#### Apollo'nius or Apollo'nius Pergaeus

surnamed PERGAEUS,from Perga in Pamphylia, his native city, a mathematician educated at Alexandria under the successors of Euclid. He was born in the reign of Ptolemy Euergetes (Eutoc.*Comm. in Ap. Con.*lib. i.), and died under Philopator, who reigned B. C. 222-205. (Hephaest. apud

*Phot.*cod. cxc.) He was, therefore, probably about 40 years younger than Archimedes. His geometrical works were held in such esteem, that they procured for him the appellation of the Great Geometer. (Eutoc.

*l.c.*) He is also mentioned by Ptolemy as an astronomer, and is said to have been called by the sobriquet of e, from his fondness for observing the moon, the shape of which was supposed to resemble that letter.

#### Works

#### Conic Sections

Apollonius' most important work, the only considerable one which has come down to our time, was a treatise on Conic Sections in eight books. Of these the first four, with the commentary of Eutocius, are extant in Greek; and all but the eighth in Arabic.
The eighth book seems to have been lost before the date of the Arabic versions. We have also introductory lemmata to all the eight, by Pappus.
The first four books probably contain little more than the substance of what former geometers had done; they treat of the definitions and elementary properties of the conic sections, of their diameters, tangents, asymptotes, mutual intersections, &c. But Apollonius seems to lay claim to originality in most of what follows. (See the introductory epistle to the first book.)
The fifth treats of the longest and shortest right lines (in other words the *normals*) which can be drawn from a given point to the curve. The sixth of the equality and similarity of conic sections; and the seventh relates chiefly to their diameters, and rectilinear figures described upon them.

We learn from Eutocius (*Comm.* in lib. i.), that Heraclius in his life of Archimedes accused Apollonius of having appropriated to himself in this work the unpublished discoveries of that great mathematician; however this may have been, there is truth in the reply quoted by the same author from Geminus: that neither Archimedes nor Apollonius pretended to have invented this branch of Geometry, but that Apollonius had introduced a real improvement into it. For whereas Archimedes, according to the ancient method, considered only the section of a *right* cone by a plane perpendicular to its side, so that the species of the curve depended upon the angle of the cone; Apollonius took a more general view, conceiving the curve to be produced by the intersection of *any* plane with a cone generated by a right line passing always through the circumference of a fixed circle and *any* fixed point.

#### Editions

*The principal edition of the Conics is that of Halley, " Apoll. Perg. Conic, lib. viii., &c.," Oxon. 1710, fol.*The eighth book is a conjectural restoration founded on the introductory lemmata of Pappus.

*The first four books were translated into Latin, and published by J. Bapt. Memus (Venice, 1537)*, and by

*Commandine (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)*.

#### Other works

Apollonius was the author of several other works. The following are described by Pappus in the 7th book of his Mathematical Collections:--

#### and Περὶ Χωρίου Ἀποτομῆς

__Περὶ Λόγου Ἀποτομῆς__and Περὶ Χωρίου Ἀποτομῆς, 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.

#### Editions

Of the first of these an Arabic version is still extant, of which a translation was edited by*Halley, with a conjectural restoration of the second. (Oxon. 1706.)*

#### 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,

*l.*c.) A solution of this problem was published by Robt. Simson.

#### Περὶ Τόπων Ἐπιπέδων

Περὶ Τόπων Ἐπιπέδων, " A Treatise in two books on*Plane Loci.*Restored by Robt. Simson," Glasg. 1749.

__Περὶ Ἐπαφῶν__, 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. Camerer. Goth. et Amst. 1795, 8vo.)

__Περὶ Νεύσεων__, 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. *De Cochlea* and *De Perturbatis Rationibus.*

####
Ptolemy (*Magn. Const.* lib. xii. init.) refers to Apollonius for the demonstration of certain propositions relative to the stations and retrogradations of the planets.

####
Eutocius, in his commentary on the Dimensio Circuli of Archimedes, mentions an arithmetical work called Ὠκυτόβοον, (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 Mathém.* 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]