This is the second in a series of discussion of selected parts of Paul Strathern’s The Other Renaissance: From 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.
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.
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.
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 manuscript. Georg 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.
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.
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.
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 x and y are variable unknowns, and a, b, and c 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 a times articulum b gives [the product n, 50 x 40 = 2000].
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.
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.
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 Stevin: Science 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. 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.
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.
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.
 Vincent Ilardi, Renaissance Vision from Spectacles to Telescopes, American Philosophical Society, 2007, pp. 80-81
 Glen van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009 pp. 260-261
 Van Brummelen p. 261
 Dijksterhuis, Stevin.