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Marine chronometer, lunar distance method or something else altogether?

Trying to find a method to determine longitude at sea was one of the greatest technical problems of the Early Modern Period. Quite a wide-range of ideas were floated of which the most were either totally impractical or simply false. In the end the two main competitors were: on the one hand the attempts to develop a clock reliable enough to carry time from a given starting point accurately enough through all the vicissitudes of a long sea voyage to be then compared with local time and thus to determine longitude, i.e. the marine chronometer. Or on the other to develop accurate tables of the Moon’s orbit respective a set of given fixed stars in order to be able to use the Moon’s position at any given time as a clock with which to calculate longitude, i.e. the lunar distant method. Both of these concepts were first presented in the sixteenth century but it took until the middle of the eighteenth century before they could be realised.

Around 1760, Tobias Mayer succeeded in delivering up a set of tables of the lunar orbit accurate enough to be used for determining longitude using the lunar distance method. Shortly after this John Harrison showed with his H4 that a solution with a chronometer was also possible. Unfortunately even with the naval almanac produced by Nevil Maskelyne to simplify the calculations the lunar distant method was mathematically difficult to execute. As I have written elsewhere although Harrison’s H4 showed that a chronometer solution was possible, the clock itself was too complex and too expensive to provide a real solution to the longitude problem. It would take well into the nineteenth century before enough affordable, accurate chronometers were available to make this a viable mass method. Many sources claim that in the mean time navigators used the lunar distant method, but did they?

It would appear that for the first fifty or so years following those breakthroughs seafarers relied on a mixture of navigational methods to help determine their longitude. Principally they relied on the old tried and trusted method of dead reckoning. This is the process of calculating the ships new position from a previous one based on compass direction, ship’s speed based on log line measurements, and knowledge of currents. In the period we are talking about, many navigators checked their dead reckoning results against chronometer or lunar distant determinations. Given the lack of reliability of the available chronometers the navigators often carried several watches, comparing or even averaging the results. Sometimes the lunar distant method was only used by landfall to correct or control the longitude determined by dead reckoning. In general it seems that the well-established dead reckoning was the principle method used, supplement by one or other or both of the new methods, although neither of them was really trusted by the navigators.

For a more detailed picture of the navigational methods used from the middle of the eighteenth century to the middle of the nineteenth by the various European sea going nations I can recommend Navigational Enterprises in Europe and its Empires, 1730–1850 (1) edited by Richard Dunn (@Lordoflongitude) and Rebekah Higgitt (@beckyfh) a set of academic papers that supplement their more popular, excellent Finding Longitude.


After an excellent general introduction to the subject by the editors follow eleven papers covering a wide range of aspects of the subject, all of which maintain a very high level of scholarship.



My only real quibble with the book is the unfortunately usual high price putting it beyond my humble resources and probably those of most others interested in reading and learning from this highly informative volume.

(1) Ricard Dunn & Rebekah Higgitt eds., Navigational Enterprises in Europe and its Empires, 1730–1850, Palsgrave Macmillan, 2015



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Dangerous Twaddle

Someone on Twitter drew my attention to a BBC4 television documentary by David Malone from 2007 about mathematics. Interested I thought I would give it a whirl, I wish I hadn’t. It’s a sort of biography of Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing so what could go wrong? It’s called Dangerous Knowledge but Dangerous Twaddle would have been more appropriate.

In my opinion it starts off with a real humdinger: atmospheric images with the following dramatic voice over:

Beneath the surface of the world are the rules of science but beneath them there is a far deeper set of rules. A matrix of pure mathematics, which explains the nature of the rules of science and how it is we can understand them in the first place.

Ignoring the fact that I don’t actually agree with this piece of trite metaphysics, the author completely blows it in my opinion because another even more dramatic voice over follows this with the following quote:

To see a World in a Grain of Sand

And a Heaven in a Wild Flower

Hold Infinity in the palm of your hand

And Eternity in an hour

This is of course one of the most well known quotes by William Blake taken from his Auguries of Innocence. Malone is obviously ignorant of Blake’s opinion of the mathematical description of the world.

God forbid that Truth should be confined to Mathematical Demonstrations! (Written as a marginal note to Reynolds’ Discourses.)

Before we get down to the real reason that I’m writing this a couple of things that annoyed me whilst watching this documentary. The author-narrator mispronounces the names of both Leibniz and Dedekind. One would think that if somebody is making a documentary about mathematics and mathematicians they would at least take the trouble to get the names of famous mathematicians right. At the end of the section about Cantor he describes him as the greatest mathematician of his century! Regular readers of this blog will know that I intensely dislike such superlatives in the history of science. Even if I didn’t, is Cantor really the greatest mathematician of the nineteenth century? There’s an awful lot of competition. In the section on Gödel, we get told about his friendship with his fellow Austrian mathematician, Albert Einstein.


Two Austrian Mathematicians!? Albert Einstein und Kurt Gödel in Princeton, circa 1948, Foto: Oskar Morgenstern; mit freundlicher Genehmigung des Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, US

Now, I know that trying to keep track of Einstein’s nationality is rather difficult for the non-historian; he had a total of eight different ones including being stateless for five years. However, he was only a citizen of the Austrian Empire from April 1911 to July 1912, as professor at the University of Prague. Eleven months out of a life of 76 years hardly justifies calling him an Austrian.

My real beef with the documentary is contained in a further piece of voice over from the introduction:

… pursued the questions to the brink of insanity and over it.

Basically Malone spends eighty minutes telling the world that if brilliant mathematicians think outside the box it can and will drive them insane! This is quite simply bullshit!

He devotes the largest part of the documentary to Georg Cantor and the invention of set theory. I found his explanations of what Cantor achieved and why he did it totally opaque and I spent quite a lot of time at university studying and understanding it. Malone gives a totally bogus explanation of the continuum hypothesis, which suggests very strongly that he simply doesn’t understand it, and then goes on to explain that it was Cantor’s inability to prove the continuum hypothesis drove him insane. I will return to this.


Georg Cantor, around 1870 Source: Wikimedia Commons

We then move on to Ludwig Boltzmann and his championing of a probablistic atomic theory when the majority of physicists and philosophers opposed the real existence of atoms. Once again Malone tells us that it was Boltzmann’s science that drove him mad and led him to commit suicide.


Ludwig Boltzmann Source: Wikimedia Commons

Although it is always dubious to make historical diagnoses of illnesses, in particular mental ones, Both Cantor and Boltzmann displayed all of the symptoms of a severe bipolar disorder. Bipolar disorder is not caused by mathematical research or any other work for that matter. A stressful working situation might well aggravate an existing bipolar disorder it won’t cause it. This is as I said, dangerous twaddle.

Malone now accelerates his gallop into the realm of total crap with his segment on Kurt Gödel. Following the usually incorrect statement of Gödel’s incompleteness theorem. People almost invariably leave off the very important final “within the system” in their accounts. What Gödel showed in that we cannot produce a formal logical system within which all true mathematical statements are provable. However this does not mean that the statements that are unprovable within the given system are fundamentally unprovable, as Malone claims in his statement of Gödel. However this is a minor quibble compared to Malone’s central claim. He states that Gödel took up the continuum hypothesis and because he like Cantor was unable to prove it, he too went insane. Now, it is well known that Gödel displayed serious symptoms of mental illness that got increasingly worse as he got older, until he quite literally starved himself to death due to his paranoid belief that somebody was trying to poison him. I’m not a clinical psychiatrist, but I’m more that willing to state that Gödel’s ability or lack of it to solve the continuum hypothesis did not cause his mental illness. Malone, however, seem to be totally unaware that Gödel in fact showed the continuum hypothesis was consistent, i.e. cannot be proved false, with the axioms of Zermelo-Fraenkel set theory. This is one of the major breakthroughs in the history of set theory; far from being frustrated by the continuum hypothesis Gödel produced one of his most important results with it.


Kurt Gödel Image Credit

Malone closes out his trip through the insane and suicidal mathematical geniuses with none other than Alan Turing. Following up on the usually false claim that Turing invented the computer and a very confusing explanation of Turing’s achievements in meta-mathematics, Malone takes us forward to Turing’s death. He has the British secret service responsible for Turing’s chemical castration following his conviction for indecency, which is just simply crap. Turing was offered a choice between a prison sentence or probation with the hormonal treatment as a condition by the court. He freely chose the later. I’m not even going to enter the discussion of whether he committed suicide or not and if he did why. There has already been enough ink spilt on that particular topic.


Alan Turing Source: Wikimedia Commons

Malone made a documentary about four major figures in the history of mathematics, logic and mathematical physics and presents the quite honestly laughable thesis that it was their intellectual audacity and the opposition that they experienced to their theories that drove them insane. This is quite simple put, bullshit. As someone who has experienced, at time quite serious, mental illness I find it quite frightening that an organisation such as the BBC is not only prepared to air such crap but to finance it with obviously comparatively large sums of money. We live in a society where it is extremely difficult to explain to people what mental illness is and such pseudo-psychological bullshit as Malone’s documentary does nothing to help with this problem.




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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.


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.


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.


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.


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.


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.


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.


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.


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.


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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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.


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.


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!


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.


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 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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