– 8 –
(3) The various arithmetical operations, addition, subtraction, and multiplication (division being omitted), also ‘short multiplication’ by numbers less than 16, which will be much quicker than long multiplication.
(4) To perform the various logical operations digit by digit. It will be sufficient to be able to do ‘and’, ‘or’, ‘not’, ‘if and only if’, ‘never’ (in symbols A & B, A v B, ∼ A, A = B, F). In other words we arrange to do the processes corresponding to x y, x + y + xy, 1 + x, 1 + (x + y)2, 0 digit by digit, modulo 2, where x and y are two corresponding digits from two particular TS (actually TS 9 and TS 10).
5. Fundamental Circuit Elements.
The electronic part of the calculator will be somewhat elaborate, and it will certainly not be feasible to consider the influence of every component on every other. We shall avoid the necessity of doing this if we can arrange that each component only has an appreciable influence on a comparatively small number of others. Ideally we would like to be able to consider the circuit as built up from a number of circuit elements, each of which has an output which depends only on its inputs, and not at all on the circuit into which it is working. Besides this we would probably like the output to depend only on certain special characteristics of the inputs. In addition we would often be glad for the output to appear simultaneously with the inputs.
These requirements can usually be satisfied, to a fairly high accuracy, with electronic equipment working at comparatively low frequencies. At megacycle frequencies however various difficulties tend to arise. The input capacities of valves prevent us from ignoring the nature of the circuit into which we are working; limiting circuits do not work very satisfactorily: capacities and transit times are bound to cause delays between input and output. These difficulties may be best resolved by bending before the storm. The delays may be tolerated by accepting them and working out a time table which takes them into account. Indefiniteness in output may be tolerated by thinking in terms of ‘classes of outputs’. Thus instead of saying ‘The inputs A and B give rise to the output C‘, we shall say ‘Inputs belonging to classes P and Q give rise to an output in class R‘. The various classes must be quite distinct and must be far from overlapping, i.e. topologically speaking we might say that they must be a finite distance apart. If we do this we shall have made a very definite division of labour between the mathematicians and the engineers, which will enable both parties to carry on without serious doubts as to whether their assumptions are in agreement with those of the other party.
For the present we shall merely ignore the difficulties because we wish to illustrate the principles. We shall assume the circuit elements to have all the most agreeable properties. It may be added that this will only affect our circuits in so far as we assume instantaneous response, and that not very seriously. The questions of stable output only involve the mathematician to the extent of a few definitions.
