Solving systems of linear equations mechanically

W. Thomson's design of a device for the solution of six linear equations for six unknowns (just one equation is shown). The picture is taken from Treatise on Natural Philosophy, Vol.1, p.487. Source: University of Michigan Library.

William Thomsons concept

In 1878 William Thomson, the later Lord Kelvin, published a concept for a mechanical device for the solution of systems of linear equations. This device was certainly a byproduct of his work on tide predicting machines where he computed weighted sums of sine functions using pulley systems.

John Wilbur's Simultaneous Calculator

From 1934 to 1936 John Wilbur developed the MIT Simultaneous Calculator. He completed a prototype for the solution of systems with two equations on October 27, 1934. The final device could solve a system of nine homogeneous equations in ten unknowns. I have collected some links to newspaper, magazine and journal articles on the web.

The most prominent user was Wassily W. Leontief who later won a Nobel prize in 1973 for his Input-Output analysis.

The Simultaneous Calculator is briefly described in the fantastic book A Computer perspective by the office of Charles and Ray Eames, and was on display in the underlying IBM exhibition in 1973.

Unfortunately, the device seems to be lost at some later point of time.

I recommend to read the essay W. Cauer and his Mathematical Device by H. Petzold in the book Exposing Electronics about the Simultaneous Calculator and other very interesting, but more or less unsuccessful devices for the solution of systems of linear equations in the pre-digital era.

The device at the Aviation Laboratory of Tokyo Imperial University

In 1944 Sasaki Tatsujiro, Shiga Makoto, Miita Junichi built a calculator that closely resembled the Simultaneous Calculator. It is on permanent display at the National Museum of Nature and Science, Tokyo.

Further descriptions and constructions

  • F. W. Sinden, Mechanisms for Linear Programs, Operation Research 7 (1959), 728-739.
  • J. Stringer, K. B. Haley, The Application of Linear Programming to a Large-Scale Transportation Problem, Proc of First International Conference on Operational Research, Oxford, 1957.

The first one deals with mechanisms where the coefficients are 0 or 1 and shows nicely how duality in linear programming is reflected in dual mechanisms. The second one describes at the end a concrete device to solve the transportation problem.