Category Archives: History of Astrology

Why Mathematicus?

“The Renaissance Mathematiwot?”

“Mathematicus, it’s the Latin root of the word mathematician.”

“Then why can’t you just write The Renaissance Mathematician instead of showing off and confusing people?”

“Because a mathematicus is not the same as a mathematician.”

“But you just said…”

“Words evolve over time and change their meanings, what we now understand as the occupational profile of a mathematician has some things in common with the occupational profile of a Renaissance mathematicus but an awful lot more that isn’t. I will attempt to explain.”

The word mathematician actually has its origins in the Greek word mathema, which literally meant ‘that which is learnt’, and came to mean knowledge in general or more specifically scientific knowledge or mathematical knowledge. In the Hellenistic period, when Latin became the lingua franca, so to speak, the knowledge most associated with the word mathematica was astrological knowledge. In fact the terms for the professors[1] of such knowledge, mathematicus and astrologus, were synonymous. This led to the famous historical error that St. Augustine rejected mathematics, whereas his notorious attack on the mathematici[2] was launched not against mathematicians, as we understand the term, but against astrologers.

The earliest known portrait of Saint Augustine in a 6th-century fresco, Lateran, Rome Source: Wikimedia Commons

The earliest known portrait of Saint Augustine in a 6th-century fresco, Lateran, Rome
Source: Wikimedia Commons

However St. Augustine lived in North Africa in the fourth century CE and we are concerned with the European Renaissance, which, for the purposes of this post we will define as being from roughly 1400 to 1650 CE.

The Renaissance was a period of strong revival for Greek astrology and the two hundred and fifty years that I have bracketed have been called the golden age of astrology and the principle occupation of our mathematicus is still very much the casting and interpretation of horoscopes. Mathematics had played a very minor role at the medieval universities but the Renaissance humanist universities of Northern Italy and Krakow in Poland introduced dedicated chairs for mathematics in the early fifteenth century, which were in fact chairs for astrology, whose occupants were expected to teach astrology to the medical students for their astro-medicine or as it was known iatro-mathematics. All Renaissance professors of mathematics down to and including Galileo were expected to and did teach astrology.

A Renaissance Horoscope Kepler's Horoskop für Wallenstein Source: Wikimedia Commons

A Renaissance Horoscope
Kepler’s Horoskop für Wallenstein
Source: Wikimedia Commons

Of course, to teach astrology they also had to practice and teach astronomy, which in turn required the basics of mathematics – arithmetic, geometry and trigonometry – which is what our mathematicus has in common with the modern mathematician. Throughout this period the terms Astrologus, astronomus and mathematicus – astrologer, astronomer and mathematician ­– were synonymous.

A Renaissance mathematicus was not just required to be an astronomer but to quantify and describe the entire cosmos making him a cosmographer i.e. a geographer and cartographer as well as astronomer. A Renaissance geographer/cartographer also covered much that we would now consider to be history, rather than geography.

The Renaissance mathematicus was also in general expected to produce the tools of his trade meaning conceiving, designing and manufacturing or having manufactured the mathematical instruments needed for astronomer, surveying and cartography. Many were not just cartographers but also globe makers.

Many Renaissance mathematici earned their living outside of the universities. Most of these worked at courts both secular and clerical. Here once again their primary function was usually court astrologer but they were expected to fulfil any functions considered to fall within the scope of the mathematical science much of which we would see as assignments for architects and/or engineers rather than mathematicians. Like their university colleagues they were also instrument makers a principle function being horologist, i.e. clock maker, which mostly meant the design and construction of sundials.

If we pull all of this together our Renaissance mathematicus is an astrologer, astronomer, mathematician, geographer, cartographer, surveyor, architect, engineer, instrument designer and maker, and globe maker. This long list of functions with its strong emphasis on practical applications of knowledge means that it is common historical practice to refer to Renaissance mathematici as mathematical practitioners rather than mathematicians.

This very wide range of functions fulfilled by a Renaissance mathematicus leads to a common historiographical problem in the history of Renaissance mathematics, which I will explain with reference to one of my favourite Renaissance mathematici, Johannes Schöner.

Joan Schonerus Mathematicus Source: Wikimedia Commons

Joan Schonerus Mathematicus
Source: Wikimedia Commons

Schöner who was a school professor of mathematics for twenty years was an astrologer, astronomer, geographer, cartographer, instrument maker, globe maker, textbook author, and mathematical editor and like many other mathematici such as Peter Apian, Gemma Frisius, Oronce Fine and Gerard Mercator, he regarded all of his activities as different aspects or facets of one single discipline, mathematica. From the modern standpoint almost all of activities represent a separate discipline each of which has its own discipline historians, this means that our historical picture of Schöner is a very fragmented one.

Because he produced no original mathematics historians of mathematics tend to ignore him and although they should really be looking at how the discipline evolved in this period, many just spring over it. Historians of astronomy treat him as a minor figure, whilst ignoring his astrology although it was this that played the major role in his relationship to Rheticus and thus to the publication of Copernicus’ De revolutionibus. For historians of astrology, Schöner is a major figure in Renaissance astrology although a major study of his role and influence in the discipline still has to be written. Historians of geography tend to leave him to the historians of cartography, these whilst using the maps on his globes for their studies ignore his role in the history of globe making whilst doing so. For the historians of globe making, and yes it really is a separate discipline, Schöner is a central and highly significant figure as the founder of the long tradition of printed globe pairs but they don’t tend to look outside of their own discipline to see how his globe making fits together with his other activities. I’m still looking for a serious study of his activities as an instrument maker. There is also, as far as I know no real comprehensive study of his role as textbook author and editor, areas that tend to be the neglected stepchildren of the histories of science and technology. What is glaringly missing is a historiographical approach that treats the work of Schöner or of the Renaissance mathematici as an integrated coherent whole.

Western hemisphere of the Schöner globe from 1520. Source: Wikimedia Commons

Western hemisphere of the Schöner globe from 1520.
Source: Wikimedia Commons

The world of this blog is at its core the world of the Renaissance mathematici and thus we are the Renaissance Mathematicus and not the Renaissance Mathematician.

[1] That is professor in its original meaning donated somebody who claims to possessing a particular area of knowledge.

[2] Augustinus De Genesi ad Litteram,

Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium, maxime dicentes vera, cavendi sunt, ne consortio daemoniorum animam deceptam, pacto quodam societatis irretiant. II, xvii, 37


Filed under History of Astrology, History of Astronomy, History of Cartography, History of Mathematics, History of science, History of Technology, Renaissance Science

Another public service announcement

Marius Book Launch

In September 2014 a conference was held in Nürnberg, as the climax of a year dedicated to celebrating the life and work of the Franconian astronomer, astrologer and mathematician Simon Marius, whose magnum opus Mundus Iovialis was published four hundred years earlier in 1614.

The papers held at that conference together with some other contributions from people who could not attend in person have now been collected together in the book Simon Marius und Seine Forschung, eds. Hans Gaab and Pierre Leich (= Acta Historica Astronomiae, Band 57) which will be official launched in the Thalia bookshop in Nürnberg on this coming Thursday, 13 October at 18:30 MET.

This volume contains papers by a wide range of scholars and could/should be of interest to anybody studying the histories of astronomy, astrology and/or mathematics in the Early Modern Period. It can be purchased online, after Thursday, directly from the publishers, Leipziger Universitätsverlag


For those who would like to know more about the book including a table of contents (Inhaltsverzeichnis) they can inform themselves on the Marius Portal here.

For those who cannot read German, an English edition of the book is in planning for next year, for which further contributions on the life and work of Simon Marius would also be welcome. If anybody has any questions regarding this volume I would be happy to answer them.


P.S. For those waiting for blogging to resume here at the Renaissance Mathematicus I can report that there is light at the end of the tunnel!





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

The Astrolabe – an object of desire

Without doubt the astrolabes is one of the most fascinating of all historical astronomical instruments.

Astrolabe Renners Arsenius 1569 Source: Wikimedia Commons

Astrolabe Renners Arsenius 1569
Source: Wikimedia Commons

To begin with it is not simply one object, it is many objects in one:


  • An astronomical measuring device
  • A timepiece
  • An analogue computer
  • A two dimensional representation of the three dimensional celestial sphere
  • A work of art and a status symbol


This Medieval-Renaissance Swiss Army penknife of an astronomical instrument had according to one medieval Islamic commentator, al-Sufi writing in the tenth century, more than one thousand different functions. Even Chaucer in what is considered to be the first English language description of the astrolabe and its function, a pamphlet written for a child, describes at least forty different functions.

The astrolabe was according to legend invented by Hipparchus of Nicaea, the second century BCE Greek astronomer but there is no direct evidence that he did so. The oldest surviving description of the planisphere, that two-dimensional representation of the three-dimensional celestial sphere, comes from Ptolemaeus in the second century CE.

Modern Planisphere Star Chart c. 1900 Source: Wikimedia Commons

Modern Planisphere Star Chart c. 1900
Source: Wikimedia Commons

Theon of Alexandria wrote a thesis on the astrolabe, in the fourth century CE, which did not survive and there are dubious second-hand reports that Hypatia, his daughter invented the instrument. The oldest surviving account of the astrolabe was written in the sixth century CE by John Philoponus. However it was first the Islamic astronomers who created the instrument, as it is known today, it is said for religious purposes, to determine the direction of Mecca and the time of prayer. The earliest surviving dated instrument is dated 315 AH, which is 927/28 CE.

The Earliest  Dated Astrolabe Source: See Link

The Earliest Dated Astrolabe
Source: See Link

It is from the Islamic Empire that knowledge of the instrument found its way into medieval Europe. Chaucer’s account of it is based on that of the eight-century CE Persian Jewish astrologer, Masha’allah ibn Atharī, one of whom claim to fame is writing the horoscope to determine the most auspicious date to found the city of Baghdad.

So-called Chaucer Astrolabe dated 1326, similar to the one Chaucer describes, British Museum Source: Wikimedia Commons

So-called Chaucer Astrolabe dated 1326, similar to the one Chaucer describes, British Museum
Source: Wikimedia Commons

However this brief post is not about the astrolabe as a scientific instrument in itself but rather the last point in my brief list above the astrolabe as a work of art and a status symbol. One of the reasons for people’s interest in astrolabes is the fact that they are simply beautiful to look at. This is not a cold, functional scientific instrument but an object to admire, to cherish and desire. A not uncommon reaction of people being introduced to astrolabes for the first time is, oh that is beautiful; I would love to own one of those. And so you can there are people who make replica astrolabes but buying one will set you back a very pretty penny.

That astrolabes are expensive is not, however, a modern phenomenon. Hand crafted brass, aesthetically beautiful, precision instruments, they were always very expensive and the principal market would always have been the rich, often the patrons of the instrument makers. The costs of astrolabes were probably even beyond the means of most of the astronomers who would have used them professionally and it is significant that most of the well know astrolabe makers were themselves significant practicing astronomers; according to the principle, if you need it and can’t afford it then make it yourself. Other astronomers would probably have relied on their employers/patrons to supply the readies. With these thoughts in mind it is worth considering the claim made by David King, one of the world’s greatest experts on the astrolabe, that the vast majority of the surviving astrolabes, made between the tenth nineteenth centuries – about nine hundred – were almost certainly never actually used as scientific instruments but were merely owned as status symbols. This claim is based on, amongst other things, the fact that they display none of the signs of the wear and tear, which one would expect from regular usage.

Does this mean that the procession of astrolabes was restricted to a rich elite and their employees? Yes and no. When European sailors began to slowly extend their journeys away from coastal waters into the deep sea, in the High Middle Ages they also began to determine latitude as an element of their navigation. For this purpose they needed an instrument like the astrolabe to measure the elevation of the sun or of chosen stars. The astrolabe was too complex and too expensive for this task and so the so-called mariners astrolabe was developed, a stripped down, simplified, cheaper and more robust version of the astrolabe. When and where the first mariner’s astrolabe was used in not known but probably not earlier than the thirteenth century CE. Although certainly not cheap, the mariner’s astrolabe was without doubt to be had for considerably less money than its nobler cousin.


Mariner’s Astrolabe Francisco de Goes 1608 Source: Istituto e Museo di Storia della Scienza, Firenze

Another development came with the advent of printing in the fifteenth century, the paper astrolabe. At first glance this statement might seem absurd, how could one possibly make a high precision scientific measuring instrument out of something, as flexible, unstable and weak as paper? The various parts of the astrolabe, the planisphere, the scales, the rete star-map, etc. are printed onto sheets of paper. These are then sold to the customer who cuts them out and pastes them onto wooden forms out of which he then constructs his astrolabe, a cheap but serviceable instrument. One well-known instrument maker who made and sold printed-paper astrolabes and other paper instruments was the Nürnberger mathematician and astronomer Georg Hartmann. The survival rate of such cheap instruments is naturally very low but we do actually have one of Hartmann’s wood and paper astrolabes.

Hartmann Paper Astrolabe Source: Oxford Museum of History of Science

Hartmann Paper Astrolabe
Source: Oxford Museum of History of Science

In this context it is interesting to note that, as far as can be determined, Hartmann was the first instrument maker to develop the serial production of astrolabes. Before Hartmann each astrolabe was an unicum, i.e. a one off instrument. Hartmann standardised the parts of his brass astrolabes and produced them, or had them produced, in batches, assembling the finished product out of these standardised parts. To what extent this might have reduced the cost of the finished article is not known but Hartmann was obviously a very successful astrolabe maker as nine of those nine hundred surviving astrolabes are from his workshop, probably more than from any other single manufacturer.

Hartmann Serial Production Astrolabe Source: Museum Boerhaave

Hartmann Serial Production Astrolabe
Source: Museum Boerhaave


If this post has awoken your own desire to admire the beauty of the astrolabe then the biggest online collection of Medieval and Renaissance scientific instruments in general and astrolabes in particular is the Epact website, a collaboration between the Museum of the History of Science in Oxford, the British Museum, the Museum of the History of Science in Florence and the Museum Boerhaave in Leiden.

This blog post was partially inspired by science writer Philip Ball with whom I had a brief exchange on Twitter a few days ago, which he initiated, on our mutual desire to possess a brass astrolabe.






Filed under History of Astrology, History of Astronomy, History of science, History of Technology, Mediaeval Science, Renaissance Science

The Reformation, Astrology, and Mathematics in Schools and Universities.

It is one of the ironies of the medieval universities that mathematics played almost no role in undergraduate education. It is ironical because the curriculum was nominally based on the seven liberal arts of which the mathematical sciences – arithmetic, geometry, music and astronomy – formed one half, the quadrivium. Although the quadrivium was officially a large part of the curriculum in reality the four mathematical disciplines were paid little attention and hardly taught at all. This only began to change in the fifteenth century with the rise of astro-medicine or iatromathematics, to give it its formal name. With the rise of this astrology-based medicine the humanist universities of Northern Italy and Kraków introduced chairs of mathematics to teach astrology to their students of medicine. This of course entailed first teaching mathematics and then astronomy in order to be able to do astrology and thus mathematics gained a first foothold in the European universities. Ingolstadt became the first German university to introduce a chair for mathematics, also for teaching astrology to medical students, in the 1470s. It became an important centre for seeding new chairs at other universities with its graduates. Stabius and Stiborius going from there to Vienna with Celtis, for example. However there was no systematic introduction of mathematics into the university curriculum as of yet, this would first come as a result of the Reformation and the educational reforms of Philip Melanchthon.

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 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 was born Philip Schwartzerdt in Bretten near Karlsruhe on 16 February 1497. A great nephew of Johann Reuchlin a leading humanist scholar Philip changed his name to Melanchthon, a literal Greek translation of his German name, which means black earth, at Reuchlin’s suggestion. Melanchthon was a child prodigy who would grow up to be Germany’s greatest humanist scholar. He studied at Heidelberg University where he was denied his master degree in 1512 on account of his youth. He transferred to Tübingen where he came under the influence of Johannes Stöffler, one of those Ingolstadt graduates, a leading and highly influential mathematician/astrologer.

Johannes Stöffler Source Wikimedia Commons

Johannes Stöffler
Source Wikimedia Commons

The cosmograph Sebastian Münster was another of Stöffler’s famous pupils. Stöffler also has a great influence on several of the Nürnberger mathematician-astronomers, especial Johannes Schöner and Georg Hartmann. Under Stöffler’s influence Melanchthon became a passionate supporter of astrology.

On Reuchlin’s recommendation Melanchthon became professor of Greek at Luther’s University of Wittenberg at the age of twenty-one and thus a central figure in the Reformation. One of the major problems faced by the reformers was the fact that the education system was totally in the hands of the Catholic Church, which meant that they had to start from scratch and create their own school and university system; this task was taken on by Melanchthon, who became Luther’s Preceptor Germania, Germany’s Schoolmaster.

Because of his own personal passion for astrology Melanchthon introduced mathematics into the curriculum of all the Lutheran schools and universities. He invented a new type of school on a level between the old Church Latin schools and the universities that were devised to prepare their pupils for a university education. The very first of these was the Eigidien Oberschule in Nürnberg, which opened in 1526 with Johannes Schöner, as its first professor for mathematics.


These type of school created by Melanchthon would become the Gymnasium, still today the highest level secondary schools in the German education system.

In Wittenberg he appointed Johannes Volmar (1480-1536) professor for the higher mathematic, music and astronomy, and Jakob Milich (1501- 1559) professor for the lower mathematic, arithmetic and geometry, in 1525. Their most famous students were Erasmus Reinhold, who followed Volmar on the chair for higher mathematics when he died in 1536, and Georg Joachim Rheticus, who followed Milich on the chair for lower mathematics, in the same year when Milich became professor for medicine. Schöner, Reinhold and Rheticus were not the only mathematicians supported by Melanchthon, who played an important role in the dissemination of the heliocentric astronomy. Although following Melanchthon’s lead these Protestant mathematicians treated the heliocentric hypothesis in a purely instrumentalist manner, i.e. it is not true but is mathematically useful, they taught it in their university courses alongside the geocentric astronomy.

As a result of Melanchthon’s passion for astrology the Lutheran Protestant schools and universities of Europe all had departments for the study of mathematics headed by qualified professors. The Catholic schools and universities would have to wait until the end of the sixteenth century before Christoph Clavius did the same for them, although his motivation was not astrology. Sadly Anglican England lagged well behind the continent with Oxford first appointing professors for geometry and astronomy in the 1620s at the bequest of Henry Savile, who had had to go abroad to receive his own mathematical education. Cambridge only followed suit with the establishment of the Lucasian Chair in 1663, whose first occupant was Isaac Barrow followed by that other Isaac, Newton. In 1705 John Arbuthnot could still complain in an essay that there was not one single school in England that taught mathematics.





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

The Arch-Humanist

The name Conrad Celtis is not one that you’ll find in most standard books on the history of mathematics, which is not surprising as he was a Renaissance humanist scholar best known in his lifetime as a poet. However, Celtis played an important role in the history of mathematics and is a good example of the fact that if you really wish to study the evolution of the mathematical sciences it is necessary to leave the narrow confines of the mathematics books.

Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507 Source: Wikimedia Commons

Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507
Source: Wikimedia Commons

Born Konrad Bickel or Pyckell, (Conrad Celtis was his humanist pseudonym) the son of a winemaker, in Franconian Wipfield am Main near Schweinfurt on 1 February 1459, he obtained his BA at the University of Cologne in 1497. Unsatisfied with the quality of tuition in Cologne he undertook the first of many study journeys, which typified his life, to Buda in 1482, where he came into contact with the humanist circle on the court of Matthias Corvinus, the earlier patron of Regiomontanus. 1484 he continued his studies at the University of Heidelberg specialising in poetics and rhetoric, learning Greek and Hebrew and humanism as a student of Rudolf Agricola, a leading Dutch early humanist scholar. Celtis obtained his MA in 1485. 1486 found him underway in Italy, where he continued his humanist studies at the leading Italian universities and in conversation with many leading humanist scholars. Returning to Germany he taught poetics at the universities of Erfurt, Rostock and Leipzig and on 18 April 1487 he was crowned Poet Laureate by Emperor Friedrich III in Nürnberg during the Reichstag. In Nürnberg he became part of the circle of humanists that produced the Nürnberger Chronicle to which he contributed the section on the history and geography of Nürnberg. It is here that we see the central occupation of Celtis’ life that brought him into contact with the Renaissance mathematical sciences.

During his time in Italy he suffered under the jibes of his Italian colleges who said that whilst Italy had perfect humanist credentials being the inheritors of the ancient Roman culture, Germany was historically a land of uncultured barbarians. This spurred Celtis on to prove them wrong. He set himself the task of researching and writing a history of Germany to show that its culture was the equal of Italy’s. Celtis’ concept of history, like that of his Renaissance contemporaries, was more a mixture of our history and geography the two disciplines being regarded as two sides of the same coin. Geography being based on Ptolemaeus’ Geographia (Geographike Hyphegesis), which of course meant cartography, a branch of the mathematical sciences.

Continuing his travels in 1489 Celtis matriculated at the University of Kraków specifically to study the mathematical sciences for which Kraków had an excellent reputation. A couple of years later Nicolaus Copernicus would learn the fundamentals of mathematics and astronomy there. Wandering back to Germany via Prague and Nürnberg Celtis was appointed professor of poetics and rhetoric at the University of Ingolstadt in 1491/92. Ingolstadt was the first German university to have a dedicated chair for mathematics, established around 1470 to teach medical students astrology and the necessary mathematics and astronomy to cast a horoscope. When Celtis came to Ingolstadt there were the professor of mathematics was Andreas Stiborius (born Stöberl 1464–1515) who was followed by his best student Johannes Stabius (born Stöberer before 1468­–1522) both of whom Celtis convinced to support him in his cartographic endeavours.

In 1497 Celtis received a call to the University of Vienna where he established a Collegium poetarum et mathematicorum, that is a college for poetry and mathematics, with Stiborius, whom he had brought with him from Ingolstadt, as the professor for mathematics. In 1502 he also brought Stabius, who had succeeded Stiborius as professor in Ingolstadt, and his star student Georg Tanstetter to Vienna. Stiborius, Stabius and Tanstetter became what is known, to historians of mathematics, as the Second Viennese School of Mathematics, the First Viennese School being Johannes von Gmunden, Peuerbach and Regiomontanus, in the middle of the fifteenth century. Under these three Vienna became a major European centre for the mathematical sciences producing many important mathematicians the most notable being Peter Apian.

Although not a mathematician himself Conrad Celtis, the humanist poet, was the driving force behind one of the most important German language centres for Renaissance mathematics and as such earns a place in the history of mathematics. A dedicated humanist, wherever he went on his travels Celtis would establish humanist societies to propagate humanist studies and it was this activity that earned him the German title of Der Erzhumanist, in English the Arch Humanist. Celtis died in 1508 but his Collegium poetarum et mathematicorum survived him by twenty-two years, closing first in 1530



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

Hans Holbein and the Nürnberg–Ingolstadt–Vienna Renaissance mathematical nexus.

There is a strong tendency, particularly in the popular history of science, to write about or present scientists as individuals. This leads to a serious distortion of the way that science develops and in particular propagates the lone genius myth. In reality science has always been a collective endeavour with its practitioners interacting in many different ways and on many different levels. In the Renaissance, when travelling from one end of Europe to the other would take weeks and letters often even longer, one might be excused for thinking that such cooperation was very low level but in fact the opposite was the truth, with scholars in the mathematical sciences exchanging information and ideas throughout Europe. A particularly strong mathematical nexus existed in the Southern German speaking area between the cities of Nürnberg, Ingolstadt and Vienna in the century between 1450 and 1550. Interestingly two of the paintings of the Northern Renaissance artist Hans Holbein the Younger open a door into this nexus.

Holbein (c. 1497–1543) was born in Augsburg the son of the painter and draughtsman Hans Holbein the Elder. As a young artist he lived and worked for a time in Basel where he became acquainted with Erasmus and worked for the printer publisher Johann Froben amongst others. Between 1526 and 1528 he spent some time in England in the household of Thomas More and it is here that he painted the second portrait I shall be discussing. The next four years find him living in Basel again before he returned to England in 1532 where he became associated with the court of Henry VIII, More having fallen out of favour. It was at the court that he painted, what is probably his most well know portrait, The Ambassadors in 1533.

Hans Holbein The Ambassadors Source: Wikimedia Commons

Hans Holbein The Ambassadors
Source: Wikimedia Commons

The painting shows two courtiers, usually identified as the French Ambassador Jean de Dinteville and Georges de Selve, Bishop of Lavaur standing on either side of a set of shelves laden with various books and instruments. There is much discussion was to what the instruments are supposed to represent but it is certain that, whatever else they stand for, they represent the quadrivium, arithmetic, geometry music and astronomy, the four mathematical sciences taught at European medieval universities. There are two globes, on the lower shelf a terrestrial and on the upper a celestial one. The celestial globe has been positively identified, as a Schöner globe and the terrestrial globe also displays characteristics of Schöner’s handwork.

Terrestrial Globe The Ambassadors Source Wikimedia Commons

Terrestrial Globe The Ambassadors
Source Wikimedia Commons

Celestial Globe The Ambassadors Source Wikimedia Commons

Celestial Globe The Ambassadors
Source Wikimedia Commons

Johannes Schöner (1477–1547) was professor for mathematics at the Egidienöberschule in Nürnberg, the addressee of Rheticus’ Narratio Prima, the founder of the tradition of printed globe pairs, an editor of mathematical texts for publication (especially for Johannes Petreius the sixteenth centuries most important scientific publisher) and one of the most influential astrologers in Europe. Schöner is a central and highly influential figure in Renaissance mathematics.

On the left hand side of the lower shelf is a copy of Peter Apian’s Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen (published in Ingolstadt in 1527) held open by a ruler. This is a popular book of commercial arithmetic, written in German, typical of the period. Peter Apian (1495–1552) professor of mathematics at the University of Ingolstadt, cartographer, printer-publisher and astronomer was a third generation representative of the so-called Second Viennese School of Mathematics. A pupil of Georg Tannstetter (1482–1535) a graduate of the University of Ingolstadt who had followed his teachers Johannes Stabius and Andreas Stiborious to teach at Conrad Celtis’ Collegium poetarum et mathematicorum, of which more later. Together Apian and Tannstetter produced the first printed edition of the Optic of Witelo, one of the most important medieval optic texts, which was printed by Petreius in Nürnberg in 1535. The Tannstetter/Apian/Petreius Witelo was one of the books that Rheticus took with him as a present for Copernicus when he visited him in 1539. Already, a brief description of the activities of Schöner and Apian is beginning to illustrate the connection between our three cities.

Apian's Arithmetic Book The Ambassadors Source: Wikimedia Commons

Apian’s Arithmetic Book The Ambassadors
Source: Wikimedia Commons

When Sebastian Münster (1488–1552), the cosmographer, sent out a circular requesting the cartographers of Germany to supply him with data and maps for his Cosmographia, he specifically addressed both Schöner and Apian by name as the leading cartographers of the age. Münster’s Cosmographia, which became the biggest selling book of the sixteenth century, was first published by Heinrich Petri in Basel in 1544. Münster was Petri’s stepfather and Petri was the cousin of Johannes Petreius, who learnt his trade as printer publisher in Heinrich’s printing shop in Basel. The Petri publishing house was also part of a consortium with Johann Amerbach and Johann Froben who had employed Hans Holbein in his time in Basel. Wheels within wheels.

The, mostly astronomical, instruments on the upper shelf are almost certainly the property of the German mathematician Nicolaus Kratzer (1487–1550), who is the subject of the second Holbein portrait who will be looking at.

Nicolas Kratzer by Hans Holbein Source: Wikimedia Commona

Nicolas Kratzer by Hans Holbein
Source: Wikimedia Commona

Born in Munich and educated at the universities of Cologne and Wittenberg Kratzer, originally came to England, like Holbein, to become part of the Thomas More household, where he was employed as a tutor for More’s children. Also like Holbein, Kratzer moved over to Henry VIII’s court as court horologist or clock maker, although the clocks he was responsible for making were more probably sundials than mechanical ones. During his time as a courtier Kratzer also lectured at Oxford and is said to have erected a monumental stone sundial in the grounds of Corpus Christi College. One polyhedral sundial attributed to Kratzer is in the Oxford Museum for the History of Science.

Polyhedral Sundial attributed to Nicolas Kratzer Source: MHS Oxford

Polyhedral Sundial attributed to Nicolas Kratzer
Source: MHS Oxford

In 1520 Kratzer travelled to Antwerp to visit Erasmus and here he met up with Nürnberg’s most famous painter Albrecht Dürer, who regular readers of this blog will know was also the author of a book on mathematics. Dürer’s book contains the first printed instructions, in German, on how to design, construct and install sundials, so the two men will have had a common topic of interest to liven there conversations. Kratzer witnessed Dürer, who was in Antwerp to negotiate with the German Emperor, painting Erasmus’ portrait and Dürer is said to have also drawn a portrait of Kratzer that is now missing. After Kratzer returned to England and Dürer to Nürnberg the two of them exchanged, at least once, letters and it is Kratzer’s letter that reveals some new connections in out nexus.

Albrecht Dürer selfportrait Source: Wikimedia Commons

Albrecht Dürer selfportrait
Source: Wikimedia Commons

In his letter, from 1524, Kratzer makes inquires about Willibald Pirckheimer and also asks if Dürer knows what has happened to the mathematical papers of Johannes Werner and Johannes Stabius who had both died two years earlier.

Willibald Pirckheimer (1470–1530) a close friend and patron of Dürer’s was a rich merchant, a politician, a soldier and a humanist scholar. In the last capacity he was the hub of a group of largely mathematical humanist scholars now known as the Pirckheimer circle. Although not a mathematician himself Pirckheimer was a fervent supporter of the mathematical sciences and produced a Latin translation from the Greek of Ptolemaeus’ Geōgraphikḕ or Geographia, Pirckheimer’s translation provided the basis for Sebastian Münster’s edition, which was regarded as the definitive text in the sixteenth century. Stabius and Werner were both prominent members of the Pirckheimer circle.

Willibald Pirckheimer by Albrecht Dürer Source: Wikimedia Commons

Willibald Pirckheimer by Albrecht Dürer
Source: Wikimedia Commons

The two Johanneses, Stabius (1450–1522) and Werner (1468–1522), had become friends at the University of Ingolstadt where the both studied mathematics. Ingolstadt was the first German university to have a dedicated chair for mathematics. Werner returned to his hometown of Nürnberg where he became a priest but the Austrian Stabius remained in Ingolstadt, where he became professor of mathematics. The two of them continued to correspond and work together and Werner is said to have instigated the highly complex sundial on the wall of the Saint Lorenz Church in Nürnberg, which was designed by Stabius and constructed in 1502.

St Lorenz Church Nürnberg Sundial 1502 Source: Astronomie in Nürnberg

St Lorenz Church Nürnberg Sundial 1502
Source: Astronomie in Nürnberg

It was also Werner who first published Stabius’ heart shaped or cordiform map projection leading to it being labelled the Werner-Stabius Projection. This projection was used for world maps by Peter Apian as well as Oronce Fine, France’s leading mathematicus of the sixteenth century and Gerard Mercator, of whom more, later. The network expands.

Mercator cordiform world map 1538 Source: American Geographical Society Library

Mercator cordiform world map 1538
Source: American Geographical Society Library

In his own right Werner produced a partial Latin translation from the Greek of Ptolemaeus’ Geographia, was the first to write about prosthaphaeresis (a trigonometrical method of simplifying calculation prior to the invention of logarithms), was the first to suggest the lunar distance method of determining longitude and was in all probability Albrecht Dürer’s maths teacher. He also was the subject of an astronomical dispute with Copernicus.

Johannes Werner Source: Wikimedia Commons

Johannes Werner
Source: Wikimedia Commons

Regular readers of this blog will know that Stabius co-operated with Albrecht Dürer on a series of projects, including his famous star maps, which you can read about in an earlier post here.

Johannes Statius Portrait by Albrecht Dürer Source: Wikimedia Commons

Johannes Statius Portrait by Albrecht Dürer
Source: Wikimedia Commons

An important non-Nürnberger member of the Pirckheimer Circle was Conrad Celtis (1459–1508), who is known in Germany as the arch-humanist. Like his friend Pirckheimer, Celtis was not a mathematician but believed in the importance of the mathematical sciences. Although already graduated he spent time in 1489 on the University of Kraków in order to get the education in mathematics and astronomy that he couldn’t get at a German university. Celtis had spent time at the humanist universities of Northern Italy and his mission in life was to demonstrate that Germany was just as civilised and educated as Italy and not a land of barbarians as the Italians claimed. His contributions to the Nuremberg Chronicle can be viewed as part of this demonstration. He believed he could achieve his aim by writing a comprehensive history of Germany including, as was common at the time its geography. In 1491/92 he received a teaching post in Ingolstadt, where he seduced the professors of mathematics Johannes Stabius and Andreas Stiborius (1464–1515) into turning their attention from astrology for medicine student, their official assignment, to mathematical cartography in order to help him with his historical geography.

Conrad Celtis Source: Wikimedia Commons

Conrad Celtis
Source: Wikimedia Commons

Unable to achieve his ends in Ingolstadt Celtis decamped to Vienna, taking Stabius and Stiborius with him, to found his Collegium poetarum et mathematicorum as mentioned above and with it the so-called Second Viennese School of Mathematics; the first had been Peuerbach and Regiomontanus in the middle of the fifteenth century. Regiomontanus spent the last five years of his life living in Nürnberg, where he set up the world’s first scientific publishing house. Stiborius’ pupil Georg Tannstetter proved to be a gifted teacher and Peter Apian was by no means his only famous pupil.

The influence of the Nürnberg–Ingolstadt–Vienna mathematicians reached far beyond their own relatively small Southern German corridor. As already stated Münster in Basel stood in contact with both Apian and Schöner and Stabius’ cordiform projection found favour with cartographers throughout Northern Europe. Both Apian and Schöner exercised a major influence on Gemma Frisius in Louvain and through him on his pupils Gerard Mercator and John Dee. As outlined in my blog post on Frisius, he took over editing the second and all subsequent editions of Apian’s Cosmographia, one of the most important textbooks for all things astronomical, cartographical and to do with surveying in the sixteenth century. Frisius also learnt his globe making, a skill he passed on to Mercator, through the works of Schöner. Dee and Mercator also had connections to Pedro Nunes (1502–1578) the most important mathematicus on the Iberian peninsular. Frisius had several other important pupils who spread the skills in cosmography, and globe and instrument making that he had acquired from Apian and Schöner all over Europe.

Famously Rheticus came to Nürnberg to study astrology at the feet of Johannes Schöner, who maintained close contacts to Philipp Melanchthon Rheticus patron. Schöner was the first professor of mathematics at a school designed by Melanchthon. Melanchthon had learnt his mathematics and astrology at the University of Tübingen from Johannes Stöffler (1452–1531) another mathematical graduate from Ingolstadt.

Kupferstich aus der Werkstatt Theodor de Brys, erschienen 1598 im 2. Bd. der Bibliotheca chalcographica Source: Wikimedia Commons

Kupferstich aus der Werkstatt Theodor de Brys, erschienen 1598 im 2. Bd. der Bibliotheca chalcographica
Source: Wikimedia Commons

Another of Stöffler’s pupils was Sebastian Münster. During his time in Nürnberg Rheticus became acquainted with the other Nürnberger mathematicians and above all with the printer-publisher Johannes Petreius and it was famously Rheticus who brought the manuscript of Copernicus’ De revolutionibus to Nürnberg for Petreius to publish. Rheticus says that he first learnt of Copernicus’s existence during his travels on his sabbatical and historians think that it was probably in Nürnberg that he acquired this knowledge. One of the few pieces of astronomical writing from Copernicus that we have is the so-called Letter to Werner. In this manuscript Copernicus criticises Werner’s theory of trepidation. Trepidation was a mistaken belief based on faulty data that the rate of the precession of the equinoxes is not constant but varies with time. Because of this highly technical dispute amongst astronomers Copernicus would have been known in Nürnberg and thus the assumption that Rheticus first heard of him there. Interestingly Copernicus includes observations of Mercury made by Bernhard Walther (1430–1504), Regiomontanus partner, in Nürnberg; falsely attributing some of them to Schöner, so a connection between Copernicus and Nürnberg seems to have existed.

In this brief outline we have covered a lot of ground but I hope I have made clear just how interconnected the mathematical practitioners of Germany and indeed Europe were in the second half of the fifteenth century and the first half of the sixteenth. Science is very much a collective endeavour and historians of science should not just concentrate on individuals but look at the networks within which those individual operate bringing to light the influences and exchanges that take place within those networks.


Filed under History of Astrology, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, Renaissance Science

Asterisms and Constellations and how not to confuse them with Tropical Signs.

If you are going to write about something, especially if you intend to lay bare somebody else’s ignorance, it pays to actually know what you are talking about otherwise you could well end up looking like a total idiot, as does Anna Culaba in her article on the RYOT website, The Stars and Your Astrological Signs Have Been Lying to You This Whole Time. I should point out that Ms Culaba is by no means the first person to publically embarrass themselves pontificating on this subject, in fact it’s a reoccurring theme much loved by scientists and science fans who want to take a cheap shot at astrology. Indeed, as we will see later Ms Culaba, in her article, is in fact just regurgitating the content of a BBC website. So what exactly does our intrepid science fan say in her blog post?

My horoscope for today (I’m a Virgo) according to reads, “Today, explore an aspect of an unfamiliar religion or culture. Today is a day to make plans and aim high.” There are only two things that are keeping me from leaving work right now: one, I don’t really believe that the stars can determine what will happen in my life and two, I wasn’t really born under the star sign that the world told me I was born into. According to the BBC, about 86 percent of people are actually born under a different sign than the one they think. This is because 2,000 years ago, when the Ancient Greeks first created the zodiacs, the star signs corresponded to the position of the sun relative to the constellations that appeared in the sky the day people were born. Unfortunately, during that time people didn’t know of the phenomenon known as the precession. Live Sciences reports that the precession is when the Earth continually wobbles around its axis in an almost 26,000-year cycle thanks to the gravitational attraction of the moon. Thanks to this phenomenon, the constellations some people live and die by have actually drifted away from us. This means that constellations are now actually off by a month. So if you were born between August 11 to September 16 you’re not the picky and critical Virgo that you thought you were — you’re really an ambitious Leo whose strength of purpose allows you to accomplish many, many things. And if you’re astrological world hasn’t been rocked enough, if you thought you had your star sign wrong, wait until some of you realize that there’s actually a 13th zodiac sign known as the Ophiuchus. According to the BBC, the Ancient Greeks deliberately left out the original zodiac so that ancient astrologers would be able to divide the sun’s 360 degree path into 12 equal parts. Where does Ophiuchus fit into the zodiac calendar? It goes between Scorpio and Sagittarius, so if you were born between November 30 and December 18 consider yourself an Ophiuchus. You’re probably very secretive and good at hide and seek.

I have reproduced the whole of Ms Culaba’s screed here to save me having to quote it in little bits, merely removing the links from the original. If you read it through you what will discover is the central claim that astrologers were too stupid to realise the astronomical phenomenon of precession and so you were not actually born under the star sign that they claim you were. There are two general points to be made here, firstly astrologers were well aware of precession and secondly Ms Culaba and the source she is quoting don’t know the fundamental difference between constellations and tropical signs. So for the benefit of Ms Culaba and all others who are confused by the topic we will have a Renaissance Mathematicus guide to asterisms, constellations, the zodiac and tropical signs.

If you go out on a dark night with a clear sky in an area with little or no light pollution (and if you have never done so you should, it’s awesome) and look up in the heavens you will see a myriad of stars looking down on you in a vast blue black vault. If you are not a trained astronomer you will probably find no means of orienting your gaze in this confusion of twinkling lights. This problem was confronted by all human cultures since the dawn of human existence. The human brain seems to be programmed for pattern recognition and so, like children with a join up the dots picture book, all cultures started to create pictures by imagining lines joining up or outlining eye-catching groups of stars and giving these pictures names. These pictures, and they exist in all human cultures, are known technically as asterisms. These asterisms help the observing eye gain orientation when traversing the vast dome of the night sky and early astronomers started compiling lists of the most prominent such join-up-the-dots-pictures or asterisms in order to use them as a scaffolding for mapping the heavens. Those asterisms contained in such formal lists are called constellations. Our modern, western list of constellations has its origins in ancient Babylonian astrology/astronomy and comes down to us via the ancient Greeks and the medieval Islamic astronomers. In his Syntaxis Mathematiké, Ptolemaeus lists 48 constellations by name. Currently, the International Astronomical Union (IAU) recognises 88 named constellations. We now need to turn our attention to the origins of the zodiac.

Viewed from the earth, and before the beginning of the so-called space age that was the only way possible to view the heavens, the sun appears to orbit the earth once every year. In fact the year is defined as the time it takes for the sun to orbit the earth. The path the sun follows on its way around the earth is called the ecliptic and is tilted at approximately 23 degrees to the earth’s equator. This tilt, known as the obliquity of the ecliptic, is the reason why we have seasons on the earth. The six planets visible to the naked eye and know in antiquity – Moon, Mercury, Venus, Mars, Jupiter and Saturn – all appear to orbit the earth in the plane of the ecliptic making this imaginary belt around the heavens very important for the study of astronomy. The earliest known mapping of the ecliptic is contained in a set of Babylonian clay tablets known as the MUL.APIN, which date from around 1000 BCE. Here the path of the moon’s orbit is described or mapped with 17 or 18 (the text is somewhat ambiguous) constellations and stars. The moon’s orbit is tilted at about five degrees to the ecliptic. This mapping was still in use around 700 BCE. By around 500 BCE the 17/18 constellations/stars had be replaced by twelve constellations of varying sizes. Circa 420 BCE the Babylonians had replaced those twelve constellations with twelve equal divisions of the ecliptic comprising 30° segments. These segments were named after the constellations they replaced and form the zodiac that was taken over by the Greeks and made its way down to us. Those segments are known technically as tropical or sun signs, form the basis of zodiacal astrology and are abstract geometrical segment of the ecliptic and not constellations. The constellations slowly circle the heavens due to precession, the tropical signs do not! If an astrologer says you were born under the sign Virgo it means that the sun was in the 30° segment of the ecliptic that bears the name Virgo at the moment of your birth. This has nothing apart from the name in common with the constellation Virgo.

It is not the astrologers who display ignorance of the precession of the equinox, to give the phenomenon its full name, but Ms Culaba who displays total ignorance of both astronomy and astrology. This is not a very good situation to be in if you are going to write about the history of science and yes we are talking about the history of science here, the zodiac with its tropical signs was originally conceived for astronomical purposes. Ms Culaba might be excused because she did not originate this particular piece of history of science rubbish but is merely regurgitating false information from what she obviously thought was a reliable source, the BBC.

Here we have the presenter of Stargazing Live, a high prestige BBC science programme, Dara O Brian presenting the world with high-grade bullshit under the BBC’s banner. O Brian and his co-presenter Brian Cox should know better and I find it a total disgrace that the fee payers money is being wasted on such rubbish under the guise of educational television, both the presenters and the Beeb should be thoroughly ashamed of themselves.


Filed under History of Astrology, History of Astronomy, Myths of Science