Merchants and Mechanics

Based on Richard Rubenstein, Aristotle’s Children (Harcourt, 2003); Olaf Pedersen, Early Physics and Astronomy (Cambridge, 1993); Edward Grant, The Foundations of Modern Science in the Middle Ages (Cambridge, 1996); A. C. Crombie, Augustine to Galileo (Harvard, 1953)

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European science before the twelfth century was rudimentary and entwined with morality.

The study of nature was not expected to lead to hypotheses…but to provide vivid symbols of moral realities. The moon was the image of the Church reflecting the divine light, the wind an image of the spirit, the sapphire bore a resemblance to divine contemplation, and the number eleven which “transgressed” ten, representing the commandments, stood itself for sin.¹

The works of the ancient Greeks were almost unknown in Christian Europe. In the fourth century the Roman empire, its borders threatened by barbarians in the west and Persians in the east, was divided into two halves that soon became linguistically isolated. Greek had always been the dominant language of the eastern empire, and it remained so. Latin had dominated the western empire, but the elite had prized knowledge of the Greek language. The elite’s scholarly accomplishments had been built on leisure, though, and after the barbarian invasions, no-one in Europe — not even the monks — had leisure. Knowledge of the Greek language dissipated. The elite first lost its ability to read the books of the ancient Greeks, and then lost the books themselves. These losses were little regretted, and by the seventh century the distinction between elite and popular culture had almost disappeared.

The Greeks’ science and philosophy survived in the west almost entirely in the form of compilations written by Boethius, Cassiodorus, Isidore of Seville, and a few others. These compilations were superficial, introducing a few important concepts and definitions but not developing them into complete systems. Mathematics was a practical discipline, “very elementary and limited to what was necessary to keep accounts, calculate the date of Easter, and measure land for the purposes of surveying.”²

European science was revived by a translation movement that began in the twelfth century, precipitated by events in Toledo and Sicily.

Muslim fighters had conquered a large part of Spain during the eighth century. In a campaign known as the Reconquista, soldiers from Christian kingdoms drove back the Muslims in stages, with the last stage being their expulsion from Granada in 1492. One of the early Christian victories was the recapture of Toledo in 1085. Toledo’s libraries contained Arabic translations of the works of the Greek philosophers, including many of Aristotle’s most significant works.

Here…were the Greek sage’s great essay on the philosophy of being, Metaphysics, and his treaties on methods of reasoning and the divisions of knowledge. Here were his scientific masterpieces: Physics, On the Heavens, History of Animals, and On Generation and Corruption. And here…were Aristotle’s world-famous treatise on the soul, De Anima; his Nicomachean Ethics; and his Politics.³

The libraries also contained commentaries and original research by Islamic scholars. Taken together, these books were of incalculable value to Europe’s scholars. Gerald of Cremona, who would become Toledo’s most famous translator, came to the city in search of a copy of Ptolemy’s Almagest. “There, seeing the abundance of books in Arabic on every subject, and regretting the poverty of the Latins in these things, he learned the Arabic language in order to be able to translate.”⁴

The translation movement in Toledo began in the twelfth century. The final language was always Latin, but there were so few Arabic-to-Latin translators that an intermediate translation (say, Arabic-to-Spanish, to be followed by Spanish-to-Latin) was often required. The early translators had little familiarity with the substance of the books that they were translating, which led to imperfect translations.

The translations were often literal, and often words whose meanings were imperfectly understood were simply transliterated from their Arabic or Hebrew form. Many of these words have survived to the present day as, for example, alkali, zircon, alembic (the upper part of a distilling vessel),…nadir, zenith, azure, zero, cipher, algebra.⁵

Sicily had been under Islamic control for two centuries when, in the eleventh century, Norman mercenaries took control of the island. As in Toledo, the retreating Muslims left behind libraries filled with precious books. The island’s people spoke Greek, Latin, and Arabic, making the island an ideal center for translations. The first direct Greek-to-Latin translations were made there.

The ability to speak multiple languages was sufficiently rare that the translation movement would have made little headway if it had relied solely on Christians. Muslims and Jews were essential to the movement’s success. Toledo and Sicily were the first translation centers, but additional centers developed in places like Provence and northern Italy, where Christians, Jews and Muslims congregated freely.

Almost all significant Greek works of science and philosophy had been translated into Latin by the middle of the thirteenth century. Many Islamic scientific and philosophic works written before the end of the twelfth century were also translated into Latin. Islamic works written after that date were rarely translated.

A new kind of educational institution, the university, appeared in Europe late in the twelfth century. Universities were established in Oxford, Bologna, and Paris before 1200, and hundreds more were established during the next century. Students began their studies in the arts program, but could move on to higher studies in theology, law, or medicine. The curriculum in the arts program was built around Aristotle’s newly translated books.

Christianity and Greek Philosophy

Aristotle’s philosophy was once again available to the scholars of Christian Europe, but it was not evident that it could be made compatible with Christianity itself. Christians imagined a God who saw the little sparrow fall; Aristotle imagined only an abstract Prime Mover whom he banished to the margins of the universe. Christians believed

… that our brief life here on earth is merely a prelude to the afterlife; that we are inveterate sinners who can be saved, if at all, only by God’s freely given grace; and that our ability to understand this fallen universe is limited by both the feebleness of our reason and the inherent deceptiveness of sense impressions. How can we be truly happy here? Aside from fleeting moments of joy, we cannot, since this realm of pain and illusion is not our true home.⁶

By contrast,

Aristotle does not appear to be aware that the world of the senses is a place of suffering and unreality, or that there is a better, more real world to come. His universe is one, and he seems to feel entirely at home in it…Human reason, far from being crippled, is perfectly adequate to secure man’s knowledge, good behavior, and happiness.⁷

Making Aristotle’s worldview compatible with Christianity would appear to be a task as difficult as squaring the circle.

The ideas of one Greek philosopher — Plato — had already been “naturalized” by Christians, but Plato’s ideas were more easily aligned with Christianity. The most influential Christian interpreter of Plato was Augustine, who was already in his thirties, intellectually dissatisfied and not yet a Christian, when he first read Plato. Plato’s ideas resonated with him, and ultimately led him to Christianity and a life in the Church.

Augustine discovered parallels between Greek and Christian thought that he had not previously recognized. Not only did Plato affirm the existence of a Supreme Good — unitary, immaterial, perfect, and timeless — but he also gave precedence of spiritual over material values, argued for the immortality of the soul, and advocated a way of life aimed at refining human existence…But his most important contribution, from Augustine’s point of view, was to insist that the world of appearances — the world of “facts” apprehended through sense impressions — is a kind of watered-down and distorted reality, a universe of imperfect copies rather than the originals.⁸

Augustine adopted Plato’s definition of evil as an absence of good — not something created by God, but a “privation of being”…brought into the universe by man’s misuse of his free will.⁹

Augustine became the Church’s leading theologian. His great work, The City of God (c. 420), incorporated ideas drawn from Plato’s works. In the end, though, Augustine had to move beyond Plato, for Plato had not envisioned any possibility of salvation.

Eight hundred years later, when Aristotle’s works were rediscovered, Augustine’s theology still reigned. Could Aristotle’s philosophy find a place in a Church that had already absorbed Plato’s more amenable philosophy? This question was decided within the universities, with the University of Paris playing a pivotal role.

Aristotelian philosophy deviated from Church doctrine in several ways. Aristotle’s belief in an eternal and uncreated world flatly contradicted the Bible’s creation account. Adam and Eve’s expulsion from the Garden of Eden was part of this account, and it was the foundation for the Church’s doctrine of original sin, which was inimical to Aristotle’s understanding of human nature. Some of Aristotle’s claims, such as the impossibility of a vacuum, implied that there are limits to God’s power — a position that no Christian theologian would consider.

Aristotle’s works were initially translated from Arabic rather than Greek, and their interpretation was guided by translations of Islamic commentaries. From the time of al-Kindi, Islamic scholars had infused Aristotle’s philosophy with elements of Neoplatonism so that it aligned more closely with Islam’s teachings. But the interpolations and interpretations that made Aristotle’s philosophy more acceptable to Muslims, made it less acceptable to Christians.

The Arab interpretation of Aristotle was strongly coloured by the Neoplatonic conception of the chain of being stretching from first matter through inanimate and animate nature, man, the angels and Intelligences to God as the origin of all. When such commentators as Alkindi, Alfarabi, Avicenna and particularly Averroes (1126-98) introduced from the Mohammedan religion into the Aristotelian system the idea of creation, they interpreted this in such a way as to deny free will to man and even to God himself. According to them the world had been created not directly by God but by a hierarchy of necessary causes starting with God and descending through the various Intelligences which moved the celestial spheres, until the Intelligence moving the moon’s sphere caused the existence of a separate Active Intellect which was common to all men and the sole source of their knowledge. The form of the human soul already existed in this Active Intellect before the creation of man, and after death each human soul merged again into it.¹⁰

Several points in this system were entirely unacceptable to the philosophers of Western Christendom in the 13th century. It denied the immortality of the individual human soul. It denied human free will…It was rigidly determinist, denying that God could have acted in any way except that indicated by Aristotle.¹¹

Debates over the content of Aristotelian philosophy were implicitly debates over a much more consequential issue: the roles of reason and revelation.

Reason was the mode of analysis in philosophy, which was often considered co-extensive with the theoretical sciences, most of which would not themselves become independent disciplines until the seventeenth century and later. The arts masters ruled over the domain of reason and, therefore, of philosophy. But theologians held sway over revelation, and it is not difficult to understand why they held the upper hand in a society dominated by religion.¹²

Evidence that the theologians “held the upper hand” is easily found. In 1210 persons affiliated with the University of Paris were prohibited from reading Aristotle’s natural philosophy, in public or private, on pain of excommunication. (Aristotle’s logic and ethics were not subject to the ban.) In 1245 the prohibition was extended to the University of Toulouse, which had proudly but unwisely declared that Aristotle could be freely studied there. In 1272 the University of Paris instituted an oath for its arts masters: they were required to swear that they would not discuss theological issues, and that if such a discussion became unavoidable, they would resolve the issue in favour of Church doctrine. In 1277 the Bishop of Paris condemned 219 propositions that the university’s arts masters were alleged to espouse.

Many of the condemned propositions were related to Aristotle’s natural philosophy. Twenty-seven separate propositions dealt with the eternity of the world. Further propositions dealt with ideas, such as the impossibility of a vacuum, that implied limits on God’s power. But many of them dealt with the relative status of the arts masters and the theologians. These propositions suggest that the arts masters did not consider themselves, the proponents of reason, to be inferior to the theologians, the guardians of revelation:

40. That there is no more excellent state than to devote oneself to philosophy.
145. That there is no question disputable by reason which a philosopher should not dispute and resolve….
150. That a man ought not to be satisfied to have certitude on any question on the basis of authority.
152. That theological discussions are based on fables.
154. That the only wise men of the world are philosophers.

Clearly, the arts masters were an uppity lot.

And yet, the “naturalization” of Aristotle’s philosophy was largely the work of theologians, most especially, of Albertus Magnus and Thomas Aquinas. For them, the fundamental problem was the relationship between reason and revelation.

In his attempt to resolve this difficulty Albertus based himself on two certainties: the realities of revealed religion and the facts which had come within his own personal experience. Albertus and St. Thomas did not regard Aristotle as an absolute authority as Averroes had done, but simply as a guide to reason. Where Aristotle, either explicitly or as interpreted by Arab commentators, conflicted with the facts either of revelation or of observation he must be wrong: that is, the world could not be eternal, the individual human soul must be immortal, both God and man must enjoy the exercise of free will…But Albertus and more definitely St. Thomas realized…that theology and natural science often spoke of the same thing from a different point of view, that something could be both the work of Divine providence and the result of a natural cause. In this way they established a distinction between theology and philosophy which assigned to each its appropriate methods and guaranteed to each its own sphere of action. There could be no real contradiction between truth as revealed by religion and truth as revealed by reason.¹³

Aquinas’s philosophical conclusions were sometimes controversial, and some of his beliefs were targeted in the condemnations of 1277. After his death, the growing acceptance of scientific thinking led to a reappraisal of his ideas. He was declared a saint in 1323, and today is considered the intellectual equal of Augustine.

The restructuring of European beliefs was still not complete. Aquinas had argued that there is a single body of knowledge that can be apprehended through either revelation or reason.¹⁴ This position was contested by Jon Duns Scotus and William of Ockham.

Thomas [Aquinas] believed that by understanding the laws governing nature, we perceive, even if dimly, the creative intentions of God…According to Duns Scotus, this was a serious mistake…We cannot look into the “mirror of nature” and reach valid conclusions about God’s intentions, desires, or plans. Nor, on the basis of what we observe in the natural universe, can we deduce the truths of religion.¹⁵

Knowledge of God, Duns Scotus argued, can only be acquired through revelation. But equally, if God is not to be perceived through the study of nature, this study can only reveal nature itself. Theology and natural philosophy are therefore separate domains.

One could arrive at the same conclusion by applying “Ockham’s razor,” which warns against unnecessarily multiplying concepts. Aquinas’s philosophy was replete with concepts that served only to link the natural to the divine. The idea that the natural and the divine are separate realms does away with all of these concepts, leaving only “specialized” concepts appropriate to a single realm.

On the science side, there are concepts and methods derived from experience and processed by reason that help us to understand the natural world and the world of human society. On the theology side, there are doctrines revealed by Scripture or the Church that help us to understand God and what he requires of us.¹⁶

The separation of the natural and divine domains meant that science was “freed of the burden of making theological sense of scientific findings,”¹⁷ facilitating the development of true empiricism.

The Nature of Medieval Science

European science needed more than a century to absorb the contents of the newly translated books. It had acquired all of Ptolemaic astronomy and Aristotelian physics. Its mathematical acquisitions included Euclid’s geometry, Appollonius’s conics, and al-Khwarizmi’s algebra and arithmetic. The progress of science was slow during the twelfth and thirteenth centuries, but then began to quicken. In optics, for example, Roger Bacon’s major accomplishment was to synthesize and interpret the work of Islamic scholars. But at the beginning of the fourteenth century, a European scholar and an Islamic scholar simultaneously deduced the cause of the rainbow. Eyeglasses, an early application of optics, were invented in Europe at about the same time.¹⁸

There were three developments in the middle ages that would have significant impacts on European science during the following centuries. First, scholars engaged in disputations on matters related to natural philosophy. These disputations were often public events, but a written version of disputation, called a question, also emerged. One question was, “Is the world eternal?” Another was, “Can there be more than one world?” Several hundred of these questions have been found in the literature of the time. A scholar would undertake to examine and resolve a question in a systematic fashion. He would first present all the arguments supporting a “yes” response and all of the arguments supporting a “no” response. Then he would give his own answer and briefly support it. Finally, he would go through all of the arguments that opposed his own answer, explaining the weakness of each of them. Disputation taught European scholars how to think both deeply and logically about matters of natural philosophy. At the same time in the Islamic world, the faithful were being discouraged from studying natural philosophy for fear that they would be led astray. European science had barely begun, but it was already diverging from Islamic science.

Second, European scholars thought deeply about what can be learned through observation. They were aware that their most important findings were not the observations themselves, but the inferences drawn from them. This awareness led to an analysis of inductive reasoning that had no counterpart in Greek or Islamic science. Robert Grosseteste argued that induction is a valuable tool, but one that must be used carefully.

Grosseteste held that it was never possible in natural science to arrive at a complete definition or an absolutely certain knowledge of the cause or form from which effects followed…The same effect might follow from more than one cause and it was never possible to know all the possibilities. By making deductions from the various theories advanced and by eliminating theories whose consequences were contradicted by experience, it was possible, Grosseteste held, to approach closer to a true knowledge of the causal principles or forms really responsible for events in the world of our observation.¹⁹

Jon Duns Scotus clarified the distinction between causal laws and empirical generalizations. Even if one could not discern the causal law underlying an empirical generalization, he argued, the empirical relationship could be relied upon because it followed from the uniformity of nature. William of Ockham

…held that the only certain knowledge about the world of experience was what he called “intuitive knowledge” gained by the perception of individual things through the senses…All the rest, all the theories constructed to explain the observed facts, comprised “rational science,” in which names stood merely for concepts and not for anything real.²⁰

God creates the universe, said William of Ockham, but the patterns that we discover by reasoning abstractly about created things are the products of our mental processes, not evidence of divine intentions. Reason is not inherent in nature…but in our own minds.²¹

William believed that causal laws could never be completely certain. Sometimes more than one “candidate” law explains a given set of observations. Some candidates have implications that are inconsistent with other observations, so they can be eliminated. Candidates that unnecessarily multiplied conceptual entities can also be eliminated.

The third development was the appearance of the idea that differences in a quality can be represented by differences in a quantity. Aristotle believed that quantity and quality were distinct concepts. A change in quantity was brought about by adding or subtracting homogeneous parts and was therefore measurable. A change in quality involved the replacement of one attribute with another and was therefore not measurable. By contrast, Jon Duns Scotus and William of Ockham believed that qualities like heat were measurable. When Galileo made the first rudimentary thermometer in 1593, he was acting upon a medieval insight. It was also recognized that motion — another “quality” — is measurable. Aristotle had argued that one body is faster than another if it travels farther in the same time or if it travels the same distance in a shorter time. The idea of forming a ratio of two “unlike” quantities, distance and time, was unnatural to him, so he did not imagine velocity to be precisely measurable.²² By contrast, European scholars used the concepts of velocity and acceleration in mathematical studies during the middle ages. Moreover, these concepts were implementable because time could be measured with a water clock or a sand glass. Galileo’s method of determining short intervals of time by weighing the water flowing through a water clock was first suggested in the fifteenth century by Nicholas Cusanus.

Along with Europe’s new understanding of measurement came a new appreciation of measurement error.

Philosophers realized more and more that although approximations were unavoidable they were no scandal to science. Cusanus strongly underlined the fact that even if the entire truth cannot be grasped it can be approached more and more closely…The uncertainty of measurements is thus, he argued, a natural condition of any kind of experience.²³

The third development was a step towards mathematical physics, and the second and third developments foreshadowed the empiricism that would characterize European science.

Although European scholars embraced Ptolemaic astronomy and Aristotelian physics, there were some interesting dissents from these paradigms. One involved the question of whether the stars rotated around the earth (as in Ptolemaic astronomy), or were fixed while the earth rotated on its axis (as suggested by Heraclides in the fourth century BC). Ptolemy had argued against the earth’s rotation on the grounds that it would be evident to humans — for example, an object thrown straight up would be “left behind” by the rotating earth and would not land on the spot from which it was thrown. The Islamic astronomer al-Tusi believed the earth to be stationary, but disputed Ptolemy’s claim that rotation would be readily discernible. With regard to a thrown object, he noted that, “The part of the air adjacent to the Earth could conceivably conform to the Earth’s motion along with whatever is joined to it.” Nicole Oresme took up the same issue in 1377. He dismissed the claim that rotation would be observable by arguing that everything on the earth shares the earth’s rotation. He preferred a rotating earth on the basis of simplicity: if the earth stood still, the stars would have to move at an incredible speed in order to cover vast distances each day. In the end, though, he could not persuade himself to break from the tradition of a stationary earth.

Oresme also questioned Aristotle’s theory of natural place. Aristotle claimed that the sublunar region consisted of nested spheres of fire, air, water and earth. Earth possessed gravity that propelled it to the center of the world, fire possessed levity that sent it outwards to the inner surface of the lunar sphere. Water and air also found their natural places. Oresme proposed a thought experiment that undermined the theory.

Aristotle had taught that the “media” water and air have, respective, no absolute gravity and levity. He assumed, for instance, that a particle of air placed in the sphere of fire would move downwards towards its natural place (the sphere of air), and as a result seem to be heavy. On the other hand, a bubble of air would appear to be light when placed in the sphere of water. By means of a thought experiment involving a canal stretching from the centre of the Earth through the four elementary spheres to the inner surface of the sphere of the Moon, Oresme takes Aristotle’s argument one step further. First the canal is supposed to be filled with water, and a small particle of air placed at the centre. This air will, he said ascend through the water. According to Aristotle it would stop somewhere inside the sphere of air, but Oresme maintained that it would continue to the upper end of the canal. Similarly, if the canal is filled with fire, a particle of air placed at the top will fall all the way down to the centre of the world. In both cases, Oresme asserted, the motion is “natural.”²⁴

Oresme also disputed Aristotle’s notion of gravity and suggested an alternative.

Oresme maintained…that the centre of the world is a purely mathematical point incapable of exerting any physical influence. He explained gravity by assuming that heavy bodies have a natural tendency or inclination to unite with each other.²⁵

Nicholas Cusanus (1401-1464) appears to have completely rejected both Ptolemaic astronomy and Aristotelian physics.

In the De docta ignorantia he emphasized that the world is an all-embracing unity in which the all necessarily is in everything. It follows, he said, that the universe has no fixed immovable centre. The world cannot be said to be infinite, yet it is not finite because it has no boundaries…Cusanus regarded the earth merely as “a noble star,” its own specific light, heat, and influence making it different from other stars. He also toyed with the idea that observers could be placed upon the Sun, Moon, Mars or other stars, maintaining that any observer would see the universe turning around himself…It was Cusanus who popularized the much-repeated Hermetic slogan “the whole machine of the universe is a sphere which has its centre everywhere, but its circumference nowhere.” Such audacious opinions make Cusanus the first supporter of the cosmological principle that everywhere, roughly speaking, the universe must have very much the same structure.²⁶

Nevertheless, it must be acknowledged that few precursors of modern science can be found in the science of the middle ages. The achievements of medieval science lie in philosophy (the splitting of the natural and divine realms, mediated by the theologians themselves) and in methodology (the critical analysis of hypotheses, the nature of induction, and the measurability of the physical world).

Most of the people who developed medieval science were clerics (including Albertus Magnus, Jon Dun Scotus, William of Ockham, Robert Grosseteste, Roger Bacon, Theodoric of Freiberg, Nicholas Cusanus, and Nicole Oresme), and most of them held positions at universities. A. C. Crombie believes that their work marked “the beginnings of the conscious understanding of the nature of the [scientific] enterprise.”²⁷ But the roles of both clerics and universities would soon diminish.

An historical gulf now began to open between the innovators on the one hand and the old centres of learning on the other. In the thirteenth century, the universities had been able to attract all the best scholars of the time. In the fourteenth century they were the obvious centres for critical discussions and alternative solutions to traditional problems. But in the fifteenth century the intellectual standard of the universities began to decline. Their growing numbers thinned the ranks of scholars inside the individual institutions, national and religious divisions became more and more apparent, and new subject matters were ignored…

The disastrous consequences of this gradual decay appeared during the sixteenth and seventeenth centuries when many of the protagonists of science simply left the universities to seek support from kings and princes, either on a private basis or as members of scientific academies. Characteristically, neither Copernicus nor Tycho Brahe occupied university chairs. The situation reached a climax when, in 1610, Galileo left his chair at Padua to become a court mathematician of the Grand Duke of Tuscany.²⁸

Copernicus

Copernicus was a revolutionary: his heliocentric theory shattered Ptolemaic astronomy. He was also a traditionalist: he had intended to reform Ptolemaic astronomy, not destroy it. Of the system’s three central assumptions — the earth lies motionless at the center of the universe, the celestial spheres are real, the motion of every heavenly body is circular and uniform — he abandoned only one.

Aristotle, and Pythagoras before him, had proposed a model of the universe in which the earth lay at the center of everything. The moon, the sun, the planets, and the fixed stars were embedded in nested crystalline spheres, all centered on the earth and rotating at constant speeds. The universe was a sort of machine: the Prime Mover drove the outermost sphere, and its motion was communicated from one sphere to the next all the way to the innermost (lunar) sphere. It was an elegant model, but the Greeks discovered that its predictions did not match the observational data. Ptolemy revised the model so that it was more consistent with the data, while retaining (in some form) the three central assumptions of Aristotle’s model. His revision introduced some extra machinery. The eccentric was a point some distance from the earth. The nested spheres were assumed to be centered on it, rather than on the earth. The equant was another point some distance from the earth. Motion was assumed to be uniform (constant speed) from the perspective of an observer at the equant, not an observer on the earth. Ptolemy’s model was not truly geocentric: motion was circular around one point and uniform with respect to another, but neither of these points was on the earth.

The epicycle, first proposed by Eudoxus (c. 408-355 BC), was a small orbit superimposed on a larger orbit. Its purpose was to solve the problem of retrograde motion. Viewed from the earth, each planet appears to move across the night sky from west to east, but periodically reverses its course, travelling from east to west for a time, before resuming its eastward travel.²⁹ This reverse or retrograde motion causes the path of the planet, when viewed against the backdrop of the stars, to display loops or zigzags. The way in which epicycles solve this problem is most easily understood by thinking of a planet’s orbit. A planet embedded in a sphere traces out a circular orbit. Eudoxus assigned this orbit, not to the planet, but to a single point that was the center of a smaller orbit. The planet itself moved around the smaller orbit. Under this scheme, the planet travels in a circular orbit around a single point that travels in a circular orbit around the eccentric. The planet’s two motions are sometimes reinforcing and sometimes opposing, which makes retrograde motion possible.

The epicycle attenuates the assumption that the planets move in circular orbits at constant speeds: their movement is produced by a combination of uniform and circular movements. Ptolemy’s model retains the original assumptions of the Aristotelian model, but only in a legalistic fashion. The elegant simplicity of the original model has been lost.

In all of this apparatus, the least pleasing bit was the equant. The rest of the model could still be imagined to be a machine, but the equant was just a reference point, unconnected with everything else. It was criticized repeatedly by al-Haytham in his evaluation of the Ptolemaic model, Doubts about Ptolemy (c. 1028). Al-Haytham’s book was the foundation document for the Maragha school of Islamic astronomy, which sought to remedy the shortcomings of Ptolemy’s model by introducing new machinery. Its most famous innovations were the Tusi couple, which generates reciprocal motion from a combination of circular motions, and the double epicycle. The culmination of the Maragha school’s research program was al-Shatir’s planetary model, which was the first truly geocentric model (no eccentrics and no equants) capable of matching the observational data.

Copernicus was also critical of the equants. His full planetary system is contained in De revolutionibus (1543), but sometime before 1514, he wrote a brief outline of his theory that he circulated among his friends and colleagues. This outline, now known as Commentariolus, began with a very short review of astronomical models. After describing eccentrics and epicycles, Copernicus wrote:

The[se] planetary theories…, although consistent with the numerical data, seemed likewise to present no small difficulty. For these theories were not adequate unless certain equants were also conceived; it then appeared that a planet moved with uniform velocity neither on its deferent nor about the center of its epicycle.³⁰ Hence a system of this sort seemed neither sufficiently absolute nor sufficiently pleasing to the mind.³¹

While the Maragha school tackled the equant problem by inventing new machinery, Copernicus dealt with it by jettisoning one of the model’s fundamental assumptions, namely, that the earth lies motionless at the center of everything.

In the Commentariolus, Copernicus explained that he had searched for “a more reasonable arrangement of circles,” and found one that obeyed the following postulates:

1. There is no one center of the celestial spheres.
2. The center of the earth is not the center of the universe, but only of gravity and of the lunar sphere.
3. The sun is the center of the universe.
4. The distance from the sun to the earth is imperceptible compared to the distance from the sun to the stars.
5. The earth rotates each day, causing the apparent daily movement of the stars.
6. The earth orbits the sun, causing the apparent annual movements of the sun.
7. The apparent retrograde motion of the planets is caused by the movement of the earth along its orbit (and the movement of the planets along their orbits).

As well as “saving” the assumption of uniform motion, these postulates simplified astronomy in several ways. First, a rotating earth provided a single explanation for three phenomena that had previously been treated separately: the daily movement of the stars, the rising and setting of the sun, and the rising and setting of the moon. Second, Copernicus was able to identify a new empirical regularity: the closer a planet is to the sun, the faster it moves. Third, retrograde motion finally made sense. It was caused by the relative movements of the earth and another planet, as shown in the figure below.³²

Note the second postulate. Copernicus’s planetary theory forced him to discard Aristotle’s ideas about gravity — ideas that Oresme and Cusanus had questioned before him.

The three basic assumptions of Ptolemaic astronomy were that the earth rested at the center of the universe, the celestial spheres existed physically, and all celestial motion was circular and uniform. Copernicus abandoned the first assumption to save the third. His abandonment of the first assumption made him the first major figure of the Scientific Revolution. His attempt to preserve the third assumption made him the last significant Ptolemaic astronomer.

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1. A. C. Crombie, Augustine to Galileo (Harvard, 1953), p. 6. ↩
2. A. C. Crombie, Augustine to Galileo (Harvard, 1953), p. 214. ↩
3. Rubenstein, Aristotle’s Children, p. 16. ↩
4. From a biography of Gerald of Cremona attached to one of his translations. Quoted by Grant, The Foundations of Modern Science in the Middle Ages, p. 24. ↩
5. A. C. Crombie, Augustine to Galileo, p. 21. ↩
6. Rubenstein, Aristotle’s Children, p. 79. ↩
7. Rubenstein, Aristotle’s Children, p. 79. ↩
8. Rubenstein, Aristotle’s Children, pp. 53-4. ↩
9. Rubenstein, Aristotle’s Children, p. 55. ↩
10. Crombie, Augustine to Galileo, pp. 39-40. ↩
11. Crombie, Augustine to Galileo, p. 40. ↩
12. Grant, The Foundations of Modern Science in the Middle Ages, p. 72. ↩
13. Crombie, Augustine to Galileo, p. 42. ↩
14. This idea was a thousand years old: it had been advanced by Philo Judaeus of Alexandria in the first century AD. ↩
15. Rubenstein, Aristotle’s Children, p. 249. ↩
16. Rubenstein, Aristotle’s Children, p. 252. ↩
17. Rubenstein, Aristotle’s Children, p. 254. ↩
18. The use of magnification is much older. Seneca, once his eyes had weakened, used water-filled globes to magnify writing. ↩
19. Crombie, Augustine to Galileo, p. 221. ↩
20. Crombie, Augustine to Galileo, p. 230. ↩
21. Rubenstein, Aristotle’s Children, p. 253. ↩
22. Marshall Clagett, Greek Science in Antiquity (Dover, 2001), pp. 67-8 ↩
23. Pedersen, Early Physics and Astronomy, p. 255. ↩
24. Pedersen, Early Physics and Astronomy, p. 207. ↩
25. Pedersen, Early Physics and Astronomy, pp. 206-7. ↩
26. Pedersen, Early Physics and Astronomy, p. 248. ↩
27. Crombie, Augustine to Galileo, pp. 218-9. ↩
28. Pedersen, Early Physics and Astronomy, pp. 255-6. ↩
29. Mars, for example, exhibits retrograde motion every 26 months, for a period of roughly two months. ↩
30. The deferent is the larger circle around which the center of the smaller orbit — the epicycle — moves.  Copernicus is essentially arguing that equants don’t preserve uniform motion in any meaningful way. ↩
31. Edward Rosen, editor and translator, Three Copernican Treatises (Octagon Books, 1971), p. 57 ↩
32. This image is from the physics site splung.com. ↩