Diophantus
Diophantus

200?-284?

Essentially nothing is known of Diophantus' life, and there has been much debate regarding the date at which he lived. The most details we have of Diophantus's life come from the Greek Anthology, compiled by Metrodorus around 500, which implies that he married at the age of 26, and had a son who died at the age of 42, 4 years before Diophantus himself died aged 84.

Diophantus is best known for his Arithmetica, the most outstanding work on algebra in Greek mathematics. It is a collection of 130 problems giving numerical solutions of both determinate and indeterminate equations. The method for solving the latter is now known as Diophantine analysis. Only 6 of the original 13 books have survived, though there are some who believe that 4 other arab books discovered in 1968 are also due to Diophantus.

This work considers the solution of many problems concerning linear and quadratic equations, but considers only positive rational solutions to these problems. Diophantus looked at 3 different types of quadratic equations, because he did not have any notion for zero and he avoided negative coefficients. He also considered simultaneous quadratic equations.

In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form 4n + 3 cannot be the sum of 2 squares. Diophantus also appears to know that every number can be written as the sum of 4 squares. If indeed he did know this result it would be truly remarkable for even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Lagrange proved it using results due to Euler.

Although Diophantus did not use sophisticated algebraic notation, he did introduce an algebraic symbolism that used an abbreviation for the unknown and for the powers of the unknown. Since an abbreviation is also employed for the word "equals", he took a fundamental step from verbal algebra towards symbolic algebra.

One thing will be clear from the examples we have quoted and that is that Diophantus is concerned with particular problems more often than with general methods. The reason for this is that although he made important advances in symbolism, he still lacked the necessary notation to express more general methods.

Fragments of another of Diophantus's books On polygonal numbers, a topic of great interest to Pythagoras and his followers, has survived. Diophantus himself refers to another work which consists of a collection of lemmas, but this book is entirely lost. We do know three lemmas contained in it since Diophantus refers to them in the Arithmetica. One such lemma is that given any numbers a, b then there exist numbers c, d such that a3 - b3 = c3 + d3.

Diophantus was not, as he has often been called, "the father of algebra", as many of the methods for solving linear and quadratic equations go back to Babylonian mathematics. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.