Category Archives: History of Mathematics

A book or many books?

If you count mathematics as one of the sciences, and I do, then without any doubt the most often reissued science textbook of all time has to be The Elements of Euclid. As B L van der Waerden wrote in his Encyclopaedia Britannica article on Euclid:

Almost from the time of its writing and lasting almost to the present, the Elements has exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, next to the Bible, the “Elements” may be the most translated, published, and studied of all the books produced in the Western world.

The Elements have appeared in numerous editions from their inceptions, supposedly in the fourth century BCE down to the present day. In recent years, Kronecker-Wallis issued a new luxury edition of Oliver Byrne’s wonderful nineteenth century, colour coded version of the first six books of The Elements, extending it to all thirteen books.

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There are far too many different editions of this fundamental geometry textbook to be able to name them all, but this automatically raises the question, are they all the same book? If we take a random example of a book with the title The Elements of Euclid, will we always find the same content between the covers? The simple answer to this question is no. The name of the author, Euclid, and the title of the book, The Elements, are much more a mantle into which, over a period of more than two thousand years, related but varying geometrical content has been poured to fit a particular time or function, never quite the same. Sometimes with minor variations sometimes major ones. The ever-changing nature of this model of mathematical literature is the subject of Benjamin Wardhaugh’s fascination volume, The Book of WonderThe Many Lives of Euclid’s Elements.[1]

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To write a detailed, complete, chronological history of The Elements, would probably produce something with the dimensions of James Frazer’s twelve volume The Golden Bough and Wardhaugh doesn’t attempt the task here. What he does do is to deliver a selective series of episodes out of the long and complex life of the book. These episodes rather than book chapters might best be described, as essays or even short stories. In total they sum up to a comprehensive, but by no means complete, overview of this fascinating mathematical tome. Wardhaugh’s essay collection is split up into four section, each of which takes a different approach to examining and presenting the history of Euclid’s opus magnum. 

The first section opens with Euclid’s Alexandria, the geometry of the period and the man himself. It clearly shows how little we actually know about the origins of this extraordinary book and its purported author. The following essays deliver a sketch of the history of the book itself. We move from the earliest surviving fragments over the first known complete manuscript from Theon in the fourth century CE. We meet The Elements in Byzantium, in Arabic, in Latin and for the first time in print. 

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In the latter case I tripped over the only seriously questionable historical claim that I was aware of in the book. Wardhaugh repeats the nineteenth century claim that Erhard Ratdolt, the printer/publisher of that first printed edition, had been apprenticed to Regiomontanus. This claim is based on the fact that Ratdolt printed and published various manuscripts that had previously belonged to Regiomontanus, including the Euclid. However, there is absolutely no other evidence to support this claim. Regiomontanus was famous throughout Europe both as a mathematicus and as a printer/publisher, people were publishing books, which weren’t from him, more than one hundred years after his death, under his name. If Ratdolt had indeed learnt the printing trade from Regiomontanus he would, with certainty, have advertised the fact, he didn’t.

The first section closes with the flood of new editions that Ratdolt’s first printed edition unleashed in the Early Modern Period. 

The second section deals with the various philosophical interpretations to which The Elements were subjected over the centuries. We start with Plato, who supposedly posted the phrase, “Let no man ignorant of geometry enter” over the entrance to his school. Up next is Proclus, whose fifth century CE commentary on The Elements was the first source that names Euclid as the author. We then have one of Wardhaugh’s strengths as a Euclid chronicler, in his book he digs out a series of women, who over the centuries have in some way engaged with The Elements; here we get the nun Hroswitha (d. c. 1000CE), whose play Sapientia included sections of Euclidian number theory. Following Levi ben Gershon and his Hebrew Euclid, we get a section that particularly appealed to me. First off Christoph Clavius’ Elements, possibly the most extensively rewritten version of the book and one of the most important seventeenth century maths textbooks. This is followed by the Chinese translation of the first six books of Clavius’ Elements by Matteo Ricci and Xu Guangqi.

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The second continues with an English stage play on geometry written for the carnival in Rome in 1635. Wardhaugh’s Euclidean research has dug very deep. Baruch Spinoza famously wrote a book on ethics in the style of Euclid’s Elements and of course it’s included here. The section closes with another woman, this time the nineteenth century landowner, Anne Lister.

The third section of the book deals with applied geometry. We start with ancient Egyptian surveyors, move onto music theory and the monochord, Roman field surveyors and the Arabic mathematician Muhammad abu al-Wafa al-Buzjani, who work on the theory of dividing up surfaces for the artisans to create those wonderful geometrical patterns so typical of Islamic ornamentation. Up next are medieval representations of the muse Geometria, which is followed by Piero della Francesca and the geometry of linear perspective. There is a brief interlude with the splendidly named seventeenth century maths teacher, Euclid Speidel before the section closes with Isaac Newton. 

The fourth section of the book traces the decline of The Elements as a textbook in the nineteenth century. We start with another woman, Mary Fairfax, later Mary Sommerville, and her battles with her parents to be allowed to read Euclid. We travel to France and François Peyrard’s attempts to create, as far as possible, a new definitive text for the Elements. Of course, Nicolai Ivanovich Lobachevsky and the beginnings of non-Euclidian geometry have to put in an appearance. Up next George Eliot’s The Mill on the Floss is brought in to illustrate the stupefying nature of Euclidian geometry teaching in English schools in the nineteenth century. We move on to teaching Euclid in Urdu in Uttar Pradesh. A survey of the decline of Euclid in the nineteenth century would no be complete without Lewis Carroll’s wonderful drama Euclid and his Modern Rivals. Carroll is followed by, in his time, one of the greatest historians of Greek mathematics, Thomas Little Heath, whose superb three volume English edition of The Elements has graced my bookshelf for several decades.

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The book closes with an excursion into the arts. Max Ernst’s Euclid’s Mask morphs into a chapter on Euclidean design, including Oliver Byrne’s colour coded Elements, mentioned earlier. The final chapter is some musing on the iconic status of Euclid and his book.

There are no foot or endnotes and the book contains something that I regard as rather inadequate. Notes on Sources, which for every chapter gives a short partially annotated reading list. Not, in my opinion the most helpful of tools. There is an extensive bibliography and a good index. The book is illustrated with the now standard grey in grey prints.

Benjamin Wardhaugh is an excellent storyteller and his collected short story approach to the history of The Elements works splendidly. He traces a series of paths through the highways and byways of the history of this extraordinary mathematics book that is simultaneously educational, entertaining and illuminating. In my opinion a highly desirable read for all those, both professional and amateur, who interest themselves for the histories of mathematics, science and knowledge or the course of mostly European intellectual history over almost two and a half millennia.  


[1] Benjamin Wardhaugh, The Book of WonderThe Many Lives of Euclid’s Elements, William Collins, London, 2020

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Filed under Book Reviews, History of Mathematics

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

The Moon is the Earth’s nearest celestial neighbour and the most prominent object in the night sky. People have been tracking, observing and recording the movements of the Moon for thousands of years, so one could assume that calculating its orbit around the Earth should be a reasonable simple matter, however in reality it is anything but.

The problem can be found in the law of gravity itself, which states that any two bodies mutually attract each other. However, that attraction is not restricted to just those two bodies but all bodies attract each other simultaneously. Given the relative masses of somebody standing next to you and the Earth, when calculating the pull of gravity on you, we can, in our calculation, neglect the pull exercised by the mass of your neighbour. With planets, however, it is more difficult to ignore multiple sources of gravitational force. We briefly touched on the gravitational effect of Jupiter and Saturn, both comparatively large masses, on the flight paths of comets, so called perturbation. In fact when calculating the Earth orbit around the Sun then the effects of those giant planets, whilst relatively small, are in fact detectable.

With the Moon the problem is greatly exacerbated. The gravitation attraction between the Earth and the Moon is the primary force that has to be considered but the not inconsiderable gravitational attraction between the Sun and the Moon also plays an anything but insignificant role. The result is that the Moon’s orbit around the Sun Earth is not the smooth ellipse of Kepler’s planetary laws that it would be if the two bodies existed in isolation but a weird, apparently highly irregular, dance through the heavens as the Moon is pulled hither and thither between the Earth and the Sun.

Kepler in fact did not try to apply his laws of planetary motion to the Moon simply leaving it out of his considerations. The first person to apply the Keplerian elliptical astronomy to the Moon was Jeremiah Horrocks (1618–1641), an early-convinced Keplerian, who was also the first person to observe a transit of Venus having recalculated Kepler’s Rudolphine Tables in order to predict to correct date of the occurrence. Horrocks produced a theory of the Moon based on Kepler’s work, which was far and away the best approximation to the Moon’s orbit that had been produced up till that time but was still highly deficient. This was the model that Newton began his work with as he tried to make the Moon’s orbit fit into his grand gravitational theory, as defined by his three laws of motion, Kepler’s three laws of planetary motion and the inverse square law of gravity; this would turn into something of a nightmare for Newton and cause a massive rift between Newton and John Flamsteed the Astronomer Royal.

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Portrait of Newton at 46 by Godfrey Kneller, 1689 Source: Wikimedia Commons

What Newton was faced with was attempting to solve the three-body problem, that is a general solution for the mutual gravitational attraction of three bodies in space. What Newton did not and could not know was that the general analytical solution simple doesn’t exist, the proof of this lay in the distant future. The best one can hope for are partial local solutions based on approximations and this was the approach that Newton set out to use. The deviations of the Moon, perturbations, from the smooth elliptical orbit that it would have if only it and the Earth were involved are not as irregular as they at first appear but follow a complex pattern; Newton set out to pick them off one by one. In order to do so he need the most accurate data available, which meant new measurement made during new observations by John Flamsteed the Astronomer Royal.

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Source: Wikimedia Commons

For Newton solving the lunar orbit was the most pressing problem in his life and he imperiously demanded that Flamsteed supply him with the data that he required to make his calculations. For Flamsteed the important task in his life, as an observational astronomer, was to complete a new star catalogue on a level of observational accuracy hitherto unknown. The principle interests of the two men were thus largely incompatible. Newton demanded that Flamsteed use his time to supply him with his lunar data and Flamsteed desired to use his time to work on his star catalogue, although to be fair he did supply Newton, if somewhat grudgingly with the desired data. As Newton became more and more frustrated by the problems he was trying to solve the tone of his missives to Flamsteed in Greenwich became more and more imperious and Flamsteed got more and more frustrated at being treated like a lackey by the Lucasian Professor. The relations between the two degenerated rapidly.

The situation was exacerbated by the presence of Edmond Halley in the mix, as Newton’s chief supporter. Halley had started his illustrious career as a protégée of Flamsteed’s when he, still an undergraduate, sailed to the island of Saint Helena to make a rapid survey of the southern night skies for English navigators. The men enjoyed good relations often observing together and with Halley even deputising for Flamsteed at Greenwich when he was indisposed. However something happened around 1686 and Flamsteed began to reject Halley. It reached a point where Flamsteed, who was deeply religious with a puritan streak, disparaged Halley as a drunkard and a heathen. He stopped referring him by name calling him instead Reymers, a reference to the astronomer Nicolaus Reimers Ursus (1551–1600). Flamsteed was a glowing fan of Tycho Brahe and he believed Tycho’s accusation that Ursus plagiarised Tycho’s system. So Reymers was in his opinion a highly insulting label.

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

Newton only succeeded in resolving about half of the irregularities in the Moon’s orbit and blamed his failure on Flamsteed. This led to one of the most bizarre episodes in the history of astronomy. In 1704 Newton was elected President of the Royal Society and one of his first acts was to call Flamsteed to account. He demanded to know what Flamsteed had achieved in the twenty-nine years that he had been Astronomer Royal and when he intended to make the results of his researches public. Flamsteed was also aware of the fact that he had nothing to show for nearly thirty years of labours and was negotiating with Prince George of Denmark, Queen Anne’s consort, to get him to sponsor the publication of his star catalogue. Independently of Flamsteed, Newton was also negotiating with Prince George for the same reason and as he was now Europe’s most famous scientist he won this round. George agreed to finance the publication, and was, as a reward, elected a member of the Royal Society.

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Prince George of Denmark and Norway, Duke of Cumberland Portrait by Michael Dahl c. 1705 Source: Wikimedia Commons

Newton set up a committee, at the Royal Society, to supervise the work with himself as chairman and the Savilian Professors of Mathematics and Astronomy, David Gregory and Edmond Halley, both of whom Flamsteed regarded as his enemies, Francis Robartes an MP and teller at the Exchequer and Dr John Arbuthnotmathematician, satirist and physician extraordinary to Queen Anne. Although Arbuthnot, a Tory, was of opposing political views to Newton, a Whig, he was a close friend and confidant. Flamsteed was not offered a place on this committee, which was decidedly stacked against him.

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David Gregory Source: Wikimedia Commons

Flamsteed’s view on what he wanted published and how it was to be organised and Newton’s views on the topic were at odds from the very beginning. Flamsteed saw his star catalogue as the centrepiece of a multi-volume publication, whereas all that really interested Newton was his data on the planetary and Moon orbits, with which he hoped to rectify his deficient lunar theory. What ensued was a guerrilla war of attrition with Flamsteed sniping at the referees and Newton and the referees squashing nearly all of Flamsteed wishes and proposals. At one point Newton even had Flamsteed ejected from the Royal Society for non-payment of his membership fees, although he was by no means the only member in arrears. Progress was painfully slow and at times virtually non-existent till it finally ground completely to a halt with the death of Prince George in 1708.

George’s death led to a two-year ceasefire in which Newton and Flamsteed did not communicate but Flamsteed took the time to work on the version of his star catalogue that he wanted to see published. Then in 1710 John Arbuthnot appeared at the council of the Royal society with a royal warrant from Queen Anne appointing the president of the society and anybody the council chose to deputise ‘constant Visitors’ to the Royal Observatory at Greenwich. ‘Visitor’ here means supervisor in the legal sense. Flamsteed’s goose was well and truly cooked. He was now officially answerable to Newton. Instead of waiting for Flamsteed to finish his star catalogue the Royal Society produced and published one in the form that Newton wanted and edited by Edmond Halley, the man Flamsteed regarded as his greatest enemy. It appeared in 1712. In 1713 Newton published the second edition of his Principia with its still defective lunar theory but with Flamsteed name eliminated as far as possible.

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John Arbuthnot Portrait by Godfrey Kneller Source: Wikimedia Commons

The farce did not end here. In 1714 Queen Anne died and the Visitor warrant thus lost its validity. The Tory government fell and the Whigs regained power. Newton’s political sponsor, Charles Montagu, 1st Earl of Halifax, died in 1715 leaving him without a voice in the new government. Flamsteed, however, was friends with the Lord Chamberlain, Lord Boulton. On 30 November 1715 Boulton signed a warrant ordering Newton and co to hand over the remaining 300 copies of their ‘pirate’ catalogue to Flamsteed.  After some procrastination and some more insults aimed at Flamsteed they finally complied on 28 March 1716. Flamsteed “made a Sacrifice of them to Heavenly truth”, that is he burnt them. Flamsteed had in the mean time published his star catalogue at his own expense and devoted the rest of his life to preparing the rest of his life’s work for publication. He died in 1719 but his widow, Margaret, and two of his former assistants, Joseph Crosthwait and Abraham Sharp, edited and published his Historia coelestis britannia in three volumes in 1725; it is rightly regarded as a classic in the history of celestial observation. Margaret also took her revenge on Halley, who succeeded Flamsteed as Astronomer Royal. Flamsteed had paid for the instruments in the observatory at Greenwich out of his own pocket, so she stripped the building bare leaving Halley with an empty observatory without instruments. For once in his life Newton lost a confrontation with a scientific colleague, of which there were quite a few, game, set and match

The bitter and in the end unseemly dispute between Newton and Flamsteed did nothing to help Newton with his lunar theory problem and to bring his description of the Moon’s orbit into line with the law of gravity. In the end this discrepancy in the Principia remained beyond Newton’s death. Mathematicians and astronomers in the eighteen century were well aware of this unsightly defect in Newton’s work and in the 1740s Leonhard Euler (1707­–1783), Alexis Clairaut (1713–1765) and Jean d’Alembert (1717–1783) all took up the problem and tried to solve it, in competition with each other.  For a time all three of them thought that they would have to replace the inverse square law of gravity, thinking that the problem lay there. Clairaut even went so far as to announce to the Paris Academy on 15 November 1747 that the law of gravity was false, to the joy of the Cartesian astronomers. Having then found a way of calculating the lunar irregularities using approximations and confirming the inverse square law, Clairaut had to retract his own announcement. Although they had not found a solution to the three-body problem the three mathematicians had succeeded in bringing the orbit of the Moon into line with the law of gravity. The first complete, consistent presentation of a Newtonian theory of the cosmos was presented by Pierre-Simon Laplace in his Traité de mécanique céleste, 5 Vol., Paris 1798–1825.

Mathematicians and astronomers were still not happy with the lack of a general solution to the three-body problem, so in 1887 Oscar II, the King of Sweden, advised by Gösta Mittag-Leffler offered a prize for the solution of the more general n-body problem.

Given a system of arbitrarily many mass points that attract each according to Newton’s law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converge uniformly.

Nobody succeeded in solving the challenge but Henri Poincaré’s attempt to find a solution although not successful, contained enough promising leads that he was awarded the prize. As stated a solution to the problem was found for three bodies by Karl F Sundman in 1912 and generalised for more than three bodies by Quidong Wang in the 1990s.

The whole episode of Newton’s failed attempt to find a lunar theory consonant with his theory of gravitation demonstrates that even the greatest of mathematicians can’t solve everything. It also demonstrates that the greatest of mathematicians can behave like small children having a temper tantrum if they don’t get their own way.

 

 

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Filed under History of Astrology, History of Mathematics, History of Physics, Newton

A scientific Dutchman

For many decades the popular narrative version of the scientific revolution started in Poland/Germany with Copernicus moving on through Tycho in Denmark, Kepler in Germany/Austria, Galileo et al in Northern Italy, Descartes, Pascal, Mersenne etc., in France and then Newton and his supporters and opponents in London. The Netherlands simply didn’t get a look in except for Christiaan Huygens, who was treated as a sort of honorary Frenchman. As I’ve tried to show over the years the Netherlands and its scholars–Gemma Frisius, Simon Stephen, Isaac Beeckman, the Snels, and the cartographers–actually played a central role in the evolution of the sciences during the Early Modern Period. In more recent years efforts have been made to increase the historical coverage of the contributions made in the Netherlands, a prominent example being Harold J Cook’s Matters of Exchange: Commerce, Medicine and Science in the Dutch Golden Age.[1]

A very strange anomaly in the #histSTM coverage concerns Christiaan Huygens, who without doubt belongs to the seventeenth century scientific elite. Whereas my bookcase has an entire row of Newton biographies, and another row of Galileo biographies and in both cases there are others that I’ve read but don’t own. The Kepler collection is somewhat smaller but it is still a collection. I have no idea how many Descartes biographies exist but it is quite a large number. But for Christiaan Huygens there is almost nothing available in English. The only biography I’m aware of is the English translation of Cornelis Dirk Andriesse’s scientific biography of Christiaan Huygens, The Man Behind the Principle.[2] I read this several years ago and must admit I found it somewhat lacking. This being the case, great expectation have been raised by the announcement of a new Huygens biography by Hugh Aldersey-Williams, Dutch Light: Christiaan Huygens and the Making of Science in Europe.[3]

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So does Aldersey-Williams fulfil those expectations? Does he deliver the goods? Yes and no, on the whole he has researched and written what is mostly an excellent biography of the Netherland’s greatest scientist[4] of the Early Modern Period but it is in my opinion marred by sloppy history of science fact checking that probably won’t be noticed by the average reader but being the notorious #histSTM pedant that I am I simply can’t and won’t ignore.[5]

My regular readers will known that I describe myself as a narrative contextual historian of science and I personally believe that if we are to understand how science has evolved historical then we have to tell that story with its complete context. This being the case I’m very happy to report that Aldersey-Williams is very much a narrative contextual historian, who tells the complete story of Christiaan Huygens life within its wider context and not just offering up a list of his scientific achievements. In fact what the reader gets for his money is not just a biography of Christiaan but also a biography of his entire family with some members being given more space than other. In particular it is a full biography of Christiaan and his father Constantijn, who played a significant and central role in shaping Christiaan’s life.

The book opens by setting the scientific scene in the early seventeenth-century Netherlands. We get introduced to those scientists, who laid the scientific foundations on which Christiaan would later build. In particular we get introduced to Simon Steven, who shaped the very practice orientated science and technology of the Early Modern Netherlands. We also meet other important and influential figures such as Hans Lipperhey, Isaac Beeckman, Willebrord Snel, Cornelius Drebbel and others.

There now follows what might be termed a book within a book as Aldersey-Williams delivers up a very comprehensive biography of Constantijn Huygens diplomat, poet, composer, art lover and patron and all round lover of knowledge. Constantijn was interested in and fascinated by almost everything both scientific and technological. His interest was never superficial but was both theoretical and practical. For example he was not only interested in the newly invented instruments, the telescope and the microscope, but he also took instruction in how to grind lenses and that from the best in the business. Likewise his love for art extended beyond buying paintings and patronising artists, such as Rembrandt, but to developing his own skills in drawing and painting. Here Aldersey-Williams introduces us to the Dutch term ‘kenner’ (which is the same in German), which refers to someone such Constantijn Huygens, whose knowledge of a subject is both theoretical and practical. Constantijn Huygens married Suzanna von Baerle for love and they had five children over ten years, four sons and a daughter, Christiaan was the second oldest, and Suzanna died giving birth to their daughter, also named Suzanna.

Constantijn Huygens brought up his children himself educating them in his own polymathic diversity with the help of tutors. When older the boys spent brief periods at various universities but were largely home educated. We now follow the young Christiaan and his older brother, also Constantijn, through their formative young years. The two oldest boys remained close and much of Christiaan’s astronomical work was carried out in tandem with his older brother. We follow Christiaan’s early mathematical work and his introduction into the intellectual circles of Europe, especially France and England, through his father’s widespread network of acquaintances. From the beginning Christiaan was set up to become either a diplomat, like his father, grandfather and brothers, or a scientist and it is the latter course that he followed.

Aldersey-Williams devotes an entire chapter to Christiaan’s telescopic observations of Saturn, with a telescope that he and Constantijn the younger constructed and his reputation making discovery of Titan the largest of Saturn’s moons, and the first discovered, and his determination that the strange shapes first observed by Galileo around Saturn were in fact rings. These astronomical discoveries established him as one of Europe’s leading astronomers. The following chapter deals with Huygens’ invention of the pendulum clock and his excursions into the then comparatively new probability theory.

Saturn and the pendulum clock established the still comparatively young Huygens as a leading light in European science in the second half of the seventeenth century and Aldersey-Williams now takes us through ups and downs of the rest of Christiaan’s life. His contact with and election to the Royal Society in London, as its first foreign member. His appointment by Jean-Baptist Colbert, the French First Minister of State, as a founding member of the Académie des sciences with a fairy generous royal pension from Louis XIV. His sixteen years in Paris, until the death of Colbert, during which he was generally acknowledged as Europe’s leading natural philosopher. His initial dispute over light with the young and comparatively unknown Newton and his tutorship of the equally young and unknown Leibniz. His fall from grace following Colbert’s death and his reluctant return to the Netherlands. The last lonely decade of his life in the Netherlands and his desire for a return to the scientific bustle of London or Paris. His partial rapprochement with Newton following the publication of the Principia. Closing with the posthumous publication of his works on gravity and optics. This narrative is interwoven with episodes from the lives of Constantijn the father and Constantijn his elder brother, in particular the convoluted politics of the Netherlands and England created by William of Orange, whose secretary was Constantijn, the younger, taking the English throne together with his wife Mary Stewart. Christiaan’s other siblings also make occasional appearances in letters and in person.

Aldersey-Williams has written a monumental biography of two generations of the Huygens family, who played major roles in the culture, politics and science of seventeenth century Europe. With a light, excellent narrative style the book is a pleasure to read. It is illustrated with 37 small grey in grey prints and 35 colour plates, which I can’t comment on, as my review proof copy doesn’t contain them. There are informative footnotes scattered through out the text and the, by me hated, hanging endnotes referring to the sources of direct quotes in the text. Here I had the experience more than once of looking up what I took to be a direct quote only to discover that it was not listed. There is an extensive bibliography of both primary and secondary sources and I assume an extensive index given the number of blank pages in my proof copy. There were several times when I was reading when I had wished that the index were actually there.

On the whole I would be tempted to give this book a glowing recommendation were it not for a series of specific history of science errors that simple shouldn’t be there and some general tendencies that I will now detail.

Near the beginning Aldersey-Williams tells us that ‘Stevin’s recommendation to use decimals in arithmetical calculations in place of vulgar fractions which could have any denominator [was] surely the sand-yacht of accountancy … Thirty years later, the Scottish mathematician John Napier streamlined Stevin’s notation by introducing the familiar comma or point to separate off the fractional part…” As is all too often the case no mention is made of the fact that Chinese and Arabic mathematicians had been using decimal fractions literally centuries before Stevin came up with the concept. In my opinion we must get away from this Eurocentric presentation of the history of science. Also the Jesuit mathematician Christoph Clavius introduced the decimal point less than ten years after Stevin’s introduction of decimal fractions, well ahead of Napier, as was its use by Pitiscus in 1608, the probable source of Napier’s use.

We also get told when discussing the Dutch vocabulary that Stevin created for science that, “Chemistry becomes scheikunde, the art of separation, an acknowledgement of the beginnings of a shift towards an analytical science, and a useful alternative to chemie that severs the etymological connections with disreputable alchemy.” This displays a complete lack of knowledge of alchemy in which virtually all the analytical methods used in chemistry were developed. The art of separation is a perfectly good term from the alchemy that existed when Stevin was creating his Dutch scientific vocabulary. Throughout his book Aldersey-Williams makes disparaging remarks about both alchemy and astrology, neither of which was practiced by any of the Huygens family, which make very clear that he doesn’t actually know very much about either discipline or the role that they played in the evolution of western science, astrology right down to the time of Huygens and Newton and alchemy well into the eighteenth century. For example, the phlogiston theory one of the most productive chemical theories in the eighteenth century had deep roots in alchemy.

Aldersey-Williams account of the origins of the telescope is a bit mangled but acceptable except for the following: “By the following spring, spyglasses were on sale in Paris, from where one was taken to Galileo in Padua. He tweaked the design, claimed the invention as his own, and made dozens of prototypes, passing on his rejects so that very soon even more people were made aware of this instrument capable of bringing the distant close.”

Firstly Galileo claimed that he devised the principle of the telescope and constructed his own purely on verbal descriptions without having actually seen one but purely on his knowledge of optics. He never claimed the invention as his own and the following sentence is pure rubbish. Galileo and his instrument maker produced rather limited numbers of comparatively high quality telescopes that he then presented as gifts to prominent political and Church figures.

Next up we have Willebrord Snel’s use of triangulation. Aldersey-Williams tells us, “ This was the first practical survey of a significant area of land, and it soon inspired similar exercises in England, Italy and France.” It wasn’t. Mercator had previously surveyed the Duchy of Lorraine and Tycho Brahe his island of Hven before Snel began his surveying in the Netherlands. This is however not the worst, Aldersey-Williams tells us correctly that Snel’s survey stretched from Alkmaar to Bergen-op-Zoom “nearly 150 kilometres to the south along approximately the same meridian.” Then comes some incredible rubbish, “By comparing the apparent height of his survey poles observed at distance with their known height, he was able to estimate the size of the Earth!”

What Snel actually did, was having first accurately determined the length of a stretch of his meridian using triangulation, the purpose of his survey and not cartography, he determined astronomically the latitude of the end points. Having calculated the difference in latitudes it is then a fairly simple exercise to determine the length of one degree of latitude, although for a truly accurate determination one has to adjust for the curvature of the Earth.

Next up with have the obligatory Leonard reference. Why do pop history of science books always have a, usually erroneous, Leonardo reference? Here we are concerned with the camera obscura, Aldersey-Williams writes: “…Leonardo da Vinci gave one of the first accurate descriptions of such a design.” Ibn al-Haytham gave accurate descriptions of the camera obscura and its use as a scientific instrument about four hundred and fifty years before Leonardo was born in a book that was translated into Latin two hundred and fifty years before Leonardo’s birth. Add to this the fact that Leonardo’s description of the camera obscura was first published late in the eighteenth century and mentioning Leonardo in this context becomes a historical irrelevance. The first published European illustration of a camera obscura was Gemma Frisius in 1545.

When discussing Descartes, a friend of Constantijn senior and that principle natural philosophical influence on Christiaan we get a classic history of mathematics failure. Aldersey-Williams tells us, “His best known innovation, of what are now called Cartesian coordinates…” Whilst Descartes did indeed cofound, with Pierre Fermat, modern algebraic analytical geometry, Cartesian coordinates were first introduced by Frans van Schooten junior, who of course features strongly in the book as Christiaan’s mathematics teacher.

Along the same lines as the inaccurate camera obscura information we have the following gem, “When applied to a bisected circle (a special case of the ellipse), this yielded a new value, accurate to nine decimal places, of the mathematical constant π, which had not been improved since Archimedes” [my emphasis] There is a whole history of the improvements in the calculation of π between Archimedes and Huygens but there is one specific example that is, within the context of this book, extremely embarrassing.

Early on when dealing with Simon Stevin, Aldersey-Williams mentions that Stevin set up a school for engineering, at the request of Maurits of Nassau, at the University of Leiden in 1600. The first professor of mathematics at this institution was Ludolph van Ceulen (1540–1610), who also taught fencing, a fact that I find fascinating. Ludolph van Ceulen is famous in the history of mathematics for the fact that his greatest mathematical achievement, the Ludophine number, is inscribed on his tombstone, the accurate calculation of π to thirty-five decimal places, 3.14159265358979323846264338327950288…

Next up we have Christiaan’s correction of Descartes laws of collision. Here Aldersey-Williams writes something that is totally baffling, “The work [his new theory of collision] only appeared in a paper in the French Journal des Sçavans in 1669, a few years after Newton’s laws of motion [my emphasis]…” Newton’s laws of motion were first published in his Principia in 1687!

Having had the obligatory Leonardo reference we now have the obligatory erroneous Galileo mathematics and the laws of nature reference, “Galileo was the first to fully understand that mathematics could be used to describe certain laws of nature…” I’ve written so much on this that I’ll just say here, no he wasn’t! You can read about Robert Grosseteste’s statement of the role of mathematics in laws of nature already in the thirteenth century, here.

Writing about Christiaan’s solution of the puzzle of Saturn’s rings, Aldersey-Williams say, “Many theories had been advanced in the few years since telescopes had revealed the planet’s strange truth.” The almost five decades between Galileo’s first observation of the rings and Christiaan’s solution of the riddle is I think more than a few years.

Moving on Aldersey-Williams tells us that, “For many however, there remained powerful reasons to reject Huygens’ discovery. First of all, it challenged the accepted idea inherited from Greek philosophers that the solar system consisted exclusively of perfect spherical bodies occupying ideal circular orbits to one another.” You would have been hard put to it to find a serious astronomer ín 1660, who still ascribed to this Aristotelian cosmology.

The next historical glitch concerns, once again, Galileo. We read, “He dedicated the work [Systema Saturnium] to Prince Leopoldo de’ Medici, who was patron of the Accademia del Cimento in Florence, who had supported the work of Huygens’ most illustrious forebear, Galileo.” Ignoring the sycophantic description of Galileo, one should perhaps point out that the Accademia del Cimento was founded in 1657 that is fifteen years after Galileo’s death and so did not support his work. It was in fact founded by a group of Galileo’s disciples and was dedicated to continuing to work in his style, not quite the same thing.

Galileo crops up again, “the real power of Huygens’ interpretation was its ability to explain those times when Saturn’s ‘handles’ simply disappeared from view, as they had done in 1642, finally defeating the aged Galileo’s attempts to understand the planet…” In 1642, the year of his death, Galileo had been completely blind for four years and had actually given up his interest in astronomy several years earlier.

Moving on to Christiaan’s invention of the pendulum clock and the problem of determining longitude Aldersey-Williams tells us, “The Alkmaar surveyor Adriaan Metius, brother of the telescope pioneer Jacob, had proposed as long ago as 1614 that some sort of seagoing clock might provide the solution to this perennial problem of navigators…” I feel honour bound to point out that Adriaan Metius was slightly more than simply a surveyor, he was professor for mathematics at the University of Franeker. However the real problem here is that the clock solution to the problem of longitude was first proposed by Gemma Frisius in an appendix added in 1530, to his highly popular and widely read editions of Peter Apian’s Cosmographia. The book was the biggest selling and most widely read textbook on practical mathematics throughout the sixteenth and well into the seventeenth century so Huygens would probably have known of Frisius’ priority.

Having dealt with the factual #histSTM errors I will now turn to more general criticisms. On several occasions Aldersey-Williams, whilst acknowledging problems with using the concept in the seventeenth century, tries to present Huygens as the first ‘professional scientist’. Unfortunately, I personally can’t see that Huygens was in anyway more or less of a professional scientist than Tycho, Kepler or Galileo, for example, or quite a long list of others I could name. He also wants to sell him as the ‘first ever’ state’s scientist following his appointment to the Académie des sciences and the accompanying state pension from the king. Once again the term is equally applicable to Tycho first in Denmark and then, if you consider the Holy Roman Empire a state, again in Prague as Imperial Mathematicus, a post that Kepler inherited. Galileo was state ‘scientist’ under the de’ Medici in the Republic of Florence. One could even argue that Nicolas Kratzer was a state scientist when he was appointed to the English court under Henry VIII. There are other examples.

Aldersey-Williams’ next attempt to define Huygens’ status as a scientist left me somewhat speechless, “Yet it is surely enough that Huygens be remembered for what he was, a mere problem solver indeed: pragmatic, eclectic and synthetic and ready to settle for the most probable rather than hold out for the absolutely certain – in other words. What we expect a scientist to be today.” My ten years as a history and philosophy of science student want to scream, “Is that what we really expect?” I’m not even going to go there, as I would need a new blog post even longer than this one.

Aldersey-Williams also tries to present Huygens as some sort of new trans European savant of a type that had not previously existed. Signifying cooperation across borders, beliefs and politics. This is of course rubbish. The sort of trans European cooperation that Huygens was involved in was just as prevalent at the beginning of the seventeenth century in the era of Tycho, Kepler, Galileo, et al. Even then it was not new it was also very strong during the Renaissance with natural philosophers and mathematici corresponding, cooperating, visiting each other, and teaching at universities through out the whole of Europe. Even in the Renaissance, science in Europe knew no borders. It’s the origin of the concept, The Republic of Letters. I suspect my history of medieval science friend would say the same about their period.

In the partial rapprochement between Huygens and Newton following the Publication of the latter’s Principia leads Aldersey-Williams to claim that a new general level of reasonable discussion had entered scientific debate towards the end of the seventeenth century. Scientists, above all Newton, were still going at each other hammer and tongs in the eighteenth century, so it was all just a pipe dream.

Aldersey-Williams sees Huygens lack of public profile, as a result of being in Newton’s shadow like Hooke and others. He suggests that popular perception only allows for one scientific genius in a generation citing Galileo’s ascendance over Kepler, who he correctly sees as the more important, as another example. In this, I agree with him, however he tries too hard to put Huygens on the same level as Newton as a scientist, as if scientific achievement were a pissing contest. I think we should consider a much wider range of scientists when viewing the history of science but I also seriously think that no matter how great his contributions Huygens can’t really match up with Newton. Although his Horologium oscillatorium sive de motu pendularium was a very important contribution to the debate on force and motion, it can’t be compared to Newton’s Principia. Even if Huygens did propagate a wave theory of light his Traité de la lumière is not on a level with Newton’s Opticks. He does have his Systema saturniumbut as far as telescopes are concerned Newton’s reflector was a more important contribution than any of Huygens refractor telescopes. Most significant, Newton made massive contributions to the development of mathematics, Huygens almost nothing.

Talking of Newton, in his discussion of Huygens rather heterodox religious views Aldersey-Williams discussing unorthodox religious views of other leading scientists makes the following comment, “Newton was an antitrinitarian, for which he was considered a heretic in his lifetime, as well as being interested in occultism and alchemy.” Newton was not considered a heretic in his lifetime because he kept his antitrinitarian views to himself. Alchemy yes, but occultism, Newton?

I do have one final general criticism of Aldersey-Williams’ book. My impression was that the passages on fine art, poetry and music, all very important aspects of the life of the Huygens family, are dealt with in much greater depth and detail than the science, which I found more than somewhat peculiar in a book with the subtitle, The Making of Science in Europe. I’m not suggesting that the fine art, poetry and music coverage should be less but that the science content should have been brought up to the same level.

Despite the long list of negative comments in my review I think this is basically a very good book that could in fact have been an excellent book with some changes. Summa summarum it is a flawed masterpiece. It is an absolute must read for anybody interested in the life of Christiaan Huygens or his father Constantijn or the whole Huygens clan. It is also an important read for those interested in Dutch culture and politics in the seventeenth century and for all those interested in the history of European science in the same period. It would be desirable if more works with the wide-ranging scope and vision of Aldersey-Williams volume were written but please without the #histSTM errors.

[1] Harold J Cook, Matters of Exchange: Commerce, Medicine and Science in the Dutch Golden Age, Yale University Press, New Haven & London, 2007

[2] Cornelis Dirk Andriesse, The Man Behind the Principle, scientific biography of Christiaan Huygens, translated from Dutch by Sally Miedem, CUP, Cambridge, 2005

[3] Hugh Aldersey-Williams, Dutch Light: Christiaan Huygens and the Making of Science in Europe, Picador, London, 2020.

[4] Aldersey-Williams admits that the use of the term scientist is anachronistic but uses it for simplicity’s sake and I shall do likewise here.

[5] I have after all a reputation to uphold

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Filed under Book Reviews, History of Astronomy, History of Mathematics, History of Navigation, History of Optics, History of Physics, History of science, Newton

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

Why wasn’t Newton’s Principia the end of the gradual emergence and acceptance of a heliocentric astronomical model for the then known cosmos? There is not one simple answer to this question, but a serious of problems created in different areas all of which had still to be addressed if there was going to be an unquestioned acceptance of heliocentricity. Some of those problems were inherent in the Principia itself, which should best be viewed as a work in progress rather than a finished concept. In fact, as we will see, Newton carried on working on improving the Principia over two further editions, expanding and correcting the first edition. Other problems arose in the philosophical rejection of key aspects of Newton’s work by highly influential and knowledgeable detractors. Finally there were still massive unsolved empirical problems outside of the scope of the Principia itself. These sets of problems run chronologically parallel to each other some of them all the way into the nineteenth century and beyond so in dealing with them I will take each one in turn following it to its conclusion and then return to the starting point for the next problem but first I will sketch in a little bit more detail the problems listed above.

To begin with we need to look at the reception of the Principia when it was first published. On a very general level that reception can be viewed as very positive. Firstly there were only a comparatively small number of experts qualified to judge the Principia, as the work is highly technical and complex. There is a famous anecdote of two men observing Newton walking in the gardens of Trinity College and one says to the other, “there goes a man, who wrote a book that is so complex that even he doesn’t understand it.” However, those, who could and did understand it all, acknowledged that the Principia was a monumental piece of mathematic physics, which had no equal at that time. They also acknowledged that Newton belonged to the very highest levels both as a natural philosopher and mathematician. However, both the Cartesians and Leibnizians rejected the whole of Newton’s work on fundamental philosophical grounds and as we will see it was a long uphill struggle to overcome their objections.

Of course the biggest obstacle to the general acceptance of a heliocentric system was the fact that there was still absolutely no empirical evidence for movement of the Earth, either diurnal rotation or annual rotation around the Sun. This was of course no small issue and could not be dismissed out of hand no matter how convincing and coherent the model that Newton was presenting appeared to be.

The final set of problems were astronomical ones that Newton had failed to solve whilst writing the Principia, open questions that still needed to be answered. There were two major ones the succeeding history of which we will examine, comets and the orbit of the Moon. As we will see showing that the orbit of the Moon obeys the law of gravity proved to be one of the biggest astronomical problems of most of the next century. In the 1680s Newton had only managed to show that the comet of 1680/81 had rounded the Sun on a parabolic orbit and extrapolated from this one result that the orbits of all comets obeyed the law of gravity. This was an unsatisfactory situation for Newton and it was here that he first began his programme to revise the Principia.

For what might be termed project comet flight path, Newton engaged Edmond Halley, who following his efforts as copyeditor, publisher, financier and midwife of the Principia became Newton’s lieutenant and most loyal supporter and one of the few fellow savants, whom Newton apparently never fell out with. Halley willingly took on the task of trying to determine the flight path of comets other than the 1680/81 comet, already included in the 1st edition of Principia.

Edmund_Halley-2

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

Starting around 1695 Halley began searching for and collecting observation data on all of the comets throughout history that he could find. Having acquired enough raw data to make a start he set about analysing it in order to try and determine flight paths. In the 1680s Newton had been the first astronomer to develop a technique for determining the flight path of a comet given three accurate observations at equal or nearly equal time differences. However, the method that he devised was anything but simple or practicable. Using his data he created a geometrical, semi-graphical plot of the flight path that he then iterated time and again, interpolating and extrapolating producing ever more accurate versions of the flight path. This method was both difficult and time consuming. Halley improved on this method, as he wrote to Newton, that having obtained the first three observations he had devised a purely numerical method for the determination of the flight path.

Halley started with the comet of 1683 and found a good fit for a parabolic orbit. This was followed by the comet of 1664, recognising some errors in Hevelius’ observations, and once again found a good fit for a parabolic orbit.

Komet_Flugschrift

The Great Comet of 1664: Johann Thomas Theyner (Frankfurt 1665) Source: Wikimedia Commons

At this point he first began to suspect that the comet of 1682,

which he had observed, was the same as the comet of 1607, observed by Thomas Harriot, William Lower and Johannes Kepler,

herlitz-von-dem-cometen_1-2

David Berlitz, Von dem Cometen oder geschwentzten newen Stern, welcher sich im September dieses 1607. Source

and the comet of 1531 observed Peter Apian amongst others.

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Halley’s Comet 1531 Peter Apian Source

He also in his correspondence with Newton on the topic began to consider the problem of perturbation, that is deviation from the flight path caused by the gravitational attraction of Saturn and Jupiter, as a comet flew passed them. Neither Halley nor Newton succeeded in solving the problem of perturbation. In 1696 Halley held talks at the Royal Society in which he presented the results of his cometary research including his belief that the comets of 1607 and 1682 were one and the same comet on an elliptical orbit, which would return in 1757 or 1758.

Over a period of ten years Halley calculated the orbits of a further twenty comets presenting the results of his researches to the Royal society in 1702. Following his appointment as Savilian Professor for Astronomy at Oxford in 1705 he published the results of his work in the Philosophical Transactions of the Royal Society, Astronomiae cometicae synopsis, and also as a separate broadsheet, with the same title, from the Sheldonian Theatre in Oxford.

halley+sinopsys

An English translation, A synopsis of the astronomy of comets, was published in London in the same year. This work contained a table of results for twenty-four comets in total. Over the years Halley continued to work on comets and a final updated version of Astronomiae cometicae synopsis in 1726.

synopsisofastron00hall

In his work Halley emphasised the problem inherent in working with inaccurate historical observations. Newton used some of Halley’s results in both the second and third editions of Principia.

PSM_V76_D021_Orbit_of_the_planets_and_halley_comet

Diagram of Halley’s orbit in the Solar System Popular Science Monthly Volume 76 Source: Wikimedia Commons

Halley would have been one hundred and one years old in 1757 meaning he had little chance of seeing whether he had been correct in his assumptions concerning the comet from 1682; in fact he died at the ripe old age of eight-five in 1742. A team of three French mathematicians–Alexis Clairaut (1713–1765), Joseph Lalande (1732–1807) and Nicole-Reine Lepaute (1723–1788)–recalculated the orbit of the comet making adjustments to Halley’s results.

clairaut

Alexis Claude Clairaut Source: MacTutor

Jérôme_Lalande

Jérôme Lalande after Joseph Ducreux Source: Wikimedia Commons

lepaute001

Taken from Winterburn The Quite Revolution of Caroline Herschel see footnote 1

The comet returned as predicted and was first observed on Christmas Day 1758 by the German farmer and amateur astronomer Johann Georg Palitzsch (1723–1788).This was a spectacular confirmation of Newton’s theory of gravity and Halley’s work. The comet was named after Halley and is officially designated 1P/Halley. It is now know that it is the comet that appeared in 1066 and is depicted on the Bayeux tapestry

Tapisserie de Bayeux - Scène 32 : des hommes observent la comète de Halley

Bayeux Tapestry depiction of Comet Halley in 1066

PSM_V76_D015_Halley_comet_in_1066_after_emergence_from_the_sun_rays

Halley comet in 1066 after emergence from the sun rays artist unknown Source: Wikimedia Commons

and it was also the comet observed by Peuerbach and Regiomontanus in 1456.

PSM_V76_D015_Halley_comet_in_1456

Comet Halley 1456 artist unknown Source: Wikimedia Commons

SS2567833

Comet Halley 1456 a prognostication!

It still caused a sensation in 1910

Halley's_Comet,_1910

An image of Halley’s Comet taken June 6, 1910. The Yerkes Observatory – Purchased by The New York Times for publication. Source: Wikimedia Commons

 

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Filed under History of Astronomy, History of Mathematics, Newton

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

 

Newton’s Principia is one of the most original and epoch making works in the history of science. There is absolutely nothing original in Newton’s Principia. These two seemingly contradictory judgements of Isaac Newton’s Philosophiæ Naturalis Principia Mathematica are slightly exaggerated versions of real judgements that have been made at various points in the past. The first was the general hagiographical view that was prevalent for much of the eighteenth, nineteenth and twentieth centuries. The second began to appear in the later part of the twentieth century as some historian of science thought that Newton, or better his reputation, needed to be cut down a bit in size. So, which, if either of them, is correct? The surprising answer is, in a way, both of them.

Royal_Society_-_Isaac_Newton’s_Philosophiae_Naturalis_Principia_Mathematica_manuscript_1

Isaac Newton’s Philosophiae Naturalis Principia Mathematica manuscript volume from which the first edition was printed. Written in the hand of Humphrey Newton, Isaac Newton’s assistant. Source: Royal Society Library via Wikimedia Commons

The Principia is a work of synthesis; it synthesises all of the developments in astronomy and physics that had taken place since the beginning of the fifteenth century. All of the elements that make up Newton’s work were, so to speak, laid out for him to integrate into the book. This is what is meant when we say that there is nothing original in the Principia, however the way that Newton integrated them and what he succeeded in creating was at the time unique and totally original. The Principia was truly a case of the whole being greater than the parts. Before we take a brief look at the contents of the Principia there are a couple of anomalies in its construction that need to be addressed.

The first concerns the general methodological structure of the book. Medieval science was dominated, not exclusively, by the theories of Aristotle and Aristotelian methodology. The developments in astronomy, physics and mathematics that we have covered up to now in this series have seen a gradual but steady deconstruction of the Aristotelian structures and theories. In this situation it comes as a bit of surprise that the methodology of the Principia is classically Aristotelian. Aristotle stated that true episteme (Greek) or scientia (Latin), what we would term scientific knowledge, is achieved by setting out a set of first principles or axioms that are perceived as being true and not in need proof and then logically deducing new knowledge from them. Ironically the most famous example of this methodology is the Elements of Euclid, ironically because Aristotle regarded mathematics as not being real knowledge because it doesn’t deal with objects in the real world. This is the methodology that Newton uses in the Principia, setting out his three laws of motion as his basic principles, which we will come back to later, and not the modern methodologies of Francis Bacon or René Descartes, which were developed in the seventeenth century to replace Aristotle.

The second anomaly concerns the mathematics that Newton uses throughout the Principia. Ancient Greek mathematics in astronomy consisted of Euclidian geometry and trigonometry and this was also the mathematics used in the discipline in both the Islamic and European Middle Ages. The sixteenth and seventeenth centuries in Europe saw the development of analytical mathematics, first algebra and then infinitesimal calculus. In fact, Newton made major contributions to this development, in particular he, together with but independently of Gottfried William Leibniz, pulled together the developments in the infinitesimal calculus extended and codified them into a coherent system, although Newton unlike Leibniz had at this point not published his version of the calculus. The infinitesimal calculus was the perfect tool for doing the type of mathematics required in the Principia, which makes it all the more strange that Newton didn’t use it, using the much less suitable Euclidian geometry instead. This raises a very big question, why?

In the past numerous people have suggested, or even claimed as fact, that Newton first worked through the entire content of the Principia using the calculus and then to make it more acceptable to a traditional readership translated all of his results into the more conventional Euclidian geometry. There is only one problem with this theory. With have a vast convolute of Newton’s papers and whilst we have numerous drafts of various section of the Principia there is absolutely no evidence that he ever wrote it in any other mathematical form than the one it was published in. In reality, since developing his own work on the calculus Newton had lost faith in the philosophical underpinnings of the new analytical methods and turned back to what he saw as the preferable synthetic approach of the Greek Euclidian geometry. Interestingly, however, the mark of the great mathematician can be found in this retrograde step in that he translated some of the new analytical methods into a geometrical form for use in the Principia. Newton’s use of the seemingly archaic Euclidian geometry throughout the Principia makes it difficult to read for the modern reader educated in modern physics based on analysis.

When referencing Newton’s infamous, “If I have seen further it is by standing on the sholders [sic] of Giants”, originally written to Robert Hooke in a letter in 1676, with respect to the Principia people today tend to automatically think of Copernicus and Galileo but this is a misconception. You can often read that Newton completed the Copernican Revolution by describing the mechanism of Copernicus’ heliocentric system, however, neither Copernicus nor his system are mentioned anywhere in the Principia. Newton was a Keplerian, but that as we will see with reservations, and we should remember that in the first third of the seventeenth century the Copernican system and the Keplerian system were viewed as different, competing heliocentric models. Galileo gets just five very brief, all identical, references to the fact that he proved the parabola law of motion, otherwise he and his work doesn’t feature at all in the book. The real giants on whose shoulders the Principia was built are Kepler, obviously, Descartes, whose role we will discuss below, Huygens, who gets far to little credit in most accounts, John Flamsteed, Astronomer Royal, who supplied much of the empirical data for Book III, and possibly/probably Robert Hooke (see episode XXXIX).

We now turn to the contents of the book; I am, however, not going to give a detailed account of the contents. I Bernard Cohen’s A Guide to Newton’s Principia, which I recommend runs to 370-large-format-pages in the paperback edition and they is a whole library of literature covering aspects that Cohen doesn’t. What follows is merely an outline sketch with some comments.

As already stated the book consists of three books or volumes. In Book I Newton creates the mathematical science of dynamics that he requires for the rest of the book. Although elements of a science of dynamics existed before Newton a complete systematic treatment didn’t. This is the first of Newton’s achievement, effectively the creation of a new branch of physics. Having created his toolbox he then goes on to apply it in Book II to the motion of objects in fluids, at first glance a strange diversion in a book about astronomy, and in Book III to the cosmos. Book III is what people who have never actually read Principia assume it is about, Newton’s heliocentric model of the then known cosmos.

Mirroring The Elements of Euclid, following Edmond Halley’s dedicatory ode and Newton’s preface, Book I opens with a list of definitions of terms used. In his scholium to the definitions Newton states that he only defines those terms that are less familiar to the reader. He gives quantity of matter and quantity of motion as his first two definitions. His third and fourth definitions are rather puzzling as they are a slightly different formulation of his first law the principle of inertia. This is puzzling because his laws are dependent on the definitions. His fifth definition introduces the concept of centripetal force, a term coined by Newton in analogy to Huygens’ centrifugal force. In circular motion centrifugal is the tendency to fly outwards and centripetal in the force drawing to the centre. As examples of centripetal force Newton names magnetism and gravity. The last three definitions are the three different quantities of centripetal force: absolute, accelerative and motive. These are followed by a long scholium explicating in greater detail his definitions.

We now arrive at the Axioms, or The Laws of Motions:

1) Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

This is the principle of inertia that Newton had taken from Descartes, who in turn had taken it from Isaac Beeckman.

2) A change of motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.

Somewhat different from the modern formulation of F=ma, this principle has its origin in the work of Huygens although there is not a one to one correspondence.

3) To any action there is always an opposite and equal reaction, in other words, the actions of two bodies upon each other are always equal and always opposite in direction.

This law originates with Newton and its source is not absolutely clear. It seems to have been inspired by Newton’s examination of Descartes laws of inelastic collision but it might also have been inspired by a similar principle in alchemy of which Newton was an ardent disciple.

Most people are aware of the three laws of motion, the bedrock of Newton’s system, in their modern formulations and having learnt them, think that they are so simple and obvious that Newton just pulled them out of his hat, so to speak. This is far from being the case. Newton actually struggled for months to find the axioms that eventually found their way into the Principia. He tried numerous different combinations of different laws before finally distilling the three that he settled on.

Having set up his definitions and laws Newton now goes on to produce a systematic analysis of forces on and motion of objects in Book I. It is this tour de force that established Newton’s reputation as one of the greatest physicist of all time. However, what interests us is of course the law of gravity and its relationship to Kepler’s laws of planetary motion. The following is ‘plagiarised’ from my blog post on the 400th anniversary of Kepler’s third law.

In Book I of Principia, the mathematics and physics section, Newton first shows, in Proposition 11[1], that for a body revolving on an ellipse the law of the centripetal force tending towards a focus of the ellipse is inversely as the square of the distance: i.e. the law of gravity but Newton is not calling it that at this point. In Proposition 14[2] he then shows that, If several bodies revolve about a commo[3]n center and the centripetal force is inversely as the square of the distance of places from the center, I say that the principal latera recta of the orbits are as the squares of the areas which bodies describe in the same time by radii drawn to the center. And Proposition 15: Under the same supposition as in prop. 14, I say the square of the periodic times in ellipses are as the cubes of the major axes. Thus Newton shows that his law of gravity and Kepler’s third law are equivalent, although in this whole section where he deals mathematically with Kepler’s three laws of planetary motion he never once mentions Kepler by name. Newton would go one to claim the rights to laws one and two as he had, in his opinion, provided their first real proof. He acknowledges, however, Kepler’s claim to the third law.

Book II as already mentioned appears to go off a tangent in that it deals with motion in a fluid medium, as a result it tends to get ignored, although it is as much a tour de force as Book I. Why this detour? The answer can be found in the theories of René Descartes and Newton’s personal relationship to Descartes and his works in general. As a young man Newton undertook an extensive programme of self-study in mathematics and physics and there is no doubt that amongst the numerous sources that he consulted Descartes stand out as his initial primary influence. At the time Descartes was highly fashionable and Cambridge University was a centre for interest in Descartes philosophy. At some point in the future he then turned totally against Descartes in what could almost be describe as a sort of religious conversion and it is here that we can find the explanation for Book II.

Descartes was a strong supporter of the mechanical philosophy that he had learnt from Isaac Beeckman, something that he would later deny. Strangely, rather like Aristotle, objects could only be moved by some form of direct contact. Descartes also rejected the existence of a vacuum despite Torricelli’s and Pascal’s proof of its existence. In his Le Monde, written between 1629 and 1633 but only published posthumously in 1664 and later in his Principia philosophiae, published in 1644, Descartes suggested that the cosmos was filled with very, very fine particles or corpuscles and that the planets were swept around their orbits on vortexes in the corpuscles. Like any ‘religious’ convert, Newton set about demolishing Descartes theories. Firstly, the title of his volume is a play upon Descartes title, whereas Descartes work is purely philosophical speculation, Newton’s work is proved mathematically. The whole of Book II exists to show that Descartes’ vortex model, his cosmos full of corpuscles is a fluid, can’t and doesn’t work.

Book III, entitled The System of the World, is as already said that which people who haven’t actually read it think that the Principia is actually about, a description of the cosmos. In this book Newton applies the mathematical physics that he has developed in Book I to the available empirical data of the planets and satellites much of it supplied by the Astronomer Royal, John Flamsteed, who probably suffered doing this phase of the writing as Newton tended to be more than somewhat irascible when he needed something from somebody else for his work. We now get the astronomical crowning glory of Newton’ endeavours, an empirical proof of the law of gravity.

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

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

In the 1st edition of Principia Newton referenced the solar system itself and the moons of Jupiter as system that could be shown empirically to Kepler’s third law and added the moons of Saturn in the 3rd edition.

Book III in the first edition closes with Newton’s study of the comet of 1680/81 and his proof that its flight path was also determined by the inverse square law of gravity showing that this law was truly a law of universal gravity.

I have gone into far more detain describing Newton’s Principia than any other work I have looked out in this series because all the various streams run together. Here we have Copernicus’s initial concept of a heliocentric cosmos, Kepler’s improved elliptical version of a heliocentric cosmos with it three laws of planetary motion and all of the physics that was developed over a period of more than one hundred and fifty years woven together in one complete synthesis. Newton had produced the driving force of the heliocentric cosmos and shown that it resulted in Kepler’s elliptical system. One might consider that the story we have been telling was now complete and that we have reached an endpoint. In fact, in many popular version of the emergence of modern astronomy, usually termed the astronomical revolution, they do just that. It starts with Copernicus’ De revolutionibus and end with Newton’s Principia but as we shall see this was not the case. There still remained many problems to solve and we will begin to look at them in the next segment of our story.

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

[2] Newton, Principia, 1999 p. 467

[3] Newton, Principia, 1999 p. 468

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

 

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

The event that would eventually lead to Isaac Newton writing and publishing his magnum opus, the Philosophiæ Naturalis Principia Mathematica (the Mathematical Principles of Natural Philosophy), took place in a London coffee house.

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Title page of ‘Principia’, first edition (1687). Source: Wikimedia Commons

This is not quite as strange as it might at first appear, shortly after their first appearance in England around 1650 coffee houses became the favourite meeting places of the English scientific intelligentsia, the astronomers, mathematicians and natural philosophers. Here, these savants would meet up to exchange ideas, discuss the latest scientific theories and pose challenges to each other. These institutions also earned the appellation Penny Universities, as some of those savants, such as William Whiston, Francis Hauksbee and Abraham de Moivre, bettered their incomes by holding lectures or demonstrating experiments to willing audiences, who paid the price of a cup of coffee, a penny, for their intellectual entertainment. Later, after he had become Europe’s most famous living natural philosopher, Isaac Newton would come to hold court in a coffee shop, surrounded by his acolytes, the original Newtonians, distributing words of wisdom and handing round his unpublished manuscripts for scrutiny. However, all that still lay in the future.

One day in January 1684 Christopher Wren, Robert Hooke and Edmond Halley were discussing the actual astronomical theories over a cup of coffee. Wren, today better known as one of England most famous architects, was a leading mathematician and astronomers, who had served both as Gresham and Savilian professor of astronomy. Newton would name him along with John Wallis and William Oughtred as one of the three leading English mathematicians of the seventeenth century.

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Wren, portrait c.1690 by John Closterman Source: Wikimedia Commons

Hooke was at the time considered to be the country’s leading experimental natural philosopher and Halley enjoyed an excellent reputation as a mathematician and astronomer.

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Portrait by Richard Phillips, before 1722 Source: Wikimedia Commons

The topic of discussion was Kepler’s elliptical, heliocentric astronomy and an inverse, squared law of gravity. All three men had arrived separately and independently at an inverse, squared law of gravity probably derived from Huygens’ formula for centrifugal force. Wren posed the question to the other two, whether they could demonstrate that such a law would lead to Kepler’s elliptical planetary orbits.

Hooke asserted that he already had such a demonstration but he would first reveal it to the others after they had admitted that they couldn’t solve the problem. Wren was sceptical of Hooke’s claim and offered a prize of a book worth forty shillings to the first to produce such a demonstration.  Hooke maintained his claim but didn’t deliver. It is worth noting that Hooke never did deliver such a demonstration. Halley, as already said no mean mathematician, tried and failed to solve the problem.

In August 1684 Halley was visiting Cambridge and went to see Newton in his chambers in Trinity College, who, as we know, he had met in 1682.

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Trinity College Cambridge, David Loggan’s print of 1690 Source: Wikimedia Commons

According the Newton’s account as told to Abraham DeMoivre, Halley asked Newton, “what he thought the Curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of the distance from it. Sir Isaac replied immediately that it would be an Ellipse…” Here was Newton claiming to know the answer to Wren’s question. Halley asked Newton how he knew it and he replied, “I have calculated it…” Newton acted out the charade of looking for the supposed solution but couldn’t find it. However he promised Halley that he would send him the solution.

In November Edward Paget, a fellow of Trinity College, brought Halley a nine page thesis entitled De motu corporum in gyrum (On the Motion of Bodies in an Orbit).

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Page of the De motu corporum in gyrum

When Halley read Newton’s little booklet he was immediately aware that he held something truly epoch making in the history of astronomy and physics in his hand. Newton had delivered up a mathematical proof that an elliptical orbit would be produced by an inverse square force situated at one of the foci of the ellipse, thus combining the inverse square law of gravity with Kepler’s first law. He went on to also derive Kepler’s second and third laws as well as laying down the beginnings of a mathematical theory of dynamics. Halley reported details of this extraordinary work to the Royal Society on 10 December 1684:

Mr Halley gave an account, that he had lately seen Mr. Newton at Cambridge, who had shewed him a curious treatise, De motu: which, upon Mr. Halley’s desire, was he said promised to be sent to the Society to be entered upon their register.

Mr. Halley was desired to put Mr. Newton in mind of his promise for securing his invention to himself till such time as he could be at leisure to publish it. Mr. Paget was desired to join with Mr. Halley.

The interest in and the demand to read Newton’s new production was very high but the author decided to improve and rewrite his first offering, triggering one of the most extraordinary episodes in his life.

Although he was Lucasian Professor and would turn forty-two on 25 December 1684, Newton remained a largely unknown figure in the intellectual world of the late seventeenth century. Following the minor debacle that resulted from the publication of his work in optics in the 1670s he had withdrawn into his shell, living in isolation within the walls of Cambridge University. He carried out his duties as Lucasian Professor but had almost no students to speak of and definitely no disciples. Thanks to the word of mouth propaganda of people like his predecessor as Lucasian Professor, Isaac Barrow, and above all the assiduous mathematics groupie, John Collins, it was rumoured that a mathematical monster slumbered in his chambers in Trinity College but he had done nothing to justify this bruited reputation. His chambers were littered with numerous unfinished scientific manuscripts, mostly mathematical but also natural philosophical and an even larger number of alchemical and theological manuscripts but none of them was in a fit state to publish and Newton showed no indication of putting them into a suitable state. Things now changed, Newton had found his vocation and his muse and the next two and a half years of his life were dedicated to creating the work that would make him into a history of science legend, the reworking of De motu into his Principia.

Over those two and a half years Newton turned his nine-page booklet into a major three-volume work of science. The modern English translation by I B Cohen runs to just over 560 large format pages, although this contains all the additions and alterations made in the second and third editions, so the original would have been somewhat shorter. Halley took over the editorship of the work, copyediting it and seeing it through the press. In 1685 the Royal Society had voted to take over the costs of printing and publishing Newton’s masterpiece, so everything seemed to be going smoothly and then disaster struck twice, firstly in the form of Robert Hooke and secondly in the form of a financial problem.

Hooke never slow to claim his priority in any matter of scientific discovery or invention stated that he alone had first discovered the inverse square law of gravity and that this fact should, indeed must, be acknowledged in full in the preface to Newton’s book. Halley, realising at once the potential danger of the situation, was the first to write to Newton outlining Hooke’s claim to priority, stating it, of course, as diplomatically as possible. Halley’s diplomacy did not work, Newton went ballistic. At first his reaction was comparatively mild, merely pointing out that he had had the inverse square law well before his exchanges with Hook in 1679 and had, in fact, discussed the matter with Wren in 1677, go ask him, Newton said. Then with more time to think about the matter and building up a head of steam, Newton wrote a new letter to Halley tearing into Hooke and his claim like a rabid dog. All of this ended with Newton declaring that he would no longer write volume three of his work. Halley didn’t know this at the time but this was in fact, as we shall see, the most important part of the entire work in which Newton presented his mathematical model of a Keplerian cosmos held together by the law of gravity. Halley remained calm and used all of his diplomatic skills to coax, flatter, persuade and cajole the prickly mathematician into delivering the book as finished. In the end Newton acquiesced and delivered but acknowledgements to Hooke were keep to a minimum and offered at the lowest level of civility.

The financial problem was of a completely different nature. In 1685 the Royal Society had taken over the cost of printing and publishing the deceased Francis Willughby’s Historia piscium as edited by John Ray.

This was an expensive project due to the large number plates that the book contained and the book was, at the time, a flop. This meant when it came time to print and publish Newton’s work the Royal Society was effectively bankrupt. One should note here that the popular ridicule poured out over Willughby’s volume, it having almost prevented Newton’s masterpiece appearing, is not justified. Historia piscium is an important volume in the history of zoology. Halley once again jumped into the breach and took over the costs of printing the volumes; on the 5 July 1687 Halley could write to Newton to inform him that the printing of his Philosophiæ Naturalis Principia Mathematica had been completed.

 

 

 

 

 

 

 

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

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

One of the most often repeated false statements in the history of science is that Isaac Newton discovered gravity. Of course he didn’t discovery it, it’s all around us. You can observe gravity every time you drop something. Making the claim more precise, by saying Newton discovered the law of gravity, doesn’t really improve the situation much. What Newton did do was he proved the law of gravity and made the fairly rational assumption based on the available evidence that this law applies universally to all bodies in the cosmos. An assumption that is not written in stone and has been questioned in the present time for the general theory of relativity, the theory that replaced the Newtonian theory of universal gravity and of which the Newtonian theory of gravity is a very good approximation for local cases. However we don’t want to take the path to modern theories of cosmology and dark matter but want to stay firmly in the seventeenth century with Newton.

We can start with a brief survey of theories of gravity before Newton. Originally gravity was the Latin term applied to Aristotle’s explanation of why, when dropped, things fall to the ground. Aristotle thought that objects did so through a form of vital attraction, returning to their natural home, consisting predominantly of the elements earth and water. Fire and air rise up. This only applied to the Earth, as things beyond the Moon were made of a fifth element, aether, the quintessence, for which the natural form of motion was uniform circular motion.

This neat model wouldn’t work, however for Copernicus’ heliocentric model, which disrupted the division between the sublunar and supralunar worlds. To get around this problem Copernicus suggested that each planet had its own gravity, like the Earth. So we have terrestrial gravity, Saturnian gravity, Venusian gravity etc. This led Alexander von Humboldt, in the 19th century, to claim that Copernicus should be honoured as the true originator of the universal theory of gravity, although it is by no means clear that Copernicus thought that he planetary gravities were all one and the same phenomenon.

The whole concept became even more questionable when the early telescopic astronomers, above all Galileo, showed that the Moon was definitely Earth like and by analogy probably the other planets too. At the end of a long line of natural philosophers stretching back to John Philoponus in the sixth century CE, Galileo also showed that gravity, whatever it might actually be, was apparently not a vitalist attraction but a force subject to mathematical laws, even if he did get the value for the acceleration due to gravity ‘g’ wrong and although he didn’t possess a clear concept of force.. Throughout the seventeenth century other natural philosophers, took up the trail and experimented with pendulums and dropped objects. A pendulum is of course an object, whose fall is controlled. Most notable were the Jesuit, natural philosopher Giovanni Battista Riccioli (1598–1671) and the Dutch natural philosopher Christiaan Huygens (1629–1695). Riccioli conducted a whole series of experiments, dropping objects inside a high tower, making a direct confirmation of the laws of fall. Both Riccioli and Huygens, who independently of each other corrected Galileo’s false value for ‘g’, experimented extensively with pendulums in particular determining the length of the one-second pendulum, i.e. a pendulum whose swing in exactly one second. As we will see later, the second pendulum played a central roll in an indirect proof of diurnal rotation. Huygens, of course, built the first functioning pendulum clock.

Turning to England, it was not Isaac Newton, who in the 1670s and 80s turned his attention to gravity but Robert Hooke (1635–1703), who was Curator of Experiments for the newly founded Royal Society. Like Riccioli and Huygens Hooke experimented extensively with dropping objects and pendulums to try and determine the nature of gravity. However his experiments were not really as successful as his continental colleagues. However, he did develop the idea that it was the force of gravity that controlled the orbits of the planets and, having accepted that comets were real solid objects and not optical phenomena, also the flight paths of comets. Although largely speculative at this point Hooke presented a theory of universal gravity, whilst Newton was still largely confused on the subject. Hooke turned to Newton in a letter with his theory in order to ask his opinion, an act that was to lead to a very heated priority dispute.

Before we handle that correspondence we need to go back to the beginnings of the 1670s and an earlier bitter dispute between the two.  In 1672 Newton announced his arrival on the European natural philosophy scene with his first publication, a letter in the Philosophical Transactions of the Royal Society (1671/72), A New Theory of Light and Colours, which described the experimental programme that he had carried out to demonstrate that white light actually consisted of the colours of the spectrum.

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Newton’s original letter. Source: Royal Society

This brilliant piece of experimental optics did not receive the universal praise that, reading it today, we might have expected, in fact it was heavily criticised and attacked. Some critics were unable to reproduce Newton’s experimental results, probably because their prisms were of too poor quality. However, others, Hooke to the fore, criticised the content. Hooke and Huygens, the two current leaders in the field of optics both criticised Newton for interpreting his results within the framework of a particle theory of light, because they both propagated a wave theory of light. Newton actually wrote a paper that showed that his conclusions were just as valid under a wave theory of light, which, however, he didn’t publish. The harshest criticism came from Hooke alone, who dismissed the whole paper stating that he had already discovered anything of worth that it might contain . This did not make Newton very happy, who following this barrage of criticism announced his intention to resign from the Royal Society, to which he had only recently been elected.  Henry Oldenburg (c. 1619–1677), secretary of the Royal Society, offered to waive Newton’s membership fees if he would stay. Newton stayed but had little or nothing more to do with the society till after Hooke’s death in 1703. Newton did, however, write a very extensive paper on all of his optical work, which remained unpublished until 1704, when it formed a major part of his Opticks.

By  1679 tempers had cooled and Robert Hooke, now secretary of the Royal Society, wrote to Isaac Newton to enquire if he would be interested in reopening his dialogue with the Royal Society. In the same letter he asked Newton’s opinion on his own hypothesis that planetary motions are compounded of a tangential motion and “an attractive motion towards the centrall body…” Hooke is here referencing his Attempt to Prove the Motion of the Earth from Observations (1674, republished 1679),

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which contains the following fascinating paragraph:

This depends on three Suppositions. First, That all Coelestial Bodies whatsoever, have an attractive or gravitating power towards their own Centers, whereby they attract not only their own parts, and keep them from flying from the, as we observe the earth to do, [here Hooke is obviously channelling Copernicus] but that they do also attract all other Coelestial Bodies that are within the sphere of their activity … The second supposition is this, That all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a streight line, till they are by some other effectual power deflected and bent into a Motion, describing a Circle, Ellipsis, or some other more compounded Curve Line. [the principle of inertia, as propounded by Descartes] The third supposition is, That these attractive powers are so much the more powerful in operating, by how much nearer the body wrought upon is to there own Centers. Now what these several degrees are I have not yet experimentally verified…

Whether or not this is truly a universal theory of gravity is a much-debated topic, but if not, it comes very close and was moving much more in that direction than anything Newton had produced at the time. As we shall see later this was to cause not a little trouble between the two rather prickly men.

Newton declined the offer of a regular exchange of ideas, claiming that he was moving away from (natural) philosophy to other areas of study. He also denied having read Hooke’s paper but referred to something else in it in a later letter to Flamsteed. However, in his reply he suggested an experiment to determine the existence of diurnal rotation involving the usually dropping of objects from high towers. Unfortunately for Newton, he made a fairly serious error in his descripting of the flight path of the falling object, which Hooke picked up on and pointed out to him, if unusually politely, in his reply. Newton of course took umbrage and ended the exchange but he did not forget it.

In our next episode we will deal with the events leading up to the writing and publication of Newton’s great masterpiece, Philosophiæ Naturalis Principia Mathematica (1687), which include the repercussions of this brief exchange between Hooke and its author.

 

 

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Mathematics and natural philosophy: Robert G socks it to GG

In my recent demolition of Mario Livio’s very pretentious Galileo and the Science Deniers I very strongly criticised Livio’s repeated claims, based on Galileo’s notorious Il Saggiatore quote on the two books, that Galileo was somehow revolutionary in introducing mathematics into the study of science. I pointed out that by the time Galileo wrote his book this had actually been normal practice for a long time and far from being revolutionary the quote was actually a common place.

Last night whilst reading my current bedtime volume, A Mark Smith’s excellent From Sight to Light: The Passage from Ancient to Modern Optics,(University of Chicago Press, 2015) I came across a wonderfully appropriate quote on the topic from Robert Grosseteste (c.1175–1253). For those that don’t know Grosseteste was an English cleric who taught at Oxford University and who became Bishop of Lincoln. He played an important and highly influential role in medieval science, particularly in helping to establish optics as a central subject in the medieval university curriculum.

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An early 14th-century portrait of Grosseteste Source: Wikimedia Commons

Of course, this is problematic for Livio, who firmly labelled the Catholic Church as anti-science and who doesn’t think there was any medieval science, remember that wonderfully wrong quote:

Galileo introduced the revolutionary departure from the medieval, ludicrous notion that everything worth knowing was already known.

If this were true then medieval science would be an oxymoron but unfortunately for Livio’s historical phantasy there was medieval science and Grosseteste was one of its major figure. If you want to know more about Grosseteste then I recommend the Ordered Universe website set up by the team from Durham University led by Giles Gasper, Hannah Smithson and  Tom McLeish

I already knew of Grosseteste’s attitude towards natural philosophy and mathematics but didn’t have a suitable quote to hand, so didn’t mention it in my review. Now I do have one. Let us first remind ourselves what Galileo actually said in Il Saggiatore:

Philosophy [i.e. natural philosophy] is written in this grand book — I mean the Universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.

And now what Grosseteste wrote four hundred years earlier in his De lineis, angulis et figuris (On lines, angles and figures) between 1220 and 1235:

“…a consideration of lines, angles and fugures is of the greatest utility because it is impossible to gain a knowledge of natural philosophy without them…for all causes of natural effects must be expressed by means of lines, angles and figures”

Remarkably similar is it not!

 

 

 

 

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How to create your own Galileo

Writing this book review caused me a great deal of of stress, even leading to sleepless night when I made the mistake of reading the offending piece of literature as bedtime reading. The review itself has become horrendously long and I must at times fight my instinct to add even more explanations, as to why this or that was wrong. It is in the words of that excellent history of science author, Matthew Cobb, ‘baggy and rambling’ and should actually be radically edited but I just can’t be arsed to do it, so I’m simply posting the whole monstrosity. For those, who don’t want to read the whole thing, and I wouldn’t blame you, the first three and the last five paragraphs offer a sort of synopsis of the whole thing.

Since I began writing book reviews on a more regular basis I have tried only to review books that I personally find good and which I think might be of interest to those who come here to read my weekly scribblings. I decided that on the whole it isn’t worth wasting time and energy writing about uninteresting, mediocre or simply bad books. However, occasionally a book come along that I feel duty bound, given my reputation as a #histSTM grouch, to debunk as a favour to my readers so that they don’t waste their time and energy reading it; today’s review is one such.

Some time back I wrote a post about the Alexandrian mathematician and philosopher Hypatia, which started with the fact that she has been used as a sort of blank slate onto which numerous people down the centuries have projected their images of what they would have wanted her to be. In the case of Hypatia this is fairly easy, as the rest of my post pointed out we know next to nothing about the lady. Another figure, who has been used extensively over the years as a silhouette, which people fill out according to their own wishes is Galileo Galilei; in his case this is more difficult as we actually know an awful lot about the Tuscan mathematician’s life and work. However, this has not prevented numerous authors from creating their own Galileos.

The latest author, who has decided to present the world with his Galileo, is the astrophysicist and very successful author of popular books on mathematics and science, Mario Livio with his Galileo and the Science Deniers.[1] I might not have bothered with this book but Livio is a very successful pop science book author, as is made very clear by the fact that the hardback and paperback were both issued simultaneously and at very low prices; the publishers expect it to sell well, so it will unfortunately have a big impact on uninformed peoples perceptions of Galileo. I say unfortunately, which, of course, gives readers of this review a very strong clue as to what I think of this book. Quite simply don’t bother, it brings nothing new to our knowledge of Galileo and in fact is full of, at times, quite serious historical errors, serious that is if you’re a historian, who takes getting the facts right seriously.

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The opening sentences starts with a couple of wonderful errors and also lays out Livio’s version of Galileo:

Being an astrophysicist myself, I have always been fascinated by Galileo. He was, after all, not only the founder of modern astronomy and astrophysics–the person who turned an ancient profession into the universe’s deepest secrets and awe-inspiring wonders–but also a symbol of the fight for intellectual freedom.

I think Copernicus, Tycho Brahe and Johannes Kepler might want a word with Livio about, who exactly is the founder of modern astronomy. Also, excuse the language, but what the fuck did Galileo ever do for astrophysics? The final half sentence tells us into which silhouette Livio has decided to pour his Galileo; Livio’s Galileo is the white knight of freedom of speech and freedom of thought, who has mounted his charger and taking up his lance sets off to kill the anti-science dragon of the Holy Roman Catholic Church. This is, of course not a new Galileo but a well-known old model, which historians of science have spent a lot of time and effort dismantling over the last fifty plus years.

Central to the problems with Livio’s book is that he completely ignores the historical context in which the Galileo story took place. His is totally a presentist view in which he applies the social rules and moral judgements of the twentieth-first century to the various occurrences he sketches in the early seventeenth century. This is quite simply very bad historiography. He compounds this error by trying to draw parallels between Galileo’s conflict with the Catholic Church and the current problems with science denialists in our times, hence the title of his book. To do this he simply denies Galileo’s critics any scientific basis for their criticism whatsoever, Galileo is science, his critics are anti-science. A rather simplistic and historically highly inaccurate presentation of the known facts.

Just to make clear what exactly the historical context was, there existed no freedom of speech or freedom of thought under any civil or religious authority anywhere in Europe at the beginning of the seventeenth century; such social concepts still lay in the future. There is a slight irony in the fact that the current wave of science denialists, against whom Livio’s book is directed, are in fact exercising their, protected by law, rights of freedom of thought and speech. More importantly the Holy Roman Catholic Church was not just a religion and a church but also a powerful political and judicial body with judicial rights over all within its dominion and this in an age of absolutism with the Pope as the most absolute of all absolute rulers. All authorities both civil and religious reserved for itself the right to determine what its subject were permitted to express in public, the Catholic Church was in no way unique in claiming and exercising this right.

Still in the preface to Livio’s book we find his first distortion of the historical scientific facts, he writes that Galileo’s telescopic discoveries, “All but destroyed the stability of the Earth-centered Ptolemaic universe.” Here Livio, and not only here, fails to differentiate between Aristotelian cosmology and Ptolemaic astronomy. All of the telescopic discoveries, with the exception of the phases of Venus, demolished aspects of Aristotelian cosmology but had no significance for Ptolemaic geocentric astronomy. The discovery of the phases of Venus, of course, refuted a pure geocentric system but was perfectly compatible with a Tychonic geo-heliocentric system, which then became the default alternative to a heliocentric system. With two notable exceptions that I will deal with later Livio makes no clear mention of the fact that the telescopic discoveries were made within the same approximately three year period not only by Galileo but simultaneous by others, so if Galileo had never used a telescope it would have made very little difference to the subsequent history of astronomy. This makes rather a mockery of Livio’s next dubious claim, “his [Galileo’s] ideas became the basis on which modern science has been erected.” This is much less true than Livio and other Galileo groupies would have us believe. Galileo made a contribution but others in the seventeenth century actually contributed significantly more.

One last comment from the preface, Livio writes:

He insisted on publishing many of his scientific findings in Italian [actually Tuscan not Italian] (rather than Latin), for the benefit of every educated rather than for a limited elite.

In the early seventeenth century almost every educated person would per definition have been able to read and write Latin; Latin was the default language of education.

Reading the opening chapter of Livio’s book, Rebel with a Cause, I constantly had the feeling that I had been transported back to the 1960s and 70s, when I first began to read books about the history of science in general and Galileo in particular. It as if the last fifty plus years of history of science research had never taken place, he even relies on Einstein and Bertrand Russell as his historical authorities, at times I shuddered. He goes so far as to tell us that the Renaissance happened because people discovered that they were individuals! I can’t remember when I last read this particular piece of inanity and I would be curious who actually put it into the world. The final page of this chapter contains all of the classic Galileo clichés.

Perhaps most important, Galileo was the pioneer and star of advancing the new art of experimental science. He realised that he could test or suggest theories by artificially manipulating various terrestrial phenomena. He as also the first scientist whose vision and scientific outlook incorporated methods and results that were applicable to all branches of science.

There is a long historical list of people who would disagree–Archimedes, Ptolemaeus, al-Haytham, Grosseteste, Roger Bacon, William Gilbert and a whole host of alchemists starting with Abū Mūsā Jābir ibn Hayyān (for Livio opinion on alchemy see below)–just to name the most prominent. Modern research has also conclusively shown that artisanal practice in the fifteenth and sixteenth centuries played a significant role in the development of empirical, experimental science. Livio’s last sentence here is also rather dubious, apart from some rather trivial aspects, there are no methods and results that are applicable to all branches of science.

…in four areas he revolutionised the field: astronomy and astrophysics; the laws of motion and mechanics; the astonishing relationship between mathematics and physical reality […]; and experimental science.

Despite everything, Galileo’s contributions to astronomy were rather minimal and he certainly didn’t revolutionise the field, others such as Kepler, whom he ignored, did. I am still trying to work out what his contributions to astrophysics could possibly be? His real major contribution was indeed to motion and mechanics but he was no means alone in this others such as Simon Stevin and Isaac Beeckman made substantial contributions to the new developments in these areas. The mathematics thing, to which Livio keeps returning, is baloney and I shall deal with it separately later. Galileo made contributions to the development of experimental science but he was by no means alone in this and to say he revolutionised it is hyperbole.

The only defense remaining to those obstinately refusing to accept the conclusions implied by the accumulating weight of empirical facts and scientific reasoning was to reject the results almost solely on the basis of religious or political ideology

Here Livio betrays his own tactic, put crudely, throughout the book he twists the historical facts in order to try and make out that there no legitimate scientific objections to Galileo’s claims, however there were.

The next chapter is the usual enthusiastic fan boy description of Galileo’s talents as an all round humanist and contains nothing particularly objectionable but does contain a strong indication of the superficiality of Livio’s historical knowledge. He writes, “First, at age twenty-two, Galileo, already had the chutzpah to challenge the great Aristotle on topics related to motion…” People had been consistently challenging the great Aristotle on topics related to motion since the sixth century CE and Galileo was merely joining a long tradition of such work. Livio also casually calls Aristotle’s theory of motion impetus! Impetus was, of course, a theory initially developed by John Philoponus in the sixth century CE when seriously challenging Aristotle’s theory of motion. On a side note Livio says that the tools to treat such variables such as velocity and acceleration, i.e. calculus, were first developed by Newton and Leibniz. Other seventeenth century mathematicians who contributed substantially to the development of the calculus such as Cavalieri, de Saint-Vincent, Fermat, Pascal, Descarte, John Wallis and Isaac Barrow would be very surprised to hear this. On the same page he repeats the myth that Christoph Clavius was “the senior mathematician on the commission that instituted the Gregorian calendar, he wasn’t, Ignazio Danti was.

Clavius turns up as one of the leading mathematicians, who the young Galileo turned to for mentorship when he was trying to establish a reputation as a mathematician and get support to find an appointment as professor of mathematics. Interestingly Galileo’s other mentor Guidobaldo del Monte (1545–1607) appears nowhere in Livio’s book. This is strange as it was del Monte, who acquired the professorship in Pisa for Galileo through his brother Cardinal Francesco Maria del Monte (1549–1627), who was the de ‘Medici cardinal and recommended Galileo to the Grand Duke. It was also del Monte, who devised the experiment that led Galileo to the parabola law, which Livio calls one of Galileo’s crowning achievements.

In the next chapter on Galileo’s work on the theory of fall Livio can’t help taking a sideswipe at alchemy and astrology:

It is certainly true that, at their inception, the sciences were not immune to false beliefs, since they are sometimes connected to fictitious fields such as alchemy and astrology. This was partly the reason why Galileo decided later to rely on mathematics, which appeared to provide a more secure foundation.

This off hand rejection ignores completely that astrology was the main driving force behind astronomy since its beginnings in antiquity down to the seventeenth century and that all the leading Renaissance astronomers, including Galileo, were practicing astrologers. The practice of astrology/astronomy, of course, requires a high level of mathematical ability. Alchemy developed virtually all of the experimental methods and the necessary equipment to carry out those experiments on which chemistry was built.

Now in Padua, where Galileo was also professor of mathematics, a position that he once again acquired with the assistance of del Monte, we get the story of Galileo’s three lectures on the nova of 1604. Livio informs us that “Christoph Clavius confirmed the null parallax determination–that is, no shift had been observed–but refused to accept its implications as compelling.”

This is once again Livio’s tactic of trying to discredit the Jesuits. The implications that he is talking about are that the heavens are not unchanging as claimed by Aristotle. Clavius observed the nova of 1572 and already in 1581 published a digression on the subject fully accepting that the nova was supralunar and that the heavens were not unchanging. He included this in his Sphaera in 1585, the most widely read astronomy textbook in the late sixteenth and early seventeenth centuries and he probably thus had the most influence in persuading others that change had occurred in the heavens. He also included the same results for the novae of 1600 and 1604, so what is Livio talking about? Clavius was unable to explain what these novae were but then again nobody else in the seventeenth century could either.

We now move on to Galileo, telescopic astronomy and the Sidereus Nuncius. Although he actually talks about other telescopic astronomers–Scheiner, Marius, Harriot, Fabricius–they are only offered bit parts in Livio’s screenplay, which follows the usual path of giving Galileo credit for everything. He attributes the discovery of Earthshine, the Moon illuminated by sunlight reflected by the Earth, to Galileo, whereas it was previously discovered by Leonardo, who didn’t publish, and Michael Mästlin, who did. He attributes the discovery of stars that can’t be seen without a telescope to Galileo, whereas this was already noted in the printed account of the first telescope demonstration in Den Hague, the source of Sarpi’s and thus Galileo’s first knowledge of the telescope. We then get one of the most bizarre claims made by Livio in the book:

Even more consequential for the future of astrophysics was Galileo’s discovery that stars varied enormously in brightness, with some being a few hundred times brighter than others.

Coming from a professional astrophysicist I find this statement mind boggling. The difference in brightness between celestial objects is obvious to anybody with reasonable eyesight, who simply looks up at the night sky in an area without light pollution. Astronomers even use a six-point scale to designate the different levels of brightness, which is termed magnitude; this was first introduced by Ptolemaeus around 150 CE!

We then get a very brief account of the star size argument as originated by Tycho, which Livio falsely claims Galileo dismissed by saying that the observed star discs are merely artefacts. They are in fact merely artefacts but Galileo didn’t say this. He accepts their existence and uses a completely different argument to try and dismiss the star size argument.

We now arrive at the Moons of Jupiter and Simon Marius. Livio mentions Marius several times in his book but insists on calling him Simon Mayr, his birth name, why? Marius issued all of his publications under the Latinised version of his name and so historian refer to him as Simon Marius. Livio doesn’t call Copernicus, Kopernik or Tycho, Tyge their birth names, so why does he call Marius, Mayr? What he writes about Marius and the Moons of Jupiter left me, as a Marius expert, totally flabbergasted:

What would have undoubtedly annoyed Galileo no end is that the Galilean satellites are known today by the names assigned to them by the German astronomer Simon Mayr rather than as the “Medici stars.” Mayr may have independently discovered the satellites before Galileo, but he failed to understand that the moons were orbiting the planet. [my emphasis]

First off, the names were suggested by Kepler not Marius, who however first published them specifically mentioning the fact that they were suggested by Kepler. Secondly Marius discovered the moons, famously, one day later than Galileo, any confusion about who discovered what when being produced by use of different calendars, Gregorian and Julian. Thirdly, the clause that I have emphasised above is pure and utter bullshit. Marius knew very well that the moons orbited Jupiter and he calculated the orbits, calculations that he published before Galileo. Marius’ calculations are also more accurate than those of Galileo. Should Livio doubt any of this I can send him scans of the relevant pages of Mundus Jovialis in the original Latin or in German and/or English translation. Livio now brings the story of Galileo hating Marius because he accused him of being behind Baldessar Capra’s plagiarism of Galileo’s proportional compass pamphlet in 1606. Marius had been Capra’s mathematics teacher earlier in Padua. Livio fails to mention that the accusations are provably false. Galileo in 1607 had himself cleared Marius of any involvement in the case and the whole episode took place a year after Marius had left Padua.

We now move on to the peculiar shape of Saturn and the discovery of the phases of Venus. In the later case we get absolutely no mention that the phases of Venus were discovered independently by Harriot, Marius, and the astronomers of the Collegio Romano, the latter almost certainly before Galileo. Livio notes correctly that the discovery of the phases definitively refutes the possibility of a pure geocentric system. However, it does not refute a geo-heliocentric Tychonic system. Livio admits this very grudgingly:

…but could not definitely dispose of Brahe’s geocentric-heliocentric compromise […]. This left a potential escape route for those Jesuit astronomers who were still determined to avoid Copernicanism.

Throughout his book Livio tries to imply that there is no real justification for supporting the Tychonic system, whereas it was not only the Jesuits, who did so but many other astronomers as well because the empirical evidence supported it more that a heliocentric one, of which more later. However, Livio consistently ignores this fact because it doesn’t fit his fairy-tale narrative.

Livio deals fairly conventionally with the telescopic discovery of sunspots and the discussion on their nature between Galileo and Christoph Scheiner and although he ends his account by noting the publication of Scheiner’s Rosa Ursina sive Sol (1626–1630) he makes no mention of the fact that the book is a masterpiece of astronomy, far better than anything Galileo published in the discipline. As should always be noted, due to the haste in which he wrote and published it, Sidereus Nuncius was closer to a press report than a scientific publication. He does however mention, what he calls “some further comments he made later in the book The Assayer, which the Jesuit astronomer took to be directed at him personally, did turn him into an unappeasable enemy.” Galileo actual vehemently and totally falsely accused Scheiner of plagiarism in The Assayer, which he later compounded by plagiarising Scheiner’s work in his own Dialogo. Scheiner’s antagonism is understandable. We now get the real reason why Livio keeps badmouthing the Jesuits; he sees them as behind Galileo’s trial in 1633. He writes, “This marked just the beginning of a conflict with the Jesuits, which would culminate in the punitive actions against Galileo in 1633.” This is an old myth and quite simply not true, the Jesuits did not come to Galileo defence but they were also not responsible for his trial.

We now come to objections to the telescopic discoveries:

How could anyone be sure that what Galileo was seeing was a genuine phenomenon and not a spurious artifact produced by the telescope itself?

Not only wasn’t there a convincing theory of optics a that could demonstrate that the telescope doesn’t deceive, they contended but also the validity of such a theory in itself based on mathematics, was questionable. [my emphasis]

 

Livio tries to imply that both objections are just anti-science nit picking but they are in fact very solid, very necessary scientific question that had to be asked and to be answered if people were going to accept the validity of the telescopic discovery. To the first general objection, although Galileo, an excellent observer, made none himself, there were numerous cases of published discoveries that turned out to be merely optical artefacts in the early years of telescopic astronomy. Not really surprising given the really poor quality of the instruments being used, Galileo’s included.

That an optical theory of the telescope didn’t exist was a very serious problem, as it would be with any new scientific instrument. If you can’t explain how the instrument works how do you expect people to accept the results? Kepler solved the problem with his Dioptrice published in 1611, which explained fully and scientifically how lenses and lens combinations function, describing various different types of telescope. Galileo dismissed and mocked, what is now regarded as a milestone in the history of geometrical optics. The last clause is, once again, Livio spouting total crap. Theories of optics had been geometrical, i.e. mathematical, since at least, in the fourth century BCE and even Aristotle classified optics as one of the mixed sciences, i.e. those such as astronomy that are dependent on mathematics for their proofs. Kepler’s book was accepted by all those qualified to pass judgement on the matter, with the notable exception of Galileo, who didn’t want to share the limelight with anybody, and together with Kepler’s earlier Pars Optica (1604) formed the foundations of modern scientific optics.

The reference to mathematics here is Livio’s attempt to create or propagate a myth that before Galileo, nobody conceived of a mathematics-based science. It is time to tackle that myth. Livio argues that Aristotle rejected mathematics in science and that Aristotelians regarded anything proof based on mathematics as not valid. He, of course, finds an obscure Aristotelian contemporary of Galileo’s to quote to prove this but does not quote any evidence to the contrary or even appear to think that some might exist. He is very wrong in this. Because, in Aristotle’s opinion, mathematics does no deal with the real world the results of mathematic are not episteme or scientia or as we would say knowledge. He however makes allowances for the so-called mixed sciences, astronomy, optics and statics. Livio acknowledges this status for astronomy but argues with the medieval Aristotelians that astronomical mathematical models are mere calculating devices and not models of reality; describing cosmological reality was the domain of the philosophers and not the mathematical astronomers. He also claims that this was still the situation in the second decade of the seventeenth century, it wasn’t. Beginning with Copernicus astronomers began to claim that their mathematical models were models of reality and by the time of Galileo’s first dispute with the Catholic Church this had become the generally accepted state of the discipline. The debate was which mathematical model describes the real cosmos?

It is a standard cliché in the history of science that one of the major factors that drove the so-called scientific revolution was the mathematization of science. Like many clichés there is more that a modicum of truth in this claim. Livio believes it is absolutely central and one of the major themes of his book is that Galileo was the first to mathematize science in his experiments on motion and the laws of fall. This is quite simply not true and Livio can only maintain his claim by steadfastly ignoring the history of mathematics in science prior to Galileo or did he even bother to look if there was any?

Starting with Galileo’s researches into motion and fall there is a three hundred year history of experimental and mathematical investigation into exactly this area starting with the Oxford Calculatores, who derived the mean speed theorem, which lies at the heart of the laws of fall and going down to Giambattista Benedetti (1530–1590), who produced all of the arguments and thought experiments on the subject for which Galileo is famous. There is much more, which I have already dealt with in an earlier post and won’t repeat here.Galileo knew of all of this work. The Archimedean renaissance in mathematics and the sciences, replacing the authority of Aristotle with that of Archimedes, in which Galileo is a major figure, does not start with Galileo but goes back at least to Regiomontanus (1436–1476).  The works of Archimedes were edited by Thomas Venatorius (1488–1551) and printed and published in a bilingual Greek and Latin edition in Basel in 1544. In general the sixteenth century saw a massive increase in the application of mathematics to a wide range of subjects, a development that was already well underway in the fifteenth century, including linear perspective in art, cartography, surveying, navigation, physics and astronomy. Galileo in no way started the mathematization but represents, together with several of his contemporaries such as Johannes Kepler, Simon Stevin, Christoph Clavius and Isaac Beeckman, a temporary high point in these developments. All four of those contemporaries were actually better mathematicians than Galileo.

On the question of the epistemological status of mathematical proofs, which Livio clearly states was still doubted in Galileo’s time, Christoph Clavius, who many people don’t realise was an excellent epistemologist, had already changed perceptions on this when Galileo was still a child. Clavius a Jesuit and thus by definition a Thomist Aristotelian used Aristotle’s own arguments to demonstrate that mathematical proofs have the same epistemological status as philosophical proofs. He even went to the extent of translating parts of the Elements of Euclid into Aristotelian syllogisms to show that mathematical proofs transport truth in the same way as philosophical, logical ones. Clavius’ influence was massive, he fought to get mathematics accepted as part of the educational reform programme of the Jesuits and then got the mathematical sciences established as a central part of the curriculum in Catholic schools, colleges and university also training the necessary teachers to carry out his programme. There is a reason why the young Galileo turned to Clavius, when seeking a mentor for his mathematical ambitions.

Taking all of this together the roll of mathematics and status of mathematical proofs in the sciences was very different in the early seventeenth century than the picture that Livio serves up. Far from being ground breaking Galileo’s (in)famous quote from The Assayer  “the book of nature is written in the language of mathematics” (which Livio offers up several times in his book) was actually stating a truth that had been generally accepted by many natural philosophers and mathematicians for many decades before Galileo put pen to paper.

Returning to Galileo’s telescope discoveries, Livio tells us that Kepler published his letter praising Galileo’s telescopic discoveries under the title Dissertio cum Nuncio Sidero (1610) then goes on to write: “Galileo was clearly pleased with its content, the letter was reprinted in Florence later in the year.” What Livio neglects to mention is that Galileo was responsible for that edition in Florence, which was a pirate edition published without Kepler’s knowledge and without his permission or consent. Livio makes it appear that the Jesuit astronomers of the Collegio Romano only reluctantly started to try and confirm Galileo’s discoveries and then only when ordered to do so. This is a complete distortion of what actually happened.

The astronomers in the Collegio Romano had their own telescopes and had been making astronomical telescopic observations well before Galileo published the Sidereus Nuncius. They immediately leapt on the pamphlet and set out to try and confirm or refute his observations. They had some difficulties constructing telescopes good enough to make the necessary observations and Christoph Grienberger (1561–1636), who was acting head of the school of mathematics due to Clavius’ advanced age, corresponded with Galileo, who provided copious advice and tips on observing and telescope construction. This was a work of friendly cooperation under fellow mathematicians. After some difficulties they succeeded in providing the necessary confirmation, which they made public and celebrated by throwing a banquet for Galileo when he visited Rome in 1611. As already stated above the Jesuit astronomers probably observed the phases of Venus before Galileo.

Livio then goes on to draw parallels with the fact that, “The current debate on global warming had to go […] through a similar painful [my emphasis] type of confirmation process.” I find this statement, quite frankly, bizarre coming from a scientist. All scientific discoveries have to be independently confirmed by other scientists, it is a central and highly important part of the whole scientific process. What the astronomers of the Collegio Romano did for Galileo was in no way “painful” but a necessary part of that scientific process for which Galileo was very thankful. I find it particularly bizarre given the very lively current debate on the significant number of scientific papers that have to be retracted because of failing confirmation. Reading Livio in the worst possible light, and not just here but at numerous other points in his narrative, he seems to be saying, if Galileo says it is so, then it must be true and anybody, who dares to criticise him, is in the wrong.

Of course, Livio cannot avoid the myth that, “First Copernicus and Galileo removed the Earth from its central position in the solar system.” Having previously quoted the “Copernicus principle”: the realisation that the Earth, and we human beings, are nothing special…” Also: “ What’s more the Copernican system was bound to be at odds with a worldview that had placed humans at the very center of creation, not only physically but also as a purpose and focus of for the universe’s existence.” Although geometrically central, the philosophers and astronomers in the Renaissance did not regard the Earth’s position as central in any special way. It was far more the bottom, the dregs of the universe. Trying to move the Earth into the heavens was moving it into an exalted place. At least Livio is honest enough to admit that Galileo remained blind to Kepler’s work, although Livio reduces it to just the discovery of elliptical orbits, whereas Kepler actually contributed more to modern astronomy than Copernicus and Galileo together.

Livio now moves on to Galileo’s entry into theology and his Letter to Castelli. As with all Galileo apologists, whist admitting that Galileo was trespassing in the territory of the theologians, he thinks that Galileo was right to do so and what he wrote was eminently sensible and should have been acknowledged and accepted. What Galileo did struck at the vey heart of the Reformation/Counter Reformation dispute that had been raging in Europe for one hundred years and just three years later would trigger the Thirty Years War, which devastated central Europe and resulted in the death of somewhere between one and two thirds of the entire population. The Catholic Church had always claimed that they and only they were permitted to interpret Holy Scripture. Luther claimed in opposition to this that every man should be allowed to interpret it for themselves. This led to schism and the Reformation. The Catholic Church confirmed, with emphasis, at the Council of Trent that only the Church’s own theologians were permitted to interpret the Bible. Now along comes a mere mathematicus, the lowest rang in the academic hierarchy, and cheerfully tells the theologians how to interpret the Holy Writ. The amazing thing is that they didn’t simply throw him into a foul dungeon and throw away the key.  I mentioned earlier that the Church was a judicial organ and the decisions of the Council of Trent were binding laws on all Catholics. Galileo knowingly and very provocatively broke that law and got mildly and unofficially admonished for doing so. Whatever a modern observer may think about the quality of Galileo’s theological arguments is completely irrelevant, it’s the fact that he made them at all that was the offence. However, in doing so he together with Foscarini provoked the Church into taking the heliocentric hypothesis under the microscope. He had been warned, as early as 1613, by various friends including Cardinal Maffeo Barberini, the future Pope Urban VIII not to do so.

Livio thinks that because he finds Galileo’s arguments in the Letter to Castelli reasonable and ‘because of science’ that the Catholic Church should have cut Galileo some slack and let him reinterpret the Bible. The Catholic Church should abandon their exclusive right to interpret Holy Writ, one of the fundaments of their entire religion, so that a nobody, and despite his celebrity status, in the grand scheme of things Galileo was a nobody, could promote an unproven astronomical hypothesis! This is the same exclusive right for which the same Church was prepared to engage in one of the most devastating wars in European history, just three years later. In his pseudo-historical narrative Livio has here completely lost touch with the historical context.  In fact Livio is not writing history at all but making presentist moral judgements with hindsight.

There is another bizarre statement by Livio where he writes:

All this notwithstanding, however, the Church might have still accommodated (albeit with difficulty) a hypotheticalsystem that would have made it easier for mathematicians to calculate orbits, positions, and appearances of planets and stars as long as such a system could be dismissed as not representing a true physical reality. The Copernican system could be accepted as a mere mathematical framework: a model invented so as to “save the appearances” of astronomical observations–that is, to fit the observed motion of the planets.

I am frankly baffled by this paragraph because that is exactly what the Church did in fact do. They fully accepted heliocentricity as a hypothesis, whilst rejecting it as a real physical description of the cosmos. This is shown very clearly by their treatment of Copernicus’ De revolutionibus, which unlike Kepler’s books, for example, was not placed on the Index of forbidden books but was only placed on it until corrected. This correction was carried out by 1620 and consisted only of changing or removing the comparatively few statement in the book claiming that heliocentricity was a real physical description of the cosmos. From 1621 Catholics were free to read the now purely hypothetical De revolutionibus. Livio relates all of this fairly accurately and then drops another clangour. He writes:

In reality, the modifications introduced by Cardinal Luigi Caetani and later by Cardinal Francesco Ingoli were indeed relatively minor and the publication of the revised version was approved in 1620. However, the new edition never reached the press, and so Copernicus’s book remained on the Index of Prohibited Books until 1835!

This is once again complete rubbish. The Catholic Church never intended to publish a new or revised edition of De revolutionibus. What they did was to issue the list of corrections deemed necessary and every Catholic owner of the book was expected to carry out the corrections in the own copies themselves. Quite a few obviously did and we have a number of surviving copies, including Galileo’s own private copy, with the corrections carried out according to the issued instructions. Interestingly almost all of the thus censored copies are in Italy or of Italian provenance, it seems that Catholics outside of Italy didn’t take much notice of the Vatican’s censorship order. De revolutionibuswas of course removed from the Index in 1620 having been corrected. Also, I know of no case of anyone being prosecuted for reading or owning an uncensored copy of the book.

Livio tries to counter the argument that I have presented above that Galileo was admonished because he meddled in theology by claiming that the motivation was one of anti-science. Livio. “[They] were trying only to convince Galileo not to meddle in theology, as a few modern scholars have concluded.” To counter this he brings statements from Grienberger and Bellarmino saying that elements of Copernicus theory contradict passages of Holy Writ. He writes:”[they] were quite intent on crushing the Copernican challenge as a representation of reality because, from their perspective, they were vindicating the authority of Scripture in determining truth.” Dear Dr Livio that is theology! As Bellarmino wrote in his letter to Foscarini, if a contradiction exists between Holy Writ and a proven scientific fact, the heliocentric hypothesis was of course at this point in time no where near being a proven scientific fact, then the theologians have to very carefully considered how to reinterpret Holy Writ; that is what theologians do!

This brings us to Roberto Bellarmino famous letter to Paolo Antonio Foscarini. Foscarini, a monk, had written a book defending heliocentricity and reinterpreting the Bible in a similar way to Galileo. Criticised, he sent his book to Roberto Bellarmino for his judgement; he hoped it would be favourable. The title contains the word Pythagorean, so Livio explains that the Pythagoreans thought Earth etc. orbited a central fire, therefore the comparison with Copernicus’ theory. Livio then writes, “Greek philosopher Heraclides of Pontus added, also in the fourth century BCE that the Earth rotated on its axis too…” As far as can be determined Heraclides proposed diurnal rotation in a geocentric system and not in a heliocentric or Pythagorean one.

Livio goes into a lot of detail about Foscarini’s text and Bellarmino’s letter but I will only mention two points. Livio quotes the paragraph that I have already paraphrased above, “…if there were a true demonstration that the sun is at the center of the world and the earth in the third heaven, and that the sun does not circle the earth but the earth circles the sun, then one would have to proceed with great care in explaining the Scriptures that appear contrary, and say rather that we do not understand them, than what is demonstrated is false.” Livio adds, “But I will not believe that there is such a demonstration, until it is shown me. Nor is it the same to demonstrate that by supposing the sun to be at the center and the earth in heaven one can save the appearances, and to demonstrate that in truth the sun is at the center and the earth in the heaven; for I believe the first demonstration may be available, but I have very great doubts about the second, and in case of doubt one must not abandon the Holy Scripture as interpreted by the Holy Fathers.”

This is of course eminently sensible and rational. If you want me to accept you scientific theory then show me the proof! Livio doesn’t accept this and goes of into a long diatribe, which demonstrates his own prejudices rather more than any faults in Bellarmino’s logic. He then comes with a totally spurious argument:

If two theories explain all the observed facts equally well, scientists would prefer to adopt, even if tentatively, the simpler one. Following Galileo’s discoveries, such a process would have definitely favoured the Copernican system over the Ptolemaic one, which was what Galileo had been championing all along. The requirement of simplicity would have also given an advantage to Copernicanism over Tycho Brahe’s hybrid geocentric-heliocentric model.

Ignoring the fact that the Ptolemaic system was dead in the water after the discovery of the phases of Venus and so the comparison is a waste of time, any alert reader will immediately spot the massive error in this argument. The two theories, Copernicus and Brahe, do not explain all the observed facts equally well. The Copernican system requires something very central that the Tychonic system does not, terrestrial motion. Livio adds this in a very off hand way, “Of course the ultimate test would have been to find direct proof for the Earth’s motion…” There was in fact absolutely no empirical proof of the Earth’s motion and wouldn’t be until Bradley discovered stellar aberration in 1725! To give the “advantage to Copernicanism over Tycho Brahe’s hybrid geocentric-heliocentric model” would be under the circumstances actually unscientific.

A little bit further on Livio delivers another highly spurious comment, he writes, “…but Bellarmino’s position was extremely rigid. He did not believe that a proof of Copernicanism could ever be found.” Livio is here putting words into Bellarmino’s mouth, who never said anything of the sort, rather he expressed doubt that that such a proof existed.  Livio finishes off his series of spurious attacks on Bellarmino by claiming to prove him theologically wrong. I find it slightly amusing that a twenty-first century astrophysicists claims that Bellarmino, who was universally regarded as the greatest living Catholic theologian and whose reputation as a theologian was such that at the end of his life he was both head of the Index and head of the Inquisition, was theologically wrong.

Things developed as they must and we now have Galileo rushing off to Rome to try and rescue the situation with his infamous theory of the tides. Livio explains the theory and its possible origins then he drops the following jewel:

Albeit wrong, Galileo’s commitment to mechanical easy-to-understand causation made his theory of tides at least plausible.

There is only one possible answer to this claim, bullshit! A theory that states there is only one high tide and one low tide at the same time every day, when there are in fact two of each of which the times travel around the clock over the lunar month (a strong indication of the correct theory of the tides) is anything but a plausible theory. It is as I said bullshit.

We now turn to the committee of consultors set up to examine the theological implications of heliocentricity. Livio of course has much to say against this. His first objection:

Ironically, the same office that had objected vehemently to scientists intruding into theology was now asking the theologians to judge on two purely scientific questions–two of the central tenets of he Copernican model.

Once again Livio appears to have no idea what theology is. The discipline of theology covers all forms of human activity in their entirety. There is absolutely nothing in human existence that doesn’t fall under the remit of theology. Secondly the function of the consultors in this case were being asked to examine the two central tenets of heliocentricity in relation to Catholic religious belief, not a scientific question at all.

Next up, Livio objects to the consultors themselves: “Not one was a professional astronomer or even an accomplished scientist in any discipline.” All of the consultors were highly educated, learned men, who would have had a solid instruction to Ptolemaic astronomy during there education and were more than capable of asking an expert for his advice if necessary.

Consultor: Is there any empirical evidence that the Earth moves and the Sun stands still?

Astronomer: No

Consultor: Is there any empirical evidence that the Sun and not the Earth is at the centre of the cosmos?

Astronomer: No

Simple wasn’t it.

 

The decisions of the consultors are well know:

On February 24 the Qualifiers delivered their unanimous report: the proposition that the Sun is stationary at the centre of the universe is “foolish and absurd in philosophy, and formally heretical since it explicitly contradicts in many places the sense of Holy Scripture”; the proposition that the Earth moves and is not at the centre of the universe “receives the same judgement in philosophy; and … in regard to theological truth it is at least erroneous in faith. (Wikipedia)

Foolish and absurd in philosophy is the scientific judgement and sounds somewhat harsh but can be simply translated as, is not supported by the available empirical evidence. Livio would disagree with both the judgement and my interpretation of it but it is historically fundamentally accurate. The second part of each judgement is of course the theological one. As is also well known the Pope commissioned Cardinal Bellarmino to inform Galileo of the decision and to instruct him not to hold or teach the heliocentric theory. Books, such as those of Kepler, claiming the physical reality of heliocentricity, were placed on the Index and De revolutionibus, as detailed above until corrected, which it was.

Bewilderingly Livio accuses Bellarmino and the Jesuits of failing to support Galileo against the Pope, which displays an incredible ignorance of the Catholic Church, the Pope and the Jesuit Order in the seventeenth century. As stated at the beginning the Catholic Church was a religious, political and judicial power in an age of absolutism and the Pope was an absolutist ruler. The Society of Jesus (Jesuits), and Bellarmino was also a Jesuit, is a religious order dedicated to and directly under the authority of the Pope. Livio’s accusations are totally insane.  He, of course, can’t resist making ahistorical and inaccurate comments about the decision, he writes:

The ruling made by officers of the Church for whom retaining authoritative power over areas totally outside their expertise took priority over open-minded critical thinking informed by scientific evidence.

Livio here continues to ignore/deny the simple fact that the scientific evidence in the early seventeenth century simply did not support an interpretation of heliocentricity as a physical reality and whilst it appears somewhat draconian the Church decision doesn’t actually say anything else.

Livio also launches the presentist moral outrage attack, “[some] argue that some of the responsibility for the prohibition of Copernicanism lies with Galileo himself, because he wouldn’t keep his mouth shut. Such claims are outrages.” Firstly the heliocentric hypothesis was never prohibited only the heliocentric theory, which given its scientific status at the time was in fact, although unnecessarily harsh, justifiable and secondly if Galileo had displayed somewhat more tact, instead of behaving like the proverbial bull in a china shop, things would never have taken the turn that they did.

We move on to the dispute over the nature of comets between the Jesuit astronomer Orazio Grassi and Galileo. Here Livio again displays his ignorance of the history of astronomy. He writes:

Grassi’s theory of comets deviated courageously from the Aristotelian view, which placed comets at about the distance of the Moon. Instead following Tycho Brahe, Grassi proposed that the comets were further out between the Moon and the Sun.

[…]

As to the actual nature of comets, many astronomers at the time were sill adopting Aristotle’s theory, which stated that these represented exhalations of the Earth that became visible above a certain height due to combustion, disappearing from view as soon as that inflammable material was exhausted. Grassi, however, again followed Brahe in suggesting that comets were some sort of “imitation planets.”

 

The modern debate on the nature of comets and whether they were sub- or supralunar began in the fifteenth century with Toscanelli (1397–1482), who tried to track the path of Comet Halley in 1456, as if it were a supralunar object. The debate continued in the work of Georg von Peuerbach (1423–1461), Toscanelli’s one time student, and Peuerbach’s student, Regiomontanus (1436–1476), who wrote a work on how to detect parallax in a moving comet. The debate continued in the 1530’s with many leading European astronomers taking part, including, Johannes Schöner (1477–1547), who published Regiomontanus’ work on comets, Peter Apian (1495–1552), after whom the law concerning comets’ tails in named, Copernicus (1473–1543), Gerolamo Cardano (1501–1576) and Jean Pena (1528–1558). The latter two both proposed a theory that comets were translucent, supralunar, bodies that focused the Sun’s rays like a lens creating the comets tail. Tycho’s comet, the great comet of 1577 was observed by astronomers all over Europe and Tycho, Michael Mästlin (1550-1631) and Thaddaeus Hagecius ab Hayek (1525–1600), three leading astronomers, all determined that comets were supralunar. Clavius accepted these results and included the fact that comets were supralunar in his Sphaera. This meant that the official view of the Catholic Church in general and the Jesuits in particular was that comets were supralunar. This view was confirmed again by astronomers throughout Europe observing Comet Halley in 1607. The was nothing courageous about Grassi’s theory of comets and in fact you would be hard put to it to find a serious European astronomer, apart from Galileo, who still adhered to Aristotelian cometary theory in 1618. In the same year Grassi’s Jesuit colleague Johann Baptist Cysat (c. 1587–1657), a student of Christoph Scheiner, became the first astronomer to observe a comet with a telescope giving the first ever description of a comet’s nucleus in his Mathemata astronomica de loco, motu, magnitudine et causis cometae qui sub finem anni 1618 et initium anni 1619 in coelo fulsit. Ingolstadt Ex Typographeo Ederiano 1619 (Ingolstadt, 1619). He followed Tycho Brahe in believing that comets orbited the sun. He also demonstrated the orbit was parabolic not circular.

Galileo, who due to ill health had not observed the comets of 1618, launched a vicious and insulting, unprovoked attack on Grassi’s publication, presenting a view of comets that was totally out of date, ignoring all of the accumulated scientific evidence from the last two centuries on the nature of comets just to put one over on the Jesuits and the supporters of Tycho’s theories. Livio does his best to defend Galileo’s disgusting behaviour but even he admits that Grassi was principally in the right and Galileo simply wrong. Livio goes as far as to claim that because comets has an elongated elliptical orbit (actually only some do) that Galileo’s claim that they travel in straight lines was more correct than Grassi’s claim that they orbit the Sun. In all other instances Livio goes out of his way to emphasise that hindsight shows that Galileo was right and his critics wrong so why the opposite tack here? Comets do orbit the Sun. Livio scrabbles around in the cesspit that is Galileo’s paper on comets looking for crumbs for which he can give Galileo credit.

Livio now criticises Grassi’s answer to Galileo’s attack because it contained sarcastic attacks on Galileo. Talk about pot calling the kettle black. He even brings up the obtuse suggestion that it was actually written by Christoph Scheiner because of his antagonism towards Galileo. This theory has a small problem; Scheiner only became antagonistic towards Galileo after Galileo had viciously insulted him in The Assayer, a publication that still lay in the future. Livio’s whole account of the affair is biased in Galileo’s favour so that it serves as a lead up to The Assayer, for the time being the last document in the dispute, because, as already mentioned, Livio sees it as the document in which Galileo established the place of mathematics in science. Livio’s account of The Assayer and its significance is more than somewhat outlandish.

With very little evidence to base this opinion upon, Galileo thought in 1623 that he knew the answer: the universe “is written in the language of mathematics.” It was this dedication to mathematics that raised Galileo above Grassi and the other scientist of his day, even when his specific arguments fell short of convincing–and even though he assigned to geometry a more important role than it seemed to deserve at the time. His opponents, he wrote, “failed to notice that to go against geometry is to deny truth in broad daylight.”

This whole paragraph contains so much that is wrong that it is difficult to know where to start.  I have already explained above that by the time Galileo wrote this infamous piece of purple prose it was widely accepted by both mathematician and natural philosophers that the future of science lay in an intensive mathematization. A process that was well under way when Galileo wrote something that was not new and sensational but a common place. A lot of contemporary scientists were dedicated to mathematics, such as Johannes Kepler, Simon Steven and Isaac Beeckman. In fact the last two both contributed at least as much to the development of mathematical physics in the seventeenth century as Galileo if not more. Unfortunately their achievements tend to get blended out on the popular level by the Galileo myth machine of which, Livio is just the latest in a long line of operators.

To raise Galileo above Grassi because of his dedication to mathematics is more than a joke; it’s grotesque. Earlier in his account of the dispute between Grassi and Galileo, Livio acknowledged that Grassi was an excellent optical physicist and an equally excellent architect both disciplines are fundamentally mathematical disciplines. He also points out that Grassi succeeded Grienberger as professor for mathematics at the Collegio Romano, who had succeeded Clavius. The chair for mathematics at the Collegio Romano was unique in European universities. Clavius had set up what we would now call an institute for advanced mathematics, a roll that both Grienberger and Grassi kept alive. This institute was dedicated to exemplifying, establishing and developing the roll of mathematics in the sciences. The Collegio Romano was quite simply the most advanced school for mathematics and its application anywhere in Europe. As far as geometry goes the standard textbook for geometry throughout most of the seventeenth century was Christoph Clavius’ Euclides Elementorum Libri XV, Rom 1574, note the date. This was not simply a new translation of Euclid’s classic but a modernised, simplified, streamlined textbook that was used extensively by both Catholic and Protestant educational establishments; the last edition was printed in 1717.

Shortly after the above passage on Galileo’s supposed revolutionary thoughts on mathematics we get the following throwaway line:

Galileo introduced the revolutionary departure from the medieval, ludicrous notion that everything worth knowing was already known.

When I read this I didn’t know whether to laugh, cry, rip my hair out (if I had any), or simply go out and throw myself off a high cliff in the face of such imbecilic drivel. I strongly suspect that any of my history of medieval science friends and colleagues will react similarly should they happen to read the above sentence. Starting at the very latest with the translation movement in the twelfth century medieval science was an evolving developing field with advances in a wide range of disciplines. The medieval scholars laid the foundations upon which Galileo built his own achievements. I would be quite happy to give Dr Livio a very long reading list of good books on medieval science to help him find a way out of his ignorance.

At the end of his chapter on The Assayer Livio warms up the old discovery of Pierto Redondi that Galileo was denounced to the Inquisition for the bits of primitive atomism contained in The Assayer. This was indeed true but the accusation was dismissed and nothing came of it, as Livio admits. Livio, however, now writes a whole paragraph about how important atomism, he actually means particle physics, is in modern physics, mentioning quarks, leptons, gage bosons etc., etc. I wonder how Livio would react if he knew that the principle source of atomism in the seventeenth century is now considered to be the German alchemist Daniel Sennert (1572–1637) reviving the theories of the thirteenth century alchemist Paul of Taranto. You remember alchemy one of those fictitious fields together with astrology that scientists sometime connected to.

Next up the Dialogo: Livio acknowledges that there were external political and social factors that affected the situation within the Vatican in the years leading up to the publication of the Dialogo. He starts with the astrological scandal. In 1630 an astrological prognostication predicting the Pope’s death was made and circulated by, to quote Livio, the abbot of Saint Praxedes in Rome. Livio then tells us, “some of Galileo’s adversaries tried to pin the blame on Galileo…” What Livio neglects to mention is that although Galileo was in this case innocent there were plausible ground for suspecting him, it was a case of guilt by association. Firstly, Galileo was known to be a practicing astrologer. Secondly, the abbot of Saint Praxedes, Orazio Morandi had been a good friend of Galileo’s since at least 1613. Thirdly, following an audience with the Pope concerning the forthcoming Dialogo in 1630, Galileo took part in a supper with Moriandi in Saint Praxedes together with Rafaello Visconti (Master of the Sacred Palace), another friend of Galileo’s, who read the manuscript of the Dialogo for Niccolò Ricardi the censor, who never actually read it, and an appraiser of the Inquisition. When Morandi was arrested for his horoscope and thrown into the Vatican’s dungeon, Visconti was also implicated and banished from the Vatican. That Galileo came under suspicion by association is hardly surprising. This was not a plot against Galileo as Livio claims.

We then have a wonderfully mangled piece of history from Livio, who write:

Unfortunately, this was not the end of the trials and tribulations Galileo had to endure for the publication of the Dialogo. Most significant of these was the sudden death on August 1, 1630, of Federico Cesi, the founder and sole source of funding for the Accademia dei Lincei. As a result the printing had to be done in Florence, outside of Riccardi’s jurisdiction. After some negotiations, it was agreed that Father Jacinto Stefani, a consultor of the Inquisition in Florence, would be in charge, but only after Riccardi approved the introduction and conclusion.

Although Cesi’s death was a serious blow to Galileo’s plans because he Cesi was supposed to finance the publication of the Dialogo, but this was not the reason why it was published in Florence and not in Rome. What actually happened is that after Galileo had returned to Florence from Rome with his manuscript the plague broke out in Florence. Restrictions on travel imposed by the authorities meant that Galileo could not return to Rome to oversee the printing and publication of his book. He requested permission from Riccardi to get the book published in Florence instead, but as already mentioned Riccardi hadn’t actually read the book intending to review the pages as they came of the printing press instead, having accepted Visconti’s recommendation. Riccardi was now in a pickle and wanted Galileo to send him a copy of the manuscript but due to the immense cost of producing such a copy, Galileo was very reluctant to do so.  Riccardi agreed to Galileo just sending the introduction and conclusion to Rome to be controlled and the rest being controlled in Florence by Stefani. Galileo and his circle of supporters now manipulated and even oppressed the two censors and played them against each other. The result was that the imprimatur was granted by Stefani under the impression that Ricarrdi had already cleared the manuscript for publication in Rome, he hadn’t, without actually controlling the text himself. Galileo had an imprimatur that had been obtained under false pretences, which meant that he didn’t actually have an imprimatur at all. All of this came out during the investigations following publication, which contributed to Galileo’s being prosecuted but did not play a role in the actual trial.

All of this, which Livio doesn’t mention at all, is important because when dealing with the trial Livio several times emphasises that the Church had given Galileo to publish the book as it was because he had not one but two imprimaturs, whereas in fact formally he didn’t have one at all.

Livio now tells us:

There is a certain sleight of hand in the title. [Dialogue Concerning the Two Chief Systems of the World, Ptolemaic and Copernican, Propounding Inconclusively in the Philosophical Reasons as Much for the One Side as for the Other] Even if one were to ignore the fact that the Aristotelian and the Ptolemaic systems were not identical, there was at least one other world system that in terms of agreement with observations was superior to the Ptolemaic: Tycho Brahe’s Hybrid system in which the planets revolved around the Sun, but the Sun itself revolved around the Earth. Galileo always regarded that system as unnecessarily complex and contrived, and he also thought that he’d found proof for the Earth’s motion through the phenomenon of the tides, so in striving to hand Copernicanism a clear victory (although formally the book was inconclusive) he probably didn’t want to confuse the issue with superfluous qualifications.

Once again so much to unpick. Livio obviously doesn’t understand that the system propagated by the Catholic Church before Copernicus was an uneasy mixture of Ptolemaic astronomy and Aristotelian cosmology, not Aristotelian astronomy, which is a whole different kettle of fish that had been revived by some in the sixteenth century and against which Clavius had fought tooth and nail. In fact he devotes much more space to refuting the Aristotelian homocentric astronomy in his Sphaera than he devotes to refuting Copernicus. The developments in astronomy since Copernicus published De revolutionibus had left Aristotelian cosmology in shreds and Clavius had been quite happy to also jettison that, so for Clavius, speak the Catholic Church, the world system was simply the Ptolemaic.

In fact Galileo’s whole title and thus his whole book is a complete sham By 1630 the two chief systems of the world were the Tychonic system and Johannes Kepler’s elliptical heliocentric system, which was regarded as separate from and as a competitor to Copernicus’ system. The Ptolemaic system had been killed off by the discovery of the phases of Venus and the plausible assumption that Mercury would also orbit the Sun as its general behaviour was identical to that of Venus; the phases of Mercury were first observed in 1639. Galileo just used Ptolemy as a fall guy for his sham Copernican victory. Copernicus’ heliocentric system had been rendered totally obsolete by Kepler’s discovery of the three laws of planetary motion, empirically based mathematical laws I would point out, which Galileo just completely ignored clinging to Copernicus’ ‘unnecessarily complex and contrived’ system of deferents and epicycles. Livio’s dismissal of the Tychonic system as ‘superfluous qualifications’ is put quite simply a joke, especially given that the Tychonic system was at the time the leading contender as the world system because of the failing evidence of terrestrial motion.

Livio without realising it now points out the central problem with the Dialogo:

The Dialogo is one of the most engaging science texts ever written. There are conflicts and drama, yes, but also philosophy, humor, cynicism, and poetic usage of language, so that the sum is much more than its parts.

All of the above is true except that as a piece of astronomy the sum is much less than its parts, which I will explain shortly. There is no doubt whatsoever that for all of his undeniably polymathic talents, Galileo’s greatest gift was as a polemicist. A friend of mine, who is a Galileo expert, calls him the first science publicist and this is a function that he carried out brilliantly. Yes, the Dialogo is a brilliant piece of literature, which is probably unequalled by any other scientific publication in the entire history of science. However, its literary brilliance appears to have blinded many of its readers to the fact that as a piece of astronomy it’s total crap.

As already mention, Galileo struts on to the stage to discuss what he calls the two chief world systems but actually delivers up is a sham battle between two obsolete and refuted systems. He clung stubbornly to his completely false theory that comets are mere optical illusions originating on the Earth against a mass of solid, empirical, scientific evidence that comets were in fact supralunar celestial objects that orbited the Sun. Something that Galileo was no prepared to accept because it was first proposed by Tycho, who saw it as supporting evidence for his system. He clung to Copernicus’ deferents and epicycles rather than acknowledge Kepler’s much simpler, empirically proven elliptical orbits. In fact, Galileo completely ignores Kepler’s three laws of planetary motion, by far and away, the best scientific supporting evidence for a heliocentric system because if he did acknowledge them he would have to hand the laurels for proving the superiority of the heliocentric system to Kepler instead of winning them for himself, his one and only aim in the whole story. Last but by no means least he structures his whole book and his argument around his totally ludicrous theory of the tides. One of the greatest mysteries in Galileo’s life is why he, an undeniably brilliant scientist, clung so tenaciously to such an obviously bankrupt theory.

Galileo’s masterwork sailed majestically past the actually astronomy debate in the 1630s and played little or no role in the ensuing astronomical discussion of the seventeenth century in which it was largely ignored being of no real relevance. It only became crowned as a classic in the late eighteenth and early nineteenth centuries when Galileo was declared to be a scientific martyr

Livio, like so many others, blinded by the radiance of Galileo’s rhetoric sees the matter somewhat differently. In a surprisingly short presentation of the book he praises Galileo’s achievements. There are a couple of minor points that I would like to pick up on, Livio delivers up once again the myth of heliocentricity removing the Earth from its central place in the cosmos:

More important, the act of removing humans from their central place in the cosmos was too brutal to be remedied by some philosophical pleasantries at the end of a debate from a very different tone.

The whole central place in the cosmos myth is one created in the late eighteenth century and I know of no seventeenth century use of it to criticise the heliocentric hypothesis. In a bit of waffle towards the end of this chapter Livio says the “He [Galileo] did his best…” If Galileo had truly done his best he would not have ignored the most compelling evidence for the heliocentric hypothesis, Kepler’s laws of planetary motion. He goes on to say that, “History has indeed proved that Galileo was right,” it hasn’t Galileo was wrong and Kepler was right.

Livio gives a fairly short and largely accurate account of Galileo’s trial by the Inquisition and the events leading up to following the publication of the book. The only major error being, as mentioned above, his insistence that the book had two imprimaturs. Livio acknowledges that the judgement of the three clerics, commissioned to read the book and determine whether Galileo taught or defended in anyway the heliocentric theory, that he had in fact done just that and thus broken the order from 1616 was correct. Although he can’t avoid a dig at Melchior Inchofer, the Jesuit under the three. This was the charge that was brought against Galileo and of which he was found guilty. He also can’t avoid turning up the emotional rhetoric, “What happened on the following day remains one of the most shameful events in our intellectual history.” Galileo deliberately and wilfully broke the law and received the standard punishment for having done so, which included abjuring. There is an old saying under criminals, if you can’t do the time don’t do the crime. Galileo was arrogant enough to think that he could put one over on the Catholic Church and get away scot-free, it turned out that he couldn’t.

We get a short, once again, rather gushing account of the Discorsi, Galileo real claim to fame but Livio rather spoils it by once again trying to claim that Galileo created modern science.

Through an ingenious combination of experimentation (for example, with inclined planes), abstraction (discovering mathematical laws), and rational generalisation (understanding that the same laws apply to all accelerated motions), Galileo established what has since become the modern approach to the study of all natural phenomena.

Although in the case of the studies presented by Galileo in the Discorsi he proved himself to be an excellent experimental scientist, all of these things had been done by others before Galileo and independently by others contemporaneously to Galileo. He was only one amongst other who helped to establish this methodology. Galileo was part of the evolution of a new scientific methodology that had started long before he was born and which he did not initiate. Like many others before him Livio also falsely attributes Newton’s first law, the principle of inertia, to Galileo. Whilst Galileo did indeed produce a version of the principle of inertia, Newton took his first law from the works of René Descartes, who in turn had taken it from Isaac Beeckman, who had formulated it independently of Galileo.

The next chapter of Livio’s book is an obtuse story of an account of the Galileo affair commissioned by the Vatican in the 1940s and then not published but then published under the name of a different author in the 1960s. The sole aim of this chapter is simply to take another gratuitous swing at the Catholic Church. The book closes with a fairly long digression on Einstein’s views on science and religion, which brings us to a major problem with the book, apart from the historical inaccuracies, it tries to be too many things at once.

One thing that I have mentioned in passing is Livio’s attempts to draw parallels between what happened to Galileo and the current crop of science deniers. The analogies simply don’t work because no matter how hard Livio tries to claim the opposite, Galileo’s critics in astronomy, especially the Jesuits, were not science deniers but just as much scientists as Galileo, who argued for an equally valid, in fact empirically more valid, system of astronomy, the Tychonic one, as Galileo’s heliocentric system. All the way through the book Livio keeps trying to disqualify the Tychonic system as unscientific but in the first half of the seventeenth century it was just as scientific as the heliocentric hypothesis. The only person practicing science denial here is Livio. He also wants to present the book as a discussion of the general relationship between science and religion but the whole time he argues from a presentist standpoint and refuses to view the relationship in Galileo’s time in its correct historical context. Lastly he actually wants to sell the book as a new biography of Galileo presented with the insights of a working astrophysicist, his own claim at the beginning of the book. Unfortunately it is here that he fails most.

He enters his story with a preconceived image of Galileo as a white knight on his mighty charger fighting for freedom of speech and freedom of thought in the sciences and as the originator and creator of modern experimental and mathematical science. With this image firmly in mind, from the start of his narrative, he fills out the picture with a classic case of confirmation bias. He completely ignores any real facts from the history of science that might force him to rethink his preconceived image of his hero. There is no mistaking the fact that is a strong element of hero worship in Livio’s vision of Galileo. Instead of describing the real state of science in the early seventeenth century, he present the reader with a comic book version of Aristotelian philosophy from the thirteenth century making it easier for him to present Galileo as some sort of superman, who dragged natural philosophy kicking and screaming into the modern world, whilst singlehandedly creating modern science. Edward Grant the eminent historian of medieval science (a discipline that Livio probably thinks doesn’t exist, because he seems to think that there was no medieval science), once very perceptively wrote that Aristotelian philosophy was not Aristotle’s philosophy and went on to point out that it is very difficult to define Aristotelian philosophy, as it kept on evolving and changing down the centuries. The Thomist philosophy of the Jesuits in the first third of the seventeenth century was a very different beast to the Aristotelian philosophy that Thomas Aquinas propagated in the thirteenth century. The historical distortions that Livio presents would be funny if they weren’t so grotesque.

On the question of Galileo being ‘a symbol of the fight for intellectual freedom, a lifetime of studying and thinking about Early Modern science has brought me to the conclusion that he wasn’t. In my opinion Galileo didn’t really care about such abstractions as freedom of thought, freedom of speech or intellectual freedom, all he cared about was his own vainglory. As Mario Biagioli clearly shows in his Galileo Courtier,[2] Galileo was a social climber. He was a relatively unknown, middle aged, professor of mathematics, who overnight became the most celebrated astronomer in Europe because of his telescopic discoveries. Alone the way he presented those discoveries shows his principle aim was to see what he could gain socially from them. Galileo loved his celebrity status and revelled in it. His engagement for heliocentricity was all motivated by the thought that if he could prove it true, then he would become even more famous and even more feted. To achieve this aim he lied, cheated and plagiarised. He attacked and viciously stomped on all those he regarded as competitors in his strivings for fame and adulations. He also deliberately ignored any evidence for heliocentricity presented by others (see Kepler’s laws of planetary motion) that might mean that they get the laurels and not he. Galileo might have been a great scientist but he was also a vain egoist. I think all of this might go someway to explaining his extraordinary blindness to the enormous inadequacies of his theory of the tides.

Reading this book made me very angry. The only positive thing I can say about it is that Livio is an excellent writer and the book is very well written and easy to read, but in the end even this must be viewed negatively. Mario Livio is a prominent scientist and the very successful author of popular books on mathematics and science. Because of his reputation non-specialist journals will have glowing reviews of his book, mostly written by people, who are neither Galileo experts and nor historians of science. If it follows the normal pattern for such books, specialist journals and professional historians of science will decline to review it, because it’s a pop book. The book will almost certainly become a genre bestseller and another generation of readers will acquire a mythical image of Galileo Galilei and a totally false impression of Renaissance science, something I have battled against in the eleven years that I have been writing this blog.

[1] Mario Livio, Galileo and the Science Deniers, Simon & Schuster paperbacks, New York, London, Toronto, Sydney, New Delhi, 2020

[2] Mario Biagioli, Galileo Courtier: The Practice of Science in the Culture of Absolutism, University of Chicago Press, Chicago & London, ppb. 1994

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May 27, 2020 · 8:35 am

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

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The seventeenth century is commonly called the scientific revolution principally for the emergence of two branches of science, although much more was actually going on. Firstly, the subject of this series, astronomy, and secondly the branch of science we now know as physics. The name physics had a significantly different meaning in the medieval Aristotelian philosophy. As we know astronomy and physics are intimately connected, in fact, intertwined with each other and this connection came into being during the seventeenth century. We have already seen in an earlier episode how the modern concepts of motion began to emerge from Aristotelian philosophy in the sixth century reaching a temporary high point in the early seventeenth century in the works of Galileo and Beeckman.

Galileo is often regarded as the initiator, founder of these developments and lauded with titles such as the father of physics, which is just so much irrelevant verbiage. In fact as we saw in the case of the laws of fall he was just following developments that had long preceded him. On a more general level the situation is no different. Kepler was apparently the first to use the concept of a physical force rather than a vitalist anima. Simon Stevin (1548–1620) resolved the forces acting on an object on an inclined plane, effectively using the parallelogram of forces to do so. In hydrostatics he also discovered the so-called hydrostatic paradox i.e. that the pressure in a liquid is independent of the shape of the vessel and the area of the base, but depends solely on its depth. Thomas Harriot (c. 1560–1621) actually developed a more advanced mechanics than Galileo but as usually didn’t publish, so his work had no impact. However, it clearly shows that Galileo was by no means the only person considering the problems. All of these early discoveries, including Galileo’s, suffered from a problem of vagueness. Nobody really knew what force was and the definitions of almost all the basic concepts–speed, velocity, acceleration etc.–were defective or simply wrong. The century saw the gradual development of a vocabulary of correctly defined terms for the emerging new physics and a series of important discoveries in different areas, mechanics, statics, hydrostatics, optics etc.

I’m not going to give a blow-by-blow history of physics in the seventeenth century, I would need a whole book for that, but I would like to sketch an aspect that in popular accounts often gets overlooked. The popular accounts tend to go Galileo–Descartes–Newton, as if they were a three-man relay team passing the baton of knowledge down the century. In reality there were a much larger community of European mathematicians and proto-physicists, who were involved, exchanging ideas, challenging discovery claims, refining definitions and contributing bits and pieces to big pictures. Each of them building on the work of others and educating the next generation. What emerged was a pan European multidimensional cooperative effort that laid the foundations of what has become known as classical or Newtonian physics, although we won’t be dealing with Newton yet. Once again I won’t be able to give all the nodes in the network but I hope I can at least evoke something of the nature of the cooperative effort involved.

I will start of with Simon Stevin, a man, who few people think of when doing a quick survey of seventeenth century physics but who had a massive influence on developments in the Netherlands and thus, through connections, in France and further afield. Basically an engineer, who also produced mathematics and physics, Stevin motivated Maurits of Nassau, Stadholder of the young Dutch Republic to establish engineering and the mathematical sciences on the new Dutch universities. Stevin’s work influenced both the Snels, Rudolph (1546–1613) and his son Willebrord (1580–1626), the latter translated Stevin’s work into French and Latin from the Dutch, making it available to a much wider audience.

Simon-stevin

Source: Wikimedia Commons

Stevin set up a school for engineering at the University of Leiden with Ludolph van Ceulen as the first professor of mathematics teaching from textbooks written by Stevin. Van Ceulen (1540–1610), who was Willebrord Snel’s teacher, was succeeded by his pupil Frans van Schooten the elder (1581–1646), whose pupils included his own son, Frans van Schooten the younger (1615–1660), Jan de Witt (1625–1672), Johann Hudde (1628–1704), Hendrick van Heuraet (1633–1660?), René-François de Sluse ((1622–1685) and Christiaan Huygens (1629–1695), all of whom would continue their mathematical development under van Schooten junior and go on to make important contributions to the mathematical sciences. An outlier in the Netherlands was Isaac Beeckman (1588–1637), a largely self taught natural philosopher, who made a point of seeking out and studying Stevin’s work. This group would actively interact with the French mathematicians in the middle of the century.

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Portrait of Frans van Schooten the younger by Rembrandt Source: Wikimedia Commons

On the French side with have a much more mixed bunch coming from varying backgrounds although Descartes and Mersenne were both educated by the Jesuits at the College of La Flèche. Nicolas-Claude Fabri de Peiresc (1580–1637), the already mentioned René Descartes (1596–1650) and Marin Mersenne (1588–1648), Pierre de Fermat (1607–1665), Pierre Gassendi (1592–1655), Ismaël Boulliau (1605–1694) and Blaise Pascal (1623–1662) are just some of the more prominent members of the seventeenth century French mathematical community.

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Pierre de Fermat artist unknown Source: Wikimedia Commons

René Descartes made several journeys to the Netherlands, the first as a soldier in 1618 when he studied the military engineering of Simon Stevin. He also got to know Isaac Beeckman, with whom he studied a wide range of areas in physics and from who he got both the all important law of inertia and the mechanical philosophy, borrowings that he would later deny, claiming that he had discovered them independently. Descarte and Beeckman believed firmly on the necessity to combine mathematics and physics. Beeckman also met and corresponded with both Gassendi and Mersenne stimulating their own thoughts on both mathematics and physics.

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René Descartes at work Source: Wikimedia Commons

On a later journey to the Netherlands Descartes met with Frans van Schooten the younger, who read the then still unpublished La Géometrié. This led van Schooten to travel to Paris where he studied the new mathematics of both living, Pierre Fermat, and dead, François Viète (1540–1603), French mathematicians before returning to the Netherlands to take over his father’s professorship and his group of star pupils. Descartes was also a close friend of Constantijn Huygens (1596–1687), Christiaan’s father.

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Source: Wikimedia Commons

Peiresc and Mersenne were both scholars now referred to as post offices. They both corresponded extensively with mathematicians, astronomers and physicists all over Europe passing on the information they got from one scholar to the others in their networks; basically they fulfilled the function now serviced by academic journals. Both had contacts to Galileo in Italy and Mersenne in particular expended a lot of effort trying to persuade people to read Galileo’s works on mechanics, even translating them into Latin from Galileo’s Tuscan to make them available to others. Mersenne’s endeavours would suggest that Galileo’s work was not as widely known or appreciated as is often claimed.

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Nicolas-Claude Fabri de Peiresc by Louis Finson Source: Wikimedia Commons

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Marin Mersenne Source: Wikimedia Commons

Galileo was, of course, by no means the only mathematician/physicist active in seventeenth century Italy. The main activist can be roughly divided in two groups the disciples of Galileo and the Jesuits, whereby the Jesuits, as we will see, by no means rejected Galileo’s physics. The disciples of Galileo include Bonaventura Francesco Cavalieri (1598–1647) a pupil of Benedetto Castelli (1578­–1643) a direct pupil of Galileo, Evangelista Torricelli (1608–­1647) another direct pupil of Galileo and Giovanni Alfonso Borelli (1608-1679) like Cavalieri a pupil of Castelli.

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Benedetto Castelli artist unknown Source: Wikimedia Commons

On the Jesuit side we have Giuseppe Biancani (1565–1624) his pupil Giovanni Battista Riccioli (1598–1671) and his one time pupil and later partner Francesco Maria Grimaldi (1618–1663) and their star pupil Giovanni Domenico Cassini (1625–1712), who as we have already seen was one of the most important telescopic astronomers in the seventeenth century. Also of interest is Athanasius Kircher (1602–1680), professor at the Jesuit University, the Collegio Romano, who like Peiresc and Mersenne was an intellectual post office, collecting scientific communications from Jesuit researchers all over the world and redistributing them to scholars throughout Europe.

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Athanasius Kircher Source: Wikimedia Commons

Looking first at the Jesuits, Riccioli carried out extensive empirical research into falling bodies and pendulums. He confirmed Galileo’s laws of fall, actually using falling balls rather than inclined planes, and determined an accurate figure for the acceleration due to gravity; Galileo’s figure had been way off. He was also the first to develop a second pendulum, a device that would later prove essential in determining variation in the Earth’s gravity and thus the globes shape.

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Riccioli as portrayed in the 1742 Atlas Coelestis (plate 3) of Johann Gabriel Doppelmayer. Source: Wikimedia Commons

Grimaldi was the first to investigate diffraction in optics even giving the phenomenon its name. Many of the people I have listed also did significant work in optics beginning with Kepler and the discovery of more and more mathematical laws in optics helped to convince the researchers that the search for mathematical laws of nature was the right route to take.

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Francesco Maria Grimaldi Source: Wikimedia Commons

As we saw earlier Borelli followed Kepler’s lead in trying to determine the forces governing the planetary orbits but he also created the field of biomechanics, applying the newly developed approaches to studies of muscles and bones.

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Giovanni Alfonso Borelli Source: Wikimedia Commons

Torricelli is, of course, famous for having invented the barometer, a device for measuring air pressure, of which more in a moment, he was trying to answer the question why a simple air pump cannot pump water to more than a height of approximately ten metres. However, most importantly his experiments suggested that in the space above the mercury column in his barometer there existed a vacuum. This was a major contradiction to traditional Aristotelian physics, which claimed that a vacuum could not exist.

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Evangelista Torricelli by Lorenzo Lippi (c. 1647) Source: Wikimedia Commons

Torricelli’s invention of the barometer was put to good use in France by Blaise Pascal, who sent his brother in law, Périer, up the Puy de Dôme, a volcano in the Massif Central, carrying a primitive barometer. This experiment demonstrated that the level of the barometer’s column of mercury varied according to the altitude thus ‘proving’ that the atmosphere had weight that lessened the higher one climbed above the earth’s surface. This was the first empirical proof that air is a material substance that has weight. One person, who was upset by Torricelli’s and Pascal’s claims that the barometer demonstrates the existence of a vacuum, was René Descartes to whom we now turn.

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Painting of Blaise Pascal made by François II Quesnel for Gérard Edelinck in 1691 Source: Wikimedia Commons

Descartes, who is usually credited with being one of, if not the, founders of modern science and philosophy, was surprisingly Aristotelian in his approach to physics. Adopting Beeckman’s mechanical philosophy he thought that things could only move if acted upon by another object by direct contact; action at a distance was definitely not acceptable. Aristotle’s problem of projectile motion, what keeps the projectile moving when the contact with the projector breaks was solved by the principle of inertia, which reverses the problem. It is not longer the question of what keeps the projectile moving but rather what stops it moving. He also, like Aristotle, adamantly rejected the possibility of a vacuum. His solution here was to assume that all space was filled by very fine particles of matter. Where this theory of all invasive particles, usually called corpusculariansim, comes from would takes us too far afield but it became widely accepted in the second half of the seventeenth century, although not all of its adherents rejected the possibility of a vacuum.

Descartes set up laws of motion that are actually laws of collision or more formally impact. He starts with three laws of nature; the first two are basically the principle of inertia and the third is a general principle of collision. From these three laws of nature Descartes deduces seven rules of impact of perfectly elastic (i.e. solid) bodies. Imagine the rules for what happens when you play snooker or billiards.  The details of Descartes rules of impact needn’t bother us here; in fact his rules were all wrong; more important is that he set up a formal physical system of motion and impact. Studying and correcting Descartes rules of impact was Newton’s introduction to mechanics.

Turning to another Frenchman, we have Ismaël Boulliau, who was a convinced Keplerian. Kepler had hypothesised that there was a force emanating from the Sun that swept the planets around their orbits, which diminished inversely with increasing distance from the Sun. Boulliau didn’t think that such a force existed but if it did then it would be an inversed square force in analogy to Kepler’s law for the propagation of light; a candidate for the first modern mathematical law of physics. The foundations of the new physics were slowly coming together.

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Ismaël Boulliau portrait by Pieter van Schuppen Source: Wikimedia Commons

Our last link between the Dutch and French mathematicians is Christiaan Huygens. Huygens initially took up correspondence with Mersenne around 1648; a correspondence that covered a wide range of mathematics and physics. In 1655 he visited Paris and was introduced to Boulliau and a year later began correspondence with Pierre Fermat. Frans van Schooten the younger continued to act as his mathematical mentor.

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Christiaan Huygens by Caspar Netscher, 1671, Museum Boerhaave, Leiden Source: Wikimedia Commons

Huygens absorbed the work of all the leading European mathematician and physicists and as an avowed Cartesian became acknowledged as Europe’s leading natural philosopher. He realised that Descartes rules of impacts were wrong and corrected them. Huygens was also the first to derive and state what is now know as Newton’s second law of motion and derived the law of centripetal force, important steps on the route to a clear understanding of forces and how they operate. Huygens also created the first functioning pendulum clock in the process of which he derived the correct formula for the period of an ideal mathematical pendulum. It is very easy to underestimate Huygens contributions to the development of modern physics; he tends to get squeezed out between Descartes and Newton.

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Spring-driven pendulum clock, designed by Huygens, built by instrument maker Salomon Coster (1657), and a copy of the Horologium Oscillatorium. Museum Boerhaave, Leiden Source: Wikimedia Commons

All the way through I have talked about the men, who developed the new physics as mathematicians and this is highly relevant. The so-called scientific revolution has been referred to as the mathematization of science, an accurate description of what was taking place. The seventeenth century is also known as the golden age of mathematics because the men who created the new physics were also at the same time creating the new mathematical tools needed to create that physics. An algebra based analytical mathematics came to replace the geometric synthetic mathematics inherited from the Greeks.

Algebra first entered Europe in the twelfth century with Robert of Chester’s translation of Muḥammad ibn Mūsā al-Khwārizmī’s ninth century Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing), the word algebra coming from the Arabic al-ğabr, meaning completion or setting together (in Spanish an algebraist is a bone setter). This introduction had little impact. It was reintroduced in the thirteenth century by Leonardo of Pisa, this time as commercial arithmetic, where it, especially with the introduction of double entry bookkeeping, had a major impact but still remained outside of academia.

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Leonardo of Pisa Liber Abaci

It was first in the sixteenth century that algebra found its way into academia through the work of Scipione del Ferro (1465–1526), Niccolò Fontana known as Tartaglia (c.1499–1557)and above all Gerolamo Cardano (1501–1576), whose Artis Magnæ, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra) published by Johannes Petreius (c. 1496–1550) in Nürnberg in 1545 is regarded as the first modern algebra textbook or even the beginning of modern mathematics (which, as should be obvious to regular readers, is a view that I don’t share).

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Source: Wikimedia Commons

Modern readers would find it extremely strange as all of the formulas and theorems are written in words or abbreviations of words and there are no symbols in sight. The status of algebra was further established by the work of the Italian mathematician Rafael Bombelli (1526–1572),

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Source: Wikimedia Commons

(1572)

Another school of algebra was the German Cos school represented by the work of the

German mathematician Michael Stifel (1487–1567), Arithmetica integra (1544),

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Source: Wikimedia Commons

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Michael Stifel’s Arithmetica Integra (1544)

Simon Stevin in the Netherlands L’arithmétique (1585)

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and Robert Recorde (c. 1512–1558) in Britain with his The Whetstone of Witte (1557).

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The passage in The Whetstone of Witte introducing the equals sign Source: Wikimedia Commons

Algebra took a new direction with the French mathematician François Viète (1540–1603),

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François Viète Source: Wikimedia Commons

who wrote an algebra text based on the work of Cardano and the late classical work of Diophantus of Alexandria (2nd century CE) In artem analyticam isagoge (1591) replacing many of the words and abbreviations with symbols.

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Algebra was very much on the advance. Of interest here is that Galileo, who is always presented as the innovator, rejected the analytical mathematics, whereas the leading Jesuit mathematician Christoph Clavius (1538–1612), the last of the staunch defenders of Ptolemaic astronomy, wrote a textbook on Viète’s algebra for the Jesuit schools and colleges.  Two further important publications on symbolic algebra in the seventeenth century were the English mathematician, William Oughtred’s Clavis Mathematicae (1631),

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which went through several editions and was read all over Europe and Thomas Harriot’s unnamed (1631), the only one of his scientific works ever published and that only posthumously.

The development of the then new analytical mathematics reach a high point with the independent invention by Pierre Fermat and René Descartes of analytical geometry, which enabled the geometrical presentation of algebraic functions or the algebraic presentation of geometrical forms; a very powerful tool in the armoury of the mathematical physicists in the seventeenth century. Fermat’s pioneering work in analytical geometry (Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum) was circulated in manuscript form in 1636 (based on results achieved in 1629) This manuscript was published posthumously in 1679 in Varia opera mathematica, as Ad Locos Planos et Solidos Isagoge (Introduction to Plane and Solid Loci).

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Descartes more famous work was published as La Géometrié, originally as an appendix to his Discours de la méthode (1637). However, much more important for the dissemination of Descartes version of the analytical geometry was the expanded Latin translation produced by Frans van Schooten the younger with much additional material from van Schooten himself, published in 1649 and the second edition, with extra material from his group of special students mentioned above, in two volumes 1659 and 1661. Van Schooten was the first to introduce the nowadays, ubiquitous orthogonal Cartesian coordinates and to extend the system to three dimensions in his Exercises (1657).

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The other branch of analytical mathematics that was developed in the seventeenth century was what we now know as infinitesimal calculus, the mathematics that is necessary to deal with rates of change, for example for analysing motion. There is a prehistory, particularly of integral calculus but it doesn’t need to interest us here. Kepler used a form of proto-integration to prove his second law of planetary motion and a more sophisticated version to calculate the volume of barrels in a fascinating but often neglected pamphlet. The Galilean mathematician Cavalieri developed a better system of integration, his indivisibles, which he published in his Geometria indivisibilibus continuorum nova quadam ratione promota, (Geometry, developed by a new method through the indivisibles of the continua) (1635) but actually written in 1627, demonstrated on the area of a parabola. This work was developed further by Torricelli, who extended it to other functions.

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The other branch of calculus the calculating of tangents and thus derivatives was worked on by a wide range of mathematicians. Significant contributions were made by Blaise Pascal, Pierre de Fermat, René Descarte, Gregoire de Saint-Vincent, John Wallis and Isaac Barrow. Fermat’s work was the most advanced and included contributions to both integral and deferential calculus, including a general method for determining tangents that is still taught in schools. The Scottish mathematician, James Gregory (1638–1675), inspired by Fermat’s work developed the second fundamental theory of calculus, which states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many anti-derivatives. Isaac Barrow (1630–1677) was the first to provide a full proof of the fundamental theorem of calculus, which is a theorem that links the concept of differentiating a function with the concept of integrating a function. Fermat’s work and John Wallis’ Arithmetica Infinitorum (1656) would be an important jumping off point for both Leibniz and Newton in the future.

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Source: Wikimedia Commons

By about 1670, the mathematicians of Europe, who knew of and built on each other’s work had made major advances in the development of both modern mathematics and physics laying the foundations for the next major development in the emergence of modern astronomy. However, before we reach that development there will be a couple of other factors that we have to consider first.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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May 6, 2020 · 8:33 am