Category Archives: History of Mathematics

Imre and me – a turning point

Today is once again the anniversary of the day I started this blog nine years ago. Nine years‽ I have difficulty believing that I have really churned out blog posts on a regular basis, with only minor breaks, for nine years now. It has become something of a tradition that on my blog anniversary I post something autobiographical and I have decided this year to maintain that tradition and explain why, when asked, I always name Imre Lakatos’ Proof and Refutations not just as my favourite book but as the most important/influential book in my life.

As regular readers might have gathered my life has been anything but the normal career path one might expect from a white, middle class, British man born and raised in Northeast Essex. It has taken many twists and turns, detoured down one or other dark alleyway, gone off the rails once or twice and generally not taken the trajectory that my parents and school teachers might have hoped or expected it to take.

In 1970 I went to university in Cardiff to study archaeology but after one year I decided that archaeology was not what I wanted to do and dropped out. I however continued to live in Cardiff apart from some time I spent living in Belgium and but that’s another story. During this period of my life I earned my living doing a myriad of different things whilst I was supposedly trying to work out what it was that I actually wanted to do. As I’ve said on several occasions I became addicted to the history of mathematics at the age of sixteen and during this phase of my life I continued to teach myself both the history of maths and more generally the history of science.

In 1976 my life took another left turn when I moved to Malmö in Sweden.


Image of Malmö (Elbogen) in Scania, Southern Sweden from a German book (Civitates orbis terrarum, Vol. IV, by G. Braun & F. Hogenberg) .1580 Source: Wikimedia Commons

This was not my first attempt to move to Sweden there had been another abortive attempt a couple of years earlier but that is also another story. This time the move was not instituted by me but by my then partner K. K was a qualified nursery nurse and had applied for a job looking after the children of a pair of doctors in Malmö and her application had been successful. The couple agreed to my accompanying K on the condition that to pay my part of the rent of the flat (that went with the job) I would look after their garden until such time as I found work.

So after witnessing the rained out but brilliant Bob Marley open air in Cardiff football stadium in the summer of 76, we set of for a new life in Malmö. Not having employment my role was to do the cleaning, shopping, cooking and looking after the garden, all things I had been doing for years so no sweat. This left me with a lot of spare time and it wasn’t long before I discovered the Malmö public library. The Swedes are very pragmatic about languages; it is a country with a comparatively small population that lives from international trade so they start learning English in kindergarten. The result in that the public library has lots and lots of English books including a good section on the history and philosophy of mathematics and science, which soon became my happy hunting ground. Card catalogues sorted by subject are a great invention for finding new reading matter on the topic of your choice.

At that point in life I was purely a historian of mathematics with a bit of history of science on the side but in Malmö public library I discovered two books that would change that dramatically. The first was Stephan Körner’s The Philosophy of Mathematics–mathematics has a philosophy I didn’t know that–and the second was Karl Popper’s collection of papers, Conjectures and Refutations: The Growth of Scientific Knowledge. Both found their way back to our flat and were consumed with growing enthusiasm. From that point in my life I was no longer a historian of mathematics and science but had become that strange two-headed beast a historian and philosopher of mathematics and science.

Given the fundamental difference between empirical science and logically formal mathematics my next move might seem to some to be somewhat strange. However, I began to consider the question whether it would be possible to construct a Popperian philosophy of mathematics based on falsification. I gave this question much thought but made little progress. In 1977, for reasons I won’t expand upon here, we returned to the UK and Cardiff.

In Cardiff I continued to pursue my interest in both the histories and philosophies of mathematics and science. In those days I bought my books in a little bookshop in the Morgan Arcade in Cardiff.


Morgan Arcade Source: Wikimedia Commons

One day the owner, whose name I can’t remember but who knew my taste in books said, “I’ve got something here that should interest you” and handed me a copy of Imre Lakatos’ Proofs and Refutations: The Logic of Mathematical Discovery[1]. I now for the first time held in my hands a Popperian philosophy of mathematics or as Lakatos puts it a philosophy of mathematics based on the theories of George Pólya, Karl Popper and Georg Hegel, a strange combination.


This is still the copy that I bought on that fateful day in the small bookshop in the Morgan Arcade forty plus years ago

Lakatos was born Imre Lipschitz in Debrecen, Hungary in 1922. He studied mathematics, physics and philosophy graduating from the University of Debrecen in 1944. Following the German invasion in 1944 he formed a Marxist resistance group with his girlfriend and later wife. During the occupation he changed his Jewish name to Molnár to avoid persecution. After the War he changed it again to Lakatos in honour of his grandmother, who had died in Auschwitz. After the War he became a civil servant in the ministry of education and took a PhD from the University of Debrecen in 1948. He also studied as a post doc at the University of Moscow. Involved in political infighting he was imprisoned for revisionism from 1950 to 1953. One should point out that in the post War period Lakatos was a hard-line Stalinist and strong supporter of the communist government. His imprisonment however changed his political views and he began to oppose the government. Out of prison he returned to academic life and translated Georg Pólya’s How to Solve It[2] into Hungarian. When the Russians invaded in 1956, Lakatos fled to the UK via Vienna. He now took a second PhD at the University of Cambridge in 1961 under R.B. Braithwaite. In 1960 he was appointed to a position at the LSE where he remained until his comparatively early death at the age of 51 in 1974.


Library of the London School of Economics and Political Science – Professor Imre Lakatos, c1960s Source: Wikimedia Commons

The book that I had acquired is a large part of Lakatos’ 1961 PhD thesis, published in book form posthumously[3], and extends Popper’s philosophy of logical discovery into the realm of mathematics. In his seminal work The Logic of Scientific Discovery, (which I had read shortly after discovering his Conjectures and Refutations) Karl Popper moved the discussion in the philosophy of science from justification to discovery. Most previous work in the philosophy of science had been devoted to attempting to justify the truth of accepted scientific theories; Popper’s work was concerned on a formal level at how we arrive at those theories. The same situation existed in the philosophy of mathematics. Philosophers of mathematics were concerned with the logical justification of proven mathematical theorems. Lakatos turned his attention instead to the historical evolution of mathematical theorem.

Proofs and Refutations is written in the form of a Socratic dialogue, although the discussion has more than two participants. A teacher and his class, the students all have Greek letters for names, who are trying to determine the relationship between the number of vertices, edges and faces in polyhedra, V-E+F = 2; a formula now known as the Euler characteristic or Euler’s Gem[4]. The discussion in the class follows and mirrors the evolution in spacial geometry that led to the discovery of this formula. Lakatos giving references to the historical origins of each step in the footnotes. The discussion takes the reader down many byways and cul de sacs and on many detours and around many corners where strange things are waiting to surprise the unwary reader.

The book is thoroughly researched and brilliantly written: erudite and witty, informative on a very high level but a delight to read. I don’t think I can express in words the effect that reading this book had on me. It inspired me to reach out to new heights in my intellectual endeavours, although I knew from the very beginning that I could never possibly reach the level on which Lakatos resided. Before reading Proofs and Refutations, history of mathematics had been a passionate hobby for me; afterwards it became the central aim in my life. I applied to go back to university in Cardiff to study philosophy, having already matriculated six years earlier to study archaeology this meant a one to one interview with a head of department. I completely blew the interview; I always do!

In 1980 I moved to Germany and in 1981 I applied to go to university in Erlangen to study mathematics, which I was able to do after having spent a year learning German. I wanted to choose philosophy as my subsidiary, which meant an interview with a professor. The man I met was Christian Thiel, a historian of logic and mathematics although I didn’t know that at the time, who was just starting his first year as professor, although he had earlier studied in Erlangen. We clicked immediately and although he no longer remembers on that day we discussed the theories of Imre Lakatos. As I documented here Christian Thiel became my mentor and is indirectly more than somewhat responsible for this blog

I have read a lot of books in my life and I continue to do so, although now much more slowly than in the past, but no book has ever had the same impact on me as Proofs and Refutations did the first time I read it. This is why I always name it when asked questions like, what was the most important book you have read or what is your all time favourite book.


[1] Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, eds. John Worrall and Elie Zahar, CUP, Cambridge etc., 1976

[2] How to Solve It is a wonderful little volume describing methods for solving mathematical problems; its methodology can also be used for a much wider range of problems and not just mathematical ones.

[3] Part of the thesis had been published as a series of four papers paper under the title Proofs and Refutations in The British Journal for the Philosophy of Science, 14 1963-64. The main part of the book is an expanded version of those original papers.

[4] I recommend David S. Richeson, Euler’s Gem, University Press Group Ltd., Reprint 2012


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Filed under Autobiographical, History of Mathematics, Uncategorized

Who cares about facts? – Make up your own, it’s much more fun!

Math Horizons is a magazine published by Taylor & Francis for the Mathematical Association of America aimed at undergraduates interested in mathematics: It publishes expository articles about “beautiful mathematics” as well as articles about the culture of mathematics covering mathematical people, institutions, humor, games, cartoons, and book reviews. (Description taken from Wikipedia, which attributes it to the Math Horizons instructions for authors January 3 2009). Apparently, however, authors are not expected to adhere to historical facts, they can, it seems, make up any old crap.

The latest edition of Math Horizons (Volume 25, Issue 3, February 2018) contains an article by a Stephen Luecking entitled Albrecht Dürer’s Celestial Geometry. As I am currently, for other reasons, refreshing my knowledge of Albrecht the mathematician I thought, oh that looks interesting I must read that. I wish I hadn’t.

Luecking’s sub-title seems innocent enough: Renaissance artist Albrecht Dürer designed a specialty compass for astronomical drawings, but when you read the article you discover that Luecking says an awful lot more and most of it is hogwash. What does he have to say?


Albrecht Dürer Self-Portrait 1500 Source: Wikimedia Commons

Albrecht Dürer (1471–1528), noted Renaissance printer and painter, twice left his native Germany for sojourns to Italy, once from 1494 to 1495 and again from 1505 to 1507. During those years his wide-ranging intellect absorbed the culture and thinking of noted artists and mathematicians. Perhaps the most important
 outcome of these journeys was his
introduction to scientific methods. 
His embrace of these methods
 went on to condition his thinking 
for the rest of his life. 

So far so good. However what Dürer absorbed on those journeys to Italy was not scientific methods but linear perspective, the mathematical method, developed in Northern Italy in the fifteenth century, to enable artists to represent three dimensional reality realistically in a two dimensional picture. Dürer played a significant role in distributing these mathematical techniques in Europe north of the Alps. His obsession with mathematics in art led to him developing the theory that the secret of beauty lay in mathematical proportion to which de devoted a large part of the rest of his life. He published the results of his endeavours in his four-volume book on human proportions, Vier Bücher von Menschlicher Proportion, in the year of his death, 1528.


Title page of Vier Bücher von menschlicher Proportion showing the monogram signature of artist Source: Wikimedia Commons

If Dürer wanted to learn scientific methods, by which, as we will see Luecking means astronomy, he could and probably did learn them at home in Nürnberg. Dürer was part of the humanist circle of Willibald Pirckheimer, he close friend and patron.


Engraving of Willibald Pirckheimer at 53 by Albrecht Dürer, 1524. We live by the spirit. The rest belongs to death. Source: Wikimedia Commons

Franconian houses are built around a courtyard; Dürer was born in the rear building of the Pirckheimer house on the market square in Nürnberg. Although his parents bought their own house a few years later Albrecht and Willibald remained close friends and possibly even lovers all of their lives. Pirckheimer was a big supporter of the mathematical sciences—astronomy, mathematics, cartography and astrology—and his circle included, amongst others, Johannes Stabius, Johannes Werner, Erhard Etzlaub, Georg Hartmann, Konrad Heinfogel and Johannes Schöner all of whom were either astronomers, mathematicians, cartographers, instrument makers or globe makers some of them all five and all of them friends of Dürer.

Next up Luecking tells us:

One notable
consequence was Dürer’s abandonment of astrological subject
matter—a big seller for a printer
and publisher such as himself—in favor of astronomy.


Albrecht Dürer Syphilis 1496 Syphilis was believed to have an astrological cause Source: Wikimedia Commons

Luecking offers no evidence or references for this claim, so I could offer none in saying that it is total rubbish, which it is. However I will give one example that shows that Albrecht Dürer was still interested in astrology in 1517. Lorenz Beheim (1457–1521) was a humanist, astrologer, physician and alchemist, who was a canon of the foundation of the St Stephan Church in Bamberg, he was a close friend of both Pirckheimer and Dürer and corresponded regularly with Pirckheimer. In a letter from 8 December 1517 he informed Pirckheimer that Johannes Schöner was coming to Nürnberg with printed celestial globes that could be used for astrology, which if his wished could be acquired by him and Albrecht Dürer. He would not have passed on the information if he thought that they wouldn’t be interested. Beheim also cast horoscopes for both Pirckheimer and Dürer.


Gores for Johannes Schöner’s Celestial Globe 1517  Source: Hans Gaab, Die Sterne Über Nürnberg: Albrecht Dürer und seine Himmelskarten von 1515, Nürnberger Astronomische Gesellschaft, Michael Imhof Verlag, 2015 p. 115


Next up Luecking starts, as he means to go on, with pure poppycock. All of the above Nürnberger mathematician, who all played significant roles in Dürer’s life, were of course practicing astrologers.

Astronomy was not to be a casual interest. Just before his second trip to Italy, Dürer published De scientia motus orbis, a cosmological treatise by the Persian Jewish astronomer Masha’Allah ibn Atharī (ca. 740–815 CE). Since Masha’Allah wrote the treatise for laymen and included ample illustrations, it was a good choice for introducing Europeans to Arabic astronomy.

The claim that Dürer published Masha’Allah’s De scientia motus orbis is so mind bogglingly wrong anybody with any knowledge of the subject would immediately stop reading the article, as it is obviously a complete waste of time and effort. The book was actually edited and published by Johannes Stabius and printed by Weissenburger in Nürnberg in 1504.

The woodcut illustrations came from the workshop of Albrecht Dürer, but probably not from Dürer himself. There were traditionally attributed to Hans Süß von Kulmbach (1480–1522), one of Dürer’s assistants, who went on to become a successful painter in his own right, but modern research has shown that Süß didn’t move to Nürnberg until 1505, a year after the book was published.


Hans Süß portrait  Source: Wikimedia Commons

Although Luecking wants Masha’Allah to be an astronomer he was in fact a very famous astrologer, who amongst other things cast the horoscope for the founding of Bagdad. De scientia motus orbis is indeed a book on Aristotelian cosmology and physics but it includes his theory that there are ten heavenly spheres not eight as claimed by Aristotle. His extra heavenly spheres play a significant role in his astrological theories. It is very common practice for astrologers, starting with Ptolemaeus, to publish their astronomy and astrology in separate books but they are seen as complimentary volumes. From their beginnings in ancient Babylon down to the middle of the seventeenth century astronomy and astrology were always seen as two sides of the same coin.


Title page De scientia motus orbis Although this woodcut is usually titled The Astronomer I personally think the figure looks more like an astrologer Source: Wikimedia Commons

In 1509 Dürer purchased the entire library of Regiomontanus (1436–1476 CE) from the estate of Nuremberg businessman Bernhard Walther. Regiomontanus was Europe’s leading astronomer,
a noted mathematician, and a designer of astronomical instruments. Walther had sponsored Regiomontanus’s residency in Nuremberg between 1471 and 1475. Part of Walther’s largesse was to provide a print shop from which Regiomontanus published the world’s first scientific texts ever printed.

Regiomontanus was of course first and foremost an astrologer and most of those first scientific texts that he published in Nürnberg were astrological texts. Walther did not sponsor Regiomontanus’ residency in Nürnberg but was his colleague and student in his endeavours in the city. An analysis of Walther’s astronomical observation activities in Nürnberg after Regiomontanus’ death show that he too was an astrologer rather than an astronomer. When Regiomontanus came to Nürnberg he brought a very large number of manuscripts with him, intending to edit and publish them. When he died these passed into Walther’s possession, who added new books and manuscripts to the collection. The story of what happened to this scientific treasure when Walther died in 1504 is long and very complicated. In fact Dürer bought not “the entire library” but a mere ten manuscripts not when he bought Walther’s house, the famous Albrecht Dürer House, in 1509 but first in 1522.

In 1515, Dürer and Austrian cartographer and mathematician Johannes Stabius produced the first map of the world portraying the earth as a sphere.


Johannes Statius portrait by Albrecht Dürer Source: Wikimedia Commons

The Stabius-Dürer world map was not “the first map of the world portraying the earth as a sphere”. The earliest know printed world map portraying the earth as a sphere is a woodcut in a Buchlein über die Kunst Corsmographia, (Booklet about the Art of Cosmographia) published in Nürnberg in about 1490. There are others that predate the Stabius-Dürer map most notably on the title page of Waldseemüller’s Die Welt Kugel (The Earth Sphere) published in Straßburg in 1509.

There are no surviving copies of the Stabius-Dürer world map from the sixteenth century so we don’t actually know what it was produced for. The woodblocks survived and were rediscovered in the 18th century.

It is however dedicated to both the Emperor Maximilian, Stabius’s employer who granted the printing licence, and Cardinal Matthäus Lang, so it might well have been commissioned by the latter. Lang commissioned the account of Magellan’s circumnavigation on which Schöner based his world map of that circumnavigation.

Afterward, Stabius proposed continuing their collaboration by publishing a star map—the first such map published in Europe. Their work relied heavily on data assembled by Regiomontanus, plus refinements from Walther.

It will probably not surprise you to discover that this was not “the first such map published in Europe. It’s the first printed one but there are earlier manuscript ones, two of which from 1435 in Vienna and 1503 in Nürnberg probably served as models for the Stabius–Dürer–Heinfogel one. Their work did not rely “heavily on data assembled by Regiomontanus, plus refinements from Walther” but was based on Ptolemaeus’ star catalogue from the Almagest. There is a historical problem in that there was not printed copy of that star catalogue available at the time so they probably work from one or more manuscripts and we don’t know which one(s). The star map contains the same dedications to Maximilian and Lang as the world map so one again might have been a commission from Lang, Stabius acting as the commissioning agent. Stabius and Lang studied together at the University of Ingolstadt.


Stabs-Dürer-Heinfogel Star Map Northern Hemisphere Source: Ian Ridpath’s Star Tales

For more details on the star maps go here

The star map required imprinting the three- dimensional dome of the heavens onto a two- dimensional surface without extreme distortions, a task that fell to Stabius. He used a stereographic projection. In this method, rays originate at the pole in the opposite hemisphere, pass through a given point in the hemisphere, and yield a point on a circular surface.

You will note that I have included the name of Konrad Heinfogel to the producers of the map and it was actually he, and not Stabius, who was responsible for the projection of the map and the location of the individual stars. In fact in this project Johannes Stabius as commissioning agent was project leader, Konrad Heinfogel was the astronomical expert and Albrecht Dürer was the graphic artist hired to draw the illustration. Does one really have to point out that in the sixteenth century star maps were as much, if not more, for astrologers than for astronomers.

Luecking now goes off on an excurse about the history of stereographic projection, which ends with the following paragraph.

As the son of a goldsmith, Dürer’s exposure to stereographic projection would have been by way of the many astrolabes being fabricated in Nuremburg, then Europe’s major center for instrument makers. As the 16th century moved on, the market grew for such scientific objects as astrology slipped into astronomy. Handcrafted brass instruments, however, were affordable only to the wealthy, whereas printed items like the Dürer-Stabius maps reached a wider market.

Nürnberg was indeed the major European centre for the manufacture of scientific instruments during Dürer’s lifetime but scientific instrument makers and goldsmiths are two distinct professional groups, so Luecking’s argument falls rather flat, although of course Dürer would have well acquainted with the astrolabes made by his mathematical friends. Astrolabes are of course both astrological and astronomical instruments and astrology did not slip into astronomy during the 16th century. In fact the 16th century is regarded by historians as the golden age of astrology.

There now follows another excurse on the epicycle-deferent model of planetary orbits as a lead up to the articles thrilling conclusion.

In his 1525 book Die Messerung (On Measurement), Dürer presents an instrument of his own design used to draw these and other more general curves. This compass for drawing circles upon circles consisted of four telescoping arms and calibrated dials. An arm attached to the first dial could rotate in a full circle, a second arm fixed to another dial mounted on the end of this first arm could rotate around the end of the first arm, and so on.


Dürer’s four arm compass


Underweysung der Messung mit dem Zirkel und Richtscheyt Title Page

The title of Dürer’ 1525 book is actually Underweysung der Messung mit dem Zirckel und Richtscheyt (Instructions for Measuring with Compass and Straightedge). It is a basic introduction to geometry and its applications, which Dürer wrote when he realised that his Vier Bücher von Menschlicher Proportion was too advanced for the artist apprentices that he thought should read it. The idea was first read and digest the Underweysung then read the Vier Bücher von Menschlicher Proportion.

Luecking tells us that:

As a trained metalsmith, Dürer possessed the expertise to craft this complex tool. Precision calibration and adjustable arms allowed its user to plot an endless number of curves by setting the length of each telescoping arm and determining the rate at which the arms turned. This, in effect, constituted manual programming by setting the parameters of each curve plotted.

As a teenager Dürer did indeed serve an apprenticeship under his father as a goldsmith, but immediately on completing that apprenticeship he undertook a second apprenticeship as a painter with Michael Wolgemut from 1486 to 1490 and dedicated his life to painting and fine art printing. Luecking has already correctly stated that Nürnberg was the major European centre for scientific instrument making and Dürer almost certainly got one of those instrument makers to produce his multi-armed compass. Luecking describes the use to which Dürer put this instrument in drawing complex geometrical curves. He then goes on to claim that Dürer might actually have constructed it to draw the looping planetary orbits produced by the epicycle-deferent model. There is absolutely no evidence for this in the Underweysung and Luecking’s speculation is simple pulled out of thin air.

To summarise for those at the back who haven’t been paying attention. Dürer did not absorb scientific methods in Italy. He did not abandon astrology for astronomy. He didn’t publish Masha’Allah’s De scientia motus orbis, Johannes Stabius did. Dürer only bought ten of Regiomontanus’ manuscripts and not his entire library. The Stabius-Dürer world map was not “the first map of the world portraying the earth as a sphere”. The Stabius–Dürer–Heinfogel star charts were the first star-charts printed in Europe but by no means the first ones published. Star charts are as much astrological, as they are astronomical. Astrology did not slip into astronomy in the 16th century, which was rather the golden age of astrology. There is absolutely no evidence that Dürer’s multi-arm compass, as illustrated in his geometry book the Underweysung, was ever conceived for drawing the looping orbits of epicycle-deferent planetary models, let alone used for this purpose.

It comes as no surprise that Stephen Luecking is not a historian of mathematics or art for that matter. He is the aged (83), retired chairman of the art department of DePaul University in Chicago.

Whenever I come across an article as terrible as this one published by a leading scientific publisher in a journal from a major mathematical organisation such as the MAA I cringe. I ask myself if the commissioning editor even bothered to read the article; it was certainly not put out to peer review, as any knowledgeable Dürer expert would have projected it in an elegant geometrical curve into his trashcan. Above all I worry about the innocent undergraduates who are subjected to this absolute crap.


















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

Exposing Galileo’s strawmanning

There is a widespread, highly erroneous, popular perception in the world, much loved by gnu atheists and supporters of scientism, that as soon as Petreius published Copernicus’s De Revolutionibus in 1543 the question as to which was the correct astronomical/cosmological system for the cosmos was as good as settled and that when Galileo published his Dialogo[1] everything was finally done and dusted and anybody who still persisted in opposing the acceptance of the heliocentric world view, did so purely on grounds of ignorant, anti-science, religious prejudice. Readers of this blog will know that I have expended a certain amount of energy and several thousand words over the years countering this totally mistaken interpretation of the history of astronomy in the early modern period and today I’m going to add even more words to the struggle.

Galileo is held up by his numerous acolytes as a man of great scientific virtue, who preached a gospel of empirical scientific truth in the face of the superstitious, faith based errors of his numerous detractors; he was a true martyr for science. The fact that Galileo was capable of scientific skulduggery does not occur to them, but not only was he capable of such, his work is littered with examples of it. One of his favourite tactics was not to present his opponents true views when criticising them but to create a strawman, claiming that this represents the views of his opponent and then to burn it down with his always-red-hot rhetorical flamethrower.

Towards the end of The First Day in the Dialogo, Galileo has Simplicio, the fall guy for geocentricity, introduce a “booklet of theses, which is full of novelties.” Salviati, who is the champion of heliocentricity and at the same time Galileo’s mouthpiece, ridicules this booklet as producing arguments full of “falsehoods and fallacies and contradictions” and 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.” Galileo goes on to do a polemical hatchet job on what he claims are the main arguments in said “booklet of theses.” Amongst others he accuses the author of “setting himself up to refute another’s doctrine while remaining ignorant of the basic foundations upon which the whole structure are supported.”

The “booklet of theses”, which Galileo doesn’t name, is in fact the splendidly titled:


English translation of the Latin title page Source: Notre Dame Press

Now most of the acolytes who fervently praise Galileo as the great defender of science against superstition probably have no idea who Johann Georg Locher was but they might well have heard of Christoph Scheiner, who was famously embroiled in a dispute with Galileo over the nature of sunspots and who first observed them with a telescope. In fact the authorship of the Mathematical Disquisitions has often falsely attributed to Scheiner and Galileo’s demolition of it seen as an extension of that dispute and it’s sequel in the pages of his Il Saggiatore.

Whereas Galileo’s Dialogo has been available to the general reader in a good English translation by Stillman Drake since 1953, anybody who wished to consult Locher’s Mathematical Disquisitions in order to check the veracity or lack thereof of Galileo’s account would have had to hunt down a 17th century Latin original in the rare books room of a specialist library. The playing field has now been levelled with the publication of an excellent modern English translation of Locher’s booklet by Renaissance Mathematicus friend, commentator and occasional guest contributor Chris Graney[2].


Graney’s translation (Christopher M. Graney, Mathematical Disquisitions: The Booklet of Theses Immortalised by Galileo, University of Notre Dame Press, Notre Dame, Indiana, 2017)  presents a more than somewhat different picture of Locher’s views on astronomy to that served up by Galileo in the Dialogo and in fact gives us a very clear picture of the definitely rational arguments presented by the opponents to heliocentricity in the early part of the seventeenth century. The translation contains an excellent explanatory introduction by Graney, extensive endnotes explaining various technical aspects of Locher’s text and also some of the specific translation decisions of difficult terms. (I should point out that another Renaissance Mathematicus friend, Darin Hayton acted as translation consultant on this volume). There is an extensive bibliography of the works consulted for the explanatory notes and an excellent index.

The book is very nicely presented by Notre Dame Press, with fine reproductions of Locher’s wonderful original illustrations.


Locher’s illustration to his discussion of diurnal rotation p. 32

Graney’s translation provides a great addition to his previous Setting Aside All Authority, which I reviewed here. Graney is doing sterling work in adjusting the very distorted view of the astronomical system discussion in the first half of the seventeenth century and anybody, who is seriously interested in learning the true facts of that discussion, should definitely read his latest contribution.




[1] By a strange cosmic coincidence the first printed copy of the Dialogo was presented to the dedicatee Ferdinando II d’Medici, Grand Duke of Tuscany 386 years ago today on 22 February 1632.

[2] At the end of my review of Setting Aside All Authority I wrote the following, which applies equally to this review; in this case I provided one of the cover blurbs for Chris’ latest book.

Disclosure; Chris Graney is not only a colleague, but he and his wife, Christina, are also personal friends of mine. Beyond that, Chris has written, at my request, several guest blogs here at the Renaissance Mathematicus, all of which were based on his research for the book. Even more relevant I was, purely by accident I hasten to add, one of those responsible for sending Chris off on the historical trail that led to him writing this book; a fact that is acknowledged on page xiv of the introduction. All of this, of course, disqualifies me as an impartial reviewer of this book but I’m going to review it anyway. Anybody who knows me, knows that I don’t pull punches and when the subject is history of science I don’t do favours for friends. If I thought Chris’ book was not up to par I might refrain from reviewing it and explain to him privately why. If I thought the book was truly bad I would warn him privately and still write a negative review to keep people from wasting their time with it. However, thankfully, none of this is the case, so I could with a clear conscience write the positive review you are reading. If you don’t trust my impartiality, fair enough, read somebody else’s review.

Addendum: The orthography of the neologism in the title was change—23,02,18— following a straw pole on Twitter


Filed under Book Reviews, Early Scientific Publishing, History of Astronomy, History of Mathematics, Myths of Science, Renaissance Science

Christmas Trilogy 2017 Part 3: Kepler’s big book

Johannes Kepler was incredibly prolific, he published over eighty books and booklets over a very wide range of scientific and mathematical topics during his life. As far as he was concerned his magnum opus was his Ioannis Keppleri Harmonices mundi libri V (The Five Books of Johannes Kepler’s The Harmony of the World) published in 1619 some twenty years after he first conceived it. Today in popular #histsci it is almost always only mentioned for the fact that it contains the third of his laws of planetary motion, the harmonic law. However it contains much, much more of interest and in what follows I will attempt to give a brief sketch of what is in fact an extraordinary book.


A brief glace at the description of the ‘five books’ thoughtfully provided by the author on the title page (1) would seem to present a mixed bag of topics apparently in some way connected by the word or concept harmonic. In order to understand what we are being presented with we have to go back to 1596 and Kepler’s first book Mysterium Cosmographicum (The Cosmographic Mystery). In this slim volume Kepler presents his answer to the question, why are there only six planets? His, to our eyes, surprising answer is that the spaces between the planets are defined by the regular so-called Platonic solids and as the are, and can only be, five of these there can only be six planets.

Using the data from the greatest and least distances between the planets in the Copernican system, Kepler’s theory produces an unexpectedly accurate fit. However the fit is not actually accurate enough and in 1598 Kepler began working on a subsidiary hypothesis to explain the inaccuracies. Unfortunately, the book that he had planned to bring out in 1599 got somewhat delayed by his other activities and obligations and didn’t appear until 1619 in the form of the Harmonice mundi.

The hypothesis that Kepler presents us with is a complex mix of ideas taken from Pythagoras, Plato, Euclid, Proclus and Ptolemaeus centred round the Pythagorean concept of the harmony of the spheres. Put very simply the theory developed by the Pythagoreans was that the seven planets (we are talking geocentric cosmology here) in their orbits form a musical scale than can, in some versions of the theory, only be heard by the enlightened members of the Pythagorean cult. This theory was developed out of the discovery that consonances (harmonious sounds) in music can be expressed in the ratio of simple whole numbers to each other (the octave for example is 1:2) and the Pythagorean belief that the integers are the building block of the cosmos.

This Pythagorean concept winds its way through European intellectual history, Ptolemaeus wrote a book on the subject, his Harmonice and it is the reason why music was one of the four disciplines of the mathematical quadrivium along with arithmetic, geometry and astronomy. Tycho Brahe designed his Uraniburg so that all the architectonic dimensions from the main walls to the window frames were in Pythagorean harmonic proportion to one another.


Tycho Brahe’s Uraniborg Blaeus Atlas Maior 1663 Source: Wikimedia Commons

It is also the reason why Isaac Newton decided that there should be seven colours in the rainbow, to match the seven notes of the musical scale. David Gregory tells us that Newton thought that gravity was the strings upon which the harmony of the spheres was played.

In his Harmony Kepler develops a whole new theory of harmony in order to rescue his geometrical vision of the cosmos. Unlike the Pythagoreans and Ptolemaeus who saw consonance as expressed by arithmetical ratios Kepler opted for a geometrical theory of consonance. He argued that consonances could only be constructed by ratios between the number of sides of regular polygons that can be constructed with a ruler and compass. The explication of this takes up the whole of the first book. I’m not going to go into details but interestingly, as part of his rejection of the number seven in his harmonic scheme Kepler goes to great lengths to show that the heptagon construction given by Dürer in his Underweysung der Messung mit dem Zirckel und Richtscheyt is only an approximation and not an exact construction. This shows that Dürer’s book was still being read nearly a hundred years after it was originally published.


In book two Kepler takes up Proclus’ theory that Euclid’s Elements builds systematically towards the construction of the five regular or Platonic solids, which are, in Plato’s philosophy, the elemental building blocks of the cosmos. Along the way in his investigation of the regular and semi-regular polyhedra Kepler delivers the first systematic study of the thirteen semi-regular Archimedean solids as well as discovering the first two star polyhedra. These important mathematical advances don’t seem to have interested Kepler, who is too involved in his revolutionary harmonic theory to notice. In the first two books Kepler displays an encyclopaedic knowledge of the mathematical literature.


The third book is devoted to music theory proper and is Kepler’s contribution to a debate that was raging under music theorist, including Galileo’s father Vincenzo Galilei, about the intervals on the musical scale at the beginning of the seventeenth century. Galilei supported the so-called traditional Pythagorean intonation, whereas Kepler sided with Gioseffo Zarlino who favoured the ‘modern’ just intonation. Although of course Kepler justification for his stance where based on his geometrical arguments. Another later participant in this debate was Marin Mersenne.


In the fourth book Kepler extends his new theory of harmony to, amongst other things, his astrology and his theory of the astrological aspects. Astrological aspects are when two or more planets are positioned on the zodiac or ecliptic at a significant angle to each other, for example 60° or 90°. In his Harmonice, Ptolemaeus, who in the Renaissance was regarded as the prime astrological authority, had already drawn a connection between musical theory and the astrological aspects; here Kepler replaces Ptolemaeus’ theory with his own, which sees the aspects are being derived directly from geometrical constructions. Interestingly Kepler, who had written and published quite extensively on astrology, rejected nearly the whole of traditional Greek astrology as humbug keeping only his theory of the astrological aspects as the only valid form of astrology. Kepler’s theory extended the number of influential aspects from the traditional five to twelve.

The fifth book brings all of the preceding material together in Kepler’s astronomical/cosmological harmonic theory. Kepler examines all of the mathematical aspects of the planetary orbits looking for ratios that fit with his definitions of the musical intervals. He finally has success with the angular velocities of the planets in their orbits at perihelion and aphelion. He then examines the relationships between the tones thus generated by the different planets, constructing musical scales in the process. What he in missing in all of this is a grand unifying concept and this lacuna if filled by his harmonic law, his third law of planetary motion, P12/P22=R13/R23.


There is an appendix, which contains Kepler’s criticisms of part of Ptolemaeus’ Harmonice and Robert Fludd’s harmony theories. I blogged about the latter and the dispute that it triggered in an earlier post

With his book Kepler, who was a devoted Christian, was convinced that he had revealed the construction plan of his geometrical God’s cosmos. His grandiose theory became obsolete within less than fifty years of its publication, ironically pushed into obscurity by intellectual forces largely set into motion by Kepler in his Astronomia nova, his Epitome astronomiae Copernicanae and the Rudolphine Tables. All that has survived of his great project are his mathematical innovations in the first two books and the famous harmonic law. However if readers are prepared to put aside their modern perceptions and prejudices they can follow one of the great Renaissance minds on a fascinating intellectual journey into his vision of the cosmos.

(1) All of the illustration from the Harmonice mundi in this post are taken from the English translation The Harmy of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aston, A.M. Duncan and J.V. Field, American Philosophical Society, 1997


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

Christmas Trilogy 2017 Part 1: Isaac the Imperator

Isaac Newton came from a fairly humble although not poor background. His father was a yeoman farmer in Lincolnshire, who unfortunately died before he was born. A yeoman farmer owned his own land and in fact the Newton’s were the occupants of the manor house of Woolsthorpe-by-Colsterworth.

Woolsthorpe Manor, Woolsthorpe-by-Colsterworth, Lincolnshire, England. This house was the birthplace and the family home of Isaac Newton.
Source: Wikimedia Commons

Destined to become a farmer until he displayed little aptitude for life on the land, his mother was persuaded by the local grammar school master to let him complete his education and he was duly dispatched off to Cambridge University in 1661. Although anything but poor, when Newton inherited the family estates they generated an income of £600 per annum, at a time when the Astronomer Royal received an income of £100 per annum, his mother enrolled him at Cambridge as a subsizar, that is a student who earned his tuition by working as a servant. I personally think this reflects the family’s puritan background rather than any meanness on the mother’s part.

In 1664 Newton received a scholarship at Trinity and in 1667 he became a fellow of the college. In 1669 he was appointed Lucasian professor of mathematics. Cambridge was in those days a small market town and a bit of a backwater. The university did not enjoy a good reputation and the Lucasian professorship even less of one. Newton lived in chambers in Trinity College and it was certainly anything but a life of luxury.

Trinity College Great Court
Source: Wikimedia Commons

There is an amusing anecdote about David Hilbert writing to the authorities of Trinity at the beginning of the twentieth century to complain about the fact that Godfrey Hardy, whom he regarded as one of the greatest living mathematicians, was living in what he regarded as a squalid room without running water or adequate heating. What Hilbert didn’t realise was that Hardy would never give up this room because it was the one that Newton had inhabited.

Newton remained an obscure and withdrawn Cambridge don until he presented the Royal Society with his reflecting telescope and published his first paper on optics in 1672. Although it established his reputation, Newton was anything but happy about the negative reactions to his work and withdrew even further into his shell. He only re-emerged in 1687 and then with a real bombshell his Philosophiæ Naturalis Principia Mathematica, which effectively established him overnight as Europe’s leading natural philosopher, even if several of his major competitors rejected his gravitational hypothesis of action at a distance.

Having gained fame as a natural philosopher Newton, seemingly having tired of the provinces, began to crave more worldly recognition and started to petition his friends to help him find some sort of appropriate position in London. His lobbying efforts were rewarded in 1696 when his friend and ex-student, Charles Montagu, 1st Earl of Halifax, had him appointed to the political sinecure, Warden of the Mint.

Newton was no longer a mere university professor but occupant of one of the most important political sinecures in London. He was also a close friend of Charles Montagu one of the most influential political figures in England. By the time Montagu fell from grace Newton was so well established that it had little effect on his own standing. Although Montagu’s political opponents tried to bribe him to give up his, now, Mastership of the Mint he remained steadfast and his fame was such that there was nothing they could do to remove him from office. They wanted to give the post to one of their own. Newton ruled the Mint with an iron hand like a despot and it was not only here that the humble Lincolnshire farm lad had given way to man of a completely different nature.

As a scholar, Newton held court in the fashionable London coffee houses, surrounded by his acolytes, for whom the term Newtonians was originally minted, handing out unpublished manuscripts to the favoured few for their perusal and edification. Here he was king of the roost and all of London’s intellectual society knew it.

He became President of the Royal Society in 1703 and here with time his new personality came to the fore. When he became president the society had for many years been served by absentee presidents, office holders in name only, and the power in the society lay not with the president but with the secretary. When Newton was elected president, Hans Sloane was secretary and had already been so for ten years and he was not about to give up his power to Newton. There then followed a power struggle, mostly behind closed doors, until Newton succeeded in gaining power in about 1610 1710, Sloane, defeated resigned from office in 1613 1713 but got his revenge by being elected president on Newton’s death. Now Newton let himself be almost literally enthroned as ruler of the Royal Society.

Isaac Newton’s portrait as Royal Society President Charles Jervas 1717
Source: Royal Society

The president of the society sat at table on a raised platform and on 20 January 1711 the following Order of the Council was made and read to the members at the next meeting.

That no Body Sit at the Table but the President at the head and the two Secretaries towards the lower end one on the one Side and the other Except Some very Honoured Stranger, at the discretion of the President.

When the society was first given its royal charter in 1660, although Charles II gave them no money he did give them an old royal mace as a symbol of their royal status. Newton established the custom that the mace was only displayed on the table when the president was in the chair. When Sloane became president his first act was to decree that the mace was to be displayed at all meetings, whether the president was present or not. Newton ruled over the meetings with the same iron hand with which he ruled over the Mint. Meeting were conducted solemnly with no chit chat or other disturbances as William Stukeley put it:

Indeed his presence created a natural awe in the assembly; they appear’d truly as a venerable consessus Naturae Consliariorum without any levity or indecorum.

Perhaps Newton’s view of himself in his London years in best reflected in his private habitat. Having lived the life of a bachelor scholar in college chambers for twenty odd years he now obtained a town house in London. He installed his niece Catherine Barton, who became a famous society beauty, as his housekeeper and lived the life of a London gentleman, albeit a fairly quiet one. However his personal furnishings seem to me to speak volumes about how he now viewed himself. When he died an inventory of his personal possessions was made for the purpose of valuation, as part of his testament. On the whole his household goods were ordinary enough with one notable exception. He possessed crimson draperies, a crimson mohair bed with crimson curtains, crimson hangings, a crimson settee. Crimson was the only colour mentioned in the inventory. He lived in an atmosphere of crimson. Crimson is of course the colour of emperors, of kings, of potentates and of cardinals. Did the good Isaac see himself as an imperator in his later life?


All the quotes in this post are taken from Richard S, Westfall’s excellent Newton biography Never at Rest.




Filed under History of Astronomy, History of Mathematics, History of Optics, History of Physics, History of science, Newton

Men of Mathematics

This is something that I wrote this morning as a response on the History of Astronomy mailing list; having written it I have decided to cross post it here.

John Briggs is the second person in two days, who has recommended Eric Temple Bell’s “Men of Mathematics”. I can’t remember who the first one was, as I only registered it in passing, and it might not even have been on this particular mailing list. Immediately after John Briggs recommended it Rudi Lindner endorsed that recommendation. This series of recommendations has led me to say something about the role that book played in my own life and my view of it now.

“Men of Mathematics” was the first book on the history of science and/or mathematics that I ever read. I was deeply passionate fan of maths at school and my father gave me Bell’s book to read when I was sixteen years old. My other great passion was history and I had been reading history books since I taught myself to read at the age of three. Here was a book that magically combined my two great passions. I devoured it. Bell has a fluid narrative style and the book is easy to read and very stimulating.

Bell showed me that the calculus, that I had recently fallen in love with, had been invented/discovered (choose the verb that best fits your philosophy of maths), something I had never even considered before. Not only that but it was done independently by two of the greatest names in the history of science, Newton and Leibniz, and that this led to one of the most embittered priority and plagiarism disputes in intellectual history. He introduced me to George Boole, whom I had never heard of before and whose work and its reception in the 19th century I would seriously study many years later in a long-year research project into the history of formal or mathematical logic, my apprenticeship as a historian of science.

Bell’s tome ignited a burning passion for the history of mathematics in my soul, which rapidly developed into a passion for the whole of the history of science; a passion that is still burning brightly fifty years later. So would I join the chorus of those warmly recommending “Men of Mathematics”? No, actually I wouldn’t.

Why, if as I say Bell’s book played such a decisive role in my own development as a historian of mathematics/science, do I reject it now? Bell’s florid narrative writing style is very seductive but it is unfortunately also very misleading. Bell is always more than prepared to sacrifice truth and historical accuracy for a good story. The result is that his potted biographies are hagiographic, mythologizing and historically inaccurate, often to a painful degree. I spent a lot of time and effort unlearning a lot of what I had learnt from Bell. His is exactly the type of sloppy historiography against which I have taken up my crusade on my blog and in my public lectures in my later life. Sorry but, although it inspired me in my youth, I think Bell’s book should be laid to rest and not recommended to new generations.



Filed under Book Reviews, History of Logic, History of Mathematics, History of science, Myths of Science

Can we please stop (mis)quoting Albert on Emmy, it’s demeaning?

Emmy Noether, whom I’ve blogged about a couple of times in the past, is without any doubt one of the greats in the history of mathematics, as is well documented by the testimonials written by some of the greatest contemporary mathematicians and physicists and collected in Auguste Dick’s slim but well research biography, Emmy Noether: 1882–1935.

Emmy Noether c. 1930
Source:Wikimedia Commons

Yesterday was World Maths Day and the Royal Society tweeted portraits of mathematicians with links to articles all day, one of those tweets was about Emmy Noether. The tweet included a paraphrase of a well known quote from Albert Einstein, after all what could be better than a quote from old Albert the greatest of the great? Well almost anything actually, as the Einstein quote is highly demeaning. As given informally by the Royal Society it read as follows:

Emmy Noether was described by Einstein as the most important woman in the history of mathematics.

What Einstein actually wrote in a letter to the New York Times on the occasion of her death in 1935 was the following:

In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered, methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.

In the same year, but before she died, Norbert Wiener wrote:

Miss Noether is… the greatest woman mathematician who has ever lived; and the greatest woman scientist of any sort now living, and a scholar at least on the plane of Madame Curie.

Now I’m sure that the Royal Society, Albert Einstein and Norbert Wiener all meant well, but take a step back and consider what all of them said in their different ways, Emmy Noether was pretty good for a woman [my emphasis].

Emmy Noether was one of the greatest mathematicians of the twentieth century, male or female, man or woman, about that there is absolutely no doubt, to qualify that praise with the term woman is quite simple demeaning.

In my mind it triggers the text of Melanie Safka’s mega pop hit from 1971, Brand New Key:

I ride my bike, I roller skate, don’t drive no car

Don’t go too fast, but I go pretty far

For somebody who don’t drive

I been all around the world

Some people say, I done all right for a girl [my emphasis]

On twitter, space archaeologist, Alice Gorman (@drspacejunk) took it one stage further, in my opinion correctly, and asked, “Dare I cite Samuel Johnson’s aphorism about the talking dog?” For those who are not up to speed on the good doctor’s witticisms:

I told him I had been that morning at a meeting of the people called Quakers, where I had heard a woman preach. Johnson: “Sir, a woman’s preaching is like a dog’s walking on his hind legs. It is not done well; but you are surprised to find it done at all.” – Boswell: Life

Can we please in future when talking about Emmy Noether resist the temptation to quote those who affix their praise of her mathematical talents with the term woman and just acknowledge her as a great mathematician?



Filed under History of Mathematics, Ladies of Science