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Renaissance garbage ­– II

This is the second in a series of discussion of selected parts of Paul Strathern’s The Other RenaissanceFrom Copernicus to Shakespeare, (Atlantic Books, 2023). For more general details on both the author and his book see the first post in this series.

Today, I turn my attention to his chapter on the fifteenth century, German philosopher and theologian, Nicholas of Cusa, a large part of which is, as we shall see, actually devoted to another fifteenth century German scholar. Right in his opening paragraph to this chapter, Strathern lets a historical bomb of major dimensions explode, he writes:

As the world of art in northern Europe began its drastic transformation, shedding the stylistic formalism and religious subject- matter of medieval art, so the northern intellectual world underwent a similar revolution. The origins of the humanistic way of thought and its empirical attitude to learning were not the sole preserve of the Italian Renaissance. Indeed, the ‘father of humanism’ is generally recognized as a German, born in 1401 in the Electorate of Trier: Nicholas of Cusa [my emphasis].

Regular readers will already know that here at the Renaissance Mathematicus we react extremely allergically to the phrase ‘father of anything’, but to title Nicholas of Cusa ‘father of humanism’ is really breath takingly stupid, further to also claim that this is ‘generally recognised’ pushes the claim into the mindboggling. As I explained in the fourth part of my series on Renaissance science, Renaissance Humanism, which originated in Northern Italy, did so almost a century before Nicholas of Cusa was born, so if he was, as Strathern claims, the ‘father of humanism’ then his birth in 1401 must have been a reincarnation. 

Nicholas of Cusa, by Master of the Life of the Virgin Source: Wikimedia Commons

Strathern writes that Nicholas was “the son of ‘a prosperous boatman and ferryman’.” A boatman is, according to the dictionary, “a man who takes people or goods somewhere in a small boat, or who has small boats that you can rent for a period of time.” The Stanford Encyclopedia of Philosophy says his father was “a prosperous merchant who became one of the landed gentry in Trier” and German Wikipedia say he was “als Schiffer ein wohlhabender Kaufmann,” that is, “as shipper a wealthy merchant”. Once again according to the dictionary “a shipper is a company that arranges for goods to be taken somewhere by ship.” There appears to be a major disparity concerning the real profession of the father and Strathern’s version.  

Nicholas was a precocious student. In his early teens he entered the University of Heidelberg, the oldest in Germany, which had been established in the middle of the previous century. Here he studied law, before transferring to the University of Padua near Venice. He graduated in 1423, but instead of becoming a lawyer he took up minor holy orders. 

Early teens was a perfectly normal age to enter university in the fifteenth century, so not necessarily precocious. Strathern seems not to be aware that law was divided into civil law and canon law, that is church law, at medieval universities and Nicholas obtained a doctorate in canon law in Padua in 1423, so it was perfectly normal for him to take minor holy orders on graduating. 

From the outset, Nicholas was an imaginative polymath, his mind fecund with novel ideas on all manner of subjects. Under normal circumstances such ideas would have been controversial, and might even have put his life in mortal danger (almost 150 years later, the Italian philosopher Giordano Bruno would be burned at the stake for expressing similar ideas). However, it seems that the sheer brilliance of Nicholas of Cusa’s mind won him friends in high places. 

Here we have an oblique reference to Nicholas’ cosmological speculation, which did include, like Bruno, the idea that the stars were other suns and there might be other inhabited planets orbiting around them; speculations also shared by Nicole Oresme in the previous century. However, unlike Bruno, Nicholas did not travel around Europe pissing off everybody who was anybody, and also did not deny the divinity of Christ or the Virgin Birth, so his life was never in danger, because of his cosmological speculations. 

In 1450 Nicholas completed a work in the form of dialogues between a layman and a priest. This was entitled Idiota de Mente (literally translated as ‘An Idiot Speaks His Mind’). Surprisingly, it is the ‘idiot’ who puts forward Nicholas’s bold proposals, which contrast sharply with the orthodox Aristotelian views proposed by the priest. It should be borne in mind that during this period Aristotle was regarded as the highest authority on intellectual matters: his word was seen as little less than Holy Writ. 

In medieval Latin Idiota means layman so the title actually translates as A Layman Speaks his Mind. Strathern actually says this in a footnote, so I really don’t understand his next sentence. It is time for my favourite Edward Grant saying, medieval “Aristotelian philosophy is not Aristotle’s philosophy” and in fact it was constantly changing and evolving. Scholars were constantly discussing, criticising, and modifying Aristotle’s thoughts throughout the Middle Ages, so no, it was not Holy Writ.

Next up we have a rather thin presentation of Nicholas’ theological philosophy and his use therein of mathematics and measurement, which is not particular accurate, but I can’t be bothered to unravel it. However, Strathern makes the following claim: 

Despite the abstract flavour of Nicholas’s mathematical pronouncements, his motives were entirely practical. Delineating discrete parts of the world by measurement was what led to knowledge, which was essentially a practical matter. Such thoughts opened the way to an entirely different method of learning. 

He then states, “In order to understand the magnitude of Nicholas’s mode of thought it is necessary for the moment to take the wider view,” and proceeds to give a very thin and not very accurate account of the supposed decline of China. Re-enter Nicholas:

Ironically, it was the very opposite to the process which was taking place in Europe. And it was Nicholas of Cusa who was giving voice to this new direction. Mathematical measurement should be applied to the world. Architecture, commerce, shipbuilding, the very nature of tools and machines – all would undergo major developments during the Renaissance era as a result of this new attitude towards the practical world. 

I wrote a whole series of blog posts about those developments in practical knowledge in the Renaissance and the theological-philosophical ramblings of Nicholas of Cusa did not in any way play a significant role in them. 

As examples of Nicholas’ practical applications of mathematics and measurement he gives the following from his attempts to square the circle (of which more later):

However, in the course of his attempts by pure geometry to solve this problem he managed to calculate the value of π as 3.1423, a figure of greater accuracy than any before – including that calculated by Archimedes, who in fact only worked out its limits of between 223/71 and 22/7 (3.14084 and 3.14285). 

I love the “only” by Archimedes’ process of calculating Pi. It is one of the puzzles of the history of maths, as to why Archimedes stopped where he did and didn’t carry out the next iteration(s)of his calculation, which would, naturally, have given him value for Pi much more accurate than that of Nicholas. Some have suggested that there was a second, now missing, book where he completed his calculations.

In fact, Nicholas’ value is no more accurate than the value used by Ptolemaeus in the second century CE and less accurate than the value calculated by the Indian mathematician Āryabhaṭīya in the sixth century. Closer to Nicholas’s time in the fourteenth century the Indian mathematician Mādhava of Sangamagrāma calculated a value for Pi accurate to eleven decimal places and in 1425, the Persian mathematician Jamshīd al-Kāshī calculated Pi accurate to sixteen decimal places. 

Next up we have Nicholas as calendar reformer:

Nicholas also argued that there was a need to calculate a new calendar, as the seasons were gradually falling out of synchronization with the dates and the months (it would be almost 150 years before his suggestion was taken up by Pope Gregory XIII). 

The recognition of the need to reform the Julian calendar, to bring it back into line with the solar year, goes back at least to the Venerable Bede in the eighth century CE. Notable mathematicians, who made reform suggestions earlier that Nicholas, include Johannes de Sacrobosco (c. 1195–c. 1256), Roger Bacon (c. 12220–c. 1290) and Johannes de Muris (c. 1290–1344). When Gregory XII finally put that reform into practice, he was not taking up the suggestion of Nicholas of Cusa.

Strathern’s next claim completely blew my mind and sent me down a major rabbit hole:

Perhaps Nicholas’s most important invention was a new type of spectacle lens. Previously, lenses had been ground to a convex shape. This was an easier process, and it enabled the viewer to achieve long-sightedness. Nicholas tried the opposite method, grinding a lens into a concave shape, and found that it enabled the viewer to achieve near-sightedness. This brought about a revolution. Old men with failing sight could continue reading, learning, making suggestions, discoveries, inventions. It is little exaggeration to say that intellectual life almost doubled over the coming century as a result of Cusa’s innovation. 

Now, the history of optics, including the history of spectacles, has been a special area of interest of mine for at least thirty years and I have a rather large literature collection on the subject, as a result, but I have never ever come across the claim that Nicholas of Cusa invented the concave spectacle lens, indeed a major development in the history of optics. I was, as I said above, mind blown. I first of all googled Cusanus and spectacles and to my amazement came up with hundreds or even thousands of websites making exactly this claim that Nicholas of Cuse invented the concave spectacle lens in 1450/1. Mostly there was just one simple sentence with no explanation, no source, no history, nothing! Still not convinced I dug deeper and consulted Vincent Ilardi an expert on the history of spectacles and found the answer to this conundrum. 

More certain in this respect, on the other hand, is the often-cited quotation from Cardinal Nicholas of Cusa’s De beryllo (On the Beryl) as the first mention of concave lenses for the correction of myopia. In this treatise, written over a five-year period and completed in in 1485, Nicholas treated the beryl metaphorically but also as a practical magnifying device:

The beryl is a clear, bright, and transparent stone, to which is given a concave as well as a convex form, and by looking through it, one attains what was previously invisible. If the intellectual beryl, which possesses both the maximum and the minimum in the same way, is adapted to the intellectual eyes, the indivisible principle of all things is attained.

Shorn of its convolution, for which Nicholas had a special aptitude, this passage seems to indicate that the beryl used in its concave shape aided distant vision (“the maximum”) whereas the convex shaped one brought short distance images into focus (“the minimum”). And in another passage from his Compendium, completed in 1463, he again cited beryl as lenses to aid vision in a celebration of human creativeness and inventiveness to remedy the deficiencies of nature and master the environment at a level for superior than the capabilities of the animals. 


For man alone discovers how to supplement weakness of light with a burning candle, so that he can see, how to aid deficient vision with lenses [berylli], and how to correct errors concerning vision with the perspectival art.


The above quotations seem to indicate that Nicholas was familiar with spectacles fitted both with concave and convex lenses just a few years before we have unequivocal proof of the former’s availability in quantity.[1]

It is very clear that Nicholas is in no way claiming to have invented the concave spectacle lens, but is merely describing the fact that they exist. It would be an interesting exercise to try and discover who first misinterpreted this passage in this way. As an interesting side note, the use of beryl to make lenses, because of the poor quality of the available glass, led to the fact that spectacles are called Brillen in German. Of course, as Ilardi says, concave lenses aided distant vision and did not as Starter writes enable, old men with failing sight to continue reading, that task had already been covered by the convex spectacle lens. Personally, I think that a historian when confronted by this claim should weigh up the probability that a cardinal and high-ranking Church diplomat ground lenses in his spare time, possible but highly improbable.

A further revolution was instigated when Nicholas turned his attention to a study of the heavens. Despite the fact that the telescope had yet to be invented, his observations enabled him to reach some highly original conclusions. While several of the Ancient Greeks had speculated on such matters, drawing their own similar conclusions, Nicholas was perhaps the first to put these together into a truly universal structure. 

Nicholas’ thoughts on cosmology were based on speculation not observations and although interesting had almost no impact on the actual astronomy/cosmology debate in the Renaissance. He was also by no means “the first to put these together into a truly universal structure.”

However, none of this accounts for the sheer originality of his thinking. Besides the subjects already mentioned, Nicholas made original contributions in fields ranging from biology to medicine. By applying his belief in rigorous measurement to the field of medicine, he would introduce the practice of taking precise pulse rates to use as an indication of a patient’s health. Previously, physicians had been in the habit of taking a patient’s pulse and using their own estimation of its rate to infer the state of their health. Nicholas of Cusa introduced an exact method, weighing the quantity of water which had run from a water clock during one hundred pulse beats. 

As far as I can see, measuring the pulse using a water clock is the only original contributions in fields ranging from biology to medicine that he made. How original it was is debateable:

Pulse rate was first measured by ancient Greek physicians and scientists. The first person to measure the heartbeat was Herophilos of Alexandria, Egypt (c. 335–280 BC) who designed a water clock to time the pulse. (Wikipedia)

In the middle of a lot of stuff about Nicholas’ role as a Church diplomat we get:

Nicholas’s scientific work would go on to influence thinkers of the calibre of the German philosopher-mathematician Gottfried Leibniz, a leading philosopher of the Enlightenment who lived two centuries later. 

It is interesting to note that Nicholas of Cusa is regarded as one of the great Renaissance thinkers and although he was very widely read, his influence on others was actually minimal. Whether or not he influenced Leibniz is actually an open question.

For whatever reason, Strathern now turns to a completely different Renaissance thinker:

The work and thought of Nicholas of Cusa is indicative of the wide-ranging re-examination of the human condition which was beginning to take place, especially amongst thinkers of the northern Renaissance. Another leading German scientific thinker from this period, who would become a friend of Nicholas of Cusa, was Regiomontanus, who was born Johannes Müller in rural Bavaria, southern Germany, in 1436. 

Regiomontanus holding up an astrolabe , see below, Woodcut from the 1493 Nuremberg Chronicle vis Wikimedia Commons

We get a long spiel about scholars adopting Latin names during the Renaissance and the use of Latin in general during the medieval period, but nowhere does he mention that Johannes Müller never actually used the name Regiomontanus, which was first coined by Philip Melanchthon in 1535, that is almost seventy years after his death. 

“[Regiomontanus] would become a friend of Nicholas of Cusa”, really‽ I can find no references whatsoever to this ‘friendship’. There is no correspondence between the two of them, no record of their having ever met. Although, a meeting would have been possible as Regiomontanus lived and worked in Italy during the last three years of Nicholas’ life (1461–64), and even lived in Rome, where Nicholas was living, for some of this time. 

Regiomontanus’ view of Nicholas of Cusa can best be taken from his analysis of Nicholas’s attempts to square the circle.  Nicholas wrote four texts on the topic–De circuli quadratura , 1450, Quadratura circuli 1450, Dialogus de circuli quadratura  1457 and De caesarea circuli quadratura , 1457–all of which he sent to Georg von Peuerbach in Vienna. Regiomontanus wrote a series of notes analysing these texts during his time in Vienna and his conclusion was far from flattering, “Cusanus makes a laughable figure as a geometer; he has, through vanity, increased the claptrap in the world.” Regiomontanus’ very negative analysis of Nicholas of Cusa geometry was first published by Johannes Schöner as an appendix to Regiomontanus’ De triangulis omnimodis in 1533.

Nicholas of Cusa was a good friend of Regiomontanus’ teacher Georg von Peuerbach (1423–1461). Georg von Peuerbach travelled through Italy between graduating BA in 1448 and when he returned to Vienna to graduate MA in 1443.  In Italy he became acquainted with the astronomers Giovanni Bianchini (1410–after 1469), Paolo dal Pozzo Toscanelli (1397–1482), and Nicholas of Cusa. In fact, he lived with Nicholas in his apartment in Rome for a time. Later Georg von Peuerbach and Nicholas corresponded with each other. During his travels in Italy Regiomontanus met Toscanelli and Bianchini and also corresponded with both of them but for Nicholas we have no record of any personal contact whatsoever. As we have seen Regiomontanus heavily criticised Nicholas’ mathematics, but this only became public long after both of them were dead.

Strathern tells us:

Regiomontanus was sent to the University of Leipzig in 1437 [my emphasis], at the age of eleven. Five years later he was studying at the University of Vienna, where he took a master’s degree and began lecturing in optics and classical literature at the age of twenty-one. 

Note there is here no mention of Georg von Peuerbach, in fact, in the whole section about Regiomontanus Georg von Peuerbach gets no mention whatsoever. This is quite incredible! Writing about Regiomontanus without mentioning Georg von Peuerbach is like writing about Robin the Boy Wonder without mentioning Batman! Peuerbach was Regiomontanus’ principal and most influential teacher in Vienna and after Regiomontanus graduated MA, the worked closely together as a team, reforming, and modernising astronomy up till Peuerbach’s death in 1461. Their joint endeavours played a massive role in the history of European astronomy. 

But be warned gentle readers there is far worse to come. If we go and search for the good Georg von Peuerbach, reported missing here, we find the following horror in the chapter on Copernicus:

He had also read the work of the Austrian Georg von Peuerbach, who had lived during the earlier years of the century (1423–61). Peuerbach had been taught by Regiomontanus [my emphasis] and had collaborated with him, using instruments which he invented to measure the passage of the stars in the heavens. 

I don’t know whether to laugh or cry or simply to don rubber gloves, pick up the offending tome, and dump it in the garbage disposal.

You might also note that in 1437, Regiomontanus was one year old not eleven!

While Regiomontanus was teaching at the University of Vienna, the city was visited by the Greek scholar Bessarion, who would play a significant role in Regiomontanus’s subsequent career. As such, it is worth examining Bessarion’s unusual background. 

This is followed by a reasonable brief synopsis of Bessarion’s life prior to his visit to Vienna but no explanation of why he was there or what he did respective Peuerbach and Regiomontanus whilst he was there. This is important in order to understand future developments. Bessarion came to Vienna in 1460 as papal legate to negotiate with the Holy Roman Emperor Frederick III. He also sought out Georg von Peuerbach, who was acknowledged as one of the leading astronomer/mathematicians in Europe, for a special commission. Earlier Bessarion had commissioned another Greek scholar, Georg of Trebizond (1395–1472) to produce a new translation Ptolemy’s Mathēmatikē Syntaxis or as it is better known the Almagest from the original Greek into Latin, providing him with a Greek manuscriptGeorg of Trebizond made a mess of the translation and Bessarion asked Georg von Peuerbach to do a new translation. Georg von Peuerbach couldn’t read Greek, but he knew the Almagest inside out and offered instead to produce an improved, modernised Epitome of it instead. Bessarion accepted the offer and Georg von Peuerbach set to work. Bessarion then asked Georg von Peuerbach if he would become part of his familia (household) and accompany him back to Italy. Georg von Peuerbach agreed on the condition that Regiomontanus could accompany them; Bessarion accepted the condition. Unfortunately, Georg von Peuerbach, only having completed six of the thirteen books of the Almagest, died in 1461, so it was only Regiomontanus, who accompanied Bessarion back to Italy as a member of his familia. A more detailed version is here.

Back to Strathern:

Under Bessarion’s guidance, many works of Ancient Greece – of which western Europe was ignorant – were translated into Latin. And it was in this way that Regiomontanus learned sufficient Greek for him to be accepted as a member of Bessarion’s entourage while he travelled through Italy. 

Most of those works were actually already known in Europe, either through poor quality translations from the Greek or translation from Arabic. This was not how Regiomontanus learnt Greek. He was part of Bessarion’s familia and Bessarion taught him Greek during their travels. 

During these years, Regiomontanus would complete a new translation of the second-century Greek Almagest by Ptolemy. 

Regiomontanus didn’t complete a translation of Ptolemaeus’ Almagest, he completed Georg von Peuerbach’s Epitome of the Almagest (Epytoma in almagesti Ptolemei), fulfilling a death bed promise to Georg von Peuerbach to do so. To quote Michael H Shank

The Epitome is neither a translation (an oft repeated error) nor a commentary but a detailed sometimes updated, overview of the Almagest. Swerdlow once called it “the finest textbook of Ptolemaic astronomy ever written.

Epytoma in almagesti Ptolemei. frontispiece Source: Wikimedia Commons

Strathern continues:

This is the work in which Ptolemy describes the movements of the sun, the moon, the planets and the stars around the earth, which was deemed to be the centre of the universe. For many centuries, such geocentric teaching had been accepted by the early Christians as Holy Writ, and as such its authority lay beyond question. 

Strathern is perpetuating a popular myth. Geocentric cosmology and the Ptolemaic version of it were very often questioned and subjected to criticism throughout the medieval period, both by Islamic and European astronomers and philosophers, as I have documented in numerous blog posts. In fact, Copernicus’ heliocentric model appeared during an intense period of criticism of the accepted astronomy, which began around 1400. Strathern himself in this chapter details Nicholas of Cusa’s unorthodox cosmological speculations!

Strathern now delivers the standard speculation that Regiomontanus was moving towards a heliocentric view of the cosmos based on an over interpretation of a couple of quotes but then tells us:

Some suspect that Regiomontanus must surely have thought through the obvious implications of these remarks, i.e. that the earth moves around the sun. But there is no evidence for this. On the contrary, despite his suspicions as to the accuracy of Ptolemy’s universe, Regiomontanus seems to have continued to use geocentric astronomical mathematics, as well as accepting the authority of Aristotle’s pronouncement that ‘comets were dry exhalations of Earth that caught fire high in the atmosphere or similar exhalations of the planets and stars’. This reliance on ‘authority’ was certainly the case when he made observations of the comet which remained visible for two months during early 1472. He calculated this comet’s distance from the earth as 8,200 miles, and its coma (the diameter at its head) as 81 miles. According to the contemporary astronomer David A. J. Seargent: ‘These values, of course, fail by orders of magnitude, but he is to be commended for this attempt at determining the physical dimensions of the comet.’* 

In the footnote indicated by the *. Strathern writes:

* This comet is visible on earth at intervals ranging from seventy-four to seventy-nine years. Its first certain observation was recorded in a Chinese chronicle dating from 240 bc. When it was observed by the English astronomer Edmond Halley in 1705 it was named after him. The justification for this is that Halley was the first to realize that it was the same comet as had appeared at 74–79-year intervals since time immemorial. Even so, Regiomontanus deserves more than a little credit for his observation of the comet, for in the words of the twentieth-century American science writer Isaac Asimov: ‘This was the first time that comets were made the object of scientific study, instead of serving mainly to stir up superstitious terror.’ 

There is quite a lot to unpack in these two paragraphs, but we can start with the very simple fact that the Great Comet of 1472 was not Comet Halley! The most important point of Regiomontanus’ comet observations is that he tried to determine its distance from the Earth using parallax, this was an important development in the history of astronomy despite his highly inaccurate results. He wrote a book De Cometae, outlining how to determine the parallax of a moving object that was published in Nürnberg in 1531 and played an important role in the attempts to determine the nature of comets in the sixteenth century. 

Comet of 1472 Woodcut from the 1493 Nuremberg Chronicle via Wikimedia Commons

Regiomontanus was not the first to make comets “the object of scientific study” that honour goes to Paolo dal Pozzo Toscanelli, who began treating comets as celestial objects and trying to track their path through the heavens beginning with the comet of 1433, and continuing with the comets of 1449-50, Halley’s comet of 1456, the comet of May, 1457, of June-July-August, 1457, and that of 1472.  He did not publish his observations, but he almost certainly showed them to Georg von Peuerbach when they met. Georg von Peuerbach went back to Vienna in thee 1440s he applied Toscanelli’s methods of comet observation to Comet Halley in 1456 together with his then twenty-year-old student Regiomontanus, as did Toscanelli in Italy. 

Paolo dal Pozzo Toscanelli Source: Wikimedia Commons

Following on to the comet disaster Strathern writes: 

However, Regiomontanus would make two contributions of lasting importance. In his work on rules and methods applicable to arithmetic and algebra, Algorithmus Demonstratus, he reintroduced the symbolic algebraic notation used by the third-century Greek mathematician Diophantus of Alexandria. He also added certain improvements of his own. Basically, this is the algebra we use today, where unknown quantities are manipulated in symbolic form, such as ax by c. Here and are variable unknowns, and ab, and are constants. 

My first reaction was basically, “Yer wot!” I am, for my sins, supposed to be something of a Regiomontanus expert and I have never heard of a book titled Algorithmus Demonstratus and I know for a fact that Regiomontanus did not introduce or reintroduce symbolic algebra, so it was rabbit hole time again.

During his travels in Italy and Hungary, Regiomontanus collected a large number of mathematical, astrological, and astronomical manuscripts, a number of which he intended to print and publish when he settled down in Nürnberg; of which more later. Unfortunately, he died before he could print more than a handful and it turns out that the Algorithmus Demonstratus was one of those manuscripts, which was then edited by Johannes Schöner and published by Johannes Petreius in Nürnberg in 1535. Although it has been falsely attributed to both Regiomontanus, and to the thirteenth century mathematician Jordanus de Nemore, it is not actually known who the author was. Although it has some very primitive attempts to introduce letters for numbers It is in no way an (re)introduction of symbolic algebra as you can judge for yourself here.

As Frank J. Swetz, an expert for Early Modern arithmetic, writes on the MAA website:

On page 10 of the Algorithmus, we find crude attempts to employ symbolic notation. For example, the third paragraph down notes that digit a multiplied by digit b will result in articulum c. An example is given in the margin: 5 x 4 = 20; also articulum times articulum b gives [the product n, 50 x 40 = 2000].

Source: MAA see link above

It is obvious that Strathern literally doesn’t know what he’s talking about and has never even bothered to take a look at the book he is describing.

The garbage continues:

Regiomontanus also made considerable advances in trigonometry, although it has since been discovered that at least part of this was plagiarized from the twelfth-century Arab writer Jabir ibn Aflah. On top of this, Regiomontanus drew up books of trigonometric tables: these lists provided ready answers in the calculation of angles and lengths of sides of right-angle triangles. 

Strathern is here referencing Regiomontanus’ De triangulis omnimodis (On Triangles of All Kinds) edited by Johannes Schöner and printed and published posthumously by Johannes Petreius in Nürnberg in 1533. This is the book he should have featured and not the spurious  Algorithmus Demonstratus. The accusation that he had plagiarised Jābir ibn Aflah was already made in the sixteenth century by the Italian polymath Gerolamo Cardano (1501–1576), whose books were also printed and published by Petreius in Nürnberg. For its role in the history of trigonometry I quote Glen van Brummelen (In his own words, he is the “best trigonometry historian, and the worst trigonometry historian” (as he is the only one)):

[…] what separates the De triangulis from its predecessors is–as the title say–its universal coverage of all cases of triangles, plane or spherical, and its demonstrations from first principles of the most important theorems. It is remarkable in the way that Euclid’s Elements is: not because its results were new, but its structure codified the subject for the future. Although not published until 1533, the De triangulis was to be the foundation of trigonometrical work for centuries, and was a source of inspiration for Copernicus, Rheticus, and Brahe, among many others.[2]

Van Brummelen follows this with a section on possible sources, which Regiomontanus might have used:

There are several possible Arabic sources that Regiomontanus might have used for the De triangulis.


Rather, as the absence of the tangent function in the De triangulis suggests, Regiomontanus’s debt seems to lie mostly in the tradition of the Toledan Tables and Jābir ibn Aflah, whose writings were still being published after Regiomontanus’s death. Several Arabic antecedents have been suggested for particular theorems in De triangulis, but the smoking gun of transmission awaits discovery.[3]

Title page of De triangulis omnimodis  Source MAA

De triangulis does not include the tangent function because Regiomontanus had already dealt with that in his earlier Tabula directionem, which was written in 1467 but first published by Erhard Ratdolt in Augsburg in 1490. This book was a Renaissance bestseller and went through eleven edition the last appearing at the beginning of the seventeenth century. 

This is followed by another piece of misinformation from Strathern:

And it is in these tables that Regiomontanus popularized yet another notational advance. Instead of fractions, which could become increasingly complex, he started using decimal point notation, which was much easier to manipulate. A simple example: the sum of 1/8 + 1/5 is much easier to calculate when these numbers are written as 0.125 + 0.2. The answer in fractional form is 13/40, but in decimal form it is simply 0.325. Furthermore, the decimal answer is much more amenable to further addition, multiplication and so forth with other numbers in decimal form. 

Regiomontanus did not use decimal point notation, to quote Wikipedia, which paraphrases E. J Dijksterhuis, Simon StevinScience in the Netherlands around 1600, 1970 (Dutch original, 1943):

Simon Stevin in his book describing decimal representation of fractions (De Thiende), cites the trigonometric tables of Regiomontanus as suggestive of positional notation.  

Decimal positional notation had existed in Arabic mathematics since the tenth century and there is a complex history of its use over the centuries. Stevin is credited with having introduced it in European mathematics in1585, although, as stated, he credits Regiomontanus as a predecessor, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions.[4]   However, Stevin did not use a decimal point, this innovation is often falsely attributed to John Napier in his Mirifici logarithmorum canonis constructio written before 1614, but first published posthumously in 1620. However, Christoph Clavius had already used the decimal point in the goniometric tables of his astrolabium text in 1593.  

After leaving Rome, Regiomontanus travelled around Europe, continuing to compile his tables and frequently constructing ingenious objects for his hosts. In Hungary, for King Matthias I, he created a handheld astrolabe. Such devices were first made by the Ancient Greeks in around 200 BC. They contain many moving parts, which mirror the movements of the planets and the stars. Astrolabes can be put to a variety of uses, including astronomy, navigation, the calculation of tides, and the determining of horoscopes for astrologers. 

Strathern seems to be under the impression that Regiomontanus spent his four years as a member of Bessarion’s familia living in Rome, whereas in fact he spent most of his time travelling around Italy visiting libraries and archives to search out manuscripts which he copied both for himself and Bessarion. He left Italy in 1465 and for the next two years we don’t know where he was. In 1467 he was on the court of János Vitéz the Archbishop of Esztergom in Hungary, about 45 kilometres northwest of Budapest, working as his librarian. It was Vitéz, who commissioned him to write his Tabula directionem. In 1468 he moved to the court of the King, Matthias Corvinus (1443–1490), again as librarian, where he stayed until 1471, when he moved to Nürnberg. 

Strathern’s few sentences on the astrolabe are amongst to worst that I have ever read on the instrument. I shall forgive him the, “Such devices were first made by the Ancient Greeks in around 200 BC,” as variation on this myth can be found everywhere, including on Wikipedia, usually crediting the invention of the astrolabe to either Hipparchus or Apollonius. I shall take the opportunity to correct this myth.

We don’t actually know where or when the astrolabe first put in an appearance. The earliest mention of the stereographic projection of the celestial sphere that is at the heart of an astrolabe was the Planisphaerium of Ptolemy written in the second century CE. This text only survived as an Arabic translation. The earliest known description of the astrolabe and how to use it was attributed to Theon of Alexandria (c. 335–c. 405 CE), it hasn’t survived but is mentioned in the Suda, a tenth century Byzantine encyclopaedia of the ancient Mediterranean world, as well as Arabic sources. The extant treatises on the astrolabe of John Philoponus (c. 490–c. 570) and of Severus Sebokht (575–667) both draw on Theon’s work. The development of the instrument is attributed to Islamic astronomers; the oldest surviving astrolabe is a tenth century Arabic instrument.

An astrolabe usually only has two moving parts, the rete, a cut out star map with the ecliptic and, in the northern hemisphere, the tropic of cancer, that rotates on the front side over the stereographic projection of the celestial sphere. On the back of the astrolabe is an alidade, a sighting device. Some astrolabes also have a rotating rule on the front to make taking readings easier.

Arabic Astrolabe 1208 Above: Cut out Rete with eccentric ecliptic and stars as points of thorns. Below right: front of astrolabe with rete Left: rear of astrolabe with alidade Source: Wikimedia commons

Regiomontanus wrote a text on the construction and use of the astrolabe, whilst he was in Vienna. He is thought to have constructed several instruments of which the most famous is one he made for and dedicated to Bessarion in 1462. The instrument he made for Corvinus has not survived.

Regiomontanus astrolabe from 1462 dedicated to Bessarion

We move on:

In Nuremberg, Regiomontanus established a novel type of printing press, the first of its kind devoted entirely to the printing of scientific and mathematical works. 

I’m not sure how to interpret this sentence. Does Strathern mean that Regiomontanus’ printing press, meaning printing house, was novel, because “it was the first of its kind devoted entirely to the printing of scientific and mathematical works,” which is true. Or does he mean that Regiomontanus had created a novel mechanical printing press, which is not true.

He also oversaw the building of the earliest astronomical observatory, in Germany. 

This is simply not true, there was no observatory. Regiomontanus and his partner Bernhard Walter made their astronomical observations with portable instruments out in the street. 

Finally returning to Rome, he constructed a portable sundial for Pope Paul II. Later he also seems to have re-established contact with his friend and mentor, Cardinal Bessarion, who was in Rome in 1471 for the conclave to elect a new pope after the death of Paul II. 

Regiomontanus remained in Nürnberg from 1471 to 1475, when he was called to Rome to assist in a calendar reform. He died there in 1476 probably in an epidemic.

To call this capital of Strathern’s shoddy would be akin to praising it. It creates the impression that he gathered together a pile of out-of-date references and debunked myths, threw them up in the air and then sent the ones that landed on his desk to the publishers. 

[1] Vincent Ilardi, Renaissance Vision from Spectacles to Telescopes, American Philosophical Society, 2007, pp. 80-81

[2] Glen van Brummelen, The Mathematics of the Heavens and the EarthThe Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009 pp. 260-261

[3] Van Brummelen p. 261

[4] Dijksterhuis, Stevin. 


Filed under Book Reviews, Renaissance Science, Uncategorized

Renaissance garbage ­– I

In case you hadn’t noticed there is a four-weekly cycle of blog posts here at the Renaissance Mathematicus. Week one is a new series post, week two a fairly random #histSTM post, week three the next new series post, and week four a book review. Sometimes I throw in a random HISTSCI_HULK post just because. Today should be the next series post but I posted the last episode of my Renaissance Science series two weeks ago. You might be pleased or dismayed to learn that I have a new series lined up, at least in my head, but I want to take a little breathing space before I start in on it. To bridge the gap, I shall be posting a series of posts, which are sort of related to the Renaissance Science series, on a book by Paul Strathern, The Other RenaissanceFrom Copernicus to Shakespeare, (Atlantic Books, 2023). The Other Renaissance is, of course, what is normally referred to as the Northern Renaissance.

I must admit that I had never heard of Paul Strathern, but a friend, who shall remain nameless, received a review copy of this book, and thought it was so bad that he decided not to review it at all. My historical garbage antenna went up and I made further inquiries, whereupon said friend was kind enough to send me a pdf of said book. And yes folks, it is truly a stinker. I don’t intend to review the whole thing, but I thought I would post a series of HISTSCI_HULK typical put downs of selected elements of the text.

However, before we get down to the nitty-gritty a few words about our author. According to his Wikipedia page, Paul Strathern (born 1940!!) is a Scots Irish writer and academic, and an incredibly prolific writer he is too. To date, he has published five novels, fifteen “academic” books, nineteen titles in his Philosophers in 90 Minutesseries, eleven titles in his Great Writers in 90 Minutes series, twelve titles in his The Big Idea: Scientists Who Changed the World series, and finally three travel guides. I wonder what he does in his spare time? Also, if his The Big Idea: Scientists Who Changed the World are as bad as his The Other RenaissanceFrom Copernicus to Shakespeare, then I’m glad that the HISTSCI_HULK hasn’t read any of them.

As I will be only commenting on selected bits of the book here is the full contents list:

Map viii–ix Timeline of Significant Events during the Northern Renaissance x-xi 

Prologue: Lifting the Lid 1 

1 Gutenberg 13 

2 Jan van Eyck 21 

3 Nicholas of Cusa 33 

4 Francis I and the French Renaissance 47 

5 A New Literature: Rabelais 63 

6 Martin Luther and the Protestant Reformation 73 

7 The Rise of England 89 

8 The Rise and Rise of the Fuggers 105 

9 Copernicus 125 

10 Erasmus 141 

11 Dürer 157

12 Straddling Two Ages: Paracelsus and Bruegel the Elder 175 

13 Versions of the True: Mercator and Viète 193 

14 Vesalius 211 

15 Catherine de’ Medici 229 

16 Montaigne 249 

17 Elizabethan England 265 

18 Brahe and Kepler 281 

19 Europe Expands 301 

Conclusion: A Last Legacy 317 

And the timeline: (The formatting is a bit weird, result of my cut and paste)


1415 Jan Hus, founder of the Hussites, burned at the stake
1432 Van Eyck completes the Ghent Altarpiece
1460s Regiomontanus oversees the building of the first observatory in Europe (Rubbish!)
1464 Death of Nicholas of Cusa
1474 Technique of painting in oils spreads from the Netherlands to Italy
1494 Signing of the Treaty of Tordesillas: Pope Alexander VI draws a line down the Atlantic Ocean, dividing the globe between Spain and Portugal
1497 John Cabot sails from Bristol and reaches North America 

1509 Erasmus writes In Praise of Folly
1512 Torrigiano is commissioned by Henry VIII to create a Renaissance tomb for Henry VII

1514  Dürer produces Melencolia I 

1515  Francis I ascends to the throne of France

1517 Martin Luther nails his Ninety-Five Theses to the door of Wittenberg Castle Church
1519 The death of Leonardo da Vinci in France
1519 Charles V is crowned Holy Roman Emperor
1525 Death of Jacob Fugger the Rich
1529 The Colloquy of Marburg fails to unite Protestants 

1533 Henry VIII breaks with Rome 

1536 John Calvin arrives in Geneva; Calvinist missionaries soon begin to spread over northern Europe 

1536 French explorer Jacques Cartier brings Chief Donnacona from the New World to see Francis I 

1541 Death of Paracelsus
1543 Copernicus is shown the first published edition of his De Revolutionibus Orbium Coelestium, describing the solar system, while on his deathbed
1543 Vesalius publishes De Fabrica, describing human anatomy
1547 The completion of the Château de Chambord in the Loire Valley 

1547 The death of Francis I of France
1553 The death of Rabelais
1553 English explorer Richard Chancellor visits the court of Ivan the Terrible
1558 The death of Holy Roman Emperor Charles V
1569 Mercator publishes his Atlas, containing his cylindrical projection map of the world
1572 The St Bartholomew’s Day massacre of Huguenots in France 

1588  The Spanish Armada fails to invade Elizabethan England 

1589  The death of Catherine de’ Medici, mother of French kings and ruler of France 

1596 Johannes Kepler publishes his laws describing the elliptical orbit of plants (Love this typo!)

1600  The founding of the East India Company in London 

1601  Tycho Brahe dies in Prague 

1608 Invention of the perspicillium in Holland, which inspires Galileo to create the telescope (Rubbish!)
1616 The death of Shakespeare
1642 Cardinal Richelieu dies in France
1648 The Peace of Westphalia ends the Thirty Years’ War 

I decided to start this week with Strathern’s chapter on Albrecht Dürer, as I have a local connection to the man. 

After a conventional biographical introduction from birth to marriage, Strathern tells us:

Dürer did not travel on his own. He is thought to have been accompanied by an ebullient rugged-faced companion named Willibald Pirckheimer, [my emphasis] whose appearance belied his acute intelligence and thirst for learning. Pirckheimer came from a distinguished family in Nuremberg, was a year older than Dürer, and was filled with patrician self-confidence. He was studying law at Padua, and during the course of their friendship Pirckheimer would fill the huge gaps in Dürer’s education, introducing him to the humanist ideas he had picked up at university and amongst his father’s intellectual circle in Nuremberg. 

When I read that Dürer, on his first journey to Italy, was accompanied by Willibald Pirckheimer, I did a double take. Having read quite a lot about Dürer and Pirckheimer and I’ve never come across any such claim. So back to the literature. My first stop was German Wikipedia, where to my surprise I read the following:

In der Folgezeit bis 1500 schuf er eine Serie von kleinen Landschaftsaquarellen mit Nürnberger Motiven bzw. mit Motiven von Stationen seiner ersten Italienreise, die er in der ersten Hälfte des Oktobers 1494, bereits drei Monate nach seiner Hochzeit, antrat. Diese Reise verstärkte sein Interesse an der Kunst des Quattrocento. Im Mai 1495 kehrte er zurück nach Nürnberg.

Von der jüngeren Forschung wird angezweifelt, dass Dürer im Rahmen dieser Reise jemals die Grenzen des deutschen Sprachgebiets überschritt, und die Indizien, die gegen einen Aufenthalt in Venedig sprechen, häufen sich: Dürer selbst erwähnte in seiner Familienchronik 1494/95 keine Reise nach Venedig. Die italienischen Züge in seinen Werken ab 1497 interpretieren manche als direkten Einfluss des paduanischen Malers Andrea Mantegna, der 1494/95 zwar nicht in Padua war, dessen Werke Dürer aber dort gesehen haben könnte. Beweisbar ist nur, dass Dürer in Innsbruck, Trient und Arco beim Gardasee war. Von Orten südlich von Arco gibt es bei Dürers Aquarellen keine Spur, also auch nicht von Venedig. Auch die Route spricht gegen die Venedig-Theorie: Für Dürer hätte es näher gelegen, den für Nürnberger (Kaufleute) üblichen Weg nach Venedig zu nehmen, der über Cortina und Treviso verlief und „Via Norimbergi“ genannt wurde. Die Bilder aus seiner späteren, nachweisbar venezianischen Zeit ab 1505 haben deutlich stärker venezianische Charakteristika.

For those who don’t read German, it basically says that recent research doubts that Dürer ever left the German language area in 1494 and thus was never in Italy on this journey. This was new to me as I have always read about and accepted that Dürer made two journeys to Italy, the first in 1494. Happily, in 2021, the National Gallery in London put on a major expedition Dürer’s JourneysTravels of a Renaissance Artist for which there is an amazing book[1], which thanks to my very generous stepmother I own a copy. Turning to this wonderful tome I discovered that it is really so that historians now believe that Dürer did not reach Italy in 1494. Apparently, the whole story of the first Italian journey is based on two very short ambiguous quotes and the rest has been built up over the years based on reading the tealeaves in Dürer’s work. 

I actually began to question these two paragraphs of Strathern because of his claim that Pirckheimer had accompanied Dürer on this journey. No journey so no Pirckheimer but there is more. Strathern correctly states that Pirckheimer was studying in law in Padua, which he did for seven years from 1488, first returning to Nürnberg on 1495. This was when Willibald and Albrecht first met!

Later Strathern turns to the “second” Italian journey, the one that really did take place and dishing up a myth that has been long debunked, he tells us:

Between 1507 and 1509 Dürer paid a second visit to Italy, passing beyond Venice to Padua and maybe even Mantua. He certainly visited Bologna, for it was here that he met Luca Pacioli, the friar mathematician and friend of Leonardo da Vinci. It was Pacioli who had taught Leonardo mathematics, and it seems that Dürer too studied with him. Dürer’s meticulous and exact art inclined him to mathematics, and it would play an increasing role in both his painting and his other intellectual interests. Pacioli is known to have taught Dürer linear perspective, which was by now widely developed amongst Italian Renaissance artists. But Pacioli probably taught Dürer much more than this useful mathematical–artistic device, for Dürer would continue to study mathematics over the coming years. 

It is simply not known from whom Dürer learnt the basics of, the then still comparatively new, linear perspective. The question has a certain historical importance, as he is credited with having introduced linear perspective into Northern European art. I have no idea who first introduced the theory that he learnt it from Luca Pacioli during his time in Bologna but that is absolutely no evidence to support it. The theory was finally totally debunked, when somebody pointed out that when Dürer was in Bologna, Pacioli was in Milano! Maybe Pacioli taught him telepathically? As for the implication that Pacioli also taught Dürer mathematics, we know from fairly solid evidence that Dürer didn’t need to go to Italy for his maths lessons, he got them at home in Nürnberg from Johannes Werner. (1468–1522)

Source: Wikimedia Commons

Next up we get a strange twist in the Dürer timeline from Strathern:

By the time Dürer returned home from his second visit to Italy, he was known by reputation throughout Europe. In 1512, the Holy Roman Emperor Maximilian I became one of his patrons. Despite this, Dürer found that he was making insufficient income from his paintings, and even abandoned this art form for several years in favour of making woodcuts and engravings – which could be reproduced and thus sold many times over. He may not have been the best painter in Europe, but his engravings were unsurpassed. 

Dürer served his apprenticeship in the studio of Michael Wolgemut between 1486 and 1490. Wolgemut specialised in producing woodblock prints as book illustrations. For example, his studio produced the illustrations for the famous  Liber Chronicarum, better known as the Nuremberg Chronicle in English and Die Schedel’sche Weltchronik in German. There are even speculations by art historians as to whether the young apprentice was responsible for some of those illustrations. From the very beginning when he set up his own workshop in 1495, Dürer specialised in woodblock printing. He also developed his skill in engraving, almost certainly learnt in his original apprenticeship under his father, a goldsmith. To illustrate a couple of quotes from Wikipedia:

Arguably his best works in the first years of the workshop were his woodcut prints, mostly religious, but including secular scenes such as The Men’s Bath House (ca. 1496). These were larger and more finely cut than the great majority of German woodcuts hitherto, and far more complex and balanced in composition.

His series of sixteen designs for the Apocalypse] is dated 1498, as is his engraving of St Michael Fighting the Dragon. He made the first seven scenes of the Great Passion in the same year, and a little later, a series of eleven on the Holy Family and saints. The Seven Sorrows Polyptych, commissioned by Frederick III of Saxony in 1496, was executed by Dürer and his assistants c. 1500. In 1502, Dürer’s father died. Around 1503–1505 Dürer produced the first 17 of a set illustrating the Life of the Virgin which he did not finish for some years. Neither these nor the Great Passion were published as sets until several years later, but prints were sold individually in considerable numbers.

In 1496 he executed the Prodigal Son, which the Italian Renaissance art historian Giorgio Vasari singled out for praise some decades later, noting its Germanic quality. He was soon producing some spectacular and original images, notably Nemesis (1502), The Sea Monster (1498), and Saint Eustace (c. 1501), with a highly detailed landscape background and animals.

Prints are highly portable and these works made Dürer famous throughout the main artistic centres of Europe within a very few years.

As you can see this is not post 1512 but we have just reached 1505 and Dürer is a highly prolific and famous producer of fine art prints. In fact, rather than being a painter, who turned to fine art printing for financial reasons, as Strathern would have us believe, Dürer was a highly successful fine art printer, who painted on the side.

We get the standard discussions of The Rhinoceros, Dürer’s most famous print, and Melencolia I, his most enigmatic and most interpreted print. I have only one question about Strathern’s waffle here, he writes about Melencolia I:

Though mathematics, especially geometry (Plato’s favourite), underlies much of the scene. 

There is no other reference to Plato anywhere in his convoluted discussion of Melencolia I, so why shove him in here? Earlier he makes the equally strange comment:

Set into the wall above the angel’s head is a four-by-four magic square, indisputable evidence of Dürer’s continuing mathematical interest. 

There was nothing to say that Dürer had ever stopped being interested in mathematics.

Having dealt with The Rhinoceros and before Melencolia I, Strathern enlightens us with the following paragraph:

As we have seen, the year 1500 marked one and a half millennia since the birth of Christ, and there was a widespread belief around this time that it heralded the Second Coming of Christ, which is mentioned in the Bible: ‘This same Jesus, which is taken up from you into heaven, shall so come in like manner as ye have seen him go into heaven.’ Such an event would precede the Last Judgement, after which our souls would be despatched to Purgatory, Hell or Heaven.

Various dates were considered by various people to signify the second coming but I personally have never come across a reference to 1500 as one of them.

Strathern also turns his spotlight on the Ehrenpforte Maximilians I, known in English as The Triumphal Arch or the Arch of Maximillian I, he writes:

Dürer created a number of works for his most important patron, Maximilian I. Amongst these is a large, highly complex woodcut of a triumphal arch, which measures almost ten feet by ten feet. Dürer spent over two years – on and off – busying himself with this work, which includes 195 separate woodcuts printed on 36 sheets of paper. The intention was that it should be hung in princely palaces and city halls throughout the Holy Roman Empire. Indeed, Maximilian I made a habit of giving away copies of this work with this intention. 

The work itself is a suitably grandiose hotchpotch of styles – resembling, if anything, an example of Indian architecture rather than any classical triumphal arch (such as Marble Arch in London, or the Washington Square Arch in New York). It stands more as a monument to Dürer’s indefatigable technical expertise than any aesthetic achievement. Such a work made him rich, allowing him independence – even if it contributed nothing to his artistic attainment and was otherwise a complete waste of his time. [my emphasis]

Ehrenpforte Maximilians I, Source: Wikimedia Commons

As is, unfortunately, all to common Strathern attributes this work to Dürer alone but it was the work of a group of people, quoting, yet again, Wikipedia:

The design program and explanations were devised by Johannes Stabius, the architectural design by the master builder and court-painter Jörg Kölderer and the woodcutting itself by Hieronymous Andreae, with Dürer as designer-in-chief. […] the flanking round towers are attributed to Albrecht Altdorfer.

The closing clause from Strathern, that I have emphasised, displays, in my opinion, his ignorance of the professional life of an artist and in particular that of his subject Albrecht Dürer. Dürer ran a highly professional, commercial fine art print studio, with which he not only earned the money on which he and his family lived but also the money with which he paid his employees. The Ehrenpforte was anything but a complete waste of time, as the commission raised the status of his studio and did in fact contribute to his artistic attainment, as it displayed his mastership in woodblock printing to the world. 

Here we have the name of the mathematician, Johannes Stabius 1450–1522), who was the Imperial Court historian, was the director of the project, as he would employ Dürer on two further commissions, neither of which Strathern considers worth mentioning, despite his continued references to Dürer’s interest in mathematics.

Johannes Stabius portrait by Albrecht Dürer Source: Wikimedia Commons

The first was the Stabius-Dürer World Map

Stabius-Dürer World Map published in 1515, a perspective representation of the earth as a globe. Source: Wikimedia Commons

and the second, and historically much more important the Stabius-Dürer-Heinfogel planispheres of the southern and northern hemispheres, the first European, printed celestial maps, which I wrote about here.

The Dürer Star Maps in a hand coloured edition Source: Ian Ridpath Star Tales

 We get accounts of Dürer’s journey to the Netherlands to get his Imperial pension renewed following the death of Maximillian, including an account of his portrait of Erasmus and Strathern’s rather bizarre interpretation of its Greek inscription. 

Almost at the end of his chapter, Strathern turns to Dürer the mathematician:

With Dürer’s eyesight fading, he devoted less of his energies to his art. Instead he concentrated on writing treatises on such subjects as ‘human proportions’ and ‘fortifications’. However, his most important work was his Four Books on Measurement. These contain the wealth of mathematical knowledge he accumulated during his life – including the geometrical construction of shadows in prints (projective geometry), as well as several ideas by the Tuscan artist Piero della Francesca which had not yet been published. (These Dürer had almost certainly learned from Luca Pacioli.) Very little of this vast compendium of work is original, but it was written in the vernacular German rather than in Latin. This established Dürer as the first figure of the northern Renaissance to outline in German Euclidean geometry and demonstrate the construction of the five Platonic solids, other Archimedean semi-regular truncated solids, and a number of constructed figures which are thought to have been of his own invention. His treatises were the first printed north of the Alps to view art in a scientific fashion, exposing the mathematical bones upon which much artistic flesh is based. 

I’m sure Dürer would be delighted to know that his Four Books on Measurement was his most important work, whereas his ‘human proportions’ was just a treatise on “such subjects”! 

About the time of his “second” journey to Italy, Dürer became obsessed with the idea that the secret of beauty lies in the mathematical theory of proportions. He began working on his Vier Bücher von menschlicher Proportion (Four Books on Human Proportion) in 1512 and the four books, written at different time over the years, deal with various aspects of exactly that, human proportion. An appendix to the book explains Dürer’s theories on ideal beauty.

Title page of Vier Bücher von menschlicher Proportion Source: Wikimedia Commons

The book was written for apprentice artists and in the middle of the 1520s, Dürer realised that the geometry of the book was too advanced for the intended readers, so he sat down and wrote his Four Books on Measurement (Underweysung der Messung mit dem Zirckel und Richtscheyt or Instructions for Measuring with Compass and Ruler), (which I wrote about here) an introductory textbook on geometry for apprentice artists. It is the book on human proportions that is his most important work, the Four Books on Measurement merely developed the mathematical tools needed to understand it. 

Underweysung der Messung mit dem Zirkel und Richtscheyt Title Page

I have no idea which not yet published ideas from Piero della Francesca Dürer’s book supposedly contains. It goes without saying that Dürer didn’t learn anything from Luca Pacioli, whom he never met and with whom, as far as we know, he didn’t correspond. However, he might have accessed della Francesca work via Pacioli, who had plagiarised it in his Divina proportione, published in 1509, which was possibly owned by one of the Nürnberg mathematicians, Werner or Stabius, but that’s just speculation. 

Far from being a vast compendium of work, Underweysung der Messung is 27cm X 18cm and probably less than 200 pages long, it’s not paginated so I had to guess. Dürer’s Underweysung der Messung is actually the very first mathematics book printed in German and like most textbooks it is of course derivative. However, it does contain one important geometrical innovation. Dürer introduced the geometry net, which is the two-dimensional figure that arises when you open a three-dimensional figure along edges.

Underweysung der Messung geometrical net

In one sense Underweysung der Messung did become more important than Vier Bücher von menschlicher Proportion both were into Latin and Underweysung der Messung was translated into several different European languages. Vier Bücher von menschlicher Proportion appealed to a very limited readership but Underweysung der Messung became a very widely read geometry textbook throughout Europe for most of the next hundred years. 

Strathern has obviously not bothered to do serious research for his book but has just thrown it together from the first sources that crossed his path without bothering to check whether they were factually correct or not. As we will see in later chapters this sloppy approach is not confined to Dürer but is characteristic of the whole book. 

[1] Susan Foister and Peter van den Brink eds., Dürer’s JourneysTravels of a Renaissance Artist, National Gallery, London, distributed by Yale University Press, 2021. 


Filed under Book Reviews, Renaissance Science, Uncategorized

Artificial Bullshit!

There has been much hot air expended in recent days over the supposed artificially intelligent program Chat-GPT, which is, in reality, a more sophisticated Internet search engine. James Maynard on his website The Cosmic Companion proudly announced that he had used it “to construct a pictorial journey exploring the story of Hypatia!” Having spent some time using my biological intelligence to survey the modern historical literature on the lady and used the information gained to write a blog post, I have decided to don my pedagogical persona and evaluate the results produced by this new program. The full text is below with my comments in italics.

Reconstructing Hypatia of Alexandria Using Artificial Intelligence

Using Chat-GPT and MidJourney to construct a pictorial journey exploring the story of Hypatia – The last great scientist of the ancient age of the western world.

We’ve used generative artificial intelligence to learn about the last great scientist of the ancient age in the western world.

Sorry but Hypatia was not a great scientist and as for being last of the ancient age in the western world, I think Proclus, Boethius, Simplicius, John Philoponus, and a couple of others might like a word. This is a variant on the, “they murdered science when they murdered Hypatia,” myth, more of which gets spewed out in the closing paragraphs.

Hypatia of Alexandria was an accomplished astronomer, mathematician, and philosopher who lived in the final days of the ancient age of science. Born sometime around 360 CE, Hypatia was raised in the cultural and intellectual center of the Mediterranean, Alexandria, by her mathematician father Theon.

We don’t have any direct proof as to how accomplished Hypatia actually was, as none of her writings have survived.

Hypatia dedicated her life to advancing science and reason in an age when dark forces were closing in on her society.

Once again, a statement with no basis in known facts, as we have no evidence to support it and what are these ominous dark forces?

The Great Library of Alexandria, founded just after 300BCE, was once the greatest storehouse of information in the ancient world. The Library was one part of a larger institution of learning, the Musaeum of Alexandria, which also included a grand university.

Hyperbole, the Great Library of Alexandria was one of the great libraries of antiquity. It was part of the Mouseion, why use Musaeum, the Latin name, for a Greek institution in a Greek city?  The Mouseion was a research institute, or one might call it an institute of advanced learning. However, there was no university, grand or otherwise.

The root of our English word museum, the term Musaeum originally referred to temples honoring the Muses. Over time, this word came to represent centers of learning. 

Nothing to criticize here, but I wonder why articles about Hypatia almost always include sections on the Library and the Mouseion, as both had ceased to exist long before Hypatia was even born?

At the start of the Fifth Century, in the final years of the Alexandrian University, Theon raised his budding scientist in the manner usually reserved for boys — in the father’s trade — in this case, math and science. History leaves us no knowledge about Hypatia’s mother.

What Alexandrian University? Anachronical use of the term scientist is here especially unnecessary as mathematician and astronomer would be more accurate.

Living her life in the cultural and intellectual center of the Mediterranean, Alexandria, Hypatia attended classes and later taught on the ancient grounds of learning, delivering understandable lessons on complex scientific subjects.

Notes based on her teachings are said to cover astronomy, geometry, the use of astrolabes, and more. Hypatia taught classes, some of them to large audiences. Ancient accounts are nearly unanimous in noting her intellectual prowess.

In the normal meaning of the term there are no surviving “notes based on her teachings.” What we have are a handful of general comments on the areas that she taught.

She was a gifted science educator and her works were reported to contain insights on astronomy, geometry, the use of astrolabes, and more.

Very general and rather vague reports, with almost no specifics.

Even her rivals often admired her talents, including John of Nikiu, who stated, “The breadth of her interests is most impressive. Within mathematics, she wrote or lectured on astronomy, geometry, and algebra, and made an advance in computational technique — all this as well as engaging in religious philosophy and aspiring to a good writing style.”

Not a bad review from someone who really doesn’t like you.

What John of Nikiû, who lived more than two hundred years after Hypatia died, actually wrote about her:

In those days a female philosopher appeared in Alexandria, a pagan named Hypatia, and she was completely devoted to magic, astrolabes and music instruments, and she deceived many people through (her) Satanic wiles.

There appears to be something of a discrepancy here between the two accounts!

As violence between Christians, Jewish residents, and Pagans grew, Hypatia assigned herself the task of updating, recording, and safeguarding the mathematical and astronomical knowledge of her age.

This is pure fantasy and has no basis in the known historical facts.

Her fate was sealed in 391 CE, when Emperor Theodosius I issued a decree directing the burning of all Pagan temples. 

Let us see what Wikipedia has to say about Theodosius and pagans: 

Although Theodosius interfered little in the functioning of traditional pagan cults and appointed non-Christians to high offices, he failed to prevent or punish the damaging of several Hellenistic temples of classical antiquity, such as the Serapeum of Alexandria, by Christian zealots. 

We appear to have a contradiction and I know which version I think is correct.

Armed with this acquiescence, Theophilus, bishop of Alexandria, ordered that the center of learning be destroyed. He and his followers carried out the decree, dealing massive destruction to the grounds. Theophilus then ordered a church to be built on the site.

More than 20 years later, in the year 412, he ordered the pillaging of the Serapeum or Temple to Serapis, the Pagan protector of Alexandria. This would prove prophetic. 

We have accounts of Theophilius destroying a hidden pagan temple and his followers mocking the pagan artifacts, which led to a riot during which the pagans withdrew to the Serapeum, which Theophilius then destroyed. I know of no center of learning that he supposedly destroyed. The Serapeum had probably earlier been a smaller daughter library to the Library of Alexandria but no longer fulfilled this function when it was destroyed by order of Theophilius, not in 412 but in 391. 

There is no center of learning involved. The Serapeum was a center for the Neoplatonism of Iamblichus a rival group to the Neoplatonism of Plotinus to which Hypatia adhered. Theophilius actually tolerated Hypatia’s school, so hardly prophetic.

Cyril, Theophilus’s nephew, was named bishop of the region, and he launched a campaign against Pagan temples and set about expelling the Jewish population from Alexandria. Civil unrest between Pagans, Christians, and the Jewish population broke out into years of violence in the city.

In March 415, followers of Cyril ransacked the remaining classrooms and study rooms, destroying what remained of the greatest institution of learning in the ancient world.

This paragraph is pure fantasy. The Mouseion, which I assume is being referenced here, had ceased to exist a couple of hundred years earlier.

The crowds ambushed Hypatia as she rode through the city. The last great scientist and science educator of the ancient western world was flailed, dismembered, and her remains were paraded through the city and burned in a mockery of Pagan funerary rites.

Hypatia was not “the last great scientist and science educator of the ancient western world.” She wasn’t even a great scientist. 

Hypatia’s brutal murder marked the end of science in the west for a thousand years. Europe soon fell into ten centuries of intellectual stagnation that would not lift until the Scientific Renaissance in the middle of the 15th Century. 

Remember that myth at the beginning? This whole paragraph is totally and utter hogwash! We have the classic myth about a thousand-year gap in the history of science from 500 CE to 1500 CE. To counter this rubbish, I could recommend several books e.g. Stephen C. McCluskey, “Astronomies and Cultures in Early Medieval Europe (CUP, 1998), Seb Falk, The Light Ages: The Surprising Story of Medieval Science, (W.W. Norton, 2020), or Edward Grant, Science & Religion 400 BC –AD 1550: From Aristotle to Copernicus (Johns Hopkins University Press, 2004).

Today, Hypatia stands as an example to all science educators to connect with their audience and with those around us. She also broke the gender barrier for science in ancient Europe, an accomplishment for feminism and women that would not be matched until the 18th Century.

We have entered the realm of Hypatia hagiography and mythology.

Hypatia of Alexandria dedicated her life to exploring the mysteries of the Cosmos, and relating her knowledge so that everyone could understand and learn. As darkness closed in on Alexandria, Hypatia spread the light of science for all humanity, and all time.

We are still in the realm of Hypatia hagiography and mythology.

She remains an inspiration to us all.

Does she?

How this was done:

We do not know exactly what Hypatia of Alexandria or the institution looked like, but there are descriptions in ancient texts, as well as modern insights based on contemporary technologies and insights into history and genetics.

The Cosmic Companion used Chat-GPT to merge information from both ancient and modern sources into the most accurate description of her we could produce.

Facts were checked and cross-referenced with accounts from reliable sources, including Encyclopedia Brittanica, The Smithsonian Institution, and National Geographic. The resulting text was translated, as closely as possible, into a prompt for the artificial intelligence graphics engine MidJourney.

Text was created by a similar AI/human process.

Given the explanation above, what this demonstrates is that you get the results based on the quality of the sources you use, a truism for all historical research, and it is painfully clear that the sources used in this case were totally crap. According to the computer programming rule GIGO­–garbage in, garbage out–here Chat-GPT has used garbage sources and produced a garbage text.

If I was grading this apology for an essay as a piece of work handed in by a student, it would of course garner a big fat for fail.Not only is it factually a total disaster area but it is from style a bizarre collection of fragmentary paragraphs that don’t even add up to a coherent whole. If this is the best that Chat-GPT can do even with human editing, then historians have nothing to fear from this AB i.e., Artificial Bullshit


Filed under History of science, Myths of Science, Uncategorized

Ptolemy the pagan

The descriptive panel below, from the Museum of the Bible in Washington DC was posted on Twitter by the historian of Chinese astrology, Jeffrey Kotyk, who posed the question, “I wonder whether Ptolemy would have considered himself “pagan”?”

Reading through the text I have several other comments and queries, but first I will address Jeffrey’s question. Ptolemy lived in the second century CE and was an Alexandrian Greek. At that point in time the Latin word pagan from pāgānus meant “villager, rustic; civilian, non-combatant”. Only in the fourth century did early Christians begin to refer to people who practiced polytheism, or ethic religions other than Judaism as pagans. The word pagan meaning “person of non-Christian or non-Jewish faith” first entered the English language around 1400 CE, so Ptolemy would definitely not have considered himself pagan.

Also referring to Ptolemy, one of the greatest mathematical polymaths of antiquity, as “scholar of the stars” is somewhat limited, not to say strange. The text then attributes a “passion for mathematics, geography, and astronomy” to him but leaves out optics, music theory, and, of course, astrology.  Strangely the opening paragraph seems to attribute those things that developed out of astronomy all to Ptolemy alone. What about all the other astronomers, geographers, mathematicians, who existed before Ptolemy, contemporaneously with him, and after him, didn’t they contribute anything?

Of the things listed, “the ability to navigate the earth, determine agricultural seasons, and organise time into days, months, and years,” only the first, navigation, can really be said to have grown out of astronomy. Systematic agriculture and with it, knowledge of the agricultural seasons predates mathematical astronomy by about six thousand years. Days are a natural phenomenon of which homo sapiens would have been aware since they first evolved, although I assume that animals are also aware of days. 

The same of course applies to the year of which every sentient creature that lives long enough becomes aware without any help from astronomers. Astronomers, of course, determined how many days there are in a solar year, but they took long enough to get it right.

Months are a completely different problem. If we are referring to lunar months, and after all the word month derives from the word for moon, then the same applies, as to days and years. Although the astronomers had the problem of how to align lunar months with solar years, they don’t fit at all, as became obvious fairly early on and you don’t really want to know about the history of early calendrics. Trust me you don’t, that way lies madness! If, however, we are referring to our current system of twelve irregular months fitted into the solar year, then, although the astronomers played a role, they are largely the result of political decisions.

As a result, the Church was able to use scripture and science to identify and commemorate holy days such as Easter.

Knowing something about the history of the determination of the so-called movable Christian holy days, I cringed when I read this very short paragraph. I will pass over it with the simple comment that these holy days are determined not identified and that determination was a very complex religio-political process stretching over several centuries and astronomers had very little to do with it, other than providing the date of the vernal equinox, which in early days was falsely considered to be the 25 March and providing lunar tables. 

I developed the most advanced geocentric model of the universe, at which I believed Earth was the center. 

This sentence is, of course, wonderfully tautological, geocentric meaning the earth is at the centre. The sentence is also, as Blake Stacey pointed out on Mastodon after I posted this, “not only redundant, it’s not even grammatical.”

My geocentric model of the universe was accepted until Copernicus, Galileo, and others introduced a heliocentric model.

Ptolemy’s model was extensively modified by a succession of Arabic astronomers and “the most advanced geocentric model of the universe” before Copernicus was that of the Austrian, Renaissance astronomer, Georg von Peuerbach (1423–1461), whose system Copernicus studied as a student. 

Galileo, who in reality contributed very little to the heliocentric model or to its acceptance, in fact by rejecting supralunar comets, which orbited the sun, and ignoring Kepler’s laws of planetary motions, he explicitly hindered that acceptance, gets a name check with Copernicus, whereas, Kepler, whose heliocentric model was the one that actually became accepted gets dumped under others!

This is a more than questionable piece of museum signage and I wish I could blame it on the religious nature of the museum but such ill researched signage is unfortunately too common. 


Filed under History of Astronomy, Uncategorized

Renaissance Science – L

The so-called scientific revolution in the seventeenth century is often characterised as throwing off the yoke of Aristotelian philosophy that had held the scholastic medieval university in a strangle hold since Albertus Magnus (c. 1200–1280) had made it compatible with Catholic doctrine in the thirteenth century. In reality rather than being thrown off, on the one hand the Aristotelian philosophy itself evolved to some extent over the centuries, as Edward Grant put it, medieval Aristotelian philosophy was not Aristotle’s philosophy and it changed over time.  On the other, it was slowly undermined from various different directions by other ways of doing things in various areas of knowledge acquisition. To some extent much of the content of the previous episodes in this series have sketched that process of undermining, in areas such as cartography, medicine, botany, zoologymineralogy, geology, and palaeontology. 

Because of the dominance of Aristotelian philosophy on the European medieval university people tend to forget that it was only one of the schools of philosophy that grew up and flourished in Ancient Greece and that in particular Aristotle’s concept as to what constitutes episteme or scientia, that is knowledge, was by no means the only game in town. We have already seen how the non-philosophical concept of knowledge produced mathematical that was propagated by Archimedes in the third century BCE came to replace Aristotle’s rejection of mathematically produced knowledge.

The most obvious competing Ancient Greek philosophy is that of Plato, Aristotle’s teacher. They shared a common cosmology, Aristotle having taken that of his teacher and elaborated it, but in other areas their views diverged substantially. Plato was much more sympathetic to mathematics that his pupil, some even labelling him a Pythagorean for his geometrical view of the world. On the whole I think Neo-Platonism had less influence on the development of modern science in the Early Modern Period than is usually attributed to it. However, much of Kepler’s work had a distinctly Platonic flavour and Neo-Platonism was a major factor in the development of the occult sciences during the Renaissance, which did much to rock the Aristotelian boat.

Another well-known competitor to Aristotle in Ancient Greece was Atomism, a philosophical movement about whose fundamental tenants Aristotle was exceedingly rude. Atomism did come to play a major role in the evolution of modern science in the seventeenth century, but less so during the period of Renaissance science that I defined for this series, ending in 1648. Atomism was initially rejected in the Early Modern Period by mainstream thinkers because the Epicurean atomism that began to re-emerge then contained at its core a belief that the cosmos was eternal, without a beginning. This, of course, contradicted the Church doctrine of the Creation and so was distinctly heretical. Even worse, Epicure was also considered to be an atheist and atomism an atheistic theory.

Portrait of Epicurus, founder of the Epicurean school. Roman copy after a lost Hellenistic original. Source: Wikimedia Commons

Several thinkers, including both Thomas Harriot (c. 1560–1621) and Galileo (1564–1642), came under suspicion for holding atomistic views. However, just as Albertus Magnus had made Aristotelianism compatible with Christian doctrine in the thirteenth century, Pierre Gassendi (1592–1655) made Atomism compatible with it in the middle of the seventeenth century, when it had been taken up by thinkers such as Isaac Beeckman (1588–1637) and René Descartes (1596–1650). Interestingly, the centuries most prominent and influential atomist, Robert Boyle (1627–1691), took his atomism not from Epicure, but from the Germany alchemist Daniel Sennert (1572–1637), who in turn had taken it from the late thirteenth and early fourteenth century alchemical work of the Pseudo-Geber, Paul of Taranto .

Another competitor to Aristotle was Stoicism founded by Zeno of Citium in the early third century BCE. Today, most people automatically think of an ethical philosophy when they confront to terms Stoic or Stoicism, and they are not wrong in doing so, the emphasis in Stoicism being living a life of virtue. However, Stoicism also had theories of scientia and cosmology that differed substantially from those of Aristotle and would come to have a major influence in the sixteenth century during a revival of Stoicism in Europe. 

Zeno Source: Wikimedia Commons

For Aristotle the cosmos was a finite sphere with the Earth at the centre, but his sphere was divided into two, the dividing line being the orbit of the Moon. Everything supralunar consisted of the fifth element, the quintessence, was eternal, perfect, and unchanging. The planets were carried around their orbits on celestial spheres. Everything sublunar was constituted from the four elements–water, earth, air, fire–was imperfect and subject to change and decay. 

The Stoics had a very different take. Their cosmos was, like that of Aristotle, also a finite sphere but there the similarity ends. Their cosmos was filled with pneuma:

In Stoic philosophy, pneuma (variously rendered ignis, aer, or spiritus in Latin) is the concept of the “breath of life,” a mixture of the elements air (in motion) and fire (as warmth).  For the Stoics, pneuma is the active, generative principle that organizes both the individual and the cosmos. In its highest form, pneuma constitutes the human soul (psychê), which is a fragment of the pneuma that is the soul of God. As a force that structures, it exists even in inanimate objects.


As the whole cosmos is filled with pneuma there is, in the Stoic cosmos, no difference between the supralunar and sublunar regions, it is all one. There are also no spheres to carry the planets, which swim through the heavens driven by the dynamic characteristic of pneuma. Whereas in Aristotelian cosmology, comets, which are definitely not eternal, perfect, and unchanging, are sublunar atmospheric phenomena. For the Stoics, however, comets are clearly supralunar, celestial phenomena. These major differences in the two cosmologies would come to play a significant role in the evolution of cosmology/astronomy in the sixteenth century.  

One figure in the sixteenth century, who launched an all-out critique of Aristotle was the French Huguenot, Peter Ramus (1515–1572).

Source: Wikimedia Commons

Unlike others, Ramus did not try to replace Aristotle with another, different ancient philosophy, but set out to reform, what he saw as corrupted Aristotelian philosophy. He radically simplified both Aristotelian logic and rhetoric and reordered them into what he saw as a correct system. Unlike other humanists, he also heavily criticised Cicero. Ramus was also a big supporter of mathematics, which he saw as having been somehow perverted by Plato and Aristotle. Although not a mathematician he presented what he saw as a purified mathematics. However, unlike the modern Archimedeans he didn’t develop a mathematics based natural philosophy. He came under heavy attack from other academics but following his death there grew up, what one could call, a cult of Ramism, throughout Europe. Both Rudolph (1546–1613) and Willebrord Snel (1580–1626), who played a leading role in the development of the mathematical sciences in the Netherlands were Ramists. François Viète (1540–1603), who played a significant role in the transition of algebra from commercial arithmetic to a proper mathematical discipline was also influenced by Ramus.

I started talking about Stoic cosmology and seemed to veer off in a different direction to Ramus but there is a connection. Jean Pena (1528 or 1530–1558 or 1568), another French Huguenot, was appointed professor of mathematics at the Collège Royal in Paris, where he had earlier worked under Ramus (who was regius professor for philosophy and eloquence) as one of a small circle of students producing new translations of classical authors in science and mathematics.  Following Ramus’ lead Pena developed an anti-Aristotelian standpoint in his own work but unlike his teacher, rather than simplifying and rationalising Aristotle he replaced some of the Aristotelian doctrines with ones from the optical tradition and from Stoicism.

Pena’s knowledge of Stoicism probably came from the works of Cicero, Seneca, and Pliny the Elder, all authors favoured by the humanists, and of Stoic cosmology from the works of Origen and Plutarch. In the preface to his De usu Optices (a new translation of Euclid’s Optics published in 1555), when discussing atmospheric refraction, Pena posits that the whole of space is filled with animabilis spiritum, a phrase used by Cicero in an exposition of Stoic cosmology, from the surface of the Earth to the fixed stars. His space has no celestial spheres and the planets swim through space. Inspired by the discovery of Peter Apian that the tails of comets always point away from the Sun, Pena hypothesised that comets were supralunar lenes that focused the sunlight. 

Christoph Rothman (c. 1555–1601), the chief astronomer of Landgrave Wilhelm IV of Hessen-Kassel, wrote a report on the comet from 1558, Scriptum de cometa, qui anni Christi 1585 mensib. Octobri et Novembri apparuit, which included much of Pena’s theories of the cosmos and of comets. This book was first published in 1618 by Willebrord Snel, but Rothmann sent a copy of the manuscript to Tycho Brahe (1546–1601) in 1586. It was this text that most likely convinced Tycho to abandon the Aristotelian cosmological theory of the celestial spheres, also adopting a Stoic theory on the nature of planetary motion.

According to Tycho Brahe, in a letter of February 1589, the substance of the heavens is a « most pure and most fluid aethereal substance » distinct from all the terrestrial elements. However, if a comparison must be made, the substance of the heavens is closest to fire, as Paracelsus teaches, although it is a fire that burns without being consumed. For Tycho Brahe stars and planets are formed from this substance. Consequently, just as birds made from the mumia of air live in the air, and fish made from the mumia of water move in the waters, for the same reason it is likely that the Sun and stars, which are made of a kind of unburning fire, carry out their revolutions in the aether which is fiery and unburning.1

Whilst proposing a different form of planetary motion, a force emanating from the Sun, Johannes Kepler (1571–1630) maintained the continuous fluid heaven of Pena, Rothmann, and Brahe. It is interesting to note the Roberto Bellermino (1542–1621), the Jesuit inquisitor, who taught astronomy in his younger days, was also a proponent of the fluid heavens theory. 

The medieval dominance of Aristotelian philosophy was not blasted away in one go by a single alternative but was slowly chipped away by diverse other models, Archimedean, Platonism, Atomism, and Stoicism in various areas of scientia, as outlined here. In some areas Aristotle continued to hold sway even into the eighteenth century. As I like to point out, Newton’s Principia (1687) is a perfect model of Aristotelian episteme, a set of evidently true axioms from which empirically true facts are deduced by formal logic. However, by the middle of the seventeenth century the fields of scientific enquiry had changed substantially from what they had been in around fourteen hundred.

1 Quoted from Peter Barker, Stoic alternatives to Aristotelian cosmologyPena, Rothmann and Brahe, Revue D’ Histoire Des Sciences, 2008/2 (Tome 61) pp. 265–286. Barker’s essay informs most of the Stoicism content of this blog post. 


Filed under Uncategorized

Leonardo and gravity

Mory Gharib an engineer from Caltech has published an article about his interpretation of some diagrams he discovered in one of the Leonardo manuscripts, which he claims are Leonardo’s attempts to determine the acceleration due to gravity. I’m not going to comment on Gharib’s work, which looks interesting, but rather on the article published in ARS TECHNICA by science writer Jennifer Ouellette describing Gharib’s work, which contains some, in my opinion, bizarre statements. 

It starts with Ouellette’s title: Leonardo noted link between gravity and acceleration centuries before Einstein! Equating an experiment of Leonardo’s, assuming Gharib is correct in his suppositions, with Einstein’s general theory of relativity is so far fetched it’s absurd. Just in case you think it’s just a hyperbolic title we get it repeated more emphatically at the end of the first paragraph:

Clip from article

Further investigation revealed that Leonardo was attempting to study the nature of gravity, and the little triangles were his attempt to draw an equivalence between gravity and acceleration—well before Isaac Newton came up with his laws of motion, and centuries before Albert Einstein would demonstrate the equivalence principle with his general theory of relativity.

Now we have Leonardo not just raised on a pedestal with Einstein, but with Newton too. I could point out that Newton didn’t come up with his laws of motion he collated them from the work of others. The comparison with Newton comes again in the next paragraph:

What makes this finding even more astonishing is that Leonardo did all this without a means of accurate timekeeping and without the benefit of calculus, which Newton invented in order to develop his laws of motion and universal gravitation in the 1660s.

Two things are wrong with this. Firstly, as I will explain shortly, lots of people investigated the acceleration due to gravity before and after Leonardo but before Newton without using calculus. Secondly, Newton did not invent calculus, he collated, and systemised the work of many other, as did Leibniz. He also didn’t do this to develop his laws of motion and universal gravitation, in fact, as I have explained once before, contrary to popular opinion, Newton did not use calculus to write the Principia, but good old fashioned Euclidian geometry. Just for the record Newton’s work in this area was done in the 1680s not 1660s. 

We get served up an old dubious claim:

Leonardo foresaw the possibility of constructing a telescope in his Codex Atlanticus (1490) when he wrote of “making glasses to see the moon enlarged”—a century before the instrument’s invention.  

Most expert on the history of the telescope follow Van Helden and don’t think Leonardo was here referring to any form of telescope but rather a single magnifying lens held at arm’s length. 

Moving on:

The concept of inertia wasn’t even known at the time; Leonardo’s earlier writings show that he accepted the Aristotelian notion that one needs a continuous force for any object to move. 

It is true that the theory of inertia wasn’t known at the time but around 1500 Leonardo almost certainly used the post-Aristotelian impulse theory.

As a historian of Renaissance mathematics, the following truly boggled my mind:

Leonardo went even further, Gharib et al. assert, and essentially tried to model the data from his experiment to find the gravitational constant using geometry—the best mathematical tool available at the time. “There was no concept of equations or math, but Leonardo had such an intuitive understanding of math in its non-equation form,” Roh told Ars. “I think that’s where he started using geometry to write out equations, in a way. 

“There was no concept of equations or math…”!!!!!!!! Just savour this statement for a moment, I can’t even begin… Leonardo’s maths teacher, Luca Pacioli, might have a few words to say about that.

To close, I wish to suggest a list of people in Europe, who in various ways investigated the acceleration of gravity, post Aristotle before Leonardo, contemporaneously with him or after him but before Newton and before the invention of calculus, with whom Ms Ouellette might have compared Leonardo’s interesting endeavours rather than Newton and Einstein. 

We start in the sixth century CE with John Philoponus. Moving on to the fourteenth century we have the Oxford Calculatores, who derived the mean speed theorem. Staying in the same century we have Nicole Oresme, who produced a geometrical representation of the mean speed theorem. Post Leonardo in the sixteenth century we have Tartaglia, and Benedetti. At the end of the sixteenth and beginning of the seventeenth centuries we have Simon Stevin and some guy called Galileo Galilei, you might have heard of him.  


Filed under History of Physics, Uncategorized

Temporarily closed




Filed under Uncategorized

An inventor of instruments

Way back at the beginning of November I wrote what was intended to be the first of a series of posts about English mathematical practitioners, who were active at the end of the sixteenth and the beginning of the seventeenth centuries. I did not think it would be two months before I could continue that series with a second post, but first illness and then my annual Christmas trilogy got in the way and so it is only now that I am doing so. The subject of this post is a man for whom a whole series of mathematical instruments are named, Edmund Gunter (1581–1626).

Unfortunately, as is all to often the case with Renaissance mathematici, we know almost nothing about Gunter’s origins. His father was apparently a Welshmen from Gunterstown, Brecknockshire in South Wales but he was born somewhere in Hertfordshire sometime in 1581. Obviously from an established family he was educated at Westminster School as a Queen’s Scholar i.e., a foundation scholar (elected on the basis of good academic performance and usually qualifying for reduced fees). He matriculated at Christ Church Oxford 25 January 1599 (os). He graduated BA 12 December 1603 and MA 2 July 1606. He took religious orders and proceeded B.D. 23 November 1615. He was appointed Rector of St. George’s, Southwark and of St Mary Magdalen, Oxford in 1615, he retained both appointments until his death. 

Whilst still a student in 1603, he wrote a New Projection of the Sphere in Latin, which remained in manuscript until it was finally published in 1623. This came to the attention of Henry Briggs (1561–1630), who had been appointed professor of geometry at the newly founded Gresham College in 1596, and as such was very much a leading figure in the English mathematical community. Briggs was impressed by the young mathematician befriending him and becoming his mentor. The two men spent much time together at Gresham College discussing topics of practical mathematics. In 1606, Gunter developed a sector, about which later, and wrote a manuscript describing it in Latin, without a known title. This circulated in manuscript for many years and was much in demand. Gunter gave into that demand and finally published it also in 1623.

When the first Gresham professor of astronomy, Edward Brerewood (c. 1556–1613) died 4 November 1613, Briggs recommended Gunter as his successor. However, Thomas Williams another Christ Church graduate, of whom little is known, was appointed just seven days later 11 November 1613. When Williams resigned from the post 4 March 1619, for reasons unknown, Briggs once again supported his friend for the position, this time with success. Gunter was appointed just two days later, 6 March 1619. Like his two rectorships, he retained the Gresham professorship until his death. 

Gresham College, engraving by George Vertue, 1740 Source: Wikimedia Commons

Apparently, he was already spending so much time at Gresham College before being appointed that when the mathematician William Oughtred (1574–1660) visited Henry Briggs there in 1618, he thought that Gunter was already professor there.

In the Spring 1618 I being at London went to see my honoured friend Master Henry Briggs at Gresham College: who then brought me acquainted with Master Gunter lately chosen Astronomical lecturer there, and was at that time in Doctor Brooks his chamber. With whom falling into speech about his quadrant, I showed him my Horizontal Instrument. He viewed it very heedfully: and questioned about the projecture and use thereof, often saying these words, it is a very good one. And not long after he delivered to Master Briggs to be sent to me mine own Instrument printed off from one cut in brass: which afterwards I understood he presented to the right Honourable the Earl of Bridgewater, and in his book of the sector printed six years after, among other projections he setteth down this.

Gunter and Oughtred would go on to become firm friends.

William Oughtred engraving by Wenceslaus Hollar Source: Wikimedia Commons There are apparently no portraits of Briggs or Gunter

We now have the known details of the whole of Gunter’s life and can turn our attention to his mathematical output but before we do so there is an anecdote from Seth Ward (1617–1689), another mathematician and astronomer, concerning a position that Gunter did not get. In 1619, Henry Savile (1549–1622) established England’s first university chairs for mathematics the Savilian chairs for geometry and astronomy at Oxford. Savile’s first choice for the chair of geometry was Edmund Gunter and he invited him to an interview, according to John Aubrey (1626–1697) relating a report from Seth Ward:

[Gunter] brought with him his sector and quadrant, and fell to resolving triangles and doing a great many fine things. Said the grave knight [Savile], “Do you call this reading of geometry? This is showing of tricks, man!”, and so dismissed him with scorn, and sent for Henry Briggs.

So, Henry Briggs became England’s first university professor of geometry and not Edmund Gunter. One should point out that Ward can only have heard the story second hand as he was only two years old in 1619.

In 1614, John Napier (1550–1617) published his Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms, 1614), a new method of simplifying calculations. Edward Wright (1561–1615) produced an English translation, which was published posthumously in 1616. Napier’s logarithms were base:

NapLog(x) = –107ln (x/107)

Henry Briggs travelled all the way to Edinburgh to meet the inventor of this new calculating tool. After discussion with Napier, he received his blessing to produce a set of base ten logarithms. His Logarithmorum chilias prima, which were publish in London sometime before Napier’s death in 1617.

Many people don’t realise that Napier’s logarithmic tables were not straight logarithms but logarithms of trigonometrical functions. These are of particular use for astronomers and navigators. It is almost certainly through Brigg’s influence that Gunter’s first publication was a set of base ten, seven figure logarithmic tables of sines and tangents. His Canon Triangulorum sive Tabulae Sinuum et Tangentium Artificialum was published in Latin in 1620. An English translation was published in the same year. The terms sine and tangent were already in use, but it was Gunter, who introduced the terms cosine and cotangent in this publication. Later, on his scale or rule he introduced the short forms sin and tan.

In 1623, Gunter finally published his New Projection of the Sphere written in his last year as an undergraduate. He also published his most important book, Description and Use of the Sector, the Crosse-staffe and other Instruments. This was one of the most important guides to the use of navigational instruments for seamen and became something of a seventeenth century best seller in various forms. David Waters in his The Art of Navigation say this, ” Gunter’s De Sectore & Radio must rank with Eden’s translation of Cortes’s Arte de Navegar and Wright’s Certain Errors as one of the three most important English books ever published for the improvement of navigation.” [1]

Waters opposite page 360

His various publications were collected into The Works of Edmund Gunter, which went through six editions by 1680. Each edition having extra content by other authors. Isaac Newton (1642-1727) bought a copy of the second edition. The title page of the fifth edition is impressive:

The Workers of Edmund Gunter 5th ed. Title page with diagrams of the sector on the fly leaf

The Works of Edmund Gunter:
Containing the description and Use of the
Sector, Cross-staff, Bow, Quadrant,
And other Instruments.
With a Canon of Artificial Sines and Tangents to a Radius of 10.00000 parts, and the Logarithms from Unite to 100000:
The Uses whereof are illustrated in the Practice of
Arithmetick, geometry, Astronomy, Navigation, Dialling and Fortification.
And some Questions in Navigation added by Mr. Henry Bond, Teacher of mathematicks in Ratcliff, near London.
To which is added,
The Description and Use of another Sector and Quadrant, both of them invented by Mr. Sam. Foster, Late Professor of Astronomy in Gresham Colledge, London, furnished with more Lines, and differing from those of Me. Gunter′s both in form and manner of Working.
The Fifth Edition,
Diligentyl Corrected, and divers necessary Things and Matters (pertinent thereunto) added, throughout the whole work, not before Printed.
By William Leybourne, Philomath.
Printed by A.C. for Francis Eglesfield at the Marigold in St. Pauls Church-yard. MDCLXXIII.

The sector, also known as a proportional compass or military compass, was a major calculating instrument in use from the end of the sixteenth century until the nineteenth century. It is an instrument consisting of two rulers of equal length joined by a hinge. A number of scales are inscribed upon the instrument which facilitate various mathematical calculations. It was used for solving problems in proportion, multiplication, and division, geometry, and trigonometry, and for computing various mathematical functions, such as square and cube roots. Its several scales permitted easy and direct solutions of problems in gunnery, surveying and navigation. The sector derives its name from the fourth proposition of the sixth book of Euclid, where it is demonstrated that similar triangles have their like sides proportional. (Wikipedia)

The sector has many alleged inventors. The earliest was Fabrizio Mordente (1532–c. 1608). The invention is often credited to Galileo (1564–1642), who marketed a very successful variant in the early seventeenth century, including selling lessons and an instruction manual in its use. However, Galileo’s instrument was a development of one created by Guidobaldo dal Monte (1545–1607). It is not known if dal Monte developed the device independently or knew of Mordent’s. Thomas Hood (1556–1620) appear to have reinvented the instrument, a description of which he published in his Making and Use of the Sector, 1596.

Waters opposite page 345

Gunter developed Hood’s instruments adding addition scales, including a scale for use with Mercator’s new projection of the sphere. 


Water opposite page 361
Waters page 361

The French Jewish scholar, Levi ben Geshon (1288–1344), published the first description of the cross staff or Jacob’s staff, used in astronomy, surveying, and navigation, in his Book of the Wars of the Lord (originally in Hebrew but also translated into Latin). 

Gunter image of a cross staff

Gunter’s book also describes the Gunter Quadrant, basically a horary quadrant for telling the time by taking the altitude of the sun but with some additional functions.

Boxwood Gunter-type sector, made by Isaac Carver and owned by George Lason; 1706 Whipple Museum
Illustration of a quadrant from Edmund Gunter’s Works (1653). Image © the Whipple Library.
Modern reproduction of the Gunter Quadrant Source

There is also a description of the crossbow an alternative to the backstaff that never became popular. 

; Navigation: an Astrolabe, a Cross-Staff, and a Back-Staff or Davis’s Sextant; Wellcome Collection;
; Navigation: a Cross-Staff or Cross-Bow, and a Sailor Using the Device; Wellcome Collection;

Gunter’s most popular instrument was his scale. The Gunter scale or rule was a rule containing trigonometrical and logarithmic scales, which could be used with a pair of dividers to carry out calculations in astronomy and in particular navigations. The Gunter scale is basically a sector folded into a straight line without the hinge.Sailors simply referred to the rule as a Gunter. William Oughtred would go on to place two Gunter rules next to each other thus creating the slide rule and eliminating the need for dividers to carry out the calculations.

Gunter scale front side
Gunter scale back side

In 1622, Gunter engraved a new sundial at Whitehall, which carried many different dial plates supplying much astronomical data. At the behest of Prince Charles, he wrote and published an explanation of the dials, The Description and Use of His Majesties Dials in Whitehall, 1624. The sundial was demolished in 1697.

Gunter’s most well-known instrument was his surveyor’s chain, which became the standard English Imperial chain. 100 links and 22 yards (66 feet) long, there are 10 chains in a furlong and 80 chains to a mile. 

Although Gunter invented, designed, and described the use of several instruments, he didn’t actually make any of them. All of his instruments were produced by the London based, instrument maker Elias Allen (c. 1588–1652). Allen was born in Kent of unknown parentage and was apprenticed in 1602 to London instrument maker Charles Whitwell (c. 1568–1611) in the Grocer’s Company, serving his master for nine years. Following Whitwell’s death in 1611, Allen set up his own business. He rapidly became the foremost instrument maker in London, working mostly in brass, but occasionally in silver. He became very successful and made instruments for various aristocratic patrons and both James I and Charles I. Allen also produced the engravings in Gunter’s books, using them also as advertising in his shop.

He worked closely with various mathematicians including both Oughtred and Gunter. His workshop became a meeting place for discussion amongst mathematical practitioners. He was the first London instrument maker, who could make a living from just making instruments without working on the side as a map engraver or surveyor. His master Whitwell subsidised his income as a map engraver. He rose in status in the Grocers’ Company, becoming its treasurer in 1636 and its master for eighteen months in 1637-38. Over the years many of his apprentices became successful instrument maker masters in the own right, most notably Ralph Greatorex (1625–1675), who was associated with Oughtred, Samuel Pepys, John Evelyn, Samuel Hartlib, Christopher Wren, Robert Boyle, and Jonas Moore, the English scientific elite of the time. 

Allen had the distinction of being one of the few seventeenth-century artisans to have his portrait painted. The Dutch artist Hendrik van der Borcht the Younger (1614–1676) produced the portrait, now lost, in about 1640. It still exists as an engraving done by the Bohemian engraver, Wenceslaus Hollar (1607­–1677).

Edmund Gunter was not a mathematician as we understand the term today, but a mathematical practitioner, who exercised a large influence on the practical side of astronomy, navigation, and surveying in the seventeenth century through the instruments that he designed and the texts he wrote explaining how to use them. 

[1] David Walters, The Art of Navigation in England in Elizabethan and Early Stuart Times, Yale University Press, 1958 p. 359

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Christmas Trilogy 2022 Part 3: Portrait of the Mathematician as a Pet Dog

The American science writer Kitty Ferguson wrote a fairly good double biography of Tycho Brahe and Johannes Kepler titled The Nobleman and His Housedog. Tycho Brahe and Johannes Kepler: the Strange Partnership that Revolutionised Science (review, 2002). Tycho is obviously the noble but Kepler as a housedog? Isn’t that rather insulting? It would be if it wasn’t for the fact that the description is from Kepler himself. In 1597, Kepler wrote an unpublished essay, in Latin, in which he describes himself in quite a lot of rather rambling detail. It is here that we find him calling himself a dog, I don’t have a copy of the Latin original but in the German translation the term is Haushund, which is what I suspect Ferguson presents in English as housedog, I think pet dog is a better translation. A year earlier, he had made some notes describing his grandfather and his parents as an appendix to their horoscopes. In what follows, I present an English translation, from the German of the passage, where he describes himself as a pet dog and excerpts from the descriptions of his immediate family.

Source: Wikimedia Commons

I have in every sense the nature of a dog. I am a spoilt lap dog. My body is agile, skinny, well proportioned. My way of life is identical: I enjoy gnawing on bones, bread crusts delight me, I am greedy, I lack discipline, I seize whatever I happen to see. I drink little. I am satisfied with the least possible. I continuously pester my superiors like a pet dog. I am always loyal to others, I serve them, I’m never cross with them if the criticise me, I am always prepared in every way to win their favour again. On my own accord, I research everything in the sciences, the state, in household affairs, in the slightest tasks. I am always on the go and strive after those who undertake something, in that I pursue and research it as well. I am impatient in my dealings and all too often I greet those, who come into the house no differently than a dog. Whenever anybody snatches even a little from me, I grumble and create a commotion like a dog. I hold on tenaciously, follow those who behave badly, and naturally I bark. I also snap and have a nasty retort on my tongue. Many therefore hate and avoid me, and my superiors are fond of me like the people living in a house are fond of a good dog. Just like a dog, I afraid of being bathed, getting wet and being washed. Put simply, within me dwells an uncontrolled recklessness but directly alongside it a fear of life. Daring in dangerous situations is far from my nature. That is more or less enough, about anger, desire, and about the things for which one is mostly admonished. 

This is not a pretty portrait that the good Johannes presents of himself but the one he presented a year earlier of his family, based on astrology, was not better. His grandfather Sebald, a judge in the Imperial City of Weil, he describes as follows:

He was eloquent when meeting the uninformed, he is more content as dictator and more content as leader than as reporter.

Of his father he has the following to report:

My father, Heinrich, was born in 1527 on the 19 January. Saturn destroyed everything, producing a heinous, brusque, argumentative, and not least one destined to die a terrible death. The position of the stars multiplied his malice, plunged him into poverty, however he found a rich wife. He was a trained artillery man, had many enemies and a marriage full of strife. False and vain striving for honour and hope drove both of them, also wanderlust.

This is followed by list of his father’s disappearances to serve as mercenary, and his numerous business failures. Johannes closes with a hard judgement. 

Just as a window shutter gets damages, so my father injured himself on his return home, he treated my mother in the same way and finally he went into exile and died.

Heinrich is followed by his younger brother, Johannes’ Uncle Sebald, who appears even worse than the father:

His brother Sebald was born 13 November 1552, was a conjuror, Jesuit priest with the first and second ordination, sordid in life: because he was Catholic but pretended to be Lutheran. He died young of dropsy after many illnesses. He found a noble and rich wife but with many children. He caught the gallic disease [syphilis], was nefarious and hated his fellow citizens.

Strangely he ends his account of this rapscallion with the following:

He was educated in the humanities and a good companion.

We now arrive at his mother:

My mother, Katharina Guldenmann was born 8 November 1547

This is then followed by a discussion that the dates of her birth and her marriage didn’t add up and the contradictory claims made by her mother and her grandfather. Now comes Johannes’ negative judgement over his mother:

She is small, lean, black, with acerbic humour, argumentative, and has an unpleasant personality. 

Before this he had already written, “she is very like me”!

He does, however, say that she had it hard:

1589, she was handled very badly by her husband. My mother was also beaten by her parents, even when she was pregnant, however, she escaped.

There now follows a claim from Kepler that is given in almost all biographies:

My conception can be calculated for 16 May at 4:37 pm. My mother had a premature birth, and my sickliness awakes the suspicion that the shortened pregnancy was the cause. 

 This is based on his calculation of the time between the wedding and his birth but there is of course the suspicion that his mother was already pregnant when she married.

All of the above taken together create the impression that in his middle twenties, at the time he wrote is Mysterium Cosmographicum, reflecting on himself and his family Kepler was not a happy puppy!


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Mathematician, astrologer, conjurer! 

It is almost impossible to imagine a modern university without a large mathematics department and a whole host of professors for an ever-increasing array of mathematical subdisciplines. Mathematics and its offshoots lie at the centre of modern society. Because popular history of science has a strong emphasis on the prominent mathematicians, starting with Euclid and Archimedes, it is common for people to think that mathematics has always enjoyed a central position in the intellectual life of Europe, but they are very much mistaken if they do so. As I have repeated on several occasions, mathematics had a very low status at the medieval European university and led a starved existences in the shadows. Some people like to point out that the basic undergraduate degree at the medieval university formally consisted of the seven liberal arts, the trivium and quadrivium, with the latter consisting of the four mathematical disciplines–arithmetic, geometry, music, and astronomy. If fact, what was largely taught was the trivium–grammar, logic, rhetoric–and large doses of, mostly Aristotelian, philosophy. A scant lip service was paid to the quadrivium at most universities, with only a very low-level introductory courses being offered in them. There were no professors for any of the mathematical disciplines.

Things only began to change during the Renaissance, when the first universities, in Northern Italy, began to establish chairs for mathematics, which were actually chairs for astrology, because of the demand for astrology for medical students. The concept of general chairs for mathematics for all educational institutions began with Philip Melanchthon (1497–1560), when he set up the school and university system for Lutheran Protestantism, to replace the previously existing Catholic education system, in the second quarter of the sixteenth century.

Melanchthon in 1526: engraving by Albrecht Dürer Translation of Latin caption: «Dürer was able to draw Philip’s face, but the learned hand could not paint his spirit».
Source: Wikimedia Commons

Melanchthon did so because he was a passionate advocate of astrology and to do astrology you need astronomy and to do astronomy you need arithmetic, geometry, and trigonometry, so he installed the full package in all Lutheran schools and universities. He also ensured that the universities provided enough young academic mathematicians to fill the created positions.  

Catholic educational institutions had to wait till the end of the sixteenth century before Christopher Clavius (1538–1612) succeeded in getting mathematics integrated into the Jesuit educational programme and installed a maths curriculum into Catholic schools, colleges, and universities throughout Europe over several decades. He also set up a teacher training programme and wrote the necessary textbooks, incorporating the latest mathematical developments.

Christoph Clavius. Engraving Francesco Villamena, 1606 Source: Wikimedia Commons

England lagged behind in the introduction of mathematics formally into its education system. Even as late as the early eighteenth century, John Arbuthnot (1667–1735) could write that there was not a single grammar school in England that taught mathematics.

John Arbuthnot, by Godfrey Kneller Source: Wikimedia Commons

This is not strictly true because The Royal Mathematical School was set up in Christ’s Hospital, a charitable institution for poor children, in 1673, to teach selected boys’ mathematics, so that they could become navigators. At the tertiary level the situation changed somewhat earlier. 

Gresham College was founded in London under the will of Sir Thomas Gresham (c. 1519–1579) in 1595 to host public lectures.

Gresham College 1740 Source: Wikimedia Commons

Sir Thomas Gresham by Anthonis Mor Rijksmuseum

Amongst other topics, professors were appointed to hold lectures in both geometry and astronomy. As with the Royal Mathematical School a century later these lectures were largely conceived to help train mariners. The instructions for the geometry and astronomy professors were as follows:

The geometrician is to read as followeth, every Trinity term arithmetique, in Michaelmas and Hilary terms theoretical geometry, in Easter term practical geometry. The astronomy reader is to read in his solemn lectures, first the principles of the sphere, and the theory of the planets, and the use of the astrolabe and the staff, and other common instruments for the capacity of mariners.

The first university professorships for mathematics were set up at Oxford University in 1619 financed by a bequest from Sir Henry Savile (1549–1622), the Savilian chairs for astronomy and geometry.

Henry Savile Source: Wikimedia Commons

Over the years it was not unusual for a Gresham professor to be appointed Savilian professor, as for example Henry Biggs (1561–1630), who was both the first Gresham professor and the first Savilian professor of geometry.

Henry Briggs

Henry Savile was motivated in taking this step by the wretched state of mathematical studies in England. Potential mathematicians at Cambridge University had to wait until a bequest from Henry Lucas (c. 1610–1663), in 1663, established the Lucasian Chair of Mathematics, whose first incumbent was Isaac Barrow (1630–1677), succeeded famously by Isaac Newton (1642–1726 os).  This was followed in 1704 with a bequest by Thomas Plume to “erect an Observatory and to maintain a studious and learned Professor of Astronomy and Experimental Philosophy, and to buy him and his successors utensils and instruments quadrants telescopes etc.” The Plumian Chair of Astronomy and Experimental Philosophy, whose first incumbent was Roger Cotes (1682–1716).

unknown artist; Thomas Plume, DD (1630-1704); Maldon Town Council;

Before the, compared to continental Europe, late founding of these university chairs for the mathematical sciences, English scholars wishing to acquire instruction in advanced mathematics either travelled to the continent as Henry Savile had done in his youth or find a private mathematics tutor either inside or outside the universities. In the seventeenth century William Oughtred (1574–1660), the inventor of the slide rule, fulfilled this function, outside of the universities, for some notable future English mathematicians. 

William Oughtred by Wenceslas Hollar 1646

One man, who fulfilled this function as a fellow of Oxford University was Thomas Allen (1542–1632), who we met recently as Kenhelm Digby’s mathematics tutor.

Thomas Allen by James Bretherton, etching, late 18th century Source: wikimedia Commons

Although largely forgotten today Allen featured prominently in the short biographies of the Alumni Oxonienses of Anthony Wood (1632–1695) and the Brief Lives of John Aubrey (1626–1697), both of them like Allen antiquaries. Aubrey’s description reads as follows: 

Mr. Allen was a very cheerful, facecious man and everybody loved his company; and every House on their Gaudy Days, were wont to invite him. The Great Dudley, Early of Leicester, made use of him for casting of Nativities, for he was the best Astrologer of his time. Queen Elizabeth sent for him to have his advice about the new star that appeared in the Swan or Cassiopeia … to which he gave his judgement very learnedly. In those dark times, Astrologer, Mathematician and Conjuror were accounted the same thing; and the vulgar did verily believe him to be a conjurer. He had many a great many mathematical instruments and glasses in his chamber, which did also confirm the ignorant in their opinion; and his servitor (to impose on Freshmen and simple people) would tell them that sometimes he should meet the spirits coming up his stairs like bees … He was generally acquainted; and every long vacation he rode into the country to visit his old acquaintances and patrons, to whom his great learning, mixed with much sweetness of humour, made him very welcome … He was a handsome, sanguine man and of excellent habit of body.

The “new star that appeared in the Swan or Cassiopeia” is the supernova of 1572, which was carefully observed by astronomers and interpreted by astrologers, often one and the same person, throughout Europe.

Star map of the constellation Cassiopeia showing the position of the supernova of 1572 (the topmost star, labelled I); from Tycho Brahe’s De nova stella. Source: Wikimedia Commons

Conjuror in the Early Modern Period meant an enchanter or magician rather than the modern meaning of sleight of hand artist and was closely associated with black magic. Allen was not the only mathematician/astrologer to be suspected of being a conjuror, the same accusation was aimed at the mathematician astronomer, and astrologer, John Dee (1527–c. 1609). At one public burning of books on black magic at Oxford university in the seventeenth century, some mathematics books were reputedly also thrown into the flames. Aubrey also relates the story that when Allen visited the courtier Sir John Scudamore (1542–1623), a servant threw his ticking watch into the moat thinking it was the devil. The anonymous author of Leicester’s Commonwealth (1584), a book attacking Elizabet I’s favourite Robert Dudley, Earl of Leicester (1532–1588) accused Allen of employing the art of “figuring” to further the earl of Leicester’s unlawful designs, and of endeavouring by the “black art” to bring about a match between his patron and the Queen. The same text accuses both Allen and Dee of being atheists. 

Anthony Wood described Allen as:

… clarrissimus vir [and] very highly respected by other famous men of his time … Bodley, Savile, Camden, Cotton, Spelman, Selden, etc. … a great collector of scattered manuscripts …  an excellent man, the father of all learning and virtuous industry, an unfeigned lover and furtherer of all good arts and sciences.

The religious controversialist Thomas Herne (d. 17722) called Allen:

… a very great mathematician and antiquary [and] a universal scholar. 

In his History of the Worthies of Britain (1662), the historian Thomas Fuller (1608–1661) wrote of Allen:

…he succeeded to the skill and scandal of Friar Bacon [and] his admirable writings of mathematics are latent with some private possessors, which envy the public profit thereof.

The jurist John Selden (1584–1654), even in comparison with the historian William Camden (1551–1623), the diplomat and librarian Thomas Bodley (1545–1613) and the Bible translator and mathematician Henry Savile, called Allen:

…the brightest ornament of the famous university of Oxford.

So, who was this paragon of scholarship and learning, whose praises were sung so loudly by his notable contemporaries?

Thomas Allen was the son of a William Allen of Uttoxeter in Staffordshire. Almost nothing is known of his background, his family, or his schooling before he went up to Oxford. It is not known how, where, when, or from whom he acquired his knowledge of mathematics. He began acquiring mathematical manuscripts very early and there is some indication that he was largely an autodidact. He went up to Trinity College Oxford comparatively late, at the age of twenty in 1561. He graduated BA in 1563 and was appointed a fellow of Trinity 1565. He graduated MA in 1567. He might have acquired his mathematical education at Merton College. There is no indication the Allen was a Roman Catholic, but he joined an exodus of Catholic scholars from Trinity, resigning his fellowship, and moving to Gloucester Hall in 1570.

In 1598 he was appointed a member of a small steering committee to supervise and assist Thomas Bodley (1535–1613) in furnishing a new university library. Allen and Bodley had both entered Oxford at around the same time, graduating BA in the same year, and remained live long friends. Allen’s patrons all played a leading role in donating to the new library. About 230 of Allen’s manuscripts are housed in the Bodleian, 12 of them donated by Allen himself when the library was founded and the rest by Kenhelm Digby, who inherited them in Allen’s will. 

Through his patron, Robert Dudley, 1st Earl of Leicester, Allen came into contact with John Dee and the two mathematician/astrologers became friends.

Robert Dudley, 1st Earl of Leicester artist disputed Source: Wikimedia Commons

The Polish noble and alchemist Olbracht Łaski (d. 1604), who took Dee with him back to Poland in 1583, also tried to persuade Allen to travel with him to the continent, but Allen declined the invitation. 

Olbracht Łaski Source: Wikimedia Commons

In this time of publish or perish for academics, where one’s status as a scholar is measured by the number of articles that you have managed to get published, it comes as a surprise to discover that Allen, who, as we have seen from the quotes, was regarded as one of the leading English mathematicians of the age, published almost nothing in his long lifetime. His reputation seems to be based entirely on his activities as a tutor and probably his skills as a raconteur. 

As a tutor, unlike a Christoph Clavius for example, there is not a long list of famous mathematicians, who learnt their trade at his feet. In fact, apart from Kenelm Digby (1603–1665) the only really well-known student of Allen’s was not a mathematician at all but the courtier and poet Sir Philip Sidney (1554–1586) for whom he probably wrote a sixty-two-page horoscope now housed in the Bodleian Library.

Sir Philip Sidney, by unknown artist, National Portrait Gallery via Wikimedia Commons

He may have taught Richard Hakluyt (1553–1616) the promotor of voyages of explorations.

Hakluyt depicted in stained glass in the west window of the south transept of Bristol Cathedral – Charles Eamer Kempe, c. 1905. Source: Wikimedia Commons

He did teach Robert Fludd (1574–1637) physician and occult philosopher

Source: Wikimedia Commons

as well as Sir Thomas Aylesbury (1576–1657), who became Surveyor of the Navy responsible for the design of the warships.

This painting by William Dobson probably represents Sir Thomas Aylesbury, 1st Baronet.
Source: Wikimedia Commons

At the end of his life, he taught and influenced the German scientific translator and communicator, Theodore Haak (1605–1690), who only studied in Oxford between 1628 and 1631.

Portrait of Theodore Haak by Sylvester Harding.Source: Wikimedia Commons

As a member of Gloucester Hall, he tutored the sons of many of the leading, English Catholic families. In this role, he tutored several of the sons of Henry Percy, 8th Earl of Northumberland the highest-ranking Catholic aristocrat in the realm. He probably recommended the Gloucester Hall scholar, Robert Widmerpoole, as tutor to the children of Henry Percy, 9th Earl of Northumberland. Percy went on to become Allen’s patron sometime in the 1580s.

HENRY PERCY, 9TH EARL OF NORTHUMBERLAND (1564-1632) by Sir Anthony Van Dyck (1599-1641). The ‘Wizard Earl’ was painted posthumously as a philosopher, hung in Square Room at Petworth. This is NT owned. Source: Wikimedia Commons

Allen became a visitor to Percy’s Syon House in Middlesex, where he became friends with the mathematician and astronomer Thomas Harriot (c. 1560–1621), who studied in Oxford from 1577 to 1580.

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

When he died Harriot left instructions in his will to return several manuscripts that he had borrowed from Allen. Percy was an avid fan of the sciences known for his enthusiasm as The Wizard Earl. He carried out scientific and alchemical experiments and assembled one of the largest libraries in England. Allen with his experience as a manuscript collector and founder of the Bodleian probably advised Percy on his library. Harriot was not the only mathematician in Percy’s circle, he also patronised Robert Hues (1553–1632), who graduated from Oxford in 1578, Walter Warner (1563–1643), who also graduated from Oxford in 1578, and Nathaniel Torporley (1564–1632), who graduated from Oxford in 1581. Torporley was amanuensis to François Viète (1540–1603) for a couple of years. Torpoley was executor of Harriot’s papers, some of which he published together with Warner. All three of them were probably recommended to Percy by Allen. 

When Allen died, he had little to leave to anybody having spent all his money on his manuscript collection, which he left to Kenelm Digby, who in turn donated them to the Bodleian Library. But as we have seen he was warmly regarded by all who remembered him and, in some way, he helped to keep the flame of mathematics alive in England, at a time when it was burning fairly low. 


Filed under History of Mathematics, Renaissance Science, Uncategorized