Colour and Terture. Deep yellow-scintillant. Gray yellow-scintillant. Deep yellow-scintillant. Black--scintillant. Porcelain. Mixture. Result. . Glass penetrating the crucible 4. A 1 Glass nly - A melted porous mass 3. The same. 4. S Remained in powder. I. 3. Glass 4. A melted porous mass, not polished in the Ś fracture 1. The same. 4:} } } 2:}Partially fused. 4:}Remained in powder. 2:}a poreds melted mass, part of the oxide re- Deep yellow—scintillant. Gray-scintillant. 1 . Alumine copper A porous half-fused mass Gray--scintillant. Gray-scintillant. TABLE III. Shewing the Action of the Vitrifying matters on the Crucibles that contain them. Substances used. Result in the Claycrucible Result in the Chalk crucible Result in the Charcoal (A). (B). crucible (C). beginning fusion where in ble. No change. No cbange. Fluor Marble. # Porcelain 11 ble (B). Porcupine. Fluor spar. Man. Result in the Clay crucible Result in the Chalk crucia Result in the Charcoal cru- Porcelain Substances used. (A.) cible (C). II Melted and ran through Melted down with the cru. Scarcely altered, except Porism, the crucible. cible to a tough slag. slight fusion at the edges. Compact, wbite and no signs Run into a hard blue clear As in A. glass. A compact mass partially A perfectly black glass. As in A. melted, A black glass covered with A semitransparent apple- A brown scoria containing grains of iron. No fusion, but the colour Completelyfused in the parts As in A. touching the crucible. As in A. as not to fall to powder on exposure to air. Agraysemitransparentglass As in A. green scoria, also with a A green glass with many grains of iron. vations, his skin appeared white and smooth, yet the skin became as it was before. His health, during his PORCH, in Architecture, a kind of vestibule sup whole life has been remarkably good ; but there is one ported by columns ; much used at the entrance of the particular of his case more extraordinary than all the ancient temples, balls, churches, &c. rest; this man has bad six children, and all of them A porch, in the ancient architecture, was a vestibule, had the same rugged covering as bimself, which came or a disposition of insulated colamps usually crowned on like his own about nine weeks after the birth. Of with a pediment, forming a covert place before the these children only one is now living, a pretty boy, principal door of a temple or court of justice. Such is who was shown with bis father. It appears therefore, that before the door of St Paul's, Covent-Garden, the as Mr Baker remarks, that a race of people might be work of Inigo Jones. When a porch bad four columns propagated by this man, as different from other men as in front, it was called a tetrastyle ; when six, hexastyle ; an African is from an Englishman ; and that if this when eight, octostyle, &c. should bave happened in any former age, and the acciPorch, in Greek close, a public portico in Athens, dental original have been forgotten, there would be the adorned with the pictures of Polygnotus and other emi same objections against their being derived from the nent painters. It was in this portico that Zeno the phi same common stock with others : it must therefore be losopher taught; and hence his followers were called admitted possible, that the differences now subsisting Stoics. See Stoics and ZENO. between one part of mankind and another nay have PORCUPINE-Man, the name by which one Edward ter the earth had been peopled by one common proge- all rising to an equal beight, which is about an on the river Halle, near Mount Jura, 22 miles south of called 입 Porism. called corollaries. But he gives the same name, by way nihil enim proficiebam. Cumqne cogitationes de hac re fue of eminence, to a particular class of propositions which multum mibi temporis consumpserint, atque tandem mohe collected in the course of his researches, and selected lestæ admodum evaserint, firmiter animum induxi dunfrom among many others on account of their great sub quam in posterum investigare ; præsertim cum optimus serviency to the business of geometrical investigation in Geometra Halleius spem omnem de iis intelligendis abgeneral.' These propositions were so named by him, jecisset. Unde quoties menti occurrebant, toties eas either from the way in which be discovered them, while arcebam. Postea tamen accidit ut improvidum et prohe was investigating something else, by which means positi immemorem invaserint, meque detinuerint donec they might be considered as gains or acquisitions, or tandem lux quædam effulserit quae spem mibi faciebat from their utility in acquiring farther knowledge as inveniendi saltem Pappi propositionem generalem,quam steps in the investigation. In this sense they are poris- quidem multa investigatione tandem restitui. Hæc aumata ; for Top 3 signifies both to investigate and to ac tem paulo post una cum Porismate primo lib. 1. in pressa quire by investigation. These propositions formed a est inter Transactiones Philosophicas anni 1723, No177." collection, which was familiarly known to the ancient Dr Simson's Restoration has all the appearance of geometers by the name of Euclid's porisons; and Pap- being just ; it precisely corresponds to Pappus's depus of Alexandria says, that it was a must ingenious col- scription of them. All the lemmas which Pappus has lection of many things conducive to the analysis or so given for the better understanding of Euclid's proposilution of the most difficult problems, and which afforded tions are equally applicable to those of Dr Simson, great delight to those who were able to understand and which are found to differ from local theorems precisely to investigate them. as Pappus affirms tbn-e of Euclid to have done. They Unfortunately for mathematical science, this valua- require a particular mode of analysis, and are of imble collection is now lost, and it still remains a doubtful mense service in geometrical investigation ; on which question in what manner the ancients conducted their account they may justly claim our attention. researches upon this curious subject. We have, how While Dr Simson was employed in this inquiry, he ever, reason to believe that their method was excellent carried on a correspondence upon the subject with the both in principle and extent; for their analysis led them late Dr M. Stewart, professor of mathematics in the to many profound discoveries, and was restricted by the university of Edinburyl ; who, besides entering into De severest logic. The only account we have of this class Simson's views, and communicating to him many curiof geometrical propositions, is in a fragment of Pappus, ous porisms, pursued the same subject in a new and very in which he attempts a general description of them as a different direction. He published the result of his inset of mathematical propositions, distinguishable in kind quiries in 1746, under the title of General Theorems, from all others; but of this description nothing remains, not wishing to give them any other name, lest he might except a criticism on a definition of them given by some appear to anticipate the labours of his friend and forgeometers, and with which he finds fault, as defining mer preceptor. The greatest part of the propositions them only by an accidental circunstance, “ A Porism contained in that work are porisms, but without deis that which is deficient in hypothesis from a local mon-trations; therefore, whoever wishes to investigate theorem." one of the most curious subjects in geometry, will there Pappus then proceeds to give an account of Euclid's find abundance of materials, and an ample field for disporisms; but the enunciations are so extremely defec cussion. tive, at the same time that they refer to a figure now Dr Simson defines a porism to be “a proposition, in lost, that Dr Halley confesses the fragment in question which it is proposed to demonstrate, that one or more to be beyond bis comprehension. things are given, between which, and every one of ioThe high encomiums given by Pappus to these pro numerable other things not given, but assumed accordpositions have excited the curiosity of the greatest geo- ing to a given law, a certain relation described in the meters of modern times, who bave attempted to dis. proposition is shown to take place." cover their nature and manner of investigation. M. This definition is not a little obscure, but will be Fermat, a French mathematician of the 17th century, plainer if expressed thus : “ A porism is a proposition attaching himself to the definition which Pappus cri- affirming the possibility of finding such conditions as ticises, published an introduction (for this is its modest will render a certain problem indeterminate, or capable title) to this subject, which many others tried to eluci. of innumerable solutions.” This definition agrees with date in vain. At length Dr Simson, Professor of Ma- Pappus's idea of these propositions, so far at least as thematics in the University of Glasgow, was so fortu they can be understood from the fragment already meitnate as to succeed in restoring the Porisnis of Euclid. tioned; for the propositions here defined, like those The account he gives of his progress and the obstacles which he describes, are, strictly speaking, neither theohe encountered will always be interesting to mathema rems nor problems, bat of an intermediate nature beticians. In the preface to his treatise De Porismatibus, tween both; for they neither simply enunciate a truth he says, " Postquam vero apud Pappum legeram Poris to be demonstrated, nor propose a question to be resolmata Euclidis Collectionem fuisse artificiosissimam mul- ved, but are affirmations of a truth in which the detertarum rerum, quæ spectant ad analysin difficiliorum et mination of an unknown quantity is involved. In as generalium problematum, magno desiderio tenebar, ali- far, therefore, as they assert that a certain problem may quid de iis cognoscendi; quare sæpius et multis variisque become indeterminate, they are of the nature of theoviis tum Pappi propositionem generalem, mancam et im rems; and, in as far as they seek to discover the condiperfectam, tum primum lib. 1. porisma, quod, ut dictum tions by which that is brought about, they are of the fuit, solum ex omnibus in tribus libris integrum adhuc nature of problems. manet, intelligere et restituere conabar; frustra tamen, We shall endeavour to make our readers understand this Pori-in this subject distinctly, by considering them in the of an intermediate nature between problems and theo- Porisin. way in which it is probable they occurred to the an rems; and that they admitted of being enunciated in a cient geometers in the course of their researches: this manner peculiarly elegant and concise. It was to such will at the same time show the nature of the analysis propositions that the ancients gave the name of porisms. peculiar to them, and their great use in the solution of This deduction requires to be illustrated by an example: problems. suppose, therefore, that it were required to resolve the It appears to be certain, that it has been the solution following problem. of problems which, in all states of the mathematical sci A circle ABC (fig. 1.), a straight line DE, and a Plate ences, has led to the discovery of geometrical truths: point F, being given in position, to find a point G in the ceccxxxvii, the first mathematical inquiries, in particular, must have straight line DE such, that GF, the line drawn from it fig. 1. occurred in the form of questions, where something was to the given point, shall be equal to GB, the line drawn given, and something required to be done ; and by the from it touching the given circle. reasoning necessary to answer these questions, or to dis Suppose G to be found, and GB to be drawn touchcover the relation between the things given and those ing the given circle ABC in B, let H be its centre, join. to be found, many truths were suggested, which came HB, and let HD be perpendicular to DE. From D afterwards to be the subject of separate demonstra draw DL, touching the circle ABC in L, and join tions. IL; also from the centre G, with the distance GĚ or The number of these was the greater, because the an GF, describe the circle BKF, meeting HD in the points cient geometers always undertook the solution of pro- K and K'. It is evident that HD and DL are given in blems, with a scrupulous and minute attention, insoniuch position and magnitude: also because GB touches the that they would scarcely suffer any of the collateral truths circle ABC, HBG is a right angle; and since G is the to escape their observation. centre of the circle BKF, IB touches that circle, and therefore HL'+DK'=DH? But HL+LD'S which, in one particular case, when a certain relation 1 Porism. in DE, the straight line drawn from G to the point K infected from these points to any point in the circum- Porin shall be equal to the straight line drawn from Ĝ touch- ference ABC, shall have to each other a given ratio , ing the given circle ABC.” which ratio is also to be found." Hence also we bave The problem which follows appears to have led to an example of the derivation of porisms from one anothe discovery of many porisins. ther; for the circle ABC, and the points D and E reFig. 2. A circle ABC (fig. 2.) and two points D, E, in a maining as before (fig. 3.), if, through D ne draw diameter of it being given, to find a point F in the cir any line whatever HDB, meeting the circle in B and and E are given, from which two lines are to be inflectThe synthetical demonstration is easily derived from ed, and a circumference ABC, in which these lines are the preceding analysis. It must, however, be remarked, to meet, as also a ratio which these lines are to have to that the construction fails when the circle LFM falls each other. Now these conditions are all independent either wholly within or wholly without the circle ABC, of one another, so that any one may be changed without so that the circumferences do not intersect; and in these any change whatever in the rest. This is true in genecases the problem cannot be solved. It is also obvious ral; but yet in one case, viz. when the points are so rethat the construction will fail in another case, viz. when lated to another that the rectangle under their distances the two circumferences LFM, ABC, entirely coincide. from the centre is equal to the square of the radius of In this case, it is farther evident, that every point in the the circle ; it follows from the preceding analysis, that circumference ABC will answer the conditions of the the ratio of the inflected lines is no longer a matter of problem, which is therefore capable of numberless solu- choice, but a necessary consequence of this disposition tions, and may, as in the former instances, be converted of the points. into a porism. We are now to inquire, therefore, in From what has been already said, we may trace the what circumstances the point L will coincide with A, imperfect definition of a porisnı which Pappus ascribes and also the point M with C, and of consequence the to the later geometers, viz. that it differs from a local circumference LFM with ABC. If we suppose that theorem, by wanting the hypothesis assumed in that they coincide, EA : AD::H:ß:: EC: CD, and EA theorem. Now, to understand this, it must be observed, :EC::AD: CD, or by conversion, EA: AC::AD: that if we take one of the propositions called loci, and CD-AD:: AD: 2DÓ, O being the centre of the make the construction of the figure a part of the hypocircle ABC, therefore, also, EA: AO:: AD: DO, thesis, we get what was called by the ancient geometers, and by composition, EO:A0::A0:DO, therefore a local theorem. If, again, in the enunciation of the EOXOD=AO”. Hence, if the given points E and theorem, that part of the hypothesis which contains the Fig 3. D (fig. 3.) be so situated that EO XODXAO', and construction be suppressed, the proposition thence arising at the same time a:8:: EA : AD:: EC:CD, the will be a porism, for it will enunciate a truth, and will problem admits of omberless solutions; and if either of require to the full understanding and investigation of the points D or E be given, the other point, and also that truth, that something should be found, viz. the cirthe ratio which will render the problem indeterminate, cumstances in the construction supposed to be omitted. may be found. Hence we have this porism: Thus, when we say, if from two given points E, D, “A circle ABC, and also a point D being given, (fig. 3.) two straight lines EF, FD, are inflected to a fig; another point E may be found, such that the two lines third point F, so as to be to one another in a given ra• : |