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


By the middle of the nineteenth century there was no doubt that the Earth rotated on its own axis, but there was still no direct empirical evidence that it did so. There was the indirect evidence provided by the Newton-Huygens theory of the shape of the Earth that had been measured in the middle of the eighteenth century. There was also the astronomical evidence that the axial rotation of the other known solar system planets had been observed and their periods of rotation measured; why should the Earth be an exception? There was also the fact that it was now known that the stars were by no means equidistant from the Earth on some sort of fixed sphere but distributed throughout deep space at varying distances. This completely destroyed the concept that it was the stars that rotated around the Earth once every twenty-four rather than the Earth rotating on its axis. All of this left no doubt in the minds of astronomers that the Earth the Earth had diurnal rotation i.e., rotated on its axis but directly measurable empirical evidence of this had still not been demonstrated.

From the beginning of his own endeavours, Galileo had been desperate to find such empirical evidence and produced his ill-fated theory of the tides in a surprisingly blind attempt to deliver such proof. This being the case it’s more than somewhat ironic that when that empirical evidence was finally demonstrated it was something that would have been well within Galileo’s grasp, as it was the humble pendulum that delivered the goods and Galileo had been one of the first to investigate the pendulum.

From the very beginning, as the heliocentric system became a serious candidate as a model for the solar system, astronomers began to discuss the problems surrounding projectiles in flight or objects falling to the Earth. If the Earth had diurnal rotation would the projectile fly in a straight line or veer slightly to the side relative to the rotating Earth. Would a falling object hit the Earth exactly perpendicular to its starting point or slightly to one side, the rotating Earth having moved on? The answer to both questions is in fact slightly to the side and not straight, a phenomenon now known as the Coriolis effect produced by the Coriolis force, named after the French mathematician and engineer Gaspard-Gustave de Coriolis (1792–1843), who as is often the case, didn’t hypothesise or discover it first. A good example of Stigler’s law of eponymy, which states that no scientific discovery is named after its original discoverer.


Gaspard-Gustave de Coriolis. Source: Wikimedia Commons

As we saw in an earlier episode of this series, Giovanni Battista Riccioli (1594–1671) actually hypothesised, in his Almagustum Novum, that if the Earth had diurnal rotation then the Coriolis effect must exist and be detectable. Having failed to detect it he then concluded logically, but falsely that the Earth does not have diurnal rotation.


Illustration from Riccioli’s 1651 New Almagest showing the effect a rotating Earth should have on projectiles.[36] When the cannon is fired at eastern target B, cannon and target both travel east at the same speed while the ball is in flight. The ball strikes the target just as it would if the Earth were immobile. When the cannon is fired at northern target E, the target moves more slowly to the east than the cannon and the airborne ball, because the ground moves more slowly at more northern latitudes (the ground hardly moves at all near the pole). Thus the ball follows a curved path over the ground, not a diagonal, and strikes to the east, or right, of the target at G. Source: Wikimedia Commons

Likewise, the French, Jesuit mathematician, Claude François Millet Deschales (1621–1678) drew the same conclusion in his 1674 Cursus seu Mondus Matematicus. The problem is that the Coriolis effect for balls dropped from towers or fired from cannons is extremely small and very difficult to detect.


The question remained, however, a hotly discussed subject under astronomers and natural philosophers. In 1679, in the correspondence between Newton and Hooke that would eventually lead to Hooke’s priority claim for the law of gravity, Newton proffered a new solution to the problem as to where a ball dropped from a tower would land under the influence of diurnal rotation. In his accompanying diagram Newton made an error, which Hooke surprisingly politely corrected in his reply. This exchange did nothing to improve relations between the two men.

Leonard Euler (1707–1783) worked out the mathematics of the Coriolis effect in 1747 and Pierre-Simon Laplace (1749–1827) introduced the Coriolis effect into his tidal equations in 1778. Finally, Coriolis, himself, published his analysis of the effect that’s named after him in a work on machines with rotating parts, such as waterwheels in 1835, G-G Coriolis (1835), “Sur les équations du mouvement relatif des systèmes de corps”. 

What Riccioli and Deschales didn’t consider was the pendulum. The simple pendulum is a controlled falling object and thus also affected by the Coriolis force. If you release a pendulum and let it swing it doesn’t actually trace out the straight line that you visualise but veers off slightly to the side. Because of the controlled nature of the pendulum this deflection from the straight path is detectable.

For the last three years of Galileo’s life, that is from 1639 to 1642, the then young Vincenzo Viviani (1622–1703) was his companion, carer and student, so it is somewhat ironic that Viviani was the first to observe the diurnal rotation deflection of a pendulum. Viviani carried out experiments with pendulums in part, because his endeavours together with Galileo’s son, Vincenzo (1606-1649), to realise Galileo’s ambition to build a pendulum clock. The project was never realised but in an unpublished manuscript Viviani recorded observing the deflection of the pendulum due to diurnal rotation but didn’t realise what it was and thought it was due to experimental error.


Vincenzo Viviani (1622- 1703) portrait by Domenico Tempesti Source: Wikimedia Commons

It would be another two hundred years, despite work on the Coriolis effect by Giovanni Borelli (1608–1679), Pierre-Simon Laplace (1749–1827) and Siméon Denis Poisson (1781–1840), who all concentrated on the falling ball thought experiment, before the French physicist Jean Bernard Léon Foucault (1819–1868) finally produced direct empirical evidence of diurnal rotation with his, in the meantime legendary, pendulum.

If a pendulum were to be suspended directly over the Geographical North Pole, then in one sidereal day (sidereal time is measured against the stars and a sidereal day is 3 minutes and 56 seconds shorter than the 24-hour solar day) the pendulum describes a complete clockwise rotation. At the Geographical South Pole the rotation is anti-clockwise. A pendulum suspended directly over the equator and directed along the equator experiences no apparent deflection. Anywhere between these extremes the effect is more complex but clearly visible if the pendulum is large enough and stable enough.

Foucault’s first demonstration took place in the Paris Observatory in February 1851. A few weeks later he made the demonstration that made him famous in the Paris Panthéon with a 28-kilogram brass coated lead bob suspended on a 67-metre-long wire from the Panthéon dome.


Paris Panthéon Source: Wikimedia Commons

His pendulum had a period of 16.5 seconds and the pendulum completed a full clockwise rotation in 31 hours 50 minutes. Setting up and starting a Foucault pendulum is a delicate business as it is easy to induce imprecision that can distort the observed effects but at long last the problem of a direct demonstration of diurnal rotation had been produced and with it the final demonstration of the truth of the heliocentric hypothesis three hundred years after the publication of Copernicus’ De revolutionibus.


Léon Foucault, Pendulum Experiment, 1851 Source


Filed under History of Astronomy, History of Physics, History of Technology

A book is a book is a book is a book


I assume that most of the people reading this would agree that a book is for reading. The writer of the book puts their words down on the page and the reader reads them; it is a form of interpersonal communication. However, if one stops to think about it books also fulfil many other functions and book historian Tom Mole has not only thought long and deeply about it but has put those thoughts down, as a series of essays, in the pages of a book to read, his delightful The Secret Life of Books: why they mean more than words[1], which has recently appeared in paperback.


I will say a bit more about Mole’s book about books not just being books to read in a bit, but first I want to sketch what books have meant in my life, thoughts provoked by his opening essay. Mole describes a university professor, he had as a student, whose office slowly disappeared under steadily increasing number of books. Ever more books meant ever more bookcases until the weight threatened the structural integrity of the building. This is a scenario that speaks volumes to me, and I suspect to many other lifelong book consumers.

I grew up in a house full of books. My father was a university teacher, and my mother was a voracious book reader. Reading books was an integral part of our family life, as long as I can remember. We, the four kids in the family, had a playroom, when we were small. In this playroom there was a book cupboard containing a collection of several hundred children’s books, a collection that grew steadily every year. I had taught myself to read by the time I was about three years old and at around the same age I acquired my first library card. Once a week the family would walk the short stretch to the village library, housed in the primary school, and each one of us would choose new reading matter for the following seven days. My mother always returned from these trips with four new novels, which would be consumed before the next outing. That library was a treasure trove; I can still remember the joy I experienced the first time I discovered Crockett Johnson’s Harold and the Purple Crayon. Later I was always excited to take home a new volume of Mary Norton’s The Borrowers or Richmal Crompton’s William Brown series.

Moving forward in time, when my mother died I, as the only child still living at home, was pushed off to boarding school, there was an excellent school library, and my father and I left our Essex village and moved to London, where my father worked. At the beginning we didn’t have a house, so we lived in the Royal Anthropological Institute on Bedford Square, which my father ran in those days. He had a small bedsitting room with an attached kitchen, that was his office and during the school holidays or weekends home I slept, on an inflatable mattress, on the floor of the Sir Richard Burton Library, that’s the nineteenth century explorer infamous for his translation of The Perfumed Garden. I can assure you that the bookshelves only contained boring tomes on geography, anthropology etc., and no porn, I checked.

When we did finally acquire a house in Colville Place, one of the most beautiful streets in London. My father and I spent several weeks lining the walls of the house with self-constructed bookshelves to house not only his books from our family home but from his office at the RAI and his office at SOAS, where he taught. That house didn’t need any wall paper. During the time that I lived there, now a maturing teenager, I perused many of the fascinating volumes on those shelves covering a bewildering range of topics.

Over the years, books continued to play a very central role in my life and I still own quite a few of the volumes that I acquired over the next decade that very much shaped the historian I am today. For example, Hofstader’s Gödel, Escher, Bach: an Eternal Golden Braid, Bronowski’s The Ascent of Man, Lakatos’ Proofs and Refutations, Criticism and the Growthof Knowledge edited by Lakatos & Musgrave, Polya’s How to Solve It, and Boyer’s A History of Mathematics. They are old friends and have shared my living spaces for more than forty years.

As regular readers of this blog will know, I moved to Germany forty years ago and one year later I started to study at the University of Erlangen. The professor, who most influenced and shaped me, Christian Thiel, is also a serious book consumer. The walls of his university office were completely covered with books and over the years his desk, the windowsills and the floor all acquired steadily growing piles of books. Thiel is the owner of a fairly large house and he is also a serious collector of logic books, he is said to own the second largest such private collection in the world. The walls of most of the rooms in his house are lined with this collection. It reached a point where his wife dictated that he could only acquire new volumes if he sold the same width in centimetres of the old ones.

The walls of my small appartement, where I am sitting typing this, are also lined with bookshelves, except for the 2,60 metres covered by my CD shelves. Those bookshelves are filled, to overflowing and the piles of not shelved books continue to grow. I keep telling myself that I must stop acquiring books or at least dispose of some of them but the thought of parting with one of them is on a par with the thought of having teeth extracted without anaesthetic and as I write, four new books are winging there way to my humble abode from various corners of the world.

My name is Thony and I am a bookaholic.

Returning to the volume that inspired this autobiographical outburst, as already mentioned above, Tom Mole’s book is really a collection of eight essays each of which deals with a different aspect of the book as not reading matter. There are also three interludes that take a look at books depicted in paintings, surely a topic for a whole book. I’m not going to go into detail because that would spoil the pleasure that the reader will get out of these carefully crafted gems, but I will list the topics as given in the essay titles: 1) Book/Book, 2) Book/Thing, 3) Book/Bookshelf[2] 4) Book/Relationship 5) Book/Life 6) Book/World 7) Book/Technology 8) Book/Future

 The book is completed with a relatively small number of endnotes for each chapter, which include bibliographical references for deeper reading on the given theme and an adequate index.

If you are a book lover then this is definitively a book you will want to own and read. Both the original hardback and the paperback are at almost throwaway prices and this small volume would make a perfect stocking filler for the bookaholic in your life. However, be warned if you do give them this book for Christmas, they probably won’t speak anymore after unpacking it, as their nose will be buried in The Secret Life of Books.


[1] Tom Mole, The Secret Life of Books: why they mean more than words, ppb., Elliot & Thompson, London 2020

[2] Mole is going to push me to buy Henry Petroski’s classic study (Mole’s term) The Book on the Bookshelf, London: Vintage, 2000. I already own Petroski’s The Pencil, Alfred A. Knopf, New York, 2004 and it’s brilliant.


Filed under Autobiographical, Book Reviews

Illuminating medieval science


There is a widespread popular vision of the Middle ages, as some sort of black hole of filth, disease, ignorance, brutality, witchcraft and blind devotion to religion. This fairly-tale version of history is actively propagated by authors of popular medieval novels, the film industry and television, it sells well. Within this fantasy the term medieval science is simply an oxymoron, a contradiction in itself, how could there possible be science in a culture of illiterate, dung smeared peasants, fanatical prelates waiting for the apocalypse and haggard, devil worshipping crones muttering curses to their black cats?

Whilst the picture I have just drawn is a deliberate caricature this negative view of the Middle Ages and medieval science is unfortunately not confined to the entertainment industry. We have the following quote from Israeli historian Yuval Harari from his bestselling Sapiens: A Brief History of Humankind (2014), which I demolished in an earlier post.

In 1500, few cities had more than 100,000 inhabitants. Most buildings were constructed of mud, wood and straw; a three-story building was a skyscraper. The streets were rutted dirt tracks, dusty in summer and muddy in winter, plied by pedestrians, horses, goats, chickens and a few carts. The most common urban noises were human and animal voices, along with the occasional hammer and saw. At sunset, the cityscape went black, with only an occasional candle or torch flickering in the gloom.

On medieval science we have the even more ignorant point of view from American polymath and TV star Carl Sagan from his mega selling television series Cosmos, who to quote the Cambridge History of Medieval Science:

In his 1980 book by the same name, a timeline of astronomy from Greek antiquity to the present left between the fifth and the late fifteenth centuries a familiar thousand-year blank labelled as a “poignant lost opportunity for mankind.” 

Of course, the very existence of the Cambridge History of Medieval Science puts a lie to Sagan’s poignant lost opportunity, as do a whole library full of monographs and articles by such eminent historians of science as Edward Grant, John Murdoch, Michael Shank, David Lindberg, Alistair Crombie and many others.

However, these historians write mainly for academics and not for the general public, what is needed is books on medieval science written specifically for the educated layman; there are already a few such books on the market, and they have now been joined by Seb Falk’s truly excellent The Light Ages: The Surprising Story of Medieval Science.[1]  


How does one go about writing a semi-popular history of medieval science? Falk does so by telling the life story of John of Westwyk an obscure fourteenth century Benedictine monk from Hertfordshire, who was an astronomer and instrument maker. However, John of Westwyk really is obscure and we have very few details of his life, so how does Falk tell his life story. The clue, and this is Falk’s masterstroke, is context. We get an elaborate, detailed account of the context and circumstances of John’s life and thereby a very broad introduction to all aspects of fourteenth century European life and its science.

We follow John from the agricultural village of Westwyk to the Abbey of St Albans, where he spent the early part of his life as a monk. We accompany some of his fellow monks to study at the University of Oxford, whether John studied with them is not known.


Gloucester College was the Benedictine College at Oxford where the monks of St Albans studied

We trudge all the way up to Tynemouth on the wild North Sea coast of Northumbria, the site of daughter cell of the great St Alban’s Abbey, main seat of Benedictines in England. We follow John when he takes up the cross and goes on a crusade. Throughout all of his wanderings we meet up with the science of the period, John himself was an astronomer and instrument maker.

Falk is a great narrator and his descriptive passages, whilst historically accurate and correct,[2] read like a well written novel pulling the reader along through the world of the fourteenth century. However, Falk is also a teacher and when he introduces a new scientific instrument or set of astronomical tables, he doesn’t just simply describe them, he teachers the reader in detail how to construct, read, use them. His great skill is just at the point when you think your brain is going to bail out, through mathematical overload, he changes back to a wonderfully lyrical description of a landscape or a building. The balance between the two aspects of the book is as near perfect as possible. It entertains, informs and educates in equal measures on a very high level.

Along the way we learn about medieval astronomy, astrology, mathematics, medicine, cartography, time keeping, instrument making and more. The book is particularly rich on the time keeping and the instruments, as the Abbott of St Albans during John’s time was Richard of Wallingford one of England’s great medieval scientists, who was responsible for the design and construction of one of the greatest medieval church clocks and with his Albion (the all in one) one of the most sophisticated astronomical instruments of all time. Falk’ introduction to and description of both in first class.


The book is elegantly present with an attractive typeface and is well illustrated with grey in grey prints and a selection of colour ones. There are extensive, informative endnotes and a good index. If somebody reads this book as an introduction to medieval science there is a strong chance that their next question will be, what do I read next. Falk gives a detailed answer to this question. There is an extensive section at the end of the book entitled Further Reading, which gives a section by section detailed annotated reading list for each aspect of the book.

Seb Falk has written a brilliant introduction to the history of medieval science. This book is an instant classic and future generations of schoolkids, students and interested laypeople when talking about medieval science will simply refer to the Falk as a standard introduction to the topic. If you are interested in the history of medieval science or the history of science in general, acquire a copy of Seb Falk’s masterpiece, I guarantee you won’t regret it.

[1] American edition: Seb Falk, The Light Ages: The Surprising Story of Medieval Science, W. W. Norton & Co., New York % London, 2020

British Edition: Seb Falk, The Light Ages: A Medieval Journey of Discover, Allen Lane, London, 2020

[2] Disclosure: I had the pleasure and privilege of reading the whole first draft of the book in manuscript to check it for errors, that is historical errors not grammatical or orthographical ones, although I did point those out when I stumbled over them.


Filed under History of Astrology, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, Mediaeval Science, Myths of Science

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


By the end of the eighteenth century, Newton’s version of the heliocentric theory was firmly established as the accepted model of the solar system. Whilst not yet totally accurate, a reasonable figure for the distance between the Earth and the Sun, the astronomical unit, had been measured and with it the absolute, rather than relative, sizes of the orbits of the known planets had been calculated. This also applied to Uranus, the then new planet discovered by the amateur astronomer, William Herschel (1738–1822), in 1781; the first planet discovered since antiquity. However, one major problem still existed, which needed to be solved to complete the knowledge of the then known cosmos. Astronomers and cosmologists still didn’t know the distance to the stars. It had long been accepted that the stars were spread out throughout deep space and not on a fixed sphere as believed by the early astronomer in ancient Greece. It was also accepted that because all attempts to measure any stellar parallax down the centuries had failed, the nearest stars must actually be at an unbelievably far distance from the Earth.

Here we meet a relatively common phenomenon in the history of science, almost simultaneous, independent, multiple discoveries of the same fact. After literally two millennia of failures to detect any signs of stellar parallax, three astronomers each succeeded in measuring the parallax of three different stars in the 1830s. This finally was confirmation of the Earth’s annual orbit around, independent of stellar aberration and gave a yardstick for the distance of the stars from the Earth.

The first of our three astronomers was the Scotsman, Thomas Henderson (1798–1844).


Thomas Henderson Source: Wikimedia Commons

Henderson was born in Dundee where he also went to school. He trained as a lawyer but was a keen amateur astronomer. He came to the attention of Thomas Young (1773-1829), the superintendent of the HM Nautical Almanac Office, after he devised a new method for determining longitude using lunar occultation, that is when a star disappears behind the Moon. Young brought him into the world of astronomy and upon his death recommended Henderson as his successor.


Copy of a portrait of Thomas Young by Henry Briggs Source: Wikimedia Commons

Henderson didn’t receive to post but was appointed director of the Royal Observatory at the Cape of Good Hope. The observatory had only opened in 1828 after several years delay in its construction. The first director Fearon Fallows (1788–1831), who had overseen the construction of the observatory had died of scarlet fever in 1831 and Henderson was appointed as his successor, arriving in 1832.


The Royal Observatory Cape of Good Hope in 1857 Illustrated London News, 21 March 1857/Ian Glass Source: Wikimedia Commons

The Cape played a major role in British observational astronomy. In the eighteenth century, it was here that Charles Mason (1728–1786) and Jeremiah Dixon (1733–1779), having been delayed in their journey to their designated observational post in Sumatra, observed the transit of Venus of 1761. John Herschel (1792–1871), the son and nephew of the astronomers William and Caroline Herschel, arrived at the Cape in 1834 and carried extensive astronomical observation there with his own 21-foot reflecting telescope. cooperating with Henderson successor Thomas Maclear. In 1847, Herschel published his Results of Astronomical Observations made at the Cape of Good Hope, which earned him the Copley Medal of the Royal Society.

Manuel John Johnson (1805–1859), director of the observatory on St Helena, drew Henderson’s attention to the fact that Alpha Centauri displayed a high proper motion.


Ladder Hill Observatory St Helena Source

Proper motion is the perceived motion of a star relative to the other stars. Although the position of the stars relative to each other appears not to change over long periods of time they do. There had been speculation about the possibility of this since antiquity, but it was first Edmund Halley, who in 1718 proved its existence by comparing the measured positions of prominent stars from the historical record with their current positions. A high proper motion is an indication that a star is closer to the Earth.

Aimed with this information Henderson began to try to determine the stellar parallax of Alpha Centauri. However, Henderson hated South Africa and he resigned his position at the observatory in 1833 and returned to Britain. In his luggage he had nineteen very accurate determinations of the position of Alpha Centauri. Back in Britain Henderson was appointed the first Astronomer Royal for Scotland in 1834 and professor for astronomy at the University of Edinburgh, position he held until his death.

Initially Henderson did not try to determine the parallax of Alpha Centauri from his observational data. He thought that he had too few observations and was worried that he would join the ranks of many of his predecessors, who had made false claims to having discovered stellar parallax; Henderson preferred to wait until he had received more observational data from his assistant William Meadows (?–?). This decision meant that Henderson, whose data did in fact demonstrate stellar parallax for Alpha Centauri, who had actually won the race to be the first to determine stellar parallax, by not calculating and publishing, lost the race to the German astronomer Friedrich Wilhelm Bessel (1784–1846).


Portrait of the German mathematician Friedrich Wilhelm Bessel by the Danish portrait painter Christian Albrecht Jensen Source: Wikimedia Commons

Like Henderson, Bessel was a self-taught mathematician and astronomer. Born in Minden as the son of a minor civil servant, at the age of fourteen he started a seven-year apprenticeship as a clerk to an import-export company in Bremen. Bessel became interested in the navigation on which the company’s ships were dependent and began to teach himself navigation, and the mathematics and astronomy on which it depended. As an exercise he recalculated the orbit of Halley’s Comet, which he showed to the astronomer Heinrich Wilhelm Olbers (1758–1840), who also lived in Bremen.


Portrait of the german astronomer Heinrich Wilhelm Matthias Olbers (lithography by Rudolf Suhrlandt Source: Wikimedia Commons

Impressed by the young man’s obvious abilities, Olbers became his mentor helping him to get his work on Halley’s Comet published and guiding his astronomical education. In 1806, Olbers obtained a position for Bessel, as assistant to Johann Hieronymus Schröter (1745–1816) in Lilienthal.


Johann Hieronymus Schröter Source: Wikimedia Commons

Here Bessel served his apprenticeship as an observational astronomer and established an excellent reputation.


Schröter’s telescope in Lilienthal on which Bessel served his apprenticeship as an observational astronomer

Part of that reputation was built up through his extensive correspondence with other astronomers throughout Europe, including Johann Carl Fried Gauss (1777–1855). It was probably through Gauss’ influence that in 1809 Bessel, at the age of 25, was appointed director of the planned state observatory in Königsberg, by Friedrich Wilhelm III, King of Prussia.


Königsberg Observatory in 1830. It was destroyed by bombing in the Second World War. Source: Wikimedia Commons

Bessel oversaw the planning, building and equipping of the new observatory, which would be his home and his workplace for the rest of his life. From the beginning he planned to greatly increase the accuracy of astronomical observations and calculation. He started by recalculated the positions of the stars in John Flamsteed’s stellar catalogue, greatly increasing the accuracy of the stellar positions. Bessel also decided to try and solve the problem of determining stellar parallax, although it would be some time before he could undertake that task.

One of the astronomers with whom Bessel took up contact was Friedrich Georg Wilhelm von Struve (1793–1864), who became a good friend and his rival in the search for stellar parallax, although the rivalry was always good natured. Struve was born the son of Jacob Struve (1755–1841), a schoolteacher and mathematician, in Altona then in the Duchy of Holstein, then part of the Denmark–Norway Kingdom and a Danish citizen.


Friedrich Georg Wilhelm von Struve Source: Wikimedia Commons

Whilst he was still a youth, his father sent him to live in Dorpat (nowadays Tartu) in Estonia with his elder brother, to avoid being drafted into the Napoleonic army. In Dorpat he registered as a student at the university to study, at the wish of his father, philosophy and philology but also registered for a course in astronomy. He financed his studies by working as a private tutor to the children of a wealthy family. He graduated with a degree in philology in 1811 and instead of becoming a history teacher, as his father wished, he took up the formal study of astronomy. The university’s only astronomer, Johann Sigismund Gottfried Huth (1763–1818), was a competent scholar but was an invalid, so Struve basically taught himself and had free run of the university’s observatory whilst still a student, installing the Dolland transit telescope that was still packed in the crates it was delivered in. In 1813 he graduated PhD and was, at the age of just twenty, appointed to the faculty of the university. He immediately began his life’s work, the systematic study of double stars.


The old observatory building in Dorpat (Tartu) Source: Wikimedia Commons

Like Bessel, Struve was determined to increase the accuracy of observational astronomy. In 1820 whilst in München, to pick up another piece of observational equipment, he visited Europe’s then greatest optical instrument maker, Joseph Fraunhofer (1787–1826), who was putting the finishing touches to his greatest telescopic creation, a refractor with a 9.5-inch lens.


Joseph Fraunhofer Source: Wikimedia Commons

Struve had found his telescope. He succeeded in persuading the university to purchase the telescope, known as the ‘Great Refractor’ and began his search for observational perfection.


Frauenhofer’s Great Refractor Source: Wikimedia Commons

Like Struve, Bessel turned to Fraunhofer for the telescope of his dreams. However, unlike Struve, whose telescope was a general-purpose instrument, Bessel desired a special purpose-built heliometer, a telescope with a split objective lens, especially conceived to accurately measure the distance between two observed objects. The first  really practical heliometer was created by John Dolland (1706–1761) to measure the variations in the diameter of the Sun, hence the name. Bessel needed this instrument to fulfil his dream of becoming the first astronomer to accurately measure stellar parallax. Bessel got his Fraunhofer in 1829.


Königsberger Heliometer Source: Wikimedia Commons

One can get a very strong impression of Bessel’s obsession with accuracy in that he devoted five years to erecting, testing, correcting and controlling his new telescope. In 1834 he was finally ready to take up the task he had set himself. However, other matters that he had to attend to prevented him from starting on his quest.

The Italian astronomer Giuseppe Piazzi (1746–1826), famous for discovering the first asteroid, Ceres, had previously determined that the star 61 Cygni had a very high proper motion, meaning it was probably relatively close to the Earth and this was Bessel’s intended target for his attempt to measure stellar parallax.


Giuseppe Piazzi pointing at the asteroid Ceres Painting by Giuseppe Velasco (1750–1826). Source: Wikimedia Commons

It was also Struve’s favoured object for his attempt but, unfortunately, he was unable in Dorpat with his telescope to view both 61 Cygni and a reference star against which to measure any observable parallax, so he turned his attention to Vega instead. In 1837, Bessel was more than somewhat surprised when he received a letter from Struve containing seventeen preliminary parallax observations of Vega. Struve admitted that they were not yet adequate to actually determine Vega’s parallax, but it was obvious that he was on his way. Whether Struve’s letter triggered Bessel’s ambition is not known but he relatively soon began a year of very intensive observations of 61 Cygni. In 1838 having checked and rechecked his calculations, and dismantled and thoroughly examined his telescope for any possible malfunctions, he went public with the news that he had finally observed a measurable parallax of 61 Cygni. He sent a copy of his report to John Herschel, President of the Royal Astronomical Society in London. After Herschel had carefully studied the report and after Bessel had answered all of his queries to his satisfaction. Herschel announced to the world that stellar parallax had finally been observed. For his work Bessel was awarded the Gold Medal of the Royal Astronomical Society. Just two months later, Henderson, who had in the meantime done the necessary calculations, published his measurement of the stellar parallax of Alpha Centauri. In 1839 Struve announced his for Vega. Bessel did not rest on his laurels but reassembling his helioscope he spent another year remeasuring 61 Cygni’s parallax correcting his original figures. 

All three measurements were accepted by the astronomical community and both Henderson and Struve were happy to acknowledge Bessel’s priority. There was no sense of rivalry between them and the three men remained good friends. Modern measurements have shown that Bessel’s figures were within 90% of the correct value, Henderson’s with in 75%, but Struve’s were only within 50%. The last is not surprising as Vega is much further from the Earth than either Alpha Centauri or Cygni 61 making it parallax angle much, much smaller and thus considerably more difficult to measure.

In the sixteenth century Tycho Brahe rejected heliocentricity because the failure to detect stellar parallax combined with his fallacious big star argument meant that in a heliocentric system the stars were for him inconceivably far away. I wonder what he would think about the fact that Earth’s nearest stellar neighbour Proxima Centauri is 4.224 lightyears away, that is 3. 995904 x 1013 kilometres!



Filed under History of Astronomy, History of Optics, History of science, History of Technology

A master instrument maker from a small town in the Fränkischen Schweiz


Eggolsheim is a small market town about twenty kilometres almost due north of Erlangen in the Fränkischen Schweiz (Franconian Switzerland).


Eggolsheim Source: Wikimedia Commons

The Fränkischen Schweiz is a hilly area with many rock faces and caves in Middle Franconia, to the north of Nürnberg that is very popular with tourists, day trippers, wanderers, rock-climbers and potholers. It also has lots of old churches and castles.


Fränkische Schweiz Source Wikimedia Commons

When I first moved to Middle Franconia the Fränkischen Schweiz had the highest density of private breweries of anywhere in the world. It also has many bierkeller that during the summer months attract large crowds of visitors at the weekend. Eggolsheim is these days probably best known for its bierkeller, but in the late fifteenth century it was the birthplace of the Renaissance mathematicus, Georg Hartmann, who would become one of the leading instrument makers in Renaissance Nürnberg in the early sixteenth century.


Georg Hartmann Source: Astronomie in Nürnberg

Hartmann was born on 9 February 1489. Unfortunately, as with so many Renaissance figures, we know nothing about his background or childhood. He matriculated at the university of Ingolstadt in 1503, which is where people from Franconia often studied as there were no University in either Nürnberg or Bamberg. Johannes Werner and Johannes Stabius, two other members of Nürnberg’s Renaissance mathematical community were graduates of Ingolstadt. In 1506, Hartmann transferred to the University of Köln, where he studied mathematics and theology, graduating in 1510. As was quite common during this period he completed his studies on a journey through Italy between 1510 and 1518. He spent several years in Rome, where he was friends with Andreas Copernicus, the older brother of Nicolas, who died in Rome, possibly of leprosy or syphilis in 1518.

In 1518 Hartmann arrived in Nürnberg, where he was appointed a vicar of the St. Sebaldus Church, one of the two parish churches of the city. Unlike the modern Anglican Church, where the vicar is the principal priest of a church, in the sixteenth century Catholic Church a vicar was a deputy or replacement priest with a special function appointed either permanently or temporarily. He might, for example, be appointed to sing a daily mass in the name of a rich deceased member of the parish, who left a stipend in his will to pay for this service, as another of Nürnberg’s mathematical community, Johannes Schöner, was appointed to do in Kirchehrenbach, also in the Fränkischen Schweiz, in 1523. We don’t know what Hartmann’s specific duties in the St. Sebaldus Church were. In 1522 he was also granted the prebend of the St. Walburga Chapel in Nürnberg.

St. Sebald von Norden

St. Sebaldus in Nürnberg Source: Wikimedia Commons

This was a sinecure. It was not unusual for mathematici to receive sinecures from the Church to enable them to carry out their activities as mathematicians, instrument makers or cartographers in the service of the Church. This was certainly the case with Johannes Schöner, who was many years paid as a member of the St Joseph Beneficence in Bamberg but worked as mathematicus, printer and bookbinder for the Bishop. If this was actually so in Hartmann’s case is not known.

When he arrived in Nürnberg he became part of the, for the time, comparatively large community of mathematici, print makers, printer/publishers and instrument makers, which included both Werner and Stabius, the latter as a regular visitor, but both of whom died in 1522. I have written about this group before here and here. It also included Schöner, who only arrived in 1525, Erhard Etzlaub, Johann Neudörffer, Johannes Petreius and Albrecht Dürer.  Central to this group was Willibald Pirckheimer, who although not a mathematicus, was a powerful local figure–humanist scholar, merchant trader, soldier, politician, Dürer’s friend and patron–who had translated Ptolemaeus’ Geographia from Greek into Latin. Hartmann was friends with both Pirckheimer and Dürer, and acted as Schöner’s agent in Nürnberg, selling his globes in the city, during the time Schöner was still living in Kirchehrenbach. Like other members of this group Hartmann also stood in contact with and corresponded with many other scholars throughout Europe; the Nürnberger mathematici were integrated into the European network of mathematici.

Hartmann established himself as one of Nürnberg’s leading scientific instrument makers; he is known to have produced sundials, astrolabes, armillary spheres and globes. None of his armillary spheres or globes are known to have survived, although a few globe gores made by him are extant, an important factor when trying to assess the impact or range of an instrument maker, we can only work with that which endures the ravages of time. We know for example that Hartmann’s friend and colleague, Schöner, produced and sold large numbers of terrestrial and celestial globes but only a small handful of his globes are preserved.

A total of nine of Hartmann’s brass astrolabes are known to have survived and here Hartmann proved to be an innovator.


Hartmann astrolabe front



Hartmann astrolabe back

As far as is known, Hartmann was the earliest astrolabe maker to introduce serial production of this instrument. It is now assumed that he designed the instruments and then commissioned some of Nürnberg’s numerous metal workers to mass produce the separate parts of the astrolabe, which he them assembled and sold. Nine astrolabes might not seem a lot but compared to other known astrolabe makers, from whom often just one or two instruments are known, this is a comparatively large number. This survival rate suggests that Hartmann made and sold a large number of his mass-produced instruments.  

With his sundials the survival rate is much higher, there are seventy-five know Hartmann sundials in collection around the world. Hartmann made sundials of every type in brass, gold and ivory but is perhaps best known for his portable diptych sundials, a Nürnberg specialty. A diptych consists of two flat surfaces, usually made of ivory, connected by a hinge that fold flat to be put into a pocket. When opened the two surfaces are at the correct angle and joined by a thread, which functions as the dial’s gnomon. The lower surface contains a compass to help the user correctly orientate his dial during use.


Hartmann diptych sundial open


Hartmann diptych sundial closed


Open diptych sundial showing string gnomon and Hartmann’s name

Hartmann also made elaborate dials such as this ivory crucifix dial.


One thing that Hartmann is noted for is his paper instruments*. These are the elements for instrument printed on sheets of paper. These can be cut out and glued to thin wood backing to construct cheap but fully functioning instruments. Of course, the survival rates of such instruments are very low and in fact only one single paper astrolabe printed by Hartmann is known to have survived.


Hartmann paper astrolabe Source:History of Science Museum Oxford

However, we are lucky that several hundred sheets of Hartmann’s printed paper instruments have survived and are now deposited in various archives. There have been discussions, as to whether these were actually intended to be cut out and mounted onto wood to create real instruments or whether there are intended as sales archetypes, designed to demonstrate to customers the instruments that Hartmann would then construct out of ivory, brass or whatever.

Hartmann_Kruzifix_1529,_AGKnr4_2004,_s12 Hartmann paper crucifix


Printed paper instrument part



Apart from designing and constructing instruments Hartman was obviously engaged in writing a book on how to design and construct instrument. Several partial manuscripts of this intended work exist but the book was never finished in his lifetime. The book however does reveal his debt as an instrument designer to Johannes Stöffler’s Elucidatio fabricae usuque astrolabii.

As a manufacturer of portable sun dials with built in compasses Hartmann also developed a strong interest in the magnetic compass. Whilst living in Rome he determined the magnetic declination of the city, i.e., how much a compass needle varies from true north in that location. Hartmann also appears to have been the first to discover magnetic dip or inclination, which information he shared with Duke Albrecht of Prussia in a letter in 1544, but he never published his discovery, so it is usually credited to the English mariner Robert Norman, who published the discovery in his The Newe Attractive, shewing The Nature, Propertie, and manifold Vertues of the Loadstone; with the declination of the Needle, Touched therewith, under the Plaine of the Horizon in 1581.

The only book that Hartmann did publish in his lifetime was an edition of John Peckham’s Perspectiva communis, the most widely used medieval optic textbook, which was printed by Johannes Petreius in 1542.


Hartmann died in Nürnberg in 1564 and was buried in the St Johannes graveyard, outside the city walls, where the graves of his friend Pirckheimer, Dürer and Petreius can also be found amongst many other prominent citizens of the Renaissance city.  


Hartmann’s grave Source: Astronomie in Nürnberg


Hartmann’s epitaph Source: Astronomie in Nürnberg

  • For a detailed description of Hartmann’s printed paper instruments see: Suzanne Karr Schmidt, Interactive and Sculptural Printmaking in the Renaissance, Brill, 2017

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

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

By the middle of the eighteenth century, Newton’s version of the heliocentric theory had been universally accepted by all of those knowledgeable enough to express a considered opinion on the subject. However, some (most?) Jesuit astronomers continued to pay lip service to a variant of the Tychonic geo-heliocentric system, as their personal allegiance to the Pope required them to. This, of course, raises the question of the Catholic Church’s stand on the subject. You can find accounts that claim that the Church only accepted that the solar system is heliocentric in the nineteenth century, whilst other go as far as to claim that this back down first really took place with the Vatican’s examination of the Galileo case in the 1980s. Neither of these views is actually correct and is mostly the case it was in reality a long drawn out process that it pays to review, beginning with a brief recap of how it all started.

The whole story started in 1615/16 when Galileo Galilei and the Carmelite theologian Paolo Antonio Foscarini (c. 1565–1616) provoked the Catholic Church into making a serious assessment of its position on the heliocentric theory of the cosmos. Famously the Church’s examiners said that the theory contradicted by Holy Scripture and the then scientific consensus. In a famous meeting with Roberto Bellarmino, Galileo was instructed that he was not allowed to hold or teach the heliocentric theory as fact, a stricture that applied to all other members of the Church. Several books that did in fact present the heliocentric theory, as fact were placed on the Index of forbidden books, including, for example, those of Johannes Kepler. Interestingly, Copernicus’ De revolutionibus was only placed on the Index until corrected. This correction was carried out and only consisted of the removal or modification of a handful of passages that stated or implied that the heliocentric theory was true.  By 1621 the thus mildly censored De revolutionibus was again accessible for Catholic astronomers to study.

Famously, Galileo then provoked the Church further in 1632 with his Dialogo that very definitely did teach the heliocentric theory as true with the well-known consequences. Although under the circumstances Galileo’s punishment was relatively mild and the Church left him in peace in his house arrest, even turning a blind eye when his Discorsi was published in 1638. However, when Galileo died the plans of the Grand Duke of Tuscany, Ferdinand II, to bury him in a specially erected marble mausoleum in his honour in the Basilica of Santa Croce in Florence, were basically stopped by the Pope and he was buried in a simple grave in a side chapel instead.

It is important to note that although the Church banned the heliocentric theory as a true model of the cosmos it was still permissible to discuss the heliocentric hypothesis. Outside of Italy the Church’s ban had very little effect even in Catholic countries and of course none in Protestant ones. In the seventeenth century, within Italy astronomers would discuss heliocentricity but starting their work with something along the lines of, the Holy Mother Church in its wisdom has ruled that the heliocentric theory is false, but it is an interesting mathematical hypothesis, which I will now elucidate. And so, both sides were happy. There are no major cases of astronomers being prosecuted for holding the heliocentric theory, although both the leading Catholic astronomers Pierre Gassendi and Giovanni Battista Riccioli were investigated by the Inquisition after being suspected of holding the heliocentric theory in their respective main astronomical works, Institutio astronomica (1653) and Almagestum novem (1651), where they discussed the heliocentric hypothesis helping to spread knowledge of it. No charges were raised in either case.

By the end of the seventeenth century, following the publication of Newton’s Principia Mathematica in 1687, it was fairly obvious that the heliocentric system had become the accepted model of the cosmos amongst astronomers, although as noted earlier empirical proof of the Earth’s movement had still not been found. In the early eighteenth century the Catholic Church’s stand on heliocentricity and Galileo began to slowly weaken. In 1718 the Inquisition’s ban on printing the works of Galileo was lifted and permission for an edition of his works was granted, which however excluded the Dialogo. In 1741, Pope Benedict XIV authorised a complete edition of his works including a lightly censored version of the Dialogo.

Meanwhile, in 1737 the Church gave permission for Galileo to be reburied. His corpse was removed from its grave in the side chapel and he was reburied in a spectacular tomb in the main body of the Basilica.


Galileo’s Tomb Source: Wikimedia Commons

Bizarrely during this process, three fingers and a tooth were removed from his body and these are now displayed like some sort of religious relics in the Museo Galileo in Florence. This is all part of the Galileo as martyr for science and/or free speech that has grown up over the centuries.


Middle finger of Galileo’s right hand Source: Wikimedia Commons

In 1758 the general prohibition against publications on the heliocentric theory was lifted by the Pope but the books that had been placed on the Index for propagating the heliocentric theory remained there. Things remained quiet until 1820, when The Master of the Sacred Palace (the Church’s chief censor), Filippo Anfossi (1748–1825), refused to licence a book by the Catholic canon, Giuseppe Settele (1770–1841), which treated the heliocentric theory as factual. Settele appealed to Pope Pius VII, who referred the matter to the Congregation of the Index and the Holy Office, who after due consideration overturned Anfossi’s decision. Following this decision, the banned books on heliocentricity were removed from the Index when it was next revised in 1835.

Many supporters of science against, what they see as, the ignorance of the Catholic Church, who have a very narrow focus, demand to know why the Church did not remove its ban much earlier and at the same time rehabilitate Galileo. These people simply ignore the fact that the Catholic Church is one of the largest religious institution in the world, which regards itself as responsible for the whole of humanities existence and its actions. In this grand scheme of things astronomy and cosmology, whilst important, play, very much, a minor role, and as long as there is no immediate need to address any problems, they might not, then the Church has more important things to occupy its attention. Also, because of its sheer size and influence the Church took its time when changing a doctrine that would have a wide impact, after all the heliocentric theory does contradict Holy Scripture. For an institution that was already fifteen hundred years old when it had its initial disagreement with Galileo, a couple of centuries is not a long time.

As opposed to popular opinion Galileo, himself, had rather drifted out of the limelight during the seventeenth and eighteenth centuries; science had moved on and left him behind. However, at the end of the eighteenth century and the beginning of the nineteenth people began to elevate him to his current mythical status, as the martyr for science and/or free speech in the supposed eternal war between science and religion, which never actually existed. From this point on the demands for a rehabilitation of Galileo by the Catholic Church began to grow in volume.

As already observed the Church moves slowly in such matters and it was first in 1979 that Pope, John Paul II expressed the hope that “theologians, scholars and historians, animated by a spirit of sincere collaboration, will study the Galileo case more deeply and in loyal recognition of wrongs, from whatever side they come.” In 1992 issued a statement concerning the deliberation of the committee he had set up to reassess the conflict between the Catholic Church and Galileo in 1979. Contrary to popular belief this was not the Church admitting that they were wrong and Galileo right but an interesting fairly even handed assessment of the mistakes made at the time by the Church, which clearly states that there was blame on both sides, although he puts it somewhat more diplomatically:

tragic mutual incomprehension [emphasis in original] has been interpreted as the reflection of a fundamental opposition between science and faith. The clarifications furnished by recent historical studies enable us to state that this sad misunderstanding now belongs to the past.

It pays to read the whole document

People will almost certainly go on discussing the conflict between Galileo and the Church for many years to come but I personally don’t think anything new can be won by doing so.







Filed under History of Astronomy

You can con all of the people some of the time, and some of the people all of the time, but you can’t con all of the people all of the time. However, you can con enough people long enough to cause a financial crisis.


The name Isaac Newton evokes for most people the discovery of the law of gravity[1] and if they remember enough of their school physics his three laws of motion. For those with some knowledge of the history of mathematics his name is also connected with the creation of calculus.[2] However, Newton lived eighty-four years and his life was very full and very complex, but most people know very little about that life. One intriguing fact is that in 1720/21 Newton lost £25,000 in the collapse of the so-called South Sea Bubble. A modern reader might think that £25,000 is a tidy sum but not the world. However, in 1720 £25,000 was the equivalent of several million ponds today. Beyond this, when he died about eight years later his estate was still worth about the same sum. Taken together this means that Isaac Newton was in his later life a vey wealthy man.

These details out of Newton’s later life raise a whole lot of questions. Amongst other, how did he become so wealthy? What was the South Sea Bubble and how did Newton come to lose so much money when it collapsed? Science writer and Renaissance Mathematicus friend,[3] Tom Levenson newest book, Money for Nothing [4], offers detailed answers to the last two questions but not the first[5].


Both Newton and the South Sea Bubble play central roles in Levenson’s book but they are actually only bit players in his story. The real theme of the book is the birth of the modern world of political and capitalist finance in which both the creation of the South Sea Company and its eventual collapse played a dominant role. You can find explanations and the origins of all the gobbledegook that gets spouted in tv, radio and print-media finance reports, derivatives, call and put options, etc. It is also here that the significance Newton as a central figure becomes clear. There were other notable figures in the early eighteenth century, who made or lost greater fortunes than the substantial loses that Newton suffered, but he is really here for different and important reasons.

One reason for Newton’s presence is, of course, his role as boss of the Royal Mint during this period and his secondary role as financial consultant and advisor. Another reason is that central feature of this new emerging world of finance was the application of mathematical modelling, parallel to the mathematical modelling in physics and astronomy, in which Newton is very much the dominant figure, not just in the very recently created United Kingdom.  

We get introduced the work of William Petty and Edmond Halley, who applied the recently created branches of mathematics, statistics and probability, to social and political problems.


I found particular interesting the work of Archibald Hutchinson, who I’d never come across before, who carried out a deep and extensive mathematical analysis of the South Sea Company scheme, basically to turn the national debt into shares of a joint stock company, which promised a dividend, could not work as it existed because the South Sea Company would never generate enough profit to fulfil its commitments to its shareholders. Whilst the South Sea Company was booming and everybody was scrabbling to obtain shares at vastly inflated prices, Hutchinson’s cool analytical warnings of doom were ignored, he was truly a prophet crying in the wilderness. After the event when he had been proved right nobody was interested in hearing, I told you so.


Another fascinating figure, who was new to me, is John Law, a brilliant mathematician and felon[6], who landed up in France and through his mathematical analysis became the most powerful figure in French financial politics. Law created the comparatively new concept of paper money (new that is in Europe, the Chinese had had printed paper money for centuries by this time) and the Mississippi Company, which served a similar function to the South Sea Company, to deal with the French national debt. The Mississippi Company collapsed just as spectacularly as the South Sea Company and Law was forced to flee France.


Levenson goes on to show how the French and UK governments each dealt with the financial disasters that their experiments in modern finance had delivered up. The French government basically returned to the old methods, whereas the UK government now moved towards the future world of capitalist finance, which gave them a financial advantage over their much greater and richer rival in the constant wars that the two colonial powers waged against each other throughout the eighteenth century.

The book features a cast that is a veritable who’s who of the great and the infamous in England in the early eighteen century. As well as Isaac Newton and Edmund Halley we have, amongst many others, Johnathan Swift, Daniel Defoe, Alexander Pope, John Gay, Georg Handel, William Hogarth, Sarah Churchill, Duchess of Marlborough (who played the market and made a fortune), Charles Montagu, 1st Earl of Halifax (Newton’s political patron), Christopher Wren and Uncle Bob Walpole and all.


The book closes with an epilogue, which draws the very obvious parallels between the financial crisis caused by the South Sea Bubble and the worldwide one caused in in 2008 but the collapse of the very rotten American derivative market based on mortgages. Echoing the adage that those who don’t know history are doomed to repeat it. History really does have its uses.

The hard back is nicely presented, with an attractive type face and the apparently, in the meantime, obligatory grey in grey prints. There are not-numbered footnotes scattered throughout the text, which explain various terms or expand on points in the narrative but otherwise the book has, what I regard as the worst option, hanging endnotes giving the sources for the direct quotes in the text. There is an extensive bibliography, which our author has very obviously read and mined and an excellent index.

Levenson has written a big in scope and complex book with multiple interwoven layers of mathematical, financial, political and social history that taken together, illuminate an interesting corner of early eighteenth-century life and outline the beginnings of our modern capitalist world. The result is a dense story that could be a challenge to read but, as one would expect of the professor for science writing at MIT, Levenson is a first class storyteller with a light touch and an excellent feel for language, who guides his readers through the tangled maze of the material with a gentle hand. There is much to ponder and digest in this fascinating and rich slice of truly interdisciplinary history, which will leave the reader, who braves its complexities, enriched and possibly wiser than they were before they entered the world of the notorious South Sea Bubble.

[1] As I have pointed out in the past, he didn’t discover the law of gravity he proved it, which is something different.

[2] As I pointed out long ago in a blog post that is no longer available, neither Newton nor Leibniz invented/discovered (choose your term according to your philosophy of mathematics) calculus, even created is as step too far.

[3] Disclosure: Several years ago, I read through Tom’s original book proposal and more recently one chapter of the book, to see if the facts about Newton were correct, but otherwise had nothing to do with this book apart from the pleasure of reading it.  

[4] Money for Nothing: The South Sea Bubble and the Invention of Modern Capitalism, Head of Zeus ltd., London, 2020.

[5] For this you will have to read other books including, perhaps, Tom’s earlier excellent Newton book, Newton and the Counterfeiters: The Unknown Detective Career of the World’s Greatest Scientist, Houghton Mifflin Harcourt, Boston & New York, 2009.

[6] Why I refer to John Law as a felon is a much too intriguing story that I’m going to spoil in in this review; for that you are going to have to read Professor Levenson’s book


Filed under Book Reviews, History of Mathematics

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

The model of the solar system that Johannes Kepler presented in his mature work had dimensions significantly larger than any of the other geocentric, heliocentric, or geo-heliocentric systems on offer in the early part of the seventeenth century. Although by 1630, Kepler’s heliocentric system with its elliptical orbits had become one of the two leading contenders for the correct model of the cosmos, the vast majority of astronomers stayed with the much smaller dimensions, as presented by the Ptolemaic, Copernican and Tychonic systems. Even such an ardent promoter of the Keplerian system as Ismaël Boulliau (1605–1694) preferred his own calculated value of c.1,500 e.r. for the astronomical unit to Kepler’s more than double as large value.

There were however two notable exceptions amongst the Keplerians, but before we look at the first one we need to look briefly at another idea of Kepler’s on the cosmic dimensions that did have a major impact throughout the seventeenth century. Kepler a major fan of Pythagorean harmony theory believed that the sizes of the planets were proportional their distances from the Sun. This concept was immensely popular in the seventeenth century and even extending into the eighteenth; Bode’s Law, which suggests that, extending outward, each planet would be approximately twice as far from the Sun as the one before is just such a concept. This led to increased efforts throughout the seventeenth century to determine both the apparent and the actual sizes of the planetary discs as viewed through telescopes. Before the invention of the micrometer later in the century these efforts produced extremely contradictory results.

In 1631 Pierre Gassendi (1592–1655) became the first person to observe a transit of Mercury, which had been predicted by Kepler in his Rudolphine Tables.


Pierre Gassendi after Louis-Édouard Rioult. Source: Wikimedia Commons

The result of his observations that stirred up the most discussion was the fact that Mercury was very much smaller than had been determined in all previous observation, whether with or without a telescope. This result caused a lot of astronomers to question or even reject Gassendi’s observations.


Kepler had also, correctly, predicted a transit of Venus for 1631, which was however not visible from Europe.

Kepler had, however, not predicted the transit of Venus that was due to take place in in 1639. The young Keplerian astronomer Jeremiah Horrocks (1618–1641), who had bought a copy of the Rudolphine Tables and both corrected and extended them realised that there would be a transit in 1639 and informed his friend and fellow Keplerian astronomer William Crabtree (1610–1644) and the two of them observed the transit. As with Gassendi and Mercury, they observed that Venus was very much smaller than had been previously believed and in his reports on their observations, Horrocks stated that they had vindicated Gassendi. Using similar arguments to those used by Kepler, Horrocks determined the solar parallax to be a maximum of fourteen minutes of arc and the astronomical unit thus 15,000 e.r. Unfortunately, Horrocks died before he could publish his findings and they only became known when published by Johannes Hevelius (1611–1687) in 1662.


Today, less well known that Horrocks is the Flemish, Keplerian astronomer Govaert Wendelen (1580–1667) (also referred to as Gottfried Wendelin).


Godefridus Wendelinus by Philip Fruytiers (1648) Source: Wikimedia Commons

He had actually used Aristarchus’ half-moon method to determine the astronomical unit in 1626. In a publication on 1644 he used an astronomical unit of c. 14,600 e.r. making him the first to put a value greater than Kepler’s in print.

When he was compiling his astronomical encyclopaedia, Almagestum Novum (1651), the Jesuit astronomer Giovanni Riccioli (1598-1671), a supporter of a semi-Tychonic system, investigated various values for the astronomical unit including Wendelen’s.


Wedelen’s value was, as stated above c. 14,600 e.r. and that of Michel Florent van Langren (1598–1675), another Lowlands astronomer, most well-known for his map of the Moon,


Michel Florent van Langren Map of the Moon 1645 Source: Wikimedia Commons

which was c. 3, 400 e.r. Riccioli took an average of these two values and presented as his own value of 7,300 e.r. Following the publication of Horrocks’ work in 1662 both Christiaan Huygens (1629–1695) and Thomas Streete (1621–1689) started arguing for the significantly larger values for the astronomical unit of Kepler and Horrocks but Huygens admitted quite freely that with his value he could err by a factor of three in either direction. As should be very clear, by this point in the century, there was no unity amongst astronomers on the value of the astronomical unit and they were very much groping around in the dark as to the true value.

In 1672 there was a return to Tycho’s attempts to determine the parallax, and thus the distance, of Mars at opposition. Kepler had already calculated the correct relative distances of the planets, so only one correct absolute distance was necessary to determine all of them and both Jean-Dominique Cassini (1625-1712), the director of the French national observatory in Paris,


Jean-Dominique Cassini (artist unknown) Source: Wikimedia Commons

and John Flamsteed (1646–1719, who would go on to be appointed the first Astronomer Royal,


Source: Wikimedia Commons

decided that their best bet lay with determining the parallax of Mars, now they had advanced telescopes with crosshairs and micrometers.

Flamsteed’s observations were a very lowkey effort made from his then home-base in London. Cassini, however, launched a major international programme to observe Mars in opposition, with a whole team of observers in Paris and the dispatch of Jean Richer (1630–1696)


Jean Richer working in French Guiana from an engaging by Sébastien Leclerc.

to Cayenne in French Guiana to make observations there. Here we come up against an interesting historical phenomenon. Both Flamsteed and Cassini came up with figures for the solar parallax and the astronomical unit that are reasonable approximations for the correct figures. Flamsteed found the parallax of Mars to be at most fifteen seconds of arc, which made the Sun’s parallax seven seconds of arc and the astronomical unit 29,000 e.r. Cassini’s figures were twenty-five seconds of arc for Mars’ parallax and 22,000 e.r. for the astronomical unit. Richer found the parallax of Mars to have a maximum of perhaps twelve or fifteen seconds of arc. The modern value is c. nine seconds of arc for solar parallax and c. 23,500 e.r. for the astronomical unit, so problem finally solved or? Why is this an interesting historical phenomenon? The answer is quite simple what Flamsteed, Cassini and co. were actually measuring, although they didn’t realise it at the time, was the limit of the measurement accuracy of the instrument that they were using.

On a sidenote, Richer was sent to Cayenne, which is very close to the equator to finally solve the problem of the atmospheric refraction. Since antiquity astronomers had been well aware of the fact that the accuracy of their observational measurements was affected by the light coming from the celestial objects under observation being refracted by the Earth’s atmosphere. From Ptolemaeus onwards they had used an error factor to correct for this, but this factor was at best an informed guess. An observation made directly overhead on the equator is free of refraction, so a comparison of the observations made by Richer in Cayenne and those made in Paris, would and did deliver an accurate figure for the necessary refraction correction.

Cassini was well aware of numerous problems in his measurements and his subsequent calculations and spent a lot of time fudging his figures. A man, who normally rushed into print with his discoveries, he took twelve years to finally publish the results of the 1672 measurements. Despite his own reservations about what exactly he had measured and how reliable those measurements were, he however remained by his conclusion that the astronomical unit lay somewhat over 20,000 e.r.

This twenty thousand plus figure, for the astronomical unit, from Cassini and Flamsteed came to be accepted by almost all European astronomers in the early eighteenth century, including Isaac Newton, who had originally determined a solar parallax of a minimum of twenty seconds of arc, much larger that Flamsteed and Cassini. The one notable exception to this general acceptance amongst astronomers was Edmond Halley (1656–1742), who did not accept the Flamsteed/Cassini determinations of the parallax of Mars and thus the solar parallax and astronomical unit based on those determinations. In his opinion the instruments used were not capable of discerning the angles that they had claimed to have measured.


Portrait of Edmond Halley painted around 1687 by Thomas Murray (Royal Society, London) Source: Wikimedia Commons

Halley did not think that the Mars parallax method was fit for purpose and suggested an alternative method for determining the astronomical unit. In 1676, Halley, whilst still a student, was sent by the English government to the South Atlantic island of St Helena to map the southern heavens as a navigation aid for English mariners. Whilst there he observed a transit of Mercury. Up to this point in time, transits of Mercury had only been used to determine the size of the planet, but Halley was aware of a proposal made by the Scottish astronomer, James Gregory (1638–1675), in his Optica Promota (1663).


James Gergory Source: Wikimedia Commons


Gregory outlined how transits could be used to determine solar parallax. Halley was able during his observations of the transit to record the both the moment of initial contact between Mercury of the Sun and the moment of final contact. On his return to England he discovered that the French astronomer, Jean Charles Gallet, in Avignon had also observed the transit. Combining Gallet’s results with his own he determined a parallax for Mercury of one minute six seconds of arc and for the Sun of forty-five seconds of arc. However, he did not regard these results as being very accurate.

Rejecting the Mars parallax method, Halley now became a propagandist for Gregory’s transit method. In 1702, in his Astronomiae physicae et geometricae elementa,


David Gregory, James’ nephew, stated that for an accurate solar parallax measurement people would have to wait for the 1761 transit of Venus but in the meantime, he accepted Newton’s values.


David Gregory Source: Wikimedia Commons

In 1716, Halley published a paper in the Philosophical Transactions of the Royal Society, Dr. Halley’s Dissertation of the Method of Determining the Parallax of the Sun by the Transit of Venus, June 6, 1761, in which he claimed that such a determination would be accurate to one part in five hundred. From this point on he continually drew astronomers’ attention to his proposal, well aware that he wouldn’t live long enough to observe the transit himself.


In 1761 and then again in 1769 astronomers from all over the world travelled to good observation points equipped with the latest in astronomical instrument and telescope technology to observe the transits of Venus. It turned out that for various reasons the observers were not actually able to achieve the accuracy that Halley had forecast not least because of the black drop effect that prevents accurate measurement of the exact moment of first contact.


James Cook’s measurements of the 1769 transit of Venus. (NASA)

Despite all of the problems, the Venus transit of 1761 was the first true determination of the astronomical unit. Over the subsequent centuries that determination was continually improved, and it meant that from 1761 the absolute dimensions of the solar system from the Sun out to Saturn were now known if not exactly accurately. The question that remained open was the distance to the fixed stars and it would be some time before that problem was finally solved.


Filed under History of Astronomy

“A sea of wild, woolly thinking!”

Today’s musings on the history of science re-examine a topic that I have already dealt with several times in the past, that of presentist judgements on the heuristic used by a historical figure to find or reach their solution to a given scientific problem. In the world of scientific investigations, a heuristic is the scaffolding consisting of assumptions and presumptions that the investigator erects to direct and guide his efforts to explain a given set of phenomena. It is not necessary for a heuristic to be factually true, whatever that may mean. What  is important is that the heuristic delivers useful developments within the phenomena under investigation. Already in the sixteenth century Christoph Clavius, an excellent logician and philosopher of science pointed out that false premisses in science can nevertheless lead to correct deductions and therefore suggested falsification as a method to check scientific hypotheses; yes, Clavius was a Popperian three and a half centuries before Popper.

Johannes KeplerKopie eines verlorengegangenen Originals von 1610

Portrait of Kepler by an unknown artist, 1610 Source: Wikimedia Commons

The particular heuristics that I’m going to examine here are those on which Johannes Kepler erected his whole astronomical planetary theories, starting with his Mysterium Cosmographicum (1596) and all the way through to his Harmonices mundi (1619) and his Epitome Astronomiae Copernicanae (1617–1621). In her Measuring the Universe, Kitty Ferguson refers to Kepler’s work as follows:

His most celebrated discoveries seem like small islands of dazzling insights in a sea of wild, woolly thinking.[1]

The sea of wild, woolly thinking that Ferguson is referring to here is the heuristic that Kepler applied to his investigations to arrive at his famous conclusions concerning the shape and laws of the cosmos and also to a large part of those conclusions, which as opposed to his three laws of planetary motions today get ignored by everybody except the historians. Let us examine the collection of assumptions and presumptions under which Kepler conducted his research. Just how wild and woolly were they?

Kepler’s first and most important assumption was his devout and unquestioning belief in his Christian God. This is, of course, like a red rag to a bull to the gnu atheists, who continue to insist that religion and science should never occupy the same building let alone the same brain. This is problematic, as his belief in his God was the principle and singular driving force in all of Kepler’s scientific work. To understand this, we need to look at some more of Kepler’s assumptions. For Kepler it was obvious that his God had created the cosmos and that he had done so specifically for mankind. In his belief that God exists, and that God had created the world, Kepler differed in no way from the vast majority of his fellow Europeans in the late sixteenth and early seventeenth centuries but Kepler and not just Kepler took it further.

What is here central to the issue is Kepler’s personal perception of his God. Kepler’s God is not one of those ancient Greek or Scandinavia goods, who seem to take great pleasure in personally dicking around in the lives of selective individuals, just for the fun of it. His God is also not the fire and brimstone god of the Old Testament, who wipes out cities or murders babies. Kepler’s God is a rational, logical entity; in fact, Kepler’s God is a mathematician, which for Kepler means he is a geometer. Kepler is by no means the only natural philosophers in the Renaissance/Early Modern Period, who held this view of God. In fact, it was a common trope in the Middle Ages that produced a corresponding iconography.


God as Architect/Builder/Geometer/Craftsman, The Frontispiece of Bible Moralisee Source: Wikimedia Commons

In Kepler’s opinion his mathematician God had created his cosmos according to a completely logical, mathematical construction plan and it was Kepler’s task as an astronomer and natural philosopher to reconstruct and explicate that construction plan. He shared this view with many others in the Early Modern Period including both Galileo and Newton.

Before we go into detail, we need to pause and take stock. Kepler believed that a mathematician god had created the cosmos on mathematical principles and therefore he needs to discover and expose the mathematical patterns of his god’s construction plan. Leave out Kepler’s god and you should realise that Kepler’s assumptions and approach are no different to those used by scientists today; i.e. the cosmos is fundamentally logical and can be analysed, described and explained using mathematical models. The fact that this approach works so well led historians and philosophers to describe the so-called scientific revolution, as the mathematisation of nature but on the other side led to Eugene Wigner’s infamous essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960).

Kepler set out with a series of open questions about the nature of the cosmos as it was known during his lifetime. One of his questions was why are there six planets in the heliocentric system that he believed in and why did their orbits have the distances to each other that they have? At the time, on the basis of the known facts, perfectly reasonable questions. He, sort of, stumbled into his answer. Whilst discussing, with a school class, the long-term cycle of the conjunctions of Saturn and Jupiter he realised that the diagrammatic presentation of those conjunctions over time is a perfectly symmetrical geometrical diagram.


Kepler’s original diagram trigon of the great conjunctions of Saturn and Jupiter

He wondered if the orbital distances of the planets also form some sort of symmetrical geometrical diagram. He tried various two-dimensional models without success then he his upon the three-dimensional, regular Euclidian solids. There are, and can only be, five of them, bingo! Six planets, five spaces, five Euclidian solids, do they fill out those spaces. Kepler positioned the five solids around and inside the spheres of the orbits of each pair of neighbouring planets and found they actually make a more than reasonable fit, not perfect but also not bad enough to immediately reject. He had the makings of a rational, geometrical construction plan for his cosmos.

thinking 3d005

Johannes Kepler Mysterium Cosmographicum

Kepler’s model was a good fit, but it wasn’t a perfect fit. In this situation the mediocre mathematical modeler simple accepts the imperfections, shrugs and moves on, but Kepler was not mediocre. In the situation, Kepler had two choices, he could abandon his model, or he could question his data. Kepler knew that the Ptolemaic/Copernican data he had inherited was inaccurate and corrupt, so he went in search of better data; a search that led him to Prague and working for Tycho Brahe, who had the best astronomical data available.

When Kepler finally got hold of some of Tycho’s data, it was to calculate the orbit of Mars that would eventually lead to his Astronomia nova. Kepler spent years trying to derive the most accurate orbit possible for Mars from Tycho’s data. His work was concentrated and precise and he developed several new approaches to orbit calculation in the process. At one point he had a circular orbit with just eight minutes of arc error in places; this was an amazing achievement in terms of the accepted levels of accuracy for the times, but it was neither accurate enough for Kepler’s personal standards, or in his opinion was it accurate enough to honour the accuracy of Tycho’s observations, so he worked further. As is well known he finally derived the correct elliptical orbit and with it his first two laws of planetary motion. The whole of this project was driven by Kepler’s desire to give accuracy to his Euclidian solids model.

In his Mysterium Cosmographicum Kepler had also floated the idea that his Euclidian solids model was fine-tuned by a second mathematical model the Pythagorean concept of celestial harmony. This is harmony in both its mathematical and musical meanings. This model said that the distances between the planetary orbits built a harmonious musical scale, the melody thus created only being audible to enlightened Pythagoreans. In choseing this particular approach Kepler was very much in tune with his times. The Pythagorean theory of celestial motion was very popular in the Middle Ages and in the Early Modern Perdiod Tycho Brahe designed and built his observatory Uraniborg entirely in Pythagorean harmonic proportions,


Tycho Brahe’s Uraniborg main building from the 1663 Blaeu’s Atlas Maior Source: Wikimedia Commons

whereas Newton built the Pythagorean theory into his analysis of white light. Kepler would, once again, spend years of his life following this mathematical trail, publishing the results of his research in his magnum opus, Harmonices Mundi (1619). He had investigated the ratios between all possible position or velocities of the orbits of the planets; the most famous result being his harmony law, his third law of planetary motion:

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

Throughout the twenty-three years between the initial publication of the Mysterium Cosmographicum and the Harmonices Mundi Kepler never lost sight of his original model and in 1621 he published a second updated edition of Mysterium Cosmographicum.

Although the mathematical models that Kepler chose for his model of the cosmos are, from our point of view, more than somewhat bizarre, throughout his entire work, Kepler’s thinking was never even remotely a sea of wild, woolly thinking, just the opposite. Kepler’s thinking was always concentrated, exact, concise, logical, mathematical thinking, which consistently followed the chosen mathematical model of the subject of his research, the cosmos. His thinking contained no contradictions, imprecisions, deviations or internal errors. We might reject his heuristic, and in fact we do, but to dismiss it as wild and woolly, in the way that Kitty Ferguson and many other do, is to do Kepler a major injustice.

[1] Kitty Ferguson, Measuring the Universe: Our Historic Quest to Chart the Horizons of Space and Time, Walker & Company, 1999, p. 70


Filed under History of Astronomy, History of science

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

One aspect central to the astronomical-cosmological discourse since antiquity was the actual size of the cosmos. This became particularly relevant to the astronomical system debate following Tycho’s star size argument. He argued given his failure to detect the stellar parallax, which should be observable in a heliocentric system, the stars must be so far away that the apparent size of the star discs would mean they must be quite literally unimaginably large and thus the system was not heliocentric. He also argued that under these circumstances there must also be an unimaginably vast distance between the orbit of Saturn and the sphere of the fixed stars. He thought it was ridiculous to suppose that there exists so much empty space, which for him also spoke against heliocentricity.

The earliest known serious attempt to determine the dimensions of the solar system was made by Aristarchus of Samos (c. 310–c. 230 BCE) infamous for proposing a heliocentric theory of the cosmos. We only have second-hand accounts of that system from Archimedes and Plutarch. However, the only manuscript attributed to him is Peri megethon kai apostematon (On the Sizes and Distances (of the Sun and Moon)). Aristarchus assumed that at half-moon the Earth, Moon and Sun form a right-angle triangle and that the angle between the Earth and the Moon is 87°.


From these assumptions he calculated that the ratio of the Earth/Sun distance to the Earth/Moon distance is approximately 1:19. In reality the ratio is approximately 1:400 because the angle is closer to 89.5° and is not differentiable by the human eye. Also, it is almost impossible to say exactly when half-moon is.

Aristarchus used a different geometrical construction based on the lunar eclipse to determine the actual sizes of the Earth, Moon and Sun.



Aristarchus’ 3rd century BCE calculations on the actual sizes of, from left, the Sun, Earth and Moon, from a 10th-century CE Greek copy Source: Wikimedia Commons

It is possible to reconstruct Aristarchus’ values (Source: Wikimedia Commons



Actual Values

Sun’s radius in Earth radii (e.r.)



Earth’s radius in Moon radii



Earth/Moon distance in e.r.)



Earth/Sun distance in e.r.)



Hipparchus (c. 190 – c. 120 BCE) used a modified version of Aristarchus’ eclipse diagram, using a solar rather than a lunar eclipse, to make the same calculations arriving at a value of between 59 and c. 67 e.r. for the Moon’s distance and 490 e.r. for the Sun.


As with almost all of Hipparchus’ other writings, his work on this topic has been lost but we have his method and results from Ptolemaeus, who also used a modified version of the solar eclipse diagram to make the same calculations. Ptolemaeus got widely different values for the furthest c. 64 e.r. and nearest c. 34 e.r. distance of the Moon from the Earth. The first is almost the correct value the second wildly off. He determined the Sun to be 1,210 e.r. distant.

In the history of astronomy literature, particularly the older literature, it is often claimed that Copernicus’ heliocentric model leads automatically to a set of relative distances for all the known planets from the Sun, which is true, but there is no equivalent set of measures for a Ptolemaic geocentric system, which is false. It is the case that in his great astronomical work, the Mathēmatikē Syntaxis (Almagest), he gives detailed epicycle-deferent models for each of the then known seven planets–Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn–but does not deal with their distances from each other or from the Earth. However, he wrote another smaller work, his Planetary Hypotheses, and here he delivers those missing dimensions. For Ptolemaeus each planetary orbit is embedded in a crystalline sphere the dimensions of which are determined by the ecliptic-deferent model for the planet. How this works is nicely illustrated in Georg von Peuerbach’s (1423–1461) Theoricae Novae Planetarum (New Planetary Theory) published by Regiomontanus in Nürnberg in 1472.


Diagram from Peuerbach’s Theoricae novae planetarum showing the orbit embedded in its crystalline sphere (green) Source: Wikimedia Commons

It was long thought that Peuerbach’s was an original work but when in 1964 the first ever know manuscript in Arabic, and till today the only one, of Ptolemaeus Planetary Hypotheses was found it was realised that it was merely a modernised version of Ptolemaeus’ work.

Ptolemaeus’ model of the cosmos was quite literally spheres within spheres, a sort of babushka doll model of the solar system. The Moon’s sphere enclosed the Earth. Mercury’s sphere began where the Moon’s sphere stopped, Venus’ sphere began where Mercury’s stopped, the Sun’s sphere began where Venus’ stopped and so on till the outer surface of Saturn’s sphere. Using this model Ptolemaeus calculated the following values and a value of 20,000 e.r. for the distance from the Earth to the sphere of fixed star and c. 1,200 e.r. for the Earth/sun distance.


Ptolemaeus’ model and at least his basic dimensions–Earth/Moon, Earth/Sun and fixed star sphere distances–remained the astronomical/cosmological norm for nearly all astronomers in the Islamic and European Middle Ages and we first begin to see new developments in the sixteenth century and the so-called astronomical revolution.

In the geocentric model the order of the orbits of Mercury, Venus and the Sun moving away from the Earth and the Moon is purely arbitrary as they all have an orbital period of one year relative the Earth. Ptolemaeus’ order was, in antiquity, only one of several; in fact, he played with different possible orders himself. In a heliocentric system the correct order of the planets moving away from the Sun is given automatically by the length of their orbits. This is, of course, the basis of Kepler’s third law of planetary motion. The relative size of those orbits is also given with respect to the distance between the Earth and the Sun, the so-called astronomical unit. This gives a new incentive to trying to find the correct value for this distance, determine the one and you have determined them all.

Copernicus determined the distances between the Earth and the other planets using his epicycle models and Ptolemaeus’ data, which produced much smaller values for those distances that by Ptolemaeus. Although he appeared to calculate the astronomical unit for himself, however, he chose parameters that gave him approximately Ptolemaeus’ value of 1,200 e.r.


Tycho Brahe’s values were also smaller than those of Ptolemaeus, but he also chose a value for the astronomical unit that was in the same area of those of Ptolemaeus and Copernicus. Tycho’s failure to detect stellar parallax led him to argue that the parallax value for the fixed stars, if it exists, must have a maximum of one minute, i.e. one sixtieth of a degree, meaning that in a Copernican cosmos the fixed stars must have a minimum distance of approximately 7,850,000 e.r. Copernicans had no choice but to accept this, for the time, literally unbelievable distance. Tycho himself set the distance of the fixed stars in his system just beyond the orbit of Saturn at 14,000 e.r.


Up till now all of those distances had been calculated based on a combination series of dubious assumptions and rathe dodgy geometrical models, this would all change with the advent of Johannes Kepler in the game. Through out his career Kepler returned several times to the problem of the distance of the planets from the Sun expressed relative to the astronomical unit. By the time he wrote and published his Harmonices Mundi containing his all-important third law of planetary motion in 1619, the values that he had obtained were largely correct, but he still had no real measure for the astronomical unit or from the distance of the fixed stars. For his own estimate of the astronomical unit Kepler turned to a parallax argument. He argued that no solar parallax was visible, not even with the recently invented telescope, so the parallax could be, at the most, one minute i.e. one sixtieth of a degree. This would give him a minimum value for the astronomical unit of c. 3,500 e.r., three times as big as the Ptolemaic/Copernican value. As a convinced Copernican Kepler was more than prepared to accept Tycho’s argument for very distant fixed stars, his minimum value was 60,000,000 e.r.


Because the astronomical unit was essential for turning his relative values for the distances of the planets into absolute values, over the years he considered various methods for determining it. He even reconsidered Aristarchus’ half-moon method, hoping that the telescope would make it possible to accurately determine the time of half-moon and measure the angle. His own attempts failed and in his ephemeris for 1618 he appeals to Galileo and Simon Marius to make the necessary observations. However, even they would not have been able to oblige, as the telescopes were still too primitive for the task.

For once Galileo did not take part in the attempts to establish the dimensions of the solar system, accepting Copernicus’ values. He did make some measurements of the size of the planets, a parallel undertaking to determining the planetary distances. He never published a systematic list of those measurements preferring instead just to snipe at other astronomers, who published different values to his.

Kepler’s work was a major game changer in the attempts to calculate the size of the cosmos and its components. His solar system has very different dimensions to everything that preceded it and for those supporting his viewpoint it meant the necessity to find new improved ways to find a value for the astronomical unit.

*  All diagrams and tables are taken from Albert van Helden, Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley, University of Chicago Press, 1985, unless otherwise stated.


Filed under History of Astronomy