Category Archives: History of Mathematics

Lists!

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.

bombelli

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.

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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.

RaphaelHasanBadge

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

Leonardo
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

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.

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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.

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

An unfortunate conclusion.

On Twitter I follow three accounts that tweet daily titbits out of the history of mathematics, the Mathematical Association of America (@maanow), the British Society for the History of Mathematics (@mathshistory) and my good friend Pat Ballew (@OnThisDayinMathematics).  Yesterday both Pat and the MAA tweeted links to a brief paragraph about the Renaissance humanist scholar Andreas Dudith. The MAA’s paragraph, which I read first, was the following:

Andreas Dudith (1533-1589), mathematician and opponent of astrology, argued in a letter that observations of the comet of 1577 proved the Aristotelian explanation fallacious (for Aristotle, comets were accidental exhalations of hot air from the earth that rise in the sublunar sphere). Dudith’s use of mathematically precise observations to criticize a general physical theory of Aristotle foreshadowed Galileo’s work fifty years later.

Pat’s almost identical offering was the following:

Andreas Dudith (1533–1589), mathematician and opponent of astrology, argued in a letter that observations of the comet of 1577 proved the Aristotelian explanation fallacious (for Aristotle, comets were accidental exhalations of hot air from the earth that rise in the sublunar sphere). Dudith’s use of mathematically precise observations to criticize a general physical theory of Aristotle betokens Galileo’s work fifty years later.

Although Pat gives his source as the maths history website from V. Frederick Rickey they obviously both have a common source, namely the Dudith article in the Dictionary of Scientific Biography written by the historian of Renaissance mathematics, Paul Lawrence Rose.

The first problem with this account as presented here is that Dudith in his letter was not referring to his own observations but to those of his friend and correspondent Thaddaeus Hagecius, personal physician to the Holy Roman Emperors in Prague who was also a correspondent of Tycho Brahe and later a friend and colleague of Kepler. The second problem is that Hagecius, and through him Dudith, were by no means the only people to accept that parallax measurements showed comets to be supra-lunar thus contradicting the Aristotelian theory of comets, as seems to be implied here. Amongst others, both Tycho and Michael Maestlin, Kepler’s teacher, who were much more influential than Dudith, had also reached this conclusion. In fact much earlier in the sixteenth century, based on their observations of the 1530s comets, Gemma Frisius, Jean Pena, Girolamo Fracastoro and Gerolamo Cardano had already reached the same conclusion. In fact the intensive observations and parallax measurements of the 1577 comet were to determine if Frisius et al. were correct or not in their deductions. In his letter from 19th January 1581 Dudith is merely joining a fairly large and influential choir.

The real problem in this brief account is to be found in the conclusion. A conclusion that is to be found in the Paul Rose original:

Dudith’s use of a mathematically precise observation to criticize a general physical theory of Aristotle’s betokens the same kind of dissatisfaction with Aristotelian physical doctrines that was most eloquently expounded in the works of Galileo fifty years later.

Whilst it is true that Galileo replaced Aristotle’s doctrine on falling objects with his own mathematical laws of fall[1] obtained or at least confirmed through ingenious physical experiments his record on comets was to say the least embarrassing making the comparison here highly questionable.

In 1618 the Jesuit astronomer Orazio Grassi showed by observation and parallax measurement that the comet of that year was indeed supra-lunar driving another nail in the coffin of the Aristotelian theory of comets. Galileo, who due to illness had been unable to observe the comet, was urged by his claque to enter the arena with his opinion on the nature of comets. Galileo then famously launched an unprovoked and extremely vitriolic attack on Grassi condemning his work and defending what was basically a version of the Aristotelian theory. It was one of Galileo’s less glorious moments, far from using mathematic to criticise a doctrine of Aristotle’s Galileo was defending Aristotle’s theory of comets against an astronomer who had used mathematic to disprove it.

 

 

 


[1] It should be pointed out that the essence of Galileo’s laws of fall can be found in the work of Giambattista Bendetti, who by a strange coincidence died on 20th January 1590, a couple of decades earlier.

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Filed under History of Astronomy, History of Mathematics

Dying to make life easier for historians.

There is a clichéd view of history encouraged by bad teaching that presents the subject as the memorising of long lists of dates that somebody has designated as being significant, 55 BC, 1066, 1492, 1687, 1859, 1914 etc., etc. Now whilst in reality history is much more concerned with what happened and why it happened than with when it happened dates are the scaffolding on which historians hang up their historical facts for inspection.

When presenting biographies of scientists two key dates that the historical biographer has to remember are those of the birth and the death of her or his subject. One of my favourite Renaissance mathematici mathematics teacher, astronomer, astrologer, cartographer and globe maker, Johannes Schöner, about whom I have already blogged in the past, made life easier for historians by dying on his seventieth birthday. He was born on 16th January 1477 in Karlstadt am Main and died on 16th January 1547 in Nürnberg.  This means one only has to remember his birthdate and them simply add seventy to get his date of death.

 

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

Christmas Trilogy 2013 Part II: On Her Majesty’s Secret Service

From the modern standpoint Charles Babbage tends to be regarded as a one trick pony, “the father” of the computer. Now whilst it is true that Babbage’s obsession with his calculating engines played a very central role in much of his life, he actually applied his immense intelligence to a wide range of topics and projects. One of theses he seems to have regarded as a distraction as he tells us in his autobiography.

Deciphering is, in my opinion, one of the most fascinating of arts, and I fear I have wasted upon it more time than it deserves. I practised it in its simplest form when I was at school. The bigger boys made ciphers, but if I got hold of a few words, I usually found out the key. The consequence of this ingenuity was occasionally painful: the owners of the detected ciphers sometimes thrashed me, though the fault really lay in their own stupidity

Babbage was a cryptologist, and not just as a school boy, there is even some circumstantial evidence that like Alan Turing and John Wallis he served the powers that be as a code maker and breaker assisting Rear Admiral Sir Francis Beaufort, naval hydrographer and creator of the Beaufort Scale, during the Crimean War.

Babbage continued his schoolboy antics as an adult. He and his friend the physicist and inventor, Charles Wheatstone, took great pleasure in enciphering deciphering the coded messages in the classified advertisements in The Times. Wheatstone even going so far as send a new encoded message to a young lady, whose Oxford student friend had by this means proposed an elopement, telling her not to. In 1854 Babbage served as an expert witness in a law case. Enciphering Deciphering a letter sent by a Captain Childe to a mysterious lady. In the same year Babbage’s activities as a cryptologist reach a highpoint in an exchange of letters in the Journal of the Society of Arts concerning the so-called Vigenère cipher.

Named after Blaise de Vigenère by Bourbonnais, who had published it in his Traité de Chiffres in 1586 it was generally considered in the nineteenth century to be unbreakable. One of the most basic ciphers is the simple alphabetic substitution, named the Caesar cipher after Julius Caesar who is said to have used it, that most of us learn some time in primary school. In this cipher, each letter is replaced by a letter obtained by sliding one alphabet against another, e.g.

ABCDEFGHIJKLMNOPQRSTUVWXYZ

BCDEFGHIJKLMNOPQRSTUVWXYZA

Here Renaissance Mathematicus would become Sfobjttbodf Nbuifnbujdvt.

Such a cipher is naturally fairly easy to encipher and it is obvious that anybody wanting to indulge in serious coding would need something somewhat more complex, enter the Vigenère cipher, a polyalphabetic substitution cipher. In this each letter in the original message is fed through a different one of the 26 substitution alphabets according to a predetermined scheme. The cipher was given its final twist in that the alphabets used were chosen using a keyword known to both parties. I won’t go into details but for those interested you can find a good description in the Wikipedia article.

Before returning to Babbage it should be noted that although named after Vigenère this polyalphabetic cipher was not invented by him and we find quite a collection of significant Renaissance figures involved in its history.  The Renaissance polymath Leon Battista Alberti, who famously published the first account of linear perspective, wrote the earliest known account of the Vigenère cipher using a cipher disk in 1466, which was however first published posthumously in 1568. In a book written in 1508 but first published posthumously in 1518, the Renaissance occultist Johannes Trithemius independently discovered the Vigenère cipher in his case using a cipher tableau.  In 1553 Giovanni Battista Belaso published a pamphlet introducing the use of a key and in a work published in 1563 polymath and friend of Galileo, Giovanni Battista della Porto showed that the methods of Alberti and Trithemius were one and the same. Back to Babbage.

Over the centuries various people had (re)discovered the Vigenère cipher and in 1854 John Hall Brock Thwaites, a Bristol surgeon and dentist, claimed in a letter to the Journal of the Society of Arts to have invented an unbreakable cipher, which was in fact the keyword variation of the Vigenère cipher. This was pointed out to him in an anonymous letter from Babbage, who had almost certainly been consulted by the Society of Arts. Babbage closed his letter in his usual snide manner with the following remark:

“It may be laid down as a principle that it is never worth the trouble of trying any inscrutable cypher unless its author has himself deciphered some very difficult cypher”.

Stung by Babbage’s remark Thwaites challenged Babbage, given clear and encrypted message to supply the keywords used in the encryption. In his next letter Babbage obliged by doing just that revealing Thwaites’ keywords, two combined. He also returned the original message re-encrypted and challenged Thwaites in his turn to find his keywords, foreign supremacy. Thwaites withdrew from the unequal contest. This exchange of letters revealed Babbage to be a master cryptologist but didn’t reveal his working methods.

In fact Babbage had developed a general analytical method not only for the Vigenère cipher in question, but also for even more complex variations, using modulo arithmetic, which itself had only been first published by Gauss in 1801. Had Babbage published his results they would not only confirmed his high status as a mathematician but also established him as a leading cryptologists. In fact he did plan and make preparations for just such a publication, to be entitled The Philosophy of Deciphering, but never carried through on the project. Had he done so, his book would have been the first work of mathematical analysis in cryptology.

We now move onto the highly speculative part of this story. The mathematical method for breaking the Vigenère cipher was first published in 1863 by the German army officer Major Friedrich Wilhelm Kasiski, who received the credit for it, as a result of which the method is known as Kasiski analysis or Kasiski examination. This opens the question as to why Babbage didn’t claim credit for his discovery. Enter Beaufort.

Francis Beaufort is credited with having invented a cipher known as the Beaufort cipher that is in fact another variation of the Vigenère cipher. Now one of the puzzling facts is that there are no papers amongst Beaufort’s vast archives related to cryptology in general or the Beaufort cipher in particular, whereas a variation on the Beaufort cipher can be found amongst Babbage’s cryptology papers. Did Babbage in fact supply the Beaufort cipher and did the Admiralty, through Francis Beaufort, have access to Babbage’s methodology for solving Vigenère ciphers during the Crimean War giving them the same strategic advantage that the Allies enjoyed in terms of Enigma during the Second World War thanks to Bletchley Park? There are suggestions in Babbage’s papers and correspondence that this might indeed have been the case but no conclusive evidence. Was Babbage engaged on Her Majesty’s Secret Service as a code breaker? We will probably never know for certain but it’s nice to think that he was.

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Christmas Trilogy 2013 Part I: The Other Isaac [1].

In a recent post on John Wallis I commented on seventeenth century English mathematicians who have been largely lost to history, obscured by the vast shadow cast by Isaac Newton. One person, who has suffered this fate, possibly more than any other, was the first Lucasian Professor of Mathematics at Cambridge, and thus Newton’s predecessor on that chair, Isaac Barrow (1630 – 1677), who in popular history has been reduced to a mere footnote in the Newton mythology.

Statue of Isaac Barrow in the Chapel of Trinity College

Statue of Isaac Barrow in the Chapel of Trinity College

He was born in London in 1630 the son of John Barrow a draper. The Barrow’s were a Cambridge family notable for its many prominent scholars and theologians. Isaac father was the exception in that he had gone into trade but he was keen that his son should follow the family tradition and become a scholar.  With this aim in view the young Isaac was originally sent to Charterhouse School where he unfortunately more renowned as the school ruffian than for his learning. His father thus placed him in Felsted School in Essex, where John Wallis was also prepared for university, and where he soon turned his hand to more scholarly pursuits. Barrow’s success at school can be judged by the fact that when his father got into financial difficulties, and could no longer pay his school fees, the headmaster of the school took him out of the boarding house and lodged him in his own private dwelling free of charge and also arranged for him to earn money as tutor to William Fairfax.

In 1643 he was due to go up to Peterhouse Cambridge, where his uncle Isaac was a fellow. However his uncle was ejected from the college by the puritans and so the plan came to nought. Cut loose in society young Barrow ended up in Norfolk at the house of Edward Walpole a former schoolfellow who on going up to Cambridge decided to take Barrow with him and pay his keep. So it was that Barrow was admitted to Trinity College in 1646. Following further trials and tribulations he graduated BA in 1649 and was elected fellow shortly after. He went on to graduate MA in 1652 displaying thereby a mastery of the new philosophy. Barrow’s scholarly success was all the more remarkable, as throughout his studies he remained an outspoken Anglican High Church man and a devout royalist, things not likely to endear him to his puritan tutors.

In the 1650s Barrow devoted much of his time and efforts to the study of mathematics and the natural sciences together with a group of young scholars dedicated to these pursuits that included John Ray and Ray’s future patron Francis Willughby who had both shared the same Trinity tutor as Barrow, James Duport. Barrow embraced the mathematical and natural science of Descartes, whilst rejecting his metaphysics, as leading to atheism. He also believed students should continue to study Aristotle and the other ancients for the refinement of their language.  During this period Barrow began to study medicine, a common choice for those interested in the natural sciences, but remembering a promise made to himself whilst still at school to devote his life to the study of divinity he dropped his medical studies.

It was during this period that Barrow produced his first mathematical studies producing epitomes of both Euclid’s Elements and his Data, as well as of the known works of Archimedes, the first four books of Apollonius’ Conics and The Sphaerics of Theodosius. Barrow used the compact symbolism of William Oughtred to produce the abridged editions of these classical works of Greek mathematics. His Elements was published in 1656 and then again together with the Data in 1657. The other works were first published in the 1670s.

In 1654 a new wave of puritanism hit the English university and to avoid conflict Barrow applied for and obtained a travel scholarship leaving Cambridge in the direction of Paris in 1655. He spent eight months in Paris, which he described as, “devoid of its former renown and inferior to Cambridge!” From Paris he travelled to Florence where he was forced to extend his stay because an outbreak of the plague prevented him continuing on to Rome. In November 1656 he embarked on a ship to Smyrna, which on route was attacked by Barbary pirates, Barrow joining the crew in defending the ship acquitted himself honourably. He stayed in Smyrna for seven months before continuing to Constantinople. Although a skilled linguist fluent in eight languages Barrow made no attempt to learn Arabic, probably because of his religious prejudices against Islam, instead deepening his knowledge of Greek in order to study the church fathers.  Barrow left Constantinople in December 1658 arriving back in Cambridge, via Venice, Germany and the Netherlands, in September 1659.  It should be noted that the Interregnum was over and the Restoration of the monarchy would take place in the very near future. Unfortunately all of Barrow’s possessions including his paper from his travels were lost on the return journey, as his ship went up in flames shortly after docking in Venice

Barrow’s career, strongly supported by John Wilkins, now took off. In 1660 he was appointed Regius Professor of Greek at Cambridge followed in 1662 by his appointment as Gresham Professor of Geometry at Gresham College in London. His Gresham lectures were unfortunately lost without being published so we know little of what he taught there.  On the creation of the Lucasian Chair for Mathematics in 1663 Barrow was, at the suggestion of Wilkins, appointed as it first occupant. In 1664 he resigned both the Regius and the Gresham professorships. Meanwhile Barrow had started on the divinity trail being granted a BD in 1661 and beginning his career as a preacher.

Barrow only retained the Lucasian Chair for six years and in this time he lectured on mathematics, geometry and optics. His attitude to mathematics was strange and rather unique at the time. He was immensely knowledgeable of the new analytical mathematics possessing and having studied intently the works of Galileo, Cavalieri, Oughtred, Fermat, Descartes and many others however he did not follow them in reducing mathematics to algebra and analysis but went in the opposite directions reducing arithmetic to geometry and rejecting algebra completely. As a result his mathematical work was at one and the same time totally modern and up to date in its content whilst being totally old fashioned in its execution. Whereas his earlier Euclid remained a popular university textbook well into the eighteenth century his mathematical work as Lucasian professor fell by the wayside superseded by those who developed the new analysis. His optics lectures were a different matter. Although they were the last to be held they were the first to be published after he resigned the Lucasian chair. Pushed by that irrepressible mathematics communicator, John Collins, to publish his Lucasian Lectures Barrow prepared his optics lectures for publication assisted by his successor as Lucasian Professor, Isaac Newton, who was at the time delivering his own optics lectures, and who proof read and corrected the older Isaac’s manuscript. Building on the work of Kepler, Scheiner and Descartes Barrow’s Optics Lectures is the first work to deal mathematically with the position of the image in geometrical optics and as such remained highly influential well into the next century.

As he had once given up the study of medicine in his youth Barrow resigned the Lucasian Professorship in 1669 to devote his life to the study of divinity. His supporters, who now included an impressive list of influential bishops, were prepared to have him appointed to a bishopric but Barrow was a Cambridge man through and through and did not want to leave the college life. To solve the problem his friends had him appointed Master of Trinity instead, an appointment he retained until his tragically early death in 1677, just forty-seven years old. Following his death his collected sermons were published and it is they, rather than his mathematical work, that remain his intellectual legacy. Throughout his life all who came into contact with him acknowledge Barrow as a great scholar.

Near the beginning of this post I described Barrow as, having “been reduced to a mere footnote in the Newton mythology”. What did I mean by this statement and what exactly was the connection between the two Isaacs, apart from Barrow’s Optics Lectures? Older biographies of Newton and unfortunately much modern popular work state that Barrow was Newton’s teacher at Cambridge and that the older Isaac in realising the younger Isaac’s vast superiority as a mathematician resigned the Lucasian Chair in his favour. Both statements are myths. We don’t actually know who Newton’s tutor was but we can say with certainty that it was not Barrow. As far as can be ascertained the older Isaac first became aware of his younger colleague after Newton had graduated MA and been elected a fellow of Trinity. The two mathematicians enjoyed cordial relations with the older doing his best to support and further the career of the younger. As we have already seen above Barrow resigned the Lucasian Professorship in order to devote his live to the study and practice of divinity, however he did recommend his young colleague as his successor and Newton was duly elected to the post in 1669. Barrow also actively helped Newton in obtaining a special dispensation from King Charles, whose royal chaplain he had become, permitting him not to have to be ordained in order to hold the post of Lucasian Professor[2].


[1] On Monday I wrote that I might not be blogging for a while following Sascha untimely death. However I spent some time and effort preparing this years usual Christmas Trilogy of post and I find that writing helps to divert my attention from thoughts of him and to stop me staring at the wall. Also it’s what Sascha as general manager of this blog would have wanted.

[2] Should anyone feel a desire to learn more about Isaac Barrow I can highly recommend Before Newton: The Life and Times of Isaac Barrow, ed. Mordechai Feingold, CUP, Cambridge, 1990 from which most of the content of this post was distilled.

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

What Isaac actually asked the apple.

Yesterday on my twitter stream people were retweeting the following quote:

“Millions saw the apple fall, but Newton asked why.” —Bernard Baruch

For those who don’t know, Bernard Baruch was an American financier and presidential advisor. I can only assume that those who retweeted it did so because they believe that it is in some way significant. As a historian of science I find it is significant because it is fundamentally wrong in two different ways and because it perpetuates a false understanding of Newton’s apple story. For the purposes of this post I shall ignore the historical debate about the truth or falsity of the apple story, an interesting discussion of which you can read here in the comments, and just assume that it is true. I should however point out that in the story, as told by Newton to at least two different people, he was not hit on the head by the apple and he did not in a blinding flash of inspiration discover the inverse square law of gravity. Both of these commonly held beliefs are myths created in the centuries after Newton’s death.

Our quote above implies that of all the millions of people who saw apples, or any other objects for that matter, fall, Newton was the first or even perhaps the only one to ask why. This is of course complete and utter rubbish people have been asking why objects fall probably ever since the hominoid brain became capable of some sort of primitive thought. In the western world the answer to this question that was most widely accepted in the centuries before Newton was born was the one supplied by Aristotle. Aristotle thought that objects fall because it was in their nature to do so. They had a longing, desire, instinct or whatever you choose to call it to return to their natural resting place the earth. This is of course an animistic theory of matter attributing as it does some sort of spirit to matter to fulfil a desire.

Aristotle’s answer stems from his theory of the elements of matter that he inherited from Empedocles. According to this theory all matter on the earth consisted of varying mixtures of four elements: earth, water, fire and air. In an ideal world they would be totally separated, a sphere of earth enclosed in a sphere of water, enclosed in a sphere of air, which in turn was enclosed in a sphere of fire. Outside of the sphere of fire the heavens consisted of a fifth pure element, aether or as it became known in Latin the quintessence. In our world objects consist of mixtures of the four elements, which given the chance strive to return to their natural position in the scheme of things. Heavy objects, consisting as they do largely of earth and water, strive downwards towards the earth light objects such as smoke or fire strive upwards.

To understand what Isaac did ask the apple we have to take a brief look at the two thousand years between Aristotle and Newton.

Ignoring for a moment the Stoics, nobody really challenged the Aristotelian elemental theory, which is metaphysical in nature but over the centuries they did challenge his physical theory of movement. Before moving on we should point out that Aristotle said that vertical, upwards or downwards, movement on the earth was natural and all other movement was unnatural or violent, whereas in the heavens circular movement was natural.

Already in the sixth century CE John Philoponus began to question and criticise Aristotle’s physical laws of motion. An attitude that was taken up and extended by the Islamic scholars in the Middle Ages. Following the lead of their Islamic colleagues the so-called Paris physicists of the fourteenth century developed the impulse theory, which said that when an object was thrown the thrower imparted an impulse to the object which carried it through the air gradually being exhausted, until when spent the object fell to the ground. Slightly earlier their Oxford colleagues, the Calculatores of Merton College had in fact discovered Galileo’s mathematical law of fall: The two theories together providing a quasi-mathematical explanation of movement, at least here on the earth.

You might be wondering what all of this has to do with Isaac and his apple but you should have a little patience we will arrive in Grantham in due course.

In the sixteenth century various mathematicians such as Tartaglia and Benedetti extended the mathematical investigation of movement, the latter anticipating Galileo in almost all of his famous discoveries. At the beginning of the seventeenth century Simon Stevin and Galileo deepened these studies once more the latter developing very elegant experiments to demonstrate and confirm the laws of fall, which were later in the century confirmed by Riccioli. Meanwhile their contemporary Kepler was the first to replace the Aristotelian animistic concept of movement with one driven by a non-living force, even if it was not very clear what force is. During the seventeenth century others such as Beeckman, Descartes, Borelli and Huygens further developed Kepler’s concept of force, meanwhile banning Aristotle’s moving spirits out of their mechanistical philosophy. Galileo, Beeckman and Descartes replaced the medieval impulse theory with the theory of inertia, which says that objects in a vacuum will either remain at rest or continue to travel in a straight line unless acted upon by a force. Galileo, who still hung on the Greek concept of perfect circular motion, had problems with the straight-line bit but Beeckman and Descartes straightened him out. The theory of inertia was to become Newton’s first law of motion.

We have now finally arrived at that idyllic summer afternoon in Grantham in 1666, as the young Isaac Newton, home from university to avoid the plague, whilst lying in his mother’s garden contemplating the universe, as one does, chanced to see an apple falling from a tree. Newton didn’t ask why it fell, but set off on a much more interesting, complicated and fruitful line of speculation. Newton’s line of thought went something like this. If Descartes is right with his theory of inertia, in those days young Isaac was still a fan of the Gallic philosopher, then there must be some force pulling the moon down towards the earth and preventing it shooting off in a straight line at a tangent to its orbit. What if, he thought, the force that holds the moon in its orbit and the force that cause the apple to fall to the ground were one and the same? This frighteningly simple thought is the germ out of which Newton’s theory of universal gravity and his masterpiece the Principia grew. That growth taking several years and a lot of very hard work. No instant discoveries here.

Being somewhat of a mathematical genius, young Isaac did a quick back of an envelope calculation and see here his theory didn’t fit! They weren’t the same force at all! What had gone wrong? In fact there was nothing wrong with Newton’s theory at all but the figure that he had for the size of the earth was inaccurate enough to throw his calculations. As a side note, although the expression back of an envelope calculation is just a turn of phrase in Newton’s case it was often very near the truth. In Newton’s papers there are mathematical calculations scribbled on shopping lists, in the margins of letters, in fact on any and every available scrap of paper that happened to be in the moment at hand.

Newton didn’t forget his idea and later when he repeated those calculations with the brand new accurate figures for the size of the earth supplied by Picard he could indeed show that the chain of thought inspired by that tumbling apple had indeed been correct.

 

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Cromwell’s Code Breaker.

People who aren’t deeply cognisant with the history of seventeenth century mathematics might be forgiven for thinking that Isaac Newton was the only significant English mathematician in this century of scientific change. This is far from the truth, a fairly large group of English mathematician, now largely under the radar, made significant contributions to the discipline throughout the century. Newton, personally, listed William Oughtred, Christopher Wren and John Wallis who was born on 23 November 1616 as, in his opinion, the most important English mathematician of the century. John Wallis’ career as a mathematician was so extraordinary that one could write a novel about it; in fact somebody did. Iain Pears’ An Instance of the Fingerpost features John Wallis as one of its central characters. Why? John Wallis was a one-man Parliamentary Bletchley Park during the English Civil War, using his extraordinary mathematical talents to decipher the intercepted coded missives of the Royalist forces; he rose to fame and fortune as Cromwell’s code breaker.

John Wallis by Sir Godfrey Kneller

John Wallis by Sir Godfrey Kneller

Wallis was the third of five children of the Reverend John Wallis minister of Ashford in Kent. He received an excellent all round education that was however completely devoid of mathematics. His only contact with mathematics, as a child, was through a book that his older brother brought home from school. Already knowledgeable in Latin, Greek and Hebrew, he went up to Emmanuel College, Oxford Cambridge in 1632 where he appears to have studied everything except mathematics. In his autobiography he wrote:

But did forward prosecute it [mathematics], as a pleasing diversion at spare hours, as books of Arithmetick or others Mathematical fell occasionally in my way, without any to direct me, what books to read, or what to seek, or in what methode to proceed. For Mathematicks were not, at that time, looked upon as Accademical Learning; but the business of Traders, Merchants, Sea-men, Carpenters, land-measurers, or the like; or perhaps some Almanak-makers in London: And of more than 200 students at that time in our College, I do not know of any two that had more of Mathematicks than myself, which was but very little; having never made it my serious studie (otherwise than as a pleasant diversion) till some little time before I was designed for a Professor in it.

This brief passage says an incredible amount about the level of mathematical education in England in the middle of the seventeenth century. A state of affairs that is confirmed in the writings of many other seventeenth century English mathematicians. Even at the beginning of the eighteenth century John Arbuthnot complained that mathematics was not taught in a single English school.

Wallis graduated BA in 1637 and MA in 1640 and upon leaving university was ordained into the priesthood. He served in various positions mostly as a private chaplain. Following the death of his mother in 1643 he inherited a substantial estate and became privately wealthy, removing the necessity to work for a living although he continued to do all of his long life.  It was at a supper party in 1642 that Wallis was first shown an encrypted letter and his talent for deciphering came to the fore; a talent that was then exploited throughout the Civil War by the Parliamentarian Forces. His detractors would later accuse him of having deciphered private letters of the Royal Family, a charge that he strenuously denied.

Some time around 1647 Wallis chanced upon a copy of William Oughtred’s Clavis Mathematicae, which according to his own account he devoured in a couple of weeks. In the absence of any real formal mathematical training within the education system, budding mathematicians were forced to teach themselves or to seek the services of a private mathematical tutor of whom Oughtred, who deserves and will receive his own blog post, was by far and away the best. His Clavis Mathematicae was justifiably considered the best algebra textbook available in Europe at the time. Oughtred tutored many of the leading English mathematicians of the period and, although he taught himself from Oughtred’s book, Wallis considered himself one of Oughtred’s pupils. Later he would dedicate one of his most important books to Oughtred and also edit and publish his posthumous writings.

In 1649 Wallis was appointed Savilian Professor of Mathematics at Oxford, the previous incumbent having been removed because of his Royalist sympathies. This was an extraordinary move, as at this point in his life Wallis had received no formal mathematical training what so ever and published no mathematical works. However he was to go on for the next forty years as one of the most able incumbents this honourable chair of mathematics has ever had. Using the facilities of the university library Wallis taught himself the whole of the then mathematical curriculum and over the years published major works on a very wide range of mathematical topics.

Wallis’ work for the Parliamentarian forces and his very obvious political appointment to the Savilian Chair might have caused him major problems at the Restoration, if it had not been for his stand at the tail of Charles I. He openly opposed the execution of the King and even signed a petition against it. The result of these actions was that Charles II confirmed his appointment to the Savilian Chair and even appointed him a Royal chaplain as well as nominating him for a committee to revise the book of common prayer.

Wallis was highly active in several of the groups that would go on to form the Royal Society of which he was a founding member. In fact his autobiography contains one of the accounts on which our knowledge of the prehistory of the Royal Society is based.

Amongst his numerous mathematical publications his most important were his Treatise on Algebra and his Arthmetica infinitorum. The former contains a detailed history of the topic as well as presenting the most complete study of the subject up till that time. The later is one of the most important works on analysis before Leibniz and Newton pulled the strands of the subject together to create the calculus. Newton, who was mostly reluctant to acknowledge any of his sources, openly admitted his debt to Wallis’ masterpiece.

Wallis was fiercely nationalist in his science, editing and promoting the posthumous works of other seventeenth century English mathematicians most notably Oughtred, Harriot and Horrocks. He even accused René Descartes of having plagiarised Harriot’s algebra; an accusation that has never been entirely disproved. (Descartes suffered badly within the European mathematical community being accused of having plagiarised the law of refraction from Snel, the mechanical philosophy from Beeckman and having Newton imply indirectly that he plagiarised the correct explanation of the rainbow from Marco Antonio de Dominis.) Descartes was not the only major European philosopher to suffer the rough edge of Wallis’ tongue. He started a major dispute with Thomas Hobbes in 1655 over Hobbes’ claim to have successfully squared the circle. The dispute rumbled on with the two heavy weight Oxford scholars firing off vitriolic pamphlets at each other at regular intervals until Hobbes’ death in 1679.

Wallis was not only the leading English mathematician of the age but he also translated and published Greek scientific works as well as writing and published extensively on a wide range of other subjects including, logic, grammar and linguistics and theology. A large and robust man with an immense intellect and a forthright manner he was both respected and loathed by his contemporaries.

Lost in the vast shadow cast by Isaac Newton, John Wallis is a towering figure of seventeenth century English intellectual history, who deserves to be much better known than he is.

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Was Madge really Mad or simply a woman?

As my contribution this year to Ada Lovelace day I am writing about a woman who wasn’t just a scientist but who also wrote extensively about natural philosophy in the seventeenth century, Margaret Cavendish née Lucas.

Margaret Lucas was born in Colchester in about 1623. (Regular readers of my ramblings will immediately recognise that I’m biased, as I was born down the road from Colchester myself and went to school there). Her family were rich landed gentry but not titled. She received the usual non-education of a gentlewoman of the period. In 1642 as the civil war was cranking into gear, her brother Charles would later be executed following the siege of Colchester, she went to live with her sister in Oxford and succeeded in becoming a maid of honour at the court of Queen Henrietta then resident in Oxford. In 1644 when the Queen withdrew to Paris Margaret accompanied her.

William Cavendish (1592 – 1676) a member of the very wealthy and influential Cavendish family was an aristocrat and courtier who worked his way up the greasy pole of privilege acquiring various titles and lands until he was finally appointed Duke of Newcastle. A gentleman of leisure he was a polymath, an excellent swordsman, equestrian and soldier given to the usual pursuits of the landed gentry but he was also poet, playwright and architect who was both a disciple and a patron of Ben Jonson as well as being patron to a whole host of poets, playwright, artists and musicians.  Both William and his younger brother Charles were devotees of natural philosophy and the mathematical sciences maintaining close contact, before the civil war, with most of the leading English mathematicians and mathematical practitioners of the period, including John Pell, William Oughtred and John Wallis.

Both William and Charles served with distinction in the royalist army during the civil war but were on the losing side at the battle of Marston Moor in 1644. Forced to flee England the Cavendish brothers joined the Queen’s court in Paris where William, who had lost his first wife, met Margaret fell in love with the much younger woman and married her in 1645 against the wishes of the Queen.

In Paris William and Charles maintained a philosophical salon whose participants included René Descartes, Marin Mersenne, Pierre Gassendi and the English philosophers Kenelm Digby and Thomas Hobbes, who had been private tutor to another branch of the Cavendish family. An unusual aspect of this august discussion circle was that Margaret was not only permitted to attend but also to participate as an equal, an almost unheard of thing for a gentlewoman in this period. In 1648 the Cavendish circus decamped to Holland setting up home in Reuben’s house in Antwerp where their circle of intellectual friends included Pell, now teaching in Holland, Descartes and Constantijn Huygens. In 1660 with the Restoration they could return to England and the life of the landed gentry.

William himself wrote plays and poetry but was outstripped by his young vivacious wife who poured out a series of volumes of poetry and plays in her own right and in her own name, a more than somewhat unusual activity for a female aristocrat. However Margaret pushed the boundaries even further. Having received an education in philosophy from some of the greatest minds in Europe she began to write and publish extensively on the philosophy of science. At first tending to support Hobbes’ materialism, in her more mature writings she rejected both the traditional Aristotelian philosophy as well as the mechanical philosophies of the moderns and developed her own version of vitalism. I’m not going to bore you with an analysis of her somewhat arcane ideas but her writings on the philosophy of science are not to be rejected out of hand. In 1667 she caused a major sensation by becoming the first, and before the 19th century, only women to attend a meeting of the Royal Society. A visit made possible more by her husband’s status and wealth than her own scientific merits. This visit is mentioned together with some rather intriguing details of her correspondence on chemistry with Constantijn Huygens in a recent BBC Radio 4 Point of View by Lisa Jardine.

Having briefly sketched the life of Margaret Cavendish I can now explain the title of this post. Although the habit seems to be dying out Margaret Cavendish was for a long time almost universally referred to, as Mad Madge and it was certainly not meant as a compliment. I know of at least two different explanations for this less than flattering sobriquet. One source has the following to say on the subject:

Margaret was viewed by her contemporaries as being rather eccentric. She was extravagent and flirtatious, accused of using speech full of ‘oaths and obscenity’, and was noted for her unusual sense of fashion. This reputation for eccentricity survives today, when Margaret is widely referred to as ‘Mad Madge’.

Now both of the Cavendish brother, Descartes and Digby were all professional soldiers and it would not surprise me if the language of their discussion, when the nights were long and the bottles almost empty, sometimes resembled that of the barrack room rather than the schools and that Margaret learnt to hold her own in this heady atmosphere. Now the above description could, with a little modification, equally be applied to Margaret’s near contemporary Edmond Halley but nobody refers to him as Loony Eddy!

The other explanation is that Margaret is so referred to because of her unladylike passion for science and its philosophy. Kenelm Digby her Paris companion, who also like Margaret ran a chemistry laboratory and at the same time as she was writing and publishing her tracts on vitalism Digby was publishing his on his strange amalgam of Aristotelian and Cartesian philosophies that enjoyed a certain vogue in the early years of the Royal society. Both philosophies are now out of style and appear to us rather strange but nobody refers to Digby as Krazy Kenelm!

I think Margaret Cavendish gets called Mad Madge for daring to compete in a man’s world. She gets denigrated not because of her outlandish behaviour or her passion for science but simply because she was a woman who had these attributes. I think we should no longer call her Mad Madge but respect and honour Margaret Cavendish as an intelligent and able woman who was a pioneering female philosopher of science at a time when this was an exclusively male occupation.

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