Category Archives: History of Astronomy

Christoph and the Calendar

The first substantive history of science post that I wrote on this blog was about the Jesuit mathematician and astronomer Christoph Clavius. I wrote this because at the time I was preparing a lecture on the life and work of Clavius to be held in his hometown Bamberg. Clavius is one of my local history of science celebrities and over the years I have become the local default Clavius expert and because of his involvement in the Gregorian calendar reform of 1572 I have also become the local default expert on that topic too.

Christoph Clavius

Christoph Clavius

All of this means that I have become very sensitive to incorrect statements about either Clavius or the Gregorian calendar reform and particularly sensitive to false statements about Clavius’ involvement in the latter. Some time back the Atlas Obscura website had a ‘time week’ featuring a series of blog post on the subject of time one of which, When The Pope Made 10 Days Disappear, was about the Gregorian calendar reform and contained the following claim:

The new lead astronomer on the project, Jesuit prodigy Christopher Clavius, considered this and other proposals for five years.

The brief statement contains three major inaccuracies, the most important of which, is that Clavius as not the lead astronomer, or lead anything else for that matter, on the project. This is a very widespread misconception and one to which I devote a far amount of time when I lecture on the subject, so I thought I would clear up the matter in a post. Before doing so I would point out that I have never come across any other reference to Clavius as a prodigy and there is absolutely nothing in his biography to suggest that he was one. That was the second major inaccuracy for those who are counting.

Before telling the correct story we need to look at the wider context as presented in the article before the quote I brought above we have the following:

A hundred years later, Pope Gregory XIII rolled up his sleeves and went for it in earnest. After a call for suggestions, he was brought a gigantic manuscript. This was the life’s work of physician Luigi Lilio, who argued for a “slow 10-day correction” to bring things back into alignment, and a new leap year system to keep them that way. This would have meant that years divisible by 100 but not by 400 (e.g. 1800, 1900, and 2100) didn’t get the extra day, thereby shrinking the difference between the solar calendar and the Earthly calendar down to a mere .00031 days, or 26 seconds.

Luigi LIlio Source: Wikimedia Commons

Luigi LIlio
Source: Wikimedia Commons

This is correct as far as it goes, although there were two Europe wide appeals for suggestions and we don’t actually know how many different suggestions were made as the relevant documents are missing from the Vatican archives. It should also be pointed out the Lilio was a physician/astronomer/astrologer and not just simply a physician. Whether or not his manuscript was gigantic is not known because it no longer exists. Having decided to consider Lilio’s proposal this was not simply passed on to Christoph Clavius, who was a largely unknown mathematicus at the time, which should be obvious to anybody who gives more than five minutes thought to the subject.

The problem with the calendar, as far as the Church was concerned, was that they were celebrating Easter the most important doctrinal festival in the Church calendar on the wrong date. This was not a problem that could be decided by a mere mathematicus, at a time when the social status of a mathematicus was about the level of a bricklayer, it was far too important for that. This problem required a high-ranking Church commission and one was duly set up. This commission did not consider the proposal for five years but for at least ten and possibly more, again we are not sure due to missing documents. It is more than likely that the membership of the commission changed over the period of its existence but because we don’t have the minutes of its meetings we can only speculate. What we do have is the signatures of the nine members of the commission who signed the final proposal that was presented to the Pope at the end of their deliberations. It is to these names that we will now turn our attention.

The names fall into three distinct groups of three of which the first consists of the high-ranking clerics who actually lead this very important enquiry into a fundamental change in Church doctrinal practice. The chairman of the committee was of course a cardinal,Guglielmo Sirleto (1514–1584) a distinguished linguist and from 1570 Vatican librarian.

Cardinal Guglielmo Sirleto Source: Wikimedia Commons

Cardinal Guglielmo Sirleto
Source: Wikimedia Commons

The vice chairman was Bishop Vincenzo Lauro (1523–1592) a Papal diplomat who was created cardinal in 1583. Next up was Ignatius Nemet Aloho Patriarch of Antioch and head of the Syriac Orthodox Church till his forced resignation in 1576. Ignatius was like his two Catholic colleagues highly knowledgeable of astronomy and was brought into the commission because of his knowledge of Arabic astronomy and also to try to make the reform acceptable to the Orthodox Churches. The last did not function as the Orthodox Churches initially rejected the reform only adopting it one after the other over the centuries with the exception of the Russian Eastern Orthodox Churches, whose church calendar is still the Julian one, which is why they celebrate Christmas on 6 7 January.

Our second triplet is a mixed bag. First up we have Leonardo Abela from Malta who functioned as Ignatius’ translator, he couldn’t speak Latin, and witnessed his signature on the commissions final report. He is followed by Seraphinus Olivarius an expert lawyer, whose role was to check that the reform did not conflict with any aspects of cannon law. The third member of this group was Pedro Chacón a Spanish mathematician and historian, whose role was to check that the reform was in line with the doctrines of the Church Fathers.

Our final triplet consists of what might be termed the scientific advisors. Heading this group is Antonio Lilio the brother of Luigi and like his brother a physician and astronomer. He was here to elucidate Luigi’s plan, as Luigi was already dead. The lead astronomer, to use the Atlas Obscura phase, was the Dominican monk Ignazio Danti (1536–1582) mathematician, astronomer, cosmographer, architect and instrument maker.

Ignazio Danti Source: Wikimedia Commons

Ignazio Danti
Source: Wikimedia Commons

In a distinguished career Danti was cosmographer to Cosimo I, Duke of Tuscany whilst professor of mathematics at the university of Pissa, professor of mathematics at the University of Bologna and finally pontifical mathematicus in Rome. For the Pope Danti painted the Gallery of Maps in the Cortile del Belvedere in the Vatican Palace and deigned and constructed the instruments in the Sundial Rome of the Gregorian Tower of Tower of Winds above the Gallery of Maps.

Map of Italy, Corsica and Sardinia - Gallery of Maps - Vatican Museums. Source: Wikimedia Commons

Map of Italy, Corsica and Sardinia – Gallery of Maps – Vatican Museums.
Source: Wikimedia Commons

After the calendar reform the Pope appointed him Bishop of Altari. Danti was one of the leading mathematical practitioners of the age, who was more than capable of supplying all the scientific expertise necessary for the reform, so what was the role of Christoph Clavius the last signer of the commission’s recommendation.

The simple answer to this question is that we don’t know; all we can do is speculate. When Clavius (1538–1612) first joined the commission he was, in comparison to Danti, a relative nobody so his appointment to this high level commission with its all-star cast is somewhat puzzling. Apart from his acknowledged mathematical skills it seems that his membership of the Jesuit Order and his status as a Rome insider are the most obvious reasons. Although relative young the Jesuit Order was already a powerful group within the Church and would have wanted one of theirs in such a an important commission. The same thought concerns Clavius’ status as a Rome insider. The Church was highly fractional and all of the other members of the commission came from power bases outside of Rome, whereas Clavius, although a German, as professor at the Collegio Romano counted as part of the Roman establishment, thus representing that establishment in the commission. It was probably a bit of all three reasons that led to Clavius’ appointment.

Having established that Clavius only had a fairly lowly status within the commission how did the very widespread myth come into being that he was somehow the calendar reform man? Quite simply after the event he did in fact become just that.

When Pope Gregory accepted the recommendations of the commission and issued the papal bull Inter gravissimas on 24 February 1582, ordering the introduction of the new calendar on 4 October of the same year,

Inter-grav

he granted Antonio Lilio an exclusive licence to write a book describing the details of the calendar reform and the modifications made to the process of calculating the date of Easter. The sales of the book, which were expected to be high, would then be the Lilio family’s reward for Luigi Lilio having created the mathematical basis of the reform. Unfortunately Antonio Lilio failed to deliver and after a few years the public demand for a written explanation of the reform had become such that the Pope commissioned Clavius, who had by now become a leading European astronomer and mathematician, to write the book instead. Clavius complied with the Pope’s wishes and wrote and published his Novi calendarii romani apologia, Rome 1588, which would become the first of a series of texts explaining and defending the calendar reform. The later was necessary because the reform was not only attacked on religious grounds by numerous Protestants, but also on mathematical and astronomical grounds by such leading mathematicians as François Viète and Michael Maestlin. Over the years Clavius wrote and published several thousand pages defending and explicating the Gregorian calendar reform and it is this work that has linked him inseparably with the calendar reform and not his activities in the commission.

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Filed under History of Astronomy, History of Mathematics, History of science, Local Heroes, Renaissance Science

The orbital mechanics of Johann Georg Locher a seventeenth-century Tychonic anti-Copernican

Our favourite guest blogger Chris Graney is back with a question. Busy translating the Disquisitiones mathematicae de controversis et novitatibus astronomicis (1614) of Johann Georg Locher, a student of Christoph Scheiner at the University of Ingolstadt, he came across a fascinating theory of orbital mechanics, which he outlines in this post. Chris’s question is how does this theory fit in with seventeenth-century force dependent orbital theories? Read the post and enlighten Chris with your history of astronomy wisdom!

Did Johann Georg Locher write something very interesting in 1614 about how the Earth could orbit the Sun under the influence of gravity? I am hoping that the RM and his many readers might be able to weigh in on this.

Who is Locher? He is the author of the 1614 Disquisitiones Mathematicae (Mathematical Disquisitions), an anti-Copernican book known primarily because Galileo made sport of it within his Dialogue Concerning the Two Chief World Systems: Ptolemaic and Copernican. It is the “booklet of theses, which is full of novelties” that Galileo has the anti-Copernican Simplicio drag out in order to defend one or another wrong-headed idea. Galileo describes the booklet’s author as producing arguments full of “falsehoods and fallacies and contradictions,” as “thinking up, one by one, things that would be required to serve his purposes, instead of adjusting his purposes step by step to things as they are,” and as being excessively bold and self-confident, “setting himself up to refute another’s doctrine while remaining ignorant of the basic foundations upon which the greatest and most important parts of the whole structure are supported.” As far as I can tell, little is known about Locher himself other than what he says in his book: he was from Munich; he studied at Ingolstadt under the Jesuit astronomer Christopher Scheiner. This is the same Scheiner who Galileo debated regarding sunspots. Some writers treat the Disquisitions as Scheiner’s work.

I became better acquainted with the Disquisitions through Dennis Danielson’s work on Milton, in which it plays a part. This prompted me to look at Locher’s work directly. Then I discovered that Locher wields Tycho Brahe’s star size argument against Copernicus, that he illustrates the Disquisitions lavishly, and that the Disquisitions is short. So I decided to read and translate it cover-to-cover.

The Disquisitions turns out to be fascinating. It is nothing like what one might expect from reading the Dialogue. And among the gems within it is this thing that Locher thinks up:

Imagine an L-shaped rod, buried in the Earth, with a heavy iron ball attached to it, as shown in the left-hand figure below. The heaviness or gravity of the ball (that is, its action of trying to reach its natural place at the center of the universe—in 1614 Newtonian physics was many decades in the future; Aristotelian physics was the rule) presses down on the rod, but the rigidity of the rod keeps the ball from falling.

Now imagine the rod being hinged at the Earth’s surface (at point A in the right-hand figure below). The heaviness of the ball will now cause the rod to pivot about the hinge. The ball will fall along an arc of a circle whose center is A, striking the Earth at B.

Fig1

Now imagine the Earth is made smaller relative to the rod. The same thing will still occur—the rod pivots; the iron ball falls in a circular arc (below left). If the Earth is imagined to be smaller still, the rod will be what hits the ground, not the ball (below right), so the ball stops at C, but the ball still falls in a circular arc whose center is A.

Fig2

If you imagine the Earth to be smaller and smaller, the ball still falls, driven by its gravity, in a circular arc (below). You can see where Locher is going! He is thinking his way toward a limiting case.

Fig3

At last Locher says to imagine the rod to be pivoting on the center of the universe itself—the Earth vanishing to a point. Surely, he says, in this situation, a complete and perpetual revolution will take place around that same pivot point A (fiet reuolutio integra & perpetua circa idem A).

Fig4

Now, he says, we have demonstrated that perpetual circular motion of a heavy body is possible. And if we imagine the Earth in the place of the iron ball, suspended over the center of the universe, now we have a thought experiment (cogitatione percipi possit—it may be able to be perceived by thought) that shows how the Earth might be made to revolve about that center (and about the sun, which would be at the center in the Copernican system). But this sort of thing does not exist, he says, and if it did exist, it would not help the Copernicans any, because no phenomena are saved—that is, no observations are explained—by means of it.

Below is Locher’s sketch of this. Curves MN, OP, and QR are the surface of the Earth, being imagined smaller and smaller. S is the iron ball. A is the center of the universe. Circle CHIC is the path of the orbiting ball.

Fig5

So it seems that in 1614 an anti-Copernican—a student of one of Galileo’s adversaries—proposed a mechanism to explain the orbit of the Earth, and that mechanism involved a fall under a central force. This is not the Newtonian explanation of Earth’s orbit, but it does have significant elements in common with Newton. And, Locher was definitely an anti-Copernican. Indeed, while he illustrates telescopic discoveries such as the phases of Venus, and states that the telescope shows that the world is structured according to the Tychonic system (sun, moon, and stars circle Earth, planets circle sun), he clearly rejects Copernicus—on the grounds of the star size problem (and the Copernican tendency to invoke the Creator’s majesty to get around that problem) and on the grounds that a moving Earth grossly complicates the motions of bodies moving over its surface.

Fig6

The history of orbital mechanics is not my bailiwick, so I ask RM readers whether they think Locher is a “first”? Is this really as interesting as it seems to me? Or do RM readers know of others who proposed the “an orbit is a fall under a central force” idea prior to Locher? Whether I search in English or in Latin I can find neither primary nor secondary sources that discuss the Disquisitions’ treatment of orbits, nor can I find primary or secondary sources that discuss orbits and central forces in general prior to the late-seventeenth century. In fact, I can find little written on the Disquisitions itself (outside of its role in the Dialogue), and what I have found typically conflicts with what is actually in the Disquisitions (for example, one author describes the Disquisitions as a book “in which the proponents of Earth’s motion were violently attacked,” but actually Locher’s worst words are for Simon Marius, a fellow supporter of the Tychonic system, while his most favorable words are for Galileo). But many of you are much more well-read than I am.

My searches did turn up one interesting item, however. Locher uses the term forced suspension to describe what is going on in an orbit (motus huius continui caussa est violenta suspensio—the cause of this continuing motion is forced suspension) and I have found that term in what appears to be another seventeenth-century Jesuit’s commentary on the work of Thomas Aquinas.

With luck the translation of Disquisitions will be published in a year or so.

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Filed under History of Astronomy, History of science

The Arch-Humanist

The name Conrad Celtis is not one that you’ll find in most standard books on the history of mathematics, which is not surprising as he was a Renaissance humanist scholar best known in his lifetime as a poet. However, Celtis played an important role in the history of mathematics and is a good example of the fact that if you really wish to study the evolution of the mathematical sciences it is necessary to leave the narrow confines of the mathematics books.

Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507 Source: Wikimedia Commons

Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507
Source: Wikimedia Commons

Born Konrad Bickel or Pyckell, (Conrad Celtis was his humanist pseudonym) the son of a winemaker, in Franconian Wipfield am Main near Schweinfurt on 1 February 1459, he obtained his BA at the University of Cologne in 1497. Unsatisfied with the quality of tuition in Cologne he undertook the first of many study journeys, which typified his life, to Buda in 1482, where he came into contact with the humanist circle on the court of Matthias Corvinus, the earlier patron of Regiomontanus. 1484 he continued his studies at the University of Heidelberg specialising in poetics and rhetoric, learning Greek and Hebrew and humanism as a student of Rudolf Agricola, a leading Dutch early humanist scholar. Celtis obtained his MA in 1485. 1486 found him underway in Italy, where he continued his humanist studies at the leading Italian universities and in conversation with many leading humanist scholars. Returning to Germany he taught poetics at the universities of Erfurt, Rostock and Leipzig and on 18 April 1487 he was crowned Poet Laureate by Emperor Friedrich III in Nürnberg during the Reichstag. In Nürnberg he became part of the circle of humanists that produced the Nürnberger Chronicle to which he contributed the section on the history and geography of Nürnberg. It is here that we see the central occupation of Celtis’ life that brought him into contact with the Renaissance mathematical sciences.

During his time in Italy he suffered under the jibes of his Italian colleges who said that whilst Italy had perfect humanist credentials being the inheritors of the ancient Roman culture, Germany was historically a land of uncultured barbarians. This spurred Celtis on to prove them wrong. He set himself the task of researching and writing a history of Germany to show that its culture was the equal of Italy’s. Celtis’ concept of history, like that of his Renaissance contemporaries, was more a mixture of our history and geography the two disciplines being regarded as two sides of the same coin. Geography being based on Ptolemaeus’ Geographia (Geographike Hyphegesis), which of course meant cartography, a branch of the mathematical sciences.

Continuing his travels in 1489 Celtis matriculated at the University of Kraków specifically to study the mathematical sciences for which Kraków had an excellent reputation. A couple of years later Nicolaus Copernicus would learn the fundamentals of mathematics and astronomy there. Wandering back to Germany via Prague and Nürnberg Celtis was appointed professor of poetics and rhetoric at the University of Ingolstadt in 1491/92. Ingolstadt was the first German university to have a dedicated chair for mathematics, established around 1470 to teach medical students astrology and the necessary mathematics and astronomy to cast a horoscope. When Celtis came to Ingolstadt there were the professor of mathematics was Andreas Stiborius (born Stöberl 1464–1515) who was followed by his best student Johannes Stabius (born Stöberer before 1468­–1522) both of whom Celtis convinced to support him in his cartographic endeavours.

In 1497 Celtis received a call to the University of Vienna where he established a Collegium poetarum et mathematicorum, that is a college for poetry and mathematics, with Stiborius, whom he had brought with him from Ingolstadt, as the professor for mathematics. In 1502 he also brought Stabius, who had succeeded Stiborius as professor in Ingolstadt, and his star student Georg Tanstetter to Vienna. Stiborius, Stabius and Tanstetter became what is known, to historians of mathematics, as the Second Viennese School of Mathematics, the First Viennese School being Johannes von Gmunden, Peuerbach and Regiomontanus, in the middle of the fifteenth century. Under these three Vienna became a major European centre for the mathematical sciences producing many important mathematicians the most notable being Peter Apian.

Although not a mathematician himself Conrad Celtis, the humanist poet, was the driving force behind one of the most important German language centres for Renaissance mathematics and as such earns a place in the history of mathematics. A dedicated humanist, wherever he went on his travels Celtis would establish humanist societies to propagate humanist studies and it was this activity that earned him the German title of Der Erzhumanist, in English the Arch Humanist. Celtis died in 1508 but his Collegium poetarum et mathematicorum survived him by twenty-two years, closing first in 1530

 

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Filed under History of Astrology, History of Astronomy, History of Cartography, History of Mathematics, Renaissance Science

The greatest villain in the history of science?

In the popular version of the so-called astronomical revolution Andreas Osiander, who was born on the 19th December 1496 or 1498, is very often presented as the greatest villain in the history of science because he dared to suggest in the ad lectorum (to the reader) that he added to the front of Copernicus’ De revolutionibus that one could regard the heliocentric hypothesis as a mere mathematical model and not necessarily a true representation of the cosmos. Is the judgement of history just and who was Andreas Osiander anyway?

Andreas Osiander portrait by Georg Pencz Source: Wikimedia Commons

Andreas Osiander portrait by Georg Pencz
Source: Wikimedia Commons

Andreas Osiander was born in Gunzenhausen, a small town to the south of Nürnberg, the son of Endres Osannder a smith and Anna Herzog. His father was also a local councillor, who later became mayor. He entered the University of Ingolstadt in 1515 where he, amongst other things, studied Hebrew under Johannes Reuchlin one of the greatest humanist scholars in Germany at that time, the great uncle from Philipp Melanchthon and the leading Hebrew scholar of the age. In 1520 Osiander was ordained a priest and called to Nürnberg to teach Hebrew at the Augustinian Cloister. This had been a major centre for reformatory debate for a number of years and it is here that Osiander became a religious reformer. In 1522 he was appointed preacher at the Saint Lorenz church in Nürnberg and became the leading voice for religious reform in the city. In 1525 Nürnberg, a city-state, became the first state to officially adopt the Lutheran Protestant religion, and Osiander became a highly influential and powerful figure. He was largely responsible for converting Albrecht of Prussia to Protestantism and also had a major influence on Thomas Cranmer, later Archbishop of Canterbury and author of the Common Book of Prayer. A trivial pursuits fact is that Cranmer married one of Osiander’s nieces.

Osiander’s first links with the printer/publisher Johannes Petreius was as the author of polemical religious tracts, which Petreius published. How he became an editor for Petreius is not know. It is also not known when and where Osiander developed his interest in and knowledge of the mathematical sciences. What is certain is that it was Osiander who, after Petreius had discovered Cardano’s books at the book fair in Frankfurt, who wrote to the Italian mathematician/physician/philosopher on Petreius’ behalf offering to publish his books in Germany; an offer that Cardano was more than willing to accept. Osiander then became the editor of those books of Cardano’s that Petreius published over the years; a service for which Cardano thanks him very warmly in the preface to one of his books, praising him highly for his abilities as an editor.

When Rheticus published his account of Copernicus’ heliocentric astronomy in his Narratio Prima, in the form of an open letter addressed to Johannes Schöner, another of Petreius’ editors, it was Osiander who wrote to Rheticus on behalf of the publishing house showing great interest in the cyclical astrological theory of history outlined by Rheticus in his little book.

After Rheticus had brought the manuscript of De revolutionibus to Nürnberg, Philipp Melanchthon pressured him to take up the professorship for mathematics in Leipzig and Osiander took over the task of seeing the text through the press. It is here that Osiander added the ad lectorum to the finished book, which has, over the centuries, pulled down so much odium on his head. Is this harsh judgement of his actions justified or have we, as I believe, been blaming the wrong man for the last four and a half centuries.

In the early days of printing there was no such thing as authors rights. The rights to a book lay with the printer/publisher, who was also the first port of call should the authorities decide that a book or pamphlet was seditious, blasphemous or in any other way unacceptable. And please remember our concepts of freedom of speech simply did not exist in sixteenth-century Europe. The ad lectorum was added to De revolutionibus certainly with Petreius’ knowledge and almost certainly at his instigation. This is confirmed by his reaction as Copernicus’ friend Bishop Tiedemann Giese complained to the city council of Nürnberg about the inclusion of the ad lectorum in his dead friend’s magnum opus. Consulted by the council on the subject Petreius basically flew off the handle and told them to get stuffed, it was his book and he’d put what the hell he liked in it.

Osiander continued to edit the books of Cardano for Petreius but in 1548 the city of Nürnberg accepted the Augsburg Interim an edict issued by Charles V, Holy Roman Emperor, who had just won a decisive victory against the Protestant forces, forcing the Protestant states within the Empire to revert to Catholicism. In a moonlight flit Osiander fled the city of Nürnberg and made his way north to Königsberg, where Albrecht appointed him professor of theology at the newly established university. This caused much bad blood, as Osiander was not a qualified theologian. In this position Osiander became embroiled in a major theological dispute with the supporters of Melanchthon in Wittenberg over the doctrine of justification. This dispute is known in German church history as the Osiander Dispute and led to a schism between the two parties, with Osiander basically forming his own branch of Protestantism.

Osiander died in 1552 a controversial figure both in the history of religion and the history of science. However as I have sketched above I think his bad reputation in the history of science is not really justified and the real villain of the piece, if there is one, is Johannes Petreius. I say if there is one, because many historians are of the opinion that the ad lectorum saved the De revolutionibus from being condemned straight away, when it was published, and allowed the heliocentric hypothesis it contained to spread relatively unhindered and become established.

 

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Filed under Early Scientific Publishing, History of Astronomy, History of science, Renaissance Science

Mensis or menstruation?

I recently stumbled upon this rather charming rant by Anglo-Danish comedian, writer, broadcaster and feminist Sandi Toksvig.

Women's Calendar

 

Now I’m a very big fan of Ms Toksvig and was very sad when she retired as presenter of BBC Radio 4’s excellent News Quiz, so I don’t want to give the impression that I’m trying to put her down, but if she had know a little bit more about the early history of the calendar then she might not have jumped to the conclusion that this supposed bone calendar must have been made by a woman.

Before I start to explain why Ms Toksvig might be mistaken in her assumption that this purported primitive calendar came from the hands of a woman I would like to waste a few words on all such artefacts. There are a number of bone and stone objects of great antiquity bearing some number of scratches, incisions, notches, indentations or other forms of apparent marking and someone almost always comes along and declares them to be purposely created mathematical artefacts with one or other function. I must say that being highly sceptical by nature I treat all such claims with more than a modicum of wariness. Even assuming that the markings were made by a human hand might they not have been made in an idle moment by a Neolithic teenager trying out his newly acquired flint knife or in the case of our incised bone by an early musician making himself scraper to accompany the evening camp fire sing-a-long? What I’m am saying is that there are often multiple possible explanations for the existence of such marked artefacts and regarding them as signs of some sort of mathematical activity is only one of those possibilities.

However, back to Ms Toksvig and her revelation. She is of course assuming that the twenty-eight incisions are the result of a women counting off the days between her periods, the menstrual cycle being roughly twenty-eight days for most women. Now if Ms Toksvig had taken her thoughts a little further she might have realised that the word menstruation derives from the Latin word for month, which is mensis: a month being originally a lunar month which, depending on how you measure it, has approximately twenty-eight days. In fact much human thought has been expended over the centuries over the fact that a lunar month and the menstrual circle have the same length.

What we have here with this incised bone could well be not a menstrual record, as Ms Toksvig seems to assume, but a mensis record or part of a lunar calendar. This supposition is lent credence by the fact that, with the very notable exception of the ancient Egyptian calendar, all early cultures and civilisations had lunar and not solar calendars including the ancient Romans before Gaius Julius Caesar borrowed the Egyptian solar calendar, the forerunner of our own Gregorian one.

Assuming that the archaeologist or anthropologist who decided that said bone was a primitive calendar was right and it is not the idle whittling of some bored stone-age teenager, we of course still have no idea whether it was the work of a man or a woman.

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A misleading book title that creates the wrong impression

A new biography of Johannes Kepler has just appeared and although I haven’t even seen it yet, let alone read it, it brings out the HistSci Hulk side of my personality. What really annoys me on David Love’s book, Kepler and the Universe[1], is the title or rather the subtitle, How One Man Revolutionised Astronomy. Now, I for one have for many years conducted a private campaign to persuade people not to claim that we live in a Copernican Cosmos, a standard cliché, but that we live in a Keplerian Cosmos, because it was the very different elliptical system of Kepler that helped heliocentricity to its breakthrough and not the system of Copernicus. However Love’s subtitle immediately evokes the spectre of the lone genius and for all his undoubted brilliance Kepler was not a lone genius and especially not in terms of his cosmology/astronomy.

A 1610 portrait of Johannes Kepler by an unknown artist Source: Wikipedia Commons

A 1610 portrait of Johannes Kepler by an unknown artist
Source: Wikipedia Commons

Even a cursory examination of Kepler’s road to his system will immediately reveal his intellectual debts and his co-conspirators, both willing and unwilling. First off is naturally Copernicus himself. Kepler did not conceive a heliocentric system from scratch but was, on his own admission a glowing admirer or even acolyte of the Ermländer scholar. This admiration is one of the principle reasons that we don’t truly acknowledge Kepler’s achievement but tend to dismiss it as having just dotted the ‘Is’ and crossed the ‘Ts’ in Copernicus’ system, a demonstrably false judgement. Kepler, of course, didn’t help the situation when he titled the most simple and readable version of his system, and the one that together with the Rudolphine Tables had the most influence, the Epitome Astronomiae Copernicanae. Not a smart move! Whatever, we are already at two men who revolutionised astronomy.

Nicolaus Copernicus 1580 portrait (artist unknown) in the Old Town City Hall, Toruń Source: Wikimedia Commons

Nicolaus Copernicus 1580 portrait (artist unknown) in the Old Town City Hall, Toruń
Source: Wikimedia Commons

Kepler did not discover Copernicus himself but was introduced to him by his teacher Michael Maestlin at the University of Tübingen. Usually Maestlin gets mentioned in passing as Kepler’s teacher and then forgotten but he played a very important role in Kepler’s early development. In reality Maestlin was himself one of the leading European astronomers and mathematicians in the latter part of the sixteenth century, as well as being by all accounts an excellent teacher. He was also one of the very few supporters of both Copernican astronomy and cosmology. This meant that he gave Kepler probably the best foundation in the mathematical sciences that he could have found anywhere at the time, as well as awakening his interest in Copernican thought. It was also Maestlin who decided Kepler would be better off becoming a teacher of mathematics and district mathematician rather than training for the priesthood; a decision that Kepler only accepted very, very reluctantly. Even after he had left Tübingen Maestlin continued to support the young Kepler, although he would withdraw from him in later years. Maestlin edited, corrected and polished Kepler’s, so important, first publication, the Mysterium Cosmographicum. In fact Maestlin’s contributions to the finished book were so great he might even be considered a co-author. Some people think that in later life Kepler abandoned the, for us, rather bizarre Renaissance hypothesis of the Cosmographicum, but he remained true to his initial flash of inspiration till the very end, regarding all of his later work as just refinements of that first big idea. Maestlin’s contribution to the Keplerian system was very substantial. And then there were three.

Michael Maestlin Source: Wikimedia Commons

Michael Maestlin
Source: Wikimedia Commons

Tycho! Without Tycho Brahe there would be no Keplerian System. Tycho and Kepler are the Siamese twins of elliptical astronomy joined at the astronomical data. Without Tycho’s data Kepler could never have built his system. This duality is recognised in many history of astronomy texts with the two, so different, giants of Renaissance astronomy being handled together. The popular history of science writer, Kitty Ferguson even wrote a dual biography, Tycho and Kepler, The Unlikely Partnership that Forever Changed our Understanding of the Heavens[2], a title that of course contradicts Love’s One Man. Her original title was The Nobleman and His Housedog, with the rest as a subtitle, but it seems to have been dropped in later editions of the book. The ‘housedog’ is a reference to Kepler characterising himself as such in the horoscope he wrote when he was twenty-five years old.

Portrait of Tycho Brahe (1596) Skokloster Castle Source: Wikimedia Commons

Portrait of Tycho Brahe (1596) Skokloster Castle
Source: Wikimedia Commons

Tycho invited Kepler to come and work with him in Prague when the Counter Reformation made him jobless and homeless. Tycho welcomed him back when Kepler went off in a huff at their first meeting. It was Tycho who assigned him the task of calculating the orbit of Mars that would lead him to discover his first two laws of planetary motion. It has been said that Tycho’s data had just the right level of accuracy to enable Kepler to determine his elliptical orbits. Any less accurate and the slight eccentricities would not have been discernable. Any more accurate and the irregularities in the orbits, thus made visible, would have made the discovery of the elliptical form almost impossible. It has also been said that of all the planets for which Tycho had observation data Mars was the one with the most easily discernable elliptical orbit. Serendipity seems to have also played a role in the discovery of Kepler’s system. The high quality of Tycho’s data also led Kepler to reject an earlier non-elliptical solution for the orbit of Mars, which another astronomer would probably have accepted, with the argument that it was not mathematically accurate enough to do honour to Tycho’s so carefully acquired observational data.

Tycho was anything but a one-man show and his observatory on the island of Hven has quite correctly been described as a research institute. A substantial number of astronomer, mathematicians and instrument maker came and went both on Hven and later in Prague over the almost thirty years that Tycho took to accumulate his data. The number of people who deserve a share in the cake that was Kepler’s system now reaches a point where it become silly to count them individually.

Our list even includes royalty. Rudolph II, Holly Roman Emperor, was the man, who, at Tycho’s request, gave Kepler a position at court, even if he was more than somewhat lax at paying his salary, official to calculate the Rudolphine Tables, a task that would plague Kepler for almost thirty years but would in the end lead to the acceptance of his system by other astronomers. Rudolph also appointed Kepler as Tycho’s successor, as Imperial Mathematicus, after the latter’s untimely death, thus giving him the chance to continue his analysis of Tycho’s data. Rudolph could just as easily have sacked him and sent him on his way. Tycho’s heirs did not assist Kepler in his struggle to maintain access to that all important data, which belonged to them and not the Emperor, causing him much heartache before they finally allowed him to use Tycho’s inheritance. After he had usurped his brother, Rudolph, in 1612, Matthias allowed Kepler to keep his official position and title as Imperial Mathematicus, although sending him away from court, a fact that certainly assisted Kepler in his work. Being Imperial Mathematicus gave him social status and clout.

Rudolph II portrait by Joseph Heinz the Elder Source: Wikimedia Commons

Rudolph II portrait by Joseph Heinz the Elder
Source: Wikimedia Commons

Kepler described his long and weary struggles with the orbit of Mars as a battle, but he did not fight this battle alone. In a long and fascinating correspondence with the astronomer, David Fabricius, Kepler tried out his ideas and results with a convinced supporter of Tycho’s system. Kepler would present his ideas and David Fabricius subjected them to high level and very knowledgeable criticism. Through this procedure Kepler honed, refined and polished his theories to perfection before he submitted them to public gaze in his Astronomia Nova, Knowing that they would now withstand high-level professional criticism. David Fabricius, who never met Kepler, nevertheless took a highly active role in the shaping of the Keplerian system[3].

Monument for David and Johann Fabricius in the Graveyard of Osteel

Monument for David and Johann Fabricius in the Graveyard of Osteel

Even after Kepler’s death the active participation of others in shaping his astronomical system did not cease. Jeremiah Horrocks corrected and extended the calculations of the Rudolphine Tables, enabling him to predict and observe a transit of Venus, an important stepping-stone in the acceptance of the elliptical astronomy. Horrocks also determined that the moon’s orbit was a Keplerian ellipse, something that Kepler had not done.

 

Stained glass roundel memorial in Much Hoole Church to Jeremiah Horrocks making the first observation and recording of a transit of Venus in 1639. The Latin reads "Ecce gratissimum spectaculum et tot votorum materiem": "oh, most grateful spectacle, the realization of so many ardent desires". It is taken from Horrocks's report of the transit

Stained glass roundel memorial in Much Hoole Church to Jeremiah Horrocks making the first observation and recording of a transit of Venus in 1639. The Latin reads “Ecce gratissimum spectaculum et tot votorum materiem”: “oh, most grateful spectacle, the realization of so many ardent desires”. It is taken from Horrocks’s report of the transit

Cassini, together with Riccioli and Grimaldi, using a heliometer determined that either the orbit of the sun around the earth or the earth around the sun, the method can’t determine which is true, is an ellipse another important empirical stepping-stone on the road to final acceptance for the system.

Giovanni Cassini Source: Wikimedia Commons

Giovanni Cassini
Source: Wikimedia Commons

Nicholas Mercator produced a new mathematical derivation of Kepler’s second law around 1670. Kepler’s own derivation was, as he himself admitted, more than a little suspect, viewed mathematically. The first and third laws had been accepted by the astronomical community fairly easily but the second law was a major bone of contention. Mercator’s new derivation basically laid the dispute to rest.

Cassini in his new role as director of the Paris observatory showed empirically that the satellite systems of both Jupiter and Saturn also obeyed Kepler’s third law extending it effectively to all orbitary systems and not just the planets of the solar system.

Lastly Newton derived Kepler’s first and second laws from his axiomatic system of dynamics giving them the true status of laws of physics. This led Newton to claim that the third law was Kepler’s but the first two were his because he, as opposed to Kepler, had really proved them

As we can see the list of people involved in revolutionising astronomy in the seventeenth century in that they replaced all the geocentric systems with a Keplerian elliptical system is by no means restricted to ‘one man’ as claimed in the subtitle to David Love’s book but is quite extensive and very diverse. There are no lone geniuses; science is a collective, collaborative enterprise.

 

 

 

 

[1] David Love, Kepler and the Universe: How One Man Revolutionized Astronomy, Prometheus Books, 2015

[2] Kitty Ferguson, Tycho and Kepler, The Unlikely Partnership that Forever Changed our Understanding of the Heavens, Walker Books, 2002

[3] For a wonderful description of this correspondence and how it contributed to the genesis of Astronmia Nova see James Voelkel’s excellent, The Composition of Kepler’s Astronomia nova, Princeton University Press, 2001

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Filed under History of Astronomy, Myths of Science, Uncategorized

Hans Holbein and the Nürnberg–Ingolstadt–Vienna Renaissance mathematical nexus.

There is a strong tendency, particularly in the popular history of science, to write about or present scientists as individuals. This leads to a serious distortion of the way that science develops and in particular propagates the lone genius myth. In reality science has always been a collective endeavour with its practitioners interacting in many different ways and on many different levels. In the Renaissance, when travelling from one end of Europe to the other would take weeks and letters often even longer, one might be excused for thinking that such cooperation was very low level but in fact the opposite was the truth, with scholars in the mathematical sciences exchanging information and ideas throughout Europe. A particularly strong mathematical nexus existed in the Southern German speaking area between the cities of Nürnberg, Ingolstadt and Vienna in the century between 1450 and 1550. Interestingly two of the paintings of the Northern Renaissance artist Hans Holbein the Younger open a door into this nexus.

Holbein (c. 1497–1543) was born in Augsburg the son of the painter and draughtsman Hans Holbein the Elder. As a young artist he lived and worked for a time in Basel where he became acquainted with Erasmus and worked for the printer publisher Johann Froben amongst others. Between 1526 and 1528 he spent some time in England in the household of Thomas More and it is here that he painted the second portrait I shall be discussing. The next four years find him living in Basel again before he returned to England in 1532 where he became associated with the court of Henry VIII, More having fallen out of favour. It was at the court that he painted, what is probably his most well know portrait, The Ambassadors in 1533.

Hans Holbein The Ambassadors Source: Wikimedia Commons

Hans Holbein The Ambassadors
Source: Wikimedia Commons

The painting shows two courtiers, usually identified as the French Ambassador Jean de Dinteville and Georges de Selve, Bishop of Lavaur standing on either side of a set of shelves laden with various books and instruments. There is much discussion was to what the instruments are supposed to represent but it is certain that, whatever else they stand for, they represent the quadrivium, arithmetic, geometry music and astronomy, the four mathematical sciences taught at European medieval universities. There are two globes, on the lower shelf a terrestrial and on the upper a celestial one. The celestial globe has been positively identified, as a Schöner globe and the terrestrial globe also displays characteristics of Schöner’s handwork.

Terrestrial Globe The Ambassadors Source Wikimedia Commons

Terrestrial Globe The Ambassadors
Source Wikimedia Commons

Celestial Globe The Ambassadors Source Wikimedia Commons

Celestial Globe The Ambassadors
Source Wikimedia Commons

Johannes Schöner (1477–1547) was professor for mathematics at the Egidienöberschule in Nürnberg, the addressee of Rheticus’ Narratio Prima, the founder of the tradition of printed globe pairs, an editor of mathematical texts for publication (especially for Johannes Petreius the sixteenth centuries most important scientific publisher) and one of the most influential astrologers in Europe. Schöner is a central and highly influential figure in Renaissance mathematics.

On the left hand side of the lower shelf is a copy of Peter Apian’s Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen (published in Ingolstadt in 1527) held open by a ruler. This is a popular book of commercial arithmetic, written in German, typical of the period. Peter Apian (1495–1552) professor of mathematics at the University of Ingolstadt, cartographer, printer-publisher and astronomer was a third generation representative of the so-called Second Viennese School of Mathematics. A pupil of Georg Tannstetter (1482–1535) a graduate of the University of Ingolstadt who had followed his teachers Johannes Stabius and Andreas Stiborious to teach at Conrad Celtis’ Collegium poetarum et mathematicorum, of which more later. Together Apian and Tannstetter produced the first printed edition of the Optic of Witelo, one of the most important medieval optic texts, which was printed by Petreius in Nürnberg in 1535. The Tannstetter/Apian/Petreius Witelo was one of the books that Rheticus took with him as a present for Copernicus when he visited him in 1539. Already, a brief description of the activities of Schöner and Apian is beginning to illustrate the connection between our three cities.

Apian's Arithmetic Book The Ambassadors Source: Wikimedia Commons

Apian’s Arithmetic Book The Ambassadors
Source: Wikimedia Commons

When Sebastian Münster (1488–1552), the cosmographer, sent out a circular requesting the cartographers of Germany to supply him with data and maps for his Cosmographia, he specifically addressed both Schöner and Apian by name as the leading cartographers of the age. Münster’s Cosmographia, which became the biggest selling book of the sixteenth century, was first published by Heinrich Petri in Basel in 1544. Münster was Petri’s stepfather and Petri was the cousin of Johannes Petreius, who learnt his trade as printer publisher in Heinrich’s printing shop in Basel. The Petri publishing house was also part of a consortium with Johann Amerbach and Johann Froben who had employed Hans Holbein in his time in Basel. Wheels within wheels.

The, mostly astronomical, instruments on the upper shelf are almost certainly the property of the German mathematician Nicolaus Kratzer (1487–1550), who is the subject of the second Holbein portrait who will be looking at.

Nicolas Kratzer by Hans Holbein Source: Wikimedia Commona

Nicolas Kratzer by Hans Holbein
Source: Wikimedia Commona

Born in Munich and educated at the universities of Cologne and Wittenberg Kratzer, originally came to England, like Holbein, to become part of the Thomas More household, where he was employed as a tutor for More’s children. Also like Holbein, Kratzer moved over to Henry VIII’s court as court horologist or clock maker, although the clocks he was responsible for making were more probably sundials than mechanical ones. During his time as a courtier Kratzer also lectured at Oxford and is said to have erected a monumental stone sundial in the grounds of Corpus Christi College. One polyhedral sundial attributed to Kratzer is in the Oxford Museum for the History of Science.

Polyhedral Sundial attributed to Nicolas Kratzer Source: MHS Oxford

Polyhedral Sundial attributed to Nicolas Kratzer
Source: MHS Oxford

In 1520 Kratzer travelled to Antwerp to visit Erasmus and here he met up with Nürnberg’s most famous painter Albrecht Dürer, who regular readers of this blog will know was also the author of a book on mathematics. Dürer’s book contains the first printed instructions, in German, on how to design, construct and install sundials, so the two men will have had a common topic of interest to liven there conversations. Kratzer witnessed Dürer, who was in Antwerp to negotiate with the German Emperor, painting Erasmus’ portrait and Dürer is said to have also drawn a portrait of Kratzer that is now missing. After Kratzer returned to England and Dürer to Nürnberg the two of them exchanged, at least once, letters and it is Kratzer’s letter that reveals some new connections in out nexus.

Albrecht Dürer selfportrait Source: Wikimedia Commons

Albrecht Dürer selfportrait
Source: Wikimedia Commons

In his letter, from 1524, Kratzer makes inquires about Willibald Pirckheimer and also asks if Dürer knows what has happened to the mathematical papers of Johannes Werner and Johannes Stabius who had both died two years earlier.

Willibald Pirckheimer (1470–1530) a close friend and patron of Dürer’s was a rich merchant, a politician, a soldier and a humanist scholar. In the last capacity he was the hub of a group of largely mathematical humanist scholars now known as the Pirckheimer circle. Although not a mathematician himself Pirckheimer was a fervent supporter of the mathematical sciences and produced a Latin translation from the Greek of Ptolemaeus’ Geōgraphikḕ or Geographia, Pirckheimer’s translation provided the basis for Sebastian Münster’s edition, which was regarded as the definitive text in the sixteenth century. Stabius and Werner were both prominent members of the Pirckheimer circle.

Willibald Pirckheimer by Albrecht Dürer Source: Wikimedia Commons

Willibald Pirckheimer by Albrecht Dürer
Source: Wikimedia Commons

The two Johanneses, Stabius (1450–1522) and Werner (1468–1522), had become friends at the University of Ingolstadt where the both studied mathematics. Ingolstadt was the first German university to have a dedicated chair for mathematics. Werner returned to his hometown of Nürnberg where he became a priest but the Austrian Stabius remained in Ingolstadt, where he became professor of mathematics. The two of them continued to correspond and work together and Werner is said to have instigated the highly complex sundial on the wall of the Saint Lorenz Church in Nürnberg, which was designed by Stabius and constructed in 1502.

St Lorenz Church Nürnberg Sundial 1502 Source: Astronomie in Nürnberg

St Lorenz Church Nürnberg Sundial 1502
Source: Astronomie in Nürnberg

It was also Werner who first published Stabius’ heart shaped or cordiform map projection leading to it being labelled the Werner-Stabius Projection. This projection was used for world maps by Peter Apian as well as Oronce Fine, France’s leading mathematicus of the sixteenth century and Gerard Mercator, of whom more, later. The network expands.

Mercator cordiform world map 1538 Source: American Geographical Society Library

Mercator cordiform world map 1538
Source: American Geographical Society Library

In his own right Werner produced a partial Latin translation from the Greek of Ptolemaeus’ Geographia, was the first to write about prosthaphaeresis (a trigonometrical method of simplifying calculation prior to the invention of logarithms), was the first to suggest the lunar distance method of determining longitude and was in all probability Albrecht Dürer’s maths teacher. He also was the subject of an astronomical dispute with Copernicus.

Johannes Werner Source: Wikimedia Commons

Johannes Werner
Source: Wikimedia Commons

Regular readers of this blog will know that Stabius co-operated with Albrecht Dürer on a series of projects, including his famous star maps, which you can read about in an earlier post here.

Johannes Statius Portrait by Albrecht Dürer Source: Wikimedia Commons

Johannes Statius Portrait by Albrecht Dürer
Source: Wikimedia Commons

An important non-Nürnberger member of the Pirckheimer Circle was Conrad Celtis (1459–1508), who is known in Germany as the arch-humanist. Like his friend Pirckheimer, Celtis was not a mathematician but believed in the importance of the mathematical sciences. Although already graduated he spent time in 1489 on the University of Kraków in order to get the education in mathematics and astronomy that he couldn’t get at a German university. Celtis had spent time at the humanist universities of Northern Italy and his mission in life was to demonstrate that Germany was just as civilised and educated as Italy and not a land of barbarians as the Italians claimed. His contributions to the Nuremberg Chronicle can be viewed as part of this demonstration. He believed he could achieve his aim by writing a comprehensive history of Germany including, as was common at the time its geography. In 1491/92 he received a teaching post in Ingolstadt, where he seduced the professors of mathematics Johannes Stabius and Andreas Stiborius (1464–1515) into turning their attention from astrology for medicine student, their official assignment, to mathematical cartography in order to help him with his historical geography.

Conrad Celtis Source: Wikimedia Commons

Conrad Celtis
Source: Wikimedia Commons

Unable to achieve his ends in Ingolstadt Celtis decamped to Vienna, taking Stabius and Stiborius with him, to found his Collegium poetarum et mathematicorum as mentioned above and with it the so-called Second Viennese School of Mathematics; the first had been Peuerbach and Regiomontanus in the middle of the fifteenth century. Regiomontanus spent the last five years of his life living in Nürnberg, where he set up the world’s first scientific publishing house. Stiborius’ pupil Georg Tannstetter proved to be a gifted teacher and Peter Apian was by no means his only famous pupil.

The influence of the Nürnberg–Ingolstadt–Vienna mathematicians reached far beyond their own relatively small Southern German corridor. As already stated Münster in Basel stood in contact with both Apian and Schöner and Stabius’ cordiform projection found favour with cartographers throughout Northern Europe. Both Apian and Schöner exercised a major influence on Gemma Frisius in Louvain and through him on his pupils Gerard Mercator and John Dee. As outlined in my blog post on Frisius, he took over editing the second and all subsequent editions of Apian’s Cosmographia, one of the most important textbooks for all things astronomical, cartographical and to do with surveying in the sixteenth century. Frisius also learnt his globe making, a skill he passed on to Mercator, through the works of Schöner. Dee and Mercator also had connections to Pedro Nunes (1502–1578) the most important mathematicus on the Iberian peninsular. Frisius had several other important pupils who spread the skills in cosmography, and globe and instrument making that he had acquired from Apian and Schöner all over Europe.

Famously Rheticus came to Nürnberg to study astrology at the feet of Johannes Schöner, who maintained close contacts to Philipp Melanchthon Rheticus patron. Schöner was the first professor of mathematics at a school designed by Melanchthon. Melanchthon had learnt his mathematics and astrology at the University of Tübingen from Johannes Stöffler (1452–1531) another mathematical graduate from Ingolstadt.

Kupferstich aus der Werkstatt Theodor de Brys, erschienen 1598 im 2. Bd. der Bibliotheca chalcographica Source: Wikimedia Commons

Kupferstich aus der Werkstatt Theodor de Brys, erschienen 1598 im 2. Bd. der Bibliotheca chalcographica
Source: Wikimedia Commons

Another of Stöffler’s pupils was Sebastian Münster. During his time in Nürnberg Rheticus became acquainted with the other Nürnberger mathematicians and above all with the printer-publisher Johannes Petreius and it was famously Rheticus who brought the manuscript of Copernicus’ De revolutionibus to Nürnberg for Petreius to publish. Rheticus says that he first learnt of Copernicus’s existence during his travels on his sabbatical and historians think that it was probably in Nürnberg that he acquired this knowledge. One of the few pieces of astronomical writing from Copernicus that we have is the so-called Letter to Werner. In this manuscript Copernicus criticises Werner’s theory of trepidation. Trepidation was a mistaken belief based on faulty data that the rate of the precession of the equinoxes is not constant but varies with time. Because of this highly technical dispute amongst astronomers Copernicus would have been known in Nürnberg and thus the assumption that Rheticus first heard of him there. Interestingly Copernicus includes observations of Mercury made by Bernhard Walther (1430–1504), Regiomontanus partner, in Nürnberg; falsely attributing some of them to Schöner, so a connection between Copernicus and Nürnberg seems to have existed.

In this brief outline we have covered a lot of ground but I hope I have made clear just how interconnected the mathematical practitioners of Germany and indeed Europe were in the second half of the fifteenth century and the first half of the sixteenth. Science is very much a collective endeavour and historians of science should not just concentrate on individuals but look at the networks within which those individual operate bringing to light the influences and exchanges that take place within those networks.

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Filed under History of Astrology, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, Renaissance Science