Like father like son

Anybody who did physics at school has almost certainly at some point stumbled across Snell’s Law; this is a law in optics that states that for a light ray travelling from one medium into another the ratio of the sines of the angles of incidence and refraction is constant; that ratio is known as the index of refraction of those two media. We’ll come back to Snell’s Law later but who was Snell? Actually there were two of them, father and son, it’s the son who gave his name to the law and actually the family name only has one ‘l’, it’s Snel not Snell. Rudolph Snel van Royan, the father, and Willebrord Snel van Royan, the son, both played a leading role in the establishment of the mathematical sciences in the then very young Dutch Republic and in what follows I will sketch the role that they played. The English misspelling of the family name comes from a false shortening of the Latinised version of the name, Snellius, used by both as a nom de plume, as was they custom in the Early Modern Period.

Rudolph Snel (1546–1613) was born in the town of Oudewater; he studied Hebrew, mathematics and philosophy at various German universities and taught for some time at the University of Marburg. He acquired a medical doctorate in Italy and settled in the town of his birth in 1575. When the seven United Provinces broke away from the Spanish Netherlands in 1568 they lost their university in Leuven, leaving the Dutch Republic without a university. The University of Leiden was founded in 1575 but didn’t have a mathematics department. In 1579 some students asked Rudolph Snel to hold maths lectures at the university. In 1580 he was given a temporary lectureship, which was converted to a professor extraordinarius in 1581. It would be twenty year before he was finally promoted to professor ordinarius. He was however, viewed historically, the first professor of mathematics of the young Dutch Republic.

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Rudolf Snell (1546-1613), Hebräist, Mathematiker; aus der Bibliotheca chalcographica von Jean-Jacques Boissard und Theodor de Bay Source: Wikimedia Commons

Rudolph Snel was not in anyway a great mathematician but he was a convinced Ramist i.e. he propagated the pedagogic theories of the French philosopher Pierre de la Ramée (1515–1572), who believed in a simplified, practical, anti-Aristotelian pedagogic. As a result Snel was regarded as modern (not in a positive sense) and controversial. In the history of science Rudolph is by no means as significant as his son but he in notable for two things. Firstly, he acted as an informal tutor in mathematics to the theology student Isaac Beeckman (1588–1637), who went on to have a major influence in the development of the mechanical philosophy. Rudolph Snel was also probably the first person to hold lectures on the newly invented telescope, doing so already in 1609. He was certainly the source of Johann Fabricius’ knowledge of the instrument; Fabricius going on to become one of the first telescopic discoverers of sunspots and the first to publish that discovery. Rudolph Snel also possibly played a role in a Dutch telescope acquired by Simon Marius.

Naturally, Willebrord Snel (1580–1626) was born in Leiden. He originally studied law but also devoted much time to mathematics and in 1600 began to teach mathematics at the university. Shortly afterward he left Leiden to go on a study tour of Europe. He visited Adriaan van Roomen the professor of medicine and mathematics in Würzburg, Tycho Brahe and Kepler in Prague and Michael Maestlin in Tübingen. In 1602 he studied law in Paris from whence he went to Switzerland with his father finally returning to the Dutch Republic in 1604.

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Willebrord Snel Source: Wikimedia Commons

Back in Leiden he devoted his life to mathematics, translating Simon Stevin’s Wisconstighe Ghedachtenissen into Latin as Hypomnemata Mathematica (this was the collected volume of Stevin’s textbooks on mathematical, engineering and science topics) and working on a reconstruction of Apollonius’s books on plane loci; both works were published in 1608. In the same year Willebrord was awarded his M.A. Over the years he assisted his father at the university lecturing on mathematics and when Rudolph died in 1613, Willebrord succeeded him as professor. Like his Father Willebrord was a Ramist and in 1613 he published Ramus’s Arithmetica with a commentary.

Snel’s main interest was geodesy and he was the first to attempt to determine the length of one degree of meridian arc, and thus the circumference of the Earth, using the method of triangulation first suggested by the Frisian mathematicus Gemma Frisius. He carried out his triangulation using instruments (including a large quadrant) designed and constructed by Willem Janszoon Blaeu.

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Quadrant built by Willem Janszoon Blaeu for Willebrord Snel Image: Museum Boerhaave, Leiden Source: Wikimedia Commons

Snel mapped the stretch between Alkmaar and Bergen op Zoom, two towns roughly on the same meridian. He published his results in his Eratosthenes batavus in Leiden in 1617. Named for the Alexandrian geographer Eratosthenes, who also determined the size of the Earth, and Batavus the Roman name for the Netherlands. Later he extended his triangulation net over most of the Dutch Republic.

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Snel’s Triangulation of the Dutch Republic from 1615 Source: Wikimedia Commons

Snel also published works on astronomy, navigations (coining the term loxodromes for Pedro Nunez’s so-called rhumb lines), and trigonometry. Using the method devised by Van Ceulen—another Leiden professor some of whose work he translated into Latin—he calculated π to thirty-four decimal places. Willebrord died comparatively young in 1626.

You will note that I’ve said nothing in the above about Snell’s Law, the law of refraction. The law of refraction was discovered independently by at least five different scholars of whom Snel was chronologically the third. Refraction plays an important role in observation astronomy, as the rays of light coming from celestial objects are refracted by the earth’s atmosphere, the amount of refraction, i.e. bending, varying with the altitude of the object under observation. The rays coming from an object on the horizon are bent differently to those of one observed overhead. Ptolemaeus was well aware of this problem but was despite extensive research unable to determine the law governing the relationship between the angle of incidence and the angle of refraction. The problem was also well known to Islamic and Medieval European astronomers.

It would appear that the first to discover the correct ratio was the tenth century Persian mathematician Abū Saʿd al-ʿAlāʾ ibn Sahl. In his optical treatise Ibn Sahl gives a geometrical equivalent to the trigonometrical law of refraction.

A Pioneer in Anaclastics: Ibn Sahl on Burning Mirrors and Lenses

Reproduction of Millī MS 867 fol. 7r, showing Ibn Sahl’s discovery of the law of refraction (from Rashed, 1990). Source: Wikipedia Commons

However his work remained unknown even to his Islamic contemporaries and so played no role in the history of optics. The next to discover the law of refraction was the English polymath Thomas Harriot.

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Portrait often claimed to be Thomas Harriot (1602), which hangs in Oriel College, Oxford. Source: Wikimedia Commons

Interestingly although Harriot discussed the problem extensively in a correspondence with Johannes Kepler he never revealed that he had the correct law and also never published it. So his discovery, like Ibn Sahl’s, remained unknown. Kepler needed the law for his, important in the history of optics, Dioptrice from 1611; not knowing it he used an approximations for his pioneering work on the function of lenses, which proved adequate.

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Source: Wikimedia Commons

The first to publish the correct law was Descartes in his Dioptrique issued as an appendix to his Discours de la méthode in 1637. In France the law is still referred to as Descartes’ Law.issued as an appendix to his Discours de la méthode in 1637. In France the law is still referred to as Descartes’ Law.

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Descartes’ explanation of the law of refraction by analogy from his Dioptrique Source: Wikimedia Commons

Christiaan Huygens first discovered that Willebrord Snel had stated the law in an unpublished manuscript from 1621 and falsely accused Descartes of plagiarism. In the end, Ibn Sahl’s and Harriot’s discoveries remaining unknown, Snel’s priority was acknowledged and the law of refraction became Snell’s Law. Strangely, somewhat later than Descartes publication, the Scottish optician James Gregory, unaware of this, rediscovered the law based on his research into Kepler’s work in optics.

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James Gregory Source: Wikimedia Commons

The whole story is a wonderful example for multiple-discovery in the history of science, which are in fact much more common than many people think. It also illustrates that important scientific discoveries can get lost and have to be rediscovered, again a not uncommon occurrence.

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The Prince and Astronomer

The outline of Galileo’s rise to fame is one of the most well-known stories in the history of science. How he heard about the invention of the telescope sometime in 1609 and then built an improved version of the instrument. How he began observing the heavens towards the end of the year and at the beginning of 1610 discovered the four largest moons of Jupiter, the most spectacular astronomical discovery since the beginnings of the discipline in antiquity. Realising what he had achieved, Galileo rushed into print, publishing his Sidereus Nuncius, containing an account of his discoveries in March 1610.

At the beginning the majority were justifiably sceptical about Galileo’s claims and independent collaboration of his discoveries was slow in coming. The telescopes available at the time were of very poor quality and observing with them extremely difficult. However, the Jesuit astronomers of the Collegio Romano in Rome were finally able to confirm all of Galileo’s discoveries and he was hailed as the greatest astronomer of the age. In March 1611 he undertook a triumphant journey to Rome to celebrate his newfound status. The Jesuits at the Collegio Romano held a great banquet in his honour praising both him and his discoveries. Shortly after a slightly different group, the Accademia dei Lincei, also held a banquet in his honour, inducting him as a member of their academy and giving his new wonderful instrument its name, the telescope.

Of course this immediately raises the question who or what is the Accademia dei Lincei. In most accounts of Galileo’s life we get the information that this was an early scientific society founded by the aristocrat Federico Cesi, which shared Galileo’s empirical views on science and thus inducted him into their society actively publishing his Istoria e dimostrazione intorno alle macchie solari (Letters on Sunspots) in 1613 and his Il Saggiatore (The Assayer) in 1623. Other members of the society might or might not be mentioned by name but apart from that very little gets said about them.

Who were they really? What were their aims? To what extent did they and Galileo share the same view of science? How did they relate to other major players of the age, the Catholic Church, the Jesuits? It is very rare for any of these questions to be even asked let alone answered. But they are interesting and important questions within the context of the evolution of science in Northern Italy in the early seventeenth century.

A recently published book—recent that is in English translation, it appeared in Italian three years ago—is Paolo Galluzzi’s The Lynx and the Telescope: The Parallel Worlds of Cesi and Galileo.[1] Galluzzi delivers what it says on the cover, a complete history of the Accademia dei Lincei with a central emphasis on their relations to Galileo.

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It turns out that although the Lincei were proud to have Galileo as a member and more than willing to support his endeavours, Cesi’s approach to science differed substantially from that of the Tuscan astronomer. Cesi was well aware of this and refrained from showing his own studies in natural history to Galileo. The Lincei were also keen to maintain good relations with the Church and especially the Jesuits and made serious efforts behind the scenes to curb some of Galileo’s more ardent attacks on his religious opponents.

Much of the original material, in particular Cesi’s own very extensive, largely unpublished writings have been lost so Galluzzi’s book is the product of a long detective search, digging up every letter, note, reference etc. that still exists that has some sort of relevance to the Lincei and Cesi their leader. As a result his book is exhaustive and I must admit that I at least found it at times exhausting. If you really want to know about the Lincei and Cesi in great detail and only want to read one book on the subject then this is without doubt the book to read (although I would also recommend reading David Freedberg’s The Eye of the Lynx[2])

However my feelings about the book are not one hundred per cent positive. Galluzzi has been director of the Museo Galileo – Institutio e Museo di Storia della Scienza in Florence since 1982 and he is a Galileo groupie. For him Galileo can do no wrong. He states several times in the book that Galileo was motivated purely by a belief in the truth and the defence of freedom of expression. He says that those, present writer included, who claim that Galileo was to a great extent the cause of his own problems are flat out wrong. It is interesting in this context that when discussing the Il Saggiatore, for example, he makes no mention of the fact that Galileo’s accusations of plagiarism against Simon Marius or Christoph Scheiner were false and that Galileo knew them to be false. Or that in the main issue being debated, the nature of comets, that Grassi was right and Galileo flat out wrong, choosing to argue as he did only to win at all cost. A man who only championed the truth?

At one point Galluzzi delivers up something illustrating Galileo’s deviousness, which was entirely new to me. It appears that Cesi had read Kepler’s Astronomia Nova and was much taken with the simplicity of Kepler elliptical orbits when compared to the deferent-epicycle model employed by both Ptolemaeus and Copernicus. In a letter Galileo says to Cesi:

We should not wish nature to accommodate itself to what seems better ordered and disposed to us; we should rather accommodate our own intellect to what nature has made, in the certainty that this is the best and only way. And since it pleased her to make the stars move around different centres, we can be sure that such an arrangement is the most perfect and admirable and that any other would be lacking in elegance, incongruous and puerile.[3]

Galluzzi then goes on to say:

One may suspect that Galileo’s brusque reaction was aimed at a target far more sensitive to Cesi than was Lagalla [the direct subject of the discussion on epicycles]. Distancing himself from the Prince’s anthropocentric conception of nature, Galileo aimed to strike at the heart of a person of a quite different scientific calibre, an author towards whom Federico had shown an inclination: Johann Kepler, who had developed the vision of a cosmos constructed in accordance with the rules of order, mathematical proportion, rationality and harmony based on the centrality of mankind in the plan of the divine Creation.[4]

 Galluzzi then reveals his own pro Galileo bias:

It is telling that, after the discovery of the satellites of Jupiter, Kepler had wondered what influence they exerted on the inhabitants of Earth. Galileo was disappointed to discover that Cesi’s wary approach to heliocentrism was by way of the obscure paths [my emphasis] of the Imperial astronomer.

It seems to have escaped Galluzzi’s notice that Kepler’s version of heliocentrism was correct and Galileo’s wrong. Also Kepler’s comments appeared in his Conversations with Galileo’s Sidereal Messenger, which Galileo did not hesitate to publish in a pirate edition in Florence upon receiving a copy from the author.

For me it appears that Galileo will resort to any rhetorical argument, not for the first or the last time, to defend his truth against all comers rather than admit that the others might have better arguments.

Galluzzi’s groupie mentality aside, this is an important book on the milieu in which Galileo worked and created his theories and one that should be read by anyone with a deep interest in the subject. It has, as would be expected excellent footnotes, index and bibliography but being from Brill a price that no normal human being would want to pay. I borrowed it from the library.

[1] Paolo Galluzzi, The Lynx and the Telescope: The Parallel Worlds of Cesi and Galileo, Trans: Peter Mason, Brill, Leiden & Boston, 2017

[2] David Freedberg, The Eye of the Lynx: Galileo, His Friend, and the Beginnings of Modern Natural History, University of Chicago Press, Chicago & London, 2002

[3] Letter to Cesi, 30 June 1612, Galluzzi p. 114

[4] Galluzzi p. 115

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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?

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Dürer’s four arm compass

title-page

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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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On’t Moor

Richard Carter lives in the picturesque, Yorkshire, market town of Hebden Bridge, which nestles in the Upper Calder Valley surrounded by large expanses of millstone grit moorland.

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Calder Valley around Hebden Bridge Scott L. Cockroft – Own work taken with a Canon Ixus 700 View of the Calder Valley looking South West from Heptonstall (above Hebden Bridge). Source: Wikimedia Commons

Richard likes walking on those moors and he has written a book about it[1].

On't Moor001

He doesn’t so much write about walking, as ramble about his ramblings. The chapters are really essays or vignettes, each of which has a simple term as title or heading, many of them single words such as Grouse, Snow, Skull and so on. Each of these signals something that Richard stumbles upon, observes or contemplates on one of his walks and serves as the starting point for several excurses extrapolating on the term into science, history, natural history, engineering or personal history. These excurses cover a bewildering range of topics but often somehow end up with Charles Darwin; Richard makes no secret of the fact that he is a devoted and very knowledgeable fan of England’s nineteenth century natural historian extraordinaire. Two topics that seem to creep in from every direction into these highly entertaining and informative digressions are evolution and entropy.

The narrative is very assertively first person singular and veers along a random zigzag path between stroppy but interesting guy in the pub and highly erudite polymath. It is by turns provocative, humorous, entertaining, fascinating and informative but never dull. Occasionally the narrative takes a hefty sideswipe at some aspect or other of pseudoscience or so-called alternative medicine. Richard is not afraid to let his readers know where he stands on such topics.

The essays are relatively short and compact and although, as indicated, there are some themes that weave their way through the whole book each essay can be read alone. This makes it an ideal bedside book. The essays are short enough that they can be read in one go before one falls asleep and not so long that they keep you awake beyond that point. If, when reading, you like to be enlightened and educated in an enjoyable and entertaining manner then buy Richard’s book; I promise you won’t regret it.

As a true denizen of the Internet age Richard has posted an extensive list of the sources for the information that the book contains on the Internet and the link is included at the back of the book.

Disclosure: Richard has long been one of my Internet friends and is a loyal fan of the Renaissance Mathematicus, who turns up fairly regularly in the comments columns. I even once spent a very enjoyable afternoon with him in his nice house in Hebden Bridge drinking Yorkshire TeaTM. If you do read his book you will also find my name amongst the acknowledgements, as I acted as expert advisor (it’s amazing what you can blag your way into with enough chutzpah) on one of the chapters; I’m not going to tell you which one, you’ll have to work it out for yourselves. For my efforts I got a free copy of the book, the one I’ve just read and reviewed here. All of this of course means that I am anything but a neutral reviewer. However, as I said when reviewing Chris Graney’s books it is not my style to do friends a favour with good reviews of their books or to pull punches just because they are friends. The above very positive review is my honest, unbiased opinion of the book that Richard has written and you are just going to have to take my word for it. If not, then read somebody else’s review.

[1] Richard Carter, On The Moor: Science, History and Nature on a County Walk, Gruts Media, Hebden Bridge, 2017

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Conversations in a sixteenth century prison cell

Science writer Michael Brooks has thought up a delightful conceit for his latest book.* The narrative takes place in a sixteenth century prison cell in Bologna in the form of a conversation between a twenty-first century quantum physicist (the author) and a Renaissance polymath. What makes this conversation particularly spicy is that the Renaissance polymath is physician, biologist, chemist, mathematician, astronomer, astrologer, philosopher, inventor, writer, auto-biographer, gambler and scoundrel Girolamo Cardano, although Brooks calls him by the English translation of his name Jerome. In case anybody is wondering why I listed autobiographer separately after writer, it is because Jerome was a pioneer in the field writing what is probably the first autobiography by a mathematician/astronomer/etc. in the Early Modern Period.

Gerolamo_Cardano_(colour)

Portrait of Cardano on display at the School of Mathematics and Statistics, University of St Andrews. Source: Wikimedia Commons

So what do our unlikely pair talk about? We gets fragments of conversation about Jerome’s current situation; a broken old man rotting away the end of his more than extraordinary life in a prison cell with very little chance of reprieve. This leads to the visitor from the future, relating episodes out of that extraordinary life. The visitor also picks up some of Jerome’s seemingly more strange beliefs and relates them to some of the equally, seemingly strange phenomena of quantum mechanics. But why should anyone link the misadventures of an, albeit brilliant, Renaissance miscreant to quantum mechanics. Because our author sees Jerome the mathematician, and he was a brilliant one, as the great-great-great-great-great-great-great-great-great-great-great-great-great grandfather of quantum mechanics!

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As most people know quantum mechanics is largely non-deterministic in the conventional sense and relies heavily on probability theory for its results. Jerome wrote the first mathematical tome on probability theory, a field he entered because of his professional gambling activities. He even included a section about how to cheat at cards. I said he was a scoundrel. The other thing turns up in his Ars Magna (printed and published by Johannes Petreius the publisher of Copernicus’ De revolutionibus in Nürnberg and often called, by maths historians, the first modern maths book); he was the first person to calculate with so-called imaginary numbers. That’s numbers using ‘i’ the square root of minus one. Jerome didn’t call it ‘i’ or the numbers imaginary, in fact he didn’t like them very much but realised one could use them when determining the roots of cubic equation, so, holding his nose, that is exactly what he did. Like probability theory ‘i’ plays a very major role in quantum mechanics.

What Michael Brooks offers up for his readers is a mixture of history of Renaissance science together with an explanation of many of the weird phenomena and explanations of those phenomena in quantum mechanics. A heady brew but it works; in fact it works wonderfully.

This is not really a history of science book or a modern physics science communications volume but it’s a bit of both served up as science entertainment for the science interested reader, lay or professional. Michael Brooks has a light touch, spiced with some irony and a twinkle in his eyes and he has produced a fine piece of science writing in a pocket-sized book just right for that long train journey, that boring intercontinental flight or for the week in hospital that this reviewer used to read it. If this was a five star reviewing system I would be tempted to give it six.

*  Michael Brooks, The Quantum Astrologer’s Handbook, Scribe, Melbourne & London, 2017

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Galileo & Roberto

One of the books that I am currently reading is Rob Iliffe’s Priest of Nature: The Religious Worlds of Isaac Newton (a full review will follow when I finish it but I can already say it will be very positive). I stumbled more than somewhat when I read the following:

…and Lucas Trelcatius’s list of some of the most significant places in Scripture, which was composed as a response to the Catholic interpretations of various texts offered by the great scholar (and scourge of Galileo [my emphasis]) Cardinal Robert Bellarmine.

Four words that caused me to draw in my breath, why? Let as first take a look at the meaning of the word scourge:

A scourge was originally a particularly nasty and extremely cruel multi-thong whip. Transferred to describe a person it means: a person that causes great trouble of suffering. Can Robert Bellarmine really be described as “scourge of Galileo”?

Robert Bellarmine (actually Roberto Bellarmino) (1542-1621) was a Jesuit scholar who was specialist for post Tridentine theology, that is the theological teachings of the Catholic Church as laid down as official church doctrine at the Council of Trent (1545-1563. He rose through the ranks to arch-bishop and then cardinal, was professor for theology at the Collegio Romano, the Jesuit University in Rome, and later the universities rector. In the early seventeenth century he was regarded as the leading Catholic authority on theology and as such he was a powerful and highly influential figure in Rome.

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Robert Bellarmine artist unknown Source: Wikimedia Commons

How did Bellarmine’s life interact with that of Galileo? The first contact was very indirect and occurred after Galileo had published his Sidereus Nuncius, making public his telescopic discoveries. Bellarmine inquired of the mathematician astronomers under Clavius’ leadership at the Collegio Romano, whether the discoveries claimed by Galileo were real. Being the first astronomers to confirm those discoveries, Clavius was able to report in the positive.

In 1615 Galileo wrote his Letter to Castelli in which he argued that those Bible passages that contradicted Copernican heliocentricity should be re-interpreted to solve the contradiction. He was stepping into dangerous territory, a mere mathematicus—the lowest of the low in the academic hierarchy—telling the theologians how to interpret the Bible. This was particularly risky, as it was in the middle of the Counter-Reformation given that the Reformation was about who is allowed to interpret the Bible. The Protestants said that everyman should be able to interpret it for themselves and the Catholic Church said that only the Church should be allowed to do so. Remember we are only three years away from the Thirty Years War the high point, or should that be the low point, of the conflict between the two religions, which led to the destruction of most of central Europe and the death of between one and two thirds of its population.

Justus_Sustermans_-_Portrait_of_Galileo_Galilei,_1636

Justus Sustermans – Portrait of Galileo Galilei, 1636 Source: Wikimedia Commons

Galileo’s suggestion in his letter came to the attention of his opponents in the Church and led the Pope, Paul V, to set up a commission of eleven theologians, known as the Qualifiers, to investigate the propositions of heliocentricity.

In the meantime Paolo Antonio Foscarini (c. 1565–June 1616), a Carmelite father, attempted to publish his Epistle concerning the Pythagorean and Copernican opinion of the Mobility of the Earth and stability of the sun and the new system or constitution of the WORLD, which basically contained the same arguments for reinterpreting the Bible as Galileo’s Letter to Castelli. The censor of Foscarini’s order rejected his tract, as too contentious. I should point out at this point something that most people ignore that is all powers both civil and religious in Europe exercised censorship; there was no such thing as free thought or freedom of speech in seventeenth century Europe. Foscarini wrote a defence of his Epistle and sent the two pieces to Bellarmine, as the leading theologian, for his considered opinion. Bellarmine’s answers the so-called Foscarini Letter is legendary and I reproduce it in full below.

My Reverend Father,

I have read with interest the letter in Italian and the essay in Latin which your Paternity sent to me; I thank you for one and for the other and confess that they are all full of intelligence and erudition. You ask for my opinion, and so I shall give it to you, but very briefly, since now you have little time for reading and I for writing.

First I say that it seems to me that your Paternity and Mr. Galileo are proceeding prudently by limiting yourselves to speaking suppositionally and not absolutely, as I have always believed that Copernicus spoke. For there is no danger in saying that, by assuming the Earth moves and the sun stands still, one saves all of the appearances better than by postulating eccentrics and epicycles; and that is sufficient for the mathematician. However, it is different to want to affirm that in reality the sun is at the center of the world and only turns on itself, without moving from east to west, and the earth is in the third heaven and revolves with great speed around the sun; this is a very dangerous thing, likely not only to irritate all scholastic philosophers and theologians, but also to harm the Holy Faith by rendering Holy Scripture false. For Your Paternity has well shown many ways of interpreting Holy Scripture, but has not applied them to particular cases; without a doubt you would have encountered very great difficulties if you had wanted to interpret all those passages you yourself cited.

Second, I say that, as you know, the Council [of Trent] prohibits interpreting Scripture against the common consensus of the Holy Fathers; and if Your Paternity wants to read not only the Holy Fathers, but also the modern commentaries on Genesis, the Psalms, Ecclesiastes, and Joshua, you will find all agreeing in the literal interpretation that the sun is in heaven and turns around the earth with great speed, and that the earth is very far from heaven and sits motionless at the center of the world. Consider now, with your sense of prudence, whether the church can tolerate giving Scripture a meaning contrary to the Holy Fathers and to all the Greek and Latin commentators. Nor can one answer that this is not a matter of faith, since it is not a matter of faith “as regards the topic”, it is a matter of faith “as regards the speaker”; and so it would be heretical to say that Abraham did not have two children and Jacob twelve, as well as to say that Christ was not born of a virgin, because both are said by the Holy Spirit through the mouth of the prophets and the apostles.

 

Third, I say that if there were a true demonstration that the sun is at the center of the world and the earth in the third heaven, and that the sun does not circle the earth but the earth circles the sun, then one would have to proceed with great care in explaining the Scriptures that appear contrary; and say rather that we do not understand them than that what is demonstrated is false. But I will not believe that there is such a demonstration, until it is shown me. Nor is it the same to demonstrate that by supposing the sun to be at the center and the earth in heaven one can save the appearances, and to demonstrate that in truth the sun is at the center and the earth in the heaven; for I believe the first demonstration may be available, but I have very great doubts about the second, and in case of doubt one must not abandon the Holy Scripture as interpreted by the Holy Fathers. I add that the one who wrote, “The sun also riseth, and the sun goeth down, and hasteth to his place where he arose,” was Solomon, who not only spoke inspired by God, but was a man above all others wise and learned in the human sciences and in the knowledge of created things; he received all this wisdom from God; therefore it is not likely that he was affirming something that was contrary to truth already demonstrated or capable of being demonstrated. Now, suppose you say that Solomon speaks in accordance with appearances, since it seems to us that the sun moves (while the earth does so), just as to someone who moves away from the seashore on a ship it looks like the shore is moving, I shall answer that when someone moves away from the shore, although it appears to him that the shore is moving away from him, nevertheless he knows that it is an error and corrects it, seeing clearly that the ship moves and not the shore; but in regard to the sun and the earth, no wise man has any need to correct the error, since he clearly experiences that the earth stands still and that the eye is not in error when it judges that the it also is not in error when it judges that the stars move. And this is enough for now.

With this I greet dearly Your Paternity, and I pray to God to grant you all your wishes.

At home, 12 April 1615.

To Your Very Reverend Paternity.

As a Brother,

Cardinal Bellarmine

 

(Source for the English transl.: M. Finocchiaro, The Galileo Affair. A Documentary History (Berkeley, CA: University of California Press, 1989), pp. 67-69.Original Italian text, G. Galilei, Opere, edited by A. Favaro (Firenze: Giunti Barbera, 1968), vol. XII, pp. 171-172.)

A, in my opinion, brilliant piece of measured, diplomatic writing. Bellarmine tactfully suggests that one should only talk of heliocentricity hypothetically, its correct scientific status in 1615, the first empirical proof for the movement of the Earth was found in 1725, when Bradley discovered stellar aberration. He, as the great Tridentine theologian, then reiterates the Church’s position on the interpretation of Holy Scripture. Finally he brings, what is without doubt, the most interesting statement in the letter.

Third, I say that if there were a true demonstration that the sun is at the center of the world and the earth in the third heaven, and that the sun does not circle the earth but the earth circles the sun, then one would have to proceed with great care in explaining the Scriptures that appear contrary; and say rather that we do not understand them than that what is demonstrated is false.

What he says is bring proof and we’ll reinterpret the Bible but until then…

On 24 February the Qualifiers delivered the results of their deliberations on the heliocentricity hypothesis:

( i ) The sun is the centre of the universe (“mundi”) and absolutely

immobile in local motion.

( ii ) The earth is not the centre of the universe (“mundi”); it is not

immobile but turns on itself with a diurnal movement.

All unanimously censure the first proposition as “foolish, absurd in philosophy { i.e. scientifically untenable] and formally heretical on the grounds of expressly contradicting the statements of Holy Scripture in many places according to the proper meaning of the words, the common exposition and the understanding of the Holy Fathers and learned theologians”; the second proposition they unanimously censured as likewise “absurd in philosophy” and theologically “at least erroneous in faith”.

It should be pointed out that although the Qualifiers called the first statement heretical, only the Pope could formally declare something heretical and no pope ever did, so heliocentricity was never officially heretical.

Pope Paul V now ordered Bellarmine to covey the judgement of the Qualifiers to Galileo and to inform him that he may not hold or teach the heliocentric theory. This he did on 26 February 1616. Bellarmine was not one of the Qualifiers and here functioned only as the messenger. By all accounts the meetings between Bellarmine and Galileo were cordial and friendly.

When Galileo returned to Florence rumours started spreading that he had been forced to recant and do penance, which was of course not true. Galileo wrote to Bellarmine requesting a letter explaining that this was not true. Bellarmine gladly supplied said letter, defending Galileo’s honour. However Galileo made the mistake in 1633 of thinking that Bellarmine’s letter was a get out of jail free card.

Bellarmine died in 1621 and between 1616 and his death there was no further contact between the Cardinal and the mathematicus. Personally I can see nothing in the three interactions, indirect and direct, between Bellarmine and Galileo that would in anyway justify labelling Bellarmine as the “scourge of Galileo”. This accusation is historically highly inaccurate and paints a wholly false picture of the relationship between the two men. I expect better of Rob Iliffe, who is without doubt one on Britain’s best historians of seventeenth century science.

NB Before somebody pops up in the comments claiming that Robert Bellarmine was one of the three Inquisition judges, who confirmed the death sentence on Giordano Bruno. He was but that has no relevance to his interactions with Galileo, so save yourself time and energy and don’t bother.

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Not so much a book review more a WHAT THE F…

This blog post is not about one of the books I read whilst in hospital but about one that I had on order through inter library loan and picked up on Monday when I came out of hospital; to this book there is a back story.

As I have related on several occasions on this blog my initial interest in the history of mathematics was awakened by reading Eric Temple Bell’s Men of Mathematics, as a teenager. What particularly stirred my interest was the co-discovery/-invention[1] of the calculus by Gottfried Leibnitz and Isaac Newton during the last third of the seventeenth century, as calculus was my favourite mathematical discipline. Over the years I intensified my interest in and knowledge of the history of calculus/analysis and at some point, I can no longer say when, I stumbled upon the very interesting so-called Kerala School of mathematics in medieval India, roughly the period 1300 to 1600 CE, who developed various infinite series and their sums well before the same series were discovered by various European mathematicians in the seventeenth century. I first read about this school in George Gheverghese Joseph’s rather shrill The Crest of the Peacock: Non-European Roots of Mathematics[2] and then later in Kim Plofker’s somewhat more sober Mathematics in India.[3] I included the Kerala School in my very brief outline of the history of calculus here.

Sometime back, somebody on the Internet, who believes that the history of science is far too Eurocentric put together a bibliography of books dealing with non-European history of science that included George Gheverghese Joseph, The Crest of the Peacock. More recently I once again stumbled across this bibliography, however, it now contained the complaints of an Indian gentleman, called C.K. Raju, that George Gheverghese Joseph had plagiarised his work on the Kerala School in The Crest of the Peacock. He also included links to a website that included the letters and emails he had sent to the University of Manchester, where George Gheverghese Joseph had completed his postgraduate studies, detailing his accusations of plagiarism. He also named his own book on the subject, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE.[4] Oh, I thought that looks interesting, especially as it is generally thought that no transmission took place between Kerala and Europe, I must see if I can get hold of a copy of that and indeed I could and this was the book that I picked up from the university library on Monday.

Eager to find out what Mr Raju had to say about transmission I turned straight to this section of his book and as I read my eyes got bigger and bigger, as it goes in the fairy tales, as Mr Raju idea of history has more to do with fairy tales than anything that happened in the real world.

I’m not going to do a blow-by-blow analysis of Mr Raju’s fairy tales but just pick out some bits and pieces to give you an idea. He starts of by claiming that during the 11th century translation movement, which he claims wrongly all took place in Toledo, the Christians who were so ashamed that they were acquiring their knowledge from Islam declared that it actually all came from white, European Greece. A trick and a lie, you get the picture. He says that “Euclid” is a mythical creature of the imagination. Further he claims that Ptolemy never existed and that the Almagest was a compendium of various astronomical texts from various, non-Greek, sources put together by the Arabs in the Middle Ages.

He repeats as factual history the Persian legend that when Alexander invaded Persia he translated all of the Persian knowledge into Greek and then destroyed the Persian originals therefore all supposed Greek knowledge is actually Persian. The following passage blew my mind completely:

For example, until just before the time of Alexander, the Greeks regarded any kind of scientific thought as an offence punishable by death, as is clear from the trials of Socrates and Anaxagoras, and the subsequent flight of Aristotle. How could any scientific thought have been produced in such an atmosphere? Why would it have been produced—to what economic process did it relate? The absence of serious answers to these questions tends to collaborate that Aristotle was at best merely a translator of books looted during Alexander’s conquests (and at worst merely a brand name used by later translators or scribes to increase the prices of their products).

So all you Aristotle scholars have been wasting your time, he didn’t even exist.

Another statement that tickled my fancy was part of his general claim that the Church controlled knowledge and forbade scholars from naming any non-Christian sources, he writes:

Thus in the days of the Inquisition there was little likelihood that even an honest European would have acknowledged knowledge from any non-Christian sources: Similar remarks, with some slight modification, would apply a fortiori to those high up in the church hierarchy like Clavius, Tycho Brahe, etc.

The vision of the solidly Lutheran Danish aristocrat, Tycho Brahe, as high up in the Catholic Church hierarchy is one to treasure.

To close I will just mention an interesting proposal from Mr Raju in the history of astronomy. There is a fairly well substantiated theory that sometime in the early part of the first millennium CE Greek astronomy was transmitted to India. It is most significant that the Indians began to use the Greek deferent-epicycle model for determining planetary orbits. Raju argues that the Greeks with their abacus arithmetic (sic) were mathematical incapable of having developed the deferent-epicycle model, which was in fact invented by the Indians with their much superior mathematics and then later included by the Arabs in the Almagest (see above) and falsely attributed to the non-existent Ptolemy! He seems to be completely unaware that the deferent-epicycle model is attributed to Apollonius, who doesn’t play any role whatsoever in his historical fantasy.

I could go on but I grow weary and I have no desire to read more of Mr Raju fantasies. At no point does he present any serious evidence for his fairy tales and it should come as no surprise that he is not a historian of mathematics or science but a statistician turned computer scientist. My biggest wonder is that was published by Pearson Longman a normally serious publishing house, maybe Pearson Longman India has other standards than their London mother house.

[1] Choose the term that best fits your personal philosophy of mathematics.

[2] George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, London, 1992

[3] Kim Plofker, Mathematics in India, Princeton University Press, Princeton and Oxford, 2009

[4] C.K. Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE = History od Science, Philosophy and Culture in Indian Civilisation Volume X Part 4, Pearson Longman, Delhi, 2007

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