**PAPPUS, OF ALEXANDRIA**, a geometer of a very high order, belongs to a time when already the Greek mathematicians of great original genius had been succeeded and replaced by a race of learned compilers and com-mentators, who confined their investigations within the limits previously attained, without adding anything to the development of mathematics. To the general mediocrity Pappus must be considered to be a remarkable exception; for, although much even of his work is of the nature of compilation (which is, however, itself of great historical value), there is yet much the discovery of which cannot well be attributed to any one else. According to Proclus, he was at the head of a school; but 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, another living power amid general stagnation. In reading the Collection of Pappus, we meet with 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 a perusal of his work, we should only know that he was later than Claudius Ptolemy, whom he quotes often and with respect. Suidas states that he was of the same age as Theon of Alexandria, who wrote commentaries on Ptolemy's great work, the Almagest, and flourished in the reign of Theodosius I. (379-395 A.D.). Suidas asserts 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. We have apparently no reason to question the statement of Suidas that Pappus wrote such a commentary. But the similarity of two such commentaries as those of Pappus and Theon may easily have led Suidas to confuse the two, and so suppose the two authors to have been contemporary. There is, then, reason to believe that Suidas may have been mistaken; we have, however, another authority, whose statement, on the supposition that it is false, is completely incomprehensible. This is the author of certain historical glosses, which are found in the margin of a MS. belonging to the beginning of the 10th century. Here it is stated, in connexion with the reign of Diocletian (284-305 A.D.), that Pappus wrote during that period. Except the two distinctly contradictory statements of Suidas and the scholiast, we have no evidence of the date of Pappus; and it seems accordingly best to accept the date indicated by the scholiast.

The work of Pappus which has come down to us bore the title _____ or Collection, as we gather from references in the work itself, and from the scholia appended to the Vatican MS. 218 of the 12th century. This collection, which consisted of eight books, we possess only in an incomplete form, there being no part remaining of the first book, and the rest also having suffered considerably. It is curious that no ancient writer, with the exception of the author of the appendix to book iii., quotes the work under its proper title, though Eutocius's reference (Archimedes, p. 139 sq., ed. Torelli), OJS IlaWos ¤v pT]xa.viK(us eto-ayojyais, is no doubt to book viii. of the Collection.

Suidas enumerates other works of Pappus as follows :— _____, _____ _____ _____, _____, _____. The question of Pappus's commentary on Ptolemy's work is discussed by Hultsch, Pappi Collectio (Berlin, 1878), vol iii. p. xiii. sq. Pappus himself refers to another commentary of his own on the _____ of Diodorus, of whom nothing is known. There are, moreover, indications that he commented on Euclid's Elements, and on Ptolemy's _____. Further, there is a doubtful work entitled Opusculum de multiiilieatione et divisione sexagesimalibus P/iophanto vel Pappo tribuendum, which has been edited by C, Henry (Halle, 1879); and, lastly, a tract, Anonymi commentaries de figuris planis isoperimetris, has been inserted by Hultsch in vol. iii. of his edition of Pappus.

The characteristics of Pappus's Collection are that it con-tains 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, many of his additions having no very decided points of connexion with the direct subject under discussion. 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, in the light of modern developments of mathematics, to be among the most important.

Of book i. the whole has been lost. We can only conjecture that it, 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 exist-ing 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 Friedlein, *Die Zahlzeichen und das elementare Rechnen der Griechen und Römer*, Erlangen, 1869.

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 doubling the cube, reduced by Hippocrates to the former. Pappus gives the solutions of this problem by Eratosthenes, Nicomedes, and Heron, and finally 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 three different means between two straight lines, the arithmetic, the geometric, and the harmonic, 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 repre-senting examples of each in whole numbers. (3) On a curious problem of the same type as Eucl. i. 21. (4) On the inscribing of each of the five regular polyhedra 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 begin-ning are various theorems on the circle, leading up to the problem of the construction of a circle wdiich shall circumscribe three given circles touching each other two and two. This and several other problems of contact 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 dis-covered most probably by Hippias of Elis about 420 B.C., and known by the name i) ______, or quadratrix, from the property that, if it could be practically constructed, it would enable us to square the circle. Having described the ordinary—the mechanical, as Pappus calls it—definition of this curve, he proceeds to show how it might be constructed by projecting orthogonally suitable plane sections of certain surfaces wdiich he calls plectoids described by means of (a) the helix described on a cylinder, (b) the plane helix, or Archimedes's spiral. From these propositions it would seem that plectoid was the Greek general term for surfaces described by the motion of a straight line always passing through a fixed straight line and a curve, and remaining parallel to a fixed plane. 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 are 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 instance of quadrature of a curved surface. The rest of the book treats of the trisection of an angle, and the solution of certain problems by means of the quadratrix and spiral.

In book v., after an interesting preface concerning regular polygons, and containing some 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 wdiich have all the same superficial area, and, lastly, a comparison of the five regular solids of Plato.

According to the preface, book vi. is intended to resolve diffi-culties occurring in the so-called ______. It accordingly comments on the Sphmrica of Theodosius, a treatise 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 Phenomena.

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, Aristeeus, 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 Porisiiis of Euclid we have an account of the relation of porism to theorem and problem. In the same preface we have enunciated (a) the famous problem known by Pappus's name—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 inclina-tions 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 ; (b) the theorems which since the 17th century have been called by the name of Guldin, but appear to have been dis-covered by Pappus himself. Book vii. contains also (1) under the head of the de determinata nectione 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 Conies of Apollonius, which is the first statement of the constant relation between the distances of any point on a conic from the focus and directrix.

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 gives a simple construction for the axes of an ellipse, when a pair of conjugate diameters are given.

Of the whole work of Pappus the best edition is that of Hultsch, bearing the title Pappi Alexandrini Collectionis quae supersunt e libris manuscriptis edidit Latina interprelatione et commentariis instruxit Fridericus Hultsch, Berlin, 1875-78. Previously the entire collection had been published only in a Latin translation, Pappi Alexandrini mathematicae collectiones a Federico Commandino Urbinate in latinum conrersae et commentariis illusti atx, Pesaro, 1588 (reprinted at Venice, 1589, and Pesaro, 1602). A second edition of this work was published by Carolas Manolessius, entitled Pappi Alexandrini mathematicx collectiones a Federico Commandino Urbinate in latinum conversx et commentariis illustrate, in hoc nostra editione innumeris quibus scatebant mendis et prxcipue in Graeco contextu diligenter vindicatx, Bologna, 1660. The merits of these two works are discussed by Hultsch, who remarks that the editor of the second edition, so far from making good the title and his boastful preface, has actually much marred the original book.

Of books which contain parts of Pappus's work, or treat incidentally of it, we may mention the following titles:—(1) Pappi Alexandrini collectiones mathematicae nunc primum Graece edidit Herrn. Jos. Eisenmann, Libri quinti pars altera, Parisiis, 1824. (2) Pappi Alexandria/. Secundi Libri Mathematicae Collectionis Fragmentum e codice MS. edidit Latinum fecit Notisque illustravit Johannes Wallis, Oxonite, 1688. (3) Apollonii Pergxi de sections rationis libri duo ex Arabico MSto latine versi, Accedunt eiusdem de sectione spatii libri duo restituti, Praemittitur Pappi Alexandrini praefatio ad Yllmum collectionis mathematicae, nunc primum graece edita: cum lemmatibus eiusdem Pappi ad hos Apollonii libros. Opera et studio Edmundi Halley, Oxonii, 1706. (4) Apollonii Pergsei conicorum libri IV. priores cum Pappi Alexandrini lemmatis ex codd. MSS. Graecis edidit Edmundus Halleius, Oxonite, 1710. (5) Der Sammlung des Pappus von Alexandrien siebentes und achtes Buch griechisch und deutsch herausgegeben von C. I. Gerhardt, Halle, 1871. (T. L. H.)

**The above article was written by:** T. L. Heath, B.A.