Our knowledge of Pappus's life is almost nil. It appears that he was born in Alexandria and lived there all his life. A reference to Pappus in Proclus's writings says that he headed a school there.

Pappus's major work in geometry is *Synagoge*, a collection of mathematical writings in 8 books thought to have been written in around 340. Obviously written with the object of reviving the classical Greek geometry, it covers practically the whole field. It is, however, a handbook
or guide to Greek geometry rather than an encyclopaedia. It was intended to be read with the original works rather than to enable them to be dispensed with.

Book 1 covered arithmetic and is now lost. Book2 is partly lost, but the remaining part deals with Apollonius's method for dealing with large numbers. The method expresses numbers as powers of 10,000.

Book 3 is divided by Pappus into four parts. The first part looks at the problem of finding two mean proportionals between two given straight lines. The second part gives a construction of the arithmetic, geometric and harmonic means. The third part describes a collection of geometrical paradoxes which Pappus says are taken from a work by Erycinus. The final part shows how each of the 5 regular polyhedra can be inscribed in a sphere.

Book 4 contains properties of curves including the spiral of Archimedes and the quadratrix of Hippias and includes his trisection methods. In Book 5 he discusses the 13 semiregular solids discovered by Archimedes. He compares the areas of figures with equal perimeters and volumes of solids with equal surface areas, proving a result due to Zenodorus that the sphere has greater volume than any regular solid with equal surface area. He also proves the related result that, for two regular solids with equal surface area, the one with the greater number of faces has the greater volume.

Books 6 and 7 consider books of other authors , such as Theodosius, Autolycus, Aristarchus, Euclid, Apollonius, Aristaeus and Eratosthenes. It is in Book 7 that the "Pappus problem" appears. This problem had a major impact on the development of geometry. In Book 8, Pappus deals with mechanics.

The whole work does not show a great deal of originality but it does show that Pappus has a deep understanding of a whole range of mathematical topics and that he had mastered all the major available mathematical techniques. He writes well, shows great clarity of thought and the this is a work of very great historical importance in the study of Greek geometry.

He also wrote many commentaries on the works of others. Other works which could have been written by Pappus include one on music and one on hydrostatics. Certainly an instrument to measure liquids is attributed to him.