PAPPUS OF ALEXANDRIA, Greek geomet&, 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 by 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 cited or of the time at which he himself wrote. Since he frequently quotes Ptolemy the astronomer (fl., say, A.D. 140), he can hardly have been earlier than the end of the 2nd century. A marginal note to a loth century ms. states, in connection with the reign of Diocletian (A.D. 284-305) that Pap pus wrote during that period ; and in the absence of better testi mony it seems best to accept this indication. Suidas, it is true, makes him contemporary with Theon of Alexandria (4th century), who wrote a commentary on Ptolemy's Syntaxis; but, as Pappus also wrote a commentary (doubtless largely assimilated by Theon), Suidas perhaps failed to disconnect the two, and so assigned the same date to both.
The great work of Pappus, in eight books and entitled avvayaryil 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 : Xcopoypackia 01 KOVIIEPLA, els Ta reacrapa 1303X/a TC1S IlroXektaLov tleyaXis 01111 T4ECOS ITOTap.OVS TOin Ev Ati307, 6vEtp0Kpl.TI.Ka. He also wrote commentaries on the Analemma of Diodorus, on Ptolemy's Planispherium and Harmonica, and on Euclid's Elements. Cita tions from the commentary on the Elements are made by Proclus and others, while fragments of the portion relating to book x. survive in the Arabic.
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 valu able 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 terminology.
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 continued multiplication coupled with the expression of large numbers in terms of "tetrads" (powers of io,000), due to Apollonius of Perga.
Book iii. contains geometrical problems, plane and solid. It may be divided into five sections: (I) On the famous problem of finding two mean proportionals between two given lines, to which Hippocrates of Chios had reduced the problem of duplicating the cube. Pappus gives several solutions of this problem, including his own, and also a method of approximating continually to a solution, the significance of which he apparently failed to ap preciate. (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 dis tinguishes ten kinds with examples. (3) On a curious problem suggested by Eucl. i. 21. (4) On the inscribing of each of the five regular polyhedra in a sphere.
Of book iv. the title and preface have been lost, so that the programme has to be gathered from the book itself. At the be ginning is the well-known generalization of Eucl. i. 47, then fol low 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 semi-circles and known as ap,3r7Xos (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, and the curve discovered by Hippias of Elis about 42o B.C., and known by the name i or quadratrix. Proposition 3o describes the construction of a curve of double curvature called by Pappus the helix on a sphere, a construction analogous to that of Archime des's spiral in a plane. The area of the surface included between this curve and its base is found by the classical method of "ex haustion" equivalent to integration. The rest of the book treats of the trisection of any 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.