Chapter XVII: Conclusion (1)

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: