APOLLONIUS, a Greek epic poet, surnamed Rhodius, from his long residence in Rhodes, though he is supposed to have been a native of Alexandria. He is known to us as the author of the Argonautica, a poem which he began while in youth, studying under the poet Callimachus. His master is supposed to have slighted the production of the youthful Apollonius, and their connection ended in the most violent enmity, though we are ignorant of the particulars of their quarrel. The disappointed youth retired to Rhodes, where he is conjectured to have polished and completed his work, supporting himself by the profession of rhetoric, and receiving from the Rhodians the freedom of their city. He was at length recalled to Alexandria to succeed Eratosthenes in the care of the great library about 194 b.c., in the reign of Ptolemy Euergetes, who had been educated by Aristarchus, and rivalled his predecessors in the munificent encouragement he gave to learning. The only extant work of Apollonius is his poem above mentioned, in four books, on the Argonautic expedition. Both Longinus and Quintilian have assigned to this work the mortifying character of mediocrity. It was first printed at Florence in 1496, with the ancient Greek Scholia, in a 4to volume, now exceedingly rare. An excellent edition was published by Brunck in 1780, and another by Beck, in 1797; but the best is that of Professor Schafer (Leipsic, 2 vols. 8vo, 1810-13). The poem was translated into English verse by Fawkes and Green in 1780; another translation in English verse, with critical notes, was published by W. Preston in 1803.

APOLLONIUS of Tralles and his brother Tauriscus were the sculptors of the famous Farnese Bull, a group representing Zethus and Amphion tying the revengeful Dirce to the tail of a wild bull. This work is now at Naples. There were several other sculptors named Apollonius.

APOLLONIUS, a grammarian of Alexandria, author of a Homeric Lexicon, Λ∈^eις t OμηpLκai, which was first published by Villoison, in two vols, fol., at Paris, in 1773.

APOLLONIUS of Perga (Pergaeus), next to Archimedes the most illustrious of the ancient Greek geometricians, was born probably about 250 b.c., and died during the reign of Ptolemy Philopator (222-205 b.c. ), flourishing thus about forty years later than Archimedes. He studied at Alexandria under the successors of Euclid, and is one of the brightest ornaments of that famous mathematical school.

But few of the mathematical works of Apollonius have escaped the ravages of time. Of the greater part we have merely the names and general description given by Pappus in his preface to book vii. of the Mathematical Collections. His treatise on the Conic Sections gained him the title of The Great Geometer, and is that by which his fame has been transmitted to modem times. It is not, indeed, to be for a moment supposed that Apollonius discovered all, or even the greater part, of the demonstrations which he gives, any more than that Euclid devised the propositions that go by his name. Pappus mentions several treatises on conics known to have existed previously—in particular the five books of Aristaeus “The Ancient” (350 b.c.) on Solid Loci; and there can be little doubt that Apollonius availed himself of these, as well as of the discoveries of Conon, Euclid, Eudoxus, Menechmus, Nicoteles, Thrasidaeus, and others, who had explored the way before him. At this distance of time we cannot distinguish the original from the borrowed propositions; but, though it is certain that he both added to and improved upon the existing theory of conics, the mere embodying in a complete and logical treatise of a number of miscellaneous theorems was in itself a work of great mathematical genius. Eutocius informs us that Apollonius was the first to show that all the three sections may be cut from the same cone, by varying the position of the intersecting plane; for previous authors had supposed the plane of section always perpendicular to the slant side of the cone,—an hypothesis which requires that the three sections be cut from cones of different species, namely, the parabola from a right-angled cone, the ellipse from one with an obtuse, and the hyperbola from one with an acute vertical angle. But Archimedes, as Ubaldus shows in his commentary on the Aequipon- derantes, had been acquainted with this fact. Pappus ascribes to Apollonius the names by which the three sections are now distinguished; the term Parabola, however, occurs in the writings of Archimedes. Of the Conicorum Libri Octo of Apollonius, unfortunately only four have reached us directly through the original Greek. Three more have been preserved in an Arabic version, but the eighth has never been found. Though many attempts had been made to discover the last four books, they continued to be regarded as lost till 1658, when Borelli, the celebrated author of the treatise De motu Animalium, discovered in the Medici library at Florence an Arabic manuscript, entitled Apollonii Pergcei Conicorum Libri Octo. With the assistance of Abraham Ecchellensis he translated and published in 1661 the fifth, sixth, and seventh books; but the eighth, notwithstanding the title, was wanting. Some years previously Golius, Oriental professor at Leyden, had returned from the East with an Arabic version of the whole seven books, and had projected the publication of a translation; but it never appeared. A note appended to the MS. of Golius stated that the eighth book had never been translated into Arabic. It was long a favourite problem with geometers to attempt to restore the lost books of Apollonius, that is, to infer from the general nature of their contents, as described by Pappus, the propositions they had contained. Maurolycus, a Sicilian geometer of the 16th century, Viviani, the last favourite pupil of Galileo, Fermat, Halley, Simson, and a number of others, all attempted this with more or less success. Halley, guided by the descriptions of Pappus, and the assertion that his preliminary lemmas to the seventh book really belonged tc the eighth, as well as by the statement of Apollonius himself that the eighth was a continuation of the seventh book, restored this book for the edition issued by the Oxford Press in 1710, the only edition of the Greek text that has as yet appeared. The last four books of the conics of Apollonius formed the chief part of the higher geometry of the ancients; and they present some elegant geometrical solutions of problems, which offer considerable difficulty even to the modern analytical method. For example, the fifth book treats of the greatest and least lines that can be 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—lsi, The Section of Ratio, or Proportional Sections; 2<∕, the Section of Space; 3<7, the Determinate Section · 4dh, 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.

Isi, De Dationis 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.

Ath, 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 Pergoei que supersunt, ac maxime Lemmata Pappi in hos Libros, cum Observationibus, &c. (Gothae, 1795, 8vo).

5 th, 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).

3th, 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 Pergoei Conicorum libri quatuor, ex versione Frederici Commanding Bononiae, 1566, fol. 2. Apollonii Pergoei Conicorum libri V. vi. vii. Paraphraste Abalphato Asphanensi nunc primum editi: Additus in calce Archimedis Assumtorum Liber, ex Codicibus Arabicis Manuscr.: Abrahamus Ecchellensis Latinos reddidit: J. Alfonsus Borellus curam in Geometricis Versioni contulit, et Notas uberiores in universum opus adjecit. Florentiae, 1661, fol. 3. Apollonii Pergoei Conicorum libri octo, et Sereni Antissensis de Sectione Cylindri et Coni libri duo. Oxoniae, 1710, fol. (This is the splendid edition of Dr Halley.) 4. The edition of the first four books of the Conics given in 1675 by Barrow. 5. Apollonii Pergoei de Sectione Rationis libri duo: Accedunt ejusdem de Sectione Spatii libri duo Restituti: Proemittitur, kc. Opera et Studio Edmundi Halley. Oxoniae, 1706, 4to.

See Bayle’s Dictionary; Bossut, Essai sur THist. Gén. des Math., tome i.; Montucla, Hist, des Math., tome i.; Vossius, De Scient. Math.; Simson’s Sectiones Conicoe, preface; and Hutton’s Mathematical Dictionary.

APOLLONIUS, surnamed Tyanaeus, a Pythagorean philosopher, born at Tyana, the capital of Cappadocia, shortly before the Christian era. According to his biographer Philostratus, he studied grammar and rhetoric at Tarsus under Eu thy demus, but he soon left that gay and luxurious city for the quiet town of Aegae in the vicinity, where he spent his time in the company of philosophers and priests within the temple of Aesculapius. Among these he met Eυxenus, 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, conversing 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 paid him everywhere by Grecian priests and oracles. He visited Crete and Rome, where he astonished the magistrates by raising to life the dead body of a noble lady. Spain, Sicily, Egypt, Ethiopia, Greece, and Asia Minor became in time the scenes of his wanderings and his wonderworkings; and so distinguished had he become that even during his life he was raised almost to the rank of a divinity.

He is supposed to have died at Ephesus at a very great age, but his biographer finds it convenient to represent his end as involved in mystery, in order to heighten the reverence due to his hero. The words of the biographer are: “Here ends the history of Apollonius the Tyanaean, as written by Damis. Concerning the manner of his death, if he did die, the accounts are various.” Like Pythagoras, his master, he left no indication of his age; and Philostratus could not ascertain whether he died at Ephesus, or vanished at Lindus, or in Crete.

After his death Apollonius was worshipped with divine honours for a period of four centuries. A temple was raised to him at Tyana, which obtained from the Romans the immunities of a sacred city. His statue was placed among those of the gods, and his name was invoked as a being possessed of superhuman powers. The defenders of paganism, at the period of its decline, placed the life and miracles of Apollonius in rivalry to those of Christ; and some moderns have not hesitated to make the same comparison. There is no reason to suppose, however, that Philostratus entertained any idea of this sort in composing his life of Apollonius. That biography was undertaken by order of Julia, wife of the Emperor Severus, more than a century after the death of the subject of it; and it is extraordinary that Apollonius, if so renowned and widely known in his day, found no place in history for 120 years after his decease. The preamble of Philostratus, in which he professes to set forth things with which men were before unacquainted, is in striking contrast with the commencement of the Gospel of Luke.

Apollonius is not to be looked upon as a shallow and vulgar impostor, though to influence men’s minds he had recourse to artifices and pretensions unworthy of a true philosopher. With some of the spirit of a moral and religious reformer, he appears to have attempted, though vainly, to animate expiring paganism with a new and purer life. See Philostrδ,tus.