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Copernicus and his Revolutions

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Scientific truth is largely determined by authority and this has always been so. Today, any new idea must be supported by the weight of existing authorities and expressed in their language. The more radical the idea the more necessary it is to blunt its impact by emphasising its similarities with shared traditions. While he was writing De revolutionibus orbium coelestium, Nicolaus Copernicus was aware of both the radical nature of what he was suggesting and the need to communicate it in a way that would be both comprehensible to and respected by his readers. This means he needed to use many different kinds of arguments to carry a readership that he anticipated would include mathematicians, humanists and scholastics. Furthermore, he was aware that he had not been able to find the definitive proof for his hypothesis that the Earth moved around the sun and so must make up this deficit with as much evidence as he could muster. So Copernicus had to provide evidence to satisfy all these constituencies as well as address the various authoritative sources that contradicted him. De revolutionibus contains several different kinds of arguments not all of which formed part of the process by which Copernicus convinced himself.

In 1543, astronomy was one of the seven liberal arts traditionally taught to undergraduates at university. It was essentially a technical subject that involved being able to figure out where the sun, moon and planets were going to be against the backdrop of fixed stars. Formulae existed for working this out and the standard text, Ptolemy’s Almagast, included exceedingly complicated models that explained how these formulae worked. Rather than have to use the formulae, pre-calculated tables gave all the information about the planets required. These were amended and replaced through the period. For instance, the Toledine Tables were ditched in favour of the Alphonsine Tables, although the latter turned out to be no more accurate [NOTE]. The professional astronomer’s job description was largely confined to checking and correcting these tables for their use by astrologers, navigators and chronologists. He could also note when the movements of the planets did not follow their predicted paths and perhaps try to improve the calculations. What he was certainly not expected to do was explain why the planets were moving in the way they were or offer a picture of how the heavens really worked. This task was firmly within the purview of the natural philosopher with his knowledge of Aristotle. The Philosopher, as the ancient Greek was called, gave a qualitative account of the heavens which was much simpler than the epicycles of Ptolemy. Indeed, it was abundantly clear that the two systems were inconsistent but the senior discipline of natural philosophy could insist on the superiority of its dialectical knowledge over the empirical and mathematical work of astronomy. This conflict seems to have been the reason that many pregnant questions and suggestions about astronomy were not addressed in the Middle Ages.

To an educated person in the first half of the sixteenth century, the belief that the Earth was a stationary sphere in the centre of the universe was as well grounded as our own heliocentric cosmology is today. While it is difficult to reconstruct the hierarchy of the thought processes that caused this certainty, it is clear that a number of factors supported a geocentric model. The first and most important reason for believing something not directly accessible to the senses in the sixteenth century was, as it still is today, that of authority. There were three forms that carried weight – the religious tradition, the philosophical tradition and experience. None of these was necessarily able to carry the field against the others, but on geocentricism they were all agreed. Religious opinion was based on a number of biblical passages, such as 'And the sun stood still, and the moon stayed... So the sun stood still in the midst of heaven, and hasted not to go down about a whole day.' [NOTE] and 'Who laid the foundations of the earth, that it should not be removed for ever' [NOTE]. Conversely, it had been established by the early Church Fathers, such as Augustine, that it was not necessary to take the bible literally at all times and that it could at times be symbolic. Some medieval commentators had taken this doctrine so far as to insist that Genesis 1 had to be interpreted as an allegory or else it would simply be absurd [NOTE].

This environment meant that Copernicus faced several challenges if he was to carry his argument. Obviously, he needed to convincingly demonstrate that the mathematics of his model was able to predict the movements of the planets in a more parsimonious manner than Ptolemy’s. Then, he had to address the physical arguments against a moving Earth that also conflicted with common sense. Finally, he needed to overcome the objections to astronomy seeking to usurp the authority of the senior discipline of natural philosophy in making pronouncements on physics. He essentially ignored any biblical questions but used a religious argument to justify his whole system.

Physical arguments

To warm up at the start of the first book of De revolutionibus, Copernicus starts with some very basic natural philosophy to try and ground his work in the established tradition. None of his readers needed to be convinced that the Earth was a sphere or that the universe was the same shape. However, by taking well-known arguments and using them at the start of his more radical book, he was able to reassure his readers that he was plugged into the same currents of thought that they were. For example, by comparing the universe to a drop of water which is also spherical by its nature, he used an argument found in Aristotle’s De caelo and thus demonstrated scholastic credentials [NOTE]. Likewise, his explanation that the Earth is a sphere by the way that the area of sky that can be seen varies uniformly with latitude is implied in De caelo [NOTE] as well and even such popular travelogues as John Mandeville’s Travels [NOTE]. Contrary to popular belief, no educated person in the Middle Ages thought the Earth was flat.

Various scientific and empirical objections to a moving Earth are dealt with in the next few chapters. Within the philosophical tradition, there existed arguments for a stationary Earth such as the fact that the fixed stars are just that – they do not move relative to each other as might be expected if the Earth was moving. It was also suggested that if the Earth was moving, we would all be swept off our feet and if it was rotating we would all be thrown outwards by the centrifugal force.

The diurnal rotation of the Earth had a long pedigree, some of which Copernicus mentions in his quotation from Plutarch in the address to Paul III [NOTE]. He restricts himself to ancient thinkers while modern scholarship has found some more names to add to his list [NOTE]. But the most exhaustive analysis of this hypothesis that existed at the time was contained in Traité du ciel et du monde (1377) by the French theologian Nicole Oresme. Building on and refuting the earlier work of John Buridan, Oresme demonstrated that if the Earth was rotating and we were all rotating with it, an arrow shot in the air would land back on the spot it was launched rather than behind as it would carry the Earth’s rotational movement as well as the motion up and down. Hence, physics alone could not demonstrate whether or not the Earth was rotating and it was necessary to consult the Scriptures to settle the matter. This turns out to be less simple than might be expected as Oresme, fresh from his insight into relative motion, realised that any reference to the motion of the sun with reference to the Earth could be a reference to its relative rather than absolute motion. He explained that the Bible was written in the common language of men using terms like ‘sunrise’. Copernicus made exactly the same point when he explained why he would continue to use everyday words to describe the movements of the sun even though he has demonstrated the sun does not actually move [NOTE]. To make a judgement as to whether the Earth really was rotating, Oresme needed a passage with a ‘God’s eye’ viewpoint. He eventually settled on "the world also is stablished, that it cannot be moved" [NOTE] as settling the matter but one cannot help thinking he has conformed to tradition rather than followed his argument to its logical conclusion [NOTE].

Copernicus used a similar argument in De revolutionibus 1:8, although it is impossible to be certain that he knew of Oresme’s work as, after all, Traité du ciel was never printed. He stated that because we all share in the Earth’s natural motion of rotation, we do not feel and cannot observe it. The motion is not violent or forced so there is nothing to feel. Copernicus needed this argument because his other one is extremely weak. He simply posed the question that if the Earth’s rotation would do violence by virtue of its speed and size, how much more violent must be the rotation of the entire stellar sphere. This argument fails both because it assumes that what holds for the Earth must also hold for the super-lunar spheres and also, as Copernicus went on to show that the Earth’s rotation is not violent at all, he leaves us wondering why the outer sphere’s should be either. In short, he is able, like Oresme, to show that the Earth’s rotation is physically possible but, also like his predecessor, cannot provide any positive proof for his assertion.

It was accepted by the scholastics and ancients that the universe was very big by comparison to the Earth. We find this explicitly spelt out by Cicero in his passage on the dream of Scipio from The Republic that became extremely well known in the Middle Ages due to the commentary of Macrobius. It is also found in Boethius who wrote in his sixth century Consolation of Philosophy ‘beside the extent of the heavens the circumference of the Earth has the size of a point’ [NOTE]. Even Dante is clear about this point and we also find that the South England Legendary, a Middle English poetry collection, helpfully gives the distance to the stars are being over 120 million miles [NOTE] (slightly more than the modern figure for the distance between the Earth and sun). But Copernicus still had to do some work to adapt this argument. Firstly, most of the authorities, while happy to admit the insignificance of the Earth, were actually making a moral rather than physical case and were concerned about showing man how pointless is his pride in the immensity of space. Copernicus supplied the argument in De revolutionibus 1:6 that everywhere on the Earth appears to be at the centre of the heavenly sphere, to give a physical justification for the Earth’s tiny size. Second, he carried out a small sleight of hand to try and slip through the problems that the lack of stellar parallax caused to his heliocentric theory. So, building on the accepted view that the Earth was a point relative to the size of the universe, Copernicus went further and said that the distance between the Earth and the centre of the universe (effectively the sun) is also immaterial. In other words, he had increased the size of the universe by an enormous amount without any justification at all. But if he was to explain why the stars appear fixed, he had to do this, however ad hoc the explanation might appear, and thus he made a radical change while claiming to remain entirely within contemporary belief.

Philosophical arguments

The most radical idea that Copernicus put forward is found in his preface addressed to Pope Paul III – that astronomical hypotheses should reflect reality. He set out the fundamental and well-known contradiction between the concentric circle cosmology of the natural philosophers who follow Aristotle and the more complicated structures of the mathematical astronomers who follow Ptolemy. As Copernicus explained, the Aristotelian model appears to be physically possible but is not borne out by observations while the Ptolemaic model does agree with observation but is physically absurd. He compared it to a monster with hands, feet and head stuck together in the wrong order. This clearly means they have gone wrong at some point because ‘if the hypotheses assumed by them were not false, everything which follows from their hypotheses would be confirmed beyond any doubt’ [NOTE]. Anyone who looks at Ptolemy’s model of the universe can agree it is extremely complicated but it is less clear why this means it must be false. We can dispose of the Aristotelian model because it does not yield correct empirical results, but why should Ptolemy also be discarded when by Copernicus’s own admission he appears ‘in large measure to have solved the problem of the apparent motions with appropriate calculations’ [NOTE]? The reason given is that the model ‘contradicts the first principles of uniform motion’ that is that motion should be constant and circular. Hence the model does not show the ‘true symmetry’ of the parts of the universe. In other words, before we even consider what hypothesis to use, we are taking on board the basic axioms of elegance and parsimony that have been accepted by Christians both on ancient authority and for theological motives. They are still accepted by even the least religious of scientists today.

Even the earliest Christians saw the work of God in the heavens. The Hebrew Scriptures praised God because he had ‘arranged all things by measure and number and weight’ [NOTE] and delighted in ‘thy heavens, the work of thy fingers, the moon and the stars, which thou hast ordained’ [NOTE]. The idea of the world as a machine with a divine artificer arrived in the Latin West with the translations of Dionysius the pseudo-Areopagite [NOTE] and remained a constant theme that became especially popular once clockwork could be used to construct armillary spheres. Copernicus clearly held to this idea and was annoyed with philosophers for not understanding ‘the movements of the world machine, created for our sake by the best and most systematic Artisan of all’ [NOTE]. For him, there simply had to be a better explanation for the heavens that justly reflected the glory of their creator hence the need for a model that produced correct results with ‘greater compactness and more becomingly’ [NOTE] than the existing alternatives. Tearing up Aristotle and Ptolemy was not a problem if it meant this most basic belief in the nature of the heavens could be upheld. For Copernicus, religion was not bolted on to make his ideas more palatable to contemporary taste, but was the very foundation on which his rational and scientific ideas were built.

Aristotle’s cosmology is far more central to the pre-modern worldview than we often realise today. To Aristotle, natural motion could be explained by each element striving to reach its allotted place. For Earth – the heaviest element – this was the centre of the universe, then came water, then air and finally fire. By transplanting the Earth to a position well off centre, Copernicus needed a new definition of gravity so, in De revolutionibus 1:3, he declared it is the God-given propensity of things to gather into lumps. This is much less satisfying than Aristotle’s scheme of things falling always towards the centre of the universe but it was the best anyone could do until Einstein.

Rhetorical arguments

Reacting to his book, natural philosophers could retort that Copernicus had no business to be sticking his nose into such weighty matters as gravity. Astronomers, as we have seen, are supposed to make predictions rather than hypothesise about the true nature of things. That this was a real issue is illustrated in the earliest critique of De revolutionibus by the Dominican Giovanni Maria Tolosani who praises the mathematical knowledge in the book while complaining that this is not sufficient. ‘The inferior science,’ he says, ‘receives principles proved by the superior’ [NOTE] meaning that astronomers should leave the loftier matter of natural philosophy to their betters. To bolster his position, it is necessary for Copernicus to talk up the status of astronomers without sounding too offensive. Copernicus used a number of rhetorical devices that were quite separate from his core arguments to try and lessen the force of these problems. The most obvious of these is the rather obsequious tone of much of De revolutionibus with Copernicus reminding his readers of his ‘mediocrity’ [NOTE]. But this was nothing unusual during the period and was a technique that could be used even when an author was putting down his rivals. Dedicating the work to a powerful patron was also usual practice and Paul III, the pope to whom De revolutionibus was addressed, had previously been the recipient of the dedication in another astronomical work. In 1538, Girolamo Fracastoro had produced his own, rather less radical, reform of Ptolemy and seen fit to offer it to the same Pope [NOTE].

The other major rhetorical feature of De revolutionibus is the frequent reference to classical authorities and the almost total lack of Christian ones. Humanism was one of the dominant features of Copernicus’s intellectual life and so the use of the ancient sources was consistent with his education. For example, he leavens his prose with snippets like the alternative names of the planets taken from Chalcidius’s commentary on Plato’s Timaeus [NOTE] (rather than from the Timaeus itself, as Copernicus claims [NOTE]). The process of authentication and correction made up a good part of the humanist project, so it is perhaps ironic that the Letter from Lysis to Hipparchus, which Copernicus had intended to use, is spurious. He mentions the letter in his address to Paul III but the actual document did not make it into the printed edition [NOTE]. The purpose of the letter had been to give De revolutionibus an air of special or privileged knowledge of the sort that the Pythagoreans allegedly handed down by word of mouth. This makes the reader an initiate into the mysteries of Pythagoras whom Copernicus believed was also a proponent of the heliocentric hypothesis. In De revolutionibus 1:10, mystic imagery presents the universe as a beautiful temple lit by the great, all-seeing lantern at its centre. The obvious parallel is to the neo-Platonic thinkers of Italy, where Copernicus was a student from 1496 to 1503, who were rediscovering their own versions of the wisdom of sages like Hermes Trismegistus and recalibrating the occult arts. A good deal of this work, while not strictly illicit, was certainly quite risqué, so it is a strange thing to be invoking if Copernicus really was carefully trying to avoid treading on the toes of any theologians. In fact, he seems to have been appealing to humanists – a large intellectual constituency who did not recognise the traditional demarcations of the sciences and for whom mathematics was a more important subject than natural philosophy. Another advantage of ancient writing was that the Greeks had thought of just about everything and hence it was inevitable that at least one or two could be found to support ones views, however out on a limb they were. That said, these authorities do not always seem to say what Copernicus claimed they were. It is interesting to note that even Philolaus the Pythagorean says the sun, as well as the moon and Earth, are orbiting some sort of ‘fire’ which is not the same model as Copernicus is proposing [NOTE]. The reference to Aristarchus of Samos, who actually did suggest a heliocentric model, was removed from De revolutionibus before printing, perhaps because he was not considered Pythagorean enough [NOTE]. Worse, Copernicus came to realise that the astronomical observations found in the ancient authorities were not as accurate as he had believed and at times appeared to have been fabricated [NOTE].

As for previous generations of scholastic thinkers, Copernicus simply did not mention them while freely borrowing their ideas whenever they helped his cause. There were good reasons for not explicitly engaging with medieval work. Humanistic bias was against the dialectical methods of scholasticism which meant humanists did not tend to look upon them as valid authorities. While a writer of the enormous prestige of Erasmus can safely indulge in biting satire, someone like Copernicus was better advised just to ignore them. Another incentive for caution is tied up again with the classification of the sciences. Keeping the book strictly astronomical was part of the way that Copernicus hoped to demonstrate that his own subject could, by itself, reveal the true workings of the universe. Hence, he did not what to dilute his message by getting involved in contemporary arguments about natural philosophy which risked detracting from his main points.

As we have seen, Copernicus never really engages with difficulties engendered by the apparent contradiction between his hypothesis and a literal reading of the bible. De revolutionibus is an astronomical book that parks its tanks on the natural philosophers’ lawn. This is bad enough but even an experienced Regent Master who has lectured in the Arts Faculties on all the works of the Philosopher and his Commentator would be ill advised to pick a fight with a theologian – indeed he was forbidden by oath to do so. Figures like Aquinas and Duns Scotus each had their followers who fought bitter ideological battles with their opponents and frequently each other. The best way not to cause any offence was simply to keep well out of it and not mention anyone at all. It is true that one did not always have to take the Bible completely literally and it was also appreciated that it was a religious rather than scientific book. However, this emphatically did not mean that anyone could start work on some amateur exegesis in order to find the meaning that one wanted to find. Instead, to support a controversial meaning for a Biblical passage, one had to be in possession of a doctorate of theology as well as plenty of suitable quotations from some of the fathers of the church and their most esteemed followers. Once again, this is a discussion that Copernicus was ill-equipped to enter and he chose the wisest course of allowing the astronomy to speak for itself, rather than engage in detailed hermeneutics. The only exception to this is his attack on the Latin writer, Lactantius also known as the Christian Cicero. Here, he finds a theological writer whom everyone can agree got something wrong in claiming the Earth is flat. Copernicus took this as a warning to meddlers who might criticise his work without understanding it in full. ‘Mathematics is written for mathematicians’ [NOTE] and everyone else would do well to mind their own business.

Mathematical arguments

Arguments that a modern thinker would describe as ‘scientific’ are rarer in De revolutionibus than might be expected. Certainly, there is no hint of Popper’s falsification or much else that qualifies as ‘scientific method’. Of the six books, more than five are taken up by a detailed geometrical and chronological analysis that successfully shows that a heliocentric model does indeed provide an alternative to the Ptolemaic model. That is to say that one can use Copernicus’s calculations to predict the positions of the heavenly bodies with about as much accuracy as one can expect from using Ptolemy’s. Of course, all this depends on which sets of observations we are using and Copernicus helpfully provided his preferred tables in his book. He even explained how to build and use an astrolabe [NOTE] and included some of his own observations, especially in book five.

Under the geocentric model of Ptolemy, the Earth was stationary and the planets orbited around it. It was taken as axiomatic that the proper motion of planets was circular but it was found that this alone could not explain why they traveled across the sky at varying speeds and sometimes even went backwards (known as retrograde motion). The solution used was for each planet to orbit in a circle around a point on another orbit around the Earth. The main orbit was called a deferent and the smaller orbit an epicycle. But this was not enough. The Earth was not at the centre of the deferent but off to one side with the planet actually orbiting around an ‘equant point’ which varied from planet to planet. There were other additional complications for the individual planets and even then, the model was unable to explain why Mercury and Venus always stayed close to the sun or why they had a periodicity that was linked to the sun’s orbit. The model was also incompatible with Aristotelian physics that said that the planets were attached to great spheres centred on the Earth and made of the fifth element of ether. Aristotle had no room for epicycles, largely because he was dead centuries before Ptolemy had formularised their use. This meant that while natural philosophers were happy to use Ptolemy’s model to make predictions and ‘save the appearances’ of the phenomena, they did not believe it reflected actual reality.

Copernicus did away with the equant points and could also make do with fewer epicycles. His model did not improve on the empirical results of Ptolemy, but it was as good and slightly simpler. He also explained why it was that Venus and Mars stayed close to the sun although not any better than the geo-heliocentric theory later developed by Tycho Brahe. The idea that Mercury and Venus were orbiting the sun while it in turn orbited the Earth had a long pedigree in the West. Copernicus cites Martianus Capella, who put forward this model in his Marriage of Philology and Mercury, probably more through mixing up his sources than due to any great conceptual breakthrough [NOTE]. The ninth century theologian, John Scotus Eriugena, in his De divisione naturae also suggests that the two inner planets moved around the sun, and perhaps the other planets as well, as the most elegant explanation for Mercury and Venus staying close to the sun at all times [NOTE]. Copernicus might have been nodding to Eriugena with his mention of ‘certain other Latin writers’ [NOTE] but did not wish to mention anyone of such questionable orthodoxy. Another widely noticed component of the celestial motions was that the moon and sun moved in a different way to the other planets. Neither of these planets (for that is how they were thought of) displayed any retrograde motion although their speed did vary. This suggested that they might have a different relationship to the Earth than the other planets. Even the Ptolemy loyalist, Girolamo Cardano commented in his Aphorisms that Copernicus might not be altogether wrong, as so distinct was the motion of the moon [NOTE]. Copernicus’s system does indeed explain the reason that neither the sun nor the moon ever travel backwards in the sky. He also accounts for the large observed variations in the brightness of Mars which are not predicted by any of the other astronomical models [NOTE].

The most innovative piece of geometry used by Copernicus has been dubbed the ‘Tūsī Couple’ by modern scholarship after it was found to have been developed by Nasir al-Din al-Tūsī at the Marāgha observatory in Iran during the thirteenth century [NOTE]. It is a clever way to model planetary paths using circles and is illustrated in the diagram referring to De revolutionibus 3:4. Copernicus would have come across this in Italy and it is an important part of his attempt to simplify the work of Ptolemy, allowing him to claim to have done away with equant points while still using only circular motion.


It is clear from De revolutionibus that the reader is supposed to view mathematical demonstration as the dominant argument. The book claims it is intended for people who can follow this mathematical demonstration for themselves. Few could do so, which means that few would be able to tell that Copernicus had not succeeded in the way that he implies or that some of the techniques he used, like the ‘Tūsī Couple’, do not require a heliocentric hypothesis to improve the model of the heavens. This makes his physical arguments very important because they had to show that a moving Earth is not absurd and imply that the proof was to be found in the technical detail. This implies that the details of his solution to the very difficult problem of producing a heliocentric model are as much exercises in concealment as enlightenment. Moreover, Copernicus joined the humanists in setting up mathematics as an authority able to counter the traditional disciplines and so included himself among the select people able to make pronouncements on astronomy. His work was only to be judged by his peers.

After a close study of his arguments, the unanswered question is why Copernicus hit on the idea of a heliocentric system in the first place. He was contradicting all the respected authorities of his time while trying to enthrone mathematics as a further font of truth that could be believed. We can surely take it for granted that this was not as a result of his combing the ancient writers in search of enlightenment – especially as he read into them what he wanted to find. Neither was it the innate mathematical superiority of the model that attracted him as it was not any more accurate and not much simpler than the Ptolemaic alternative. His own stated reasons for wanting to improve the symmetry and elegance of the model were tripped up by the necessary mathematical consequences of needing a fit to the observed data. It seems as if he started from a simple archetype to which he found he had to add more and more convoluted enhancements. The original elegance of his solution lived on only in his imagination.

Perhaps the best hint to an answer is found in De revolutionibus 1:10 where Copernicus asked of the sun: ‘who would place this lamp in another or better position’ than the motionless centre [NOTE]. Certainly not God who, even if he had not quite designed the machine of astounding simplicity that Copernicus had hoped to find, did at least manage to put the sun where, despite all the evidence, it was supposed to be.

Notes in the article above appear in the status bar of your browser if you have Javascript enabled. All quotations from or references to this essay should be accompanied by a link back to this page and the name of the author.  This essay may be reproduced only with permission of the author although such permission will not normally be declined. 


Africa, Thomas 'Copernicus's relation to Aristarchus and Pythagoras' Isis 52:3 1961

Boethius The Consolation of Philosophy trans. VE Watts, Penguin 1999

Copernicus, Nicolaus The Revolutions of the Heavenly Spheres trans. Charles Glenn Wallis, Prometheus 1995

Grafton, Anthony Cardano's Cosmos Harvard 1999

Grant, Edward God and Reason in the Middle Ages Cambridge 2001

Huff, Toby The Rise of Early Modern Science Cambridge 1993

Kuhn, Thomas The Copernican Revolution Harvard 1957

Lewis CS The Discarded Image Canto 1994

Mandeville, John The Travels trans. CWRD Moseley, Penguin 1983

Martianus Capella The Marriage of Philology and Mercury trans. Stahl and Johnson, Columbia 1977

McColley, Grant 'The Theory of the Diurnal Rotation of the Earth' Isis 26:2 1937

Pedersen, Olaf 'Astronomy' in ed. David Lindberg Science in the Middle Ages Chicago 1978

Swerdlow and Neugebauer Mathematical Astronomy in Copernicus's De Revolutionibus Springer-Verlag 1984

Westman, Robert 'The Copernicans and the Churches' in eds. David Lindberg and Ronald Numbers God and Nature: Historical Essays on the Encounter between Christianity and Science California 1986

Westman, Robert S 'Proof, Poetics and Patronage: Copernicus's Preface to De Revolutionibus' in eds. David Lindberg and Robert Westman Reappraisals of the Scientific Revolution Cambridge 1990

Notes in the article above appear in the status bar of your browser if you have Javascript enabled. All quotations from or references to this essay should be accompanied by a link back to this page and the name of the author.  This essay may be reproduced only with permission of the author although such permission will not normally be declined. 

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© James Hannam 2003.
Last revised: 08 December, 2009 .