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Pappus of Alexandria

Pappus of Alexandria (290 - 350), was a Hellenized Egyptian born in Alexandria. Very little is known about his life, but the written records suggest he was probably a teacher. His main contribution to mathematics was primarily as an encyclopedist. Pappus summarized and enlarged all of Greek mathematics in his work Synagoge (The (Mathematical) Collection) of which eight volumes survived (perhaps originally in twelve) of which the first and part of the second are missing. The Collection is sometimes the only source for our knowledge of his predecessors' achievements. Pappus' other works include also a commentary on Ptolemy’s Almagest. For instance, in his commentary to Almagest he determined the eclipse in 320 AD. For his own originality, even if his chief importance is as the preserver of Greek scientific knowledge, Pappus stands (with Diophantus) as the last of the long and distinguished line of Alexandrian mathematicians (Hutchinson dictionary of scientific biography).

The Collection contains supplements to earlier treatises including geometry, recreational mathematics, doubling the cube, polygons and polyhedra, astronomy, and mechanics. It dates from the late third century A.D. (325-340  conjectured) and is the last important work of Greek mathematics. Pappus not only reproduces known solutions to geometric problems, but he frequently gives own solutions, or improvements and extensions to existing solutions and theorem.

For instance, Pappus handles the problem of inscribing five regular solids in a sphere in a way quite different from Euclid. Gives a generalization to the famous Pythagorean theorem, and provides a demonstration of squaring the circle which is quite different from the method of Archimedes (who used a spiral) or that of Nicomedes (who used the conchoid).

A Collection manuscript is in the papal library from the 13th century, and is the archetype of all later copies, of which none is earlier than the sixteenth century.

The six and a half extant books (as mentioned Book I and Book II are partly lost) were first edited and translated into Latin by Commandinus in 1588 (PAPPUS of Alexandria Mathematicae Collectiones a Federico Commandino Urbinate in latinum conversae, et commentariis illustratae. Pesaro, Girolamo Concordia, 1588).

Commandinus edition stimulated a revival of geometry in the 17th century. The most interesting part of the Collection, measured by its influence on modern mathematics, is Book VII.

The collection was reedited by Frederick Hultsch (1876-78) who gave definite Greek text with Latin translation. [1] ; French translation [2] ; German translation [3] .

In a passage on Apollonius' Conics, the attempt to conceive of the product of three, four, five, six or more than six lines as geometrical entities, known as Pappus' Problem Descartes devoted a major part of his own Géométrie to this, and solved it by the use of algebraic notation. Thus Descartes demonstrated that the difficulties which Pappus was unable to overcome could be got round by the use of his new algebraic method. Pappus thus came to play a catalytic, if minor, role in the founding of Cartesian analytical geometry.

In his Principia (1687) Newton also found inspiration in Pappus; he proved in a purely geometrical manner that the locus with respect to four lines is a conic section, which may degenerate into a circle.

Pappus formally defined analysis and synthesis as they are still commonly applied in the solution of geometrical riders. Pappus stumbled upon the projective invariance of the cross-ratio of four collinear points and other related results reclaimed by modern projective geometry; and he gave the first recorded statement of the focus-directrix property of the three conic sections. He formulated the "centrobaric" theorems, frequently attributed to Paul Guldin (1577-1643), for calculating the volume and surface generated by a plane figure rotating about an axis in its own plane. He discussed theoretical mechanics, the equilibrium of a heavy body on an inclined plane, the use of the mechanical powers, and the construction of mechanical toys (Biographical dictionary of scientists).


[1]  Pappi Alexandrini. (1875-1878). Collectiones quae supersunt (ed. F. Hultsch) 3 Vols.. Berlin (reprint Amsterdam: Hakkert 1965).

[2]  Pappus d’Alexandrie. (1933). La Collection Mathématique (French transl. and comments  P. ver Ecke). Paris: Brügge.

[3]  (1871) Die  Sammlung des Pappus von Alexandrien (Buch VII und VIII). Greek and German ed. by C.J.Gerhardt). Halle.

Cite this web-page as:

Štefan Porubský: Pappus of Alexandria.

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