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

The publication of Galileo’s Sidereus Nuncius was by no means the end of the spectacular and game changing telescopic astronomical discoveries during that first hot phase, which spanned 1610 to 1613. There were to be three further major discoveries, one of which led to a bitter priority dispute that would in the end play a role in Galileo’s downfall and another of which would sink Ptolemaeus’ geocentric model of the cosmos for ever.

The first new discovery post Sidereus Nuncius was the rather strange fact that Saturn appeared to have ears or as Galileo put it, it was three bodies “accompanied by two attendants who never leave his side.”


Galileo’s drawings of Saturn

What Galileo had in fact observed were the rings of Saturn, which however because of the relative positions of Saturn and the Earth were not discernable as rings but as strange semi-circular projections on either side of the planet. What exactly the strange protrusions visible on Saturn were would remain a mystery until Christiaan Huygens solved the problem much later in the century. The astronomers of the Collegio Romano claimed priority on the Saturn discovery. Whether they or Galileo saw the phenomenon first cannot really be determined but it demonstrates once again that Galileo was by no means the only one making these new telescopic discoveries. Saturn’s two “attendants” didn’t really play a role in the ongoing astronomy/cosmology debate but the next discovery did in a very major way.

Probably stimulated by a letter from his one time student Benedetto Castelli (1578–1643) Galileo turned his attention to Venus and its potential phases.


Benedetto Castelli Source: Wikimedia Commons

If Venus was indeed lit by the sun then in both Ptolemaeus’ geocentric system and in a heliocentric system it would, like the moon, display phases but these phases would differ according to whether Venus orbited the Earth in a geocentric system or the sun in either a heliocentric or a geo-heliocentric one. Galileo’s observations clearly showed that the phases of Venus were consistent with a solar orbit and not a terrestrial one.


The Phases of Venus in both systems

The pure Ptolemaic geocentric system was irredeemably sunk but not, and that must be strongly emphasised, a number of geo-heliocentric systems. As already mentioned earlier, because they never strayed far from the sun’s vicinity and in a geocentric system even shared the sun orbital period, Mercury and Venus had since antiquity been assumed, by some, to orbit the sun whereas the sun orbited the earth in what is known as the Capellan system; a system that was very popular in the Middle Ages and had been praised as such by Copernicus in his De revolutionibus. Phases of Venus indicating a solar orbit were, of course, also consistent with a full Tychonic system in which the planets, apart from the moon, orbited the sun, which in turn together with the moon orbited the earth, as well as several variant semi-Tychonic systems. It was assumed that Mercury also orbited the sun, although its phases were first observed by  Pierre Gassendi (1592–1655) in 1631. The heliocentric phases of Venus were also discovered independently by Thomas Harriot, who, as always, didn’t publish, by Simon Marius, whose discovery was published by Kepler, and by the Collegio Romano astronomers, who also didn’t published but announced their discovery in their correspondence.

The other major telescopic discovery was the presence of blemishes or spots on the surfaces of the sun, again something that contradicted Aristotle’s assumption of the perfection of the celestial bodies. This discovery led to one of Galileo’s biggest priority disputes. This whole sorry episode began with a communication from the Augsburger banker and science fan, Marcus Welser(1558–1614), who was also a close friend of the Jesuits.


Marcus Welser Source: Wikimedia Commons

This communication contained three letters on sunspots written by the Ingolstädter Jesuit Christoph Scheiner (1573 or 75–1650) under the pseudonym, Appeles.


Christoph Scheiner (artist unknown)

Welser wanted to hear Galileo’s opinion on Scheiner’s discovery. Galileo was deeply offended, the heavens were his territory and only he was allowed to make discoveries there! The dispute was carried on two levels, the first was the question of priority and the second was the question of how to interpret what had been observed. Although, during the whole dispute Galileo kept changing the date when he first observed sunspots, in order to establish his priority and to claim the discovery as his, viewed with hindsight the priority dispute was a bit of a joke. We now know that Thomas Harriot  had recorded observations of sunspot before either Galileo or Scheiner but because he never published his observations, they were blissfully unaware of his priority. Even stranger, Johannes Fabricius (1587–1616), the son of Kepler’s intellectual sparing partner David Fabricius, had brought home a telescope from university in Leiden, where Rudolph Snell (1546–1613) was already holding lectures on the telescope in 1610, and together with his father had not only been observing sunspots but had already published a pamphlet on his observation in Wittenberg in 1611, where he was now studying.


The second part of the dispute was by far and away the more important. Scheiner had initially interpreted the sunspots as shadows cast upon the surface of the sun by small satellites orbiting it. It is was possible that through this interpretation he wished to preserve the Aristotelian perfection of this celestial body. Galileo opposed this interpretation and was convinced, correctly as it turned out, that the sunspots were actually some sort of blemishes on the surface of the sun.

Galileo answered Scheiner’s letters with three of his own, in the process stepping up his observation of the sunspots, as well as gathering observational reports from other astronomers. He was able to show through the quality of his observations and through mathematical analysis that the sunspots must be on the surface of the sun and that the sun must be revolving about its axis. With time Scheiner came to accept Galileo’s conclusions. Scheiner published three more sunspot letters under the title Accuratior Disquisitio in 1612.


The Accademia dei Lincei, which had elected Galileo a member when he came to Rome to celebrate the Jesuit’s confirmation of his telescopic discoveries, published Scheiner’s original three letters together with Galileo’s three answering letters in a book titled, Istoria e Dimontrazioni, in 1613.


Having in his opinion won the priority dispute and proved that the sunspots were on the surface of the sun, Galileo basically gave up on his solar observations; Scheiner did not. Having built what was effectively the first Keplerian or astronomical telescope with two convex lenses, instead of one convex and one concave, as in the Dutch or Galilean telescope, giving a much wider field of vision and a much clearer and stronger image, Scheiner set out on a programme of solar astronomy.


Scheiner Observing the Sun

The astronomical telescope provided an inverted image but this was irrelevant as Scheiner was projecting the image onto paper in order to simplify the drawing on the sunspots and also to protect his eyes. A method also used by Fabricius and Galileo. He mounted his telescope on a special holder that allowed him to follow the sun in its journey across the heavens.


Scheiner’s Helioscope

The end of this programme was his Rosa Ursina sive Sol, published in 1626-30, which remained the most important book on solar astronomy until the nineteenth century.


Scheiner’s Sunspot Observations

Galileo’s and Scheiner’s priority dispute entails a strong sense of historical irony. Not only did Harriot begin observing sunspots earlier than both of them and Johannes Fabricius publish on the subject before either of them but Chinese and Korean astronomers had been recording naked-eye observations of sunspots since the first millennium BCE. There are also scattered observations of sunspots beginning with the ancient Greeks and down through the Middle Ages in Europe.


A drawing of a sunspot in the Chronicles of John of Worcester 1129 Source: Wikimedia Commons

Famously Kepler recorded observations of a large sunspot that he made in 1607 mistakenly believing that he was observing a transit of Mercury.

1613 marks the end of the first phase of astronomical telescopic discoveries, partially because the observers continued to use Dutch or Galilean telescopes instead of changing to the vastly superior Keplerian or astronomical telescopes, largely influenced by Galileo’s authority, he publicly rubbished astronomical telescopes, basically because he hadn’t started using them first; the transition to the better instruments would take a couple of decades to be completed.









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

Calculus for the curious

Some weeks ago I got involved in a discussion on Twitter about, which books to recommend on the history of calculus. Somebody chimed in that Steven Strogatz’s new book would tell you all that you needed to know about the history of calculus. I replied that I couldn’t comment on this, as I hadn’t read it. To my surprise Professor Strogatz popped up and asked me if I would like to have a copy of his book. Never one to turn down a freebee, I naturally said yes. Very soon after a copy of Infinite Powers: The Story of Calculus The Language of the Universe arrived in the post and landed on my to read pile. Having now read it I can comment on it and intend to do so.

For those, who don’t know Steven Strogatz, he is professor of applied mathematics at Cornell University and the successful author of best selling popular books on mathematics.


First off, Infinite Powers is not a history of calculus. It is a detailed introduction to what calculus is and how it works, with particular emphasis on its applications down the centuries, Strogatz is an applied mathematician, presented in a history-light frame story. Having said this, I’m definitely not knocking, what is an excellent book but I wouldn’t recommend it to anybody, who was really looking for a history of calculus, maybe, however, either as a prequel or as a follow up to reading a history of calculus.

The book is centred on what Strogatz calls The Infinity Principle, which lies at the heart of the whole of calculus:

To shed light on any continuous shape, object, motion, process, or phenomenon–no matter how wild and complicated it may appear–reimagine it as an infinite series of simpler parts, analyse those, and then add the results back together to make sense of the original whole.

Following the introduction of his infinity principle Strogatz gives a general discussion of its strengths and weakness before moving on in the first chapter proper to discuss infinity in all of its guises, familiar material and examples for anybody, who has read about the subject but a well done introduction for those who haven’t. Chapter 2 takes us  into the early days of calculus, although it didn’t yet have this name, and introduces us to The Man Who Harnessed Infinity, the legendary ancient Greek mathematician Archimedes and the method of exhaustion used to determine the value of π and the areas and volumes of various geometrical forms. Astute readers will have noticed that I wrote early days and not beginning and here is a good example of why I say that this is not a history of calculus. Although Archimedes put the method of exhaustion to good use he didn’t invent it, Eudoxus did. Strogatz does sort of mention this in passing but whereas Archimedes gets star billing, Eudoxus gets dismissed in half a sentence in brackets. The reader is left completely in the dark as to who, why, what Eudoxus is/was. OK here, but not OK in a real history of calculus. This criticism might seem petty but there are lots of similar examples throughout the book that I’m not going to list in this review and this is why the book is not a history of calculus and I don’t think Strogatz intended to write one; the book he has written is a different one and it is a very good one.

After Archimedes the book takes a big leap to the Early Modern Period and Galileo and Kepler with the justification that, “When Archimedes died, the mathematical study of nature nearly died along with him. […] In Renaissance Italy, a young mathematician named Galileo Galilei picked up where Archimedes had left off.” My inner historian of mathematics had an apoplectic fit on reading these statements. They ignore a vast amount of mathematics, in particular the work in the Middle Ages and the sixteenth century on which Galileo built the theories that Strogatz then presents here but I console myself with the thought that this is not a history of calculus let alone a history of mathematics. However, I’m being too negative, let us return to the book. The chapter deals with Galileo’s terrestrial laws of motion and Kepler’s astronomical laws of planetary motion. Following this brief introduction to the beginnings of modern science Strogatz moves into top gear with the beginnings of differential calculus. He guides the reader through the developments of seventeenth century mathematics, Fermat and Descartes and the birth of analytical geometry bringing together the recently introduced algebra and the, by then, traditional geometry. Moving on he deals with tangents, functions and derivatives. Strogatz is an excellent teacher he introduces a new concept carefully, explains it, and then shows how it can be applied to an everyday situation.

Having laid the foundations Strogatz move on naturally to the supposed founders of modern calculus, Leibnitz and Newton and their bringing together of the strands out of the past that make up calculus as we know it and how they fit together in the fundamental theorem of calculus. This is interwoven with the life stories of the two central figures. Again having introduced concepts and explained them Strogatz illustrates them with applications outside of pure mathematics.

Having established modern calculus the story moves on into the eighteenth century.  Here I have to point out that Strogatz perpetuates a couple of myths concerning Newton and the writing of his Principia. He writes that Newton took the concept of inertia from Galileo; he didn’t, he took it from Descartes, who in turn had it from Isaac Beeckman. A small point but as a historian I think an important one. Much more important he seems to be saying that Newton created the physics of Principia using calculus then translated it back into the language of Euclidian geometry, so as not to put off his readers. This is a widely believed myth but it is just that, a myth. To be fair it was a myth put into the world by Newton himself. All of the leading Newton experts have over the years very carefully scrutinised all of Newton’s writings and have found no evidence that Newton conceived and wrote Principia in any other form than the published one. Why he rejected the calculus, which he himself developed, as a working tool for his magnum opus is another complicated story that I won’t go into here but reject it he did[1].

After Principia, Strogatz finishes his book with a random selection of what might be termed calculus’ greatest hits, showing how it proved its power in solving a diverse series of problems. Interestingly he also addresses the future. There are those who think that calculus’ heyday is passed and that other, more modern mathematical tools will in future be used in the applied sciences to solve problems, Strogatz disagrees and sees a positive and active future for calculus as a central mathematical tool.

Despite all my negative comments, and I don’t think my readers would expect anything else from me, given my reputation, I genuinely think that this is on the whole an excellent book. Strogatz writes well and fluidly and despite the, sometimes, exacting content his book is a pleasure to read. He is also very obviously an excellent teacher, who is very good at clearly explaining oft, difficult concepts. I found it slightly disappointing that his story of calculus stops just when it begins to get philosophical and logically interesting i.e. when mathematicians began working on a safe foundation for the procedures that they had been using largely intuitively. See for example Euler, who made great strides in the development of calculus without any really defined concepts of convergence, divergence or limits, but who doesn’t appear here at all. However, Strogatz book is already 350-pages-long and if, using the same approach, he had continued the story down to and into the twentieth century it would probably have weighed in at a thousand plus pages!

Despite my historical criticisms, I would recommend Strogatz’s book, without reservations, to anybody and everybody, who wishes to achieve a clearer, deeper and better understanding of what calculus is, where it comes from, how it functions and above all, and this is Strogatz’s greatest strength, how it is applied to the solution of a wide range of very diverse problems in an equally wide and diverse range of topics.


[1] For a detailed analysis of Newton’s rejection of analytical methods in mathematics then I heartily recommend, Niccolò Guicciardini, Reading the Principia, CUP, 1999, but with the warning that it’s not an easy read!



Filed under Book Reviews, History of Mathematics

Why, FFS! why?

On Twitter this morning physicist and science writer Graham Farmelo inadvertently drew my attention to a reader’s letter in The Guardian from Sunday by a Collin Moffat. Upon reading this load of old cobblers, your friendly, mild mannered historian of Renaissance mathematics instantly turned into the howling-with-rage HISTSCI_HULK. What could possibly have provoked this outbreak? I present for your delectation the offending object.

I fear Thomas Eaton (Weekend Quiz, 12 October) is giving further credence to “fake news” from 1507, when a German cartographer was seeking the derivation of “America” and hit upon the name of Amerigo Vespucci, an obscure Florentine navigator. Derived from this single source, this made-up derivation has been copied ever after.

The fact is that Christopher Columbus visited Iceland in 1477-78, and learned of a western landmass named “Markland”. Seeking funds from King Ferdinand of Spain, he told the king that the western continent really did exist, it even had a name – and Columbus adapted “Markland” into the Spanish way of speaking, which requires an initial vowel “A-”, and dropped “-land” substituting “-ia”.

Thus “A-mark-ia”, ie “America”. In Icelandic, “Markland” may be translated as “the Outback” – perhaps a fair description.

See Graeme Davis, Vikings in America (Birlinn, 2009).

Astute readers will remember that we have been here before, with those that erroneously claim that America was named after a Welsh merchant by the name of Richard Ap Meric. The claim presented here is equally erroneous; let us examine it in detail.

…when a German cartographer was seeking the derivation of “America” and hit upon the name of Amerigo Vespucci, an obscure Florentine navigator.

It was actually two German cartographers Martin Waldseemüller and Matthias Ringmann and they were not looking for a derivation of America, they coined the name. What is more, they give a clear explanation as to why and how the coined the name and why exactly they chose to name the newly discovered continent after Amerigo Vespucci, who, by the way, wasn’t that obscure. You can read the details in my earlier post. It is of interest that the supporters of the Ap Meric theory use exactly the same tactic of lying about Waldseemüller and Ringmann and their coinage.

The fact is that Christopher Columbus visited Iceland in 1477-78, and learned of a western landmass named “Markland”.

Let us examine what is known about Columbus’ supposed visit to Iceland. You will note that I use the term supposed, as facts about this voyage are more than rather thin. In his biography of Columbus, Felipe Fernandez-Armesto, historian of Early Modern exploration, writes:

He claimed that February 1477–the date can be treated as unreliable in such a long –deferred recollection [from 1495]–he sailed ‘a hundred leagues beyond’ Iceland, on a trip from Bristol…

In “Christopher Columbus and the Age of Exploration: An Encyclopedia”[1] edited by the American historian, Silvio A. Bedini, we can read:

The possibility of Columbus having visited Iceland is based on a passage in his son Fernando Colón’s biography of his father. He cites a letter from Columbus stating that in February 1477 he sailed “a hundred leagues beyond the island of Til” (i.e. Thule, Iceland). But there is no evidence to his having stopped in Iceland or spoken with anyone, and in any case it is unlikely that anyone he spoke to would have known about the the Icelandic discovery of Vinland.

This makes rather a mockery of the letter’s final claim:

Seeking funds from King Ferdinand of Spain, he told the king that the western continent really did exist, it even had a name – and Columbus adapted “Markland” into the Spanish way of speaking, which requires an initial vowel “A-”, and dropped “-land” substituting “-ia”.

Given that it is a well established fact that Columbus was trying to sail westward to Asia and ran into America purely by accident, convinced by the way that he had actually reached Asia, the above is nothing more than a fairly tale with no historical substance whatsoever.

To close I want to address the question posed in the title to this brief post. Given that we have a clear and one hundred per cent reliable source for the name of America and the two men who coined it, why oh why do people keep coming up with totally unsubstantiated origins of the name based on ahistorical fantasies? And no I can’t be bothered to waste either my time or my money on Graeme Davis’ book, which is currently deleted and only available as a Kindle.

[1] On days like this it pays to have one book or another sitting around on your bookshelves.

Felipe Fernández-Armesto, Columbus, Duckworth, London, ppb 1996, p. 18. Christopher Columbus and the Age of Exploration: An Encyclopedia, ed. Silvio A. Bedini, Da Capo Press, New York, ppb 1992, p. 314


Filed under History of Cartography, History of Navigation, Myths of Science, Renaissance Science

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

A widespread myth in the popular history of astronomy is that Galileo Galilei (1564–1642) was the first or even the only astronomer to realise the potential of the newly invented telescope as an instrument for astronomy. This perception is very far from the truth. He was just one of a group of investigator, who realised the telescopes potential and all of the discoveries traditionally attributed to Galileo were actually made contemporaneously by several people, who full of curiosity pointed their primitive new instruments at the night skies. So why does Galileo usually get all of the credit? Quite simply, he was the first to publish.


Galileo’s “cannocchiali” telescopes at the Museo Galileo, Florence

Starting in the middle of 1609 various astronomers began pointing primitive Dutch telescopes at the night skies, Thomas Harriot (1560–1621) and his friend and student William Lower (1570–1615) in Britain, Simon Marius (1573–1625) in Ansbach, Johannes Fabricius (1587–1616) in Frisia, Odo van Maelcote (1572–1615) and Giovanni Paolo Lembo (1570–1618) in Rome, Christoph Scheiner (1573 or 1575–1650) in Ingolstadt and of course Galileo in Padua. As far as we can ascertain Thomas Harriot was the first and the order in which the others took up the chase is almost impossible to determine and also irrelevant, as it was who was first to publish that really matters and that was, as already stated, Galileo.

Harriot made a simple two-dimensional telescopic sketch of the moon in the middle of 1609.


Thomas Harriot’s initial telescopic sketch of the moon from 1609 Source: Wikimedia Commons

Both Galileo and Simon Marius started making telescopic astronomical observations sometime late in the same year. At the beginning Galileo wrote his observation logbook in his Tuscan dialect and then on 7 January 1610 he made the discovery that would make him famous, his first observation of three of the four so-called Galilean moons of Jupiter.


It was on this page that Galileo first noted an observation of the moons of Jupiter. This observation upset the notion that all celestial bodies must revolve around the Earth. Source: Wikimedia Commons

Galileo realised at once that he had hit the jackpot and immediately changed to writing his observations in Latin in preparation for a publication. Simon Marius, who made the same discovery just one day later, didn’t make any preparations for immediate publication. Galileo kept on making his observations and collecting material for his publication and then on 12 March 1610, just two months after he first saw the Jupiter moons, his Sidereus Nuncius (Starry Messenger of Starry Message, the original Latin is ambiguous) was published in Padua but dedicated to Cosimo II de Medici, Fourth Grand Duke of Tuscany. Galileo had already negotiated with the court in Florence about the naming of the moons; he named them the Medicean Stars thus taking his first step in turning his discovery into personal advancement.


Title page of Sidereus nuncius, 1610, by Galileo Galilei (1564-1642). *IC6.G1333.610s, Houghton Library, Harvard University Source: Wikimedia Commons

What exactly did Galileo discover with his telescope, who else made the same discoveries and what effect did they have on the ongoing astronomical/cosmological debate? We can start by stating quite categorically that the initial discoveries that Galileo published in his Sidereus Nuncius neither proved the heliocentric hypothesis nor did they refute the geocentric one,

The first discovery that the Sidereus Nuncius contains is that viewed through the telescope many more stars are visible than to the naked-eye. This was already known to those, who took part in Lipperhey’s first ever public demonstration of the telescope in Den Haag in September 1608 and to all, who subsequently pointed a telescope of any sort at the night sky. This played absolutely no role in the astronomical/cosmological debate but was worrying for the theologians. Christianity in general had accepted both astronomy and astrology, as long as the latter was not interpreted deterministically, because the Bible says  “And God said, Let there be lights in the firmament of the heaven to divide the day from night; and let them be for signs, and for seasons, and for days, and years:” (Gen 1:14). If the lights in the heavens are signs from God to be interpreted by humanity, what use are signs that can only be seen with a telescope?

Next up we have the fact that some of the nebulae, indistinct clouds of light in the heavens, when viewed with a telescope resolved into dense groups of stars. Nebulae had never played a major role in Western astronomy, so this discovery whilst interesting did not play a major role in the contemporary debate. Simon Marius made the first telescopic observations of the Andromeda nebula, which was unknown to Ptolemaeus, but which had already been described by the Persian astronomer, Abd al-Rahman al-Sufi (903–986), usually referred to simply as Al Sufi. It is historically interesting because the Andromeda nebula was the first galaxy to be recognised outside of the Milky Way.


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

Galileo’s next discovery was that the moon was not smooth and perfect, as required of all celestial bodies by Aristotelian cosmology, but had geological feature, mountains and valleys, just like the earth i.e. the surface was three-dimensional and not two-dimensional, as Harriot had sketched it. This perception of Galileo’s is attributed to the fact that he was a trained painter used to creating light and shadows in paintings and he thus recognised that what he was seeing on the moons surface was indeed shadows cast by mountains.

As soon as he read the Sidereus Nuncius, Harriot recognised that Galileo was correct and he went on to produce the first real telescopic map of the moon.


Thomas Harriot’s 1611 telescopic map of the moon Source: Wikimedia Commons

Galileo’s own washes of the moon, the most famous illustrations in the Sidereus Nuncius, are in fact studies to illustrate his arguments and not accurate illustrations of what he saw.


Galileo’s sketches of the Moon from Sidereus Nuncius. Source: Wikimedia Commons

That the moon was earth like and for some that the well-known markings on the moon, the man in the moon etc., are in fact a mountainous landscape was a view held by various in antiquity, such as Thales, Orpheus, Anaxagoras, Democritus, Pythagoras, Philolaus, Plutarch and Lucian. In particular Plutarch (c. 46–c. 120 CE) in his On the Face of the Moon in his Moralia, having dismissed other theories including Aristotle’s wrote:

Just as our earth contains gulfs that are deep and extensive, one here pouring in towards us through the Pillars of Herakles and outside the Caspian and the Red Sea with its gulfs, so those features are depths and hollows of the Moon. The largest of them is called “Hecate’s Recess,” where the souls suffer and extract penalties for whatever they have endured or committed after having already become spirits; and the two long ones are called “the Gates,” for through them pass the souls now to the side of the Moon that faces heaven and now back to the side that faces Earth. The side of the Moon towards heaven is named “Elysian plain,” the hither side, “House of counter-terrestrial Persephone.”

So Galileo’s discovery was not so sensational, as it is often presented. However, the earth-like, and not smooth and perfect, appearance of the moon was yet another hole torn in the fabric of Aristotelian cosmology.

Of course the major sensation in the Sidereus Nuncius was the discovery of the four largest moons of Jupiter.


Galileo’s drawings of Jupiter and its Medicean Stars from Sidereus Nuncius. Image courtesy of the History of Science Collections, University of Oklahoma Libraries. Source: Wikimedia Commons

This contradicted the major premise of Aristotelian cosmology that all of the celestial bodies revolved around a common centre, his homo-centricity.  It also added a small modicum of support to a heliocentric cosmology, which had suffered from the criticism, if all the celestial bodies revolve around the sun, why does the moon continue to revolve around the earth. Now Jupiter had not just one but four moons, or satellites as Johannes Kepler called them, so the earth was no longer alone in having a moon. As already stated above Simon Marius discovered the moons of Jupiter just one day later than Galileo but he didn’t publish his discovery until 1614. A delay that would later bring him a charge of plagiarism from Galileo and ruin his reputation, which was first restored at the end of the nineteenth century when an investigation of the respective observation data showed that Marius’ observations were independent of those of Galileo.

The publication of the Sidereus Nuncius was an absolute sensation and the book quickly sold out. Galileo went, almost literally overnight, from being a virtually unknown, middle aged, Northern Italian, professor of mathematics to the most celebrated astronomer in the whole of Europe. However, not everybody celebrated or accepted the truth of his discoveries and that not without reason. Firstly, any new scientific discovery needs to be confirmed independently by other. If Simon Marius had also published early in 1610 things might have been different but he, for whatever reasons, didn’t publish his Mundus Jovialis (The World of Jupiter) until 1614. Secondly there was no scientific explanation available that explained how a telescope functioned, so how did anyone know that what Galileo and others were observing was real? Thirdly, and this is a very important point that often gets ignored, the early telescopes were very, very poor quality suffering from all sorts of imperfections and distortions and it is almost a miracle that Galileo et al discovered anything with these extremely primitive instruments.

As I stated in the last episode, the second problem was solved by Johannes Kepler in 1611 with the publication of his Dioptrice.


A book that Galileo, always rather arrogant, dismissed as unreadable. This was his triumph and nobody else was going to muscle in on his glory. The third problem was one that only time and improvements in both glass making and the grinding and polishing of lenses would solve. In the intervening years there were numerous cases of new astronomical discoveries that turned out to be artefacts produced by poor quality instruments.

The first problem was the major hurdle that Galileo had to take if he wanted his discoveries to be taken seriously. Upon hearing of Galileo discoveries, Johannes Kepler in Prague immediately put pen to paper and fired off a pamphlet, Dissertatio cum Nuncio Sidereo (Conversation with the Starry Messenger) congratulating Galileo, welcoming his discoveries and stating his belief in their correctness, which he sent off to Italy. Galileo immediately printed and distributed a pirate copy of Kepler’s work, without even bothering to ask permission, it was after all a confirmation from the Imperial Mathematicus and Kepler’s reputation at this time was considerably bigger than Galileo’s.

Johannes Kepler, Dissertatio cum Nuncio sidereo… (Frankfurt am Main, 1611)

A reprint of Kepler’s letter to Galileo, originally issued in Prague in 1610

However, Kepler’s confirmations were based on faith and not personal confirmatory observations, so they didn’t really solve Galileo’s central problem. Help came in the end from the Jesuit astronomers of the Collegio Romano.

Odo van Maelcote and Giovanni Paolo Lembo had already been making telescopic astronomical observations before the publication of Galileo’s Sidereus Nuncius. Galileo also enjoyed good relations with Christoph Clavius (1538–1612), the founder and head of the school of mathematics at the Collegio Romano, who had been instrumental in helping Galileo to obtain the professorship in Padua. Under the direction of Christoph Grienberger (1561–1636), soon to be Clavius’ successor as professor for mathematics at the Collegio, the Jesuit astronomers set about trying to confirm all of Galileo’s discoveries. This proved more than somewhat difficult, as they were unable, even with Galileo’s assistance via correspondence, to produce an instrument of sufficient quality to observe the moons of Jupiter. In the end Antonio Santini (1577–1662), a mathematician from Venice, succeeded in producing a telescope of sufficient quality for the task, confirmed for himself the existence of the Jupiter moons and then sent a telescope to the Collegio Romano, where the Jesuit astronomers were now also able to confirm all of Galileo’s discovery. Galileo could not have wished for a better confirmation of his efforts, nobody was going to doubt the word of the Jesuits.

In March 1611 Galileo travelled to Rome, where the Jesuits staged a banquet in his honour at which Odo van Maelcote held an oration to the Tuscan astronomer. Galileo’s strategy of dedicating the Sidereus Nuncius to Cosimo de Medici and naming the four moons the Medicean Stars paid off and he was appointed court mathematicus and philosophicus in Florence and professor of mathematics at the university without any teaching obligations; Galileo had arrived at the top of the greasy pole but what goes up must, as we will see, come down.





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Filed under Early Scientific Publishing, History of Astronomy, Renaissance Science

Mathematical aids for Early Modern astronomers.

Since its very beginnings in the Fertile Crescent, European astronomy has always involved a lot of complicated and tedious mathematical calculations. Those early astronomers described the orbits of planets, lunar eclipses and other astronomical phenomena using arithmetical or algebraic algorithms. In order to simplify the complex calculations needed for their algorithms the astronomers used pre-calculated tables of reciprocals, squares, cubes, square roots and cube roots.


Cuniform reciprocal table Source

The ancient Greeks, who inherited their astronomy from the Babylonians, based their astronomical models on geometry rather than algebra and so needed other calculation aids. They developed trigonometry for this work based on chords of a circle. The first chord tables are attributed to Hipparkhos (c. 190–c. 120 BCE) but they did not survive. The oldest surviving chord tables are in Ptolemaeus’ Mathēmatikē Syntaxis written in about 150 CE, which also contains a detailed explanation of how to calculate such a table in Chapter 10 of Book I.


Ptolemaeus’ Chord Table taken from Toomer’s Almagest translation. The 3rd and 6th columns are the interpolations necessary for angles between the given ones

Greek astronomy travelled to India, where the astronomers replaced Ptolemaeus’ chords with half chords, that is our sines. Islamic astronomers inherited their astronomy from the Indians with their sines and cosines and the Persian astronomer Abū al-Wafāʾ (940–998 CE) was using all six of the trigonometrical relations that we learnt at school (didn’t we!) in the tenth century.


Abū al-Wafāʾ Source: Wikimedia Commons

Astronomical trigonometry trickled slowly into medieval Europe and Regiomontanus (1536–1576)  (1436–1476) was the first European to produce a comprehensive work on trigonometry for astronomers, his De triangulis omnimodis, which was only edited by Johannes Schöner and published by Johannes Petreius in 1533.

Whilst trigonometry was a great aid to astronomers calculating trigonometrical tables was time consuming, tedious and difficult work.

A new calculating aid for astronomers emerged during the sixteenth century, prosthaphaeresis, by which, multiplications could be converted into additions using a series of trigonometrical identities:

Prosthaphaeresis appears to have first been used by Johannes Werner (1468–1522), who used the first two formulas with both sides multiplied by two.

However Werner never published his discovery and it first became known through the work of the itinerant mathematician Paul Wittich (c. 1546–1586), who taught it to both Tycho Brahe (1546–1601) on his island of Hven and to Jost Bürgi (1552–1632) in Kassel, who both developed it further. It is not known if Wittich learnt the method from Werner’s papers on one of his visits to Nürnberg or rediscovered it for himself. Bürgi in turn taught it to Nicolaus Reimers Baer (1551–1600) in in exchange translated Copernicus’ De revolutionibus into German for Bürgi, who couldn’t read Latin. This was the first German translation of De revolutionibus. As can be seen the method of prosthaphaeresis spread throughout Europe in the latter half of the sixteenth century but was soon to be superceded by a superior method of simplifying astronomical calculations by turning multiplications into additions, logarithms.

As is often the case in the histories of science and mathematics logarithms were not discovered by one person but almost simultaneously, independently by two, Jost Bürgi and John Napier (1550–1617) and both of them seem to have developed the idea through their acquaintance with prosthaphaeresis. I have already blogged about Jost Bürgi, so I will devote the rest of this post to John Napier.


John Napier, artist unknown Source: Wikimedia Commons

John Napier was the 8th Laird of Merchiston, an independently owned estate in the southwest of Edinburgh.


Merchiston Castle from an 1834 woodcut Source: Wikimedia Commons

His exact date of birth is not known and also very little is known about his childhood or education. It is assumed that he was home educated and he was enrolled at the University of St. Andrews at the age of thirteen. He appears not to have graduated at St. Andrews but is believed to have continued his education in Europe but where is not known. He returned to Scotland in 1571 fluent in Greek but where he had acquired it is not known. As a laird he was very active in the local politics. His intellectual reputation was established as a theologian rather than a mathematician.

It is not known how and when he became interested in mathematics but there is evidence that this interest was already established in the early 1570s, so he may have developed it during his foreign travels. It is thought that he learnt of prosthaphaeresis through John Craig (d. 1620) a Scottish mathematician and physician, who had studied and later taught at Frankfurt an der Oder, a pupil of Paul Wittich, who knew Tycho Brahe. Craig returned to Edinburgh in 1583 and is known to have had contact with Napier. The historian Anthony à Wood (1632–1695) wrote:

one Dr. Craig … coming out of Denmark into his own country called upon John Neper, baron of Murcheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus as ’tis said) to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportionable numbers. [Neper is the Latin version of his family name]

Napier is thought to have begum work on the invention of logarithms about 1590. Logarithms exploit the relation ship between arithmetical and geometrical series. In modern terminology, as we all learnt at school, didn’t we:

Am x An = Am+n

Am/An = Am-n

These relationships were discussed by various mathematicians in the sixteenth century, without the modern notation, in particularly by Michael Stefil (1487–1567) in his Arithmetica integra (1544).


Michael Stifel Source: Wikimedia Commons


Michael Stifel’s Arithmetica Integra (1544) Source: Wikimedia Commons

What the rules for exponents show is that if one had tables to convert all numbers into powers of a given base then one could turn all multiplications and divisions into simple additions and subtractions of the exponents then using the tables to covert the result back into a number. This is what Napier did calling the result logarithms. The methodology Napier used to calculate his tables is too complex to deal with here but the work took him over twenty years and were published in his Mirifici logarithmorum canonis descriptio… (1614).


Napier coined the term logarithm from the Greek logos (ratio) and arithmos (number), meaning ratio-number. As well as the logarithm tables, the book contains seven pages of explanation on the nature of logarithms and their use. A secondary feature of Napier’s work is that he uses full decimal notation including the decimal point. He was not the first to do so but his doing so played an important role in the acceptance of this form of arithmetical notation. The book also contains important developments in spherical trigonometry.

Edward Wright  (baptised 1561–1615) produced an English translation of Napier’s Descriptio, which was approved by Napier, A Description of the Admirable Table of Logarithmes, which was published posthumously in 1616 by his son Samuel.


Gresham College was quick to take up Napier’s new invention and this resulted in Henry Briggs (1561–1630), the Gresham professor of geometry, travelling to Edinburgh from London to meet with Napier. As a result of this meeting Briggs, with Napier’s active support, developed tables of base ten logarithms, Logarithmorum chilias prima, which were publish in London sometime before Napier’s death in 1617.


He published a second extended set of base ten tables, Arithmetica logarithmica, in 1624.


Napier’s own tables are often said to be Natural Logarithms, that is with Euler’s number ‘e’ as base but this is not true. The base of Napierian logarithms is given by:

NapLog(x) = –107ln (x/107)

Natural logarithms have many fathers all of whom developed them before ‘e’ itself was discovered and defined; these include the Jesuit mathematicians Gregoire de Saint-Vincent (1584–1667) and Alphonse Antonio de Sarasa (1618–1667) around 1649, and Nicholas Mercator (c. 1620–1687) in his Logarithmotechnia (1688) but John Speidell (fl. 1600–1634), had already produced a table of not quite natural logarithms in 1619.


Napier’s son, Robert, published a second work by his father on logarithms, Mirifici logarithmorum canonis constructio; et eorum ad naturales ipsorum numeros habitudines, posthumously in 1619.


This was actually written earlier than the Descriptio, and describes the principle behind the logarithms and how they were calculated.

The English mathematician Edmund Gunter (1581–1626) developed a scale or rule containing trigonometrical and logarithmic scales, which could be used with a pair of compasses to solve navigational problems.


Table of Trigonometry, from the 1728 Cyclopaedia, Volume 2 featuring a Gunter’s scale Source: Wikimedia Commons

Out of two Gunter scales laid next to each other William Oughtred (1574–1660) developed the slide rule, basically a set of portable logarithm tables for carry out calculations.

Napier developed other aids to calculation, which he published in his Rabdologiae, seu numerationis per virgulas libri duo in 1617; the most interesting of which was his so called Napier’s Bones.


These are a set of multiplication tables embedded in rods. They can be used for multiplication, division and square root extraction.


An 18th century set of Napier’s bones Source: Wikimedia Commons

Wilhelm Schickard’s calculating machine incorporated a set of cylindrical Napier’s Bones to facilitate multiplication.

The Swiss mathematician Jost Bürgi (1552–1632) produced a set of logarithm tables independently of Napier at almost the same time, which were however first published at Kepler’s urging as, Arithmetische und Geometrische Progress Tabulen…, in 1620. However, unlike Napier, Bürgi delivered no explanation of the how his table were calculated.


Tables of logarithms became the standard calculation aid for all those making mathematical calculations down to the twentieth century. These were some of the mathematical tables that Babbage wanted to produce and print mechanically with his Difference Engine. When I was at secondary school in the 1960s I still carried out all my calculations with my trusty set of log tables, pocket calculators just beginning to appear as I transitioned from school to university but still too expensive for most people.


Not my copy but this is the set of log tables that accompanied me through my school years

Later in the late 1980s at university in Germany I had, in a lecture on the history of calculating, to explain to the listening students what log tables were, as they had never seen, let alone used, them. However for more than 350 years Napier’s invention served all those, who needed to make mathematical calculations well.














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

Robert Hunter (June 23 1941–September 23 2019)

If you don’t like the Grateful Dead then don’t read this. The Grateful Dead and especially the songs of Jerry Garcia and Robert Hunter have been the soundtrack of my life for the last fifty years. Those songs have given me hope when I was down and transported me to the stars when I was up. They have accompanied me through all the up and downs, along the twisting and turning highway that has been my life, the strange diversions and dead ends. They were always there a mental bedrock to which I could cling whatever happened.

Robert Hunter was one of the truly great lyricists of the rock era, with all of the literary and high art implications that lyricist rather than simple songwriter carries. The breadth and depth of emotional colours that his words could and do magic into existence are seemingly infinite. The music and words of Garcia and Hunter are attuned to my soul in a way no other music is, was or ever will be and I own and listen to a very wide spectrum of music. Robert Hunter’s lyrics melded perfectly with Jerry Garcia’s liquid gold guitar lines.

I listen to music when I write and about eighty per cent of the time it’s the Grateful Dead. Hundred Year Hall, to which Hunter wrote some very beautiful sleeve notes, is blasting out of the stereo system, as I write these inadequate words.

I cried when I heard that Jerry Garcia had died fourteen years ago, something that surprised more than a little but which I accepted. I’m crying now having heard of the passing of Robert Hunter. I, and I suspect many others, own him an unpayable debt for all of the joy, sustenance in dark times and peace of mind that he has given me through his wonderful songs.

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Filed under Autobiographical

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

It is not an exaggeration to say that the invention of telescope was a very major turning point in the general history of science and in particular the history of astronomy. Basic science is fundamentally empirical; people investigating the world make observations with their senses–taste, sight, touch, smell, hearing–then try to develop theories to describe and explain what has been observed and recorded. The telescope was the first ever instrument that was capable of expanding or strengthening one of those senses that of sight. The telescope made it possible to see things that had never been seen before.

The road to the telescope was a long one and one of the questions is why it wasn’t invented earlier. There are various legends or myths about devices to enable people to see things at a distance throughout antiquity and various lens shaped objects also from the distant past that might or might not have been lenses. Lenses in scientific literature in antiquity and the early middle ages were burning lenses used to focus sunlight to ignite fires. The first definite use of lenses to improve eyesight were the so-called reading stones, which emerged around 1000 CE, approximately hemispherical lenses, placed on documents to help those suffering from presbyopia, weakening of the ability of the eye to focus due to aging.


Source: Zeiss

Reading glasses utilising plano-convex lenses first appeared around 1290.


The earliest pictorial evidence for the use of eyeglasses is Tommaso da Modena’s 1352 portrait of the cardinal Hugh de Provence reading in a scriptorium Source: Wikimedia Commons

The current accepted theory of the discovery of simple lenses is that in the Middle Ages monks cutting gems to decorate reliquary discovered the simple magnifying properties of the gemstones they were grinding and polishing.


Reliquary Cross, French, c. 1180 Source: Wikimedia Commons

By the middle of the fifteenth century eye glasses utilising both convex and concave lenses were being manufactured and traded, so why did it take until 1608 before somebody successfully combined a concave lens and convex lens to create a simple so-called Dutch telescope?

There are in fact earlier in the sixteenth century in the writings of Girolamo Fracastoro (ca. 1476–1553) and Giambattista della Porta (1532–1615) descriptions of the magnifying properties of such lens combinations but these are now thought to refer to special eyeglasses rather than telescopes.

Della Porta Telescope Sketch

The early lenses were spherical lenses, which were hand ground and polished and as a result were fairly inaccurate in their form tending to deviate from their ideal spherical form the further out one goes from the centre.  These deformations caused distortions in the images formed and combining lenses increased the level of distortion making such combinations next to useless. It is now thought that the breakthrough came through the use of a mask to stop down the diameter of the eyepiece lens cutting out the light rays from the periphery, restricting the image to the centre of the lens and thus massively reducing the distortion. So who made this discovery? Who first successfully developed a working telescope?

This question has been hotly discussed and various claims just as hotly disputed since at least the middle of the seventeenth century. However, there now exists a general consensus amongst historian of optics.

[To see the current stand on the subject read the bog post that I wrote at this time last year, which I don’t intend to repeat here]

Popular accounts of the early use of the telescope in astronomy almost always credit Galileo Galilei, at the time a relatively unknown professor for mathematics in Padua, with first recognising the potential of the telescope for astronomy; this is a myth.

As can be seen from the quote from the French newsletter AMBASSADES DV ROY DE SIAM ENVOYE’ A L’ECELence du Prince Maurice, arriué à la Haye le 10. Septemb.1608., recording the visit of the ambassador of the King of Siam (Thailand), who was also present at the first demonstration of the telescope the potential of this new instrument, as a tool for astronomy was recognised from the very beginning:

even the stars which normally are not visible for us, because of the scanty proportion and feeble sight of our eyes, can be seen with this instrument.

In fact the English polymath Thomas Harriot (1560–1621) made the earliest known telescopic, astronomical observations but, as with everything else he did, he didn’t publish, so outside of a small group of friends and acquaintances his work remained largely unknown. Also definitely contemporaneous with, if not earlier than, Galileo the Franconian court mathematicus, Simon Marius (1573–1625), began making telescopic observations in late 1609. However, unlike Galileo, who as we will see published his observations and discoveries as soon as possible, Marius didn’t publish until 1614, which would eventually bring the accusation of having plagiarised Galileo.  At the Collegio Romano, the Jesuit University in Rome, Odo van Maelcote (1572–1615) and Giovanni Paolo Lembo (1570–1618) were also making telescopic observations within the same time frame. There were almost certainly others, who didn’t make their observations public.

Before we turn to the observations and discoveries that these early telescopic observers made, we need to look at a serious technical problem that tends to get ignored by popular accounts of those discovery, how does a telescope work? In 1608 when the telescope first saw the light of day there existed absolutely no scientific explanation of how it worked. The group of early inventors almost certainly discovered its magnifying effect by accident and the first people to improve it and turn it into a viable scientific instrument, again almost certainly, did so by trial and error. At this point the problem is not to find the optical theory needed to develop better telescopes systematically but to find the optical theory necessary to justify the result the telescope produced. Using any sort of instrument in science requires a scientific explanation of how those results are achieved and as already stated at the beginning no such theory existed. The man, who came to the rescue, was Johannes Kepler in the second of his major contributions to the story of heliocentric astronomy.

Already in 1604 in his Ad Vitellionem Paralipomena Astronomiae pars optica, Kepler had published the first explanation of how lenses focus light rays and how eyeglasses work to compensate for short and long sightedness so he already had a head start on explaining how the telescope functions.



Francesco Maurolico (1494–1575) had covered much of the same ground in his Theoremata de lumine et umbra earlier than Kepler but this work was only published posthumously in 1611, so the priority goes to Kepler.


In 1611 Kepler published his very quickly written Dioptrice, in which he covered the path of light rays through single lenses and then through lens combinations. In this extraordinary work he covers the Dutch or Galilean telescope, convex objective–concave eyepiece, the astronomical or Keplerian telescope, convex objective–convex eyepiece, the terrestrial telescope, convex objective–convex eyepiece–convex–field–lens to invert image, and finally for good measure the telephoto lens! Galileo’s response to this masterpiece in the history of geometrical optics was that it was unreadable!


Source: Wikimedia Commons

In the next section we will turn to the discoveries that the various early telescopic astronomical observers made and the roles that those various discoveries played in the debates on, which was the correct astronomical model of the cosmos. A much more complicated affair than it is often presented.







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