Diophantus and his Arithmetica
Diophantus was a Greek mathematician from Alexandria who lived in the 3rd century AD. He is known to us through his work Arithmetica, a series of texts containing over 100 mathematical problems that survived the destruction of the Library of Alexandria in 641AD.
Arithmetica has had a resounding influence on the development of mathematics, particularly on number theory and the solution of algebraic equations. It was translated into Arabic in the 10th century by the Persian mathematician Abu al-Wafa' Buzjani (بوژگانی) and spread through Europe after the Italian mathematician Rafael Bombelli's 1572 book L'Algebra, and the publication of Arithmetica in full in 1621 by French mathematician Claude Gaspar Bachet de Méziriac.
Arithmetica contains the earliest known use of algebraic symbolism, which had a huge influence on Islamic mathematics and the subsequent development of algebra as we know and love it. Indeed, the word Algebra itself comes from the Arabic الجبر (al-jabr), which itself came from the title of an early 9th century book Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala (The Compendious Book on Calculation by Completion and Balancing) by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c780–850). The development of algebra as a discipline independent of arithmetic and geometry by al-Khwārizmī was taken further by the French mathematician François Viète (1540-1603), and — with contributions to notation from Italian, German, Dutch and Welsh mathematicians — led to the algebra we take for granted today.
The Four Square Theorem
Diophantus was undoubtedly ahead of his time. He knew, for example, that every number can be written as the sum of at most four squares:
\[\begin{array}{c}\begin{align}1 &= {1^2}\\2 &= {1^2} + {1^2}\\ &\vdots \\43 &= {5^2} + {4^2} + {1^2} + {1^2}\\ &\vdots \\999,999,999 &= {30985^2} + {6319^2} + {3^2} + {2^2}\\ &\vdots \end{align}\end{array}\]
It was remarkable that Diophantus knew this, given that it took another 1500 years before it was proven — by Joseph-Louis Lagrange in 1770. Even Leonhard Euler, widely considered to be the greatest mathematician in history, and certainly the most prolific, was unable to prove that it was true. (You can use this applet to try some numbers out for yourself (maybe try your birthdate), and if you'd like to see and explore Lagrange's proof of the Four Square Theorem, this Wikipedia article isn't a bad place to start.)
As is always the case in mathematics, Diophantus' work provoked more questions that paved the way for more profound developments. For example, with respect to the four square theorem: what numbers can be written as the sum of four squares in only one way, excluding \({0^2}\)? (In fact, there are only 138 such numbers.) How many ways can each number be written as the sum of at most four squares (see here)? Do these numbers have certain characteristics? What numbers can be written as the sum of five squares, excluding \({0^2}\)? (In fact, only 12 numbers can't!) What about cubes...? And so on.
Arithmetica and Fermat's Last Theorem
Artihmetica is also renowned in the history of mathematics because it led to Fermat's Last Theorem, and the awe-inspiring story of its eventual proof by the Abel Prize Winner Sir Andrew Wiles (see Simon Singh's wonderful book, short video, and documentary). Fermat had a copy of Arithmetica and when working through it he wrote a tantalising note in the margin next to one problem: 'I have discovered a truly remarkable proof [but] this margin is too small to contain it.' Fermat was asserting that he could prove that there are no positive integers for which \({x^n} + {y^n} = {z^n}\) when n is greater than 2, but he died before he gave his proof. Fermat's son published his father's note in a 1670 edition of Arithmetica, after his father's death, and the mathematical world was subsequently transfixed for centuries.
The Absurd Equation
Diophantus was also the first Greek mathematician to recognise fractions as numbers. It may seem incredible to us to say this today, but before Diophantus, fractions simply weren't 'allowed' to be solutions to problems. Only positive integers were 'allowed'. This was because it was felt that numbers had to have a geometric sense; they were representations of lengths, areas, and volumes.
But the incredible Diophantus, the genius who opened the mathematical door for so many who followed, would not 'allow' negative numbers to be solutions to problems. He described the concept of \(\lambda \varepsilon \iota \psi \iota \varsigma \), meaning 'deficiency,' and the rules that 'deficiency multiplied by deficiency yields availability' (the product of two negative numbers is positive), and 'deficiency multiplied by availability yields deficiency' (the product of a negative and a positive number is negative). But even though he was happy to manipulate negative numbers in order to get to a solution, as a solution themselves Diophantus considered negative numbers 'useless'. By way of illustration he described the equation \(4 = 4x + 20\) as 'absurd,' because it leads to the 'useless' negative solution:
\[\begin{array}{c}\begin{align}4 &= 4x + 20\\ - 16 &= 4x\\ - 4 &= x\end{align}\end{array}\]
The Problem of Diophantus' Age
Although we lack information about Diophantus’ life, we can work out his age upon his death from an algebraic problem inscribed on his tombstone. There are several versions of the epitaph, including this from Sir Thomas Little Heath's 1910 study of Diophantus:
'His boyhood lasted \(\frac{1}{6} \) of his life; his beard grew after \(\frac{1}{12}\) more; after \(\frac{1}{7}\) more he married, and his son was born five years later; the son lived to half his father’s age, and the father died four years after his son.'
\[\begin{array}{c}\begin{align}x &= \frac{x}{6} + \frac{x}{{12}} + \frac{x}{7} + 5 + \frac{x}{2} + 4\\x &= \frac{{14x}}{{84}} + \frac{{7x}}{{84}} + \frac{{12x}}{{84}} + 5 + \frac{{42x}}{{84}} + 4\\x &= \frac{{75x}}{{84}} + 9\\\frac{{9x}}{{84}} &= 9\\9x &= 756\\x &= 84\end{align}\end{array}\]
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