Category Archives: History of Mathematics

Reading Euclid

This is an addendum to yesterday review of Reading Mathematics in Early Modern Europe. As I noted there the book was an outcome of two workshops held, as part of the research project Reading Euclid that ran from 2016 to 2018. The project, which was based at Oxford University was led by Benjamin Wardhaugh, Yelda Nasifoglu (@YeldaNasif) and Philip Beeley.

The research project has its own website and Twitter account @ReadingEuclid. As well as Benjamin Wardhaugh’s The Book of Wonders: The Many Lives of Euclid’s Elements, which I reviewed here:

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And Reading Mathematics in Early Modern EuropeStudies in the Production, Collection, and Use of Mathematical Books, which I reviewed yesterday.

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There is also a third online publication Euclid in print, 1482–1703: A catalogue of the editions of the Elements and other Euclidian Works, which is open access and can be downloaded as a pdf for free.

All of this is essential reading for anybody interested in the history of the most often published mathematics textbook of all times.

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There’s more to reading than just looking at the words

When I first became interested in the history of mathematics, now literally a lifetime ago, it was dominated by a big events, big names approach to the discipline. It was also largely presentist, only interested in those aspects of the history that are still relevant in the present. As well as this, it was internalist history only interested in results and not really interested in any aspects of the context in which those results were created. This began to change as some historians began to research the external circumstances in which the mathematics itself was created and also the context, which was often different to the context in which the mathematics is used today. This led to the internalist-externalist debate in which the generation of strictly internalist historians questioned the sense of doing external history with many of them rejecting the approach completely.

As I have said on several occasions, in the 1980s, I served my own apprenticeship, as a mature student, as a historian of science in a major research project into the external history of formal or mathematical logic. As far as I know it was the first such research project in this area. In the intervening years things have evolved substantially and every aspect of the history of mathematics is open to the historian. During my lifetime the history of the book has undergone a similar trajectory, moving from the big names, big events modus to a much more open and diverse approach.

The two streams converged some time back and there are now interesting approaches to examining in depth mathematical publications in the contexts of their genesis, their continuing history and their use over the years. I recently reviewed a fascinating volume in this genre, Benjamin Wardhaugh’s The Book of Wonder: The Many Lives of Euclid’s Elements. Wardhaugh was a central figure in the Oxford-based Reading Euclid research project (2016–2018) and I now have a second volume that has grown out of two workshops, which took place within that project, Reading Mathematics in Early Modern Europe: Studies in the Production, Collection, and Use of Mathematical Books[1]. As the subtitle implies this is a wide-ranging and stimulating collection of papers covering many different aspects of how writers, researchers, and readers dealt with the mathematical written word in the Early Modern Period.

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In general, the academic standard of all the papers presented here is at the highest level.  The authors of the individual papers are all very obviously experts on the themes that they write about and display a high-level of knowledge on them. However, all of the papers are well written, easily accessible and easy to understand for the non-expert. The book opens with a ten-page introduction that explains what is being presented here is clear, simple terms for those new to the field of study, which, I suspect, will probably the majority of the readers.

The first paper deals with Euclid, which is not surprising given the origin of the volume. Vincenzo De Risi takes use through the discussion in the 16th and 17th centuries by mathematical readers of the Elements of Book 1, Proposition 1 and whether Euclid makes a hidden assumption in his construction. Risi points out that this discussion is normally attributed to Pasch and Hilbert in the 19th century but that the Early Modern mathematicians were very much on the ball three hundred years earlier.

We stay with Euclid and his Elements in the second paper by Robert Goulding, who takes us through Henry Savile’s attempts to understand and maybe improve on the Euclidean theory of proportions. Savile, best known for giving his name and his money to establish the first chairs for mathematics and astronomy at the University of Oxford, is an important figure in Early Modern mathematics, who largely gets ignored in the big names, big events history of the subject, but quite rightly turns up a couple of times here. Goulding guides the reader skilfully through Savile’s struggles with the Euclidean theory, an interesting insight into the thought processes of an undeniably, brilliant polymath.

In the third paper, Yelda Nasifoglu stays with Euclid and geometry but takes the reader into a completely different aspect of reading, namely how did Early Modern mathematicians read, that is interpret and present geometrical drawings? Thereby, she demonstrates very clearly how this process changed over time, with the readings of the diagrams evolving and changing with successive generations.

We stick with the reading of a diagram, but leave Euclid, with the fourth paper from Renée Raphael, who goes through the various reactions of readers to a problematic diagram that Tycho Brahe used to argue that the comet of 1577 was supralunar. It is interesting and very informative, how Tycho’s opponents and supporters used different reading strategies to justify their standpoints on the question. It illuminates very clearly that one brings a preformed opinion to a given text when reading, there is no tabula rasa.

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We change direction completely with Mordechai Feingold, who takes us through the reading of mathematics in the English collegiate-humanist universities. This is a far from trivial topic, as the Early Modern humanist scholars were, at least superficially, not really interested in the mathematical sciences. Feingold elucidates the ambivalent attitude of the humanists to mathematical topics in detail. This paper was of particular interest to me, as I am currently trying to deepen and expand my knowledge of Renaissance science.

Richard Oosterhoff, in his paper, takes us into the mathematical world of the relatively obscure Oxford fellow and tutor Brian Twyne (1581–1644). Twyne’s manuscript mathematical notes, complied from various sources open a window on the actual level and style of mathematics’ teaching at the university in the Early Modern Period, which is somewhat removed from what one might have expected.

Librarian William Poole takes us back to Henry Savile. As well as giving his name and his money to the Savilian mathematical chairs, Savile also donated his library of books and manuscripts to be used by the Savilian professors in their work. Poole takes us on a highly informative tour of that library from its foundations by Savile and on through the usage, additions and occasional subtractions made by the Savilian professors down to the end of the 17th century.

Philip Beeley reintroduced me to a recently acquired 17th century mathematical friend, Edward Bernard and his doomed attempt to produce and publish an annotated, Greek/Latin, definitive editions of the Elements. I first became aware of Bernard in Wardhaugh’s The Book of Wonder. Whereas Wardhaugh, in his account, concentrated on the extraordinary one off, trilingual, annotated, Euclid (Greek, Latin, Arabic) that Bernard put together to aid his research and which is currently housed in the Bodleian, Beeley examines Bernard’s increasing desperate attempts to find sponsors to promote the subscription scheme that is intended to finance his planned volume. This is discussed within the context of the problems involved in the late 17th and early 18th century in getting publishers to finance serious academic publications at all. The paper closes with an account of the history behind the editing and publishing of David Gregory’s Euclid, which also failed to find financial backers and was in the end paid for by the university.

Following highbrow publications, Wardhaugh’s own contribution to this volume goes down market to the world of Georgian mathematical textbooks and their readers annotations. Wardhaugh devotes a large part of his paper to the methodology he uses to sort and categorise the annotations in the 366 copies of the books that he examined. He acknowledges that any conclusions that he draws from his investigations are tentative, but his paper definitely indicates a direction for more research of this type.

Boris Jardine takes us back to the 16th century and the Pantometria co-authored by father and son Leonard and Thomas Digges. This was a popular book of practical mathematics in its time and well into the 17th century. Jardine examines how such a practical mathematics text was read and then utilised by its readers.

Kevin Tracey closes out the volume with a final contribution on lowbrow mathematical literature and its readers with an examination of John Seller’s A Pocket Book, a compendium of a wide range of elementary mathematical topics written for the layman. Following a brief description of Seller’s career as an instrument maker, cartographer and mathematical book author, Tracey examines marginalia in copies of the book and shows that it was also actually used by university undergraduates.

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The book is nicely presented and in the relevant papers illustrated with the now ubiquitous grey in grey prints. Each paper has its own collection of detailed, informative, largely bibliographical endnotes. The books referenced in those endnotes are collected in an extensive bibliography at the end of the book and there is also a comprehensive index.

As a whole, this volume meets the highest standards for an academic publication, whilst remaining very accessible for the general reader. This book should definitely be read by all those interested in the history of mathematics in the Early Modern Period and in fact by anybody interested in the history of mathematics. It is also a book for those interested in the history of the book and in the comparatively new discipline, the history of reading. I would go further and recommend it for general historians of the Early Modern Period, as well as interested non experts.

[1] Reading Mathematics in Early Modern Europe: Studies in the Production, Collection, and Use of Mathematical Books, eds. Philip Beeley, Yelda Nasifoglu and Benjamin Wardhaugh, Material Readings in Early Modern Culture, Routledge, New York and London, 2021

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Illuminating medieval science

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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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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.

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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.

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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,

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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.

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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.

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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.

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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.

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Alexis Claude Clairaut Source: MacTutor

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Jérôme Lalande after Joseph Ducreux Source: Wikimedia Commons

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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

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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.

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Comet Halley 1456 artist unknown Source: Wikimedia Commons

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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.

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

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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