Online Encyclopedia
Search over 40,000 articles from the original, classic Encyclopedia Britannica, 11th Edition.
PORISM
Online EncyclopediaEncyclopedia Home :: POL-PRE
del.icio.us it!
|
PORISM . The subject of porisms is perplexed by the multitude of different views which have been held by geometers as to what a porism really was and is. The See also:treatise which has given rise to the controversies on this subject is the Porisms of See also:Euclid, the author of the Elements. For as much as we know of this lost treatise we are indebted to the Collection of Pappus of See also:Alexandria, who mentions it along with other geometrical See also:treatises, and gives a number of lemmas necessary for under-See also:standing it. Pappus states that the porisms of Euclid are neither theorems nor problems, but are in some sort intermediate, so that they may be presented either as theorems or as problems; and they were regarded accordingly by many geometers, who looked merely at the See also:form of the enunciation, as being actually theorems or problems, though the See also:definitions given by the older writers showed that they better understood the distinction between the three classes of propositions. The older geometers regarded a theorem as directed to proving what is proposed, a problem as directed to constructing what is proposed, and finally a porism as directed to finding what is proposed (eh ropur by avro"u ro"u 7rpomtvo EVOV). Pappus goes on to say that this last See also:definition was changed by certain later geometers, who defined a porism on the ground of an accidental characteristic as rb Xe'irov ixoKo-et roirocou OEwpitµaros, that which falls See also:short of a See also:locus-theorem by a (or in its) See also:hypothesis. See also:Proclus points out that the word was used in two senses. One sense is that of " corollary," as a result unsought, as it were, but seen to follow from a theorem. On the "porism " in the other sense he adds nothing to the definition of " the older geometers " except to say (what does not really help) that the finding of the center of a circle and the finding of the greatest See also:common measure are porisms (Proclus, ed. Friedlein, p.•301). Pappus gives a See also:complete enunciation of a porism derived from Euclid, and an See also:extension of it to a more See also:general See also:case. This porism, expressed in See also:modern See also:language, asserts that—given four straight lines of which three turn about the points in which they meet the See also:fourth, if two of the pcints of intersection of these lines See also:lie each on a fixed straight See also:line, the remaining point of inter-See also:section will also lie on another straight line. The general enunciation applies to any number of straight lines, say (n-{-1), of which n can turn about as many points fixed on the (n+1)th. These n straight lines cut, two and two, in in (n–1) points, in (n–1) being a triangular number whose See also:side is (n–1). If, then, they are made to turn about the n fixed points so that any (n–1) of their In (n–1) points of intersection, chosen subject to a certain See also:limitation, lie on (n–1) given fixed straight lines, then each of the remaining points of intersection, 2 (n–1) (n–z) in number, describes a straight line. Pappus gives also a complete enunciation of one porism of the first See also:book of Euclid's treatise. This may be expressed thus: If about two fixed points P, Q we make turn two straight lines See also:meeting on a given straight line L, and if one of them cut off a segment AM from a fixed straight line AX, given in position, we can determine another fixed straight line BY, and a point B fixed on it, such that the segment BM' made by the second moving line on this second fixed line measured from B has a given ratio X to the first segment AM. The See also:rest of the enunciations given by Pappus are incomplete, and he merely says that he gives See also:thirty-eight lemmas for the three books of porisms; and these include 171 theorems. The lemmas which Pappus gives in connexion with the porisms are interesting historically, because he gives (1) the fundamental theorem that the See also:cross or an See also:harmonic ratio of a See also:pencil of four straight lines meeting in a point is See also:constant for all transversals; (2) the See also:proof of the harmonic properties of a complete See also:quadrilateral; (3) the theorem that, if the six vertices of a hexagon lie three and three on two straight lines, the three points of concourse 9f opposite sides lie on a straight line. During the last three centuries this subject seems to have had See also:great See also:fascination for mathematicians, and many geometers have attempted to restore the lost porisms. Thus See also:Albert See also:Girard says in his Traite de trigonometrie (1626) that he hopes to publish a restoration. About the same See also:time P. de See also:Fermat wrote a short See also:work under the See also:title Porismatum euclidaeorum renovata doctrina et sub forma isagoges recentioribus geometeis exhibita (see Oeuvres de Fermat, i., See also:Paris, 1891); but two at least of the five examples of porisms which he gives do not fall within the classes indicated by Pappus. See also:Robert See also:Simson was the first to throw real See also:light upon the subject. He first succeeded in explaining the only three propositions which Pappus indicates with any completeness. This explanation was published in the Philosophical Transactions in 1723. Later he investigated the subject of porisms generally in a work entitled De porismatibus
traclatus; quo doctrinam porisrnatum satis explicatam, et in posterum ab oblivion tutam fore sperat auctor, and published after his See also:death in a See also:volume, Roberti Simson See also:opera quaedam reliqua (See also:Glasgow, 1776). Simson's treatise, De porismatibus, begins with definitions of theorem, problem, datum, porism and locus. Respecting the porism Simson says that Pappus's definition is too general, and therefore he will substitute for it the following: Porisma est propositio in qua proponitur demonstrate rem aliquam vel plures Batas ease, cui vel quibus, ut et cuilibet ex See also:rebus innumeris non quidem datis, sed quae ad See also:ea quae data sunt eandem habent relationem, convenire ostendendum est affectionem quandam communem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ex quibus data demonstranda aunt, invenienda proponantur." A locus (says Simson) is a See also:species of porism. Then follows a Latin See also:translation of Pappus's See also:note on the porisms, and the propositions which form the bulk of the treatise. These are Pappus's thirty-eight lemmas See also:relating to the porisms, ten cases of the proposition concerning four straight lines, twenty-nine porisms, two problems in See also:illustration and some preliminary lemmas. See also: See also:Playfair remarked that the careful investigation of all possible particular cases of a proposition would show that (1) under certain conditions a problem becomes impossible; (2) under certain other conditions, indeterminate or capable of an See also:infinite number of solutions. These cases could be enunciated separately, were in a manner intermediate between theorems and problems, and were called " porisms." Playfair accordingly defined a porism thus: " A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions." Though this definition of a porism appears to be most favoured in See also:England, Simson's view has been most generally accepted abroad, and has the support of the great authority of See also:Michael See also:Chasles. However, in Liouville's See also:Journal de mathematiques pures et appliquees (vol. xx., See also:July, 1855), P. See also:Breton published Recherches nouvelles sur See also:les porismes d'Euclide, in which he gave a new translation of the See also:text of Pappus, and sought to See also:base thereon a view of the nature of a porism more closely conforming to the definitions in Pappus. This was followed in the same journal and in La See also:Science by a controversy between Breton and A. J. H. See also:Vincent, who disputed the See also:interpretation given by the former of the text of Pappus, and declared himself in favour of the See also:idea of Schooten, put forward in his Mathematicae exercitationes (1657), in which he gives the name of "porism" to one section. According to F. See also:van Schooten, if the various relations between straight lines in a figure are written down in the form of equations or proportions, then the See also:combination of these equations in all possible ways, and of new equations thus derived from them leads to the discovery of innumerable new properties of the figure, and here we have " porisms." The discussions, however, between Breton and Vincent, in which C. See also:Housel also joined, did not carry forward the work of restoring Euclid's Porisms, which was See also:left for Chasles. His work (Les See also:Troia livres de porismes d'Euclide, Paris, 186o) makes full use of all the material found in Pappus. But we may doubt its being a successful See also:reproduction of Euclid's actual work. Thus, in view of the See also:ancillary relation in which Pappus's lemmas generally stand to the See also:works to which they refer, it seems incredible that the first seven out of thirty-eight lemmas should be really See also:equivalent (as Chasles makes them) to Euclid's first seven Porisms. Again, Chasles seems to have been wrong in making the ten cases of the four-line Porism begin the book, instead of the intercept-Porism fully enunciated by Pappus, to which the "lemma to the first Purism " relates intelligibly, being a particular case of it. An' inter-eating hypothesis as to the Porisms was put forward by H. G. Zeuthen (See also:Die Lehre von den Kegelschnitten See also:im Altertum, 1886, ch. viii.). Observing, e.g., that the intercept-Porism is still true if the two fixed points are points on a conic, and the straight lines See also:drawn through them intersect on the conic instead of on a fixed straight line, Zeuthen conjectures that the Porisms were a by-product of a fully See also:developed projective See also:geometry of conics. It is a fact that Lemma 31 (though it makes no mention of a conic) corresponds exactly to See also:Apollonius's method of determining the foci of a central conic (Conics, iii. 45–47 with 42). The three porisms stated by See also:Diophantus in his Arithmetica arepropositions in the theory of See also:numbers which can all be enunciated in the form " we can find numbers satisfying such and such conditions "; they are sufficiently analogous therefore to the geometrical porism as defined in Pappus and Proclus. A valuable See also:chapter on porisms (from a philological standpoint) is included in J. L. See also:Heiberg's Litterargeschichtliche Studien fiber Euklid (See also:Leipzig, 1882) ; and the following books or tracts may also be mentioned : Aug. See also:Richter, Porismen nach Simson bearbeitet (See also:Elbing, 1837); M. Cantor, " Ueber die Porismen See also:des Euklid and deren Divinatoren," in Schlomilch's Zeitsch. f. Math. u. Phy. (1857), and Literaturzeitung (1861), p. 3 seq.; Th. Leidenfrost, Die Porismen des Euklid (Programm der Realschule zu See also:Weimar, 1863) ; Fr. See also:Bach-binder, Euclids Porismen and Data (Programm der kgl. Landesschule See also:Pforta, 1866). (T. L. Additional information and CommentsThere are no comments yet for this article.
» Add information or comments to this article.
Please link directly to this article:
Highlight the code below, right click, and select "copy." Then paste it into your website, email, or other HTML. Site content, images, and layout Copyright © 2006 - Net Industries, worldwide. |
|
|
[back] PORFIRIUS, PUBLILIUS OPTATIANUS |
[next] POROS |