Footnotes

• 'PLEASE DON'T FEEL THERE IS ANY NEED TO ANSWER THESE RAVINGS OF MINE': ALAN TURING'S ANALYSIS OF THE MONTE-CARLO GAMBLING SYSTEM & WORK ON A NEW METHOD OF LIGHTING.
  
  Turing's analysis of the Monte-Carlo Gambling System is couched in the language of mathematics and equations but put simply, by working out the probability of winning at increasing increments of 150, 1,520, 4,560 and 30,400 spins he proved mathematically that the longer you play, the probability of losing money increases – "...for short runs one most probably wins or else one loses an unexpectedly large sum. As the length of run is increased the chances of winning becomes more remote...", he confirms.
  
  The recipient of our letter, Alfred William Beuttell (1880-1965) was the father of Turing's schoolfriend Victor Beuttell (see the following lot). After leaving Sherborne, the boys remained close and Turing spent much time with the Beuttell family. Alfred had made his fortune by inventing and patenting the Linolite electric strip reflector lamp (better known now as the striplight) in 1901, his first important contract being a system of continuous lighting for the Coronation of Edward VII, manufactured by the Edison and Swan Electric Company with whom he had a long association.
  
  Buoyed by his company's success pre-war he enjoyed the life of a wealthy bachelor – flying (he met Bleriot and watched the Wright brothers), motor racing and sailing. He spent several months at Monte Carlo operating with some success his own gambling system. Stories of having lived for a month on his winnings piqued the young Turing's interest whilst staying with the family over Christmas 1932 after the death of Victor's mother in November: '...He showed Alan his gambling system, which Alan took back to Cambridge and studied. On 2 February 1933 he wrote back with the result of his analysis, which was that the system yielded an expected gain of exactly zero, and that accordingly Mr Beuttell's winning had been entirely due to luck and not to skill...' (Hodges, A., Alan Turing: The Enigma, 2014, p.89). Our letter is hitherto unpublished, but his biographer Andrew Hodges mentions it with reference to a letter Turing wrote to his friend Norman Routledge in 1953 mentioning the odds of being arrested for homosexuality: '...Alan Turing used logarithms of betting odds as the key to the work he had done for the 'racket' of cryptography, and his sustained fascination with probability is illustrated by that reference [in Routledge's letter] to a one-in-ten chance of being caught. In his 1953 stoic humour there is a link with innocent Anti-War undergraduate days of twenty years earlier when he analysed Alfred Beuttell's Monte Carlo gambling system...' (Hodges, p.xxxi). Whilst his analysis here was a light-hearted exercise, Turing's work on probability had wider implications, and his research paper, The Applications of Probability to Cryptology, in which he applied vigorous probability analysis to a wide range of cryptoanalytic problems, was key to the decryption work undertaken at Bletchley Park.
  
  Turing also helped Beuttell in his pioneering work in the scientific measurement of illumination. In 1927 Beuttell had taken out patents on a new invention, the 'K-ray Lighting System', designed to allow uniform illumination of pictures of posters within a glass box and in Turing he found the perfect person to find the right formula for the curvature of the glass. Turing '...suddenly produced the formula, without being able to explain it, which agreed with Alfred Beuttell's calculation. But Alan went further, and pointed out the complication which arose through the thickness of the glass, which would cause a second reflection on the front surface. This made necessary a change in the curve of the K-ray System which was soon put into application for exterior hanging signs, the first contract being with J. Lyons and Co., Ltd, the catering chain... it always delighted Alan that a mathematical formula could actually work in the physical world. He had always liked practical demonstrations, even though he was not good at them...' (Hodges, p.73-4). By the early 1930's Beuttell's work as a specialist lighting consultant brought him large projects such as the illumination of the Freemason's new headquarters in London, which Turing also advised on. Our letter provides a formula for the illumination of the floor of a hemispheric room lit from its centre, a problem which they had previously discussed ("...I just thought this one might be of use to you..."). During the Second World War, the Linolite company became the main supplier of hose clips for the de-icing systems of bomber aircraft but reverted back their main activity in 1945. The company remained in Malmesbury until its closure in 1993.
  
  A photograph of Alfred Beuttell at the time this letter was written and a copy of his biography The Man Who Made Linolite is included in the lot.
  
  Provenance: Alfred Beuttell (1880-1965); Victor Beuttell (1915-1993); and thence by descent.