Category Archives: History of Mathematics

You shouldn’t believe everything you read

One of the things that I have been reading recently is a very interesting paper by John N. Crossley, the Anglo-Australian logician and historian of mathematics, about the reception and adoption of the Hindu-Arabic numbers in medieval Europe.[1]Here I came across this wonderful footnote:[2]

[…]

It is interesting to note that Richard Lemay in his entry “Arabic Numerals,” in Joseph Reese Strayer, ed., Dictionary of the Middle Ages(New York, 1982–89) 1:382–98, at 398 reports that in the University of Padua in the mid-fifteenth century, prices of books should be marked “non per cifras sed per literas claras.” He gives a reference to George Gibson Neill Wright, The Writing of Arabic Numerals(London, 1952), 126. Neill Wright in turn gives a reference to a footnote of Susan Cunnigton, The Story of Arithmetic: A Short History of Its Origin and Development(London, 1904), 42, n. 2. She refers to Rouse Ball’s Short History of Mathematics, in fact this work is: Walter William Rouse Ball, A Short Account of the History of Mathematics, 3rded. (London, 1901), and there one finds on p. 192: “…in 1348 the authorities of the university of Padua directed that a list should be kept of books for sale with the prices marked ‘non per cifras sed per literas claras’ [not by cyphers but by clear letters].” I am yet to find an exact reference for this prohibition. (There is none in Rouse Ball.) Chrisomalis Numerical Notations, p. 124, cites J. Lennart Berggren, “Medieval Arithmetic: Arabic Texts and European Motivations,” in Word, Image, Number: Communication in the Middle Ages, ed. John J. Contreni and Santa Casciani (Florence, 2002), 351–65, at 361, who does not give a reference.

Here we have Crossley the historian following a trail of quotes, references and footnotes; his hunt doesn’t so much terminate in a dead-end as fizzle out in the void, leaving the reader unsure whether the university of Padua really did insist on its book prices being written in Roman numerals rather than Hindu-Arabic ones or not. What we have here is a succession of authors writing up something from a secondary, tertiary, quaternary source with out bothering to check if the claim it makes is actually true or correct by looking for and going back to the original source, which in this case would have been difficult as the trail peters out by Rouse Ball, who doesn’t give a source at all.

This habit of writing up without checking original sources is unfortunately not confined to this wonderful example investigated by John Crossley but is seemingly a widespread bad habit under historians and others who write historical texts.

I have often commented that I served my apprenticeship as a historian of science in a DFG[3]financed research project on Case Studies into a Social History of Formal Logic under the direction of Professor Christian Thiel. Christian Thiel was inspired to launch this research project by a similar story to the one described by Crossley above.

Christian Thiel’s doctoral thesis was Sinn und Bedeutung in der Logik Gottlob Freges(Sense and Reference in Gottlob Frege’s Logic); a work that lifted him into the elite circle of Frege experts and led him to devote his academic life largely to the study of logic and its history. One of those who corresponded with Frege, and thus attracted Thiel interest, was the German meta-logician Leopold Löwenheim, known to students of logic and meta-logic through the Löwenheim-Skolem theorem or paradox. (Don’t ask!) Being a thorough German scholar, one might even say being pedantic, Thiel wished to know Löwenheim’s dates of birth and death. His date of birth was no problem but his date of death turned out to be less simple. In an encyclopaedia article Thiel came across a reference to c.1940; the assumption being that Löwenheim, being a quarter Jewish and as a result having been dismissed from his position as a school teacher in 1933, had somehow perished during the holocaust. In another encyclopaedia article obviously copied from the first the ‘circa 1940’ had become a ‘died 1940’.

Thiel, being the man he is, was not satisfied with this uncertainty and invested a lot of effort in trying to get more precise details of the cause and date of Löwenheim’s death. The Red Cross information service set up after the Second World War in Germany to help trace people who had died or gone missing during the war proved to be a dead end with no information on Löwenheim. Thiel, however, kept on digging and was very surprised when he finally discovered that Löwenheim had not perished in the holocaust after all but had survived the war and had even gone back to teaching in Berlin in the 1950s, where he died 5. May 1957 almost eighty years old. Thiel then did the same as Crossley, tracing back who had written up from whom and was able to show that Löwenheim’s death had already been assumed to have fallen during WWII, as he was still alive and kicking in Berlin in the early 1950s!

This episode convinced Thiel to set up his research project Case Studies into a Social History of Formal Logic in order, in the first instance to provide solid, verified biographical information on all of the logicians listed in Church’s bibliography of logic volume of the Journal of Symbolic Logic, which we then proceeded to do; a lot of very hard work in the pre-Internet age. Our project, however, was not confined to this biographical work, we also undertook other research into the history of formal logic.

As I said above this habit of writing ‘facts’ up from non-primary sources is unfortunately very widespread in #histSTM, particularly in popular books, which of course sell much better and are much more widely read than academic volumes, although academics are themselves not immune to this bad habit. This is, of course, the primary reason for the continued propagation of the myths of science that notoriously bring out the HISTSCI_HULK in yours truly. For example I’ve lost count of the number of times I’ve read that Galileo’s telescopic discoveries proved the truth of Copernicus’ heliocentric hypothesis. People are basically to lazy to do the legwork and check their claims and facts and are much too prepared to follow the maxim: if X said it and it’s in print, then it must be true!

[1]John N. Crossley, Old-fashioned versus newfangled: Reading and writing numbers, 1200–1500, Studies in medieval and Renaissance History, Vol. 10, 2013, pp.79–109

[2]Crossley p. 92 n. 42

[3]DFG = Deutsche Forschungsgemeinschaft = German Research Foundation

 

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Hypatia – What do we really know?

The fourth century Alexandrian mathematician and philosopher Hypatia has become a feminist icon. She is probably the second most well known woman in #histSTM after Marie Curie. Unfortunately, down the centuries she has been presented more as a legend or a myth intended to fulfil the teller’s purposes rather than a real human being. As Alan Cameron puts it in his excellent essay, Hypatia: Life, Death, and Works:[1]

A pagan in the Christian city of Alexandria, she is one of those figures whose tragic death inspired a legend which could take almost any form because so few facts are known. As a pagan martyr, she has always been a stick to beat Christians with, a symbol in the continuing struggle between science and revealed religion. The memorable account in Gibbon begins wickedly “On a fatal day in the holy season of lent.” As a woman she can be seen as a feminist as well as a pagan martyr. Her name has been a feminist symbol down the centuries more recently a potent name in lesbian and gay circles. As an Egyptian, she has also been claimed as a black woman martyr. There is an asteroid named after her, a crater on the moon, and a journal of feminist studies. As early as 1886, the women of Wichita Kansas, familiar from the movies of our youth as a lawless western cattle town, formed a literary society called the Hypatia Club. Lake Hypatia in Alabama is a retreat for freethinkers and atheists. Rather less in tune with her scholarly activity, there is Hypatia Capital, a merchant bank whose strategy focuses on the top female executives in the Fortune 1000.

A few minutes’ googling will produce countless eulogies of Hypatia as a uniquely gifted philosopher, mathematician and scientist, the second female scientist after Marie Curie, the only woman in antiquity appointed to a university chair, a theorist who anticipated Copernicus with the heliocentric hypothesis. The 2009 movie Agoragoes even further in this direction. A millennium before Kepler, Hypatia discovered that earth and its sister planets not only go round the sun but do so in ellipses, not circles. She remained unmarried, and could therefore be seen as a model of pagan virginity. Alternatively, since the monks are said to have killed her because of her influence on the prefect of Egypt, she could be seen as a slut. It is fascinating to observe how down the centuries she served as a lay figure for the prejudices of successive generations.

So what do we know about the real Hypatia? The answer is almost nothing. We know that she was the daughter of Theon (c.335–c.405) an Alexandrian mathematician and philosopher, most well known for his edition of The Elements of Euclid. We don’t know her birth date with estimates ranging from 350 to 370 CE. Absolutely nothing is known about her mother to whom no references whatsoever exist. It is assumed that she was educated by her father but once again, whilst highly plausible, no real evidence exists for this assumption. If we take a brief looked at the available sources for her biography the reason for all of this uncertainty becomes very clear.

The only source we have from somebody who actually knew Hypatia is Synesius of Cyrene (c.373–probably 413), who was one of her Christian students around 393 CE. In 410 CE he was appointed Bishop of Ptolemais. There was an edition of his letters, which contains seven letters to Hypatia and some to others that mention her. Unfortunately his letters tell us nothing about he death as he predeceased her. His last letter to her was written from his deathbed in 413 CE. Two of his letters, however, request her assistance for acquaintances in civil matters, which indicates that she exercised influence with the civil authorities.

Our second major source is Socrates of Constantinople (c.380–died after 439) a Christian church historian, who was a contemporary but who did not know her personally. He mention her and her death in his Historia Ecclesiastica:

There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner which she had acquired in consequence of the cultivation of her mind, she not infrequently appeared in public in the presence of the magistrates. Neither did she feel abashed in going to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more.

The third principle source is Damascius (c.458–after 538) a pagan philosopher, who studied in Alexandria but then moved to Athens where he succeeded his teacher Isidore of Alexandria (c.450–c.520) as head of the School of Athens. He mentions Hypatia in his Life of Isidore, which has in fact been lost but which survives as a fragment that has been reconstructed.

We also have the somewhat bizarre account of the Egyptian Coptic Bishop John of Nikiû (fl. 680–690):

And in those days there appeared in Alexandria a female philosopher, a pagan named Hypatia, and she was devoted at all times to magic, astrolabes and instruments of music, and she beguiled many people through her Satanic wiles. And the governor of the city honoured her exceedingly; for she had beguiled him through her magic. And he ceased attending church as had been his custom… And he not only did this, but he drew many believers to her, and he himself received the unbelievers at his house.

It is often claimed that she was head of The Neo-Platonic School of philosophy in Alexandria. This is simply false. There was no The Neo-Platonic School in Alexandria. She inherited the leadership of her father’s school, one of the prominent schools of mathematics and philosophy in Alexandria. She however taught a form of Neo-Platonic philosophy based mainly on Plotonius, whereas the predominant Neo-Platonic philosophy in Alexandria at the time was that of Iamblichus.

If we turn to her work we immediately have problems. There are no known texts that can be directly attributed to her. The Suda, a tenth-century Byzantine encyclopaedia of the ancient Mediterranean world list three mathematical works for her, which it states have all been lost. The Suda credits her with commentaries on the Conic Sections of the third-century BCE Apollonius of Perga, the “Astronomical Table” and the Arithemica of the second- and third-century CE Diophantus of Alexandria.

Alan Cameron, however, argues convincingly that she in fact edited the surviving text of Ptolemaeus’ Handy Tables, (the second item on the Suda list) normally attributed to her father Theon as well as a large part of the text of the Almagest her father used for his commentary.  Only six of the thirteen books of Apollonius’ Conic Sections exist in Greek; historians argue that the additional four books that exist in Arabic are from Hypatia, a plausible assumption.

All of this means that she produced no original mathematics but like her father only edited texts and wrote commentaries. In the history of mathematics Theon is general dismissed as a minor figure, who is only important for preserving texts by major figures. If one is honest one has to pass the same judgement on his daughter.

Although the sources acknowledge Hypatia as an important and respected teacher of moral philosophy there are no known philosophical texts that can be attributed to her and no sources that mention any texts from her that might have been lost.

Of course the most well known episode concerning Hypatia is her brutal murder during Lent in 414 CE. There are various accounts of this event and the further from her death they are the more exaggerated and gruesome they become. A rational analysis of the reports allows the following plausible reconstruction of what took place.

An aggressive mob descended on Hypatia’s residence probably with the intention of intimidating rather than harming her. Unfortunately, they met her on the open street and things got out of hand. She was hauled from her carriage and dragged through to the streets to the Caesareum church on the Alexandrian waterfront. Here she was stripped and her body torn apart using roof tiles. Her remains were then taken to a place called Cinaron and burnt.

Viewed from a modern standpoint this bizarre sequence requires some historical comments. Apparently raging mobs and pitched battles between opposing mobs were a common feature on the streets of fourth-century Alexandria. Her murder also followed an established script for the symbolic purification of the city, which dates back to the third-century. There was even a case of a pagan statue of Separis being subjected to the same fate. There is actually academic literature on the use of street tiles in street warfare[2]. What is more puzzling is the motive for the attack.

The exact composition of the mob is not known beyond the fact that it was Christian. There is of course the possibility that she was attacked simply because she was a woman. However, she was not the only woman philosopher in Alexandria and she enjoyed a good reputation as a virtuous woman. It is also possible that she was attacked because she was a pagan. Once again there are some contradictory facts to this thesis. All of her known students were Christians and she had enjoyed good relations with Theophilus the Patriarch of Alexandria (384–412), who was responsible for establishing the Christian dominance in Alexandria. Theophilus was a mentor of Synesius. Also the Neoplatonic philosophy that she taught was not in conflict with the current Christian doctrine, as opposed to the Iamblichan Neoplatonism. The most probably motive was Hypatia’s perceived influence on Orestes (fl. 415) the Roman Prefect of Egypt who was involved in a major conflict with Cyril of Alexandria (c.376–444), Theophilis’ nephew and successor as Patriarch of Alexandria. This would make Hypatia collateral damage in modern American military jargon. In the end it was probably a combination of all three factors that led to Hypatia’s gruesome demise.

Hypatia’s murder has been exploited over the centuries by those wishing to bash the Catholic Church but also by those wishing to defend Cyril, who characterise her as an evil woman. Hypatia was an interesting fourth-century philosopher and mathematician, who deserves to acknowledged and remembered for herself and not for the images projected on her and her fate down the centuries.

[1]Alan Cameron, Hypatia: Life, Death, and Works, in Wandering Poets and Other Essays on Late Greek Literature and Philosophy, OUP, 2016 pp. 185–203 Quote pp. 185–186

[2]You can read all of this in much more detail in Edward J. Watts’ biography of Hypatia, Hypatia: The Life and Legend of an Ancient Philosopher, OUP, 2017, which I recommend with some reservations.

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Christmas Trilogy 2018 Part 3: Johannes’ battle with Mars

It would be entirely plausible to claim the Johannes Kepler’s Astronomia Nova was more important to the eventual acceptance of a heliocentric view of the cosmos than either Copernicus’ De revolutionibus or Galileo’s Sidereus Nuncius. As with most things in Kepler’s life the story of the genesis of the Astronomia Nova ΑΙΤΙΟΛΟΓΗΤΟΣ seu physica coelestis, tradita commentariis de motibus stellae Martis ex observationibus G.V. Tychonis Brahe, to give it its full title, is anything but simple.

Astronomia_Nova

Astronomia Nova title page Source: Wikimedia Commons

From his studies at Tübingen University Kepler was sent, in 1594, by the Lutheran Church to Graz as mathematics teacher in the local Lutheran school and as district mathematicus, responsible for surveying, cartography and above all yearly astrological prognostica. His situation in Graz, a predominantly Catholic area, was anything but easy and as the Counter Reformation gained pace the Lutheran school was closed and the Protestants were given the choice of converting to Catholicism or leaving the area. Largely because of the success of his prognostica Kepler was granted an exemption. However, by 1600 things got very tight even for him and he was desperately seeking a way out. All of his efforts to obtain employment failed, including, somewhat surprisingly, an appeal to his teacher Michael Mästlin in Tübingen.

During his time in Graz he had published his first book, Mysterium Cosmographicum (The Cosmographic Mystery), in which he attempted to prove, the for us today bizarre hypothesis, that in a heliocentric cosmos there were and could only be six planets because their obits were separated by the ratios of the volumes of the five regular Platonic solids. He realized that regular polygons bound one inscribed and one circumscribed circle at definite ratios, which, he reasoned, might be the geometrical basis of the universe. However, when he actually did the maths he realised that although he had a good approximation, it wasn’t good enough; he blamed the failing accuracy on the poor quality of the data he had available. He needed to obtain better data.

Kepler-solar-system-1

Kepler’s Platonic solid model of the solar system, from Mysterium Cosmographicum (1596) Source: Wikimedia Commons

Kepler was well aware of the fact that, for the last thirty years or so, Tycho Brahe had been accumulating vast quantities of new, comparatively accurate data, so he decided to go visit the Danish aristocratic astronomer in Prague, maybe there was also a chance of employment. There was, however, a small problem with this plan. As a young, first time author, he had sent complimentary copies of his publication to all of the leading European astronomers. This had led to the first correspondence with Galileo, who had received his copy rather more by accident than by design but also to a very tricky situation with Tycho. Kepler had sent a copy to Nicolaus Reimers Baer (1551–1600), also called Ursus, at the time Tycho’s predecessor as Imperial Mathematicus to the Emperor Rudolph II; this was accompanied by the usual highly flattering Renaissance letter of introduction. Ursus was engaged in a very bitter priority dispute with Tycho about the so-called Tychonic system, Tycho had accused him of having stolen it during a visit to Hven, and Ursus printed Kepler’s flattering letter in a highly insulting answer to Tycho’s accusations. Kepler was very definitely not Tycho’s favourite astronomer, as a result. Despite all of this Tycho, also in desperate need of new assistants since moving to Prague, actually invited the young Kepler to come and visit him. By a strange twist of fate the letter of invitation arrived after Kepler had already left Graz for Prague.

Kepler duly arrived in Prague and one of the most fateful meetings in the history of astronomy took place. That first meeting was a monumental disaster. Kepler took umbrage and departed to sulk in a Prague hotel, convinced that his journey to Prague had been in vain. However, thanks to the diplomatic efforts of the Bohemian physician Johannes Jessenius (1566–1621) the two hot heads settled their differences and Tycho offered Kepler at least temporary employment. Having no better offers Kepler agreed, returned to Graz, packed up his home and together with his wife and children returned to Prague.

Unfortunately, there was no way that Tycho was going to trust a comparative stranger with the accumulated treasures of thirty years of observations and Kepler had to be satisfied with the tasks that Tycho gave him. First of all, maybe as a form of punishment, Tycho set him to work writing a vindication of Tycho’s claims against Ursus. Although Kepler did not produce the stinging condemnation of Ursus that Tycho wanted, he did produce a fascinating philosophical analysis of the role of hypothesis in the history of astronomy, A Defence of Tycho against Ursus, which was not published at the time but which historian Nicholas has described in the title of his scholarly edition of the work as ‘the birth of history and philosophy of science.[1]’ On the astronomical front, Tycho gave Kepler what would prove to be a task of immeasurable importance in the history of astronomy, the determination of the orbit of Mars.

When Kepler initially arrived in Prague to work with Tycho, Longomontanus, Tycho’s chief assistant, was working on the orbit of Mars. With Kepler’s arrival Tycho moved Longomontanus onto his model of the Moon’s orbit and put Kepler onto Mars. When first assigned Kepler famously claimed that he would knock it off in a couple of weeks, in the end he took the best part of six years to complete the task. This fact out of Kepler’s life often gets reported, oft with the false claim that he took ten years, but what the people almost never add is that in those six years the Astronomia Novawas not the only thing that occupied his time.

Not long after Kepler began his work Tycho died and he inherited the position of Imperial Mathematicus. This, however, had a major snag. Tycho’s data, the reason Kepler had come to Prague, was Tycho’s private property and that inherited his children including his daughter Elizabeth and her husband Frans Gansneb genaamd Tengnagel van de Camp. Being present when Tycho had died, Kepler secured the data for himself but was aware that it didn’t belong to him. There followed long and weary negotiations between Kepler and Frans Tengnagel, who claimed that he intended to continue Tycho’s life’s work. However, Tengnagel was a diplomat and not an astronomer, so in the end a compromise was achieved. Kepler could retain the data and utilise it but any publications that resulted from it would have Tengnagel named as co-author! In the end he contributed a preface to the Astronomia Nova. Kepler also spent a lot of time and effort haggling with the bureaucrats at Rudolph’s court, attempting to get his salary paid. Rudolph was good at appointing people and promising a salary but less good at paying up.

Apart from being distracted by bureaucratic and legal issues during this period Kepler also produced some other rather important scientific work. In 1604 he published his Astronomiae Pars Optica, written in 1603, which was the most important work in optics published since the Middle Ages and laid the foundations for the modern science. It included the first explanations of how lenses work to correct short and long sight and above all the first-ever correct explanation of how the image is formed in the eye. The latter was confirmed empirically by Christoph Scheiner.

C0194687-Kepler_s_Astronomiae_Pars_Optica_1604_

Astronomiae Pars Optica title page

In 1604 a supernova appeared in the skies and Kepler systematically observed it, confirmed it was definitively supralunar (i.e. above the moon’s orbit) and wrote up and published his findings, De Stella nova in pede Serpentarii, in Prague in 1606.

The Astronomia Nova is almost unique amongst major scientific publications in that it appears to outline in detail the work Kepler undertook to arrive at his conclusions, including all of the false turnings he took, the mistaken hypotheses he used and then abandoned and the failures he made in his calculations. Normally scientific researchers leave these things in their notebooks, sketchpads and laboratory protocols; only presenting to their readers a sanitised version of their results and the calculations or experiments necessary to achieve them. Why did Kepler act differently with his Astronomia Nova going into great detail on his six-year battle with Mars? The answer is contained in my ‘it appears’ in the opening sentence of this paragraph. Kepler was to a large extent pulling the wool over his readers’ eyes.

Kepler was a convinced Copernicaner in a period where the majority of astronomers were either against heliocentricity, mostly with good scientific reasons, or at best sitting on the fence. Kepler was truly revolutionary in another sense, he believed firmly in a physical cause for the structure of the cosmos and the movement of the planets. This was something that he had already propagated in his Mysterium Cosmographicum and for which he had been strongly criticised by his teacher Mästlin. The vast majority of astronomers still believed they were creating mathematical models to save the phenomena, irrespective of the actually physical truth of those models. The true nature of the cosmos was a question to be answered by philosophers and not astronomers.

Kepler structured the rhetoric of the Astronomia Nova to make it appear that his conclusions were inevitable; he had apparently no other choice, the evidence led him inescapably to a heliocentric system with real physical cause. Of course, he couldn’t really prove this but he did his best to con his readers into thinking he could. He actually road tested his arguments for this literary tour de force in a long-year correspondence with the Frisian astronomer David Fabricius. Fabricius was a first class astronomer and a convinced Tychonic i.e. he accepted Tycho’s geo-heliocentric model of the cosmos. Over the period of their correspondence Kepler would present Fabricius with his arguments and Fabricius would criticise them to the best of his ability, which was excellent. In this way Kepler could slowly build up an impression of what he needed to do in order to convince people of his central arguments. This was the rhetoric that he then used to write the final version of Astronomia Nova[2].

To a large extent Kepler failed in both his main aims when the book was published in 1609. In fact it would not be an exaggeration to say that it was initially a flop. People weren’t buying either his heliocentricity or his physical cause arguments. But the book contains two gems that in the end would prove very decisive in the battle of the cosmological systems, his first and second laws of planetary motion:

1) That planets orbit the Sun on elliptical paths with the Sun situated at one focus of the ellipse

2) That a line connecting the planet to the Sun sweeps out equal areas in equal periods of time.

Kepler actually developed the second law first using it as his primary tool to determine the actually orbit of Mars. The formulation of this law went through an evolution, that he elucidates in the book, before it reached its final form. The first law was in fact the capstone of his entire endeavour. He had known for sometime that the orbit was oval and had even at one point considered an elliptical form but then rejected it. When he finally proved that the orbit was actually an ellipse he knew that his battle was over and he had won. Today school kids learn the first two laws together with the third one, discovered thirteen years later when Kepler was working on his opus magnum the Harmonice mundi, but they rarely learn of the years of toil that Kepler invested in their discovery during his battle with Mars.

[1]Nicholas Jardine, The Birth of History and Philosophy of Science: Kepler’s ‘A Defence of Tycho against Ursus’ with Essays on its Provenance and Significance, CUP, ppb. 2009

[2]For an excellent account of the writing of Astronomia Novaand in particular the Kepler-Fabricius correspondence read James R. Voelkel, The Composition of Kepler’s Astronomia Nova, Princeton University Press, 2001

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Christmas Trilogy 2018 Part 2: A danseuse and a woven portrait

I had decided some time ago to give up my attempts to rescue Charles Babbage’s reputation from the calumnies of the acolytes of Saint Ada, as a lost cause. However, the recent attempt by said acolytes to heave her onto the planned new British £50 banknote combined with the vast amounts of crap posted all over the Internet on her birthday on 10 December this year convinced me to return to the foray. I shan’t be writing about Ada per see but analysing two quotes that he supporters claim show her superior understanding of the potential of the computer over the, in their opinion, pitifully inadequate Babbage.

Two things should be born in mind when assessing the Notes to the translation of Menabrea’s essay on the Analytical Engine. Firstly, everything that Lovelace knew about the Analytical Engine she had learnt from Babbage and secondly, it is an established fact that Babbage co-authored those notes. The supporters of Lovelace as some sort of computing prophet always state, without giving any sort of proof for their claim, that Babbage was only interested in his Analytical Engine as a sort of super number cruncher and that anything that goes beyond that must per definition come from Lovelace. One should never forget that any computer is in fact just a super number cruncher; everything that one does on a computer, typing this post for example, has first to be translated in mathematical algorithms in binary code so that the computer can understand them. Babbage was, of course, first and foremost interested in producing a machine or automata capable of reading and carrying out the widest possible range of mathematical functions to give it maximum flexibility.

We now turn to one of the favourite Ada fan club quotes:

[The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine…Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.

It would not be false to claim that music is in fact applied mathematics. Both rhythm and pitch can be and are expressed by mathematical functions. For about two thousand years music was part of the mathematical curriculum, as one of the four disciplines of the quadrivium. That a mathematician of Babbage’s stature wouldn’t think of the possibility of programming his computer to play or even create music is asking a lot. However, we have very direct proof that Babbage was well aware of the relationship between music and automata.

Chapter three of Babbage’s autobiography opens thus:

“During my boyhood my mother took me to several exhibitions of machinery. I well remember one of them in Hanover Square, by a man who called himself Merlin. I was so greatly interested in it, that the Exhibiter remarked the circumstance, and after explain some of the objects to which the public had access, proposed to my mother to take me up to his workshop. Where I would see still more wonderful automata. We accordingly ascended to the attic. There were two uncovered female figures of silver, about twelve inches high.

[…]

The other silver figure was an admirable danseuse, with a bird on the fore finger of her right hand, which wagged its tail, flapped its wings, and opened its beak. This lady attitudinized in a most fascinating manner. Her eyes were full of imagination, and irresistible.[1]

Following Merlin’s death in 1803, his automata were acquired by another showman Thomas Weeks, who in turn having gone out of business died in 1834. Babbage, now a grown man and very wealthy, attended the auction of Week’s possessions and for £35 acquired the danseuse. He restored the model and having had clothes made for her displayed the danseuse on a glass pedestal in his salon[2]. Babbage’s passion, and it was truly a passion, for machines was sparked by a musical automata, his silver danseuse, an image somewhat far from that of the boring mathematician only interested in numbers. In fact many of the most famous model produced in the golden age of automata in the late 18thand early 19thcenturies were musical, something which Babbage, who became a great expert if not ‘the great expert’ on, would have well aware of. That Babbage probably did play with the thought of his super automata, his Analytical Engine, producing music is a more than plausible concept.

The all time favourite quote of the Ada acolytes that they flourish like a hand of four aces in poker is:

“The Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves.”

Here the turn of phrase might well be Ada’s but the concept is with certainty Babbage’s. Anybody who thinks otherwise has never read anything by or on Babbage or the Analytical Engine or even the Notes supposedly written alone by Ada. Babbage’s greatest stroke of genius in his conception of his Analytical Engine was the idea of programming it with punch cards; an idea that he borrowed from Joseph Marie Jacquard 1752–1834), who had used it to program his silk weaving loom.

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Detail of Jaquard loom at TextielMuseum Tilburg showing punch cards Source: Wikimedia Commons

Jacquard in turn had borrowed from Jacques de Vaucanson (1709–1782), arguably the greatest of the automata builders in that great age of automata. Vaucanson produced two famous musical automata, a flute player with a repertoire of twelve tunes and a tambourine player. The role of the punch cards and their origin are discussed extensively in the Notes. Turning once again to Babbage’s autobiography we find the following:

It is a known fact that the Jacquard loom is capable of weaving any design which the imagination of man may conceive. It is also the constant practice for skilled artists to be employed by manufacturers in designing patterns. These patterns are then sent to a peculiar artist, who, by means of a certain machine, punches holes in a set of pasteboard cards in such a manner that when the cards are placed in a Jacquard loom, it will then weave upon its produce the exact pattern designed by the artist.

Now the manufacturer may use, for the warp and weft of his work, threads which are all of the same colour; let us suppose them to be unbleached or white threads. In this case the cloth will be woven all of one colour; but there will be a damask pattern upon it such as the artist designed.

But the manufacturer might use the same cards, and put into the warp threads of any other colour. Every thread might even be of a different colour, or of a different shade of colour; but in all these cases the form of the pattern will be precisely the same—the colours only will differ.

The Analogy of the Analytical Engine with this well-known process is nearly perfect.

[…]

Every formula which the Analytical Engine can be required to compute consists of certain algebraic operations to be performed upon given letters, and of certain other modifications depending on the numerical values assigned to those letters.

There are therefore two sets of cards, the first to direct the nature of the operations to be performed—these are called operation cards: the other to direct the particular variable on which those cards are required to operate—these latter are called variable cards

[…]

Under this arrangement, when any formula is required to be computed, a set of operation cards must be strung together, which contain the series of operations in the order in which they occur. Another set of cards must then be strung together, to call in the variables into the mill, the order in which they are required to be acted upon.

[…]

Thus the Analytical Engine will possess a library of its own. Every set of cards once made will at any future time reproduce the calculations for which it was first arranged. The numerical value of its constants may then be inserted[3].

 

This may not have the poetical elegance of Ada’s pregnant phrase but Babbage here clearly elucidates (I’ve left out a lot of the details) how the Analytical Engine will weave algebraical patterns.

Of interest in the whole story of the punch cards, the Jacquard loom and the Analytical Engine is the story of the portrait. As a demonstration of the versatility of his system, in 1839 a portrait of Jacquard was woven in silk on a Jacquard loom; it required 24,000 punch cards to create.

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Portrait of Jacquard woven in silk Source: Science Museum via Wikimedia Commons

Charles Babbage acquired one of these woven portraits for the then enormous sum of £800 and displayed it in his salon along with his silver danseuse and the ‘miracle performing’ unit of his Difference Engine. Having astounded his guests with performances of the danseuse and his Difference Engine he would then unveil the portrait and challenge his guests to guess how it had been produced. Babbage was as much a showman as he was a mathematician.

[1]Charles Babbage, Passages from The Life of a Philosopher, Longman, Green, Longman, Roberts, & Green, London, 1864 p. 17

[2]For the full story of Merlin, Weeks and much more see Simon Schaffer, Babbage’s Dancer

[3]Babbage, Passages, pp. 116–118

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Christmas Trilogy 2018 Part 1: The Harmonic Isaac

Isaac Newton is often referred to, as the ‘father’ of modern science but then again so is Galileo Galilei. In reality modern science has many fathers and some mothers as well. Those who use this accolade tend to want to sweep his theological studies and his alchemy under the carpet and pretend it doesn’t really count. Another weird aspect of Newton’s intellectual universe was his belief in prisca theology. This was the belief that in the period following the creation humankind had perfect knowledge of the natural world that got somehow lost over the centuries. This meant for Isaac that in his own scientific work he wasn’t making discoveries but rediscovering once lost knowledge. Amongst, what we would now regard as his occult beliefs, Isaac also subscribed to the Pythagorean belief in Harmonia (harmony), as a unifying concept in the cosmos.

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Robert Fludd’s Pythagorean Monocord

Although he was anything but a fan of music, he was a dedicated student of Harmonia, the mathematical theory of proportions that was part of the quadrivium. According to the legend Pythagoras was the first to discover that musical interval can be expressed as simple ratios of whole numbers related to a taut string: 1:1 (unison), 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). Unfortunately, anybody who has studied the theory of music knows these ratios don’t quite work. If you start on a given tone and move up in steps of a perfect fifth you don’t actually arrive back at the original tone seven octaves higher after twelve fifths but slightly off. This difference is known as the Pythagorean comma. This disharmony was well known and in the sixteenth and seventeenth centuries a major debate developed on how to ‘correctly’ divide up musical scale to avoid this problem. The original adversaries were Gioseffo Zarlino (1570–1590) and Vincenzo Galilei (1520–1591) (Galileo’s father) and Kepler made a contribution in his Harmonice Mundi; perhaps the most important contribution being made by Marin Mersenne (1588–1648) in his Harmonie universelle, contenant la théorie et la pratique de la musique.

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Harmonie Universelle title page

Here he elucidated Mersenne’s Laws:

Frequency is:

  1. Inversely proportional to the length of the string (this was known to the ancients; it is usually credited toPythagoras)
  2. Proportional to the square root of the stretching force, and
  3. Inversely proportional to the square root of the mass per unit length.
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Source: Gouk p. 115

As a student Newton took up the challenge in one of his notebooks and we don’t need to go into his contribution to that debate here, however it is the first indication of his interest in this mathematics, which he would go on to apply to his two major scientific works, his optics and his theory of gravity.

After he graduated at Cambridge Newton’s first serious original research was into various aspects of optics. This led to his first published paper:

A Letter of Mr. Isaac Newton, Professor of the Mathematicks in the University of Cambridge; Containing His New Theory about Light and Colors: Sent by the Author to the Publishee from Cambridge, Febr. 6. 1671/72; In Order to be Communicated to the R. Society

In which he described his experiments with a prism that showed that white light consists of blended coloured light and that the spectrum that one produces with a prism is the splitting up of the white light into its coloured components. Previous theories had claimed that the spectrum was produced by the dimming or dirtying of the white light by the prism. Newton wrote an extensive paper expanding on his optical research, An hypothesis explaining the properties of light, but due to the harsh criticism his first paper received he withheld it from publication. This expanded work only appeared in 1704 in his book, Opticks: A Treatise of the Reflections, Refractions, Inflections & Colours of Light. Here we can read:

In the Experiments of the fourth Proposition of the first Part of this first Book, when I had separated the heterogeneous Rays from one another, the Spectrum ptformed by the separated Rays, did in the Progress from its End p, on which the most refrangible Rays fell, unto its other End t, on which the most refrangible Rays fell, appear tinged with this Series of Colours, violet, indigo, blue, green, yellow, orange, red, together with all their intermediate Degrees in a continual Succession perpetually varying . So that there appeared as many Degrees of Colours, as there were sorts of Rays differing in Refrangibility.

This is of course the list of seven colours that we associate with the rainbow today. Before Newton researchers writing about the spectrum listed only three, four or at most five colours, so why did he raise the number to seven by dividing the blue end of the spectrum into violet, indigo and blue? He did so in order to align the number of colours of the spectrum with the notes on the musical scales. In the Queries that were added at the end of the Opticks over the years and the different editions we find the following:

Qu. 13. Do not several sorts of Rays make Vibrations of several bigness, which according to their bignesses excite Sensations of several Colours, much after the manner that the Vibrations of the Air, according to their several bignesses excite Sensations of several Sounds? And particularly do not Vibrations for making a Sensation of deep violet, the least refrangible the largest for making a Sensation of deep red, and several intermediate sorts of Rays, Vibrations of several intermediate bignesses to make Sensations of the several intermediate Colours?

Qu. 14. May not the harmony and discord of Colours arise from the proportions of the Vibrations propagated through the Fibres of the optick Nerves into the Brain, as the harmony and discord of Sounds arise from the proportions of the Vibrations of the Air? And some Colours, if they be view’d together, are agreeable to one another, as those of Gold and Indigo and other disagree.

In the An Hypothesis, Newton published a diagram illustrated the connection he believed to exist between the colours of the spectrum and the notes of the scale.

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Source: Gouk p. 118

Interestingly Voltaire presented Newton’s theory in his Elemens de la philosophie de Newton (1738), again as a diagram.

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Source: Gouk p. 119

Turning now to Newton’s magnum opus we find the even more extraordinary association between his theory of gravity and the Pythagorean theory of harmony. Newton’s Law of Gravity is probably the last place one would expect to meet with Pythagorean harmony but against all expectations one does. In unpublished scholia on Proposition VIII of Book III of the Principia(the law of gravity) Newton claimed that Pythagoras had known the inverse square law. He argued that Pythagoras had discovered the inverse-square relationship in the vibration of strings (see Mersenne above) and had applied the same principle to the heavens.

…consequently by comparing those weights with the weights of the planets , and the lengths of the strings with the distances of the planets, he understood by means of the harmony of the heavens that the weights of the planets towards the Sun were reciprocally as the squares of their distances from the Sun.[1]

Although Newton never published this theory David Gregory (1661–1708) did. David Gregory was a nephew of the physicist James Gregory who in 1684 became professor of mathematics at the University of Edinburgh, where he became “the first to openly teach the doctrines of the Principia, in a public seminary…in those days this was a daring innovation.”[2]

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Davis Gregory bust Source: Wikimedia Commons

In 1691, with Newton’s assistance, he was appointed Savilian Professor of Astronomy at Oxford going on to become an important mathematician, physicist and astronomer. He worked together with Newton on the planned second edition of the Principia, although he did not edit it, dying in 1708; the second edition appearing first in 1713 edited by Richard Bentley. In his Astronomiae physicae et geometricae elementa, a semi-popular presentation of Newton’s theories first published in Latin in 1702

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Gregory wrote the following:

The Elements of Astronomy, Physical and Geometrical By David Gregory M.D. SavilianProfessor of Astronomy at Oxfordand Fellow of the Royal Society (1615)

The Author’sPreface

As it is manifest that the Ancients were apprized of, and had discover’d the Gravity of all Bodies towards one another, so also they were not unacquainted with the Law and Proportion which the action of Gravity observ’d according to the different Masses and Distances. For that Gravity is proportional to the Quantity of Matter in the heavy Body, Lucretiusdoes sufficiently declare, as also that what we call light Bodies, don’t ascend of their own accord, but by action of a force underneath them, impelling them upwards, just as a piece of Wood is in Water; and further, that all Bodies, as well the heavy as the light, do descend in vacuo, with an equal celerity. It will be plain likewise, from what I shall presently observe, that the famous Theorem about the proportion whereby Gravity decreases in receding from the Sun, was not unknown at least to Pythagoras. This indeed seems to be that which he and his followers would signify to us by the Harmony of the Spheres: That is, they feign’d Apolloplaying on a Harp of seven Strings, by which Symbol, as it is abundantly evident from Pliny, Macrobiusand Censorinus, they meant the Sun in Conjunction with the seven planets, for they made him the leader of that Septenary Chorus, and Moderator of Nature; and thought that by his Attractive force he acted upon the Planets (and called it Jupiter’s Prison, because it is by this Force that he retains and keeps them in their Orbits, from flying off in Right Lines) in the Harmonical ration of their Distances. For the forces, whereby equal Tensions act upon Strings of different lengths (being equal in other respects) are reciprocally as the Squares of the lengths of the Strings.

I first came across this theory, as elucidated by Gregory, years ago in a book, which book I have in the meantime forgotten, where it was summarised as follows:

Gravity is the strings upon which the celestial harmony is played.

 

 

 

 

 

 

 

 

[1]Quoted from Penelope Gouk, The harmonic roots of Newtonian science, in John Fauvel, Raymond Flood, Michael Shortland & Robin Wilson eds., Let Newton Be: A new perspective on his life and works, OUP, Oxford, New York, Tokyo, ppb. 1989 The inspiration and principle source for this blog post.

[2]Quoted from Significant Scots: David Gregory

https://www.electricscotland.com/history/other/gregory_david.htm

 

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Filed under History of Astronomy, History of Mathematics, History of Optics, History of science, Newton

Printing the Hindu-Arabic numbers

Arte dell’Abbaco, a book that many consider the first-ever printed mathematics book, was dated four hundred and forty years ago on 10 December 1478. I say many consider because the book, also known as the Treviso Arithmetic, is a commercial arithmetic textbook and some historians regard commercial arithmetic as a separate discipline and not really mathematics.

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Calculation from the Arte dell’Abbaco

The unknown author explains his book thus:

I have often been asked by certain youths in whom I have much interest, and who look forward to mercantile pursuits, to put into writing the fundamental principles of arithmetic, commonly called abbacus.

The Treviso Arithmetic is actually an abbacus book, those books on calculating with the Hindu-Arabic numerals that derive their existence from Leonardo Pisano’s Liber Abbaci. Like most abbacus books it is written in the vernacular, which in this case is the local Venetian dialect. If you don’t read 15thcentury Venetian there is an English translation by Frank J. Swetz, Capitalism and Arithmetic: The New Math of the 15thCentury Including the Full Text of the Treviso Arithmetic of 1478, Open Court, 1987.

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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.

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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]

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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.

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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]

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“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]

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Quadrivium

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?

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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

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