Chapter 12

SOCIAL CONTROL:

MACHINES THAT THINK AND HOW SOCIETY MAY CONTROL THEM

It is often easier for men to create a device than to guide it well afterwards: it is often easier for a scientist to study his science than to study the results for good or evil that his discoveries may lead to. But it is not right nor proper for a scientist, a man who is loyal to truth as an ideal, to have no regard for what his discoveries may lead to.

This principle is now being widely recognized. Many scientists today—both as individuals and as groups, and especially the atomic scientists—are considering the results of their scientific discoveries; and they are sharing in the effort to render those results truly useful to humanity.

It would be easy to leave out of this book any discussion of how machines that think may be controlled, any consideration of how they may be made truly useful to humanity. But that would be hardly right or proper. In concluding a book such as this one, that touches on many aspects of machines that think, we need to consider what can and should be done to make such machines of true benefit to all of humanity.

So, we come to the most important of all our questions: What sort of control over machines that think do we need in human society?

MACHINE THAT BOTH THINKS AND ACTS

From a narrow point of view, a machine that only thinks produces only information. It takes in information in one state, and it puts out information in another state. From this viewpoint, information in itself is harmless; it is just an arrangement of marks; and accordingly, a machine that thinks is harmless, and no control is necessary.

Although it is true that the information produced only becomes good or evil after other machinery or human beings act on the information, in reality a machine with the power to produce information is constructed only for the reason of its use. We want to know what such machines can tell us only because we can then proceed to act much more efficiently than before. For example, a guided missile needs a mechanical brain only because then it can reach its target. In all cases mechanical brains are inseparable from their uses.

For the purposes of this chapter, the narrow view will be rejected because it dodges the issue. We shall be much concerned with the combination of a machine that thinks with another machine that acts; and we shall often call this combination the _robot machine_.

READING THIS CHAPTER

Now, before launching further into the discussion, we need to say that the conclusions suggested in this chapter are not final. Even if they are expressed a little positively in places, they are nevertheless subject to change as more information is discovered and as the appraisal of information changes with time. Also, almost any conclusions about social control—including, certainly, the conclusions in this chapter—are subject to controversy. But controversy is good: it leads to thought. The more minds that go to work on solving the problem of social control over robot machines and other products of the new technology—which is rushing upon us from the discoveries of the scientists—the better off we all will be. If, while stimulating disagreement, the ideas expressed in this chapter should succeed in stimulating thought and deliberation, the purpose of this chapter will be well fulfilled.

Up to this point in this book, the emphasis has been on possibilities of benefits to humanity that may arise from machines that think. In this chapter, devoted as it is to the subject of control, the emphasis is on possibilities for harm. Both possibilities are valid, and the happening of either depends upon the actions of men. In much the same way, atomic energy is a great possibility for benefit and for harm. It is the nature of control to put a fence around danger; and so it is natural in this chapter that the weight of attention should shift to the dangerous aspects of machines that think.

Perhaps a reader may feel that a chapter of this kind is rather out of place in a book, such as this one, that seeks to be scientific. If so, he is reminded that, in accordance with the general suggestions for reading this book stated in the preface, he should omit this chapter.

FRANKENSTEIN

Perhaps the first study of the consequences of a machine that thinks is a prophetic novel called _Frankenstein_, written more than a hundred years ago, in 1818. The author, then only 21 years old, was Mary W. Godwin, who became the wife of the poet Percy Bysshe Shelley.

According to the story, a young Swiss, an ardent student of physiology and chemistry, Victor Frankenstein, finds the secret of life. He makes an extremely ugly, clever, and powerful monster, with human desires. Frankenstein promptly flees from his laboratory and handiwork. The monster, after seeking under great hardships for a year or two to earn fair treatment among men, finds himself continually attacked and harmed on account of his ugliness, and he becomes embittered. He begins to search for his creator for either revenge or a bargain. When they meet:

“I expected this reception,” said the daemon.

“All men hate the wretched; how then must I be hated who am miserable beyond all living things! Yet you my creator detest and spurn me, thy creature, to whom thou art bound by ties only dissoluble by the annihilation of one of us. You purpose to kill me. How dare you sport thus with life? Do your duty towards me, and I will do mine towards you and the rest of mankind. If you will comply with my conditions, I will leave them and you at peace; but if you refuse, I will glut the maw of death, until it be satiated with the blood of your remaining friends.”

Frankenstein starts to comply with the main condition, which is to make a mate for the monster; but Frankenstein cannot bring himself to do it. So the monster causes the death one after another of all Frankenstein’s family and closest friends; and the tale finally ends with the death of Frankenstein and the disappearance of the monster.

As the dictionary says about Frankenstein, “The name has become a synonym for one destroyed by his own works.”

ROSSUM’S UNIVERSAL ROBOTS

Perhaps the next study of the consequences of a machine that thinks is a remarkable play called _R.U.R._ (for Rossum’s Universal Robots), first produced in Prague in 1921. Karel Čapek, the Czech dramatist who wrote it, was then only 31. The word “robot” comes from the Czech word “robota,” meaning compulsory service.

According to the play, Rossum the elder, a scientist, discovered a “method of organizing living matter” that was “more simple, flexible, and rapid” than the method used by nature. Rossum the younger, an engineer, founded a factory for the mass production of artificial workmen, robots. They had the form of human beings, intelligence, memory, and strength; but they were without feelings.

In the first act, the factory under Harry Domin, General Manager, is busy supplying robots to purchasers all over the world—for work, for fighting, for any purpose at all, to anyone who could pay for them. Domin declares:

“... in ten years, Rossum’s Universal Robots will produce so much corn, so much cloth, so much everything that things will be practically without price. There will be no poverty. All work will be done by living machines. Everybody will be free from worry and liberated from the degradation of labor. Everybody will live only to perfect himself.... It’s bound to happen.”

In the second act, ten years later, it turns out that Domin and the others in charge of the factory have been making some robots with additional human characteristics, such as the capacity to feel pain. The newer types of robots, however, have united all the robots against man, for the robots declare that they are “more highly developed than man, stronger, and more intelligent, and man is their parasite.”

In the last act, the robots conquer and slay all men except one—an architect, Alquist, who in the epilogue provides a final quirk to the plot.

FACT AND FANCY

Now what is fact and what is fancy in these two warnings given to us a hundred years apart?

Of course, it is very doubtful that a Frankenstein monster or a Rossum robot will soon be constructed with nerves, flesh, and blood like an animal body. But we know that many types of robot machines can even now be constructed out of hardware—wheels, motors, wires, electronic tubes, etc. They can handle many kinds of information and are able to perform many kinds of actions, and they are stronger and swifter than man.

Of course, it is doubtful that the robot machines, by themselves and of their own “free will,” will be dangerous to human beings. But as soon as antisocial human beings have access to the controls over robot machines, the danger to society becomes great. We want to escape that danger.

Escape from Danger

A natural longing of many of us is to escape to an earlier, simpler life on this earth. Victor Frankenstein longed to undo the past. He said:

“Learn from me, if not by my precepts, at least by my example, how dangerous is the acquirement of knowledge, and how much happier that man is who believes his native town to be the world, than he who aspires to become greater than his nature will allow.”

Any sort of return to the past is, of course, impossible. It is doubtful that men could, even if they wanted to, stop the great flood of technical knowledge that science is now producing. We all must now face the fact that the kind of world we used to live in, even so recently as 1939, is gone. There now exist weapons and machines so powerful and dangerous in the wrong hands that in a day or two most of the people of the earth could be put to death. Giant brains are closely related to at least two of these weapons: scientists have already used mechanical brains for solving problems about atomic explosives and guided missiles. In addition, thinking mechanisms designed for the automatic control of gunfire were an important part of the winning of World War II. They will be a still more important part of the fighting of any future war.

Nor can we escape to another part of the earth which the new weapons will not reach. At 300 miles an hour, any spot on earth can be reached from any other in less than 48 hours. A modern plane exceeds this speed; a rocket or guided missile doubles or trebles it.

Nor can we trust that some kind of good luck will pull us through and help men to escape the consequences of what men do. Both Frankenstein and Domin reaped in full the consequences of what they did. The history of life on this earth that is recorded in the rocks is full of evidence of races of living things that have populated the earth for a time and then become extinct, such as the dinosaurs. In that long history, rarely does a race survive. In our own day, insects and fungi rather than men have shown fitness to survive and spread over the earth: witness the blight that destroyed the chestnut trees of North America, in spite of the best efforts of scientists to stop it.

There seems to be no kind of escape possible. It is necessary to grapple with the problem: How can we be safe against the threat of physical harm from robot machines?

UNEMPLOYMENT

The other chief threat from robot machines is against our economic life. Harry Domin, in _R.U.R._, you remember, prophesied: “All work will be done by living machines.” As an example, in the magazine _Modern Industry_ for Feb. 15, 1947, appeared a picture of a machine for selling books, and under the picture were the words:

_Another new product in robot salesmen_—Latest in the parade of mechanical vending machines is this book salesman.... It is designed for use in hospitals, rail terminals, and stores. It offers 15 different titles, selected manually, and obtained by dropping quarter in slot. Cabinet stores 96 books.

Can you feel the breath of the robot salesman, workman, engineer,—--, on the back of your neck?

At the moment when we combine automatic producing machinery and automatic controlling machinery, we get a vast saving in labor and a great increase in technological unemployment. In extreme cases, perhaps, the effect of robot machines will be the disappearance of men from a factory. Such a factory will be like a modern power plant that turns a waterfall into electricity: once the machinery is installed, only one watchman is ordinarily needed. But, in most cases, this will be the effect: in a great number of factories, mines, farms, etc., the labor force needed will be cut by a great proportion. The effect is not different in quality, because the new development is robot machinery; but the amount of technological unemployment coming from robot machines is likely to be considerably greater than previously.

The robot machine raises the two questions that hang like swords over a great many of us these days. The first one is for any employee: What shall I do when a robot machine renders worthless all the skill I have spent years in developing? The second question is for any businessman: How shall I sell what I make if half the people to whom I sell lose their jobs to robot machines?

SOCIAL CONTROL AND ITS TWO SIDES

The two chief harmful effects upon humanity which are to be expected from robot machines are physical danger and unemployment. These are serious risks, and some degree of social control is needed to guard against them.

There will also be very great advantages from robot machines. The monster in _Frankenstein_ is right when he says, “Do your duty towards me, and I will do mine towards you and the rest of mankind.” And Harry Domin in _R.U.R._ is right as to possibility when he says, “There will be no poverty.... Everybody will be free from worry.” Social control must also be concerned with how the advantages from robot machines are to be shared.

The problem of social control over men and their devices has always had two sides. The first side deals with what we might plan for control if men were reasonable and tolerant. This part of the problem seems relatively easy. The other side deals with what we must ordinarily arrange, since most men are often unreasonable and prejudiced and, as a result, often act in antisocial ways. This part of the problem is hard. Let us begin with the easier side first.

TYPES OF CONTROL—IF MEN WERE REASONABLE

In seeking to fulfill wants and achieve safety, men have used hundreds of types of control. The main types are usually called political and economic systems, but there are always great quantities of exceptions. The more mature and freer the society, the greater the variety of types of control that can be found in it.

Probably the most widely used type of control in this country is private and public control working together, as private ownership and public regulation—for example, railroads, banks, airlines, life insurance companies, telephone systems, and many others. It would be reasonable to expect private ownership and public regulation of a great many classes of robot machines, to the end that they would never threaten the safety of people.

Another common type of control is public ownership and operation; examples are toll bridges, airports, city transit systems, and water-supply systems. Atomic energy was so clearly fraught with serious implications that in 1946 the Congress of the United States placed it entirely under public control expressed as the Atomic Energy Commission. There is a class of robot machinery which has already reached the stage of acute public concern: guided missiles and automatic fire-control. It would be reasonable that in this country all activity in this subdivision should be under close control by the Department of Defense.

In the international arena, again, the problem becomes soluble if we assume men to be reasonable. An international agency, such as an organ of the United Nations, would take over inspection and control of robot machine activities closely affecting the public safety anywhere in the world. Particularly, this agency would concern itself with guided missiles, robot pilots for planes, automatic gunfire control, etc. Much manufacturing skill is needed to make such products as these: the factories where they could be manufactured would thereby be determined. Also, a giant brain is a useful device for solving scientific problems about weapons of mass destruction. So the agency would need to inspect the problems being solved on such machines. This agency would be responsible to a legislature or an executive body representing all the people in the world—if men were reasonable.

In regard to the effects of robot machines on unemployment, again, if men were reasonable, the problem would be soluble. The problem is equivalent to the problem of abundance: how should men distribute the advantages of a vast increase in production among all the members of society in a fair and sensible way? A vast increase in production is not so impossible as it may seem. For example, in 1939, with 45 million employed, the United States index of industrial production was at 109, and, in 1943, with 52½ million employed, the index of production was at 239.

If men were reasonable, the net profits from robot machinery would be divided among (1) those who had most to do with devising the new machinery, and (2) all of society. A rule would be adopted (probably it could be less complicated than some existing tax rules) which would take into account various factors such as rewards to the inventors, incentives to continue inventing, adequate assistance to those made unemployed by the robot machines, reduction of prices to benefit consumers, and contributions to basic and applied scientific research.

In fact, under the assumption “if men were reasonable,” it would hardly be necessary to devote a chapter to the problem of social control over robot machines!

OBSTACLES

The discussion above of how robot machines could be controlled supposing that men were reasonable, seems, of course, to be glaringly impractical. Men are not reasonable on most occasions most of the time. If we stopped at this point, again we would be dodging the issue. What are the obstacles to reasonable control?

There are, it seems, two big obstacles and one smaller one to reasonable types of social control over robot machines. The smaller one is ignorance, and the two big obstacles are prejudice and a narrow point of view.

Ignorance

By ignorance we mean lack of knowledge and information. Now mechanical brains are a new and intricate subject. A great many people will, through no fault of their own, naturally remain uninformed about mechanical brains and robot machines for a long time. However, there is a widespread thirst for knowledge these days: witness in magazines, for example, the growth of the article and the decline of the essay. There is also a fairly steady surge of knowledge from the austere scientific fountain of new technology. We can thus see both a demand and a supply for information in such fields as mechanical brains and robot machines. We can expect, therefore, a fairly steady decline in ignorance.

Prejudice

Prejudice is a much more serious obstacle to reasonable control over robot machines. It will be worth our while to examine it at length.

Prejudice is frequent in human affairs. For example, in some countries, but not in all, there is conflict among men, based on their religious differences. Again, in other countries, but not in all, there is wide discrimination among men, based on the color of their skin. Over the whole world today, there is a sharp lack of understanding between conservatives, grading over to reactionaries, on the one hand, and liberals, grading over to radicals, on the other hand. All these differences are based on men’s attitudes, on strongly held sets of beliefs. These attitudes are not affected by “information”; the “information” is not believed. The attitudes are not subject to “judgment”; they come “before judgment”: they are prejudices. Even in the midst of all the science of today, prejudice is widespread. In Germany, from 1933 to 1939, we saw one of the most scientific of countries become one of the most prejudiced.

Prejudice is often difficult to detect. We find it hard to recognize even in ourselves. For a prejudice always seems, to the person who has it, the most natural attitude in the world. As we listen to other people, we are often uncertain how to separate information, guesses, humor, prejudice, etc. Circumstances compel us to accept provisionally quantities of statements just on other people’s say-so. A good test of a statement for prejudice, however, is to compare it with the scientific view.

Prejudice is most dangerous for society. Its more extreme manifestations are aggressive war, intolerance (especially of strange people and customs), violence, race hatred, etc. In the consuming hatred that a prejudiced man has towards the object of his prejudice, he is likely to destroy himself and destroy many more people besides. In former days, the handy weapon was a sword or a pistol; not too much damage could be done when one man ran amuck. But nowadays a single use of a single weapon has slain 70,000 people (the atom bomb dropped at Hiroshima), and so a great many people live anxious and afraid.

What is prejudice? How does it arise? How can it be cured, and thus removed from obstructing reasonable control over robot machines and the rest of today’s amazing scientific developments?

Prejudice is a disease of men’s minds. It is infectious. The cause and development of the disease are about as follows: Deprive someone of something he deeply needs, such as affection, food, or opportunity. In this way hurt him, make him resentful, hostile; but prevent him from expressing his resentments in a reasonable way, giving him instead false outlets, such as other people to hurt, myths to believe, hostile behavior patterns to imitate. He will then break out with prejudices as if they were measles. The process of curing the disease of prejudice is about as follows: Make friends with the patient; win his trust. Encourage him to pour out his half-forgotten hates. Help him to talk them over freely, by means of questions but not criticisms, until finally the patient achieves insight, sees through his former prejudices, and drops them.

In these days prejudice is a cardinal problem of society. It is perhaps conservative to say that a chief present requirement for the survival of human society—with the atom bomb, bacterial warfare, guided missiles, etc., near at hand—is cure of prejudice and its consequences, irrational and unrestrained hate.

Narrow Point of View

A narrow point of view regarding what is desirable or good is the third obstacle to rational control over robot machines. What do we mean by this?

Our point of view as a two-year-old is based on pure self-interest. If we see a toy, we grab it. There is no prejudice about this; it is entirely natural—for two-year-olds. As we grow older, our point of view concerning what is good or desirable rapidly broadens: we think of others and their advantage besides our own. For example, we may become interested in a conservation program to conserve birds, or soil, or forests, and our point of view expands, embraces these objectives, which become part of our personality and loyalties.

Unfortunately, it seems to be true that the expanding point of view, the expanding loyalties, of most people as they grow up are arrested somewhere along the line of: self, family, neighborhood, community, section of country, nation. An honorable exception is the scientists’ old and fine tradition of world-wide unity and loyalty in the search for objective truth.

Now the problem of rational control over robot machines and other parts of the new technology is no respecter of national boundaries. To be solved it requires a world-wide point of view, a loyalty to human society and its best interests, a social point of view.

Almost all that you and I have and do and think is the result of a long history of human society on this earth. All men on the earth today are descendants of other men who lived 1000, 2000, 3000 ... years ago, whether they were Romans or Chinese or Babylonians or Mayas or members of any other race. To ride in a subway or an airplane, to talk on the telephone, to speak a language, to calculate, to survive smallpox or the black death, etc.—all these privileges are our inheritance from countless thousands of other human beings, of many countries, and nearly all of whom are now dead. During our lives we pass on to our own children an inheritance in which our own contribution is remarkably small. Since each person is the child of two others, the number of our forefathers is huge, and we are all undoubtedly blood cousins. Because of this relationship, and because we owe to the rest of society nearly all that we are, we have a social responsibility—we need to hold a social point of view. Each of us needs to accept and welcome a world-wide social responsibility, as a member of human society, as a beneficiary and trustee of our human inheritance. Otherwise we are drones, part of the hive without earning our keep. The social point of view is equitable, it is inspiring, and it is probably required now in order for human beings to survive. We need to let go of a narrow point of view.

CONCLUSION

We have now outlined the problem of social control over robot machines, supposing that human beings were reasonable. We have also discussed the practical obstacles that obstruct reasonable control.

It is not easy to think of any yet organized group of people anywhere that would have both the strength and the vision needed to solve this problem through its own efforts. For example, a part of the United Nations might have some of the vision needed, but it does not have the power. Consequently, it is necessary and desirable for individuals and groups everywhere to take upon themselves an added load of social responsibility—just as they tend to do in time of war. People often “want to do their share.” Through encouragement and education, the basic attitude of a number of people can contain more of “This is our business; we have a responsibility for helping to solve this problem.” We also need public responsibility; we need a public body responsible for study, education, advice, and some measure of control. It might be something like an Atomic Energy Commission, Bacterial Defense Commission, Mental Health Commission, and Robot Machine Commission, all rolled into one.

When, at last, there is an effective guarantee of the two elements physical safety and adequate employment, then at last we shall all be free from the threat of the robot machine. We can then welcome the robot machine as our deliverer from the long hard chores of many centuries.

Supplement 1

WORDS AND IDEAS

The purpose of this book is to explain machines that think, without using technical words any more than necessary. This supplement is a digression. Its purposes are to consider how to explain in this way and to discuss the attempt made in this book to achieve simple explanation.

WORDS AS INSTRUMENTS FOR EXPLAINING

Words are the chief instruments we use for explaining. Of course, many other devices—pictures, numbers, charts, models, etc.—are also used; but words are the prime tools. We do most of our explaining with them.

Words, however, are not very good instruments. Like a stone arrow-head, a word is a clumsy weapon. In the first place, words mean different things on different occasions. The word “line,” for example, has more than fifty meanings listed in a big dictionary. How do we handle the puzzle of many meanings? As we grow older we gather experience and we develop a truly marvelous capacity to listen to a sentence and then fit the words together into a pattern that makes sense. Sometimes we notice the time lag while our brain hunts for the meaning of a word we have heard but not grasped. Then suddenly we guess the needed meaning, whereupon we grasp the meaning of the sentence as a whole in much the same way as the parts of a puzzle click into place when solved.

Another trouble with words is that often there is no good way to tell someone what a word means. Of course, if the word denotes a physical object, we can show several examples of the object and utter the word each time. In fact, several good illustrations of a word denoting a physical thing often tell most of its meaning. But the rest of its meaning we often do not learn for years, if ever. For instance, two people would more likely disagree than agree about what should be called a “rock” and what should be called a “stone,” if we showed them two dozen examples.

In the case of words not denoting physical objects, like “and,” “heat,” “responsibility,” we are worse off. We cannot show something and say, “That is a ···.” The usual dictionary is of some help, but it has a tendency to tell us what some word _A_ means by using another word _B_, and when we look up the other word _B_ we find the word _A_ given as its meaning. Mainly, however, to determine the meanings of words, we gather experience: we soak up words in our brains and slowly establish their meanings. We seem to use an unconscious reasoning process: we notice how words are used together in patterns, and we conclude what they must mean. Clearly, then, words being clumsy instruments, the more experience we have had with a word, the more likely we are to be able to use it, work with it, and understand it. Therefore explanation should be based chiefly on words with which we have had the most experience. What words are these? They will be the well-known words. A great many of them will be short.

SET OF WORDS FOR EXPLAINING

Now what is the set of all the words needed to explain simply a technical subject like machines that think? For we shall need more words than just the well-known and short ones. This question doubtless has many answers; but the answer used in this book was based on the following reasoning. In a book devoted to explanation, there will be a group of words (1) that are supposed to be known already or to be learned while reading, and (2) that are used as building blocks in later explanation and definitions. Suppose that we call these words the _words for explaining._ There are at least three groups of such words:

_Group 1._ Words not specially defined that are so familiar that every reader will know all of them; for example, “is,” “much,” “tell.”

_Group 2._ Words not specially defined that are familiar, but perhaps some reader may not know some of them; for example, “alternative,” “continuous,” “indicator.”

_Group 3._ Words that are not familiar, that many readers are not expected to know, and that are specially defined and explained in the body of the book; for example, “abacus,” “trajectory,” “torque.”

In writing this book, it was not hard to keep track of the words in the third group. These words are now listed in the index, together with the page where they are defined or explained. (The index, of course, also lists phrases that are specially defined.)

But what division should be made between the other two groups? A practical, easy, and conservative way to separate most words between the first and second groups seemed to be on the basis of number of syllables. All words of one syllable—if not specially defined—were put in Group 1. Also, if a word became two syllables only because of the addition of one of the endings “-es,” “-ed,” “-ing,” it was kept in Group 1, for these endings probably do not make a word any harder to understand. In addition, there were put into Group 1:

1. Numbers; for example, “186,000”; “³/₁₀”.

2. Places: “Philadelphia”; “Massachusetts”.

3. Nations, organizations, people, etc.: “Swedish”; “Bell”.

4. Years and dates: “February”; “1946”.

5. Names of current books or articles and their authors.

Of course, not all these words would be familiar to every reader (for example, “Maya”), but in the way they occur, they are usually not puzzling, for we can tell from the context just about what they must mean.

All remaining words for explaining—chiefly, words of two or more syllables and not specially defined—were put in Group 2 and were listed during the writing of this book. Many Group 2 words, of course, would be entirely familiar to every reader; but the list had several virtues. No hard words would suddenly be sprung like a trap. The same word would be used for the same idea. Every word of two or more syllables was continually checked: is it needed? can it be replaced by a shorter word? It is perhaps remarkable that there were fewer than 1800 different words allowed to stay in this list. This fact should be a comfort to a reader, as it was to the author.

Now there are more words in this book than _words for explaining_. So we shall do well to recognize:

_Group 4._ Words that do not need to be known or learned and that are not used in later explanation and definitions.

These words occur in the book in such a way that understanding them, though helpful, is not essential. One subdivision of Group 4 are names that appear just once in the book, as a kind of side remark, for example, “a chemical, called _acetylcholine_.” Such a name will also appear in the index, but it is not a _word for explaining_. Another subdivision of Group 4 are words occurring only in quotations. For example, in the quotation from _Frankenstein_ on page 198, a dozen words appear that occur nowhere else in the book, including “daemon,” “dissoluble,” “maw,” “satiate.” Clearly we would destroy the entire flavor of the quotation if we changed any of these words in any way. But only the general drift of the quotation is needed for understanding the book, and so these words are Group 4 words.

In this way the effort to achieve simple explanation in this book proceeded. But even supposing that we could reach the best set of words for explaining, there is more to be done. How do we go from simple explanation to understanding?

UNDERSTANDING IDEAS

_Understanding_ an idea is basically a standard process. First, we find the name of the idea, a word or phrase that identifies it. Then, we collect true statements about the idea. Finally, we practice using them. The more true statements we have gathered, and the more practice we have had in applying them, the more we understand the idea.

For example, do you understand zero? Here are some true statements about zero.

1. Zero is a number. 2. It is the number that counts none or nothing. 3. It is marked 0 in our usual numeral writing. 4. The ancient Romans, however, had no numeral for it. Apparently, they did not think of zero as a number. 5. 0 is what you get when you take away 17 from 17, or when you subtract any number from itself. 6. If you add 0 to 23, you get 23; and if you add 0 to any number, you get that number unchanged. 7. If you subtract 0 from 48, you get 48; and if you subtract 0 from any number, you get that number unchanged. 8. If you multiply 0 by 71, you get 0; and if you multiply together 0 and any number, you get 0. 9. Usually you are not allowed to divide by 0: that is against the rules of arithmetic. 10. But if you do, and if you divide 12 by 0, for example—and there are times when this is not wrong—the result is called _infinity_ and is marked ∞, a sign that is like an 8 on its side.

This is not all the story of zero; it is one of the most important of numbers. But, if you know these statements about zero, and have had some practice in applying them, you have a good _understanding_ of zero. Incidentally, a mechanical brain knows all these statements about zero and a few more; they must be built into it.

For us to understand any idea, then, we pursue three aims:

1. We find out what it is called. 2. We collect true statements about it. 3. We apply those statements—we use them in situations.

We can do this about any idea. Therefore, we can understand any idea, and the degree of our understanding increases as the number of true statements mastered increases.

Perhaps this seems to be a rash claim. Of course, it may take a good deal of time to collect true statements about many ideas. In fact, a scientist may spend thirty years of his life trying to find out from experiment the truth or falsehood of one statement, though, when he has succeeded, the fact can be swiftly told to others. Also, we all vary in the speed, perseverance, skill, etc., with which we can collect true statements and apply them. Besides, some of us have not been taught well and have little faith in our ability to carry out this process: this is the greatest obstacle of all. But, there is in reality no idea in the field of existing science and knowledge which you or I cannot understand. The road to understanding lies clear before us.

Supplement 2

MATHEMATICS

In the course of our discussion of machines that think, we have had to refer without much explanation to a number of mathematical ideas. The purpose of this supplement is to explain a few of these ideas a little more carefully than seemed easy to do in the text and, at the end of the supplement, to put down briefly some additional notes for reference.

DEVICES FOR MULTIPLICATION

Suppose that we have to multiply 372 by 465. With the ordinary school method, we write 465 under the 372 and proceed about as follows: 5 times 2 is 10, put down the 0 and carry the 1; 5 times 7 is 35, 35 and 1 is 36, put down the 6 and carry the 3; 5 times 3 is 15, 15 and 3 is 18, put down the 8 and carry the 1; ... The method is based mainly on a well-learned subroutine of continually changing steps:

1. Select a multiplicand digit. 2. Select a multiplier digit. 3. Refer to the multiplication table with these digits. 4. Obtain the value of their product, called a _partial product_. 5. Add the preceding carry. 6. Set down the right-hand digit. 7. Carry the left-hand digit.

We can, however, simplify this subroutine for a machine by delaying the carrying. We collect in one place all the right-hand digits of partial products, collect in another place all the left-hand digits, and delay all addition until the end.

For example, let us multiply 372 by 465 with this method:

RIGHT-HAND LEFT-HAND USUAL METHOD, DIGITS DIGITS FOR COMPARISON

372 372 372 × 465 × 465 × 465 —————— —————— —————— 550 131 1860 822 141 2232 288 120 1488 ————— ————— —————— 37570 13541 172980

FINAL ADDITION

37570 + 13541 ———————— 172980

37570 is called the _right-hand component_ of the product. It is convenient to fill in with 0 the space at the end of 13541 and to call 135410 the _left-hand component_ of the product.

This process is called _multiplying by right- and left-hand components_. It has the great advantage that no carrying is necessary to complete any line of the original multiplications. Some computing machines use this process. Built into the hardware of the machine is a multiplication table up to 9 × 9. The machine, therefore, can find automatically the right-hand digit and the left-hand digit of any partial product. In a computing machine that uses this process, all the left-hand digits are automatically added in one register, and all the right-hand digits are added in another register. The only carrying that is needed is the carrying as the right-hand digits are accumulated and as the left-hand digits are accumulated. At the end of the multiplication, one of the registers is automatically added into the other, giving the product.

Another device used in computing machines for multiplying is to change the multiplier into a set of digits 0 to 5 that are either positive or negative. For example, suppose that we want to multiply 897 by 182. We note that 182 equals 200 minus 20 plus 2, and so we can write it as

_ 222.

The minus over the 2 marks it as a _negative digit_ 2. Then to multiply we have:

897 _ 222 ———— 1794 - 1794 1794 —————— 163254

The middle 1794 is subtracted. This process is usually called _short-cut multiplication_. Everybody discovers this trick when he decides that multiplying by 99 is too much work, that it is easier to multiply by 100 and subtract once.

BINARY OR TWO NUMBERS

We are well accustomed to decimal notation in which we use 10 decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and write them in combinations to designate decimal numbers. In _binary notation_ we use two binary digits 0, 1 and write them in combinations to designate _binary numbers_. For example, the first 17 numbers, from 0 to 16 in the decimal notation, correspond with the following numbers in binary notation:

DECIMAL BINARY DECIMAL BINARY 0 0 1 1 9 1001 2 10 10 1010 3 11 11 1011 4 100 12 1100 5 101 13 1101 6 110 14 1110 7 111 15 1111 8 1000 16 10000

In decimal notation, 101 means one times a hundred, no tens, and one. In binary notation, 101 means one times four, no twos, and one. The successive digits in a decimal number from right to left count 1, 10, 100, 1000, 10000, ...—successive _powers_ of 10 (for this term, see the end of this supplement). The successive digits in a binary number from right to left count 1, 2, 4, 8, 16, ...—powers of 2.

The decimal notation is convenient when equipment for computing has ten positions, like the fingers of a man, or the positions of a counter wheel. The binary notation is convenient when equipment for computing has just two positions, like “yes” or “no,” or current flowing or no current flowing.

Addition, subtraction, multiplication, and division can all be carried out unusually simply in binary notation. The addition table is simple and consists only of four entries.

+ 0 1 +—————— 0 | 0 1 | 1 | 1 10

The multiplication table is also simple and contains only four entries.

× 0 1 +—————— 0 | 0 0 | 1 | 0 1

Suppose that we add in binary notation 101 and 1001:

BINARY ADDITION CHECK

101 5 + 1001 9 —————— ——— 1110 14

We proceed: 1 and 1 is 10; write down 0 and carry 1; 0 and 0 is 0, and 1 to carry is 1; and 1 and 0 is 1; and then we just copy the last 1. To check this we can convert to decimal and see that 101 is 5, 1001 is 9, and 1110 is 14, and we can verify that 5 and 9 is 14.

One of the easiest ways to subtract in binary notation is to add a _ones complement_ (that is, the analogue of the nines complement) and use end-around-carry (for these two terms, see the end of this supplement). A ones complement can be written down at sight by just putting 1 for 0 and 0 for 1. For example, suppose that we subtract 101 from 1110:

SUBTRACTION BY DIRECT ADDING ONES SUBTRACTION CHECK COMPLEMENT

1110 14 1110 - 101 -5 + 1010 ————— ———— —————— 1001 9 (1)1000 ↓ ⎯→ 1 —————— 1001

Multiplication in the binary notation is simple. It amounts to (1) adding if the multiplier digit is 1 and not adding if the multiplier digit is 0, and (2) moving over or shifting. For example, let us multiply 111 by 101:

BINARY MULTIPLICATION CHECK

111 7 × 101 × 5 —————— 111 111 —————— ——— 100011 35

The digit 1 in the 6th (or _n_th) _binary_ place from the right in 100011 stands for 1 times 2 to the 5th (or _n_-1 th) power, 2 × 2 × 2 × 2 × 2 = 32. The result 100011 is translated into 32 plus 2 plus 1, which equals 35 and verifies.

Division in the binary notation is also simple. It amounts to (1) subtracting (yielding a quotient digit 1) or not subtracting (yielding a quotient digit 0), and (2) shifting. We never need to try multiples of the divisor to find the largest that can be subtracted yet leave a positive remainder. For example, let us divide 1010 (10 in decimal) into 10001110 (142 in decimal):

1110 (14 in decimal) —————————— 1010)10001110 1010 —————— 1111 1010 ————— 1011 1010 ————— 10 (remainder, 2 in decimal)

In decimal notation, digits to the right of the decimal point count powers of ⅒. In binary notation, digits to the right of the _binary point_ count powers of ½: ½, ¼, ⅛, ¹/₁₆.... For example, 0.1011 equals ½ + ⅛ + ¹/₁₆, or ¹¹/₁₆.

If we were accustomed to using binary numbers, all our arithmetic would be very simple. Furthermore, binary numbers are in many ways much better for calculating machinery than any other numbers. The main problem is converting numbers from decimal notation to binary. One method depends on storing the powers of 2 in decimal notation. The rule is: subtract successively smaller powers of 2; start with the largest that can be subtracted, and count 1 for each power that goes and 0 for each power that does not. For example, 86 in decimal becomes 1010110 in binary:

86 64 64 goes 1 ——— 22 32 does not go 0 16 16 goes 1 ——— 6 8 does not go 0 4 4 goes 1 ——— 2 2 goes 1 2 1 does not go 0 ——— 0

It is a little troublesome to remember long series of 1’s and 0’s; in fact, to write any number in binary notation takes about 3⅓ times as much space as decimal notation. For this reason we can separate binary numbers into triples beginning at the right and label each triple as follows:

TRIPLE LABEL 000 0 001 1 010 2 011 3 100 4 101 5 110 6 111 7

For example, 1010110 would become 1 010 110 or 126. This notation is often called _octal notation_, because it is notation in the scale of eight.

BIQUINARY OR _TWO-FIVE_ NUMBERS

Another kind of notation for numbers is _biquinary notation_, so called because it uses both 2’s and 5’s. Essentially this notation is very like Roman numerals, ancient style. By ancient style we mean, for example, VIIII instead of IX. In the following table we show the first two dozen numbers in decimal, biquinary, and ancient Roman notation:

DECIMAL BIQUINARY ROMAN 0 0 1 1 I 2 2 II 3 3 III 4 4 IIII 5 10 V 6 11 VI 7 12 VII 8 13 VIII 9 14 VIIII 10 100 X 11 101 XI 12 102 XII 13 103 XIII 14 104 XIIII 15 110 XV 16 111 XVI 17 112 XVII 18 113 XVIII 19 114 XVIIII 20 200 XX 21 201 XXI 22 202 XXII 23 203 XXIII

The biquinary columns alternate in going from 0 to 4 and from 0 to 1. The digits from 0 to 4 are not changed. The digits from 5 to 9 are changed into 10 to 14. We see that the _biquinary digits_ are 0 to 4 in odd columns and 0, 1 in even columns, counting from the right.

This is the notation actually expressed by the _abacus_. The beads of the abacus show by their positions groups of 2 and 5 (see Fig. 1).

SOME OPERATIONS OF ALGEBRA

One of the operations of algebra that is important for a mechanical brain is _approximation_, the problem of getting close to the right value of a number. Take, for example, finding _square root_ (see the end of this supplement). The ordinary process taught in school is rather troublesome. We can set down another process, however, using a desk calculator to do division, which gives us square root with great speed.

Suppose that we want to find the square root of a number _N_, and suppose that we have _x_₀ as a guessed square root correct to one figure. For example, _N_ might be 67.2 and _x_₀ might be 8, chosen because 8 × 8 is 64, and 9 × 9 is 81, and it seems as if 8 should be near the square root of 67.2. Here is the process:

1. Divide _x_₀ into _N_, and obtain _N_/_x_₀.

2. Multiply _x_₀ + _N_/_x_₀ by 0.5 and call the result _x_₁.

Now repeat:

1. Divide _x_₁ into _N_ and obtain _N_/_x_₁.

2. Multiply _x_₁ + _N_/_x_₁ by 0.5 and call the result _x_₂.

Every time this process is repeated, the new _x_ comes a great deal closer to the correct square root. In fact it can be shown that, if _x_₀ is correct to one figure, then:

APPROXIMATION IS CORRECT TO ··· FIGURES _x_₁ 2 _x_₂ 4 _x_₃ 8 _x_₄ 16

Let us see how this actually works out with 67.2 and a 10-column desk calculator.

Round 1: 8 divided into 67.2 gives 8.4. One half of 8 plus 8.4 is 8.2. This is _x_₁.

Round 2: 8.2 divided into 67.2 gives 8.195122. One half of 8.2 plus 8.195122 is 8.197561. This is _x_₂.

Round 3: 8.197561 divided into 67.2 gives 8.197560225. One half of 8.197561 and 8.197560225 is 8.1975606125. This is _x_₃.

Checking _x_₃, we find that 8.1975606125 divided into 67.2 gives 8.1975606126 approximately.

In this case, then, _x_₃ is correct to more than 10 figures. In other words, with a reasonable guess and two or three divisions we can obtain all the accuracy we can ordinarily use. This process is called _rapid approximation_, or _rapidly convergent approximation_, since it _converges_ (points or comes together) very quickly to the number we are seeking.

Another important operation of algebra is _interpolation_, the problem of putting values smoothly in between other values. For example, suppose that we have the table:

_x y_ 5 26 6 37 7 50 8 65 9 82

Suppose that we want to find the value that _y_ (or _yₓ_) ought to have when _x_ has the value of 7.2. This is the problem of _interpolating y_ so as to find _y_ at the value of 7.2, _y_₇ˌ₂.

One way of doing this is to discover the formula that expresses _y_ and then to put _x_ into that formula. This is not always easy. Another way is to take the difference between _y_₇ and _y_₈, 15, and share the difference appropriately over the distance 7 to 7.2 and 7.2 to 8. We can, for example, take ²/₁₀ of 15 = 3, add that to _y_₇ = 50, and obtain an estimated _y_₇ˌ₂ = 53. This is called _linear interpolation_, since the difference 0.2 in the value of _x_ is used only to the first power. It is a good practical way to carry out most interpolation quickly and approximately.

Actually here _y_ = _x_² + 1, and so the true value of _y_₇ˌ₂ is (7.2 × 7.2) + 1, or 52.84, which is rather close to 53. Types of interpolation procedures more accurate than linear interpolation will come much nearer still to the true value.

ALGEBRA OF LOGIC

We turn now to the _algebra of logic_. The first half of Chapter 9, “Reasoning” (through the section “Logical-Truth Calculation by Algebra”), introduces this subject. There the terms _truth values_, _truth tables_, _logical connectives_, and _algebra of logic_ are explained. The part of Chapter 3, “A Machine That Will Think,” that discusses the operations _greater-than_ and _selection_, also explains some of the algebra of logic. It introduces, for example, the formula

_p_ = _T_(_a_ > _b_) = 1, 0

This is a way of saying briefly that the truth value of the statement “_a_ is greater than _b_” equals _p_; _p_ is 1 if the statement is true and 0 if the statement is false. The truth value 1 corresponds with “yes.” The truth value 0 corresponds with “no.”

With mechanical brains we are especially interested in handling mathematics and logic without any sharp dividing line between them. For example, suppose that we have a register in which a ten-digit number like 1,765,438,890 may be stored. We should be able to use that register to store a number consisting of only 1’s and 0’s, like 1,100,100,010. Such a number may designate the answers to 10 successive questions: yes, yes, no, no, yes, no, no, no, yes, no. Or it may tell 10 successive binary digits. The register then is three times as useful: it can store either decimal numbers or truth values or binary digits. We need, of course, a way to obtain from the register any desired digit. For this purpose we may have two instructions to the machine: (1) read the left-hand end digit; (2) shift the number around in a circle. The second instruction is the same as multiplying by 10 and then putting the left-most digit at the right-hand end. For example, suppose that we want the 3rd digit from the left in 1,100,100,010. The result of the first circular shift is 1,001,000,101; the result of the second circular shift is 0,010,001,011; and reading the left-most digit gives 0. A process like this has been called _extraction_ and is being built into the newest mechanical brains.

Using truth values, we can put down very neatly some truths of ordinary algebra. For example:

(the _absolute value_ of _a_) = _a_ × (the truth of _a_ greater than or equal to 0) - _a_ × (the truth of _a_ less than 0)

⎮_a_⎮ = _a_ · _T_(_a_ ≥ 0) - _a_ · _T_(_a_ < 0)

For another example:

Either _a_ is greater than _b_, or else _a_ equals _b_, or else _a_ is less than _b_

_T_(_a_ > _b_) + _T_(_a_ = _b_) + _T_(_a_ < _b_) = 1

Many common logical operations, like selecting and comparing, and the behavior of many simple mechanisms, like a light or a lock, can be expressed by truth values. Chapter 4, on punch-card mechanisms, contains a number of examples.

* * * * *

pronoun, variable

In ordinary language, a _pronoun_, like “he,” “she,” “it,” “the former,” “the latter,” is a word that usually stands for a noun previously referred to. A pronoun usually stands for the last preceding noun that the grammar allows. In mathematics, a _variable_, like “_a_,” “_b_,” “_x_,” “_m₁_,” “_m₂_” closely resembles a pronoun in ordinary language. A variable is a symbol that usually stands for a number previously referred to, and usually it stands for the same number throughout a particular discussion.

multiplicand, dividend, augend, etc.

IN THE THE NAME THE NAME THE NAME EQUATION: OF _a_ IS: OF _b_ IS: OF _c_ IS:

_a_ + _b_ = _c_ augend addend sum _a_ - _b_ = _c_ minuend subtrahend remainder _a_ × _b_ = _c_ multiplicand multiplier product _a_ ÷ _b_ = _c_ dividend divisor quotient

_Augend_ and _addend_ are names of registers in the Harvard Mark II calculator (see Chapter 10).

subtraction by adding, nines complement

Two digits that add to 9 (0 and 9, 1 and 8, 2 and 7, 3 and 6, 4 and 5) are called _nines complements_ of each other. The _nines complement_ of a number _a_ is the number _b_ in which each digit of _b_ is the nines complement of the corresponding digit of _a_; for example, the nines complement of 173 is 826. Ordinary subtraction is the same as addition as of the nines complement, with a simple correction; for example, 562 less 173 (equal to 389) is the same as 562 plus 826 (equal to 1388) less 1000 plus 1.

end-around-carry

The correction “less 1000 plus 1” of the foregoing example may be thought of as carrying the 1 (in the result 1388) around from the left-hand end to the right-hand end, where it is there added. So the 1 is called _end-around-carry_.

tens complement

Two digits that add to 10 are called _tens complements_ of each other. The _tens complement_ of a number _a_, however, is equal to the nines complement of the number plus 1. For example, the tens complement of 173 is 827. When subtracting by adding a tens complement, the left-most digit 1 in the result is dropped. For example, 562 less 173 (equal to 389) is the same as 562 plus 827 (equal to 1389) less 1000.

power, square, cube, reciprocal, etc.

A _power_ of any number _a_ is _a_ multiplied by itself some number of times. _a_ × _a_ × _a_ ... × _a_ where _a_ appears _b_ times is written _a_ᵇ and is read _a_ to the _b_th power. _a_², a to the 2nd power, is _a_ × _a_ and is called _a squared_ or the _square_ of _a_. _a_³, _a_ to the 3rd power, _a_ × _a_ × _a_, is called _a cubed_, or the _cube_ of _a_. _a_⁰, _a_ to the zero power, is equal to 1 for every _a_. _a_¹, _a_ to the power 1, is _a_ itself. The first power is often called _linear_. _a_ to some negative power is the same as 1 divided by that power; that is, _a_⁻ᵇ = 1/_a_ᵇ. _a_⁻¹, _a_ to the power minus 1, is 1/_a_, and is called the _reciprocal_ of _a_. _a_¹ᐟ², _a_ to the one-half power, is a number _c_ such that _c_ × _c_ = _a_, and is called the _square root_ of _a_ and often denoted by √_a_.

table, tabular value, argument, etc.

An example of a _table_ is:

0.025 0.03 +—————————————————— 1 | 1.02500 1.03000 | 2 | 1.05063 1.06090 | 3 | 1.07689 1.09273

The numbers in the body of the table, called _tabular values_, depend on or are determined by the numbers along the edge of the table, called _arguments_. In this example, if 1, 2, 3 are choices of a number _n_, and if 0.025, 0.03 are choices of a number _i_, then each tabular value _y_ is equal to 1 plus _i_ raised to the _n_th power. _n_ and _i_ are also called _independent variables_, and _y_ is called the _dependent variable_. The table expresses a _function_ or _formula_ or _rule_. The rule could be stated as: add _i_ to 1; raise the result to the _n_th power.

constant

A number is said to be a _constant_ if it has the same value under all conditions. For example, in the formula: (area of a circle) = π × (radius)², π is a constant, equal to 3.14159 ..., applying equally well to all circles.

infinity

Mathematics recognizes several kinds of infinity. One of them occurs when numbers become very large. For example, the quotient of 12 divided by a number _x_, as _x_ becomes closer and closer to 0, becomes indefinitely large, and the limit is called _infinity_ and is denoted ∞.

equation, simultaneous, linear

An example of two linear simultaneous _equations_ is:

7_x_ + 8_t_ = 22

3_x_ + 5_t_ = 11

_x_ and _t_ are called _unknowns_—that is, unknown variables—because the objective of solving the equations is to find them. These equations are called _simultaneous_ because they are to be solved together, at the same time, for values of _x_ and _t_ which will fit in both equations. The equations are called _linear_ because the only powers of the unknowns that appear are the first power. Values that solve equations are said to _satisfy_ them. It is easy to solve these two equations and find that _x_ = 2 and _t_ = 1 is their solution. But it is a long process to solve 10 linear simultaneous equations in 10 unknowns, and it is almost impossible (without using a mechanical brain) to solve 100 linear simultaneous equations in 100 unknowns.

derivative, integral, differential equation, etc.

See the sections in Chapter 5 entitled “Differential Equations,” “Physical Problems,” and “Solving Physical Problems.” There these ideas and, to some extent, also the following ideas were explained: formula, equation, function, differential function, instantaneous rate of change, interval, inverse, integrating. See also a textbook on calculus. If _y_ is a function of _x_, then a mathematical symbol for the derivative of _y_ with respect to _x_ is _Dₓ y_, and a symbol for the integral of _y_ with respect to _x_, is ∫_y dx_. An integral with given initial conditions (see p. 83) is a _definite integral_.

exponential

A famous mathematical function is the _exponential_. It equals the constant _e_ raised to the _x_ power, _eˣ_, where _e_ equals 2.71828.... The exponential lies between the powers of 2 and the powers of 3. It can be computed from:

_x_² _x_³ _eˣ_ = 1 + _x_ + —————— + ————————— + ... 1 · 2 1 · 2 · 3

It is a solution of the differential equation _Dₓy_ = _y_. See also a textbook on calculus. The _exponential to the base 10_ is 10ˣ.

logarithm

Another important mathematical function is the _logarithm_. It is written log _x_ or logₑ _x_ and can be computed from the two equations:

log _uv_ = log _u_ + log _v_

_x_² _x_³ log(1 + _x_) = _x_ - —————— + —————— - ..., _x_² < 1 2 3

It is a solution of the differential equation _Dₓy_ = 1/_y_. If _y_ is the logarithm of _x_, then _x_ is the _antilogarithm_ of _y_. The _logarithm to the base 10_ of _x_, log₁₀ _x_, equals the _logarithm to the base e_ of _x_, logₑ _x_, divided by logₑ 10. See also textbooks on algebra and calculus.

sine, cosine, tangent, antitangent

These also are important mathematical functions. The _sine_ and _cosine_ are solutions of the differential equation _Dₓ_(_Dₓy_) =-_y_ and are written as sin _x_ and cos _x_. They can be computed from

_x_³ _x_⁵ sin _x_ = _x_ - —————— + ————————— - ... 1·2·3 1·2·3·4·5

_x_² _x_₄ cos _x_ = 1 - —————— + —————— - ... 1·2 1·2·3·4

The _tangent_ of _x_ is simply sine of _x_ divided by cosine of _x_. If _y_ is the tangent of _x_, then _x_ is the _antitangent_ of _y_. See also references on trigonometry and on calculus. _Trigonometric_ tables include sine, cosine, tangent, and related functions.

Bessel functions

These are mathematical functions that were named after Friedrich W. Bessel, a Prussian astronomer who lived from 1784 to 1846. Bessel functions are found as some of the solutions of the differential equation

_x_² _Dₓ_(_Dₓy_) + x _Dₓy_ + (_x_² - _n_²)_y_ = O

This equation arises in a number of physical problems in the fields of electricity, sound, heat flow, air flow, etc.

matrix

A _matrix_ is a table (or _array_) of numbers in rows and columns, for which addition, multiplication, etc., with similar tables is specially defined. For example, the matrix

⎮1 2⎮ ⎮ ⎮ ⎮3 4⎮

plus the matrix

⎮5 20⎮ ⎮ ⎮ ⎮60 100⎮

equals the matrix

⎮6 22⎮ ⎮ ⎮ ⎮63 104⎮.

(Can you guess the rule defining addition?)

Calculations using matrices are useful in physics, engineering, psychology, statistics, etc. To add a _square matrix_ of 100 terms in an array of 10 columns and 10 rows to another such matrix, 100 ordinary additions of numbers are needed. To multiply one such matrix by another, 1000 ordinary multiplications and 900 ordinary additions are needed. See references on matrix algebra and matrix calculus.

differences, smoothness, checking

On p. 221, a sequence of values of _y_ is shown: 26, 37, 50, 65, 82. Suppose, however, the second value of _y_ was reported as 47 instead of 37. Then the _differences_ of _y_ as we pass down the sequence would not be 11, 13, 15, 17 (which is certainly regular or _smooth_) but 21, 3, 15, 17 (which is certainly not smooth). The second set of differences would strongly suggest a mistake in the reporting of _y_. The _smoothness_ of differences is often a useful check on a sequence of reported values.

Supplement 3

REFERENCES

A book like the present one can cover only a part of the subject of machines that think. To obtain more information about these machines and other topics to which they are related there are many references that may be consulted. There are still few books directly on the subject of machines that think, but there are many articles and papers, most of them rather specialized.

The purpose of this supplement is to give a number of these references and to provide a brief, general introduction to some of them. The references are subdivided into groups, each dealing with a branch of the subject. The references in each group are in alphabetical order by name of author (with “anonymous” last), and under each author they are in chronological order by publication date. Some publications, especially a forum or symposium, are listed more than once, according as the topic discussed falls in different groups. In this supplement, the sign three dots ( ...) next to the page numbers for an article indicates that the article is continued on later, nonconsecutive pages.

It seemed undesirable to try to make the group of references dealing with a subject absolutely complete, so long as enough were given to provide a good introduction to the subject. It proved impractical to try to make the citation of every single reference technically complete, so long as enough citation was given so that the reference could certainly be found. Furthermore, in a list of more than 250 references, errors are almost certain to occur. If any reader should send me additions or corrections, I shall be more than grateful.

THE HUMAN BRAIN

No one yet knows specifically how particular ideas are thought about in the human brain. The references listed in this section, however, contain some information about such topics as:

The structural differences, development, and evolution of the brains of animals, apes, primitive man, and modern man. The effect on the brain of blood composition, body temperature, supply of oxygen, and other biochemical factors. The structure and physiology of the brain, the nervous system, and nerve impulses. The theory of learning, intelligence, and memory.

BARCROFT, JOSEPH, _The Brain and Its Environment_, New Haven: Yale University Press, 1948, 117 pp.

BEACH, FRANK A., Payday for Primates, _Natural History_, vol. 56, no. 10, Dec. 1947, pp. 448-451.

BEACH, FRANK A., Can Animals Reason? _Natural History_, vol. 57, no. 3, Mar. 1948, pp. 112-116 ...

BERRY, R. J. A., _Brain and Mind, or the Nervous System of Man_, New York: The Macmillan Co., 1928, 608 pp.

BORING, EDWIN G., _A History of Experimental Psychology_, New York: Century Co., 1929, 699 pp.

FRANZ, SHEPHERD I., _The Evolution of an Idea; How the Brain Works_, Los Angeles: University of California, 1929, 35 pp.

HERRICK, C. JUDSON, _The Thinking Machine_, Chicago: University of Chicago Press, 1929, 374 pp.

HERRICK, C. JUDSON, _Brains of Rats and Men_, Chicago: University of Chicago Press, 1930, 382 pp.

LASHLEY, KARL S., _Brain Mechanisms and Intelligence_, Chicago: University of Chicago Press, 1929, 186 pp.

PIERON, HENRI, _Thought and the Brain_, London: Kegan, Paul, Trench, Trübner & Co., 1927, 262 pp. Also New York: Harcourt, Brace & Co.

SCHRÖDINGER, ERWIN, _What is Life?_, New York: The Macmillan Co., 1945, 90 pp.

SHERRINGTON, CHARLES S., _The Brain and Its Mechanism_, Cambridge, England: The University Press, 1933, 35 pp.

TILNEY, FREDERICK, _The Brain from Ape to Man_, New York: P. B. Hoeber, Inc., 1928, 2 vol., 1075 pp.

WIENER, NORBERT, _Cybernetics, or Control and Communication in the Animal and the Machine_, New York: John Wiley & Sons, 1948, 194 pp.

ANONYMOUS, Ten Billion Relays, _Time_, Feb. 14, 1949, p. 67.

MATHEMATICAL BIOPHYSICS

There has recently been another approach to the problem: How does a brain think? A group of men, many of them in and near Chicago, have been saying: “We know the properties of nerves, nerve impulses, and simple nerve networks. We know the activity of the brain. What mathematical model of nerve networks is necessary to account for the activity of the brain?” These men have used mathematics, statistics, and mathematical logic in the effort to attack this problem, and they support a _Bulletin of Mathematical Biophysics_.

HOUSEHOLDER, ALSTON S., A Neural Mechanism for Discrimination, _Psychometrika_, vol. 4, no. 1, Dec. 1939, pp. 45-58.

HOUSEHOLDER, ALSTON S., and Herbert D. Landahl, _Mathematical Biophysics of the Central Nervous System_, Bloomington, Ind.: Principia Press, 1945.

LANDAHL, HERBERT D., Contributions to the Mathematical Biophysics of the Central Nervous System, _Bulletin of Mathematical Biophysics_, vol. 1, no. 2, June 1939, pp. 95-118.

LANDAHL, HERBERT D., WARREN S. MCCULLOCH, and WALTER PITTS, A Statistical Consequence of the Logical Calculus of Nervous Nets, _Bulletin of Mathematical Biophysics_, vol. 5, no. 4, Dec. 1943, pp. 135-137.

LANDAHL, HERBERT D., A Note on the Mathematical Biophysics of Central Excitation and Inhibition, _Bulletin of Mathematical Biophysics_, vol. 7, no. 4, Dec. 1945, pp. 219-221.

LETTVIN, JEROME Y., and WALTER PITTS, A Mathematical Theory of the Affective Psychoses, _Bulletin of Mathematical Biophysics_, vol. 5, no. 4, Dec. 1943, pp. 139-148.

MCCULLOCH, WARREN S., and WALTER PITTS, A Logical Calculus of the Ideas Immanent in Nervous Activity, _Bulletin of Mathematical Biophysics_, vol. 5, no. 4, Dec. 1943, pp. 115-133.

RASHEVSKY, N., _Mathematical Biophysics_, Chicago: University of Chicago Press. Revised edition, 1948, 669 pp.

RASHEVSKY, N., Mathematical Biophysics of Abstraction and Logical Thinking, _Bulletin of Mathematical Biophysics_, vol. 7, no. 3, Sept. 1945, pp. 133-148.

RASHEVSKY, N., Some Remarks on the Boolean Algebra of Nervous Nets in Mathematical Biophysics, _Bulletin of Mathematical Biophysics_, vol. 7, no. 4, Dec. 1945, pp. 203-211.

RASHEVSKY, N., A Suggestion for Another Statistical Interpretation of the Fundamental Equations of the Mathematical Biophysics of the Central Nervous System, _Bulletin of Mathematical Biophysics_, vol. 7, no. 4, Dec. 1945, pp. 223-226.

RASHEVSKY, N., The Neural Mechanism of Logical Thinking, _Bulletin of Mathematical Biophysics_, vol. 8, no. 1, Mar. 1946, pp. 29-40.

LANGUAGES: WORDS AND SYMBOLS FOR THINKING

Hardly any field of techniques for thinking is more fascinating than language. The following list of references, of course, is short; it is meant chiefly as an introduction pointing out a number of different paths into the field of language and languages. Such topics as the following are introduced by the references in this list:

The origin of languages and alphabets. The languages of the world, and speech communities. The comparison of words and structure from language to language. The significance of grammar and syntax. The problem of clear meanings. Writing and speaking that is easy to understand.

BLOOMFIELD, LEONARD, _Language_, New York: Henry Holt & Co., 1933, 564 pp.

BODMER, FREDERICK, and LAUNCELOT HOGBEN, _The Loom of Language_, New York: W. W. Norton & Co., 1944, 692 pp.

FLESCH, RUDOLF, _The Art of Plain Talk_, New York: Harper & Brothers, 1946, 210 pp.

GRAFF, WILLEM L., _Language and Languages: An Introduction to Linguistics_, New York: D. Appleton & Co., 1932, 487 pp.

HAYAKAWA, S. I., _Language in Action_, New York: Harcourt, Brace & Co., 1941, 345 pp.

JESPERSEN, OTTO, _The Philosophy of Grammar_, New York: Henry Holt & Co., 1929 (third printing), 359 pp.

JESPERSEN, OTTO, _Analytic Syntax_,

In this book, by means of a well-contrived system of letters and signs, the great linguistic scholar Jespersen depicts all the important inter-relations of English words and parts of words in connected speech.

OGDEN, C. K., _The System of Basic English_, New York: Harcourt, Brace & Co., 1934, 320 pp.

SCHLAUCH, MARGARET, _The Gift of Tongues_, New York: Modern Age Books, 1942, 342 pp.

WALPOLE, HUGH R., _Semantics: The Nature of Words and Their Meanings_, New York: W. W. Norton & Co., 1941, 264 pp.

LANGUAGES: MACHINES FOR THINKING

For many years, nearly all references about machines as a language for thinking have been specialized and limited. Colleges with scholars who write textbooks usually have not had a variety of expensive and versatile computing machinery. As a result, the main environment for stimulating possible authors has until recently been missing. The list of references is accordingly brief.

AIKEN, HOWARD H., and others, _Proceedings of a Symposium on Large-Scale Digital Calculating Machinery_, Cambridge, Mass.: Harvard University Press, 1948, 302 pp.

COMRIE, JOHN LESLIE, The Application of Commercial Calculating Machines to Scientific Computing, _Mathematical Tables and Other Aids to Computation_, vol. 2, no. 16, Oct. 1946, pp. 149-159.

CREW, E. W., Calculating Machines, _The Engineer_, vol. 172, Dec. 1941, pp. 438-441.

FRY, MACON, _Designing Computing Mechanisms_, Cleveland, Ohio: Penton Publishing Co., 1946, 48 pp. (Reprinted from _Machine Design_, Aug. 1945 through Feb. 1946.)

HARTREE, D. R., _Calculating Machines: Recent and Prospective Developments and Their Impact on Mathematical Physics_, Cambridge, England: The University Press, 1947, 40 pp.

HORSBURGH, E. H., _Modern Instruments and Methods of Calculation_, London: G. Bell and Sons, Ltd., 1914, 343 pp.

LILLEY, S., Mathematical Machines, _Nature_, vol. 149, Apr. 25, 1942, pp. 462-465.

MURRAY, FRANCIS J., _The Theory of Mathematical Machines_, New York: King’s Crown Press, 1947, 116 pp.

The author states that a mathematical machine is a mechanism that provides information concerning the relationships among a specified set of mathematical concepts.

TURCK, J. A. V., _The Origin of Modern Calculating Machines_, Chicago: Western Society of Engineers, 1921.

Recently, however, some magazine and newspaper publishers have seen news value in machines that think, and some good general articles with appeal to a wide audience have appeared. For the references to these articles, see the section of this supplement entitled “Digital Machines—Miscellaneous.”

PUNCH-CARD CALCULATING MACHINES

There are a few general references on punch-card calculating machines:

BAEHNE, G. WALTER, editor, and others, _Practical Applications of the Punched Card Method in Colleges and Universities_, New York: Columbia University Press, 1935, 442 pp.

This is a collection of many contributions from a number of authors, describing various applications, chiefly educational.

ECKERT, W. J., _Punched-Card Methods in Scientific Computation_, New York: Columbia University, The Thomas J. Watson Astronomical Computing Bureau, 1940, 136 pp.

This is a scientific treatise, chiefly relating to the computation of orbits in astronomy.

HARTKEMEIER, HARRY PELLE, _Principles of Punch-Card Machine Operation_ (Subtitle: _How to Operate Punch-Card Tabulating and Alphabetic Accounting Machines_), New York: Thomas Y. Crowell Co., 1942, 269 pp.

This is based on the author’s experience in teaching statistical analysis using IBM tabulators. The book does not deal with the collator or multiplying punch.

HEDLEY, K. J., _The Development of the Punched-Card Method_, Actuarial Society of Australasia, 1946, 20 pp.

INTERNATIONAL BUSINESS MACHINES CORPORATION, _International Business Machines_ (form no. A-4036-6-45), New York: International Business Machines Corporation, 1945, 65 pp.

Pages 6 to 31 show pictures and brief descriptions of about 20 punch-card machines, available in 1945.

SCHNACKEL, H. G., and H. C. LANG, _Accounting by Machine Methods_, New York: Ronald Press Co., 1939, 53 pp.

WOLF, ARTHUR W., and EDMUND C. BERKELEY, _Advanced Course in Punched Card Operations_, Newark, N. J.: Prudential Insurance Company of America, 1942, 98 pp.

A useful and authoritative description of IBM punch-card calculating machinery is the following:

INTERNATIONAL BUSINESS MACHINES CORPORATION, DEPARTMENT OF EDUCATION, _Machine Methods of Accounting_, Endicott, N. Y.: International Business Machines Corporation, 1936-41, 385 pp.

This is a collection of 28 separate booklets telling the detailed operation of IBM punch-card machinery. They were written for employees of IBM and users of IBM equipment. The following list of the booklets is useful in locating them:

NO. OF TITLE FORM NO. DATE PAGES Machine Methods of Accounting—Foreword AM 1936 6 Development of IBM Corporation AM-1-1 1936 14 Principles of the Electric Accounting Machine Method AM-2 1936 12 The Tabulating Card AM-3-1 1936 20 Design of Tabulating Cards AM-4-1 1936 16 Preparation and Use of Codes AM-5 1936 28 Organization and Supervision of the Tabulating Department AM-6 1936 16 Selection and Training of Key Punch Operators AM-7 1936 12 Accounting Control AM-8 1936 8 Punches AM-9 1936 12 Alphabetic Printing Punches AM-10 1936 7 Facts to Know about Key Punches AM-11 1936 4 Verifiers AM-12 1936 4 Gang Punches AM-13 1936 8 Card-Operated Sorting Machines AM-14 1936 12 Facts to Know about Sorters AM-14a 1936 4 Electric Tabulating Machines AM-15 1936 20 Electric Accounting Machines (Type 285 and Type 297) AM-16 1936 16 Alphabetic Direct Subtraction Accounting Machine AM-17 1936 28 Numerical Interpreters AM-18 1936 8 Electric Punched-Card Interpreter (Type 552) AM-18a 1941 8 Reproducing Punches (Type 512) AM-19 1936 16 Automatic Summary Punches for Use with the Numerical Accounting Machines (Type 285-297) AM-20 1936 16 Automatic Summary Punches for Use with the Alphabetic Accounting Machines (Type 405) AM-20a 1940 16 Multiplying Punches AM-21 1936 16 Application of Machines to Accounting Functions AM-22 1936 24 Other International Products AM-23-2 1936 19 The International Automatic Carriage (Type 921) AM-24 1938 15

The Department of Education of IBM has begun a second series of booklets on the principles of operation of punch-card calculating machinery:

INTERNATIONAL BUSINESS MACHINES CORPORATION, DEPARTMENT OF EDUCATION, _Principles of Operation_, Endicott, N. Y.: International Business Machines Corporation, 1942 and later (except for one published in 1939).

Many of the booklets in this series have good examples of machine operation and applications. Also, for the first time, letters and numbers have been used as coordinates to label the hubs on the plugboards. This series includes the following:

NO. OF TITLE FORM NO. DATE PAGES

CARD PUNCHES AND VERIFIERS Card-Punching and Verifying Machines 52-3176-0 1946 21 Alphabetical Verifier, Type 055 52-3295-1 1946 4

INTERPRETERS Card Interpreters, Type 550, 551, and 552 52-3178-0 1946 14

REPRODUCERS Automatic Reproducing Punch, Type 513 52-3180-0 1945 22 End Printing Reproducing Punch, Type 519 52-3292-1 1946 26

Electric Document-Originating Machine, June Type 519 52-3292-2 1948 26

COLLATORS Collator AM-25 1943 31 Collator Counting Device C.R. 9178 1942 12

CALCULATING PUNCHES Electric Multiplier, Type 601 52-3408-1 1947 47 Calculating Punch, Type 602 52-3409-0 1946 83 Calculating Punch, Type 602 52-3409-5 1947 93 Calculating Punch, Type 602-A (Preliminary Manual) 22-5489-0 1948 59 Electronic Multiplier, Type 603 52-3561-0 1946 5 Electronic Calculating Punch, Type 604 22-5279-0 1948 51

TABULATORS Accounting Machine, Type 402 and 403 (Preliminary Manual) 22-5654-0 1949 146 Alphabetical Accounting Machine, Type 404 52-3395-1 1946 96 Typical Applications, Alphabetical Accounting Machine, Type 404, with Multiple Line Printing 22-3771-1 1947 47 Alphabetical Accounting Machine, Type 405 AM 17 (1), 1943 90 Revised 1/1/43 Nov. Alphabetical Accounting Machine, Type 405 52-3179-2 1948 81

AUTOMATIC PRINTING CARRIAGES Bill Feed, Type 920 52-3184-0 1945 21 Form Feeding Device 52-3235-0 1946 11 Automatic Carriage, Type 921 52-3183-0 1945 36 Tape-Controlled Carriage (Preliminary Manual, Revised) 22-5415-1 1948 27

TEST SCORING MACHINE Test Scoring Machine 94-2333-0 1939 19 May Test Scoring Machine 32-9145-1 1946 20 Published Tests Adapted for Use with June the IBM Electric Test Scoring Machine 27-4286-9 1948 8

In addition to the new types of punch-card machines referred to in the above list, an elaborate punch-card calculating machine is described in the following reference:

ECKERT, W. J., The IBM Pluggable Sequence Relay Calculator, _Mathematical Tables and Other Aids to Computation_, vol. 3, no. 23, July 1948, pp. 149-161.

A description of punch-card machinery in rather a light vein is contained in:

ANONYMOUS, Speaking of Pictures: New Mechanical Monsters Ease _Life’s_ Growing Pains, _Life_, Sept. 15, 1947, pp. 15-16.

ANONYMOUS, _540_, Chicago: Time-Life-Fortune Magazine, Subscription Fulfillment Office, 1948, 15 pp.

New types of punch-card machinery are continually coming into use. Among them are: machines that take in punch cards and make punched paper tape (such as teletype tape), and vice versa—useful for transmitting punch-card information over wires; an electric typewriter operated by punch cards—useful for preparing almanacs for sea and air navigation, etc.; a calculator programmed by punch cards, consisting of an assembly of a tabulator, an electronic calculating punch, and an auxiliary storage unit, all cabled together—useful for some types of long calculation; etc. For information about such machinery, the manufacturers may be consulted.

PUNCH-CARD CALCULATING MACHINERY: APPLICATIONS

There are many articles in scientific journals on applications of punch-card calculating machinery to technical problems. The fields of engineering, education, indexing, mathematics, surveying, statistics, and others are all represented in the following list of sample references:

ALT, FRANZ L., Multiplication of Matrices, _Mathematical Tables and Other Aids to Computation_, vol. 2, no. 13, Jan. 1946, pp. 12-13.

BAILEY, C. F., and others, Punch Cards for Indexing Scientific Data, _Science_, vol. 104, Aug. 23, 1946, p. 181.

BOWER, E. C., On Subdividing Tables, _Lick Observatory Bulletin_, vol. 16, no. 455, Nov. 1933, pp. 143-144.

BOWER, E. C., Systematic Subdivision of Tables, _Lick Observatory Bulletin_, vol. 17, no. 467, Apr. 1935, pp. 65-74.

CLEMENCE, G. M., and PAUL HERGET, Optimum-Interval Punched-Card Tables, _Mathematical Tables and Other Aids to Computation_, vol. 1, no. 6, Apr. 1944, pp. 173-176.

CULLEY, FRANK L., Use of Accounting Machines for Mass-Transformation from Geographic to Military-Grid Coordinates, Washington, D. C.: National Research Council, _American Geophysical Union Transactions of 1942_, part 2, pp. 190-197.

DEMING, W. EDWARDS, and MORRIS H. HANSEN, On Some Census Aids to Sampling, _Journal of the American Statistical Association_, vol. 38, no. 225, Sept. 1943, pp. 353-357.

DUNLAP, JACK W., The Computation of Means, Standard Deviations, and Correlations by the Tabulator When the Numbers Are Both Positive and Negative, _Proceedings of the Educational Research Forum_, International Business Machines Corporation, Aug. 1940, pp. 16-19.

DWYER, PAUL S., The Use of Tables in the Form of Prepunched Cards, _Proceedings of the Educational Research Forum_, International Business Machines Corporation, Aug. 1940, pp. 125-127.

DWYER, PAUL S., Summary of Problems in the Computation of Statistical Constants with Tabulating and Sorting Machines, _Proceedings of the Educational Research Forum_, International Business Machines Corporation, Aug. 1940, pp. 20-28.

DWYER, PAUL S., and ALAN D. MEACHAM, The Preparation of Correlation Tables on a Tabulator Equipped with Digit Selection, _Journal of the American Statistical Association_, vol. 32, 1937, pp. 654-662.

DYER, H. S., Making Test Score Data Effective in the Admission and Course Placement of Harvard Freshmen, _Proceedings of the Research Forum_, International Business Machines Corporation, 1946, pp. 55-62.

ECKERT, W. J., and RALPH F. HAUPT, The Printing of Mathematical Tables, _Mathematical Tables and Other Aids to Computation_, vol. 2, no. 17, Jan. 1947, pp. 196-202.

FEINSTEIN, LILLIAN, and MARTIN SCHWARZCHILD, Automatic Integration of Linear Second-Order Differential Equations by Means of Punched-Card Machines, _Review of Scientific Instruments_, vol. 12, no. 8, Aug. 1941, pp. 405-408.

HOTELLING, HAROLD, Some New Methods in Matrix Calculation, _The Annals of Mathematical Statistics_, vol. 14, no. 1, Mar. 1943, pp. 1-34.

INTERNATIONAL BUSINESS MACHINES CORPORATION, editor, and others, _Proceedings of the Educational Research Forum_, Endicott, N. Y.: International Business Machines Corporation, 1941.

INTERNATIONAL BUSINESS MACHINES CORPORATION, editor, and others, _Proceedings of the Research Forum_, Endicott, N. Y.: International Business Machines Corporation, 1946, 94 pp.

KING, GILBERT W., Punched-Card Tables of the Exponential Function, _Review of Scientific Instruments_, vol. 15, no. 12, Dec. 1944, pp. 349-350.

KING, GILBERT W., and GEORGE B. THOMAS, Preparation of Punched-Card Tables of Logarithms, _Review of Scientific Instruments_, vol. 15, no. 12, Dec. 1944, p. 350.

KORMES, MARK, A Note on the Integration of Linear Second-Order Differential Equations by Means of Punched Cards, _Review of Scientific Instruments_, vol. 14, no. 4, Apr. 1943, p. 118.

KORMES, MARK, Numerical Solution of the Boundary Value Problem for the Potential Equation by Means of Punched Cards, _Review of Scientific Instruments_, vol. 14, no. 8, Aug. 1943, pp. 248-250.

KORMES, MARK, and JENNIE P. KORMES, Numerical Solution of Initial Value Problems by Means of Punched-Card Machines, _Review of Scientific Instruments_, vol. 16, no. 1, Jan. 1945, pp. 7-9.

KUDER, G. FREDERIC, Use of the IBM Scoring Machine for Rapid Computation of Tables of Intercorrelations, _Journal of Applied Psychology_, vol. 22, no. 6, Dec. 1938, pp. 587-596.

MAXFIELD, D. K., Library Punched Card Procedures, _Library Journal_, vol. 71, no. 12, June 15, 1946, pp. 902-905 ...

MCLAUGHLIN, KATHLEEN, Adding Machines Nip AEF Epidemics, New York: _New York Times_, Apr. 27, 1945.

MCPHERSON, JOHN C., On Mechanical Tabulation of Polynomials, _Annals of Mathematical Statistics_, Sept. 1941, pp. 317-327.

MCPHERSON, JOHN C., Mathematical Operations with Punched Cards, _Journal of the American Statistical Association_, vol. 37, June 1942, pp. 275-281.

MILLIMAN, WENDELL A., Mechanical Multiplication by the Use of Tabulating Machines, _Transactions of the Actuarial Society of America_, vol. 35, part 2, Oct. 1934, pp. 253-264; for discussion see also vol. 36, part 1, May 1935, pp. 77-84.

ROYER, ELMER B., A Machine Method for Computing the Biserial Correlation Coefficient in Item Validation, _Psychometrika_, vol. 6, no. 1, Feb. 1941, pp. 55-59.

WHITTEN, C. A., Triangulation Adjustment by International Business Machines, Washington, D. C.: National Research Council, _American Geophysical Union Transactions of 1943_, part 1, p. 31.

The following bibliography may be obtained on request to the Watson Scientific Computing Laboratory, Columbia University, 612 West 116 Street, New York 27, N. Y.:

WATSON SCIENTIFIC COMPUTING LABORATORY, _Bibliography: The Use of IBM Machines in Scientific Research, Statistics, and Education_, New York: International Business Machines Corporation (form no. 50-3813-0), Sept. 1947, 25 pp.

The organization and equipment of this laboratory are described in:

ECKERT, W. J., Facilities of the Watson Scientific Computing Laboratory, _Proceedings of the Research Forum_, International Business Machines Corporation, 1946, pp. 75-80.

THE DIFFERENTIAL ANALYZER

The basic scientific articles on the two differential analyzers at Massachusetts Institute of Technology are:

BUSH, VANNEVAR, The Differential Analyzer: A New Machine for Solving Differential Equations, _Journal of the Franklin Institute_, vol. 212, no. 4, Oct. 1931, pp. 447-488.

BUSH, VANNEVAR, and SAMUEL H. CALDWELL, A New Type of Differential Analyzer, _Journal of the Franklin Institute_, vol. 240, no. 4, Oct. 1945, pp. 255-326.

Some of the less technical articles about the second differential analyzer at M.I.T. are:

CALDWELL, SAMUEL H., Educated Machinery, _Technology Review_, vol. 48, no. 1, Nov. 1945, pp. 31-34.

GENET, N., 100-Ton Brain at M.I.T., _Scholastic_, vol. 48, Feb. 4, 1946, p. 36.

ANONYMOUS, Mathematical Machine; New Electronic Differential Analyzer, _Science News Letter_, vol. 48, Nov. 10, 1945, p. 291.

ANONYMOUS, Robot Einstein: Differential Analyzer at M.I.T., _Newsweek_, vol. 26, Nov. 12, 1945, p. 93.

ANONYMOUS, M.I.T.’s 100-Ton Mathematical Brain is Now to Tackle Problems of Peace, _Popular Science_, vol. 148, Jan. 1946, p. 81.

ANONYMOUS, The Great Electro-Mechanical Brain; M.I.T.’s Differential Analyzer, _Life_, vol. 20, Jan. 14, 1946, pp. 73-74 ...

ANONYMOUS, All the Answers at Your Fingertips; in the Laboratory of M.I.T., _Popular Mechanics_, vol. 85, Mar. 1946, pp. 164-167 ...

A differential analyzer was built at the Moore School of Electrical Engineering:

TRAVIS, IRVEN, Differential Analyzer Eliminates Brain Fag, _Machine Design_, July 1935, pp. 15-18.

A differential analyzer was built at the General Electric Company, Schenectady, N. Y. Instead of using a mechanical or electrical amplifier of the motion of the little turning wheel riding on the disc, this machine follows the motion using polarized light. This machine is described in:

BERRY, T. M., Polarized Light Servo System, _Transactions of the American Institute of Electrical Engineers_, vol. 63, Apr. 1944, pp. 195-197.

KUEHNI, H. P., and H. A. PETERSON, A New Differential Analyzer, _Transactions of the American Institute of Electrical Engineers_, vol. 63, May 1944, pp. 221-227.

A differential analyzer has been put into use at the University of California:

BOELTER, L. M. K., and others, _The Differential Analyzer of the University of California_, Los Angeles: University of California, 1947, 25 pp.

A differential analyzer was built at Manchester University, England. It was built first from “Meccano” parts, at a total cost of about 20 pounds, and later refined for more exact work. Some articles dealing with this differential analyzer are:

HARTREE, D. R., The Differential Analyzer, _Nature_, vol. 135, June 8, 1935, p. 940.

HARTREE, D. R., The Mechanical Integration of Differential Equations, _The Mathematical Gazette_, vol. 22, 1938, pp. 342-364.

HARTREE, D. R., and A. PORTER, The Construction of a Model Differential Analyser, _Memoirs and Proceedings of the Manchester Literary and Philosophical Society_, vol. 79, July 1935, pp. 51-72.

Other small scale differential analyzers built in England are covered in:

BEARD, R. E., The Construction of a Small Scale Differential Analyser and Its Application to the Calculation of Actuarial Functions, _Journal of the Institute of Actuaries_, vol. 71, 1942, pp. 193-227.

MASSEY, H. S. W., J. WYLIE, and R. A. BUCKINGHAM, A Small Scale Differential Analyser: Its Construction and Operation, _Proceedings of the Royal Irish Academy_, vol. 45, 1938, pp. 1-21.

A differential analyzer constructed in Germany is briefly described in the following:

SAUER, R., and H. POESCH, Integrating Machine for Solving Ordinary Differential Equations, _Engineers Digest_ (American Edition), vol. 1, May 1944, pp. 326-328.

From the historical point of view there are some interesting papers on a machine for solving differential equations by Sir William Thomson (Lord Kelvin), including one by his brother James Thomson. They are in the _Proceedings of the Royal Society_, vol. 24, Feb. 1876, pp. 262-275. The method of integration by a machine is described, but the state of machine tools at the time was such that no accurate mechanism was constructed. Another interesting paper foreshadowing the differential analyzer is:

WAINWRIGHT, LAWRENCE L., _A Ballistic Engine_, Chicago: University of Chicago, thesis for Master’s Degree, 1923, 28 pp.

Some of the applications and mathematical limitations of differential analyzers are covered in:

BUSH, V., and S. H. CALDWELL, Thomas-Fermi Equation Solution by the Differential Analyzer, _Physical Review_, vol. 38, no. 10, 1931, pp. 1898-1902.

HARTREE, D. R., A Great Calculating Machine: the Bush Differential Analyser and Its Applications in Science and Industry, _Proceedings of the Royal Institution of Great Britain_, vol. 31, 1940, pp. 151-170.

HARTREE, D. R., and A. PORTER, The Application of the Differential Analyser to Transients on a Distortionless Transmission Line, _Journal of the Institute of Electrical Engineering_, vol. 83, no. 503, Nov. 1938, pp. 648-656.

HARTREE, D. R., and J. R. WOMERSLEY, A Method for the Numerical or Mechanical Solution of Certain Types of Partial Differential Equations, _Proceedings of the Royal Society of London_, series A, vol. 161, 1937, pp. 353-366.

MAGINNISS, F. J., Differential Analyzer Applications, _General Electric Review_, vol. 48, no. 5, May 1945, pp. 54-59.

SHANNON, CLAUDE E., Mathematical Theory of the Differential Analyzer, _Journal of Mathematics and Physics_, Cambridge, Mass.: Massachusetts Institute of Technology, vol. 20, no. 4, 1941, pp. 337-354.

HARMONIC ANALYZERS AND SYNTHESIZERS

Another branch of the analogue calculating machine is the harmonic analyzer and synthesizer. These are machines that study wave motions and related physical and mathematical functions. A brief list of articles on this type of machine follows:

ARCHER, R. M., Projecting Apparatus for Compounding Harmonic Vibrations, _Journal of Scientific Instruments_, vol. 14, 1937, pp. 408-410.

BROWN, S. L., A Mechanical Harmonic Synthesizer-Analyzer, _Journal of the Franklin Institute_, vol. 228, 1939, pp. 675-694.

BROWN, S. L., and L. L. WHEELER, A Mechanical Method for Graphical Solution of Polynomials, _Journal of the Franklin Institute_, vol. 231, 1941, pp. 223-243.

BROWN, S. L., and L. L. WHEELER, Use of the Mechanical Multiharmonograph for Graphing Types of Functions and for Solution of Pairs of Non-Linear Simultaneous Equations, _Review of Scientific Instruments_, vol. 13, Nov. 1942, pp. 493-495.

BROWN, S. L., and L. L. WHEELER, The Use of a Mechanical Synthesizer to Solve Trigonometric and Certain Types of Transcendental Equations, and for the Double Summations Involved in Patterson Contours, _Journal of Applied Physics_, vol. 14, Jan. 1943, pp. 30-36.

FÜRTH, R., and R. W. PRINGLE, A New Photo-Electric Method for Fourier Synthesis and Analysis, _London, Edinburgh and Dublin Philosophical Magazine and Journal of Science_, vol. 35, series 7, 1944, pp. 643-656.

INTERNATIONAL HYDROGRAPHIC BUREAU, _Tide Predicting Machines_, International Hydrographic Bureau, Special Publication 13, July 1926.

KRANZ, FREDERICK W., A Mechanical Synthesizer and Analyzer, _Journal of the Franklin Institute_, vol. 204, 1927, pp. 245-262.

MARBLE, F. G., An Automatic Vibration Analyzer, _Bell Laboratories Record_, vol. 22, Apr. 1944, pp. 376-380.

MAXWELL, L. R., An Electrical Method for Compounding Sine Functions, _Review of Scientific Instruments_, vol. 11, Feb. 1940, pp. 47-54.

MILLER, DAYTON C., A 32-Element Harmonic Synthesizer, _Journal of the Franklin Institute_, vol. 181, 1916, pp. 51-81.

MILLER, DAYTON C., The Henrici Harmonic Analyzer and Devices for Extending and Facilitating Its Use, _Journal of the Franklin Institute_, vol. 182, 1916, pp. 285-322.

MILNE, J. R., A “Duplex” Form of Harmonic Synthetiser and Its Mathematical Theory, _Proceedings of the Royal Society of Edinburgh_, vol. 39, 1918-19, pp. 234-242.

MONTGOMERY, H. C., An Optical Harmonic Analyzer, _Bell System Technical Journal_, vol. 17, no. 3, July 1938, pp. 406-415.

RAYMOND, W. J., An Harmonic Synthesizer Having Components of Incommensurable Period and Any Desired Decrement, _Physical Review_, vol. 11, series 2, 1918, pp. 479-481.

ROBERTSON, J. M., A Simple Harmonic Continuous Calculating Machine, _London, Edinburgh and Dublin Philosophical Magazine and Journal of Science_, vol. 13, 1932, pp. 413-419.

SOMERVILLE, J. M., Harmonic Synthesizer for Demonstrating and Studying Complex Wave Forms, _Journal of Scientific Instruments_, vol. 21, Oct. 1944, pp. 174-177.

STRAITON, A. W., and G. K. TERHUNE, Harmonic Analysis by Photographic Method, _Journal of Applied Physics_, vol. 14, 1943, pp. 535-536.

WEGEL, R. L., and C. R. MOORE, An Electrical Frequency Analyzer, _Bell System Technical Journal_, vol. 3, 1924, pp. 299-323.

NETWORK ANALYZERS

A third branch of the analogue calculating machine is the network analyzer. To solve problems, this machine uses the laws governing a network of electrical circuits. For example, an electric power company with a system of power lines over hundreds of miles may have a problem about electrical power: will an accident or a sudden demand cause a breakdown anywhere in the system? In the General Electric Company in Schenectady, N. Y., there is a machine called the A.C. Network Analyzer. All the properties of the power company’s network of lines can be fed on a small scale into the analyzer. Certain dials are turned and certain plugwires are connected. Then various kinds of “accidents” and “sudden demands” are fed into the machine, and the response of the system is noted. The answers given by the machine are multiplied by the proper scale factor, and in this way the problem of the power company is solved.

There are two kinds of problems that network analyzers are built to solve: the steady state conditions and the transient conditions. For example, you may not overload a fuse with an electric iron when it is plugged in and being used, but as you pull out the cord, you may blow the fuse: the steady state does not overstrain the system, but the transient does.

Some articles on network analyzers are:

ENNS, W. E., A New Simple Calculator of Load Flow in A.C. Networks, _Transactions of the American Institute of Electrical Engineers_, vol. 62, 1943, pp. 786-790.

HAZEN, H. L., and others, _The M.I.T. Network Analyzer_, Cambridge, Mass.: Massachusetts Institute of Technology, Department of Electrical Engineering, Serial No. 69, Apr. 1931.

KUEHNI, H. P., and R. G. LORRAINE, A New A.C. Network Analyzer, _Transactions of the American Institute of Electrical Engineers_, vol. 57, 1938, pp. 67-73.

PARKER, W. W., Dual A.C. Network Calculator, _Electrical Engineering_, May 1945, pp. 182-183.

PARKER, W. W., The Modern A.C. Network Calculator, _Transactions of the American Institute of Electrical Engineers_, vol. 60, Nov. 1941, pp. 977-982.

PETERSON, H. A., An Electric Circuit Transient Analyzer, _General Electric Review_, Sept. 1939, pp. 394-400.

VARNEY, R. N., An All-Electric Integrator for Solving Differential Equations, _Review of Scientific Instruments_, vol. 13, Jan. 1942, pp. 10-16.

Some of the articles on applications of network analyzers to various problems are:

KRON, GABRIEL, Equivalent Circuits of the Elastic Field, _Journal of Applied Mechanics_, vol. A11, Sept. 1944, pp. 146-161.

KRON, GABRIEL, Tensorial Analysis and Equivalent Circuits of Elastic Structures, _Journal of the Franklin Institute_, vol. 238, Dec. 1944, pp. 399-442.

KRON, GABRIEL, Numerical Solution of Ordinary and Partial Differential Equations by Means of Equivalent Circuits, _Journal of Applied Physics_, vol. 16, 1945, pp. 172-186.

KRON, GABRIEL, Electric Circuit Models for the Vibration Spectrum of Polyatomic Molecules, _Journal of Chemical Physics_, vol. 14, no. 1, Jan. 1946, pp. 19-31.

KRON, G., and G. K. CARTER, A.C. Network Analyzer Study of the Schrödinger Equation, _Physical Review_, vol. 67, 1945, pp. 44-49.

KRON, G., and G. K. CARTER, Network Analyzer Tests of Equivalent Circuits of Vibrating Polyatomic Molecules, _Journal of Chemical Physics_, vol. 14, no. 1, Jan. 1946, pp. 32-34.

PETERSON, H. A., and C. CONCORDIA, Analyzers for Use in Engineering and Scientific Problems, _General Electric Review_, vol. 48, no. 9, Sept. 1945, pp. 29-37.

MACHINES FOR SOLVING ALGEBRAIC EQUATIONS

Another branch of the analogue calculating machine is a type of machine that will solve various kinds of algebraic equations (see Supplement 2). A list of some articles follows. The article by Mallock describes a machine for solving up to 10 linear simultaneous equations in 10 unknowns, and the article by Wilbur, a machine for solving up to 9.

DIETZOLD, ROBERT L., The Isograph—A Mechanical Root-Finder, _Bell Laboratories Record_, vol. 16, no. 4, Dec. 1937, pp. 130-134.

DUNCAN, W. J., Some Devices for the Solution of Large Sets of Simultaneous Linear Equations, _London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science_, vol. 35, series 7, 1944, pp. 660-670.

FRAME, J. SUTHERLAND, Machines for Solving Algebraic Equations, _Mathematical Tables and Other Aids to Computation_, vol. 1, no. 9, Jan. 1945, pp. 337-353.

HART, H. C., and IRVEN TRAVIS, Mechanical Solution of Algebraic Equations, _Journal of the Franklin Institute_, vol. 225, Jan. 1938, pp. 63-72.

HERR, D. L., and R. S. GRAHAM, An Electrical Algebraic Equation Solver, _Review of Scientific Instruments_, vol. 9, Oct. 1938, pp. 310-315.

MALLOCK, R. R. M., An Electrical Calculating Machine, _Proceedings of the Royal Society_, series A, vol. 140, 1933, pp. 457-483.

MERCNER, R. O., The Mechanism of the Isograph, _Bell Laboratories Record_, vol. 16, no. 4, Dec. 1937, pp. 135-140.

STIBITZ, GEORGE R., Electric Root-finder, _Mathematical Tables and Other Aids to Computation_, vol. 3, no. 24, Oct. 1948, pp. 328-329.

WILBUR, J. B., The Mechanical Solution of Simultaneous Equations, _Journal of the Franklin Institute_, vol. 222, Dec. 1936, pp. 715-724.

ANALOGUE MACHINES—MISCELLANEOUS

Some articles referring to various other kinds of analogue machines and their applications are here listed together:

BUSH, V., F. D. GAGE, and R. R. STEWART, A Continuous Integraph, _Journal of the Franklin Institute_, vol. 203, 1927, pp. 63-84.

GRAY, T. S., A Photo-Electric Integraph, _Journal of the Franklin Institute_, vol. 212, 1931, pp. 77-102.

HAZEN, H. L., G. S. BROWN, and W. R. HEDEMAN, The Cinema Integraph: A Machine for Evaluating a Parametric Product Integral (two parts and appendix), _Journal of the Franklin Institute_, vol. 230, July 1940, pp. 19-44, and Aug. 1940, pp. 183-205.

MCCANN, G. D., and H. E. CRINER, Mechanical Problems Solved Electrically, _Westinghouse Engineer_, vol. 6, no. 2, March 1946, pp. 49-56.

MYERS, D. M., An Integraph for the Solution of Differential Equations of the Second-Order, _Journal of Scientific Instruments_, vol. 16, 1939, pp. 209-222.

PEKERIS, C. L., and W. T. WHITE, Differentiation with the Cinema Integraph, _Journal of the Franklin Institute_, vol. 234, July 1942, pp. 17-29.

SMITH, C. E., and E. L. GOVE, An Electromechanical Calculator for Directional-Antenna Patterns, _Transactions of the American Institute of Electrical Engineers_, vol. 62, 1943, pp. 78-82.

YAVNE, R. O., High Accuracy Contour Cams, _Product Engineering_, vol. 19, part 2, Aug. 1948, 3 pp.

ANONYMOUS, Electrical Gun Director Demonstrated, _Bell Laboratories Record_, vol. 22, no. 4, Dec. 1943, pp. 157-167.

ANONYMOUS, Development of the Electric Director, _Bell Laboratories Record_, vol. 22, no. 5, Jan. 1944, pp. 225-230.

ANONYMOUS, Old Field Fortune Teller: Electronic Oil Pool Analyzer, _Popular Mechanics_, vol. 86, Sept. 1946, p. 154.

HARVARD IBM AUTOMATIC SEQUENCE-CONTROLLED CALCULATOR

The basic scientific description of this machine as of September 1, 1945, is contained in:

AIKEN, HOWARD H., and STAFF OF THE COMPUTATION LABORATORY, _A Manual of Operation for the Automatic Sequence-Controlled Calculator_, Cambridge, Mass.: Harvard University Press, 1946, 561 pp.

The machine has changed rather a good deal since Sept. 1, 1945. Some circuits have been removed. Other circuits have been added. The capacity of the machine to do problems has been greatly increased. The Computation Laboratory at Harvard University is cordial towards scientific inquiries, and some unpublished, mimeographed information is available at the laboratory regarding the details of these changes.

Some shorter scientific and technical descriptions of the machine are contained in:

AIKEN, HOWARD H., and GRACE M. HOPPER, The Automatic Sequence Controlled Calculator (3 parts), _Electrical Engineering_, vol. 65, nos. 8, 9, and 10, Aug. to Nov. 1946, p. 384 ... (21 pp.).

BLOCH, RICHARD M., Mark I Calculator, _Proceedings of a Symposium on Large-Scale Digital Calculating Machinery_, Harvard University Press, 1948, pp. 23-30.

HARRISON, JOSEPH O., JR., The Preparation of Problems for the Mark I Calculator, _Proceedings of a Symposium on Large-Scale Digital Calculating Machinery_, Harvard University Press, 1948, pp. 208-210.

INTERNATIONAL BUSINESS MACHINES CORPORATION, _IBM Automatic Sequence-Controlled Calculator_, Endicott, N. Y.: International Business Machines Corporation, 1945, 6 pp.

Some of the less technical articles regarding the machine are:

GENET, N., Got a Problem? Harvard’s Amazing New Mathematical Robot, _Scholastic_, vol. 45, Sept. 18, 1944, p. 35.

TORREY, V., Robot Mathematician Knows All the Answers, _Popular Science_, vol. 145, Oct. 1944, pp. 86-89....

ANONYMOUS, Giant New Calculator, _Science News Letter_, vol. 46, Aug. 12, 1944, p. 111.

ANONYMOUS, Mathematical Robot Presented to Harvard, _Time_, vol. 44, Aug. 14, 1944, p. 72.

ANONYMOUS, World’s Greatest Machine for Automatic Calculation, _Science News Letter_, vol. 46, Aug. 19, 1944, p. 123.

ANONYMOUS, Superbrain, _Nation’s Business_, vol. 32, Sept. 1944, p. 8.

ANONYMOUS, Robot Works Problems Never Before Solved, _Popular Mechanics_, vol. 82, Oct. 1944, p. 13.

ENIAC, THE ELECTRONIC NUMERIC INTEGRATOR AND CALCULATOR

There is as yet no full-scale, published scientific account of the Eniac. At the Ballistic Research Laboratories, Aberdeen, Md., where the machine now is, there are a few copies of some long mimeographed reports on the machine and the way it works. These were prepared by H. H. Goldstine and others when at the Moore School of Electrical Engineering, as a part of the contract under which the machine was constructed for the U. S. Government. It is possible that these reports might be consulted on request by serious students.

Some scientific descriptions of the machine and its properties are:

BURKS, ARTHUR W., Electronic Computing Circuits of the ENIAC, _Proceedings of the Institute of Radio Engineers_, vol. 35, no. 8, Aug. 1947, pp. 756-767.

CLIPPINGER, R. F., _A Logical Coding System Applied to the Eniac_, B. R. L. Report No. 673, Aberdeen, Md.: Ballistic Research Laboratories, Sept. 29, 1948, 41 pp.

ECKERT, J. PRESPER, JR., JOHN W. MAUCHLY, HERMAN H. GOLDSTINE, and J. G. BRAINERD, Description of the ENIAC and Comments on Electronic Digital Computing Machines, Applied Mathematics Panel Report 171.2R, Washington, D. C.: National Defense Research Committee, Nov. 1945, 78 pp.

GOLDSTINE, HERMAN H., and ADELE GOLDSTINE, The Electronic Numerical Integrator and Computer (ENIAC), _Mathematical Tables and Other Aids to Computation_, vol. 2, no. 15, July 1946, pp. 97-110.

HARTREE, D. R., The ENIAC, an Electronic Computing Machine, _Nature_, vol. 158, Oct. 12, 1946, pp. 500-506.

HARTREE, D. R., _Calculating Machines: Recent and Prospective Developments and Their Impact on Mathematical Physics_, Cambridge, England: The University Press, 1947, 40 pp. (Pages 14 to 27 are devoted to the Eniac.)

TABOR, LEWIS P., Brief Description and Operating Characteristics of the ENIAC, _Proceedings of a Symposium on Large-Scale Digital Calculating Machinery_, Harvard University Press, 1948, pp. 31-39.

Some of the less technical articles on Eniac are:

ROSE, A., Lightning Strikes Mathematics: ENIAC, _Popular Science_, vol. 148, Apr. 1946, pp. 83-86.

ANONYMOUS, Robot Calculator: ENIAC, All Electronic Device, _Business Week_, Feb. 16, 1946, p. 50 ...

ANONYMOUS, Answers by ENY: Electronic Numerical Integrator and Computer, ENIAC, _Newsweek_, vol. 27, Feb. 18, 1946, p. 76.

ANONYMOUS, Adds in ¹/₅₀₀₀ Second: Electronic Computing Machine at the University of Pennsylvania, _Science News Letter_, vol. 49, Feb. 23, 1946, p. 113 ...

ANONYMOUS, ENIAC: at the University of Pennsylvania, _Time_, vol. 47, Feb. 25, 1946, p. 90.

ANONYMOUS, It Thinks with Electrons; the ENIAC, _Popular Mechanics_, vol. 85, June 1946, p. 139.

ANONYMOUS, Electronic Calculator: ENIAC, _Scientific American_, vol. 174, June 1946, p. 248.

BELL LABORATORIES RELAY COMPUTERS

As yet no full-scale scientific report is available on the Bell Laboratories general-purpose relay computers that went to Aberdeen and Langley Field. However, there is some information about these and other Bell Laboratories relay computing machines in the following articles:

ALT, FRANZ L., A Bell Telephone Laboratories’ Computing Machine (two parts), _Mathematical Tables and Other Aids to Computation_, vol. 3, no. 21, Jan. 1948, pp. 1-13, and vol. 3, no. 22, Apr. 1948, pp. 69-84.

CESAREO, O., The Relay Interpolator, _Bell Laboratories Record_, vol. 24, no. 12, Dec. 1946, pp. 457-460.

JULEY, JOSEPH, The Ballistic Computer, _Bell Laboratories Record_, vol. 25, no. 1, Jan. 1947, pp. 5-9.

WILLIAMS, SAMUEL B., A Relay Computer for General Application, _Bell Laboratories Record_, vol. 25, no. 2, Feb. 1947, pp. 49-54.

WILLIAMS, SAMUEL B., Bell Telephone Laboratories’ Relay Computing System, _Proceedings of a Symposium on Large-Scale Digital Calculating Machinery_, Harvard University Press, 1948, pp. 40-68.

ANONYMOUS, Complex Computer Demonstrated, _Bell Laboratories Record_, vol. 19, no. 2, Oct. 1940, pp. v-vi.

ANONYMOUS, _Computer Mark 22 Mod. 0: Development and Description_, Navord Report No. 178-45, Washington, D. C.: Navy Department, Dec. 6, 1945, 225 pp.

ANONYMOUS, Relay Computer for the Army, _Bell Laboratories Record_, vol. 26, no. 5, May 1948, pp. 208-209.

THE KALIN-BURKHART LOGICAL-TRUTH CALCULATOR

As yet there are no published references on the Kalin-Burkhart Logical-Truth Calculator.

Some books covering a good deal of mathematical logic are:

QUINE, W. V., _Mathematical Logic_, New York: W. W. Norton & Co., 1940, 348 pp.

REICHENBACH, HANS, _Elements of Symbolic Logic_, New York: The Macmillan Co., 1947, 444 pp.

TARSKI, ALFRED, _Introduction to Logic_, New York: Oxford University Press, 1941, 239 pp.

WOODGER, J. H., _The Axiomatic Method in Biology_, Cambridge, England: The University Press, 1937, 174 pp.