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The emergence of modern astronomy – a complex mosaic: Part II

You can read Part I here

Before we progress we need to take stock and deal with a couple of points that came up in a comment to Part I. This series is about the factors that led to the emergence of heliocentricity in Europe in the Early Modern Period. It doesn’t deal with any of the factors from earlier periods and other cultures that also explicitly and implicitly flowed into European astronomy. If one were to include all of those, it would be a total history of western astronomy that doesn’t even start in the West but in Babylon in about 2000 BCE. That is not what I intend to write and I won’t be doing so.

The other appears to contradict what I said above. At my starting point circa 1400 CE people became aware of a need to increase their usage of mathematical astronomy for a number of reasons that I sketched in Part I. Ptolemaic mathematical astronomy had been available in Europe in two Latin translations, the first from Greek the second from Arabic, since the twelfth century. However, medieval Europeans in general lacked the mathematical knowledge and to some extent the motivation to engage with this highly technical work. The much simpler available astronomical tables, mostly from Islamic sources, fulfilled their needs at that time. It was only really at the beginning of the fifteenth century that a need was seen to engage more fully with real mathematical astronomy. Having said that, at the beginning the users were not truly aware of the fact that the models and tables that they had inherited from the Greeks and from Islamic culture were inaccurate and in some cases defective. Initially they continued to use this material in their own endeavours, only gradually becoming aware of its deficiencies and the need to reform. As in all phases of the history of science these changes do not take place overnight but usually take decades and sometimes even centuries. Science is essential conservative and has a strong tendency to resist change, preferring to stick to tradition. In our case it would take about 150 years from the translation of Ptolemaeus’ Geographiainto Latin, my starting point, and the start of a full-scale reform programme for astronomy. Although, as we will see, such a programme was launched much earlier but collapsed following the early death of its initiator.

Going into some detail on points from the first post. I listed Peuerbach’s Theoricarum novarum planetarum(New Planetary Theory), published by Regiomontanus in Nürnberg in 1472, as an important development in astronomy in the fifteenth century, which it was. For centuries it was thought that this was a totally original work from Peuerbach, however, the Arabic manuscript of a cosmology from Ptolemaeus was discovered in the 1960s and it became clear that Peuerbach had merely modernised Ptolemaeus’ work for which he must have had a manuscript that then went missing. Many of the improvements in Peuerbach’s and Regiomontanus’ epitome of Ptolemaeus’ Almagest also came from the work of Islamic astronomers, which they mostly credit. Another work from the 1st Viennese School was Regiomontanus’ De Triangulis omnimodis Libri Quinque (On Triangles), written in 1464 but first edited by Johannes Schöner and published by Johannes Petreius in Nürnberg in1533.


Title page of a later edition of Regiomontanus’ On Triangle

This was the first comprehensive textbook on trigonometry, the mathematics of astronomy, published in Europe. However, the Persian scholar Abū al-Wafā Būzhjānī (940–988) had already published a similar work in Arabic in the tenth century, which of course raises the question to what extent Regiomontanus borrowed from or plagiarised Abū al-Wafā.

These are just three examples but they should clearly illustrate that in the fifteenth and even in the early sixteenth centuries European astronomers still lagged well behind their Greek and Islamic predecessors and needed to play catch up and they needed to catch up with those predecessors before they could supersede them.

After ten years of travelling through Italy and Hungary, Regiomontanus moved from Budapest to Nürnberg in order to undertake a major reform of astronomy.


City of Nürnberg Nuremberg Chronicles Workshop of Michael Wohlgemut Printed by Aton Koberger and published in Nürnberg in 1493

He argued that astrological prognostications were inaccurate because the astronomical data on which they were based was also inaccurate, which it indeed was. He had an ambitious two part programme; firstly to print and publish critical editions of the astronomical and astrological literature, the manuscripts of which he had collected on his travels, and secondly to undertake a new substantial programme of accurate astronomical observations. He tells us that he had chosen Nürnberg because it made the best scientific instruments and because as a major trading centre it had an extensive communications network. The latter was necessary because he was aware that he could not complete this ambitious programme alone but would need to cooperate with other astronomers.

Arriving in Nürnberg, he began to cooperate with a resident trading agent, Bernhard Walther, the two of them setting up the world’s first printing press for scientific literature. The first publication was Peuerbach’s Theoricae novae planetarum (New Planetary Theory)


followed by an ambitious catalogue of planned future publications from the astrological and astronomical literature. Unfortunately they only managed another seven publications before Regiomontanus was summoned to Rome by the Pope to work on a calendar reform in 1475, a journey from which he never returned dying under unknown circumstances, sometime in 1476. The planned observation programme never really got of the ground although Walther continued making observations, a few of which were eventually used by Copernicus in his De revolutionibus.

Regiomontanus did succeed in printing and publishing his Ephemerides in 1474, a set of planetary tables, which clearly exceeded in accuracy all previous planetary tables that had been available and went on to become a scientific bestseller.


However he didn’t succeed in printing and publishing the Epytoma in almagesti Ptolemei; this task was left to another important early publisher of scientific texts, Erhard Ratdolt (1447–1528, who completed the task in Venice twenty years after Regiomontanus’ death. Ratdolt also published Regiomontanus’ astrological calendars an important source for medical astrology.


Calendarius by Regiomontanus, printed by Erhard Ratdolt, Venice 1478, title page with printers’ names Source: Wikimedia Commons

The first printed edition of Ptolemaeus’ Geographia with maps was published in Bologna in 1477; it was followed by several other editions in the fifteenth century including the first one north of the Alps in Ulm in 1482.

The re-invention of moveable type printing by Guttenberg in about 1450 was already having a marked effect on the revival and reform of mathematical astronomy in Early Modern Europe.







Filed under Early Scientific Publishing, History of Astronomy, History of Cartography, History of Mathematics, Uncategorized

Christmas at the Renaissance Mathematicus – A guide for new readers


Being new to the Renaissance Mathematicus one might be excused if one assumed that the blogging activities were wound down over the Christmas period. However, exactly the opposite is true with the Renaissance Mathematicus going into hyper-drive posting its annual Christmas Trilogy, three blog posts in three days. Three of my favourite scientific figures have their birthday over Christmas–Isaac Newton 25thDecember, Charles Babbage 26thDecember and Johannes Kepler 27thDecember–and I write a blog post for each of them on their respective birthdays. Before somebody quibbles I am aware that the birthdays of Newton and Kepler are both old style, i.e. on the Julian Calendar, and Babbage new style, i.e. on the Gregorian Calendar but to be honest, in this case I don’t give a shit. So if you are looking for some #histSTM entertainment or possibly enlightenment over the holiday period the Renaissance Mathematicus is your number one address. In case the new trilogy is not enough for you:

The Trilogies of Christmas Past

Christmas Trilogy 2009 Post 1

Christmas Trilogy 2009 Post 2

Christmas Trilogy 2009 Post 3

Christmas Trilogy 2010 Post 1

Christmas Trilogy 2010 Post 2

Christmas Trilogy 2010 Post 3

Christmas Trilogy 2011 Post 1

Christmas Trilogy 2011 Post 2

Christmas Trilogy 2011 Post 3

Christmas Trilogy 2012 Post 1

Christmas Trilogy 2012 Post 2

Christmas Trilogy 2012 Post 3

Christmas Trilogy 2013 Post 1

Christmas Trilogy 2013 Post 2

Christmas Trilogy 2013 Post 3

Christmas Trilogy 2014 Post 1

Christmas Trilogy 2014 Post 2

Christmas Trilogy 2014 Post 3

Christmas Trilogy 2015 Post 1

Christmas Trilogy 2015 Post 2

Christmas Trilogy 2015 Post 3

Christmas Trilogy 2016 Post 1

Christmas Trilogy 2016 Post 2

Christmas Trilogy 2016 Post 3

Christmas Trilogy 2017 Post 1

Christmas Trilogy 2017 Post 2

Christmas Trilogy 2017 Post 3



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Filed under Odds and Ends, Uncategorized

It’s Solstice Time Again!

We are deep in what is commonly called the holiday season. For personal reasons I don’t celebrate Christmas and as I explained in this post starting the New Year on 1 January on the Gregorian Calendar is/was a purely arbitrary decision. I wrote there that I consider the winter solstice to be the best choice to celebrate the end and beginning of a solar cycle in the northern hemisphere.


Stonehenge Winter Solstice

Today at 22:23 UTC the sun will turn at the Tropic of Capricorn and begin its journey northwards to the Tropic of Cancer and the summer solstice.  Tropic comes from the Latin tropicus “pertaining to a turn,” from Greek tropikos “of or pertaining to a turn or change.”

I wish all of my readers a happy solstice and may the next 365 days, 5 hours, 48 minutes and 45 seconds bring you much light, joy, peace and wisdom. We can only hope that they will be better than the last 365 days, 5 hours, 48 minutes and 45 seconds (length of the mean tropical or solar year).

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Filed under Autobiographical, Odds and Ends, Uncategorized

Internalism vs. Externalism?

This is one of those blog posts where I do some thinking out loud[1]. I not really sure where it’s going and it might not end up where I intended it to. I shall be skating on the thin ice of historiography. The dictionary defines historiography as follows:

  1. The wring of history
  2. The study of the development of historical method, historical research, and writing
  3. Any body of historical literature[2]

I’m using the term in the sense of definition (2) here. Formulated slightly differently historiography is the methodology of doing history, i.e. historical research and the reporting of that research in writing. Maybe unfortunately there isn’t just one historiography or methodology for doing history there are historiographies, plural that often conflict or even contradict each other, dividing historians into opposing camps indulging in trench warfare with each other through their monographs and journals.

On the whole I tend to view historiographies with a jaundiced eye. I have a maxim for historiographies: ‘Historiography becomes dogma and dogma blinds.’ I like to mix and match my methodologies according to what I happen to be engaged in at any given moment. A single methodology or historiography is just one perspective from which to view a given historical topic and it is often useful to view it from several different perspectives simultaneously, even seemingly contradictory ones.

Since I have been involved in the history of science, and I realise with somewhat horror that is a good half century now, one of the on going historiography debates, or even disputes, within the disciple has been Internalism vs. Externalism.


Definitions are very slippery things but if I was asked to explain what this means my first simple answer would be internalism is the historical study of the facts, hypotheses, theories etc. that science has produced and externalism is the historical study of the contexts in which those facts, hypotheses, theorems etc. were discovered, developed, formulated etc.

To give an abstract example from the history of mathematics an internalist would be interested in when mathematician X first proved theorem Y and the technical method that he used to do so. They might investigate on whose or which work X built his own work  and also possibly, who picked up on X’s proof and extended it mathematically; anything extraneous to that wouldn’t not be the concern of our imaginary internalist. An externalist would, however, be at least as interested in the context in which X carried out his mathematical endeavours. They would possibly look at X’s biography, how X came to be doing this work at all, what were X’s motivations for this particular piece of research, in which context (university, court mathematicus, insurance mathematician etc.) X was carrying out this work, who was financing it and why etc., etc. From this brief description it should be clear that the perspective of the internalist is a very narrow, very focused one, whereas that of the externalist is a very broad, very sweeping one, although any given externalist investigation might only concentrate on one or two of the various perspectives that I have listed.

Extreme internalism assumes that just presenting the ‘facts’ in the history of science is adequate because science is somehow independent of the world/society/culture in which it arose/developed/originated. Science is totally objective in some way and doesn’t need a context. Extreme internalism also tends to be highly presentist. That is it looks back through history and selects those events/developments in science that can be identified within science, as it exists today. It sees science as cumulative and progressive even teleological. It’s destination being some sort of complete truth.

Externalism sees science at any given point in time as a product of the world/society/culture in which it arose/developed/originated. The externalist historical picture includes all the bits the researchers of the period got wrong and were subsequently jettisoned somewhere down the line on the way to the present. Externalism sees any period of science, as not just embedded in its world/society/culture but as an integral part of the whole of that world/society/culture that cannot and should not be viewed independently.

To give just a couple of very simple examples out of my own main personal historical area of interest: An internalist is only interested in Kepler’s three laws of planetary motion as results that are still valid today. They are not interested in the complex twists and turns of Kepler’s battle to find the first two laws, which he outlines in great detail and great depth in his Astronomia nova. As for the third law, they take it gladly and ignore all of the remaining five hundred pages of the Harmonice Mundi, with its bizarre theories of consonance and dissonance, and cosmic harmony. As for Kepler’s distinctly unscientific motivations, the internalist shudders in horror. For the externalist everything that the internalist rejects is an interesting field of study. They are not just interested in Kepler’s laws as results but in how he arrived at them and what was driving him to search for them in the first place.

Turning to Newton, it is now a commonplace that he devoted far more time and energy to studying alchemy and theology that he did to either physics or mathematics. For the internalist these ‘non-scientific’ areas are an irrelevance to be ignored, all that matters are the scientific results, the law of gravity, the calculus etc. Externalists have shown that the various diffuse areas of Newton’s thoughts and endeavours are intertwined into a complex whole and if one really wants to understand the man and his science then one must regard and attempt to understand that whole.

Where do I stand on this issue? I think it should be obvious to anybody who regularly reads this blog that I am a convinced externalist. I am, however, happy to admit that when I first became interested in the history of mathematics as a teenager I was to all intents and purposes an internalist. Who discovered this or that theorem and when? Who developed this or that method of solving this or that type of problem? These were the questions that initially interested me. I also had strong presentist and even Whiggish tendencies. For those who have forgotten or maybe don’t know yet, the Whig theory of history is the belief that human existence or in this case science, is progressing towards some sort of final truth. Over the years, as I learnt more, my views changed and I became slowly but surely an externalist. This change was, at the latest, completed as I worked for many years, my apprenticeship, in a research project into the history of formal logic. This project was official called, Case Studies into a Social History of Formal Logic, where social is a synonym for external.

As I see it extreme internalism is not just too narrow, too focused but is actually distorting. The internalist history of mathematics, for example, when considering antiquity tends to concentrate on what could be called higher mathematics–the Euclids, Archimedes et al– who only represent a very small minority of those engaged in mathematical pursuits in their period and whose results were only interesting to an equally small minority. In doing so they ignore the vast majority of mathematical practitioners surveyors, bookkeepers extra, whose work actually contributed more in real terms to their societies than that of the ‘star’ mathematicians. A good example is the much-touted Babylonian mathematics, which was largely developed by clerks doing administration not by mathematicians. This fact is simply ignored by internalist historians of mathematics, who are only interested in the results.

Turning to the High Middle Ages and Renaissance, traditional internalist history of mathematics tend to simply ignore this period as having no mathematics worth mentioning. In reality it was the mathematical practitioners of this period–astrologers, astronomers, geographers, cartographers, surveyors, architects, engineers, instrument designers and makers, globe makerset al.–who created the mathematics that drove the so-called scientific revolution.

Having being very rude about internalist history of science I should point out that I by no means reject it totally, in fact exactly the opposite. Anybody who opens Newton’s Principia for the first time, even in the excellent modern English translation by Cohen and Whitman would probably understand very little of the mathematics and physics that they would find there. They have a choice either to spend several months chewing through Newton’s masterpiece or alternatively to turn to Cohen excellent internalist guide to the contents. The same is true of virtually any historical STEM text. Close internalist readings and interpretations help the historian to comprehension. Having gained that internalist comprehension they should, in my opinion, embed that comprehension into its wider externalist context.

Historians of science should be simply historian, in the first instance, investigating the breadth and depth of a discipline within its social context. However this also implies a solid understanding of the science involved, i.e. the internal aspects. You can’t investigate the role of a scientific discipline within a social context if you don’t understand the science. This means for me, that a good historian of science must be both an internalist and an externalist, weaving together both approaches into a coherent whole.

All of the above is of course my own subjective take on the dichotomy and they are certainly other viewpoints and other opinions on the issue. As always, readers are welcome to ventilate their views in the comments.

For any future historian, who might be interested in my motivation for writing this post, it was inspired by a request from a reader to write something on the ‘conflict’ between internalist and externalist histories of science and illustrate it with examples of the two different approaches with reference to my own blog posts. I’m not sure if that which I have written really fulfils their request and as should be obvious I, as a convinced externalist, can’t really supply the desired examples. However I am grateful to the reader for having motivated me to write something on the topic even if it not really what they wanted.

[1]If I was being pretentious I might have said, “Where I philosophise” but I don’t regard my stream of consciousness meanderings as rigorous enough to be dignified with the term philosophy.

[2]Collins English Dictionary online.



Filed under History of science, Uncategorized

Printing the Hindu-Arabic numbers

Arte dell’Abbaco, a book that many consider the first-ever printed mathematics book, was dated four hundred and forty years ago on 10 December 1478. I say many consider because the book, also known as the Treviso Arithmetic, is a commercial arithmetic textbook and some historians regard commercial arithmetic as a separate discipline and not really mathematics.

0380b Large

Calculation from the Arte dell’Abbaco

The unknown author explains his book thus:

I have often been asked by certain youths in whom I have much interest, and who look forward to mercantile pursuits, to put into writing the fundamental principles of arithmetic, commonly called abbacus.

The Treviso Arithmetic is actually an abbacus book, those books on calculating with the Hindu-Arabic numerals that derive their existence from Leonardo Pisano’s Liber Abbaci. Like most abbacus books it is written in the vernacular, which in this case is the local Venetian dialect. If you don’t read 15thcentury Venetian there is an English translation by Frank J. Swetz, Capitalism and Arithmetic: The New Math of the 15thCentury Including the Full Text of the Treviso Arithmetic of 1478, Open Court, 1987.


Filed under Early Scientific Publishing, History of Mathematics, Renaissance Science, Uncategorized

The Seven Learned Sisters

I have suffered from a (un)healthy[1]portion of imposter syndrome all of my life. This is the personal feeling in an academic context that one is just bluffing and doesn’t actually know anything and then any minute now somebody is going to unmask me and denounce me as an ignorant fraud. I always thought that this was a personal thing, part of my general collection of mental and emotional insecurities but in more recent years I have learned that many academics, including successful and renowned ones, suffer from this particular form of insecurity. On related problem that I have is the belief that anything I do actually know is trivial, generally known to everyone and therefore not worth mentioning[2]. I experienced an example of this recently on Twitter when I came across the following medieval illustration and its accompanying tweet.


Geometria Source: Wikimedia Commons

Woman teaching geometry to monks. In the Middle Ages, it is unusual to see women represented as teachers, in particular when the students appear to be monks. Euclid’s Elementa, in the translation attributed to Adelard of Bath, 1312.

I would simply have assumed that everybody knew what this picture represents and not commented. It is not a “women teaching geometry to monks” as the tweeter thinks but a typical medieval personification of Geometria, one of the so-called Seven Learned Sisters. The Seven Learned Sisters are the personifications of the seven liberal arts, the trivium (grammar, rhetoric and dialectic)

and the quadrivium (arithmetic, geometry, music and astronomy),

which formed the curriculum in the lower or liberal arts faculty at the medieval university. The seven liberal arts, however, have a history that well predates the founding of the first universities. In what follows I shall only be dealing with the history of the quadrivium.

As a concept this four-fold division of the mathematical sciences can be traced back to the Pythagoreans. The mathematical commentator Proclus (412–485 CE) tells us, in the introduction to his commentary on the first book of Euclid’s Elements:

The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving.

The earliest know written account of this division can be found at the beginning of the late Pythagorean Archytas’ book on harmonics, where he identifies a set of four sciences: astronomy, geometry, logistic (arithmetic) and music. Archytas’ dates of birth and death are not known but he was, roughly speaking, a contemporary of Plato. He was the teacher of Eudoxus (c.390–c.337 BCE) Harmonics, by the way, is the discipline that later became known as music in the quadrivium.

Without mentioning Archytas, Plato (428/427 or 424/423 – 348/347 BCE), who was highly influenced by the Pythagoreans,takes up the theme in his Republic (c.380 BCE). In a dialogue with Glaucon, Plato explains the merits of learning the “five” mathematical sciences; he divides geometry into plane geometry (two dimensional) and solid geometry (three dimensional). He also refers to harmonics and not music.

In the CE period the first important figure is the Neo-Pythagorean, Nicomachus of Gerasa (c.60–c.120 CE), who wrote an Introduction to Arithmeticand a Manual of Harmonics, which are still extant and a lost Introduction to Geometry. The four-fold division of the mathematical sciences only acquired the name quadrivium in the works of Boethius (c.477–524 CE), from whose work the concept of the seven liberal arts was extracted as the basic curriculum for the medieval university. Boethius, who saw it as his duty to rescue the learning of the Greeks, heavily based his mathematical texts on the work of Nicomachus.

Probably the most influential work on the seven liberal arts is the strange De nuptiis Philologiae et Mercurii (“On the Marriage of Philology and Mercury“) of Martianus Capella (fl.c. 410-420). The American historian H. O. Taylor (1856–1941) claimed that On the Marriage of Philology and Mercurywas “perhaps the most widely used schoolbook in the Middle Ages,” quoted from Martianus Capella and the Seven Liberal Artsby William Harris Stahl.[3]Stahl goes on to say, “It would be hard to name a more popular textbook for Latin reads of later ages.”

Martianus introduces each of the members of the trivium and quadrivium as bridesmaids of the bride Philology.

“Geometry enters carrying a radius in her right hand and a globe in her left. The globe is a replica of the universe, wrought by Archimedes’ hand. The peplos she wears is emblazoned with figures depicting celestial orbits and spheres; the earth’s shadow reaches into the sky, giving a purplish hue to the golden globes of the sun and moon; there are gnomons of sundials and figures showing intervals weights, and measures. Her hair is beautifully groomed, but her feet are covered with grime and her shoes are worn to shreds with treading across the entire surface of the earth.”[4]


A 16th century Geometria in a printed copy of the Margarita Philosophica

“As she enters the celestial hall, Arithmetic is even more striking in appearance than was Geometry with her dazzling peplos and celestial globe. Arithmetic too wears a robe, hers concealing an “intricate undergarment that holds clues to the operations of universal nature.” Arithmetic’s stately bearing reflects the pristine origin, antedating the birth of the Thunder God himself. Her head is an awesome sight. A scarcely perceptible whitish ray emanates from her brow; then another ray, the projection of a line, as it were, coming from the first. A third ray and a fourth spring out, and so on, up to a ninth and a tenth ray–all radiating from her brow in double and triple combinations. These proliferate in countless numbers and in a moment are miraculously retracted into the one.”[5]An allusion to the Pythagorean decade.


Gregor Reisch Margarita Philosophica: Arthimetica presiding over a computing competition between Hindu-Arabic numerals and a reckoning board

“Astronomy like her sister Geometry, is a peregrinator of the universe. She has traversed all the heavens and can reveal the constellations lying beneath the celestial arctic circle. […] Astronomy tells us that she is also familiar with the occult lore of Egyptian priest, knowledge hoarded in their sanctums; she kept herself in seclusion in Egypt for nearly forty thousand years, not wishing to divulge those secrets. She is also familiar with antediluvian Athens.”[6]


“Harmony herself is ineffably dazzling and Martianus is stricken in his efforts to describe her. A lofty figure, her head aglitter with gold ornaments, she walks along between Apollo and Athena. Her garment is tiff with incised and laminated gold; it tinkles softly and soothingly with every measured step She carries in her right hand what appears to be a shield, circular in form. It contains many concentric circles, and the whole is embroidered with striking figure. The circular chords encompass one another and from them pours forth a concord of all tones: Small models of theatrical instruments, wrought of gold, hang suspended from Harmony’s left hand. No know instrument produces sounds to compare with those coming from the strange rounded form.”[7]



I have included Stahl’s passages of Martianus’ descriptions of the quadrivium to make clear then when I talk of the disciplines being personified as women I don’t just mean that they get a female name but are fully formed female characters. This of course raises the question, at least for me, why the mathematical disciplines that were taught almost exclusively to men in ancient Greece, the Romano-Hellenistic culture and in the Middle Ages should be represented by women. Quite honestly I don’t know the answer to my own question. I assume that it relates to the nine ancient Greek Muses, who were also women and supposedly the daughters of Zeus and Mnemosyne (memory personified). This however just pushes the same question back another level. Why are the Muses female?


Having come this far it should be noted that although the quadrivium was officially part of the curriculum on medieval universities it was on the whole rather neglected. When taught the subjects were only taught at a very elementary level, arithmetic based on the primer of Boethius, itself an adaption of Nicomachus, geometry from Euclid but often only Book One and even that only partially, music again based on Boethius and astronomy on the very elementary Sphere of Sacrobosco. Often the mathematics courses were not taught during the normal classes but only on holidays, when there were no normal lectures. At most universities the quadrivium disciplines were not part of the final exams and often a student who had missed a course could get the qualification simple by paying the course fees. Mathematics only became a real part of the of the university curriculum in the sixteenth century through the efforts of Philip Melanchthon for the protestant universities and somewhat later Christoph Clavius for the Catholic ones. England had to wait until the seventeenth century before there were chairs for mathematics at Oxford and Cambridge.

[1]On the one hand imposter syndrome can act as a spur to learn more and increase ones knowledge of a given subject. On the other it can lead one to think that one needs to know much more before one closes a given research/learn/study project and thus never finish it.

[2]To paraphrase some old Greek geezer, the older I get and the more I learn, the more I become aware that what I know is merely a miniscule fraction of that which I could/should know and in reality I actually know fuck all.

[3]William Harris Stahl, Martianus Capella and the Seven Liberal Arts: Volume I The Quadrivium of Martianus Capella. Latin Traditions in the Mathematical Science, With a Study of the Allegory and the Verbal Disciplines by Richard Johnson with E. L. Burge, Columbia University Press, New York & London, 1971, p. 22

[4]Stahl pp. 125–126

[5]Stahl pp. 149–150

[6]Stahl p. 172

[7]Stahl p. 203


Filed under History of Astrology, History of Astronomy, History of Mathematics, Mediaeval Science, Renaissance Science, Uncategorized

Carl Sagan Skewered

I didn’t have time this week to write a proper blog post, so I thought I would pass on something I read recently. Not necessarily here on the blog but I tend to annoy people when I make rude comments about the American astrophysicist and science populariser Carl Sagan. Many people grew up watching his 1980s TV series Cosmos and regarded him as some sort of science saint. However, whatever his abilities to communicate science Sagan’s presentation of the history of science was terrible. Another thing that is likely to bring out the HIST_SCI HULK is mention of the biopic Agora, supposedly the life story of the ancient Greek mathematician Hypatia. Unfortunately the story line of Agora has more in common with a fairy tale than real history of science.

The medieval volume of the Cambridge History of Science[1]skewers both Sagan and Agora in just one paragraph and one footnote.

Many otherwise well-educated people have long taken this picture for granted. [Complete lack of science in the Middle Ages] No one has diffused it more widely than astronomer Carl Sagan (1934–1996), whose television series Cosmos drew an audience estimated at half a billion. In his 1980 book by the same name, a timeline of astronomy from Greek antiquity to the present left between the fifth and the late fifteenth centuries a familiar thousand-year blank labelled as a “poignant lost opportunity for mankind.” (a) The timeline reflected not the state of knowledge in 1980 but Sagan’s own “poignant lost opportunity” to consult the library of Cornell University, where he taught. In it, Sagan would have discovered large volumes devoted to the medieval history of his own field, some of them two hundred years old. He would also have learnt that the alleged medieval vacuum spawned the two institutions in which he spent his life: the observatory as a research institution (Islamic civilization) and the university (Latin Europe).

(a) Carl Sagan, Cosmos (New York: Random House, 1980), p. 335. Sagan’s outlook recently regained currency thanks to Alejandro Amenábar’s spectacular and spectacularly anachronistic film “Agor” (2009), which portrays Hypatia (d. 415) as on the verge of discovering the law of free fall and heliocentric planetary ellipses before she is murdered by fanatical monks.

[1]The Cambridge History of Science: Volume 2 Medieval Science, ed. David C. Lindberg & Michael H. Shank, CUP, New York, ppb. 2015 pp.9-10


Filed under Myths of Science, Uncategorized