a famous architect under Trajan and Adrian, was born at Damascus. He had the direction of the bridge of stone which Trajan ordered to be built over the Danube in the year 104, which was esteemed the most magnificent of all the works of that emperor. Adrian, one day as Trajan was discoursing with this architect upon the buildings he had raised at Rome, would needs give his judgment, and showed he understood nothing of the matter. Apollodorus turned upon him bluntly, and said to him, "Go paint gourds, for you are very ignorant of the subject we are talking upon." Adrian at this time boasted of his painting gourds well. The insult cost Apollodorus his life.

a celebrated painter of Athens about 408 years before the birth of Christ, was the first who invented the art of mingling colours, and of expressing lights and shades. He was admired also for his judicious choice of subjects, and for beauty and strength of colouring surpassed all the masters that went before him. He excelled likewise in statuary.

APOLLODORUS the Athenian, a famous grammarian, the son of Asclepiades and disciple of Aristarchus. He wrote many works not now extant; but his most famous production is his Bibliotheca, which treats of the gods and the heroic ages. It is supposed by some that this is only an abridgement by another hand, and not the original work of Apollodorus. In any view it is, however, of value in mythological inquiries. The best edition is that of Heyne, in 2 vols. 8vo, published in 1803. A French translation, with notes, was published at Paris in 1805, in 2 vols. 8vo.

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 Apollonius its author, that they bestowed on him the title of the 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 of 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, Eudoxus of Cnidus, Menechmus, Euclid, Conon, Thrasydeus, Nicoteles, 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 Equiponderantes 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 Viviani, 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 instructed 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 Pergai Conicorum Libri Octo. Borelli obtained permission to carry this manuscript to Rome, where, with the assistance of Abraham Ecclellensis, 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:—1st, The Section of Ratio, or Proportional Sections; 2d, the Section of Space; 3d, the Determinate Sections; 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. 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..." 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 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 geometers. 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 Marius Ghetaldus, a patrician of Ragusa, afterwards by Hugo de Ormetique 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 difficult 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 Apollonius 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 Apolloni Pergaei quae supersunt, ac maxime Lemmata Pappi in hos libros, cum Observationibus, &c. Gothe, 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 merit of 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 reaction of self-love, which leads them to contest the most legitimate titles. But whatever may have been the case with regard to the per- Apollonius' zonal 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 Pergei Conicorum libri quatuor, ex versione Frederici Commandini.* Bononiae, 1568, fol. 2. *Apollonii Pergei Conicorum libri v. vi. vii.* Paraphraste Abalphtato Asphaniensi nunc primum editi: Additus in colo Archimedis Assumptum liber, ex Codicebus Arabiciis Manuser. Abrahamus Eccellenlis Latinos reddidit: J. Alfonso Borellus curam in Geometricis Versioni contulit, et Notas uberiores in universum opus adject. Florentiae, 1691, fol. 3. *Apollonii Pergei 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 Pergei de Sectione Spatii libri duo Restituti:* Accedunt ejusdem de Sectione Spatii libri duo Restituti: Præmittitur, &c. Opera et Studio Edmundi Halley. Oxoniae, 1706, 4to.

See Bayle's Dictionary; Bossut, Essai sur l'Hist. Géom. des Math., tome i.; Montucla, Hist. des Math., tome i.; Vossius de Scient. Math.; Simson's Sectiones Conice, preface; and Hutton's Mathematical Dictionary.