The Dutch ambassador to the court of Henri IV ironically suggested to the French king that there was no mathematician in his kingdom who could solve this equation. Henri summoned Viète. Viète was already a well-known cryptographer and mathematician, he had recently published a book on trigonometric formulae, and - fortunately for la gloire de France - recognised the equation as an expansion of some powers of trigonometric functions. In a few minutes he was able to determine the twenty-three positive solutions of van Roomen's problem, and the following day he provided the remaining 22 negative solutions as well (although he disdained them).
Van Roomen was extremely impressed, travelled to Paris, met Viète, and they remained buddies for the rest of their lives. Aww, you might say. Geek man-crush and all that.
Looking at this equation, I have no idea where to begin solving it. Descartes' rule of signs tells me that there are twenty-three sign changes in the equation, indicating the existence 23 positive roots (not all necessarily distinct, of course). By the fundamental theorem of algebra, we know that an algebraic equation in the 45th power will have forty-five roots. So we can deduce that there will be 22 negative roots. But what the devil are they?
Recourse to the programming environment R's polynom package is easily done, but as that's a numerical solver, the nested square roots in the constant part of the above equation can create instabilities rendering the final answer somewhat unlikely. Indeed, I find that I get a bunch of complex roots, whereas Viète determined real-valued solutions.
I thought I might use Wolfram Alpha to solve the equation symbolically. Unfortunately, the polynomial definition is too long for the input box on that site.
Another problem is that there appear to be different versions written down of the equation: some of the coefficients are different (e.g. I've seen 740,259x35 and 740,459x35; there are other discrepancies as well.)
Perhaps I should use the clue that Viète used a trigonometric identity to solve the equation. By the sixteenth century, several of these were already known in Europe. For example,
If we let , we can get a cubic equation , where . Were a solver to recognise the left-hand-side as an expansion for a trigonometric function, he or she would be able to use the identity to solve the algebraic equation. Similarly, Viète realised that using the above substitution for x, that large 45th degree equation reduced to , which he could then solve using a table of sines.1
Check out Philippe Henry's paper (in French)2 - it's got as comprehensive a treatment of the problem and Viète's approaches to solving it as you might want. I gotta tell you - it has less to do with algebra than geometry. And indeed, the reason Viète rejected the negative solutions is that they did not make sense in 'real life'. Or, that is to say, geometry.
In Janwillem van der Vetering's Outsider in Amsterdam the chief inspector is dismissive about pseudo-Buddhism and nuts who want to improve the world.
"... He buys an old rackety house at the Haarlemmer Houttuinen, fixes it up a little and whitewashes all its walls. He buys a second-hand imitation of an Asiatic statue and puts it in the hall, lights an incense stick and sells health food. Unwashed tomatoes and grains. The kind that sticks in your throat. A rat couldn't digest it. And carrot juice."
He interrogated the detectives with his eyes. Both nodded.
It was clear that the chief inspector had no liking for carrot juice. They knew what he liked. He liked Dutch gin, and shrimp cocktails, snails and peppersteak. Pineapple with whipped cream. And cognac.
"There's a bar as well," Grijpstra said.
The chief inspector looked surprised.
"A what?"
"A bar," repeated Grijpstra, "downstairs, as you go in, on the right, a bar where they sell gin and beer."
"Good idea," the chief inspector said. "With a glass of jenever you can get through to the other nuts. And when you have weakened their defences you can make them eat unpeeled rice."
Sometime before 1653, a Dutchman named Willem Lithgouw was seized of a religious afflatus to go to the Grand Turk and save a Christian soul - preferably a virgin, or, failing that, a widow. In cahoots with a French naval officer, he arrived in Stamboul 'to render service unto God.' With great interest he and the Frenchman inspected one 'staerk naked' soul after another at the 'Weibermarket'. After having checked out the wares on offer, they sadly decided that a virgin was beyond their means, and settled for a Dalmatian widow, purchasing her for thirty-six ducats.
They took her across the Bosphorus to the far shore and put her up for the night. The next morning, the Dutchman was incensed to discover that the 'old lecher', the dastardly French Papist had happily bonked the woman all night, and, indeed, was plotting to sell her on again.
Infuriated, Lithgouw created a major scene, threatening to involve the French ambassador to the Ottoman court. He managed to extricate the woman, and even found her a job in a tavern. He earned her everlasting gratitude, but didn't presume to accept any favours in kind, and went back to Holland feeling rather pleased with himself.
Being a Protestant, Lithgouw's stereotypical view of Catholics was very likely vindicated by the perfidy of the Frenchman. But what brought him even more satisfaction was the knowledge that that one 'night of fornication' had cost his erstwhile partner the heavy sum of thirty-six ducats.
Welcome to the twenty-fourth instalment of the Giant’s Shoulders, the monthly carnival of science and history-of-science blogging from around the planet. In this issue, there’s a bit of a preponderance of historical vignettes, which I trust you’ll find interesting.
Shall we start with a timeline of scientific discoveries, analyses and inventions? How about this – from years that end in 24 in every century? I’m missing several years, so if you’d like to fill those in, please be my guest.
I can’t think of any particular grouping in this carnival. You know, by subject and all that. It’s pretty motley. Chronological order is somewhat unsatisfying, like arranging books on your shelf in alphabetical order of title. Let us, therefore, be whimsical.
From New at LacusCurtius & Livius, we have Bill Thayer’s introduction and a link to a classic paper about how people developed effective ways to compute the heights of mountains. It’s a wide-ranging work, starting from the Ancient Greeks and working its way through Kepler and The paper is from 1929, so expect nothing on GPS in it!
Continuing with mensuration, Ethan Siegel at ScienceBlogs has a neat little piece about how to determine distances to the stars. In How Far to the Stars, he explains the principle of parallax, notes that Copernicus thought he could use it to tell distances, and then goes on to Christiaan Huygens’s idea of using luminosity to compute how far Sirius is from us.
So we are able to measure distances with excellent accuracy. But what of Heisenberg’s Uncertainty Principle? Surely it’s impossible to measure simultaneously (to arbitrary precision) both the position and the momentum of an object? This is certainly true on the scale of the truly small, but why is it not so evident on the scale of our senses? Why does quantum mechanics explain to remarkable accuracy the weird behaviour of the minute but loses that weirdness at the macroscopic? On homunculus, Philip Ball presents Big Quantum, where he talks about this explanatory lacuna in the transition between the very small and the large. It’s all due to decoherence!
For another example of the genre, we have Caroline Rance at The Quack Doctor with an interesting little piece about one type of alternative medicine – uroscopy. This is the use of urine’s colour, consistency, taste and smell as a diagnostic tool. By the 19th century, professional medics had discounted this as quackery, but common people continued to frequent piss-doctors. In Cameron the Piss-Prophet, we learn how one such worthy was hoist by his own petard.
From one aspect of quackery, we move on to another. In 1726, a woman named Mary Toft (or Tofts) was a sensation in England for having allegedly given birth to rabbits.
She had supporters and deriders in equal measure; scientists argued both in her favour (e.g. a medical doctor, John Maubray) and against (Sir Richard Manningham, a male midwife). William Hogarth’s Cunicularii at the Wellcome Library Item of the Month (besides analysing some lovely etchings by the great artist) raises a subtle question.
Leaving quackery, we ponder another form of obfuscation. Via Thony C. we have an article titled “Follies of the present day”: Scriptural Geology from 1817 to 1857 by J.M. Lynch at A Simple Prop. Who were scriptural geologists? These were scientists who tried somehow to unify vast geological time with their faith in the Bible’s six days of creation. Note that these people did not believe (pace Ussher) that the Universe was formed in 4004 BC.
Why did these Christian scientists not pay closer attention to James Hutton? In 1785, he had published an article Concerning the System of the Earth, Its Duration, and Stability in which he began the study of deep geological time. Via Thony C., we have John F. Ptak’s Raising Greatness: James Hutton’s Deep View of Time, 1795 posted at Ptak Science Books talking about Hutton’s remarkable intuition, as he saw the entirety of Earth’s history explained in the sloping red sandstone at Siccar Point.
While on the subject of deep time, a fossil of the Archaeopteryx was first discovered in 1861. Since then palaeontologists have argued whether this transitional creature (between dinosaurs and true birds) was capable of flight. Ed Yong presents First Birds Were Poor Fliers at the Discover Magazine Blog. There were some reasons to believe that the beast could fly but also good reasons why not.
Dorset Ancestors have a biography of Francis Glisson, a native of that English county, who went on to great success as a doctor in 17th century London. When he heard of a new debilitating disease among children of Dorset, he studied it carefully and concluded that this bone-deformative condition (which we know as ‘rickets’) was caused by malnutrition.
A sesquicentennial after Glisson plied his investigative mind, another Englishman began to consider the various causes of deafness in people. As a diagnostic tool, John Harrison Curtis invented the Cephaloscope, which is described in Jaipreet Virdi’s post The Cephaloscope on From the Hands of Quacks.
John Harrison Curtis also insisted that aurists – students of auditory illnesses – should also be attached to medical hospitals. Jaipreet Virdi’s related post The London Asylum for the Deaf and Dumb tells this story.
From illnesses of the body, we segue to a illness of the mind. The dreaded African disease, sleeping sickness, when in an advanced stage, often resulted in violent mania. Was this true insanity on the part of the natives, or just a manifestation of a fear of isolation in a camp? From Sleeping Sickness and Lunacy at the Colonial Psychiatry Hub, we learn that much documentation of the effects of sleeping sickness on the local population is available from the British staffers of the Colonial Office (Uganda).
This video is an overview of the life, times, and accomplishments of Antony van Leeuwenhoek (1632-1723), the Dutch scientist who used hand-made single-lens microscopes to become the first human to see protozoa, bacteria, sperm, and red blood cells, among many other things.
And although that video’s only 7 minutes long, it promises more riches to come. Can there be any more to van Leeuwenhoek, then? Indeed there can, and Thony C. presents Lego-nhoek? posted at Antoni van Leeuwenhoek Centraal.
van Leeuwenhoek, of course, was a contemporary, competitor and Royal Society colleague of that polymath Robert Hooke. I found several blogs on this brilliant Englishman, and planned to chuck them in here, but closer scrutiny revealed that at least one of those pieces was a copy of a Wikipedia article. I hope this is not a similar copy: Leland Velazquez discusses the Physics of Bungee Jumping. Hooke’s work on elasticity leading to his eponymous law is the underpinning of this extreme sport.
If Robert Hooke is around, Christopher Wren can’t be far behind. Another brilliantly multifaceted man, Wren involved himself in astronomy, mathematics, architecture, art, and – we learn from Christopher Wren and the Bees by Gene Kritsky at Wonders & Marvels – beekeeping. Why is this scientifically relevant? He also developed the transparent beehive that was suitable for the scientific observation of bee behaviour.
The bees were able to move between the various layers of the hive, and glass panels set into the structure allowed an observer to see the honey cascading down inside it. Although Wren’s construction had not been an immediate success (due to a failure to realise that bees worked downwards), it offered the prospect of an ever-increasing stock of bees and honey within the same hive. [from here]
Speaking of polymaths, there was Florence Fenwick Miller in the 19th century. She was a suffragette, a journalist, an educator, and a qualified medical doctor. From Peacay at the absolutely ravishing BibliOdyssey, we have An Atlas of Anatomy.
From Peacay at BibliOdyssey again, we have Fungis Danicis, a lovely set of illustrations of fungi studied by a Danish lawyer-turned-botanist named Theodor Holmskjold in the 18th century. This worthy actually identified over fifty new species during two years of observation in the countryside near Aarhus.
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Now for some controversy! When scientists without training in history try to develop works in the history of science, clearly feathers are ruffled. Via Thony C., we have Renaissance Art or Neuroanatomy? Part 1 and part 2, posted by Darin at PACHSmörgåsbord. It appears that there have been a series of papers over the past 20 years by various scientists trying to find hidden anatomically precise messages in Michelangelo’s art.
R. Donald Fields wrote up a guest blog at Scientific American recently that discusses the oeuvre; as it turns out, none of the scientists involved is a trained historian. As the rebuttal at PACHSmörgåsbord makes clear, this is dangerous practice:
Why does this matter? Because such articles attract the attention of editors and contributors at places like Scientific American. These contributors then post a redaction of the article on the website, where the post becomes one of the most popular. Such redactions misrepresent the history of science, suggest that anybody can write the history of science, and deny that historians of science have a discrete expertise. That’s why it matters.
But just to show that not all scientific speculation on art is ill-founded, check out this interesting bit of detective work by a bunch of astronomers at Texas State University. Via Thony C., we have a post by Roger Sinnott on Sky And Telescope titled Walt Whitman’s Meteor-Procession, we learn about the great poet’s imagery (“the strange huge meteor-procession" that went "shooting over our heads" with "its balls of unearthly light”) and an interesting astronomical phenomenon.
I’m not sure if this next piece fits within the purview of our Carnival (as it may be more aligned to sociology) but I think it relates to a seminal work of behavioural psychology, so perhaps I can present it? Christopher Greene’s Clarks’ Black-White Doll Experiment Replicated at Advances in the History of Psychology blog discusses an even more controversial subject than that in the preceding paragraph, and highlights its most recent replication under the aegis of CNN (full report here in PDF).
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On to more jolly subjects. Remember Light Amplification by the Stimulated Emission of Radiation? It is 50 years since the invention of the (optical) laser. Via Thony C., we have Happy Birthday! posted by Matt Springer at Built on Facts.
Another anniversary is that of Vannevar Bush’s visionary piece sixty-five years ago in the Atlantic Review about associative links in information, which one can easily see as an early precursor to hypertext and all that. Via gg, we have Simon Harper’s review ‘As We May Think’ at 65 from his blog Thinking Out Loud.
We end on a luminous, numinous note. What links a certain 1980s teen film, a song by John Parr, the Chinese sea-goddess Mazu, and – wait for it – ionised air? In St Elmo’s Fire, Captain Skellet of A Schooner of Science explains the whys and the hows of the gorgeous blue glow that appears on the tips of masts on sailing ships.
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[And there you have it. Thanks for stopping by. I hope this encourages you to write up your own pieces on science and its history and philosophy. Do consider submitting your blog article to the next edition of the giant's shoulders at The Dispersal of Darwin. You can use the carnival submission form. Past posts and future hosts can be found on the blog carnival index page.]
Following KM's pointer to Futility Closet, I came across an assertion at that site that the following sentence makes sense in Dutch. Never willing to take things at face value (and also because it's a slow morning), I undertook the following correspondence with a Dutch colleague:
Does this sentence make sense in Dutch? What does it mean? Als In Bergen, bergen bergen bergen bergen bergen bergen bergen bergen bergen bergen.
Hmmmm, direct translation: As in Mountains, mountains, mountains, mountains etc. your guess as good as mine...
So it doesn't mean: "If in Bergen, heaps of mountains salvage heaps of mountains, then heaps of mountains salvage heaps of mountains"?
There would need to be comma after the sixth bergen.
So there you go, folks. The Dutchman has spoken and it is verily so.
Ever since my organisation was taken over by les braves Belges, the quality of food in the kitchen has undergone a vast improvement - both in taste and in variety. A few days ago, for instance, one of my colleagues decided to celebrate his birthday with some Dutch treats, and brought in a fine selection of bakery. Before he could send out his invite to the rest of the floor, I had raided the larder and come away with sumptuous examples of stroopwafels, gevulde koeken, krakelingen, and some other biscuits whose name escape me.
Caramel filled syrup waffles, known for short as stroopwafels, date from the 18th century, and are possibly Holland's best known sweet treats. They are made from vanilla and cinnamon dough, filled with syrup, pressed and baked. Ostensibly made from secret recipes known only within families of stroopwafel chefs, these originated in the Gouda region of the Netherlands.
The gevulde koeken are almond-filled pastries made from butter-dough, with an egg-based glaze atop to give it a shiny effect. It's usually had with coffee or hot chocolate or tea, or wolfed down ravenously if you are like me and impatient. You can get oblong varieties of this delicacy, called rondos (or Amsterdammertje), in case you long for change but don't want to give up the essence of almond and butter.
Krakelingen are puff pastry cookies, famed for being chucked at revellers in the Flemish town of Geraardsbergen (where, soon after, the revels culminate in live fish immersed in wine being hastily consumed), but in other aspects are not quite so lively: they have flaky interiors and a glazed sweet top.
Are you envious/drooling/checking your cholesterol yet?
Here's a brilliant white creation by some intrepid Spanish architects. The Carabanchel housing project, 102 dwellings in all, has been designed and executed by Desmasuno.[Via Yatzer]
But - again - I have to say that there's nothing new under the sun. About 11 years ago, the WOZOCO homes for the elderly was constructed by MVRDV Architects in Holland.[Photo by Roryrory.]
I happened to read A. C. Baantjer's DeKok and the Geese of Death recently. Baantjer is a famous Dutch writer of crime fiction, and his police inspector DeKok is as erudite a man as Sherlock Holmes, with knowledge of far more recondite things than the great detective. This book is rather weak, though. It's fairly obvious early on what the plot is, and the rather literal translation from the Dutch makes for a slightly jarring read.
But one interesting thing I came across is Bargoens, the Dutch argot of sundry low-life, criminals and tramps. In the translation of Baantjer's book, it was rendered into a sort of pidgin Italian-English, a sort of Cockney-mixed-with-Brooklyn (at least, that is what it looked like in English). Bargoens, itself, though, has a long and immodest history. It was fairly widespread till about the 1980s (the book is from 1983), but fewer and fewer people spoke or understood it even then. For instance, Baantjer makes the case that DeKok understood it well, although he didn't like to speak it at all. His colleagues seemed to have a slightly trickier time comprehending it.
Bargoens has a large complement of Yiddish loan-words in its vocabulary. (See here, for instance.) It appears to have originated in the 16th century, but became popular in the late 19th and early 20th centuries. Jewish travelling salesmen, ostracised by the general population, were large contributors to the slang, explaining thereby the Yiddish influence. Traces of the tongue now survive in the Amsterdam dialect of Dutch, but mainly in cuss-words, and terms for money or sex.
There are a few Dutch people in my office. Well, usually, there are. Today, for some reason, most of them are absconding. There's only one around: Loes. And she hasn't heard of Bargoens. Perhaps not known in my part of Holland? she says.
Man, I gotta find me some real Dutch toughies.
Paul Van Hauwermeiren, a researcher who is writing a book on the argot, makes the following remarks 1, (paraphrased by me).
"Dutch Bargoens dates from 1504. You can find it, for instance, in the work of the 16th century poet Anna Bijns of Antwerp. Over time, some Bargoens words transferred to general Dutch.
"It is hardly spoken any longer. In Flanders and in the Netherlands, there are a few places where Bargoens lives on. In Roermond (Dutch Limburg), for instance... In Flanders, it is spoken by traders in scrap and used cars. They call themselves 'travellers', indigenous Flemings descended from travelling salesmen and artisans. Many of them still live in mobile homes. They speak Bargoens, but they agree that their kids grow up not speaking it much. When these people meet another who they think is a 'traveller', they attempt a few words of Bargoens, and if he responds, they know that he is a 'traveller', too.
"Roermond is the only place where Bargoens is heard in daily use... The district is a former working class neighbourhood with great solidarity...They mixed Bargoens with their own dialect to create a Begoons. Many locals think that strange Bargoens words in their speech are authentic old Roermondese words... But there are families in Roermond who use Bargoens words like premerik (priest) and drejjerik, instead of the Roermondese pesjtoor and vlaaj
"It is not very easy to collect the words and their meanings. The Bargoens speakers in 't Veld are not forthcoming with information about their Begoons. This is understandable, because if they explain the meaning of a word to everyone, it no longer remains a secret. Fortunately, I have an informant in t' Veld who occasionally feeds me a few words.
"Other countries had their own secret languages as well. Around 1450, the French poet Villon was writing ballads in the argot of the infamous Coquillards. In England, around the same time, began the Thieves' Cant. The Germans had Rotwelsch, which is very related to Bargoens. Travellers of either Bargoens or Rotwelsch country were mutually intelligible in a broad area of Europe, from coastal Flanders through Denmark to Poland.
"...Anywhere that Bargoens was spoken, you would have a basic core of words, around which there were words that were typical for that particular area. There is Yiddish influence, influence from the Romany in the south, and from French in Belgium. Dutch words were distorted. New words were formed by derivation and composition.
"It was never a full language with its own grammar. In the oldest lists of words, Bargoens only has nouns and verbs. Other parts of speech were added as its range extended: adjectives, a few adverbs, and also secret counting words. There are many nouns for money. It was more a vocabulary, a series of words to be inserted in a given language or dialect. In the 19th century, the average speaker used about three to four hundred Bargoens words. But if you counted all the words used by the speakers in Roeselare in West Flanders, Zele in East Flanders, Brussels or Antwerp, you already had a list of two thousand different words.
"Bargoens was grafted onto the local dialects. Each speaker was basically bilingual. Besides Bargoens, he spoke the local dialect of the national language. One would switch to Bargoens only when necessary. For example, when doing 'business in the margins,' when there were outsiders present, crucial words would be uttered in Bargoens. Thus, with minimal effort, the conversation was rendered impenetrable to outsiders. It was a shield. Remember that if you had no fixed place to live, you were a nobody, treated with suspicion. There was a large group of people without permanent rights to residence. They only had their secret language to guard against the inimical society around them.
Reference
1. Interview with Paul Van Hauwermeiren. Taalschrift, 29/01/2007.
I have modestly given myself the name of the greatest of the Eldar: Fëanor, the maker of the Silmarils and the creator of the Tengwar. He was a Master of many things. I am merely a Jack.
, we can get a cubic equation 
, where 
. Were a solver to recognise the left-hand-side as an expansion for a trigonometric function, he or she would be able to use the identity to solve the algebraic equation. Similarly, Viète realised that using the above substitution for x, that large 45th degree equation reduced to 
, which he could then solve using a table of sines.1
