# Did you know? Notes from the history of Maths

Mathematics – a cure for madness?

Augusta Ada King, Countess of Lovelace, was the daughter of the poet Lord Byron. Her parents separated shortly after her birth and her mother, detesting her ex-husbands excesses, “hoped that instruction in Mathematics might root out any madness could have inherited from her father”[1]. Ada was privately tutored (as women were not allowed to attend university) by Mary Sommerville, a polymath who translated LaPlace’s work on astronomy, and later by Augustus de Morgan, who worked on symbolic logic such as “de Morgan’s laws”: ( AB )’ = A’ ∩ B’ . In 1833, Sommerville introduced Ada to Charles Babbage. She worked with him on the design of his machines, especially the Analytical Engine: “the Engine weaves Algebraic patterns, just as the Jacquard loom weaves flowers and leaves”, she once said[2]. In particular, Ada wrote an algorithm for this theoretical machine (it was never built) to calculate Bernoulli numbers. These occur as the coefficients in the summation of powers of consecutive integers:

where Bk are the Bernoulli numbers. So, for example:

m =1 gives the triangle numbers:

The sum of squares

requires B2 = ⅙ to give the formula ⅓ n3 – ½ n2 + ⅙ n for the sum of squares.

All sums of powers can be expressed as polynomials with coefficients involving the Bernoulli numbers. They are also used in the Taylor expansion for tan(x) and for the Riemann zeta function. Many consider Ada’s algorithm to be the first computer program.  Alan Turing studied her work as part of his work on programming.

Did the study of Mathematics cure Ada? It did cause her to lose a lot of money when she created what she thought was a sure-win mathematical model for gambling. Though her mother forbade her ever to see a portrait of Lord Byron, when she died, at the tragically early age of 36, she specified that she should be buried next to him in Hucknall, Nottinghamshire.

Don Hoyle

[1] The Story of Mathematics, Anne Rooney, Arcturus Publishing, 2009,p45

[2] The music of the primes, Marcus du Sautoy, Harper, 2004, p190

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