PAPPUS OF ALEXANDRIA

flourished AD 320 the last great Greek geometer, whose Synagoge (c. AD 340; Collection) incorporates a wealth of mathematical writings, many of them no longer available in any other form. Pappus' efforts to arrest the general decay of mathematics in the late Roman Empire were unsuccessful, however. The Synagoge was made up of eight books, of which the first and part of the second are lost. His other works include commentaries on the Analemma (on an astronomical instrument) of Diodorus; on Ptolemy's Almagest (the classical astronomical work of his day), Planisphaerium, and Harmonica; and on Euclid's Elements. One of Pappus' own theorems is still cited as the basis of modern projective geometry. The Synagoge contains a systematic account of the most important works in ancient Greek mathematics, with historical annotations, improvements and alterations of existing theorems and propositions, and original material. This work was intended as a guide to be used with the original works. Included are systematic introductions to each book, setting forth clearly the contents and general scope of the topics to be treated. Book 1 covered arithmetic; the existing fragment of Book 2 sets forth a system of continued multiplication coupled with the expression of large numbers in terms of tetrads (powers of 10,000). Book 3 contains problems in plane and solid geometry, including that of finding two mean proportionals between two given lines. Pappus gave several solutions to this problem, one of them his own, and included a method of approximating continually to a solution, the significance of which he apparently failed to appreciate. The study of the arithmetic, geometric, and harmonic means and the problem of representing all three in one geometric figure served as an introduction to a general theory of means. He distinguished among 10 kinds of means with examples. Book 3 also reveals how each of the five regular polyhedra may be inscribed in a sphere. Included in Book 4 are various theorems on the circle that circumscribes three given circles tangent to one another. Also considered are certain properties of various curves, including the Spiral of Archimedes, the conchoid of Nicomedes (fl. c. 240 BC), and the quadratrix of Hippias of Elis (fl. c. 430 BC). Proposition 30 describes the construction of a curve of double curvature called by Pappus the helix on a sphere. The area of the surface included between this curve and its base is found by the classical method of exhaustion equivalent to integration. The rest of the book concerns the trisection of any angle and the solution of problems by means of special curves. Book 5 concerns the areas of different plane figures and the volumes of different solids; the 13 semiregular polyhedra discovered by Archimedes; and the surface and volume of a sphere. Book 6 comments on problems of geometry and astronomy treated by Theodosius of Bithynia, Autolycus of Pitane, Aristarchus, and Euclid. Book 7 explains the terms analysis and synthesis and the distinction between theorem and problem. Also the works of Euclid, Apollonius of Perga, Aristaeus, and Eratosthenes of Cyrene, 33 in all, are enumerated, as well as the famous problem of Pappus, which inspired Ren Descartes and the theorems rediscovered by and named after Paul Guldin (15771643) of Switzerland. Book 8 deals principally with mechanics; interspersed are some questions on pure geometry. Pappus' commentary on Euclid's theory of irrational numbers is extant in an Arabic translation and traces the historical development of the theory of irrationals.