Pappus

Alexandrian mathematician

• Born: c. 300
• Birthplace: Alexandria, Egypt
• Died: c. 350
• Place of death: Unknown

Pappus provided a valuable compilation of the contributions of earlier mathematicians and inspired later work on algebraic solutions to geometric problems.

Early Life

Almost nothing is known about Pappus’s life, including the dates of his birth and death. A note written in the margin of a text by a later Alexandrian geometer states that Pappus wrote during the time of Diocletian (284-305 c.e.). The earliest biographical source is a tenth century Byzantine encyclopedia compiled by Suidas. This work lists the writings of Pappus and describes him as a “philosopher,” which suggests that he may have held some official position as a teacher of philosophy. Nevertheless, this reference to philosophy may be no more than an indication of his interest in natural science. The geometer had at least one child, a son, since he dedicated one of his books to him. In addition, Pappus mentions two of his contemporaries in his texts: a philosopher, Hierius, although the connection between the two is not clear; and Pandrosian, a woman who taught mathematics. Pappus addressed one of his works to her, not as a tribute, but because he found several of her students deficient in their mathematical education.

Pappus lived at a time when the main course of Greek mathematics had been in decline for more than five hundred years; although geometry continued to be studied and taught, there were few original contributions to the subject. To alleviate this lack, he attempted to compile all available sources of earlier geometry and made several significant contributions to the subject. As the first author in this new tradition, sometimes called the silver age of mathematics, Pappus provides a valuable resource for all of ancient Greek geometry.

Life’s Work

Throughout his life, Pappus maintained a lively interest in a number of areas dealing with mathematics and natural science. The bulk of his surviving works can be found in the Synagogē (c. 340 c.e.; partial translation The Collection, 1986). Other works either are in fragmentary form or else are no longer extant, although mentioned by other writers. There exists part of a commentary on the mechanics of Archimedes which considers problems associated with mean proportions and constructions using straightedge and compass. There are two remaining books of a commentary on Ptolemy’s Mathēmatikē syntaxis (c. 150 c.e.; Almagest, 1948) explaining some of the finer points of the text to the inexperienced reader. Pappus continued his interest in the popularization of difficult texts in a work, of which only a fragment survives, on Euclid’s Stoicheia (c. 300 b.c.e.; The Elements of Geometrie of the Most Auncient Philosopher Euclide of Megara, 1570, commonly known as the Elements), in which Pappus explains the nature of irrational magnitudes to the casual reader. The lost works include a geography of the inhabited world, a description of rivers in Libya, an interpretation of dreams, several texts on spherical geometry and stereographic projection, an astrological almanac, and a text on alchemical oaths and formulas. Pappus was more than a geometer; he was a person who lived in a world where the search for new knowledge was rapidly declining and where political instability was the order of the day. Yet he expressed a continuing interest in the education of those less fortunate than himself and showed a lively interest in affairs outside his city.

Pappus’s claim to historical and mathematical significance is found in a compendium of eight books on geometry. This collection covers the entire range of Greek geometry and has been described as a handbook or guide to the subject. In several of the books, when the classical texts are available, Pappus shows how the original proof is accomplished as well as alternative methods to prove the theorem. In other books, where the classical sources are not easily accessible, Pappus provides a history of the problems as well as different attempts at finding a solution. An overall assessment of these books shows few moments of great originality; rather, a capable and independent mind sifts through the entire scope of Greek geometry while demonstrating fine technique and a clear understanding of his field of study.

A summary of the contents of the eight books shows that some are of only historical interest, providing information on or elucidation of classical texts. Other books, particularly book 7, have been a source of inspiration for later mathematicians. All of book 1 and the first part of book 2 are lost. The remainder of book 2 deals with the problems of multiplying all the numbers between 1 and 800 together and expressing the product in words using the myriad (10,000) as base. Pappus refers to a lost work by Apollonius of Perga which seems to be part of the problem of expressing large numbers in words that began with Archimedes’ Psiammites (c. 230 b.c.e.; The Sand-Reckoner, 1897). Book 3 deals with construction problems using straightedge and compass: finding a mean proportion between two given straight lines, finding basic means between two magnitudes (arithmetic, geometric, and harmonic), constructing a triangle within another triangle, and constructing solids within a sphere. Book 4 consists of a collection of theorems, including several famous problems in Greek mathematics: a generalization of Pythagoras’s theorem, the squaring of the circle, and the trisection of an angle. Book 5 begins with an extensive introduction on the hexagonal cells of honeycombs and suggests that bees could acquire geometric knowledge from some divine source. This discussion leads to the question of the maximum volume that can be enclosed by a superficial area and to a sequence of theorems that prove that the circle has the greatest area of figures of equal parameter. His proof appears to follow those formulated by an earlier Hellenistic geometer named Zenodorus, whose work is lost. In a later section of this book, Pappus introduces a section on solids with a Neoplatonist statement that God chose to make the universe in a sphere because it is the noblest of figures. It has been asserted but not proved that the sphere has the greatest surface of all equal surface figures. Pappus then proceeds to examine the sphere and regular solids. Book 6 is sometimes called “Little Astronomy”; it deals with misunderstandings in mathematical technique and corrects common misrepresentations.

Book 7 is by far the most important, both because it had a direct influence on modern mathematics and because it gives an account of works in the so-called Treasury of Analysis or Domain of Analysis, of which a large number are lost. These are works by Euclid, Apollonius, and others that set up a branch of mathematics that provides equipment for the analysis of theorems and problems. Classical geometry uses the term “analysis” to mean a reversal of the normal procedure called “synthesis.” Instead of taking a series of steps through valid statements about abstract objects, analysis reverses the procedure by assuming the validity of the theorem and working back to valid statements. Through the preservation of Pappus’s account of these works it is possible to reconstruct most of them.

His most original contribution to modern mathematics comes in a section dealing with Apollonius’s Cōnica (Treatise on Conic Sections, 1896; best known as Conics), where Pappus attempts to demonstrate that the product of three or four straight lines can be written as a series of compounded ratios and is equal to a constant. This came to be known as the “Pappus problem.” Book 8 is the last of the surviving books of Synagoge, although there is internal evidence that four additional books existed. In this book, Pappus takes on the subject of mechanical problems, including weights on inclined planes, proportioning of gears, and the center of gravity.

There exist substantial references to various lost books of Pappus; among the lost works is a commentary on Euclid’s Elements, although a two-part section does exist in Arabic. Several other works fit into this category, surviving only in commentary by later writers or in fragments of questionable authorship in Arabic. One of the more interesting Arabic manuscripts (discovered in 1860) shows that Pappus may have invented a volumeter similar to one invented by Joseph-Louis Gay-Lussac (1778-1850). Pappus was not merely a geometer; he was a conserver of classical tradition, a popularizer of Greek geometry, and an inventor as well.

Significance

The works of Pappus have provided later generations with a storehouse of ancient Greek geometry, both as an independent check against the authenticity of other known sources and as a valuable source of lost texts. For modern mathematics, Pappus offers more than merely historical interest. In 1631, Jacob Golius pointed out to René Descartes the “Pappus problem,” and six years later this became the centerpiece of Descartes’s Des matières de la géométrie, which was a section of his Discours de la méthode (1637; Discourse on Method, 1649). Descartes realized that his new algebraic symbols could easily replace Pappus’s more difficult geometric methods and that the product of the locus of straight lines generated from conic sections could generate equations of second, third, and higher orders.

In 1687 Sir Isaac Newton found a similar inspiration in the “Pappus problem” using purely geometric methods. Nevertheless, it was Descartes’s algebraic methods that would be utilized in the future. Pappus also anticipated the well-known “Guldin’s theorem,” dealing with figures generated by the revolution of plane figures about an axis. It can be argued that Pappus was the only geometer who possessed the ability to work out such a theorem during the silver age of Greek mathematics.

Bibliography

Bulmer-Thomas, I. “Guldin’s Theorem—or Pappus’s?” Isis 75 (1984): 348-352. There exists some question whether the Pappus text is original or if the text was corrupted at a later date. A less significant issue here is the interpretation of the Pappus manuscript—a historical problem concerned with the extent to which Pappus anticipated Guldin.

Cuomo, Serafina. Pappus of Alexandria and the Mathematics of Late Antiquity. New York: Cambridge University Press, 2000. Sees Pappus’s work as part of a wider context and relates it to other contemporary cultural practices, opening new avenues of research into the understanding of mathematics in antiquity.

Descartes, René. The Geometry of René Descartes. Translated by Davis E. Smith and Marcia L. Latham. Chicago: Open Court, 1952. It is possible to follow from Descartes’s own text the relevant passages from Pappus’s work, seeing how Descartes develops his new symbols and why this method would later become the preferred method.

Fried, Michael N. Apollonius of Perga’s “Conica”: Text, Context, and Subtext. Boston: E. J. Brill, 2001. An extensive discussion of this work, from which arose the “Pappus problem.”

Heath, Sir Thomas. From Aristarchus to Diophantus. Vol. 2 in A History of Greek Mathematics. Reprint. New York: Dover Publications, 1981. This edition contains several long sections from the Collection as well as commentaries on the history and contents of these theorems.

Pappus. Book 7 of the Collection. Edited by Alexander Jones. 2 vols. New York: Springer-Verlag, 1986. These two volumes contain the most complete rendition of book 7; in addition, there are exhaustive commentaries and notes on every aspect of this text. Contains a detailed account of the history of various Pappus manuscripts and notes on the problems of translating ancient Greek text.