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# Mechanical Integrators and Differential Analyzers

Analog devices – instruments in which numbers are represented by continuous quantities like length rather than discrete units like digits – have long played an important role in everyday computation. Drawing instruments like protractors, compasses, and rules are analog. So are pocket sundials, mechanical clocks and watches, and slide rules. In the nineteenth century, engineers, scientists, and mathematicians devised analog instruments to explore quantities associated with curves. The largest number of these objects were planimeters, instruments that made it possible to find the area bounded by a closed curve in a plane by tracing around it. Planimeters in the Smithsonian mathematics collections are discussed in a separate object group. A related group of objects, called integrators, when traced along a curve, gave the integral of the curve, that is to say the area between the curve and a given straight line. Integrators also were incorporated into more general computing devices, such as those the Ford Instrument Company made for the U.S. Navy. Related instruments actually drew the integral, rather than giving output on a dial or dials. These were called integraphs. The electrical engineer Vannevar Bush, who taught at MIT, devised an instrument for finding and plotting the integral of the product of two functions. His “product integraph” used modified forms of standard electrical apparatus.

In the early nineteenth century, the French mathematical physicist Joseph Fourier showed that many curves could be represented as the sum of a series of trigonometric functions or harmonics. If one knew, for example, the terms that combined to produce the tides, one could estimate times of high and low tides. William Thomson in England and later William Ferrel in the United States designed instruments that predicted tides. Ferrel’s first tide predictor is in the NMAH collections. This approach was called “harmonic synthesis” and was represented for demonstration purposes in apparatus of the physicist Carl Barus. Others proposed instruments in which one could reduce an observed curve to the sum harmonics – these was called harmonic analyzers. The harmonic analyzers that sold were elegant instruments.

Vannevar Bush and his colleagues made more complex, room-sized electrically powered analog machines for doing complex calculations. These were called differential analyzers. Analog computing by no means ended with the introduction of instruments with electronic representation of numbers. However, these were no longer “mechanical.”