# Pappus of Alexandria (3rd century AD)

Diagram to illustrate Pappus's theorem.

Pappus of Alexandria was the last of the great Greek geometers whose eight-volume *Mathematical
Collection* summarized the bulk of mathematics known at that time. In
this compendium, Pappus added a considerable number of his own explanations
and amplifications of the earlier work of Euclid, Archimedes, Apollonius,
and others.

The dates when he lived are not accurately known. Suidas states that Pappus
was a contemporary of Theon, thus placing him toward the end of the 4th
century, and ascribes several treatises to him. These treatises have not
survived, and the only work by which Pappus is now known, his *Mathematical
Collection*, receives no mention from Suidas. This work consisted of
eight books, the first and the earlier part of the second of which are lost,
and its interest is mainly, though not exclusively, historical. From what
remains of the second book, it is conjectured that the first two books were
arithmetical, The third book explains some of the methods for duplicating
the cube, and deals with progressions and the five regular polyhedra.
The fourth book discusses the figure called the arbelos,
the Archimedean spiral, the conchoid of Nicomedes, and the quadratrix
of Hippias. The fifth book contains some theorems regarding isoperimetrical
figures, plane and solid, and a short account of Archimedean
solids. The sixth book comments on some of the works of Theodosius, Aristarchus of Samos, and Euclid.
From the seventh book, which is the longest and most valuable of the *Collection*,
is derived a large part of our knowledge of Greek geometry. Many of the
writings here analyzed are no longer extant, and it is on the indications
(in the notable instance of Euclid's *Porisms*, the very obscure
indications) which Pappus gives of the object or the contents of them that
the geometers of the seventeenth and eighteenth centuries relied for their restorations
of these writings. The eighth book is devoted mainly to mechanics.

The mathematical interest of the *Collection* does not equal the
historical, but several of the books contain important theorems, the discovery
of some of which may be due to Pappus himself. The last six books of the *Mathematical Collections* were translated into Latin by Commandinus,
an Italian geometer, and were published in 1588; another edition appeared
in 1660.

## Pappus's theorem

On two sides of a triangle *ABC* (see diagram) construct arbitrary parallelograms *ACDE* and *BCFG*; extend the sides *ED* and *ED* and *FG* to meet in the point *H*; draw lines through *A* and *B* parallel to *HC* meeting *HD* and *HF* in *K* and *L* respectively; join *KL*. We now have a third parallelogram *ABLK* whose area is equal to the sum of the areas of the two original parallelograms.