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Casting the Yarrow: An 18th Century Method for Calculating Pi

The views expressed are those of the author and are not necessarily those of Scientific American.


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Happy Pi Day! In honor of the occasion, Jen-Luc Piquant has dug up an archival post on a lesser-known historical figure you devised an ingenious method of calculating Pi.

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In Phillip Pullman’s The Amber Spyglass, fictional Oxford physicist Mary Malone finds she can communicate with the mysterious, conscious “Dust” using the yarrow casting methods of the I Ching (it’s also possible to use coins and other symbolic units). For those who scoff that a physicist would never express any appreciation for a pagan method of divination, consider this: when he was knighted, Neils Bohr included the tai chi symbol in the design for his coat of arms, to reflect his appreciation for the I Ching’s ingenious use of probabilistic concepts.

Mary Malone’s divination method actually has a real-world counterpart in one of the oldest problems in geometrical probability, known as Buffon’s Needle. This experiment in probability was the brainchild of a French naturalist and mathematician named Georges-Louis Leclerc, Comte de Buffon.

Portrait of Georges-Louis Leclerc, comte de Buffon (1753). Musée Buffon, Montbard. Public domain.

Buffon had quite the interesting life as the son of Benjamin Leclerc, lord of Dijon and Montbard. Born and raised on the Cote d’Or, the young George-Louis started off studying law before getting sidetracked by mathematics and science. It’s not clear that he ever earned a degree, however, because he was forced to leave the university after getting tangled up in a duel. He toured Europe for awhile, only returning when he heard his father had remarried — not so much out of familial devotion as concern over his inheritance of the title and estates.

Buffon fils is best known for writing the Histoire Naturelle, a whopping 44 volumes of encyclopedic knowledge that covered everything known to date about the natural world (originally there were 36 volumes; 8 more were published after Buffon’s death). A full 100 years before Charles Darwin’s Origin of Species, Buffon noted the similarities between humans and apes and mused on the possibility of a common ancestry, concluding that species  must have evolved since that common point.

He never progressed beyond those musings to propose an actual mechanism for this evolution, but his tome was translated into numerous languages, and certainly influenced Darwin, who described Buffon — in the foreword to the 6th edition of Origin — as “the first author who in modern times has treated it in a scientific spirit.”

But we’re more interested in a paper he published in 1777 entitled, Sur le jeu de franc-carreau, in which he first considered a small coin (an “ecu,” for all you crossword puzzle buffs) thrown randomly on a square-tiled floor. It was all the rage in Buffon’s social circles to place bets on whether the coin would land entirely within the bounds of a single tile, or across the boundaries of two tiles right next to each other. Buffon had a bit of an advantage  over his peers thanks to his mathematical interests. He realized he could figure out the odds of the wager using calculus — making him the first person to introduce calculus into probability theory.

Buffon was a pretty sharp cookie: he knew his geometry, noting that the coin would land entirely within a tile whenever the exact center of the coin landed within a smaller square — and that smaller square’s side was equal to the side of a floor tile, minus the diameter of the coin used in the toss. Ergo, he concluded that the probability of the coin landing entirely inside a single tile could be expressed mathematically as the ratio of the area of the tile to the area of the smaller square.

It marked the beginning of “geometric probability,” wherein one could determine probabilities by comparing measurements, instead of the tediously time-consuming method of identifying and counting all the other alternative, yet equally probable, outcomes or events — a specific hand in poker, for example, or a roll of the dice in craps.

Source: Wolfram MathWorld. http://mathworld.wolfram.com/BuffonsNeedleProblem.html

You can perform the same basic experiment using a sewing needle and a checkerboard — hence the name “Buffon’s Needle.” Drop the needle onto the checkerboard, and one of two things happens: either the needle crosses or touches one of the lines, or it doesn’t cross any lines. (It’s worth noting this is an idealization, and it assumes parallel lines or squares spaced about 1 inch apart, and the use of a needle 1 inch long.)

Buffon dropped the needle over and over again, keeping track of how the needle randomly landed each time. Buffon’s key insight was that the probability that a dropped needle (or tossed coin) would cross a line  is basically 2 divided by Pi.

He simply divided the number of crossing needles by the total number of needles, and realized that the more times one drops the needle, the closer one would approach the value of the probability — i.e., the closer one would come to the value of Pi.

There’s tons of online versions of the experiment, employing the Monte Carlo method, wherein the user can repeat the “toss” as many times as s/he wishes: 500, 1000, even 100,000 times.

The essence of probability theory is that the more times you repeat the experiment — the more hands of poker you play, or the more times you roll the dice at the craps table, or spin the roulette wheel — the more closely you will approach the calculated textbook probability.

There may be winning or losing streaks in the short term, but the more you play, the more predictable things become. It’s just a quirky little oddity that the value relates to Pi. In fact, with an infinite number of tosses, the value will be exactly Pi assuming one could ever reach infinity. The mathematician Pierre La Place definitively proved this in 1812. This is also the essence of what Mary Malone discovers in The Amber Spyglass.

So there you have it: a seemingly random scattering of needles (or yarrow sticks) over a sheet of lined paper can nonetheless give you a very precise number. Such is the power of calculus. And via the wonderful YouTube series, Numberphile, we now have a more modern (and delicious!) method of calculating pi — using actual pies (and note the length of the video):

Jennifer Ouellette About the Author: Jennifer Ouellette is a science writer who loves to indulge her inner geek by finding quirky connections between physics, popular culture, and the world at large. Follow on Twitter @JenLucPiquant.

The views expressed are those of the author and are not necessarily those of Scientific American.



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  1. 1. Robin Lloyd 5:01 am 03/16/2013

    For more on Buffon, check out this peculiar 2011 SciAm feature by Lee Dugatkin on the search for a moose to combat a European prejudice against the United States in the late 18th century/early 19th century. http://www.scientificamerican.com/article.cfm?id=jeffersons-moose

    Link to this

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