Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/202

Rh 188 APOLLONIUS drawn from given points to the peripheries of conies, and contains the chief properties of normals and radii of curva ture. The other treatises of Apollonius mentioned by Pappus are Is;, The Section o* Ratio, or Proportional Sections ; 2d, the Section of Space ; 3d, the Determinate Section ; kth, the Tangencies ; 5th, the Inclinations ; Gth, the Plane Loci. Each of these was divided into two books, and, with the Data of Euclid and the Porisms, they formed the eight treatises which, according to Pappus, constituted the body of the ancient analysis. 1st, De Eationis Sectione had for its subject the resolution of the following problem : Given two straight lines and a point in each to draw through a third given point a straight line cutting the two fixed lines, so that the parts intercepted between the given points in them and the points of inter section of this third line, may have a given ratio. 2d, De Spatii Sectione discussed the similar problem, which requires that the space contained by the three lines shall be equal to a given rectangle. Dr Halley published in 1706 a restoration of these two treatises, founded on the indications of their contents given by Pappus. An Arabic version of the first had previously been found in the Bodleian library at Oxford by Dr Edward Bernard, who began a translation of it, but broke off on account of the extreme inaccuracy of the MS. 3d, De Sectione Determinata resolved the problem : In a given straight line to find a point, the rectangles or squares of whose distances from given points in the given straight line shall have a given ratio. Several restorations of the solution have been attempted, one by Snellius, another by Alex. Anderson of Aberdeen, in the supplement to his Apollonius Redivivus (Paris, 1612), but by far the most complete and elegant by Dr Simson of Glasgow. 4th, De Tactionibus embraced the following general pro blem : Given three things (points, straight lines, or circles) in position, to describe a circle passing through the given points, and touching the given straight lines or circles. The most difficult case, and the most interesting from its historical associations, is when the three given things are circles. This problem, though now regarded as elementary, was proposed by Vieta in the 16th century to Adrianus Romanus, who gave a very clumsy solution. Vieta there upon proposed a simpler construction, and restored the whole treatise of Apollonius in a small work, which he entitled Apollonius Gallus (Paris, 1660.) Both Descartes and Newton have discussed this problem, though they failed to give it that simplicity of character which it has since been shown to possess. A very full and interesting historical account of the problem is given in the preface to a small work of Camerer, entitled Apollonii Pergoei que supersunt, ac maxime Lemmata Pappi in hos Libros, cum Observationibus, &c. (Gothse, 1795, 8vo). 5th, De Inclinationibus had for its object to insert a given straight line, tending towards a given point, between two given (straight or circular) lines. Restorations have been given by Marinus Ghetalclus, by Hugo de Omerique (Geometrical Analysis, Cadiz, 1698), and elegantly by Dr Horsley (1770). 6th, De Locis Plants is merely a collection of properties of the straight line and circle, and corresponds to the con struction of equations of the first and second degrees. It has been successfully restored by Dr Simson. The great estimation in which Apollonius was held by the ancients, and the great value attached to his productions, are manifest from the number and celebrity of the commentators who undertook to explain them. Among these we find the names of Pappus, the learned and unfortunate Hypatia, Serenus, Eutocius, Borelli, Halley, Barrow, and others. Various discoveries in other departments of mathematical science were also ascribed to him by the ancients. Pappus says that he made improvements on the modes of re presenting and multiplying large arithmetical numbers. The invention of the method of projections has been attri buted to him ; and he has the honour of being the first to found astronomical observations on the principles of geometry. The best editions of the works of Apollonius are the following : 1. Apollonii Pergcei Conicorum libri quatuor, ex versione Frederici Commandini. Bononise, 1566, fol. 2. Apollonii Pergcei Coni corum libri v. vi. vii. Paraphraste Abalphato Asphanensi nunc primum editi : Additus in calce Archimcdis Assumtorum Liber, ex Codicibus Arabicis Manuscr. : Abrahamus Ecchellensis Latinos rcddidit : J. Alfonsus JBorellus curam in Gcometricis Versioni contulit, et Notas uberiores in univcrsum opus adjecit. Florentine, 1661, fol. 3. Apollonii Pergcei Conicorum libri octo, et Sereni Antissensis de Sectione Cylindri ct Coni libri duo. Oxoniae, 1710, fol. (This is the splendid edition of Dr Halley.) 4. The edition of the first four books of the Conies given in 1675 by Barrow. 5. Apollonii Pergcei de Sectione Eationis libri duo: Accedunt cjusdem de Sectione Spatii libri duo Restituti: Prccmittitur, &c. Opera, et Studio Edmundi Halley. Oxonise, 1706, 4to. See Bayle s Dictionary ; Bossut, Essaisur I Hist. G^n. des Math., tome i. ; Montucla, Hist, des Math., tome i. ; Yossius, De Scicnt. Math. ; Simson s Scctioncs Conicce, preface ; and Button s Mathe matical Dictionary. APOLLONIUS, surnamed TYAN^US, a Pythagorean philosopher, born at Tyana, the capital of Cappadocia, shortly before the Christian era. According to his bio grapher Philostratus, he studied grammar and rhetoric at Tarsus under Euthydemus, but he soon left that gay and luxurious city for the quiet town of JEgse in the vicinity, where he spent his time in the company of philosophers and priests within the temple of ^Esculapius. Among these he met Euxenus, one of the followers of Pythagoras, and from him he learned with enthusiasm the doctrines of the Samian sage. While yet a mere youth he renounced all the ordinary pleasures of life. Abjuring the use of flesh and wine, he lived on the simple fruits of the soil, wore no clothing but linen and no sandals on his feet, suffered his hair to grow, and slept on the hard ground. He strictly observed the Pythagorean penance of five years silence, suffering often the most painful trials of his patience without a murmur. Philostratus relates so many wonderful stories of his hero how on one occasion, for instance, he awed an excited populace to silence by the mere waving of his hands, how he performed many miracles with a word, and how he knew all tongues without ever having learned them that some have questioned the very existence of Apollonius ; while others, admitting with reason the fact of some such ascetic having lived about this time, regard him as a compound of magician, impostor, and religious fanatic. After spending some time in the cities of Cilicia and Pamphylia, Apollonius extended his travels into the East, and wandered on foot over Assyria, Persia, and India, con versing with Magi, Brahmins, Gymnosophists, and priests, visiting the temples, preaching a purer morality and religion than he found, and attracting wherever he went admiration and reverence. At Nineveh he met with Damis, who became his adoring disciple and the companion of his journeyings, and left those doubtful records of his life which Philostratus made use of, and probably improved upon. The account of his exploits during his wanderings in India reads like the tales of the Arabian Nights ; and where Damis cannot vouch for having seen the prodigies he mentions, he unhesitatingly adduces in support of them the authority of his master. From his visit to the Hill of Sages (described in the third book), Apollonius returned an accomplished sage himself, able to foretell earthquakes and eclipses, to cure the plague, to summon spirits from the unseen world, and to restore the dead to life. On his return from the East he had the Greatest reverence