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# Diophantus

Diophantus was another mathematician who lived at Alexandria and about whose life very little is known. We are not even sure of his nationality (though he was probably not a Greek) or his dates, though it is generally accepted that he lived in the third century AD.

However, there is one reference, an epitaph to Diophantus, which can be found in the book known as the *Palatine Anthology* written by the grammarian Metrodorus in the sixth century. This is a paraphrase of what it says:

‘One sixth of Diophantus’ life was spent in childhood, one twelfth as a youth and one seventh more before he married. Five years after his marriage, a son was born. The son died four years before the father. The son’s age when he died was half of Diophantus’ age when he died. How old was Diophantus when he died?’

It’s quite easy to solve this problem, with the result that Diophantus was 84 years old when he died.

Diophantus is famous for the introduction of what is known as **syncopated algebra**. The evolution of algebra has been characterised by Nesselmann as having three stages:

1. **Rhetorical algebra**. This is when the solution to a problem is given using only words, with no abbreviations or symbols. Before Diophantus, all algebra was rhetorical.

2. **Syncopated algebra**. This is when abbreviations are used for some of the more common operations and quantities.

3. **Symbolic algebra**. This is when almost all the steps in the solution to a problem are given in shorthand using mathematical symbols for both operations and quantities. This is the type of algebra that we are familiar with today.

Here is an example of Diophantus’ use of syncopated algebra. He used:

Diophantus wrote three works. One, *Porisms* , is completely lost, and only a fragment remains of another, *On Polygonal Numbers* . However six books from thirteen of his most important work,* Arithmetica* , are still in existence through a Latin translation of the original Greek text by Xlander in the sixteenth century.

*Arithmetica* contains 130 problems in number theory, of various types, all leading to equations of the first or second degree, some with as many as three unknowns. Diophantus does not employ any general methods for the solution of the problems, but tackles each problem individually often using ingenious methods. All Diophantus’ solutions were rational numbers and he usually only supplied one answer for each problem (even though there were often more).

[As a footnote, it should be added that the great French number theorist Pierre de Fermat (1601 – 1665) was influenced by the work of Diophantus. Fermat is said to have written in the margin of his copy of *Arithmetica* that he had a solution for the famous theorem of mathematics known as Fermat’s Last Theorem i.e There are no integer solutions of the equation x^{n} + y^{n} = z^{n} for n > 2.]

**A contribution of Diophantus to mathematics**

The following is a statement of *Arithmetica* Book II, problem 28 and its solution. For simplicity, modern notation is used, but the method is due to Diophantus.

**Problem** Find two square numbers such that the sum of the product of the two numbers with either number is also a square number.

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