Category Archives: Book Reviews

3 into 2 does go!

It would of course be totally unethical for me to review a book of which I am one of the authors. However, as my contribution is only six of two-hundred pages, of which three are illustrations, and the book is one that could/would/should interest some (many) of my readers, I’m going to be unethical and review it anyway.

Thinking 3D is an intellectual idea, it is a website, it is exhibitions and other events, it is a book but above all it is two people, whose idea it is: Daryl Green, who was Fellow Librarian of Magdalen College, Oxford and is now Special Collections Librarian of the University of Edinburg and Laura Moretti, who is Senior Lecturer in Art History at the University of St Andrews. The Thinking 3D idea is the historical investigation of the representation of the three-dimensional world on the two-dimensional page particular, but not exclusively, in print.

The Thinking 3D website explains in great detail what it is all about and contains a full description of the activities that have been carried out. For those quarantined there is a fairly large collection of essays on various topics from the project.

In 2019 Thinking 3D launched a major exhibition with The Bodleian Libraries Oxford as part of the commemorations of the 500th anniversary of Leonardo da Vinci’s death, Thinking 3D From Leonardo to the Present, which ran from March 2019 to February 2020 and which I have been told was quite exceptional.

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As an extension and permanent record of that exhibition Bodleian Libraries published a book, Thinking 3D: Books, Images and Ideas from Leonardo to the Present[1], which appeared in autumn 2019. This is both a coffee table book but also, at the same time, a piece of serious academic literature.

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The book opens with a long essay by Green and Moretti, The history of thinking 3D in forty books, which delivers exactly what the title says. This is an excellent survey of the topic and it is worth reading the book just for this. However, it does contain one historical error that I, in my alter ego of the HIST_SCI HULK, simply cannot ignore, at least not if I want to maintain my hard won reputation. Having introduced the topic of Copernicus’ De revolutionibus the authors write:

As mentioned above, the oft-published heliocentric diagram, and its theoretical propositions, are what launched this book into infamy (the book was immediately put on the Catholic Church’s Index of Prohibited Books [my emphasis]), but the execution of this relational illustration is simple and reductive.

De revolutionibus was published in 1543 but was first placed on the Index sixty-three years later in 1616 and more importantly, as I wrote very recently, not for the first time, it was placed on the Index until corrected. These corrections, which were fairly minimal, were carried out surprisingly quickly and the book became available to be studied by Catholics already in 1621.

Other than this I noticed no other errors in the highly informative introductory essay, which is followed by an essay from Matthew Landrus, Leonardo da Vinci, 500 years on, which examines Leonardo’s three-dimensional perception of the world and everything in it. It was for me an interesting addition to my previous readings on the Tuscan polymath.

The main body of the book is taken up by sixteen fairly short essays in four categories: Geometry, Astronomy, Architecture and Anatomy.

Geometry starts off with Ken Saito’s presentation of a ninth century manuscript of The Elements of Euclid, where he demonstrates very clearly that the author has no real consistent, methodology for presenting a 3D space on a 2D page.   This is followed by Renzo Baldasso’s essay on Luca Pacioli’s De divina proportione (1509). Here the three dimensional solids are presented perfectly by Pacioli’s friend, colleague and one time pupil Leonardo. We return to Euclid for Yelda Nasifoglu’s investigation of the English translation of The Elements by Henry Billingsley in 1570. This volume is totally fascinating as three-dimensional figures are present as pop-up figure like those that we all know from our children’s books. The geometry section closes with a book that I didn’t know, Max Brückner’s Vielecke und Vielflache (1900) presented by George Hart. This is a vast collection of photographs of paper models of three-dimensional figures, which I learnt also influenced M. C. Escher a master of the third dimension.

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Luca Pacioli De divina proportione

 

Karl Galle, Renaissance Mathematicus friend and guest blogger, kicks of the astronomy section with Johannes Kepler’s wonderfully bizarre presentation of the planetary orbits embedded in the five regular Platonic solids from his Mysterium Cosmographicum (1596). Yours truly is up next with an account of Galileo’s Sidereus Nuncius (1610) and it’s famous washes of the Moon displaying three-dimension features. Also covered are the later pirate editions that screwed up those illustrations. Stephanie O’Rourke presents one of the most extraordinary nineteenth century astronomy books James Nasmyth’s and James Carpenter’s The Moon: Considered as a Planet, a World, and a Satellite(1874). This contains stunningly realistic photographic plates of the Moon’s surface but which are not actually real. The two Jameses constructed plaster models that they then lit and photographed to achieve the desired effect. We close the astronomy section with Thinking 3D’s co-chef, Daryl Green, taking on a survey of the surface of Mars with the United Stated Geological Survey, Geological Map of Mars (1978).

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Johannes Kepler Mysterium Cosmographicum

Turning our attention to architecture, we travel back to the twelfth century, with Karl Kinsella as our guide, to Richard of St Victor’s In visionen Ezekielis; a wonderfully modern in its presentation but somewhat unique medieval architectural manuscript. The other half of the Thinking 3D team, Laura Moretti now takes us up to the sixteenth century and Sebastiano Serlio’s catalogue of the buildings of Rome (1544), which has an impossibly long Italian title that I’m not going to repeat here. We remain in the sixteenth century for Jacques Androuet du Cerceau’s Le premier [et second] volume des plus excellent bastiment de France (1576–9), where our guide is Frédérique Lemerle. Moving forward a century we close out the architecture section with Francesco Marcorin introducing us to Hans Vredeman de Vries’s absolutely stunning Perspective (1604–5).

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Hans Vredeman de Vries Perspective

It would not be too difficult to guess that the anatomy section opens with one of the greatest medical books of all time, Andreas Vesalius’ De fabrica but not with the full version but the shorter (cheaper?) De humani corporis fabrica libroum epitome, like the full version published in 1543 in Basel. Our guide to Vesalius’ masterpiece is Mark Samos. Camilla Røstvik introduces us to William Hunter’s The Anatomy of the Human Gravid Uterus (1774), as she makes very clear a milestone in the study of women’s bodies with its revolutionary and controversial study of the pregnant body. For me this essay was a high point in a collection of truly excellent essays. We stay in the eighteenth century for Jacques Fabien Gautier D’Agoty’s Exposition anatomique des organes des sens (1775). Dániel Margócsy present a fascinating guide to the controversial work of this pioneer of colour printing. Anatomy, and the book as a whole, closes with Denis Pellerin’s essay on Arthur Thomson’s Anatomy of the Human Eye (1912). Thomson’s book was accompanied by a collection of stereoscopic images of the anatomy of the eye together with a stereoscope with which to view the 3D images thus created; a nineteenth century technology that was already dying out when Thomson published his work.

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William Hunter The Anatomy of the Human Gravid Uterus

The book closes with a bibliography of five books for further reading for each essay, brief biography of each of the authors, a glossary of technical terms and a good general index. All sixteen of the essays are short, informative, light to read, easily accessible introductions to the volumes that they present and maintain a high academic quality throughout the entire book.

I said at the outset that this is also a coffee table book and that was not meant negatively. It measures 24X26 cm and is printed on environmentally friendly, high gloss paper. The typeface is attractive and light on the eyes and the illustrations are, as is to be expected for a book about the history of book illustration, spectacularly beautiful. The publishing team of the Bodleian Libraries are to be congratulated on an excellent publication. If you leave this on your coffee table then your visitors will soon be leafing though it admiring the pictures, whether they are interested in book history or not. I don’t usually mention the price of books that I review but at £35 this beautifully presented and wonderfully informative volume is very good value for money.

[1] Thinking 3D: Books, Images and Ideas from Leonardo to the Present, edited by Daryl Green and Laura Moretti, Bodleian Library, Oxford, 2019.

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Filed under Book Reviews, Early Scientific Publishing, History of Astronomy, History of Mathematics, History of medicine

My name is Bond, Jamie Bond.

Today we have a first at the Renaissance Mathematicus, a book review of two interrelated books that have nothing, or at least very little, to do with the histories of science and mathematics. They, however, both deal with England during the Revolution (Civil War) and Interregnum in the middle of the seventeenth century, so very much home territory for this blog.

The word spy is one that for most people instantly evokes a male figure, for someone of my generation, a man in a dinner jacket with a martini glass in one hand and a Beretta pistol in the other. Very few people would immediately associate the word spy with a woman, although there have been some notable female spies throughout history. Dutch historian of early modern English literature Nadine Akkerman, Reader at Leiden University, stumbled across a female spy during her research into the correspondence of Elizabeth Stuart, Queen of Bohemia (1596–1662), who lived out the last forty years of her life in The Hague. Inspired by this discovery Akkerman, who believed that female spies were perhaps not so rare as one might suspect, began to systematically search archives for traces of other women involved in espionage in the seventeenth century. The result of her researches appeared in a book two years ago, Invisible Agents: Women and Espionage in Seventeenth-Century Britain. [1] The paperback, that I’ve been reading, was published just this month.

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Akkerman’s book is a truly excellent piece of historical scholarship. Her, apparently tireless, excavations of the archives have turned up a large amount of evidence for the existence and activities of female spies, or as she prefers to call them she-intelligencers, as they were then commonly known, in the three decades of the seventeenth century, 1640s to1660s, in Britain. She has sorted, analysed and interpreted this flood of data to produce a coherent narrative about her she-intelligencers. From the start she explains that there is both too much data and too much of it fragmentary to produce a complete picture of the women involved in espionage in this period so instead she presents the reader with a series of case studies.

The first chapter deals with the mostly aristocratic women who worked as she-intelligencers for Charles I during his imprisonment by the parliamentary forces acting as couriers in the various plots to free the king. These women, on the whole, engaged in these activities out of loyalty to king and country. This is contrasted in the second chapter with accounts of the largely working class women, who sold information to Thurloe the parliamentary spymaster. Here we should note in particular, for later, her account of Diana Stewart, who appears to have supplied information to both sides, a double agent perhaps, or was she simply some sort of early modern con artist?

The third chapter is dedicated to the story of Susan Hyde, the sister of Sir Edward Hyde, a prominent royalist politician, who became 1st Earl of Clarendon and Lord Chancellor under Charles II. Susan Hyde was an active royalist she-intelligencer but has till now remained under the radar and Akkerman is the first to entangle and tell her story, giving it the attention it deserves. The next two chapters deal with Elizabeth Murray, who unlike Susan Hyde is a well-documented historical figure. Here Akkerman displays her analytical talents to the full. In the first chapter she deconstructs the accepted historical narrative about Murray and shows why it is at best dubious and at worst false. In the second chapter she reconstructs Murray’s story using the sources that she has excavated in her research.

Following Murray we have another Elizabeth, Elizabeth Carey, Lady Mordaunt. A she-intelligencer, who together with her husband was involved in espionage during the late phases of the Interregnum. Of particular interest here is Akkerman’s analysis of Carey through her correspondence with the gardener and diarist John Evelyn. Next up, is Anne, Lady Halkett and another deconstruction by Akkerman. This time she deconstructs the interpretations by other literary historians of Halkett’s own extensive written account of her espionage activities. The final figure in the book is probably the most well known female English author of the seventeenth century Aphra Behn, who was also a she-spy, or was she? Another deconstruction job by Akkerman.

Each subject in the book is presented in the full political and social context of her times. Her activities are described in as much detail as the sources allow and we, the readers, are introduced to the full array of early modern espionage activities. The post offices and the post routes the coded letters, the cyphers used, the secret societies, the counter espionage activities of the other side and the fate of those, who were trapped by those counter espionage activities. After having read Akkerman’s book one comes away with a rich knowledge of the activities of the seventeenth century English she-intelligencers.

Akkerman’s book is a masterpiece in the assimilation, ordering and interpretation of archival sources within a given historical area and can be held up as an example of how to do and present historical research. The book bristles with extensive footnotes, no endnotes, and has an equally extensive bibliography of both primary and secondary sources. The index is first class and is followed by an Index Occultus, a key to all the code names used in the original source documents for the historical characters in the book.

At the beginning I said this was a review of two interrelated book, the second is the novel Killing Beauties by Pete Langman[2].

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As far as I know this is Langman’s first novel but it is not, by a long chalk, his first artistic endeavour. A one time rock guitarist and then music teacher, he has worked as a music journalist, is a cricketer and along the way acquired a doctorate in early modern literature with a thesis about Francis Bacon. On a side note he gently and politely corrects me when I say something stupid about the Viscount St. Alban. Pete Langman also suffers from early-onset Parkinson’s disease and is the author of the highly acclaimed Slender Threads: A young person’s guide to Parkinson’s Disease.[3]

Pete Langman is also Nadine Akkerman’s partner and has borrowed two of the central figures from Invisible Agents, Diana Stewart and Susan Hyde, to weave a semi-fictional tale of espionage in the Interregnum, imaginatively filling out the gaps in Akkerman’s research.

Langman takes his readers on a fascinating journey through the streets, alleyways, drinking holes, apothecaries and seats of espionage of Interregnum London, evoking an authentic picture of life in the capital city in Cromwell’s time. A wide cast of fascinating and captivating characters lead the readers through the twists and turns of a risky espionage coup and the counter espionage moves to prevent that coup from being put into effect. None of the characters is entirely good or entirely bad but each of them is a real human being with all the normal faults and virtues, meaning that one doesn’t end up rooting for one side or the other, or at least I didn’t. There are enough twists and turns in the narrative to delight Agatha Christie fans and things don’t necessarily turn out, as you might have expected during the earlier chapters.

Langman’s voice is the authoritative voice of the seventeenth century historian but is is the voice of the story teller and not the lecturer, the artist and not the teacher. He recreates a visceral and authentic picture of a period of English history when the populous was torn between two philosophies of life and politics and some paid for their beliefs in one or other of those systems with their honour and even their lives.

His book is both an excellent historical novel and an excellent espionage novel that should delight fans of both genres and is also a wonderful companion to Akkerman’s historical presentation of the material. I would recommend both books to anybody interested in seventeenth century Britain, the history of espionage or simply just good writing. According to taste a potential reader can choose one or the other, but if you should choose to read both then I would recommend first reading Langman’s novel and then Akkerman’s historical presentation as the back story.

Disclosure: As should be obvious from various comments in this review, Pete Langman is an Internet friend, known as @elegantfowl on Twitter, with whom I share a mutual interest in the guitar playing of Gary Lucas and the history of seventeenth century science, amongst other things. Unbound is a crowd funding book publisher and when Pete announced on Twitter that he was trying to publish a novel on Unbound I became a subscriber, which is why I came to read Killing Beauties. Having read it, I was intrigued enough to acquire Invisible Agents when it appeared in paperback. Some might therefore not regard me as a neutral reviewer but as I have said in the past in similar circumstances if I didn’t like the book then I wouldn’t have reviewed it.

[1] Nadine Akkerman, Invisible Agents: Women and Espionage in Seventeenth-Century Britain, OUP, Oxford, 2018, ppb. 2020

[2] Pete Langman, Killing Beauties, Unbound, London, 2020.

[3] Pete Langman, Slender Threads: A young person’s guide to Parkinson’s Disease, Self Published, 2013. On a personal note, Pete said some very sensible and comforting things when I discussed my own problems with coming to terms with my brother’s Parkinson’s with him.

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Mathematics at the Meridian

Historically Greenwich was a village, home to a royal palace from the fifteenth to the seventeenth centuries, that lay to the southeast of the city of London on the banks of the river Thames, about six miles from Charing Cross. Since the beginning of the twentieth century it has been part of London. With the Cutty Sark, a late nineteenth century clipper built for the Chinese tea trade, the Queen’s House, a seventeenth-century royal residence designed and built by Inigo Jones for Anne of Denmark, wife of James I & VI, and now an art gallery, the National Maritime Museum, Christopher Wren’s Royal Observatory building and of course the Zero Meridian line Greenwich is a much visited, international tourist attraction.

Naturally, given that it is/was the home of the Royal Observatory, the Zero Meridian, the Greenwich Royal Hospital School, the Royal Naval College (of both of which more later), and most recently Greenwich University, Greenwich has been the site of a lot mathematical activity over the last four hundred plus years and now a collection of essays has been published outlining in detail that history: Mathematics at the Meridian: The History of Mathematics at Greenwich[1]

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This collection of essays gives a fairly comprehensive description of the mathematical activity that took place at the various Greenwich institutions. As a result it also function as an institutional history, an often-neglected aspect of the histories of science and mathematics with their concentration on big names and significant theories and theorems. Institutions play an important role in the histories of mathematic and science and should receive much more attention than they usually do.

The first four essays in the collection cover the history of the Royal Observatory from its foundation down to when it was finally closed down in 1998 following several moves from its original home in Greenwich. They also contain biographies of all the Astronomers Royal and how they interpreted their role as the nation’s official state astronomer.

This is followed by an essay on the mathematical education at the Greenwich Royal Hospital School. The Greenwich Royal Hospital was established at the end of the seventeenth century as an institution for aged and injured seamen. The institution included a school for the sons of deceased or disabled sailors. The teaching was centred round seamanship and so included mathematics, astronomy and navigation.

When the Greenwich Royal Hospital closed at the end of the nineteenth century the buildings were occupied by the Royal Naval College, which was moved from Portsmouth to Greenwich. The next three chapters deal with the Royal Naval College and two of the significant mathematicians, who had been employed there as teachers and their contributions to mathematics.

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Another institute that was originally housed at Greenwich was The Nautical Almanac office, founded in 1832. The chapter dealing with this institute concentrates on the life and work of Leslie John Comrie (1893–1950), who modernised the production of mathematical tables introducing mechanisation to the process.

Today, apart from the Observatory itself and the Meridian line, the biggest attraction in Greenwich is the National Maritime Museum, one of the world’s leading science museums and there is a chapter dedicated to the scientific instruments on display there.

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Also today, the buildings that once housed the Greenwich Royal Hospital and then the Royal Naval College now house the University of Greenwich and the last substantial chapter of the book brings the reader up to the present outlining the mathematics that has been and is being taught there.

The book closes with a two-page afterword, The Mathematical Tourist in Greenwich.

Each essay in the book is written by an expert on the topic and they are all well researched and maintain a high standard throughout the entire book. The essays covers a wide and diverse range of topics and will most probably not all appeal equally to all readers. Some might be more interested in the history of the Royal Observatory, whilst the chapters on the mathematical education at the Greenwich Royal Hospital School and on its successor the Royal Naval College should definitely be of interest to the readers of Margaret Schotte’s Sailing School, which I reviewed in an earlier post.

Being the hopelessly non-specialist that I am, I read, enjoyed and learnt something from all of the essays. If I had to select the four chapters that most stimulated me I would chose the opening chapter on the foundation and early history of the Royal Observatory, the chapter on George Biddel Airy and his dispute with Charles Babbage over the financing of the Difference Engine, which I blogged about in December, the chapter on Leslie John Comrie, as I’ve always had a bit of a thing about mathematical tables and finally, one could say of course, the chapter on the scientific instruments in the National Maritime Museum.

The book is nicely illustrated with, what appears to have become the standard for modern academic books, grey in grey prints. There are extensive endnotes for each chapter, which include all of the bibliographical references, there being no general bibliography, which I view as the books only defect. There is however a good, comprehensive general index.

I can thoroughly recommend this book for anybody interested in any of the diverse topic covered however, despite what at first glance, might appear as a somewhat specialised book, I can also recommend it for the more general reader interested in the histories of mathematics, astronomy and navigation or those perhaps interested in the cultural history of one of London’s most fascinating district. After all mathematics, astronomy and navigation are all parts of human culture.

[1] Mathematics at the Meridian: The History of Mathematics at Greenwich, eds. Raymond Flood, Tony Mann, Mary Croarken, CRC Press, Taylor & Francis Group, Bacon Raton, London, New York, 2020.

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Filed under Book Reviews, History of Astronomy, History of Mathematics, History of Navigation

The Swinging 1660s

Readers of my occasional autobiographical posts will know that I came of age in the late 1960’s and early 1970s and was a fully-fledged member of the drug freak generation. Indulging freely in a wide range of illicit substances, something I neither regret nor overly value; it was how it was. However, always the born historian, when my drug freak colleagues were busy lighting up that spliff or dropping that tab, I was also busy reading up on the report of the 1894 Indian Hemp Drugs Commission or the Scythian shamans use of cannabis or Albert Hofmann’s synthesis of LSD at Sandoz or the medieval outbreaks of St Anthony’s Fire caused by ergot-based drugs. In other words I didn’t just want to get high but also to discover the history of humans getting high.

Later in my life during the time that I managed the monthly #histsci blog carnival On Giants’ Shoulders and then ran the weekly #histsci journal Whewell’s Gazette I regularly read a lot of blogs and one blog that I very much enjoyed was Benjamin Breen’s Res Obscura. Though not strictly a #histsci blog Res Obscura is a wonderful cornucopia of erudite, entertaining, enlightening and educational essays about, well, obscure things as the blog name says.

Given this two rather disparate aspects of my life I was delighted when I discovered that Benjamin Breen had written and published a book with the title, The Age Of Intoxication: Origins of the Global Drug Trade*. I knew that this was a book that I wanted to read and read it I have and it has fulfilled all my expectations.

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Now it might seem at first glance that my youthful adventures in the age of sex and drugs and rock’n’roll and Breen’s academic opus about the beginnings of the global drug trade in the early modern period would have little or nothing in common but appearances can be deceptive and in this case they are. One of Breen’s central themes in his book is that the dichotomies that characterised the world of drugs in the 1960s and 70s, medical–recreational, legal–illicit, natural–synthetic were in fact created during the European confrontation with exotic new drugs from South America and Asia during the Early Modern Period, which shaped the way we see intoxicants today.

Early in his book Breen explains to the reader, or in my case reminds him, that the word drug originally meant dry goods, as is still obvious in the North American drugstore or the German Drogeriemarkt. This meant that the “drugs” that the early European trader–explorer brought back home from all over the world included not only what we would now call drugs but also a very diverse range of other goods, including herbs and spices, dyes, soaps, incenses, pigments or even jewels. Although, one should add than many of these non drug dry good were often also regarded as medicines. One should also remember that three of our everyday commodities, coffee, chocolate and tobacco, were originally viewed as medicinal drugs.

Breen narrative centres around two of the early European empires the Portuguese and the English, as the main sources for the introduction and establishment of intoxicant drugs into European culture. The book is divided into two sections. The first of these, entitled Invention of Drugs, begins with the Portuguese search of new drugs in the jungles of Brazil, inspired by the discovery of quinine, the ground up bark of the cinchona tree, by the Jesuits in Peru. We then move on to the selling of the new drugs in the Apothecaries of Europe. This section closes with a fascinating discussing Fetishizing Drugs about the relationship between drug use, religion and magic in Early Modern Africa.

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The second section, Altered States, tackles the whole concept of intoxication. It opens with the strange, under the counter so to speak, relationship between the Portuguese, oft Jesuit, discoverers and importers of drugs and the natural philosophers of the English Royal Society. This exchange of information and knowledge, whilst for a period highly active, remained largely clandestine because of the religious, political and philosophical clash that existed publically between the two parties. But the exchange did take place and was highly fruitful. Historians of science in the know will perhaps be aware of Robert Hooke’s dope smoking activities but as Breen shows there was very much more. We now move on to the problems involved in trying to describe and classify states of intoxication. The only real reference point for the Europeans was getting drunk on alcohol, whereas the highs produced by the alkaloids contained in the drugs imported from South America and Asia are very different. I know this from personal experience.  Try explaining an acid trip to somebody whose only experience of deliberately losing control of ones mental facilities is getting pissed!

The second section closes with what might within the context of the book be described as a case study. Entitled Three Ways of Looking at Opium it chronicles how the perception and acceptance of opium changed between the seventeenth and nineteenth centuries. Breen starts with a fact that was completely new to me, the opium poppy is actually a native European plant and the perception that opium comes into Europe from Asia is one of those changes that took place in the early modern period. Breen relates how a fairly positive image of opium as a medicinal drug gradually changes to a negative one, a process accelerated in the nineteenth century by the successful synthesis of the of first morphine and later heroine from raw opium; the synthetic forms becoming the acceptable medical drugs, whereas raw opium becomes an unacceptable illicit substance.

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The book closes with a meditation on our attitude to drugs then and now under the title, Drugs Past and Present.

This is a truly polymathic, historical achievement; Breen weaves together a world history out of elements of the social, cultural and core histories of exploration, discovery, botany, chemistry, medicine, pharmacology, trade, economics, magic, religion and philosophy. As was to be expected from the author of Res Obscura this book is beautifully written and is a real pleasure to read. It is well presented with a wide range of grey in grey illustrations. There are extensive, highly informative endnotes, requiring the somewhat tiresome two bookmarks method of reading, a useful bilingual (Portuguese and English) glossary, a very comprehensive bibliography and an excellent index.

Whatever your historical interests, if you like reading good quality, excellently researched and equally excellently written history, then do yourself a favour and read Breen’s fascinating academic excursion through the world of the Early Modern drug trade.

*Benjamin Breen, The Age Of Intoxication: Origins of the Global Drug Trade, University of Pennsylvania Press, Philadelphia, 2019

 

 

 

 

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Finding your way on the Seven Seas in the Early Modern Period

I spend a lot of my time trying to unravel and understand the complex bundle that is Renaissance or Early Modern mathematics and the people who practiced it. Regular readers of this blog should by now be well aware that the Renaissance mathematici, or mathematical practitioners as they are generally known in English, did not work on mathematics as we would understand it today but on practical mathematics that we might be inclined, somewhat mistakenly, to label applied mathematics. One group of disciplines that we often find treated together by one and the same practitioner consists of astronomy, cartography, navigation and the design and construction of tables and instruments to aid the study of these. This being the case I was delighted to receive a review copy of Margaret E. Schotte’s Sailing School: Navigating Science and Skill, 1550–1800[1], which deals with exactly this group of practical mathematical skills as applied to the real world of deep-sea sailing.

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Schotte’s book takes the reader on a journey both through time and around the major sea going nations of Europe, explaining, as she goes, how each of these nations dealt with the problem of educating, or maybe that should rather be training, seamen to become navigators for their navel and merchant fleets, as the Europeans began to span the world in their sailing ships both for exploration and trade.

Having set the course for the reader in a detailed introduction, Schotte sets sail from the Iberian peninsular in the sixteenth century. It was from there that the first Europeans set out on deep-sea voyages and it was here that it was first realised that navigators for such voyages could and probably should be trained. Next we travel up the coast of the Atlantic to Holland in the seventeenth century, where the Dutch set out to conquer the oceans and establish themselves as the world’s leading maritime nation with a wide range of training possibilities for deep-sea navigators, extending the foundations laid by the Spanish and Portuguese. Towards the end of the century we seek harbour in France to see how the French are training their navigators. Next port of call is England, a land that would famously go on, in their own estimation, to rule the seven seas. In the eighteenth century we cross the Channel back to Holland and the advances made over the last hundred years. The final chapter takes us to the end of the eighteenth century and the extraordinary story of the English seaman Lieutenant Riou, whose ship the HMS Guardian hit an iceberg in the Southern Atlantic. Lacking enough boats to evacuate all of his crew and passengers, Riou made temporary repairs to his vessel and motivating his men to continuously pump out the waters leaking into the rump of his ship, he then by a process of masterful navigation, on a level with his contemporaries Cook and Bligh, brought the badly damaged frigate to safety in South Africa.

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In each of our ports of call Schotte outlines and explains the training conceived by the authorities for training navigators and examines how it was or was not put into practice. Methods of determining latitude and longitude, sailing speeds and distances covered are described and explained. The differences in approach to this training developed in each of the sea going European nations are carefully presented and contrasted. Of special interest is the breach in understanding of what is necessary for a trainee navigator between the mathematical practitioners, who were appointed to teach those trainees, and the seamen, who were being trained, a large yawning gap between theory and practice. When discussing the Dutch approach to training Schotte clearly describes why experienced coastal navigators do not, without retraining, make good deep-sea navigators. The methodologies of these two areas of the art of navigation are substantially different.

The reader gets introduced to the methodologies used by deep-sea navigators, the mathematics developed, the tables considered necessary and the instruments and charts that were put to use. Of particular interest are the rules of thumb utilised to make course corrections before accurate methods of determining longitude were developed. There are also detailed discussions about how one or other aspect of the art of navigation was emphasised in the training in one country but considered less important in another. One conclusion the Schotte draws is that there is not really a discernable gradient of progress in the methods taught and the methods of teaching them over the two hundred and fifty years covered by the book.

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As well as everything you wanted to know about navigating sailing ships but were too afraid to ask, Schotte also delivers interesting knowledge of other areas. Theories of education come to the fore but an aspect that I found particularly fascinating were her comments on the book trade. Throughout the period covered, the teachers of navigation wrote and marketed books on the art of navigation. These books were fairly diverse and written for differing readers. Some were conceived as textbooks for the apprentice navigators whilst others were obviously written for interested, educated laymen, who would never navigate a ship. Later, as written exams began to play a greater role in the education of the aspirant navigators, authors and publishers began to market books of specimen exam questions as preparation for the exams. These books also went through an interesting evolution. Schotte deals with this topic in quite a lot of detail discussing the authors, publishers and booksellers, who were engaged in this market of navigational literature. This is detailed enough to be of interest to book historians, who might not really be interested in the history of navigation per se.

Schotte is excellent writer and the book is truly a pleasure to read. On a physical level the book is beautifully presented with lots of fascinating and highly informative illustrations. The apparatus starts with a very useful glossary of technical terms. There is a very extensive bibliography and an equally extensive and useful index. My only complaint concerns the notes, which are endnotes and not footnotes. These are in fact very extensive and highly informative containing lots of additional information not contained in the main text. I found myself continually leafing back and forth between main text and endnotes, making continuous reading almost impossible. In the end I developed a method of reading so many pages of main text followed by reading the endnotes for that section of the main text, mentally noting the number of particular endnotes that I wished to especially consult. Not ideal by any means.

This book is an essential read for anybody directly or indirectly interested in the history of navigation and also the history of practical mathematics. If however you are generally interested in good, well researched, well written history then you will almost certainly get a great deal of pleasure from reading this book.

[1] Margaret E. Schotte, Sailing School: Navigating Science and Skill, 1550–1800, Johns Hopkins University Press, Baltimore, 2019.

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Calculus for the curious

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

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

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

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

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

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

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

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

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

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

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

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

 

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

 

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The role of celestial influence in the complex structure of medieval knowledge.

My entire life has followed a rather strange and at time confusing path that bears no relationship to the normal career path of a typical, well educated, middle class Englishman. It has taken many twists and turns over the years but without doubt one of the most bizarre was how I got to know historian of astrology Darrel Rutkin. We met on a bus, when he a total stranger commented that he knew the author of the book that I was reading, Monica Azzolini’s excellent, The Duke and the Stars: Astrology and Politics in Renaissance Milan. You can read the story in full here. At the time Darrel was a fellow at the International Consortium for Research in the Humanities: Fate, Freedom and Prognostication. Strategies for Coping with the Future in East Asia and Europe in Erlangen, where he was working on his book on the history of European astrology. Darrel and I became friends, talking about Early Modern science and related topics over cups of coffee and he twice took part in my History of Astronomy tour of Nürnberg. Before he left Erlangen he asked me if I would be interested in reading and reviewing his book when he finished writing it. I, of course, said yes. Some weeks ago I received my review copy of H. Darrel Rutkin, Sapientia Astrologica: Astrology, Magic and Natural Knowledge, ca. 1250–1800: I.Medieval Structures (1250–1500): Conceptual, Institutional, Socio-Political, Theologico-Religious and Cultural and this is my review.

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As should be obvious from the impressive title this is not in anyway a popular or even semi-popular presentation but a very solid piece of hard-core academic research. What I have, and will discuss here, is just volume one of three, which weighs in at over six hundred pages. In his work Rutkin present two theses the first of which he explicates in Volume I of his epos and the second of which forms the backbone of the two future volumes. The central thesis of Volume I is summed up in the slightly intimidating twelve-word term “astrologizing Aristotelian natural philosophy with its geometrical-optical model of celestial influences.” A large part of the book is devoted to constructing this object and I will now attempt to produce a simplified description of what it means and how it operated in medieval Europe.

It is common in the history of astrology to treat it as a separate object, as if it had little or nothing to do with the rest of the contemporary knowledge complex. It is also very common to lump astrology together with magic and the other so-called occult sciences. For the High Middle Ages, the period that his book covers, Rutkin rejects both of these approaches and instead proposes that astrology was an integral and important part of the accepted scientific knowledge of the period. His book is divided into five sections each of which I will now outline.

The first section is an eighty-nine-page introduction, which contains a detailed road map of the author’s intentions including a brief summary of what he sees as the current situation in various aspects of the study of the subject under investigation. This also includes an excursion: Astrological Basics: Horoscopes and Practical Astrology. This section is not based on the author’s own work but on that of Roger Bacon, one of the central figures of the book, so if you want to know how a leading medieval astrologer set up and worked with a horoscope then this is the right place to come.

The first section of the book proper deals with the relationship between astrology and natural philosophy in the thirteenth century and it is this section that defines and explains our intimidating twelve-word term from above. Rutkin’s analysis is based on four primary sources; these are an anonymous astrological text the Speculum Astronomiae, written around 1260 and often attributed to Albertus Magnus, an attribution that Rutkin disputes, the writings of Albertus Magnus (before 1200–1280), those of Thomas Aquinas (1225–1274) and those of Roger Bacon (ca. 1220­–1292), as well as numerous other sources from antiquity, and both the Islamic and Christian Middle Ages. In this first section he first presents those writings of Aristotle that contain his thoughts on celestial influence, which form the philosophical foundations for the acceptance of astrology as a science. He then demonstrates how the Speculum Astronomiae, Bacon and Albertus expanded Aristotle’s thoughts to include the whole of horoscope astrology and imbedded it into medieval Aristotelian natural philosophy, this is our “astrologizing Aristotelian natural philosophy.” He also shows how Thomas, whilst not so strongly astrological, as the others, also accepts this model. The technical astrology that is considered here is a highly mathematical, read geometrical, one based on the radiation theories of the Arabic scholar al-Kindi in his De radiis stellarum, as originally introduced into European thought by Robert Grosseteste (1175–1253) in his optical theories and adopted by Bacon. This explains how every geographical point on the earth at every point in time has a unique horoscope/astrological celestial influence: the “geometrical-optical” part of our intimidating twelve-word term. This also ties in with Aristotle’s geographical theories of the influence of place on growth and change. What comes out of this analysis is an astrological-geographical-mathematical-natural philosophical model of knowledge based on Aristotle’s natural philosophy, Ptolemaeus’ astronomy and astrology, and al-Kindi’s radiation theory at the centre of thirteenth century thought.

Rutkin does not simple state an interpretation of Albertus’, Bacon’s or Aquinas’ views but analyses their actual writings in fine detail. First he outlines one step in a given thought process then he quotes a paragraph from their writings in English translation, with the original in the footnotes, including original terms in brackets in the translation if they could possible be considered ambiguous. This is followed by a detailed analysis of the paragraph showing how it fits into the overall argument being discussed. He proceeds in this manner paragraph for paragraph cementing his argument through out the book. This makes hard work for the reader but guarantees that Rutkin’s arguments are as watertight as possible.

The second section of the book proper deals with the subject of theology, a very important aspect of the medieval knowledge complex. Rutkin shows that both Albertus and Thomas accepted astrology within their theology but were careful to show that celestial influence did not control human fate, providence or free will these being the dominion of their Christian God. This is of course absolutely central for the acceptance of astrology by Christian theologians. Bacon’s attitude to astrology and theology is completely different; he builds a complete history of the world’s principle religions based on the occurrence of planetary conjunctions, explaining why, as a result, Christianity is the best religion and addressed to the Pope, for whom he is writing, how one needs to combat the religion of the Anti-Christ.

The third section of the book proper now turns to the vexed question of the relationship between astrology and magic. Rutkin shows that both the Speculum Astronomiae and Albertus in his writing accept that astrology can be used to create magical images or talisman for simple tasks such as killing snakes. However, this is the limit of the connection between the two areas, other aspects of magic being worked by evil spirits or demons. Thomas, not surprisingly rejects even this very circumscribed form of astrological magic regarding all of magic to have its roots in evil. Bacon is much more open to a wider range of connections between the areas of astrology and magic.

Having set up the place of astrology in the medieval knowledge complex of the thirteenth century, the fourth and final section of the book proper takes brief looks at the evidence for its use in various fields within Europe in the period up to 1500. Fields sketched rather than covered in great detail included mathematics, medicine, teaching in the various faculties at the universities, annual prognostications at the universities and to close astrology in society, politics and culture.

Does Rutkin succeed in proving his central thesis for this his first volume? History is not like mathematics and does not deliver conclusive proofs but Rutkin’s thesis is argued in great detail with an impressive array of very convincing evidence. His work is rock solid and anybody wishing to refute his thesis is going to have their work cut out for them. That is not to say that with time, new research and new evidence his thesis will not undergo modification, refinement and improvement but I think its foundations will stand the test of time.

His second main thesis, which will be presented in the two future volumes of his work, is to explain how and why the medieval, mathematics based (read mathematical astrology), Aristotelian natural philosophy that had been created in the High Middle Ages came to replaced by a very different mathematics based, system of natural philosophy in the seventeenth and eighteenth centuries. Having ploughed my way through Volume I, I very much look forward to reading both future volumes.

It goes without saying that the book has an impressively long bibliography of both primary and secondary sources that the author has consulted. I consider myself reasonably well read on the history of European astrology but if I were to sit down and read all of the new, interesting titles I discovered here, I would be very busy for a number of years to come. There is also a first class index and I’m very happy to report that the book also has excellent footnotes, many of which I consulted whilst reading, rather than the unfortunately ubiquitous endnotes that plague modern publishing.

Before I move to a conclusion I wish to point out a second way to read this book. As it stands this is not a book that I would necessarily dump on an undergraduate or a historian, whose interest in the fine detail of Rutkin’s argument was peripheral but that is not necessary or at least not in its totality. I have already mentioned that the introduction contains a detailed road map to the whole volume and as well as this, each of the four sections has an introduction outlining what the section sets out to show and a conclusion neatly summarising what has been demonstrated in the section. By reading main introduction and the introductions and conclusions to the sections a reader could absorb the essence of Rutkin’s thesis without having to work through all of the documentary proof that he produces.

In general I think that Rutkin has set standards in the historiography of medieval astrology and that his book will become a standard work on the topic, remaining one for a long time. I also think that anybody who wishes to seriously study medieval European astrology and/or medieval concepts of knowledge will have to read and digest this fundamental and important work.

I’m posting this today, having pulled it up from the back of a list of planned blog posts because today Darrel’s book is being formally presented at the University of Venice, where he is currently working in a research project, this afternoon with Monica Azzolini as one of those discussing the book and so a circle closes. I shall be there with them in spirit.

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