DIFFERENCE ENGINE NO. 1.
“Oh no! we never mention it,
Its name is never heard.”
Difference Engine No. 1 — First Idea at Cambridge, 1812 — Plan for
Dividing Astronomical Instruments — Idea of a Machine to calculate
Tables by Differences — Illustrations by Piles of Cannon-balls.
Calculating Machines comprise various pieces of mechanism for assisting
the human mind in executing the operations of arithmetic. Some few of
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
self-acting.
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.
〈ORIGIN OF DIFFERENCE ENGINE.〉
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.
〈ADDITION AND CARRIAGE.〉
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.
〈UNEXPECTED DIFFICULTY EXPLAINED.〉
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.
〈VERIFICATION OF TYPE.〉
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.
〈MOULDS AND COPPER-PLATE.〉
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.
〈‘EDINBURGH REVIEW.’〉
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
kind.
〈MR. DEACON—MR. SCHEUTZ.〉
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.
〈ARITHMETICAL TABLES.〉
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
untrue.
〈DIFFERENCES.〉
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.
〈GROUPS OF MARBLES.〉
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—
[Illustration]
He might then, beginning with the second, place two other marbles under
each, thus—
[Illustration]
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—
[Illustration]
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.
〈SECOND DIFFERENCE CONSTANT.〉
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
thus:—
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.
〈TRIANGULAR 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
it.
“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.”
〈SQUARE NUMBERS.〉
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
numbers
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.
〈CANNON BALLS.〉
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
5 35 21
6 56
He will at once perceive that this Table of the number of cannon balls
contained in a triangular pyramid can be carried to any extent by
simply adding successive differences, the third of which is constant.
The next step will naturally be to inquire how any number in this Table
can be calculated by itself. A little consideration will lead him to a
fair guess; a little industry will enable him to confirm his conjecture.
It will be observed at p. 49 that in order to find {56} independently
any number of the Table of the price of butchers’ meat, the following
rule was observed:—
〈NUMBER IN EACH PILE.〉
Take the number whose tabular number is required.
Multiply it by the first difference.
This product is equal to the required tabular number.
Again, at p. 53, the rule for finding any triangular number was:—
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 is the number of marbles in the 5th group.
Now let us make a bold conjecture respecting the Table of cannon balls,
and try this rule:—
Take the number whose tabular number is
required, say 5
Add 1 to that number 6
Add 1 more to that number 7
─────
Multiply all three numbers together 2)210
─────
Divide by 2 105
─────
The real number in the 5th pyramid is 35. But the number 105 at which
we have arrived is exactly three times as great. If, therefore, instead
of dividing by 2 we had divided by 2 and also by 3, we should have
arrived at a true result in this instance.
The amended rule is therefore— {57}
Take the number whose tabular number is
required, say _n_
Add 1 to it _n_ + 1
Add 1 to this _n_ + 2
Multiply these three numbers
together _n_ × (_n_ + 1) × (_n_ + 2)
Divide by 1 × 2 × 3.
The result is (_n_(_n_ + 1)(_n_ + 2))/6
This rule will, upon trial, be found to give correctly every tabular
number.
By similar reasoning we might arrive at the knowledge of the number of
cannon balls in square and rectangular pyramids. But it is presumed
that enough has been stated to enable the reader to form some general
notion of the method of calculating arithmetical Tables by differences
which are constant.
〈ASTRONOMICAL TABLES.〉
It may now be stated that mathematicians have discovered that all the
Tables most important for practical purposes, such as those relating
to Astronomy and Navigation, can, although they may not possess any
constant differences, still be calculated in detached portions by that
method.
Hence the importance of having machinery to calculate by differences,
which, if well made, cannot err; and which, if carelessly set, presents
in the last term it calculates the power of verification of every
antecedent term.
_Of the Mechanical Arrangements necessary for computing Tables by the
Method of Differences._
From the preceding explanation it appears that all Tables may be
calculated, to a greater or less extent, by the method of Differences.
That method requires, for its successful {58} execution, little beyond
mechanical means of performing the arithmetical operation of Addition.
Subtraction can, by the aid of a well-known artifice, be converted into
Addition.
〈ADDITION.〉
The process of Addition includes two distinct parts—1st. The first
consists of the addition of any one digit to another digit; 2nd. The
second consists in carrying the tens to the next digit above.
Let us take the case of the addition of the two following numbers, in
which no carriages occur:—
6023
1970
────
7993
It will be observed that, in making this addition, the mind acts by
successive steps. The person adding says to himself—
0 and 3 make three,
7 and 2 make nine,
9 and 0 make nine,
1 and 6 make seven.
〈CARRIAGE.〉
In the following addition there are several carriages:—
2648
4564
────
7212
The person adding says to himself—
4 and 8 make 12: put down 2 and carry one.
1 and 6 are 7 and 4 make 11: put down 1 and carry one.
1 and 5 are 6 and 6 make 12: put down 2 and carry one.
1 and 4 are 5 and 2 make 7: put down 7.
Now, the length of time required for adding one number to another is
mainly dependent upon the number of figures to {59} be added. If we
could tell the average time required by the mind to add two figures
together, the time required for adding any given number of figures to
another equal number would be found by multiplying that average time by
the number of digits in either number.
When we attempt to perform such additions by machinery we might follow
exactly the usual process of the human mind. In that case we might take
a series of wheels, each having marked on its edges the digits 0, 1, 2,
3, 4, 5, 6, 7, 8, 9. These wheels might be placed above each other upon
an axis. The lowest would indicate the units’ figure, the next above
the tens, and so on, as in the Difference Engine at the Exhibition, a
woodcut of which faces the title-page.
Several such axes, with their figure wheels, might be placed around a
system of central wheels, with which the wheels of any one or more axes
might at times be made to gear. Thus the figures on any one axis might,
by means of those central wheels, be added to the figure wheels of any
other axis.
But it may fairly be expected, and it is indeed of great importance
that calculations made by machinery should not merely be exact, but
that they should be done in a much shorter time than those performed by
the human mind. Suppose there were no tens to carry, as in the first
of the two cases; then, if we possessed mechanism capable of adding
any one digit to any other in the units’ place of figures, a similar
mechanism might be placed above it to add the tens’ figures, and so on
for as many figures as might be required.
But in this case, since there are no carriages, each digit might be
added to its corresponding digit at the same time. Thus, the time of
adding by means of mechanism, any two numbers, however many figures
they might consist of, would {60} not exceed that of adding a single
digit to another digit. If this could be accomplished it would render
additions and subtractions with numbers having ten, twenty, fifty, or
any number of figures, as rapid as those operations are with single
figures.
〈SUCCESSIVE CARRIAGE.〉
Let us now examine the case in which there were several carriages. Its
successive stages may be better explained, thus—
2648
4584
Stages. ────
1 Add units’ figure = 4 2642
2 Carry 1
────
2652
3 Add tens’ figure = 8 8
2632
4 Carry 1
────
2732
5 Add hundreds’ figure = 5 5
────
2232
6 Carry 1
────
3232
7 Add thousands’ figure = 4 4
────
7232
8 Carry 0. There is no carr.
Now if, as in this case, all the carriages were known, it would then be
possible to make all the additions of digits at the same time, provided
we could also record each carriage as it became due. We might then
complete the addition by adding, at the same instant, each carriage in
its proper place. The process would then stand thus:— {61}
Stages
2648
4564
────
│ 6102 Add each digit to the digit above.
1 │ 111 Record the carriages.
────
2 │ 7212 Add the above carriages.
Now, whatever mechanism is contrived for adding any one digit to any
other must, of course, be able to add the largest digit, nine, to that
other digit. Supposing, therefore, one unit of number to be passed
over in one second of time, it is evident that any number of pairs of
digits may be added together in nine seconds, and that, when all the
consequent carriages are known, as in the above case, it will cost one
second more to make those carriages. Thus, addition and carriage would
be completed in ten seconds, even though the numbers consisted each of
a hundred figures.
But, unfortunately, there are multitudes of cases in which the
carriages that become due are only known in successive periods of time.
As an example, add together the two following numbers:—
8473
1528
Stages ─────
1 Add all the digits 9991
2 Carry on tens and warn next car. 1
─────
9901
3 Carry on hundreds, and ditto 1
─────
9001
4 Carry on thousands, and ditto 1
─────
00001
5 Carry on ten thousands 1
─────
10001
{62}
In this case the carriages only become known successively, and they
amount to the number of figures to be added; consequently, the mere
addition of two numbers, each of fifty places of figures, would require
only nine seconds of time, whilst the possible carriages would consume
fifty seconds.
The mechanical means I employed to make these carriages bears some
slight analogy to the operation of the faculty of memory. A toothed
wheel had the ten digits marked upon its edge; between the nine and the
zero a projecting tooth was placed. Whenever any wheel, in receiving
addition, passed from nine to zero, the projecting tooth pushed over
a certain lever. Thus, as soon as the nine seconds of time required
for addition were ended, every carriage which had _become due_ was
indicated by the altered position of its lever. An arm now went round,
which was so contrived that the act of replacing that lever caused the
carriage which its position indicated to be made to the next figure
above. But this figure might be a nine, in which case, in passing to
zero, it would put over its lever, and so on. By placing the arms
spirally round an axis, these successive carriages were accomplished.
Multitudes of contrivances were designed, and almost endless drawings
made, for the purpose of economizing the time and simplifying the
mechanism of carriage. In that portion of the Difference Engine in
the Exhibition of 1862 the time of carriage has been reduced to about
one-fourth part of what was at first required.
〈ANTICIPATING CARRIAGE.〉
At last having exhausted, during years of labour, the principle of
successive carriages, it occurred to me that it might be possible
to teach mechanism to accomplish another mental process, namely—to
foresee. This idea occurred to me in October, 1834. It cost me
much thought, but the {63} principle was arrived at in a short
time. As soon as that was attained, the next step was to teach the
mechanism which could foresee to act upon that foresight. This was
not so difficult: certain mechanical means were soon devised which,
although very far from simple, were yet sufficient to demonstrate the
possibility of constructing such machinery.
The process of simplifying this form of carriage occupied me, at
intervals, during a long series of years. The demands of the Analytical
Engine, for the mechanical execution of arithmetical operations, were
of the most extensive kind. The multitude of similar parts required by
the Analytical Engine, amounting in some instances to upwards of fifty
thousand, rendered any, even the simplest, improvement of each part
a matter of the highest importance, more especially as regarded the
diminished amount of expenditure for its construction.
_Description of the existing portion of Difference Engine No. 1._
That portion of Difference Engine, No. 1, which during the last twenty
years has been in the museum of King’s College, at Somerset House, is
represented in the woodcut opposite the title page.
It consists of three columns; each column contains six cages; each cage
contains one figure-wheel.
The column on the right hand has its lowest figure-wheel covered by a
shade which is never removed, and to which the reader’s attention need
not be directed.
The figure-wheel next above may be placed by hand at any one of the ten
digits. In the woodcut it stands at zero.
The third, fourth, and fifth cages are exactly the same as the second.
The sixth cage contains exactly the same as the four just {64}
described. It also contains two other figure-wheels, which with a
similar one above the frame, may also be dismissed from the reader’s
attention. Those wheels are entirely unconnected with the moving part
of the engine, and are only used for memoranda.
It appears, therefore, that there are in the first column on the right
hand five figure-wheels, each of which may be set by hand to any of the
figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
The lowest of these figure-wheels represents the unit’s figure of
any number; the next above the ten’s figure, and so on. The highest
figure-wheel will therefore represent tens of thousands.
Now, as each of these figure-wheels may be set by hand to any digit,
it is possible to place on the first column any number up to 99999. It
is on these wheels that the Table to be calculated by the engine is
expressed. This column is called the Table column, and the axis of the
wheels the Table axis.
The second or middle column has also six cages, in each of which a
figure-wheel is placed. It will be observed that in the lowest cage,
the figure on the wheel is concealed by a shade. It may therefore be
dismissed from the attention. The five other figure-wheels are exactly
like the figure-wheels on the Table axis, and can also represent any
number up to 99999.
This column is called the First Difference column, and the axis is
called the First Difference axis.
The third column, which is that on the left hand, has also six cages,
in each of which is a figure-wheel capable of being set by hand to any
digit.
The mechanism is so contrived that whatever may be the numbers placed
respectively on the figure-wheels of each of {65} the three columns,
the following succession of operations will take place as long as the
handle is moved:—
1st. Whatever number is found upon the column of first differences will
be added to the number found upon the Table column.
2nd. The same first difference remaining upon its own column, the
number found upon the column of second differences will be added to
that first difference.
It appears, therefore, that with this small portion of the Engine any
Table may be computed by the method of differences, provided neither
the Table itself, nor its first and second differences, exceed five
places of figures.
If the whole Engine had been completed it would have had six orders of
differences, each of twenty places of figures, whilst the three first
columns would each have had half a dozen additional figures.
This is the simplest explanation of that portion of the Difference
Engine No. 1, at the Exhibition of 1862. There are, however, certain
modifications in this fragment which render its exhibition more
instructive, and which even give a mechanical insight into those higher
powers with which I had endowed it in its complete state.
As a matter of convenience in exhibiting it, there is an arrangement
by which the _three_ upper figures of the second difference are
transformed into a small engine which counts the natural numbers.
By this means it can be set to compute any Table whose second
difference is constant and less than 1000, whilst at the same time it
thus shows the position in the Table of each tabular number.
In the existing portion there are three bells; they can be respectively
ordered to ring when the Table, its first difference {66} and its
second difference, pass from positive to negative. Several weeks
after the machine had been placed in my drawing-room, a friend came
by appointment to test its power of calculating Tables. After the
Engine had computed several Tables, I remarked that it was evidently
finding the root of a quadratic equation; I therefore set the bells to
watch it. After some time the proper bell sounded twice, indicating,
and giving the two positive roots to be 28 and 30. The Table thus
calculated related to the barometer and really involved a quadratic
equation, although its maker had not previously observed it. I
afterwards set the Engine to tabulate a formula containing impossible
roots, and of course the other bell warned me when it had attained
those roots. I had never before used these bells, simply because I did
not think the power it thus possessed to be of any practical utility.
Again, the lowest cages of the Table, and of the first difference,
have been made use of for the purpose of illustrating three important
faculties of the finished engine.
1st. The portion exhibited can calculate any Table whose third
difference is constant and less than 10.
2nd. It can be used to show how much more rapidly astronomical Tables
can be calculated in an engine in which there is no constant difference.
3rd. It can be employed to illustrate those singular laws which might
continue to be produced through ages, and yet after an enormous
interval of time change into other different laws; each again to exist
for ages, and then to be superseded by new laws. These views were first
proposed in the “Ninth Bridgewater Treatise.”
〈CURIOUS QUESTIONS.〉
Amongst the various questions which have been asked respecting the
Difference Engine, I will mention a few of the most remarkable:—One
gentleman addressed me thus: {67} “Pray, Mr. Babbage, can you explain
to me in two words what is the principle of this machine?” Had the
querist possessed a moderate acquaintance with mathematics I might in
four words have conveyed to him the required information by answering,
“The method of differences.” The question might indeed have been
answered with six characters thus—
Δ^{7 }_u__{_x_} = 0.
but such information would have been unintelligible to such inquirers.
On two occasions I have been asked,—“Pray, Mr. Babbage, if you put into
the machine wrong figures, will the right answers come out?” In one
case a member of the Upper, and in the other a member of the Lower,
House put this question. I am not able rightly to apprehend the kind of
confusion of ideas that could provoke such a question. I did, however,
explain the following property, which might in some measure approach
towards an answer to it.
It is possible to construct the Analytical Engine in such a manner
that after the question is once communicated to the engine, it may be
stopped at any turn of the handle and set on again as often as may be
desired. At each stoppage every figure-wheel throughout the Engine,
which is capable of being moved without breaking, may be moved on to
any other digit. Yet after each of these apparent falsifications the
engine will be found to make the next calculation with perfect truth.
The explanation is very simple, and the property itself useless. The
whole of the mechanism ought of course to be enclosed in glass, and
kept under lock and key, in which case the mechanism necessary to give
it the property alluded to would be useless.
{68}