400 Years of The Third Law–An overlooked and neglected revolution in astronomy

Four hundred years ago today Johannes Kepler rediscovered his most important contribution to the evolution of astronomy, his third law of planetary motion.


Portrait of Johannes Kepler 1610 by unknown artist. Source: Wikimedia Commons

He had originally discovered it two months earlier on 8 March but due to a calculation error rejected it. On 15 May he found it again and this time recognised that it was correct. He immediately added it to his Harmonices Mundi:


For when the true distances between the spheres were found, through the observations of Brahe, by continuous toil for a very long time, at last, at last, the genuine proportion of the periodic times to the proportion of the spheres –

Only at long last did she look back at him as she lay motionless,

But she look back and after a long time she came [Vergil, Eclogue I, 27 and 29.]

And if you want the exact moment in time, it was conceived mentally on the 8th of March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely exact that proportion between the periodic times of any two planets is precisely the sesquialterate[1] proportion of their mean distances, that is of the actual spheres, though with this in mind, that the arithmetic mean between the two diameters of the elliptical orbit is a little less than the longer diameter. Thus if one takes one third of the proportion from the period, for example, of the Earth, which is one year, and the same from the period of Saturn, thirty years, that is, the cube roots, and one double that proportion, by squaring the roots, he has in the resulting numbers the exactly correct proportion of the mean distances of the Earth and Saturn from the Sun.[2]

writing a few days later:

Now, because eighteen months ago the first dawn, three months ago the broad daylight, but a very few days ago the full sun of a most remarkable spectacle has risen, nothing holds me back. Indeed, I give myself up to a sacred frenzy.

He finished the book on 27 May although the printing would take a year.

In modern terminology:


The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit: i.e. for two planets with P = orbital period and R = semi-major axis P12/P22=R13/R23

Kepler’s third law is probably the most important discovery on the way to the establishment of a heliocentric astronomy but its importance was initially overlooked and its implications were somehow neglected until Isaac Newton displayed its significance in his Principia Mathematica, published in 1687 sixty-eight years after the third law first appeared in print.

What the third law gives us is a direct mathematical relationship between the size of the orbits of the planets and their duration, which only works in a heliocentric system. In fact as we will see later it’s actually equivalent to the law of gravity. There is nothing comparable for either a full geocentric system or for a geo-heliocentric Tychonic or semi-Tychonic system. It should have hit the early seventeenth-century astronomical community like a bomb but it didn’t, which raises the question why it didn’t.

The main answer lies in Kepler’s own writings. Although he viewed its discovery as the crowning glory of his work on the Harmonices Mundi Kepler didn’t give it any prominence in that work. The Harmonices Mundi is a vast sprawling book explicating Kepler’s version of the Pythagorean theory of the harmony of the spheres in five books. After four introductory books covering plane geometry, music theory and astrology Kepler gets down to harmonic planetary theory in the fifth and final book. Book V, 109 pages in the English translations, contains lots of musical relationships between various aspects of the planetary orbits, with the third law presented as just one amongst the many with no particular emphasis. The third law was buried in what is now regarded as a load of unscientific dross. Or as Carola Baumgardt puts it, somewhat more positively,  in her Johannes Kepler life and letters (Philosophical Library, 1951, p. 124):

Kepler’s aspirations, however, go even much higher than those of modern scientific astronomy. As he tried to do in his “Mysterium Cosmographicum” he coupled in his “Harmonice Mundi” the precise mathematical results of his investigations with an enormous wealth of metaphysical, poetical, religious and even historical speculations. 

Although most of Kepler’s contemporaries would have viewed his theories with more sympathy than his modern critics the chances of anybody recognising the significance of the harmony law for heliocentric astronomical theory were fairly minimal.

The third law reappeared in 1620 in the second part of Kepler’s Epitome Astronomiae Copernicanae, a textbook of heliocentric astronomy written in the form of a question and answer dialogue between a student and a teacher.

How is the ratio of the periodic times, which you have assigned to the mobile bodies, related to the aforesaid ratio of the spheres wherein, those bodies are borne?

The ration of the times is not equal to the ratio of the spheres, but greater than it, and in the primary planets exactly the ratio of the 3/2th powers. That is to say, if you take the cube roots of the 30 years of Saturn and the 12 years of Jupiter and square them, the true ration of the spheres of Saturn and Jupiter will exist in those squares. This is the case even if you compare spheres that are not next to each other. For example, Saturn takes 30 years; the Earth takes one year. The cube root of 30 is approximately 3.11. But the cube root of 1 is 1. The squares of these roots are 9.672 and 1. Therefore the sphere of Saturn is to the sphere of the Earth as 9.672 is to 1,000. And a more accurate number will be produced, if you take the times more accurately.[3]

Here the third law is not buried in a heap of irrelevance but it is not emphasised in the way it should be. If Kepler had presented the third law as a table of the values of the orbit radiuses and the orbital times and their mathematical relationship, as below[4], or as a graph maybe people would have recognised its significance. However he never did and so it was a long time before the full impact of the third law was felt in astronomical community.

third law001

The real revelation of the significance of the third law came first with Newton’s Principia Mathematica. By the time Newton wrote his great work the empirical truth of Kepler’s third law had been accepted and Newton uses this to establish the empirical truth of the law of gravity.

In Book I of Principia, the mathematics and physics section, Newton first shows, in Proposition 11[5], that for a body revolving on an ellipse the law of the centripetal force tending towards a focus of the ellipse is inversely as the square of the distance: i.e. the law of gravity but Newton is not calling it that at this point. In Proposition 14[6] he then shows that, If several bodies revolve about a common center and the centripetal force is inversely as the square of the distance of places from the center, I say that the principal latera recta of the orbits are as the squares of the areas which bodies describe in the same time by radii drawn to the center. And Proposition 15[7]: Under the same supposition as in prop. 14, I say the square of the periodic times in ellipses are as the cubes of the major axes. Thus Newton shows that his law of gravity and Kepler’s third law are equivalent, although in this whole section where he deals mathematically with Kepler’s three laws of planetary motion he never once mentions Kepler by name.

Having established the equivalence, in Book III of The Principia: The System of the World Newton now uses the empirical proof of Kepler’s third law to establish the empirical truth of the law of gravity[8]. Phenomena 1: The circumjovial planets, by radii drawn to the center of Jupiter, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center. Phenomena 2: The circumsaturnian planets, by radii drawn to the center of Saturn, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center. Phenomena 3: The orbits of the five primary planets—Mercury, Venus, Mars, Jupiter, and Saturn—encircle the sun. Phenomena 4: The periodic times of the five primary planets and of either the sun about the earth or the earth about the sun—the fixed stars being at rest—are as the 3/2 powers of their mean distances from the sun. “This proportion, which was found by Kepler, is accepted by everyone.”

Proposition 1: The forces by which the circumjovial planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the center of Jupiter and are inversely as the squares of the distances of their places from that center. “The same is to be understood for the planets that are Saturn’s companions.” As proof he references the respective phenomena from Book I. Proposition 2: The forces by which the primary planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the sun and are inversely as the squares of the distances of their places from its center. As proof he references the respective phenomenon from Book I:

One of the ironies of the history of astronomy is that the general acceptance of a heliocentric system by the time Newton wrote his Principia was largely as a consequence of Kepler’s Tabulae Rudolphinae the accuracy of which convinced people of the correctness of Kepler’s heliocentric system and not the much more important third taw of planetary motion.

[1] Sesquialterate means one and a half times or 3/2

[2] The Harmony of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aiton, A.M. Duncan & J.V. Field, Memoirs of the American Philosophical Society Held at Philadelphia for Promoting Useful Knowledge, Volume 209, 1997 pp. 411-412

[3] Johannes Kepler, Epitome of Copernican Astronomy & Harmonies of the World, Translated by Charles Glenn Wallis, Prometheus Books, New York, 1995 p. 48

[4] Table taken from C.M. Linton, From Eudoxus to Einstein: A History of Mathematical Astronomy, CUP, Cambridge etc., 2004 p. 198

[5] Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy, A New Translation by I: Bernard Cohen and Anne Whitman assisted by Julia Budenz, Preceded by A Guide to Newton’s Principia, by I. Bernard Cohen, University of California Press, Berkley, Los Angeles, London, 1999 p. 462

[6] Newton, Principia, 1999 p. 467

[7] Newton, Principia, 1999 p. 468

[8] Newton, Principia, 1999 pp. 797–802



Filed under History of Astronomy, History of science, Newton, Renaissance Science, Uncategorized

“…like birds in the air and fish in the sea.”

In popular accounts of the transition from geocentric to heliocentric cosmology and astronomy it is often stated that in Aristotle’s geocentric cosmology the orbs of the planets were solid crystalline spheres, the existence of which were disproved when Tycho Brahe demonstrated that comets were not sublunar meteorological phenomena, as claimed by Aristotle, but supralunar astronomical objects. ‘Tycho’s comet of 1577 shattered Aristotle’s crystalline spheres’ is a common trope in such writings, but how true is it?

Aristotle’s cosmology divides the cosmos into the sublunar and supralunar spheres, i.e. below the moon and above the moon. The sublunar sphere, i.e. the earth, consists of the four elements earth, water, air and fire. The supralunar sphere consists of aether, the fifth element or quintessence. For his astronomy he adopted the homocentric planetary spheres of Eudoxus with the earth at their centre. As everything in the supralunar sphere consists of aether so do the planetary orbs. However Aristotle doesn’t actually say what aether is or what its qualities or characteristics are, it just is.

In his Mathēmatikē Syntaxis, Ptolemaeus adopted Aristotle’s cosmology putting it together with the deferent/epicycle model of the planetary orbits developed by Apollonius. He also offers no details as to the nature of aether. The pattern repeats itself with the astronomers of the Islamic Empire, who largely adopted the cosmology of Aristotle and the astronomy of Ptolemaeus without offering an explanation of the nature of aether.

It is first in the High Middle ages that the European, Christian, Aristotelian scholars first begin to ask about the nature of the aether and its properties; they were at least as motivated by the Bible as by the works of Aristotle and Ptolemaeus. The relevant Bible text is from the account of creation in Genesis:

6 And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters.

7 And God made the firmament and, and divided the waters which were under the firmament from the waters which were above the firmament: and it was so.

8 And God called the firmament Heaven.

Combining Aristotle’s aether and the waters above the firmament those European, Christian, Aristotelian scholars in the thirteenth century said that the planetary orbs must be fluid. However they were worried about the waters above the firmament would rain down on the earth so they thought of the orbs as having firm boundaries, like chocolate with soft centres or balloons full of water (my analogies not those of the medieval scholars). The orbs are described as solid (Latin: solidum), but this originally means that they are three-dimensional structures rather than flat disks and does not mean that they are hard.

During the fourteenth and fifteenth centuries the opinions changed and there developed a view that the orbs were not fluid but hard, however we are still far away from the crystalline spheres smashed by Tycho’s comet. In fact it is not actually known when they first appeared in the debate, although Tycho is convinced that they are propagated by his opponents. They don’t play any role in Copernicus’ astronomy so it is thought that they come into the debate somewhere between Copernicus and Tycho.

The story goes that following Tycho’s proof that the comet of 1577 was definitely supralunar the debate reverts to the possibility of fluid rather than hard planetary orbs but there are a couple of problems with this story line. Firstly, that comets were supralunar was being discussed well before Tycho’s 1577 measurement of cometary parallax. Already in the early fifteenth century Paolo dal Pozzo Toscanelli wrote a thesis in which he treated comets as astronomical phenomena and not meteorological ones. He didn’t publish his work but he did have contact with Georg Peuerbach and it can’t be just coincidence that Peuerbach and his pupil Regiomontanus also considered comets to be astronomical. Regiomontanus even wrote a paper on the problems of measuring the parallax of a moving comet, a paper that was discussed in correspondence between Tycho and John Dee. In the 1530’s there was a lively discussion on the supralunar nature of comets in which various notable European astronomers, including Copernicus, took part. When the comet of 1577 appeared it was observed very carefully and its parallax was measured by astronomers all over Europe exactly because of the earlier discussions. Although the results of those attempted measurements were hotly disputed by the various fractions, Tycho was by no means the only astronomer to determine that the comet was supralunar. The determination made by Kepler’s teacher, Michael Maestlin, probably had more impact on the debate than that of Tycho. The biggest impact, however, was made by Christoph Clavius, the leading Jesuit astronomer, who although by definition an Aristotelian scholar accepted that the comet was definitely supralunar.

If Clavius accepted that the comet was supralunar, and he did, how does this square with the fact that as a Jesuit he was required to follow a fairly strict Thomist, Aristotelian philosophy of nature. In fact this was less of a problem than one might imagine. Roberto Bellarmino, who would go on to become the most important Jesuit authority of the age, had already rejected the crystalline spheres before the appearance of the 1577 comet. In his astronomy lectures at the University of Leuven between 1570 and 1574 he taught that the whole deferent/epicycle model was just an abstract construct, which didn’t exist in reality and that the planets moved freely through a fluid medium “like birds in the air and fish in the sea.”

Tycho’s comet smashing Aristotle’s crystalline spheres is a nice story but a closer examination of the historical facts shows it to be just that a nice story but not really a true one.





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As easy as 1,2,3…

In every day life we all do our calculations, whether for the taxman, our purchases, paying the household bills or in some academic discipline, using the place value decimal number system. It consists of just ten symbols (numerals) – 1,2,3,4,5,6,7,8,9,0 – with which we can express any number, of any size that we may require. The value of the symbol changes according to its position – place – within the number that we write. This is an incredibly powerful and efficient method of writing numbers and the algorithms that it uses also make it a very efficient system for conducting calculations. The numerals are usually referred to as Arabic numerals or more correctly as Hindu-Arabic numerals because we Europeans inherited them and the entire system of how to use them from the Islamic Empire in the High Middle Ages, which in turn had inherited them from India in the Early Middle Ages, where they originated. In what follows I shall sketch the path that this number system took from India to medieval Europe, a path that has several twist and turns.

The history of the early development of the place value decimal number system is long, complicated and full of holes and I shan’t be dealing with it here. It also throws up some important and unanswered questions. The Babylonians developed a place value number system as early as the beginning of the second millennium BCE but it was a sexagesimal or base sixty number system rather than a decimal base ten one. The Babylonian system even had a placeholder zero in its later versions. This poses the question whether the Indians got the idea of a place value system from the Babylonians but it is simply not known. The Chinese also had a place value decimal number system but whether the Chinese influenced the Indians, the Indians the Chinese or both developed their systems independently is also not known.

There are three principle figures, who played a central role in the transmission of the place value decimal number system and the first of these is the Indian astronomer Brahmagupta (c.598–c.668 CE), who lived most of his life in Bhillamala (modern Bhinmal) in North-western India. He wrote his Brāhma-sphuṭa-siddhānta a treatise on astronomy written in verse, with 24 chapters and 1008 verses, in 628 CE. Writing scientific works in verse in ancient cultures was probably in order to make them easier to memorise in predominantly oral societies. Although an astronomical work Brahmagupta devotes several chapters to mathematics. Chapter 12 is devoted to arithmetic and introduces the basic arithmetical operations. In chapter eighteen he deals with negative numbers and with zero, not as a placeholder but as a number. He defines zero as that which results from subtracting a number from itself and gives the correct rules for addition, subtraction and multiplication with zero. Unfortunately he defines zero divided by zero as zero and gives a term for a number divided by zero without saying what the result would be. We, of course, now say division by zero is not defined. Brahmagupta’s use of zero as a number is the earliest known such use but this doesn’t mean that he invented zero as a number. His description suggests that this is already common usage. We know that zero as a number doesn’t appear in the astronomical text Aryabhatiya of Aryabhata (476–550 CE), which Brahmagupta criticises, so we can assume that zero as a number was developed in the period between the two works. The Brāhma-sphuṭa-siddhānta also contains description of what we would call algebra the details of which needn’t interest us here although we will meet them again. The Brāhma-sphuṭa-siddhānta was translated into Arabic in the eighth century CE and became one of the principle sources in the Islamic Empire for the Indian number system.

Our second principle figure is the eighth-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c.780–c.850 CE), who produced two works influenced by Brahmagupta’s Brāhma-sphuṭa-siddhānta, one on algebra and one on arithmetic. The more famous is his Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing) from which we get the word algebra (al-ğabr) and from his name we also get the term algorithm (a corruption of al-Khwārizmī). However it is his second work on arithmetic that interests us here. There is no known extant Arabic original of this work but it was translated into Latin in the twelfth century, possibly by Adelard of Bath,[1] under the title Algorithmo de Numero Indorum. This was the first introduction of the Hindu-Arabic numerals and the place value decimal number system into Europe. This introduction was realised at the early medieval universities, where the place value decimal number system was taught under the name algorism, as part of the discipline of computos, the calculation of the date of Easter, an important branch of mathematics at the Catholic universities. John of Sacrobosco wrote a widely read text book Algorismus aka De Arte Numerandi aka De Arithmetica in the early thirteenth century. However the use of the Hindu-Arabic numerals did not spread outside of the university.

In the Arabic world the books on algebra and arithmetic, and al-Khwārizmī’s were by no means the only ones, were largely aimed at merchants and traders. They were what we would term books on commercial arithmetic teaching bookkeeping, calculation of interest, calculation of profit shares in joint business ventures, division of property in testaments etc. and it is from this area that the Hindu-Arabic numbers and the place value decimal number system was finally introduced into Europe by the third of our principle figures Leonardo Pisano or Leonardo of Pisa (c.1175–c.1250).

Leonardo is more generally incorrectly known today by the name Fibonacci. This name, which translates as the son of Bonacci, was, however the creation of the French historian, Guilluame Libri in in 1838. Leonardo’s father Guilichmus or Guilielmo was a merchant who became a customs official. Bonacci was a general family name and not the name of his father his book the Liber Abbaci, to which we will turn shortly, starts:

Here begins the Book of Calculations

Composed by Leonardo Pisano, Family Bonacci

In the Year 1202

As with both Brahmagupta and al-Khwārizmī we know next to nothing about Leonardo personally, the only information that we have is in the introduction to the Liber Abbaci:

As my father was a public official away from our homeland in the Bugia [Now Béjaïa in Algeria] customshouse established for the Pisa merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me in the study of mathematics and to be taught for some days; there from a marvellous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learned from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily, and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learned from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method. Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle geometrical art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum. [i.e. with no knowledge of this method] If by chance, something less or more proper or necessary I omitted, your indulgence for me in entreated, as there is no one who is without fault, and in all things is altogether circumspect.

Leonardo obviously used numerous sources for his extensive book but, which sources he used is not known for certain; it is not even known if he could read Arabic and used original Arabic sources or whether he relied on the Latin translations that already existed. We do however know from textual analysis that he did use al-Khwārizmī’s book on algebra as one of his sources.

The Liber Abbaci is a book written by a merchant for merchants and it is as commercial arithmetic that the Hindu-Arabic numerals finally made it onto the big stage in medieval Europe. Abbaci, with two ‘bs’, and not one as it is often falsely written, comes from abbaco meaning to reckon or calculate in Italian and has nothing to do with abacus. Leonardo’s book might not have had the impact that it did if it had not appeared at roughly the same time as another innovation, double entry bookkeeping. The combination of the Hindu-Arabic numerals and double entry bookkeeping become the engine room to the so-called medieval economic revolution that saw the invention of banking and the rise of large scale international trading centred round the economic power house of Northern Italy. Leonardo’s book triggered a whole abbaco industry in Northern Italy.

To teach the new Indian arithmetic small abbaco schools (scuole d’abbaco or botteghe d’abbco) were established in the towns, where teenagers, who were apprentice traders or merchants, were taught commercial arithmetic and double entry bookkeeping. The teachers, who ran these establishments, maestri d’abbaco, wrote their own textbooks, a genre known as Libri d’abbaco, (abbacus books). The first ever printed mathematics book was an abbacus book, the so-called Treviso Arithmetic or Arte dell’Abbaco written in vernacular Venetian and published in Treviso in 1478. These schools and their textbooks spread to the trading cities of Southern Germany, such as Augsburg, Regensburg and Nürnberg, and from there throughout Europe. In German we have Rechenmeister, Rechenschule and Rechenbucher, in English reckoning masters, reckoning schools and reckoning books. Arithmetic and algebra remained in the province of the traders and merchants as commercial arithmetic until the middle of the fifteenth century. Gerolemo Cardano is credited with bringing algebra into the realm of mathematics with his Artis magnae, sive de regulis algebraicis liber unus published by Johannes Petreius in Nürnberg in 1545 but he also started his career as a mathematical author with an abbacus book, his Practica arithmetice et mensurandi singularis published in Milano in 1538.

The introduction of the Indian numerals into Northern Italy didn’t go entirely unopposed. In 1299 a local law was passed in Florence banning the use of them in bookkeeping, Statuto dell’Arte del Cambio, with the argument that they were easier to change, thus falsifying the accounts, than Roman numerals or written number words. Many modern authors claim that reckoning with the Hindu-Arabic numerals was faster and simpler than using the abacus or reckoning board but I don’t think this is true and I strongly suspect that most merchants continued to do their reckoning on a counting board reserving the new arithmetic for their written bookkeeping.

Leonardo was not just the man, who introduced the place value decimal number system into Europe with his Liber Abbaci, but was also the author of several other important mathematical works establishing him as an important mathematician in thirteenth-century Italy. In 1240 he was even invited to an audience with the Holy Roman German Emperor Frederick II, who was an avid patron of the sciences. The most famous judgement on the introduction of the place value decimal number system is that of the eighteenth-century French polymath Simon Laplace:

“It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”

It is Leonardo Pisano to whom we own our thanks for having introduced this invention into Europe. If you want to know more about the man and his book then I recommend Keith Devlin, The Man of Numbers: Fibonacci’s Arithmetic Revolution, Bloomsbury, London, 2011 from which the long quote from the Liber Abbaci is taken.

The theme of this post was requested by one of my anonymous €30 plus GoFundMe donors. It’s slightly different to what he suggested but I hope he’s satisfied with the end result. I wait for other donors to claim their right to negotiate a post theme.

[1] The secondary sources I have consulted say, unknown translator, probably Adelard of Bath, Robert of Chester (who definitely did translate the algebra) and John of Seville, so take your pick. Interestingly several of them name Adelard of Bath but my biography of Adelard says that the attribution is probably false.


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The influence of a Renaissance mathematics teacher.

Recent posts here have been all over the #histSTM map so I thought it was time to return to the roots and write something about a Renaissance mathematicus. It was first during the fifteenth century that Medieval European universities began to create dedicated chairs for the study of the mathematical disciplines — arithmetic, geometry, astrology, astronomy, surveying, cartography, designing and constructing sundials and mathematical instruments. The first such chair to be established in Germany was at the University of Ingolstadt in about 1470. Like its predecessors in Northern Italy this was principally a chair for teaching astrology and the mathematics and astronomy necessary to cast horoscopes to medical students. Those teaching in Ingolstadt, however, extended their activities to cover the full range of Renaissance mathematical studies. As well as producing medical students Ingolstadt also created full blood mathematical scholars, who would carry the seeds of mathematical studies to other towns and regions. One of those Ingolstadt seeds was Johannes Stöffler.


Contemporary Author’s Portrait Stöfflers from his 1534 published Commentary on the Sphaera of the Pseudo-Proklos (actually Geminos) Source: Wikimedia Commons

Johannes Stöffler was born in Blaubeuren in the Swabian Jura 10 December 1452. He received his first education in Blaubeuren and matriculated at the University of Ingolstadt on 21 April 1472 graduating BA in September 1473 and MA in January 1476. As many of the other contemporary mathematical scholars Stöffler entered a career in the Church rising to parish priest in Justingen in 1481. Parallel to his clerical work he became a highly active astrologer, astronomer, clock, globe and instrument maker. He was a very successful mathematicus and enjoyed a widespread good reputation. He constructed a, still extant, celestial globe for Daniel Zehender auxiliary Bishop of Konstanz in 1593, a clock for the Minster in Konstanz in 1596 and later another celestial globe Johann von Dalberg, Bishop of Worms. For Johannes Reuchlin, Germany’s leading Hebraist and prominent humanist scholar, he constructed an equitorium to determine the orbits of the Sun and the Moon.


Celestial Globe, Johannes Stöffler, 1493; Landesmuseum Württemberg Source: Wikimedia Commons

In 1507, the already fifty-five year old, Stöffler was appointed by Duke Ulrich I of Württemberg to the newly created chair of mathematics at the University of Tübingen. He extended his reputation as an instrument and globe maker as an academic with a successful series of technical publications.


Portrait of Johannes Stöffler produced for the Tübingen Professors’ Gallery 1614 Source: Wikimedia Commons

When he set up the first scientific publishing house in Nürnberg, Regiomontanus’ most successful publication was his ephemerides, sets of tables enabling the user to determine the position of the planets at any given time. Produced principally for astrologers they were also useful for astronomers, navigators and cartographers. There had been earlier manuscript ephemerides but Regiomontanus’ were the first printed ones and were distinguished from earlier ones by their high level of accuracy, leading to many pirated editions. Ephemerides are only calculated for a given number of years and Stöffler, together with the Ulm parish priest Jakob Pflaum, extended Regiomontanus’ ephemerides to 1531 and in a later posthumously published edition to 1551. The Regiomontanus/Stöffler/Pflaum ephemerides dominated the market and established Stöffler and Pflaum as the leading astrologers of the age.

In 1512 Stöffler published a text on the construction and use of the astrolabe, Elucidatio fabricae ususque astrolabii, which went through 16 editions up to 1620 and was highly influential.


The leading astrolabe maker in the early sixteenth century was the Nürnberger Georg Hartmann, who was probably the first instrument maker to mass-produce astrolabes in series. It has been shown that Hartmann’s work was based on Stöfffler’s book.


Astrolabe from Georg Hartmann, Yale Source: Wikimedia Commons

As university teacher Stöffler exercised a major influence on his student the most famous of which were Sebastian Münster and Philipp Melanchthon. From 1514 to 1518 Sebastian Münster, already a fan of the Renaissance mathematical sciences, studied under Stöffler.

Later Münster would publish his Cosmographia in the publishing house of his step-son Heinrich Petri in Basel. The Cosmographia, “a description of the whole world with everything it contain”, an atlas but so much more was the biggest selling book of the sixteenth century. In an age where the edition of a book was usually counted in hundred the Cosmographia is estimated to have sold in excess of 120,000 in its German and Latin editions over a period of about one hundred years.


Title page of the Cosmographia first edition Source: Wikimedia Commons

Stöffler’s biggest influence on the history of mathematics was, without doubt through Philipp Melanchthon. Melanchthon a nephew of Reuchlin was something of a child prodigy.


Portrait of Philipp Melanchthon from an oil painting on wood by Lucas Cranach d. Ä. 1543 Source: Wikimedia Commons

He entered the University of Heidelberg in 1509 and graduated BA in 1511 just thirteen years old. In 1512 he changed to the University of Tübingen, where he came under the influence of Stöffler. Under Stöffler he studied the mathematical disciplines and became a passionate supporter of the art of astrology. He graduated MA in 1514. In 1518 he, just twenty-one years old, was appointed professor of Greek, on Reuchlin’s recommendation, at the University of Wittenberg.

In Wittenberg Melanchthon became Luther’s friend and supporter and during the Reformation as Luther’s “Præceptor Germaniae” (Germany’s schoolmaster) he was charged with designing, organising and establishing the new Lutheran Protestant education system. Melanchthon now had the chance to promote his love of astrology won as a student of Stöffler. Melanchthon established chairs for mathematics on all of the new Protestant Gymnasia (high schools) and university, choosing the ablest mathematical scholars available to fill the new positions. Thus Johannes Schöner became professor for mathematics on the new gymnasium in Nürnberg and Georg Joachim Rheticus and Erasmus Reinhold the mathematics professors in Wittenberg. Melanchthon’s aim was to produce new generations of professionally educated astrologers. Through his actions the Protestant education system became an active supporter of the mathematical sciences at a time when they were largely neglected within the Catholic education system. Melanchthon’s system would go on to produce many leading sixteenth century mathematical practitioners.

Stöffler is a good example of a Renaissance mathematicus who tends not to feature in the mainstream history of mathematics but who from the second row behind the big names still had a major influence on the evolution of the discipline through various channels.

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Alea iacta est

On Monday I booked and paid for my flights to and from sunny California to attend SciFoo18 at the end of June. I shall be in San Francisco Wednesday and Thursday 20 & 21 June in case anybody wants to meet up for a coffee, a joint or a meal. On the Friday I shall be travelling down to Mountain View for SciFoo, where I shall be till the Sunday. The following week, Monday 25 to Friday 29, I shall be in LA where I will be holding a lecture on Albrecht Dürer as a Renaissance Mathematicus at the Project AWE on Thursday 28 June at 7:00 pm. Again I’m up for meetings for whatever if you are, whoever you are! I’m also up for another lecture or two during the week if you interested, whoever you are.

 The following wonderful people have contributed to my GoFundMe and helped to make this round trip to The Golden State possible for which I am eternally grateful. I am, to put it mildly, totally blown away by your generosity and thank all of you from the depths of my heart. Eighteen Nineteen of those who have donated, up till now, have acquired the right, by donating €30 or more, to negotiate a blog post subject with me if they so choose. This also applies to those who donated anonymously, I know who you are! To contact me, my email address is at the top of the blog under Contact!


The Renaissance Mathematicus SciFoo18 Wall of Fame

 1) Jonathan Dresner 2) John Kane 3) Anonymous

4) Luke Dury 5) Charles Hueneman 6) Andreas Sommer

7) Cornelis J. Schilt 8) Kevin Quiggle 9) William Connolley

10) Gypsy Van Melle Seaton 11) Anonymous

12) Fawn Nguyen 13) Anonymous 14) Anonymous

15) Gavin Moodie 16) Seb Falk 17) Anonymous

18) Ash Jogalekar 19) Noah Greenstein 20) Anonymous

21) Anonymous 22) Gene Dannen 23) Anonymous

24) Anonymous 25) Anonymous 26) Anonymous

27) Anonymous 28) Michael Traynor 29) Anonymous

30) Gerald Cummins 31) Heribert Watzke

32) Anders Ehrnberg 33) Friends of Charles Darwin

34) Anonymous 35) Anonymous 36) Anonymous

37) Sally Osborn 38) Paul Coxon 39) Matthew Cobb

40) Tim O’Neill 41) Tony Mann 42) Meg Rosenburg

43) Isaac Cowan 44) Timoer Frelink


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Taking the airs.

The eighteenth century was the century of the pneumatic chemists and of the discovery of gases or as they termed it different kinds of air. Following the invention of the pneumatic trough by Stephan Hales it became possible for researchers to produce, isolate and study gases. To quote an early post:

Most European eighteenth-century chemist accepted and worked within the framework of the phlogiston theory and produced a great deal of new important chemical knowledge. Most notable in this sense are the, mostly British, so-called pneumatic chemists. Working within the phlogiston theory Joseph Black (1728–1799), professor for medicine in Edinburgh, isolated and identified carbon dioxide whilst his doctoral student Daniel Rutherford (1749–1819) isolated and identified nitrogen. The Swede Carl Wilhelm Scheele (1742–1786) produced, identified and studied oxygen for which he doesn’t get the credit because although he was first, he delayed in publishing his results and was beaten to the punch by Joseph Priestley (1733–1804), who had independently also discovered oxygen labelling it erroneously dephlogisticated air. Priestley by far and away the greatest of the pneumatic chemists isolated and identified at least eight other gases as well as laying the foundations for the discovery of photosynthesis, perhaps his greatest achievement.

Henry Cavendish (1731–1810) isolated and identified hydrogen, which he thought for a time might actually be phlogiston, before going on to make the most important discovery within the framework of the phlogiston theory, the structure of water.


Joseph Priestley (1733-1804) – Frontispisce of Experiments and Observations on Different Kinds of Air Pneumatic trough, and other equipment, used by Joseph Priestley Source: Wikimedia Commons

Perhaps the highpoint of all this gas activity or at least the most bizarre outgrowth of it was the Pneumatic Institute set up in Bristol by Thomas Beddoes in 1799 to study the medical effects of the recently discovered and isolated gases.


Thomas Beddoes, pencil drawing by Edward Bird Source: Wikimedia Commons

Beddoes was born in Shropshire on 13 April 1760 and after being educated at Bridgnorth Grammar School and Pembroke College Oxford he entered the University of Edinburgh to study medicine in the 1780s, where he study chemistry under Joseph Black.


Mezzotint engraving by James Heath after Sir Henry Raeburn Source: Wikimedia Commons

He took his medical degree at Pembroke in 1786. After 1786 he visited Lavoisier in Paris. Beddoes was appointed professor of chemistry at Oxford University in 1788. His lectures were very popular but he was regarded as a political radical because of his sympathy for the French Revolution. He resigned from Oxford in 1792.

Between 1793 and 1799 he ran a clinic for the treatment of tuberculosis in Bristol. This led to him setting up the Pneumatic Institute in 1799 to investigate the treatment of diseases with gases. At first Beddoes’ idea might seem a little bizarre given how recent the discovery of most gases had been and basically how little was actually known about them. However, two strong indications inspired Beddoes’ lines of inquiry. Firstly carbon dioxide was known to prevent decay in organic materials. This led to trials by the navy in dosing seamen with carbonated water, invented by Priestley, to try and prevent scurvy. Today, in our age of worldwide trade, fresh fruit and vegetables are transported in a carbon dioxide atmosphere to prevent spoilage. The other was the known effects of oxygen. Priestley had amply demonstrated the life sustaining properties of oxygen and also recorded the mild high obtained from breathing the pure gas. Lavoisier and Simon Laplace had demonstrated experimentally the role that oxygen plays in mammalian respiration. On the basis of this primary knowledge Beddoes set out to see if other gases possibly possessed medical properties.

L0001624 Bristol Pneumatic Institute.

Bristol Pneumatic Institute Source: Wellcome Institute via Wikimedia Commons

Beddoes could not afford to finance his planned institute himself and failed to find a single sponsor so he took a route that now seems very 21st century. We see crowd funding as a product of the Internet age but it existed already in the eighteenth century under the name subscription. Beddoes found enough subscribers under his circle of friends to finance his endeavour. Amongst the principle subscribers were several members of the Birmingham Lunar Society to whose wider circles Beddoes belonged. James Watt, like Beddoes, was a protégé of Joseph Black and Joseph Priestly shared Beddoes’ political views. Watt designed and built the technical equipment for the Institute.


Apparatus designed by James Watt in preparation of the Pneumatic Institution Source: Wikimedia Commons

The Watt family provided assistance in another way as well. Beddoes needed a superintendent/laboratory assistant for his Pneumatic Institute and Watt’s son Gregory recommend his friend Humphry Davy (1778-1829), a young self taught chemist for the post. Beddoes was impressed by Davy’s, at that point unpublished, researches and appointed him to the post just twenty-one years old; a decision he possibly came to regret.


Sir Humphry Davy, Bt, by Thomas Phillips Source: National Portrait Gallery via Wikimedia Commons

Davy was a fearless practical researcher and set out to investigate the effects of various gases by testing them on himself, writing detailed protocols of the results of these experiments. He proceeded to test the effects of inhaling the recently discovered carbon monoxide. Now we know that carbon monoxide is highly poisonous but Davy didn’t and his career as a professional chemist almost ended before it had really begun. He inhaled pure carbon monoxide, which resulted in his becoming comatose. Fortunately he had taken the precaution of filling several balloons with pure oxygen and instructed his assistant to revive him if he should lose consciousness. Revived he wrote a detailed report of the experiment and its results, and then he proceeded to repeat it. The young Davy knew how to live dangerously.

His next experiment was considerably less dangerous but would prove far more fateful. He began to test the properties of nitrous oxide, known colloquially as laughing gas, a name coined by Davy. Nitrous oxide was one of the gases discovered and investigated by Joseph Priestly. Davy inhaled pure nitrous oxide and got high! In fact he got very high and he liked it. He liked it very much. Davy effectively became addicted to nitrous oxide inhaling it several times a day, everyday. He also began to subject other people to nitrous oxide highs and recording their reactions and behaviour whilst under the influence. Things got a little out of hand.

Davy, a very talented young man, was not just a chemist but also a recognised romantic poet well connected to Robert Southey and Samuel Taylor Coleridge. Davy invited his poetical friends down to Bristol for what were, in reality, drug parties. These drug orgies combined with the fact that Davy experimented with nitrous oxide on female subjects led to the ruin of the Institute’s reputation and the end of the whole of Beddoes research programme. There were accusations of impropriety with the female subjects; what had taken place whilst they were under the influence?

Beddoes faded into the background but Davy was able to rescue his reputation and get appointed to the post of assistant lecturer in chemistry, director of the chemical laboratory, and assistant editor of the journals of the recently established Royal Institution in London in 1801, where he went on to become one of the greatest research scientists of the nineteenth century. Although Gillray’s legendary cartoon from 1802 show that his laughing gas reputation had not been forgotten.


1802 satirical cartoon by James Gillray showing a Royal Institution lecture on pneumatics, with Davy holding the bellows and Count Rumford looking on at extreme right. Dr Thomas Garnett is the lecturer, holding the victim’s nose. Source: Wikimedia Commons

One probably other casualty of Davy’s drug trips was the medical use of nitrous oxide. Although Davy had, in his protocols, recorded the anaesthetic effects of the gas it had become so disreputable that it would be another fifty years before it was actually used as an anaesthetic.


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From decimal fractions to sand yachts – the unbelievably fertile mind of Simon Stevin

Of all the people who contributed to the evolution of modern science at the beginning of the seventeenth century and who have disappeared from popular perception under the over dimensioned shadow cast by Galileo, one of the most fascinating is the Netherland’s engineer Simon Stevin. Stevin is usually referred to as an engineer but in reality he was a jack-of-all-trades, mathematician, physicist, astronomer, engineer, inventor, music theorist, political advisor and army quartermaster.


Simon Stevin Source: Wikimedia Commons

Simon Stevin was the illegitimate son of Antheunis Stevin and Cathelijne von van der Poort probably born in Bruges around 1548. This and the fact that he seems to have come from an affluent background and was apparently well educated are all we know about his origins. In 1571 he became a merchants clerk in Antwerp and between 1571 and 1577 he travelled to Prussia, Poland, Denmark, Norway and Sweden. In 1577 he returned to Bruges where he was appointed city clerk a position he held until 1581 when he moved north to Leiden. It is not known why Stevin left Bruges for Leiden; his motive might have been religious, political or something else altogether. In Leiden he enrolled in a Latin school in 1581 and then the university in 1583 where he stayed until 1590 but appears never to have graduated. At the university he became friends with Maurits of Nassau the son of Willem I, Prince of Orange, who led the Dutch revolt against the Spanish Habsburgs. Maurits, who later became Stadtholder of the Dutch Republic and commander of the Dutch army, and Stevin remained close friends until Stevin’s death in 1620 and Stevin became Maurits’ technical and scientific advisor and tutor. Initially he was simply engineer but in 1604 he was appointed quartermaster-general of Dutch army.


Portrait of Maurits, Prince of Orange-Nassau School of Michiel Jansz. van Mierevelt Source: Wikimedia Commons

For Maurits he wrote textbooks on a wide range of mathematical and technical subjects, as well as organising and setting up a school of engineering at the university in Leiden. He wrote original scientific works as well as general surveys of the science and technology of the age. His published works include books on mathematics, mechanics, astronomy, navigation, military science, engineering, music theory, civics, dialectics, bookkeeping, geography and house building. He wrote his books in the vernacular and in doing so coined much of the necessary Dutch vocabulary for science and technology, some of which has been replaced since his times but much of which is still in use. However, much of his work was translated into Latin and/or French and was so available and known to other researchers in Europe.

Probably his most well known work is De Thiende (Tenths), a twenty-nine-page booklet in which he explained how to use decimal fractions. He did not originate the concept, Chinese and Islamic mathematician had already been using decimal fractions for several centuries before Stevin but he did introduce and make popular the idea in Europe. In mathematics he also wrote interesting forward-looking textbooks oh arithmetic and algebra, as well as linear perspective.


In physics he worked on statics some fifty years before Galileo developing and continuing the work of Archimedes. His most famous discovery here was the law of the inclined plane, which he demonstrated using a chain of wreaths. His demonstration shows that the effective component of gravity is inversely proportional to the length of the inclined plane. What Stevin is in principle using here is the theory of the parallelogram of forces, something learnt in schools today using vector algebra but Stevin is using it a couple of centuries before vector algebra existed.


Stevin’s proof of the law of equilibrium on an inclined plane, known as the “Epitaph of Stevinus”. Source: Wikimedia Commons

Of historical interest is that Stevin, unlike Galileo, actually did drop balls from a church tower and also hypothesised that objects in a vacuum would fall at the same rate irrespective of weight.

Having dealt with statics, Stevin next turned to hydrostatics, another discipline that he inherited from Archimedes, and here he was the first to demonstrate the so-called hydrostatic paradox i.e. that the pressure in a liquid is independent of the shape of the vessel and the area of the base, but depends solely on its depth. This discovery is often falsely attributed to Blaise Pascal.

In his book on astronomy published in 1608, Stevin revealed himself to be an unconditional supporter of Copernican heliocentricity at a time when very few were prepared to make such a commitment. He also accepted that the tides were caused by the moon, also a forward-looking commitment for the times. Being a Dutchman he of course wrote on the principles of navigation giving a clear explanation of steering a ship along a loxodrome or rhumb line as originally propagated by the Portuguese mathematician Pedro Nunez and used by Mercator in his famous Mercator projection.

As an engineer Stevin wrote on military fortification. Of course, as a Netherlander he also wrote on hydraulic engineering designing new types of sluices and locks, as well as better windmills for drainage work. In many areas his work was of a very practical nature but always looking for ways to improve machines or find better solutions for mechanical tasks.

In music theory, a very hot topic at the time, Stevin rejected a couple of thousand years of highly emotional debate on the subject of the intervals of the scale and proposed what is now known as equal temperament. He was not the first to do so, he was anticipated by Galileo’s father Vincenzo amongst others, but was almost certainly not aware of the fact.

On a somewhat more frivolous level he designed and built sand or land yachts (Zeilwagen) for Maurits. Wind driven carriages had existed in China for a thousand years before Stevin built his and illustrated in the Theatrum Orbis Terrarum of Abraham Ortelius in 1584 and in Mercator’s Atlas slightly later and it can be assumed that this was the source of inspiration for Stevin’s own vehicles.


Wind chariot or land yacht (Zeilwagen) designed by Simon Stevin for Prince Maurice of Orange (Engraving by Jacques de Gheyn). Source: Wikimedia Commons

Stevin collected all of his mathematical writings into his two volume Wisconstighe Ghedachtenissen in 1608, which were published simultaneously in both French and Latin; the latter translation being carried out by Willebrord Snel. A modern edition of The Principle Works of Simon Stevin in 5 volumes was published in Amsterdam 1955–1968.

Stevin wrote extensively over a very wide ranch of scientific, mathematical and technological subjects. His writings were always lucid, up to date and very often-contributed new concepts, ideas, methods and discoveries, some of which were very significant. He was in his own lifetime highly influential, both of the Snels knew him personally and Willebrord did much to spread his work. Isaac Beeckman consulted his unpublished papers from which he much profited.

I have one personal puzzle concerning Stevin’s work. When Hans Lipperhey demonstrated his newly invented telescope to Maurits in The Hague in September 1608, Simon Stevin had already been Maurits’ scientific advisor for more than twenty years and was without doubt the leading scientific and technological authority in the young Dutch Republic, but I know of no reaction, comment, statement or whatever from Stevin on this sensational new discovery. For me as a historian of the telescope his silence is deafening.




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