# Turing's Treatise on the Enigma

replacing each pair of pencil letters by the corresponding entry on Fig 51, i.e. by the position of the L.H.W. at which that pair of letters occurs as L.H.W. output. Now the whole of the pencil square can be obtained from its top line simply by filling in along diagonals. Translated into terms of the green ink entries this means to say that we only need to be given the positions at which to start copying from the short catalogue.

Actually we copy out the diagonals of the short catalogue onto staircase shaped strips (known as ‘Christmas decorations’ or ‘hand frills’) in reversed order, with the position in the short catalogue written above each square. These hand frills are numbered by the (constant) distance apart on ‘the diagonal’ of the pair of letters on them: e.g. in the hand frill no 5 shown in position for copying in Fig 56 I and F are at distance 5 on qwertzu and so are D and K. Instead of actually filling in the whole square with pairs of pencil letters we take the entries which might have been made in the top line, and write them in the top margin, and also put the entries which might have gone in the left hand column into the left hand margin. In order to find what hand frill to use for a particular diagonal the distances apart along qwertzu of the letters along the top are calculated. This should be done quite independently, to give a check on incorrectly copied letters (see ‘Mystic numbers’).

The reason for having the imaginary rod squares implied in the construction inverted is in order that the writing of diagonals may be from left to right and downwards, which is considered easier than from right to left and downwards.

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