Showing posts with label medieval astronomy. Show all posts
Showing posts with label medieval astronomy. Show all posts

Friday, December 27, 2024

The Light Ages: The Surprising Story of Medieval Science

It is difficult to know when to accept history on its own terms. We tend to not condemn the ancient Greeks, Romans, or Jews for having slaves, though we hold the ante-bellum South culpable. I was not surprised by Seb Falk’s story about Brother John Westwyk’s productive labors in mathematical astronomy and the consequential instrumentation of measurement for those works. But I was enlightened by the search for and discovery of the man who wrote an astronomy text commonly credited to Geoffrey Chaucer. 

Chaucer did author A Treatise on the Astrolabe. It was Dompnus Johannes de Westwyke (Brother John of Westwick) who created Equatorie of the Planetis, a book that had been credited to Chaucer. So, Seb Falk interleaves two stories here: the life of Brother John as best it can be built from scant records; and the development of astronomy (and, generally, science) in the Middle Ages. 

Reading this book the first time through, I knew that I would annotate post-its to bookmark passages. The second time through the book, it soon became clear that I should just copy the whole thing here—which, of course, is not allowed.

“Far from the stereotype of a stagnant scientific environment which did no more than preserve the ideas of the ancients, computists in the twelfth and thirteenth centuries continued to refine their astronomical models, with ever more accurate estimates of the solar and lunar cycles. Scholars became more outspoken in their criticism of the increasingly unrealistic ecclesiastical calendar. In the 1260s Franciscan friar and proponent of empirical science Roger Bacon wrote, at the Pope’s request, a series of tracts on educational reform.” (page 74) 

To compute, you must have a computer. God gave you ten fingers, each with two knuckles. Supported and enhanced with recipes for rapid mental arithmetic, you could calculate the rising and setting of the Sun for your locale—and you could measure musical harmonies, also. If you were enrolled at a university in 1325, to complete a bachelor’s degree required completing the Trivium of Logic, Rhetoric, and Grammar; the master’s degree required the Quadrivium of Geometry, Arithmetic, Music, and Astronomy. But attending one of the newly founded universities was expensive. So, monasteries most often paid for a limited education after which the motivated monk studied (and wrote) on his own—and sometimes her own (page 73). 

 

One result of that was that no two books are exactly the same. Copyists used the intentionally available spaces to add their own amendments and emendations, expanding, explaining, and correcting. Books lived. (ref. esp. pages 77 and 124). Knowing this, and expecting that copies will be carried to other places, Brother John Westwick cautioned the next one building an equatorie from his plans: “Nota I conseile the ne write no names of signes til that thow hast proved this commune centre defferent is trewli and justli set.” (Illustration 7.9).


The Light Ages: The Surprising Story of Medieval Science by Seb Falk; W. W. Norton & Company, New York, 2020. (Published in the UK as The Light Ages: A Medieval Journey of Discovery; W. W. Norton & Company, London, 2019.) W. W. Norton's webpage mentions the Best Book awards from The Times, The Telegraph, and BBC History Magazine. I learned of it because it garnered the AAS Historical Astronomy Division Osterbrock Book Award for 2024.


One thread not followed here, which I considered important, is that the manuscript under discussion was first rediscovered by Derek Price deSolla. Price deSolla is also credited with the first rigorous examination of the Antikythera Mechanism (NecessaryFacts here.) No mention of that appears in this book.

 

Previously on Necessary Facts

Science in the Middle Ages 

Astronomical Symbols on Ancient and Medieval Coins 

Galileo’s Two Sciences 

Rescuing Aristotle and the Church 

Copernicus On the Revolution of Heavenly Bodies 

 

 

Tuesday, February 25, 2020

Rescuing Aristotle and the Church

Last night, we started watching Cosmos: A Spacetime Odyssey on DVD. Still waiting to finish Episode 1, the narrative has problems. The error in beatifying Giordano Bruno as a martyr to science was quickly pointed out by others. (See “Did Cosmos Pick the Wrong Hero?” by Corey S. Powell, Discover magazine online here and “Why Did Cosmos Focus on Giordano Bruno?” by Josh Rosenau at the National Center for Science Education here.) Part of that false narrative included a swipe at Aristotle which no one else caught. Aristotle always takes a lot of criticism, mostly by misunderstanding and misattribution. 

Reviewing Episode 6: Deeper and Deeper Still, (which is still in my future), Jennifer Ouellette of the LA Times wrote:
“Anyway, the notion of atoms dates back to an ancient Greek philosopher named Democritus, who first proposed that it just wasn’t possible to keep dividing matter into smaller and smaller bits; at some point, you would reach the smallest possible piece, which he dubbed “atomos” (“not to be cut”). His contemporaries, including Aristotle, didn’t take Democritus seriously, and why should they? They didn’t have the tools to probe such a small scale. But eventually modern physics proved him right.” ("Cosmos’ recap: Deeper and deeper still" April 14, 2014 5:20 AM here.)
First, the physical reality of atoms was doubted by serious scientists such as Ernst Mach as late as 1900. The concept was considered a convenient construct. The reality of very small particles was easy to accept because lenses and microscopes revealed them. However, the concept of “atom” raised unsolvable problems. Of course, those metaphysical objections go back to the Greeks. The problem was argued without resolution. 

What is between atoms? Nothing? “Nothing” is not a different kind of “something” but rather “nothing” truly “does not exist.” What then is between atoms? 

If nothing is between atoms, then why does it take any time at all for atoms go from one place to another? How do atoms interact across the void to form molecules? 

Nature abhors a vacuum. Every space must be filled with something.  If the void of nothing does not exist, and if all matter consists of uncuttable objects, how do we move at all?  
The clepsydra
from Cosmos (1980, page 179)
Obviously, we do move. Just as obviously, all matter is made of something, or perhaps four or five different kinds of something. The lack of a solution to such problems did not prevent social and material progress. And the atomic theory was supported by some evidence. Empedocles of Acragas used it to explain how the “clepsydra” worked. Ultimately, in the modern 19th century, John Dalton, Dmitri Mendeleev, and others put the idea of the atom to good use.

That being so, the discovery of the electron and eventually other subatomic existents defeated the idea of an ultimately “uncuttable” object. Now, we have fields and statistical probabilities. 
 
Detail of Raphael's School at Athens. 
Plato points to the sky, while
Aristotle reaches for the world.
http://www.museivaticani.va/
Aristotle placed the Earth at the center of the universe for both logical and empirical reasons. A century later, Aristarchus of Samos put the sun at the center of our system. Archimedes attempted to settle question by measuring the parallax. He could not do it. He concluded that either the Earth is the center of the universe, or else the universe is far larger than anyone could imagine. We know now that the problem was his instrumentation. Fine as it was for the times, it was not up to the task. Friedrich Bessel achieved the first measurement of stellar parallax in 1838 using a spectroscope. Similarly, the rotation of the Earth on its axis was not proved by experiment until 1789 by Giovanni Battista Guglielmini and most dramatically demonstrated in 1851 by Léon Foucault. 

Aristotle’s embryology of the chick is one of the greatest experimental observations. He also knew that dolphins are mammals, breathing air, bearing their young alive and feeding them milk. 

In addition to being a careful observer, Aristotle attacked the philosophical (“scientific”) problems of his time by comparing and contrasting what others wrote before and then analyzing those against reason and fact. Even so, most of what we have from Aristotle was reconstructed. The great corpus of his original work was lost when the Macedonian royal family fought over his library and the scrolls were buried. Worms got to them. The damaged manuscripts were reconstructed with egregious errors. Other works from Aristotle’s students (primarily Theophrastus) were copied as if from Aristotle himself. 

The Catholic Church of the early Middle Ages found Platonism to its liking. In the 13th century, Thomas Aquinas argued well for Aristotle. By the 16th century—ten generations later—Aristotle was not just the norm, but literal truth. It was the method of men who sought literal truth in one book. We see this today when fundamentalist Christians attack Darwin. Darwin's Origin opens with citations to others before himself who proposed theories of evolution. But fundamentalists take one book as their source and therefore another single book as their target. 

The Catholic church of the Middle Ages was very supportive of astronomy in particular and science in general. Despite our modern focus on Christmas, for Christians the most important day of the calendar is Easter: the first Sunday, after the first full moon, after the first day of spring. Calculating Easter required bringing the solar and lunar calendars into agreement. They called the practice computas. It was a mathematical prediction tested by observation—thanks to the astrolabe imported from Islamic Spain in the late 11th century. The churchmen of the Counter-Reformation would have been out of step with Catholic education of the 12th century.

As Dr. Tyson says: “To make this journey, we’ll need imagination. But imagination alone is not enough because the reality of nature is far more wondrous than we can imagine. This adventure is made possible by generations of searchers strictly adhering to a simple set of rules: Test ideas by experiment and observation. Build on those ideas that pass the test. Reject the ones that fail. Follow the evidence where it leads and question everything.”

It is a nice statement of the scientific method that works for television as it streams past you. Myself, to encapsulate the scientific method, I would have bulleted these points:
  • Ask questions.
  • Explain observations with logically consistent theories.
  • Test theories with different observations and new predictions.
  • Publicize your findings and your methods.

The rules can be variously stated. Those four mandates could be expanded to 14 steps. That is how truth works. It is why over 300 proofs of the Pythagorean Theorem have been published. 

As easy as it would be to excoriate Neil deGrasse Tyson, the series was written by Ann Druyan and astrophysicist Steven Soter, both of whom worked on the original CosmosA Personal Journey with Carl Sagan. Druyan was married to Sagan. Sagan was one of Soter’s dissertation advisors. 

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Tuesday, October 23, 2018

Scorpio and the Precession of the Equinox

Sun sign astrology labels me a Scorpio, which is cool. However, right now, the sun is in Virgo. 

Ephemeris for 22 October 2018
for Austin, Texas 200 meters ASL
(US Naval Observatory)
Even astronomers mark the ecliptic and note the peripoint of Aries, which was the first day of Spring about 2000 years ago when all of this was laid down in the Roman West.  (The sun was already moving into Pisces, giving rise to other myths.) Now, it is the peripoint of Aquarius. It takes about 22,500 years for the stars to come full circle, as seen from the Earth. (Relativity, known even to the Greeks, lets us talk about it that way, from our arbitrarily chosen but very convenient inertial frame of reference.). It is intriguing to suggest that the Sphinx of Egypt is really very much older than our academics tell us; and it was built when the first day of spring was in Leo. 


Claws extend into Libra
(Texas hill country scorpion flikr John Morton)

 Modern astrologers never face the problem of Libra. Before the Romans, what we call Libra was really the claws of the Scorpion. But the Romans liked 12: 12 gods of Olympus, 12 ounces the pound, 12 inches to the foot, 12 sestertii to the denarius.  (Yes, denarius means “tenth.” They had to adjust their monetary system to pay for winning all those wars.) So, they made 12 months out of the Lunar year of 13 and gave each one a Zodiac sign.  (“…. 13.37 sidereal months, but about 12.37 synodic months, occur in a Gregorian year.” saith Wikipedia.)

 
Gold stater of Croesus (Kroisos) of Lydia c. 550 BCE
(Harlan Berk Gemini III Auction)
Lion confronts Bull.
The symbology is variously debated
but the symbols are obvious.
Originally, the ancients of Sumer recognized three constellations along the path of the sun and the moon—the three that actually look like what they are: the Lion, the Scorpion, and the Bull. (They also knew the Giant, Orion. O-names for heroes and gods are surprisingly common across many ancient cultures. But that is another story…)

Previously on Necessary Facts


Monday, November 27, 2017

Joy Hakim's "Aristotle Leads the Way"

Intended for children, The Story of Science: Aristotle Leads the Way by Joy Hakim has many small problems throughout but remains valuable for its sense of life. The author encourages understanding, exploration, discovery, and the integration of knowledge. You can find it remaindered online at prices low enough to gift an entire class of 5th graders, if you choose. Though intended for youngsters, nothing is dumbed down. So the book is enjoyable at an adult level.

The Story of Science: Aristotle Leads the Way 
by Joy Hakim, Smithsonian Books, 2004.
 The consistent problem is the lack of nuance and insight. Joy Hakim just repeats common claims about the ancient Greeks and science in the Middle Ages. She mentions Hypatia of Alexandria, but says nothing about her being the likely last and best editor of Ptolemy’s work. She never acknowledges Aspasia of Miletus. And like almost everyone else, she accepts and asserts that the ancient philosophers did not think it necessary to test theories but attempted all knowledge through pure logic.

Despite those problems and their consequences in presentations from cover to cover, the rich array of integrated facts should deliver years of engagement and encouragement to a young learner. Hakim does more than note the milestones; she reminds the reader of the road just traveled; and she looks to the next horizon.

Detail of Raphael's School at Athens.
Plato points to the sky,
Aristotle reaches for the world.
http://www.museivaticani.va/
Through lavish illustrations, and insightful narratives, she presents the mathematics needed at a level that can be grasped with arithmetic and geometry. Among the very many are the Pythagorean theorem (of course), how Aristarchus estimated the distance to the Moon, Democritus’s formula for the volume of a cone, the true nature of the cone as explained by Apollonius of Perga, and the work done by the lever of Archimedes. Occasional timelines remind us that knowledge is passed across generations to those who cared to learn or rediscover.

Writing about Thomas Aquinas, “An ‘Ox’ Who Bellowed” (Chapter 20), Hakim says:
“In the thirteenth century, Paris is the place to be, if you like tumult and activity. While most of Europe is still feudal, Paris is the center of an emerging market economy. Old ideas are being blown away. … Change is both energizing and upsetting. The feudal world was known. What a free, capitalist world might be like is unknown. It seems to offer little security or control. But there is no stopping the new forces. … In the monasteries, clerics are focused on saving their souls through prayer study and isolation. When it comes to science, they quote Pythagoras, Plato, and Augustine. That trio all concentrated, in one way or another, on ideal forms in nature, which often kept them from considering the real world. But at the budding universities, new scholars, inspired by the rediscovery of Greek science, are interested in understanding the forces of nature. Those new scholars are fascinated by Aristotle. Aristotle looked at the world about him and observed, made notes, and classified its inhabitants—plant and animal.” (page 229-230)
—NSTA Recommends
Teacher’s Guide

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Monday, November 20, 2017

COPERNICUS ON THE REVOLUTION OF HEAVENLY BODIES

It was not just that Copernicus put the sun at the center of the heavenly system. His book asserted modern ideas about the relativity of motion, the acceleration of falling bodies, and the nature of the planets. It is also true that many of his arguments are archaic, typical of the medieval scholasticism that rested on the ancient Greeks. Nonetheless, his heliocentric model changed not just how people viewed the universe, but how they thought about it.
That motion is relative was known to the Greeks. Copernicus cites the Aeneid on the point-of-view that the land slips away as the ship moves past it. (page 519, EB “Great Books” edition). Fire burns brighter at lower altitudes; and when carried to higher altitudes, the fire dies down. (Book I Section; 8. 520, EB). Copernicus says as a matter of fact, not argument, that the planets are “dark bodies” that shine by reflected light. (Book I Section 10; Page 521 EB).

In our era, in school, we learn that Galileo discovered that bodies accelerate when they fall, contrary to Aristotle’s claim that they fall at a constant velocity, determined by their weight.  In fact, Copernicus acknowledged the acceleration of free fall.  In the original edition, Copernicus wrote: “Et quaecunquae decidunt a principio lentum facientia motu velocitatem augent cadendo.” That is rendered today as: "And those which fall downward possess a slow movement at the beginning but increase their velocity as they fall."(Book I Section 8. Page 520 EB)  See also Book III Section 3 where he mentions the motion of a pendulum being faster at the bottom of the arc and slower at the tops.

The acceleration of a body in free fall apparently was known to the “Oxford calculators” of the Merton School who worked about 1325-1350.  They developed a geometric solution for the “mean rate.” The “mean rate” is the average of the initial velocity and the final velocity. Galileo cited their results in The Two New Sciences, Third Day, Theorem I, Proposition I.

Most of On the Revolution of Heavenly Spheres is tedious for anyone not passionate about the geometric arguments. Copernicus had to prove his claims to the thinkers of his time in the language of their culture. The same was true of Newton’s Principia. Richard Feynman attempted to recreate a proof for his own lecture and found that he could not because he did not know enough geometry: we do it all with calculus now. Copernicus provides extensive ephemeris tables of his own measurements. But he also worked out trigonometric tables of “half-sines.” (We call it the “sine” in school, but it is actually the half.)  And he showed how his sun-centered circles explain all of the apparent motions in longitude, oppositions, retrograde, etc. And he often cited Ptolemy, who also made good measurements of the same events.

Copernicus’s On the Revolution of Heavenly Spheres is easily available in modern translations. Several archives have full scans of the original 1543 publication.
Made available electronically by the NASAAstrophysics Data System (ADS) 
The ADS wishes to acknowledge the Lehigh University, Digital Library ("The Problem of the Planets") for providing the scans of the book. (http://digital.lib.lehigh.edu/planets/ )

The Smithsonian also has one of the 1543 editions in its Dibner collection. (See https://airandspace.si.edu/exhibitions/explore-the-universe/online/kiosks/dibner/) The image title says: Nicholas Copernicus, De Revolutionibus Orbium Coelestium Libri VI (Basel, 1543), but the title page says “Norimbergae” i.e., “Nuremberg.”

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Friday, August 3, 2012

Science in the Middle Ages

Objectivists value the scientific method as the cornerstone of the engineering achievements of our civilization from structural trusses and direct current to alternating current and cybernetics.  We too easily see the Middle Ages as a time of ignorance and barbarism in which learning was chained to (and by) theology.  The reality is more complicated.

“The Middle Ages was a period ruled by the Witch Doctor, in a firm, if mutually jealous alliance with Attila. The Witch Doctor controlled every aspect of human life and thought, while the feudal Attilas looted one another’s domains, collected material tributes from serfs – who worked, lived, and served in subhuman conditions – and maintained the power to burn heretics at the stake.
“Philosophy, in that era, existed as a “handmaiden of theology,” and the dominant influence was, appropriately, Plato, in the form of Plotinus and Augustine. Aristotle’s works were lost to the scholars of Europe for centuries. The prelude to the Renaissance was the return of Aristotle via Thomas Aquinas.” – Ayn Rand, “For the New Intellectual.”

Astronomy, in particular was not dormant, nor could it be.  The problem of Easter required bringing lunar and solar calendars into alignment.  Whether biology, botany, and medicine had any hint of modernism is a difficult question, but can only be answered with direct citations to contemporary works.  Like the revolution of the Earth on its axis and the orbit of the Earth about the Sun, proof contrary to spontaneous generation did not come until the 1840s. As with astrology versus astronomy, it is too easy to dismiss alchemy as not being "real" chemistry. While its paradigms are not ours, the practices were utilitarian: dying wool and leather were important crafts. Paints, pigments, and finishes also were consequential.

“Historians have long recognized that the rebirth of science in twelfth-century Europe flowed from a search for ancient scientific texts. But this search presupposes knowledge and interest; we only seek what we know to be valuable. The emergence of scholarly interest after centuries of apparent stagnation seems paradoxical. This book resolves that seeming contradiction by describing four active traditions of early medieval astronomy: one divided the year by observing the Sun; another computed the date of Easter Full Moon; the third determined the time for monastic prayers by watching the course of the stars; and the classical tradition of geometrical astronomy provided a framework for the cosmos. Most of these astronomies were practical; they sustained the communities in which they flourished and reflected and reinforced the values of those communities. These astronomical traditions motivated the search for ancient learning that led to the Scientific Renaissance of the twelfth century.” (Astronomies and Cultures in Early Medieval Europe by Stephen C. McCluskey. Cambridge University Press,  1997.) 

“Measurements by clepsydras prove … that although the earth is at the center of the universe, it is eccentric to the sun's orbit. At times the sun is borne at a greater distance from the earth than at other times. When the sun is climbing upwards in Cancer and Gemini, in the steeper tracts of its course, it takes longer, lingering 32 days in Gemini; but it requires less time in the lower tracts, 28 days in Sagittarius, the elapsed time for the other signs varying between those extremes (848-849) . “Dominant Traditions in Early Medieval Latin Science” by William H. Stahl, Isis, Vol. 50, No. 2 (Jun., 1959), pp. 95-124. 

Mean Speed from the Oxford
Calculators used by Galileo
"The sorry state of scientific studies at the close of the Roman Empire in the fifth century reflected Roman, not medieval, failures and short-comings." "How Science Survived: Medieval Manuscripts as Fossils” by Sharon Larimer Gilman and Florence Eliza Glaze. Science, New Series, Vol. 307, No. 5713 (Feb. 25, 2005), pp. 1208-1209.

“The dominant explanation of human behavior at this time was astrology.  Charles, like most of his contemporaries, ruled with the advice of the recognized social scientists of the day, the court astrologers. In this vein Oresme was ordered by Charles to translate Ptolemy's Quadripartitum from Latin into French. This order discharged, Oresme then attempted to debunk the popular conceptions in an attack on judicial astrology, Contra judiciaros astronomos (1360), which he later translated into French. There is no hint in the historical record that Oresme's efforts altered the predominance of astrology in the determinance of social policy; indeed, he returned to the attack ten years later with a treatise entitled Contra divinatores horoscopios, and in a series of Quaestiones (a stylized form of question and answer popular with academicians of the period).”
“Nicole Oresme and Medieval Social Science: The 14th Century Debunker of Astrology Wrote anEarly Monetary Treatise” by Kevin B. Bales, American Journal of Economics and Sociology, Vol. 42, No. 1 (Jan., 1983), pp. 101-111.

“WHEN historians and historians of science flatly state that Ptolemy's Almagest and Geography dominated the fields of astronomy and geography for fourteen centuries, they are apt to mislead unwary readers into supposing that Ptolemy was the supreme authority in Latin science during that period. Quite the contrary, his works might almost as well never have been written for all the influence they had in the Latin West until translations were produced from Greek or Arabic texts in Toledo and Sicily in the twelfth century.' If there was any dominant tradition of Latin science in the first thirteen centuries of the Christian Era, it was a stream of encyclopedic literature, the main course of which may be traced backwards through the Latin encyclopedist Varro and the Platonizing Stoic Posidonius; to trace sources beyond them is difficult indeed. In any case it is well to bear in mind that this stream of encyclopedic works skirts around Ptolemy without being appreciably influenced by him. We can trace its course through the extant writings of Pliny, Theon of Smyrna, Cleomedes, Geminus, and numerous others.

“Of the three, Martianus Capella offers the best account of encyclopedic science and will be discussed first. He seems to have flourished in the first half of the fifth century.  Martianus Capella wanted to produce an encyclopedia in the Varronian tradition and, by excluding two of Varro's disciplines, medicine and architecture, laid the foundation of the medieval trivia and quadrivia.

The first work of Archimedes translated into Latin was the Measurement of the Circle. It was translated from the Arabic twice in the twelfth century.' The first translation, which I have argued (but not surely) was done by Plato of Tivoli, was most inferior; just three manuscripts are known, only one of which is medieval. Apparently not long after this first translation the great translator Gerard of Cremona again used the Arabic text and rendered the Measurement of the Circle into Latin. This time the translation was quite accurate, and so before 1187 a faithful version of this short but important treatise became available. We are fortunate that this version was included in MS Bibliotheque Nationale, Fonds latin 9335, a handsome codex of Gerard translations. Incidentally, this manuscript is one of the best examples of intelligent copying of scientific works. It has marginal variant readings which cite alternate copies. The drawings are carefully made. Even more important the transcription of numbers - even of six places - is almost perfect. “The Impact of Archimedes on Medieval Science,” by Marshall Clagett, Isis, Vol. 50, No. 4 (Dec., 1959), pp. 419-429. 

"Among late thirteenth and fourteenth century philosophers, the Averroists have been particularly noticed by historians as advocates of the autonomy of the sciences. In arguing, for example, that physics might give an answer to the question of the eternity of the world different from the answer given in accordance with Christian belief, Boethius of Dacia based himself largely on a conception of physics as an independent discipline with its own principles, rational methods, and conclusions. This approach to the autonomy of the sciences was not, however, the only influential one in this period. Another basis for the autonomy of physics is found in the work of certain Oxford commentators on Aristotle's Physics, most prominently in William of Ockham's Expositio super octo libros Physicorum. This second approach makes physics autonomous by placing greatest confidence not in deductively and causally prior principles of physics, which some thought could be proved or made known by metaphysics, but rather in propositions accepted on the basis of experience, if not known in themselves. The primary evidence cited in the following paper." “The a Posteriori Foundations of Natural Science: Some Medieval Commentaries on Aristotle's Physics, Book I, Chapters 1 and 2” by Edith Dudley Sylla, Synthese, Vol. 40, No. 1, Jan., 1979, pp. 147-187.
 In The Logical Leap, Objectivist physicist and philosopher David Harriman denigrates medieval science. In Chapter 3 "The Mathematical Universe" under the subhead "The Birth of Celestial Physics" on page 85 (ppb), Harriman says that with the Ptolemaic Model, the relative sizes of the orbits of the planets could not be calculated.  That leads to an interesting contradiction.  If it is true that the Geocentric model prevents such calcuations, then they must have used some other model, because the relative sizes of the orbits were known.  On the other hand, perhaps the geometry and observations of the time did, indeed, allow them to make those calculations, even assuming the Geocentric model.  My reference for that is Astronomies and Cultures in Early Medieval Europe by Stephen McCluskey (Cambridge, 1998).  In fact, because of the religious viewpoint, the very scale of the measurable universe and the comparatively small size of the (spherical; not flat) Earth, were substantiating evidence to the relative unimportance of Earthly affairs. Saturn's orbit was estimated to be 72 million miles from Earth. (McCluskey, page 203).