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Translation originally published in 1843 in the *Scientific Memoirs*, *3*, 666-731.

(Go to the article to which the following Notes were appended)

[Augusta Ada Byron King, Countess of Lovelace]

the constants *a*, *b*, *c*, &c. are represented
on the seven columns of discs, of which the engine consists. It can therefore
tabulate *accurately *and to an *unlimited extent*, all series
whose general term is comprised in the above formula; and it can also tabulate
*approximately *between *intervals* *of greater or less extent*,
all other series which are capable of tabulation by the Method of Differences.

The Analytical Engine, on the contrary, is not merely adapted for *tabulating
*the results of one particular function and of no other, but for *developing
and tabulating *any function whatever. In fact the engine may be described
as being the material expression of any indefinite function of any degree
of generality and complexity, such as for instance,

which is, it will be observed, a function of all other possible functions of any number of quantities.

In this, which we may call the *neutral *or *zero *state of
the engine, it is ready to receive at any moment, by means of cards constituting
a portion of its mechanism (and applied on the principle of those used
in the Jacquard-loom), the impress of whatever *special *function
we may desire to develope or to tabulate. These cards contain within themselves
(in a manner explained in the Memoir itself, pages 11 and 12) the law of
development of the particular function that may be under consideration,
and they compel the mechanism to act accordingly in a certain corresponding
order. One of the simplest cases would be, for example, to suppose that

is the particular function

which the Difference Engine tabulates for values of *n *only up
to 7. In this case the cards would order the mechanism to go through that
succession of operations which would tabulate

where *n *might be any number whatever.

These cards, however, have nothing to do with the regulation of the
particular *numerical *data. They merely determine the *operations*[1]
to be effected, which operations may of course be performed on an infinite
variety of particular numerical values, and do not bring out any definite
numerical results unless the numerical data of the problem have been impressed
on the requisite portions of the train of mechanism. In the above example,
the first essential step towards an arithmetical result would be the substitution
of specific numbers for *n*, and for the other primitive quantities
which enter into the function.

Again, let us suppose that for F we put two complete equations of the
fourth degree between *x *and *y*. We must then express on the
cards the law of elimination for such equations. The engine would follow
out those laws, and would ultimately give the equation of one variable
which results from such elimination. Various *modes* of elimination
might be selected; and of course the cards must be made out accordingly.
The following is one mode that might be adopted. The engine is able to
multiply together any two functions of the form

This granted, the two equations may be arranged according to the powers
of *y*, and the coefficients of the powers of *y *may be arranged
according to powers of *x*. The elimination of *y *will result
from the successive multiplications and subtractions of several such functions.
In this, and in all other instances, as was explained above, the particular
*numerical *data and the *numerical *results are determined by
means and by portions of the mechanism which act quite independently of
those that regulate the *operations*.

In studying the action of the Analytical Engine, we find that the peculiar
and independent nature of the considerations which in all mathematical
analysis belong to *operations*, as distinguished from *the objects
operated upon *and from the *results* of the operations performed
upon those objects, is very strikingly defined and separated.

It is well to draw attention to this point, not only because its full
appreciation is essential to the attainment of any very just and adequate
general comprehension of the powers and mode of action of the Analytical
Engine, but also because it is one which is perhaps too little kept in
view in the study of mathematical science in general. It is, however, impossible
to confound it with other considerations, either when we trace the manner
in which that engine attains its results, or when we prepare the data for
its attainment of those results. It were much to be desired, that when
mathematical processes pass through the human brain instead of through
the medium of inanimate mechanism, it were equally a necessity of things
that the reasonings connected with *operations *should hold the same
just place as a clear and well-defined branch of the subject of analysis,
a fundamental but yet independent ingredient in the science, which they
must do in studying the engine. The confusion, the difficulties, the contradictions
which, in consequence of a want of accurate distinctions in this particular,
have up to even a recent period encumbered mathematics in all those branches
involving the consideration of negative and impossible quantities, will
at once occur to the reader who is at all versed in this science, and would
alone suffice to justify dwelling somewhat on the point, in connexion with
any subject so peculiarly fitted to give forcible illustration of it as
the Analytical Engine. It may be desirable to explain, that by the word
*operation*, we mean *any process which alters the mutual relation
of two or more things*, be this relation of what kind it may. This is
the most general definition, and would include all subjects in the universe.
In abstract mathematics, of course operations alter those particular relations
which are involved in the considerations of number and space, and the *results*
of operations are those peculiar results which correspond to the nature
of the subjects of operation. But the science of operations, as derived
from mathematics more especially, is a science of itself, and has its own
abstract truth and value; just as logic has its own peculiar truth and
value, independently of the subjects to which we may apply its reasonings
and processes. Those who are accustomed to some of the more modern views
of the above subject, will know that a few fundamental relations being
true, certain other combinations of relations must of necessity follow;
combinations unlimited in variety and extent if the deductions from the
primary relations be carried on far enough. They will also be aware that
one main reason why the separate nature of the science of operations has
been little felt, and in general little dwelt on, is the *shifting *meaning
of many of the symbols used in mathematical notation. First, the symbols
of *operation *are frequently *also *the symbols of the *results
*of operations. We may say that these symbols are apt to have both a
*retrospective *and a *prospective *signification. They may signify
either relations that are the consequence of a series of processes already
performed, or relations that are yet to be effected through certain processes.
Secondly, figures, the symbols of *numerical magnitude*, are frequently
*also * the symbols of *operations*, as when they are the
indices of powers. Wherever terms have a shifting meaning, independent
sets of considerations are liable to become complicated together, and reasonings
and results are frequently falsified. Now in the Analytical Engine, the
operations which come under the first of the above heads are ordered and
combined by means of a notation and of a train of mechanism which belongs
exclusively to themselves; and with respect to the second head, whenever
numbers meaning *operations *and not *quantities *(such as the
indices of powers) are inscribed on any column or set of columns, those
columns immediately act in a wholly separate and independent manner, becoming
connected with the *operating mechanism* exclusively, and re-acting
upon this. They never come into combination with numbers upon any other
columns meaning *quantities*; though, of course, if there are numbers
meaning *operations* upon *n *columns, these may *combine amongst
each other*, and will often be required to do so, just as numbers meaning
*quantities *combine with each other in any variety. It might have
been arranged that all numbers meaning *operations* should have appeared
on some separate portion of the engine from that which presents numerical
*quantities*; but the present mode is in some cases more simple, and
offers in reality quite as much distinctness when understood.

The operating mechanism can even be thrown into action independently
of any object to operate upon (although of course no *result *could
then be developed). Again, it might act upon other things besides *number*,
were objects found whose mutual fundamental relations could be expressed
by those of the abstract science of operations, and which should be also
susceptible of adaptations to the action of the operating notation and
mechanism of the engine. Supposing, for instance, that the fundamental
relations of pitched sounds in the science of harmony and of musical composition
were susceptible of such expression and adaptations, the engine might compose
elaborate and scientific pieces of music of any degree of complexity or
extent.

The Analytical Engine is an *embodying of the science of operations,
*constructed with peculiar reference to abstract number as the subject
of those operations. The Difference Engine is the embodying of *one particular
and very limited set of operations*, which (see the notation used in
Note B) may be expressed thus (+, +, +, +, +, +), or thus
6(+). Six repetitions of the one operation, +, is, in fact, the whole sum
and object of that engine. It has seven columns, and a number on any column
can add itself to a number on the next column to its *right-hand*.
So that, beginning with the column furthest to the left, six additions
can be effected, and the result appears on the seventh column, which is
the last on the right-hand. The *operating *mechanism of this engine
acts in as separate and independent a manner as that of the Analytical
Engine; but being susceptible of only one unvarying and restricted combination,
it has little force or interest in illustration of the distinct nature
of the *science of operations*. The importance of regarding the Analytical
Engine under this point of view will, we think, become more and more obvious
as the reader proceeds with M. Menabrea's clear and mastery article. The
calculus of operations is likewise in itself a topic of so much interest,
and has of late years been so much more written on and thought on than
formerly, that any bearing which that engine, from its mode of constitution,
may possess upon the illustration of this branch of mathematical science
should not be overlooked. Whether the inventor of this engine had any such
views in his mind while working out the invention, or whether he may subsequently
ever have regarded it under this phase, we do not know; but it is one that
forcibly occurred to ourselves on becoming acquainted with the means through
which analytical combinations are actually attained by the mechanism. We
cannot forbear suggesting one practical result which it appears to us must
be greatly facilitated by the independent manner in which the engine orders
and combines its *operations*: we allude to the attainment of those
combinations into which *imaginary quantities *enter. This is a branch
of its processes into which we have not had the opportunity of inquiring,
and our conjecture therefore as to the principle on which we conceive the
accomplishment of such results may have been made to depend, is very probably
not in accordance with the fact, and less subservient for the purpose than
some other principles, or at least requiring the cooperation of others.
It seems to us obvious, however, that where operations are so independent
in their mode of acting, it must be easy, by means of a few simple provisions
and additions in arranging the mechanism, to bring out a *double *set
of *results*, viz. - 1st, the *numerical magnitudes *which are
the results of operations performed on *numerical data*. (These results
are the *primary *object of the engine.) 2ndly, the *symbolical
results *to be attached to those numerical results, which symbolical
results are not less the necessary and logical consequences of operations
performed upon *symbolical data*, than are numerical results when
the data are numerical[2].

If we compare together the powers and the principles of construction of the Difference and of the Analytical Engines, we shall perceive that the capabilities of the latter are immeasurably more extensive than those of the former, and that they in fact hold to each other the same relationship as that of analysis to arithmetic. The Difference Engine can effect but one particular series of operations, viz. that required for tabulating the integral of the special function

and as it can only do this for values of *n *up to 7[3],
it cannot be considered as being the most *general *expression even
of *one particular *function, much less as being the expression of
any and all possible functions of all degrees of generality. The Difference
Engine can in reality (as has been already partly explained) do nothing
but *add*; and any other processes, not excepting those of simple
subtraction, multiplication and division, can be performed by it only just
to that extent in which it is possible, by judicious mathematical arrangement
and artifices, to reduce them to a *series of additions*. The method
of differences is, in fact, a method of additions; and as it includes within
its means a larger number of results attainable by *addition *simply,
than any other mathematical principle, it was very appropriately selected
as the basis on which to construct *an Adding Machine*, so as to give
to the powers of such a machine the widest possible range. The Analytical
Engine, on the contrary, can either add, subtract, multiply or divide with
equal facility; and performs each of these four operations in a direct
manner, without the aid of any of the other three. This one fact implies
everything; and it is scarcely necessary to point out, for instance, that
while the Difference Engine can merely *tabulate*, and is incapable
of *developing*, the Analytical Engine can *either tabulate or develope*.

The former engine is in its nature strictly *arithmetical*, and
the results it can arrive at lie within a very clearly defined and restricted
range, while there is no finite line of demarcation which limits the powers
of the Analytical Engine. These powers are co-extensive with our knowledge
of the laws of analysis itself, and need be bounded only by our acquaintance
with the latter. Indeed we may consider the engine as the *material and
mechanical representative *of analysis, and that our actual working
powers in this department of human study will be enabled more effectually
than heretofore to keep pace with our theoretical knowledge of its principles
and laws, through the complete control which the engine gives us over the
*executive manipulation *of algebraical and numerical symbols.

Those who view mathematical science, not merely as a vast body of abstract and immutable truths, whose intrinsic beauty, symmetry and logical completeness, when regarded in their connexion together as a whole, entitle them to a prominent place in the interest of all profound and logical minds, but as possessing a yet deeper interest for the human race, when it is remembered that this science constitutes the language through which alone we can adequately express the great facts of the natural world, and those unceasing changes of mutual relationship which, visibly or invisibly, consciously or unconsciously to our immediate physical perceptions, are interminably going on in the agencies of the creation we live amidst : those who thus think on mathematical truth as the instrument through which the weak mind of man can most effectually read his Creator's works, will regard with especial interest all that can tend to facilitate the translation of its principles into explicit practical forms.

The distinctive characteristic of the Analytical Engine, and that which
has rendered it possible to endow mechanism with such extensive faculties
as bid fair to make this engine the executive right-hand of abstract algebra,
is the introduction into it of the principle which Jacquard devised for
regulating, by means of punched cards, the most complicated patterns in
the fabrication of brocaded stuffs. It is in this that the distinction
between the two engines lies. Nothing of the sort exists in the Difference
Engine. We may say most aptly, that the Analytical Engine *weaves algebraical
patterns* just as the Jacquard-loom weaves flowers and leaves. Here,
it seems to us, resides much more of originality than the Difference Engine
can be fairly entitled to claim. We do not wish to deny to this later all
such claims. We believe that it is the only proposal or attempt ever made
to construct a calculating machine *founded on the principle of successive
orders of differences*, and capable of *printing off its own results*;
and that this engine surpasses its predecessors, both in the extent of
the calculations which it can perform, in the facility, certainty and accuracy
with which it can effect them, and in the absence of all necessity for
the intervention of human intelligence *during the performance of its
calculations*. Its nature is, however, limited to the strictly arithmetical,
and it is far from being the first or only scheme for constructing *arithmetical
*calculating machines with more or less of success.

The bounds of *arithmetic *were however outstepped the moment the
idea of applying the cards had occurred; and the Analytical Engine does
not occupy common ground with mere "calculating machines." It holds a position
wholly its own; and the considerations it suggests are most interesting
in their nature. In enabling mechanism to combine together *general *symbols
in successions of unlimited variety and extent, a uniting link is established
between the operations of matter and the abstract mental processes of the
*most abstract *branch of mathematical science. A new, a vast, and
a powerful language is developed for the future use of analysis, in which
to wield its truths so that these may become of more speedy and accurate
practical application for the purposes of mankind than the means hitherto
in our possession have rendered possible. Thus not only the mental and
the material, but the theoretical and the practical in the mathematical
world, are brought into more intimate and effective connexion with each
other. We are not aware of its being on record that anything partaking
in the nature of what is so well designated the *Analytical *Engine
has been hitherto proposed, or even thought of, as a practical possibility,
any more than the idea of a thinking or of a reasoning machine.

We will touch on another point which constitutes an important distinction
in the modes of operating of the Difference and Analytical Engines. In
order to enable the former to do its business, it is necessary to put into
its columns the series of numbers constituting the first terms of the several
orders of differences for whatever is the particular table under consideration.
The machine then works *upon *these as its data. But these data must
themselves have been already computed through a series of calculations
by a human head. Therefore that engine can only produce results depending
on data which have been arrived at by the explicit and actual working out
of processes that are in their nature different from any that come within
the sphere of its own powers. In other words, an *analysing *process
must have been gone through by a human mind in order to obtain the data
upon which the engine then *synthetically *builds its results. The
Difference Engine is in its character exclusively *synthetical*, while
the Analytical Engine is equally capable of analysis or of synthesis.

It is true that the Difference Engine can calculate to a much greater
extent with these few preliminary data, than the data themselves required
for their own determination. The table of squares, for instance, can be
calculated to any extent whatever, when the numbers *one *and *two
*are furnished; and a very few differences computed at any part of a
table of logarithms would enable the engine to calculate many hundreds
or even thousands of logarithms. Still the circumstance of its requiring,
as a previous condition, that any function whatever shall have been numerically
worked out, makes it very inferior in its nature and advantages to an engine
which, like the Analytical Engine, requires merely that we should know
the *succession and distribution of the operations* to be performed;
without there being any occasion[4], in order to obtain
data on which it can work, for our ever having gone through either the
same particular operations which it is itself to effect, or any others.
Numerical data must of course be given it, but they are mere arbitrary
ones; not data that could only be arrived at through a systematic and necessary
series of previous numerical calculations, which is quite a different thing.

To this it may be replied, that an analysing process must equally have
been performed in order to furnish the Analytical Engine with the necessary
*operative *data; and that herein may also lie a possible source of
error. Granted that the actual mechanism is unerring in its processes,
the *cards* may give it wrong orders. This is unquestionably the case;
but there is much less chance of error, and likewise far less expenditure
of time and labour, where operations only, and the distribution of these
operations, have to be made out, than where explicit numerical results
are to be attained. In the case of the Analytical Engine we have undoubtedly
to lay out a certain capital of analytical labour in one particular line;
but this is in order that the engine may bring us in a much larger return
in another line. It should be remembered also that the cards, when once
made out for any formula, have all the generality of algebra, and include
an infinite number of particular cases.

We have dwelt considerably on the distinctive peculiarities of each
of these engines, because we think it essential to place their respective
attributes in strong relief before the apprehension of the public; and
to define with clearness and accuracy the wholly different nature of the
principles on which each is based, so as to make it self-evident to the
reader (the mathematical reader at least) in what manner and degree the
powers of the Analytical Engine transcend those of an engine, which, like
the Difference Engine, can only work out such results as may be derived
from *one restricted and particular series of processes*, such as
those included in D* ^{n}u_{z}*=
0. We think this of importance, because we know that there exists considerable
vagueness and inaccuracy in the mind of persons in general on the subject.
There is a misty notion amongst most of those who have attended at all
to it, that

We are desirous likewise of removing two misapprehensions which we know
obtain, to some extent, respecting these engines. In the first place it
is very generally supposed that the Difference Engine, after it had been
completed up to a certain point, *suggested* the idea of the Analytical
Engine; and that the second is in fact the improved offspring of the first,
and *grew out *of the existence of its predecessor, through some natural
or else accidental combination of ideas suggested by this one. Such a supposition
is in this instance contrary to the facts; although it seems to be almost
an obvious inference, wherever two inventions, similar in their nature
and objects, succeed each other closely in order of *time*, and strikingly
in order of *value*; more especially when the same individual is the
author of both. Nevertheless the ideas which led to the Analytical Engine
occurred in a manner wholly independent of any that were connected with
the Difference Engine. These ideas are indeed in their own intrinsic nature
independent of the latter engine, and might equally have occurred had it
never existed nor been even thought of at all.

The second of the misapprehensions above alluded to relates to the well-known
suspension, during some years past, of all progress in the construction
of the Difference Engine. Respecting the circumstances which have interfered
with the actual completion of either invention, we offer no opinion; and
in fact are not possessed of the data for doing so, had we the inclination.
But we know that some persons suppose these obstacles (be they what they
may) to have arisen *in consequence* of the subsequent invention of
the Analytical Engine while the former was in progress. We have ourselves
heard it even *lamented *that an idea should ever have occurred at
all, which had turned out to be merely the means of arresting what was
already in a course of successful execution, without substituting the superior
invention in its stead. This notion we can contradict in the most unqualified
manner. The progress of the Difference Engine had long been suspended,
before there were even the least crude glimmerings of any invention superior
to it. Such glimmerings, therefore, and their subsequent development, were
in no way the original *cause *of that suspension; although, where
difficulties of some kind or other evidently already existed, it was not
perhaps calculated to remove or lessen them that an invention should have
been meanwhile thought of, which, while including all that the first was
capable of, possesses powers so extended as to eclipse it altogether.

We leave it for the decision of each individual (*after he has possessed
himself *of competent information as to the characteristics of each
engine) to determine how far it ought to be matter of regret that such
an accession has been made to the powers of human science, even if it *has
*(which we generally doubt) increased to a certain limited extent some
already existing difficulties that had arisen in the way of completing
a valuable but lesser work. We leave it for each to satisfy himself as
to the wisdom of desiring the obliteration (were that now possible) of
all records of the more perfect invention, in order that the comparatively
limited one might be finished. The Difference Engine would doubtless fulfil
all those practical objects which it was originally destined for. It would
certainly calculate all the tables that are more directly necessary for
the physical purposes of life, such as nautical and other computations.
Those who incline to very strictly utilitarian views may perhaps feel that
the peculiar powers of the Analytical Engine bear upon questions of abstract
and speculative science, rather than upon those involving every-day and
ordinary human interests. These persons being likely to possess but little
sympathy, or possibly acquaintance, with any branches of science which
they do not find to be *useful *(according to *their *definition
of that word), may conceive that the undertaking of that engine, now that
the other one is already in progress, would be a barren and unproductive
laying out of yet more money and labour; in fact, a work of supererogation.
Even in the utilitarian aspect, however, we do not doubt that very valuable
practical results would be developed by the extended faculties of the Analytical
Engine; some of which results we think we could now hint at, had we the
space; and others, which it may not yet be possible to foresee, but which
would be brought forth by the daily increasing requirements of science,
and by a more intimate practical acquaintance with the powers of the engine,
were it in actual existence.

On general grounds, both of an *a priori *description as well as
those founded on the scientific history and experience of mankind, we see
strong presumptions that such would be the case. Nevertheless all will
probably concur in feeling that the completion of the Difference Engine
would be far preferable to the non-completion of any calculating engine
at all. With whomsoever or wheresoever may rest the present causes of difficulty
that apparently exist towards either the completion of the old engine,
or the commencement of the new one, we trust they will not ultimately result
in this generation's being acquainted with these inventions through the
medium of pen, ink and paper merely; and still more do we hope, that for
the honour of our country's reputation in the future pages of history,
these causes will not lead to the completion of the undertaking by some
*other *nation or government. This could not but be matter of just
regret; and equally so, whether the obstacles may have originated in private
interests and feelings, in considerations of a more public description,
or in causes combining the nature of both such solutions.

We refer the reader to the 'Edinburgh Review' of July 1834, for a very
able account of the Difference Engine. The writer of the article we allude
to has selected as his prominent matter for exposition, a wholly different
view of the subject from that which M. Menabrea has chosen. The former
chiefly treats it under its mechanical aspect, entering but slightly into
the mathematical principles of which that engine is the representative,
but giving, in considerable length, many details of the mechanism and contrivances
by means of which it tabulates the various orders of differences. M. Menabrea,
on the contrary, exclusively developes the analytical view; taking it for
granted that mechanism is able to perform certain processes, but without
attempting to explain *how*; and devoting his whole attention to explanations
and illustrations of the manner in which analytical laws can be so arranged
and combined as to bring every branch of that vast subject within the grasp
of the assumed powers of mechanism. It is obvious that, in the invention
of a calculating engine, these two branches of the subject are equally
essential fields of investigation, and that on their mutual adjustment,
one to the other, must depend all success. They must be made to meet each
other, so that the weak points in the powers of either department may be
compensated by the strong points in those of the other. They are indissolubly
connected, though so different in their intrinsic nature, that perhaps
the same mind might not be likely to prove equally profound or successful
in both. We know those who doubt whether the powers of mechanism will in
practice prove adequate in all respects to the demands made upon them in
the working of such complicated trains of machinery as those of the above
engines, and who apprehend that unforeseen practical difficulties and disturbances
will arise in the way of accuracy and of facility of operation. The Difference
Engine, however, appears to us to be in a great measure an answer to these
doubts. It is complete as far as it goes, and it does work with all the
anticipated success. The Analytical Engine, far from being more complicated,
will in many respects be of simpler construction; and it is a remarkable
circumstance attending it, that with very *simplified *means it is
so much more powerful.

The article in the 'Edinburgh Review' was written some time previous
to the occurrence of any ideas such as afterwards led to the invention
of the Analytical Engine; and in the nature of the Difference Engine there
is much less that would invite a writer to take exclusively, or even prominently,
the mathematical view of it, than in that of the Analytical Engine; although
mechanism has undoubtedly gone much further to meet mathematics, in the
case of this engine, than of the former one. Some publication embracing
the *mechanical *view of the Analytical Engine is a desideratum which
we trust will be supplied before long.

Those who may have the patience to study a moderate quantity of rather dry details will find ample compensation, after perusing the article of 1834, in the clearness with which a succinct view will have been attained of the various practical steps through which mechanism can accomplish certain processes; and they will also find themselves still further capable of appreciating M. Menabrea's more comprehensive and generalized memoir. The very difference in the style and object of these two articles makes them peculiarly valuable to each other; at least for the purposes of those who really desire something more than a merely superficial and popular comprehension of the subject of calculating engines. A. A. L.

In the Difference Engine there are seven of these columns placed side by side in a row, and the working mechanism extends behind them: the general form of the whole mass of machinery is that of a quadrangular prism (more or less approaching to the cube); the results always appearing on that perpendicular face of the engine which contains the columns of discs, opposite to which face a spectator may place himself. In the Analytical Engine there would be many more of these columns, probably at least two hundred. The precise form and arrangement which the whole mass of its mechanism will assume is not yet finally determined.

We may conveniently represent the columns of discs on paper in a diagram like the following:-

The V's are for the purpose of convenient reference to any column,
either in writing or speaking, and are consequently numbered. The reason
why the letter V is chosen for this purpose in preference to any other
letter, is because these columns are designated (as the reader will find
in proceeding with the Memoir) the *Variables*, and sometimes the
*Variable columns, *or the *columns of Variables*. The origin
of this appellation is, that the values on the columns are destined to
change, that is to *vary*, in every conceivable manner. But it is
necessary to guard against the natural misapprehension that the columns
are only intended to receive the values of the *variables* in an analytical
formula, and not of the *constants*. The columns are called Variables
on a ground wholly unconnected with the *analytical *distinction between
constants and variables. In order to prevent the possibility of confusion,
we have, both in the translation and in the notes, written Variable with
a capital letter when we use the word to signify a *column of the engine*,
and variable with a small letter when we mean the *variable of a formula*.
Similarly, *Variable-cards *signify any cards that belong to a column
of the engine.

To return to the explanation of the diagram: each circle at the top
is intended to contain the algebraic sign + or -, either of which can be
substituted[1] for the other, according as the number
represented on the column below is positive or negative. In a similar manner
any other purely *symbolical *results of algebraical processes might
be made to appear in these circles. In Note A. the practicability
of developing *symbolical *with no less ease than *numerical *results
has been touched on. The zeros beneath the *symbolic *circles represent
each of them a disc, supposed to have the digit 0 presented in front. Only
four tiers of zeros have been figured in the diagram, but these may be
considered as representing thirty or forty, or any number of tiers of discs
that may be required. Since each disc can present any digit, and each circle
any sign, the discs of every column may be so adjusted[2]
as to express any positive or negative number whatever within the limits
of the machine; which limits depend on the *perpendicular *extent
of the mechanism, that is, on the number of discs to a column.

Each of the squares below the zeros is intended for the inscription
of any *general *symbol or combination of symbols we please; it being
understood that the number represented on the column immediately above
is the numerical value of that symbol, or combination of symbols. Let us,
for instance, represent the three quantities *a*, *n*, *x*,
and let us further suppose that *a* = 5, *n*= 7*, *x*
= 98*. We should have[3] -

Let us select the first. It would stand as follows, previous to calculation:-

First, six multiplications in order to get *x ^{n}* (= 98

Secondly, one multiplication in order then to get

In all, seven multiplications to complete the whole process. We may thus represent them:-

The multiplications would, however, at successive stages in the solution
of the problem, operate on pairs of numbers, derived from *different*
columns. In other words, the *same operation *would be performed on
different *subjects of operation*. And here again is an illustration
of the remarks made in the preceding Note on the independent
manner in which the engine directs its *operations*. In determining
the value of *ax ^{n}*, the

There are *two varieties *of the *Supplying *Variable-cards,
respectively adapted for fulfilling two distinct subsidiary purposes: but
as these modifications do not bear upon the present subject, we shall notice
them in another place.

In the above case of *ax ^{n}*, the Operation-cards merely
order seven multiplications, that is, they order the mill to be in the

. . .

The engine might be made to calculate all these in succession. Having
completed *ax ^{n}*, the function

Let us now suppose that we have *two *expressions whose values
have been computed by the engine independently of each other (each having
its own group of columns for data and results). Let them be *ax ^{n}*,
and

This example is introduced merely to show that we may, if we please,
retain separately and permanently any *intermediate *results (like
*ax ^{n}*,

Any group of columns may be considered as representing a *general
*function, until a *special *one has been implicitly impressed
upon them through the introduction into the engine of the Operation and
Variable-cards made out for a *particular *function. Thus, in the
preceding example, V_{1}, V_{2}, V_{3}, V_{5},
V_{6}, V_{7}, represent the *general *function f(*a*,
*n*, *b*, *p*, *x*, *y*) until the function (*ax ^{n}*)
/ (

The further we analyse the manner in which such an engine performs its processes and attains its results, the more we perceive how distinctly it places in a true and just light the mutual relations and connexion of the various steps of mathematical analysis; how clearly it separates those things which are in reality distinct and independent, and unites those which are mutually dependent. A. A. L.

The mode of application of the cards, as hitherto used in the art of
weaving, was not found, however, to be sufficiently powerful for all the
simplifications which it was desirable to attain in such varied and complicated
processes as those required in order to fulfil the purposes of an Analytical
Engine. A method was devised of what was technically designated *backing
*the cards in certain groups according to certain laws. The object of
this extension is to secure the possibility of bringing any particular
card or set of cards into use *any number* *of times successively
*in the solution of one problem. Whether this power shall be taken advantage
of or not, in each particular instance, will depend on the nature of the
operations which the problem under consideration may require. The process
is alluded to by M. Menabrea in page 16, and it is a very important simplification.
It has been proposed to use it for the reciprocal benefit of that art,
which, while it has itself no apparent connexion with the domains of abstract
science, has yet proved so valuable to the latter, in suggesting the principles
which, in their new and singular field of application, seem likely to place
*algebraical *combinations not less completely within the province
of mechanism, than are all those varied intricacies of which *intersecting
threads *are susceptible. By the introduction of the system of *backing
*into the Jacquard-loom itself, patterns which should possess symmetry,
and follow regular laws of any extent, might be woven by means of comparatively
few cards.

Those who understand the mechanism of this loom will perceive that the
above improvement is easily effected in practice, by causing the prism
over which the train of pattern-cards is suspended to revolve *backwards*
instead of *forwards*, at pleasure, under the requisite circumstances;
until, by so doing, any particular card, or set of cards, that has done
duty once, and passed on in the ordinary regular succession, is brought
back to the position it occupied just before it was used the preceding
time. The prism then resumes its *forward *rotation, and thus brings
the card or set of cards in question into play a second time. This process
may obviously be repeated any number of times.
A. A. L.

1st. Those on which the data are inscribed:

2ndly. Those intended to receive the final results:

3rdly. Those intended to receive such intermediate and temporary combinations
of the primitive data as are not to be permanently retained, but are merely
needed for *working with*, in order to attain the ultimate results.
Combinations of this kind might properly be called *secondary data*.
They are in fact so many *successive stages* towards the final result.
The columns which receive them are rightly named *Working-Variables*,
for their office is in its nature purely *subsidiary *to other purposes.
They develope an intermediate and transient class of results, which unite
the original data with the final results.

The Result-Variables sometimes partake of the nature of Working-Variables. It frequently happens that a Variable destined to receive a final result is the recipient of one or more intermediate values successively, in the course of the processes. Similarly, the Variables for data often become Working-Variables, or Result-Variables, or even both in succession. It so happens, however, that in the case of the present equations the three sets of offices remain throughout perfectly separate and independent.

It will be observed, that in the squares below the *Working*-Variables
nothing is inscribed. Any one of these Variables is in many cases destined
to pass through various values successively during the performance of a
calculation (although in these particular equations no instance of this
occurs). Consequently no *one fixed *symbol, or combination of symbols,
should be considered as properly belonging to a merely *Working*-Variable;
and as a general rule their squares are left blank. Of course in this,
as in all other cases where we mention a *general *rule, it is understood
that many particular exceptions may be expedient.

In order that all the indications contained in the diagram may be completely
understood, we shall now explain two or three points, not hitherto touched
on. When the value on any Variable is called into use, one of two consequences
may be made to result. Either the value may *return *to the Variable
after it has been used, in which case it is ready for a second use if needed;
or the Variable may be made zero. (We are of course not considering a third
case, of not unfrequent occurrence, in which the same Variable is destined
to receive the *result *of the very operation which it has just supplied
with a number.) Now the ordinary rule is, that the value *returns *to
the Variable; unless it has been foreseen that no use for that value can
recur, in which case zero is substituted. At the *end *of a calculation,
therefore, every column ought as a general rule to be zero, excepting those
for results. Thus it will be seen by the diagram, that when *m*, the
value on V_{0}, is used for the second time by Operation 5, V_{0
}becomes 0, since *m *is not again needed; that similarly, when
(*mn*^{' }- *m*^{'}*n*), on V_{12},
is used for the third time by Operation 11, V_{12 }becomes zero,
since (*mn*^{' }- *m*^{'}*n*) is not again
needed. In order to provide for the one or the other of the courses above
indicated, there are *two *varieties of the *Supplying *Variable-cards.
One of these varieties has provisions which cause the number given off
from any Variable to *return *to that Variable after doing its duty
in the mill. The other variety has provisions which cause *zero* to
be substituted on the Variable, for the number given off. These two varieties
are distinguished, when needful, by the respective appellations of the
*Retaining *Supply-cards and the *Zero *Supply-cards. We see
that the *primary *office (see Note B.) of both these
varieties of cards is the same; they only differ in their *secondary
*office.

Every Variable thus has belonging to it *one *class of *Receiving
*Variable-cards and *two *classes of *Supplying *Variable-cards.
It is plain however that only *one *or the *other *of these two
latter classes can be used by any one Variable for *one *operation;
never *both *simultaneously; their respective functions being mutually
incompatible.

It should be understood that the Variable-cards are not placed in *immediate
contiguity *with the columns. Each card is connected by means of wires
with the column it is intended to act upon.

Our diagram ought in reality to be placed side by side with M. Menabrea's corresponding table, so as to be compared with it, line for line belonging to each operation. But is was unfortunately inconvenient to print them in this desirable form. The diagram is, in the main, merely another manner of indicating the various relations denoted in M. Menabrea's table. Each mode has some advantages and some disadvantages. Combined, they form a complete and accurate method of registering every step and sequence in all calculations performed by the engine.

No notice has yet been taken of the *upper *indices which are added
to the left of each V in the diagram; an addition which we have also taken
the liberty of making to the V's in M. Menabrea's tables of pages 14, 16,
since it does not *alter *anything therein represented by him, but
merely *adds *something to the previous indications of those tables.
The *lower *indices are obviously indices of *locality *only,
and are wholly independent of the operations performed or of the results
obtained, their value continuing unchanged during the performance of calculations.
The *upper *indices, however, are of a different nature. Their office
is to indicate any *alteration *in the value which a Variable represents;
and they are of course liable to changes during the processes of a calculation.
Whenever a Variable has only zeros upon it, it is called ^{0}V;
the moment a value appears on it (whether that value be placed there arbitrarily,
or appears in the natural course of a calculation), it becomes ^{1}V.
If this value gives place to another value, the Variable becomes ^{2}V,
and so forth. Whenever a *value *again gives place to *zero*,
the Variable again becomes ^{0}V, even if it have been * ^{n}*V
the moment before. If a

There are several advantages in having a set of indices of this nature;
but these advantages are perhaps hardly of a kind to be immediately perceived,
unless by a mind somewhat accustomed to trace the successive steps by means
of which the engine accomplishes its purposes. We have only space to mention
in a general way, that the whole notation of the tables is made more consistent
by these indices, for they are able to mark a *difference *in certain
cases, where there would otherwise be an apparent *identity *confusing
in its tendency. In such a case as V* _{n }*= V

The substitutions in the preceding equations happen to be of little value towards illustrating the power and uses of the upper indices, for, owing to the nature of these particular equations, the indices are all unity throughout. We wish we had space to enter more fully into the relations which these indices would in many cases enable us to trace.

M. Menabrea incloses the three centre columns of his table under the
general title *Variable-cards*. The V's however in reality all represent
the actual *Variable-columns *of the engine, and not the cards that
belong to them. Still the title is a very just one, since it is through
the special action of certain Variable-cards (when *combined *with
the more generalized agency of the Operation-cards) that every one of the
particular relations he has indicated under that title is brought about.

Suppose we wish to ascertain how often any *one *quantity, or combination
of quantities, is brought into use during a calculation. We easily ascertain
*this*, from the inspection of any vertical column or columns of the
diagram in which that quantity may appear. Thus, in the present case, we
see that all the data, and all the intermediate results likewise, are used
twice, excepting (*mn*^{' }- *m*^{'}*n*),
which is used three times.

The *order *in which it is possible to perform the operations for
the present example, enables us to effect all the eleven operations of
which it consists with only *three Operation cards*; because the problem
is of such a nature that it admits of each *class *of operations being
performed in a group together; all the multiplications one after another,
all the subtractions one after another, &c. The operations are {6 (x),
3 (-), 2 (÷)}.

Since the very definition of an operation implies that there must be
*two *numbers to act upon, there are of course *two Supplying *Variable-cards
necessarily brought into action for every operation, in order to furnish
the two proper numbers. (See Note B.) Also, since every
operation must produce a *result*, which must be placed *somewhere*,
each operation entails the action of a *Receiving *Variable-card,
to indicate the proper locality for the result. Therefore, at least three
times as many Variable-cards as there are *operations *(not *Operation-cards*,
for these, as we have just seen, are by no means always as numerous as
the *operations*) are brought into use in every calculation. Indeed,
under certain contingencies, a still larger population is requisite; such,
for example, would probably be the case when the same result has to appear
on more than one Variable simultaneously (which is not unfrequently a provision
necessary for subsequent purposes in a calculation), and in some other
cases which we shall not here specify. We see therefore that a great disproportion
exists between the amount of *Variable *and of *Operation*-cards
requisite for the working of even the simplest calculation.

*All *calculations do not admit, like this one, of the operations
of the same nature being performed in groups together. Probably very few
do so without exceptions occurring in one or other stage of the progress;
and some would not admit it at all. The *order *in which the operations
shall be performed in every particular case is a very interesting and curious
question, on which our space does not permit us fully to enter. In almost
every computation a great *variety* of arrangements for the succession
of the processes is possible, and various considerations must influence
the selection amongst them for the purposes of a Calculating Engine. One
essential object is to choose that arrangement which shall tend to reduce
to a minimum the *time* necessary for completing the calculation.

It must be evident how multifarious and how mutually complicated are
the considerations which the workings of such an engine involve. There
are frequently several distinct *sets of effects *going on simultaneously;
all in a manner independent of each other, and yet to a greater or less
degree exercising a mutual influence. To adjust each to every other, and
indeed even to perceive and trace them out with perfect correctness and
success, entails difficulties whose nature partakes to a certain extent
of those involved in every question where *conditions *are very numerous
and inter-complicated; such as for instance the estimation of the mutual
relations amongst *statistical *phaenomena, and of those involved
in many other classes of facts.
A. A. L.

We have not subjoined any diagram of its development for this new example, as we did for the former one, because this is unnecessary after the full application already made of those diagrams to the illustration of M. Menabrea's excellent tables.

It may be remarked that a slight discrepancy exists between the formulae

given in the Memoir as the *data *for calculation, and the *results
*of the calculation as developed in the last division of the table which
accompanies it. To agree perfectly with this latter, the data should have
been given as

The following is a more complicated example of the manner in which the engine would compute a trigonometrical function containing variables. To multiply

by
B + B_{1 }cos q.

Let the resulting products be represented under the general form

This trigonometrical series is not only in itself very appropriate for illustrating the processes of the engine, but is likewise of much practical interest from its frequent use in astronomical computations. Before proceeding further with it, we shall point out that there are three very distinct classes of ways in which it may be desired to deduce numerical values from any analytical formula.

First. We may wish to find the collective numerical value of the *whole
formula*, without any reference to the quantities of which that formula
is a function, or to the particular mode of their combination and distribution,
of which the formula is the result and representative. Values of this kind
are of a strictly arithmetical nature in the most limited sense of the
term, and retain no trace whatever of the processes through which they
have been deduced. In fact, any one such numerical value may have been
attained from an *infinite variety *of data, or of problems. The values
for *x *and *y *in the two equations (see Note D.)
come under this class of numerical results.

Secondly. We may propose to compute the collective numerical value of
*each term *of a formula, or of a series, and to keep these results
separate. The engine must in such a case appropriate as many columns to
*results *as there are terms to compute.

Thirdly. It may be desired to compute the numerical value of various
*subdivisions* *of each term*,* *and to keep all these results
separate. It may be required, for instance, to compute each coefficient
separately from its variable, in which particular case the engine must
appropriate *two *result-columns to *every term that contains both
a variable and coefficient*.

There are many ways in which it may be desired in special cases to distribute
and keep separate the numerical values of different parts of an algebraical
formula: and the power of effecting such distributions to any extent is
essential to the *algebraical *character of the Analytical Engine.
Many persons who are not conversant with mathematical studies, imagine
that because the business of the engine is to give its results in *numerical
notation*, the *nature of its processes *must consequently be *arithmetical
*and *numerical*, rather than *algebraical *and *analytical*.
This is an error. The engine can arrange and combine its numerical quantities
exactly as if they were *letters *or any other *general *symbols;
and in fact it might bring out its results in algebraical *notation*,
were provisions made accordingl*y*. It might develope three sets of
results simultaneously, viz. *symbolic *results (as already alluded
to in Notes A. and B.); *numerical
*results (its chief and primary object); and *algebraical* results
in *literal *notation. This latter however has not been deemed a necessary
or desirable addition to its powers, partly because the necessary arrangements
for effecting it would increase the complexity and extent of the mechanism
to a degree that would not be commensurate with the advantages, where the
main object of the invention is to translate into *numerical *language
general formulae of analysis already known to us, or whose laws of formation
are known to us. But it would be a mistake to suppose that because its
*results *are given in the *notation *of a more restricted science,
its *processes *are therefore restricted to those of that science.
The object of the engine is in fact to give the *utmost practical efficiency
*to the resources of *numerical interpretations *of the higher
science of analysis, while it uses the processes and combinations of this
latter.

To return to the trigonometrical series. We shall only consider the
first four terms of the factor (A + A_{1 }cos q
+ &c.), since this will be sufficient to show the method. We
propose to obtain separately the numerical value of *each coefficient
*C_{o}, C_{1}, &c. of
(1.). The direct multiplication of the two factors gives

a result which would stand thus on the engine :-

First Series of
Second Series of
Third Series, which con-

Operations.
Operations.
tains only one (final) operation.

^{1}V_{10} x ^{1}V_{0 }= ^{1}V_{20}
^{1}V_{11} x ^{1}V_{0} = ^{1}V_{31}
^{1}V_{21} + ^{1}V_{31} = ^{2}V_{21},
and
^{1}V_{10} x ^{1}V_{1} = ^{1}V_{21}
^{1}V_{11} x ^{1}V_{1} = ^{1}V_{32}
V_{31} becomes = 0.
^{1}V_{10} x ^{1}V_{2} = ^{1}V_{22}
^{1}V_{11} x ^{1}V_{2} = ^{1}V_{33}
^{1}V_{10} x ^{1}V_{3} = ^{1}V_{23}
^{1}V_{11} x ^{1}V_{3} = ^{1}V_{34}

We do not enter into the same detail of *every *step of the processes
as in the examples of Notes D. and G.,
thinking it unnecessary and tedious to do so. The reader will remember
the meaning and use of the upper and lower indices, &c., as before
explained.

To proceed: we know that

Consequently, a slight examination of the second line of (2.) will show that by making the proper substitutions, (2.) will become

We shall perceive, if we inspect the particular arrangement of the results
in (2.) on the Result-columns as represented in the diagram, that, in order
to effect this transformation, each successive coefficient upon V_{32},
V_{33}, &c. (beginning with V_{32}), must through means
of proper cards be divided by *two*[2]; and
that one of the halves thus obtained must be added to the coefficient on
the Variable which precedes it by ten columns, and the other half to the
coefficient on the Variable which precedes it by twelve columns; V_{32},
V_{33}, &c. themselves becoming zeros during the process.

This series of operations may be thus expressed[3]:-

The substitution made by the engine of the processes in the second side of (3.) for those in the first side is an excellent illustration of the manner in which we may arbitrarily order it to substitute any function, number, or process, at pleasure, for any other function, number or process, on the occurrence of a specified contingency.

We will now suppose that we desire to go a step further, and to obtain
the numerical value of each *complete *term of the product (1.); that
is, of each *coefficient and variable united*, which for the (*n
*+ 1)th term would be C* _{n}* .cos

We must for this purpose place the variables themselves on another set
of columns, V_{41}, V_{42}, &c., and then order their
successive multiplication by V_{21}, V_{22}, &c., each
for each. There would thus be a final series of operations as follows:-

(N.B. that V_{40} being intended to receive the coefficient
on V_{20} which has *no *variable, will only have cos 0 q
(=1) inscribed on it, preparatory to commencing the fifth series
of operations.)

From the moment that the fifth and final series of operations is ordered,
the Variables V_{20}, V_{21}, &c. then in their turn
cease to be *Result*-Variables and become mere *Working*-Variables;
V_{40}, V_{41}, &c. being now the recipients of the
ultimate results.

We should observe, that if the variables cos q,
cos 2 q, cos 3 q,
&c. are furnished, they would be placed directly upon V_{41},
V_{42}, &c., like any other data. If not, a separate computation
might be entered upon in a separate part of the engine, in order to calculate
them, and place them on V_{41}, &c.

We have now explained how the engine might compute (1.) in the most
direct manner, supposing we knew nothing about the *general *term
of the resulting series. But the engine would in reality set to work very
differently, whenever (as in this case) we *do *know the law for the
general term.

The first two terms of (1.) are

and the general term for all these is

which is the coefficient of the (*n *+ 1)th term. The engine would
calculate the first two terms by means of a separate set of suitable Operation-cards,
and would then need another set for the third term; which last set of Operation-cards
would calculate all the succeeding terms* ad infinitum*, merely requiring
certain new Variable-cards for each term to direct the operations to act
on the proper columns. The following would be the successive sets of operations
for computing the coefficients of *n *+ 2 terms:-

Or we might represent them as follows, according to the numerical order of the operations:-

The brackets, it should be understood, point out the relation in which
the operations may be *grouped*, while the comma marks *succession*.
The symbol + might be used for this latter purpose, but this would be liable
to produce confusion, as + is also necessarily used to represent one class
of the actual operations which are the subject of that succession. In accordance
with this meaning attached to the comma, care must be taken when any one
group of operations recurs more than once, as is represented above by *n
*(11 . . .15), not to insert a comma after the number or letter prefixed
to that group. *n*, (11 . . . 15) would stand for *an operation
n*, *followed by the group of operations *(11 . . . 15); instead
of denoting *the number of groups which are to follow each other*.

Wherever a *general term *exists, there will be a *recurring
group *of operations, as in the above example. Both for brevity and
for distinctness, a *recurring group *is called a *cycle*. A
*cycle *of operations, then, must be understood to signify any *set
of operations *which is repeated *more than once*. It is equally
a *cycle*, whether it be repeated *twice *only, or an infinite
number of times; for it is the fact of a *repetition occurring at all
*that constitutes it such. In many cases of analysis there is a *recurring
group *of one or more *cycles*; that is, a *cycle of a cycle*,
or a *cycle of cycles*. For instance: suppose we wish to divide a
series by a series,

(1.)
*a *+ *bx *+ *cx*^{2}__+ . .
. . . __,

*a*' + *b*'*x *+ *c*'*x*^{2} + . . . .
.

it being required that the result shall be developed, like the dividend
and the divisor, in successive powers of *x*. A little consideration
of (1.), and of the steps through which algebraical division is effected,
will show that (if the denominator be supposed to consist of *p *terms)
the first partial quotient will be completed by the following operations:-

(2.)
{ (÷ ), *p *(x, -)}
or {(1), *p* (2, 3)},

that the second partial quotient will be completed by an exactly similar
set of operations, which acts on the remainder obtained by the first set,
instead of on the original dividend. The whole of the processes therefore
that have been gone through, by the time the *second *partial quotient
has been obtained, will be,-

(3.)
2{(÷ ), *p *(x, -)} or
2{(1), *p *(2, 3)},

which is a cycle that includes a cycle, or a cycle of the second order.
The operations for the *complete *division, supposing we propose to
obtain *n *terms of the series constituting the quotient, will be,-

(4.)
*n *{(÷ ), *p *(x, -)} or
*n *{(1), *p *(2, 3)}.

It is of course to be remembered that the process of algebraical division
in reality continues *ad infinitum*, except in the few exceptional
cases which admit of an exact quotient being obtained. The number *n
*in the formula (4.) is always that of the number of terms we propose
to ourselves to obtain; and the *n*th partial quotient is the coefficient
of the (*n *- 1)th power of *x*.

There are some cases which entail *cycles of cycles of cycles*,
to an indefinite extent. Such cases are usually very complicated, and they
are of extreme interest when considered with reference to the engine. The
algebraical development in a series of the *n*th function of any given
function is of this nature. Let it be proposed to obtain the *n*th
function of

(5.)
f(*a*, *b*, *c* . . . . . . *x*),
*x *being the variable.

We should premise, that we suppose the reader to understand what is
meant by an *n*th function. We suppose him likewise to comprehend
distinctly the difference between developing *an n*th *function
algebraically*, and merely *calculating an n*th *function arithmetically*.
If he does not, the following will be by no means very intelligible; but
we have not space to give any preliminary explanations. To proceed: the
law, according to which the successive functions of (5.) are to be developed,
must of course first be fixed on. This law may be of very various kinds.
We may propose to obtain our results in successive *powers *of *x*,
in which case the general form would be

or in successive powers of *n *itself, the index of the function
we are ultimately to obtain, in which case the general form would be

and *x *would only enter in the coefficients. Again, other functions
of *x *or of *n *instead of *powers *might be selected.
It might be in addition proposed, that the coefficients themselves should
be arranged according to given functions of a certain quantity. Another
mode would be to make equations arbitrarily amongst the coefficients only,
in which case the several functions, according to either of which it might
be possible to develope the *n*th function of (5.), would have to
be determined from the combined consideration of these equations and of
(5.) itself.

The *algebraical *nature of the engine (so strongly insisted on
in a previous part of this Note) would enable it to follow out any of these
various modes indifferently; just as we recently showed that it can distribute
and separate the numerical results of any one prescribed series of processes,
in a perfectly arbitrary manner. Were it otherwise, the engine could merely
*compute the arithmetical n*th *function*, a result which, like
any other purely arithmetical results, would be simply a collective number,
bearing no traces of the data or the processes which had led to it.

Secondly, the *law *of development for the *n*th function
being selected, the next step would obviously be to develope (5.) itself,
according to this law. This result would be the first function, and would
be obtained by a determinate series of processes. These in most cases would
include amongst them one or more *cycles *of operations.

The third step (which would consist of the various processes necessary
for effecting the actual substitution of the series constituting the *first
function*, for the variable itself) might proceed in either of two ways.
It might make the substitution either wherever *x *occurs in the original
(5.), or it might similarly make it wherever *x *occurs in the first
function itself which is the equivalent of (5.). In some cases the former
mode might be best, and in others the latter.

Whichever is adopted, it must be understood that the result is to appear
arranged in a series following the law originally prescribed for the development
of the *n*th function. This result constitutes the second function;
with which we are to proceed exactly as we did with the first function,
in order to obtain the third function, and so on, *n *- 1 times, to
obtain the *n*th function. We easily perceive that since every successive
function is arranged in a series *following the same law*, there would
(after the *first *function is obtained) be a *cycle of a cycle*,
&c. of operations[6], one, two, three, up to
*n *- 1 times, in order to get the *n*th function. We say, *after
the first function is obtained*, because (for reasons on which we cannot
here enter) the *first *function might in many cases be developed
through a set of processes peculiar to itself, and not recurring for the
remaining functions.

We have given but a very slight sketch of the principal *general *steps
which would be requisite for obtaining an *n*th function of such a
formula as (5.). The question is so exceedingly complicated, that perhaps
few persons can be expected to follow, to their own satisfaction, so brief
and general a statement as we are here restricted to on this subject. Still
it is a very important case as regards the engine, and suggests ideas peculiar
to itself, which we should regret to pass wholly without allusion. Nothing
could be more interesting than to follow out, in every detail, the solution
by the engine of such a case as the above; but the time, space and labour
this would necessitate, could only suit a very extensive work.

To return to the subject of *cycles* of operations: some of the
notation of the integral calculus lends itself very aptly to express them:
(2.) might be thus written:-

(6.)
(÷), S( + 1 )* ^{p}* ( x,
- ) or (1), S( + 1 )

where *p *stands for the variable; ( + 1 )* ^{p }*for
the function of the variable, that is, for f

(7.)
S( + 1 )* ^{n }*{(÷), S(
+ 1 )

the limits of *n *being from 1 to *n*, or from 0 to *n *-
1,

(8.)
or S( + 1 )* ^{n }*{ (1),
S( + 1 )

Perhaps it may be thought that this notation is merely a circuitous
way of expressing what was more simply and effectually expressed before;
and, in the above example, there may be some truth in this. But there is
another description of cycles which *can *only effectually be expressed,
in a condensed form, by the preceding notation. We shall call them *varying
cycles*. They are of frequent occurrence, and include successive cycles
of operations of the following nature:-

(9.)
*p *(1, 2, . . *m*), (*p *- 1) (1, 2, . . . *m*), (*p*
- 2) (1, 2, . . *m*) . . . (*p *- *n*) (1, 2 . . *m*),
[7]

where each cycle contains the same group of operations, but in which the number of repetitions of the group varies according to a fixed rate, with every cycle. (9.) can be well expressed as follows:-

(10.)
S *p *(1, 2, . . *m*), the limits
of *p *being from *p *- *n *to *p*.

Independent of the intrinsic advantages which we thus perceive to result
in certain cases from this use of the notation of the integral calculus,
there are likewise considerations which make it interesting, from the connections
and relations involved in this new application. It has been observed in
some of the former Notes, that the processes used in analysis form a logical
system of much higher generality than the applications to number merely.
Thus, when we read over any algebraical formula, considering it exclusively
with reference to the processes of the engine, and putting aside for the
moment its abstract signification as to the relations of quantity, the
symbols +, x, &c. in reality represent (as their immediate and proximate
effect, when the formula is applied to the engine) that a certain prism
which is a part of the mechanism (see Note C.) turns a
new face, and thus presents a new card to act on the bundles of levers
of the engine; the new card being perforated with holes, which are arranged
according to the peculiarities of the operation of addition, or of multiplication,
&c. Again, the *numbers *in the preceding formula (8.), each of
them really represents one of these very pieces of card that are hung over
the prism.

Now in the use made in the formulae (7.), (8.) and (10.), of the notation
of the integral calculus, we have glimpses of a similar new application
of the language of the *higher *mathematics. S,
in reality, here indicates that when a certain number of cards have acted
in succession, the prism over which they revolve must *rotate backwards*,
so as to bring those cards into their former position; and the limits 1
to *n*, 1 to *p*, &c., regulate how often this backward rotation
is to be repeated.

A. A. L.

The power of *repeating *the cards, alluded to by M. Menabrea in
page 16, and more fully explained in Note C., reduces
to an immense extent the number of cards required. It is obvious that this
mechanical improvement is especially applicable wherever *cycles *occur
in the mathematical operations, and that, in preparing data for calculations
by the engine, it is desirable to arrange the order and combination of
the processes with a view to obtain them as much as possible *symmetrically
*and in cycles, in order that the mechanical advantages of the *backing
*system may be applied to the utmost. It is here interesting to observe
the manner in which the value of an *analytical *resourse is *met
*and *enhanced* by an ingenious *mechanical *contrivance.
We see in it an instance of one of those mutual *adjustments *between
the purely mathematical and the mechanical departments, mentioned in Note
A. as being a main and essential condition of success in the invention
of a calculating engine. The nature of the resources afforded by such adjustments
would be of two principal kinds. In some cases, a difficulty (perhaps in
itself insurmountable) in the one department would be overcome by facilities
in the other; and sometimes (as in the present case) a strong point in
the one would be rendered still stronger and more available by combination
with a corresponding strong point in the other.

As a mere example of the degree to which the combined systems of cycles
and of backing can diminish the *number *of cards requisite, we shall
choose a case which places it in strong evidence, and which has likewise
the advantage of being a perfectly different *kind *of problem from
those that are mentioned in any of the other Notes. Suppose it be required
to eliminate nine variables from ten simple equations of the form -

We should explain, before proceeding, that it is not our object to consider
this problem with reference to the actual arrangement of the data on the
Variables of the engine, but simply as an abstract question of the *nature
*and *number *of the *operations* required to be performed
during its complete solution.

The first step would be the elimination of the first unknown quantity
*x*_{0 }between the first two equations.

This would be obtained by the form -

for which the operations 10 (x, x, -) would be needed. The second step
would be the elimination of *x*_{0} between the second and
third equations, for which the operations would be precisely the same.
We should then have had altogether the following operations:-

Continuing in the same manner, the total number of operations for the
complete elimination of *x*_{0 }between all the successive
pairs of equations would be -

We should then be left with nine simple equations of nine variables
from which to eliminate the next variable *x*_{1}, for which
the total of the processes would be -

We should then be left with eight simple equations of eight variables
from which to eliminate *x*_{2}, for which the processes would
be -

and so on. The total operations for the elimination of all the variables would thus be -

So that *three *Operation-cards would perform the office of 330
such cards.

If we take *n *simple equations containing *n *- 1 variables,
*n *being a number unlimited in magnitude, the case becomes still
more obvious, as the same three cards might then take the place of thousands
or millions of cards.

We shall now draw further attention to the fact, already noticed, of
its being by no means necessary that a formula proposed for solution should
ever have been actually worked out, as a condition for enabling the engine
to solve it. Provided we know the *series of operations* to be gone
through, that is sufficient. In the foregoing instance this will be obvious
enough on a slight consideration. And it is a circumstance which deserves
particular notice, since herein may reside a latent value of such an engine
almost incalculable in its possible ultimate results. We already know that
there are functions whose numerical value it is of importance for the purposes
both of abstract and of practical science to ascertain, but whose determination
requires processes so lengthy and so complicated, that, although it is
possible to arrive at them through great expenditure of time, labour and
money, it is yet on these accounts practically almost unattainable; and
we can conceive there being some results which it may be *absolutely
impossible *in practice to attain with any accuracy, and whose precise
determination it may prove highly important for some of the future wants
of science, in its manifold, complicated and rapidly-developing fields
of inquiry, to arrive at.

Without, however, stepping into the region of conjecture, we will mention
a particular problem which occurs to us at this moment as being an apt
illustration of the use to which such an engine may be turned for determining
that which human brains find it difficult or impossible to work out unerringly.
In the solution of the famous problem of the Three Bodies, there are, out
of about 295 coefficients of lunar perturbations given by M. Clausen (Astro^{e}.
Nachrichten, No. 406) as the result of the calculations by Burg, of two
by Damoiseau, and of one by Burckhardt, fourteen coefficients that differ
in the nature of their algebraic sign; and out of the remainder there are
only 101 (or about one-third) that agree precisely both in signs and in
amount. These discordances, which are generally small in individual magnitude,
may arise either from an erroneous determination of the abstract coefficients
in the development of the problem, or from discrepancies in the data deduced
from observation, or from both causes combined. The former is the most
ordinary source of error in astronomical computations, and this the engine
would entirely obviate.

We might even invent laws for series or formulae in an arbitrary manner, and set the engine to work upon them, and thus deduce numerical results which we might not otherwise have thought of obtaining; but this would hardly perhaps in any instance be productive of any great practical utility, or calculated to rank higher than as a kind of philosophical amusement. A. A. L.

The Analytical Engine has no pretensions whatever to *originate *anything.
It can do whatever we *know how to order it *to perform. It can *follow
*analysis; but it has no power of *anticipating *any analytical
relations or truths. Its province is to assist us in making *available
*what we are already acquainted with. This it is calculated to effect
primarily and chiefly of course, through its executive faculties; but it
is likely to exert an *indirect *and reciprocal influence on science
itself in another manner. For, in so distributing and combining the truths
and the formulae of analysis, that they may become most easily and rapidly
amenable to the mechanical combinations of the engine, the relations and
the nature of many subjects in that science are necessarily thrown into
new lights, and more profoundly investigated. This is a decidedly indirect,
and a somewhat *speculative*, consequence of such an invention. It
is however pretty evident, on general principles, that in devising for
mathematical truths a new form in which to record and throw themselves
out for actual use, views are likely to be induced, which should again
react on the more theoretical phase of the subject. There are in all extensions
of human power, or additions to human knowledge, various *collateral
*influences, besides the main and primary object attained.

To return to the executive faculties of this engine: the question must
arise in every mind, are they *really *even able to *follow *analysis
in its whole extent? No reply, entirely satisfactory to all minds, can
be given to this query, excepting the actual existence of the engine, and
actual experience of its practical results. We will however sum up for
each reader's consideration the chief elements with which the engine works:-

1. It performs the four operations of simple arithmetic upon any numbers whatever.

2. By means of certain artifices and arrangements (upon which we cannot
enter within the restricted space which such a publication as the present
may admit of), there is no limit either to the *magnitude *of the
*numbers *used, or to the *number *of *quantities *(either
variables or constants) that may be employed.

3. It can combine these numbers and these quantities either algebraically or arithmetically, in relations unlimited as to variety, extent, or complexity.

4. It uses algebraic *signs *according to their proper laws, and
developes the logical consequences of these laws.

5. It can arbitrarily substitute any formula for any other; effacing the first from the columns on which it is represented, and making the second appear in its stead.

6. It can provide for singular values. Its power of doing this is referred
to in M. Menabrea's memoir, page 17, where he mentions the passage of values
through zero and infinity. The practicability of causing it arbitrarily
to change its processes at any moment, on the occurrence of any specified
contingency (of which its substitution of (1/2 cos .(*n *+ 1) q
+ 1/2 cos .(*n *- 1) q)[1]
for (cos .*n *q .cos *.*q),
explained in Note E., is in some degree an illustration),
at once secures this point.

The subject of integration and of differentiation demands some notice. The engine can effect these processes in either of two ways:-

First. We may order it, by means of the Operation and of the Variable-cards,
to go through the various steps by which the required *limit *can
be worked out for whatever function is under consideration.

Secondly. It may (if we know the form of the limit for the function
in question) effect the integration or differentiation by direct[2]
substitution. We remarked in Note B., that any *set
*of columns on which numbers are inscribed, represents merely a *general
*function of the several quantities, until the special function have
been impressed by means of the Operation and Variable-cards. Consequently,
if instead of requiring the value of the function, we require that of its
integral, or of its differential coefficient, we have merely to order whatever
particular combination of the ingredient quantities may constitute that
integral or that coefficient. In *ax ^{n}*, for instance, instead
of the quantities

They would then stand thus:-

An interesting example for following out the processes of the engine would be such a form as

The methods in Arbogast's *Calcul des Dérivations *are peculiarly
fitted for the notation and the processes of the engine. Likewise the whole
of the Combinatorial Analysis, which consists first in a purely numerical
calculation of indices, and secondly in the distribution and combination
of the quantities according to laws prescribed by these indices.

We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we shall deduce it) a rather complicated example of its powers. The simplest manner of computing these numbers would be from the direct expansion of

2 2.3 2.3.4

which is in fact a particular case of the development of

mentioned in Note E. Or again, we might compute them from the well-known form

or from the form

If in the equation

(in which B_{1}, B_{3} . . . ., &c. are the Numbers
of Bernoulli), we expand the denominator of the first side in powers of
*x*, and then divide both numerator and denominator by *x*, we
shall derive

If this latter multiplication be actually performed, we shall have a series of the general form

in which we see, first, that all the coefficients of the powers of *x
*are severally equal to zero; and secondly, that the general form for
D_{2n}, the coefficient of the 2*n *+ 1th *term *(that
is of *x*^{2n }any *even *power of *x*),
is the following:-

A

We might have derived a form nearly similar to (8.), from D_{2n
-1}the coefficient of any *odd *power of *x *in (6.);
but the general form is a little different for the coefficients of the
*odd *powers, and not quite so convenient.

On examining (7. ) and (8.), we perceive that, when these formulae are
isolated from (6.), whence they are derived, and considered in themselves
separately and independently, *n *may be any whole number whatever;
although when (7.) occurs *as one of the *D's in (6.), it is obvious
that *n *is then not arbitrary, but is always a certain function of
the *distance of that *D *from the beginning*. If that distance
be = *d*, then

It is with the *independent *formula (8.) that we have to do. Therefore
it must be remembered that the conditions for the value of *n *are
now modified, and that *n *is a perfectly *arbitrary *whole number.
This circumstance, combined with the fact (which we may easily perceive)
that whatever *n *is, every term of (8.) after the (*n + *1)th
is = 0, and that the (*n *+ 1)th term itself is always = B_{2n
- 1}**. **1/1 = B_{2n - 1}, enables us to find
the value (either numerical or algebraical) of any *n*th Number of

Bernoulli B_{2n -1}, *in terms of all the preceding
ones*, if we but know the values of B_{1}, B_{3} . .
. . B_{2 n -3}. We append to this Note a Diagram and Table,
containing the details of the computation for B_{7}, (B_{1},
B_{3}, B_{5} being supposed given).

On attentively considering (8.), we shall likewise perceive that we
may derive from it the numerical value of *every *Number of Bernoulli
in succession, from the very beginning, *ad infinitum*, by the following
series of computations:-

1st Series. - Let *n *= 1, and calculate (8.) for this value of
*n*. The result is B_{1}.

2nd Series.- Let *n *= 2. Calculate (8.) for this value of *n*,
substituting the value of B_{1 }just obtained. The result is B_{3}.

3rd Series. - Let *n *= 3. Calculate (8.) for this value of *n*,
substituting the values of B_{1}, B_{3} before obtained.

The result is B_{5}. And so on, to any extent.

The diagram[3] represents the columns of the
engine when just prepared for computing B_{2n - 1} (in the
case of *n *= 4); while the table beneath them presents a complete
simultaneous view of all the successive changes which these columns then
severally pass through in order to perform the computation. (The reader
is referred to Note D. for explanations respecting the
nature and notation of such tables.)

Six numerical *data* are in this case necessary for making the
requisite combinations. These data are 1, 2, *n*(=4), B_{1},
B_{3}, B_{5}. Were *n *= 5, the additional datum B_{7}
would be needed. Were *n *= 6, the datum B_{9} would be needed;
and so on. Thus the actual *number of data *needed will always be
*n *+ 2, for *n *= *n*; and out of these *n *+ 2 data,
((*n + *2) - 3)[4]

of them are successive Numbers of Bernoulli. The reason why the Bernoulli
Numbers used as data are nevertheless placed on *Result*-columns in
the diagram, is because they may properly be supposed to have been previously
computed in a succession by the *engine *itself; under which circumstances
each B will appear as a *result*, previous to being used as a *datum*
for computing the succeeding B. Here then is an instance (of the kind alluded
to in Note D.) of the same Variables filling more than
one office in turn. It is true that if we consider our computation of B_{7}
as a perfectly *isolated *calculation, we may conclude B_{1},
B_{3}, B_{5} to have been arbitrarily placed on the columns;
and it would then perhaps be more consistent to put them on V_{4},
V_{5,} V_{6} as data and not results. But we are not taking
this view. On the contrary, we suppose the engine to be* in the course
of *computing the Numbers to an indefinite extent, from the very beginning;
and that we merely single out, by way of example, *one amongst *the
successive but distinct series' of computations it is thus performing.
Where the B's are fractional, it must be understood that they are computed
and appear in the notation of *decimal *fractions. Indeed this is
a circumstance that should be noticed with reference to all calculations.
In any of the examples already given in the translation and in the Notes,
some of the *data*, or of the temporary or permanent results, might
be fractional, quite as probably as whole numbers. But the arrangements
are so made, that the nature of the processes would be the same as for
whole numbers.

In the above table and diagram we are not considering the *signs *of
any of the B's, merely their numerical magnitude. The engine would bring
out the sign for each of them correctly of course, but we cannot enter
on *every *additional detail of this kind as we might wish to do.
The circles for the signs are therefore intentionally left blank in the
diagram.

Operation-cards 1, 2, 3, 4, 5, 6, prepare - __1__ **.
**__2 n - 1__. Thus, Card 1 multiplies

2 2

we see that the upper indices of the two Variables used, during Operation 1, remain unaltered.

We shall not go through the details of every operation singly, since the table and diagram sufficiently indicate them; we shall merely notice some few peculiar cases.

By Operation 6, a *positive *quantity is turned into a *negative
*quantity, by simply subtracting the quantity from a column which has
only zero upon it. (The sign at the top of V_{8} would become -
during this process.)

Operation 7 will be unintelligible, unless it be remembered that if
we were calculating for *n *= 1 instead of *n *= 4, Operation
6 would have completed the computation of B_{1 }itself; in which
case the engine, instead of continuing its processes, would have to put
B_{1 }on V_{21} ; and then either to stop altogether, or
to begin Operations 1, 2 . . . . 7 all over again for value of *n *(=2),
in order to enter on the computation of B_{3}; (having however
taken care, previous to this recommencement, to make the number on V_{3}
equal to *two*, by the addition of unity to the former *n *=
1 on that column). Now Operation 7 must either bring out a result equal
to zero (if *n *= 1); or a result *greater *than *zero*,
as in the present case; and the engine follows the one or the other of
the two courses just explained, contingently on the one or the other result
of Operation 7. In order fully to perceive the necessity of this *experimental
*operation, it is important to keep in mind what was pointed out, that
we are not treating a perfectly isolated and independent computation, but
one of a series of antecedent and prospective computations.

Cards 8, 9, 10 produce - __1__ **.** __2 n - 1__ + B

2 2

which again becomes a value after having passed from preceding values to zero. V

Card 12 has to perform the same office as Card 7 did in the preceding
section; since, if *n *had been = 2, the 11th operation would have
completed the computation of B_{3}.

Cards 13 to 20 make A_{3}. Since A_{2 n -1} always
consists of 2 *n *- 1 factors, A_{3 }has three factors; and
it will be seen that Cards 13, 14, 15, 16 make the second of these factors,
and then multiply it with the first; and that 17, 18, 19, 20 make the third
factor, and then multiply this with the product of the two former factors.

Card 23 has the office of Cards 11 and 7 to perform, since if *n *were
= 3, the 21st and 22nd operations would complete the computation of B_{5}.
As our case is B_{7}, the computation will continue one more stage;
and we must now direct attention to the fact, that in order to compute
A_{7}, it is merely necessary precisely to repeat the group of
Operations 13 to 20; and then, in order to complete the computation of
B_{7}, to repeat Operations 21, 22.

It will be perceived that every unit added to *n *in B_{2 n
- 1}, entails an additional repetition of operations (13 . . .
23) for the computation of B_{2 n -1}. Not only are all
the *operations *precisely the same however for every such repetition,
but they require to be respectively supplied with numbers from the very
*same pairs of columns*; with only the one exception of Operation
21, which will of course need B_{5} (from V_{23}) instead
of B_{3 }(from V_{22}). This identity in the *columns
*which supply the requisite numbers must not be confounded with identity
in the *values *those columns have upon them and give out to the mill.
Most of those values undergo alterations during a performance of the operations
(13 . . . 23), and consequently the columns present a new set of values
for the *next *performance of (13 . . . 23) to work on.

At the termination of the *repetition *of operations (13 . . .
23) in computing B_{7}, the alterations in the values on the Variables
are, that

V_{6} = 2*n *- 4 instead of 2*n *-2.

V_{7} = 6 . . . . . . . . . . . 4.

V_{10} = 0 . . . . . . . . . . 1.

V_{13} = A_{0 }+ A_{1 }B_{1} + A_{3
}B_{3 }+ A_{5 }B_{5 }instead of A_{0
}+ A_{1 }B_{1 }+ A_{3 }B_{3}.

In this state the only remaining processes are, first, to transfer the
value which is on V_{13 }to V_{24}; and secondly, to reduce
V_{6}, V_{7}, V_{13} to zero, and to add[5]
*one *to V_{3}, in order that the engine may be ready to commence
computing B_{9}. Operations 24 and 25 accomplish these purposes.
It may be thought anomalous that Operation 25 is represented as leaving
the upper index of V_{3} still = unity; but it must be remembered
that these indices always begin anew for a separate calculation, and that
Operation 25 places upon V_{3 }the *first *value *for the
new calculation*.

It should be remarked, that when the group (13 . . . 23) is *repeated*,
changes occur in some of the *upper *indices during the course of
the repetition: for example, ^{3}V_{6} would become ^{4}V_{6}
and ^{5}V_{6}.

We thus see that when *n *= 1, nine Operation-cards are used; that
when *n *= 2, fourteen Operation-cards are used; and that when *n
*> 2, twenty-five Operation-cards are used; but that no *more *are
needed, however great *n *may be; and not only this, but that these
same twenty-five cards suffice for the successive computation of all the
Numbers from B_{1 }to

B_{2 n - 1} inclusive. With respect to the number of
*Variable*-cards, it will be remembered, from the explanations in
previous Notes, that an average of three such cards to each *operation*
(not however to each Operation-*card*) is the estimate. According
to this, the computation of B_{1 }will require twenty-seven Variable-cards;
B_{3 }forty-two such cards; B_{5 }seventy-five; and for
every succeeding B after B_{5}, there would be thirty-three additional
Variable-cards (since each repetition of the group (13 . . . 23) adds eleven
to the number of operations required for computing the previous B). But
we must now explain, that whenever there is a *cycle of operations*,
and if these merely require to be supplied with numbers from the *same
pairs of columns*, and likewise each operation to place its *result
*on the *same *column for every repetition of the whole group,
the process then admits of a *cycle of Variable-cards *for effecting
its purposes. There is obviously much more symmetry and simplicity in the
arrangements, when cases do admit of repeating the Variable as well as
the Operation-cards. Our present example is of this nature. The only exception
to a *perfect identity* in *all *the processes and columns used,
for every repetition of Operations (13 . . . 23), is, that Operation 21
always requires one of its factors from a new column, and Operation 24
always puts its result on a new column. But as these variations follow
the same law at each repetition (Operation 21 always requiring its factor
from a column *one *in advance of that which it used the previous
time, and Operation 24 always putting its result on the column *one *in
advance of that which received the previous result), they are easily provided
for in arranging the recurring group (or cycle) of Variable-cards.

We may here remark, that the average estimate of three Variable-cards
coming into use to each operation, is not to be taken as an absolutely
and literally correct amount for all cases and circumstances. Many special
circumstances, either in the nature of a problem, or in the arrangements
of the engine under certain contingencies, influence and modify this average
to a greater or less extent; but it is a very safe and correct *general
*rule to go upon. In the preceding case it will give us seventy-five
Variable-cards as the total number which will be necessary for computing
any B after B_{3}. This is very nearly the precise amount really
used, but we cannot here enter into the minutiae of the few particular
circumstances which occur in this example (as indeed at some one stage
or other of probably most computations) to modify slightly this number.

It will be obvious that the very *same *seventy-five Variable-cards
may be repeated for the computation of every succeeding Number, just on
the same principle as admits of the repetition of the thirty-three Variable-cards
of Operations (13 . . . 23) in the computation of any *one *Number.
Thus there will be a *cycle of a cycle *of Variable-cards.

If we now apply the notation for cycles, as explained in Note E., we may express the operations for computing the Numbers of Bernoulli in the following manner:-

Again,

represents the total operations for computing every number in succession,
from B_{1 }to B_{2 n - 1 }inclusive.

In this formula we see a *varying cycle *of the *first *order,
and an ordinary cycle of the *second *order. The latter cycle in this
case includes in it the varying cycle.

On inspecting the ten Working-Variables of the diagram, it will be perceived,
that although the *value *on any one of them (excepting V_{4 }and
V_{5}) goes through a series of changes, the *office *which
each performs is in this calculation *fixed *and *invariable*.
Thus V_{6 }always prepares the *numerators *of the factors
of any A; V_{7} the *denominators*. V_{8} always receives
the (2*n *- 3)th factor of A_{2n - 1 }, and V_{9
}the (2* n *- 1)th. V_{10 }always decides which of two
courses the succeeding processes are to follow, by feeling for the value
of *n *through means of a subtraction; and so on; but we shall not
enumerate further. It is desirable in all calculations so to arrange the
processes, that the *offices* performed by the Variables may be as
uniform and fixed as possible.

Supposing that it was desired not only to tabulate B_{1}, B_{3},
&c., but A_{0}, A_{1}, &c.; we have only then to
appoint another series of Variables, V_{41}, V_{42}, &c.,
for receiving these latter results as they are successively produced upon
V_{11}. Or again, we may, instead of this, or in addition to this
second series of results, wish to tabulate the value of each successive
*total *term of the series (8.), viz. A_{0}, A_{1}
B_{1}, A_{3 }B_{3}, &c. We have then merely
to multiply each B with each corresponding A, as produced, and to place
these successive products on Result-columns appointed for the purpose.

The formula (8.) is interesting in another point of view. It is one particular case of the general Integral of the following Equation of Mixed Differences:-

for certain special suppositions respecting

The *general *integral itself is of the form,

and it is worthy of remark, that the engine might (in a manner more
or less similar to the preceding) calculate the value of this formula upon
most *other *hypotheses for the functions in the integral with as
much, or (in many cases) with more ease than it can formula (8.).
A. A. L.

[2] In fact, such an extension as we allude to would
merely constitute a further and more perfected development of any system
introduced for making the proper combinations of the signs *plus *and
*minus*. How ably M. Menabrea has touched on this restricted case
is pointed out in Note B.

[3] The machine might have been constructed so as
to tabulate for a higher value of *n *than seven. Since, however,
every unit added to the value of *n *increases the extent of the mechanism
requisite, there would on this account be a limit beyond which it could
not be practically carried. Seven is sufficiently high for the calculation
of all ordinary tables.

The fact that, in the Analytical Engine, the same extent of mechanism
suffices for the solution of D* ^{n }u_{z
}*= 0, whether

[4] This subject is further noticed in Note F.

[2] This adjustment is done by hand merely.

[3] It is convenient to omit the circles whenever the signs + or - can be actually represented.

[2] We recommend the reader to trace the successive substitutions backwards from (1.) to (4.), in Mons. Menabrea's Table. This he will easily do by means of the upper and lower indices, and it is interesting to observe how each V successively ramifies (so to speak) into two other V's in some other column of the Table, until at length the V's of the original data are arrived at.

[3] See note, page 34. [immediately above]

[2] This division would be managed by ordering
the number 2 to appear on any separate new column which should be conveniently
situated for the purpose, and then directing this column (which is in the
strictest sense a *Working*-*Variable*) to divide itself successively
with V_{32}, V_{33}, &c.

[3] It should be observed, that were the rest of
the factor (A + A cos q + &c.) taken into
account, instead of *four *terms only, C_{3} would have the
additional term 1/2 B_{1} A_{4}; and C_{4} the
two additional terms, B A_{4}, 1/2 B_{1} A_{5}.
This would indeed have been the case had even *six *terms been multiplied.

[4] *Classics* Editor's note: In the original
text, a vinculum appears over "B A_{1} + B_{1} A + 1/2
B_{1 }A_{2}" in the equation "( B A + 1/2 B_{1}
A_{1}) + (B A_{1 }+ B_{1 }A + 1/2 B_{1}
A_{2} . cos q)". In the present text,
the vinculum has been substituted with parentheses.

[5] *Classics* Editor's note: In the original
text, a vinculum appears over "A_{n - 1} + A_{n +
2}" in the equation "(B A* _{n }*+ 1/2 B

[6] A cycle that includes *n *other cycles,
successively *contained one within another*, is called a cycle of
the *n *+ 1th order. A cycle may simply *include* many other
cycles, and yet only be of the second order. If a series follows a certain
law for a certain number of terms, and then another law for another number
of terms, there will be a cycle of operations for every new law; but these
cycles will not be *contained one within another*, - they merely *follow
each other*. Therefore their number may be infinite without influencing
the *order *of a cycle that includes a repetition of such a series.

[7] *Classics* Editor's note: In the original
text, vinculums appear over "*p* - 1" and "*p *- *n*" in
the equation "*p* (1, 2, . . *m*), *p* - 1 (1, 2 . . .*m*),
*p* - 2 (1, 2 . . *m*) . . . *p* - *n* (1, 2 . . *m*).
In the present text, the vinculums have been substituted with parentheses.

[2] The engine cannot of course compute limits
for perfectly *simple *and *uncompounded *functions, except in
this manner. It is obvious that it has no power of representing or of manipulating
with any but *finite *increments or decrements; and consequently that
wherever the computation of limits (or of any other functions) depends
upon the *direct *introduction of quantities which either increase
or decrease *indefinitely*, we are absolutely beyond the sphere of
its powers. Its nature and arrangements are remarkably adapted for taking
into account all *finite *increments or decrements (however small
or large), and for developing the true and logical modifications of form
or value dependent upon differences of this nature. The engine may indeed
by considered as including the whole Calculus of Finite Differences; many
of whose theorems would be especially and beautifully fitted for development
by its processes, and would offer peculiarly interesting considerations.
We may mention, as an example, the calculation of the Numbers of Bernoulli
by means of the *Differences of Zero*.

[3] See the diagram at the
end of these Notes.

[4] *Classics* Editor's note: In the original
text, a vinculum appears over "*n* + 2" in the equation "(*n*
+ 2 - 3)". In the present text, the vinculum has been substituted with
parentheses.

[5] It is interesting to observe, that so complicated
a case as this calculation of the Bernoullian Numbers nevertheless presents
a remarkable simplicity in one respect; viz. that during the processes
for the computation of *millions* of these Numbers, no other arbitrary
modification would be requisite in the arrangements, excepting the above
simple and uniform provision for causing one of the data periodically to
receive the finite increment unity.