In a paper published in the Journal of the Franklin Institute in 1931, Vannevar Bush described a machine (Fig. 1) that had been constructed under his direction at M.I.T. for the purpose of solving ordinary differential equations. He christened the machine a differential analyzer. This was what would now be called an "analog" computer, and was based on the use of mechanical integrators that could be interconnected in any desired manner. The integrator was in essence a variable-speed gear, and took the form of a rotating horizontal disk on which a small knife-edged wheel rested. The wheel was driven by friction, and the gear ratio was altered by varying the distance of the wheel from the axis of rotation of the disk. The principle is illustrated in Fig. 2.

The use of mechanical integrators for solving differential equations had been suggested by Kelvin, and various special-purpose integrating devices were constructed at various times. Bush's differential analyzer was, however, the first device of sufficiently general application to meet a genuine need, and in the period immediately before and during World War II quite a number of these devices were constructed. The one shown in Fig. 4 was installed at the Mathematical Laboratory in Cambridge, England.

In order to make a practical device, it is necessary to have some means of amplifying the small amount of torque available from the rotating wheel. Bush used a torque amplifier, working on the principle of the ship's capstan, but adapting it for continuous rotation. Fig. 3 is taken from his report (1931) and sufficiently indicates the principle. The friction drums are rotated in opposite directions by a continuously running motor of sufficient power. When the input shaft is turned, one of the cords attached to the input arm begins to tighten on the friction drum round which it is wrapped. Which cord tightens depends on the direction of rotation of the input shaft. A very small tightening, and hence a very small tension in the end of the cord attached to the input arm, is sufficient, in view of the friction of the rotating drum, to produce a large tension in the end attached to the output arm. A small torque applied to the input shaft is thus capable of producing a much larger torque in the output shaft.

The integrators and torque amplifiers can be clearly seen in Fig. 4, together with the system of shafting used for effecting the connections. Changing the problem was a job for someone who did not mind hands covered in oil. The output table on which the results were plotted directly in graphical form can be seen in Fig. 4, which also shows a number of similar tables that were used for input, an operator being employed to turn a handle so that a cursor followed a curve. It is a comment on the primitive state of automatic control in the period in question that automatic curve-following devices were not provided until later. The accuracy attainable in a single integrator was about one part in three thousand, but of course a lower accuracy was to be expected in the solution.

Fig. 5 shows the notation that was used for an integrator and Fig. 6 shows how two integrators could be interconnected to solve a simple differential equation. It was not difficult to arrive at a diagram such as Fig. 6, even for a complicated equation, but working out the gear ratios required was a distinctly tedious task calling for some experience, particularly as accuracy required that full use should be made of the available range of integrator motion.

In 1945, Bush and S. H. Caldwell described a new differential analyzer in which interconnection between the integrators was effected electrically instead of mechanically. However, during the decade that followed, competition from electronic analog computers and from digital computers began to build up, and, although the new machine ran for a number of years at M.I.T., by 1955 the mechanical differential analyzer was already obsolete.

Digital Differential Analyzer

This device is based on the use of a rate multiplier as an integrator. In a rate multiplier, a constant quantity y is held in a register and, on the receipt of an input pulse, is added to the number standing in an accumulator. If input pulses arrive at a rate R, overflow pulses will emerge from the most significant end of the accumulator at a rate proportional to yR. If y now varies and if input pulses arrive whenever a certain other variable x increases by x, the number of output pulses emerging is proportion to y x or, approximately to y dx. Thus, the device serves as an integrator. Normally, x is equal to one unit in the least significant place, and continuously updated values of the variable x can be obtained by feeding the pulses into an accumulator.

The first digital differential analyzer was the MADDIDA developed in 1949 at the Northrop Aircraft Corporation. It had 44 integrators implemented using a magnetic drum for storage, the addition being done serially. There were six tracks in all on the drum, one being used for synchronizing purposes. The problem was specified by writing an appropriate pattern of bits onto one of the tracks. Compared with the digital computers then being built, the MADDIDA was on an impressively small scale. It lost some of its simplicity, however, when adequate input and output devices were added, and in the end competition from general-purpose digital computers proved too much for it. The MADDIDA and its descendants did not, therefore, have the bright future in scientific computation that was predicted for them. However, digital differential analyzers of a simple kind continue to have a place in certain control applications.


1931. Bush, V. J. Frank. Inst. 212: 447.

1945. Bush, V. and Caldwell, S. H. J. Frank. Inst. 240: 255.

1947. Crank, J. The Differential Analyser. London: Longmans, Green and Co.

1962. Huskey, H. D., and Korn, G. A. Computer Handbook. New York: McGraw-Hill.