Giant brains; or, Machines that think

by Edmund Callis Berkeley

SOLVING PHYSICAL PROBLEMS

Now we were able to change the equation about floor area into other forms, if we wanted to find length or width instead of floor area. When we did this, we ran into the inverse or opposite of multiplication: division.

In the same way, we can change the equation about speed into other forms, if we want to find distance or time instead of speed. If we do this, we run into a new idea, the inverse or opposite of the derivative, called integral. The two new equations are:

(distance) equals the integral of (speed) with respect to (time)

(time) equals the integral of [one divided by (speed)] with respect to (distance)

These equations may also be called differential equations.

An integral is the result of a process called integrating. To integrate speed and get distance is the result of three steps: (1) breaking up an interval of time into a large number of small bits, (2) adding up all the small distances that we get by taking each bit of time and multiplying by the speed which applied in that bit of time, and (3) letting the bits of time get smaller and smaller, and letting the number of them get larger and larger, indefinitely.

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.

Fig. 9. Increasing turning force;
winch, or torque amplifier.

By 1930, Dr. Bush and his group had finished the first differential analyzer. It was entirely mechanical, having no electrical parts except the motors. It was so successful that a number of engineering schools and manufacturing businesses have since then built other machines of the Bush type. Each time, some improvements were made in accuracy and capacity for solving problems. But, if you changed from one problem to another on this type of machine, you had to do a lot of work with screwdrivers and wrenches. You had to undo old mechanical connections between shafts and set up new ones. Accordingly, in 1935, the men at MIT started designing a second differential analyzer. In this one you could make all the connections electrically.

MIT finished its second differential analyzer in 1942, but the fact was not published during World War II, for the machine was put to work on important military problems. In fact, a rumor spread and was never denied that the machine was a white elephant and would not work. The machine was officially announced in October 1945. It was the most advanced and efficient differential analyzer ever built. We shall talk chiefly about it for the rest of this chapter. A good technical description of this machine is in a paper, “A New Type of Differential Analyzer,” by Vannevar Bush and Samuel H. Caldwell, published in the Journal of the Franklin Institute for October 1945.

GENERAL ORGANIZATION OF
MIT DIFFERENTIAL ANALYZER NO. 2

A differential analyzer is basically made up of shafts that turn. When we set up the machine to solve a differential equation, we assign one shaft in the machine to each quantity referred to in the equation. It is the job of that shaft to keep track of that quantity. The total amount of turning of that shaft at any time while the problem is running measures the size of that quantity at that time. If the quantity decreases, the shaft turns in the opposite direction. For example, if we have speed, time, and distance in a differential equation, we label one shaft “speed,” another shaft “time,” and another shaft “distance.” If we wish, we may assign 10 turns of the “time” shaft to mean “one second,” 2 turns of the “distance” shaft to mean “one foot,” and 4 turns of the “speed” shaft to mean “one foot per second.” These are called scale factors. We could, however, use any other convenient units that we wished.

By just looking at a shaft or a wheel, we can tell what part of a full turn it has made—a half, or a quarter, or some other part—but we cannot tell by looking how many full turns it has made. In the machine, therefore, there are mechanisms that record not only full turns but also tenths of turns. These are called counters. We can connect a counter to any shaft. When we want to know some quantity that a shaft and counter are keeping track of, we read the counting mechanism.

The second differential analyzer, which MIT finished in 1942, went a step further than any previous one. In this machine, a varying number can be expressed either (1) mechanically as the amount of turning of a shaft, or (2) electrically as the amount of two voltages in a pair of wires. The MIT men did this by means of a mechanism called an angle-indicator.

Angle indicators have essentially three parts: a transmitter, a receiver, and switches. The transmitter ([Fig. 10]) can sense the exact amount that a shaft has turned and give out a voltage in each of two wires which tells exactly how much the shaft has turned ([Fig. 11]). The receiving device ([Fig. 12]), which has a motor, can take in the voltages in the two wires and drive a second shaft, making it turn in step with the first shaft. By means of the switchboard ([Fig. 13]), the two wires from the transmitter of any angle-indicator can lead anywhere in the machine and be connected to the receiver of any other angle indicator.

Fig. 10. Scheme of angle-indicator transmitter.

Fig. 11. Indication of angle.

Fig. 12. Scheme of angle-indicator receiver.

In a differential analyzer, we can connect the shafts together in many different ways. For example, suppose that we want one shaft b to turn twice as much as another shaft a. For this to happen we must have a mechanism that will connect shaft a to shaft b and make shaft b turn twice as much as shaft a. We can draw the scheme of this mechanism in [Fig. 14]: a box, standing for any kind of simple or complicated mechanism; a line going into it, standing for input of the quantity a; a line going out of it, standing for output of the quantity b; and a statement saying that b equals 2a.

Fig. 13. Switchboard.

One mechanism that will make shaft b turn twice as much as shaft a is a pair of gears such that: (1) they mesh together and (2) the gear on shaft a has twice as many teeth as the gear on shaft b ([Fig. 15]). On the mechanical differential analyzer that MIT finished in 1930, a pair of gears was the mechanism actually used for doubling. To make one shaft turn twice as much as another by this device, we would: go over to the machine with a screwdriver; pick out from a box two gears, one with twice as many teeth as the other; slide them onto the shafts that are to be connected; make the gears mesh together; and screw them tight on their shafts.

Fig. 14. Scheme of a doubling
mechanism.

Fig. 15. Example of a doubling
mechanism.

On the MIT differential analyzer No. 2, however, we are better off. A much more convenient device for doubling is used. We make use of: a gearbox in whichthere are two shafts that may be geared so that one turns twice as much as the other, and two angle-indicator transmitters and receivers. Looking at the drawing ([Fig. 16]), we can see that: shaft a drives shaft c to turn in step, shaft c drives shaft d to turn twice as much, and shaft d drives shaft b to turn in step. Here we can accomplish doubling by closing the pairs of switches that connect to the gearbox shafts.

Angle indicators: T, transmitters, and R, receivers

Fig. 16. Another example of a doubling mechanism.

Above, we have talked about a mechanism with gears that would multiply the amount of turning by the constant ratio 2. But, of course, in a calculation, any ratio, say 7.65, 3.142, ···, might be needed, not only 2. In order to handle various constant ratios, gearboxes of two kinds are in differential analyzer No. 2. The first kind is a one-digit gearbox. It can be set to give any of 10 ratios, 0.1, 0.2, 0.3, ···, 1.0. The second kind is a four-digit gearbox. It can be set to give any one of more than 11 thousand ratios, 0.0000, 0.0001, 0.0002, ···, 1.1109, 1.1110. We can thus multiply by constant ratios.

Adders

We come now to a new mechanism, whose purpose is to add or subtract the amount of turning of two shafts. It is called an adder. The scheme of it is shown in [Fig. 17]: an input shaft with amount of turning a, another input shaft with amount of turning b, and an output shaft with amount of turning a + b. The adder essentially is another kind of gearbox, called a differential gear assembly. This name is confusing: the word “differential” here has nothing to do with the word “differential” in “differential analyzer.” This mechanism is very closely related to the “differential” in the rear axle of a motor car, which distributes a driving thrust from the motor to the two rear wheels of the car.

Fig. 17. Scheme of an adder mechanism.

Fig. 18. Example of an adding mechanism
(differential gear assembly).

A type of differential gear assembly that will add is shown in [Fig. 18]. This is a set of 5 gears A to E. The 2 gears A and B are input gears. The amount of their turning is a and b, respectively. They both mesh with a third gear, C, free to turn, but the axis of C is fastened to the inside rim of a fourth, larger gear, D. Thus D is driven, and the amount of its turning is (a + b)/2. This gear meshes with a gear E with half the number of teeth, and so the amount of turning of E is a + b.

We can subtract the turning of one shaft from the turning of another simply by turning one of the input shafts in the opposite direction.

Integrators

Another mechanism in a differential analyzer, and the one that makes it worth while to build the machine, is called an integrator. This mechanism carries out the process of integrating, of adding up a very large number of small changing quantities. [Figure 19] shows what an integrator is. It has three chief parts: a disc, a little wheel, and a screw. The round disc turns horizontally on its vertical shaft. The wheel rests on the disc and turns vertically on its horizontal shaft. The screw goes through the support of the disc; when the screw turns, it changes the distance between the edge of the wheel and the center of the disc.

Fig. 19. Mechanism of integrator.

Now let us watch this mechanism move. If the disc turns a little bit, the wheel pressing on it must turn a little bit. If the screw turns a small amount, the distance between the edge of the wheel and the center of the disc changes. The amount that the wheel turns is doubled if its distance from the center of the disc is doubled, and halved if that distance is halved. So we see that:

(the total amount that the wheel turns) equals the sum of all the small (amounts of turning), each equal to: a bit of (disc turning) multiplied by the (distance from the center of the disc to the edge of the wheel) applying to that bit

If we look back at our discussion of integrating ([p. 72]), we see that the capital words here are just the same as those used there. Thus we have a mechanism that expresses integration:

(the total amount that the wheel turns) equals the integral of (the distance from the center of the disc to the wheel) with respect to (the amount that the disc turns)

The scheme of this mechanism is shown in [Fig. 20].

For example, suppose that the screw measures the speed at which a car travels and that the disc measures time. The wheel, consequently, will measure distance traveled by the car. The mechanism integrates speed with respect to time and gives distance.

Fig. 20. Scheme of integrator.

This mechanism is the device that Lord Kelvin talked about in 1879 and that Dr. Bush made practical in 1925. The mechanical difficulty is to make the friction between the disc and the wheel turn the wheel with enough force to do other work. In the second differential analyzer, the angle indicator set on the shaft of the wheel solves the problem very neatly.

Fig. 21. Graph of air resistance coefficient.

Function Tables

The behavior of some physical quantities can be described only by a series of numbers or a graphic curve. For example, the resistance or drag of the air against a passing object is related to the speed of the object in a rather complicated way. Part of the relation is called the drag coefficient or resistance coefficient; a rough graph of this is shown in [Fig. 21]. This graph shows several interesting facts: (1) when the object is still, there is no air resistance; (2) as it travels faster and faster, air resistance rapidly increases; (3) when the object travels with the speed of sound, resistance is very great and increases enormously; (4) but, when the object starts traveling with a speed about 20 per cent faster than sound, the drag coefficient begins to decrease. This drawing or graph shows in part how air resistance depends on speed of object; in other words, it shows the drag coefficient as a function of speed ([see Supplement 2]).

Fig. 22. Pointer following graph.

Now we need a way of putting any function we wish into a differential analyzer. To do this, we use a mechanism called a function table. We draw a careful graph of the function according to the scale we wish to use, and we set the graph on the outside of a large drum ([Fig. 22]). For example, we can put the resistance coefficient graph on the drum; the speed (or independent variable) goes around the drum, and the resistance coefficient (or dependent variable) goes along the drum. The machine slowly turns the drum, as may be called for by the problem. A girl sits at the function table and watches, turning a handwheel that keeps the sighting circle of a pointer right over the graph. The turning of the handwheel puts the graphed function into the machine. Instead of employing a person, we can make one side of the graph black, leaving the other side white, and put in a phototube (an electronic tube sensitive to amount of light) that will steer from pure black or pure white to half and half ([see Fig. 23]).

We do not need many function tables to put in information, because we can often use integrators in neat combinations to avoid them. We shall say more about this possibility later.

We can also use a function table to put out information and to draw a graph. To do so, we disconnect the handwheel; we connect the shaft of the handwheel to the shaft that records the function we are interested in; we take out the pointer and put in a pen; and we put a blank sheet of graph paper around the drum.

Fig. 23. Phototube following graph.

We have now described the main parts of the second MIT differential analyzer. They are the parts that handle numbers. We can now tell the capacity of the differential analyzer by telling the number of main parts that it holds:

Shafts	About 130
One-digit gearboxes	12
Four-digit gearboxes	16
Adders	About 16
Integrators	18
Function tables	3

On a simpler level, we can say that the machine holds these physical parts:

Miles of wire	About 200
Relays	About 3000
Motors	About 150
Electronic tubes	About 2000

INSTRUCTING THE MIT
DIFFERENTIAL ANALYZER NO. 2

Besides the function tables for putting information into the machine, there are three mechanisms that read punched paper tape. The three tapes are called the A tape, the B tape, and the C tape. From these tapes the machine is set up to solve a problem.

Suppose that we have decided how the machine is to solve a problem. Suppose that we know the number of integrators, adders, gearboxes, etc., that must be used, and know how their inputs and outputs are to be connected. To carry out the solution, we now have to put the instructions and numbers into the machine.

The A tape contains instructions for connecting shafts in the machine. Each instruction connects a certain output of one type of mechanism (adder, etc.) to a certain input of another type of mechanism. When the machine reads an instruction on this tape, it connects electrically the transmitting angle-indicator of an output shaft to the receiving angle-indicator of another input shaft.

Now the connecting part of the differential analyzer behaves as if it were very intelligent: it assigns an adder or an integrator or a gearbox, etc., to a new problem only if that mechanism is not busy. For example, if a problem tape calls for adder 3 (in the list belonging to the problem), the machine will assign the first adder that is not busy, perhaps adder 14 (in the machine), to do the work. Each time that adder 3 (in the problem list) is called for in the A tape, the machine remembers that adder 14 was chosen and assigns it over again. This ability, of course, is very useful.

The B tape contains the ratios at which the gearboxes are to be set. For example, suppose that we want gearbox 4 (in the problem list) to change its input by the ratio of 0.2573. The machine, after reading the A tape, has assigned gearbox 11 (in the machine list). Then, when the machine reads the B tape, it sets the ratio in gearbox 11 to 0.2573.

The answer to a differential equation is different for different starting conditions. For example, when we know speed and time and wish to find distance traveled and where we have arrived, we must know the point at which we started. We therefore need to arrange the machine so that we can put in different starting conditions (or different initial conditions, as the mathematician calls them).

The C tape puts the initial conditions into the machine. For example, reading the C tape for the problem, the machine finds that 3000 should, at the start of the problem, stand in counter 4. The machine then reads the number at which counter 4 actually stands, say 6728.3. It subtracts the two numbers and remembers the difference, -3728.3. And whenever the machine reads that counter later, finding, say, 9468.4 in it, first the 3728.3 is subtracted, and then the answer 5740.1 is printed.