The Role of Imagination in Bringing About the Copernican Revolution


Science often advances because scientists gain access to new data, or even entirely new types of data, which their theories must then explain.  New technologies can open up entirely new perspectives on the physical world.  For example, in astronomy the introduction of the telescope, the spectroscope, and the radio dish allowed astronomers to gaze at the world in entirely new ways.  The wealth of data generated by these new instruments led to important breakthroughs in our understanding of the physical world.  As a result of examples like these, we may come to think that science only progresses when new data becomes available, perhaps generated by new instruments.  This is not so.  The human imagination is also a vital tool for the advancement of science, and in some important cases science has progressed through the workings of the human mind without the benefit of new data.

The human imagination can lead to the advancement of science by providing a new way to think about old data.  Scientific theories provide us with a way of envisioning the functioning of the universe and predicting the data we would find if we made various kinds of measurements.  If the predictions match with the data we actually find, then that counts as a success for the theory.  A new theory, however, can give us a different vision of the universe that could produce the same data.  Such a new theory gives new meaning to the old data.

Why, though, would we bother with a new theory if we already have one that works?  Perhaps the new theory does a better job of matching the observed data, but that is not the only possibility.  Even a new theory that is not any better than the old theory, from an empirical standpoint, might be preferred.  New theories can help us to answer questions that the old theory was unable to answer.  Perhaps more importantly, new theories can generate new and interesting questions that the old theory did not even allow us to ask.  The possibility of future knowledge may lure some scientists to cast off even well-established theories in pursuit of a new and potentially groundbreaking perspective on the universe.

New scientific theories do not arise from nothing.  They are built on existing ideas, though sometimes combined in new ways or coupled with unique insights.  Moreover, to gain any consideration a new scientific theory must arise within a space in which the old theory can be questioned.  Only then can the possible advantages of the new theory over the old even be recognized.

Here we will explore the role played by imagination in the so-called “Copernican Revolution.”  The transition from a geocentric to a heliocentric worldview was driven, at its outset, much more by new ideas than by new data.[1]  Our aim here is to examine how the ability to envision the functioning of the universe in a new way led to the introduction of the heliocentric system by Copernicus and its alteration into an astrophysical system by Kepler.

Before Copernicus

Before we can begin our exploration of the new ideas introduced by Copernicus and Kepler we must begin by discussing the old ideas: the uneasy combination of Aristotelian cosmology and Ptolemaic astronomy that medieval Europe inherited, largely by way of the Middle East, from the ancient Greeks.  Ancient observers were well aware that the entire sky appeared to move from east to west, completing a full rotation in one day.  Close observation showed that although the “fixed stars” seemed to remain in place on the sky as it rotated, a handful of objects failed to keep their spots among these stars.  Most obviously, the Sun moved along an eastward path through the fixed stars over the course of a year.  The Moon, similarly, moved eastward through the stars but in a somewhat irregular way.  Less obvious, but more striking once noticed, the five “wandering stars” moved mostly eastward through the fixed stars but occasionally stopped their eastward motion, moved west for a time, and then resumed their eastward trek.  Three of these wanderers (Mars, Jupiter, and Saturn) always engaged in this westward “retrograde” motion when they were in the opposite part of the sky from the Sun.  These came to be known as the superior planets.  In contrast, the inferior planets (Mercury and Venus), were always found relatively near the Sun and were found in the same part of the sky as the Sun when in retrograde.

The ancient Greek astronomers accounted for the motion of the fixed stars by proposing that they lay upon a giant celestial sphere which spun about a fixed axis.  The Earth lay at the center of this tremendous sphere.  Meanwhile other spheres were assigned to the wandering bodies.  These spheres were in some sense attached to the celestial sphere and thus turned with its daily motion, but they also possessed independent motions of their own which caused the “planets” (which included the Sun and Moon) to move relative to the fixed stars.  Aristotle adopted this general cosmological structure and unified it with his ideas about physics.  He proposed that the celestial spheres rotated about the stationary Earth because they were composed of a celestial substance, the ether, for which such a motion was natural.  The Earth and its environs, in contrast, were composed of four elements (earth, water, air, and fire) whose natural motion was either toward or away from the center of the celestial sphere.  Since the Earth was made of “heavy” elements like earth and water it was only natural that it should come to rest in the form of a sphere centered on this universal central point.[2]

The spheres which carried around the celestial bodies were often assumed to be solid but transparent.  The order of the spheres, though, was uncertain.  There was a general agreement that the sphere of fixed stars lay farthest from the Earth.  For the other spheres, it was often assumed that those whose motion most deviated from that of the fixed stars must lie closest to Earth.  Thus Saturn, with its nearly thirty-year cycle of motion relative to the fixed stars, lay nearest the stars, while the Moon, which circles the stars in less than thirty days, lay closest to Earth.  This principle, however, could not distinguish among the Sun, Mercury, and Venus which all complete their motion through the stars in one year’s time (on average).  Even so, Aristotle proposed a definite order.  Working from the center outward: Earth, the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and finally the sphere of the fixed stars.

Although this structure fit well with Aristotle’s physics, it did not do a very good job of matching the detailed motions of the planets relative to the stars.  Eventually this general structure was merged, to some degree, with a more accurate theory for the planetary motions.  This theory reached its most complete form in the work of Claudius Ptolemy.[3]  Ptolemy proposed that the wandering stars each moved on a pair of conjoined circles: a deferent circle that was roughly centered on Earth, and an epicycle circle whose center moved along the deferent while the planet moved along the epicycle.  This model could account for the observed retrograde motion, but special synchronizations were required between these motions and that of the Sun to ensure that retrograde took place in the proper location, relative to the Sun.  For superior planets the motion of the epicycle had to be linked to the Sun, while for inferior planets it was the deferent motion which had to be so linked.

Ptolemy’s full theory was more complicated and included a device called the “equant” which caused the center of the epicycle to move in a non-uniform way along the deferent.  The equant violated the principle of uniform circular motion that Aristotle and others believed should govern the motion of all celestial bodies.  That wasn’t the only tension between the Ptolemaic astronomical models and the Aristotelian cosmology: the Ptolemaic deferents were not centered on Earth and the Earth lay entirely outside of the Ptolemaic epicycles.  What is more, Ptolemy was unable to solve the ordering problem without bringing in non-astronomical arguments along the lines of those used by Aristotle.  His astronomical theory alone could not specify any relations between the circles for one planet and those for another.  Indeed, his theory did not provide a system, but rather a collection of separate theories, one for each planet.

The Revolutions of Copernicus

In spite of these challenges, Ptolemaic astronomy, housed uneasily within an Aristotelian cosmology, worked.  It worked extremely well.  So well that it remained largely unchallenged (though not uncriticized) for roughly 1400 years.  The first to pose a serious challenge to this established system was Nicolaus Copernicus, whose 1534 De Revolutionibus Orbium Coelestium presented the world with a heliocentric system to explain the apparent motions of the planets and the fixed stars.[4]  We don’t really know what led Copernicus to his new vision of the universe.  He says he hated Ptolemy’s equant, but he may have had astrological motivations or he may have wanted to explain the odd synchronizations of the planetary motions with the Sun which had been noted by his predecessors Georg Puerbach and Johann Regiomontanus.  Perhaps he was seeking a reason for the distinction between the inferior and superior planets.

Whatever his initial motivation, Copernicus worked out a detailed theory in which the Earth rotated on an axis while also orbiting around a stationary Sun.  While we don’t know what led him to this theory, we do know that somewhere along the way he realized that all of the annual motions that were synced to the Sun could be attributed to a single motion, that of an orbiting Earth.  Likewise, the daily motions of every object could be attributed to a single daily rotation of the Earth.  Copernicus was able to envision what it must be like to look out from a moving platform and he recognized that the motion of the observer shows up as an apparent motion in every object observed.  Therefore, the common annual and daily apparent motions present in all celestial bodies (except the Moon, which lacks an annual motion) could most efficiently be attributed to actual motions of the Earth.

Once he began to envision the universe in this way things must have fallen into place quickly.  If Earth orbits a stationary Sun, perhaps so too do the wandering stars.  Should their orbits be larger or smaller than that of Earth?  A smaller orbit leads to the wanderer staying near the Sun on the sky, as seen from Earth, and retrograding when close to the Sun.  These are the inferior planets Mercury and Venus.  A larger orbit leads to retrogrades when the planet is opposite the Sun on the sky.  These are the superior planets Mars, Jupiter, and Saturn.  Not only did Copernicus’ theory explain the inferior/superior distinction, but it explained why, even demanded that, the retrograde motions be correlated to the Sun.

Armed with this new perspective, Copernicus could explain not just the exact order of the planets but also their relative distances from the Sun.  Copernicus’ theory provided a unified system: the behavior of each planet was intimately linked to that of Earth and, through Earth, to all of the other planets.  Within this system, Copernicus’ theory revealed a pleasing harmony with more distant planets taking longer to orbit the Sun.  The separate pieces of the Ptolemaic system could never be united into such a pleasing whole.

Although we don’t know what led Copernicus to his radical new idea, we do know what did not lead him to this view: data.  It was not that Copernicus made no new measurements.  He measured planetary positions using naked-eye instruments like the triquetrum and the armillary, but neither his instruments nor his use of them were different from those of the ancient Greeks.  Indeed, his instruments may well have been inferior to those used by Ptolemy and Hipparchus.  He used his new data to obtain updates and more accurate values for some astronomical periods, but nothing in his data required a new view of the cosmos.  His measurements could more easily have been used to simply update Ptolemy’s model.  Thus it was Copernicus’ imagination, not his instruments or his data, that led to his revolutionary new model of what could now be called the “solar system.”

The description above paints a beautiful picture of the Copernican theory, but the devil was in the details.  Copernicus’ planets (a term that now included the Earth, but not the Moon or Sun) mostly orbited the Sun in circles, but they still retained a small epicycle that served as a replacement for the hated Ptolemaic equant.  As a result, Copernicus’ theory was just as complicated as Ptolemy’s.  Meanwhile, Copernicus gave each planet tiny and subtle motions that were synchronized to Earth’s annual motion, so he did not eliminate the strange synchronizations that bothered Puerbach and Regiomontanus.  The Earth, on the other hand, had a unique orbit with no epicycle (just as Ptolemy had denied the Sun an equant).  At a detailed level, Copernicus’ theory was just as weird and abstractly geometrical as Ptolemy’s.

These technical issues did not bother Copernicus’ contemporaries.  They were used to such things from Ptolemy.  The motions of the Earth, however, did bother them.  Very few astronomers adopted the Copernican theory in the 16th century.[5]  But although the Copernican theory was not embraced, Aristotle’s cosmology came under attack.  Tycho Brahe used his observations of the nova of 1572 to show that change could occur in the heavens.[6]  His observations of the 1577 comet indicated that the comet moved through the heavens in such a way that it must slice through any solid celestial spheres supposedly carrying the planets around.  If there were no solid spheres, as Tycho argued, then what caused the planets to move around as they did, whether around the Earth or around the Sun as Copernicus claimed?  Did they simply swim through the heavens like fish, or fly through the celestial regions like birds?

Whatever moved the planets, Tycho set out to measure their movements with unprecedented precision using enormous custom-built instruments.  Although this wealth of new data would play an important role in the advancement of astronomy, Tycho’s instruments were not truly new but were simply improved versions of existing instruments.  Although his measurements were more accurate than those of his predecessors, his data was of the same type that had been gathered previously.  While Tycho’s data may have supplied a key, only a brilliant imagination could use that key to unlock the door to a new astronomy.  The man who supplied that imagination, and opened that door, was Tycho’s assistant Johannes Kepler.

Kepler’s Physical Astronomy

Kepler learned about the Copernican theory from his teacher Michael Maestlin at the University of Tübingen. [7]  Kepler quickly became a devotee of the new theory and as a student he gave a disputation in which he displayed his remarkable ability to imagine what things would look like from another point of view.  The point of view Kepler chose was that of the Moon, the one object that Copernicus and Ptolemy could agree on since they both had it orbiting the Earth.  In his disputation Kepler described lunar astronomy.  The lunar astronomer would observe the stars circling the sky once a month.  The Sun and planets would share this motion but would also display an annual motion, with the planets having an additional motion unique to each planet.  The clear lesson here was that all objects seen from the Moon would appear to have a monthly motion due to the Moon’s monthly rotation (synchronized to its orbit around Earth), in just the same way that Copernicus explained the daily motion of the heavens with a daily rotation for Earth.  Likewise, the annual motions seen from the Moon could be attributed to the fact that the Moon accompanies the Earth on its annual trek around the Sun.

The appearance of the Earth itself would be truly remarkable as seen from the Moon.  For some lunar observers the Earth would hang in the sky at all times, exhibiting a small oscillatory motion with a period of a month.  Other observers would never see the Earth in their sky.  Kepler made great use of this feature when, much later in life, he turned his disputation into what some consider to be the first science fiction novel (Somnium) about a journey from Earth to the Moon.[8]  Kepler described the Moon and its inhabitants, dividing the lunar geography into two great regions: one from which Earth is visible and the other with no Earth in its skies.

When Kepler graduated from Tübingen he accepted a position as a mathematics teacher and local astronomer/astrologer.  He began to ask himself new questions: Why are there six planets?  Why do they have the orbits they have?  Why do they move as they do along those orbits?  While giving a lecture on the great conjunctions of Jupiter and Saturn, Kepler had an interesting idea.  He realized that one could use geometric figures to set a size ratio for two circles.  For example, the radius of a circle circumscribed around an equilateral triangle has a set ratio to the radius of a circle inscribed within that triangle.  Perhaps this principle explained the relative sizes of the planetary orbits?

Kepler quickly realized that this idea would not work with regular polygons, but then he had an insight that would set the course of his career.  Kepler thought perhaps he should be thinking about spheres rather than circles, and regular polyhedrons rather than polygons.  And there were exactly five regular polyhedrons to fill the five spaces between the six planetary spheres!  Kepler became convinced that he had deduced God’s plan for the universe and he set to work trying to determine if the orbit ratios matched the geometrical ratios for spheres circumscribed around and inscribed within the regular polyhedrons.  He had to alter the octahedron ratio a bit, and even then the match wasn’t perfect, but it was good enough that Kepler could doubt the data rather than his theory.  He quickly published his new vision of the universe as the Mysterium Cosmographicum (Secret of the Universe) in 1596.[9]

In that short book Kepler also made some observations about the motions of the planets.  He noticed that, in the Copernican system, not only did planets farther from the Sun have longer orbital periods but they also moved more slowly than planets closer to the Sun.  It would have been impossible to make this observation using the Ptolemaic theory, but it proved to be a critical step for Kepler.  He began to think that the planets might be moved by a physical force whose origin lay in the Sun.  If that force grew weaker with increasing distance then that could explain why more distant planets moved slower.

Kepler became even more convinced that the Sun moved the planets when he considered the motion of a single planet.  Copernicus had replaced the Ptolemaic equant with a tiny epicycle, but Kepler returned to the equant because he felt that it gave a more “physical” description of the planet’s motion.  He realized that in an orbit with an equant and the Sun off-center, the planet would move fastest when it was closest to the Sun and slowest when it was farthest away.  This fit perfectly with Kepler’s growing intuition about the physics of the heavens, paving the way for what we now call astrophysics.  No astronomer before Kepler tried to use physical principles to explain astronomical motions, but Kepler embraced this idea fully and used it to completely change the way astronomy was done.  And Kepler was prompted to this new approach by his imaginative insight rather than any new data.

But Kepler wanted new data.  Specifically, he wanted Tycho’s accurate planetary observations.  Through a series of dramatic twists and turns Kepler managed to secure a position as Tycho’s assistant in Prague, where the great Danish astronomer had relocated to become Imperial Astronomer to the Holy Roman Emperor after losing royal patronage in his native Denmark.  Tycho set Kepler to work on the orbit of Mars.  When Tycho died shortly thereafter, Kepler inherited Tycho’s title and continued his work on Mars, which culminated in the publication of his 1609 book entitled Astronomia Nova.[10]  The extended title of the book translates loosely as “New Astronomy based on causes, or celestial physics, treated by means of commentaries on the motions of the star Mars, from the observations of Tycho Brahe.”  In this book Kepler gave full form to his vision of planetary motions governed by physical forces from the Sun.

Although Kepler was already fully convinced that planets were moved by physical forces, he went to great lengths to convince his readers that this must be true.  He imagined being a planet himself and trying to steer his way around the heavens.  He could not move in a circle around some empty point in space because he would have no guidepost by which to steer.  He could not move on a circle with the Sun off-center because although the Sun might serve as a guidepost it would be impossible to steer around, toward, and away from the Sun in just the right way to produce a circular path.  Kepler further visualized the motions through space generated by the Ptolemaic deferent and epicycle and quickly dismissed them as physically impossible due to their pretzel-like complexity.  Right from the start of the Astronomia Nova it was clear that Kepler was thinking about planetary motions in terms of real movements through space rather than in terms of geometrical constructions.

Based on this kind of physical thinking Kepler discarded Copernicus’ tiny epicycles and returned to the Ptolemaic equant, but now applied to circular orbits around an off-center Sun.  However, Kepler referred the motion to the body of the Sun which lay opposite the center from the equant point, rather than to the equant point.  As noted previously, a planet on such an orbit would move faster when close to the Sun and slower when far away.  Kepler envisioned a force emanating from the Sun and sweeping the planet around in its orbit as the Sun rotated (even though there was not yet any evidence that the Sun did rotate).  Influenced by William Gilbert’s De Magneteof 1600,[11] Kepler supposed that this force might be a magnetic force that could act across empty space but with diminishing strength farther from the Sun, so that the planet’s speed would be inversely proportional to its distance from the Sun.

This species motrix, as Kepler called it, might explain why the planets move around the Sun, but what could explain why the planets moved closer to, or farther from, the Sun?  Kepler imagined this situation as something like a boat moving on a circular river.  The current might carry the boat around the circle but to move toward one shore or the other there must be some kind of celestial oar to steer the boat in or out.  Kepler proposed a second type of magnetic force to carry out this task.  He thought that each planet must be a magnet (as Gilbert had claimed for Earth) with two poles.  He thought the Sun might be a magnet as well, but with only one magnetic pole spread out over its surface.  Suppose the Sun’s surface is the north pole of this unusual magnetic monopole.  When the south pole of a planet’s magnet is directed toward the Sun the planet would be pulled in toward the Sun, but as the planet orbits its orientation will change and eventually its north pole will be directed toward the Sun, resulting in a repulsive force that pushes the planet back out again.

Guided by this kind of physical thinking, Kepler considered several different orbits for Mars.  Although he used geometry to construct the orbits, he tested each orbit against his ideas about physical forces.  If it seemed unlikely that a given orbit could be produced by the combination of magnetic forces described above, then Kepler deemed that orbit unlikely to be the true orbit of Mars.  Of course, Kepler also tested his orbits against Tycho’s data, insisting that any deviations between the predictions from his orbit and Tycho’s observations had to be smaller than the estimated error in Tycho’s data.

After many trials (and tribulations) this combination of physical intuition and empiricism led Kepler to abandon circular orbits and embrace elliptical orbits governed by his so-called area law (which was in turn an approximation to the idea that the speed of a planet must be inversely proportional to its distance from the Sun).  Two thousand years of astronomical tradition were overturned.  The result was the most accurate astronomical theory ever devised.

Kepler’s revolutionary astrophysical insights were not readily accepted by his contemporaries, including his teacher and mentor Maestlin.  But eventually the accuracy of his predictions outweighed the objections to mixing physics and astronomy, two disciplines that had traditionally been kept separate.  The idea of physical forces guiding planetary motions was here to stay, and would be put to gloriously successful use by Isaac Newton later in the 17th century.

Imagination and Scientific Revolutions

What motivates a revolution in science?  Thomas Kuhn, in his Structure of Scientific Revolutions, put forth the notion that revolutions in science are brought on by anomalies – observations or measurements that simply cannot be explained by current theories.[12]  The anomalous data may appear because of the development of new instruments or simply because scientists decide to measure new things, but Kuhn placed new data squarely at the forefront of generating scientific revolutions.  We have seen, though, in the examples of Copernicus and Kepler, two tremendous forward leaps in the science of astronomy that were generated almost entirely by new ideas rather than by new instruments or new data.  This is not to deny that new data can spark scientific revolutions.  It certainly can.  Sometimes, though, revolutions arise in the absence of new data.  Why?

One thing that may motivate scientists to introduce new theories, without the prodding of new data, is the desire to answer previously unanswered (or poorly answered) questions.  Sometimes a theory is successful enough that most scientists are content to ignore its few failures, but eventually someone will demand that these failures be addressed.  The Ptolemaic theory worked quite well at predicting planetary positions, but it could not explain why there were two classes of planets (inferior and superior) nor could it provide a definite ordering for the planets.  Copernicus found a way to explain the things that Ptolemy could not.

Another motivation for the introduction of new ideas in science is the destruction of an old idea.  For millennia astronomers had imagined giant, solid, transparent sphere spinning around to create the celestial motions.  Tycho’s comet observations relegated that idea to the dustbin of history and opened a space for Kepler’s new ideas about physical forces from the Sun moving the planets through space.

Sometimes scientists are driven to new ideas by asking questions that nobody has ever asked before.  Why are there six planets?  Why do they have the orbits they have and why do they move as they do on those orbits?  Nobody had asked these questions before Kepler, in part because they mixed together the separate disciplines of astronomy and natural philosophy (or physics).  Asking these questions led Kepler deeper into the frontier where the boundary between astronomy and physics was not so clear, and the result was an astrophysical approach that changed astronomy forever.

Finally, scientists may be led to new ideas out of sheer stubbornness.  Commitments to certain theoretical principles may drive scientists to feats of imagination in order to hold on to these cherished ideas.  Copernicus may have come up with this heliocentric theory as a way to avoid the non-uniform motions of Ptolemy’s equant.  Kepler chose to abandon circular orbits and try ellipses rather than give up on his physically-motivated area law. Success in science may be partly a matter of choosing to be stubborn about the right things.

Imagination can open up revolutionary new avenues for scientific thought even without new data.  New ideas can give us new ways of picturing the universe that help us answer previously unanswerable questions and ask previously unaskable questions, thus driving scientific progress.  These new ideas can also serve as the building blocks for even newer ideas and theories.  We have seen that Kepler drew on the Copernican planetary theory, Ptolemy’s equant, and Gilbert’s ideas about magnetic forces to develop his ideas on celestial physics.  Although we now reject Kepler’s specific ideas about magnetic forces, his approach to understanding the motion of the planets through the action of physical forces from the Sun would, when combined with new ideas about terrestrial physics from Galileo, Descartes, and others, lead the way to Newton’s theory of universal gravitation.

Of course, imagination can also lead us astray.  That is why it is important to constrain our imagination with the requirement that our new ideas match our observations and measurements.  Good data, like Tycho’s planetary observations, is very important.  Data is great at telling us what is not happening, but we need imagination to help us see what might be happening.

It is particularly ironic that Copernicus’ new heliocentric theory, and Kepler’s astrophysical re-imagining of that theory, were both accomplished before the introduction of the telescope.  The first astronomical uses of the telescope date to 1609, just months after Kepler published his Astronomia Nova.  This amazing new instrument soon provided supporting evidence that helped win widespread acceptance for the ideas of Copernicus and Kepler, but it played no role in the genesis of those ideas.  Although the telescope allowed human eyes to see farther into space than ever before, even the telescope could not reveal the depths that could be seen with the eye of the mind

by Todd K. Timberlake, Berry College, PO Box 495004, Mount Berry, GA 30149

[1] For an overview of the Copernican Revolution, see Timberlake, T. and Wallace, P. (2019).  Finding Our Place in the Solar System: The Scientific Story of the Copernican Revolution.  Cambridge University Press, Cambridge, UK.

[2] Aristotle’s cosmology is discussed in his Physics and De Caelo.  For the Physicssee Aristotle (2009). The Basic Works of Aristotle, translated by R. McKeon.  Modern Library, New York.  For De Caelo see Aristotle (n.d.). On the Heavens, translated by J. L. Stock.   Generic NL Freebook Publisher.

[3] Ptolemy’s astronomy is detailed in Ptolemy, C. (1998).  Ptolemy’s Almagest, translated by G. J. Toomer.  Princeton University Press, Princeton, NJ.

[4] Copernicus, N. (1992).  Nicholas Copernicus: On the Revolutions, translated by E. Rosen.  Johns Hopkins Press, Baltimore, MD.

[5] A survey of the responses to Copernicus, based on annotations in copies of De Revolutionibus, is found in Gingerich, O. (2004).  The Book Nobody Read: Chasing the Revolutions of Nicolaus Copernicus.  Penguin Books, New York.

[6] Tycho’s work is discussed in Toren, V. E. (1990).  The Lord of Uraniborg: A Biography of Tycho Brahe.  Cambridge University Press, Cambridge, UK.

[7] Kepler’s life is detailed in Caspar, M. (1993).  Kepler, translated by C. D. Hellman.  Dover Publications, New York.

[8] Kepler, J.  (2003).  Kepler’s Somnium: The Dream, or Posthumous Work on Lunar Astronomy, translated by E. Rosen.  Dover Publications, New York.

[9] Kepler, J. (1981).  Mysterium Cosmographicum: The Secret of the Universe, translated by A. M. Duncan.  Abaris Books, New York.

[10] Kepler, J. (2015).  Astronomia Nova, translated by W. H. Donahue.  Green Lion Press, Santa Fe, NM.

[11] Gilbert, W. (1958).  De Magnete, translated by P. F. Mottelay.  Dover Publications, New York.

[12] Kuhn, T. S. (1970).  The Structure of Scientific Revolutions (2nd ed.).  University of Chicago Press, Chicago.