Differential equations, in particular

Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...

, rose in prominence during World War II in calculating the accurate trajectory of ballistics, both rocket-propelled and gun or cannon type projectiles. Originally, mathematicians used the simpler

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

of earlier centuries to determine velocity, thrust, elevation, curve, distance, and other parameters. New weapons, however, such as Germany's giant cannons, the " Paris Gun" (Encyclopedia Astronautica) and " Big Bertha," and the

V-2 rocket The V-2 (german: Vergeltungswaffe 2, lit=Retaliation Weapon 2), with the technical name ''Aggregat 4'' (A-4), was the world’s first long-range guided ballistic missile. The missile, powered by a liquid-propellant rocket engine, was developed ...

, meant that projectiles would travel hundreds of miles in distance and dozens of miles in height, in all weathers. As a result, variables such as diminished wind resistance in thin atmospheres and changes in

gravitational pull In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

reduced accuracy using the historic methodology. There was the additional problem of planes that could now fly hundreds of miles an hour. Differential equations were applied to

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

es. Developing machines that could speed up human calculation of differential equations led in part to the creation of the modern computer through the efforts of

Vannevar Bush Vannevar Bush ( ; March 11, 1890 – June 28, 1974) was an American engineer, inventor and science administrator, who during World War II headed the U.S. Office of Scientific Research and Development (OSRD), through which almost all wartime ...

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

and others. According to Mary Croarken in her paper "Computing in Britain During World War II," by 1945, the Cambridge Mathematical Laboratory created by

John Lennard-Jones Sir John Edward Lennard-Jones (27 October 1894 – 1 November 1954) was a British mathematician and professor of theoretical physics at the University of Bristol, and then of theoretical science at the University of Cambridge. He was an imp ...

utilized the latest computing devices to perform the equations. These devices included a model "

differential analyser The differential analyser is a mechanical analogue computer designed to solve differential equations by integration, using wheel-and-disc mechanisms to perform the integration. It was one of the first advanced computing devices to be used operat ...

," and the

Mallock machine The Mallock machine is an electrical analog computer built in 1933 to solve simultaneous linear differential equations. It uses coupled transformers, with numbers of turns digitally set up to +/-1000 and solved sets of up to 10 linear differential e ...

, described as "an electrical simultaneous equation solver." According to Croarken, the Ministry was also interested in the new arrival of a differential analyzer accommodating eight integrators. This exotic computing device built by Metropolitan-Vickers in 1939 consisted of wheel and disk mechanisms that could provide descriptions and solutions for differential equations. Output resulted in a plotted graph. At the same time, in the United States,

analog computer An analog computer or analogue computer is a type of computer that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities (''analog signals'') to model the problem being solved. In c ...

pioneer Vannevar Bush took on a similar role to that of Lennard-Jones in the military effort after President

Franklin Delano Roosevelt Franklin Delano Roosevelt (; ; January 30, 1882April 12, 1945), often referred to by his initials FDR, was an American politician and attorney who served as the 32nd president of the United States from 1933 until his death in 1945. As the ...

entrusted him with the bulk of wartime research into automatic control of fire power using machines and computing devices. According to

Sarah Bergbreiter Sarah Bergbreiter is a professor of Mechanical Engineering at Carnegie Mellon University, previously a professor at the University of Maryland. Her research specifically has focused on microrobotics, with projects influencing the medicine and cons ...

in her paper "Moving from Practice to Theory: Automatic Control after World War II," fire control for the downing of enemy aircraft by anti-aircraft guns was the priority. The analog electro-mechanical computing machines plotted the differential firing data while servos created by H.L. Hazen adapted the data to the guns for precise firing control and accuracy. Other improvements of a similar type by Bell Labs increased firing stability so that output from the differential engines could be fully used to compensate for stochastic behaviors of enemy aircraft and large guns. A new age of intelligent warfare had begun. This work at

MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...

and Bell Labs would later lead to

Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...

's development of the electronic computer and the science of

cybernetics Cybernetics is a wide-ranging field concerned with circular causality, such as feedback, in regulatory and purposive systems. Cybernetics is named after an example of circular causal feedback, that of steering a ship, where the helmsperson m ...

for the same purpose, speeding the differential calculation process exponentially and taking one more giant step toward the creation of the modern digital computer using

von Neumann architecture The von Neumann architecture — also known as the von Neumann model or Princeton architecture — is a computer architecture based on a 1945 description by John von Neumann, and by others, in the ''First Draft of a Report on the EDVAC''. The ...

. Dr. von Neumann was one of the original mathematicians employed in the development of differential equations for ballistic warfare.

References

{{Reflist * Croarken, Mary. "Computing in Britain During World War II," IEE History of Technology Summer Meeting 6 July 2002

* Bergbreiter, Sarah. "Moving from Practice to Theory: Automatic Control after World War II." Student paper: HIS 285S: History of Science, University of California, Berkeley

* MacRae, Norman. ''John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More.'' N.Y.: Pantheon Books, 1992. Numerical differential equations History of computing

See also

References