Of Alexandria (date uncertain, between 150 bc and ad 280), mathematician, wrote an algebraic work on indeterminate equations, Arithmētika, in thirteen books, of which six survive in Greek and four more in Arabic. The latter are numbered 4–7, and certainly represent Diophantus' original books 4–7. The Greek books are numbered 1–6 in the MSS, but of these only 1–3 represent Diophantus' original numbering, while ‘4–6’ must be made up of extracts from the original 8–13. In the Greek (but not the Arabic) MSS the words for the unknown (arithmos) and its powers up to the sixth degree are represented by symbols, as is the operation for minus, so that the equations appear in a primitive algebraical notation, but it is likely that this was introduced in Byzantine times rather than by the author. Diophantus' method is to propose a problem, e.g. ‘to find three numbers such that the product of any two of them plus their sum is a square’ (2. 34), and then to go through every step of finding a single solution, in rational but not necessarily integer numbers. The method for finding more solutions is only implied by the example given. This procedure, using specific numbers, puts Diophantus in a tradition going back ultimately to Babylonian mathematics, and is in stark contrast to the abstract methods of classical Greek geometry. He does not recognize negative or irrational numbers as solutions. Books 1–3 contain linear or quadratic indeterminate equations, many of them simultaneous. Beginning with book 4 cubes and higher powers are found. The solutions often demonstrate great ingenuity. A small treatise by Diophantus on polygonal numbers is preserved, but a work on porisms to which he refers and which may be his own is lost.
G. J. Toomer
Subjects: Classical Studies — Mathematics.