1911 Encyclopædia Britannica/Pappus of Alexandria

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20821711911 Encyclopædia Britannica, Volume 20 — Pappus of AlexandriaThomas Little Heath

PAPPUS OF ALEXANDRIA, Greek geometer, flourished about the end of the 3rd century A.D. In a period of general stagnation in mathematical studies, he stands out as a remarkable exception. How far he was above his contemporaries, how little appreciated or understood by them, is shown by the absence of references to him in other Greek writers, and by the fact that his work had no effect in arresting the decay of mathematical science. In this respect the fate of Pappus strikingly resembles that of Diophantus. In his Collection, Pappus gives no indication of the date of the authors whose treatises he makes use of, or of the time at which he himself wrote. If we had no other information than can be derived from his work, we should only know that he was later than Claudius Ptolemy whom he often quotes. Suidas states that he was of the same age as Theon of Alexandria, who wrote commentaries on Ptolemy’s great work, the Syntaxis mathematica, and flourished in the reign of 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 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 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 10th-century MS., where it is stated, in connexion with the reign of 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 συναγωγή or Collection, we possess only in an incomplete form, the first book being lost, and the rest having suffered considerably. Suidas enumerates other works of Pappus as follows: Χωρογραφία οἰκουμενική, εἰς τὰ τέσσαρα βιβλία τῆς Πτολεμαίου μεγάλης ουντάξεως ὑπόμνημα, ποταμοὺς τοὺς ἐν Λιβύῃ, ὀνειροκριτικά. The question of Pappus’s commentary on Ptolemy’s work is discussed by Hultsch Pappi collectio (Berlin, 1878), vol. iii. p. xiii. seq. Pappus himself refers to another commentary of his own on the Ἀνάλημμα of Diodorus, of whom nothing is known. He also wrote commentaries on Euclid’s Elements (of which fragments are preserved in Proclus and the Scholia, while that on the tenth Book has been found in an Arabic MS.), and on Ptolemy’s Ἁρμονικά.

The characteristics of Pappus’s Collection are that it contains an 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 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 scope of the subjects to be treated. From these introductions we are able to judge of the style of Pappus’s writing, which is excellent and even elegant the moment he is 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 arithmetic, book iii. being clearly introduced as beginning a new subject.

The whole of book ii. (the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition) related to a system of multiplication due to Apollonius of Perga. On this subject see Nesselmann, Algebra der Griechen (Berlin, 1842), pp. 125–134; and M. Cantor, Gesch. d. Math, i.² 331.

Book iii. contains geometrical problems, 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 cube, reduced by Hippocrates to the former. Pappus gives several solutions of this problem, including a method of making successive approximations to the solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the side of a cube whose content is in any given ratio to that of a given one. (2) On the arithmetic, geometric and 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 numbers. (3) On a curious problem suggested by Eucl. i. 21. (4) On the inscribing of each of the five regular polyhedral in a sphere. (5) An addition by a later writer on another solution of the first problem of the book.

Of book iv. the title and preface have been lost, so that the programme 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 ἄρβηλος (shoemaker’s knife) form the first division of the book. Pappus turns then to a consideration of certain properties of Archimedes’s spiral, the conchoid of Nicomedes (already mentioned in book i. as supplying a method of doubling the cube), and the curve discovered most probably by Hippias of Elis about 420 B.C., and known by the name ἡ τετραγωνίζουσα, or quadratrix. Proposition 30 describes the construction of a curve of double curvature called by Pappus the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its diameter uniformly, the point describing a quadrant and the great circle a complete revolution in the same time. The area of the surface included between this curve and its base is found—the first known instance of a quadrature of a curved surface. The rest of the book treats of the trisection of an angle, and the solution of more general problems of the same kind by means of the quadratrix and spiral. In one solution of the former problem is the first recorded use of the property of a conic (a hyperbola) with reference to the 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 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 Plato. Incidentally Pappus describes the thirteen other polyhedral bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.

According to the preface, book vi. is intended to resolve difficulties occurring in the so-called μικρὸς ἀστρονομούμενος. It accordingly comments on the Sphaerica of Theodosius, the Moving Sphere of Autolycus, Theodosius’s book on Day and Night, the treatise of Aristarchus On the Size and Distances of the Sun and Moon, and Euclid’s Optics and Phaenomena.

The preface of book vii. explains the terms analysis and synthesis, and the distinction between theorem and problem. Pappus then enumerates works of Euclid, Apollonius, Aristaeus and Eratosthenes, 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 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 locus of a point such that the lengths of the perpendiculars upon, or {more generally) the lines drawn from it obliquely at given inclinations to, the given lines satisfy the condition that the product of certain of them may bear a constant ratio to the product of the remaining ones; (Pappus does not express it in this form but by means of 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 odd one, if any, to a given straight line, the point will lie on a curve given in position); (b) the theorems which were rediscovered by and named after Paul Guldin, but appear to have been discovered by Pappus himself. Book vii. contains also (I), under the head of the de determinate 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 parabola, ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than I (the first recorded proofs of the properties, which do not appear in Apllonius).

Lastly, book viii. treats principally of mechanics, the properties of the centre of gravity, and some mechanical powers. Interspersed are some questions of pure geometry. Proposition 14 shows how to draw an ellipse through five given points, and Prop. 15 gives a simple construction for the axes of an ellipse when a pair of conjugate diameters are given.

Authorities.Of the whole work of Pappus the best edition is that of Hultsch, bearing the title Pappi alexandrini colleclionis quae supersunt e libris manuscript is edidit latina interpretatione et commentaries inslruxit Fridericus Hultsch (Berlin, 1876-1878). Previously the entire collection had been published only in a Latin translation, Pappi alexandrini mathematical collectiones a Federico Commandino Urbinate in latinum converse et commentariis illustrate (Pesaro, 1588) (reprinted at Venice, 1589, and Pesaro, 1602). A second (inferior) edition of this work was published by Carolus Manolessius.

Of books which contain parts of Pappus’s work, or treat incidentally of it, we may mention the following titles: (1) Pappi alexandrini collection es mathematical nunc primum graece edidit Herm. Jos. Eisenmann, libri quinti pars altera (Parisiis, 1824). (2) Pappi alexandrini secundi libri mathematical colleclionis fragment um e codice MS. edidit latinum fecit fibtisque illustravil Johannes Wallis (Oxonii, 1688). (3) Apollonii pergaei de sectione rationis libri duo ex arabico MSto latine versi, accedunt eiusdem de sectione spatii libri duo restitiiti, praemittitur Pappi alexandrini praefatio ad VIImum colleclionis mathematical, nunc primum graece edita: cum lemmatibus eiusdem Pappi ad hos Apollonii libros, opera et studio Edmundi Halley (Oxonii. 1706). (4) Der Sammlung des Pappus von Alexandrien siebentes und achtes Buck griechisch und deutsch, published by C. I. Gerhardt, Halle, 1871. (5) The portions relating to Apllonius are reprinted in Heiberg’s Apollonius, ii. 101 sqq.  (T. L. H.)