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general indications of their contents, by later. writers, especially Pappus. After the Conics in eight Books had been written in a first edition, Apollonius brought out a second edition, considerably revised as regards Books i.-ii., at the instance of one Eudemus of See also:

PERGAMUM, or PERGAMTS (mod. Bergama)

Pergamum; the first three books were sent to Eudemus at intervals, as revised, and the later books were dedicated (after Eudemus' See also:

DEATH

death) to See also:

King Attalus I. (241–197 B.C.). Only four Books have survived in Greek; three more are extant in Arabic; the eighth has never been found. Although a fragment has been found of a Latin See also:

TRANSLATION

translation from the Arabic made in the 13th See also:

CENTURY (from Lat. centuria, a division of a hundred men)

century, it was not until 1661 that a Latin translation of Books v.-vii. was available. This was made by Giovanni Alfonso See also:

BORELLI, GIOVANNI ALFONSO (1608-1679)

Borelli and See also:

ABRAHAM, or ABRAM (Hebrew for " father is high ")

Abraham See also:

ECCHELLENSIS (or ECHELLENSIS), ABRAHAM (d. 1664)

Ecchellensis from the See also:

FREE

free version in Arabic made in 983 by See also:

Abu '1-Fath of Ispahan and preserved in a See also:

Florence MS. But the best Arabic translation is that made as regards Books i.-iv. by Hilal See also:

ibn Abi Hilal (d. about 883), and as regards Books v.-vii. by See also:

TOBIT, THE BOOK OF

Tobit See also:

BEN (from Old Eng. bennan, within)

ben Korra (836–901). See also:

HALLEY, EDMUND (1656–1742)

Halley used for his translation an See also:

Oxford MS. of this translation of Books v.-vii., but the best MS. (Bodl.

943) he only referred to in See also:

order to correct his translation, and it is still unpublished except for a fragment of See also:

BOOK

Book v. published by L. Nix with See also:

German translation (Drugulin, See also:

LEIPZIG

Leipzig, 1889). Halley added in his edition (1710) a restoration of Book viii., in which he was guided by the fact that Pappus gives lemmas " to the seventh and eighth books " under that one heading, as well as by the statement of Apollonius himself that the use of the seventh book was illustrated by the problems solved in the eighth. The degree of originality of the Conics can best be judged from Apollonius' own prefaces. Books i.-iv. See also:

FORM (Lat. forma)

form an " elementary introduction," i.e. contain the essential principles; the See also:

REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest are specialized investigations in particular directions. For Books i.-iv. he claims only that the See also:

GENERATION (from Lat. generare, to beget, procreate; genus, stock, race)

generation of the curves and their fundamental properties in Book i. are worked out more fully and generally than they were in earlier treatises, and that a number of theorems in Book iii. and the greater See also:

PART

part of Book iv. are new. That he made the fullest use of his predecessors' See also:

WORKS

works, such as See also:

Euclid's four Books on Conics, is clear from his allusions to Euclid, See also:

CONON

Conon and Nicoteles. The generality of treatment is indeed remarkable; he gives as the fundamental See also:

PROPERTY

property of all the conics the See also:

EQUIVALENT

equivalent of the Cartesian See also:

EQUATION (from Lat. aequatio, aequare, to equalize)

equation referred to oblique axes (consisting of a See also:

DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter and the tangent at its extremity) obtained by cutting an oblique circular See also:

CONE (Gr. Kwvos)

cone in any manner, and the axes appear only as a particular See also:

case after he has shown that the property of the conic can be expressed in the same form with reference to any new diameter and the tangent at its extremity. It is clearly the form of the fundamental property (expressed in the terminology of the " application of areas ") which led him to See also:

CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

call the curves for the first See also:

time by the names See also:

PARABOLA

parabola, See also:

ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

ellipse, See also:

HYPERBOLA

hyperbola. Books v.-vii. are clearly See also:

ORIGINAL

original. Apollonius' See also:

GENIUS (from Lat. genere, gignere)

genius takes its highest See also:

FLIGHT

flight in Book v., where he treats of normals as minimum and maximum straight lines See also:

DRAWN

drawn from given points to the See also:

CURVE (Lat. curvus, bent)

curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic. The other treatises of Apollonius mentioned by Pappus are -1st, AOyov aaoro a', Cutting off a Ratio; 2nd, Xwplou airoroµil, Cutting off an See also:

AREA

Area; 3rd, t.twplvpEVq ropil, Determinate See also:

Section; 4th,'Eira¢ai, Tangencies; 5th, See also:

NEBO, or NABU (" the proclaimer ")

Nebo-es, Inclinations; 6th, Tolroi E7rl7rEb01, See also:

PLANE

Plane Loci.

Each of these was divided into two books, and, with the Data, the Porisms and See also:

SURFACE

Surface-Loci of Euclid and the Conics of Apollonius were, according to Pappus, included in the See also:

BODY

body of the See also:

ancient See also:

ANALYSIS (Gr. avci and Meta, to break up into parts)

analysis. 1st. De Rationis Sectione had for its subject the See also:

RESOLUTION

resolution of the following problem: Given two straight lines and a point in each, to draw through a third given point a straight See also:

LINE

line cutting the two fixed lines, so that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio. 2nd. De Spatii Sectione discussed the similar problem which requires the rectangle contained by the two intercepts to be equal to a given rectangle. An Arabic version of the first was found towards the end of the 17th century in the Bodleian library by Dr See also:

EDWARD

Edward See also:

Bernard,.187 who began a translation of it; Halley finished it and published it along with a restoration of the second treatise in r7o6. 3rd. De Sectione Determinata resolved the problem: Given two, three or four points on a straight line, to find another point on it such that its distances from the given points satisfy the See also:

CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition that the square on one or the rectangle contained by two has to the square on the remaining one or the rectangle contained by the remaining two, or to the rectangle contained by the remaining one and another given straight line, a given ratio. Several restorations of the See also:

SOLUTION (from Lat. solvere, to loosen, dissolve)

solution have been attempted, one by W. Snellius (See also:

LEIDEN

Leiden, 1698), another by Alex. See also:

Anderson of See also:

Aberdeen, in the supplement to his Apollonius Redivivus (See also:

Paris, 1622), but by far the best is by See also:

Robert See also:

Simson, See also:

OPERA (Italian for " work ")

Opera quaedam reliqua (See also:

GLASGOW

Glasgow, 1776). 4th.

De Tactionibus embraced the following general problem: Given three things (points, straight lines or circles) in position, to describe a circle passing through the given points, and touching the given straight lines or circles. The most difficult case, and the most interesting from its See also:

HISTORICAL

historical associations, is when the three given things are circles. This problem, which is sometimes known as the Apollonian Problem, was proposed by See also:

VIETA (or VIETE), FRANCOIS, SEIGNEUR DE LA BIGOTIERE (1540-1603)

Vieta in the 16th century to Adrianus See also:

ROMANUS

Romanus, who gave a solution by means of a hyperbola. Vieta thereupon proposed a simpler construction, and restored the whole treatise of Apollonius in a small See also:

WORK

work, which he entitled Apollonius See also:

Gallus (Paris, 2600). A very full and interesting historical See also:

account of the problem is given in the See also:

PREFACE (Med. Lat. prefatia, for classical praefatio, praefari, to speak beforehand)

preface to a small work of J. W. Camerer, entitled A pollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in hos Libros, cum Observationibus, &c. (Gothae, 1795, 8vo). 5th. De Inclinationibus had for its See also:

OBJECT

object to insert a straight line of a given length, tending towards a given point, between two given (straight or circular) lines. Restorations have been given by See also:

MARINO

Marino Ghetaldi, by See also:

Hugo d'Omerique (Geometrical Analysis, See also:

Cadiz, 2698), and (the best) by See also:

SAMUEL

Samuel See also:

Horsley (1770). 6th.

De Locis Planis is a collection of propositions See also:

RELATING

relating to loci 'which are either straight lines or circles. Pappus gives somewhat full particulars of the propositions, and restorations were attempted by P. See also:

FERMAT, PIERRE DE (1601-1665)

Fermat (Euvres, i., 2891, pp. 3-51), F. Schooten (Leiden,. 1656) and, most successfully of all, by R. Simson (Glasgow, 1749). Other works of Apollonius are referred to by ancient writers, viz. (1) HEpi Tor) 7rvplou, On the Burning-See also:

Glass, where the See also:

FOCAL

focal properties of the parabola probably found a See also:

PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place; (2) Hepi rout Koxkiov, On the Cylindrical See also:

HELIX (Gr. € tE, a spiral or twist)

Helix (mentioned by See also:

PROCLUS, or PROCULUS (A.D. 410-485)

Proclus); (3) a comparison of the See also:

DODECAHEDRON (Gr. Melia, twelve, and ESpa, a face or base)

dodecahedron and the See also:

ICOSAHEDRON (Gr. e'IKOVL, twenty, and Ebpa, a face or base)

icosahedron inscribed in the same See also:

SPHERE (Gr. vclsa?pa, a ball or globe)

sphere; (4) `H KaObXou 7rpay sareia, perhaps a work on the general principles of See also:

MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics in which were included Apollonius' criticisms and suggestions for the improvement of Euclid's Elements; (5) 'Slevrbi ov (See also:

QUICK

quick bringing-to-See also:

BIRTH (a word common in various forms to Teutonic languages from the root of the verb " to bear ")

birth), in which, according to Eutocius, he showed how to find closer limits for the value of it than the 3 7 and 3+7- of Archimedes; (6) an arithmetical work (as to which see PAPPUS) on a See also:

SYSTEM

system of expressing large See also:

numbers in See also:

LANGUAGE (adapted from the Fr. langage, from langue, tongue, Lat. lingua)

language closer to that of See also:

COMMON

common See also:

LIFE

life than that of Archimedes' See also:

Sand-reckoner, and showing how to multiply such large numbers; (7) a great See also:

EXTENSION (Lat. ex, out ; tendere, to stretch)

extension of the theory of irrationals expounded in Euclid, Book x., from See also:

BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term)

binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' See also:

COMM

comm. on Eucl. x., preserved in Arabic and published by Woepcke, 1856). Lastly, in See also:

ASTRONOMY (from Gr. &arpov, a star, and v ,usiv, to classify or arrange)

astronomy he is 'credited by Ptolemy with an explanation of the See also:

motion of the See also:

PLANETS, MINOR

planets by a. system of epicycles; he also made researches in the lunar theory, for which he is said to have been called Epsilon (E). The best See also:

EDITIONS

editions of the works of Apollonius are the following : (I) Apollonii Pergaei Conicorum libri quatuor, ex version Frederici Commandini (Bononiae, 1566), fol. ;.

(2) A pollonii Pergaei Conicorum libri octo, et Sereni Antissensis de Sectione Cylindri et Coni libri duo (Oxoniae, 171o), fol. (this is the monumental edition of See also:

Edmund Halley) ; (3) the edition of the first four books of the Conics given in 1675 by See also:

Barrow; (4) Apollonii Pergaei de Sectione Rationis libri duo: Accedunt ejusdem de Sectione Spatii libri duo Restituti: Praemittitur, &c., Opera et Studio Edmundi Halley (Oxoniae, 1706), 4to; (5) a German translation of the Conics by H. See also:

BALSAM (from Gr. f &X raµov, through Lat. balsamum, contracted by popular use to O. Fr. basme, mod. Fr. bdme; Eng. balm)

Balsam (See also:

Berlin, 1861); (6) the definitive Greek See also:

TEXT (Lat. textum, woven fabric, from texere, to weave)

text of See also:

HEIBERG, JOHAN LUDVIG (1791-186o)

Heiberg (A pollonii Pergaeiquae Graece exstant Opera, Leipzig, 1891–1893); (7) T. L. See also:

Heath, Apollonius, Treatise on Conic Sections (See also:

Cambridge, 1896) ; see also H. G. Zeuthen, See also:

Die Lehre von den Kegelschnitten See also:

IM(K2+a2+h2—2 ah cos 0)

im Altertum (See also:

COPENHAGEN (Danish Kjobenhavn)

Copenhagen, 1886 and 1902). (T. L.

End of Article: APOLLONIUS OF PERGA [PERGAEUS]

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