Page:EB1911 - Volume 22.djvu/117

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During the last three centuries this subject seems to have had great fascination for mathematicians, and many geometers have attempted to restore the lost porisms. Thus Albert Girard says in his Traité de trigonométrie (1626) that he hopes to publish a restoration. About the same time P. de Fermat wrote a short work under the title Porismatum euclidaeorum renovata doctrina et sub forma isagoges recentioribus geometris exhibita (see Oeuvres de Fermat, i., Paris, 1891); but two at least of the five examples of porisms which he gives do not fall within the classes indicated by Pappus. Robert Simson was the first to throw real 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 tractatus; quo doctrinam porismatum satis explicatam, et in posterum ab oblivione tutam fore sperat auctor, and published after his death in a volume, Roberti Simson opera quaedam reliqua (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 datas esse, cui vel quibus, ut et cuilibet ex rebus innumeris non quidem datis, sed quae ad ea quae data sunt eandem habent relationem, convenire ostendendum est affection em quandam communem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ex quibus data demonstranda sunt, invenienda proponantur.” A locus (says Simson) is a species of porism. Then follows a Latin translation of Pappus's note on the porisms, and the propositions which form the bulk of the treatise. These are Pappus's thirty-eight lemmas relating to the porisms, ten cases of the proposition concerning four straight lines, twenty-nine porisms, two problems in illustration and some preliminary lemmas. John Playfair's memoir (Trans. Roy. Soc. Edin., 1794, vol. iii.), a sort of sequel to Simson's treatise, had for its special object the inquiry into the probable origin of porisms—that is, into the steps which led the ancient geometers to the discovery of them. 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 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 England, Simson's view has been most generally accepted abroad, and has the support of the great authority of Michael Chasles. However, in Liouville's Journal de mathémaliques pures et appliquées (vol. xx., July, 1855), P. Breton published Recherches nouvelles sur les porismes d'Euclide, in which he gave a new translation of the text of Pappus, and sought to 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 Science by a controversy between Breton and A. J. H. Vincent, who disputed the interpretation given by the former of the text of Pappus, and declared himself in favour of the idea of Schooten, put forward in his Mathematicae exercitationes (1657), in which he gives the name of “porism” to one section. According to F. van Schooten, if the various relations between straight lines in a figure are written down in the form of equations or proportions, then the 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. Housel also joined, did not carry forward the work of restoring Euclid's Porisms, which was left for Chasles. His work (Les Trois livres de porismes d'Euclide, Paris, 1860) makes full use of all the material found in Pappus. But we may doubt its being a successful reproduction of Euclid's actual work. Thus, in view of the ancillary relation in which Pappus's lemmas generally stand to the works to which they refer, it seems incredible that the first seven out of thirty-eight lemmas should be really 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 Porism” relates intelligibly, being a particular ease of it. An interesting hypothesis as to the Porisms was put forward by H. G. Zeuthen (Die Lehre von den Kegelschnitten 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 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 developed projective geometry of conics. It is a fact that Lemma 31 (though it makes no mention of a conic) corresponds exactly to Apollonius's method of determining the foci of a central conic (Conics, iii. 45-47 with 42).

The three porisms stated by Diophantus in his Arithmetica are propositions in the theory of 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 chapter on porisms (from a philological standpoint) is included in J. L. Heiberg's Litterargeschichtliche Studien über Euklid (Leipzig, 1882); and the following books or tracts may also be mentioned: Aug. Richter, Porismen nach Simson bearbeitet (Elbin, 1837); M. Cantor, “Ueber die Porismen des Euklid und deren Divinatoren,” in Schlömilch's Zeitsch. f. Math. u. Phy. (1857), and Literaturzeitung (1861), p. 3 seq.; Th. Leidenfrost, Die Porismen des Euklid (Programm der Realschule zu Weimar, 1863); Fr. Buchbinder, Euclids Porismen und Data (Programm der kgl. Landesschule Pforta, 1866). (T. L. H.)

POROS, or Poro (“the Ford”), an island off the east coast of the Morea, separated at its western extremity by only a narrow channel from the mainland at Troezen, and consisting of a mass of limestone rock and of a mass of trachyte connected by a slight sandy isthmus. The town looks down on the beautiful harbour between the island and the mainland on the south.

The ancient Calauria, with which Poros is identified, was given, according to the myth, by Apollo to Poseidon in exchange for Delos; and it became in historic times famous for a temple of the sea-god, which formed the centre of an amphictyony of seven maritime states—Hermione, Epidaurus, Aegina, Athens, Prasiae, Nauplia, and Orchomenus. Here Demosthenes took sanctuary with “gracious Poseidon,” and, when this threatened to fail him, sought death. The building was of Doric architecture and lay on a ridge of the hill commanding a fine view of Athens and the Saronic Gulf, near the middle of the limestone part of the island. The site was excavated in 1894, and traces of a sacred agora with porticoes and other buildings, as well as the temple, have been found. In the neighbourhood of Poros-Calauria are two small islands, the more westerly of which contains the ruins of a small temple, and is probably the ancient Sphaeria or Hiera mentioned by Pausanias as the seat of a temple of Athena Apaturia. The English, French, and Russian plenipotentiaries met at Poros in 1828 to discuss the basis of the Greek government.

See Chandler, Travels; Leake, Morea; Le Bas, Voyage archéologique; Curtius, Peloponnesos; Pouillon-Boblaye, Recherches; Bursian, Geographie von Griechenland; Rangabé “Ein Ausflug nach Poros,” in Deutsche Revue (1883); and S. Wide, in Mitteilungen d. deutsch. Inst. Athen. (1895), vol. xx.

PORPHYRIO, POMPONIUS, Latin grammarian and commentator on Horace, possibly a native of Africa, flourished during the 2nd century A.D. (according to others, much later). His scholia on Horace, which are still extant, mainly consist of rhetorical and grammatical explanations. It is not probable that we possess the original work, which must have suffered from alterations and interpolations at the hands of the copyists of the middle ages, but on the whole the scholia form a valuable aid to the student of Horace.

Ed. W. Meyer (1874); A. Holder (1894); see also C. F. Urba, Meletemata porphyrionea (1885); E. Schweikert, De Porphyrionis . . . scholiis Horatianis (1865); F. Pauly, Quaestiones criticae de . . . Porphyrionis commentariis Horatianis (1858).

PORPHYRY (Πορφύριος) (A.D. 233-c. 304), Greek scholar, historian, and Neoplatonist, was born at Tyre, or Batanaea in Syria. He studied grammar and rhetoric under Cassius Longinus (q.v.). His original name was Malchus (king), which was changed by his tutor into Porphyrius (clad in purple), a jesting allusion to the colour of the imperial robes (cf. porphyrogenitus, born in the purple). In 262 he went to Rome, attracted by the reputation of Plotinus, and for six years devoted himself to the study of Neoplatonism. Having injured his health by overwork, he went to live in Sicily for five years. On his return to Rome, he lectured on philosophy and endeavoured to render the obscure doctrines of Plotinus (who had died in the meantime) intelligible to the ordinary understanding. His most distinguished pupil was Iamblichus. When advanced in years he married Marcella, a widow with seven children and an enthusiastic student of philosophy. Nothing more is known of his life, and the date of his death is uncertain.

Of his numerous works on a great variety of subjects the following are extant: Life of Plotinus and an exposition of his teaching in the