Giant brains; or, Machines that think

In the previous chapter we talked about machines that move information expressed as holes in cards. In this chapter we shall talk about machines that move information expressed as measurements.

ANALOGUE MACHINES

A simple example of a device that uses a measurement to handle information is a doorpost. Here the height of a child may be marked from year to year as he grows (Fig. 1). Or, suppose that we have a globe of the world and wish to find the shortest path between Chicago and Moscow. We may lay a piece of string on the globe, pull it tight between those points, and then measure the string on a scale to see about what distance it shows (Fig. 2).

Machines that handle information as measurements of physical quantities are called analogue machines, because the measurement is analogous to, or like, the information. A common example of analogue machine is the slide rule. With this we calculate by noting the positions of ruled lines on strips that slide by each other. These strips are made of fine wood, or of plastic, or of steel, in such fashion that the ruled lines will hold true positions and not warp. If we space the rulings so that 1, 2, 3, 4, 5, 6 ··· are equally spaced, then the slide rule is useful for addition (Fig. 3). But if we space the rulings so that powers (for example, powers of two—1, 2, 4, 8, 16, 32 ···) (Fig. 4) are equally spaced, we can do multiplication. The spacings are then according to the logarithms of numbers (see Supplement 2). Multiplication is more troublesome than addition, and so more slide rules are made for multiplication than for addition.

During World War II, the aiming and firing of guns against hostile planes was done by machine. After sighting a plane, these machines automatically calculated how to direct fire against it. They were much better and faster than any man. These fire-control instruments were analogue machines with steel and electrical parts built to fine tolerances. With care we can get accuracy of 1 part in 10,000 with analogue machines, but greater accuracy is very hard to get.

PHYSICAL QUANTITIES

Suppose that we wish to make an analogue machine. We need to represent information by a measurement of something. What should we select? What physical thing to be measured should we choose to put into the machine? Different amounts of this physical quantity will match with different amounts of the measurement being expressed. In the case of the doorpost, the string, and the slide rule, the physical quantity is distance. In many fire-control instruments, the physical quantity is the amount of turning of a shaft (Fig. 5). Many other physical quantities have from time to time been used in analogue machines, such as electrical measurements. The speedometer of an automobile tells distance traveled and speed. It is an analogue machine. It uses the amount of turning of a wheel, and some electrical properties. It handles information by means of measurements. The basic physical quantity that it measures is the amount of turning of a shaft.

DIFFERENTIAL ANALYZER

The biggest and cleverest mechanical brain of the analogue type which has yet been built is the differential analyzer finished in 1942 at Massachusetts Institute of Technology in Cambridge, Mass. The fundamental physical quantity used in this machine is the amount of turning of a shaft. The name analyzer means an apparatus or machine for analyzing or solving problems. It happens that the word “analyzer” has been used rather more often in connection with analogue machines, and so in many cases the word “analyzer” carries the meaning “analogue” as well. The word “differential” in the phrase “differential analyzer” refers to the main purpose of the machine: it is specially adapted for solving problems involving differential equations. Now what is a differential equation?

DIFFERENTIAL EQUATIONS

In order to explain what a differential equation is, we need to use certain ideas. These ideas are: equation; formula; function; rate of change; interval; derivative; and integral. In the next few paragraphs, we shall introduce these ideas briefly, with some explanation and examples. It is entirely possible for anyone to understand these ideas rather easily, by collecting true statements about them; no one should feel that because these ideas may be new they cannot be understood readily.

PHYSICAL PROBLEMS

In physics, chemistry, mechanics, and other sciences there are many problems in which the behavior of distance, of time, of speed, heat, volume, electrical current, weight, acceleration, pressure, and many other physical quantities are related to each other. Examples of such problems are:

In physical problems like these, the answer is not a single number but a formula. What we want to do in any one of these problems is find a formula so that any one of the quantities may be calculated, given the behavior of the others. For example, here is a familiar problem in which the answer is a formula and not a number:

How are the floor area of a room, its length, and its width related to each other? (See Fig. 8.)

The answer is told in any one of three equations:

The first equation shows that the floor area depends on the length of the room and also on the width of the room. So we say floor area is a function of length and width. This particular function happens to be product, the result of multiplication. In other words, floor area is equal to the product of length and width.

Now there is another kind of function called a differential function or derivative. A differential function or derivative is an instantaneous rate of change. An instantaneous rate of change is the result of two steps: (1) finding a rate of change over an interval and then (2) letting the interval become smaller and smaller indefinitely. For example, suppose that we have the problem:

How are speed, distance, and time related to each other?

One of the answers is:

(speed) equals the instantaneous rate of change of (distance) with respect to (time)

Or we can say, and it is just the same thing in other words:

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

Now we can tell what a differential equation is. It is simply an equation in which a derivative occurs, such as the last example. Perhaps the commonest kind of equation in physical problems is the differential equation.

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:

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.

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.

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.

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.

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.

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.

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.

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:

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 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.

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.

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).

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.

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:

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

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.

ANSWERS

Information may come out of the machine in either one of two ways: in printed numbers or in a graph. In fact, the same quantity may come out of the machine in both ways at the same time. To obtain a graph, we change a function table from input to output, put a pen on it, and have it draw the graph.

The machine has 3 electric typewriters. The machine will take numerical information out of the counters at high speed even while they are turning, and it will put the information into relays. Then it will read from the relays into the typewriter keys one by one while they type from left to right across the page.

HOW THE DIFFERENTIAL ANALYZER CALCULATES

Up to this point in this chapter, the author has tried to tell the story of the differential analyzer in plain words. But for reading this section, a little knowledge of calculus is necessary. (See also Supplement 2.) If you wish, skip this section, and go on to the next one.

We have described how varying quantities, or variables, are operated on in the machine in one way or another: adding, subtracting, multiplying by a constant, referring to a table, and integrating. What do we do if we wish to multiply 2 variables together? A neat trick is to use the formula:

To multiply in this way requires 2 integrators and 1 adder. The connections that are made between them are as follows:

A product of 2 variables under the integral sign can be obtained a little more easily, because of the curious powers of the differential analyzer. Thus, if it is desired to obtain ∫ xy dt, we can use the formula:

and this operation does not require an adder. The connections are as follows:

In order to get the quotient of 2 variables, x/y, we can use some more tricks. First, the reciprocal 1/y can be obtained by using the two simultaneous equations:

The connections are as follows:

In order to get x/y, we can then multiply x by 1/y. We see that this setup gives us log y for nothing, that is, without needing more integrators or other equipment. Clearly, other tricks like this will give sin x, cos x, eˣ, x², and other functions that satisfy simple differential equations.

An integral of a reciprocal can be obtained even more directly. Suppose that

Then

The connections therefore are:

The light wheel then drives the heavy disc. Clearly only the angle-indicator device makes this possible at all. Naturally, the closer the wheel gets to the center of the disc, that is, x approaching zero, the greater the strain on the mechanism, and the more likely the result is to be off. Mathematically, of course, the limit of 1/x as x approaches zero equals infinity, and this gives trouble in the machine.

There is no standard mathematical method for solving any differential equation. But the machine provides a standard direct method for solving all differential equations with only one independent variable. First: assign a shaft for each term that appears in the equation. For example, the highest derivative that appears and the independent variable are both assigned shafts. The integral of the highest derivative is easily obtained, and the integral of that integral, etc. Second: connect the shafts so that all the mathematical relations are expressed. Both explicit and implicit equations may be expressed. Third: for any shaft there must be just one drive, or source of torque. A shaft may, however, drive more than one other shaft. Fourth: choose scale factors so that the limits of the machine are not exceeded yet at the same time are well used. For example, the most that an integrator or a function table can move is 1 or 2 feet. Also, the number of full turns made by a shaft in representing its variable should be large, often between 1000 and 10,000.

Of course, as with all these large machines, anyone would need some months of actual practice before he could put on a problem and get an answer efficiently.

AN APPRAISAL OF THE MACHINE

The second MIT differential analyzer is probably the best machine ever built for solving most differential equations. It regularly has an accuracy of 1 part in 10,000. This is enough for most engineering problems. If greater accuracy is needed, the second differential analyzer cannot provide it. Once in a while the machine can reach an accuracy of 1 part in 50,000; but, to balance this, it is sometimes less accurate than 1 part in 10,000.

The MIT differential analyzer No. 2 can find answers to problems very quickly. The time for setting up a problem to be run on the machine ranges from 5 to 15 minutes. The time for preparing the tapes that set up the problem is, of course, longer; but the punch for preparing the tape is a separate machine and does not delay the differential analyzer. The time for the machine to produce a single solution to a problem ranges usually from 3 minutes to a half-hour. It is easy to put on a problem, run a few solutions, take the problem off, study the results, change a few numbers, and then put the problem back on again. This virtue is a great help in a search in a new field. While the study is going on, time is not wasted, for the machine can be busy with a different problem.

Running a problem a second time is a good check on the reliability of an answer. For, when the problem is run the second time, we can arrange that the machine will route the problem to other mechanisms.

The machine has a control panel. Here the operator watching the machine can tell what units are doing what parts of what problems. If a unit gives trouble or needs to be inspected, the clerk can throw a “busy” switch. Then the machine cannot choose that unit for work to be done. The machine contains many protecting signals and alarms. It is idle for repairs less than 5 per cent of the time.

It is not easy to determine the total cost of the machine, for it was partially financed by several large gifts. Also, much of the labor was done by graduate students in return for the instruction that they gained. The actual out-of-pocket cost was about $125,000. If the machine were to be built by industry, the cost would likely be more than 4 times as much. A simple differential analyzer, however, can be cheap. Small scale differential analyzers have been built for less than $1000; their accuracy has been about 1 part in 100.

There are many things that this machine cannot do; it was not built to do them. (1) It cannot choose methods of solution. (2) It cannot perform steps in solving a problem that depend on results as they are found. (3) It cannot solve differential equations containing two or more independent variables. Such equations are called partial differential equations; they appear in connection with the flow of heat or air or electricity in 2 or 3 dimensions, and elsewhere. (4) It cannot solve problems requiring 6 or more digits of accuracy. (5) The machine, while running, can store numbers only for an instant, since it operates on the principle of smoothly changing quantities; however, when the machine stops, the number last held by each device is permanently stored.

None of these comments, however, are criticisms of the machine. Instead they show avenues of development for future machines. As was said before, for solving most differential equations, this machine has no equal to date. The range of problems which any differential analyzer can do depends mostly on the number of its integrators. The second differential analyzer has 18 and provides for expansion to 30. The machine is constructed, also, so that it can be operated in 3 independent sections, each working to solve a different problem. The differential analyzer can operate unattended. After it has been set up and the first few results examined, it can be left alone to grind out large blocks of answers.

An interesting example of the experimental use of a differential analyzer in a commercial business is the following: In Great Britain, R. E. Beard of the Pearl Assurance Company built a differential analyzer with 6 integrators. He applied this machine to compute to 3 figures certain insurance values known as continuous annuities and continuous contingent insurances. He has described the machine and the application he made in a paper published in the Journal of the Institute of Actuaries, Vol. 71, 1942, pp. 193-227.