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The Analog Procedure

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##The Analog Procedure

In an analog machine each number is represented by a suitable physical quantity, whose values, measured in some pre-assigned unit, is equal to the number in question. This quantity may be the angle by which a certain disk has rotated, or the strength of a certain current, or the amount of a certain (relative) voltage, etc. To enable the machine to compute, i.e. to operate on these numbers according to a predetermined plan, it is necessary to provide organs (or components) that can perform on these representative quantities the basic operations of mathematics.

###The Conventional Basic Operations

These basic operations are usually understood to be the “four species of arithmetic”: addition (the operation \(x + y\)), subtraction \((x - y)\), multiplication \((xy)\), division \((x / y)\).

Thus it is obviously not difficult to add or to subtract two currents (by merging them in parallel or in antiparallel directions). Multiplication (of two currents) is more difficult, but there exist various kinds of electrical componentry which will perform this operation. The same is true for division (of one current by another). (For multiplication as well as for division - but not for addition and subtraction - of course the unit in which the current is measured is relevant).

###Unusual Basic Operations

A rather remarkable attribute of some analog machines, on which I will have to comment a good deal further, is this. Occasionally the machine is built around other “basic” operations than the four species of arithmetic mentioned above. Thus the classical “differential analyzer”, which expresses numbers by the angles by which certain disks have rotated, proceeds as follows. Instead of addition, \(x + y\), and subtraction, \(x - y\), the operations \(\frac{x + y}{2}\) and \(\frac{x - y}{2}\) are offered, because a readily available simple component, the “differential gear” (the same one that is used on the back axle of an automobile) produces these. Instead of multiplication, \(xy\), an entirely different procedure is used: In the differential analyzer all quantities appear as functions of time, and the differential analyzer makes use of an organ called the “integrator”, which will, for two such quantities \(x(t), y(t)\) form the (“Stieltjes”) integral \(z(t) = \int^t x(t) dy(t)\).

The point in this scheme is threefold:

First: the three above operations will, in suitable combinations, reproduce three of the four usual basic operations, namely addition, subtraction and multiplication.

Second: in combination with certain “feedback” tricks, they will also generate the fourth operation, division. I will not discuss the feedback principle here, except by saying that while it has the appearance of a device for solving implicit relations, it is in reality a particularly elegant short-circuited iteration and successive approximation scheme.

Third, and this is the true justification of the differential analyzer: its basic operations \(\frac{x + y}{2}\), \(\frac{x - y}{2}\) and integration are, for wide classes of problems, more economical than the arithmetic ones \(x + y,\ x - y,\ xy,\ x / y\). More specifically: any computing machine that is to solve a complex mathematical problem must be “programmed” for this task. This means that the complex operation of solving that problem must be replaced by a combination of the basic operations of the machine. Frequently it means something even more subtle: approximation of that operation - to any desired (prescribed) degree - by such combinations. Now for a given class of problems one set of basic operations may be more efficient, i.e. allow the use of simpler, less extensive, combinations than another such set. Thus, in particular, for systems of total differential equations - for which the differential analyzer was primarily designed - the above mentioned basic operations of that machine are more efficient than the previously mentioned arithmetical basic operations \(x + y,\ x - y,\ xy,\ x / y\).

Taken from The Computer and the Brain by John von Neumann.