e Rhodians to Rome. He was the first Greek who addressed the senate without the aid of an interpreter. Cicero renewed his studies under him when he afterwards visited Rhodes on his return from Asia. The works of Apollonius have perished.

Another rhetorician of the same name, likewise a native of Alabanda, and an inhabitant of Rhodes, was surnamed the Effeminate (ὁ Μακαρώτα). Both are mentioned by Cicero with high respect.—Cicero, Brutus, 89, 90, 91; De Inv. i. 56; De Orat. i. 17, 28; Quintil. iii. 1. § 16; xii. 6, § 7, &c.

APOLLONUS, surnamed Dyscolus (Δυσκόλος), or the Crabbed, was a native of Alexandria, and lived in the reigns of Hadrian and Antoninus Pius. Priscian calls him "grammaticorum princeps." He was the first systematic writer on grammar, and his extant treatises on various branches of the science, display great ingenuity and vigour of thought. These are, 1. On Syntax; 2. On the Pronoun; 3. On Conjunctions; 4. On Adverbs. The best edition of the Syntax is that of Bekker, Berlin, 1817.

APOLLONUS 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 Rhodus, was regarded with so much admiration by the contemporaries of disciple of the illustrious Galileo, was employed on the Apollonius, 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 Alphonsus 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 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:—1st, The Section of Ratio, or Proportional Sections; 2nd, the Section of Space; 3rd, 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 branch- ed 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..." Apollonius 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 hyperbolae. 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 Galles. 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 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 Pergei qua supersunt, ac maxime Lemmata Pappi in hos libros, cum Observationibus, &c. Gothæ, 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' 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 Pergei Conicorum libri quatuor, ex versione Frederici Commandini.* Bononiae, 1566, fol. 2. *Apollonii Pergei Conicorum libri v. vii.* Paraphrase Abalphto Asaphenisi nunc primum editi: Additus in eadem Archimedis Assumptorum liber, ex Codicibus Arabicis Manuscr.: Abrahamus Echellensis Latinos reddidit: J. Alfonso Borellus curam in Geometricis Versioni contulit, et Notas uberiores in universum opus adjectit.* Florentiae, 1661, fol. 3. *Apollonii Pergei Conicorum libri octo, et Sereni Antissensis de Sectione Cylindri et Coni libri duo.* Oxomiae, 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 Rationis libri duo: Accedunt ejusdem de Sectione Spati libri duo Restituti: Premittitur, &c. Opera et Studio Edmundo Halley.* Oxomiae, 1706, 4to.

See Bayle's Dictionary; Bossut, Essai sur l'Hist. Gén. des Math., tome i.; Montucla, Hist. des Math., tome i.; Vossius de Scient. Math.; Simson's Sectiones Conicæ, preface; and Hutton's Mathematical Dictionary.author of the Argonautics, and surmamed 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 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 Eratothenes 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 Schäfer, printed at Leipzig in 2 vols. 8vo, in 1810-13.

The Argonautics have been well 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 and Tauriscus of Tralleis, 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.

Another Apollonius, a sophist, wrote a Homeric Lexicon, which was first published by Villeison. His *Oxyrhynchus* appeared in two vols. 4to, at Paris, in 1773.

Apollonius of Tyana, a celebrated Pythagorean philosopher, was born at Tyana, the capital of Cappadocia, a few years before the Christian era. At the age of 14 he was sent by his father to Tarsus, to study grammar and rhetoric under the Phoenician Euthydemus; but finding the vain and luxurious manners of that city unfavourable to serious study, he retired, with his father's permission, to the neighbouring town of Æga, where he spent most of his time in the temple of Æsculapius, in the company of priests and philosophers. Here he fell in with Euxenus, a philosopher who professed the principles of Pythagoras, but followed in his practice the less rigid maxims of Epicurus. Apollonius received with enthusiasm the Pythagorean instructions of Euxenus, and with more consistency than his master, determined at the age of 16 to mould his future life by the precepts of the Samian sage. Thenceforth renouncing the pleasures and luxuries of life, he devoted himself to the cultivation of his soul, and the improvement of his fellow-men. Abjuring the use of animal flesh and of wine, he fed on the simple fruits of the earth, wore no clothing but linen, and no sandals on his feet, suffered his hair to grow, and slept on the hard ground. The still more difficult penance of a five years' silence, prescribed by Pythagoras to his disciples, was a few years after strictly observed by Apollonius. This period he passed chiefly in Pamphylia and Cilicia, enduring the painful trial of passing through scenes of violence and disorder without suffering even a murmur to pass his lips; successful, however, if his admiring biographer may be credited, in restraining tumult and excess by the look of his countenance, or the silent eloquence of his hand. At the city of Aspendus, he found the inhabitants rising in mutiny against the governor, whom they unjustly blamed for the general scarcity of corn. The presence of Apollonius awed them into silence, and the governor urged in his own defence that the wealthy citizens who hoarded their corn were the authors of the famine. The excited populace were hurrying to take vengeance on the monopolizers, and plunder their stores, when Apollonius, by silent signs, hushed their fury, and the hoarders of the corn were brought into his presence. With difficulty restraining his voice, amid the tears of perishing women and children and old men, he wrote on a tablet the following words:—"Apollonius to the monopolizers of corn in Aspendus, greeting! The earth is the common mother of all, for she is just. You are unjust, for you have made her only the mother of yourselves: and if you will not cease from thus doing, I will not suffer you to remain upon her." This just rebuke, says Philostratus, had the desired effect: the market was filled with grain, and the city recovered from its distress. This story, whether true or not, may be taken as a fair indication of the real character of Apollonius, both as to the general tenor of his moral teaching, and his high and mysterious pretensions.

After spending some time at Antioch, Apollonius extended his travels into the East, and wandered 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 disciple and the companion of his journeyings, and left those doubtful records of his life which Philostratus made use of, and probably improved upon. He afterwards visited Egypt, Greece, and Italy, and died, as is supposed, at Ephesus, at a very advanced age. His biographer has represented his death as involved in doubt and mystery, with the view of heightening the reverence due to his hero.

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 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 modern Deists have not disdained to make the same unworthy comparison.

The life of Apollonius by Philostratus, composed about 120 years after the philosopher's death, by order of Julia, wife of the Emperor Severus, is the only source of our information regarding him. Founded, as it is, on the insufficient testimony of Damis, combined with vague and exaggerated traditions, we are left to draw our estimate of Apollonius from mere probability. See Philostratus. It would seem, then, that though he may very possibly have been led, in his desire to strengthen his influence over men's minds, to use artifices and pretensions unworthy of a true sage, he was far from having been a vulgar or shallow impostor. With some of the spirit of a moral and religious reformer, he appears to have attempted, though vainly, to animate with a new and purer life the expiring breath of Paganism. His journey to the East was probably directed by the wish to trace the original traditions of the human race to their native source, believing, as he seems to have done, that these had been corrupted by the impure mythology of an artful priesthood. The story already quoted illustrates his doctrine that the earth was the common home of the human family, and that its inhabitants could attain true happiness only in the recognition of their mutual brotherhood. He inculcated the uselessness of supplication and of sacrifice, as unworthy of the Divine majesty, proclaiming the only acceptable offering to be a pure and devout heart.

In a philosophical relation, Apollonius may be regarded as the precursor of the Alexandrian philosophy, by bringing philosophy and religion into union, and attempting to combine the philosophical spirit of Greece with the mystical religion of the East. Of the authentic works of Apollonius none are extant but his Apology, preserved along with the life by Philostratus.

APOLLO, in Scripture History, a Jew of Alexandria, who came to Ephesus during the absence of St Paul at Jerusalem (Acts xviii. 24). Apollos was an eloquent man, and well versed in the Scriptures, and preached in the synagogue, with zeal and fervour the doctrine of a Messiah, knowing as yet "only the baptism of John." Aquila and Priscilla having heard him, took him home with them, instructed him more fully in the doctrines of the Gospel, and baptized him.

Some time after this he travelled into Achaea; and, having come to Corinth, was there very useful in convincing the Jews out of the Scriptures, and demonstrating to them that Jesus was the Christ. Thus he watered what St Paul had planted in this city. (1 Cor. iii. 6.) But the great affection which his disciples entertained for him threatened to produce a schism, some saying, I am of Paul, others, I am of Apollos, I am of Cephas. This division, however, which St Paul speaks of in the chapter last quoted, did not prevent that apostle and Apollos from being closely united by the bonds of charity. Apollos hearing that the apostle was at Ephesus, went to meet him, and was there when St Paul wrote the first epistle to the Corinthians; wherein he testifies that he had earnestly entreated Apollos to return to Corinth, but hitherto had not been able to prevail with him; that nevertheless he gave him room to hope that he would go when he had an opportunity. St Jerome says that Apollos was so dissatisfied with these divisions at Corinth, that he retired into Crete with Zenas, a doctor of the law; and that this disturbance having been appeased by St Paul's letter to the Corinthians, Apollos returned to this city, and became its bishop. The Greeks made him bishop of Duras, Apollos others say he was bishop of Iconium in Phrygia, and others that he was bishop of Cesarea.