APOLLONIUS of Perga, in Pamphylia, is one of the most illustrious of the ancient Greek geometricians. The date of his birth has not been precisely ascertained; but as he flourished under Ptolemy Philopater, who died in the year 205 b c., after a reign of 16 years, it is conjectured that he was born about the middle of the third century before our era, and that he was about 40 years posterior to Archimedes. He studied at Alexandria under the successors of Euclid, and is pre-eminently distinguished among the disciples of that illustrious school, in which the mathematical sciences were at all times held in the highest estimation.

Few of the numerous compositions of Apollonius have escaped the ravages of time; but the description which has been given of them by Pappus, in the preface to the seventh book of the Mathematical Collections, explains their nature and value, and gives the admirers of the ancient analysis great reason to regret the loss of those which have perished. The most celebrated of his productions was the treatise on the Conic Sections ; a work∙which, according to the testimony of Geminius Rhodius, was regarded with so much admiration by the contemporaries of [7:3:288].its author, that they bestowed on him the title of the z Great Geometrician. This title is justified by the excellence of the work, which is unquestionably far superior to any treatise on the subject which we know to have existed among the ancients, and which has not indeed, in some respects, been surpassed in modern times. But while we bestow this praise on the treatise of Apollonius, we are not to suppose that he was the inventor of all, or even the greater part, of the properties which are demonstrated in it. Several treatises on the conic sections are known to have existed previously, in which the theory or these curves seems to have been prosecuted to a very considerable length. Pappus mentions, in particular, in terms of the highest eulogy, the five books on Solid Loci, or the conic sections, which were composed by Aristaeus the ancient, who lived about 350 years b. c.; and the construction given by Menechmus, of the problem to find two mean proportionals between two given straight lines, which leads to the duplication of the cube, shows that the disciples of Plato had advanced far in the same department of geometry. In fact, as it was the object of Apollonius to give a complete treatise of the conic sections, he did not scruple to avail himself of the discoveries of his predecessors, and, accordingly, embodied in his own work what had previously been done by Aristaeus, Kudoxus of Cnidus, Mcnechmus, Euclid, Conon, Thrasidaeus, Nicoeles, and others. It is now impossible to distinguish the propositions which were borrowed from his predecessors from those which were invented by himself; but it is certain that he both made great additions to the theory of the conic sections, and improvements in the manner of treating it. Eutocius informs us that Apollonius was the first who showed that all the three sections may be cut from the same cone, by varying the position of the intersecting plane; for previous authors had always supposed the plane of the section perpendicular to the side of the cone,—an hypothesis which requires that the three sections be cut from three cones of different species, namely, the parabola from a right-angled cone, the ellipse from an obtuse, and the hyperbola from an acute. It has, however, been established by Guido Ubaldus, in his commentary on the second book of the Aequiponderantes of Archimedes, that the Syracusan geometer was acquainted with the fact, that the three sections may be derived from the same cone. The generalization is indeed so very obvious, that we can scarcely persuade ourselves that it was not previously made by the more ancient writers; and it is probable, that if they usually assumed the cutting plane to have a particular position with reference to the cone, it was only on account of some facilities which that hypothesis afforded them in establishing the fundamental properties of the sections. Pappus ascribes to Apollonius the names by which the three sections are now distinguished and characterized: the term Parabola, however, occurs in the writings of Archimedes.

Of the eight books which Apollonius composed on the conic sections, the first four only have reached us in the original Greek. Three more have been preserved through the medium of an Arabic version; the last is unfortunately lost. A Latin translation of the first four was published by Memmius, at Venice, in 1537; and another, much more accurate, by Commandine, in 1568, with the addition of the commentary of Eutocius, and the lemmas of Pappus to all the eight books. Hitherto the last four books had not been discovered; but as the nature of their contents was sufficiently known from the indications of Pappus, several attempts were made to supply their loss by a restoration. Maurolycus, a Sicilian geometer of the 16th century, successfully commenced the theory of the fifth and sixth books; and Vivianĩ, the last and favourite disciple of the illustrious Galileo, was employed on the, same subject, when two different manuscript versions' of the work of Apollonius were accidentally brought to light. Among a number of Arabic manuscripts brought from the East by Golius, one was found which contained seven books of the conics. Golius was sufficiently i∏. structed in geometry to be aware of the value of his discovery; he hastened to communicate it to the mathematicians of that time, and proposed to publish a translation of the work. This project, however, failed from some cause which has not been explained; and, notwithstanding the intimation which had been given, the last four books still continued to be regarded as lost, till the year 1658, when Alphonso Borelli, the celebrated author of the treatise De Motu Animalium, happened to discover, in the library of the Medici at Florence, an Arabic manuscript with the following inscription: Apollonii Pergoei Conicoum Libri Octo. Borelli obtained permission to carry this manuscript to Rome, where, with the assistance of Abraham Ecchellensis, he translated the fifth, sixth, and seventh books. Notwithstanding the inscription, the eighth book was wanting; and as this was also the case with regard to the manuscript of Golius, it seems probable that it had not been translated into Arabic.

The last four books of the conics of Apollonius formed a considerable part of what may be termed the transcendental geometry of the ancients; and they exhibit some of the most elegant and successful applications of the geometrical analysis. The fifth book, for example, which treats of the greatest and least lines that can be drawn from given points to the peripheries of the curves, contains nearly all the properties of normals and radii of curvature which are now generally investigated by the aid of the differential calculus, and almost anticipates the admirable theory of involutes and evolutes which confers so brilliant a lustre on the name of Huygens. The seventh book also contains some theorems which, although they have now passed into the elements, are sufficiently difficult and remote to afford scope for the exercise of address and ingenuity, even when their investigation is attempted by the modern analysis. Dr Halley, guided by the description of Pappus, divined the contents of the eighth book, and published a magnificent edition of the whole at Oxford in 1710.

The other treatises of Apollonius which are mentioned by Pappus are the following :—Is/, The Section of Ratio, or Proportional Sections; 2d , the Section of Space; 3d , the Determinate Sections; ⅛th, the Tangencies; 5⅛ 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 tø Pappus, constituted the body of the ancient analysis. Of the above treatises of Apollonius, the first only has reached us through an Arabic translation. It was discovered in Arabic among the Selden manuscripts in the Bodleian Library at Oxford, by Dr Edward Bernard, who commenced a translation of it, from which, however, he was deterred by the difficulties occasioned by the extreme inaccuracy of the manuscript before he had finished a tenth part. This small portion of the translation was revised by Dr David Gregory; the rest was translated, or more properly speaking, divined, by Dr Halley, who published it in 1706, together with the analogous treatise on the Section of Space, which he had restored after the indication of its contents given by Pappus. The general problem resolved in the first treatise, although it is branched out into a great variety of cases, may be comprehended in the following enunciation: “Two straight lines being given by position, together with a point in each, it is required to draw through a third given point a straight [7:3:289] line intersecting the two former straight lines, so that the 'segments intercepted between the given points and the points of intersection with the third line may be to each other in a given ratio.” The problem which forms the subject of the second treatise differs from the above only in requiring that the intercepted segments on the two straight lines given by position shall contain a given rectangle.

The object of the treatise on the Determinate Sections .was “to find a point in a straight line given by position, the rectangles or squares of whose distances from given points in the given straight line shall have a given ratio.” A restoration of this and the two preceding treatises was attempted by Snellius; but although he certainly resolved the problems which had been proposed by Apollonius, his solutions were far inferior in point of elegance to those of the Greek geometer. The discovery of the treatise on the Section of Ratio enabled a comparison to be made of the restored with the original work. Some cases of the Determinate Section were also resolved by Alexander Anderson of Aberdeen, in his supplement to the Apollonius Redivivus, published at Paris in 1612. But by far the most complete and ; elegant restoration of the problem was given by Dr Simson of Glasgow, with two additional books on the same subject. It has been published among his posthumous works.

. The treatise on Inclinations, — the object of which was to insert a straight line of a given length, and tending to a given point, between two lines (straight lines or circles) given by position,—was first investigated by Marinus Ghetaldus, a patrician of Ragusa, afterwards by Hugo de Omerique in his ingenious treatise on the Geometrical Analysis, published at Cadiz in 1698. The different cases ,of the problem have been resolved in a very elegant manner by Dr Horsley, who published his restoration in

• 1770.

The treatise de Tactionibus, which relates to the contact of circles and straight lines, has afforded exercise for the ingenuity of many modern mathematicians. The general problem which it embraces may be enunciated as follows: Three things (points, straight lines, or circles) being given by position, it is required to describe a circle which may pass through the given points and touch the given straight lines and circles. The most difficult case of the problem is that in which the three things given are circles; the question' being then to determine the centre and radius of a circle, which shall touch these circles given in magnitude and position. This problem, which is now considered as quite elementary, possesses an historical interest on account of the great names connected with its solution. It was proposed by Vieta, the most skilful geometrician of the 16th century, to Adrianus Romanus, who, in constructing it, employed the very obvious consideration of the intersection of two hyperbolas. Such a • solution of a plane problem,.which ought to be constructed by means of straight lines and circles only, was very far from being satisfactory to Vieta: he therefore himself proposed a more geometrical construction, and restored the whole treatise of Apollonius, in a small work which he published at Paris in 1660 under the title of Apollonius Gallus. The∙treatise of Vieta is entitled to the praise of great ingenuity, but it falls far short of the geometrical elegance of the known productions of Apollonius; and simpler solutions have since been found of the more dif- ∙∙ficult cases of the general problem. An algebraic solution of the same question was attempted by Descartes; -but the equations at which he arrived were so exceedingly complicated, that he himself ingenuously confessed that he should not be able to construct one of them in a shorter time than three months. The Princess Elizabeth of Bohemia, who carried on an epistolary correspondence with Descartes, gave a solution of the same kind. Newton himself, in his Universal Arithmetic, condescended to consider this problem; but he succeeded little better than Vieta, whose method he followed. In the 16th lemma of the first book of the Principia, he has, however, given a different and simpler investigation, and reduced with great skill the two hyperbolic loci of Adrianus Romanus to the intersection of two straight lines. Simple geometrical solutions, since that of Dr Simson was published, are to be found in every elementary work. In speaking of this problem, Montucla observes, that it is one of those to which the algebraic analysis applies with difficulty. His opinion, however, would have been different had he lived to see the extremely simple and elegant algebraic investigation given by Gergonne in the Annales des Mathématiques, not only of this, but of the analogous problem in space which was proposed by Descartes to Fermat, viz. to describe a sphere touching four spheres given by position. In fact, it would be difficult to select a problem in elementary geometry better calculated to display the resources and pliability of the algebraic calculus, than this very one which had been considered as belonging so exclusively to the analysis of the ancients. A very full and interesting historical account of this problem is given in the preface to a little work of Camerer, entitled Apollonii Pergaci quae supersunt, ac maxime Lemmata Pappi in hos libros, cum Observationibus, &c. Gothae, 1795, 8vo∙

The last of the treatises mentioned by Pappus,— de Locis Planis, — is only a collection of properties of the straight line and circle, and corresponds to the construction of equations of the first and second degree. It has been restored in the true spirit of the ancient geometry by Dr Simson, whose treatise well deserves the attention of the student.

Besides the works which we have now enumerated, we are informed, by the fragment of the second book of Pappus, published among the works of Dr Wallis, that Apollonius occupied himself with arithmetical researches, and composed a treatise on the multiplication of large numbers. Astronomy is also indebted to him for the discovery, or at least for the demonstration, of the method of representing, by means of epicycles and deferents, the phenomena of the stations and retrogradations of the planets. He appears also to have been the inventor of the method of projections, and has the distinguished meritof having been the first who attempted to found astronomy on the principles of geometry, and establish an alliance between these two sciences which has been productive of the greatest benefit to both.

Of the personal character of this most assiduous and inventive geometrician, nothing is known excepting what may be gathered from a few unfavourable hints thrown out by Pappus. Pappus describes him as vain, arrogant, envious of the reputation of others,, and inclined to depreciate their merit; and contrasts him with the amiable and disinterested Euclid, who was always ready to allow to every one his just share of praise, and who manifested on every occasion the most benevolent feelings towards all men, especially towards those who laboured to improve or extend the science of geometry. The charge of (appropriating to himself the discoveries of Archimedes, which was brought against-Apollonius by Heraclius, had probably no other foundation than the boastful manner in which he spoke of his own discoveries, and affected to despise those of other mathematicians; for, as has been well remarked, pretensions pushed too far excite in the rest of mankind a sort of re-action of self-love, which leads them to contest the most legitimate titles. But whatever may have been the case with regard to the per [7:3:290]sonal qualities of Apollonius, the powers of his mind and his unwearied industry command universal admiration. The great value attached to his productions by the ancients is 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, &c.

The remarkable editions of the works of Apollonius are the following :—1. Apollonii Pergaci Conicoum libri quatuor, ex versione Erederici Commandini. Bononiae, 1566, fol. 2. Apollonii Pergaci Conicoum 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 Ver-sioni contulit, et Notas uberiores in universum opus adjecit. Florentiae, 1661, fol. 3. Apollonii Pergaci Conicoum 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 Pergaci de Sectione Rationis libri duo : Accedirnt ejusdem de Sectione Spatii libri duo Restituti: Praemittitur, &c. Opera et Studio Edmundi Halley. Oxoniae, 1706, 4to.

See Bayle’s Dictionary ; Bossut, Essai sur l’Hist. Gem des Math., tome i.; Montucla, Hist, des Math., tome i.; Vossius de Scient. Math.; Simson’s Sectiones Conicae, preface; and Hutton’s Mathematical Dictionary. (s.)

Apollonius, the author of the Argonautics, and surnamed the Rhodian, from the place of his residence, is supposed to have been a native of Alexandria, where he is said to have recited some portion of his poem while he was yet a youth. Finding it ill received by his countrymen, he 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 at length returned with considerable honour to the place of his birth, succeeding Eratosthenes in the care of the Alexandrian library in the reign of Ptolemy Euergetes, who ascended the throne of Egypt in the year before Christ 246. That prince had been educated by the famous Aristarchus, and rivalled the preceding sovereigns of his liberal family in the munificent encouragement of learning. Apollonius was a disciple of the poet Callimachus; but their connection ended in the most violent enmity, which was probably owing to some degree of contempt expressed by Apollonius for the light compositions of his master. The learned have vainly endeavoured to discover the particulars of their quarrel. The only work of Apollonius which has descended to modern times 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 published for the first time at Florence in 1496, with the ancient Greek Scholia, in a 4to volume, now exceedingly rare. There is an excellent edition by Brunck, published in 1780, and another by Beck, published in 1797; but the best is that of Professor Schafer, printed at Leipsic in 2 vols. 8vo, in 1810-13.

Apollonius, a Pythagorean philosopher, born at Tyana in Cappadocia about the beginning of the first century. At 16 years of age he became a strict observer of Pythagoras’s rules, renouncing wine and all sorts of flesh; not wearing shoes, letting his hair grow, and wearing nothing but linen. He soon after set up for a reformer of mankind, and chose his habitation in a temple of Aesculapius, where he is said to have performed many wonderful cures. Philostratus has written the life of Apollonius, in which there are numberless fabulous stories recounted of him. We are told that he went five years without speaking, and yet during that time stopped many seditions in Cilicia and Pamphylia; that he travelled, and set up for a legislator; that he gave out he understood all languages without having ever learned them; and that he could tell the thoughts of men and understood the oracles which birds gave by their singing. The heathens were fond of opposing the pretended miracles of this man to those of our Saviour; and by a treatise which Eusebius wrote against one Hierocles, we find that the object of the latter, in the treatise which Eusebius refutes, seems to have been to draw a parallel between Jesus Christ and Apollonius, in which he gives the preference to this philosopher. Μ. Dupin wrote a confutation of Philostratus’s life of Apollonius.

Apollonius wrote four books of Judicial Astrology; a Treatise upon the Sacrifices, showing what was proper to be offered to each deity; and a great number of Letters; all of which are now lost.