Category Archives: History of Mathematics

Bertrand Russell did not write Principia Mathematica

Yesterday would have been Bertrand Russell’s 144th birthday and numerous people on the Internet took notice of the occasion. Unfortunately several of them, including some who should know better, included in their brief descriptions of his life and work the fact that he was the author of Principia Mathematica, he wasn’t. At this point some readers will probably be thinking that I have gone mad. Anybody who has an interest in the history of modern mathematics and logic knows that Bertrand Russell wrote Principia Mathematica. Sorry, he didn’t! The three volumes of Principia Mathematica were co-authored by Alfred North Whitehead and Bertrand Russell.


Now you might think that I’m just splitting hairs but I’m not. If you note the order in which the authors are named you will observe that they are not listed alphabetically but that Whitehead is listed first, ahead of Russell. This is because Whitehead being senior to Russell, in both years and status within the Cambridge academic hierarchy, was considered to be the lead author. In fact Whitehead had been both Russell’s teacher, as an undergraduate, and his examiner in his viva voce, where he in his own account gave Russell a hard time because he knew that it was the last time that he would be his mathematical superior.

Alfred North Whitehead

Alfred North Whitehead

Both of them were interested in metamathematics and had published books on the subject: Whitehead’s A Treatise on Universal Algebra (1898) and Russell’s The Principles of Mathematics (1903). Both of them were working on second volumes of their respective works when they decided to combine forces on a joint work the result of the decision being the monumental three volumes of Principia Mathematica (Vol. I, 1910, Vol. II, 1912, Vol. III, 1913). According to Russell’s own account the first two volumes where a true collaborative effort, whilst volume three was almost entirely written by Whitehead.

Bertrand Russell 1907 Source: Wikimedia Commons

Bertrand Russell 1907
Source: Wikimedia Commons

People referring to Russell’s Principia Mathematica instead of Whitehead’s and Russell’s Principia Mathematica is not new but I have the feeling that it is becoming more common as the years progress. This is not a good thing because it is a gradual blending out, at least on a semi-popular level, of Alfred Whitehead’s important contributions to the history of logic and metamathematics. I think this is partially due to the paths that their lives took after the publication of Principia Mathematica.

The title page of the shortened version of the Principia Mathematica to *56 Source: Wikimedia Commons

The title page of the shortened version of the Principia Mathematica to *56
Source: Wikimedia Commons

Whilst Russell, amongst his many other activities, remained very active at the centre of the European logic and metamathematics community, Whitehead turned, after the First World War, comparatively late in life, to philosophy and in particular metaphysics going on to found what has become known as process philosophy and which became particularly influential in the USA.

In history, as in academia in general, getting your facts right is one of the basics, so if you have occasion to refer to Principia Mathematica then please remember that it was written by Whitehead and Russell and not just by Russell and if you are talking about Bertrand Russell then he was co-author of Principia Mathematica and not its author.


Filed under History of Logic, History of Mathematics

Isaac and the apple – the story and the myth

The tale of Isaac Newton and the apple is, along with Archimedes’ bath time Eureka-ejaculation and Galileo defiantly mumbling ‘but it moves’ whilst capitulating before the Inquisition, is one of the most widely spread and well known stories in the history of science. Visitors to his place of birth in Woolsthorpe get to see a tree from which the infamous apple is said to have fallen, inspiring the youthful Isaac to discover the law of gravity.

The Woolsthorpe Manor apple tree Source:Wikimedia Commons

The Woolsthorpe Manor apple tree
Source:Wikimedia Commons

Reputed descendants of the tree exist in various places, including Trinity College Cambridge, and apple pips from the Woolsthorpe tree was taken up to the International Space Station for an experiment by the ‘first’ British ISS crew member, Tim Peake. Peake’s overalls also feature a Principia patch displaying the apple in fall.

Tim Peake's Mission Logo

Tim Peake’s Mission Logo

All of this is well and good but it leads automatically to the question, is the tale of Isaac and the apple a real story or is it just a myth? The answer is that it is both.

Modern historians of Early Modern science tend to contemptuously dismiss the whole story as a myth. One who vehemently rejects it is Patricia Fara, who is an expert on Newtonian mythology and legend building having researched and written the excellent book, Newton: The Making of Genius[1]. In her Science: A Four Thousand Year History she has the following to say about the apple story[2]:

More than any other scientific myth, Newton’s falling apple promotes the romantic notion that great geniuses make momentous discoveries suddenly and in isolation […] According to simplistic accounts of its [Principia’s] impact, Newton founded modern physics by introducing gravity and simultaneously implementing two major transformations in methodology: unification and mathematization. By drawing a parallel between an apple and the Moon, he linked an everyday event on Earth with the motion of the planets through the heavens, thus eliminating the older, Aristotelian division between the terrestrial and celestial realms.


Although Newton was undoubtedly a brilliant man, eulogies of a lone genius fail to match events. Like all innovators, he depended on the earlier work of Kepler, Galileo, Descartes and countless others […]


The apple story was virtually unknown before Byron’s time. [Fara opens the chapter with a Byron poem hailing Newton’s discovery of gravity by watching the apple fall].

Whilst I would agree with almost everything that Fara says, here I think she is, to quote Kepler, guilty of throwing out the baby with the bath water. But before I explain why I think this let us pass review of the myth that she is, in my opinion, quite rightly rejecting.

The standard simplistic version of the apple story has Newton sitting under the Woolsthorpe Manor apple tree on a balmy summer’s day meditation on mechanics when he observes an apple falling. Usually in this version the apple actually hits him on the head and in an instantaneous flash of genius he discovers the law of gravity.

This is of course, as Fara correctly points out, a complete load of rubbish. We know from Newton’s notebooks and from the draughts of Principia that the path from his first studies of mechanics, both terrestrial and celestial, to the finished published version of his masterpiece was a very long and winding one, with many cul-de-sacs, false turnings and diversions. It involved a long and very steep learning curve and an awful lot of very long, very tedious and very difficult mathematical calculations. To modify a famous cliché the genius of Principia and the theories that it contains was one pro cent inspiration and ninety-nine pro cent perspiration.

If all of this is true why do I accuse Fara of throwing out the baby with the bath water? I do so because although the simplistic story of the apple is a complete myth there really was a story of an apple told by Newton himself and in the real versions, which differ substantially from the myth, there is a core of truth about one step along that long and winding path.

Having quoted Fara I will now turn to, perhaps Newton’s greatest biographer, Richard Westfall. In his Never at Rest, Westfall of course addresses the apple story:

What then is one to make of the story of the apple? It is too well attested to be thrown out of court. In Conduitt’s version one of four independent ones, …

Westfall tells us that the story is in fact from Newton and he told to on at least four different occasions to four different people. The one Westfall quotes is from John Conduitt, who was Newton’s successor at the Royal Mint, married his niece and house keeper Catherine Barton and together with her provided Newton with care in his last years. The other versions are from the physician and antiquarian William Stukeley, who like Newton was from Lincolnshire and became his friend in the last decade of Newton’s life, the Huguenot mathematician Abraham DeMoivre, a convinced Newtonian and Robert Greene who had the story from Martin Folkes, vice-president of the Royal Society whilst Newton was president. There is also an account from Newton’s successor as Lucasian professor, William Whiston, that may or may not be independent. The account published by Newton’s first published biographer, Henry Pemberton, is definitely dependent on the accounts of DeMoivre and Whiston. The most well known account is that of Voltaire, which he published in his Letters Concerning the English Nation, London 1733 (Lettres philosophiques sur les Anglais, Rouen, 1734), and which he says he heard from Catherine Conduitt née Barton. As you can see there are a substantial number of sources for the story although DeMoivre’s account, which is very similar to Conduitt’s doesn’t actually mention the apple, so as Westfall says to dismiss it out of hand is being somewhat cavalier, as a historian.

To be fair to Fara she does quote Stukeley’s version before the dismissal that I quoted above, so why does she still dismiss the story. She doesn’t, she dismisses the myth, which has little in common with the story as related by the witnesses listed above. Before repeating the Conduitt version as quoted by Westfall we need a bit of background.

In 1666 Isaac, still an undergraduate, had, together with all his fellow students, been sent down from Cambridge because of an outbreak of the plague. He spent the time living in his mother’s house, the manor house in Woolsthorpe, teaching himself the basics of the modern terrestrial mechanics from the works of Descartes, Huygens and the Salisbury English translation of Galileo’s Dialogo. Although he came nowhere near the edifice that was the Principia, he did make quite remarkable progress for a self-taught twenty-four year old. It was at this point in his life that the incident with the apple took place. We can now consider Conduitt’s account:

In the year 1666 he retired again from Cambridge … to his mother in Lincolnshire & whilst he was musing in a garden it came to his thought that the power of gravity (wch brought an apple from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much further than was normally thought. Why not as high as the moon said he to himself & if so that must influence her motion & and perhaps retain her in her orbit, where-upon he fell to calculating what would be the effect of this supposition but being absent from books & taking common estimate in use among Geographers & our seamen before Norwood had measured the earth, that 60 English miles were contained in one degree latitude on the surface of the Earth his computation did not agree with his theory & inclined him to entertain a notion that together with the force of gravity there might be a mixture of that force wch the moon would have if it was carried along in a vortex…[3]

As you can see the account presented here by Conduitt differs quite substantially from the myth. No tree, no apple on the head, no instantaneous discovery of the theory of gravity. What we have here is a young man who had been intensely studying the theory of forces, in particular forces acting on a body moving in a circle, applying what he had learnt to an everyday situation the falling apple and asking himself if those forces would also be applicable to the moon. What is of note here is the fact that his supposition didn’t work out. Based on the data he was using, which was inaccurate, his calculations showed that the forces acting on the apple and those acting on the moon where not the same! An interesting thought but it didn’t work out. Oh well, back to the drawing board. Also of note here is the reference to a vortex, revealing Newton to be a convinced Cartesian. By the time he finally wrote the Principia twenty years later he had turned against Descartes and in fact Book II of Principia is devoted to demolishing Descartes’ vortex theory.

In 1666 Newton dropped his study of mechanics for the meantime and moved onto optics, where his endeavours would prove more fruitful, leading to his discoveries on the nature of light and eventually to his first publication in 1672, as well as the construction of his reflecting telescope.

The Newtonian Reflector Source: Wikimedia Commons

The Newtonian Reflector
Source: Wikimedia Commons

Over the next two decades Newton developed and extended his knowledge of mechanics, whilst also developing his mathematical skills so that when Halley came calling in 1684 to ask what form a planetary orbit would take under an inverse squared law of gravity, Newton was now in a position to give the correct answer. At Halley’s instigation Newton now turned that knowledge into a book, his Principia, which only took him the best part of three years to write! As can be seen even with this briefest of outlines there was definitely nothing instantaneous or miraculous about the creation of Newton’ masterpiece.

So have we said all that needs to be said about Newton and his apple, both the story and the myth? Well no. There still remains another objection that has been raised by historians, who would definitely like to chuck the baby out with the bath water. Although there are, as noted above, multiple sources for the apple-story all of them date from the last decade of Newton’s life, fifty years after the event. There is a strong suspicion that Newton, who was know to be intensely jealous of his priorities in all of his inventions and discoveries, made up the apple story to establish beyond all doubt that he and he alone deserved the credit for the discovery of universal gravitation. This suspicion cannot be simply dismissed as Newton has form in such falsification of his own history. As I have blogged on an earlier occasion, he definitely lied about having created Principia using the, from himself newly invented, calculus translating it back into conventional Euclidian geometry for publication. We will probably never know the final truth about the apple-story but I for one find it totally plausible and am prepared to give Isaac the benefit of the doubt and to say he really did take a step along the road to his theory of universal gravitation one summer afternoon in Woolsthorpe in the Year of Our Lord 1666.

[1] Patricia Fara, Newton: The Making of Genius, Columbia University Press, 2002

[2] Patricia Fara, Science: A Four Thousand Year History, ppb. OUP, 2010, pp. 164-165

[3] Richard S. Westfall, Never at Rest: A Biography of Isaac Newton, ppb. CUP, 1980 p. 154


Filed under History of Astronomy, History of Mathematics, History of Optics, History of Physics, History of science, Myths of Science, Newton

Well no, actually he didn’t.

Ethan Siegel has written a reply to my AEON Galileo opinion piece on his Forbes blog. Ethan makes his opinion very clear in the title of his post, Galileo Didn’t Invent Astronomy, But He DID Invent Mechanical Physics! My response is also contained in my title above and no, Galileo did not invent mechanical physics. For a change we’ll start with something positive about Galileo, his inclined plane experiments to determine the laws of fall, the description of which form the bulk of Ethan’s post, are in fact one of the truly great pieces of experimental physics and are what makes Galileo justifiably famous. However the rest of Ethan’s post leaves much to be desired.

Ethan starts off by describing the legendary Leaning Tower of Pisa experiment, in which Galileo supposedly dropped two ball of unequal weight of the tower and measured how long they took to fall. The major problem with this is that Galileo almost certainly never did carry out this experiment, however both John Philoponus in the sixth century CE and Simon Stevin in 1586 did so, well before Galileo considered the subject. The laws of fall were also investigated theoretically by the so-called Oxford Calculatores, who developed the mean speed theory, the foundation of the laws of fall, and the Paris Physicists, who represented the results graphically, both in the fourteenth century CE. Galileo knew of the work of John Philoponus, the Oxford Calculatores and the Paris Physicists, even using the same graph to represent the laws of fall in his Two New Sciences, as Oresme had used four hundred years earlier. In the sixteenth centuries the Italian mathematician Tartaglia investigated the path of projectiles, publishing the results in his Nova Scientia, his work was partially validated, partially refuted by Galileo. His landsman Benedetti anticipated most of Galileo’s results on the laws of fall. With the exception of Stevin’s work Galileo knew of all this work and built his own researches on it thus rather challenging Ethan’s claim that Galileo invented mechanical physics.

Galileo’s central achievement was to provide empirical proof of the laws of fall with his ingenious ramp experiments but even here there are problems. Galileo’s results are simply too good, not displaying the expected experimental deviations, leading Alexander Koyré, the first great historian of Galileo’s work, to conclude that Galileo never did the experiments at all. The modern consensus is that he did indeed do the experiments but probably massaged his results, a common practice. The second problem is that any set of empirical results requires confirmation by other independent researchers. Mersenne, a great supporter and propagator of Galileo’s physics, complains of the difficulties of reproducing Galileo’s experimental results and it was first Riccioli, who finally succeeded in doing so, publishing the results in 1651.

A small complaint is Ethan’s claim that Galileo’s work on the laws of fall “was the culmination of a lifetime of work”. In fact although Galileo first published his Two New Sciences in 1638 his work on mechanics was carried out early in his life and completed well before he made his telescopic discoveries.

The real problem with Ethan’s post is what follows the quote above, he writes:

…and the equations of motion derived from Newton’s laws are essentially a reformulation of the results of Galileo. Newton indeed stood on the shoulders of giants when he developed the laws of gravitation and mechanics, but the biggest titan of all in the field before him was Galileo, completely independent of what he contributed to astronomy.

This is quite simply wrong. After stating his first two laws of motion in the Principia Newton writes:

The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds. By means of the first two laws and the first two corollaries Galileo found that the decent of heavy bodies is the squared ratio of the time that the motion of projectiles occurs in a parabola, as experiment confirms, except insofar as these motion are somewhat retarded by the resistance of the air.

As Bernard Cohen points out, in the introduction to his translation of the Principia from which I have taken the quote, this is wrong because, Galileo certainly did not know Newton’s first law. As to the second law, Galileo would not have known the part about change in momentum in the Newtonian sense, since this concept depends on the concept of mass which was invented by Newton and first made public in the Principia.

I hear Galileo’s fans protesting that Newton’s first law is the law of inertia, which was discovered by Galileo, so he did know it. However Galileo’s version of the law of inertia is flawed, as he believes natural unforced motion to be circular and not linear. In fact Newton takes his first law from Descartes who in turn took it from Isaac Beeckman. Newton’s Principia, or at least his investigation leading up to it, are in fact heavily indebted to the work of Descartes rather than that of Galileo and Descartes in turn owes his greatest debts in physics to the works of Beeckman and Stevin and not Galileo.

An interesting consequence of Newton’s false attribution to Galileo in the quote above is that it shows that Newton had almost certainly never read Galileo’s masterpiece and only knew of it through hearsay. Galileo’s laws of fall are only minimally present in the Principia and then only mentioned in passing as asides, whereas the parabola law occurs quite frequently whenever Newton is resolving forces in orbits but then only as Galileo has shown.

One small irony remains in Ethan’s post. He loves to plaster his efforts with lots of pictures and diagrams and videos. This post does the same and includes a standard physics textbook diagram showing the force vectors of a heavy body sliding down an inclined plane. You can search Galileo’s work in vain for a similar diagram but you will find an almost identical one in the work of Simon Stevin, who worked on physical mechanics independently of and earlier than Galileo. Galileo made some very important contributions to the development of mechanical physics but he certainly didn’t invent the discipline.


Filed under History of Mathematics, History of Physics, Myths of Science, Newton

The Reformation, Astrology, and Mathematics in Schools and Universities.

It is one of the ironies of the medieval universities that mathematics played almost no role in undergraduate education. It is ironical because the curriculum was nominally based on the seven liberal arts of which the mathematical sciences – arithmetic, geometry, music and astronomy – formed one half, the quadrivium. Although the quadrivium was officially a large part of the curriculum in reality the four mathematical disciplines were paid little attention and hardly taught at all. This only began to change in the fifteenth century with the rise of astro-medicine or iatromathematics, to give it its formal name. With the rise of this astrology-based medicine the humanist universities of Northern Italy and Kraków introduced chairs of mathematics to teach astrology to their students of medicine. This of course entailed first teaching mathematics and then astronomy in order to be able to do astrology and thus mathematics gained a first foothold in the European universities. Ingolstadt became the first German university to introduce a chair for mathematics, also for teaching astrology to medical students, in the 1470s. It became an important centre for seeding new chairs at other universities with its graduates. Stabius and Stiborius going from there to Vienna with Celtis, for example. However there was no systematic introduction of mathematics into the university curriculum as of yet, this would first come as a result of the Reformation and the educational reforms of Philip Melanchthon.

Melanchthon in 1526: engraving by Albrecht Dürer Translation of Latin caption: «Dürer was able to draw Philip’s face, but the learned hand could not paint his spirit». Source: Wikimedia Commons

Melanchthon in 1526: engraving by Albrecht Dürer Translation of Latin caption: «Dürer was able to draw Philip’s face, but the learned hand could not paint his spirit».
Source: Wikimedia Commons

Melanchthon was born Philip Schwartzerdt in Bretten near Karlsruhe on 16 February 1497. A great nephew of Johann Reuchlin a leading humanist scholar Philip changed his name to Melanchthon, a literal Greek translation of his German name, which means black earth, at Reuchlin’s suggestion. Melanchthon was a child prodigy who would grow up to be Germany’s greatest humanist scholar. He studied at Heidelberg University where he was denied his master degree in 1512 on account of his youth. He transferred to Tübingen where he came under the influence of Johannes Stöffler, one of those Ingolstadt graduates, a leading and highly influential mathematician/astrologer.

Johannes Stöffler Source Wikimedia Commons

Johannes Stöffler
Source Wikimedia Commons

The cosmograph Sebastian Münster was another of Stöffler’s famous pupils. Stöffler also has a great influence on several of the Nürnberger mathematician-astronomers, especial Johannes Schöner and Georg Hartmann. Under Stöffler’s influence Melanchthon became a passionate supporter of astrology.

On Reuchlin’s recommendation Melanchthon became professor of Greek at Luther’s University of Wittenberg at the age of twenty-one and thus a central figure in the Reformation. One of the major problems faced by the reformers was the fact that the education system was totally in the hands of the Catholic Church, which meant that they had to start from scratch and create their own school and university system; this task was taken on by Melanchthon, who became Luther’s Preceptor Germania, Germany’s Schoolmaster.

Because of his own personal passion for astrology Melanchthon introduced mathematics into the curriculum of all the Lutheran schools and universities. He invented a new type of school on a level between the old Church Latin schools and the universities that were devised to prepare their pupils for a university education. The very first of these was the Eigidien Oberschule in Nürnberg, which opened in 1526 with Johannes Schöner, as its first professor for mathematics.


These type of school created by Melanchthon would become the Gymnasium, still today the highest level secondary schools in the German education system.

In Wittenberg he appointed Johannes Volmar (1480-1536) professor for the higher mathematic, music and astronomy, and Jakob Milich (1501- 1559) professor for the lower mathematic, arithmetic and geometry, in 1525. Their most famous students were Erasmus Reinhold, who followed Volmar on the chair for higher mathematics when he died in 1536, and Georg Joachim Rheticus, who followed Milich on the chair for lower mathematics, in the same year when Milich became professor for medicine. Schöner, Reinhold and Rheticus were not the only mathematicians supported by Melanchthon, who played an important role in the dissemination of the heliocentric astronomy. Although following Melanchthon’s lead these Protestant mathematicians treated the heliocentric hypothesis in a purely instrumentalist manner, i.e. it is not true but is mathematically useful, they taught it in their university courses alongside the geocentric astronomy.

As a result of Melanchthon’s passion for astrology the Lutheran Protestant schools and universities of Europe all had departments for the study of mathematics headed by qualified professors. The Catholic schools and universities would have to wait until the end of the sixteenth century before Christoph Clavius did the same for them, although his motivation was not astrology. Sadly Anglican England lagged well behind the continent with Oxford first appointing professors for geometry and astronomy in the 1620s at the bequest of Henry Savile, who had had to go abroad to receive his own mathematical education. Cambridge only followed suit with the establishment of the Lucasian Chair in 1663, whose first occupant was Isaac Barrow followed by that other Isaac, Newton. In 1705 John Arbuthnot could still complain in an essay that there was not one single school in England that taught mathematics.





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

Christoph and the Calendar

The first substantive history of science post that I wrote on this blog was about the Jesuit mathematician and astronomer Christoph Clavius. I wrote this because at the time I was preparing a lecture on the life and work of Clavius to be held in his hometown Bamberg. Clavius is one of my local history of science celebrities and over the years I have become the local default Clavius expert and because of his involvement in the Gregorian calendar reform of 1572 I have also become the local default expert on that topic too.

Christoph Clavius

Christoph Clavius

All of this means that I have become very sensitive to incorrect statements about either Clavius or the Gregorian calendar reform and particularly sensitive to false statements about Clavius’ involvement in the latter. Some time back the Atlas Obscura website had a ‘time week’ featuring a series of blog post on the subject of time one of which, When The Pope Made 10 Days Disappear, was about the Gregorian calendar reform and contained the following claim:

The new lead astronomer on the project, Jesuit prodigy Christopher Clavius, considered this and other proposals for five years.

The brief statement contains three major inaccuracies, the most important of which, is that Clavius as not the lead astronomer, or lead anything else for that matter, on the project. This is a very widespread misconception and one to which I devote a far amount of time when I lecture on the subject, so I thought I would clear up the matter in a post. Before doing so I would point out that I have never come across any other reference to Clavius as a prodigy and there is absolutely nothing in his biography to suggest that he was one. That was the second major inaccuracy for those who are counting.

Before telling the correct story we need to look at the wider context as presented in the article before the quote I brought above we have the following:

A hundred years later, Pope Gregory XIII rolled up his sleeves and went for it in earnest. After a call for suggestions, he was brought a gigantic manuscript. This was the life’s work of physician Luigi Lilio, who argued for a “slow 10-day correction” to bring things back into alignment, and a new leap year system to keep them that way. This would have meant that years divisible by 100 but not by 400 (e.g. 1800, 1900, and 2100) didn’t get the extra day, thereby shrinking the difference between the solar calendar and the Earthly calendar down to a mere .00031 days, or 26 seconds.

Luigi LIlio Source: Wikimedia Commons

Luigi LIlio
Source: Wikimedia Commons

This is correct as far as it goes, although there were two Europe wide appeals for suggestions and we don’t actually know how many different suggestions were made as the relevant documents are missing from the Vatican archives. It should also be pointed out the Lilio was a physician/astronomer/astrologer and not just simply a physician. Whether or not his manuscript was gigantic is not known because it no longer exists. Having decided to consider Lilio’s proposal this was not simply passed on to Christoph Clavius, who was a largely unknown mathematicus at the time, which should be obvious to anybody who gives more than five minutes thought to the subject.

The problem with the calendar, as far as the Church was concerned, was that they were celebrating Easter the most important doctrinal festival in the Church calendar on the wrong date. This was not a problem that could be decided by a mere mathematicus, at a time when the social status of a mathematicus was about the level of a bricklayer, it was far too important for that. This problem required a high-ranking Church commission and one was duly set up. This commission did not consider the proposal for five years but for at least ten and possibly more, again we are not sure due to missing documents. It is more than likely that the membership of the commission changed over the period of its existence but because we don’t have the minutes of its meetings we can only speculate. What we do have is the signatures of the nine members of the commission who signed the final proposal that was presented to the Pope at the end of their deliberations. It is to these names that we will now turn our attention.

The names fall into three distinct groups of three of which the first consists of the high-ranking clerics who actually lead this very important enquiry into a fundamental change in Church doctrinal practice. The chairman of the committee was of course a cardinal,Guglielmo Sirleto (1514–1584) a distinguished linguist and from 1570 Vatican librarian.

Cardinal Guglielmo Sirleto Source: Wikimedia Commons

Cardinal Guglielmo Sirleto
Source: Wikimedia Commons

The vice chairman was Bishop Vincenzo Lauro (1523–1592) a Papal diplomat who was created cardinal in 1583. Next up was Ignatius Nemet Aloho Patriarch of Antioch and head of the Syriac Orthodox Church till his forced resignation in 1576. Ignatius was like his two Catholic colleagues highly knowledgeable of astronomy and was brought into the commission because of his knowledge of Arabic astronomy and also to try to make the reform acceptable to the Orthodox Churches. The last did not function as the Orthodox Churches initially rejected the reform only adopting it one after the other over the centuries with the exception of the Russian Eastern Orthodox Churches, whose church calendar is still the Julian one, which is why they celebrate Christmas on 6 7 January.

Our second triplet is a mixed bag. First up we have Leonardo Abela from Malta who functioned as Ignatius’ translator, he couldn’t speak Latin, and witnessed his signature on the commissions final report. He is followed by Seraphinus Olivarius an expert lawyer, whose role was to check that the reform did not conflict with any aspects of cannon law. The third member of this group was Pedro Chacón a Spanish mathematician and historian, whose role was to check that the reform was in line with the doctrines of the Church Fathers.

Our final triplet consists of what might be termed the scientific advisors. Heading this group is Antonio Lilio the brother of Luigi and like his brother a physician and astronomer. He was here to elucidate Luigi’s plan, as Luigi was already dead. The lead astronomer, to use the Atlas Obscura phase, was the Dominican monk Ignazio Danti (1536–1582) mathematician, astronomer, cosmographer, architect and instrument maker.

Ignazio Danti Source: Wikimedia Commons

Ignazio Danti
Source: Wikimedia Commons

In a distinguished career Danti was cosmographer to Cosimo I, Duke of Tuscany whilst professor of mathematics at the university of Pissa, professor of mathematics at the University of Bologna and finally pontifical mathematicus in Rome. For the Pope Danti painted the Gallery of Maps in the Cortile del Belvedere in the Vatican Palace and deigned and constructed the instruments in the Sundial Rome of the Gregorian Tower of Tower of Winds above the Gallery of Maps.

Map of Italy, Corsica and Sardinia - Gallery of Maps - Vatican Museums. Source: Wikimedia Commons

Map of Italy, Corsica and Sardinia – Gallery of Maps – Vatican Museums.
Source: Wikimedia Commons

After the calendar reform the Pope appointed him Bishop of Altari. Danti was one of the leading mathematical practitioners of the age, who was more than capable of supplying all the scientific expertise necessary for the reform, so what was the role of Christoph Clavius the last signer of the commission’s recommendation.

The simple answer to this question is that we don’t know; all we can do is speculate. When Clavius (1538–1612) first joined the commission he was, in comparison to Danti, a relative nobody so his appointment to this high level commission with its all-star cast is somewhat puzzling. Apart from his acknowledged mathematical skills it seems that his membership of the Jesuit Order and his status as a Rome insider are the most obvious reasons. Although relative young the Jesuit Order was already a powerful group within the Church and would have wanted one of theirs in such a an important commission. The same thought concerns Clavius’ status as a Rome insider. The Church was highly fractional and all of the other members of the commission came from power bases outside of Rome, whereas Clavius, although a German, as professor at the Collegio Romano counted as part of the Roman establishment, thus representing that establishment in the commission. It was probably a bit of all three reasons that led to Clavius’ appointment.

Having established that Clavius only had a fairly lowly status within the commission how did the very widespread myth come into being that he was somehow the calendar reform man? Quite simply after the event he did in fact become just that.

When Pope Gregory accepted the recommendations of the commission and issued the papal bull Inter gravissimas on 24 February 1582, ordering the introduction of the new calendar on 4 October of the same year,


he granted Antonio Lilio an exclusive licence to write a book describing the details of the calendar reform and the modifications made to the process of calculating the date of Easter. The sales of the book, which were expected to be high, would then be the Lilio family’s reward for Luigi Lilio having created the mathematical basis of the reform. Unfortunately Antonio Lilio failed to deliver and after a few years the public demand for a written explanation of the reform had become such that the Pope commissioned Clavius, who had by now become a leading European astronomer and mathematician, to write the book instead. Clavius complied with the Pope’s wishes and wrote and published his Novi calendarii romani apologia, Rome 1588, which would become the first of a series of texts explaining and defending the calendar reform. The later was necessary because the reform was not only attacked on religious grounds by numerous Protestants, but also on mathematical and astronomical grounds by such leading mathematicians as François Viète and Michael Maestlin. Over the years Clavius wrote and published several thousand pages defending and explicating the Gregorian calendar reform and it is this work that has linked him inseparably with the calendar reform and not his activities in the commission.


Filed under History of Astronomy, History of Mathematics, History of science, Local Heroes, Renaissance Science

The Arch-Humanist

The name Conrad Celtis is not one that you’ll find in most standard books on the history of mathematics, which is not surprising as he was a Renaissance humanist scholar best known in his lifetime as a poet. However, Celtis played an important role in the history of mathematics and is a good example of the fact that if you really wish to study the evolution of the mathematical sciences it is necessary to leave the narrow confines of the mathematics books.

Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507 Source: Wikimedia Commons

Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507
Source: Wikimedia Commons

Born Konrad Bickel or Pyckell, (Conrad Celtis was his humanist pseudonym) the son of a winemaker, in Franconian Wipfield am Main near Schweinfurt on 1 February 1459, he obtained his BA at the University of Cologne in 1497. Unsatisfied with the quality of tuition in Cologne he undertook the first of many study journeys, which typified his life, to Buda in 1482, where he came into contact with the humanist circle on the court of Matthias Corvinus, the earlier patron of Regiomontanus. 1484 he continued his studies at the University of Heidelberg specialising in poetics and rhetoric, learning Greek and Hebrew and humanism as a student of Rudolf Agricola, a leading Dutch early humanist scholar. Celtis obtained his MA in 1485. 1486 found him underway in Italy, where he continued his humanist studies at the leading Italian universities and in conversation with many leading humanist scholars. Returning to Germany he taught poetics at the universities of Erfurt, Rostock and Leipzig and on 18 April 1487 he was crowned Poet Laureate by Emperor Friedrich III in Nürnberg during the Reichstag. In Nürnberg he became part of the circle of humanists that produced the Nürnberger Chronicle to which he contributed the section on the history and geography of Nürnberg. It is here that we see the central occupation of Celtis’ life that brought him into contact with the Renaissance mathematical sciences.

During his time in Italy he suffered under the jibes of his Italian colleges who said that whilst Italy had perfect humanist credentials being the inheritors of the ancient Roman culture, Germany was historically a land of uncultured barbarians. This spurred Celtis on to prove them wrong. He set himself the task of researching and writing a history of Germany to show that its culture was the equal of Italy’s. Celtis’ concept of history, like that of his Renaissance contemporaries, was more a mixture of our history and geography the two disciplines being regarded as two sides of the same coin. Geography being based on Ptolemaeus’ Geographia (Geographike Hyphegesis), which of course meant cartography, a branch of the mathematical sciences.

Continuing his travels in 1489 Celtis matriculated at the University of Kraków specifically to study the mathematical sciences for which Kraków had an excellent reputation. A couple of years later Nicolaus Copernicus would learn the fundamentals of mathematics and astronomy there. Wandering back to Germany via Prague and Nürnberg Celtis was appointed professor of poetics and rhetoric at the University of Ingolstadt in 1491/92. Ingolstadt was the first German university to have a dedicated chair for mathematics, established around 1470 to teach medical students astrology and the necessary mathematics and astronomy to cast a horoscope. When Celtis came to Ingolstadt there were the professor of mathematics was Andreas Stiborius (born Stöberl 1464–1515) who was followed by his best student Johannes Stabius (born Stöberer before 1468­–1522) both of whom Celtis convinced to support him in his cartographic endeavours.

In 1497 Celtis received a call to the University of Vienna where he established a Collegium poetarum et mathematicorum, that is a college for poetry and mathematics, with Stiborius, whom he had brought with him from Ingolstadt, as the professor for mathematics. In 1502 he also brought Stabius, who had succeeded Stiborius as professor in Ingolstadt, and his star student Georg Tanstetter to Vienna. Stiborius, Stabius and Tanstetter became what is known, to historians of mathematics, as the Second Viennese School of Mathematics, the First Viennese School being Johannes von Gmunden, Peuerbach and Regiomontanus, in the middle of the fifteenth century. Under these three Vienna became a major European centre for the mathematical sciences producing many important mathematicians the most notable being Peter Apian.

Although not a mathematician himself Conrad Celtis, the humanist poet, was the driving force behind one of the most important German language centres for Renaissance mathematics and as such earns a place in the history of mathematics. A dedicated humanist, wherever he went on his travels Celtis would establish humanist societies to propagate humanist studies and it was this activity that earned him the German title of Der Erzhumanist, in English the Arch Humanist. Celtis died in 1508 but his Collegium poetarum et mathematicorum survived him by twenty-two years, closing first in 1530


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

Christmas Trilogy 2015 Part 3: Roll out the barrel.

The village master taught his little school

The village all declared how much he knew,

‘Twas certain he could write, and cipher too;

Lands he could measure, times and tides presage,

And e’en the story ran that he could gauge

Oliver Goldsmith – The Deserted Village

As I have commented on a number of occasions in the past, although most people only know Johannes Kepler, if they have heard of him at all, as the creator of his eponymous three laws of planetary motion in fact he published more than eighty books and pamphlets in his life covering a very wide range of scientific and mathematical subjects. One of those publications, which often brings a smile to the faces of those not aware of its mathematical significance, is his Nova stereometria doliorum vinariorum (which translates as The New Art of Measuring the Contents of Wine Barrels) published in 1615. A whole book devoted to determining the volume of wine barrels! Surely not a suitable subject for a man who determined the laws of the cosmos and helped lay the foundations of modern optics, had the good Johannes taken to drink in the face of his personal problems?

Title page of Kepler's 1615 Nova stereometria doliorum vinariorum (image used by permission of the Carnegie Mellon University Libraries)

Title page of Kepler’s 1615 Nova stereometria doliorum vinariorum (image used by permission of the Carnegie Mellon University Libraries)

Because he is now regarded as one of the earliest ‘modern’ mathematicians people tend to forget that Kepler lived not in the age of the mathematician but in that of the mathematical practitioner. This means that as district mathematician in Graz, and later in Linz, Kepler would have been expected to carry out a large range of practical mathematical tasks including surveying, cartography, dialling (that is the design and construction of sundials), writing astrological prognostica, almanacs and calendars and gauging amongst others. We know that Kepler carried out a lot of these tasks but as far as I know he was never employed as a gauger, that is a man responsible for measuring and/or calculating the volume of barrels and their contents.

Nowadays with the wooden barrel degraded to the role of garden ornament in the forecourts of kitschy country pubs it is hard for people to imagine that for more than half a millennium the art of gauging and the profession of the gauger were a widespread and important part of the political and business life of Europe. Wooden barrels first made their appearance during the iron age, that is sometime during the first millennium BCE, iron making it possible to make tools with which craftsmen could work and shape the hard woods used to make barrels. It seems that we owe the invention of the barrel to the Celtic peoples of Northern Europe, who were making wooden barrels at least as early as five hundred BCE, although wooden buckets go back much earlier, with the earliest known one being from Egypt, 2690 BCE. The early wooden buckets were carved from single blocks of wood unlike barrels that are made from staves assembled and held together with hoops of saplings, rope or iron.

Source: Wood, Whiskey and Wine: A History of Barrels by Henry H. Work

Source: Wood, Whiskey and Wine: A History of Barrels by Henry H. Work

The ancient Greeks and Romans used large clay vessels called amphora to transport goods, in particular liquids such a wine and oil.

Roman Amphorae Source: Wikimedia Commons

Roman Amphorae
Source: Wikimedia Commons

However by about two hundred to three hundred CE the Romans, to whom we owe our written knowledge (supported by archaeological finds) of the Celtic origins of barrel making, were transporting wine in barrels. Wooden barrels appear to be a uniquely European invention appearing first in other parts of the world when introduced by Europeans.

By the Middle Ages wooden barrels had become ubiquitous throughout Europe used for transporting and storing a bewildering range of both dry and wet goods including books and corpses, the latter conserved in alcohol. With the vast increase in trade, both national and international, came the problem of taxes and custom duties on borders or at town gates. Wine, beer and spirits were taxed according to volume and the tax officials were faced with the problem of determining the volumes of the diverse barrels that poured daily across borders or through town gates, enter the gauger and the gauging rod.

Gauger with gauging rod Source:

Gauger with gauging rod

The simplest method of determining the volume of liquid contained in a barrel would be to pour out contents into a measuring vessel. This was of course not a viable choice for tax or customs official, so something else had to be done. Because of its shape determining the volume of a barrel-shaped container is not a simple geometrical exercise like that of determining the volume of a cylinder, sphere or cube so the mathematicians had to find another way. The solution was a gauging rod. This is a rod marked with a scale that was inserted diagonally into the barrel through the bung hole and by reading off the number on the scale the gauger could then calculate a good approximation of the volume of fluid in the barrel and then calculate the tax or custom’s duty due. From some time in the High Middle ages through to the nineteenth century gaugers and their gauging rods and gauging slide rules were a standard part of the European trade landscape.

A gauging slide rule Source

A gauging slide rule

The mathematical literature on the art of gauging, particularly from the Early Modern Period is vast. As a small side note Antonie van Leeuwenhoek, the famous seventeenth-century microscopist, also worked for a time as gauger for the City of Delft.

A Cooper Jan Luyken Source

A Cooper Jan Luyken

However after this brief excursion into the history of barrels and barrel gauging it is time to turn attention back to Kepler and his Nova stereometria doliorum vinarioru. In 1613, now living in Linz, Kepler purchased some barrels to lay in a supply of wine for his family. The wine dealer filled the casks and proceeded to measure the volume they contained using a gauging rod. Kepler being a notoriously exacting mathematician was horrified by the inaccuracy of this method of measurement and set about immediately to see if he produce a better mathematical method of determining the volume of barrels. Returning to the Eudoxian/Archimedian method of exhaustion that he had utilized to determine his second law of planetary motion he presented the volume of the barrel as the sum of a potentially infinite sum of a series of slices through the barrels. In modern terminology he used integral calculus to determine the volume. Never content to do half a job Kepler extended his mathematical investigations to determining the volumes of a wide range of three-dimensional containers and his efforts developed into a substantial book. Because he lacked the necessary notions of limits and convergence when summing infinite series, Kepler’s efforts lack mathematical rigour, as had his determination of his second law, a fact that Kepler was more than aware of. However, as with his second law he was prepared to sacrifice rigour for a practical functioning solution and to leave it to prosperity posterity to clean up the mess.

Having devoted so much time and effort to the task Kepler decided to publish his studies and immediately ran into new problems. There was at the time no printer/publisher in Linz so Kepler was forced to send his manuscript to Markus Welser, rich trader and science patron from Augsburg, who initiated the sunspot dispute between Galileo and Christoph Scheiner, to get his book published there. Unfortunately none of the printer/publishers in Augsburg were prepared to take on the risk of publishing the book and when Welser died in 1614 Kepler had to retrieve his manuscript and make other arrangements. In 1615 he fetched the printer Johannes Plank from Erfurt to Linz and paid him to print the book at his own cost. Unfortunately it proved to be anything but a best seller leaving Kepler with a loss on his efforts. In order to make his new discoveries available to a wider audience Kepler edited a very much simplified German edition in the same year under the title Ausszug auss der Vralten Messkunst Archimedis (Excerpts from the ancient art of mensuration by Archimedes). This book is important in the history of mathematics for provided the first German translations of numerous Greek and Latin mathematical terms. Plank remained in Linz and became Kepler’s house publisher during his time there.

Ausszug auss der Vralten Messkunst Archimedis title page Source

Ausszug auss der Vralten Messkunst Archimedis title page

Although not one of his most successful works Kepler’s Nova stereometria doliorum is historically important for two different reasons. It was the first book to present a systematic study of the volumes of barrels based on geometrical principles and it also plays an important role in the history of infinitesimal calculus.



Filed under History of Computing, History of Mathematics