by Ehsan Masood
Using this idea, al-Tusi was able to simplify the Ptolemaic system and get rid of the problematic equants for the sun and the ‘upper’ planets (Saturn, Jupiter and Mars). Yet he couldn’t get rid of them for Mercury. The moon was even more of a problem. The Mercury problem was partly answered at the turn of the 14th century by al-Tusi’s student and colleague Qutb al-Din al-Shirazi, by combining al-Tusi’s ideas with those of another 13th-century Arab astronomer, Mu’ayyad al-Din al-’Urdi. Half a century later, ibn al-Shatir, who worked as a muwaqqit in the Great Mosque of Damascus, went further and found a way to get rid of all extra motions but the epicycles, including those for the moon.
So by the late 14th century, the Islamic astronomers had completely overhauled Ptolemy’s system to produce a model that not only predicted the motions of the heavenly bodies with a high degree of accuracy, but also made sense in terms of the contemporary understanding of how the real world worked. This was an enormous achievement. The problem was that it was wrong, as we now know.
Moving the earth
With the benefit of hindsight, it is easy to see that the Islamic astronomers’ basic assumptions were flawed. Of course Copernicus showed in the mid-16th century that the earth does move, circling around the sun with the planets. But even this concept failed to give correct predictions until Kepler showed that the paths of the planets through space are not perfectly circular, but slightly elliptical. And it would have made no sense in terms of existing theories of how the celestial machine held together. It required the addition of Newton’s theory of gravity to complete the picture and show how it all worked.
In conventional accounts, the narrative seems to leap straight from Ptolemy to Copernicus, and to show how Copernicus had the great insight to see that the earth is not fixed, as Ptolemy said it was, but circles around the sun and spins on its axis. In this narrative, the ultimate Islamic contribution to the big picture seems comparatively small or even misguided. The Arab astronomers may have been diligent and ingenious, it seems, but they were barking up the wrong tree in backing the fixed-earth model, and it required Copernicus’s brilliant insight to set things right.
Copernicus acknowledged that some of the data he needed to prove his theory came from the charts of al-Battani and al-Bitruji, but that was all that apparently came from the Arab astronomers. Yet there are clues that this is not the full story.
Islamic source
In 1957, the historian Otto Neugebauer noticed a similarity beween an illustration in Copernicus’s first key book Commentariolus (1514), in which he first set out his idea that the earth moves, and one in ibn al-Shatir’s book in which he answered the problems of the moon’s motion. The similarity was so striking that it seemed hard not to believe that Copernicus had seen ibn al-Shatir’s book. Intrigued, Neugebauer delved deeper for connections between Copernicus and the Islamic astronomers, and soon found another apparent illustration match in Copernicus, this time with al-Tusi’s 1260 Tadhkira, in which he explains the Tusi Couple. Again the similarity was marked, even including an apparent mistake in the copying of an Arabic letter in al-Tusi’s illustration.
Many historians now believe that Copernicus drew directly from the work of the Islamic astronomers in providing proofs for his theories. Recent research has suggested that West European astronomers were far more aware of Arabic work at the time than was imagined. Indeed many may actually have spoken, or at least read, Arabic, including Guillaume Postel, a lecturer at Paris University in the early 16th century, whose highly technical notes in Arabic can clearly be seen on an Arabic astronomical text in the Vatican library.
The Arab contribution
Of course, Copernicus made the great breakthrough suggestion that the earth moved, but the argument is that it was simply yet another step down the road away from the Ptolemaic model. Indeed, at the time, in some ways it seemed like a backward step, since ibn al-Shatir’s work had matched a believably real theory with observations to a remarkable degree. Yet Copernicus’s idea did not. No one at the time could explain how the universe could possibly fit together without the earth at its centre – and Copernicus’s model made considerably less accurate predictions than ibn al-Shatir’s. These problems, as much as any theological problems that the Roman Catholic Church might have had, needed to be solved before most astronomers could accept that the earth moves.
There is no doubt that Copernicus’s idea of a heliocentric (sun-centred) universe was a seismic shift in scientific thinking. But it was a revolution waiting to happen. The way was paved by the gradual chipping away at the edifice of the Ptolemaic system over the centuries by countless Arabic astronomers, both with their observations and their often ingenious theories.
10
Number: The Living Universe of Islam
In Greek mathematics, the numbers could expand only by the laborious process of addition and multiplication. Khwarizmi’s algebraic symbols for numbers contain within themselves the potentialities of the infinite. So we might say that the advance from arithmetic to algebra implies a step from being to ‘becoming’, from the Greek universe to the living universe of Islam.
George Sarton, Introduction to the History of Science, 1927
In many areas of science, the contribution of early Islam is sometimes open to interpretation and shifts of opinion, but when it comes to numbers and mathematics the legacy is immense and indisputable. The very numbers in use in our world every day for everything from buying food to calculating the spin on an atomic particle are called Arabic numerals, because they came to the West from scholars who wrote in Arabic. What’s more, with al-Khwarizmi’s algebra, these scholars provided us with the single most important mathematical tool ever devised, and one that underpins every facet of science, as well as more everyday processes.
Abu Ja’far Muhammad ibn-Musa al-Khwarizmi is the great hero of Arabic mathematics. Like so many of the early Islamic scholars, his interests were wide-ranging, but it is in the world of numbers that his legacy lies. Little is known about the man, and much of his reported life story could be speculative. It seems likely that he was born in what is now Uzbekistan south of the Aral Sea in Central Asia.
Some scholars say his father was a Zoroastrian, and that he was brought up in this faith which dates back to the time of ancient Sumeria. Others say that this is to completely misinterpret the records. All we do know is that al-Khwarizmi was born about 786, the year Harun al-Rashid came to power, and that when Harun’s son al-Mamun set up the House of Wisdom, al-Khwarizmi was there studying. There is a story that he was summoned to al-Mamun’s sickbed to make an astrological prediction about his health. Sensibly, al-Khwarizmi predicted that the caliph would live another 50 years. In fact, al-Mamun lived just ten more days. Al-Khwarizmi lived much longer. Other accounts say that he was actually one of al-Mamun’s key advisors.
Numbers from India
One of al-Khwarizmi’s greatest contributions was to provide a comprehensive guide to the numbering system which originated in India about 500 CE. It is this system, later called the Arabic system because it came to Europe from al-Khwarizmi, that became the basis for our modern numbers. It was first introduced to the Arabic-speaking world by al-Kindi, but it was al-Khwarizmi who brought it into the mainstream with his book on Indian numerals, in which he describes the system clearly.
The system, as explained by al-Khwarizimi, uses only ten digits from 0 to 9 to give every single number from zero up to the biggest number imaginable. The value given to each digit varies simply according to its position. So the 1 in the number ‘100’ is 10 times the 1 in the number ‘10’ and 100 times the 1 in the number ‘1’. An absolutely crucial element of this system was the concept of zero.
2. Numerals through the ages: Brahmi numerals from India in the 1st century CE, the medieval Arabic-Indic system, and the symbols used today.
This was a significant advance on previous numbering systems, which were often cumbersome with any large numbers. The Roman system, for instance, needs seven digits to g
ive a number as small as, for example, 38: XXXVIII. Arabic numbering can give even very large numbers quite compactly. Seven digits in Arabic numerals can, of course, be anything up to 10 million. What’s more, by standardising units, Arabic numerals made multiplication, division and every other form of mathematical calculation simpler.
This system quickly caught on, and has since spread around the world to become a truly global ‘language’. Along with the numbers, English also gained another word, ‘algorithm’, for a logical step-by-step mathematical process, based on the spelling of al-Khwarizmi’s name in the Latin title of his book, Algoritmi de numero Indorum. The new numbers took some time to embed themselves in the Islamic world, however, as many people continued with their highly effective and fast method of finger-reckoning.
The discovery of algebra
Al-Khwarizmi’s other major contribution also introduced a new word to the language, ‘algebra’, and a whole new branch of mathematics. What is interesting is that in developing algebra, al-Khwarizmi had something eminently religious in mind, not just abstract theory. According to one report, he wrote his book on algebra in response to a request from the caliph to come up with a simple method for calculating Islamic rules on inheritance, legacies and so on. In his introduction to the book in which he describes algebra, he says that the aim is to work with ‘what is easiest and most useful in mathematics, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or when measuring lands, digging canals and making geometrical calculations’. Al-Khwarizmi would typically introduce a problem like this:
Suppose that a man who is terminally ill allows two of his slaves to buy their freedom. The price of one slave is 300 dirhams. This slave dies, leaving a daughter and two sons. He also leaves property worth 400 dirhams. Then his former master dies, and he leaves three sons and three daughters. How much do each of the children receive in inheritance?
Although we now associate algebra entirely with the idea of symbols replacing unknown numbers in calculations, al-Khwarizmi did not actually use symbols, for he wrote everything out fully in words, and for the unknown quantity he would not use ‘x’ or ‘y’ but the word ‘shay’. It was in his way of handling equations that he created algebra.
Completing and balancing
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, for instance, x2 = 40x – 4x2 to just 5x2 = 40x. Al-muqabala means ‘balancing’, and involves reducing all the postive terms to their simplest form. Al-muqabala reduces, for instance, 50 + 3x + x2 = 29 + 10x to just 21 + x2 = 7x.
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.
Universal solutions
By completing and balancing, al-Khwarizmi could reduce every equation to six simple, standard forms and then show a method of solving each. He went on to provide geometrical proofs for each of his methods, which is where the debt to Euclid comes in. So what he was saying was that he could use his notation and the rules of al-jabr and al-muqabala to simplify any kind of problem, especially ones involving the tricky quadratic. Any problem – including things not yet thought of – could be reduced into one of his special six categories. It is for these reasons that later mathematicians such as Galileo and Fibonacci held him in such high regard.
Simplifying quantities into symbols (even quadratics) dates back to the time of mathematicians such as Diophantus and Pythagoras from Greece, as well as Brahmagupta in India. But Roshdi Rashed, a historian of mathematics at the National Scientific Research Centre in Paris, says that al-Khwarizmi’s contribution represents a forward step for several reasons: although people were working on solutions to quadratics before him, al-Khwarizmi helped to find a suite of solutions that could solve all conceivable kinds of quadratics. No mathematician had done this before him.
Higher maths
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.’
The poetic mathematician
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.
Like so many scholars in these later troubled times, Khayyam spent much of his life moving from patron to patron, unable to avoid the turmoil of the age, as rulers rose and fell and political and religious factions wrangled. No wonder that in his famous Rubaiyat he is fatalistic:
What we shall be is written, and we are so.
Heedless of Good or Evil, pen, write on!
By the first day all futures were decided
(From the translation of Khayyam’s Rubaiyat by Omar Ali Shah and Robert Graves)
Euclid’s Fifth
Khayyam was one of the many Arab scholars, including al-Tusi and ibn al-Haitham, who tried to work out a proof for what is called Euclid’s Fifth Postulate, or the Parallel Postulate. The Fifth Postulate is about parallel lines. If part of a line crosses two other lines so that the inner angles on the same side add up to exactly two right angles, then the two lines it crosses must be parallel. This postulate is at the heart of basic geometric construction, and has countless practical applications.
But it is surprisingly hard to prove. Omar Khayyam came close, but it was ultimately unprovable. Euclid’s geometry works well for flat, two- or three-dimensional surfaces and most everyday situations. But just as the earth’s surface is not actually flat, however much it appears to be, so space is actually curved and has many more than three dimensions, including those of time. Euclid’s parallel postulate means that only one line can be drawn parallel to another through a given point. But if space is curved and multi-dimensional, many other parallel lines can be drawn. This is why mathematicians such as Gauss began to realise such limitations of Euclidean geometry in the 19th century and developed a new geometry for curved and multi-dimensional space.
Triangulating faith
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.
Geometric designs
Yet another example of the interplay of mathematics and faith can be found in the decorative geometric patterns adorning some of the world’s most famous mosques. These designs are known in the Western world as ‘Islamic’ geometric patterns and are characterised by a single, often complex geometric design which seems to endlessly repeat itself while fitting inside a confined space. Their development (and popularity) is sometimes ascribed to the fact that in early Islamic societies, drawing or painting the human form was frowned upon – especially in the context of religious buildings. The designs were produced by craftsmen often using no more than a ruler and a fixed compass, and with little formal mathematical training. However, some Islamic-era mathematicians attempted to describe the patterns using mathematics. Among them was the rationalist philosopher al-Farabi (from the 9th century), whose book on the subject is called Spiritual crafts and natural secrets in the details of geometrical figures. Another book by the 10th-century mathematician Abul Wafa is entitled On those parts of geometry needed by craftsmen.