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 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

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.

 

 

 

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Ohm Sweet Ohm

This is the story of two brothers born into the working class in a small town in Germany in the late eighteenth century. Both of them were recognised as mathematically gifted whilst still teenagers and went on to study mathematics at university. The younger brother was diligent and studious and completed his doctorate in mathematics with a good grade. There followed a series of good teaching jobs before he obtained a lectureship at the then leading university of Berlin, ten years after graduating. In due course, there followed positions as associate and the full professor. As professor he contributed some small but important proofs to the maths cannon, graduated an impressive list of doctoral students and developed an interesting approach to maths textbooks. He became a respected and acknowledged member of the German mathematical community.

The elder brother’s life ran somewhat differently. He started at the local university but unlike his younger brother he was anything but studious preferring a life of dancing, ice -skating and playing billiards to learning mathematics. His father a hard working craftsman was disgusted by this behaviour and forced him to leave the university and take up a teaching post in Switzerland. On the advice of his mathematics professor he taught himself mathematics by reading the greats. He returned to his home university and obtained his doctorate in the same year as his brother. There then followed a series of dead end jobs first as a badly paid university lecturer with little prospect of promotion and then a series of deadbeat jobs as a schoolteacher. In the last of these he had access to a good physics laboratory and began a series of investigations in a relatively new area of physics. At the age of thirty-eight, something of a failure, he published the results of his investigations in a book, which initially failed to make any impact. At the age of forty-four he obtained an appointment as professor at a polytechnic near to his home town and things began to finally improve in his life. At the age of fifty-two his work received acknowledgement at the highest international levels and finally at the age of sixty-three he was appointed professor for physics at a leading university.

The younger brother whose career path had been so smooth, fairly rapidly disappeared from the history of mathematics after his death in 1872, remembered by only a handful of specialists, whereas the much plagued elder brother went on to lend the family name to one of the most frequently used unit of measure in the physical sciences; a name that can be found on multiple appliances in probably every household in the western world.

The two bothers of my story are Georg Simon Ohm (1789–1854), the discover of Ohm’s Law, and his younger brother, the mathematician, Martin Ohm, who was born on 6 May 1792 and the small German town where they were born is Erlangen where I (almost) live.

The Ohm House, Fahrstraße 11, Erlangen
Source: Wikimedia Commons

Georg Simon and Martin were the sons of the locksmith Johann Wolfgang Ohm and his wife Maria Elizabeth Beck, who died when Georg Simon was only ten. Not only did the father bring up his three surviving, of seven, children alone after the death of their mother but he also educated his two sons himself. The son of a locksmith he had enjoyed little formal education but had taught himself philosophy and mathematics, which he now imparted to his sons with great success. As Georg Simon was fifteen he and Martin were examined by the local professor of mathematics, Karl Christian von Langsdorf, who, as already described above, found both boys to be highly gifted and spoke of an Erlanger Bernoulli family.

Plaque on the Ohm House

The plaque reads: The locksmith Johann Wolfgang Ohm (1753–1823) brought up and taught in this house as a true master his later famous sons

Georg Simon Ohm (1789–1854) the great physicist and Martin Ohm (1792–1872) the mathematician

I’ve already outlined the lives of the two Ohm brothers so I’m not going to repeat myself but I will fill in some detail.

Martin Ohm
Source: Wikimedia Commons

As above I’ll start with Martin, the mathematician. He made no great discoveries as such and in the world of mathematics his main claim to fame is probably his list of doctoral students several of whom became much more famous than their professor. It was as a teacher that Martin Ohm made his mark, writing a nine volume work that attempted a systematic introduction to the whole of elementary mathematics his, Versuch eines vollkommenen, consequenten Systems der Mathematik (1822–1852) (Attempt at a complete consequent system of mathematics); a book that predates the very similar, but far better known, attempt by Bourbaki by one hundred years and which deserves far more attention than it gets. Martin Ohm also wrote several other elementary textbooks for his students. In his time in Berlin Martin Ohm also taught mathematics for many years at both the School for Architecture and the Artillery Academy.

I first stumbled across Martin Ohm whilst researching nineteenth-century algebraic logics. When it was first published George Boole’s Laws of Thought (1864) received very little attention from the mathematical community. With the exception of a small handful of relatively unknown mathematicians who wrote brief papers on it, it went largely unnoticed. One of that handful was Martin Ohm who wrote two papers in German (the first works in German on Boole’s logic). Thus introducing Boole’s ground-breaking work to the German mathematical public. Boole had written and published other mathematical work in German so he was already known in Germany. Later Ernst Schröder would go on to become the biggest proponent of Boolean logic with his three volume Vorlesungen über die Algebra der Logik (1890-1905). It is perhaps worth noting that Boole like the Ohm brothers was the son of a self-educated tradesman who gave his son his first education.

Martin Ohm has one further claim to notoriety; he is thought to have been the first to use the term “golden section” (goldener Schnitt in German) thus opening the door for hundreds of aesthetic loonies who claim to find evidence of this wonderful ration all over the place.  

Georg Simon Ohm

We now move on to the man in whose shadow Martin Ohm will always stand, his elder brother Georg Simon.

House in Erlangen just around the corner from their birth house, where both Georg Simon and Martin worked as poorly paid lecturers for physics
Photo: Thony Christie

Plaque on house
Photo: Thony Christie

The Plaque reads: In this house the physicist Georg Simon Ohm (1789–1854) taught physics in the years 1811 to 1812 and the mathematician Martin Ohm (1792–1872) in the years 1812 to 1817

The school where Georg Simon began the research work into the physics of electricity was the Jesuit Gymnasium in Cologne, which even granted him a sabbatical in 1826 to intensify his researches. He published those researches as Die galvanische Kette, mathematisch bearbeitet (The Galvanic Circuit Investigated Mathematically) in 1827. It was the Royal Society who started his climb out of obscurity awarding him the Copley Medal, its highest award, in 1842 and appointing him a foreign member in the same year. Membership of other international scientific societies, such as Turin followed. Georg Simon’s first professorial post was at the Königlich Polytechnische Schule (Royal Polytechnic) in Nürnberg in 1833. He became the director of the Polytechnic in 1839 and today the school is a technical university, which bears the name Georg Simon Ohm. Georg Simon ended his career as professor of physics at the University of Munich.

The town of Erlangen is proud of Georg Simon and we have an Ohm Place, with an unfortunately rather derelict fountain, the subject of a long political debate concerning the cost of renovation and one of the town’s high schools is named the Ohm Gymnasium. The city of Munich also has a collection of plaques and statues honouring him. Ohm Straße in Berlin, however, is named after his brother Martin.

Statue of Georg Simon Ohm at the Technical University in Munich
Source: Wikimedia Commons

Any fans of the history of science with a sweet tooth should note that if they come to Erlangen one half of the Ohm House is now a sweet shop specialising in Gummibärs.

Henry and Isaac invade Oxford.

There is subject well known to all blog owners that I have never talked about, spam; I get two different varieties here at The Renaissance Mathematicus. The first is spam comments, which turn up in a never ending stream but which mostly end up in Word Press’ apparently efficient spam filter. Very occasionally one or two get through and I have to weed these out from underneath whichever post they have chosen to enrich with their presence. Otherwise the only real problem I have is remembering to regularly check the spam filter for non-spam and send the rest of its contents off to rot in cyber-hell until that day dawns when the Internet is turned off forever. Maybe I shouldn’t say this but I think that the spammers might be more successful if they didn’t have email addresses such as purchase@cheapviagra.com, just a thought.

The second type of spam I receive as a blogger is in the form of emails. These are emails from people trying to get me to either let them advertise or publish something on my blog or link something to it. Again these people might be more successful if the things that they were offering and which they are so convinced that I will find interesting were actually related in anyway to the content of my blog, they never are. As a blogger I get another type of email, ones that are invariably addressed to Professor or Doctor or even in German style to Professor Doctor. I wouldn’t mind them awarding me illusionary titles that I don’t possess, and almost certainly never will, if only they would show a little imagination in addressing me, after all professors and doctors are two a penny. Were I to get an email addressed to Our Glorious, Benevolent, Gracious, Omniscient and Wise Leader in this Age of Darkness I might just be tempted to respond, but they never do and so I don’t. The emails addressing me with imaginary academic titles usually invite me to contribute articles to their prestigious academic journal that well-known rival to Nature and Science, The East Krakatoa Journal for Island Approaches to the Philosophy of Renaissance Mathematics. Dear editors, to paraphrase Groucho Marx, “I would never submit an article to a journal that would publish anything written by me”. All of these spam emails get dispatched forthwith to cyber-hell unread, unanswered and with all links left strictly unlinked. I can spread my own viruses, thank you.

Today I received an unsolicited email asking me to advertise something, help with publicizing was the actual phrase used, and I’m actually going to do so, a first as far as I can remember here at RM. The email came from David Norbrook, Merton Professor of English Literature at the University of Oxford and he asked me very nicely to spread the word about his up coming conference Scholarship, Science, and Religion in the Age of Isaac Casaubon (1559-1614) and Henry Savile (1549-1622) at the T. S. Eliot Theatre, Merton College, Tuesday 1^st – Thursday 3^rd July 20124

For those not in the know Henry and Isaac are two of the Renaissance scholars who make you turn green with envy. Each of them was brainy enough to win a round of University Challenge on their own without teammates and each of them mastered enough academic disciplines to fill a small encyclopaedia on his own.

Isaac Casaubon was a French Huguenot classical scholar, philologist, historian and theologian born to refugee parents in Geneva. Home educated until he entered the University of Geneva aged seventeen were he studied Greek and was recommended for the chair in Greek only four years later. He was a consummate classical scholar and philologist whose main occupation was the translation, editing and publication of classical Greek text. He worked most of his life in Switzerland and France torn and troubled by the religious conflicts of the age. Regarded at his intellectual peak as one of the most learned men in the whole of Europe the Catholics, Lutherans, Calvinists and Anglicans all competed with offers of jobs, money and other inducements to win him as a propagandist for their cause. The situation has strong similarities to the attempts today of leading European football clubs to induce a star striker to sign for them and not one of their rivals. During the religious upheaval in Europe in the Early Modern Period a star polemicist was regarded as a good catch by the rival religious communities. In the end the political pressure in France caused him to move to England in 1610 were he died four years later. As a historian of science my main interest in Casaubon is his De rebus sacris et ecclesiasticis exercitationes XVI published in 1614 in which he proved by philological analysis that the Corpus Hermeticus, one of the most influential collection of texts in the Renaissance, was not as ancient as claimed but was in fact a product of late antiquity. This was a key moment in the evolution of the discipline of history, applying scientific, philological analysis to texts to determine their age.

I can’t leave even this brief account of Isaac Casaubon without mentioning his son Méric, who was the man responsible for ruining John Dee’s reputation. Despite all of the misfortunes that befell him in later life, in the early seventeenth-century Dee still enjoyed a good reputation in England for his work in the mathematical sciences. Around 1650 more and more people were starting to question the existence of ghosts, witches and other aspects of the occult. Deeply religious people, of whom Méric was one, were worried that this was the thin edge of the wedge that would inevitably lead to atheism. To counter this tendency Méric published John Dee’s Angel Diaries, his account of his conversations with angels, which up till then had remained largely unknown. Méric’s intention was that Dee’s accounts should act as a proof, from a reputable scholar, that the world of spirits is real and not to be questioned. Méric’s attempt backfired ruining Dee’s reputation causing people to forget the mathematicus and only remember the notorious Renaissance magus that he now became for the next four hundred years down to the present day.

Henry Savile was educated at Oxford and, self-taught, began to lecture there on astronomy at the age of 21 in 1570. He not only lectured on Ptolemaeus but also on the works of Regiomontanus and Copernicus, real cutting edge at the time. In 1578 he went on a grand tour of Europe meeting with and learning from the leading continental mathematicians; a necessary move for anyone interested in the mathematical sciences in England at that time as England was an intellectual backwater in terms of mathematics. On his return to England, in 1582, Savile was appointed Greek tutor to Queen Elizabeth. Later he became both Warden of Merton College Oxford and Provost of Eaton. Like Casaubon, with whom he was acquainted, Savile was a classical scholar and philologist but it is for his contributions to mathematics that he is best remembered. Appalled by the primitive level of mathematics teaching in England in comparison to the continent he established the first two university chairs for the mathematical sciences in England in 1619, the Savilian Chairs for Geometry and Astronomy at Oxford. In the seventeenth-century many of England’s leading mathematicians occupied one or other of these chairs including such figures as Henry Briggs, John Wallis and Edmund Halley, whose adventures sailing around the Atlantic you can follow on Twitter (@HalleysLog).

Both Casaubon and Savile are fascinating figures, who lived in and contributed to a period of great intellectual change in European history and I’m sure the Merton College conference on these two intellectual giants will be a stimulating and informative experience. If I had the time and the money, and I don’t have either, I personally would love to take part and I can only recommend that those who do have the time and the money to do so.

Unfortunately, I only got the information on the conference today and if you want to take advantage of the early booker rebate you only have until tomorrow to do so!

 

 

Doctor, doctor, doctorate!

In my first post on Alchemical confusion I think I was a little bit too hasty in my dismissal of Campbell’s comment about medicine, doctors and doctorates. To save you having to go look I quoted the following passage with my own embellishments:

Even more bizarre in the following paragraph:

You know which group was also ridiculed, even until the mid-1800s? Medical doctors. Many people thought Harvard was out of its mind creating a medical school in 1782, but they took it seriously and within a few generations medicine had put quacks on the fringes and adopted evidence-based practices. Before then, legitimate doctors were Ph.D.s and Medical Doctors were that other thing that didn’t count. Today, though, if you say doctor people assume you are an M.D.

The whole of this is so mind bogglingly wrong that it is not even worth criticising except to say that if it were written on paper I would flush it down the toilet.

Now I haven’t change my mind on the quality of this little turd of a paragraph but it occurred to me that some of my readers might not understand why I’m so negative about it, not possessing the requisite knowledge of the history of universities and the award of doctorates. However before I explain that I will address the charge of medical doctors being ridiculed.

Now it is true that medical practitioners have been ridiculed throughout history but then again so have scientists and philosophers as well as many others. One of the legends about Thales, supposedly both the first philosopher and the first scientist in ancient Greece, tells how one day he was so obsessed with observing the stars that he didn’t look where he was going and fell down a well. This story was not told to illustrate Thales’ dedication to science. There are many tales by prominent thinkers in ancient Greece and Rome ridiculing physicians but many of those physicians were also highly regarded, had significant social influence and were often wealthy. The same is true of physicians in the High Middle Ages, the Early Modern Period etc. etc. Even today ridiculing medical practitioners is a popular sport amongst journalists and other social commentators. However such ridicule has never had a negative impact on the academic status of medical doctors, which brings us to the real reason why the offending paragraph is total crap.

Universities were a European creation beginning in the twelfth century CE. There had been earlier institutes of higher education in other cultures but what I’m discussing here applies specifically to the European universities as they emerged in the High Middle Ages.  When fully developed the medieval university had four faculties the first of which was the lower or philosophical faculty. Students began their studies in the philosophical faculty whose curriculum was officially based on the seven liberal arts, the trivium – logic, rhetoric and grammar – and the quadrivium – arithmetic, geometry, music (the mathematical theory of proportions) and astronomy. In reality the course consisted almost entirely of the trivium taught on the basis of the works of Aristotle and his commentators. Students who completed this course were awarded a BA degree. Some would now leave the university, those that stayed continued in the philosophy faculty doing an advance course of study much the same as the previous one, which closed with an MA degree. These students were now qualified to teach the BA courses in the philosophy faculty. Those that stayed at the university to continue their studies would now teach the undergraduates whilst pursuing a course of study at one of the three higher faculties, these being the theology faculty, the law faculty and the medicine faculty. Here the advanced student would follow a long course of study passing through BA and MA degrees to finally graduate with a doctorate. The doctoral degrees were respectively doctor of divinity or theology, doctor of law (civil or canonical), and doctor of medicine. The medieval university awarded no other doctoral degrees and certainly no doctorates in philosophy.

This model remained basically unchanged, although with some changes in course content, up to the beginning of the eighteenth century. Newton’s Cambridge, for example, was in essence a medieval university. Being created by Europeans the American universities, of which Harvard was the first, followed this European model. Like their medieval European predecessors they tended to start as a theological seminaries intended to train people for the church.

At the beginning of the nineteenth century the German universities undertook a radical reform based on the pedagogical concepts of Wilhelm von Humboldt, as part of this reform they introduced a new doctorate in philosophy the Ph.D.  The first three American Ph.D.s were awarded by Yale University on July 25^th 1861.

I can’t comment on Campbell’s claims of opposition to the Harvard School of Medicine and an intensive Internet search has failed to turn up anything on a controversy or similar before, on, or after the founding of this august institution.  There were almost certainly some who were opposed to the project, as there are always some opposed to any major new project that any university undertakes; the money would be better spent on… etc. etc. Whatever the case maybe I’m fairly certain that there was no significant opposition to the proposed school of medicine based on a poor opinion of medical practitioners.