John Wallis (1616 - 1703)

Most people count sheep when trying to get to sleep. But not John Wallis. He spent his sleepless nights performing complex mental calculations.

On one remarkable night he calculated the square root of a number with 53 digits. And in the morning, he recited from memory the 26 digit answer!

His life is extraordinary for more reasons than this. He became an ace code breaker in what was a very tumultuous time.

Britain had a civil war raging and an era called the enlightenment was just beginning.

This would bring a new vision to peoples’ lives and the way that they thought about the world around them.

Nowhere did this have more impact than in science and mathematics. Wallis was at its centre.

Wallis became the most influential English mathematician before Newton. He chanced upon a book by Oughtred and was entranced and caught by mathematics from then on. He said that mathematics ‘suited my humour so well that I did thenceforth prosecute it, not as a formal study, but as a pleasing diversion at spare hours’.

He obviously had a keen mind for the subject and a way of applying this to practical and theoretical problems. While working as a minor clergyman at the height of the civil war an encrypted dispatch came to a dinner party he was at.

Wallis surprised everyone by deciphering it in two hours. He went to work as a Roundhead code breaker and was rewarded by a Chair at Oxford University in 1649.

That was his practical side. But he made great theoretical contributions as well - the award of his chair may have been for political service, but Wallis repaid this in mathematics.

He was a good politician - arguing against the execution of Charles 1. This ensured that he kept his job on the restoration of Charles 2. He remained at Oxford for 50 years.

Thinkers of the time rarely took prisoners in their arguments with one another. Here is an exchange between Wallis and Hobbes; the latter claimed to have discovered a method to square the circle which Wallis claimed was flawed.

Hobbes responded: ‘Of those who with me have written something about these matters, either I alone am mad, or I alone am not mad. No third option can be maintained, unless (as perchance it may seem to some) we are all mad.’

Wallis reply was equally witty: ‘If he is mad, he is not likely to be convinced by reason; on the other hand, if we be mad, we are in no position to attempt it.’

Wallis was also in at the beginning of the Royal Society which would do so much to advance science and scientific thinking thereafter. He also wrote the first history of mathematics in English.

Wallis’ mathematics

Wallis developed formulae for the areas under various curves in a book that he dedicated to Oughtred. He developed the result we would now write in terms of an integral,

\int\limits_0^1 x^{m}dx={1\over{m+1}}

This paved the way for Newton’s later work on calculus. He used the same means to find an infinite expansion for \pi. He did this by trying to evaluate the area ‘under’ a circle:

This is the integral,

\int\limits_0^1{\left(1-x^2\right)^{1\over{2}}}dx

Wallis couldn’t evaluate this, but he hit on the idea of trying to interpolate its value from other values; so he looked at the successive values of integer powers,

\int\limits_0^1{\left(1-x^2\right)^{0}}dx=1, \int\limits_0^1{\left(1-x^2\right)^{1}}dx={2\over{3}}, \int\limits_0^1{\left(1-x^2\right)^{2}}dx={8\over{15}},\ldots

This provided him with a sequence,

1,{2\over{3}},{8\over{15}},{16\over{35}},\ldots

and this he could interpolate. The end result is that he found that,

{\pi\over{2}}={2.2.4.4.6.6.8.\ldots\over{1.3.3.5.5.7.7.\ldots}}