Pappus of Alexandria

Pappus of Alexandria (flourished ad 320) was the most important mathematical author writing in Greek during the later Roman Empire, known for his Synagoge (“Collection”), a voluminous account of the most important work done in ancient Greek mathematics. Other than that he was born at Alexandria in Egypt and that his career coincided with the first three decades of the 4th century ad, little is known about his life. Judging by the style of his writings, he was primarily a teacher of mathematics. Pappus seldom claimed to present original discoveries, but he had an eye for interesting material in his predecessors’ writings, many of which have not survived outside of his work. As a source of information concerning the history of Greek mathematics, he has few rivals.

Pappus wrote several works, including commentaries on Ptolemy’s Almagest and on the treatment of irrational magnitudes in Euclid’s Elements. His principal work, however, was the Synagoge (c. 340), a composition in at least eight books (corresponding to the individual rolls of papyrus on which it was originally written). The only Greek copy of the Synagoge to pass through the Middle Ages lost several pages at both the beginning and the end; thus, only Books 3 through 7 and portions of Books 2 and 8 have survived. A complete version of Book 8 does survive, however, in an Arabic translation. Book 1 is entirely lost, along with information on its contents. The Synagoge seems to have been assembled in a haphazard way from independent shorter writings of Pappus. Nevertheless, such a range of topics is covered that the Synagoge has with some justice been described as a mathematical encyclopedia.

The Synagoge deals with an astonishing range of mathematical topics; its richest parts, however, concern geometry and draw on works from the 3rd century bc, the so-called Golden Age of Greek mathematics. Book 2 addresses a problem in recreational mathematics: given that each letter of the Greek alphabet also serves as a numeral (e.g., α = 1, β = 2, ι = 10), how can one calculate and name the number formed by multiplying together all the letters in a line of poetry. Book 3 contains a series of solutions to the famous problem of constructing a cube having twice the volume of a given cube, a task that cannot be performed using only the ruler-and-compass methods of Euclid’s Elements. Book 4 concerns the properties of several varieties of spirals and other curved lines and demonstrates how they can be used to solve another classical problem, the division of an angle into an arbitrary number of equal parts. Book 5, in the course of a treatment of polygons and polyhedra, describes Archimedes’ discovery of the semiregular polyhedra (solid geometric shapes whose faces are not all identical regular polygons). Book 6 is a student’s guide to several texts, mostly from the time of Euclid, on mathematical astronomy. Book 8 is about applications of geometry in mechanics; the topics include geometric constructions made under restrictive conditions, for example, using a “rusty” compass stuck at a fixed opening.

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Numbers and Mathematics

The longest part of the Synagoge, Book 7, is Pappus’s commentary on a group of geometry books by Euclid, Apollonius of Perga, Eratosthenes of Cyrene, and Aristaeus, collectively referred to as the “Treasury of Analysis.” “Analysis” was a method used in Greek geometry for establishing the possibility of constructing a particular geometric object from a set of given objects. The analytic proof involved demonstrating a relationship between the sought object and the given ones such that one was assured of the existence of a sequence of basic constructions leading from the known to the unknown, rather as in algebra. The books of the “Treasury,” according to Pappus, provided the equipment for performing analysis. With three exceptions the books are lost, and hence the information that Pappus gives concerning them is invaluable.

Pappus’s Synagoge first became widely known among European mathematicians after 1588, when a posthumous Latin translation by Federico Commandino was printed in Italy. For more than a century afterward, Pappus’s accounts of geometric principles and methods stimulated new mathematical research, and his influence is conspicuous in the work of René Descartes (1596–1650), Pierre de Fermat (1601–1665), and Isaac Newton (1642 [Old Style]–1727), among many others. As late as the 19th century, his commentary on Euclid’s lost Porisms in Book 7 was a subject of living interest for Jean-Victor Poncelet (1788–1867) and Michel Chasles (1793–1880) in their development of projective geometry.