If you can’t tell your Cassini from your Huygens then you shouldn’t be writing about the history of astronomy.

There I was, mild mannered historian of early modern science, enjoying my first cup of tea on a lazy Sunday morning, whilst cruising the highway and byways of cyberspace, when I espied a statement that caused an explosion of indignation, transforming me into the much feared, fire spitting HISTSCI_HULKTM. What piece of histSTM crap had unleashed the pedantic monster this time and sent him off on a stamping rage?

The object of HSH’s rage was contained in an essay by Vahe Peroomian (Associate Professor of Physics and Astronomy, University of Southern California – Dornsife College of Letters, Arts and Sciences) A brief astronomical history of Saturn’s amazing rings, published simultaneously on both The Conversation and PHYS.ORG 15 August 2019. Peroomian writes:

I am a space scientist with a passion for teaching physics andastronomy, and Saturn’s rings have always fascinated me as they tell the story of how the eyes of humanity were opened to the wonders of our solar system and the cosmos.

He continues:

When Galileo first observed Saturn through his telescope in 1610, he was still basking in the fame of discovering the four moons of Jupiter. But Saturn perplexed him. Peering at the planet through his telescope, it first looked to him as a planet with two very large moons, then as a lone planet, and then again through his newer telescope, in 1616, as a planet with arms or handles.


Galileo Portrait by Ottavio Leoni Source: Wikimedia Commons

Galileo actually observed Saturn three times. The first time in 1610 he thought that the rings were handles or large moons on either side of the planet, “I have observed the highest planet [Saturn] to be triple bodied. This is to say to my very great amazement Saturn was seen to me to be not a single star, but three together, which almost touch each other.”


Galileo’s 1610 sketch of Saturn and its rings

The second time was in 1612 and whatever it was that he observed in 1610 had simply disappeared, “I do not know what to say in a case so surprising, so unlooked for and so novel.” The Earth’s position relative to Saturn had changed and the rings were no longer visible but Galileo did not know this. In 1616 the rings were back but with a totally altered appearance, “The two companions are no longer two small perfectly round globes … but are present much larger and no longer round … that is, two half eclipses with two little dark triangles in the middle of the figure and contiguous to the middle globe of Saturn, which is seen, as always, perfectly round.” [1]


Galileo’s 1616 sketch of Saturn and its rings

There is no mention of a new telescope and it is fairly certain that all three periods of observation were either carried out with the same or very similar telescopes. The differences that Galileo observed were due to the changing visibility of Saturn’s rings caused by its changing relative position to Earth and not to any change of instrument on Galileo’s part.

Although sloppy and annoying, the minor errors in Peroomian’s account of Galileo’s observations of Saturn are in themselves not capable of triggering the HSH’s wrath but what he wrote next is:

Four decades later, Giovanni Cassini first suggested that Saturn was a ringed planet, and what Galileo had seen were different views of Saturn’s rings. Because of the 27 degrees in the tilt of Saturn’s rotation axis relative to the plane of its orbit, the rings appear to tilt toward and away from Earth with the 29-year cycle of Saturn’s revolution about the Sun, giving humanity an ever-changing view of the rings.


Giovanni Cassini (artist unknown) Source: Wikimedia Commons

Now, Giovanni Cassini did record some important observations of Saturn; he discovered four of Saturn’s largest moons and also the gap in the rings that is named after him. Although, Giuseppe Campani, Cassini’s telescope maker, observed the gap before he did without realising that it was a gap. However, it was not Cassini who first suggested that what people had been observing were rings but Christiaan Huygens.

Christiaan Huygens first proposed that Saturn was surrounded by a solid ring in 1655, “a thin, flat ring, nowhere touching, and inclined to the ecliptic.” In 1659 he published his book, Systema Saturnium : sive, De causis mirandorum Saturni phaenomenôn, et comite ejus Planeta Novo detailing how the appearance of the rings varied as the Earth and Saturn orbited the sun.


Plate from Huygens’ Systema Saturnium showing the various recorded observations of Saturn made by astronomers before his own times


Plate from Huygens’ Systema Saturnium explaining why the appearance of Saturn and its rings changes over time and that all those different appearances can be explained by assuming the existence of the rings

Confusing Cassini and Huygens, two of the greatest observational astronomers of the seventeenth century, who were scientific rivals, is not a trivial error and shouldn’t be made anywhere by anyone. However, to make this error in an essay that is published  on two major Internet websites borders on the criminal. I have no idea what the reach of PHYS.ORG is but The Conversation claims to have a readership of ten million plus. This means that a lot of people are being fed false history of astronomy facts by a supposed expert.

If the good doctor Peroomian had bothered to check his facts, a thing that I thought all scientists were taught to do when receiving their mother milk, he could have easily discovered his crass error and corrected it, even the much maligned Wikipedia gets it right, but apparently he didn’t consider it necessary to do so, after all it’s just history and not real science.

[1]The Galileo and Huygens quotes are taken from Ron Baalke’s excellent time line, Historical Background of Saturn’s Rings.



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

The emergence of modern astronomy – a complex mosaic: Part XVII

As I stated earlier in this series only a comparatively small number of astronomers accepted the whole of Copernicus’ theory, both cosmology and astronomy. More interestingly almost none of them had any lasting impact during the final decades of the sixteenth century on the gradual acceptance of heliocentrism. Although he appears to have abandoned Copernicus’ astronomy later in life, Rheticus did have a strong impact with his Narratio Prima(1540), which through its various editions was the first introduction to the heliocentric hypothesis for many readers. Two others, whose impact was principally in the seventeenth century, were Kepler and Galileo, who will be dealt with later. However, one astronomer who did play an important role in the sixteenth century was Michael Mästlin.


Michael Mästlin portrait 1619 artist unknown

Michael Mästlin (1550-1631) stood at the end of a long line of important Southern German astronomers and mathematicians. A graduate of the University of Tübingen he was a student of Philipp Apian (1531–1589),


Philipp Apian, artist unknown Source: Wikimedia Commons


who was a student of his more famous father Peter Apian (1495–1552) in Ingolstadt. Peter Apian had studied under Georg Tannstetter (1482–1535) in Vienna, who had studied under Andreas Stiborius (c. 1464–1515) and Johannes Stabius (1450–1522) first in Ingolstadt then in Vienna. In 1584 Mästlin succeeded his teacher Philipp Apian as professor for astronomy and mathematics at Tübingen. An active astronomer since the beginning of the 1570s Mästlin was regarded as a leading German astronomer and consulted by the Protestant princes on matters astronomical, astrological and mathematical.

Mästlin represents the transitional nature of the times probably better than any other astronomer. His Epitome Astronomiae (1582), a university textbook, which went through a total of seven editions, was a standard Ptolemaic geocentric text that he continued to teach from until his death in 1631.


However, at the same time he taught selected students the fundaments of Copernican heliocentric astronomy. Earlier accounts claimed that he did this in secret but all of the available evidence suggests that he did so quite openly. This quasi revolutionary act of teaching famously produced one significant result in that Mästlin introduced Copernican astronomy to the young Johannes Kepler, who would go on to become the most important propagator of heliocentric astronomy in the early seventeenth century.

One subject on, which the German Protestant princes consulted Mästlin was the proposed Gregorian calendar reform from 1582. Mästlin launched a vitriolic polemic against it largely on religious grounds with his Gründtlicher Bericht von der allgemeinen und nunmehr bei 1600 Jahren von dem ersten Kaiser Julio bis jetzt gebrauchten jarrechnung oder kalender (Rigorous report on the general and up till now for 1600 years used calculation of years or calendar from the first Caesar Julio) (1583). The Protestant princes accepted his advice and as a result didn’t adopt the new calendar until 1700.

On the other side of the religious divide the man charged by the Pope to promote and defend the new calendar was the Jesuit professor of astronomy and mathematics at the Collegio Romano, Christoph Clavius (1538–1612).


Christoph Clavius. Engraving Francesco Villamena, 1606 Source: Wikimedia Commons

Although Clavius was a convinced defender of the Ptolemaic system until his death, he did play a central role in the developments that led to the eventual acceptance of the heliocentric system. The Catholic universities in the last quarter of the sixteenth century still didn’t really pay the mathematical disciplines much attention and their teaching of astronomy had not really progressed beyond the High Middle Ages. Clavius introduced modern mathematics and astronomy into the Jesuit educational reform programme, following the fundamental principle of that programme, if you want to win the debate with your non-Catholic opponents you need to be better educated than them. Many Jesuit and Jesuit educated mathematicians and astronomers, who came out of the pedagogical programme established by Clavius, would, as we shall see, make significant and important contributions to the developments in astronomy in the seventeenth century.

Clavius was also the author of a number of excellent up to date textbooks on a full range of mathematical topics. His astronomy textbook In Sphaeram Ioannis de Sacro Bosco commentarius, the first edition appearing in 1570 and further updated editions appearing in 1581, 1585, 1593, 1607, 1611 and posthumously in 1618, was the most widely read astronomy textbook in the last decades of the sixteenth and early decades of the seventeenth centuries. It was strictly Ptolemaic but he presented, described and commented upon Copernicus’ heliocentric hypothesis. Although he showed great respect for Copernicus as a mathematical astronomer, he of course rejected the hypothesis. However, anybody who read Clavius’ book would be informed of Copernicus work and could if interested go looking for more information. One should never underestimate the effect of informed criticism, and Clavius’ criticism was well informed, for disseminating a scientific hypothesis. Many people certainly had their first taste of the heliocentric hypothesis through reading Clavius.

Another group who had a positive impact on the propagation of the heliocentric hypothesis in the last quarter of the sixteenth century was the so-called English School of Mathematics. Whilst Robert Recorde (1510–1558) and John Dee (1527–c. 1608) were not committed supporters of Copernicus, they did much to spread knowledge of the heliocentric hypothesis. As we have already seen John Feild (c. 1520–1587) was a declared supporter of Copernicus but as his Copernican ephemerides proved no more accurate than the Ptolemaic ones his influence diminished. Not so Dee’s foster son Thomas Digges (c. 1546–1595).

His 1576 edition of his father’s A Prognostication everlastingcontained an appendix A Perfit Description of the Caelestiall Orbes according to the most aunciente doctrine of the Pythagoreans, latelye revived by Copernicus and by Geometricall Demonstrations approved, which is an annotated translation of part of the cosmological first book of De revolutionibus into English, which continued to have an impact on English readers long after Digges’ demise.


Source: Linda Hall Library

Thomas Harriot (c. 1560–1621) was another, who was committed to the heliocentric hypothesis.


Portrait often claimed to be Thomas Harriot (1602), which hangs in Oriel College, Oxford. Source: Wikimedia Commons

His biggest problem was that he published none of his scientific or mathematical work but he was well networked and contributed extensively to the debate through correspondence. The influence of this group would, as we will see, have an impact on the early acceptance of Kepler’s work inEngland.

Another figure in the last quarter of the sixteenth century, who, although not an astronomer, made a very important contribution to the cosmological debate, was the physician William Gilbert (1544–1603).


William Gilbert (1544–1603) artist unknown. Source: Wellcome Library via Wikimedia Commons

Gilbert is well known in the history of science as the author of the first modern scientific investigation of magnetism in his De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure (On the Magnet and Magnetic Bodies, and on That Great Magnet the Earth).


Gilbert carried out many of his experiments with spherical magnets, which he called terella, from which he deduced his belief that the Earth itself is a spherical magnet. Based on his erroneous belief that a suspended terella rotates freely about its axis he came to accept and propagate diurnal rotation. Book VI of De magnete, the final book, is devoted to an analysis of the Earth as a spherical magnet based on the results of Gilbert’s experiments with his terella.

In Chapter III of Book VI, On the Daily Magnetic Revolution of the Globes, as Against the Time-Honoured Opinion of a primum mobile: A Probable Hypothesis, Gilbert gives a detailed review of the history of a geocentric system with diurnal rotation starting with Heraclides of Pontus and going through to Copernicus. Gilbert rejects the whole concept of celestial spheres, dismissing them as a human construction with no real existence. He brings the standard physical arguments that it is more logical that the comparatively small Earth rotates once in twenty-four hours rather than the vastly larger sphere of the fixed stars. In the following chapter he then argues that magnetism is the origin of this rotation. In Chapter V he discusses the arguments for and against movement of the Earth. At the end of Chapter III Gilbert writes, “I pass by the earth’s other movements, for here we treat only of the diurnal rotation…” so what he effectively promotes is a geocentric system with diurnal rotation. Later in his De Mundo Nostro Sublunari Philosophia Nova (New Philosophy about our Sublunary World), Gilbert propagated a full heliocentric system but this book was first published posthumously in 1651 and had no real influence on the astronomical discussion.


Diagram of the cosmos De Mundo p. 202 Source: Wikimedia Commons

Gilbert’s De magnete was a widely read and highly influential book in the first half of the seventeenth century. Galileo praised it but criticised its lack of mathematics. As we shall see it had a massive influence on Kepler. Because of its status the book definitely had a major impact on the acceptance of geo-heliocentric systems with diurnal rotation rather than without later in the seventeenth century.

We will stop briefly and take stock in 1593, fifty years after the publication of De revolutionibus. We have seen that within Europe astronomers had already begun to question the inherited Ptolemaic system during the fifteenth century. In the sixteenth century a major debate developed about both the astronomical and cosmological models. The Aristotelian theories of comets, the celestial spheres and celestial immutability all came under attack and were eventually overturned. Alternative models–Aristotelian homocentricity, the Capellan system and geocentricity with diurnal rotation–were promoted.  With the publication of Copernicus’ De revolutionibus with its heliocentric hypothesis the debates went into overdrive. Only a comparatively small number of astronomers propagated the heliocentric system and an even smaller number of them actually went on to have a real impact on the discussion. A much larger number showed an initial strong interest in the mathematical models in De revolutionibus and the planetary tables and ephemerides based on them, in the hope they would generate better, more accurate data for applications such as astrology, cartography and navigation. This proved not to be the case as Copernicus’ work was based on the same inaccurate and corrupted ancient data, as Ptolemaic geocentric tables. Recognising this both Wilhelm IV in Kassel and Tycho Brahe on Hven began programmes of extensive new astronomical observations. However, this very necessary new data only became generally available well into the seventeenth century. Other astronomers partially convinced by Copernicus’ arguments turned to Capellan models with Mercury and Venus orbiting the Sun rather than the Earth and full geo-heliocentric models with the Moon and the Sun orbiting the Earth and all the other five planets orbiting the Sun. This was the situation at the beginning of the 1590s but a young Johannes Kepler (1571–1630), who would have a massive impact on the future astrological and cosmological models, was waiting in the wings.







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

Vienna and Astronomy the beginnings.

Vienna and its university played a very central role in introducing the study of mathematics, cartography and astronomy into Northern Europe in the fifteenth and sixteenth century. In early blog posts I have dealt with Georg von Peuerbach and Johannes Regiomontanus, Conrad Celtis and his Collegium poetarum et mathematicorum, Georg Tannstetter and the Apians, and Emperor Maximilian and his use of the Viennese mathematici. Today, I’m going to look at the beginnings of the University of Vienna and the establishment of the mathematical science as a key part of the university’s programme.

The University of Vienna was founded in 1365 by Rudolf IV, Duke of Austria (1339–1365) and his brothers Albrecht III, (c. 1349–1395) and Leopold III (1351–1386) both Dukes of Austria.


Rudolf IV, Duke of Austria Source: Wikimedia Commons

Like most young universities it’s early decades were not very successful or very stable. This began to change in 1384 when Heinrich von Langenstein (1325–1397) was appointed professor of theology.


Presumably Heinrich von Langenstein (1325-1397), Book miniature in Rationale divinorum officiorum of Wilhelmus Durandus, c. 1395

Heinrich von Langenstein studied from 1358 in Paris and in 1363 he was appointed professor for philosophy on the Sorbonne advancing to Vice Chancellor. He took the wrong side during the Western Schism (1378–1417) and was forced to leave the Sorbonne and Paris in 1382. Paris’ loss was Vienna’s gain. An excellent academic and experienced administrator he set the University of Vienna on the path to success. Most important from our point of view is the study of mathematics and astronomy at the university. We tend to think of the curriculum of medieval universities as something fixed: a lower liberal arts faculty teaching the trivium and quadrivium and three higher faculties teaching law, medicine and theology. However in their early phases new universities only had a very truncated curriculum that was gradually expanded over the early decades; Heinrich brought the study of mathematics and astronomy to the young university.

Heinrich was a committed and knowledgeable astronomer, who established a high level of tuition in mathematics and astronomy. When he died he left his collection of astronomical manuscripts and instruments to the university. Henry’s efforts to establish astronomy as a discipline in Vienna might well have come to nothing if a successor to teach astronomy had not been found. However one was found in the person of Johannes von Gmunden (c. 1380–1442).


Initial from British Library manuscript Add. 24071 Canones de practica et utilitatibus tabularum by Johannes von Gmunden written 1437/38 by his student Georg Prunner Possibly a portrait of Johannes Source: Johannes von Gmunden (ca. 1384–1442) Astronom und Mathematiker Hg. Rudolf Simek und Kathrin Chlench, Studia Medievalia Septentrionalia 12

Unfortunately, as is often the case with medieval and Renaissance astronomers and mathematicians, we know almost nothing personal about Johannes von Gmunden. There is indirect evidence that he comes from Gmunden in Upper Austria and not one of the other Gmunden’s or Gmund’s. His date of birth is an estimate based on the dates of his studies at the University of Vienna and everything else we know about him is based on the traces he left in the archives of the university during his life. He registered as a student at the university in 1400, graduating BA in 1402 and MA in 1406.

His MA was his licence to teach and he held his first lecture in 1406 on the Theoricae planetarum by Gerhard de Sabbioneta (who might well not have been the author) a standard medieval astronomy textbook, establishing Johannes’ preference for teaching astronomy and mathematics. In 1407, making the reasonable assumption that Johannes Kraft is Johannes von Gmunden, thereby establishing that his family name was Kraft, he lectured on Euclid. 1408 to 1409 sees him lecturing on non-mathematical, Aristotelian texts and 1410 teaching Aristotelian logic using the Tractatus of Petrus Hispanus. In the same year he also taught Euclid again. 1411 saw a return to Aristotle but in 1412 he taught Algorismus de minutiis i.e. sexagesimal fractions. The Babylonian sexagesimal number system was used in European astronomy down to and including Copernicus in the sixteenth century, Aristotelian logic again in 1413 but John Pecham’s Perspectiva in 1414.


Johannes von Gmunden Algorismus de minutiis printed by Georg Tannstetter 1515 Source: Johannes von Gmunden (ca. 1384–1442) Astronom und Mathematiker Hg. Rudolf Simek und Kathrin Chlench, Studia Medievalia Septentrionalia 12

Around this time Johannes took up the study of theology, although he never proceeded past BA, and 1415 and 16 see him lecturing on religious topics although he also taught Algorismus de minutiis again in 1416. From 1417 till 1434, with breaks, he lectured exclusively on mathematical and astronomical topics making him probably the first dedicated lecturer for the mathematical disciplines at a European university. Beyond his lectures he calculated and wrote astronomical tables, taught students how to use astronomical instruments (for which he also wrote instruction manuals), including the construction of cheap paper instruments.


Johannes von Gmunden instructions for constructing an astrolabe rete Wiener Codex ÖNB 5296 fol. 6r Source: Johannes von Gmunden (ca. 1384–1442) Astronom und Mathematiker Hg. Rudolf Simek und Kathrin Chlench, Studia Medievalia Septentrionalia 12

He collected and also wrote extensive astronomical texts. As well as his teaching duties, Johannes served several times a dean of the liberal arts faculty and even for a time as vice chancellor of the university. His influence in his own time was very extensive; there are more than four hundred surviving manuscripts of Johannes Gmunden’s work in European libraries and archives.

When he died Johannes willed his comparatively large collection of mathematical and astronomical texts and instruments to the university establishing a proper astronomy department that would be inherited with very positive results by Georg von Peuerbach and Johannes Regiomontanus. Perhaps the most fascinating items listed in his will are an Albion and an instruction manual for it.


Albion front side Source: Seb Falk’s Twitter feed


Albion rear Source: Seb Falk’s Twitter feed

The Albion is possibly the most fascinating of all medieval astronomical instruments. Invented by Richard of Wallingford (1292–1336), the Abbot of St Albans, mathematician, astronomer, horologist and instrument maker, most well known for the highly complex astronomical clock that he designed and had constructed for the abbey.


Richard of Wallingford Source: Wikimedia Commons

The Albion, ‘all by one’, was a highly complex and sophisticated, multi-functional astronomical instrument conceived to replace a whole spectrum of other instruments. Johannes’ lecture from 1431 was on the Albion.

Johannes von Gmunden did not stand alone in his efforts to develop the mathematical sciences in Vienna in the first half of fifteenth century; he was actively supported by Georg Müstinger (before 1400–1442), the Prior of the Augustinian priory of Klosterneuburg.



Müstinger became prior of Klosterneuburg in 1418 and worked to turn the priory into an intellectual centre. In 1421 he sent a canon of the priory to Padua to purchase books for over five hundred florins, a very large sum of money. The priory became a centre for producing celestial globes and cartography. It produced a substantial corpus of maps including a mappa mundi, of which only the coordinate list of 703 location still exist. Scholar who worked in the priory and university fanned out into the Southern German area carrying the knowledge acquired in Vienna to other universities and monasteries.

Johannes’ status and influence are nicely expressed in a poem about him and Georg von Peuerbach written by Christoph Poppenheuser in 1551:

The great Johannes von Gmunden, noble in knowledge, distinguished in spirit, and dignified in piety                                                                                                                                         And you Peuerbach, favourite of the muses, whose praise nobody can sing well enough                                                                                                                                           And Johannes, named after his home town, known as far away as the stars for his erudition

The tradition established in Vienna by Heinrich von Langenstein, Johannes von Gmunden and Georg Müstinger was successfully continued by Georg von Peuerbach (1423–1461), who contrary to some older sources was not a direct student of Johannes von Gmunden arriving in Vienna only in 1443 the year after Johannes death. However Georg did find himself in a readymade nest for the mathematical disciplines, an opportunity that he grasped with both hands developing further Vienna’s excellent reputation in this area.







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

The emergence of modern astronomy – a complex mosaic: Part XVI

One of the things attributed to Tycho Brahe is the geo-heliocentric model of the cosmos. In this system the Earth remains at the centre and the Moon and the Sun both orbit the Earth, whereas the other five planets orbit the Sun. This system combines most of the advantages of Copernicus’ heliocentric system without the problems caused by a moving Earth. As such, as we shall see, the Tychonic system became one of the two leading contenders later in the seventeenth century. The only problem is that although it is named after him, Tycho wasn’t the only person to suggest this model and he almost certainly wasn’t the first to think of it.


A 17th century illustration of the Hypothesis Tychonica from Hevelius’ Selenographia, 1647 page 163, whereby the Sun, Moon, and sphere of stars orbit the Earth, while the five known planets (Mercury, Venus, Mars, Jupiter, and Saturn) orbit the Sun. Source: Wikimedia Commons

The first to publish a version of the geo-heliocentric model was Nicolaus Reimers Baer (1551–1600), known as Ursus, in his Nicolai Raymari Ursi Dithmari Fundamentum astronomicum (Straßburg 1588). Ursus’ system differed from Tycho’s in that he included diurnal rotation.


Nicolaus Reimers Baer, Fundamentum Astronomicum 1588 geo-heliocentric planetary model Source: Wikimedia Commons

Ursus was a self-taught astronomer, who in his youth had worked as a pig-herd until Heinrich Rantzau (1526–1598), a humanist scholar and astrologer, recognised his talents and employed him as a mathematician.


Heinrich Rantzau Source: Wikimedia Commons

There followed a period as a private tutor and a year, 1586–87, in Kassel with Wilhelm. During his time in Kassel he translated De revolutionibus into German for Jost Bürgi, who couldn’t read Latin. In exchange Bürgi taught Ursus prosthaphaeresis, a method of using trigonometrical formulas to turn multiplications into sums to simplify calculations. From 1591 till his death, in 1600, Ursus was Imperial Mathematicus to Rudolf II in Prague.

Tycho was outraged that somebody published “his system” before he did and immediately accused Ursus of plagiarism, both of the geo-heliocentric system and of prosthaphaeresis, citing an earlier visit to Hven together with Rantzau, when Ursus was in his service. The two astronomers delivered a very unseemly public squabble through a series of publications; Tycho emphasising Ursus’ lowly birth and lack of formal qualifications and Ursus giving as good as he got in return. However, when Tycho left Hven and approached Prague, Ursus fled fearing the aristocrat’s wrath. When Kepler came to Prague to work with Tycho the first task that Tycho gave him was to write an account of the dispute, naturally expecting Kepler to find in his favour. Kepler wrote his report but didn’t ever publish it. Nicholas Jardine published a heavily annotated English translation in his The Birth of History and Philosophy of Science. Kepler’s ‘A Defence of Tycho against Ursus’ with Essays on its Provenance and Significance, CUP (2nd rev. ed. 1988)[1].

Tycho’s false accusation of theft of the trigonometrical method of prosthaphaeresis is, however, very revealing. Tycho was not the discoverer/inventor[2] of prosthaphaeresis. As far as can be ascertained, the method was originally discovered by Johannes Werner (1468–1522) but was actually taught to Tycho by the itinerant mathematician/astronomer from Breslau, Paul Wittich (c. 1546–1586). It turns out that that Wittich was probably the inspiration for both Tycho’s and Ursus’ decision to adopt a geo-heliocentric system. Wittich played around with the Capellan system, in which Mercury and Venus orbit the Sun in a geocentric system. He sketches of his thoughts are contained in his copy of De revolutionibus.


Paul Wittich’s 1578 Capellan geoheliocentric planetary model – as annotated in his copy of Copernicus’s De revolutionibus in February 1578 Source: Wikimedia Commons

Following Wittich’s, comparatively early, death Tycho went to a lot of trouble and expense to obtain both of Wittich’s copies of Copernicus’ book, suggesting he was desperately trying to cover up the origins of “his system.” Another indication of Wittich’s possible or even probable influence is the fact that David Origanus (1558–1629), who had been influenced by Wittich at the University of Frankfurt an der Oder, also “independently” invented a geo-heliocentric system but with diurnal rotation like Ursus’ system.


David Origanus Source: Wikimedia Commons

The route from a Capellan system to a full geo-heliocentric system was probably the route taken by both the physician and astrologer Helisaeus Roeslin (1545–1616) and the court mathematicus Simon Marius (1573–1625), who both claimed independent discovery of the system.


Simon Marius Source: Wikimedia Commons

Geoheliocentric cosmology, 16th century

I think it should be clear by now that a geo-heliocentric system, whether with or without diurnal rotation was seen as a logical development by several astronomers following the publication of De revolutionibus, for it combined most of the advantages of Copernicus’ system, whilst not requiring the Earth to orbit the Sun, solving as it did the problem of the missing, or better said undetectable, solar stellar parallax. Such a system also solved another perceived, empirical problem, which has been largely forgotten today, that of star size.

If the cosmos were heliocentric then the lack of detectable parallax would mean that the so-called fixed stars were absurdly distant and much worse, given the naked-eye false perception the size of the star discs, all the more absurdly immense. Tycho used this as a valid empirical argument alongside religious ones to categorically reject a heliocentric system. Because the geo-heliocentric system didn’t require stellar parallax then the distance to the fixed stars was considerably shorter and thus the star size also much smaller. The apparent star size argument would continue to play a significant role in the astronomical system debate until the end of the seventeenth century.

Tycho, naturally, hoped to use his vast quantity of freshly won, comparatively accurate celestial data to prove the empirical reality of his system. Unfortunately, he died before he could really set this project in motion. On his deathbed he extracted the promise from Johannes Kepler, his relatively new assistant, to use the data to prove the validity of his system. As is well known, Kepler did nothing of the sort but actually used Tycho’s hard won data to develop his own totally novel heliocentric system, of which more later.

However, a geo-heliocentric model of the cosmos, with or without diurnal rotation, remained, as we shall see later, one of the leading contenders amongst astronomers right up to about 1660-70. The definitive version based on Tycho’s own data was produced by Christen Sørensen, known as Longomontanus, (1562-1647),


Tycho’s longest serving and most loyal assistant, in his Astronomia Danica (1622).


Longomontanus’ system was published in direct opposition to Kepler’s heliocentric one. Unlike Tycho’s, Longomontanus’ system had diurnal rotation.

Today we tend to view the various geo-heliocentric systems, with hindsight, as more than somewhat bizarre, but they provided an important and probably necessary bridge between a pure geocentric model and a pure heliocentric one, delivering many of the perceived advantages of heliocentricity, without having to solve the problems created by an Earth flying at high speed around the Sun.

[1]A highly recommended read

[2]Chose your word according to your philosophy of mathematics


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

Mathematics or Physics–Mathematics vs. Physics–Mathematics and Physics

Graham Farmelo is a British physicist and science writer. He is the author of an excellent and highly praised biography of the British physicist P A M Dirac, The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius(Faber and Faber, 2009), which won a couple of book awards. He is also the author of a book Winston Churchill role in British war time nuclear research, Churchill’s Bomb:A hidden history of Britain’s first nuclear weapon programme (Faber and Faber, 2014), which was also well received and highly praised. Now he has published a new book on the relationship between mathematics and modern physics, The Universe Speaks in Numbers: How Modern Maths Reveals Nature’s Deepest Secrets (Faber and Faber, 2019).


I must admit that when I first took Farmelo’s new book into my hands it was with somewhat trepidation. Although, I studied mathematics to about BSc level that was quite a few years ago and these days my active knowledge of maths doesn’t extend much beyond A-Level and I never studied physics beyond A-Level and don’t ask what my grade was. However, I did study a lot of the history of early twentieth century physics before I moved back to the Renaissance. Would I be able to cope with Farmelo’s book? I needn’t have worried there are no complex mathematical or physical expressions or formulas. Although I would point out that this is not a book for the beginner with no knowledge; if your mind baulks at terms like gauge theory, string theory or super symmetry then you should approach this text with caution.

The book is Farmelo’s contribution to the debate about the use of higher mathematics to create advanced theories in physics that are not based on experimental evidence or even worse confirmable through experiment. It might well be regarded as a counterpoint to Sabine Hossenfelder’s much discussed Lost in Math: How Beauty Leads Physics Astray(Basic Books, 2018), which Farmelo actually mentions on the flyleaf to his book; although he obviously started researching and writing his volume long before the Hossenfelder tome appeared on the market. The almost concurrent appearance of the two contradictory works on the same topic shows that the debate that has been simmering just below the surface for a number of years has now boiled over into the public sphere.

Farmelo’s book is a historical survey of the relationship between advanced mathematics and theoretical physics since the seventeenth century, with an emphasis on the developments in the twentieth century. He is basically asking the questions, is it better when mathematics and physics develop separately or together and If together should mathematics or physics take the lead in that development. He investigated this questions using the words of the physicists and mathematicians from their published papers, from public lectures and from interviews, many of which for the most recent developments he conducted himself. He starts in the early seventeenth century with Kepler and Galileo, who, although they used mathematics to express their theories, he doesn’t think really understand or appreciate the close relationship between mathematics and physics. I actually disagree with him to some extent on this, as he knows. Disclosure: I actually read and discussed the opening section of the book with him, at his request, when he was writing it but I don’t think my minuscule contribution disqualifies me from reviewing it.

For Farmelo the true interrelationship between higher mathematics and advanced theories in physics begins with Isaac Newton. A fairly conventional viewpoint, after all Newton did title his magnum opus The Mathematical Principles of Natural Philosophy. I’m not going to give a decade by decade account of the contents, for that you will have to read the book but he, quite correctly, devotes a lot of space to James Clerk Maxwell in the nineteenth century, who can, with justification, be described as having taken the relationship between mathematics and physics to a whole new level.

Maxwell naturally leads to Albert Einstein, a man, who with his search for a purely mathematical grand unification theory provoked the accusation of having left the realm of experiment based and experimentally verifiable physics; an accusation that led many to accuse him of having lost the plot. As the author of a biography of Paul Dirac, Farmelo naturally devote quite a lot of space to the man, who might be regarded as the mathematical theoretical physicist par excellence and who, as Farmelo emphasises, preached a gospel of the necessity of mathematically beautiful theories, as to some extent Einstein had also done.

Farmelo takes us through the creation of quantum mechanics and the attempts to combine it with the theories of relativity, which takes the reader up to the early decades following the Second World War, roughly the middle of the book. Here the book takes a sharp turn away from the historical retelling of the emergence of modern theoretical physics to the attempts to create a fundamental theory of existence using purely mathematical methods, read string theory, M theory, supersymmetry and everything associated with them. This is exactly the development in modern physics that Hossenfelder rejects in her book.

Farmelo is very sympathetic to the mathematicians and physicists, who have taken this path but he is in his account very even handed, letting the critics have their say and not just the supporters. His account is very thorough and documents both the advances and the disappointments in the field over the most recent decades. He gives much emphasis to the fruitful co-operations and exchanges that have taken place between mathematicians and theoretical physicists. I must say that as somebody who has followed the debate at a distance, having read Farmelo’s detailed account I came out of it more sympathetic to Hossenfelder’s standpoint than his.

As always with his books Farmelo’s account is excellently researched, much of the more recent material is based on interviews he conducted with the participants, and very elegantly written. Despite the density of the material he is dealing with, his prose is light and often witty, which makes it easier to grapple with the complex themes he is discussing. I would certainly recommend this book to anybody interested in the developments in modern theoretical physics; maybe to be read together with Hossenfelder’s volume. I would also make an excellent present for any young school leaver contemplating studying physics or one that had already started on down that path.


Filed under Book Reviews, History of Mathematics, History of Physics

The emergence of modern astronomy – a complex mosaic: Part XV

Before continuing with Tycho Brahe’s contributions to the development of modern astronomy it pays to take stock of the existing situation in the last quarter of the sixteenth century. The Middle Ages had cobbled together a model of the cosmos that consisted of three separate but interlocking blocks: Aristotelian cosmology, Ptolemaic astronomy and Aristotelian physics, whereby it should be noted that the medieval Aristotelian physics was, to paraphrase Edward Grant, not Aristotle’s physics. In order for a new astronomy to come into use, as we shall see, the whole model had to dissembled and each of the three blocks replaced with something new.

As we saw at the beginning, some aspects of Aristotelian cosmology–supralunar perfection and cometary theory–were already under scrutiny well before Copernicus published his De revolutionibus. They now fell following the European wide observations of the supernova in 1572 and the great comet of 1577; the Aristotelian crystalline spheres went with them, although Clavius, the leading Ptolemaic astronomer of the age, whilst prepared to sacrifice supralunar perfection and Aristotelian cometary theory, was not yet prepared to abandon the crystalline spheres. The model was beginning to crumble at the edges.

The acceptance of Copernicus’ heliocentric system had been very meagre but the interest in his mathematical models, his astronomical data and the planetary tables and ephemerides based on them had originally been very great. However, it quickly became clear that they were no more accurate or reliable than those delivered by the Ptolemaic system and the initial interest and enthusiasm gave way to disappointment and frustration. Out of this situation both Wilhelm IV in Kassel and Tycho Brahe in Denmark, following Regiomontanus’ initiative from a century earlier, decided that what was needed was to go back to basics and produce new star catalogues and planetary tables based on new accurate observations and set about doing just that. We have already looked to Wilhelm’s efforts; we now turn to Tycho’s.

Granted the island of Hven and the necessary financial support to carry out his project by Frederick II, the Danish king, Tycho set to work.


Frederick II of Denmark Portrait by Hans Knieper or Melchior Lorck, 1581.

Whereas it is theoretically possible to question the claim that Wilhelm IV had built an observatory, no such doubt exists in Tycho’s case. What he erected on his island was not so much an observatory, as a research institute the like of which had never existed before in Europe.

The centrepiece of Tycho’s establishment was his palace Uraniborg, a magnificent purpose built red brick residence and observatory. The structure included a large mural quadrant and outer towers on the balconies of which a large array of self designed and constructed instruments were situated.


Source: Wikimedia Commons


Engraving of the mural quadrant from Brahe’s book Astronomiae instauratae mechanica (1598) Source: WIkimedia Commons

As it turned out that the accuracy of the tower-mounted instrument was affected by vibration caused by the wind, Tycho constructed a second observatory, Stjerneborg. This observatory was effectively situated underground in a large pit to reduce wind vibration of the instruments.


Drawing of an above ground view of Stjerneborg Willem Blaeu – Johan Blaeu, Atlas Major, Amsterdam, 1662 Source: Wikimedia Commons


Schematic of Stjerneborg showing underground chambers: Woodcut from F.R. Friis “Tyge Brahe”, Copenhagen, 1871 Source: Wikimedia Commons

As well as his two state of the art observatories, Tycho also constructed alchemical laboratories in the cellars of Uraniborg, to carry out experiments in Paracelsian pharmacology. To publish the results of his researches Tycho constructed his own printing press and to ensure that he would have enough paper for those publications, he also constructed a water powered paper mill.

Whereas Wilhelm’s astronomical activities were a side project to his main occupation of ruling Hesse-Kassel and the work on his star catalogue was carried out by just two people, Rothmann and Bürgi, Tycho’s activities on Hven were totally dedicated to astronomy and he employed a small army of servants and assistants. Alongside the servants he needed to run his palace and its extensive gardens Tycho employed printers and papermakers and a large number of astronomical observers. Some of those who worked as astronomers on Hven and later in Prague, such as Longomontanus, who later became professor for astronomy in Copenhagen, did so for many years. Others came to work for him for shorter periods, six or nine months or a year. These shorter-term periods working for Tycho worked like a form of postgrad internship for those thus employed. Good examples of this are the Dutch cartographer and Globemaker Willem Janszoon Blaeu (1571–1638), who spent six months on Hven in 1595-96


Willem Janszoon Blaeu Source: Wikimedia Commons

and the Franconian mathematician and astronomer Simon Marius (1573–1625) who spent six months in Tycho’s observatory in Prague in 1601 shorty before Tycho’s death.

Tycho’s observation programme was massive and very much for the duration, starting in the mid 1570s and continuing up to his death in 1601[1]. His teams spent every night of the year, weather permitting, systematically observing the heavens. Two teams, one in Uraniborg and the other in Stjerneborg, made the same observations parallel to but completely independent of each other, allowing Tycho to compare the data for errors. They not only, over the years, compiled a star catalogue of over 700 stars[2] with an accuracy of several factors higher than anything produced earlier but also systematically tracked the orbits of the planets producing the data that would later prove so crucial for Johannes Kepler’s work.

When Tycho was satisfied with the determination of the position of a given star then it was engraved on a large celestial globe that he had had constructed in Germany on one of his journeys. When Willem Janszoon Blaeu was on Hven, Tycho allowed him to make a copy of this globe with the new more accurate stellar positions, which he took with him when he returned to The Netherlands. So from the very beginning Blaeu’s commercial celestial spheres, which dominated the market in the seventeenth century, were based on the best astronomical data available.

Tycho not only systematically observed using instruments and methods known up to his times but devoted much time, effort and experimentation to producing ever better observing instruments with improved scales for more accurate readings. He also studied and developed methods for recognising and correcting observational errors. It is not an exaggeration to say that Tycho dedicated his life to producing observational astronomical data on a level and of a quality never before known in European astronomy.

In 1588 Tycho’s patron and benefactor Frederick II died and after a period of regency his son, who was only eleven years old when he died, was crowned king as Christian IV in 1596.


Portrait Christian IV by Pieter Isaacsz 1612 Source: Wikimedia Commons

Due to a mixture of court intrigue and his own arrogance, Tycho fell into disfavour and Christian cut off his finances from the crown. Still a wealthy man, from his private inheritances, Tycho packed up his home and some of his instruments and left Denmark heading south through Germany in 1597, looking for a new patron. In 1599 he settled in Prague under the patronage of Rudolf II as Imperial Mathematicus,


Rudolf II Portrait by Martino Rota Source: Wikimedia Commons

erecting a new observatory in a castle in Benátky nad Jizerou about fifty kilometres from Prague.


Benátky Castle Source: Wikimedia Commons

Tycho’s biggest problem was that he had vast quantities of, for the time, highly accurate astronomical data that now needed to be processed and he was in desperate need of a mathematician who was capable of carrying out the work. Fate intervened in the form of the still relatively young Johannes Kepler ((1571–1630), who turned up in Prague in 1600 frantically looking for employment.


Johannes Kepler Source: Wikimedia Commons

This was a partnership made in hell rather than heaven but it did not last long as Tycho died under unclear circumstances[3] in October 1601, with Kepler inheriting his position as Imperial Mathematicus. I will deal with Kepler’s leading role in the story of modern astronomy in later episodes but we still need to look at Tycho’s last contribution, the so-called Tychonic system.

[1]In his Bibliographical Directory of Tycho Brahe’s Artisans, Assistants, Clients, Students, Coworkers and Other Famuli and Associates, pages 251–309 in his On Tycho’s Island: Tycho Brahe, Science, and Culture in the Sixteenth Century, John Robert Christianson list 96 names.

[2]When he left Hven Tycho increased his star catalogue to 1000, taking the missing stars from the Ptolemaic star catalogue

[3]Anybody who brings up, in the comments, the harebrained theory that Kepler murdered Tycho in order to obtain his astronomical data will not only get banned from the Renaissance Mathematicus in perpetuity but will be cursed by demons, who will visit them in their sleep every night for the rest of their pathetic lives.


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

Chilli’s story

Sacha will always remain Honorary Editor in Chief in perpetuity but we now have a new Supervisory Editor here at the Renaissance Mathematicus, her name is Chilli and this is her story.

When Sascha died, having been a dog owner for nigh on thirty years, I started looking round for a new dog. Not a replacement, you can’t replace a dog, as each one has a unique personality. I failed to find the right one, as a dog becomes your life partner, especially if you live alone, it has to feel right and none of the dogs I considered did. I then figured that all three of my dogs had found me, so if I were destined to have a new dog, it would also find me. In the mean time I became a dog sitter. I looked after other peoples dogs in the times when they couldn’t, took them for walk, fed them, provided them with a home for weekends when their owners were away. This is how I came to know Chilli.


Chilli practicing being a rock

Chilli was a family dog living not very far away from me, who I already knew from walks with Sascha in the woods behind my flat. She lived together with a woman and her husband and their two kids. It was the kids I mostly met when walking in the woods. The couple got divorced and the kids grew up and got jobs and it came about that the mother, who worked full time, was left alone with Chilli. She would rush home during her lunch breaks to take Chilli out for a quick walk. One day, having learnt that I was a dog sitter, she asked me if I would take Chilli out at lunch times during the week; I work at home. I suggested instead that I pick Chilli up in the mornings, walk her, feed her, play with her and then return her home in the evenings. And so it came about that Chilli became my daytime, working week dog.


Chilli checking out the airfield at a local air show

After about nine months the daughter, who was living in Nürnberg, became pregnant and having stopped working decided to take Chilli to live with her and her new family. And so Chilli and I parted company. Fast-forward about fifteen months, the first baby is now a toddler and the daughter is pregnant for the second time. Chilli, who is now an old lady, does not like the high-speed toddler, who zooms around the family home and starts growling softly whenever said infant gets too close. The mother is justifiably worried that Chilli might snap at her child and injure it, so I got a telephone call from her mother for whom I had originally dog sat Chilli. She explained the situation and asked if I would, under the circumstances, be prepared to give Chilli a new home. After due consideration I agreed and so Chilli has now moved in with me. We are two old folks who just want a little bit of peace and quiet.


Filed under Autobiographical