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).

[Illustration: FIG. 1. Abacus and notations.]

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: