Category Archives: History of Mathematics

Mega inanity

Since the lead up to the Turing centennial in 2012 celebrating the birth of one of the great meta-mathematicians of the twentieth century, Alan Mathison Turing, I have observed with increasing horror the escalating hagiographic accounts of Turing’s undoubted historical achievements and the resulting perversion of the histories of twentieth-century science, mathematics and technology and in particular the history of computing.

This abhorrence on my part is not based on a mere nodding acquaintance with Turing’s name but on a deep and long-time engagement with the man and his work. I served my apprenticeship as a historian of science over many years in a research project on the history of formal or mathematical logic. Formal logic is one of the so-called formal sciences the others being mathematics and informatics (or computer science). I have spent my whole life studying the history of mathematics with a special interest in the history of computing both in its abstract form and in its technological realisation in all sorts of calculating aids and machines. I also devoted a substantial part of my formal study of philosophy to the study of the philosophy of mathematics and the logical, meta-logical and meta-mathematical problems that this discipline, some would say unfortunately, generates. The history of all of these intellectual streams flow together in the first half of the twentieth century in the work of such people as Leopold Löwenheim, Thoralf Skolem, Emil Post, Alfred Tarski, Kurt Gödel, Alonso Church and Alan Turing amongst others. These people created a new discipline known as meta-mathematics whilst carrying out a programme delineated by David Hilbert.

Attempts to provide a solid foundation for mathematics using set theory and logic had run into serious problems with paradoxes. Hilbert thought the solution lay in developing each mathematical discipline as a strict axiomatic systems and then proving that each axiomatic system possessed a set of required characteristics thus ensuring the solidity and reliability of a given system. This concept of proving theories for complete axiomatic systems is the meta- of meta-mathematics. The properties that Hilbert required for his axiomatic systems were consistency, which means the systems should be shown to be free of contradictions, completeness, meaning that all of the theorems that belong to a particular discipline are deductible from its axiom system, and finally decidability, meaning that for any well-formed statement within the system it should be possible to produced an algorithmic process to decide if the statement is true within the axiomatic system or not. An algorithm is like a cookery recipe if you follow the steps correctly you will produce the right result.

The meta-mathematicians listed above showed by very ingenious methods that none of Hilbert’s aims could be fulfilled bringing the dream of a secure foundation for mathematics crashing to the ground. Turing’s solution to the problem of decidability is an ingenious thought experiment, for which he is justifiably regarded as one of the meta-mathematical gods of the twentieth century. It was this work that led to him being employed as a code breaker at Bletchley Park during WW II and eventually to the fame and disaster of the rest of his too short life.

Unfortunately the attempts to restore Turing’s reputation since the centenary of his birth in 2012 has led to some terrible misrepresentations of his work and its consequences. I thought we had reach a low point in the ebb and flow of the centenary celebrations but the release of “The Imitation Game”, the Alan Turing biopic, has produced a new series of false and inaccurate statements in the reviews. I was pleasantly pleased to see several reviews, which attempt to correct some of the worst historical errors in the film. You can read a collection of reviews of the film in the most recent edition of the weekly histories of science, technology and medicine links list Whewell’s Gazette. Not having seen the film yet I can’t comment but I was stunned when I read the following paragraph from the abc NEWS review of the film written by Alyssa Newcomb. It’s so bad you can only file it under; you can’t make this shit up.

The “Turing Machine” was the first modern computer to logically process information, running on interchangeable software and essentially laying the groundwork for every computing device we have today — from laptops to smartphones.

Before I analyse this train wreck of a historical statement I would just like to emphasise that this is not the Little Piddlington School Gazette, whose enthusiastic but slightly slapdash twelve-year-old film critic got his facts a little mixed up, but a review that appeared on the website of a major American media company and as such totally unacceptable however you view it.

The first compound statement contains a double whammy of mega-inane falsehood and I had real problems deciding where to begin and finally plumped for the “first modern computer to logically process information, running on interchangeable software”. Alan Turing had nothing to do with the first such machine, the honour going to Konrad Zuse’s Z3, which Zuse completed in 1941. The first such machine in whose design and construction Alan Turing was involved was the ACE produced at the National Physical Laboratory, in London, in 1949. In the intervening years Atanasoff and Berry, Tommy Flowers, Howard Aikin, as well as Eckert and Mauchly had all designed and constructed computers of various types and abilities. To credit Turing with the sole responsibility for our digital computer age is not only historically inaccurate but also highly insulting to all the others who made substantial and important contributions to the evolution of the computer. Many, many more than I’ve named here.

We now turn to the second error contained in this wonderfully inane opening statement and return to the subject of meta-mathematics. The “Turing Machine” is not a computer at all its Alan Turing’s truly genial thought experiment solution to Hilbert’s decidability problem. Turing imagined a very simple machine that consists of a scanning-reading head and an infinite tape that runs under the scanning head. The head can read instructions on the tape and execute them, moving the tape right or left or doing nothing. The question then reduces to the question, which set of instructions on the tape come eventually to a stop (decidable) and which lead to an infinite loop (undecidable). Turing developed this idea to a machine capable of computing any computable function (a universal Turing Machine) and thus created a theoretical model for all computers. This is of course a long way from a practical, real mechanical realisation i.e. a computer but it does provide a theoretical measure with which to describe the capabilities of a mechanical computing device. A computer that is the equivalent of a Universal Turing Machine is called Turing complete. For example, Zuse’s Z3 was Turing complete whereas Colossus, the computer designed and constructed by Tommy Flowers for decoding work at Bletchley Park, was not.

Turing’s work played and continues to play an important role in the theory of computation but historically had very little effect on the development of real computers. Attributing the digital computer age to Turing and his work is not just historically wrong but is as I already stated above highly insulting to all of those who really did bring about that age. Turing is a fascinating, brilliant, and because of what happened to him because of the persecution of homosexuals, tragic figure in the histories of mathematics, logic and computing in the twentieth century but attributing achievements to him that he didn’t make does not honour his memory, which certainly should be honoured, but ridicules it.

I should in fairness to the author of the film review, that I took as motivation from this post, say that she seems to be channelling misinformation from the film distributors as I’ve read very similar stupid claims in other previews and reviews of the film.


Filed under History of Computing, History of Logic, History of Mathematics, Myths of Science

The Queen of Science – The woman who tamed Laplace.

In a footnote to my recent post on the mythologizing of Ibn al-Haytham I briefly noted the inadequacy of the terms Arabic science and Islamic science, pointing out that there were scholars included in these categories who were not Muslims and ones who were not Arabic. In the comments Renaissance Mathematicus friend, the blogger theofloinn, asked, Who were the non-muslim “muslim” scientists? And (aside from Persians) who were the non-Arab “arab” scientists? And then in a follow up comment wrote, I knew about Hunayn ibn Ishaq and the House of Wisdom, but I was not thinking of translation as “doing science.” From the standpoint of the historian of science this second comment is very interesting and reflects a common problem in the historiography of science. On the whole most people regard science as being that which scientists do and when describing its history they tend to concentrate on the big name scientists.

This attitude is a highly mistaken one that creates a falsified picture of scientific endeavour. Science is a collective enterprise in which the ‘scientists’ are only one part of a collective consisting of scientists, technicians, instrument designers and makers, and other supportive workers without whom the scientist could not carry out his or her work. This often includes such ignored people as the secretaries, or in earlier times amanuenses, who wrote up the scientific reports or life partners who, invisible in the background, often carried out much of the drudgery of scientific investigation. My favourite example being William Herschel’s sister and housekeeper, Caroline (a successful astronomer in her own right), who sieved the horse manure on which he bedded his self cast telescope mirrors to polish them.

Translators very definitely belong to the long list of so-called helpers without whom the scientific endeavour would grind to a halt. It was translators who made the Babylonian astronomy and astrology accessible to their Greek heirs thus making possible the work of Eudoxus, Hipparchus, Ptolemaeus and many others. It was translators who set the ball rolling for those Islamic, or if you prefer Arabic, scholars when they translated the treasures of Greek science into Arabic. It was again translators who kicked off the various scientific Renaissances in the twelfth and thirteenth-centuries and again in the fifteenth-century, thereby making the so-called European scientific revolution possible. All of these translators were also more or less scientists in their own right as without a working knowledge of the subject matter that they were translating they would not have been able to render the texts from one language into another. In fact there are many instances in the history of the transmission of scientific knowledge where an inadequate knowledge of the subject at hand led to an inaccurate or even false translation causing major problems for the scholars who tried to understand the texts in the new language. Translators have always been and continue to be an important part of the scientific endeavour.

The two most important works on celestial mechanics produced in Europe in the long eighteenth-century were Isaac Newton’s Philosophiæ Naturalis Principia Mathematica and Pierre-Simon, marquis de Laplace’s Mécanique céleste. The former was originally published in Latin, with an English translation being published shortly after the author’s death, and the latter in French. This meant that these works were only accessible to those who mastered the respective language. It is a fascinating quirk of history that the former was rendered into French and that latter into English in each case by a women; Gabrielle-Émilie Le Tonnelier de Breteuil, Marquise du Châtelet translated Newton’s masterpiece into French and Mary Somerville translated Laplace’s pièce de résistance into English. I have blogged about Émilie de Châtelet before but who was Mary Somerville? (1)


Mary Somerville by Thomas Phillips

Mary Somerville by Thomas Phillips

She was born Mary Fairfax, the daughter of William Fairfax, a naval officer, and Mary Charters at Jedburgh in the Scottish boarders on 26 December 1780. Her parents very definitely didn’t believe in education for women and she spent her childhood wandering through the Scottish countryside developing a lifelong love of nature. At the age of ten, still semi-illiterate, she was sent to Miss Primrose’s boarding school at Musselburgh in Midlothian for one year; the only formal schooling she would ever receive. As a young lady she received lessons in dancing, music, painting and cookery. At the age of fifteen she came across a mathematical puzzle in a ladies magazine (mathematical recreation columns were quite common in ladies magazines in the 18th and 19th-centuries!) whilst visiting friends. Fascinated by the symbols that she didn’t understand, she was informed that it was algebra, a word that meant nothing to her. Later her painting teacher revealed that she could learn geometry from Euclid’s Elements whilst discussing the topic of perspective. With the assistance of her brother’s tutor, young ladies could not buy maths-books, she acquired a copy of the Euclid as well as one of Bonnycastle’s Algebra and began to teach herself mathematics in the secrecy of her bedroom. When her parents discovered this they were mortified her father saying to her mother, “Peg, we must put a stop to this, or we shall have Mary in a strait jacket one of these days. There is X., who went raving mad about the longitude.” They forbid her studies, but she persisted rising before at dawn to study until breakfast time. Her mother eventually allowed her to take some lessons on the terrestrial and celestial globes with the village schoolmaster.

In 1804 she was married off to a distant cousin, Samuel Grieg, like her father a naval officer but in the Russian Navy. He, like her parents, disapproved of her mathematical studies and she seemed condemned to the life of wife and mother. She bore two sons in her first marriage, David who died in infancy and Woronzow, who would later write a biography of Ada Lovelace. One could say fortunately, for the young Mary, her husband died after only three years of marriage in 1807 leaving her well enough off that she could now devote herself to her studies, which she duly did. Under the tutorship of John Wallace, later professor of mathematics in Edinburgh, she started on a course of mathematical study, of mostly French books but covering a wide range of mathematical topic, even tacking Newton’s Principia, which she found very difficult. She was by now already twenty-eight years old. During the next years she became a fixture in the highest intellectual circles of Edinburgh.

In 1812 she married for a second time, another cousin, William Somerville and thus acquired the name under which she would become famous throughout Europe. Unlike her parents and Samuel Grieg, William vigorously encouraged and supported her scientific interests. In 1816 the family moved to London. Due to her Scottish connections Mary soon became a member of the London intellectual scene and was on friendly terms with such luminaries as Thomas Young, Charles Babbage, John Herschel and many, many others; all of whom treated Mary as an equal in their wide ranging scientific discussions. In 1817 the Somervilles went to Paris where Mary became acquainted with the cream of the French scientists, including Biot, Arago, Cuvier, Guy-Lussac, Laplace, Poisson and many more.

In 1824 William was appointed Physician to Chelsea Hospital where Mary began a series of scientific experiments on light and magnetism, which resulted in a first scientific paper published in the Philosophical Transactions of the Royal Society in 1826. In 1836, a second piece of Mary’s original research was presented to the Académie des Sciences by Arago. The third and last of her own researches appeared in the Philosophical Transactions in 1845. However it was not as a researcher that Mary Somerville made her mark but as a translator and populariser.

In 1827 Henry Lord Brougham and Vaux requested Mary to translate Laplace’s Mécanique céleste into English for the Society for the Diffusion of Useful Knowledge. Initially hesitant she finally agreed but only on the condition that the project remained secret and it would only be published if judged fit for purpose, otherwise the manuscript should be burnt. She had met Laplace in 1817 and had maintained a scientific correspondence with him until his death in 1827. The translation took four years and was published as The Mechanism of the Heavens, with a dedication to Lord Brougham, in 1831. The manuscript had been refereed by John Herschel, Britain’s leading astronomer and a brilliant mathematician, who was thoroughly cognisant with the original, he found the translation much, much more than fit for the purpose. Laplace’s original text was written in a style that made it inaccessible for all but the best mathematicians, Mary Somerville did not just translate the text but made it accessible for all with a modicum of mathematics, simplifying and elucidating as she went. This wasn’t just a translation but a masterpiece. The text proved too vast for Brougham’s Library of Useful Knowledge but on the recommendation of Herschel, the publisher John Murray published the book at his own cost and risk promising the author two thirds of the profits. The book was a smash hit the first edition of 750 selling out almost instantly following glowing reviews by Herschel and others. In honour of the success the Royal Society commissioned a bust of Mrs Somerville to be placed in their Great Hall, she couldn’t of course become a member!

At the age of fifty-one Mary Somerville’s career as a science writer had started with a bang. Her Laplace translation was used as a textbook in English schools and universities for many years and went through many editions. Her elucidatory preface was extracted and published separately and also became a best seller. If she had never written another word she would still be hailed as a great translator and science writer but she didn’t stop here. Over the next forty years Mary Somerville wrote three major works of semi-popular science On the Connection of the Physical Sciences (1st ed. 1834), Physical Geography (1st ed. 1848), (she was now sixty-eight years old!) and at the age of seventy-nine, On Molecular and Microscopic Science (1st ed. 1859). The first two were major successes, which went through many editions each one extended, brought up to date, and improved. The third, which she later regretted having published, wasn’t as successful as her other books. Famously, in the history of science, William Whewell in his anonymous 1834 review of On the Connection of the Physical Sciences first used the term scientist, which he had coined a year earlier, in print but not, as is oft erroneously claimed, in reference to Mary Somerville.

Following the publication of On the Connection of the Physical Sciences Mary Somerville was awarded a state pension of £200 per annum, which was later raised to £300. Together with Caroline Herschel, Mary Somerville became the first female honorary member of the Royal Astronomical Society just one of many memberships and honorary memberships of learned societies throughout Europe and America. Somerville College Oxford, founded seven years after her death, was also named in her honour. She died on 28 November 1872, at the age of ninety-one, the obituary which appeared in the Morning Post on 2 December said, “Whatever difficulty we might experience in the middle of the nineteenth century in choosing a king of science, there could be no question whatever as to the queen of science.” The Times of the same date, “spoke of the high regard in which her services to science were held both by men of science and by the nation”.

As this is my contribution to Ada Lovelace day celebrating the role of women in the history of science, medicine, engineering, mathematics and technology I will close by mentioning the role that Mary Somerville played in the life of Ada. A friend of Ada’s mother, the older women became a scientific mentor and occasional mathematics tutor to the young Miss Byron. As her various attempts to make something of herself in science or mathematics all came to nought Ada decided to take a leaf out of her mentor’s book and to turn to scientific translating. At the suggestion of Charles Wheatstone she chose to translate Luigi Menabrea’s essay on Babbage’s Analytical Engine, at Babbage’s suggestion elucidating the original text as her mentor had elucidated Laplace and the rest is, as they say, history. I personally would wish that the founders of Ada Lovelace Day had chosen Mary Somerville instead, as their galleon figure, as she contributed much, much more to the history of science than her feted protégée.

(1) What follows is largely a very condensed version of Elizabeth  C. Patterson’s excellent Somerville biography Mary Somerville, The British Journal for the History of Science, Vol. 4, 1969, pp. 311-339



Filed under History of Astronomy, History of Mathematics, History of Physics, History of science, Ladies of Science

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, 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 1st – Thursday 3rd 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!




Filed under History of Mathematics, Renaissance Science, University History

Oh please!

The latest move in the canonisation of Alan Turing is an opera, or whatever, written by the Pet Shop Boys, which is being heavily promoted by a PR campaign launched yesterday. As part of this press onslaught this magazine cover appeared on my Twitter stream today.


For the record, as a fan and one time student of meta-mathematics I was aware of and to some extent in awe of Alan Turing long before most of the people now trying to elevate him into Olympus even knew he existed. He was without a shadow of a doubt one of the most brilliant logicians of the twentieth-century and he along with others of his ilk, such as Leopold Löwenheim, Thoralf Skolem, Emil Post, Kurt Gödel, Alonzo Church etc. etc., who laid the theoretical foundations for much of the computer age, all deserve to be much better known than they are, however the attempts to adulate Turing’s memory have become grotesque. The Gay Man Who Saved the World is hyperbolic, hagiographic bullshit!

Turing made significant contributions to the work of Bletchley Park in breaking various German codes during the Second World War. He was one of nine thousand people who worked there. He did not work in isolation; he led a team that cracked one version of the Enigma Code. To what extent the work of Bletchley Park contributed to the eventual Allied victory is probably almost impossible to assess or quantify.

Alan Turing made significant contributions to the theories of meta-mathematics and an equally significant contribution to the British war effort. He did not, as is frequently claimed by the claqueur, invent the computer and he most certainly did not “save the world”. Can we please return to sanity in our assessment of our scientific heroes?


Filed under History of Computing, History of Mathematics, Myths of Science


People appear to love list. The Internet is full of lists. The 10 most popular dog breeds, the 10 biggest waves ever ridden by a surfer, the 10… you get the idea. The lists very often have ten entries, it’s a shame that we all have the same number of fingers otherwise we could a bit more variation, the 7 biggest… or the 11 smartest… Science and its history are far from immune from this cyber cancer, lists of all sorts being produced and posted with gay abandon. We recently even had the Top 10 scientists of the 13th century! Apart from the fact that the use of the word scientist here is highly anachronistic any such selection is of course subjective and disputable. However the subject of this post is not medieval scholars, tempting though it is, but a list of “17 Equations That Changed the World”

17 Equations that Changed the World

17 Equations that Changed the World

Although it claims to be by Ian Stewart I have no idea of the original source of this list but I have stumbled across it several times in the last few months. Now when I was a young mathematical acolyte and budding historian of maths I devoured Ian Stewart’s books at the same rate as those of Martin Gardner and Isaac Asimov. Put another way Ian Stewart was a major influence on my development. As I got older, but probably not wiser, I came to realise that Stewart, a mathematician and populariser, wasn’t very accurate in his historical attributions, in fact he is down right sloppy. This list is no exception.

Don’t worry I’m not going to go through all seventeen entries but the first time I read it I immediately noticed that the first five all have significant problems and I thought it would make an interesting exercise to explain why.

We start off with what is possibly the most well known theorem in the whole of mathematics Proposition 47 from Book I of Euclid’s Elements. The correct attribution of this theorem is actually an exercise in history of mathematics 101.

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

Now Euclid is thought to have written his Elements around 300 BCE and he doesn’t attribute this theorem to anybody. The first to putatively attach Pythagoras’ name to Euclid’s Proposition 47 was Proclus in his commentary on the Elements written in the fifth century CE. However Proclus doesn’t sound very convinced by his own attribution.

If we listen to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discovery. But for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the Elements, not only because he made it fast by a more lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in the sixth Book. For in that Book he proves generally that, in right-angled triangles, the figure on the side subtending the right angle is equal to the similar and similarly situated figures described on the sides about the right angle.

Proclus would seem to want to award the credits to Euclid not Pythagoras. Those who wished to recount ancient history were Cicero writing in the first century BCE and Plutarch writing in the first century CE. One thing that makes this anecdote from antiquity somewhat dubious is the fact that the Pythagoreans rejected animal sacrifice. There is no actual contemporary evidence that associates either the Pythagoreans or Pythagoras to the theorem that we name after him. However all of this is rather academic, as the theorem existed more than a thousand years before the Pythagoreans.

There is clear evidence that the Babylonians knew of the theorem in the Old Babylonian period around 1700 BCE. However although we have several instances of them using the theorem we don’t have a Babylonia proof of the theorem Maybe they didn’t have one but there are still literally tons of Babylonian clay tablets that have never been transcribed let alone translated. It could well be that somewhere the Babylonian Pythagoras is still waiting to be discovered.

The Babylonians were not the only ones to have the theorem independently of the Greeks. A clear example of the theorem can be found in the Indian Sulba Sutras. Unfortunately the dating of early Indian texts is very problematic and the best we can do is to say the Sulba Sutras date from between 800 and 200 BCE, so if the Indian Pythagoras predated the Greek one is almost impossible to determine.

Never to be left out when it comes to ancient invention and discovery the Chinese also had their own Pythagoras. The greatest Chinese mathematical classic The Nine Chapters of the Mathematical Arts contains problems that require use of the theorem in Chapter 9. In Chinese it is known as the Gougu rule. Once again dating is a major problem, the earliest existing manuscript dates from 179 CE but the contents are probably much earlier in origin, currently thought to date to 300 to 200 BCE. A simple and elegant pictorial proof of the theorem turns up in another Chinese classic the Zhou Bi Suan

Chinese Pythagoras

Chinese Pythagoras

Jing. Also very difficult to date but probably originating around 300BCE. As can be seen this theorem doesn’t have a simple history.

Stewart now takes a massive leap into the seventeenth century CE and the invention of logarithms. Once again his simple attribution to John Napier is exactly that, simplistic and historically misleading. We can find the principle on which logarithms are based in the work of several earlier mathematicians. We can find forms of proto-logarithms in both Babylonian and Indian mathematics and also in the system that Archimedes invented to describe very large numbers. In the fifteenth century Triparty, of the French mathematician Nicolas Chuquet we find the comparison between the arithmetical and geometrical progressions that underlay the concept of logarithms but if Chuquet ever took the next step is not clear. In the sixteenth century the German mathematician Michael Stifel studied the same comparison of progressions in his Arithmetica integra and did take the next step outlining the principle of logarithms but doesn’t seem to have developed the idea further.

It was in fact John Napier who took the final step and published the first set of logarithmic tables in his book Mirifici Logarithmorum Canonis Descriptio in 1614. However the Swiss clockmaker and mathematician, Jost Bürgi developed logarithms independently of Napier during the same period although his book of tables, Arithmetische und Geometrische Progress Tabulen, was first published in 1620.

We stay in the seventeenth century for Stewart’s next equation, which is the production of a first derivative using the so-called h-method confusingly labelled calculus, confusing that is because calculus is a branch of mathematics and not an equation, and attributed to Newton 1668. To say that this line has a lot of issues would be a mild understatement. I will try to keep it relatively short. Anybody with half an idea of the history of calculus will already be asking themselves, what about Leibniz? Newton and Leibniz both developed their ideas of the calculus independently in the same period with Newton probably developing his ideas first but Leibniz being the first in print. This situation led to what is probably the most notorious priority dispute in the whole of the history of mathematics and science. What makes Stewart’s statement even more piquant is that he attributes the discovery to Newton but his equation for the first derivative is written in Leibniz’ notations. Of course there is an about two thousand year long history to the development of the calculus that I outlined in an earlier post, so I won’t repeat it now. I will however point out that the h-method to determine the first derivative is not from either Newton or Leibniz but Pierre Fermat.

Newton gets a second bite of the cherry, this time, with the equation for gravity. I’ve lost count of the number of time that I’ve pointed out that the basics of the law of gravity, the inverse square relationship, does not originate with Newton. A very quick rundown.

The first to suggest that the planets were kept in their courses by a force was Kepler who suggested a directly proportional relationship based on Gilbert’s investigations of the magnet. Borelli also speculated on forces driving the planets in his Theoricae Mediceorum Planetarum ex Causius Physicus Deductae published in 1666 and known to Newton. The first to suggest an inverse square relationship was Ismael Boulliau, a story that I’ve already told here, although I there claim erroneously that Newton admits his knowledge of Boulliau’s priority in Principia, he doesn’t, it’s in the letters he exchanged with Halley in his dispute with Hooke. In the middle of the seventeenth century Wren, Halley, Hooke and Newton all independently came to the conclusion that the force governing the planetary orbits was probably inversely proportional to the square of the distance, i.e. the law of gravity. Newton’s achievement was to show that this law was equivalent to Kepler’s third law of planetary motion and that it also allowed the deduction of Kepler’s first two laws.

Stewart’s fifth equation is his simplest i = √-1, which he attributes to Euler. Now whilst it is probably true that Euler introduced the letter “i” as the symbol for the square root of minus one, by the time he did so mathematicians had been playing with and cursing the concept for a couple of hundred years.

The first person to consciously use imaginary or complex numbers was the sixteenth century polymath Girolamo Cardano in his Ars magna, the first systematic study of the solution of polynomials published by Petreius in Nürnberg in 1545. Cardano solved cubic equation in which during the solution so-called conjugate pairs of complex numbers turned up, which when multiplied together lost their imaginary parts thus delivering real solutions. (Conjugate pairs of complex numbers are ones of the form a + b√-c and a – b√-c which when multiplied together become a2 +b2c.) Cardano thought the complex numbers were nonsensical but the solutions worked so he left them in.

Later in the century the Italian mathematician Rafael Bombelli worked quite rationally with complex numbers developing the rule for their manipulation in his Algebra published in 1572. If any one name should be attached to this equation then it’s Bombelli’s.


Of course not everybody was as happy with these very strange entities as Bombelli and it was Descartes who gave them the name imaginary in 1637. It was intended to be derogatory. Euler did much to develop the theory of complex numbers in the eighteenth century but it was first the development, independently by three mathematicians, Caspar Wessel in 1799, Jean-Robert Argand in 1806 and of course Gauss, of the geometrical interpretation of complex numbers that they became finally universally accepted.

Having ploughed your way through the historical thickets of this post some of you might be thinking that I’m just nit picking, but there is a deeper point that I’m trying to make. It is very rare in the history of mathematics or science that a theorem, theory, invention, discovery, idea, concept or hypothesis emerges in its final form, like Athena born fully armed from Zeus’ forehead. Almost always there is an, often long, period of evolution involving many thinkers and often taking long and devious routes. Very often they occur as multiple discoveries with more than one progenitor, frequently leading to priority disputes. The idea of a simple list of discoveries with one date and one name whilst superficially attractive leads inevitably to a false concept of the evolution of science and of scientific methodology. Let us get away from such lists and let students of science and mathematics really learn just how messy, complex and, I think, fascinating the histories of their disciplines really are.


Filed under History of Mathematics, Myths of Science

Luca, Leonardo, Albrecht and the search for the third dimension.

Many of my more recent readers will not be aware that I lost a good Internet friend last year with the unexpected demise of the history of art blogger, Hasan Niyazi. If you want to know more about my relationship with Hasan then read the elegy I wrote for him when I first heard the news. Hasan was passionate about Renaissance art and his true love was reserved for the painter Raffaello Sanzio da Urbino, better known as Raphael. Today, 6th April is Raphael’s birthday and Hasan’s partner Shazza (Sharon) Bishop has asked Hasan’s friends in the Internet blogging community to write and post something today to celebrate his life, this is my post for Hasan.


I’m not an art historian but there were a couple of themes that Hasan and I had in common, one of these was, for example, the problem of historical dating given differing calendars. Another shared interest was the history of linear perspective, which is of course absolutely central to the history of Renaissance art but was also at the same time an important theme in Renaissance mathematics and optics. I have decided therefore to write a post for Hasan about the Renaissance mathematicus Luca Pacioli who played an important role in the history of linear perspective.


Luca Pacioli artist unknown

Luca Pacioli
artist unknown

Luca Pacioli was born in Sansepolcro in the Duchy of Urbino in 1445.

Duchy of Urbino  Henricus Hondius 1635

Duchy of Urbino
Henricus Hondius 1635

Almost nothing is known of his background or upbringing but it can be assumed that he received at least part of his education in the studio of painter and mathematician Piero della Francesca (1415 – 1492), who like Pacioli was born in Sansepolcro.

Piero della Francesca Self Portrait

Piero della Francesca
Self Portrait

Pacioli and della Francesca were members of what is now known as the Urbino school of mathematics, as was Galileo’s patron Guidobaldo del Monte (1545 – 1607). These three Urbino mathematicians together with, Renaissance polymath, Leone Battista Alberti (1404 – 1472) all played an important role in the history of linear perspective.


Leon Battista Alberti  Artist unknown

Leon Battista Alberti
Artist unknown

Whilst still young Pacioli left Sansepolcro for Venice where he work as a mathematics tutor. Here he wrote his first book, an arithmetic textbook, around 1470. Around this time he left Venice for Rome where he lived for several months in the house of Alberti, from whom he not only learnt mathematics but also gained good connections within the Catholic hierarchy. Alberti was a Papal secretary.

In Rome Pacioli studied theology and became a Franciscan friar. From 1477 Pacioli became a peripatetic mathematics teacher moving around the courts and universities of Northern Italy, writing two more arithmetic textbooks, which like his first one were never published.

Ludovico Sforza became the most powerful man in Milan in 1476, at first as regent for his nephew Gian Galeazzo, and then, after his death in 1494, Duke of Milan.

Ludovico Sforza Zanetto Bugatto

Ludovico Sforza
Zanetto Bugatto

Ludovico was a great patron of the arts and he enticed Leonardo to come and serve him in Milan in 1482. In 1496 Pacioli became Ludivico’s court mathematicus. Leonardo and Pacioli became colleges and close friends stimulating each other over a wide range of topics.


Leonardo Francesco Melzi

Francesco Melzi

Before he went to Milan Pacioli wrote his most famous and influential book his Summa de arithmetica, geometria, proportioni et proportionalità, which he published in Venice in 1494. The Summa, as it is generally known, is a six hundred-page textbook that covers the whole range of practical mathematics, as it was known in the fifteenth-century. Pacioli was not an original mathematician and the Summa is a collection of other peoples work, however it became the most influential mathematics textbook in Europe and remained so for almost the whole of the sixteenth-century. As well as the basics of arithmetic and geometry the Summa contains the first printed accounts of double entry bookkeeping and probability, although Pacioli’s account of determining odds is wrong. From our point of view the most important aspect of the Summa is that it also contains the first extensive printed account of the mathematics of linear perspective.


Pacioli Summa Title Page

Pacioli Summa
Title Page

According to legend linear perspective in painting was first demonstrated by Fillipo Brunelleschi (1377 – 1446) in Florence early in the fifteenth-century. Brunelleschi never published an account of his discovery and this task was taken up by Alberti, who first described the construction of linear perspective in his book De pictura in 1435. Piero della Francesca wrote three mathematical treatises one on arithmetic, one on linear perspective and one on the five regular Euclidian solids. However della Francesca never published his books, which seem to have been written as textbooks for the Court of Urbino where they existed in the court library only in manuscript. Della Francesca treatment of perspective was much more comprehensive than Alberti’s.

During his time in Milan, Pacioli wrote his second major work his Divina proportione, which contains an extensive study of the regular geometrical solids with the illustrations famously drawn by his friend Leonardo.


Leonardo Polyhedra


These two books earned Pacioli a certain amount of notoriety as the Summa contains della Francesca’s book on linear perspective and the Divina proportione his book on the five regular solids both without proper attribution. In his Lives of the Most Excellent Italian Painters, Sculptors, and Architects, from Cimabue to Our Timesthe Italianartist and art historian, Giorgio Vasari (1511 – 1574)


Giorgio Vasari Self Portrait

Giorgio Vasari
Self Portrait

accused Pacioli of having plagiarised della Francesca, a not entirely fair accusation, as Pacioli does acknowledge that the entire contents of his works are taken from other authors. However whether he should have given della Francesca more credit or not Pacioli’s two works laid the foundations for all future mathematical works on linear perspective, which remained an important topic in practical mathematics throughout the sixteenth and seventeenth centuries and even into the eighteenth with many of the leading European mathematicians contributing to the genre.

With the fall of Ludovico in 1499 Pacioli fled Milan together with Leonardo travelling to Florence, by way of Mantua and Venice, where they shared a house. Although both undertook journeys to work in other cities they remained together in Florence until 1506. From 1506 until his death in his hometown in 1517 Pacioli went back to his peripatetic life as a teacher of mathematics. At his death he left behind the unfinished manuscript of a book on recreational mathematics, De viribus quantitatis, which he had compiled together with Leonardo.

Before his death Pacioli possibly played a last bit part in the history of linear perspective. This mathematical technique for providing a third dimensional to two dimensional paintings was discovered and developed by the Renaissance painters of Northern Italy in the fifteenth century, one of the artists who played a very central role in bringing this revolution in fine art to Northern art was Albrecht Dürer, who coincidentally died 6 April 1528, and who undertook two journeys to Northern Italy explicitly to learn the new methods of his Italian colleagues.

Albrecht Dürer Self Portrait

Albrecht Dürer
Self Portrait

On the second of these journey’s in 1506-7, legend has it, that Dürer met a man in Bologna who taught him the secrets of linear perspective.  It has been much speculated as to who this mysterious teacher might have been and one of the favoured candidates is Luca Pacioli but this is highly unlikely. Dürer was however well acquainted with the work of his Italian colleagues including Leonardo and he became friends with and exchanged gifts with Hasan’s favourite painter Raphael.


Filed under History of Mathematics, History of Optics, Renaissance Science, Uncategorized

Sliding to mathematical fame.

William Oughtred born on the 5th March 1575, who Newton regarded along with Christopher Wren and John Wallis as one of the three best seventeenth-century English mathematicians, was the epitome of the so-called English School of Mathematics. The English School of Mathematics is a loose historical grouping of English mathematicians stretching over several generations in the sixteenth and seventeenth centuries who propagated and supported the spread of mathematics, mostly in the vernacular, through teaching and writing at a time when the established educational institutions, schools and universities, offered little in the way of mathematical tuition. These men taught each other, learnt from each other, corresponded with each other, advertised each other in their works, borrowed from each other and occasionally stole from each other building an English language mathematical community that stretched from Robert Recorde (c. 1512 – 1558) who is regarded as its founder to Isaac Newton at the close of the seventeenth century who can be regarded as a quasi member.  Oughtred who died in 1660 spanned the middle of this period and can be considered to be one of its most influential members.

Oughtred was born at Eton College where his father Benjamin was a writing master and registrar and baptised there on 5th March 1575, which is reputedly also his birthdate. He was educated at Eton College and at King’s College Cambridge where he graduated BA in 1596 and MA in 1600. It was at Cambridge that he says he first developed his interest for mathematics having been taught arithmetic by his father.  Whilst still at Cambridge he also started what was to become his vocation, teaching others mathematics.  He was ordained priest in 1603 and appointed vicar of Shalford in Surry. In 1610 he was appointed rector of nearby Albury where he remained for the rest of his life. He married Christgift Caryll in 1606, who bore him twelve or possibly thirteen children, accounts differ. All in all Oughtred lived the life of a simple country parson and would have remained unknown to history if it had not been for his love of mathematics.

William Oughtred by Wenceslas Hollar 1646

William Oughtred
by Wenceslas Hollar 1646

Oughtred’s first claim to fame as a mathematician was as a pedagogue. He worked as a private tutor and also wrote and published one of the most influential algebra textbooks of the century his Clavis Mathematicae first published in Latin in 1631. This was a very compact introduction to symbolic algebra and was one of the first such books to be written almost exclusively in symbols, several of which Oughtred was the first to use and which are still in use today. Further Latin edition appeared in 1648, 1652, 1667 and 1698 with an English translation appearing in 1647 under the title The Key to Mathematics.

The later editions were produced by a group of Oxford mathematicians that included Christopher Wren, Seth Ward and John Wallis. Seth Ward lived and studied with Oughtred for six months and Wallis, Wren and Jonas Moore all regarded themselves as disciples, although whether they studied directly with Oughtred is not known. Wallis probably didn’t but claimed to have taught himself maths using the Clavis.

Title page Clavis Mathematicae 5th ed 1698  Ed John Wallis

Title page Clavis Mathematicae 5th ed 1698
Ed John Wallis

The Latin editions of the Clavis were read throughout Europe and Oughtred enjoyed a very widespread and very high reputation as a mathematician.

Although he always preached the importance of theory before application Oughtred also enjoyed a very high reputation as the inventor of mathematical instruments and it is for his invention of the slide rule that he is best remembered today. The international society for slide rule collectors is known as the Oughtred Society. I realise that in this age of the computer, the tablet, the smart phone and the pocket calculator there is a strong chance that somebody reading this won’t have the faintest idea what a slide rule is. I’m not going to explain although I will outline the historical route to the invention of the slide rule but will refer those interested to this website.

The Scottish mathematician John Napier and the Swiss clock and instrument maker Jobst Bürgi both invented logarithms independently of each other at the beginning of the seventeenth century although Napier published first in 1614. The basic idea had been floating around for sometime and could be found in the work of the Frenchman Nicolas Chuquet in the fifteenth century and the German Michael Stifel in the sixteenth. In other words it was an invention waiting to happen. Napier’s logarithms were base ‘e’ now called natural logarithms (that’s the ln key on your pocket calculator) and the English mathematician Henry Briggs (1561 – 1630), Gresham Professor of Geometry, thought it would be cool to have logarithms base 10 (that’s the log key on your pocket calculator), which he published in 1620. Edmund Gunter (1581 – 1626), Gresham Professor of Astronomy, who was very interested in cartography and navigation, produced a logarithmic scale on a ruler, known, not surprisingly, as the Gunter Scale or Rule, which could be read off using a pair of dividers to enable navigators to make rapid calculations on sea charts.

Briggs introduced his good friend Oughtred to Gunter, remember that bit above about teaching, learning etc. from each other, and it was Oughtred who came up with the idea of placing two Gunter Scales next to each other to facilitate calculation by sliding the one scale up and down against the other and thus the slide rule was born. Oughtred first published his invention in a pamphlet entitled The Circles of Proportion and the Horizontal Instrument in 1631, which actually describes an improved circular slide rule with the scales now on circular discs rotating about a central pin. This publication led to a very nasty dispute with Richard Delamain, a former pupil of Oughtred’s who claimed that he had invented the slide rule and not his former teacher. This led to one of those splendid pamphlet priority wars with both antagonists pouring invective over each other by the bucket load. Oughtred won the day both in his own time and in the opinion of the historians and is universally acknowledged as the inventor of the slide rule, which became the trusty companion of all applied mathematicians, engineers and physicist down the centuries. Even when I was at secondary school in the 1960s you would never see a physicist without his trusty slide rule.

It still seems strange to me that more than a whole generation has grown up with no idea what a slide rule is or what it could be used for and that Oughtred’s main claim to fame is slowly but surely sliding into the abyss of forgetfulness.


Filed under History of Computing, History of Mathematics, Renaissance Science