Category Archives: History of science

Tycho’s last bastion

In the history of science, scholars who end up on the wrong side of history tend to get either forgotten and/or vilified. What do I mean by ‘end up on the wrong side of history’? This refers to scholars who defend a theory that in the end turns out to be wrong against one that in the end turns out to be right. My very first history of science post on this blog was about just such a figure, Christoph Clavius, who gets mocked by many as the last Ptolemaic dinosaur in the astronomy/cosmology debate at the beginning of the seventeenth century. In fact there is much to praise about Clavius, as I tried to make clear in my post and he made many positive contributions to the evolution of the mathematical sciences. Another man, who ended up on the wrong side of history in the same period is the Danish astronomer, Christen Sørensen, better known, if at all, by the name Longomontanus, the Latinised toponym based on Lomborg, the Jutland village where he was born on 4 October 1562 the son of a poor labourer, who died when he was only eight years old.


Longomontanus Source: Wikimedia Commons

Tycho Brahe backed the wrong astronomical theory in this period, a theory that is generally named after him although several people seem to have devised it independently of each other in the closing quarter of the sixteenth century. However, Tycho has not been forgotten because he delivered the new data with which Johannes Kepler created his elliptical model of the solar system. However, what people tend to ignore is that Tycho did not produce that data single-handedly, far from it.

The island of Hven, Tycho’s fiefdom, was a large-scale research institute with two observatories, an alchemy laboratory, a paper mill and a printing workshop.


Map of Hven from the Blaeu Atlas 1663, based on maps drawn by Tycho Brahe in the previous century Source: Wikimedia Commons

This enterprise was staffed by a veritable army of servants, technicians and research assistant with Tycho as the managing director and head of research.


Engraving of the mural quadrant from Brahe’s book Astronomiae instauratae mechanica (1598) Showing Tyco direction observations Source: Wikimedia Commons

Over the years the data that would prove so crucial to Kepler’s endeavours was collected, recorded and analysed by a long list of astronomical research assistants; by far and away the most important of those astronomical research assistants was Christen Sørensen called Longomontanus, who also inherited Tycho’s intellectual mantle and continued to defend his system into the seventeenth century until his death in 1647.

Christen Sørensen came from a very poor background so acquiring an education proved more than somewhat difficult. After the death of his father he was taken into care by an uncle who sent him to the village school in Lemvig. However, after three years his mother took him back to work on the farm; she only allowed him to study with the village pastor during the winter months. In 1577 he ran away to Viborg, where he studied at the cathedral school, supporting himself by working as a labourer. This arrangement meant that he only entered the university in Copenhagen in 1588, but with a good academic reputation. It was here at the university that he acquired his toponym, Longomontanus. In 1589 his professor recommended him to Tycho Brahe and he entered into service on the island of Hven.


Tycho Brahe’s Uraniborg main building from the 1663 Blaeu’s Atlas Major Centre of operations Source: Wikimedia Commons

He was probably instructed in Tycho’s methods by Elias Olsen Morsing, who served Tycho from 1583 to 1590, and Peter Jacobsen Flemløse, who served from 1577-1588 but stayed in working contact for several years more and became a good friend of Longomontanus. Longomontanus proved to be an excellent observer and spent his first three years working on Tycho’s star catalogue.


Stjerneborg Tycho Brahe’s second observatory on Hven: Johan Blaeu, Atlas Major, Amsterdam Source: Wikimedia Commons

Later he took on a wider range of responsibilities. In 1597, Tycho having clashed with the new king, the entire research institute prepared to leave Hven. Longomontanus was put in charge of the attempt to bring Tycho’s star catalogue up from 777 stars to 1,000. When Tycho left Copenhagen, destination unknown, Longomontanus asked for and received his discharge from Tycho’s service.

While Tycho wandered around Europe trying to find a new home for his observatory, Longomontanus also wandered around Europe attending various universities–Breslau, Leipzig and Rostock–and trying to find a new patron. He graduated MA in Rostock. During their respective wanderings, Tycho’s and Longomontanus’ paths crossed several times and the corresponded frequently, Tycho always urging Longomontanus to re-enter his service. In January 1600 Longomontanus finally succumbed and joined Tycho in his new quarters in Prague, where Johannes Kepler would soon join the party.

When Kepler became part of Tycho’s astronomical circus in Prague, Longomontanus the senior assistant was working on the reduction of the orbit of Mars. Tycho took him off this project putting him instead onto the orbit of the Moon and giving Mars to Kepler, a move that would prove history making. As should be well known, Kepler battled many years with the orbit of Mars finally determining that it was an ellipse thereby laying the foundation stone for his elliptical astronomy. The results of his battle were published in 1609, together with his first two laws of planetary motion, in his Astronomia nova.


Portrait of Johannes Kepler. Source: Wikimedia Commons

Meanwhile, Longomontanus having finished Tycho’s lunar theory and corrected his solar theory took his final departure from Tycho’s service, with letters of recommendation, on 4 August 1600.  When Tycho died 24 October 1601 it was thus Kepler, who became his successor as Imperial Mathematicus and inherited his data, if only after a long dispute with Tycho’s relatives, and not Longomontanus, which Tycho would certainly have preferred.

Longomontanus again wandered around Northern Europe finally becoming rector of his alma mater the cathedral school in Viborg in 1603. In 1605, supported by the Royal Chancellor, Christian Friis, he became extraordinary professor for mathematics at the University of Copenhagen, moving on to become professor for Latin literature in the same year. In 1607 he became professor for mathematics, and in 1621 his chair was transformed into an extraordinary chair for astronomy a post he held until his death.

As a professor in Copenhagen he was a member of an influential group of Hven alumni: Cort Aslakssøn (Hven 159-93) professor for theology, Christian Hansen Riber (Hven 1586-90) professor for Greek, as well as Johannes Stephanius (Hven 1582-84) professor for dialectic and Gellius Sascerides (Hven 1585-86) professor for medicine.

Kepler and Longomontanus corresponded for a time in the first decade of the seventeenth century but the exchange between the convinced supporter of heliocentricity and Tycho’s most loyal lieutenant was not a friendly one as can be seen from the following exchange:

Longomontanus wrote to Kepler 6th May 1604:

These and perhaps all other things that were discovered and worked out by Tycho during his restoration of astronomy for our eternal benefit, you, my dear Kepler, although submerged in shit in the Augean stable of old, do not scruple to equal. And you promise your labor in cleansing them anew and even triumph, as if we should recognise you as Hercules reborn. But certainly no one does, and prefers you to such a man, unless when all of it has been cleaned away, he understands that you have substituted more appropriate things in the heaven and in the celestial appearances. For in this is victory for the astronomer to be seen, in this, triumph. On the other hand, I seriously doubt that such things can ever be presented by you. However, I am concerned lest this sordid insolence of yours defile the excellent opinion of all good and intelligent men about the late Tycho, and become offensive.

Kepler responded early in 1605:

The tone of your reference to my Augean stable sticks in my mind. I entreat you to avoid chicanery, which is wont to be used frequently with regard to unpopular things. So that you might see that I have in mind how the Augean stable provided me with the certain conviction that I have not discredited astronomy – although you can gather from the present letter – I will use it with the greatest possible justification. But it is to be used as an analogy, not for those things that you or Tycho were responsible for constructing – which either blinded by rage or perverted by malice you quite wrongfully attributed to me – but rather in the comparison of the ancient hypotheses with my oval path2. You discredit my oval path. I hold up to you the hundred-times-more-absurd spirals of the ancients (which Tycho imitated by not setting up anything new but letting the old things remain). If you are angry that I cannot eliminate the oval path, how much more ought you to be angry with the spirals, which I abolished. It is as though I have sinned with the oval I have left, even though to you all the rest of the ancients do not sin with so many spirals. This is like being punished for leaving behind one barrow full of shit although I have cleaned the rest of the Augean stables. Or in your sense, you repudiate my oval as one wagon of manure while you tolerate the spirals which are the whole stable, to the extent that my oval is one wagon. But it is unpleasant to tarry in rebutting this most manifest slander.

 Whereas, as already mentioned above, Kepler presented his heliocentric theory to the world in 1609, Longomontanus first honoured Tycho’s memory with his Astronomia Danica in 1622. Using Tycho’s data Longomontanus provided planetary models and planetary tables for Tycho’s geo-heliocentric system. Longomontanus, however, differed from Tycho in that he adopted the diurnal rotation of Helisaeus Roeslin, Nicolaus Raimarus and David Origanus.


The Astronomia Danica saw two new editions in 1640 and 1663. For the five decades between 1620 and 1670 Kepler’s elliptical astronomy and the Tychonic geo-heliocentric system with diurnal rotation competed for supremacy in the European astronomical community with Kepler’s elliptical system finally triumphing.

 In 1625 Longomontanus suggested to the King, Christian IV, that he should build an observatory to replace Tycho’s Stjerneborg, which had been demolished in 1601. The observatory, the Rundetaarn (Round Tower), was conceived as part of the Trinitatis Complex: a university church, a library and the observatory. The foundation stone was laid on 7 July 1637 and the tower was finished in 1642. Longomontanus was appointed the first director of the observatory, after Leiden 1632 only the second national observatory in Europe.


Copenhagen – Rundetårn Source: Wikimedia Commons

Both Kepler and Longomontanus, who lost their fathers early, started life as paupers Both of them worked they way up to become leading European astronomers. Kepler has entered the pantheon of scientific gods, whereas Longomontanus has largely been assigned to the dustbin of history. Although Longomontanus cannot be considered Kepler’s equal, I think he deserves better, even if he did back the wrong theory.








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

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

Galileo & Roberto

One of the books that I am currently reading is Rob Iliffe’s Priest of Nature: The Religious Worlds of Isaac Newton (a full review will follow when I finish it but I can already say it will be very positive). I stumbled more than somewhat when I read the following:

…and Lucas Trelcatius’s list of some of the most significant places in Scripture, which was composed as a response to the Catholic interpretations of various texts offered by the great scholar (and scourge of Galileo [my emphasis]) Cardinal Robert Bellarmine.

Four words that caused me to draw in my breath, why? Let as first take a look at the meaning of the word scourge:

A scourge was originally a particularly nasty and extremely cruel multi-thong whip. Transferred to describe a person it means: a person that causes great trouble of suffering. Can Robert Bellarmine really be described as “scourge of Galileo”?

Robert Bellarmine (actually Roberto Bellarmino) (1542-1621) was a Jesuit scholar who was specialist for post Tridentine theology, that is the theological teachings of the Catholic Church as laid down as official church doctrine at the Council of Trent (1545-1563. He rose through the ranks to arch-bishop and then cardinal, was professor for theology at the Collegio Romano, the Jesuit University in Rome, and later the universities rector. In the early seventeenth century he was regarded as the leading Catholic authority on theology and as such he was a powerful and highly influential figure in Rome.


Robert Bellarmine artist unknown Source: Wikimedia Commons

How did Bellarmine’s life interact with that of Galileo? The first contact was very indirect and occurred after Galileo had published his Sidereus Nuncius, making public his telescopic discoveries. Bellarmine inquired of the mathematician astronomers under Clavius’ leadership at the Collegio Romano, whether the discoveries claimed by Galileo were real. Being the first astronomers to confirm those discoveries, Clavius was able to report in the positive.

In 1615 Galileo wrote his Letter to Castelli in which he argued that those Bible passages that contradicted Copernican heliocentricity should be re-interpreted to solve the contradiction. He was stepping into dangerous territory, a mere mathematicus—the lowest of the low in the academic hierarchy—telling the theologians how to interpret the Bible. This was particularly risky, as it was in the middle of the Counter-Reformation given that the Reformation was about who is allowed to interpret the Bible. The Protestants said that everyman should be able to interpret it for themselves and the Catholic Church said that only the Church should be allowed to do so. Remember we are only three years away from the Thirty Years War the high point, or should that be the low point, of the conflict between the two religions, which led to the destruction of most of central Europe and the death of between one and two thirds of its population.


Justus Sustermans – Portrait of Galileo Galilei, 1636 Source: Wikimedia Commons

Galileo’s suggestion in his letter came to the attention of his opponents in the Church and led the Pope, Paul V, to set up a commission of eleven theologians, known as the Qualifiers, to investigate the propositions of heliocentricity.

In the meantime Paolo Antonio Foscarini (c. 1565–June 1616), a Carmelite father, attempted to publish his Epistle concerning the Pythagorean and Copernican opinion of the Mobility of the Earth and stability of the sun and the new system or constitution of the WORLD, which basically contained the same arguments for reinterpreting the Bible as Galileo’s Letter to Castelli. The censor of Foscarini’s order rejected his tract, as too contentious. I should point out at this point something that most people ignore that is all powers both civil and religious in Europe exercised censorship; there was no such thing as free thought or freedom of speech in seventeenth century Europe. Foscarini wrote a defence of his Epistle and sent the two pieces to Bellarmine, as the leading theologian, for his considered opinion. Bellarmine’s answers the so-called Foscarini Letter is legendary and I reproduce it in full below.

My Reverend Father,

I have read with interest the letter in Italian and the essay in Latin which your Paternity sent to me; I thank you for one and for the other and confess that they are all full of intelligence and erudition. You ask for my opinion, and so I shall give it to you, but very briefly, since now you have little time for reading and I for writing.

First I say that it seems to me that your Paternity and Mr. Galileo are proceeding prudently by limiting yourselves to speaking suppositionally and not absolutely, as I have always believed that Copernicus spoke. For there is no danger in saying that, by assuming the Earth moves and the sun stands still, one saves all of the appearances better than by postulating eccentrics and epicycles; and that is sufficient for the mathematician. However, it is different to want to affirm that in reality the sun is at the center of the world and only turns on itself, without moving from east to west, and the earth is in the third heaven and revolves with great speed around the sun; this is a very dangerous thing, likely not only to irritate all scholastic philosophers and theologians, but also to harm the Holy Faith by rendering Holy Scripture false. For Your Paternity has well shown many ways of interpreting Holy Scripture, but has not applied them to particular cases; without a doubt you would have encountered very great difficulties if you had wanted to interpret all those passages you yourself cited.

Second, I say that, as you know, the Council [of Trent] prohibits interpreting Scripture against the common consensus of the Holy Fathers; and if Your Paternity wants to read not only the Holy Fathers, but also the modern commentaries on Genesis, the Psalms, Ecclesiastes, and Joshua, you will find all agreeing in the literal interpretation that the sun is in heaven and turns around the earth with great speed, and that the earth is very far from heaven and sits motionless at the center of the world. Consider now, with your sense of prudence, whether the church can tolerate giving Scripture a meaning contrary to the Holy Fathers and to all the Greek and Latin commentators. Nor can one answer that this is not a matter of faith, since it is not a matter of faith “as regards the topic”, it is a matter of faith “as regards the speaker”; and so it would be heretical to say that Abraham did not have two children and Jacob twelve, as well as to say that Christ was not born of a virgin, because both are said by the Holy Spirit through the mouth of the prophets and the apostles.


Third, I say that if there were a true demonstration that the sun is at the center of the world and the earth in the third heaven, and that the sun does not circle the earth but the earth circles the sun, then one would have to proceed with great care in explaining the Scriptures that appear contrary; and say rather that we do not understand them than that what is demonstrated is false. But I will not believe that there is such a demonstration, until it is shown me. Nor is it the same to demonstrate that by supposing the sun to be at the center and the earth in heaven one can save the appearances, and to demonstrate that in truth the sun is at the center and the earth in the heaven; for I believe the first demonstration may be available, but I have very great doubts about the second, and in case of doubt one must not abandon the Holy Scripture as interpreted by the Holy Fathers. I add that the one who wrote, “The sun also riseth, and the sun goeth down, and hasteth to his place where he arose,” was Solomon, who not only spoke inspired by God, but was a man above all others wise and learned in the human sciences and in the knowledge of created things; he received all this wisdom from God; therefore it is not likely that he was affirming something that was contrary to truth already demonstrated or capable of being demonstrated. Now, suppose you say that Solomon speaks in accordance with appearances, since it seems to us that the sun moves (while the earth does so), just as to someone who moves away from the seashore on a ship it looks like the shore is moving, I shall answer that when someone moves away from the shore, although it appears to him that the shore is moving away from him, nevertheless he knows that it is an error and corrects it, seeing clearly that the ship moves and not the shore; but in regard to the sun and the earth, no wise man has any need to correct the error, since he clearly experiences that the earth stands still and that the eye is not in error when it judges that the it also is not in error when it judges that the stars move. And this is enough for now.

With this I greet dearly Your Paternity, and I pray to God to grant you all your wishes.

At home, 12 April 1615.

To Your Very Reverend Paternity.

As a Brother,

Cardinal Bellarmine


(Source for the English transl.: M. Finocchiaro, The Galileo Affair. A Documentary History (Berkeley, CA: University of California Press, 1989), pp. 67-69.Original Italian text, G. Galilei, Opere, edited by A. Favaro (Firenze: Giunti Barbera, 1968), vol. XII, pp. 171-172.)

A, in my opinion, brilliant piece of measured, diplomatic writing. Bellarmine tactfully suggests that one should only talk of heliocentricity hypothetically, its correct scientific status in 1615, the first empirical proof for the movement of the Earth was found in 1725, when Bradley discovered stellar aberration. He, as the great Tridentine theologian, then reiterates the Church’s position on the interpretation of Holy Scripture. Finally he brings, what is without doubt, the most interesting statement in the letter.

Third, I say that if there were a true demonstration that the sun is at the center of the world and the earth in the third heaven, and that the sun does not circle the earth but the earth circles the sun, then one would have to proceed with great care in explaining the Scriptures that appear contrary; and say rather that we do not understand them than that what is demonstrated is false.

What he says is bring proof and we’ll reinterpret the Bible but until then…

On 24 February the Qualifiers delivered the results of their deliberations on the heliocentricity hypothesis:

( i ) The sun is the centre of the universe (“mundi”) and absolutely

immobile in local motion.

( ii ) The earth is not the centre of the universe (“mundi”); it is not

immobile but turns on itself with a diurnal movement.

All unanimously censure the first proposition as “foolish, absurd in philosophy { i.e. scientifically untenable] and formally heretical on the grounds of expressly contradicting the statements of Holy Scripture in many places according to the proper meaning of the words, the common exposition and the understanding of the Holy Fathers and learned theologians”; the second proposition they unanimously censured as likewise “absurd in philosophy” and theologically “at least erroneous in faith”.

It should be pointed out that although the Qualifiers called the first statement heretical, only the Pope could formally declare something heretical and no pope ever did, so heliocentricity was never officially heretical.

Pope Paul V now ordered Bellarmine to covey the judgement of the Qualifiers to Galileo and to inform him that he may not hold or teach the heliocentric theory. This he did on 26 February 1616. Bellarmine was not one of the Qualifiers and here functioned only as the messenger. By all accounts the meetings between Bellarmine and Galileo were cordial and friendly.

When Galileo returned to Florence rumours started spreading that he had been forced to recant and do penance, which was of course not true. Galileo wrote to Bellarmine requesting a letter explaining that this was not true. Bellarmine gladly supplied said letter, defending Galileo’s honour. However Galileo made the mistake in 1633 of thinking that Bellarmine’s letter was a get out of jail free card.

Bellarmine died in 1621 and between 1616 and his death there was no further contact between the Cardinal and the mathematicus. Personally I can see nothing in the three interactions, indirect and direct, between Bellarmine and Galileo that would in anyway justify labelling Bellarmine as the “scourge of Galileo”. This accusation is historically highly inaccurate and paints a wholly false picture of the relationship between the two men. I expect better of Rob Iliffe, who is without doubt one on Britain’s best historians of seventeenth century science.

NB Before somebody pops up in the comments claiming that Robert Bellarmine was one of the three Inquisition judges, who confirmed the death sentence on Giordano Bruno. He was but that has no relevance to his interactions with Galileo, so save yourself time and energy and don’t bother.

1 Comment

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

Really! – Did the artist have a Tardis?

Those who read the occasional bursts of autobiographical information that appear here on the blog might be aware that I went to university at the tender age of eighteen as an archaeology student. I actually dropped out after one year but continued to work as a professional field archaeologist (that’s a digger to you mate) for several years. Given that I was already interested in the history of astronomy in those days and would eventually abandon archaeology for it, it would seem logical that I would be interested in archaeoastronomy, in particular because I studied under Richard Atkinson who together with Stuart Piggott carried out the first extensive, modern excavation of Stonehenge, the world’s most famous archaeoastronomical monument, in the 1950s. In fact my father also worked on that excavation. This assumption would be correct with reservations. There has been some excellent work in archaeoastronomy but unfortunately there has also been a large amount of highly dubious speculation on the topic.

In my opinion an example of the latter appeared in articles in The Guardian and on the Hyperallergic website a couple of days ago. The Guardian article was entitled, Two suns? No, it’s a supernova drawn 6,000 years ago, say scientists. This article tells us:

For decades, stone carvings unearthed in the Himalayan territory of Kashmir were thought to depict a hunting scene. But the presence of two celestial objects in the drawings has piqued the interest of a group of Indian astronomers.


Source: The Guardian

They have proposed another theory. According to a study published in the Indian Journal of History of Science, the Kashmir rock drawings may be the oldest depiction of a supernova, the final explosion of a dying star, ever discovered.

 “Our first argument was, there cannot be two suns,” Vahia said. “We thought it must have been an object that appeared and attracted the attention of the artists.”

 They settled on Supernova HB9, a star that exploded around 4,600BC.

Rewinding the map of the sky back that far revealed more clues.

Viewed from Kashmir, the supernova would have occurred somewhere near the Orion constellation. “Which is known as the scene of a hunter,” said Vahia.

“The supernova also went off just above the constellation of Taurus, the bull, which is also seen in the drawing,” Vahia added.


Source: The Guardian

So to summarise a group of astrophysicists decide that the rock drawing depicts a supernova from around 4,600 BCE that was visible in the sky in the area of the constellations Orion the hunter and Taurus the bull, which according to the researchers are also depicted in the drawing. It is by the latter claim that my bullshit detectors went off at full volume. I will explain.

The chosen supernova occurred in 4600 BCE, now I’m not an expert on prehistoric Indian asterisms, I don’t even know anybody who is, but I do know something about the Babylonian and ancient Greek ones. Taurus is indeed one of the oldest known asterisms but the earliest known mention of a bull asterism is in the Sumerian record, the Heaven’s Bull, in the third millennium BCE, that’s a couple of thousand years after the chosen supernova. Even worse it is not known whether the Sumerian asterism is the same one as the later Babylonian/Greek asterism Taurus. With Orion we have even more problems. The Sumerian asterism involving the stars of Orion was a sheep. For the ancient Egyptians the stars depicted their god Osiris. It was first the Greeks who created the asterism Orion although some mythologists see Orion as a representation of the Sumerian King Gilgamesh, who also fought a bull. This is of course highly speculative.

So we have astrophysicists identifying a rock drawing in India that is dated to the fifth millennium BCE with the constellations of Orion fighting Taurus, asterisms which don’t appear to have been identified till several thousand years later. Excuse me if I am somewhat sceptical about this identification. Just as a minor point I don’t think that the animal in the drawing actually looks like a bull, more like a stag in my opinion.






Filed under History of Astrology, History of Astronomy, History of science, Myths of Science, Uncategorized

Andromeda – From nebula to galaxy

The word galaxy derives from the Greek word galaxias meaning milky one, which was the ancient Greek term for the Milky Way that indistinct band of stars visible across the night sky in areas that don’t suffer from too much light pollution. Today galaxy is used as the general term for the very large groups of stars scattered around the universe. Current estimates of the total number of galaxies range from 2×1011 to 2×1012 or even more. Confronted by these vast numbers it is oft easy to forget that less than one hundred years ago we still thought that our galaxy, the Milky Way Galaxy, was the entire universe. This changed on 1 January 1925 when H.N. Russell read a paper by Edwin Hubble to the American Association for the Advancement of Science, which established that spiral nebulae were in fact separate galaxies. The path through the history of astronomy leading up to that epoch defining paper in 1925 goes back almost one thousand years and in what follows I shall briefly outline some of the important stations, nearly all of which concern our nearest galactic neighbour Andromeda, along that path.

The word nebula comes from the Latin and means a cloud, mist, fog, smoke, vapour, exhalation, as you can see the definition is fairly nebulous. In astronomy it can be traced back to Ptolemaeus’ Mathēmatikē Syntaxis or as it is more commonly known The Almagest. In this founding work of Western astronomy Ptolemaeus lists a total of six astronomical nebulae without giving them any great attention. All of Ptolemaeus’ nebulae were in fact indistinct star clusters too far away to be resolved with the naked eye. The first so-to-speak true nebula, the Andromeda nebula, was recorded by the Persian astronomer Abd al-Rahman al-Sufi, usually just referred to as Al Sufi, in his Book of Fixed Stars (Arabic: kitab suwar al-kawakib) around 964 CE. He describes and illustrated the Andromeda nebula as a little cloud before the mouth of the Arabic constellation Fish.


Al Sufi’s drawing of the constellation Fish with the Andromeda nebula in fount of it mouth

Amongst his other early telescopic observations Galileo showed that the Ptolemaic nebulae resolved into many unseen stars when viewed through the telescope. In 1612, it was, however, Galileo’s telescopic rival, Simon Marius who first turned his telescope on the Andromeda nebula and saw that it didn’t resolve into stars when viewed through his telescopic lenses. In his Mundus Iovialis (1614) Marius described what he saw as follows:

Among them the first is that with the spy-glass, from 15 December 1612 I discovered and observed a fixed star with a certain wonderful shape that I cannot find in the entire heavens. It is near the third and northernmost [star] in the belt of Andromeda. Without the instrument the same is seen as some sort of little cloud; and with the instrument no distinct stars are seen as in the nebular star in Cancer and other nebular stars, but rather only white rays, which the closer to the centre the brighter they come out; in the centre there is a dull and pale light; and its diameter is about a quarter of a degree. About the same brilliance appears when a bright candle is observed through a clear lantern from a long distance.


Simon Marius from the frontispiece of the Mundus Iovialis Source: Wikimedia Commons

The research into nebulae came of age first in the eighteenth century with the work of the French comet hunter Charles Messier (1730–1817). In order to make it easier for comet hunters to distinguish potential comet sightings from other indistinct and nebulous object in the night sky, Messier began to compile a catalogue of the positions and appearance of all such objects that he detected during his nightly vigils. His work, the final version of which was published in 1781 and is now known as the Messier Catalogue, contains a list of 110 Messier objects, in his time nebulae and star clusters. The Messier objects are now known to be 39 galaxies, 5 planetary nebulae, 7 other types of nebulae and 55 star clusters. The Andromeda nebula, the discovery of which Messier, ignorant of Al Sufi’s book, falsely attributes to Marius, is Messier object M31.


Charles Messier, French astronomer, at the age of 40 Source: Wikimedia Commons

Although Messier’s catalogue was compiled to assist comet hunters in differentiating potential comets from other faint celestial objects it is usually regarded as an early example of so-called deep sky astronomy; that is the study of objects well outside the solar system. The man who first practiced deep sky astronomy systematically was William Herschel, who together with his sister Caroline, methodically map the heavens quadrant for quadrant recording with his 20 foot reflecting telescope all of the non-stellar objects he could find. Caroline and he recorded 2400 nebulae in three catalogues.


William and Caroline Herschel polishing a telescope lens, 1896 Lithograph. Source: Wellcome Collection via Wikimedia Commons

They categorised the objects that they recorded into eight classes: (I) bright nebulae, (II) faint nebulae, (III) very faint nebulae, (IV) planetary nebulae, (V) very large nebulae, (VI) very compressed and rich clusters of stars, (VII) compressed clusters of small and large [faint and bright] stars and (VIII) coarsely scattered clusters of stars. Extended by his son and later John Dreyer, Herschel’s catalogue became the New General Catalogue (NGC) of 7840 deep sky objects in 1888. The NGC numbering is still used for most of the objects recorded therein. In 1785 Herschel observed a faint reddish hue in the core region of Andromeda. He believed Andromeda to be the nearest of all the great nebula.

In 1750 the English astronomer Thomas Wright (1711–1786) published his An Original Theory on New Hypothesis of the Universe in which he was the first to correctly describe the shape of the Milky Way Galaxy. He also speculated that the faint nebulae where in fact distant galaxies. However, his very perceptive thoughts remained speculations that he was unable to verify.


Thomas Wright in 1737 Source: Wikimedia Commons


Illustration of groups of stars, from An original theory or new hypothesis of the Universe, plate XVII Source: Wikimedia Commons

Interestingly his speculations were taken up by the German philosopher Immanuel Kant (1724–1804) and further developed in his anonymously published Allgemeine Naturgeschichte und Theorie des Himmels (Universal Natural History and Theory of Heaven) (1755). At the time neither Wright’s nor Kant’s theories received much credence but with hindsight both have been praised for their perceptiveness.


Title page of Kant’s Allgemeine Naturgeschichte und Theorie des Himmels Source: Wikimedia Commons

In 1850, William Parsons, using the largest reflecting telescope constructed in the nineteenth century the Leviathan of Parsonstown, was able to identify the spiral structure of the Andromeda nebula for the first time. This was just one of a series of spiral nebula, in reality galaxies, that he was able to identify.


The largest telescope of the 19th century, the Leviathan of Parsonstown. Source: Wikimedia Commons

In 1864 William Huggins, a pioneer in stellar spectroscopy, noted that the spectrum of Andromeda differs from that of a gaseous nebula. The spectrum, as observed by Huggins, had the same characteristics as the spectrum of individual stars leading he to conclude that Andromeda was in fact stellar in nature.


Sir William Huggins, by John Collier Source: Wikimedia Commons

We have already come a long way from Al Sufi’s first record of a small cloud. In 1887, Isaac Roberts, who thought that spiral nebula were solar systems in the process of forming, took the first-ever photograph of Andromeda.


Isaac Roberts’ picture of the Great Nebula in Andromeda Source: Wikimedia Commons

In 1912 the American astronomer, Vesto Slipher, measured the rotational velocity of Andromeda using spectroscopy at 300kilometres per second the highest yet measured velocity.


V.M. Slipher, astronomer at Lowell Observatory from 1901 to 1954. Source: Wikimedia Commons

In 1917 Heber Curtis observed a nova in Andromeda and discovered eleven more in the photographic record. These were on average ten magnitudes weaker that others observed in the heavens. Based in this data he estimated that Andromeda was 500,000 light-years distant. Curtis now proposed the island universes hypothesis i.e. spiral nebulae are actually independent galaxies.


Heber Doust Curtis poses before the Crossley telescope. Source: Wikimedia Commons

On 26 April 1920 Heber Curtis and Harlow Shapley held the so-called great debate at the Smithsonian Museum of Natural History on the nature of spiral nebulae. Curtis argued that they were distant independent galaxies, Shapley that they were much smaller and much nearly and thus within the Milky Way galaxy, which was the entire universe. This debate raised the question to the priority question in astronomy.


Portrait of Harlow Shapely Source: Wikimedia Commons

In 1922 Ernst Öpik measured the distance of Andromeda using the velocity of stars. His estimate was 1,500,00 light-years.


Ernst Julius Öpik Source: Wikimedia Commons

As I said in the opening paragraph Edwin Hubble finally settled the mater when he measured the distance of Andromeda using Cepheid variable stars and proved conclusively that Andromeda was not a nebula inside the Milky Way but a separate galaxy. With this result the age of galactic astronomy was born.


Studio Portrait of Edwin Powell Hubble. Photographer: Johan Hagemeyer Source: Wikimedia Commons

Of interest the method of determining distances using Cepheids was developed by Henrietta Swan Leavitt, one of the Harvard computers, investigating thousands of variable stars in the Magellanic Clouds in 1908; she published her results in 1912.


Henrietta Swan Leavitt working at her desk in the Harvard College Observatory Source: Wikimedia Commons


Early photograph of ‘Pickering’s Harem’, as the group of women assembled by Harvard astronomer Edward Charles Pickering, who were dubbed as his “computers”. The group included Leavitt, Annie Jump Cannon, Williamina Fleming, and Antonia Maury. Source: Wikimedia Commons

The story of Andromeda’s historical journey from Al Sufi’s nebula to Curtis’ galaxy illustrates very nicely how scientific knowledge grows over time with generations of researchers with differing interests and motivations contributing directly and indirectly to that growth.

Post amended 11 January 2018



Filed under astrophysics, History of Astronomy, History of science, Uncategorized

Christmas Trilogy 2017: Bonus!

Yesterday was Johannes Kepler’s nominal birthday (as he was born before the calendar reform in a Protestant state his birthday on the Gregorian calendar would be 6 January!) and as in my wont, I posted a birthday post for the good Johannes. Of course I was far from being the only person to acknowledge his birthday and amongst many others somebody linked to the 2016 article on the website of the popular science magazine, Physics Today. Upon reading this brief tribute to my favourite seventeenth century polymath I cringed inwardly and didn’t know whether to let out a prolonged #histsigh or to turn loose the HistSci_Hulk; I have decided on the latter. Below the complete text of the offending document:

Born on 27 December 1571 in Weil der Stadt in the Holy Roman Empire, Johannes Kepler was an astronomer whose careful measurements led him to develop his three laws of planetary motion. He received a Lutheran education at the University of Tübingen and originally planned to be a theologian. Then one of his teachers gave him a copy of a book by Nicolaus Copernicus, sparking Kepler’s interest in astronomy. In 1600 Danish astronomer Tycho Brahe invited Kepler to Prague to help amass a precise set of astronomical measurements. Brahe died the following year, and Kepler inherited his mentor’s data and position as imperial mathematician to the Holy Roman emperor. In 1609 Kepler published Astronomia Nova, which included his first two laws of planetary motion; his third law was published in 1619. Kepler observed a supernova (though he called it a “new star”) and completed the detailed astronomical tables Brahe had been so determined to produce. Kepler also contributed research in optics and vision. Later in the century Isaac Newton would prove his law of universal gravitation by showing that it could produce Kepler’s orbits.

Born … in Weil der Stadt in the Holy Roman Empire… This contains something about which I have had bitter disputes on Wikipedia. There is a famous quip that the Holy Roman Empire was neither holy nor Roman nor an empire, it was also neither a country nor a state. The Holy Roman Empire was a loose feudal conglomeration of autonomous and semi-autonomous states. Weil der Stadt, Kepler’s birthplace was at the time of his birth in the autonomous Duchy of Württemberg.


Map of the Duchy of Württemberg 1619 by Pieter van den Keere. You can see Weyl (Weil der Stadt) roughly in the middle. Source: Wikimedia Commons

…Johannes Kepler was an astronomer whose careful measurements led him to develop his three laws of planetary motion. Kepler was a theorist, who didn’t on the whole take measurements careful or otherwise. The measurements that he used to derive his three laws were, of course, made very carefully by Tycho Brahe.

Kepler did not originally plan to be a theologian. He was on an educational tack designed to produce Lutheran Protestant pastors and schoolteachers. He would have become a pastor but was appointed to a position as a maths teacher instead.


Then one of his teachers gave him a copy of a book by Nicolaus Copernicus, sparking Kepler’s interest in astronomy. One of Kepler’s professors in Tübingen was Michael Maestlin, who in his courses taught Copernican heliocentric astronomy alongside the then dominant geocentric astronomy. Kepler took this course and developed an interest in heliocentrism. It was Maestlin who recognised Kepler’s aptitude for mathematics and recommended that he be appointed to a teaching post rather than a village church.

In 1600 Danish astronomer Tycho Brahe invited Kepler to Prague to help amass a precise set of astronomical measurements. Tycho Brahe invited Kepler to Prague not to help amass a precise set of astronomical measurements but to use his mathematical skills to turn the already amassed measurements into calculated orbits, ephemerides etc.

Brahe died the following year, and Kepler inherited his mentor’s data and position as imperial mathematician to the Holy Roman emperor. Kepler didn’t inherit his mentor’s data, Tycho’s daughter Elizabeth and her husband Frans Gansned Genaamd Tengnagel van de Camp did. This caused Kepler no end of problems, as he needed that data to realise his vision of a heliocentric astronomy. After tough negotiations, Tengnagel allowed Kepler to use the data but only if his name was included as co-author on any books that Kepler published based on it; a condition that Kepler duly fulfilled. Given my own inabilities to spell or write grammatically I’m not usually a grammar fetishist but, as I’m putting the boot in, Imperial Mathematician is a title and should be written with capital letters as in the emperor in Holy Roman Emperor.

Kepler observed a supernova (though he called it a “new star”). Well yes, as the term supernova was only coined in 1931 Kepler could hardly have used it. However, the nova part of the name, which simple means new, comes from Kepler’s term Stellar Nova, his being the most recent supernova observed with the naked eye.

…and completed the detailed astronomical tables Brahe had been so determined to produce. Kepler didn’t just complete them he produced them single-handedly, calculating, writing, typesetting, printing, publishing and selling them. This was the task assigned to him by Tycho and to which he was official appointed by the Emperor Rudolph II.

Physics Today is a fairly major popular science magazine but it would appear that they don’t really care enough about the history of science to indulge in a modicum of fact checking.





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

Christmas Trilogy 2017 Part 3: Kepler’s big book

Johannes Kepler was incredibly prolific, he published over eighty books and booklets over a very wide range of scientific and mathematical topics during his life. As far as he was concerned his magnum opus was his Ioannis Keppleri Harmonices mundi libri V (The Five Books of Johannes Kepler’s The Harmony of the World) published in 1619 some twenty years after he first conceived it. Today in popular #histsci it is almost always only mentioned for the fact that it contains the third of his laws of planetary motion, the harmonic law. However it contains much, much more of interest and in what follows I will attempt to give a brief sketch of what is in fact an extraordinary book.


A brief glace at the description of the ‘five books’ thoughtfully provided by the author on the title page (1) would seem to present a mixed bag of topics apparently in some way connected by the word or concept harmonic. In order to understand what we are being presented with we have to go back to 1596 and Kepler’s first book Mysterium Cosmographicum (The Cosmographic Mystery). In this slim volume Kepler presents his answer to the question, why are there only six planets? His, to our eyes, surprising answer is that the spaces between the planets are defined by the regular so-called Platonic solids and as the are, and can only be, five of these there can only be six planets.

Using the data from the greatest and least distances between the planets in the Copernican system, Kepler’s theory produces an unexpectedly accurate fit. However the fit is not actually accurate enough and in 1598 Kepler began working on a subsidiary hypothesis to explain the inaccuracies. Unfortunately, the book that he had planned to bring out in 1599 got somewhat delayed by his other activities and obligations and didn’t appear until 1619 in the form of the Harmonice mundi.

The hypothesis that Kepler presents us with is a complex mix of ideas taken from Pythagoras, Plato, Euclid, Proclus and Ptolemaeus centred round the Pythagorean concept of the harmony of the spheres. Put very simply the theory developed by the Pythagoreans was that the seven planets (we are talking geocentric cosmology here) in their orbits form a musical scale than can, in some versions of the theory, only be heard by the enlightened members of the Pythagorean cult. This theory was developed out of the discovery that consonances (harmonious sounds) in music can be expressed in the ratio of simple whole numbers to each other (the octave for example is 1:2) and the Pythagorean belief that the integers are the building block of the cosmos.

This Pythagorean concept winds its way through European intellectual history, Ptolemaeus wrote a book on the subject, his Harmonice and it is the reason why music was one of the four disciplines of the mathematical quadrivium along with arithmetic, geometry and astronomy. Tycho Brahe designed his Uraniburg so that all the architectonic dimensions from the main walls to the window frames were in Pythagorean harmonic proportion to one another.


Tycho Brahe’s Uraniborg Blaeus Atlas Maior 1663 Source: Wikimedia Commons

It is also the reason why Isaac Newton decided that there should be seven colours in the rainbow, to match the seven notes of the musical scale. David Gregory tells us that Newton thought that gravity was the strings upon which the harmony of the spheres was played.

In his Harmony Kepler develops a whole new theory of harmony in order to rescue his geometrical vision of the cosmos. Unlike the Pythagoreans and Ptolemaeus who saw consonance as expressed by arithmetical ratios Kepler opted for a geometrical theory of consonance. He argued that consonances could only be constructed by ratios between the number of sides of regular polygons that can be constructed with a ruler and compass. The explication of this takes up the whole of the first book. I’m not going to go into details but interestingly, as part of his rejection of the number seven in his harmonic scheme Kepler goes to great lengths to show that the heptagon construction given by Dürer in his Underweysung der Messung mit dem Zirckel und Richtscheyt is only an approximation and not an exact construction. This shows that Dürer’s book was still being read nearly a hundred years after it was originally published.


In book two Kepler takes up Proclus’ theory that Euclid’s Elements builds systematically towards the construction of the five regular or Platonic solids, which are, in Plato’s philosophy, the elemental building blocks of the cosmos. Along the way in his investigation of the regular and semi-regular polyhedra Kepler delivers the first systematic study of the thirteen semi-regular Archimedean solids as well as discovering the first two star polyhedra. These important mathematical advances don’t seem to have interested Kepler, who is too involved in his revolutionary harmonic theory to notice. In the first two books Kepler displays an encyclopaedic knowledge of the mathematical literature.


The third book is devoted to music theory proper and is Kepler’s contribution to a debate that was raging under music theorist, including Galileo’s father Vincenzo Galilei, about the intervals on the musical scale at the beginning of the seventeenth century. Galilei supported the so-called traditional Pythagorean intonation, whereas Kepler sided with Gioseffo Zarlino who favoured the ‘modern’ just intonation. Although of course Kepler justification for his stance where based on his geometrical arguments. Another later participant in this debate was Marin Mersenne.


In the fourth book Kepler extends his new theory of harmony to, amongst other things, his astrology and his theory of the astrological aspects. Astrological aspects are when two or more planets are positioned on the zodiac or ecliptic at a significant angle to each other, for example 60° or 90°. In his Harmonice, Ptolemaeus, who in the Renaissance was regarded as the prime astrological authority, had already drawn a connection between musical theory and the astrological aspects; here Kepler replaces Ptolemaeus’ theory with his own, which sees the aspects are being derived directly from geometrical constructions. Interestingly Kepler, who had written and published quite extensively on astrology, rejected nearly the whole of traditional Greek astrology as humbug keeping only his theory of the astrological aspects as the only valid form of astrology. Kepler’s theory extended the number of influential aspects from the traditional five to twelve.

The fifth book brings all of the preceding material together in Kepler’s astronomical/cosmological harmonic theory. Kepler examines all of the mathematical aspects of the planetary orbits looking for ratios that fit with his definitions of the musical intervals. He finally has success with the angular velocities of the planets in their orbits at perihelion and aphelion. He then examines the relationships between the tones thus generated by the different planets, constructing musical scales in the process. What he in missing in all of this is a grand unifying concept and this lacuna if filled by his harmonic law, his third law of planetary motion, P12/P22=R13/R23.


There is an appendix, which contains Kepler’s criticisms of part of Ptolemaeus’ Harmonice and Robert Fludd’s harmony theories. I blogged about the latter and the dispute that it triggered in an earlier post

With his book Kepler, who was a devoted Christian, was convinced that he had revealed the construction plan of his geometrical God’s cosmos. His grandiose theory became obsolete within less than fifty years of its publication, ironically pushed into obscurity by intellectual forces largely set into motion by Kepler in his Astronomia nova, his Epitome astronomiae Copernicanae and the Rudolphine Tables. All that has survived of his great project are his mathematical innovations in the first two books and the famous harmonic law. However if readers are prepared to put aside their modern perceptions and prejudices they can follow one of the great Renaissance minds on a fascinating intellectual journey into his vision of the cosmos.

(1) All of the illustration from the Harmonice mundi in this post are taken from the English translation The Harmy of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aston, A.M. Duncan and J.V. Field, American Philosophical Society, 1997


Filed under Early Scientific Publishing, History of Astrology, History of Astronomy, History of Mathematics, History of science, Renaissance Science, Uncategorized