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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}




CHAPTER VI.

Statement relative to the Difference Engine, drawn up by the late Sir
H. Nicolas from the Author’s Papers.


The following statement was drawn up by the late Sir Harris Nicolas,
G.S.M. & G., from papers and documents in my possession relating to the
Difference Engine. I believe every paper I possessed at all bearing on
the subject was in his hands for several months.

* * * * *

For some time previous to 1822, Mr. Babbage had been engaged in
contriving machinery for the execution of extensive arithmetical
operations, and in devising mechanism by which the machine that made
the calculations might also print the results.

On the 3rd of July, 1822, he published a letter to Sir Humphry Davy,
President of the Royal Society, containing a statement of his views
on that subject; and more particularly describing an Engine for
calculating astronomical, nautical, and other Tables, by means of the
“method of differences.” In that letter it is stated that a small
Model, consisting of six figures, and capable of working two orders of
differences, had been constructed; and that it performed its work in a
satisfactory manner.

The concluding paragraph of that letter is as follows:—

“Whether I shall construct a larger Engine of this kind, and bring
to {69} perfection the others I have described, will, in a great
measure, depend on the nature of the encouragement I may receive.

“Induced, by a conviction of the great utility of such Engines, to
withdraw, for some time, my attention from a subject on which it has
been engaged during several years, and which possesses charms of
a higher order, I have now arrived at a point where success is no
longer doubtful. It must, however, be attained at a very considerable
expense, which would not probably be replaced, by the works it might
produce, for a long period of time; and which is an undertaking I
should feel unwilling to commence, as altogether foreign to my habits
and pursuits.”

The Model alluded to had been shown to a large number of Mr. Babbage’s
acquaintances, and to many other persons; and copies of his letter
having been given to several of his friends, it is probable that one of
the copies was sent to the Treasury.

On the 1st of April, 1823, the Lords of the Treasury referred that
Letter to the Royal Society, requesting—

“The opinion of the Royal Society on the merits and utility of this
invention.”

On the 1st of May the Royal Society reported to the Treasury, that—

“Mr. Babbage has displayed great talent and ingenuity in the
construction of his Machine for Computation, which the Committee
think fully adequate to the attainment of the objects proposed by the
inventor; and they consider Mr. Babbage as highly deserving of public
encouragement, in the prosecution of his arduous undertaking.”[15]

On the 21st of May these papers were ordered to be printed by the House
of Commons.

In July, 1823, Mr. Babbage had an interview with the Chancellor of the
Exchequer (Mr. Robinson[16]), to ascertain if it was the wish of the
Government that he should construct a large Engine of the kind, which
would also print the results it calculated. {70}

[15] Parliamentary Paper, No. 370, printed 22nd May, 1823.

[16] Afterwards Lord Goderich, now Earl of Ripon.

From the conversation which took place on that occasion, Mr. Babbage
apprehended that such was the wish of the Government. The Chancellor of
the Exchequer remarked that the Government were in general unwilling
to make grants of money for any inventions, however meritorious;
because, if they really possessed the merit claimed for them, the
sale of the article produced would be the best, as well as largest
reward of the inventor: but that the present case was an _exception_;
it being apparent that the construction of such a Machine could not
be undertaken with a view to profit from the sale of its produce; and
that, as mathematical Tables were peculiarly valuable for nautical
purposes, it was deemed a fit object of encouragement by the Government.

The Chancellor of the Exchequer mentioned two modes of advancing



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