Merchants and Mechanics

Based on David Lindberg, Theories of Vision from al-Kindi to Kepler (University of Chicago Press, 1976); Jim al-Khalili, The House of Wisdom (Penguin, 2011).

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Islamic science made major contributions to optics, mathematics, and astronomy. In these areas, Islamic scholars built upon the discoveries of the Greeks and the Hindus, and their discoveries (as well as those of the Greeks) became the foundation for European scholarship.

Optics

The nature of sight was a difficult issue for medieval scholars because it involved two subjects, physics and human physiology, neither of which was well understood. The first fairly complete explanation, put forward by Johannes Kepler in 1604, was the culmination of research that had begun with the ancient Greeks. Islamic scholars, especially Hasan ibn al-Haytham, made significant contributions to this research.

The Islamic scholars, as was their custom, carefully considered what the Greeks had written about the subject. What they found was little more than speculation. The atomists reduced everything to the jostling of atoms, so for them, explaining sight was about explaining the nature of the contact. According to Theophrastus, Democritus believed that “the air between the eye and the object of sight is contracted and stamped by the object seen and the seer; for from everything there is always a sort of effluence proceeding.”¹ Epicurus explained that this “effluence” takes the form of coherent images that are perceived intact by the eye:

For particles are continually streaming off from the surface of bodies, though no diminution of the bodies is observed, because other particles take their place. And those given off for a long time retain the position and arrangement which their atoms had when they formed part of the solid bodies.²

Lucretius called these images simulacra, and argued that they were not unlike other phenomena observed in the natural world:

Amongst visible things many throw off bodies; sometimes loosely diffused abroad, as wood throws off smoke and fire heat; sometimes more close-knit and condensed, as…when the slippery serpent casts off his vesture among the thorns.³

The atomists’ theory was a theory of intromission — sight occurs because something from the object comes into the eye. By contrast, Euclid proposed a theory of extramission — something from the eye goes out to the object. Plato opted for a combination theory: “fire” from the eye goes out to the object, coalesces with daylight, then brings back to the eye an image emanated by the object. Aristotle dismissed both extramission and intromission, and yet believed that vision required an intermediary: “It is not true that the beholder sees, and the object is seen, in virtue of some merely abstract relationship between them.”⁴ This intermediary — light — fills the space between the object and the viewer; it is not an emanation from either one.

Al-Kindi was the first Islamic scholar to carefully consider optics. Like Euclid, he believed in extramission.⁵ Euclid had assumed that both luminous rays and visual rays (the rays emitted by the eye) were rectilinear. Al-Kindi tried to prove this assumption by conducting experiments involving a luminous object and a second object to block its light. Intervening objects that were smaller or larger than the luminous object cast shadows that were smaller or larger. Al-Kindi showed that tangent lines from the luminous object to the intervening object determined the size of the shadow. Tangent lines also explained the length of the shadows cast upon the ground by luminous objects held at differing heights behind an intervening object. Al-Kindi nevertheless rejected Euclid’s idea that the rays had the same character as the geometer’s lines, for how could a ray with no cross-section transfer an image? He instead argued that the rays emitted by the eye form a “single continuous radiant cone.”

Hasan Ibn al-Haytham (965-1039) entirely dismissed extramission. He observed that “when the eye looks into exceedingly bright lights, it suffers greatly because of them and is injured.”⁶ The body does not injure itself, so the eye must be injured by some agent that comes into it. This agent, al-Haytham argued, is light itself. Looking at bright lights generates an afterimage, which shows that light has an impact on the eye. Al-Haytham concluded “that it is a property of light to act on the eye and that it is the nature of the eye to be affected by light.”⁷

Vision occurs because light enters the eye. This statement itself was revolutionary, but al-Haytham went further. He proposed a theory of intramission, but it wasn’t the intramission of coherent forms or simulacra. According to Al-Haytham,

From each point of every coloured body, illuminated by any light, issue light and colour along every straight line that can be drawn from that point.⁸

The eye sees not because it receives a coherent form, but because it assembles the information contained in all of the individual rays of light that enter it.

Al-Haytham’s hypothesis has a serious problem. Since light is emitted in all directions from each point on the surface of the observed object, a cone of light emanating from a single point will pass through the pupil and register on a circular area of the eye. Likewise, each point on the eye registers the light rays emanating from an area on the surface of the observed object. How is clear vision obtained from this mass of contradictory information?

Al-Haytham needed the information from each point on the observed object to be registered at a single point in the eye, so he argued that the light doesn’t stop at the surface of the eye — it passes into it. The light rays that are not perpendicular to the surface of the eye are (in accordance with rules already understood) refracted when they pass into the eye. At each point on the eye, exactly one ray will be perpendicular to the eye’s surface and pass through without refraction. Al-Haytham claimed that only the information in these unrefracted rays is used to assemble the image. He justified this claim by arguing that refraction “weakens” the rays.

Al-Haytham must have been aware that this is a terrible argument. A ray that is almost perpendicular when it enters the eye will be almost unrefracted, and therefore almost as strong as the perpendicular ray. His theory requires the refracted rays to matter not at all, not just more weakly. Al-Haytham — like other scientists before and after him — papered over a serious flaw in a promising theory in the hope that the puzzle would be resolved later. It was, but it took six centuries.⁹

Al-Haytham conducted his own experiments on refraction. He hypothesized that light always takes the “easier and quicker” path, anticipating Fermat’s principle (also known as the least time principle).

Al-Haytham sometimes used glass globes filled with water in his refraction experiments. He did not imagine these globes to be simulated raindrops, and he did not relate his experiments to the rainbow. He appears to have subscribed to Aristotle’s theory that the rainbow is caused by reflection from the irregular surfaces of clouds. But, centuries later, al-Haytham’s experiments inspired research that explained the rainbow, a phenomenon that had puzzled scholars for centuries. This research was carried out independently and almost simultaneously by Theodoric of Freiberg, a Dominican monk, and by Kamal al-Din al-Farisi. They found that the rainbow is the aggregate of effects produced within individual raindrops. As Theodoric explained in On the Rainbow (1304),

Rays of the sun strike the upper portion of the globe of water, are refracted into the sphere, are reflected backward at the inner concave surface at the rear of the globe or drop, and then are refracted once more on leaving the lower front portion of the spherical surface to travel toward the eye of the observer.¹⁰

Rainbows are arched because only some paths take light from the sun to the observer’s eye. The light from each raindrop sends only one colour to the observer’s eye, with the colour dependent on the degree of refraction required for the light to reach the observer.

Mathematics

Muhammad ibn Musa al-Khwarizmi (c. 780-850), a Persian who converted to Islam from Zoroastrianism, is Islam’s most influential mathematician. Like Euclid, he is as much known for synthesizing existing ideas as for developing new ones.

Al-Khwarizmi is most closely associated with algebra, but he was not its inventor — not by several thousand years. The Mesopotamians discussed problems involving quadratic equations and described the solutions to particular problems. Their explanations involved “sequences of unjustified calculations, and only the correctness of the answer suggests the existence of an underlying method.”¹¹ Diophantus (c. 200-284), a Hellenized Persian who lived in Alexandria, presented a more thorough analysis in Arithmetica. His book did not reach Europe until after the fall of Constantinople in 1453. It was translated into Latin in 1575, after which it became a vital textbook. Fermat (1601-1665) used it as his starting point for mathematical studies.

Al-Khwarizmi’s book, Compendium on Calculation by Completion and Balancing, reconstructed algebra by shifting it away from the discussion of particular problems and toward general solutions. The word “algebra” itself comes from the book’s Arabic title: it is a corruption of the word al-jabr, which Al-Khwarizmi used to represent his concept of completion.¹² Compendium was translated into Latin during the late twelfth century, and like Arithmetica, it was read in European universities as late as the seventeenth century.

Diophantus used a limited number of symbols in Arithmetica; al-Khwarizmi (who was unaware of Diophantus’s work) used none in Compendium. The reasoning, the mathematical operations, and even the numbers themselves were written in words.

The pivotal contribution of Compendium was its discussion of quadratic equations. Al-Khwarizmi divided them into six types, and provided a solution for each type. The book’s novelty lay in its use of geometry to demonstrate the solutions. As an illustration, here is one of al-Khwarizmi’s sample problems and its solution:

What must be the square which, when increased by ten of its own roots, amounts to thirty-nine? The solution is this: you halve the number of the roots, which in the present case yields five. This you multiply by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half the number of the roots, which is five; the remainder is three. This is the root of the square which you sought for; the square itself is nine.¹³

These manipulations work, but why do they work? Only the geometry provides a rationale for them. A modern account of the geometry associated with this particular problem is shown below.¹⁴

‘Abd al-Hamid ibn Turk wrote Logical Necessities in Mixed Equations at about the same time. Its contents are similar to those of Compendium, and it also includes geometric demonstrations. The similarity of the two books suggests that both books, like Euclid’s Elements, are works of synthesis.

Al-Khwarizmi also played a significant role in introducing the decimal number system to the Islamic world and, later, to Europe. This system had developed in India over a period of centuries.

The prototypes of the number symbols we use today all come from India. They are found in the Ashoka inscriptions from the third century BCE, the Nana Ghat inscriptions about a century later and in the Nasik Caves from the first and second centuries CE…

A positional, or place-value, notation is a numeral system in which each position is related to the next by a constant multiplier called the base. Our decimal system of course has a base of ten and credit for its development can be traced back to two great medieval Indian mathematicians, Aryabhata (476-55), who developed the place-value notation itself, and Brahmagupta a century later.¹⁵

The system likely became known to the Abbasids during al-Mansur’s reign. Al-Khwarizmi described it in The Book of Addition and Subtraction According to the Hindu Calculation (825). At the time, the Arabs were using the Greek and Roman number systems and, for astronomy, the Babylonian sexagesimal (base 60) system. Replacing one number system with another proved difficult, and for the next five centuries, “the decimal system remained no more than a curiosity.”¹⁶

The decimal system was first brought to Europe by the Italian mathematician Fibonacci (c. 1170-1250), who had travelled to the east to study with prominent Islamic mathematicians. He described the system in Liber Abaci (1202). Al-Khwarizmi’s book was then translated into Latin as Algoritmi de numero Indorum. “Algoritmi” is a latinized version of “al-Khwarizmi”: the modern word “algorithm” is derived from it.¹⁷ As in the Islamic world, the replacement of one number system with another was the work of centuries.

The zero is an essential part of the decimal system. It has two functions: it is the placeholder that distinguishes 11 from 101, and it is a quantity (for example, it is the quantity of eggs you have if you start with one egg and break one). The former function was accepted much more readily than the latter. The Hindus had recognized both functions by the time of Brahmagupta. In the Islamic world the second function was initially approached with caution, as it had been in India and would be Europe. “Even al-Khwarizmi avoided ever having to equate his algebraic quantities to zero. Instead, he would always have non-zero quantities on both sides of his equations.”¹⁸

Al-Khwarizmi was just one of a multitude of Islamic scholars who made significant contributions to mathematics. For example, Archimedes’ technique of computing the sum of infinitesimals (now known as integration) was further developed by Thabit ibn Qurra in the ninth century and by al-Haytham in the eleventh century. Ibrahim ibn Sinan and al-Biruni contributed to the theory underlying the stereographic projections embossed on the surface of astrolabes. Even the poet Omar Khayyam contributed: he wrote a book about cubic equations.

Astronomy

Islamic scholars inherited two well-developed astronomical systems. The Hindu system was adopted in the eighth century, and the Ptolemaic system was adopted in the tenth century, mostly as a result of the work of al-Battani. These two systems were not entirely compatible, and over time, the Ptolemaic system came to dominate. Nevertheless, Hindu astronomy had a significant impact on the course of Islamic astronomy. Its measurement techniques involved basic trigonometry. Islamic astronomers developed these techniques and the underlying theory, eventually converting trigonometry into an autonomous field of study.

By the ninth century the six modern trigonometric functions — sine and cosine, tangent and cotangent, secant and cosecant — had been identified…Of the six, five seem to be essentially Arabic in origin; only the sine function was introduced into Islam from India.¹⁹

Spherical trigonometry (the study of triangles laid out on the surface of a sphere) had been pioneered by the ancient Greeks, but the credit for its development belongs mostly to Islamic astronomers. It became one of their most fundamental tools.

Ptolemaic astronomy was an outgrowth of Aristotelian astronomy. This astronomy imagined that the earth sat placidly in the center of everything. The sun, moon, planets, and fixed stars were embedded in nested crystalline spheres that rotated about the earth at constant speeds. The model was geometrically perfect, but as observations accumulated, its inability to match the data became apparent. Ptolemy expanded the model’s mechanics in order to obtain a better fit with the data. The two most important additions are shown in the figure below, which shows a cross-section of a single sphere.

The first addition was used to solve the problem of retrograde motion. A planet generally moves across the sky from west to east; but it periodically reverses direction, travelling for a time from east to west, before resuming its eastward course. This phenomenon creates loops in the planet’s apparent path.

Ptolemy’s model of this behaviour begins with the eccentric (X in the figure), a point some distance from the earth. The deferent is a large ring centered on the eccentric. It carries, not the planet itself, but a point that travels around the deferent at a constant speed. A smaller ring called the epicycle is centered on this point and moves with it. The planet moves around the epicycle at a constant speed. The apparent path of the planet is the product of two motions: the movement of the point around the deferent, and the movement of the planet around the epicycle. The loops arise because these motions are sometimes reinforcing and sometimes opposing.

The second addition is a bit of a cheat. An observer on earth finds that the eastward movement of the planets is sometimes faster and sometimes slower — a clear violation of the assumption of uniform speed. The Ptolemaic model deals with this problem by introducing another point, the equant (the black dot in the figure). The equant is the same distance from the eccentric as the earth, and directly opposite the earth. The speed of the planets is assumed to be constant when seen from the equant, but it continues to be variable when seen from the earth.

In Ptolemy’s model, the motion of a planet is circular with respect to one point and constant speed with respect to another point, but neither of these points is the earth. The basic assumptions of the older model — circular motion and constant speed — are maintained in some fashion, but its elegant geometry has been abandoned.

Observational Astronomy

The need to choose between the Ptolemaic and Hindu systems of astronomy was the first driver of Islamic observational astronomy.

There were differences in numerical parameters, as well as in the geometrical structures of the planetary models. Fresh observations were necessary before a correct choice could be made; repeated solstice observations from Baghdad proved mutually incompatible. So a large-scale programme for observing the heavenly bodies (in particular the Sun and the Moon) every day for a whole year was set up at a new observatory at Mount Qasiyun in Damascus. To ensure high precision the observatory was equipped with large instruments, including a marble mural quadrant with a radius of 5 metres. Thus a tradition was founded for building observatories to carry out programmes of observation ideally covering 30 years to take in a complete revolution of slowly orbiting Saturn. Most famous among the numerous later observatories are the thirteenth-century Maragha institution founded by Nasir al-Din al-Tusi in the reign of Hulagu Khan, and Ulugh Beg’s fifteenth-century observatory at Samarkand.²⁰

The need for observation became no less acute once the astronomers had settled on the Ptolemaic system. Pythagorus and Aristotle had been interested in the basic structure of the universe, but Ptolemy wanted something more. His goal was to devise a model that would track the movements of the celestial bodies, and to achieve this end, he needed to calibrate his model. The quality of the model’s predictions would depend upon the accuracy of its parameters, so Ptolemy recognized the need for careful observation. He also recognized the need for other astronomers to verify his parameters, or to produce more accurate estimates of their own. His works include “careful instructions on how to establish the parameters from a limited number of selected observations.”²¹ The Islamic astronomers took on this task, but soon found that they weren’t just dealing with measurement error.

Trying to reconcile the data in the Almagest with the evidence of fresh observations created long-term problems of an unexpected complexity at the very root of astronomical science. The Arabic astronomers could not confirm the Almagest’s values of basic parameters, naturally thought of as constants of nature. These included the duration of the (tropical) solar year, the obliquity of the ecliptic and the rate of precession.²²

The astronomers found that in the seven hundred years since Ptolemy’s observations, the obliquity of the ecliptic²³ seemed to have fallen (it had) and the rate of precession²⁴ seemed to have risen (it hadn’t). If these parameters weren’t actually constants, revising the Ptolemaic system would require more than just nudging a few parameters in one direction or another.

Thabit ibn Qurra (d. 901), an adherent to an astral religion called the Sabians of Harran, was one of the astronomers who confronted this problem. He was able to identify a new constant.

Behind the seemingly secular variations of the tropical year…he found a virtually constant value of the sidereal year, which measures the solar motion against the background of the fixed stars. Here was the foundation for a “sidereal” doctrine of astronomy with the sphere of fixed stars as basic reference against which steady motions and constant parameters could be deduced.²⁵

He attributed the changes in both the obliquity of the ecliptic and the rate of precession to oscillations in the sphere of the fixed stars. His model of these oscillations became known as the theory of trepidation. It would not be superseded until Copernicus put the earth in motion, giving its axis both a tilt and a wobble.

Ptolemy’s observations came to the Islamic astronomers in the form of his Handy Tables, which “contained all the data needed to compute the positions of the sun, moon and planets, the rising and setting of the stars as well as eclipses of the sun and moon.”²⁶ Handy Tables became the model for the Islamic astronomical handbook known as the zij. Two zijes, one compiled by al-Khwarizmi and the other by al-Battani, became the basis for the Toledan Tables that later spread through Europe. Toledan Tables and its offshoots remained the most authoritative sources of astronomical data until the time of Kepler.

The focus of the observatories was relatively narrow, so that, for example, the supernova of 1054 was widely observed in China but “virtually unrecorded” in the Islamic world.²⁷ The observatories’ failure to branch out in new directions diminished their value. If their decline can be encapsulated by a single event, it is the demolition of Instanbul’s observatory by Sultan Murad III. The observatory was destroyed in 1580, only three years after its completion.

A naval ship approached the towering structure along the coast and, with its artillery, pummelled the masonry to the ground, as the ulema cheered in triumph and pious cries of joy rose from the crowd…Two thousand miles to the northwest in Denmark, Tycho Brahe was building his Uranoborg observatory on the island of Hven.²⁸

Planetary Theory

Islamic astronomers were quick to recognize that Ptolemy’s geometrical model of the universe was not consistent with Aristotle’s physical model. The most comprehensive study of the problem is al-Haytham’s Doubts about Ptolemy (c. 1028). Al-Haytham believed that Ptolemy’s model was a product of necessity rather than conviction.

We must elucidate the method that was followed by Ptolemy for determining the configurations of the planets. That is, he had gathered together all the motions of the individual planets that he could verify with his own observations, or the observations of those who had preceded him. He then sought a configuration that could possibly exist for real bodies that moved with those motions, and was not able to achieve it. He then assumed an imaginary configuration with imaginary lines and circles that could move in those motions, even though only some of those motions could indeed take place in [real] bodies…He was obliged to follow that route for he could not devise another.²⁹

Aristotle had assumed that the celestial spheres were physical spheres and that their movement was uniform and circular. The essence of al-Haytham’s critique was that Ptolemy’s model routinely violated this assumption. In particular, every equant violated the assumption of uniform motion.

Since the equant sphere was a fictitious sphere, and thus could not produce any perceptible motion of its own,…the only sphere that could produce a real motion was that of the deferent, and that was now proved to be moving non-uniformly around its own center. This contradicts the assumption of uniform motion.³⁰

Al-Haytham likewise objected to the “seesawing motion of the inclined planes which carried the epicycles of…Mercury and Venus,”³¹ because this motion violated the assumption that all celestial motion was circular.

These observations led al-Haytham to conclude that a planetary model had to be built on principles other than those laid out by Ptolemy.

It became clear…that the configuration, which Ptolemy had established for the five planets, was a false configuration, and that the motions of these planets must have a correct configuration, which includes bodies moving in a uniform, perpetual, and continuous motion, without having to suffer any contradiction, or be blemished by any doubt. That configuration must be other than the one established by Ptolemy.³²

Al-Haytham’s conclusion was widely accepted. It led to the development of two schools of Islamic astronomy.

One of these schools was the Andalusian school of the twelfth century. It abandoned Ptolemy’s innovations and attempted to resurrect the Aristotelian model. Since Ptolemy’s innovations were a response to the Aristotelian model’s failure to match the data, the Andalusian school made little progress.

The other school was the Maragha school, named for the Iranian observatory. The school’s most important contributors were Mu’ayyad al-Din al-‘Urdi (d. 1266), Nasir al-Din al-Tusi (1201-1274), and Ibn al-Shatir (1304-1375). Al-‘Urdi and al-Tusi worked at the observatory; al-Shatir was the timekeeper at a mosque in Damascus.

Al-‘Urdi’s main contribution was the “Urdi Lemma,” an extension of a lemma by Apollonius that allowed equants to be replaced by epicycles. Al-Tusi devised the “Tusi couple,” which allows reciprocating motion to be generated from circular motion. There are several forms of the couple, each designed for a particular purpose (for example, to replace Ptolemy’s seesawing planes with circular motion.) The simplest form involves a circle that rolls around the inner perimeter of a circle twice its diameter. The rolling of the inner circle causes any given point on the inner circle to move back and forth along the diameter of the outer circle. This device allowed al-Tusi to

… to achieve his goal of generating the nonuniform motions of the planets by combinations of uniformly rotating circles. The centers of the deferents, however, were still displaced from the earth.³³

The task of reforming the Ptolemaic model was completed by al-Shatir, who was able to remove all of the eccentrics and equants through the use of secondary epicycles.

The Ptolemaic model had the earth at its center but was not truly geocentric. Motion was circular around the eccentric and uniform around the equant, neither of which was the earth’s axis. Al-Shatir’s model was the first genuinely geocentric model of the universe.

Al-Shatir’s achievement would be overshadowed by events occurring in Europe. Al-Haytham had not recognized that the problems of the Ptolemaic model lay not in the model’s mechanics, but in the basic assumptions of the Aristotelian model: the spheres are real; the earth is motionless at the center of all things; the celestial bodies move in circular orbits at uniform speeds. European scholars would refute all of these assumptions between 1543 and 1619.

At roughly the same time, Islamic astronomy acquired a passivity that it would not shed until the nineteenth century.

All of the problems had been solved, some many times over. Much of the innovative activity had led into a cul-de-sac, from which it would not emerge until modern times…From Morocco to India the same old texts were copied and studied, recopied and restudied, usually different texts in each of the main regions. But there was no new input of any consequence.³⁴

Copernicus and the Maragha School

There is evidence that some of the mechanics of Copernicus’s model — but not its revolutionary assumption of heliocentrism — originated with the Maragha school. The extent of Copernicus’s “borrowings” is uncertain because few historians have a thorough understanding of the relevant models, and also because the historians themselves display an unusual degree of partisanship.

One example of this partisanship is a finding by Willy Hartner that is routinely presented as a “gotcha” moment by advocates of Islamic influence.³⁵ It involves a comparison of the figures used by al-Tusi and Copernicus in their proofs of the Tusi couple. These figures, as presented by Hartner, are shown below. Here is George Saliba’s description of Hartner’s finding:

By comparing Tusi’s proof, which was completed in 1260-1, with that of Copernicus, which was published in 1543, Hartner discovered that the two proofs carried the same alphabetic designators for the essential geometric points. That is, where Tusi’s proof designated a specific point with the Arabic letter “alif,” Copernicus’s proof signalled the same point with the equivalent phonetic Latin letter “A,” where Tusi had “ba,” Copernicus had “B,” etc., except in one case where Tusi had “zain” and Copernicus has “F.” On the basis of the letter correspondences, letter to letter, Hartner ventured to say that Copernicus must have known about Tusi’s work while in Italy…As far as we know neither Copernicus could read Arabic, nor was Tusi’s text, in which the theorem appeared, ever translated into Latin. To Hartner, it meant that Copernicus must have recruited someone who could explain to him the diagram, while he took notes and used those notes later when he came to write the De Revolutionibus.³⁶

Saliba then improves Hartner’s story by arguing that the letter “zain” looks much like the letter “fa,” and that the translator mistook the former for the latter. This mistake caused Copernicus to replace “zain” with “F.” So, really, the match was perfect!

But a quick look at Copernicus’s labelling shows that he has (i) labelled the endpoints of the diameter with the letters A and B, (ii) labelled three successive points lying on the diameter with the letters C, D and E, (iii) labelled the vertices of a triangle, in clockwise fashion, with the letters F, G and H. In short, he has used the first eight letters of the alphabet to identify eight points of interest in an orderly fashion.

Why does this matter? “Ockham’s razor” tells us to avoid the unnecessary multiplication of entities. A hypothesis involving a never-found manuscript, an imagined translator, and an alleged reconstruction cannot be the preferred explanation for the labelling of Copernicus’s diagram because a much simpler hypothesis — that Copernicus worked in an orderly fashion — explains the labelling at least as well.

Overreach of this kind seems to be unusually common in this literature. I found Jamil Ragep to be the most reliable narrator among the historians advocating Islamic influence.³⁷ Victor Blasjo makes a strong (one might even say, ferocious) case against Islamic influence.³⁸

André Goddu has suggested that the historians arguing this question have taken one of three positions:³⁹

• Blueprint copying: Copernicus had enough access to the planetary models of the Maragha school that he only needed to recast an Islamic geocentric model as a heliocentric model.
• Independent development: The work of the Maragha school was effectively unknown to Copernicus; De revolutionibus is Copernicus’s own invention.
• Idea diffusion: Concepts from the Maragha school, but not planetary models, had filtered into Europe over the preceding centuries and were used by Copernicus’s predecessors as well as by Copernicus himself. The concepts came to Europe in informal ways. European scholars did not acknowledge their origins because they were unaware of them.

George Saliba argued for blueprint copying, with al-Shatir’s planetary theory as the source:

All that someone like Copernicus had to do was to take any of Ibn al-Shatir’s models, hold the sun fixed and then allow the Earth’s sphere, together with all the other planetary spheres that were centered on it, to revolve around the sun instead. As we shall soon see, that was the very step that was taken by Copernicus.⁴⁰

[Al-Shatir’s] lunar model was identical to that of Copernicus, and his technical treatment of the motion of the planet Mercury used the same Tusi Couple that was used by Copernicus as well. His model for the upper planets, which was also adopted by Copernicus after shifting the center of the universe to the sun, also included the use of ‘Urdi’s Lemma, and continues to be at the center of the ongoing research that will one day determine the routes by which Copernicus knew of this astronomer’s work.⁴¹

Noel Swerdlow is another historian who argued for blueprint copying:

How Copernicus learned of the models of his [Arabic] predecessors is not known — a transmission through Italy is the most likely path — but the relation between the models is so close that independent invention by Copernicus is all but impossible.⁴²

The difficulty with this position is that blueprint copying requires the transmission to Europe of a substantial part of the Maragha tradition. However, there is no evidence that any significant Maragha works were available in Europe, in either Latin or Greek, during Copernicus’s time.⁴³

There are several reasons why the absence of these manuscripts is unsurprising. Religious tension between Muslims and Christians was high. Also, the Ottomans had captured Constantinople in 1453, and their military aggressiveness continued through the next century — they laid siege to Vienna in 1529. These factors created a cultural wall between Europe and the Muslim world. Moreover, there is no evidence that Europeans continued to think of the Muslim world as a source of interesting ideas. The translation movement brought into Europe the works of many great Islamic scholars, including a number of astronomers. Copernicus cited five of these astronomers in De revolutionibus, but the last of them, al-Bitruji, had died in 1204. Arabic-to-Latin scientific translations were uncommon after this time. Europeans did not attach to contemporary Greece the glory of Aristotle and Ptolemy, and there is no evidence that they attached to the contemporary Muslim world the glory of al-Khwarizmi and al-Haytham. Finally, the Muslims themselves did not attach great significance to the Maragha planetary models, reducing the likelihood that they would be transmitted to European scholars.⁴⁴

In the absence of any significant evidence of transmission, the blueprint hypothesis becomes a judgment call. How similar do things have to be to be “identical”? How difficult does a task have to be to be “impossible”? Neither of these words should be taken literally.

On the latter issue, the opponents of blueprint copying argue that Copernicus and the Maragha school had the same goal (modelling the heavens), using essentially the same set of data (European astronomical tables were largely based on Islamic zijes) and a limited set of tools (uniform circular motion, equants and epicycles from Ptolemy, but also Tusi couples and double epicycles, which Europeans likely learned from the Muslims). Given these constraints, it is almost inevitable that independent solutions to the problem will contain many similarities.

Now consider this quote from Swerdlow and Neugebauer:

The planetary models for longitude in the Commentariolus are all based upon the models of Ibn al-Shatir — although the arrangement for the inferior planets is incorrect — while those for the superior planets in De revolutionibus use the same arrangement as ‘Urdi’s and Shirazi’s model, and for the inferior planets the smaller epicycle is converted into an equivalent rotating eccentricity that constitutes a correct adaptation of Ibn al-Shatir’s model. In both the Commentariolus and De revolutionibus the lunar model is identical to Ibn al-Shatir’s…

The question therefore is not whether, but when, where, and in what form he learned of Maragha theory. ⁴⁵

They find some things that don’t match — that are “incorrect” or “converted” — but otherwise are able to match the components of Copernicus’s model to components in two different Islamic models. They are persuaded that blueprint copying has occurred. But this conclusion was based on the assumption that Copernicus had access to these two models, and forty years later, there is still no evidence that he had access to even one of them. In the absence of confirmatory evidence, the argument that the likeness reflects independent solutions to the same problem becomes more compelling. Some historians, including Owen Gingerich and Mario di Bono, take this position.

Blueprint copying, like cold fusion, is an interesting hypothesis that is not supported by objective evidence. By contrast, independent development (as defined above) is a hypothesis that has evidence arrayed against it. The Tusi couple was an integral part of Copernicus’s model. There is evidence that various forms of the Tusi couple were known in Europe as early as the fourteenth century. Europeans described it and used it, but none of them claimed to have invented it and none could identify its provenance. It is most likely that the device passed from Persia to Byzantium and then from Byzantium to Europe.⁴⁶ This one example is enough to dismiss independent development as a viable hypothesis.

Idea diffusion is the “last man standing.” It’s also the “muddy middle,” and there is much about it that is still not understood. But idea diffusion is actually a simpler problem than blueprint copying. Blueprint copying requires Copernicus to have had detailed knowledge of Maragha’s astronomy while the rest of Europe’s astronomers somehow remained unaware of it (there is no sign of it in their work). Idea diffusion abandons the search for this “secret knowledge,” instead focussing on the gradual transmission of knowledge from one culture to another.

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1. Quoted by Lindberg, Theories of Vision from al-Kindi to Kepler, p. 2. ↩
2. Quoted by Lindberg, Theories of Vision from al-Kindi to Kepler, p. 2. ↩
3. Quoted by Lindberg, Theories of Vision from al-Kindi to Kepler, pp. 2-3. ↩
4. Quoted by Lindberg, Theories of Vision from al-Kindi to Kepler, p. 7. ↩
5. One sometimes encounters the claim that al-Kindi believed in intromission. In fact, al-Kindi specifically rejected all forms of intromission and fully explained his reasons for doing so. For details, see David Lindberg, “Alkindi’s Critique of Euclid’s Theory of Vision,” Isis (1971). ↩
6. Quoted by Lindberg, Theories of Vision from al-Kindi to Kepler, pp. 61-2. ↩
7. Quoted by Lindberg, Theories of Vision from al-Kindi to Kepler, p. 62. ↩
8. Quoted by Lindberg, Theories of Vision from al-Kindi to Kepler, p. 73. ↩
9. The anatomy of the eye was not well understood in al-Haytham’s time, and he took the lens to be an essentially passive receptor of light (see Lindberg, Theories of Vision, p. 69 and p. 71). Kepler, using still tentative ideas about the anatomy of the eye, argued that the purpose of the lens was to focus all of the light rays coming from a single point on the observed object onto a single point on the retina, which was the true receptor of light. This hypothesis also had its problems: Kepler’s theory implied that the image on the retina was upside down and reversed. Unable to solve this puzzle, Kepler suggested (correctly) that the problem was sorted out by the optic nerve or the brain. ↩
10. Quoted by Carl Boyer, The Rainbow: From Myth to Mathematics (Princeton University Press, 1987), p. 114. ↩
11. Jacques Sesiano, An Introduction to the History of Algebra (American Mathematical Society, 2009), p. 17. ↩
12. Al-jabr was originally a medical term, meaning the setting of broken bones. ↩
13. Al-Khwarizmi. From Frederic Rosen, editor and translator, The Algebra of Mohammed Ben Musa (1831), p. 8. ↩
14. This presentation was taken from the website Story of Mathematics. ↩
15. Al-Khalili, The House of Wisdom, pp. 99-100. ↩
16. Al-Khalili, The House of Wisdom, p. 102. ↩
17. Algorithms themselves date back at least as far as the Greeks, for example, the Sieve of Eratosthenes is a process for quickly identifying prime numbers. ↩
18. Al-Khalili, The House of Wisdom, p. 105. ↩
19. Owen Gingerich, “Islamic Astronomy,” Scientific American (1986), p. 77. ↩
20. K. P. Moesgaard, “Astronomy,” in I. Grattan-Guiness, ed., Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (Routledge, 1994), p. 244. ↩
21. Gingerich, “Islamic Astronomy,” p. 79. ↩
22. Moesgaard, “Astronomy,” p. 244. ↩
23. The obliquity of the ecliptic is now understood to be a measure the earth’s tilt. This fact was not understood in earlier times, but it could still be measured by comparing the angles to the sun at noon on the summer and winter solstices. ↩
24. Precession occurs because the earth’s axis changes its orientation (not its tilt), tracing out a cone-shaped area over the course of 26,000 years. The early astronomers recognized it by the slow shifting of the constellations. The ancient Greeks estimated the shift to be one degree per century. ↩
25. Moesgaard, “Astronomy,” p. 245. ↩
26. Jim al-Khalili, The House of Wisdom, p. 83. ↩
27. Gingerich, “Islamic Astronomy,” pp. 78-9. ↩
28. Livingston, The Rise of Science in Islam and the West, p. 125. ↩
29. Al-Haytham, quoted by Saliba, Islamic Science and the Making of the European Renaissance, p. 103. ↩
30. Saliba, Islamic Science and the Making of the European Renaissance, p. 99. ↩
31. Saliba, Islamic Science and the Making of the European Renaissance, p. 99. ↩
32. Al-Haytham. Quoted by Saliba, Islamic Science and the Making of the European Renaissance, p. 100. ↩
33. Gingerich, “Islamic Astronomy,” p. 83. ↩
34. David King, “Islamic Astronomy,” in Christopher Walker, ed., Astronomy before the Telescope (British Museum Press, 1996), p. 171. ↩
35. Willy Hartner, “Copernicus, the Man, the Work, and Its History”, Proceedings of the American Philosophical Society (1973). ↩
36. George Saliba, Islamic Science and the Making of the European Renaissance, (MIT Press, 2007), p. 200. ↩
37. Specifically, see Jamil Ragep, “Copernicus and His Predecessors: Some Historical Remarks,” History of Science (2007). ↩
38. Viktor Blasjo, “A Critique of the Arguments for Maragha Influence on Copernicus,” Journal for the History of Astronomy (2014). ↩
39. André Goddu, “The (likely) Last Edition of Copernicus’ Libri revolutionum,” Variants (2009). ↩
40. Saliba, Islamic Science and the Making of the European Renaissance, p. 193. ↩
41. Saliba, Islamic Science and the Making of the European Renaissance, p. 190. ↩
42. Swerdlow, quoted by Ragep, “Copernicus and his Islamic Predecessors,” p. 68. ↩
43. On the evidence, see Saliba, Islamic Science and the Making of the European Renaissance, p. 214, or Ragep, “Copernicus and his Islamic Predecessors,” p. 68. ↩
44. With reference to “writings of the Maragha school or of Ibn al-Shatir,” Mario di Bono states that “these writings remained practically unknown in the Maghreb, a region that usually carried out the function of intermediary for the transmission of scientific information to the West.” (di Bono, “Copernicus, Amico, Fracastoro and Tusi’s Device,” Journal for the History of Astronomy, 1995, p. 144.) Also, referring to “later astronomers in Damascus and Cairo,” David King writes that “none…appear to have been particularly interested in Ibn al-Shaṭir’s non-Ptolemaic models.” Instead, al-Shatir’s posthumous reputation rested on his zij and his instrument-making. (David King, “Ibn al-Shatir,” in The Biographical Encyclopedia of Astronomers (Springer Reference), 2007, last paragraph.) ↩
45. Swerdlow and Neugebauer in 1984. Quoted by Ragep, “Copernicus and his Islamic Predecessors,” p. 68. ↩
46. Jamil Ragep, “From Tun to Torun: the Twists and Turns of the Tusi-Couple,” in Ragep, Islamic Astronomy and Copernicus (Turkish Academy of Sciences, 2022). ↩