**Pappus** is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry. He wrote commentaries on Euclid's Elements and Ptolemy's *Almagest*.

- Our knowledge of Pappus's life is almost nil.
- There appear in the literature one or two references to dates for Pappus's life which must be wrong.
- This would seem convincing but there is a chronological table by Theon of Alexandria which, when being copied, has had inserted next to the name of Diocletian (who ruled 284 AD - 305 AD) "at that time wrote Pappus".
- Clearly both of these cannot be correct, and the known inaccuracy of the Suda led historians to favour dates for Pappus which would have him writing in the period 284 AD - 305 AD, as suggested by the insertion into Theon's chronological table.
- This fixes clearly the date of 320 for Pappus's commentary on Ptolemy's Almagest Ⓣ(The major thesis: from the Arabic 'al-majisti' -- the Arabic translation of the Greek 'Mathematike Syntaxis' later translated into Latin as 'Magna Syntaxis').
- Other than this accurate date we know little else about Pappus.
- We know that he dedicated works to Hermodorus, Pandrosion and Megethion but other than knowing that Hermodorus was Pappus's son, we have no further knowledge of these men.
- Again Pappus refers to a friend who was also a philosopher, named Hierius, but other than knowing that he encouraged Pappus to study certain mathematical problems, we know nothing else about him either.
- Finally a reference to Pappus in Proclus's writings says that he headed a school in Alexandria.
- Pappus's major work in geometry is Synagoge or the Mathematical Collection which is a collection of mathematical writings in eight books thought to have been written in around 340 (although some historians believe that Pappus had completed the work by 325 AD).
- The third part describes a collection of geometrical paradoxes which Pappus says are taken from a work by Erycinus.
- Pappus introduces some of the ideas of Book V by describing how bees construct honeycombs.
- Also in Book V Pappus discusses the thirteen semiregular solids discovered by Archimedes.
- As well as reviewing these works, Pappus points out errors which have somehow entered the texts.
- 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.
- Of Pappus's commentary on Ptolemy's Almagest Ⓣ(The major thesis: from the Arabic 'al-majisti' -- the Arabic translation of the Greek 'Mathematike Syntaxis' later translated into Latin as 'Magna Syntaxis') only the part on Books 5 and 6 has survived.
- We cannot be certain that Pappus wrote a commentary which extended to the whole 13 books, but it seems highly probable that he did.
- This commentary seems to be of much poorer quality to Pappus's geometrical work.
- When Ptolemy in the chapter on the apparent diameter of the sun, moon and shadow simply remarks that the tangential cones in question contact the spheres within a negligible error in great circles, then Pappus refers to Euclid's "Optics" to show that the circle of contact has a smaller diameter than the sphere, only to add a lengthy argument to demonstrate that the error committed in Ptolemy's construction is nevertheless negligible.
- Or, when Ptolemy says that some phenomenon cannot take place, neither for the same clima nor for different geographical latitudes, Pappus feels obliged to explain "same clima" by "either in clima 3, or in 4, or in any other clima", and to illustrate "different" by referring to "Rome or Alexandria".
- Neugebauer also points out that, in addition to these pointless comments, there are also comments by Pappus which are simply incorrect.
- Of course Pappus did not write "Almagest" but the Greek title of the work.
- Other commentaries which Pappus wrote include one on Euclid's Elements.
- Proclus, in his own commentary on the Elements refers three times to Pappus's commentary and Eutocius also refers to Pappus's commentary.
- Part of Pappus's commentary may exist in an Arabic translation, namely that on Book X of the Elements.
- However, the commentary is very different in style to that of the Mathematical Collection and if indeed Pappus is the author it is a commentary which fails to show the depth of understanding that he shows in other parts of his work.
- Marinus claims that Pappus also wrote a commentary on Euclid's Data of which nothing has survived.
- That Pappus wrote on Geography is stated in the Suda and a work which claims to be written by Moses of Khoren in the fifth century seems to be largely based on Pappus's Geography.
- Other works which could have been written by Pappus include one on music and one on hydrostatics.

Born about 290, Alexandria, Egypt. Died about 350.

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African, Ancient Arab, Ancient Greek, Astronomy, Geometry, Origin Egypt, Puzzles And Problems, Special Numbers And Numerals

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive