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PAPPUS OF ALEXANDRIA

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Originally appearing in Volume V20, Page 741 of the 1911 Encyclopedia Britannica.

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PAPPUS OF See also:

ALEXANDRIA
ALEXANDRIA (Arab. Iskenderia)

ALEXANDRIA , See also:

Greek geometer, flourished about the end of the 3rd See also:

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

century A.D. In a See also:

period of See also:

general stagnation in mathematical studies, he stands out as a remark-able exception. How far he was above his contemporaries, how little appreciated or understood by them, is shown by the See also:

ABSENCE (Lat. absentia)

absence of references to him in other Greek writers, and by the fact that his See also:

WORK

work had no effect in arresting the decay of mathematical See also:

SCIENCE (Lat. scientia, from scire; to learn, know)

science. In this respect the See also:

FATE

fate of Pappus strikingly resembles that of See also:

DIOPHANTUS

Diophantus. In his Collection, Pappus gives no indication of the date of the authors whose See also:

TREATISES

treatises he makes use of, or of the See also:

time at which he himself wrote. If we had no other See also:

INFORMATION (from Lat. informare, to give shape or form to, to represent, describe)

information than can be derived from his work, we should only know that he was later than See also:

Claudius See also:

PTOLEMY (CLAUDIUS PTOLEMAEUS)

Ptolemy whom he often quotes. Suidas states that he was of the sameage as See also:

Theon of Alexandria, who wrote commentaries on Ptolemy's See also:

GREAT

great work, the Syntaxis mathematica, and flourished in the reign of See also:

THEODOSIUS

Theodosius I. (A.D. 379—395). Suidas says also that Pappus wrote a commentary upon the same work of Ptolemy. But it would seem incredible that two contemporaries should have at the same time and in the same See also:

style composed commentaries upon one and the same work, and yet neither should have been mentioned by the other, whether as friend or opponent. It is more probable that Pappus's commentary was written See also:

long before Theon's, but was largely assimilated by the latter, and that Suidas, through failure to disconnect the two commentaries, assigned a like date to both.

A different date is given by the marginal notes to a loth-century MS., where it is stated, in connexion with the reign of See also:

Diocletian (A.D. 284—305), that Pappus wrote during that period; and in the absence of any other testimony it seems best to accept the date indicated by the scholiast. The great work of Pappus, in eight books and entitled owa'ywyil or Collection, we possess only in an incomplete See also:

FORM (Lat. forma)

form, the first See also:

BOOK

book being lost, and 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 having suffered considerably. Suidas enumerates other See also:

WORKS

works of Pappus as follows: Xwpoyparga otKOVgez'u , ets Ta T ogapa , 343X11 T'1~S IITOAe;aatov u ya?rts vuvrb €ws irvoµvmua, iroraµous rows iv Ac/3u?7, ovecpoKperLKa. The question of Pappus's commentary on Ptolemy's work is discussed by Hultsch, Pa ppi collectio (See also:

Berlin, 1878), vol. iii. p. xiii. seq. Pappus himself refers to another commentary of his own on the 'AvaXrlµua of Diodorus, of whom nothing is known. He also wrote commentaries on See also:

Euclid's Elements (of which fragments are preserved in See also:

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

Proclus and the Scholia, while that on the tenth Book has been found in an Arabic MS.), and on Ptolemy's `Apuovith. The characteristics of Pappus's Collection are that it contains an See also:

account, systematically arranged, of the most important results obtained by his predecessors, and, secondly, notes explanatory of, or extending, previous discoveries. These discoveries form, in fact, a See also:

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

text upon which Pappus enlarges discursively. Very valuable are the systematic introductions to the various books which set forth clearly in outline the contents and the general See also:

SCOPE (through Ital. scopo, aim, purpose, intent, from Gr. o'KOaos, mark to shoot at, aim, o ic07reiv, to see, whence the termination in telescope, microscope, &c.)

scope of the subjects to be treated. From these introductions we are able to See also:

JUDGE (Lat. judex, Fr. juge)

judge of the style of Pappus's See also:

writing, which is excellent and even elegant the moment he is See also:

FREE

free from the shackles of mathematical formulae and expressions. At the same time, his characteristic exactness makes his collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us.

We proceed to summarize briefly the contents of that portion of the Collection which has survived, mentioning separately certain propositions which seem to be among the most important. We can only conjecture that the lost book i., as well as book ii., was concerned with See also:

ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number)

arithmetic, book iii. being clearly introduced as beginning a new subject. The whole of book ii. (the former See also:

PART

part of which is lost, the existing fragment beginning in the See also:

MIDDLE

middle of the 14th proposition) related to a See also:

SYSTEM

system of multiplication due to See also:

APOLLONIUS

Apollonius of See also:

PERGA (mod. Murtana)

Perga. On this subject see Nesselmann, See also:

ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

Algebra der Griechen (Berlin, 1842), pp. 125–134; and M. Cantor, Gesch. d. Math. i.2 331. Book iii. contains geometrical problems, See also:

PLANE

plane and solid. It may be divided into five sections: (1) On the famous problem of finding two mean proportionals between two given lines, which arose from that of duplicating the See also:

CUBE (Gr. K46os, a cube)

cube, reduced by See also:

HIPPOCRATES

Hippocrates to the former. Pappus gives several solutions of this problem, including a method of making successive approximations to the See also:

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

solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the See also:

side of a cube whose content is in any given ratio to that of a given one. (2) On the arithmetic, geometric and See also:

HARMONIC

harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure.

This serves as an introduction to a general theory of means, of which Pappus distinguishes ten kinds, and gives a table representing examples of each in whole See also:

numbers. (3) On a curious problem suggested by Eucl. i. 21. (4) On the inscribing of each of the five See also:

REGULAR

regular polyhedra in a See also:

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

sphere. (5) An addition by a later writer on another solution of the first problem of the book. Of book iv. the See also:

TITLE (0. Fr. title, mod. titre, from Lat. titulus)

title and See also:

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

preface have been lost, so that the See also:

pro-gramme has to be gathered from the book itself. At the beginning is the well-known generalization of Eucl. i. 47, then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two. This and several other propositions on contact, e.g. cases of circles touching one another and inscribed in the figure made of three semicircles and known as fpi3 Xos (See also:

SHOE (a word appearing in the Teutonic languages in various forms, as Ger. Schuh, Swed. and Dan. sko, sometimes supposed to come from an unknown root ska or sku, cover)

shoe-maker's See also:

KNIFE (0. E. cuff, a word appearing in different forms in many Teutonic languages, cf. Du. knijf, Ger. Kneif, a shoe-maker's knife, Swed. knif; the ultimate origin is unknown; Skeat finds the origin in the root of " nip," formerly " knip "; Fr. canif is a

knife) form the first See also:

DIVISION (from Lat. dividere, to break up into parts, separate)

division of the book: Pappus turns then to a See also:

CONSIDERATION (from Lat. considerare, to look at closely, examine, generally taken to be from con-, and the base seen in sidus, sideris, a star, the word being supposed to be originally an astrological or astronomical term)

consideration of certain properties of See also:

Archimedes'.s See also:

SPIRAL

spiral, the See also:

CONCHOID (Gr. «oyXn, shell, and ethos, form)

conchoid of Nicomedes (already mentioned in book i. as supplying a method of doubling the cube), and the See also:

CURVE (Lat. curvus, bent)

curve discovered most probably by Hippias of See also:

Elis about 420 B.C., and known by the name s} r€rpaiewvii-ouva, or See also:

QUADRATRIX (from Lat. quadrator, squarer)

quadratrix. Proposition 3o describes the construction of a curve of See also:

double curvature called by Pappus the See also:

HELIX (Gr. € tE, a spiral or twist)

helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its See also:

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

diameter uniformly, the point describing a quadrant and the great circle a See also:

COMPLETE

complete revolution in the same time. The See also:

AREA

area of the See also:

SURFACE

surface included between this curve and its See also:

BASE

base is found—the first known instance of a See also:

QUADRATURE (from Lat. quadratura, a making square)

quadrature of a curved surface. The rest of the book treats of the trisection of an See also:

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle, and the solution of more general problems of the same See also:

KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind by means of the quadratrix and spiral.

In one solution of the former problem is the first recorded use of the See also:

PROPERTY

property of a conic (a See also:

HYPERBOLA

hyperbola) with reference to the See also:

FOCUS (Latin for " hearth " or " fireplace ")

focus and directrix. In book v., after an interesting preface concerning regular polygons, and containing remarks upon the hexagonal form of the cells of honeycombs, Pappus addresses himself to the comparison of the areas of different plane figures which have all the same perimeter (following Zenodorus's See also:

TREATISE

treatise on this subject), and of the volumes of different solid figures which have all the same superficial area, and, lastly, a comparison of the five regular solids of See also:

PLATO

Plato. Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and See also:

VOLUME

volume of a sphere. According to the preface, book vi. is intended to resolve difficulties occurring in the so-called 1.10cpf3 harpovoµobµevos. It accordingly comments on the Sphaerica of Theodosius, the Moving Sphere of See also:

AUTOLYCUS

Autolycus, Theodosius's book on See also:

Day and See also:

NIGHT

Night, the treatise of See also:

ARISTARCHUS

Aristarchus On the See also:

SIZE

Size and Distances of the See also:

Sun and See also:

Moon, and Euclid's See also:

OPTICS

Optics and Phaenomena. The preface of book vii. explains the terms See also:

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

analysis and See also:

SYNTHESIS (Gr. a6vOeacs, from avvrcBivac, to put together)

synthesis, and the distinction between theorem and problem. Pappus then enumerates works of Euclid, Apollonius, See also:

ARISTAEUS

Aristaeus and Eratosthenes, See also:

THIRTY

thirty-three books in all, the substance of which he intends to give, with the lemmas necessary for their elucidation. With the mention of the Porisms of Euclid we have an account of the relation of See also:

PORISM

porism to theorem and problem. In the same preface is included (a) the famous problem known by Pappus's name, often enunciated thus: Having given a number of straight lines, to find the geometric See also:

LOCUS (Lat. for " place "; in Gr. rinros)

locus of a point such that the lengths of the perpendiculars upon, or (more generally) the lines See also:

DRAWN

drawn from it obliquely at given inclinations to, the given lines 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 product of certain of them may See also:

bear a See also:

CONSTANT, BENJAMIN (1845-1902)

constant ratio to the product of the remaining ones; (Pappus does not See also:

EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express it in this form but by means of See also:

composition of ratios, saying that if the ratio is given which is compounded of the ratios of pairs—one of one set and one of another—of the lines so drawn, and of the ratio of the See also:

ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

odd one, if any, to a given straight See also:

LINE

line, the point will See also:

lie on a curve given in position) ; (b) the theorems which were rediscovered by and named after See also:

Paul Guldin, but appear to have been discovered by Pappus himself. Book vii. contains also (1), under the See also:

head of the de determinata sectione of Apollonius, lemmas which, closely examined, are seen to be cases of the involution of six points; (2) important lemmas on the Porisms of Euclid (see PORISM) ; (3) a lemma upon the Surface Loci of Euclid which states that the locus of a point such that its distance from a given point bears a constant ratio to its distance from a given straight line is a conic, and is followed by proofs that the conic is a See also:

PARABOLA

parabola, See also:

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

ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than 1 (the first recorded proofs of the properties, which do not appear in Apollonius). Lastly, book viii. treats principally of See also:

MECHANICS

mechanics, the properties of the centre of gravity, and some See also:

MECHANICAL

mechanical See also:

POWERS, HIRAM (1805-1873)

powers. Interspersed are some questions of pure See also:

GEOMETRY

geometry.

Proposition 14 shows how to draw an ellipse through five given points, and Prop. 15 gives a See also:

SIMPLE

simple construction for the axes of an ellipse when a pair of conjugate diameters are given. Of books which contain parts of Pappus's work, or treat incidentally of it, we may mention the following titles: (I) Pappi alexandrini collectiones mathematicae See also:

NUN (O. Eng. nunne, from Lat. nonnus, nonna, familiar terms for an old man or woman)

nun. primum graece edidit Herm. 1os. Eisen-mane, libri quint%pars altera (Parisiis, 1824). (2) Pappi alexandrini secundi libri mathematicae collectionis fragmentum e codice MS. edidit latinum fecit notisque illustravit Johannes See also:

WALLIS, JOHN (1616-1703)

Wallis (Oxonii, 1688). (3) Apollonis pergaei de sectione rationis libri duo ex arabico MS,. See also:

WINE (Lat. vinum, Gr. oivos)

Wine versi, accedunt esusdem de sectione spatii libri duo restituti, praemitlitur Pappi alexandrini praefatio ad VIP'""'' collections mathematicae, nunc primum graece eaita : cum lemmatibus eiusdem Pappi ad hos Apollonii libros, See also:

OPERA (Italian for " work ")

opera et studio Edmundi See also:

HALLEY, EDMUND (1656–1742)

Halley (Oxonii. 1706). (4) Der Sammlung See also:

des Pappus von Alexandrien siebentes and achtes See also:

Buch griechisch and See also:

DEUTSCH, IMMANUEL OSCAR MENAHEM (1829-1873)

deutsch, published by C. I. See also:

Gerhardt, See also:

Halle, 1871.

(5) The portions See also:

RELATING

relating to Apollonius are reprinted in See also:

HEIBERG, JOHAN LUDVIG (1791-186o)

Heiberg's Apollonius, ii. iox sqq. (T. L.

End of Article: PAPPUS OF ALEXANDRIA

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