Towards a New Theory of Calculating Pi

Mathematical Paper


English by Professor Hardy Carlson, 1918

This rather dry and focused mathematical book with information about the potential for calculating pi out several decimal points.

Synopsis: The Middle East, exposed to an influx of ideas from India, was the focal point of mathematical progress during the middle ages. Carlson is partiularly interested in al Kashi, d. 1436, a prominent Arab mathematician. Al Kashi wrote a treatise in which he calculated pi out to 16 places, a few years before his death. Al Kashi utilized the Archimedean method of pi calculation, which involves approximating the circumference of a circle by circumscribing a polygon into a circle and calculating the circumference of the polygon. The more sides to the polygon, the closer one comes to the true value of pi. A six-sided hexagon, for example, only yields a value of 3. Al Kashi painstakingly used a polygon with over eight billion sides to calculate pi out to 16 places. In 1596, Ludolph van Ceulen, a mathematician from Leiden, used this method to calulcate pi out to 35 places. When van Ceulen died in 1610, he had three further places inscribed on his tombstone.

The modern method (as of the time of Carlson’s paper) is to calculate pi by means of an infinite equation. One simple equation, known as Gregory’s equation, equates pi to 4 – 4/3 + 4/5 – 4/7 + 4/9 – 4/11 … A variation on Gregory’s equation, devised by Machin, is the state of the art method for calculating pi circa 1920.

Carlson was seeking to improve on Machin’s equation by finding one that would calculate pi out to more places without having to make as many calculations. Carlson posits that the snwer lies in non-Euclidean geometry. Much of the reset of Carlson’s paper discusses non-Euclidean geometric principles and models. Carlson’s paper reaches no particular conclusions, but suggests that a reexamination of medieval Islamic mathematical technique, integrating it with non-Euclidean theory,might hold the key.

Anyone reading the paper realizes that calculating pi beyond 16 places or so has no practically scientific or engineering application in the 1920s and might wonder why someone would bother calculating it out any further.


In the Charing Cross College Library.

Towards a New Theory of Calculating Pi

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