This remark seems to require further comment, since it is in some
degree calculated to strike the mind as being at variance with the
subsequent passage, where it is
explained that *an engine which can effect these four* operations can in
fact effect *every species of calculation*. The apparent discrepancy is
stronger too in the translation than in the original, owing to its
being impossible to render precisely into the English tongue all the
niceties of distinction which the French idiom happens to admit of in
the phrases used for the two passages we refer to. The explanation
lies in this: that in the one case the execution of these four
operations is the *fundamental starting-point*, and the object proposed
for attainment by the machine is the *subsequent combination of these*
in every possible variety; whereas in the other case the execution of
some *one* of these four operations, selected at pleasure, is the
*ultimatum*, the sole and utmost result that can be proposed for
attainment by the machine referred to, and which result it cannot any
further combine or work upon. The one *begins* where the other *ends*.
Should this distinction not now appear perfectly clear, it will become
so on perusing the rest of the Memoir, and the Notes that are appended
to it.—NOTE BY TRANSLATOR.

The idea that the one engine is the offspring and has grown out of the
other, is an exceedingly natural and plausible supposition, until reflection
reminds us that no *necessary* sequence and connexion need exist between
two such inventions, and that they may be wholly independent. M.
Menabrea has shared this idea in common with persons who have not his
profound and accurate insight into the nature of either engine.
In Note A.
(see the Notes at the end of the Memoir) it will be found sufficiently
explained, however, that this supposition is unfounded. M. Menabrea's
opportunities were by no means such as could be adequate to afford him
information on a point like this, which would be naturally and almost
unconsciously *assumed*, and
would scarcely suggest any inquiry with
reference to it.—NOTE BY TRANSLATOR.

This must not be understood in too unqualified a manner. The engine is capable under certain circumstances, of feeling about to discover which of two or more possible contingencies has occurred, and of then shaping its future course accordingly.—NOTE BY TRANSLATOR.

Zero is not *always* substituted when a number is
transferred to the mill. This is explained further on in the memoir,
and still more fully in Note D.—NOTE
BY TRANSLATOR.

Not having had leisure to discuss with Mr. Babbage the manner of introducing into his machine the combination of algebraical signs, I do not pretend here to expose the method he uses for this purpose; but I considered that I ought myself to supply the deficiency, conceiving that this paper would have been imperfect if I had omitted to point out one means that might be employed for resolving this essential part of the problem in question.

For an explanation of the upper left-hand indices attached to the V's in this and in the preceding Table, we must refer the reader to Note D, amongst those appended to the memoir.—NOTE BY TRANSLATOR.

This sentence has been slightly altered in the translation in order to express more exactly the present state of the engine.—NOTE BY TRANSLATOR.

The notation here alluded to is a most interesting and important subject, and would have well deserved a separate and detailed Note upon it amongst those appended to the Memoir. It has, however, been impossible, within the space allotted, even to touch upon so wide a field.—NOTE BY TRANSLATOR.

We do not mean to imply that the *only* use made of the
Jacquard cards is that of regulating the algebraical
*operations*; but we mean to explain that *those* cards
and portions of mechanism which regulate these *operations* are
wholly independent of those which are used for other purposes. M.
Menabrea explains that there are *three* classes of cards used
in the engine for three distinct sets of objects, viz.
*Cards of
the Operations*, *Cards of the Variables*, and certain
*Cards of Numbers*.

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.

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
,
whether *n*=7, *n*=100,000, or *n*=any
number whatever, at once suggests how entirely distinct must be the *nature
of the principles* through whose application matter has been enabled to
become the working agent of abstract mental operations in each of these
engines respectively, and it affords an equally obvious presumption, that
in the case of the Analytical Engine, not only are those principles in
themselves of a higher and more comprehensive description, but also such
as must vastly extend the *practical* value of the engine whose basis they
constitute.

A fuller account of the manner in which
the signs are regulated is
given in M. Menabrea's Memoir. He himself expresses
doubts (in a note of his own) as to his having
been likely to hit on the precise methods really adopted; his
explanation being merely a conjectural one. That it *does*
accord precisely with the fact is a remarkable circumstance, and
affords a convincing proof how completely M. Menabrea has been imbued
with the true spirit of the invention. Indeed the whole of the
above Memoir is a striking production, when we consider that M.
Menabrea had had but very slight means for obtaining any
adequate ideas respecting the Analytical Engine. It requires however
a considerable acquaintance with the abstruse and complicated nature
of such a subject, in order fully to appreciate the penetration of the
writer who could take so just and comprehensive a view of it upon
such limited opportunity.

This adjustment is done by hand merely.

It is convenient to omit the circles whenever the signs + or − can be actually represented.

We recommend the reader to trace the successive substitutions backwards from (1) to (4), in M. 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.

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.

It should be observed, that were the rest of the factor
(A + A cos θ + &c.)
taken into account, instead of *four* terms only, C_{3}
would have the additional term ½B_{1}A_{4}; and
C_{4} the two additional terms, BA_{4},
½B_{1}A_{5}. This would indeed have been the case had
even *six* terms been multiplied.

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.

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 be
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*.

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.