Pathfinders

by Jim Al-Khalili

  By the time Ptolemy published his Mathematical Treatise, around 150 CE, later to be known of course by its Arabic title of Almagest, Aristotle’s model had been extended, tweaked and modified in order to match the observed motion of the heavens. But it had strayed from the ideal of perfect concentric spheres that Aristotle had proposed. Even the simplest of motions, that of the sun, was not straightforward. One way of modelling its ‘orbit’ around the earth was for it to move around a circle, known as the ‘eccentric’, the centre of which is shifted away from the earth. In this way, the Greeks could model what we now know to be the earth’s elliptical path around the sun. Another model, equivalent to the ‘eccentric’ idea, was for the sun to move around a small circle called an epicycle, whose centre itself moves around a circle centred on the earth. Both models, epicycles and eccentrics, were shown by Ptolemy to be equivalent (see Plate 22).

  Ptolemy’s greatest achievement was his theory of planetary motion. There does not appear to have been any satisfactory theoretical model to explain the rather complicated motions of the other five known planets before the Almagest. What he did was to combine together the two alternative methods for describing the motion of the sun. The path of a planet therefore consisted of circular motion on an epicycle with the centre of this epicycle moving round a circle, the centre of which was itself offset from the earth. Even this did not quite work, however, and Ptolemy’s innovation was to take a further step by introducing the concept of the equant, which deserves a brief explanation.

  The equant is an imaginary point in space that is the mirror of the earth, displaced equally on the other side of the centre of the deferent about which a planet’s epicycle rotates. Crucially, it is the point about which the centre of the epicycle moves at uniform angular velocity. According to Ptolemy, the equant is the point in space at which the planet’s epicycle centre is ‘seen’ to move round at a constant angular rate, even though it is not located at the centre of this orbit. So from the vantage point of the true centre of the deferment, the epicycle centre would be moving round in a circle, but speeding up and slowing down. And there you have it. Simple really!

  When it came to modelling the motion of the moon, and the variation of the positions of the eclipses, things became really messy, and I shall not try to go into them here. Ptolemy ended up with even more circles moving around circles. The whole thing became convoluted and unwieldy. And yet, it worked.

  Things fared little better in Indian astronomy in the period between Ptolemy and the birth of Islam. Early Indian astronomers had also proposed a heliocentric model. But the two giants of medieval Indian science, Āryabhata and Brahmagupta, seem to have considered and abandoned it in favour of a geocentric model, although they had correctly proposed a spherical earth that rotated on its axis. On the whole, Indian astronomy developed out of pre-Ptolemaic Greek ideas,8 most probably from the much earlier contact with Greek knowledge at the time of Alexander the Great and the Indo-Greek kingdom that followed. Both Āryabhata and Brahmagupta were to have a huge influence on astronomy in early Islam but, ultimately, it was Ptolemy more than anyone else who was to lay the foundations of medieval astronomy.

  The Ptolemaic model of planetary motion around the earth.

  And so we fast-forward to the eleventh century. We have seen how Ibn al-Haytham had set the shukūk ball rolling by casting doubts on Ptolemy’s astronomy, separating observational astronomy from cosmology – in a sense, doing what Newton was to do more famously in separating physics from philosophy more than six hundred years later. But Ibn al-Haytham’s shukūk movement, while inspiring many scholars, from Central Asia to Spain, to really question the mathematical models of the Greeks, had a major flaw: it was still utterly wedded to the geocentric model. While they laboured to get rid of messy equants and epicycles, they did so solely in order to return to the Aristotelian ideal of perfect concentric spheres, with the earth at the very centre of all rotations.

  Ibn al-Haytham argued that Ptolemy’s cosmology was just not physically possible: one cannot have a sphere that rotates about an axis that does not pass through the sphere’s centre – this would cause a ‘wobbling’ motion that would surely need explaining. Of course, Ibn al-Haytham had no knowledge of the law of gravity or the concept of the centre of mass, yet he was intuitively on the right track in insisting that a mathematical model should reflect physical reality. The whole basis of the shukūk movement that he started in astronomy can be summarized by saying that he could not accept a cosmological model that was physically impossible.

  A few lone voices could be heard feebly arguing for a real revolution in astronomy, for the heliocentric model had not disappeared entirely. One interesting character I have not mentioned thus far is Abū Ma’shar al-Balkhi (Albumasar) (c. 787–886), a Persian scholar who studied the texts of Ptolemy and Aristotle. He is notable for two reasons. First, many of his works were translated into Latin in the first half of the twelfth century, before much of Europe had been introduced to Aristotelian cosmology. Al-Balkhi was thus their first encounter with many of Aristotle’s philosophical ideas. Secondly, al-Balkhi had cleverly proposed a ‘halfway house’ cosmological model, with all the planets orbiting around the sun, apart from the earth, which still sat in its special position at the very centre of the universe with the sun revolving around it. Al-Balkhi’s work was known to al-Bīrūni, who, it seems, took it quite seriously, but only as a philosophical problem rather than a scientific one. It is amusing to note that what drove al-Balkhi was his obsession not with astronomy but with astrology. This is the main reason why his work was of such interest to medieval European scholars: for them, it was still the fascination with astrology that drove astronomy.

  We should not forget another astronomer, al-Sijzi, who, like his more illustrious contemporary, al-Bīrūni, was influenced by al-Balkhi’s work. Al-Sijzi also seems to have proposed a heliocentric model, but there are few details about his work other than that (according to al-Bīrūni) he built a heliocentric astrolabe.

  The important distinction to make here is that all these pre-Copernican heliocentric models were more metaphysical than mathematical. The real advances were being made in theoretical astronomy, particularly in the mathematical models of celestial mechanics pushed forward by Ibn al-Haytham. We therefore need to take a closer look at some of these advances, for the little-known yet huge debt owed by Copernicus to Muslim astronomers cannot be merely that he was not the first to propose a heliocentric model.

  By any measure, the golden age of Islam and its wonderful scientific achievements were beginning to wane by the eleventh and twelfth centuries. It might be argued that this had more to do with the anti-climax of no longer finding anyone to match the brilliance of Ibn al-Haytham, Ibn Sīna and al-Bīrūni. But this is somewhat unfair, for one could equally claim that Greek science ended with Aristotle in the fourth century BCE. To admit to a scientific decline in the Islamic Empire is one thing, but we must not forget that this was a slow process from the heady heights of the early eleventh century – and it was a long way down. In astronomy, by contrast, the best was yet to come. Despite the Mongol conquests of the Muslim East in the mid-thirteenth century, another two unlikely giants of astronomy emerged: a thirteenth-century Persian polymath who became a member of a religious sect known as the Assassins, and a fourteenth-century timekeeper in a Damascus mosque. More than any others, these two men would progress Ibn al-Haytham’s programme.

  Nasr al-Dīn al-Tūsi was born in 1201 in the city of Tūs (as his name suggests) in Khurasan in eastern Persia. The city was at the time one of the largest and most important in the whole of Persia and was the place of birth of other notables such as Jābir ibn Hayyān and the poet al-Firdawsi. Even by the standards of some of the more colourful characters we have encountered on our journey, al-Tūsi was a fascinating individual. He studied theology with his father, a prominent Shi’a cleric, but also learnt logic, philosophy and mathematics from his uncle. He completed his education in the city of Nishapūr and
quickly gained a reputation as an outstanding scholar. But these were troubled times in Khurasan, for the Mongols were advancing from the east and al-Tūsi could not settle into academic life with this looming shadow. So he accepted the invitation to join a secretive religious sect known as the Hashashīn in the relative security of their mountain stronghold.

  The Hashashīn were an offshoot of Isma’īli Shi’ism that had split from the Fātimid dynasty and initially gained widespread support east of the Fātimid capital Cairo, in what is now northern Syria, Iraq and Iran. However, they had quickly found themselves isolated and marginalized. In 1090, under their charismatic leader Hassan-i-Sabbāh, they retreated to the Alborz mountains of northern Iran, gained control of a number of strategically important fortresses and set up their headquarters in the mountaintop castle of Alamūt (‘Eagle’s Nest’).

  It was at Alamūt that al-Tūsi settled, some even say was imprisoned, for it is not clear whether he would have been able to leave if he had wished. The Hashashīn most probably derived their name from the Arabic/Persian word hashīsh, meaning ‘grass’, because of their cultivation of a wide variety of herbs on the lush farmlands around their forts (and not, as one story goes, because they would come down from the mountains to fight their enemies high on marijuana – hence the term ‘hashish’). Because they did not have a conventional army to fight those in the Abbāsid caliphate who opposed them, they would carry out covert raids and assassinations, spreading terror among the population. Indeed, the modern word ‘assassin’ comes directly from hashashīn, for even after their political power was lost, they continued to be used by the Mamlūk dynasty as hired killers for a fixed fee.

  Al-Tūsi was a lucky man. He was to escape, not once but twice, the fate of so many countless hundreds of thousands who were killed by the Mongols. The cities of Tūs and Nishapūr were completely destroyed around 1220 by the Mongol army, under Hūlāgū Khan (c. 1217–65), grandson of Genghis Khan. In Alamūt, al-Tūsi was able to pursue his studies in relative peace for almost thirty years. He built an observatory and a library and was even able to attract other scholars to work with him. But this tranquillity was shattered in 1256 when Hūlāgū’s army finally arrived at the foothills of Alamūt. Visiting the ruins of the fort today, it is hard to imagine how any army could have breached its walls, which sit isolated and inaccessible atop craggy rocks. But breach them the Mongols did and the Hashashīn were defeated. Al-Tūsi of course had other plans and quickly convinced the Mongol leader that he was worth sparing. Depending on which side of the story one believes, he was either abducted by or rescued by the Mongols. Some even accuse him of selling out to them and betraying his Isma’īli faith too readily.

  Whatever his motives, the world should be grateful to al-Tūsi for his survival instincts. He convinced Hūlāgū to employ him as his scientific adviser and that he could predict his astrological chart for him. However, in order to do this he would need the resources to build a new observatory where he could make a range of astronomical measurements. Construction began on the observatory at Marāgha to the east of Tehran in 1259 and it quickly became the world’s greatest centre for astronomy. Its centrepiece – instead of today’s giant telescopes – was an enormous brick construct on which was laid a metal arc, aligned with the meridian, known as a mural sextant (or Fakhri sextant), many feet high, and on which were marked degrees, minutes and even seconds of arc. Astronomers would line up the celestial object under study using a sighting arm, or dioptra, called the alidade (from the Arabic al-’idada, meaning ‘marked ruler’) and then make a reading from the markings on the arc, giving the definitive, accurate position of the object in the sky. A system of counter-weights and pulleys was used to allow the observer to manipulate the huge alidade.9

  Al-Tūsi gathered around him a large group of talented astronomers. His reputation by this time had spread far and wide and he attracted scholars from as far away as China to work with him. The completed zīj, known as the ‘Ilkhānī Tables’, was a masterpiece. But then so was so much of al-Tūsi’s work. For example, his book The Transversal Figure (Shakl al-Qitā’) completed and extended Islamic mathematicians’ work on trigonometry and is regarded as the very first book devoted to trigonometry as an independent branch of mathematics, as opposed to specific ‘techniques’ in the service of astronomy. In it, he extended for the first time a well-known theorem called the ‘sine rule’ from plane triangles to spherical triangles, and extended the work of mathematicians such as Omar Khayyām on number theory. He also wrote extensively on philosophy and logic, and was instrumental in keeping alive the spirit of scientific enquiry in the Islamic world following the Mongol destruction of so many great centres of learning, including the great libraries of Baghdad.

  Marāgha under al-Tūsi became much more than just an observatory, for it would also play a major role in the revival of many of the sciences. Most importantly, it lends its name to what historians today refer to as the Marāgha Revolution, a school of thought that took on the challenge first laid down by Ibn al-Haytham to overhaul Ptolemaic astronomy.

  Al-Tūsi’s most influential book was his Memoir on Astronomy (al-Tathkira fi ‘Ilm al-Hay’a), which is widely regarded as the most important and original book on astronomy to be written in the medieval world. In it, al-Tūsi describes a geometric construction, now known as a Tūsi-couple,10 which involves a small circle revolving around the inner rim of a larger circle twice its diameter. The clever feature is that a point on the circumference of the smaller circle will be seen to oscillate back and forth in linear motion along a diameter of the larger circle. By means of this construction, al-Tūsi succeeded in reforming the Ptolemaic planetary models, by doing away entirely with the awkward equants.

  There is nothing much left to see at the ruins of the Marāgha complex today other than the base of the enormous armillary arm that was its centrepiece. But there had been offices, libraries, even a mosque, in what was the equivalent of a complete research institution such as one might see today. Alongside al-Tūsi were a number of other notable astronomers who also began to tackle the mathematical models of Ptolemaic astronomy, such as al-’Urdi and al-Shirāzi. But when we refer today to the astronomers of the Marāgha School we do not restrict ourselves to those who worked at the observatory, which brings me to the second personality I must introduce before I can return to Copernicus.

  By the twelfth and thirteenth centuries an interesting reorientation was taking place in the way astronomy was perceived and funded by Islamic rulers, who were increasingly suspicious of its links to astrology. Thus, unlike the case of al-Tūsi and Hūlāgū, in which funding for the new observatory was secured only on the pretext of producing astrological charts, many Islamic astronomers became increasingly reliant on religious patronage for work solely in the service of religion. In a way, this freed them from the pressure of indulging their political patrons with unscientific and superstitious pursuits (al-Bīrūni in particular was known to be unhappy about having to work on astrology to supplement his income). And so we find that, outside Marāgha, most astronomical work was carried out by muwaqqits (the timekeepers of the mosques), whose job it was to determine the precise time for prayer from astronomical measurements and sundial readings.

  The most famous of all these muwaqqits worked in the great Umayyad mosque in the centre of Damascus. His name was Ibn al-Shātir (1304–75) and he is regarded as the world’s greatest astronomer of the fourteenth century. In popular culture, he was famous for building the most accurate and sophisticated sundial ever seen at the time. When it was completed, its ceremonial erection on an outside ledge near the top of one of the mosque’s minarets was said to be an event marked by grand celebrations among the people of Damascus. In this way, Ibn al-Shātir could take his measurements and then signal to the mu’azzin at the top of the minaret at the precise time for him to begin his call for prayer. The original sundial, now kept at the National Museum of Damascus, was damaged when a nineteenth-century muwaqqit by the name of al-Tantāwi tr
ied to move it, wrongly claiming that it was not aligned correctly. He then replaced it with an inferior replica that survives to this day (see Plate 25).

  But Ibn al-Shātir’s true legacy to astronomy was in his use of al-Tūsi’s mathematical trick of overhauling Ptolemy’s clunky models by replacing them with new solar and lunar theories that were far more advanced. In this sense, Ibn al-Shātir is regarded as the last great Islamic astronomer of the Marāgha School.

  By the time of Copernicus, European scholars had already mastered Ptolemaic astronomy. Two in particular, the Austrian Georg Peurbach (1423–61) and the German Regiomontanus (1436–76), wrote several texts that served as the main sources of Copernicus’ education. In particular, their co-authored Epitome of the Almagest is regarded as the finest textbook on Ptolemaic astronomy ever written11 and was studied very carefully by Copernicus together with his copy of a Latin translation of the Almagest (a 1515 Venice printing of Gerard of Cremona’s translation). However, it was more than just Greek astronomy that interested Copernicus. He learnt from the Epitome about the work of early Arabic astronomers such as Thābit ibn Qurra and al-Battāni as well as studying the Toledan Tables, and would later refer to some of this work in his De revolutionibus.

  The famous comparison between the Tūsi-couple diagrams of al-Tūsi in his work of 1261 (on the right) and that of Copernicus in 1543. It is not the similarity of the shapes that is remarkable but the identity of the letters labelling the points. Where al-Tūsi has an alif Copernicus has an A, where there is a bā he has a B, jīm becomes G, dāl becomes D, and so on following the order of the letters of the Arabic alphabet.

  None of this is particularly surprising. Far more telling is a cursory comparison of the geometric diagrams of planetary models showing the Tūsi-couple in the De revolutionibus and al-Tūsi’s Memoir on Astronomy, which are extraordinarily similar, even to the extent of agreement in the labelling of the different points on the circles: al-Tūsi’s Arabic letter denoting each point having been replaced by its Latin counterpart.12 Most dramatically of all, Copernicus’ lunar and solar models, and the model for the motion of Mercury, are exactly those developed by Ibn al-Shātir and al-Tūsi.