APOLLONIUS of PERGA (PERG-EUS), next to Archi- by the descriptions of Pappus, and the assertion that his medes the most illustrious of the ancient Greek geome- preliminary lemmas to the seventh book really belonged tc tricians, was born probably about 250 B.C., and died during the eighth, as well as by the statement of Apollonius him-the reign of Ptolemy Philopater (222-205 n.c.), flourishing self that the eighth was a continuation of the seventh book, thus about forty years later than Archimedes. He studied restored this book for the edition issued by the Oxford at Alexandria under the successors of Euclid, and is one Press in 1710, the only edition of the Greek text that has of the brightest ornaments of that famous mathematical as yet appeared. The last four books of the conics of school. Apollonius formed the chief part of the higher geometry of But few of the mathematical works of Apollonius have the ancients ; and they present some elegant geometrical escaped the ravages of time. Of the greater part we have solutions of problems, which offer considerable difficulty merely the names and general description given by Pappus even to the modern analytical method. For example, the in his preface to book vii. of the Matltematical Collections. fifth book treats of the greatest and least lines that can b© drawn from given points to the peripheries of conics, and contains the chief properties of normals and radii of curvature.

The other treatises of Apollonius mentioned by Pappus are - lst, The Section of Ratio, or Proportional Sections ; 2d, the Section of Space ; 3d, the Determinate Section ; 4th, the Tangencies ; 5th, the Inclinations ; 6th, 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 Rationis 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 intersection 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 problem : 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 thereupon 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 Pergcei que supersunt, ac maxime Lemmata Pappi in hos Libros, cum Observationibus, 62;c. (Gotha, 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 Ghetaldus, 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 construction 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 representing and multiplying large arithmetical numbers. The invention of the method of projections has been attributed 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 Pergxi Conicorum libri quatuor, ex versione Frcdcrici Commandini. Bononiee, 1566, fol. 2. Apollonii Pergcel Contort= libri v. vi. vii. Paraphraste Abalphato Asphanensi nuns primum editi Additus is calce Arehimedis Assztmtorum Liber, ex Codicibus Arabieis Manuscr. Abrahamus Eechellensis Latinos rcddidit : J. Alfonsus Borellus curam in Geometricis Versioni contulit, et Notas uberiores is UniVerS26717, opus adfccit. Florentim, 1661, fol. 3. Apollonii Perycet Conicorum libri octo, et Sercni Antissensis de Sections Cylindri el Coni libri duo. Oxonix, 1710, fol. (This is the splendid edition of Dr Ilalhey.) 4. The edition of the first four books of the Conics given in 1675 by Barrow. 5. Apollonii Pergesi de Sectione Rationis libri duo: Accedunt ejusdem de Sections Spatii libri duo Restituti: Brsemittitur, &e. Opera et Studio Edmundi Halley. Oxonim, 1706, 4to.

See Bayle's Dictionary; Bossut, Essai sun l' Hist. Gen. des Math., tome i. ; Montuela, Hist. des Math., tome i. ; Vossius, Dc Scient. Math. ; Simson's Scctioncs Conicce, preface ; and Hutton's Mathematical Dictionary.