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these perform the whole operation without any mental attention when
once the given numbers have been put into the machine.

Others require a moderate portion of mental attention: these latter are
generally of much simpler construction than the former, and it may also
be added, are less useful.

The simplest way of deciding to which of these two classes any
calculating machine belongs is to ask its maker—Whether, when the
numbers on which it is to operate are placed in the instrument, it is
capable of arriving at its result by the mere motion of a spring, a
descending weight, or any other constant force? If the answer be in the
affirmative, the machine is really automatic; if otherwise, it is not

Of the various machines I have had occasion to examine, many of those
for Addition and Subtraction have been found {42} to be automatic.
Of machines for Multiplication and Division, which have fully come
under my examination, I cannot at present recall one to my memory as
absolutely fulfilling this condition.


The earliest idea that I can trace in my own mind of calculating
arithmetical Tables by machinery arose in this manner:—

One evening I was sitting in the rooms of the Analytical Society, at
Cambridge, my head leaning forward on the Table in a kind of dreamy
mood, with a Table of logarithms lying open before me. Another member,
coming into the room, and seeing me half asleep, called out, “Well,
Babbage, what are you dreaming about?” to which I replied, “I am
thinking that all these Tables (pointing to the logarithms) might be
calculated by machinery.”

I am indebted to my friend, the Rev. Dr. Robinson, the Master of the
Temple, for this anecdote. The event must have happened either in 1812
or 1813.

About 1819 I was occupied with devising means for accurately dividing
astronomical instruments, and had arrived at a plan which I thought was
likely to succeed perfectly. I had also at that time been speculating
about making machinery to compute arithmetical Tables.

One morning I called upon the late Dr. Wollaston, to consult him about
my plan for dividing instruments. On talking over the matter, it turned
out that my system was exactly that which had been described by the
Duke de Chaulnes, in the Memoirs of the French Academy of Sciences,
about fifty or sixty years before. I then mentioned my other idea of
computing Tables by machinery, which Dr. Wollaston thought a more
promising subject.

I considered that a machine to execute the mere isolated {43}
operations of arithmetic, would be comparatively of little value,
unless it were very easily set to do its work, and unless it executed
not only accurately, but with great rapidity, whatever it was required
to do.


On the other hand, the method of differences supplied a general
principle by which _all_ Tables might be computed through limited
intervals, by one uniform process. Again, the method of differences
required the use of mechanism for Addition only. In order, however,
to insure accuracy in the printed Tables, it was necessary that the
machine which computed Tables should also set them up in type, or else
supply a mould in which stereotype plates of those Tables could be cast.

I now began to sketch out arrangements for accomplishing the several
partial processes which were required. The arithmetical part must
consist of two distinct processes—the power of adding one digit to
another, and also of carrying the tens to the next digit, if it should
be necessary.

The first idea was, naturally, to add each digit successively. This,
however, would occupy much time if the numbers added together consisted
of many places of figures.

The next step was to add all the digits of the two numbers each to each
at the same instant, but reserving a certain mechanical memorandum,
wherever a carriage became due. These carriages were then to be
executed successively.

Having made various drawings, I now began to make models of some
portions of the machine, to see how they would act. Each number was to
be expressed upon wheels placed upon an axis; there being one wheel for
each figure in the number operated upon.

Having arrived at a certain point in my progress, it became necessary
to have teeth of a peculiar form cut upon these {44} wheels. As my own
lathe was not fit for this job, I took the wheels to a wheel-cutter at
Lambeth, to whom I carefully conveyed my instructions, leaving with him
a drawing as his guide.


These wheels arrived late one night, and the next morning I began
putting them in action with my other mechanism, when, to my utter
astonishment, I found they were quite unfit for their task. I examined
the shape of their teeth, compared them with those in the drawings, and
found they agreed perfectly; yet they could not perform their intended
work. I had been so certain of the truth of my previous reasoning, that
I now began to be somewhat uneasy. I reflected that, if the reasoning
about which I had been so certain should prove to have been really
fallacious, I could then no longer trust the power of my own reason.
I therefore went over with my wheels to the artist who had formed
the teeth, in order that I might arrive at some explanation of this
extraordinary contradiction.

On conferring with him, it turned out that, when he had understood
fully the peculiar form of the teeth of wheels, he discovered that his
wheel-cutting engine had not got amongst its divisions that precise
number which I had required. He therefore had asked me whether another
number, which his machine possessed, would not equally answer my
object. I had inadvertently replied in the affirmative. He then made
arrangements for the precise number of teeth I required; and the new
wheels performed their expected duty perfectly.

The next step was to devise means for printing the tables to be
computed by this machine. My first plan was to make it put together
moveable type. I proposed to make metal boxes, each containing 3,000
types of one of the ten digits. These types were to be made to pass out
one by one from the {45} bottom of their boxes, when required by the
computing part of the machine.


But here a new difficulty arose. The attendant who put the types into
the boxes might, by mistake, put a wrong type in one or more of them.
This cause of error I removed in the following manner:—There are
usually certain notches in the side of the type. I caused these notches
to be so placed that all the types of any given digit possessed the
same characteristic notches, which no other type had. Thus, when the
boxes were filled, by passing a small wire down these peculiar notches,
it would be impeded in its passage, if there were included in the row a
single wrong figure. Also, if any digit were accidentally turned upside
down, it would be indicated by the stoppage of the testing wire.

One notch was reserved as common to every species of type. The object
of this was that, before the types which the Difference Engine had
used for its computation were removed from the iron platform on which
they were placed, a steel wire should be passed through this common
notch, and remain there. The tables, composed of moveable types,
thus interlocked, could never have any of their figures drawn out by
adhesion to the inking-roller, and then by possibility be restored in
an inverted order. A small block of such figures tied together by a bit
of string, remained unbroken for several years, although it was rather
roughly used as a plaything by my children. One such box was finished,
and delivered its type satisfactorily.

Another plan for printing the tables, was to place the ordinary
printing type round the edges of wheels. Then, as each successive
number was produced by the arithmetical part, the type-wheels would
move down upon a plate of soft composition, upon which the tabular
number would be {46} impressed. This mould was formed of a mixture
of plaster-of-Paris with other materials, so as to become hard in the
course of a few hours.


The first difficulty arose from the impression of one tabular number on
the mould being distorted by the succeeding one.

I was not then aware that a very slight depth of impression from
the type would be quite sufficient. I surmounted the difficulty
by previously passing a roller, having longitudinal wedge-shaped
projections, over the plastic material. This formed a series of small
depressions in the matrix between each line. Thus the expansion
arising from the impression of one line partially filled up the small
depression or ditch which occurred between each successive line.

The various minute difficulties of this kind were successively
overcome; but subsequent experience has proved that the depth necessary
for stereotype moulds is very small, and that even thick paper,
prepared in a peculiar manner, is quite sufficient for the purpose.

Another series of experiments were, however, made for the purpose of
punching the computed numbers upon copper plate. A special machine was
contrived and constructed, which might be called a co-ordinate machine,
because it moved the copper plate and steel punches in the direction
of three rectangular co-ordinates. This machine was afterwards found
very useful for many other purposes. It was, in fact, a general shaping
machine, upon which many parts of the Difference Engine were formed.

Several specimens of surface and copper-plate printing, as well as
of the copper plates, produced by these means, were exhibited at the
Exhibition of 1862.

I have proposed and drawn various machines for the purpose of
calculating a series of numbers forming Tables {47} by means of a
certain system called “The Method of Differences,” which it is the
object of this sketch to explain.

The first Difference Engine with which I am acquainted comprised a
few figures, and was made by myself, between 1820 and June 1822. It
consisted of from six to eight figures. A much larger and more perfect
engine was subsequently commenced in 1823 for the Government.

It was proposed that this latter Difference Engine should have six
orders of differences, each consisting of about twenty places of
figures, and also that it should print the Tables it computed.

The small portion of it which was placed in the International
Exhibition of 1862 was put together nearly thirty years ago. It was
accompanied by various parts intended to enable it to print the results
it calculated, either as a single copy on paper—or by putting together
moveable types—or by stereotype plates taken from moulds punched by the
machine—or from copper plates impressed by it. The parts necessary for
the execution of each of these processes were made, but these were not
at that time attached to the calculating part of the machine.

A considerable number of the parts by which the printing was to be
accomplished, as also several specimens of portions of tables punched
on copper, and of stereotype moulds, were exhibited in a glass case
adjacent to the Engine.


In 1834 Dr. Lardner published, in the ‘Edinburgh Review,’[12] a very
elaborate description of this portion of the machine, in which he
explained clearly the method of Differences.

[12] ‘Edinburgh Review,’ No. cxx., July, 1834.

It is very singular that two persons, one resident in London, the
other in Sweden, should both have been struck, on reading this review,
with the simplicity of the mathematical principle {48} of differences
as applied to the calculation of Tables, and should have been so
fascinated with it as to have undertaken to construct a machine of the


Mr. Deacon, of Beaufort House, Strand, whose mechanical skill is well
known, made, for his own satisfaction, a small model of the calculating
part of such a machine, which was shown only to a few friends, and of
the existence of which I was not aware until after the Swedish machine
was brought to London.

Mr. Scheutz, an eminent printer at Stockholm, had far greater
difficulties to encounter. The construction of mechanism, as well as
the mathematical part of the question, was entirely new to him. He,
however, undertook to make a machine having four differences, and
fourteen places of figures, and capable of printing its own Tables.

After many years’ indefatigable labour, and an almost ruinous expense,
aided by grants from his Government, by the constant assistance
of his son, and by the support of many enlightened members of the
Swedish Academy, he completed his Difference Engine. It was brought
to London, and some time afterwards exhibited at the great Exhibition
at Paris. It was then purchased for the Dudley Observatory at Albany
by an enlightened and public-spirited merchant of that city, John F.
Rathbone, Esq.

An exact copy of this machine was made by Messrs. Donkin and Co., for
the English Government, and is now in use in the Registrar-General’s
Department at Somerset House. It is very much to be regretted that this
specimen of English workmanship was not exhibited in the International
Exhibition. {49}

_Explanation of the Difference Engine._

Those who are only familiar with ordinary arithmetic may, by following
out with the pen some of the examples which will be given, easily
make themselves acquainted with the simple principles on which the
Difference Engine acts.


It is necessary to state distinctly at the outset, that the Difference
Engine is not intended to answer special questions. Its object is to
calculate and print a _series_ of results formed according to given
laws. These are called Tables—many such are in use in various trades.
For example—there are collections of Tables of the amount of any number
of pounds from 1 to 100 lbs. of butchers’ meat at various prices per
lb. Let us examine one of these Tables: viz.—the price of meat 5 _d._
per lb., we find

Number. Table.
Lbs. Price.
_s._ _d._
1 0 5
2 0 10
3 1 3
4 1 8
5 2 1

There are two ways of computing this Table:—

1st. We might have multiplied the number of lbs. in each line by 5, the
price per lb., and have put down the result in _l._ _s._ _d._, as in
the 2nd column: or,

2nd. We might have put down the price of 1 lb., which is 5 _d._, and
have added five pence for each succeeding lb.

Let us now examine the relative advantages of each plan. We shall find
that if we had multiplied each number of lbs. in {50} the Table by
5, and put down the resulting amount, then every number in the Table
would have been computed independently. If, therefore, an error had
been committed, it would not have affected any but the single tabular
number at which it had been made. On the other hand, if a single error
had occurred in the system of computing by adding five at each step,
any such error would have rendered the whole of the rest of the Table


Thus the system of calculating by differences, which is the easiest,
is much more liable to error. It has, on the other hand, this great
advantage: viz., that when the Table has been so computed, if we
calculate its last term directly, and if it agree with the last term
found by the continual addition of 5, we shall then be quite certain
that every term throughout is correct. In the system of computing each
term directly, we possess no such check upon our accuracy.

Now the Table we have been considering is, in fact, merely a Table
whose first difference is constant and equal to five. If we express it
in pence it becomes—

Table. 1st Difference.
1 5 5
2 10 5
3 15 5
4 20 5
5 25

Any machine, therefore, which could add one number to another, and at
the same time retain the original number called the first difference
for the next operation, would be able to compute all such Tables.


Let us now consider another form of Table which might readily occur to
a boy playing with his marbles, or to a young lady with the balls of
her solitaire board. {51}

The boy may place a row of his marbles on the sand, at equal distances
from each other, thus—


He might then, beginning with the second, place two other marbles under
each, thus—


He might then, beginning with the third, place three other marbles
under each group, and so on; commencing always one group later, and
making the addition one marble more each time. The several groups would
stand thus arranged—


He will not fail to observe that he has thus formed a series of
triangular groups, every group having an equal number of marbles in
each of its three sides. Also that the side of each successive group
contains one more marble than that of its preceding group.

Now an inquisitive boy would naturally count the numbers in each group
and he would find them thus—

1 3 6 10 15 21

He might also want to know how many marbles the thirtieth or any other
distant group might contain. Perhaps he might go to papa to obtain
this information; but I much fear papa would snub him, and would tell
him that it was nonsense—that it was useless—that nobody knew the
number, and so forth. If the boy is told by papa, that he is not able
to answer the question, then I recommend him to pay careful attention
to whatever that father may at any time say, for he has overcome two of
the greatest obstacles to the acquisition {52} of knowledge—inasmuch
as he possesses the consciousness that he does not know—and he has the
moral courage to avow it.[13]

[13] The most remarkable instance I ever met with of the
distinctness with which any individual perceived the exact
boundary of his own knowledge, was that of the late Dr. Wollaston.

If papa fail to inform him, let him go to mamma, who will not fail
to find means to satisfy her darling’s curiosity. In the meantime
the author of this sketch will endeavour to lead his young friend to
make use of his own common sense for the purpose of becoming better
acquainted with the triangular figures he has formed with his marbles.


In the case of the Table of the price of butchers’ meat, it was obvious
that it could be formed by adding the same _constant_ difference
continually to the first term. Now suppose we place the numbers of
our groups of marbles in a column, as we did our prices of various
weights of meat. Instead of adding a certain difference, as we did in
the former case, let us subtract the figures representing each group
of marbles from the figures of the succeeding group in the Table. The
process will stand thus:—

Table. 1st Difference. 2nd Difference.

Number of Number of Marbles Difference between the
the Group. in each Group. number of Marbles in
each Group and that
in the next.

1 1 1 1
2 3 2 1
3 6 3 1
4 10 4 1
5 15 5 1
6 21 6
7 28 7

It is usual to call the third column thus formed _the column of_ {53}
_first differences_. It is evident in the present instance that that
column represents the natural numbers. But we already know that the
first difference of the natural numbers is constant and equal to unity.
It appears, therefore, that a Table of these numbers, representing
the group of marbles, might be constructed to any extent by mere
addition—using the number 1 as the first number of the Table, the
number 1 as the first Difference, and also the number 1 as the second
Difference, which last always remains constant.

Now as we could find the value of any given number of pounds of meat
directly, without going through all the previous part of the Table, so
by a somewhat different rule we can find at once the value of any group
whose number is given.

Thus, if we require the number of marbles in the fifth group, proceed

Take the number of the group 5
Add 1 to this number, it becomes 6
Multiply these numbers together 2)30
Divide the product by 2 15

This gives 15, the number of marbles in the 5th group.

If the reader will take the trouble to calculate with his pencil the
five groups given above, he will soon perceive the general truth of
this rule.

We have now arrived at the fact that this Table—like that of the price
of butchers’ meat—can be calculated by two different methods. By the
first, each number of the Table is calculated independently: by the
second, the truth of each number depends upon the truth of all the
previous numbers.


Perhaps my young friend may now ask me, What is the use of such Tables?
Until he has advanced further in his {54} arithmetical studies, he
must take for granted that they are of some use. The very Table about
which he has been reasoning possesses a special name—it is called a
Table of Triangular Numbers. Almost every general collection of Tables
hitherto published contains portions of it of more or less extent.

Above a century ago, a volume in small quarto, containing the first
20,000 triangular numbers, was published at the Hague by E. De
Joncourt, A.M., and Professor of Philosophy.[14] I cannot resist
quoting the author’s enthusiastic expression of the happiness he
enjoyed in composing his celebrated work:

[14] ‘On the Nature and Notable Use of the most Simple Trigonal
Numbers.’ By E. De Joncourt, at the Hague. 1762.

“The Trigonals here to be found, and nowhere else, are exactly
elaborate. Let the candid reader make the best of these numbers, and
feel (if possible) in perusing my work the pleasure I had in composing

“That sweet joy may arise from such contemplations cannot be denied.
Numbers and lines have many charms, unseen by vulgar eyes, and only
discovered to the unwearied and respectful sons of Art. In features the
serpentine line (who starts not at the name) produces beauty and love;
and in numbers, high powers, and humble roots, give soft delight.

“Lo! the raptured arithmetician! Easily satisfied, he asks no Brussels
lace, nor a coach and six. To calculate, contents his liveliest
desires, and obedient numbers are within his reach.”


I hope my young friend is acquainted with the fact—that the product of
any number multiplied by itself is called the square of that number.
Thus 36 is the product of 6 multiplied by 6, and 36 is called the
square of 6. I would now recommend him to examine the series of square

1, 4, 9, 16, 25, 36, 49, 64, &c.,

{55} and to make, for his own instruction, the series of their first
and second differences, and then to apply to it the same reasoning
which has been already applied to the Table of Triangular Numbers.


When he feels that he has mastered that Table, I shall be happy to
accompany mamma’s darling to Woolwich or to Portsmouth, where he will
find some practical illustrations of the use of his newly-acquired
numbers. He will find scattered about in the Arsenal various heaps of
cannon balls, some of them triangular, others square or oblong pyramids.

Looking on the simplest form—the triangular pyramid—he will observe
that it exactly represents his own heaps of marbles placed each
successively above one another until the top of the pyramid contains
only a single ball.

The new series thus formed by the addition of his own triangular
numbers is—

Number. Table. 1st Difference. 2nd Difference. 3rd Difference.
1 1 3 3 1
2 4 6 4 1
3 10 10 5 1
4 20 15 6

Online LibraryCharles BabbagePassages from the Life of a Philosopher → online text (page 4 of 36)