Posts Tagged ‘trigonometry’
“We say the cows laid out Boston. Well, there are worse surveyors.”*…
On plotting the relationships of things in space…
Written by Leonhard Zubler, a Swiss goldsmith and instrument maker who is credited with popularizing the use of the plane table as a tool for surveying, Novum instrumentum geometricum illuminates the shared history of land-surveying and militaristic range-finding technologies. The text is intercut with elaborate copperplate engravings that showcase the might of trigonometry and triangulation in the immediacy of conflict. Bombardiers pack canons that are aimed with advanced precision at distant towers; the construction of ornate fortifications are planned with ease thanks to geometric instruments; and seemingly insurmountable crags are brought down to earth through the surveyor’s sightline. Readers are promised that they will learn how to measure the width of a moat or the height of wall in order to breach them more efficiently.
Novum instrumentum geometricum mainly features images related to an early modern instrument known as the triquetrum or Dreistab, a three-armed ruler, with two pivot points, used for charting angles in the heavens and on earth. Zubler most often showcases a two-armed variation known as the Zweistab, which includes a “finely divided scale and micrometer slide for exact settings”, writes Uta Lindgren. As if to show the versatility of this technology, the instrument is wielded on the masts of ships, balconies, and by a man perched atop the stump of a felled tree — even comically enlarged to depict its arms stretching out to touch the objects of their reconnaissance. Frequently two instruments are employed in parallel, by a pair of figures a fixed distance apart, which would allow the surveyors to estimate the distance to a faraway point using trigonometry.
Little is known about Leonhard Zubler (b. 1565), aside from his divorce in 1604, and probable death by plague circa 1611. He once created an extensive plan for modernizing the cityscape of Zurich, which was subsequently lost. During his lifetime, Zubler’s instruments were so desired that he was able to open a commercial outlet in Frankfurt am Main in 1608…
More mesmerizing illustrations: “Angles of Reconnaissance: Novum Instrumentum Geometricum (1607)” from @PublicDomainRev.
* Ralph Waldo Emerson
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As we triangulate, we might recall that it was on this date in 1667, about a year after the Great Fire of London, that Robert Hooke, a physicist (“natural philosopher”), astronomer, geologist, meteorologist, and architect was sub-contracted by friend and fellow architect Christopher Wren to conduct a survey of the fire-damaged area to establish ownership and facilitate reconstruction. As Lisa Jardine observed, “in the four weeks from the 4th of October, [Hooke] helped map the fire-damaged area, began compiling a Land Information System for London, and drew up building regulations for an Act of Parliament to govern the rebuilding.” Hooke also designed some of the buildings that made up the “new” London, among them: the Monument to the Great Fire of London (1672), Montagu House in Bloomsbury )1674 and Bethlem Royal Hospital (1674), which became known as “Bedlam.”
In 1670, Hooke was appointed Surveyor of the Royal Works. Together with Scottish cartographer and printer John Ogilby, he made precise and detailed surveys that led to the production in 1677 of a large-scale map of London, the first-known to be of a specific scale (1:1200).
“Chance, too, which seems to rush along with slack reins, is bridled and governed by law”*…
And the history of our understanding of those laws is, as Tom Chivers explains (in an excerpt from his book, Everything is Predictable), both fascinating and illuminating…
Traditionally, the story of the study of probability begins in French gambling houses in the mid-seventeenth century. But we can start it earlier than that.
The Italian polymath Gerolamo Cardano had attempted to quantify the maths of dice gambling in the sixteenth century. What, for instance, would the odds be of rolling a six on four rolls of a die, or a double six on twenty-four rolls of a pair of dice?
His working went like this. The probability of rolling a six is one in six, or 1/6, or about 17 percent. Normally, in probability, we don’t give a figure as a percentage, but as a number between zero and one, which we call p. So the probability of rolling a six is p = 0.17. (Actually, 0.1666666… but I’m rounding it off.)
Cardano, reasonably enough, assumed that if you roll the die four times, your probability is four times as high: 4/6, or about 0.67. But if you stop and think about it for a moment, that can’t be right, because it would imply that if you rolled the die six times, your chance of getting a six would be one-sixth times six, or one: that is, certainty. But obviously it’s possible to roll six times and have none of the dice come up six.
What threw Cardano is that the average number of sixes you’ll see on four dice is 0.67. But sometimes you’ll see three, sometimes you’ll see none. The odds of seeing a six (or, separately, at least one six) are different.
In the case of the one die rolled four times, you’d get it badly wrong—the real answer is about 0.52, not 0.67—but you’d still be right to bet, at even odds, on a six coming up. If you used Cardano’s reasoning for the second question, though, about how often you’d see a double six on twenty-four rolls, it would lead you seriously astray in a gambling house. His math would suggest that, since a double six comes up one time in thirty-six (p ≈ 0.03), then rolling the dice twenty-four times would give you twenty-four times that probability, twenty-four in thirty-six or two-thirds (p ≈ 0.67, again).
This time, though, his reasonable but misguided thinking would put you on the wrong side of the bet. The probability of seeing a double six in twenty-four rolls is 0.49, slightly less than half. You’d lose money betting on it. What’s gone wrong?
A century or so later, in 1654, Antoine Gombaud, a gambler and amateur philosopher who called himself the Chevalier de Méré, was interested in the same questions, for obvious professional reasons. He had noticed exactly what we’ve just said: that betting that you’ll see at least one six in four rolls of a die will make you money, whereas betting that you’ll see at least one double six in twenty-four rolls of two dice will not. Gombaud, through simple empirical observation, had got to a much more realistic position than Cardano. But he was confused. Why were the two outcomes different? After all, six is to four as thirty-six is to twenty-four. He recruited a friend, the mathematician Pierre de Carcavi, but together they were unable to work it out. So they asked a mutual friend, the great mathematician Blaise Pascal.
The solution to this problem isn’t actually that complicated. Cardano had got it exactly backward: the idea is not to look at the chances that something would happen by the number of goes you take, but to look at the chances it wouldn’t happen…
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… Pascal came up with a cheat. He wasn’t the first to use what we now call Pascal’s triangle—it was known in ancient China, where it is named after the mathematician Yang Hui, and in second-century India. But Pascal was the first to use it in problems of probability.
It starts with 1 at the top, and fills out each layer below with a simple rule: on every row, add the number above and to the left to the number above and to the right. If there is no number in one of those places, treat it as zero…
… Now, if you want to know what the possibility is of seeing exactly Y outcomes, say heads, on those seven flips:
It’s possible that you’ll see no heads at all. But it requires every single coin coming up tails. Of all the possible combinations of heads and tails that could come up, only one—tails on every single coin—gives you seven heads and zero tails.
There are seven combinations that give you one head and six tails. Of the seven coins, one needs to come up heads, but it doesn’t matter which one. There are twenty-one ways of getting two heads. (I won’t enumerate them all here; I’m afraid you’re going to have to trust me, or check.) And thirty-five of getting three.
You see the pattern? 1 7 21 35—it’s row seven of the triangle…
Pascal’s triangle is only one way of working out the probability of seeing some number of outcomes, although it’s a very neat way. In situations where there are two possible outcomes, like flipping a coin, it’s called a “binomial distribution.”
But the point is that when you’re trying to work out how likely something is, what we need to talk about is the number of outcomes— the number of outcomes that result in whatever it is you’re talking about, and the total number of possible outcomes. This was, I think it’s fair to say, the first real formalization of the idea of “probability.”..
On the historical origins of the science of probability and statistics: “Rolling the Dice: What Gambling Can Teach Us About Probability,” from @TomChivers in @lithub.
See also: Against the Gods, by Peter Bernstein.
And for a look at how related concepts shape thinking among quantum physicists, see “The S-Matrix Is the Oracle Physicists Turn to in Times of Crisis.”
* Boethius, The Consolation of Philosophy
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As we roll the bones, we might send carefully-calculated birthday greetings to a central player in this saga, Abraham de Moivre; he was born on this date in 1667. A mathematician, he’s known for de Moivre’s formula, which links complex numbers and trigonometry, and (more relevantly to the piece above) for his work on the normal distribution and probability theory. de Moivre was the first to postulate the central limit theorem (TLDR: the probability distribution of averages of outcomes of independent observations will closely approximate a normal distribution)– a cornerstone of probability theory. And in his time, his book on probability, The Doctrine of Chances, was prized by gamblers.
“No part of mathematics is ever, in the long run, ‘useless’.”*…
The number 1 can be written as a sum of distinct unit fractions, such as 1/2 + 1/3 + 1/12 + 1/18 + 1/36…
Number theorists are always looking for hidden structure. And when confronted by a numerical pattern that seems unavoidable, they test its mettle, trying hard — and often failing — to devise situations in which a given pattern cannot appear.
One of the latest results to demonstrate the resilience of such patterns, by Thomas Bloom of the University of Oxford, answers a question with roots that extend all the way back to ancient Egypt.
“It might be the oldest problem ever,” said Carl Pomerance of Dartmouth College.
The question involves fractions that feature a 1 in their numerator, like 1/2, 1/7 or 1/122. These “unit fractions” were especially important to the ancient Egyptians because they were the only types of fractions their number system contained: With the exception of a single symbol for 23, they could only express more complicated fractions (like 3/4) as sums of unit fractions (1/2 + 1/4).
The modern-day interest in such sums got a boost in the 1970s, when Paul Erdős and Ronald Graham asked how hard it might be to engineer sets of whole numbers that don’t contain a subset whose reciprocals add to 1. For instance, the set {2, 3, 6, 9, 13} fails this test: It contains the subset {2, 3, 6}, whose reciprocals are the unit fractions 1/2, 1/3 and 1/6 — which sum to 1.
More exactly, Erdős and Graham conjectured that any set that samples some sufficiently large, positive proportion of the whole numbers — it could be 20% or 1% or 0.001% — must contain a subset whose reciprocals add to 1. If the initial set satisfies that simple condition of sampling enough whole numbers (known as having “positive density”), then even if its members were deliberately chosen to make it difficult to find that subset, the subset would nonetheless have to exist.
“I just thought this was an impossible question that no one in their right mind could possibly ever do,” said Andrew Granville of the University of Montreal. “I didn’t see any obvious tool that could attack it.”…
Bloom, building on work by Ernie Croot, found that tool. The ubiquity of ways to represent whole numbers as sums of fractions: “Math’s ‘Oldest Problem Ever’ Gets a New Answer,” by Jordana Cepelewicz (@jordanacep) in @QuantaMagazine.
* “No part of mathematics is ever, in the long run, ‘useless.’ Most of number theory has very few ‘practical’ applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role.” – C. Stanley Ogilvy, Excursions in Number Theory
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As we recombine, we might send carefully-calculated birthday greetings to Ulugh Beg (or, officially, Mīrzā Muhammad Tāraghay bin Shāhrukh); he was born on this date in 1394. A Timurid sultan with a hearty interest in science and the arts, he is better remembered as an astronomer and mathematician.
The most important observational astronomer of the 15th century, he built the great Ulugh Beg Observatory in Samarkand between 1424 and 1429– considered by scholars to have been one of the finest observatories in the Islamic world at the time and the largest in Central Asia. In his observations he discovered a number of errors in the computations of the 2nd-century Alexandrian astronomer Ptolemy, whose figures were still being used. His star map of 994 stars was the first new one since Hipparchus. Among his contributions to mathematics were trigonometric tables of sine and tangent values correct to at least eight decimal places.
“Mathematics is the art of giving the same name to different things”*…
A 3,700-year-old clay tablet has proven that the Babylonians developed trigonometry 1,500 years before the Greeks and were using a sophisticated method of mathematics which could change how we calculate today.
The tablet, known as Plimpton 332, was discovered in the early 1900s in Southern Iraq by the American archaeologist and diplomat Edgar Banks, who was the inspiration for Indiana Jones.
The true meaning of the tablet has eluded experts until now but new research by the University of New South Wales, Australia, has shown it is the world’s oldest and most accurate trigonometric table, which was probably used by ancient architects to construct temples, palaces and canals…
More of the remarkable story at “3,700-year-old Babylonian tablet rewrites the history of maths – and shows the Greeks did not develop trigonometry.”
* Henri Poincaré
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As we struggle to remember the difference between a sine and a cosine, we might recall that it was on this date in 1842 that the United States Naval Observatory was authorized by an act of Congress. One of the oldest scientific agencies in the U.S., its primary task was to care for the Navy’s charts, navigational instruments, and chronometers, which were calibrated by timing the transit of stars across the meridian. It’s now probably best known as the home of the “Master Clock“, which provides precise time to the GPS satellite constellation run by the United States Air Force… and for its non-scientific mission: a house located within the Naval Observatory complex serves as the official residence of the Vice President of the United States.
Initially located at Foggy Bottom in the District of Columbia (near the current location of the State Department), the observatory moved in 1893 to its present near Embassy Row.
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