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TABLE III. Shewing the Action of the Vitrifying matters on the Crucibles that contain them.

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For an account of some valuable experiments of a similar nature, which were made by the celebrated Klaproth, in crucibles of clay and charcoal, in which the differences of the results are very striking, the reader is referred to his Analyt. Essays, or to Aikin's Dictionary of Chemistry and Mineralogy.

PORCELAIN-Shell, a species of CYPREA. See CYPRÆEA, CONCHOLOGY Index.

PORCH, in Architecture, a kind of vestibule supported by columns; much used at the entrance of the ancient temples, halls, churches, &c.

A porch, in the ancient architecture, was a vestibule, or a disposition of insulated columns usually crowned with a pediment, forming a covert place before the principal door of a temple or court of justice. Such is that before the door of St Paul's, Covent-Garden, the work of Inigo Jones. When a porch had four columns in front, it was called a tetrastyle; when six, hexastyle; when eight, octostyle, &c.

PORCH, in Greek laa, a public portico in Athens, adorned with the pictures of Polygnotus and other eminent painters. It was in this portico that Zeno the philosopher taught; and hence his followers were called Stoics. See STOICS and ZENO.

PORCUPINE. See HYSTRIX, MAMMALIA Index. PORCUPINE-Man, the name by which one Edward Lambert, who had a distempered skin, went in London.

We have the following account of him in the Philosophical Transactions for 1755, by Mr Henry Baker, F. R. S. "He is now (says he) 40 years of age, and it is 24 years since he was first shown to the society. The skin of this man, except on his head and face, the palins of his hands, and the soles of his feet, is covered with excrescences that resemble an innumerable company of warts, of a brown colour and cylindrical figure; all rising to an equal height, which is about an inch, and growing as close as possible to each other at their basis; but so stiff and elastic as to make a rustling noise when the hand is drawn over them. These excrescences are annually shed, and renewed in some of the autumn or winter months. The new ones, which are of a paler colour, gradually rise up from beneath as the old ones fall off; and at this time it has been found necessary for him to lose a little blood, to prevent a slight

ble (B). Melted down with the crucible to a tough slag.

Run into a hard blue clear glass.

A perfectly black glass.

A semitransparent applegreen glass. Completely fused in the parts

touching the crucible. The whole crucible was penetrated with a scoria so as not to fall to powder on exposure to air.

cible (C).

Scarcely altered, except

slight fusion at the edges. As in A.

As in A.

A brown scoria containing grains of iron. As in A.

As in A.

Agraysemitransparentglass As in A.

A green scoria, also with a A green glass with many

crust of iron.

grains of iron.

Of

sickness which he had been used to suffer before this precaution was taken. He bas had the small-pox, and he has been twice salivated, in hopes to get rid of this disagreeable covering; but though just when the pustules of the smallpox had scaled off, and immediately after his salivations, his skin appeared white and smooth, yet the excrescences soon returned by a gradual increase, and his skin became as it was before. His health, during his whole life has been remarkably good; but there is one particular of his case more extraordinary than all the rest; this man has bad six children, and all of them had the same rugged covering as bimself, which came on like his own about nine weeks after the birth. these children only one is now living, a pretty boy, who was shown with his father. It appears therefore, as Mr Baker remarks, that a race of people might be propagated by this man, as different from other men as an African is from an Englishman; and that if this should have happened in any former age, and the accidental original have been forgotten, there would be the same objections against their being derived from the same common stock with others: it must therefore be admitted possible, that the differences now subsisting between one part of mankind and another may have been produced by some such accidental cause, long after the earth had been peopled by one common progenitor."

PORE, in Anatomy, a little interstice or space between the parts of the skin, serving for perspiration. PORELLA, a genus of plants belonging to the cryptogamia class. See BOTANY Index.

PORENTRU, a town of Switzerland, in Elsgaw, and capital of the territory of the bishop of Basle, which is distinguished only by its castle and cathedral. The bishop was formerly a prince of the empire. It is seated on the river Halle, near Mount Jura, 22 miles south of Basle. E. Long. 7. 2. N. Lat. 47. 34.

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

Porism,

Porism. called corollaries. But he gives the same name, by way nihil enim proficiebam. Cumque cogitationes de hac re forism of eminence, to a particular class of propositions which multum mihi temporis consumpserint, atque tandem mohe collected in the course of his researches, and selected lestæ admodum evaserint, firmiter animum induxi nunfrom 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 he 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 quæ spem mihi 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 og signifies both to investigate and to ac- tem paulo post una cum Porismate primo lib. 1. impressa quire by investigation. These propositions formed a est inter Transactiones Philosophicas anni 1723, N°177." 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 defec-
tive, 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 pro-
positions have excited the curiosity of the greatest geo-
meters of modern times, who have attempted to dis-
cover their nature and manner of investigation. M.
Fermat, a French mathematician of the 17th century,
attaching himself to the definition which Pappus cri-
ticises, published an introduction (for this is its modest
title) to this subject, which many others tried to eluci-
date in vain. At length Dr Simson, Professor of Ma-
thematics in the University of Glasgow, was so fortu-
nate 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 mathema-
ticians. In the preface to his treatise De Porismatibus,
he says, 66
Postquam vero apud Pappum legeram Poris-
mata Euclidis Collectionem fuisse artificiosissimam mul-
tarum rerum, quæ spectant ad analysin difficiliorum et
generalium problematum, magno desiderio tenebar, ali-
quid de iis cognoscendi; quare sæpius et multis variisque
viis tum Pappi propositionem generalem, mancam et im-
perfectam, tum primum lib. 1. porisma, quod, ut dictum
fuit, solum ex omnibus in tribus libris integrum adhuc
manet, intelligere et restituere conabar; frustra tamen,

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 greatest 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

Parim 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, insomuch 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 will 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 pereeived, that these cases formed very curious propositions

of an intermediate nature between problems and theo- Porism. rems; and that they admitted of being enunciated in a manuer 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.

Plate

A circle ABC (fig. 1.), a straight line DE, and a point F, being given in position, to find a point G in the ceccxxxvII straight line DE such, that GF, the line drawn from it fig. 1. 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 HB or HL2=KII×HK'; but because KK' is bisected in D, KHxHK'+ DK'=DH2; therefore HL2+DK2= DH2. But HL'+LD2= DH', therefore DK' DL and DK=DL. 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 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

Porism. 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."

Fig. 2.

Fig 3.

The problem which follows appears to have led to the discovery of many porisins.

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 a to ß, 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 a to 8; produce EF towards B, bisect the angle EFD by FL, and DFB by FM: therefore EL: LD :: EF: 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 DFB 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 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 a to 8, and produce ED also to M, so that EM may be to MD in the same given ratio of a to ß; 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, EA; AD::«:ß:: 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 EOXOD=AO". Hence, if the given points E and D (fig. 3.) be so situated that EO× OD× AO3, and at the same time aẞ:: 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 circum- Porism. ference 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 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 paints; 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, (fig. 3.) two straight lines EF, FD, are inflected to a Fig. 3. third point F, so as to be to one another in a given ra

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