From the Archives – As Easy As PI(E) by Ted Nicholls
This is the first posting that delves into the MIRA archives for interesting contributions from members past and present. Hopefully, this will become a regular feature of the site.
In this post, we revisit an interesting article in MIRA 7 from 1984 submitted by Ted Nicholls about the fascinating history of PI and attempts to define it. I was particularly interested to learn of the fraction 355/113 being used in the 16th Century – in my days at school, we used the less accurate 22/7 (3.142857…) which seemed good enough in the North West of England in the early 1970’s but then again, I’m not sure how many famous mathematicians hail from Oldham….
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As Easy As PI(E) by Ted Nicholls
MIRA Volume 7, 1984
To a cook, pie means pastry and rhubarb, apples or some sort of fruit or meat. To a printer it means that someone has upset his type. But to a mathematician it is the Greek letter symbol referred to in this article as Pi. It is the ratio of the length of the perimeter of a circle to the length of its diameter. About 1600 the English mathematician, William Oughtred, used the Greek letter (π) Pi to symbolise the perimeter, (perimetron being Greek for “the measurement around”), and Delta for the diameter (diametron – “the measurement through”). Nowadays it is customary to use the Latin word “circumference” when speaking of circles – thus successfully obscuring the reason for Pi. The first top-flight man to use this symbol was the Swiss Leonard Euler in 1757 – and what was good enough for Euler was good enough for everyone else.
Having settled how pi came to be used, let us consider how the value was obtained. To find the circumference of a wheel, wheelwrights presumably wrapped some sort of cord around it, marked it where a circle was completed, and then straightened it out and re-measured it. Modern theoretical mathematicians frown at this and make haughty remarks like – “but you are making an unwarranted assumption that the line is the same length when straight as when curved”. The above honest workman would probably have solved matters by throwing the objector into the nearest cut.
When it came to the actual value, this varied enormously in ancient times. The Hebrews refused to be bothered with troublesome fractions and considered Pi to be equal to exactly 3. (Some years ago a bill was introduced in Tennessee to make Pi legally equal to 3 inside the state borders: fortunately it didn’t pass!) Mostly the ancients used 3, which is only 1 part in 2,500 high. However, along came the Greeks – and they would have nothing to do with the vile lay-down-a-string-and-measure-it-with-a-ruler business. Archimedes used the “Method of Exhaustion”, trapping the circumference of a circle between a polygon of 96 sides with the corners touching the circle inside it, and a 96 sided polygon with the sides tangential to the circle outside it. The figure he obtained, 3123/994, is only high by 1 part in 12,500.
Nothing better than this was obtained until the 16th century, when the fraction 355/113 was first used – and this is the best approximation that can be expressed as a simple fraction, being high by 1 part in 12,500,000. If we imagined that the Earth was a perfect sphere of exactly 8,000 miles diameter, by using 355/115 we get an equator of 25,132.7433 miles. The true value of Pi gives us 25,132.7412 miles – an error therefore of about 11 feet. This is good enough for most people, but mathematicians weren’t satisfied!
The key step was taken by Francois Vieta, a 16th century French mathematician considered the Father of Algebra. He performed the algebraic equivalent of Archimedes geometric method of exhaustion, involving square roots, square roots of square roots, and square roots of square roots of square roots. In 1593 he used his own series to calculate Pi to 17 decimal places. In 1615 the German Ludolf von Ceulen used an infinite series to calculate Pi to 35 places (Pi is sometimes called “Ludolf’s Number” in German textbooks). 1717 the.Englishman Abraham Sharp went several times better by working it out to 72 decimal places.
Was this really necessary? Well, if we drew a circle 10 billion miles across, with the Sun at the centre to enclose the entire Solar System – using 355/113 as Pi the circumference of over 31 billion miles would be out by less than 3,000. Using Ludolf’s 35 places the error would be equivalent to a millionth of the diameter of a proton. If we want the circumference of the known Universe (approximate diameter 80 thousand million light years) then this value of roughly 150,000,000,000,000.000,000,000 (150 sextillion) miles, using Ludolf’s value to 55 places, would be off by a millionth of an inch! Obviously Sharp’s value to 72 places was far beyond the accuracy needed now or in the future.
Did this stop mathematicians working it out to further decimal places? Not a bit of it: the improvers being George Vega (140 places); Zacharias Dase (200 places); Recher (500 places). Finally, in 1873 William Shanks reported the value of Pi to 707 places. It was the record until 1949 ~ and small wonder that it took him 15 years to make the calculation. However, in 1949; with computers coming into their own, one of the unending series was pumped into Eniac, and after 70 hours Pi was worked out to 2,035 places. In the process it was found that Shanks had made an error and well over a hundred of his last digits were wrong.
It seems that no-one had ever taken the trouble to check his calculations!
