, in confequence of the lines which by pounded of the ratios of AF to BL, AE to BE, and their intersection, or the points which by their polía DB to DA. In like manner, because E" is a point in tion, were to determine the problem required, happen the line DE and AE”, BE" are inflected to it, the ing to coincide with one another. À porism may ratio of FH to LM is compounded of the same ratios therefore be deduced from the problem to which it bé. of AF to BL, AE' to BE', and DB to DA; therelongs, just as propositions concerning the maxima and fore FH:LM::FG:NL (and consequently) :: HG minima of quantities are deduced from the problems of : MN; but the ratio of HG to MN is given, being the which they form limitations; and such is the most natu- same as that of a to B; the ratio of FH to LM is ral and obvious analysis of which this class of propofi- therefore also given, and FH being given, IM is given tions admits. in magnitude. Now LM is parallel to BE, a line The following poriím is the first of Euclid's, and the given in position; therefore M is in a line QM, parallel first also which was restored. It is given here to exem to AB, and given in position-; therefore the point M, plify the advantage which, in investigations of this kind, and also the line KLM, drawn through it parallel to may be derived from employing the law of continuity BE, are given in pofition, which were to be found. in its utmost extent, and pursuing porisms to those ex Hence this construction : From A draw. AE paralick treme cases where the indeterminate magnitudes increase to FK, so as to meet DE in E'; join BE', and take in ad infinitum. it BQ, so that a : 8::HF: BQ, and through draw This porism may be considered as having occurred in QM parallel to AB: Let HA be drawn, and produ the solution of the following problem: Two points A, B, ced till it meet DE in E", and draw BE", meeting QM Plate (fig. 4.) and also three straight lines DE, FK, KL, be in M; through M draw KML parallel to BE, then is CCCCXilling given in position, together with two points Hand M KML the line and M the point which were to be in two of these lines, to inflect from A and B to a point found. . There are two lines which will answer the con- analysis of porisms is, that it often happens, as in the 5 rilin, a a ALE L.BD"; but, the points A, B This double flatement, however, by the preceding lenina, WP.AD:+44BD ' R N Plate AB R R AB.DL'=(DG' R a a Po iím rism, he affumes not only E, any point in the line DE, AD'; and for the same reason DE'= Porifm. ' but also another point O, anywhere in the same line, N to both of which he supposes lines to be inflected from LB LB , , ' N N N . and complicated ; nor is it even heceffary, for it may be ·AL:+41-BL’+AB-DL"; that is, DE*+DF-2 ?'' = N N having to simpler to loci, or to propofitions of the data. The following porism Loʻ+LM'+ AB DL'. Join LG then by hypothes , is given as an example where this is done with some , difficulty, but with conliderable advantage both with fis LO'+LM', as to LG’, the same ratio as DF + regard to the simplicity and fhortness of the demonftra- DE- has to DG'; let it be that of R to N, then LO * tion. It will be proper to premise the following lemma. LM=LG“; and therefore DE+DF=LG+ RLG*; and therefore DE+DP•=LG+ N : , N N N let CL be any straight line. LB.AD'+LA.BD'= . LG! BA.DL'DG,and AB. N N :+ '; LB pofitionR:: : = ? LG, is given in magnitude. The point G is there- CL which is the same with that of AB to N. Now CL : LB :: LA: LE::(EK) LD: KH, and The construction easily follows from the analysis, but CL:LA:: LB:LE:: (EK) LD: KG; therefore, it may be rendered more simple ; for fince AH?: AB: (V. 24.) CL:AB :: (LD: GH ::) LD" : EKXGH: :AL:N, and BK*: AB?: : BL: N; therefore AH :: AB.LD"; therefore LB.LA'+ LA AB ' CE +BK?: AB2 : : AB:N. Likewise, if AG, BG, be CL CL CL: CL joined, AB:N:: AH' : AGʻ, and AB:N::BK2 : •LD'=ABXLE+EKXGH. Again, CL:LA:: BG?; wherefore AB:N:: AK2+ BK?: AG+BG{: : Ꭰ: : (LB : LE :: DB : DG : :) DB : DBXDG=LA and AGʻ+BG=AB2; therefore the angle AGB is a : CL right one, and AL:LG::LG: LB. If therefore AB •DD', and CL : LB :: (LA: LE::DA : DH::) be divided in L, so that AL:LB :: AH’: BK2; and DA”: DAXDH=LEDA; therefore CIDA'+ placed perpendicular to AB, G will be the point reLB LB , , CL quired. 1 he step in the analysis, by which a second intro duction of the general hypothesis is avoided, is that in LB LA LB DA? + ' which follows from DG-GL, having a given ratio to LD’, at the same timethat LD is of no determinate magCL CL nitude. For, if poffible, let GLD be obtufe (fig. 6.), Let there be three straight lines AB, AC, CB and let the perpendicular from G to AB meet it in V, given in position (fig. 5.); and from any point what. therefore V is given : and since GD-LG-=LDP+ ever in one of them, as Ó, let perpendiculars be drawn 2DLXLV; therefore, by the supposition, LD + 2DL to the other two, as DF, DE, a point G may be found, XLV must have a given ratio to LDP, therefore the fuch, that if GD be drawn from it to the point D, the ratio of LD- to DLXVL, that is, of LD to VL, is square of that line shall have a given ratio to the sum of given, so that VL being given in magnitude, LD is althe squares of the perpendiculars DF and DE, which fo given. But this is contrary to the supposition ; for ratio is to be found. LD is indefinite by hypothesis, and therefore GLD Draw AH, BK perpendicular to BC and AC; and cannot be obtuse, nor any other than a right angle. in AB take L, so that AL:LB :: AH’: BK’:: The conclusion here drawn immediately from the indeAC':CB'. The point L is therefore given ; and if termination of LD would be deduced, according to N be taken, so as to have to AL the same ratio that Dr Simfon's method, by assuming another point D' AB: has to AH, N will be given in magnitude. Al any how, and from the supposition that GD-GL’: fo, fince AH': BK'::AL:LB, and XH': AB':: LD::GD-GL’: LD, it would easily appear that ?AH LB problems from which they are derived. For example, let poli : : : LA LB' + : |