The differential analyzer is a type of analog computer that was developed to solve difficult mathematical problems. The first known example of this machine was developed in the late nineteenth century by William Thomson, later known as Lord Kelvin. While the differential analyzer was a useful tool for calculation, it was limited in many ways. An example of this would be the difficulty of configuring the machine for a new calculation. Limitations aside, the invention of the differential analyzer was a significant step forward in computing history.

The design that eventually led to the invention of the first differential analyzer began with a paper written by Lord Kelvin and his brother, Professor James Thomson. This paper discussed a mechanical integrator, invented by Prof. Thomson, which was capable of solving first order differential equations. This invention was what led Lord Kelvin to the design of the first general-purpose analog differential analyzer capable of solving differential equations of second degree and greater.

In 1876, Lord Kelvin published a paper describing how two of the devices his brother had invented (the mechanical integrators) could be linked together to solve second order differential equations. The key insight of this paper was to demonstrate how differential equations of nth degree could be solved by chaining a sufficient number of integrators together, with the output of each integrator feeding the input of the next. On paper, this solution worked. However, Lord Kelvin never succeeded in building his proposed second-degree differential analyzer.

In order for the two-stage integrator to work, the torque of the output shaft of the first integrator had to be sufficient to drive the input of the second integrator. This turned out not to be the case. There was no alternative solution to be found, as Victorian technology was insufficient to the task. It wasn't until later that Vannevar Bush and Harold Hazen would find a solution to this problem. Kelvin did go on to create a Tide Predictor, which performed differential analysis on a complicated single order differential equation.

Another important effort to create an accurate differential analyzer was that of Albert Michelson, a scientist famous for his work on experiments regarding the speed of light (including one now known as the Michelson-Morley experiment). Michelson was very competent at finding sources of error in mechanical devices. His analyzer used a model similar to Kelvin's Tide Predictor, but instead of using cords between devices, he used springs. This eliminated errors that would have accumulated over time, as the cord stretched out under the stress of tension. Michelson's machine was very accurate for an analog machine, with an average error of less than a percent. However, there was still no machine capable of solving an equation of a degree greater than one.

The answer to this problem was provided by the next important effort in the development of a differential analyzer, which was that of Bush at the Massachusetts Institute of Technology. Bush worked in MIT's Department of Electrical Engineering, and saw the need for a calculating machine to solve these difficulties. He first designed solution was a device called a "continuous integraph," which he also called a "product integraph." While this machine was able to solve first-order differential equations, it was no technological advance. It was after his assistant Harold Hazen suggested adding a second level of integration that Bush redesigned the machine, using two different sorts of integrators. The two integrator machine was an improvement, and it further inspired Bush to design the multilevel mechanical integrator that became the Differential Analyzer.

In 1930, the machine was completed. It had six wheel-and-disk integrators, similar to the wheel-and-globe models used by Kelvin, and therefore solve up to sixth-order differential equations, or a number of nth order equations in parallel, when n was one, two or three(the machine could perform 6/n of these calculations at once). In designing the machine, Hazen and Bush overcame the problem that stopped Kelvin; that of disk shaft torque being lost between integrators. This was solved by the use of C. W. Niemann's Torque Amplifier, using a servomotor to translate the precise rotation of one output shaft to the next input shaft, while at the same time amplifying the torque approximately ten thousand times. This Differential Analyzer was the model for the next several large scale differential analyzers constructed at universities across the country, and even in other countries, such as the one constructed at Manchester University by Douglas R Hartree (which was, unbelievably, constructed mostly of Meccano).^{1}

At first, these analyzers were used primarily for civilian and scientific calculation. The analyzers were used to solve problems dealing with atomic structure, power distribution across electrical networks, in making railway train timetables, and to generate ballistics tables. This last use accounted for the majority of work done by these machines use during World War II, since use of the machines during the war was taken over by the army.

While the Differential Analyzer was in many ways a great boon for engineers of its day, it had several limitations. The most severe of these was the difficulty in reprogramming the device. Instead of being able to use paper tapes to program the machine, users of Bush's early Analyzers had to engage the it with hand tools in order to prepare it for a calculation. The RDA (Rockefeller Differential Analyzer), a later design of Bush's, was capable of using programs fed in on tapes, which solved this limitation. However, due to the enormous cost of these machines, the construction of analyzers using Bush's original design continued into the 1950s.

Another problem when using the Differential Analyzers was that programmers had to construct their mathematical problems in ways that took advantage of the way the machine worked. Unlike modern computers, the analyzer worked in a very specific way, and since it was impossible to modify the machine to suit one's problem, scientists using the machine were forced to modify the expression of their problem to make economical use of the machine's capabilities.

More problems common to the analyzers were those from which most mechanical devices suffer. One problem is that as the capabilities of the machines increased, their size did as well. The more sophisticated Differential Analyzers became prohibitively large. Another problem was that the speed of the machines was limited by the mass and inertia of their components, which doomed them to be eventually outpaced by less massive analog computers. As the size of the machines increased, the power to operate them increased proportionally. As the power consumption rose, so did the waste energy lost to heat, so that the more complicated the machines became, the hotter they ran. Last, with so many moving parts, the machines were subject to frequent mechanical failure, requiring expensive maintenance.

The Differential Analyzers are historically important as the first general purpose analog computers. They are also important for their sophistication, which would never be matched by electronic analog computers, and was even difficult at first for vacuum tube-based digital computers like ENIAC and its successors.

From my class notes during RC 379 - History of Computing at U. of Mich and casual reading on the subject since.

^{1}Thanks to wertperch for this interesting fact.