Page images
PDF
EPUB
[blocks in formation]

POREE (Charles), a French Jesuit, and dramatic writer, born in 1675. He entered the society in 1692. He wrote some Latin poems and dramatic pieces; and died in 1741.

PORELLA, in botany, a genus of the natural order of musci and cryptogamia class of plants. The antheræ are multilocular, full of natural pores, with an operculum; there is no calyptra, nor pedicle; the capsules contain a powder like those of the other mosses; and their manner of shedding this powder is not by separating into two parts, like those of the selago and lycopoaim, but by opening into several holes on all sides.

PORISM, in geometry, is a name given by the ancient geometers to two classes of mathematical propositions. Euclid gives it 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 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 among many others on account of their great subserviency to the business of geometrical investigation. These propositions were so named by him either from the way in which he discovered them, while he was investigating something else, by which means they might be considered as gains or acquisitions, or from their utility in acquiring farther knowledge as steps in the investigation. In this sense they are porismata; for Topic signifies both to investigate and to acquire by investigation. These propositions formed a 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, however, this valuable collection is now lost, and it still remains doubtful in what manner the ancients conducted their researches into this curious subject. We have 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 best account we have of this class of propositions is a fragment of Pappus, in which he attempts a general definition of them as a set of mathematical propositions distinguishable in kind from all others; but of this distinction 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, Porisma est quod deficit hypothesi a theoremate locali.' Pappus then proceeds to give an account of Euclid's porisms; but the enunciations are so extremely

defective, while the figure to which they refer is now lost, that Dr. Halley confesses the fragment in question to be beyond his comprehension. The high encomiums given by Pappus to these propositions have excited the curiosity of the greatest geometers of modern times, who have attempted to discover their nature and manner of investigation. M. Fermat, a French mathematician of the seventeenth century, attaching himself to the definition which Pappus criticises, published an introduction (for this is its modest title) to this subject, which many others tried to elucidate in vain. At length Dr. Simson of Glasgow, after patient enquiry, suggested a restoration of the porisms of Euclid, which 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 enquiry 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 enquiries in 1746, under the title of General Theorems, not caring to give them any other name lest he might appear to anticipate the labors of his friend. The greater part of the propositions contained in that work are porisms, 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; it will be plainer expressed thus: A porism is a position 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 frag ment 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 theurems; 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 subject distinctly, by considering them in the way in which it is probable they occurred to the ancient geometers: 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 science, has led to the discovery of geometrical truths: the first mathematical enquiries, 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 enquiry, 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 quanities which were given. Such problems were said to become impossible; and it was soon found 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 is evident 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 abovementioned 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 enquiry it would be found that this proceeded from some condition of the problem having been involved in another, so that, in fact, there was but one which did not leave a sufficient number of independent conditions to limit the problem to a single or any determinate number

[blocks in formation]

F, being given in opposition, 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 A B C in B, let II be its centre, join II B, and let HD be perpendicular to DE. From D draw D L, touching the circle ABC in L and join HL; also from the centre G, with the distance GB or G F, describe the circle BK F, meeting HD in the points K and K'. Then HD and D L are given in position and magnitude; and, because G B touches the circle A BC, HBG is a right angle; and, since G is the centre of the circle BK F, therefore H B touches the circle BK F, and H B2 = the rectangle K'HK; which rectangle + D K2 = H D2, because K'K is bisected in D; therefore HL2 + K D2 — D H2 = H L2 and LD2; therefore DK2 = DL, and DK DL; and, since DL is given in magnitude, DK is also given, and K is a given point: for the same reason K' is a given point, and, the point F being given by hypothesis, the B KP is given in position. The point G, the centre of the circle, is therefore given, which was to be found. Hence this construction: having drawn HD perpendicular to DE, and DL touching the circle ABC, make DK and D K' 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 M N, which meets DE in G; G will be the point required; that is, if GB be drawn touching the circle ABC, and GF to the given point, G B is equal to GF. The synthetical demonstration is easily derived

from the preceding analysis; but in some cases this construction fails. For, first, if F fall anywhere in D II, as at F', the line M N becomes parallel to D E, 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 coincides with K, then M N 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 G K, wherever the point G be 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 porisin, 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 A BC. The problem which follows appears to have led to the discovery of many porisms. A circle ABC and two points fig. 2.

circumferences, and therefore is found. Hence, this construction: Divide ED in L, so that E L may be to LD in the given ratio of a to 3, 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 N L, describe the semicircle LF M; 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. But the construction fails when the circle LFM falls either wholly within or wholly without the circle A BC, so that the circumferences do not intersect; and in these cases the problem cannot be solved. The construction also fails when the two circumferences L FM, ABC, entirely coincide. In this case, every point in the circumference A B C 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 now enquire, 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:: a:ẞ

EC: CD, and EA: EC:: AD: CD, or by conversion EA: AC:: AD: C D-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 EOX ODA O2. Hence if the given points E and I be so situated that EO

[merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

:

D, F, 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 3, 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 3; produce E F 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 because DFB is bisected by F M, EM: MD:: EF: FD, that is, in a given ratio, and therefore M is given. Since D F L is half of DFE, and DFM half of DF B, therefore L F M is half of (DFEDF B), 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 L M 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

× OD=A02, and at the same time a: B:: 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 A BC, 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 A BC, and the points D and E remaining as before (fig. 3.), if, through D, we draw any line whatever II D B, meeting the circle in B and

; and if the lines EB, EH, be also drawn, these lines will cut off equal circumferences BF, liG. 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 A B, the angle DBE being bisected by B A, it is evident that the angle AOF is equal to AO H, and therefore the angle FOB to HOG, that is, the arch FB to the arch HG. This proposition appears to have been the last but one in the third book of Euclid's Porisms, and the manner of its enunciation in the porismatic form is obvious. The preceding proposition 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. These conditions are all independent on 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 one another that their 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 all this, 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. 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 E F, 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 A BC, of which the centre is O, be given by position, as also a point E; and if D be taken in the line E O, so that EOX ODA O'; and if from E and D the lines E F, D F, be inflected to any point of the circumference A B C, the ratio of EF to DF will be given, viz. the same with that of E A to AD, we have a local theorem. Lasily, when it is said, if a circle A B C be given by position, and also a point E, a point D may be found, such that if EF, F D, be inflected from E and D to any point F in the circumference A BC, 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, was imperfect. To confirm the truth of the preceding theory, professor Dr. Stewart, in a paper read many years ago before the Philosophical Society of Edinburgh, defines a porism to be A proposition affirming the possibility of finding one or more conditions of an indeterminate theorem ;' where, by an indeterminate theorem, he meant one which expresses a relation between certain quantities that are determinate and certain others that are indeterminate; a definition which evidently agrees with the explanations above given. If the idea which is given of these propositions be just, then 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 may be made on the analysis of porisms: 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 Dr. Simson frequently employs two statements of the general hypothesis, which he compares together. As, for instance, in his analysis of the last porism, he assumes not only E, any point in the line D E, but also another point O, anywhere in the same line, to both of which he supposes lines to be inflected from the points A, B. 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. 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 any inconsistency can 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 connexion 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 on experiment or observation. For, when a problem is to be resolved by help of data furnished by experiment or observation, the first thing to be considered is, whether the data so obtained be suthcient 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 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 a 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 given 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. Walls 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 in.umerable 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 the abbe 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. Besides the geometrical there is also an algebraical analysis belonging to porisms; which, however, does not belong to this place, because we give this account of them merely as an article of ancient geometry; and the ancients never employed algebra in their investigations. Mr. Playfair, professor of mathematics in the university of Edinburgh, has written a paper on the origin and geometrical investigation of porisms, which is published in the third volume of the Transactions of the Royal Society of Edinburgh, from which this account of the subject is taken. He has there promised a second part to his paper, in which the algebraical investigation of porismis is to be considered. This will no doubt throw considerable light upon the subject, as we may readily judge from that gentleman's known abilities, and from the specimen he has already given us in the first part. For more on this subject, see a very ingenious paper on Porisms by Henry Brougham, esq., in the Philosophical Transactions for 1798, or New Abridgment, vol. xviii. p. 345-355.

PORISMATIC, of or belonging to the mathematical doctrine of porisms.

[blocks in formation]

Will serve thee in winter, moreover than that, To shut up thy porklings thou meanest to fat. TuSSET. You are no good member of the commonwealth: for, in converting Jews to Christians, you raise the price of pork. Shakspeare.

This making of Christians will raise the price of pork; if we grow all to be porkeaters, we shall not shortly have a rasher on the coals for money. Id. Merchant of Venice. A priest appears,

And off rings to the flaming altars bears; A porket and a lamb that never suffered shears. Dryden. All flesh full of nourishment, as beef and pork, increase the matter of phlegm.

Flover on the Humours. Strait to the lodgment of his herd he run, Where the fat porkers slept beneath the sun.

Thus saith the prophet of the Turk, Good mussulman abstain from pork; There is a part in every swine, No friend or follower of mine May taste.

Pope.

Couper.

PORK. See Sus. The hog is the only domestic animal that we know of no use to man when alive, and therefore seems properly designed for food. The Jews, however, the Egyptians, and other inhabitants of warm countries, and all the Mahometans at present, reject the use of pork. The Greeks gave great commendation to this food, and their Athlete were fed with it. The Romans considered it as one of their delicacies. With regard to its alkalescency, no proper experiments have yet been made; but, as it is of a gelatinous and succulent nature, it is probably less so than many others. Upon the whole it appears to be a very valuable nutriment. The reason is obvious why it was forbidden to the Jews: their whole ceremonial dispensation was typical. Filth was held as an emblem or type of sin. Hence the many laws respecting frequent washings; and no animal feeds so filthily as swine. Mahomet borrowed this prohibition, as well as circumcision and many other parts of his system, from the law of Moses. But it is absurd to suppose, as some do, that Moses borrowed any thing of this kind from the Egyptians.

PORO ISLE, an island on the south-western coast of Sumatra, north-west of the Poggy Islands, and inhabited by a similar race of people. It is also denominated Pulo Siporah, or Good Fortune Island, and contains four villages, in which there are about 1000 inhabitants. In length this island is estimated at thirty-three miles, by eight the average breadth; and described as being covered with wood. Long. 99° 15′ E., lat. 2 12' S.

POROMPHALON, in medicine, from wpog, a callus, and ouçaλog, the navel, a hard piece of PORISTIC METHOD, Gг. πoρizikog, in mathe-flesh or stone growing out from the navel. matics, is that which determines when, by what means, and how many different ways a problem may be solved.

POROMUSHIR, the second of the Kurile islands, in the North Pacific, is about forty-four miles in length, and twelve in breadth. The

« PreviousContinue »