Vannevar Bush's Differential Analyzer
More than two centuries after Gottfried Wilhelm Leibniz’s reflections on the mechanization of mathematical reasoning, a dream that had long remained in the realm of fantasy took shape in the laboratories of the Massachusetts Institute of Technology (MIT). One man, Vannevar Bush, born in 1890 in Everett, Massachusetts, undertook this colossal task during the interwar period.
He created the differential analyzer, completed in 1931. A machine resulting from a collaborative effort with Frank D. Gage, Harold L. Hazen, King E. Gould, and Samuel H. Caldwell. The idea was not entirely new, however. Sir William Thomson had suggested fifty years earlier that the integrators designed by his brother could, if connected together, solve differential equations. But the technical constraints of the Victorian era made this dream unattainable.
Bush succeeded where others had failed. His differential analyzer solved sixth-order differential equations or three second-order equations simultaneously, a genuine technical feat. The machine relied on torque amplifiers that supported considerable mechanical loads, a system of transmission rods (bus shafts) that connected the various units, and impressive dimensions to maximize plotting precision. A bold concept.
The heart of this invention aimed at three qualities rarely combined: extreme flexibility, mechanical robustness, and acceptable precision. Under normal conditions, the machine achieved precision of one thousandth for each individual unit. Momentary errors naturally compensated during the integration process, recalling the behavior of a planimeter whose deviations eventually balance out.
To use this machine, several hours were needed to configure it after completing the necessary plots and determining the connection diagram. The actual solving generally took about ten minutes for each set of boundary conditions. Operators needed to acquire some experience but gained in return an intimate understanding of the differential equations they were manipulating.
Vannevar Bush and his collaborators had to overcome significant technical obstacles. Mechanical backlash and integrator slippage were complex issues. They developed an ingenious system called lashlock to eliminate backlash in worm gears and designed two-stage torque amplifiers producing very high torque ratios with minimal input torque.
To validate the reliability of their invention, they conducted forty rigorous tests on a complete integration unit. These tests, performed with loads varying from zero to one foot-pound of output torque, at different positions and in both directions of rotation, revealed an average deviation of only 0.032 % from the calibration constant. The maximum deviation, observed during a single test, did not exceed 0.12 %.
This machine marked the history of scientific computing. By positioning itself midway between rudimentary mechanical calculators and future electronic computers, the differential analyzer demonstrated that complex mathematical calculations could be mechanized. This breakthrough opened unprecedented perspectives for solving challenging problems in physics and engineering.