Category Archives: History of Astrology

The Seven Learned Sisters

I have suffered from a (un)healthy[1]portion of imposter syndrome all of my life. This is the personal feeling in an academic context that one is just bluffing and doesn’t actually know anything and then any minute now somebody is going to unmask me and denounce me as an ignorant fraud. I always thought that this was a personal thing, part of my general collection of mental and emotional insecurities but in more recent years I have learned that many academics, including successful and renowned ones, suffer from this particular form of insecurity. On related problem that I have is the belief that anything I do actually know is trivial, generally known to everyone and therefore not worth mentioning[2]. I experienced an example of this recently on Twitter when I came across the following medieval illustration and its accompanying tweet.


Geometria Source: Wikimedia Commons

Woman teaching geometry to monks. In the Middle Ages, it is unusual to see women represented as teachers, in particular when the students appear to be monks. Euclid’s Elementa, in the translation attributed to Adelard of Bath, 1312.

I would simply have assumed that everybody knew what this picture represents and not commented. It is not a “women teaching geometry to monks” as the tweeter thinks but a typical medieval personification of Geometria, one of the so-called Seven Learned Sisters. The Seven Learned Sisters are the personifications of the seven liberal arts, the trivium (grammar, rhetoric and dialectic)

and the quadrivium (arithmetic, geometry, music and astronomy),

which formed the curriculum in the lower or liberal arts faculty at the medieval university. The seven liberal arts, however, have a history that well predates the founding of the first universities. In what follows I shall only be dealing with the history of the quadrivium.

As a concept this four-fold division of the mathematical sciences can be traced back to the Pythagoreans. The mathematical commentator Proclus (412–485 CE) tells us, in the introduction to his commentary on the first book of Euclid’s Elements:

The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving.

The earliest know written account of this division can be found at the beginning of the late Pythagorean Archytas’ book on harmonics, where he identifies a set of four sciences: astronomy, geometry, logistic (arithmetic) and music. Archytas’ dates of birth and death are not known but he was, roughly speaking, a contemporary of Plato. He was the teacher of Eudoxus (c.390–c.337 BCE) Harmonics, by the way, is the discipline that later became known as music in the quadrivium.

Without mentioning Archytas, Plato (428/427 or 424/423 – 348/347 BCE), who was highly influenced by the Pythagoreans,takes up the theme in his Republic (c.380 BCE). In a dialogue with Glaucon, Plato explains the merits of learning the “five” mathematical sciences; he divides geometry into plane geometry (two dimensional) and solid geometry (three dimensional). He also refers to harmonics and not music.

In the CE period the first important figure is the Neo-Pythagorean, Nicomachus of Gerasa (c.60–c.120 CE), who wrote an Introduction to Arithmeticand a Manual of Harmonics, which are still extant and a lost Introduction to Geometry. The four-fold division of the mathematical sciences only acquired the name quadrivium in the works of Boethius (c.477–524 CE), from whose work the concept of the seven liberal arts was extracted as the basic curriculum for the medieval university. Boethius, who saw it as his duty to rescue the learning of the Greeks, heavily based his mathematical texts on the work of Nicomachus.

Probably the most influential work on the seven liberal arts is the strange De nuptiis Philologiae et Mercurii (“On the Marriage of Philology and Mercury“) of Martianus Capella (fl.c. 410-420). The American historian H. O. Taylor (1856–1941) claimed that On the Marriage of Philology and Mercurywas “perhaps the most widely used schoolbook in the Middle Ages,” quoted from Martianus Capella and the Seven Liberal Artsby William Harris Stahl.[3]Stahl goes on to say, “It would be hard to name a more popular textbook for Latin reads of later ages.”

Martianus introduces each of the members of the trivium and quadrivium as bridesmaids of the bride Philology.

“Geometry enters carrying a radius in her right hand and a globe in her left. The globe is a replica of the universe, wrought by Archimedes’ hand. The peplos she wears is emblazoned with figures depicting celestial orbits and spheres; the earth’s shadow reaches into the sky, giving a purplish hue to the golden globes of the sun and moon; there are gnomons of sundials and figures showing intervals weights, and measures. Her hair is beautifully groomed, but her feet are covered with grime and her shoes are worn to shreds with treading across the entire surface of the earth.”[4]


A 16th century Geometria in a printed copy of the Margarita Philosophica

“As she enters the celestial hall, Arithmetic is even more striking in appearance than was Geometry with her dazzling peplos and celestial globe. Arithmetic too wears a robe, hers concealing an “intricate undergarment that holds clues to the operations of universal nature.” Arithmetic’s stately bearing reflects the pristine origin, antedating the birth of the Thunder God himself. Her head is an awesome sight. A scarcely perceptible whitish ray emanates from her brow; then another ray, the projection of a line, as it were, coming from the first. A third ray and a fourth spring out, and so on, up to a ninth and a tenth ray–all radiating from her brow in double and triple combinations. These proliferate in countless numbers and in a moment are miraculously retracted into the one.”[5]An allusion to the Pythagorean decade.


Gregor Reisch Margarita Philosophica: Arthimetica presiding over a computing competition between Hindu-Arabic numerals and a reckoning board

“Astronomy like her sister Geometry, is a peregrinator of the universe. She has traversed all the heavens and can reveal the constellations lying beneath the celestial arctic circle. […] Astronomy tells us that she is also familiar with the occult lore of Egyptian priest, knowledge hoarded in their sanctums; she kept herself in seclusion in Egypt for nearly forty thousand years, not wishing to divulge those secrets. She is also familiar with antediluvian Athens.”[6]


Astronomia or possibly the Muse Urania 15th century


“Harmony herself is ineffably dazzling and Martianus is stricken in his efforts to describe her. A lofty figure, her head aglitter with gold ornaments, she walks along between Apollo and Athena. Her garment is tiff with incised and laminated gold; it tinkles softly and soothingly with every measured step She carries in her right hand what appears to be a shield, circular in form. It contains many concentric circles, and the whole is embroidered with striking figure. The circular chords encompass one another and from them pours forth a concord of all tones: Small models of theatrical instruments, wrought of gold, hang suspended from Harmony’s left hand. No know instrument produces sounds to compare with those coming from the strange rounded form.”[7]



I have included Stahl’s passages of Martianus’ descriptions of the quadrivium to make clear then when I talk of the disciplines being personified as women I don’t just mean that they get a female name but are fully formed female characters. This of course raises the question, at least for me, why the mathematical disciplines that were taught almost exclusively to men in ancient Greece, the Romano-Hellenistic culture and in the Middle Ages should be represented by women. Quite honestly I don’t know the answer to my own question. I assume that it relates to the nine ancient Greek Muses, who were also women and supposedly the daughters of Zeus and Mnemosyne (memory personified). This however just pushes the same question back another level. Why are the Muses female?


Having come this far it should be noted that although the quadrivium was officially part of the curriculum on medieval universities it was on the whole rather neglected. When taught the subjects were only taught at a very elementary level, arithmetic based on the primer of Boethius, itself an adaption of Nicomachus, geometry from Euclid but often only Book One and even that only partially, music again based on Boethius and astronomy on the very elementary Sphere of Sacrobosco. Often the mathematics courses were not taught during the normal classes but only on holidays, when there were no normal lectures. At most universities the quadrivium disciplines were not part of the final exams and often a student who had missed a course could get the qualification simple by paying the course fees. Mathematics only became a real part of the of the university curriculum in the sixteenth century through the efforts of Philip Melanchthon for the protestant universities and somewhat later Christoph Clavius for the Catholic ones. England had to wait until the seventeenth century before there were chairs for mathematics at Oxford and Cambridge.

[1]On the one hand imposter syndrome can act as a spur to learn more and increase ones knowledge of a given subject. On the other it can lead one to think that one needs to know much more before one closes a given research/learn/study project and thus never finish it.

[2]To paraphrase some old Greek geezer, the older I get and the more I learn, the more I become aware that what I know is merely a miniscule fraction of that which I could/should know and in reality I actually know fuck all.

[3]William Harris Stahl, Martianus Capella and the Seven Liberal Arts: Volume I The Quadrivium of Martianus Capella. Latin Traditions in the Mathematical Science, With a Study of the Allegory and the Verbal Disciplines by Richard Johnson with E. L. Burge, Columbia University Press, New York & London, 1971, p. 22

[4]Stahl pp. 125–126

[5]Stahl pp. 149–150

[6]Stahl p. 172

[7]Stahl p. 203



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

Maximilian and the Mathematici–astrology as political propaganda

For a long time most historians of science tried their best to ignore the history of astrology, basically sweeping it under the carpet where and when it poked its nose into their area of study. More recently this began to change with more and more historians acknowledging that astrology played a role in a large part of human history, although  most of them still treated it as some sort of largely irrelevant side issue that one could mention in passing, if necessary, and then safely ignore. However in large phases of European history astrology permeated all levels of society and was just as much a central factor of life as religion or politics. This was certainly very much the case in the Renaissance. A number of historians have begun to examine in depth the role that astrology played and present their findings in books and articles; one such book is Darin Hayton’s The Crown and the Cosmos: Astrology and the Politics of Maximilian I.[1]


Maximilian I (1486–1519) was an Austrian Habsburg, who was King of the Romans (also known as King of the Germans) from 1486 and Holy Roman Emperor from 1508 until his death.


Albrecht Dürer – Portrait of Maximilian I Source: Wikipedia Commons

Through marriage he became Duke of Burgundy and his son Philip the Handsome through his marriage to Joanna of Castile, arranged by Maximilian, established the Spanish Habsburg dynasty. As such Maximilian played a very important role in late medieval European history. Throughout his life Maximilian was involved in complex and protracted political and military campaigns and Hayton’s book illustrates in detail how Maximilian used astrology as political propaganda to further his aims in those multifarious campaigns.

Throughout his life Maximilian was associated with and actively promoted a significant number of well-known astrological mathematici, several of whom have over the years featured in various blog posts here. As Hayton explains, through his active promotion of the astrologers Maximilian wanted to present himself as a knowledgeable man of science, as erudite and educated. Maximilian’s close connection with astrology began with his birth, when his parents, the Holy Roman Emperor, Frederick III and Eleanor, infanta of Portugal, requested Regiomontanus to cast Maximilian’s natal horoscope. Regiomontanus was only twenty-three years old at the time. Regiomontanus’ teacher Peuerbach had been an astrological advisor to Frederick for some time and had cast Eleanor’s horoscope before the royal marriage.

In the early phase of his career Maximilian used the humanist scholars, Joseph Grünpeck (c. 1473–after 1530) (author of one of the first texts on the French Disease aka syphilis)


Only known portrait of Joseph Grünpeck – artist unknown

and Sebastian Brant (1457–1521) (author of Das Narrenschiff (Ship of Fools))


Sebastian Brant by Albrecht Dürer Source: Wikimedia Commons

to employ their poetical and astrological skills in helping him to create idealised works of autobiography presenting Maximilian as he whished to be viewed as a future Holy Roman Emperor. This was part of a much wider astrological propaganda campaign presenting Maximilian, as the ideal candidate for the position of power.

In a second element of his campaign Maximilian revitalised the University of Vienna, returning it to the high status it had when Georg Peuerbach (1423–1461) and Johannes Regiomontanus (1436–1476) represented the first Viennese School of Mathematics, as the heirs of Johannes von Gmunden (c. 1380–1442). A period, which had ended in 1561 when Peuerbach died and Regiomontanus left Vienna for Italy with Basilios Bessarion (c. 1400–­1472).

At the beginning of the sixteenth century Maximilian brought Conrad Celtis (1495–1508), the Arch-Humanist, from Ingolstadt to Vienna and established for him the Collegium poetarum et mathematicorum. Two professors for mathematics were installed Andreas Stiborius (c. 1464–1515) and Johannes Stabius (1450–1522), both also from Ingolstadt. Stabius was however soon promoted to court historian.


Albrecht Dürer’s portrait of Johannes Stabius Source: Wikimedia Commons

The two also brought their favourite pupil with them, Georg Tannstetter (1482–1535), who would go on to make a long and successful career in Vienna.


Georg Tannstetter Portrait ca. 1515, by Bernhard Strigel Source: Wikimedia Commons

Tannstetter would be succeeded by his own pupil Andreas Perlach (1490–1551). These men constitute the so-called second Viennese School of Mathematics.

Having dealt with Maximilian’s use of astrology in his autobiographies and his political propaganda in the opening chapters, Hayton deals in successive chapters with the various aspects of astrology–teaching of the subject, astrological instruments, wall calendars and practica, ephemerides, prognostications–and how these were used by their producers to support and enable Maximilian’s political aims and ambitions. This is all down in substantive detail illustrating nicely how the work of the mathematici and their patron created a symbiosis serving the needs of both sides. In the chapter on Perlach and his ephemerides Hayton gives a very nice analysis of Perlach’s readers, based on the hand written marginalia found in the surviving copies of his texts.

It should be noted that this service of the Viennese mathematicians did not end with Maximilian’s death in 1519. Both Tannstetter and Perlach carried on producing their astrological publications in the political interest of the Habsburgs for Maximilian’s grandsons, Ferdinand Archduke of Austria (1503–1564) and Charles V (1500–1558) Holy Roman Emperor and Emperor of the Spanish Empire, Maximilian was predeceased by their father, his son Philip the Handsome.


Ferdinand Archduke of Austria Portrait by Hans Bocksberger the Older Source: Wikimedia Commons


Charles V by Juan Pantoja de la Cruz Source: Wikimedia Commons

The book is nicely illustrated with grey tone reproductions of the texts and their illustration from the various publications. There are extensive, informative endnotes, an equally extensive bibliography of primary and secondary sources and a useful index.

Hayton has written an important study on the political use of astrology by those in a centre of power during the Renaissance that can be profitably be read in tandem with Monica Azzolini’s The Duke and the Stars, which I reviewed some time ago. As Hayton says in his introduction historians of the period need to include the history of astrology in their studies and historians of astrology need to look more closely at the general historical picture. Hayton has excellently fulfilled his own demand.




[1]Darin Hayton, The Crown and the Cosmos: Astrology and the Politics of Maximilian I, University of Pittsburgh Press, Pittsburgh, 2015


Filed under Book Reviews, History of Astrology, Renaissance Science

Renaissance mathematics and medicine

Anyone who read my last blog post might have noticed that the Renaissance mathematici Georg Tannstetter and Philipp Apian were both noted mathematicians and practicing physicians. In our day and age if someone was both a practicing doctor of medicine and a noted mathematician they would be viewed as something quite extraordinary but here we have not just one but two. In fact in the Renaissance the combination was quite common. Jakob Milich, who studied under Tannstetter in Vienna, was called to Wittenberg by Philipp Melanchthon in 1524, as professor for mathematics, where he taught both Erasmus Reinhold and Georg Joachim Rheticus. In 1536 he became professor for anatomy in Wittenberg and was succeeded by Rheticus as professor for mathematics. Rheticus in turn would later become a practicing physician in Krakow. The man, who Rheticus called his teacher, Nicolaus Copernicus, was another mathematical physician. My local Renaissance astronomer Simon Marius was another mathematician who studied and practiced medicine. That this was not a purely Germanic phenomenon is shown by the Welsh mathematicus and physician Robert Recorde and most notably by the Italian Gerolamo Cardano, who is credited with having written the first modern maths book, his Ars magna, and who was one of the most renowned physicians in Europe in his day.

These are only a few well-known examples but in fact it was very common for Renaissance mathematician to also be practicing physicians, so what was the connecting factor between these, for us, very distinct fields of study? There are in two interrelated factors that have to be taken into consideration, the first of which is astrology. The connection between medicine and astrology has a long history.

Greek legend says that Babylonian astrology was introduced into Greece by the Babylonian priest Berossus, who settled on the island of Kos in the third century BCE. Kos was the home of the Hippocratic School of medicine and astrology soon became an element in the Hippocratic Corpus. At the same time the same association between astrology and medicine came into Greek culture from Egypt in the form of the Greek-Egyptian god Hermes Trismegistos. Both the Egyptians and Babylonians had theories of lucky/unlucky, propitious/propitious days and these were integrated into the mix in the Greek lunar calendar. The Greeks developed the theory of the zodiac man, assigning the signs of the zodiac to the various part of the body. If a given part of the body was afflicted it would then be treated with the plants and minerals associated with its zodiac sign. The central role of astrology in medicine can be found in both the Hippocratic Corpus, in Airs, Waters, Placesit is stated that “astronomy is of the greatest assistance to medicine”and in Ptolemaeus’ Tetrabibloswe read, “The nature of the planets produce the forms and causes of the symptoms, since of the most important parts of man, Saturn is lord of the right ear, the spleen, the bladder, phlegm and the bones; Jupiter of touch, the lungs, the arteries and the seed; Mars of the left ear, the kidneys, the veins and the genitals; the sun of sight, the brain, the heart, the sinews and all on the right side; Venus of smell, the liver and muscles; Mercury of speech and thought, and the tongue, the bile and the buttocks; and the Moon of taste and of drinking, the mouth, the belly, the womb and all on the left side.” The connection between astrology was firmly established in Greek antiquity and was known as iatromathematica, health mathematics.

The theory of astrological medicine disappeared in Europe along with the rest of early science in the Early Medieval Period but was revived in the eighth century in the Islamic Empire when they took over the accumulated Greek Knowledge. The basic principles were fully accepted by the Islamic scholars and propagated down the centuries. When the translators moved into Spain and Sicily in the twelfth century they translated the Greek astrology and astrological medicine into Latin from Arabic along with rest of the Greek and Arabic sciences.

During the High Middle Ages, Christian scholars carried on an energetic debate about the legitimacy, or lack of it, of astrology. This debate centred on judicial astrology, this included natal astrology, mundane astrology, horary astrology, and electional astrology but excluded so called natural astrology, which included astrometeorology and astro-medicine both of which were regarded as scientific. To quote David Lindberg, “…no reputable physician of the later Middle Ages would have imagined that medicine could be successfully practiced without it.”


Woodcut of the Homo Signorum, or Zodiac Man, from a 1580 almanac. Source: Wikipedia Commons

Beginning in the fifteenth century during the humanist renaissance astrological medicine became the mainstream school medicine. It was believed that the cause, course and cure of an illness could be determined astrologically. In the humanist universities of Northern Italy and Poland dedicated chairs of mathematics were established, for the first time, which were actually chairs for astrology with the principle function of teaching astrology to medical students. Germany’s first dedicated chair for mathematics was founded at the University of Ingolstadt in about 1470 for the same reason.


Zodiac Man The Très Riches Heures du Duc de Berry c. 1412 Source: Wikimedia Commons

With the advent of moving type printing another role for mathematicians was producing astronomical/astrological calendars incorporating the phases of the moon, eclipses and other astronomical and astrological information needed by physicians to determine the correct days to administer blood lettings, purges and cuppings. These calendars were printed both as single sheet wall calendars and book form pocket calendars.


Renaissance Wall Calendar, 1544 Source: Ptak Science Books

These calendars were a major source of income for printer/publishers and for the mathematici who compiled them. Before he printed his legendary Bible, Johannes Guttenberg printed a wall calendar. Many civil authorities appointed an official calendar writer for their city or district; Johannes Schöner was official calendar writer for Nürnberg, Simon Marius for the court in Ansbach, Peter Apian for the city of Ingolstadt and Johannes Kepler for the city of Graz. Official calendar writers were still being employed in the eighteenth century. As I explained in an earlier post the pocket calendars led to the invention of the pocket diary.

Marius: [Alter und Newer Schreibcalender], [1602]  (Scan Nr. 150)

Simon Marius: Alter und Newer SchreibCalender auf das Jahr 1603 Title page Source: Deutsches Museum

With mainstream medicine based on astrology it was a short step for mathematicians to become physicians. Here we also meet the second factor. As a discipline, mathematics had a very low status in the Early Modern Period; in general mathematicians were regarded as craftsmen rather than academics. Those who worked in universities were at the very bottom of the academic hierarchy. At the medieval university it was only possible for graduates to advance to a doctorate in three disciplines, law, theology and medicine. It was not possible to do a doctorate in mathematics. With the dominance of iatromathematica, which depended on astrology, for which one in turn needed astronomy, for which one needed mathematics it was logical for mathematicians who wished to take a university doctorate, in order to gain a higher social status, to do so in medicine. The result of this is a fascinating period in European history from about 1400 to middle of the seventeenth century, where many of the leading mathematicians were also professional physicians. When astrology lost its status as a science this period came to an end.





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

A multi-functional book for a multi-functional instrument

Probably the most talked about astronomical instrument in recent years is the so-called Antikythera Mechanism, several corroded chunks of bronze gear work found in the sea of the coast of the Greek island of Antikythera at the end of the nineteenth century.


The Antikythera mechanism (Fragment A – front); visible is the largest gear in the mechanism, approximately 140 millimetres (5.5 in) in diameter Source: Wikimedia Commons

Historian of ancient astronomy, Alexander Jones, who was a member of one of the teams investigating and interpreting the mechanism, has now written a book about it, A Portable Cosmos.[1]


I say that he has written a book but in fact it is really several books in one. The first two chapters deal with the story of the original discovery and recovery of the mechanism. They also sketch the history of the succession of investigations and interpretations of the mechanism that have taken place between its discovery and the present. The longest section of the book deals with a detailed description of the external aspects of the mechanism, its dials, scales and pointers. The penultimate chapter is an examination of the physical aspects of the mechanism, its gears and gear shafts. The final chapter, an afterword, is titled The Meaning of the Mechanism. For me, the most fascinating element of the book is that Jones in his explanations of the functions of the dials and pointers delivers up a comprehensive introduction to the histories of astronomy, astrology and cosmology of ancient Babylon and Greece, in fact I would rate it as the best such introduction that I have ever read.

Despite his very obviously high level command of the material Jones does not baffle with science but writes in a light and very accessible style and I for one found the book highly readable. Of interest is the fact that because large parts of the mechanism are missing and what is there is highly damaged there is not a general agreement under the experts, who have worked on the mechanism, about how to interpret the function or purpose of numerous aspects of it. Jones doesn’t just express his own well-informed and well-reasoned explanations but draws his readers’ attention to alternative suggestions and interpretations, explaining why he prefers his own chosen one. Having said this archaeoastronomer Doris Vickers, who recommended the book to me suggested also consulting the official Greek Antikythera Mechanism Research Project website, which has more information and other viewpoints to those of Jones.

The book has a very useful glossary of technical terms, endnotes (regular readers already know my views on endnotes contra footnotes), a comprehensive bibliography so you can read up on those interpretations that deviate from Jones’ and a good index.

To quote a cliché, if you only read one book on the Antikythera Mechanism, then it really should be this one. It kept me occupied and entertained during my recent four days in hospital and proved to be an excellent companion for that period and I would whole heartedly recommended for happier circumstances as well.

[1] Alexander Jones, A Portable Cosmos: Revealing the Antikythera Mechanism, Scientific Wonder of the Ancient World, OUP, Oxford, 2007

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Filed under Book Reviews, History of Astrology, History of Astronomy, History of Technology, Uncategorized

Who cares about facts? – Make up your own, it’s much more fun!

Math Horizons is a magazine published by Taylor & Francis for the Mathematical Association of America aimed at undergraduates interested in mathematics: It publishes expository articles about “beautiful mathematics” as well as articles about the culture of mathematics covering mathematical people, institutions, humor, games, cartoons, and book reviews. (Description taken from Wikipedia, which attributes it to the Math Horizons instructions for authors January 3 2009). Apparently, however, authors are not expected to adhere to historical facts, they can, it seems, make up any old crap.

The latest edition of Math Horizons (Volume 25, Issue 3, February 2018) contains an article by a Stephen Luecking entitled Albrecht Dürer’s Celestial Geometry. As I am currently, for other reasons, refreshing my knowledge of Albrecht the mathematician I thought, oh that looks interesting I must read that. I wish I hadn’t.

Luecking’s sub-title seems innocent enough: Renaissance artist Albrecht Dürer designed a specialty compass for astronomical drawings, but when you read the article you discover that Luecking says an awful lot more and most of it is hogwash. What does he have to say?


Albrecht Dürer Self-Portrait 1500 Source: Wikimedia Commons

Albrecht Dürer (1471–1528), noted Renaissance printer and painter, twice left his native Germany for sojourns to Italy, once from 1494 to 1495 and again from 1505 to 1507. During those years his wide-ranging intellect absorbed the culture and thinking of noted artists and mathematicians. Perhaps the most important
 outcome of these journeys was his
introduction to scientific methods. 
His embrace of these methods
 went on to condition his thinking 
for the rest of his life. 

So far so good. However what Dürer absorbed on those journeys to Italy was not scientific methods but linear perspective, the mathematical method, developed in Northern Italy in the fifteenth century, to enable artists to represent three dimensional reality realistically in a two dimensional picture. Dürer played a significant role in distributing these mathematical techniques in Europe north of the Alps. His obsession with mathematics in art led to him developing the theory that the secret of beauty lay in mathematical proportion to which de devoted a large part of the rest of his life. He published the results of his endeavours in his four-volume book on human proportions, Vier Bücher von Menschlicher Proportion, in the year of his death, 1528.


Title page of Vier Bücher von menschlicher Proportion showing the monogram signature of artist Source: Wikimedia Commons

If Dürer wanted to learn scientific methods, by which, as we will see Luecking means astronomy, he could and probably did learn them at home in Nürnberg. Dürer was part of the humanist circle of Willibald Pirckheimer, he close friend and patron.


Engraving of Willibald Pirckheimer at 53 by Albrecht Dürer, 1524. We live by the spirit. The rest belongs to death. Source: Wikimedia Commons

Franconian houses are built around a courtyard; Dürer was born in the rear building of the Pirckheimer house on the market square in Nürnberg. Although his parents bought their own house a few years later Albrecht and Willibald remained close friends and possibly even lovers all of their lives. Pirckheimer was a big supporter of the mathematical sciences—astronomy, mathematics, cartography and astrology—and his circle included, amongst others, Johannes Stabius, Johannes Werner, Erhard Etzlaub, Georg Hartmann, Konrad Heinfogel and Johannes Schöner all of whom were either astronomers, mathematicians, cartographers, instrument makers or globe makers some of them all five and all of them friends of Dürer.

Next up Luecking tells us:

One notable
consequence was Dürer’s abandonment of astrological subject
matter—a big seller for a printer
and publisher such as himself—in favor of astronomy.


Albrecht Dürer Syphilis 1496 Syphilis was believed to have an astrological cause Source: Wikimedia Commons

Luecking offers no evidence or references for this claim, so I could offer none in saying that it is total rubbish, which it is. However I will give one example that shows that Albrecht Dürer was still interested in astrology in 1517. Lorenz Beheim (1457–1521) was a humanist, astrologer, physician and alchemist, who was a canon of the foundation of the St Stephan Church in Bamberg, he was a close friend of both Pirckheimer and Dürer and corresponded regularly with Pirckheimer. In a letter from 8 December 1517 he informed Pirckheimer that Johannes Schöner was coming to Nürnberg with printed celestial globes that could be used for astrology, which if his wished could be acquired by him and Albrecht Dürer. He would not have passed on the information if he thought that they wouldn’t be interested. Beheim also cast horoscopes for both Pirckheimer and Dürer.


Gores for Johannes Schöner’s Celestial Globe 1517  Source: Hans Gaab, Die Sterne Über Nürnberg: Albrecht Dürer und seine Himmelskarten von 1515, Nürnberger Astronomische Gesellschaft, Michael Imhof Verlag, 2015 p. 115


Next up Luecking starts, as he means to go on, with pure poppycock. All of the above Nürnberger mathematician, who all played significant roles in Dürer’s life, were of course practicing astrologers.

Astronomy was not to be a casual interest. Just before his second trip to Italy, Dürer published De scientia motus orbis, a cosmological treatise by the Persian Jewish astronomer Masha’Allah ibn Atharī (ca. 740–815 CE). Since Masha’Allah wrote the treatise for laymen and included ample illustrations, it was a good choice for introducing Europeans to Arabic astronomy.

The claim that Dürer published Masha’Allah’s De scientia motus orbis is so mind bogglingly wrong anybody with any knowledge of the subject would immediately stop reading the article, as it is obviously a complete waste of time and effort. The book was actually edited and published by Johannes Stabius and printed by Weissenburger in Nürnberg in 1504.

The woodcut illustrations came from the workshop of Albrecht Dürer, but probably not from Dürer himself. There were traditionally attributed to Hans Süß von Kulmbach (1480–1522), one of Dürer’s assistants, who went on to become a successful painter in his own right, but modern research has shown that Süß didn’t move to Nürnberg until 1505, a year after the book was published.


Hans Süß portrait  Source: Wikimedia Commons

Although Luecking wants Masha’Allah to be an astronomer he was in fact a very famous astrologer, who amongst other things cast the horoscope for the founding of Bagdad. De scientia motus orbis is indeed a book on Aristotelian cosmology and physics but it includes his theory that there are ten heavenly spheres not eight as claimed by Aristotle. His extra heavenly spheres play a significant role in his astrological theories. It is very common practice for astrologers, starting with Ptolemaeus, to publish their astronomy and astrology in separate books but they are seen as complimentary volumes. From their beginnings in ancient Babylon down to the middle of the seventeenth century astronomy and astrology were always seen as two sides of the same coin.


Title page De scientia motus orbis Although this woodcut is usually titled The Astronomer I personally think the figure looks more like an astrologer Source: Wikimedia Commons

In 1509 Dürer purchased the entire library of Regiomontanus (1436–1476 CE) from the estate of Nuremberg businessman Bernhard Walther. Regiomontanus was Europe’s leading astronomer,
a noted mathematician, and a designer of astronomical instruments. Walther had sponsored Regiomontanus’s residency in Nuremberg between 1471 and 1475. Part of Walther’s largesse was to provide a print shop from which Regiomontanus published the world’s first scientific texts ever printed.

Regiomontanus was of course first and foremost an astrologer and most of those first scientific texts that he published in Nürnberg were astrological texts. Walther did not sponsor Regiomontanus’ residency in Nürnberg but was his colleague and student in his endeavours in the city. An analysis of Walther’s astronomical observation activities in Nürnberg after Regiomontanus’ death show that he too was an astrologer rather than an astronomer. When Regiomontanus came to Nürnberg he brought a very large number of manuscripts with him, intending to edit and publish them. When he died these passed into Walther’s possession, who added new books and manuscripts to the collection. The story of what happened to this scientific treasure when Walther died in 1504 is long and very complicated. In fact Dürer bought not “the entire library” but a mere ten manuscripts not when he bought Walther’s house, the famous Albrecht Dürer House, in 1509 but first in 1522.

In 1515, Dürer and Austrian cartographer and mathematician Johannes Stabius produced the first map of the world portraying the earth as a sphere.


Johannes Statius portrait by Albrecht Dürer Source: Wikimedia Commons

The Stabius-Dürer world map was not “the first map of the world portraying the earth as a sphere”. The earliest know printed world map portraying the earth as a sphere is a woodcut in a Buchlein über die Kunst Corsmographia, (Booklet about the Art of Cosmographia) published in Nürnberg in about 1490. There are others that predate the Stabius-Dürer map most notably on the title page of Waldseemüller’s Die Welt Kugel (The Earth Sphere) published in Straßburg in 1509.

There are no surviving copies of the Stabius-Dürer world map from the sixteenth century so we don’t actually know what it was produced for. The woodblocks survived and were rediscovered in the 18th century.

It is however dedicated to both the Emperor Maximilian, Stabius’s employer who granted the printing licence, and Cardinal Matthäus Lang, so it might well have been commissioned by the latter. Lang commissioned the account of Magellan’s circumnavigation on which Schöner based his world map of that circumnavigation.

Afterward, Stabius proposed continuing their collaboration by publishing a star map—the first such map published in Europe. Their work relied heavily on data assembled by Regiomontanus, plus refinements from Walther.

It will probably not surprise you to discover that this was not “the first such map published in Europe. It’s the first printed one but there are earlier manuscript ones, two of which from 1435 in Vienna and 1503 in Nürnberg probably served as models for the Stabius–Dürer–Heinfogel one. Their work did not rely “heavily on data assembled by Regiomontanus, plus refinements from Walther” but was based on Ptolemaeus’ star catalogue from the Almagest. There is a historical problem in that there was not printed copy of that star catalogue available at the time so they probably work from one or more manuscripts and we don’t know which one(s). The star map contains the same dedications to Maximilian and Lang as the world map so one again might have been a commission from Lang, Stabius acting as the commissioning agent. Stabius and Lang studied together at the University of Ingolstadt.


Stabs-Dürer-Heinfogel Star Map Northern Hemisphere Source: Ian Ridpath’s Star Tales

For more details on the star maps go here

The star map required imprinting the three- dimensional dome of the heavens onto a two- dimensional surface without extreme distortions, a task that fell to Stabius. He used a stereographic projection. In this method, rays originate at the pole in the opposite hemisphere, pass through a given point in the hemisphere, and yield a point on a circular surface.

You will note that I have included the name of Konrad Heinfogel to the producers of the map and it was actually he, and not Stabius, who was responsible for the projection of the map and the location of the individual stars. In fact in this project Johannes Stabius as commissioning agent was project leader, Konrad Heinfogel was the astronomical expert and Albrecht Dürer was the graphic artist hired to draw the illustration. Does one really have to point out that in the sixteenth century star maps were as much, if not more, for astrologers than for astronomers.

Luecking now goes off on an excurse about the history of stereographic projection, which ends with the following paragraph.

As the son of a goldsmith, Dürer’s exposure to stereographic projection would have been by way of the many astrolabes being fabricated in Nuremburg, then Europe’s major center for instrument makers. As the 16th century moved on, the market grew for such scientific objects as astrology slipped into astronomy. Handcrafted brass instruments, however, were affordable only to the wealthy, whereas printed items like the Dürer-Stabius maps reached a wider market.

Nürnberg was indeed the major European centre for the manufacture of scientific instruments during Dürer’s lifetime but scientific instrument makers and goldsmiths are two distinct professional groups, so Luecking’s argument falls rather flat, although of course Dürer would have well acquainted with the astrolabes made by his mathematical friends. Astrolabes are of course both astrological and astronomical instruments and astrology did not slip into astronomy during the 16th century. In fact the 16th century is regarded by historians as the golden age of astrology.

There now follows another excurse on the epicycle-deferent model of planetary orbits as a lead up to the articles thrilling conclusion.

In his 1525 book Die Messerung (On Measurement), Dürer presents an instrument of his own design used to draw these and other more general curves. This compass for drawing circles upon circles consisted of four telescoping arms and calibrated dials. An arm attached to the first dial could rotate in a full circle, a second arm fixed to another dial mounted on the end of this first arm could rotate around the end of the first arm, and so on.


Dürer’s four arm compass


Underweysung der Messung mit dem Zirkel und Richtscheyt Title Page

The title of Dürer’ 1525 book is actually Underweysung der Messung mit dem Zirckel und Richtscheyt (Instructions for Measuring with Compass and Straightedge). It is a basic introduction to geometry and its applications, which Dürer wrote when he realised that his Vier Bücher von Menschlicher Proportion was too advanced for the artist apprentices that he thought should read it. The idea was first read and digest the Underweysung then read the Vier Bücher von Menschlicher Proportion.

Luecking tells us that:

As a trained metalsmith, Dürer possessed the expertise to craft this complex tool. Precision calibration and adjustable arms allowed its user to plot an endless number of curves by setting the length of each telescoping arm and determining the rate at which the arms turned. This, in effect, constituted manual programming by setting the parameters of each curve plotted.

As a teenager Dürer did indeed serve an apprenticeship under his father as a goldsmith, but immediately on completing that apprenticeship he undertook a second apprenticeship as a painter with Michael Wolgemut from 1486 to 1490 and dedicated his life to painting and fine art printing. Luecking has already correctly stated that Nürnberg was the major European centre for scientific instrument making and Dürer almost certainly got one of those instrument makers to produce his multi-armed compass. Luecking describes the use to which Dürer put this instrument in drawing complex geometrical curves. He then goes on to claim that Dürer might actually have constructed it to draw the looping planetary orbits produced by the epicycle-deferent model. There is absolutely no evidence for this in the Underweysung and Luecking’s speculation is simple pulled out of thin air.

To summarise for those at the back who haven’t been paying attention. Dürer did not absorb scientific methods in Italy. He did not abandon astrology for astronomy. He didn’t publish Masha’Allah’s De scientia motus orbis, Johannes Stabius did. Dürer only bought ten of Regiomontanus’ manuscripts and not his entire library. The Stabius-Dürer world map was not “the first map of the world portraying the earth as a sphere”. The Stabius–Dürer–Heinfogel star charts were the first star-charts printed in Europe but by no means the first ones published. Star charts are as much astrological, as they are astronomical. Astrology did not slip into astronomy in the 16th century, which was rather the golden age of astrology. There is absolutely no evidence that Dürer’s multi-arm compass, as illustrated in his geometry book the Underweysung, was ever conceived for drawing the looping orbits of epicycle-deferent planetary models, let alone used for this purpose.

It comes as no surprise that Stephen Luecking is not a historian of mathematics or art for that matter. He is the aged (83), retired chairman of the art department of DePaul University in Chicago.

Whenever I come across an article as terrible as this one published by a leading scientific publisher in a journal from a major mathematical organisation such as the MAA I cringe. I ask myself if the commissioning editor even bothered to read the article; it was certainly not put out to peer review, as any knowledgeable Dürer expert would have projected it in an elegant geometrical curve into his trashcan. Above all I worry about the innocent undergraduates who are subjected to this absolute crap.


















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

Conversations in a sixteenth century prison cell

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


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

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


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

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

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

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


Filed under Book Reviews, Early Scientific Publishing, History of Astrology, History of Astronomy, History of Physics, Renaissance Science, Uncategorized

Really! – Did the artist have a Tardis?

Those who read the occasional bursts of autobiographical information that appear here on the blog might be aware that I went to university at the tender age of eighteen as an archaeology student. I actually dropped out after one year but continued to work as a professional field archaeologist (that’s a digger to you mate) for several years. Given that I was already interested in the history of astronomy in those days and would eventually abandon archaeology for it, it would seem logical that I would be interested in archaeoastronomy, in particular because I studied under Richard Atkinson who together with Stuart Piggott carried out the first extensive, modern excavation of Stonehenge, the world’s most famous archaeoastronomical monument, in the 1950s. In fact my father also worked on that excavation. This assumption would be correct with reservations. There has been some excellent work in archaeoastronomy but unfortunately there has also been a large amount of highly dubious speculation on the topic.

In my opinion an example of the latter appeared in articles in The Guardian and on the Hyperallergic website a couple of days ago. The Guardian article was entitled, Two suns? No, it’s a supernova drawn 6,000 years ago, say scientists. This article tells us:

For decades, stone carvings unearthed in the Himalayan territory of Kashmir were thought to depict a hunting scene. But the presence of two celestial objects in the drawings has piqued the interest of a group of Indian astronomers.


Source: The Guardian

They have proposed another theory. According to a study published in the Indian Journal of History of Science, the Kashmir rock drawings may be the oldest depiction of a supernova, the final explosion of a dying star, ever discovered.

 “Our first argument was, there cannot be two suns,” Vahia said. “We thought it must have been an object that appeared and attracted the attention of the artists.”

 They settled on Supernova HB9, a star that exploded around 4,600BC.

Rewinding the map of the sky back that far revealed more clues.

Viewed from Kashmir, the supernova would have occurred somewhere near the Orion constellation. “Which is known as the scene of a hunter,” said Vahia.

“The supernova also went off just above the constellation of Taurus, the bull, which is also seen in the drawing,” Vahia added.


Source: The Guardian

So to summarise a group of astrophysicists decide that the rock drawing depicts a supernova from around 4,600 BCE that was visible in the sky in the area of the constellations Orion the hunter and Taurus the bull, which according to the researchers are also depicted in the drawing. It is by the latter claim that my bullshit detectors went off at full volume. I will explain.

The chosen supernova occurred in 4600 BCE, now I’m not an expert on prehistoric Indian asterisms, I don’t even know anybody who is, but I do know something about the Babylonian and ancient Greek ones. Taurus is indeed one of the oldest known asterisms but the earliest known mention of a bull asterism is in the Sumerian record, the Heaven’s Bull, in the third millennium BCE, that’s a couple of thousand years after the chosen supernova. Even worse it is not known whether the Sumerian asterism is the same one as the later Babylonian/Greek asterism Taurus. With Orion we have even more problems. The Sumerian asterism involving the stars of Orion was a sheep. For the ancient Egyptians the stars depicted their god Osiris. It was first the Greeks who created the asterism Orion although some mythologists see Orion as a representation of the Sumerian King Gilgamesh, who also fought a bull. This is of course highly speculative.

So we have astrophysicists identifying a rock drawing in India that is dated to the fifth millennium BCE with the constellations of Orion fighting Taurus, asterisms which don’t appear to have been identified till several thousand years later. Excuse me if I am somewhat sceptical about this identification. Just as a minor point I don’t think that the animal in the drawing actually looks like a bull, more like a stag in my opinion.






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