Taking some time off!


Just spent a very tiring and frustrating day undergoing test and filling in forms in order to be admitted to hospital tomorrow morning at 6:30am! This means there will no blog post this week and possible not next week either, we’ll have to see.

To fill in the time you could read Karl Galle’s excellent Copernicus guest post if you have already done so, or my guest post on Forbidden Histories on astronomy & astrology.

In the mean time I wish all my readers a good time and promise that I will be back in the not too distant future.



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Publish and Perish

As I announced yesterday I am playing away this week with a tasty post on astrology and astronomy on the excellent Forbidden Histories blog site. However the substitute bank here at The Renaissance Mathematicus is filled with the finest from the fine. This week stepping up to the plate is mega historian of Early Modern science and Renaissance Mathematicus friend, Karl Galle, brought to you all the way from the sunny streets of Cairo (we are truly international here).

 Some time ago I realised that although I specialise in the history of Renaissance astronomy, I have up till now written no substantive biographical post about Nicolaus Copernicus. Now potted biography posts are one of my specialities and I have written them about almost everyone of significance in the Renaissance astronomy crew but not of the good old Nicky. Whilst I was pondering how I could best correct this omission, It occurred to me that my #histsci buddy Karl is currently engaged in researching and writing a modern biography of the Cannon of Frombork Cathedral. Knowing no shame, I immediately contacted Karl and suggested that he could take on the task in hand as a Renaissance Mathematicus guest blogger. With enough arm-twisting and the promise of an undisclosed number of free beers next time he is in Nürnberg he graciously agreed to write an authoritative blog post on Warmia’s most famous son. Read and enjoy!


 On May 24, 1543, Nicholas Copernicus achieved every writer’s great dream of finally holding in his hands a published copy of the book he had worked on for most of his life. Having done so, he then died that same day. Sadly, he was probably unconscious when the book was placed in his hands, as we learn from the only surviving account of his death in a letter from his best friend, Bishop Tiedemann Giese, to his only student, Georg Rheticus.

I’d like to thank Thony very much for inviting me to commemorate this year’s 475th anniversary of Copernicus’s death and the publication of his book De revolutionibus orbium coelestium (“On the Revolutions of the Heavenly Spheres”) with a guest post. While I won’t attempt to match Thony’s polymathic virtuosity on all things Mathematicus related, I thought I would try and channel a bit of the blog’s myth-busting spirit by using this opportunity to look at a few of the stories traditionally told in connection with Copernicus’s remarkable idea that the Earth, formerly assumed to be resting naturally at the center of the cosmos, is in fact both rotating on its axis and moving at high speed around the sun.


Figure 1: Copernicus’s manuscript diagram of the cosmos, showing the Sun at the center and the Earth and Moon in sphere #5, from the digitized autograph copy at the Jagiellonian University Library in Cracow.

Myth #1: Debates over the Copernican theory are central to understanding Copernicus’s own life and work.

It should be an obvious point given the timing of Copernicus’s death, but virtually the entire debate over the heliocentric theory took place posthumously and without further participation by the theory’s original author. The most famous episodes occurred more than half a century later. While many writers did discuss Copernicus’s mathematical models – usually with high praise – in the early years after De revolutionibus appeared, the next really detailed defense of the heliocentric theory didn’t appear in print until 1596 with Johannes Kepler’s Mysterium Cosmographicum. The Catholic Church only declared heliocentrism theologically heretical and suspended Copernicus’s book “until corrected” in 1616 (with a list of corrections approved in 1620), and Galileo’s trial for advocating the heliocentric theory took place in 1633.

All of these and many other less well-remembered episodes are fantastically interesting in their own right. For understanding the so-called Copernican revolution, however, they are the historiographical equivalent of studying the Bolshevik movement for insight into the composition of the Communist Manifesto. They provide useful lessons about the transformation and application of radical new ideas but are often profoundly misleading in regard to those ideas’ original contexts.

Kepler’s education and worldview were shaped by Protestant university reforms that had barely begun when Copernicus died, and Galileo’s trial took place under Counter-Reformation pressures dramatically different from the political and theological environment of 1543. This is before one even discusses the invention of the telescope, the huge observational programs of Tycho Brahe and others, and the extraordinary proliferation of mathematical texts and practitioners in the century after De revolutionibus, all of which were profoundly new developments from the time in which Copernicus lived and worked (1473-1543).


Figure 2: Statue of at the base of Frombork’s cathedral hill, where Copernicus lived for most of his professional life (author photograph).

Myth #2: We don’t know very much about Copernicus’s life.

One reason why historians have often spent more time examining posthumous debates over the heliocentric theory rather than Copernicus’s own era is the assumption that we don’t have much information about his life, and this is at least a moderately defensible point. Kepler, Galileo, and other canonical giants like Darwin all enjoyed the good fortune of having not just voluminous correspondence networks but great fame while they were still alive. When they died, the bulk of their manuscripts were gathered and preserved well enough to eventually become happy hunting grounds for generations of historians. By contrast, Copernicus’s posthumous renown arrived much later and after a good share of his books and personal papers had likely been dispersed. The Giese-Rheticus letter providing the date of Copernicus’s death was one of a few surviving manuscripts that were found and published by the Cracow professor Jan Brożek after he made a pilgrimage to Warmia in 1618 in search of information about Copernicus.

Nevertheless, when people say we know little about Copernicus’s life, what they really mean is we have few documents pertaining to his life as an astronomer, and therein lies one of the key differences between Copernicus and his successors. Brahe, Brożek, Galileo, Kepler, and other notable contemporaries like Christoph Clavius and Michael Maestlin occupied a diverse range of positions across universities, courts, and church institutions. What they all had in common, however, was an ability to earn a living working on subjects related to astronomy or mathematics for most of their professional careers and to have a large and technically accomplished peer group while they did so.

By contrast, as far as we know Copernicus never earned a single schilling specifically for his work in astronomy. His professional rank was as a canon serving the prince-bishopric of Warmia, and the largest portion of his surviving papers thus derive from administrative work for the church or correspondence with regional political figures. Surveying these materials, one gets the impression of a skilled but unpretentious professional who was frequently relied on to handle some of the chapter’s most challenging tasks. If you needed a contentious land dispute settled, a sensitive diplomatic communiqué drafted, or a castle’s defenses organized during a siege by invading Teutonic Knights, Copernicus was the guy who would get it done and probably not ask for a promotion when it was all over. While these documents therefore tell us almost nothing about his astronomy, they do hint at a rather rich and interesting life.


Figure 3: Remains of the castle at Olsztyn, where Copernicus organized the defenses during an invasion by the Teutonic Knights (author photograph).

Myth #3: We should still think of Copernicus as a professional astronomer.

Astronomy was unquestionably Copernicus’s main intellectual passion and the subject to which he devoted the bulk of his private study. Even when he was called on for scholarly rather than administrative tasks, however, it was probably not what his colleagues most valued. The oldest manuscript evidence of interest in his mathematical pursuits is a set of letters in 1510 from a spy who was attempting to steal one of Copernicus’s maps on behalf of the Teutonic Knights during a period of tense territorial negotiations in the years before their armies invaded and overran most of Warmia. (The fact that this spy later became Copernicus’s boss and effectively a head of state despite being a paid agent of a hostile foreign power is only one of the remarkable stories that virtually every Copernican biographer ignores simply because it doesn’t relate to astronomy.) This particular map doesn’t survive, but other non-espionage correspondence confirms that Copernicus’s map-making abilities were called on throughout his life for political and also economic purposes like delineating fishing rights.

Sometime around 1514, Copernicus wrote a now lost commentary on calendar reform, and in 1517 he finished a first draft of a treatise on currency reform that was later revised and submitted to Polish and Prussian authorities in late 1525 or early 1526. Throughout his career in Warmia he was also in demand as a personal physician to successive bishops and other patients. The point is not that any of these other responsibilities or pursuits eclipsed (so to speak) his interest in astronomy, but that if we are going to speak in anachronistic terms, it makes at least as much sense to think of Copernicus professionally not as an astronomer but as a government functionary who occasionally wore the hat of technical specialist or senior policy advisor, all while pursuing a longstanding intellectual hobby that was only indirectly relevant to his career.

Significantly, this is very much what most of his peer group looked like as well. To list only a few examples, a rare surviving letter that mentions Copernicus’s astronomical work is one from 1535 that accompanied a set of his planetary tables. The recipient of the tables, Sigismund von Herberstein, was a life-long Habsburg diplomat who published a lengthy geography and ethnography of Russia near the end of his life based on his travels to that country. The sender of the tables, Bernard Wapowski, is best remembered as a cartographer and therefore closer to Copernicus in having mathematical interests, but he served the Polish crown for most of his life and left behind a long unpublished manuscript on Polish history. Johannes Albrecht Widmanstetter, who discussed Copernicus’s theory in the Vatican gardens in 1533, spent much of his career as a papal secretary before publishing his magnum opus, a dictionary of the Syriac language, shortly before he died.

One could multiply these cases many times to illustrate how common it was for late medieval figures to produce major scholarly works while following varied careers as public officials or church leaders rather than solely university-based teachers. Even Albert de Brudzewo – frequently cited as a likely influence on Copernicus’s early astronomical studies – left his teaching post at Cracow University in order to take up a position with the Jagiellonian Grand Duke Alexander in Vilnius. The fact that Widmanstetter was invited by Pope Clement VII to explain Copernicus’s ideas, and then rewarded with a costly manuscript for doing so, also points toward one of the most persistent misperceptions about how the heliocentric theory was received during its earliest years.

Myth #4: Church leaders were unanimously horrified and opposed to Copernicus’s theory as soon as it appeared.

The 1543 letter between Rheticus and Bishop Giese also includes details about the only actual controversy that publication of Copernicus’s book sparked immediately, namely that both Giese and Rheticus were furious about an anonymous preface Andreas Osiander had attached to the work. The background behind this preface and the complaint that Giese made to the Nuremberg city council are a fascinating story of their own, but let’s pause for a moment just to consider the nature of the participants. You have on one side a Catholic bishop allying himself with a former professor from Wittenberg university (literally the birthplace of the Protestant Reformation) in a conflict with the copy editor of De revolutionibus (Osiander), a firebrand Protestant minister who had previously been reprimanded by Nuremberg’s council after publishing a pamphlet declaring the pope to be the anti-Christ. All of this was over a book that was dedicated to the pope, written by a Catholic church official, only came into existence because Wittenberg allowed one of their professors to take extended faculty leave to help bring it out, and was solicited and issued by one of the era’s greatest printers, a man renowned for publishing not just scientific but Protestant theological and musical works. One can argue all you like about the nature of Osiander’s preface, but short of throwing in a laudatory poem by Ulrich Zwingli or a posthumous endorsement by Jan Huss, it’s hard to imagine how Copernicus’s book could have featured a broader array of church figures who might have disagreed over certain aspects of the book’s merits but had very little problem supporting its appearance.

This is not to say there weren’t a few early rumblings of concern. Martin Luther is reputed to have made disparaging verbal remarks before De revolutionibus appeared about how certain people wanted to seem clever and turn astronomy upside down. However, this only counts as sharp criticism in Luther’s world if you’ve never read any of his published texts on Jews, Turks, papists, or pretty much anyone else he considered truly theologically dangerous. In Italy during the late 1540s, at least a couple of Dominican writers previewed some of the Catholic church’s later objections to Copernicus on the grounds of illogical physics and contradictions with scriptural passages, but these criticisms seem to have gained little traction at the time. As for Copernicus, other correspondence suggests that when he wrote of his fears that some people might mock his ideas, he was referencing not simply church authorities so much as “Peripatetics,” or Aristotelian philosophers whom he correctly feared might point out among other things that he hadn’t really answered all questions that would arise from the physics of a moving Earth.

The challenge of rewriting terrestrial physics to account for complex motions and then connect with the movements of the heavens would in fact occupy natural philosophers for the next century and a half. During that same period, much of Europe would tear itself apart in increasingly apocalyptic wars inflamed by religious tensions, and the potential grounds for heresy would expand to occupy philosophical domains including astronomy that had only occasionally been considered dangerous territory in centuries past. The condemnation of Galileo and the censorship of De revolutionibus were two consequences of this expanded politicization of knowledge, but this is not something that would have necessarily been predicted when Copernicus’s book first appeared in 1543. Ironically Nuremberg’s council seems to have been entirely unconcerned with the subject matter of heliocentrism, but they did investigate and censor another book that came out that same year because the Vatican’s Copernicus expert Widmanstetter wanted to publish a selection of Latin excerpts from the Qur’an. (See my comment above about episodes that are entirely ignored by Copernican biographers because anything that doesn’t explicitly mention astronomy is considered too boring to write about.)


Figure 4: Modern memorial to Copernicus in Frombork cathedral; his coffin is below the glass tile at the base (author photograph).

The afterlife of Copernicus

Copernicus died as a liked and well-respected figure to his colleagues, but not yet an unusually famous or controversial thinker among other scholars. As a new generation of astronomers worked through the lengthy text of De revolutionibus and began trying to fit its models to more accurate observations of the heavens, however, they also increasingly acclaimed Copernicus, hailing him repeatedly as “another Ptolemy” in recognition of his great mathematical abilities despite the fact that the heliocentric theory threatened to overturn Ptolemy’s old geocentric cosmos. A memorial plaque was belatedly erected in 1581 at Frombork cathedral, and Brożek copied out its text during his visit there in 1618. The exact location of Copernicus’s burial site was nevertheless forgotten until recent years when a research team located a set of remains and ingeniously matched them to Copernicus through genetic comparison with hairs found inside one of Copernicus’s former books now at Uppsala University. (Lesson to librarians – protect your rare books, but don’t clean them too well!) He was reburied beneath a tasteful modern monument in 2010, and I had the good fortune of visiting the site last fall.

One might justifiably ask why I’ve spent so much time harping about Copernicus’s era and social context rather than going into more detail about his astronomy and mathematics. There are indeed numerous interesting things to say on the latter subject, from the ongoing debates over exactly how Copernicus arrived at the heliocentric theory to the very tangible advantages his theory offered even during an era when astronomical observations were not yet precise enough to prove the empirical advantages of his individual planetary models over comparable models derived from Ptolemy.

As much as I enjoy the details of Copernicus’s astronomy, though, I think there’s a point at which exclusively focusing on the mathematics of De revolutionibus risks becoming the late medieval equivalent of writing a micro-history of free-return trajectories as if it’s the only subject worth talking about in regard to the US-Soviet space race. To say there are other topics worth discussing is not an either-or declaration of how to do history but a simple recognition that we need to understand more realistically how new knowledge takes shape and what transformations happen when it’s applied. Especially now, it’s worth resisting the regular incorporation of figures like Copernicus and Galileo into broader societal myths about how progress only happens when a tiny number of under-appreciated geniuses, working in isolation and free of interference from Big Government, acquire their wisdom through flashes of insight that spring fully formed like Athena from the head of Zeus, after which it only remains for the rest of us to appreciate their greatness rather than numbering among the benighted peasants and medieval reactionaries if we ask too many questions. If there is a lesson to be culled from the life of Copernicus and the period that followed, perhaps it is instead that understanding the cosmos is difficult, but sometimes even a few mild-mannered professionals who work well with their colleagues might get there in the end.

Karl Galle (@GalleKarl) is working on a new biography of Nicholas Copernicus that he hopes will be completed in less time than it took to write De revolutionibus. You can browse more photos from his Copernicus-related travels here.



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The German verb fremdgehen means to be unfaithful or to have a bit on the side in English. Regular visitors to this blog will know that the Renaissance Mathematicus is from time to time unfaithful and posts his scribblings in other places on the Internet. This has happened once again and I am actually pleased to announce that I have a post up on Andreas Sommer’s excellent blog Forbidden Histories. Andreas specialises in exposing the non-scientific underbelly of the history of science. Andreas asked me if I could write something on the history of astrology and I put together an overview of the common histories of astronomy and astrology: Astronomy and Astrology: The Siamese Twins of the Evolution of Science, which you can go over and read as of now. There is probably nothing new for those, who have read all of my previous astrology posts but I hope that what I have written is a short and informative introduction to the topic.


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400 Years of The Third Law–An overlooked and neglected revolution in astronomy

Four hundred years ago today Johannes Kepler rediscovered his most important contribution to the evolution of astronomy, his third law of planetary motion.


Portrait of Johannes Kepler 1610 by unknown artist. Source: Wikimedia Commons

He had originally discovered it two months earlier on 8 March but due to a calculation error rejected it. On 15 May he found it again and this time recognised that it was correct. He immediately added it to his Harmonices Mundi:


For when the true distances between the spheres were found, through the observations of Brahe, by continuous toil for a very long time, at last, at last, the genuine proportion of the periodic times to the proportion of the spheres –

Only at long last did she look back at him as she lay motionless,

But she look back and after a long time she came [Vergil, Eclogue I, 27 and 29.]

And if you want the exact moment in time, it was conceived mentally on the 8th of March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely exact that proportion between the periodic times of any two planets is precisely the sesquialterate[1] proportion of their mean distances, that is of the actual spheres, though with this in mind, that the arithmetic mean between the two diameters of the elliptical orbit is a little less than the longer diameter. Thus if one takes one third of the proportion from the period, for example, of the Earth, which is one year, and the same from the period of Saturn, thirty years, that is, the cube roots, and one double that proportion, by squaring the roots, he has in the resulting numbers the exactly correct proportion of the mean distances of the Earth and Saturn from the Sun.[2]

writing a few days later:

Now, because eighteen months ago the first dawn, three months ago the broad daylight, but a very few days ago the full sun of a most remarkable spectacle has risen, nothing holds me back. Indeed, I give myself up to a sacred frenzy.

He finished the book on 27 May although the printing would take a year.

In modern terminology:


The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit: i.e. for two planets with P = orbital period and R = semi-major axis P12/P22=R13/R23

Kepler’s third law is probably the most important discovery on the way to the establishment of a heliocentric astronomy but its importance was initially overlooked and its implications were somehow neglected until Isaac Newton displayed its significance in his Principia Mathematica, published in 1687 sixty-eight years after the third law first appeared in print.

What the third law gives us is a direct mathematical relationship between the size of the orbits of the planets and their duration, which only works in a heliocentric system. In fact as we will see later it’s actually equivalent to the law of gravity. There is nothing comparable for either a full geocentric system or for a geo-heliocentric Tychonic or semi-Tychonic system. It should have hit the early seventeenth-century astronomical community like a bomb but it didn’t, which raises the question why it didn’t.

The main answer lies in Kepler’s own writings. Although he viewed its discovery as the crowning glory of his work on the Harmonices Mundi Kepler didn’t give it any prominence in that work. The Harmonices Mundi is a vast sprawling book explicating Kepler’s version of the Pythagorean theory of the harmony of the spheres in five books. After four introductory books covering plane geometry, music theory and astrology Kepler gets down to harmonic planetary theory in the fifth and final book. Book V, 109 pages in the English translations, contains lots of musical relationships between various aspects of the planetary orbits, with the third law presented as just one amongst the many with no particular emphasis. The third law was buried in what is now regarded as a load of unscientific dross. Or as Carola Baumgardt puts it, somewhat more positively,  in her Johannes Kepler life and letters (Philosophical Library, 1951, p. 124):

Kepler’s aspirations, however, go even much higher than those of modern scientific astronomy. As he tried to do in his “Mysterium Cosmographicum” he coupled in his “Harmonice Mundi” the precise mathematical results of his investigations with an enormous wealth of metaphysical, poetical, religious and even historical speculations. 

Although most of Kepler’s contemporaries would have viewed his theories with more sympathy than his modern critics the chances of anybody recognising the significance of the harmony law for heliocentric astronomical theory were fairly minimal.

The third law reappeared in 1620 in the second part of Kepler’s Epitome Astronomiae Copernicanae, a textbook of heliocentric astronomy written in the form of a question and answer dialogue between a student and a teacher.

How is the ratio of the periodic times, which you have assigned to the mobile bodies, related to the aforesaid ratio of the spheres wherein, those bodies are borne?

The ration of the times is not equal to the ratio of the spheres, but greater than it, and in the primary planets exactly the ratio of the 3/2th powers. That is to say, if you take the cube roots of the 30 years of Saturn and the 12 years of Jupiter and square them, the true ration of the spheres of Saturn and Jupiter will exist in those squares. This is the case even if you compare spheres that are not next to each other. For example, Saturn takes 30 years; the Earth takes one year. The cube root of 30 is approximately 3.11. But the cube root of 1 is 1. The squares of these roots are 9.672 and 1. Therefore the sphere of Saturn is to the sphere of the Earth as 9.672 is to 1,000. And a more accurate number will be produced, if you take the times more accurately.[3]

Here the third law is not buried in a heap of irrelevance but it is not emphasised in the way it should be. If Kepler had presented the third law as a table of the values of the orbit radiuses and the orbital times and their mathematical relationship, as below[4], or as a graph maybe people would have recognised its significance. However he never did and so it was a long time before the full impact of the third law was felt in astronomical community.

third law001

The real revelation of the significance of the third law came first with Newton’s Principia Mathematica. By the time Newton wrote his great work the empirical truth of Kepler’s third law had been accepted and Newton uses this to establish the empirical truth of the law of gravity.

In Book I of Principia, the mathematics and physics section, Newton first shows, in Proposition 11[5], that for a body revolving on an ellipse the law of the centripetal force tending towards a focus of the ellipse is inversely as the square of the distance: i.e. the law of gravity but Newton is not calling it that at this point. In Proposition 14[6] he then shows that, If several bodies revolve about a common center and the centripetal force is inversely as the square of the distance of places from the center, I say that the principal latera recta of the orbits are as the squares of the areas which bodies describe in the same time by radii drawn to the center. And Proposition 15[7]: Under the same supposition as in prop. 14, I say the square of the periodic times in ellipses are as the cubes of the major axes. Thus Newton shows that his law of gravity and Kepler’s third law are equivalent, although in this whole section where he deals mathematically with Kepler’s three laws of planetary motion he never once mentions Kepler by name.

Having established the equivalence, in Book III of The Principia: The System of the World Newton now uses the empirical proof of Kepler’s third law to establish the empirical truth of the law of gravity[8]. Phenomena 1: The circumjovial planets, by radii drawn to the center of Jupiter, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center. Phenomena 2: The circumsaturnian planets, by radii drawn to the center of Saturn, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center. Phenomena 3: The orbits of the five primary planets—Mercury, Venus, Mars, Jupiter, and Saturn—encircle the sun. Phenomena 4: The periodic times of the five primary planets and of either the sun about the earth or the earth about the sun—the fixed stars being at rest—are as the 3/2 powers of their mean distances from the sun. “This proportion, which was found by Kepler, is accepted by everyone.”

Proposition 1: The forces by which the circumjovial planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the center of Jupiter and are inversely as the squares of the distances of their places from that center. “The same is to be understood for the planets that are Saturn’s companions.” As proof he references the respective phenomena from Book I. Proposition 2: The forces by which the primary planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the sun and are inversely as the squares of the distances of their places from its center. As proof he references the respective phenomenon from Book I:

One of the ironies of the history of astronomy is that the general acceptance of a heliocentric system by the time Newton wrote his Principia was largely as a consequence of Kepler’s Tabulae Rudolphinae the accuracy of which convinced people of the correctness of Kepler’s heliocentric system and not the much more important third taw of planetary motion.

[1] Sesquialterate means one and a half times or 3/2

[2] The Harmony of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aiton, A.M. Duncan & J.V. Field, Memoirs of the American Philosophical Society Held at Philadelphia for Promoting Useful Knowledge, Volume 209, 1997 pp. 411-412

[3] Johannes Kepler, Epitome of Copernican Astronomy & Harmonies of the World, Translated by Charles Glenn Wallis, Prometheus Books, New York, 1995 p. 48

[4] Table taken from C.M. Linton, From Eudoxus to Einstein: A History of Mathematical Astronomy, CUP, Cambridge etc., 2004 p. 198

[5] Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy, A New Translation by I: Bernard Cohen and Anne Whitman assisted by Julia Budenz, Preceded by A Guide to Newton’s Principia, by I. Bernard Cohen, University of California Press, Berkley, Los Angeles, London, 1999 p. 462

[6] Newton, Principia, 1999 p. 467

[7] Newton, Principia, 1999 p. 468

[8] Newton, Principia, 1999 pp. 797–802


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“…like birds in the air and fish in the sea.”

In popular accounts of the transition from geocentric to heliocentric cosmology and astronomy it is often stated that in Aristotle’s geocentric cosmology the orbs of the planets were solid crystalline spheres, the existence of which were disproved when Tycho Brahe demonstrated that comets were not sublunar meteorological phenomena, as claimed by Aristotle, but supralunar astronomical objects. ‘Tycho’s comet of 1577 shattered Aristotle’s crystalline spheres’ is a common trope in such writings, but how true is it?

Aristotle’s cosmology divides the cosmos into the sublunar and supralunar spheres, i.e. below the moon and above the moon. The sublunar sphere, i.e. the earth, consists of the four elements earth, water, air and fire. The supralunar sphere consists of aether, the fifth element or quintessence. For his astronomy he adopted the homocentric planetary spheres of Eudoxus with the earth at their centre. As everything in the supralunar sphere consists of aether so do the planetary orbs. However Aristotle doesn’t actually say what aether is or what its qualities or characteristics are, it just is.

In his Mathēmatikē Syntaxis, Ptolemaeus adopted Aristotle’s cosmology putting it together with the deferent/epicycle model of the planetary orbits developed by Apollonius. He also offers no details as to the nature of aether. The pattern repeats itself with the astronomers of the Islamic Empire, who largely adopted the cosmology of Aristotle and the astronomy of Ptolemaeus without offering an explanation of the nature of aether.

It is first in the High Middle ages that the European, Christian, Aristotelian scholars first begin to ask about the nature of the aether and its properties; they were at least as motivated by the Bible as by the works of Aristotle and Ptolemaeus. The relevant Bible text is from the account of creation in Genesis:

6 And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters.

7 And God made the firmament and, and divided the waters which were under the firmament from the waters which were above the firmament: and it was so.

8 And God called the firmament Heaven.

Combining Aristotle’s aether and the waters above the firmament those European, Christian, Aristotelian scholars in the thirteenth century said that the planetary orbs must be fluid. However they were worried about the waters above the firmament would rain down on the earth so they thought of the orbs as having firm boundaries, like chocolate with soft centres or balloons full of water (my analogies not those of the medieval scholars). The orbs are described as solid (Latin: solidum), but this originally means that they are three-dimensional structures rather than flat disks and does not mean that they are hard.

During the fourteenth and fifteenth centuries the opinions changed and there developed a view that the orbs were not fluid but hard, however we are still far away from the crystalline spheres smashed by Tycho’s comet. In fact it is not actually known when they first appeared in the debate, although Tycho is convinced that they are propagated by his opponents. They don’t play any role in Copernicus’ astronomy so it is thought that they come into the debate somewhere between Copernicus and Tycho.

The story goes that following Tycho’s proof that the comet of 1577 was definitely supralunar the debate reverts to the possibility of fluid rather than hard planetary orbs but there are a couple of problems with this story line. Firstly, that comets were supralunar was being discussed well before Tycho’s 1577 measurement of cometary parallax. Already in the early fifteenth century Paolo dal Pozzo Toscanelli wrote a thesis in which he treated comets as astronomical phenomena and not meteorological ones. He didn’t publish his work but he did have contact with Georg Peuerbach and it can’t be just coincidence that Peuerbach and his pupil Regiomontanus also considered comets to be astronomical. Regiomontanus even wrote a paper on the problems of measuring the parallax of a moving comet, a paper that was discussed in correspondence between Tycho and John Dee. In the 1530’s there was a lively discussion on the supralunar nature of comets in which various notable European astronomers, including Copernicus, took part. When the comet of 1577 appeared it was observed very carefully and its parallax was measured by astronomers all over Europe exactly because of the earlier discussions. Although the results of those attempted measurements were hotly disputed by the various fractions, Tycho was by no means the only astronomer to determine that the comet was supralunar. The determination made by Kepler’s teacher, Michael Maestlin, probably had more impact on the debate than that of Tycho. The biggest impact, however, was made by Christoph Clavius, the leading Jesuit astronomer, who although by definition an Aristotelian scholar accepted that the comet was definitely supralunar.

If Clavius accepted that the comet was supralunar, and he did, how does this square with the fact that as a Jesuit he was required to follow a fairly strict Thomist, Aristotelian philosophy of nature. In fact this was less of a problem than one might imagine. Roberto Bellarmino, who would go on to become the most important Jesuit authority of the age, had already rejected the crystalline spheres before the appearance of the 1577 comet. In his astronomy lectures at the University of Leuven between 1570 and 1574 he taught that the whole deferent/epicycle model was just an abstract construct, which didn’t exist in reality and that the planets moved freely through a fluid medium “like birds in the air and fish in the sea.”

Tycho’s comet smashing Aristotle’s crystalline spheres is a nice story but a closer examination of the historical facts shows it to be just that a nice story but not really a true one.





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As easy as 1,2,3…

In every day life we all do our calculations, whether for the taxman, our purchases, paying the household bills or in some academic discipline, using the place value decimal number system. It consists of just ten symbols (numerals) – 1,2,3,4,5,6,7,8,9,0 – with which we can express any number, of any size that we may require. The value of the symbol changes according to its position – place – within the number that we write. This is an incredibly powerful and efficient method of writing numbers and the algorithms that it uses also make it a very efficient system for conducting calculations. The numerals are usually referred to as Arabic numerals or more correctly as Hindu-Arabic numerals because we Europeans inherited them and the entire system of how to use them from the Islamic Empire in the High Middle Ages, which in turn had inherited them from India in the Early Middle Ages, where they originated. In what follows I shall sketch the path that this number system took from India to medieval Europe, a path that has several twist and turns.

The history of the early development of the place value decimal number system is long, complicated and full of holes and I shan’t be dealing with it here. It also throws up some important and unanswered questions. The Babylonians developed a place value number system as early as the beginning of the second millennium BCE but it was a sexagesimal or base sixty number system rather than a decimal base ten one. The Babylonian system even had a placeholder zero in its later versions. This poses the question whether the Indians got the idea of a place value system from the Babylonians but it is simply not known. The Chinese also had a place value decimal number system but whether the Chinese influenced the Indians, the Indians the Chinese or both developed their systems independently is also not known.

There are three principle figures, who played a central role in the transmission of the place value decimal number system and the first of these is the Indian astronomer Brahmagupta (c.598–c.668 CE), who lived most of his life in Bhillamala (modern Bhinmal) in North-western India. He wrote his Brāhma-sphuṭa-siddhānta a treatise on astronomy written in verse, with 24 chapters and 1008 verses, in 628 CE. Writing scientific works in verse in ancient cultures was probably in order to make them easier to memorise in predominantly oral societies. Although an astronomical work Brahmagupta devotes several chapters to mathematics. Chapter 12 is devoted to arithmetic and introduces the basic arithmetical operations. In chapter eighteen he deals with negative numbers and with zero, not as a placeholder but as a number. He defines zero as that which results from subtracting a number from itself and gives the correct rules for addition, subtraction and multiplication with zero. Unfortunately he defines zero divided by zero as zero and gives a term for a number divided by zero without saying what the result would be. We, of course, now say division by zero is not defined. Brahmagupta’s use of zero as a number is the earliest known such use but this doesn’t mean that he invented zero as a number. His description suggests that this is already common usage. We know that zero as a number doesn’t appear in the astronomical text Aryabhatiya of Aryabhata (476–550 CE), which Brahmagupta criticises, so we can assume that zero as a number was developed in the period between the two works. The Brāhma-sphuṭa-siddhānta also contains description of what we would call algebra the details of which needn’t interest us here although we will meet them again. The Brāhma-sphuṭa-siddhānta was translated into Arabic in the eighth century CE and became one of the principle sources in the Islamic Empire for the Indian number system.

Our second principle figure is the eighth-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c.780–c.850 CE), who produced two works influenced by Brahmagupta’s Brāhma-sphuṭa-siddhānta, one on algebra and one on arithmetic. The more famous is his Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing) from which we get the word algebra (al-ğabr) and from his name we also get the term algorithm (a corruption of al-Khwārizmī). However it is his second work on arithmetic that interests us here. There is no known extant Arabic original of this work but it was translated into Latin in the twelfth century, possibly by Adelard of Bath,[1] under the title Algorithmo de Numero Indorum. This was the first introduction of the Hindu-Arabic numerals and the place value decimal number system into Europe. This introduction was realised at the early medieval universities, where the place value decimal number system was taught under the name algorism, as part of the discipline of computos, the calculation of the date of Easter, an important branch of mathematics at the Catholic universities. John of Sacrobosco wrote a widely read text book Algorismus aka De Arte Numerandi aka De Arithmetica in the early thirteenth century. However the use of the Hindu-Arabic numerals did not spread outside of the university.

In the Arabic world the books on algebra and arithmetic, and al-Khwārizmī’s were by no means the only ones, were largely aimed at merchants and traders. They were what we would term books on commercial arithmetic teaching bookkeeping, calculation of interest, calculation of profit shares in joint business ventures, division of property in testaments etc. and it is from this area that the Hindu-Arabic numbers and the place value decimal number system was finally introduced into Europe by the third of our principle figures Leonardo Pisano or Leonardo of Pisa (c.1175–c.1250).

Leonardo is more generally incorrectly known today by the name Fibonacci. This name, which translates as the son of Bonacci, was, however the creation of the French historian, Guilluame Libri in in 1838. Leonardo’s father Guilichmus or Guilielmo was a merchant who became a customs official. Bonacci was a general family name and not the name of his father his book the Liber Abbaci, to which we will turn shortly, starts:

Here begins the Book of Calculations

Composed by Leonardo Pisano, Family Bonacci

In the Year 1202

As with both Brahmagupta and al-Khwārizmī we know next to nothing about Leonardo personally, the only information that we have is in the introduction to the Liber Abbaci:

As my father was a public official away from our homeland in the Bugia [Now Béjaïa in Algeria] customshouse established for the Pisa merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me in the study of mathematics and to be taught for some days; there from a marvellous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learned from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily, and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learned from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method. Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle geometrical art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum. [i.e. with no knowledge of this method] If by chance, something less or more proper or necessary I omitted, your indulgence for me in entreated, as there is no one who is without fault, and in all things is altogether circumspect.

Leonardo obviously used numerous sources for his extensive book but, which sources he used is not known for certain; it is not even known if he could read Arabic and used original Arabic sources or whether he relied on the Latin translations that already existed. We do however know from textual analysis that he did use al-Khwārizmī’s book on algebra as one of his sources.

The Liber Abbaci is a book written by a merchant for merchants and it is as commercial arithmetic that the Hindu-Arabic numerals finally made it onto the big stage in medieval Europe. Abbaci, with two ‘bs’, and not one as it is often falsely written, comes from abbaco meaning to reckon or calculate in Italian and has nothing to do with abacus. Leonardo’s book might not have had the impact that it did if it had not appeared at roughly the same time as another innovation, double entry bookkeeping. The combination of the Hindu-Arabic numerals and double entry bookkeeping become the engine room to the so-called medieval economic revolution that saw the invention of banking and the rise of large scale international trading centred round the economic power house of Northern Italy. Leonardo’s book triggered a whole abbaco industry in Northern Italy.

To teach the new Indian arithmetic small abbaco schools (scuole d’abbaco or botteghe d’abbco) were established in the towns, where teenagers, who were apprentice traders or merchants, were taught commercial arithmetic and double entry bookkeeping. The teachers, who ran these establishments, maestri d’abbaco, wrote their own textbooks, a genre known as Libri d’abbaco, (abbacus books). The first ever printed mathematics book was an abbacus book, the so-called Treviso Arithmetic or Arte dell’Abbaco written in vernacular Venetian and published in Treviso in 1478. These schools and their textbooks spread to the trading cities of Southern Germany, such as Augsburg, Regensburg and Nürnberg, and from there throughout Europe. In German we have Rechenmeister, Rechenschule and Rechenbucher, in English reckoning masters, reckoning schools and reckoning books. Arithmetic and algebra remained in the province of the traders and merchants as commercial arithmetic until the middle of the fifteenth century. Gerolemo Cardano is credited with bringing algebra into the realm of mathematics with his Artis magnae, sive de regulis algebraicis liber unus published by Johannes Petreius in Nürnberg in 1545 but he also started his career as a mathematical author with an abbacus book, his Practica arithmetice et mensurandi singularis published in Milano in 1538.

The introduction of the Indian numerals into Northern Italy didn’t go entirely unopposed. In 1299 a local law was passed in Florence banning the use of them in bookkeeping, Statuto dell’Arte del Cambio, with the argument that they were easier to change, thus falsifying the accounts, than Roman numerals or written number words. Many modern authors claim that reckoning with the Hindu-Arabic numerals was faster and simpler than using the abacus or reckoning board but I don’t think this is true and I strongly suspect that most merchants continued to do their reckoning on a counting board reserving the new arithmetic for their written bookkeeping.

Leonardo was not just the man, who introduced the place value decimal number system into Europe with his Liber Abbaci, but was also the author of several other important mathematical works establishing him as an important mathematician in thirteenth-century Italy. In 1240 he was even invited to an audience with the Holy Roman German Emperor Frederick II, who was an avid patron of the sciences. The most famous judgement on the introduction of the place value decimal number system is that of the eighteenth-century French polymath Simon Laplace:

“It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”

It is Leonardo Pisano to whom we own our thanks for having introduced this invention into Europe. If you want to know more about the man and his book then I recommend Keith Devlin, The Man of Numbers: Fibonacci’s Arithmetic Revolution, Bloomsbury, London, 2011 from which the long quote from the Liber Abbaci is taken.

The theme of this post was requested by one of my anonymous €30 plus GoFundMe donors. It’s slightly different to what he suggested but I hope he’s satisfied with the end result. I wait for other donors to claim their right to negotiate a post theme.

[1] The secondary sources I have consulted say, unknown translator, probably Adelard of Bath, Robert of Chester (who definitely did translate the algebra) and John of Seville, so take your pick. Interestingly several of them name Adelard of Bath but my biography of Adelard says that the attribution is probably false.


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The influence of a Renaissance mathematics teacher.

Recent posts here have been all over the #histSTM map so I thought it was time to return to the roots and write something about a Renaissance mathematicus. It was first during the fifteenth century that Medieval European universities began to create dedicated chairs for the study of the mathematical disciplines — arithmetic, geometry, astrology, astronomy, surveying, cartography, designing and constructing sundials and mathematical instruments. The first such chair to be established in Germany was at the University of Ingolstadt in about 1470. Like its predecessors in Northern Italy this was principally a chair for teaching astrology and the mathematics and astronomy necessary to cast horoscopes to medical students. Those teaching in Ingolstadt, however, extended their activities to cover the full range of Renaissance mathematical studies. As well as producing medical students Ingolstadt also created full blood mathematical scholars, who would carry the seeds of mathematical studies to other towns and regions. One of those Ingolstadt seeds was Johannes Stöffler.


Contemporary Author’s Portrait Stöfflers from his 1534 published Commentary on the Sphaera of the Pseudo-Proklos (actually Geminos) Source: Wikimedia Commons

Johannes Stöffler was born in Blaubeuren in the Swabian Jura 10 December 1452. He received his first education in Blaubeuren and matriculated at the University of Ingolstadt on 21 April 1472 graduating BA in September 1473 and MA in January 1476. As many of the other contemporary mathematical scholars Stöffler entered a career in the Church rising to parish priest in Justingen in 1481. Parallel to his clerical work he became a highly active astrologer, astronomer, clock, globe and instrument maker. He was a very successful mathematicus and enjoyed a widespread good reputation. He constructed a, still extant, celestial globe for Daniel Zehender auxiliary Bishop of Konstanz in 1593, a clock for the Minster in Konstanz in 1596 and later another celestial globe Johann von Dalberg, Bishop of Worms. For Johannes Reuchlin, Germany’s leading Hebraist and prominent humanist scholar, he constructed an equitorium to determine the orbits of the Sun and the Moon.


Celestial Globe, Johannes Stöffler, 1493; Landesmuseum Württemberg Source: Wikimedia Commons

In 1507, the already fifty-five year old, Stöffler was appointed by Duke Ulrich I of Württemberg to the newly created chair of mathematics at the University of Tübingen. He extended his reputation as an instrument and globe maker as an academic with a successful series of technical publications.


Portrait of Johannes Stöffler produced for the Tübingen Professors’ Gallery 1614 Source: Wikimedia Commons

When he set up the first scientific publishing house in Nürnberg, Regiomontanus’ most successful publication was his ephemerides, sets of tables enabling the user to determine the position of the planets at any given time. Produced principally for astrologers they were also useful for astronomers, navigators and cartographers. There had been earlier manuscript ephemerides but Regiomontanus’ were the first printed ones and were distinguished from earlier ones by their high level of accuracy, leading to many pirated editions. Ephemerides are only calculated for a given number of years and Stöffler, together with the Ulm parish priest Jakob Pflaum, extended Regiomontanus’ ephemerides to 1531 and in a later posthumously published edition to 1551. The Regiomontanus/Stöffler/Pflaum ephemerides dominated the market and established Stöffler and Pflaum as the leading astrologers of the age.

In 1512 Stöffler published a text on the construction and use of the astrolabe, Elucidatio fabricae ususque astrolabii, which went through 16 editions up to 1620 and was highly influential.


The leading astrolabe maker in the early sixteenth century was the Nürnberger Georg Hartmann, who was probably the first instrument maker to mass-produce astrolabes in series. It has been shown that Hartmann’s work was based on Stöfffler’s book.


Astrolabe from Georg Hartmann, Yale Source: Wikimedia Commons

As university teacher Stöffler exercised a major influence on his student the most famous of which were Sebastian Münster and Philipp Melanchthon. From 1514 to 1518 Sebastian Münster, already a fan of the Renaissance mathematical sciences, studied under Stöffler.

Later Münster would publish his Cosmographia in the publishing house of his step-son Heinrich Petri in Basel. The Cosmographia, “a description of the whole world with everything it contain”, an atlas but so much more was the biggest selling book of the sixteenth century. In an age where the edition of a book was usually counted in hundred the Cosmographia is estimated to have sold in excess of 120,000 in its German and Latin editions over a period of about one hundred years.


Title page of the Cosmographia first edition Source: Wikimedia Commons

Stöffler’s biggest influence on the history of mathematics was, without doubt through Philipp Melanchthon. Melanchthon a nephew of Reuchlin was something of a child prodigy.


Portrait of Philipp Melanchthon from an oil painting on wood by Lucas Cranach d. Ä. 1543 Source: Wikimedia Commons

He entered the University of Heidelberg in 1509 and graduated BA in 1511 just thirteen years old. In 1512 he changed to the University of Tübingen, where he came under the influence of Stöffler. Under Stöffler he studied the mathematical disciplines and became a passionate supporter of the art of astrology. He graduated MA in 1514. In 1518 he, just twenty-one years old, was appointed professor of Greek, on Reuchlin’s recommendation, at the University of Wittenberg.

In Wittenberg Melanchthon became Luther’s friend and supporter and during the Reformation as Luther’s “Præceptor Germaniae” (Germany’s schoolmaster) he was charged with designing, organising and establishing the new Lutheran Protestant education system. Melanchthon now had the chance to promote his love of astrology won as a student of Stöffler. Melanchthon established chairs for mathematics on all of the new Protestant Gymnasia (high schools) and university, choosing the ablest mathematical scholars available to fill the new positions. Thus Johannes Schöner became professor for mathematics on the new gymnasium in Nürnberg and Georg Joachim Rheticus and Erasmus Reinhold the mathematics professors in Wittenberg. Melanchthon’s aim was to produce new generations of professionally educated astrologers. Through his actions the Protestant education system became an active supporter of the mathematical sciences at a time when they were largely neglected within the Catholic education system. Melanchthon’s system would go on to produce many leading sixteenth century mathematical practitioners.

Stöffler is a good example of a Renaissance mathematicus who tends not to feature in the mainstream history of mathematics but who from the second row behind the big names still had a major influence on the evolution of the discipline through various channels.

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