Context and sources: Cambridge’s differential analyzer

Recently there has been a striking development… Dr. Bush of the Massachusetts Institute of Technology has invented a machine which will effectively carry out processes of integration. The importance of this development can scarcely be exaggerated. It was the hope of Leibniz that the invention of the calculus would be but a step in the direction of mechanical reasoning, which would enable scientists to concentrate on the essential processes of analysis without being baffled and distracted by the complexity of the equations they employ. The idea for the integrator which has made recent developments possible was due in the first instance to Lord Kelvin, but he failed to overcome the mechanical difficulties involved. The machine constructed by Dr. Bush has proved itself capable of solving ordinary differential equations of any order up to the sixth and so brings within practical reach the solutions of an enormous number of problems in physics, chemistry and engineering.

Recently Dr Bush of the Massachusetts Institute of Technology, using modern engineering devices, has succeeded in constructing such a machine as Lord Kelvin had in mind and in coupling together integrating units in such a way as to solve differential equations. A machine of twelve units has proved itself capable of solving differential equations of one variable of any order up to the sixth, and so brings within practical reach the solutions of many problems in physics, chemistry and engineering hitherto regarded as intractable. Impressed with the remarkable promise of this machine as an instrument of research, Manchester University have recently installed eight integrating units, constructed by Messrs Metropolitan Vickers, and these have been applied successfully to the solution of problems of atomic physics. A working model of the Bush machine has also been constructed in the Chemical Department of this University. While not so robust as a full-scale machine, it can solve differential equations with remarkable accuracy, and has proved to the satisfaction of the Faculty Board of Mathematics that the claims made for the full-scale machine are justified.

The important feature of the machines to which reference has been made is the speed with which definite answers can be produced. The Bush machine, for example, can produce in a few days results which would otherwise require months of laborious computation. In many problems it is essential that reliable results should be obtained speedily if further developments, which may be based on them, are not to be held up. Such problems, if they can only be solved slowly and with great labour, cannot usefully be attempted at all. For this reason the Council of the School of the Physical Sciences are convinced that the machines have become essential in modern research and that the time has come when in the interests of the physical sciences in Cambridge a computing laboratory should be established.

Many problems in physics, chemistry and other subjects give rise to differential equations which are difficult or impossible to treat mathematically. It has long been possible to obtain numerical solutions by arithmetic methods, and the development of adding and multiplying machines to their present state of perfection has made these much simpler to carry out. The amount of work involved is, however, still very great.

The differential analyser is a machine for integrating differential equations mechanically, and the principle of its operation is becoming well known. Briefly, it consists of a series of mechanical integrating units, together with input and output tables, which can be interconnected by means of a flexible system of shafting with removable gears. The components are connected together in an appropriate way for the particular differential equation to be solved, and the solution is given in the form of a curve drawn mechanically on the output table by a moving pencil. …

While not claiming that such a high degree of accuracy can always be guaranteed, … the machine is not simply a toy, but is capable of serious work. Its limitations arise from the light form of construction adopted. More maintenance work is required than would be the case for a machine of more robust design and the machine must be run fairly slowly to avoid jerkiness. This trouble becomes more pronounced as the number of integrators in use in the set-up is increased. Our machine has five integrating units, and we consider that for general use this is the greatest number which could be satisfactorily worked, though in special set-ups it might be possible to use more.

Largely as a result of our experiences in 1937 we concluded that the scope of the machine would be much increased by the addition of a fifth integrator. Prof. Lennard-Jones very kindly made arrangements for a grant for this purpose and in the autumn Mr Wilkes and I drew up specifications. The existing machine was split down the middle about three inches and the torque amplifier motor removed to make room for the new unit in a central position. The motor was refixed elsewhere with a rubber cushion between it and the table and rubber washers under all the bolt heads, which was deemed to effect an incidental improvement in reducing the vibration. … After several delays, (due partly to a propensity on the part of the makers for covering a multitude of sins with a very undesirable coat of paint,) we were able during the Easter vacation (1938) to put the unit in working order. It was quickly pressed into service in Mr Wilson’s problem… and we felt its existence justified.

In June 1937, Lennard-Jones asked me if I would help one of his research students, Miss Elizabeth Monroe, to try out the capability of the machine on a differential equation arising in the two-centre problem in wave machines. Here I provided only the technical assistance, since, as Elizabeth observed in her thesis, I had little personal interest in the problem itself; however, she did thank me for listening “with apparent patience” to occasional monologues on it and I certainly learnt much as her work proceeded. The problem strained the model differential analyser to the limit. As left by Bratt, the machine had only four integrators, but there was a space in the middle where, with a little re-arrangement, an additional one could be inserted. Elizabeth was very anxious that we should build the additional integrator, since she saw that it would make all the difference as far as her problem was concerned. Lennard-Jones agreed to provide the necessary funds and by the autumn we had drawn up the specification and ordered the necessary parts. I had spent many happy hours when young building things with Meccano, this being an English boy’s birthright. Elizabeth, who was American and a girl to boot, had been underprivileged in this respect. I therefore gallantly ceded to her the job of assembling the Meccano parts, only intervening to make sure that the nuts were tight. It is satisfactory to record that the new integrator was a great success.

The set-up was designed by Miss E. Monroe and Mr M. V. Wilkes of the Mathematical Laboratory, to whom I am much indebted. I also wish to thank Mr Wilkes for help in using the machine.