In his work on algebra, al-Khwarizmi worked with both what we now call linear equations – that is, equations that involve only units without any squared figures – and quadratic equations, which involve squares and square roots. His advance was to reduce every equation to its simplest possible form by a combination of two processes: al-jabr and al-muqabala.
Al-jabr means ‘completion’ or ‘restoration’ and involves simply taking away all negative terms. Using modern symbols, al-jabr means simplifying. Al-muqabala means ‘balancing’, and involves reducing all the postive terms to their simplest form.
In developing algebra, al-Khwarizmi built on the work of early mathematicians from India, such as Brahmagupta, and from the Greeks such as Euclid, but it was al-Khwarizmi who turned it into a simple, all-embracing system, which is why he is dubbed the ‘father of algebra’. The very word algebra comes from the title of his book, al-Kitab al-mukhtasar fi hisab al-jabr wa’l muqabala or The Compendious Book on Calculating by Completion and Balancing.
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Beyond al-Khwarizmi, many other Arabic-speaking scholars explored mathematics. Indeed, it was fundamental to so many things, from calculating tax and inheritance to working out the direction of Mecca, that it is hard to find a scholar who did not at some time or other work in mathematics. But it wasn’t just practical applications that fascinated many of them, and they began to push mathematics to its limits.
In the early 11th century in Cairo, Hassan ibn al-Haitham, for instance, laid many of the foundations for integral calculus, which is used for calculating areas and volumes. Half a century later, the brilliant poet/mathematician Omar Khayyam found solutions to all thirteen possible kinds of cubic equations – that is, equations in which numbers are cubed. He regretted that his solutions could only be worked out geometrically rather than algebraically. ‘We have tried to work these roots by algebra, but we have failed’, he says ruefully. ‘It may be, however, that men who come after us will succeed.’
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Omar Khayyam is one of the most extraordinary figures in Islamic science, and tales of his mathematical brilliance abound. In 1079, for instance, he calculated the length of the year to 365.24219858156 days. That means that he was out by less than the sixth decimal place – fractions of a second – from the figure we have today of 365.242190, derived with the aid of radio telescopes and atomic clocks. And in a highly theatrical demonstration involving candles and globes, he is said to have proved to an audience that included the Sufi theologian al-Ghazali that the earth rotates on its axis.
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Trigonometry was first developed in ancient Greece, but it was in early Islam that it became an entire branch of mathematics, as it was aligned to astronomy in the service of faith. Astronomical trigonometry was used to help determine the qibla, the direction of the Ka’bah in Mecca. Modern historians such as David King have discovered that the Ka’bah itself is astronomically inclined. On one side it points towards Canopus, the brightest star in the southern sky. The axis that is perpendicular to its longest side points towards midsummer sunrise.
Mecca’s significance is such that when a deceased person is to be buried, contemporary Islamic tradition determines that his or her body must face Mecca. When the famous call to prayer is announced, it must be done facing Mecca. And when animals are slaughtered, slaughtermen must also turn in the direction of the holy city. Islamic-era astronomers began to compute the direction of Mecca from different cities from around the 9th century. One of the earliest known examples of the use of trigonometry (sines, cosines and tangents) for locating Mecca can be found in the work of the mathematician al-Battani, which, according to David King, was in use until the 19th century.
