In other words,
(total distance) equals the sum of all the small (distances), each equal to: a bit of (time) multiplied by the (speed) applying to that bit
This is another way of saying as before,
(distance) equals the integral of (speed) with respect to (time)
To solve a differential equation, we almost always need to integrate one or more quantities.
ORIGIN AND DEVELOPMENT OF THE
DIFFERENTIAL ANALYZER
For at least two centuries, solving differential equations to answer physical problems has been a main job for mathematicians. Mathematics is supposed to be logical, and perhaps you would think this would be easy. But mathematicians have been unable to solve a great many differential equations; only here and there, as if by accident, could they solve one. So they often wished for better methods in order to make the job easier.
A British mathematician and physicist, William Thomson (Lord Kelvin), in 1879 suggested solving differential equations by a machine. He went further: he described mechanisms for integrating and other mathematical processes, and how these mechanisms could be connected together in a machine. No such machine was then built; engineering in those years was not equal to it. In 1923, a machine of this type for solving the differential equations of trajectories was proposed by L. Wainwright.
In 1925, at Massachusetts Institute of Technology, the problem of a machine to solve differential equations was again being studied by Dr. Vannevar Bush and his associates. Dr. Bush experimented with mechanisms that would integrate, add, multiply, etc., and methods of connecting them together in a machine. A major part of the success of the machine depended on a device whereby a very small turning force would do a rather large amount of work. He developed a way in which the small turning force, about as small as a puff of breath, could be used to tighten a string around a drum already turning with a considerable force, and thus clutch the drum, bring in that force, and do the work that needed to be done. You may have watched a ship being loaded, seen a man coil a rope around a winch, and watched him swing a heavy load into the air by a slight pull on the rope ([Fig. 9]). If so, you have seen this same principle at work. The turning force (or torque) that pulls on the rope is greatly increased (or amplified) by such a mechanism, and so we call it a torque amplifier.
