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(Bologna, 1566). The 5th, 6th, and 7th were translated from an Arabic manuscript in the Medicean library by Abraham Echellensis and Borelli, and edited in Latin (Florence, 1661); and by Ravius (Kilonii, 1669).
Hep} Aoyov 'attoto^s- and Tlepl Xwptov JA?ro-to,iwjs, in which it was shewn how to draw a line through a given point so as to cut segments from two given lines, 1st. in a given ratio, 2nd. containing a given rectangle.
Of the first of these an Arabic version is still extant, of which a translation was edited by Hal-ley, with a conjectural restoration of the second. (Oxon. 1706.)
Tlepl Aid}pt(Tfji€Vifjs to/xtjs. To find a point in a given straight line such, that the rectangle of its distances from two given points in the same should fulfil certain conditions. (See Pappus, I. c.) A solution of this problem was published by Robt. Simson. Tlepl ToVwp 'ETrnrMwp, " A Treatise in two books on Plane Loci. Restored by Robt. Simson," Glasg. 1749.
TIfpl 'E7ra<£coi/, in which it was proposed to draw a circle fulfilling any three of the conditions of passing through one or more of three given points, and touching one or more of three given circles and three given straight lines. Or, which is the same thing, to draw a circle touching three given circles whose radii may have any magnitude, including zero and infinity. (Ap. de Tactionibus quae supers., ed. J. G. Canierer." Goth, et Amst. 1795, 8vo.)
Ilepl Nei$<rewj>. To draw through a given point a right line so that a given portion of it should be intercepted between two given right lines. (Restored by S. Horsley, Oxon. 1770.)
Proclus, in his commentary on Euclid, mentions two treatises, dq Cochlea and De Perturbatis Rationibus.
Eutocius, in his commentary on the Dimensio Circuli of Archimedes, mentions an arithmetical work called 'tiKvroSoov, (see Wallis, Op. vol. iii. p. 559,) which is supposed to be referred to in a fragment of the 2nd book of Pappus, edited by Wallis. (Op. vol. iii. p. 597.) (Montucla, Hist, des Mathem. vol. i.; Halley, Praef. ad Ap. Conic.; Wenrich, de auct. Graec. versionibus et comment. Syriacis, Arab. Armen. Persicisque^ Lips. 1842; Pope Blount, Censur. Celeb. Auth.) [W. F. D.]
APOLLONIUS TYANAEUS ('AnoMctvios TvavaTos), a Pythagorean philosopher, born at Tyana in Cappadocia about four years before the Christian era. Much of his reputation is to be attributed to the belief in his magical or supernatural powers, and the parallel which modern and ancient writers have attempted to draw between his character and supposed miracles, and those of the Author of our religion. His life by Philostratus is a mass of incongruities and fables : whether it have any groundwork of historical trutl\ and whether it were written wholly or partly with a controversial aim, are questions we shall be better prepared to discuss after giving an account of the contents of the work itself.
Apollonius, according to the narrative of his biographer, was of noble ancestry, and claimed kindred with the founders of the city of Tyana. We need not stop to dispute the other story of the incarnation of the god Proteus, or refer it, with Tillemont, to demoniacal agency. At the age of fourteen he was placed under the care of Euthyde-mus, a rhetorician of Tarsus ; but, being disgusted at the luxury of the inhabitants, he obtained leave of his father and instructor to retire to the neighbouring town of Aegae. Here he is said to have studied the whole circle of the Platonic, Sceptic, Epicurean, and Peripatetic philosophy, and ended by giving his preference to the Pythagorean, in which he had been trained by Euxenus of Hera-clea. (Phil. i. 7.) Immediately, as if the idea of treading in the footsteps of Pythagoras had seized him in his earliest youth, he began to exercise himself in the severe asceticism of the sect; abstained from animal food and woollen clothing, foreswore wine and the company of women, suffered his hair to grow, and betook himself to the temple of Aesculapius at Aegae, who was supposed to regard him with peculiar favour. He was recalled to Tyana, in the twentieth year of his age, by his father's death : after dividing his inheritance with a brother whom he is said to have reclaimed from dissolute living, and giving the greater part of what remained to his poorer relatives (Phil. i. 13), he returned to the discipline of Pythagoras. and for five years preserved the mystic silence, during which alone the secret truths of philosophy were disclosed. At the end of the five years, he travelled in Asia Minor, going from city to city and everywhere disputing, like Pythagoras, upor divine rites. There is a blank in his biography at this period of his life, of about twenty years during which we must suppose the same employ ment to have continued, unless indeed we hav reason to suspect that the received date of his birtl has been anticipated twenty years. He was be tween forty and fifty years old when he set out o: his travels to the east; and here Philostratu sends forth his hero on a voyage of discovery, i which we must be content rapidly to follow hin From Aegae he went to Nineveh, where he me Damis, the future chronicler of his actions, am proceeding on his route to India, he discoursed i Babylon with Bardanes, the Parthian king, an consulted the magi and Brahmins, who were su] posed to have imparted to him some theurgic s< crets. He next visited Taxila, the capital • Phraortes, an Indian prince, where he met larcha the chief of the Brahmins, and disputed with Ii dian Gjonnosophists already versed in Alexandria philosophy. (Phil. iii. 51.) This eastern journc lasted five years : at its conclusion, he returned the Ionian cities, where we first hear of his pi tensions to miraculous power, founded, as it wou seem, on the possession of some divine knowled derived from the east. If it be true that t honours of a god were decreed to him at tl period of his life, we are.of course led to susp< some collusion with the priests (iv. 1), who i said to have referred the sick to him for reli From Ionia he crossed over into Greece (iv. 1 visited the temples and oracles which lay lin '. way, everywhere disputing about religion, a assuming the authority of a divine legislator, the Eleusinian mysteries he was rejected as a r gician, and did not obtain admission to them ui