THE
                   EDINBURGH REVIEW,




                      JULY, 1834.

                        No. CXX.




                 THE CALCULATING ENGINE


                           BY

                     CHARLES BABBAGE


Art I.--1. _Letter to Sir Humphry Davy, Bart. P.R.S., on the application
of Machinery to Calculate and Print Mathematical Tables_. By CHARLES
BABBAGE, Esq. F.R.S. 4to. Printed by order of the House of Commons.

2. _On the Application of Machinery to the Calculation of Astronomical and
Mathematical Tables_. By CHARLES BABBAGE, Esq. Memoirs Astron. Soc.
Vol. I. Part 2. London: 1822.

3. _Address to the Astronomical Society, by Henry Thomas Colebrooke,
Esq. F.R.S. President, on presenting the first gold medal of the Society
to Charles Babbage, Esq. for the invention of the Calculating Engine_.
Memoirs Astron. Soc. Vol. I. Part 2. London: 1822.

4. _On the determination of the General Term of a new Class of Infinite
Series_. By CHARLES BABBAGE, Esq. Transactions Camb. Phil. Soc.
Cambridge: 1824.

5. _On Errors common to many Tables of Logarithms_. By CHARLES BABBAGE,
Esq. Memoirs Astron. Soc. London: 1827.

6. _On a Method of Expressing by Signs the Action of Machinery_.
By CHARLES BABBAGE, Esq. Phil. Trans. London: 1826.

7. _Report by the Committee appointed by the Council of the Royal
Society to consider the subject referred to in a Communication received
by them from the Treasury, respecting Mr Babbage's Calculating Engine,
and to report thereupon_. London: 1829.


THERE is no position in society more enviable than that of the few who
unite a moderate independence with high intellectual qualities.
Liberated from the necessity of seeking their support by a profession,
they are unfettered by its restraints, and are enabled to direct the
powers of their minds, and to concentrate their intellectual energies on
those objects exclusively to which they feel that their powers may be
applied with the greatest advantage to the community, and with the most
lasting reputation to themselves. On the other hand, their middle
station and limited income rescue them from those allurements to
frivolity and dissipation, to which rank and wealth ever expose their
possessors. Placed in such favourable circumstances, Mr Babbage selected
science as the field of his ambition; and his mathematical researches
have conferred on him a high reputation, wherever the exact sciences are
studied and appreciated. The suffrages of the mathematical world have
been ratified in his own country, where he has been elected to the
Lucasian Professorship in his own University--a chair, which, though of
inconsiderable emolument, is one on which Newton has conferred
everlasting celebrity. But it has been the fortune of this mathematician
to surround himself with fame of another and more popular kind, and
which rarely falls to the lot of those who devote their lives to the
cultivation of the abstract sciences. This distinction he owes to the
announcement, some years since, of his celebrated project of a
Calculating Engine. A proposition to reduce arithmetic to the dominion
of mechanism,--to substitute an automaton for a compositor,--to throw
the powers of thought into wheel-work could not fail to awaken the
attention of the world. To bring the practicability of such a project
within the compass of popular belief was not easy: to do so by bringing
it within the compass of popular comprehension was not possible. It
transcended the imagination of the public in general to conceive its
possibility; and the sentiments of wonder with which it was received,
were only prevented from merging into those of incredulity, by the faith
reposed in the high attainments of its projector. This extraordinary
undertaking was, however, viewed in a very different light by the small
section of the community, who, being sufficiently versed in mathematics,
were acquainted with the principle upon which it was founded. By
reference to that principle, they perceived at a glance the
practicability of the project; and being enabled by the nature of their
attainments and pursuits to appreciate the immeasurable importance of
its results, they regarded the invention with a proportionately profound
interest. The production of numerical tables, unlimited in quantity and
variety, restricted to no particular species, and limited by no
particular law;--extending not merely to the boundaries of existing
knowledge, but spreading their powers over the undefined regions of
future discovery--were results, the magnitude and the value of which the
community in general could neither comprehend nor appreciate. In such a
case, the judgment of the world could only rest upon the authority of
the philosophical part of it; and the fiat of the scientific community
swayed for once political councils. The British Government, advised by
the Royal Society, and a committee formed of the most eminent
mechanicians and practical engineers, determined on constructing the
projected mechanism at the expense of the nation, to be held as national
property.

Notwithstanding the interest with which this invention has been regarded
in every part of the world, it has never yet been embodied in a written,
much less in a published form. We trust, therefore, that some credit
will be conceded to us for having been the first to make the public
acquainted with the object, principle, and structure of a piece of
machinery, which, though at present unknown (except as to a few of its
probable results), must, when completed, produce important effects, not
only on the progress of science, but on that of civilisation.

The calculating machinery thus undertaken for the public gratuitously
(so far as Mr Babbage is concerned), has now attained a very advanced
stage towards completion; and a portion of it has been put together, and
performs various calculations;--affording a practical demonstration
that the anticipations of those, under whose advice Government has
acted, have been well founded.

There are nevertheless many persons who, admitting the great ingenuity
of the contrivance, have, notwithstanding, been accustomed to regard it
more in the light of a philosophical curiosity, than an instrument for
purposes practically useful. This mistake (than which it is not possible
to imagine a greater) has arisen mainly from the ignorance which
prevails of the extensive utility of those numerical tables which it is
the purpose of the engine in question to produce. There are also some
persons who, not considering the time requisite to bring any invention
of this magnitude to perfection in all its details, incline to consider
the delays which have taken place in its progress as presumptions
against its practicability. These persons should, however, before they
arrive at such a conclusion, reflect upon the time which was necessary
to bring to perfection engines infinitely inferior in complexity and
mechanical difficulty. Let them remember that--not to mention the
_invention_ of that machine--the _improvements_ alone introduced into the
steam-engine by the celebrated Watt, occupied a period of not less than
twenty years of the life of that distinguished person, and involved an
expenditure of capital amounting to L.50,000.[1] The calculating
machinery is a contrivance new even in its details. Its inventor did not
take it up already imperfectly formed, after having received the
contributions of human ingenuity exercised upon it for a century or
more. It has not, like almost all other great mechanical inventions,
been gradually advanced to its present state through a series of
failures, through difficulties encountered and overcome by a succession
of projectors. It is not an object on which the light of various minds
has thus been shed. It is, on the contrary, the production of solitary
and individual thought,--begun, advanced through each successive stage
of improvement, and brought to perfection by one mind. Yet this creation
of genius, from its first rude conception to its present state, has cost
little more than half the time, and not one-third of the expense,
consumed in bringing the steam-engine (previously far advanced in the
course of improvement) to that state of comparative perfection in which
it was left by Watt. Short as the period of time has been which the
inventor has devoted to this enterprise, it has, nevertheless, been
demonstrated, to the satisfaction of many scientific men of the first
eminence, that the design in all its details, reduced, as it is, to a
system of mechanical drawings, is complete; and requires only to be
constructed in conformity with those plans, to realize all that its
inventor has promised.

[Footnote 1: Watt commenced his investigations respecting the
steam-engine in 1763, between which time, and the year 1782 inclusive,
he took out several patents for improvements in details. Bolton and Watt
had expended the above sum on their improvements before they began to
receive any return.]

With a view to remove and correct erroneous impressions, and at
the same time to convert the vague sense of wonder at what seems
incomprehensible, with which this project is contemplated by the public
in general, into a more rational and edifying sentiment, it is our
purpose in the purpose in the present article.

_First_, To show, the immense importance of any method by which numerical
tables, absolutely accurate in every individual copy, may be produced
with facility and cheapness. This we shall establish by conveying to the
reader some notion of the number and variety of tables published in
every country of the world to which civilisation has extended, a large
portion of which have been produced at the public expense; by showing
also, that they are nevertheless rendered inefficient, to a greater or
less extent, by the prevalence of errors in them; that these errors
pervade not merely tables produced by individual labour and enterprise,
but that they vitiate even those on which national resources have been
prodigally expended, and to which the highest mathematical ability,
which the most enlightened nations of the world could command, has been
unsparingly and systematically directed.

_Secondly_, To attempt to convey to the reader a general notion of the
mathematical principle on which the calculating machinery is founded,
and of the manner in which this principle is brought into practical
operation, both in the process of calculating and printing. It would be
incompatible with the nature of this review, and indeed impossible
without the aid of numerous plans, sections, and elevations, to convey
clear and precise notions of the details of the means by which the
process of reasoning is performed by inanimate matter, and the arbitrary
and capricious evolutions of the fingers of typographical compositors
are reduced to a system of wheel-work. We are, nevertheless, not without
hopes of conveying, even to readers unskilled in mathematics, some
satisfactory notions of a general nature on this subject.

_Thirdly_, To explain the actual state of the machinery a the present
time; what progress has been made towards its completion; and what are
the probable causes of those delays in its progress, which must be a
subject of regret to all friends of science. We shall indicate what
appears to us the best and most practicable course to prevent the
unnecessary recurrence of such obstructions for the future, and to bring
this noble project to a speedy and successful issue.


Viewing the infinite extent and variety of the tables which have been
calculated and printed, from the earliest periods of human civilisation
to the present time, we feel embarrassed with the difficulties of the
task which we have imposed on ourselves;--that of attempting to convey
to readers unaccustomed to such speculations, any thing approaching to
an adequate idea of them. These tables are connected with the various
sciences, with almost every department of the useful arts, with commerce
in all its relations; but above all, with Astronomy and Navigation. So
important have they been considered, that in many instances large sums
have been appropriated by the most enlightened nations in the production
of them; and yet so numerous and insurmountable have been the
difficulties attending the attainment of this end, that after all, even
navigators, putting aside every other department of art and science,
have, until very recently, been scantily and imperfectly supplied with
the tables indispensably necessary to determine their position at sea.

The first class of tables which naturally present themselves, are those
of Multiplication. A great variety of extensive multiplication tables
have been published from an early period in different countries; and
especially tables of _Powers_, in which a number is multiplied by itself
successively. In Dodson's _Calculator_ we find a table of multiplication
extending as far as 10 times 1000.[2] In 1775, a still more extensive
table was published to 10 times 10,000. The Board of Longitude
subsequently employed the late Dr Hutton to calculate and print various
numerical tables, and among others, a multiplication table extending as
far as 100 times 1000; tables of the squares of numbers, as far as
25,400; tables of cubes, and of the first ten powers of numbers, as far
as 100.[3] In 1814, Professor Barlow, of Woolwich, published, in an
octavo volume, the squares, cubes, square roots, cube roots, and
reciprocals of all numbers from 1 to 10,000; a table of the first ten
powers of all numbers from 1 to 100, and of the fourth and fifth powers
of all numbers from 100 to 1000.

[Footnote 2: Dodson's _Calculator_. 4to. London: 1747.]

[Footnote 3: Hutton's _Tables of Products and Powers_. Folio.
London; 1781.]

Tables of Multiplication to a still greater extent have been published
in France. In 1785, was published an octavo volume of tables of the
squares, cubes, square roots, and cube roots of all numbers from 1 to
10,000; and similar tables were again published in 1801. In 1817,
multiplication tables were published in Paris by Voisin; and similar
tables, in two quarto volumes, in 1824, by the French Board of
Longitude, extending as far as a thousand times a thousand. A table of
squares was published in 1810, in Hanover; in 1812, at Leipzig; in 1825,
at Berlin; and in 1827, at Ghent. A table of cubes was published in
1827, at Eisenach; in the same year a similar table at Ghent; and one of
the squares of all numbers as far as 10,000, was published in that year,
in quarto, at Bonn. The Prussian Government has caused a multiplication
table to be calculated and printed, extending as far as 1000 times 1000.
Such are a few of the tables of this class which have been published in
different countries.

This class of tables may be considered as purely arithmetical, since the
results which they express involve no other relations than the
arithmetical dependence of abstract numbers upon each other. When
numbers, however, are taken in a concrete sense, and are applied to
express peculiar modes of quantity,--such as angular, linear,
superficial, and solid magnitudes,--a new set of numerical relations
arise, and a large number of computations are required.

To express angular magnitude, and the various relations of linear
magnitude with which it is connected, involves the consideration of a
vast variety of Geometrical and Trigonometrical tables; such as tables
of the natural sines, co-sines, tangents, secants, co-tangents, &c. &c.;
tables of arcs and angles in terms of the radius; tables for the
immediate solution of various cases of triangles, &c. Volumes without
number of such tables have been from time to time computed and
published. It is not sufficient, however, for the purposes of
computation to tabulate these immediate trigonometrical functions. Their
squares[4] and higher powers, their square roots, and other roots, occur
so frequently, that it has been found expedient to compute tables for
them, as well as for the same functions of abstract numbers.

[Footnote 4: The squares of the sines of angles are extensively used in
the calculations connected with the theory of the tides. Not aware that
tables of these squares existed, Bouvard, who calculated the tides for
Laplace, underwent the labour of calculating the square of each
individual sine in every case in which it occurred.]

The measurement of linear, superficial, and solid magnitudes, in the
various forms and modifications in which they are required in the arts,
demands another extensive catalogue of numerical tables. The surveyor,
the architect, the builder, the carpenter, the miner, the ganger, the
naval architect, the engineer, civil and military, all require the aid
of peculiar numerical tables, and such have been published in all
countries.

The increased expedition and accuracy which was introduced into the art
of computation by the invention of Logarithms, greatly enlarged the
number of tables previously necessary. To apply the logarithmic method,
it was not merely necessary to place in the hands of the computist
extensive tables of the logarithms of the natural numbers, but likewise
to supply him with tables in which he might find already calculated the
logarithms of those arithmetical, trigonometrical, and geometrical
functions of numbers, which he has most frequent occasion to use. It
would be a circuitous process, when the logarithm of a sine or co-sine
of an angle is required, to refer, first to the table of sines, or
co-sines, and thence to the table of the logarithms of natural numbers.
It was therefore found expedient to compute distinct tables of the
logarithms of the sines, co-sines, tangents, &c., as well as of various
other functions frequently required, such as sums, differences, &c.

Great as is the extent of the tables we have just enumerated, they bear
a very insignificant proportion to those which remain to be mentioned.
The above are, for the most part, general in their nature, not belonging
particularly to any science or art. There is a much greater variety of
tables, whose importance is no way inferior, which are, however, of a
more special nature: Such are, for example, tables of interest,
discount, and exchange, tables of annuities, and other tables necessary
in life insurances; tables of rates of various kinds necessary in
general commerce. But the science in which, above all others, the most
extensive and accurate tables are indispensable, is Astronomy; with the
improvement and perfection of which is inseparably connected that of the
kindred art of Navigation. We scarcely dare hope to convey to the
general reader any thing approaching to an adequate notion of the
multiplicity and complexity of the tables necessary for the purposes of
the astronomer and navigator. We feel, nevertheless, that the truly
national importance which must attach to any perfect and easy means of
producing those tables cannot be at all estimated, unless we state some
of the previous calculations necessary in order to enable the mariner to
determine, with the requisite certainty and precision, the place of his
ship.

In a word, then, all the purely arithmetical, trigonometrical, and
logarithmic tables already mentioned, are necessary, either immediately
or remotely, for this purpose. But in addition to these, a great number
of tables, exclusively astronomical, are likewise indispensable. The
predictions of the astronomer, with respect to the positions and motions
of the bodies of the firmament, are the means, and the only means, which
enable the mariner to prosecute his art. By these he is enabled to
discover the distance of his ship from the Line, and the extent of his
departure from the meridian of Greenwich, or from any other meridian to
which the astronomical predictions refer. The more numerous, minute, and
accurate these predictions can be made, the greater will be the
facilities which can be furnished to the mariner. But the computation of
those tables, in which the future position of celestial objects are
registered, depend themselves upon an infinite variety of other tables
which never reach the hands of the mariner. It cannot be said that there
is any table whatever, necessary for the astronomer, which is
unnecessary for the navigator.

The purposes of the marine of a country whose interests are so
inseparably connected as ours are with the improvement of the art of
navigation, would be very inadequately fulfilled, if our navigators were
merely supplied with the means of determining by _Nautical Astronomy_ the
position of a ship at sea. It has been well observed by the Committee of
the Astronomical Society, to whom the recent improvement of the Nautical
Almanac was confided, that it is not by those means merely by which the
seaman is enabled to determine the position of his vessel at sea, that
the full intent and purpose of what is usually called _Nautical Astronomy_
are answered. This object is merely a part of that comprehensive and
important subject; and might be attained by a very cheap publication,
and without the aid of expensive instruments. A not less important and
much more difficult part of nautical science has for its object to
determine the precise position of various interesting and important
points on the surface of the earth,--such as remarkable headlands,
ports, and islands; together with the general trending of the coast
between well-known harbours. It is not necessary to point out here how
important such knowledge is to the mariner. This knowledge, which may be
called _Nautical Geography_, cannot be obtained by the methods of
observation used on board ship, but requires much more delicate and
accurate instruments, firmly placed upon the solid ground, besides all
the astronomical aid which can be afforded by the best tables, arranged
in the most convenient form for immediate use. This was Dr Maskelyne's
view of the subject, and his opinion has been confirmed by the repeated
wants and demands of those distinguished navigators who have been
employed in several recent scientific expeditions.[5]

[Footnote 5: Report of the Committee of the Astronomical Society prefixed
to the Nautical Almanac for 1834.]

Among the tables _directly_ necessary for navigation, are those which
predict the position of the centre of the sun from hour to hour. These
tables include the sun's right ascension and declination, daily, at
noon, with the hourly change in these quantities. They also include the
equation of time, together with its hourly variation.

Tables of the moon's place for every hour, are likewise necessary,
together with the change of declination for every ten minutes. The lunar
method of determining the longitude depends upon tables containing the
predicted distances of the moon from the sun, the principal planets, and
from certain conspicuous fixed stars; which distances being observed by
the mariner, he is enabled thence to discover the _time_ at the meridian
from which the longitude is measured; and, by comparing that time with
the time known or discoverable in his actual situation, he infers his
longitude. But not only does the prediction of the position of the moon,
with respect to these celestial objects, require a vast number of
numerical tables, but likewise the observations necessary to be made by
the mariner, in order to determine the lunar distances, also require
several tables. To predict the exact position of any fixed star,
requires not less than ten numerical tables peculiar to that star; and
if the mariner be furnished (as is actually the case) with tables of the
predicted distances of the moon from one hundred such stars, such
predictions must require not less than a thousand numerical tables.
Regarding the range of the moon through the firmament, however, it will
readily be conceived that a hundred stars form but a scanty supply;
especially when it is considered that an accurate method of determining
the longitude, consists in observing the extinction of a star by the
dark edge of the moon. Within the limits of the lunar orbit there are
not less than one thousand stars, which are so situated as to be in the
moon's path, and therefore to exhibit, at some period or other, those
desirable occultations. These stars are also of such magnitudes, that
their occultations may be distinctly observed from the deck, even when
subject to all the unsteadiness produced by an agitated sea. To predict
the occultations of such stars, would require not less than ten thousand
tables. The stars from which lunar distances might be taken are still
more numerous; and we may safely pronounce, that, great as has been the
improvement effected recently in our Nautical Almanac, it does not yet
furnish more than a small fraction of that aid to navigation (in the
large sense of that term), which, with greater facility, expedition, and
economy in the calculation and printing of tables, it might be made to
supply.

Tables necessary to determine the places of the planets are not less
necessary than those for the sun, moon, and stars. Some notion of the
number and complexity of these tables may be formed, when we state that
the positions of the two principal planets, (and these the most
necessary for the navigator,) Jupiter and Saturn, require each not less
than one hundred and sixteen tables. Yet it is not only necessary to
predict the position of these bodies, but it is likewise expedient to
tabulate the motions of the four satellites of Jupiter, to predict the
exact times at which they enter his shadow, and at which their shadows
cross his disc, as well as the times at which they are interposed
between him and the Earth, and he between them and the Earth.

Among the extensive classes of tables here enumerated, there are several
which are in their nature permanent and unalterable, and would never
require to be recomputed, if they could once be computed with perfect
accuracy on accurate data; but the data on which such computations are
conducted, can only be regarded as approximations to truth, within
limits the extent of which must necessarily vary with our knowledge of
astronomical science. It has accordingly happened, that one set of
tables after another has been superseded with each advance of
astronomical science. Some striking examples of this may not be
uninstructive. In 1765, the Board of Longitude paid to the celebrated
Euler the sum of L.300, for furnishing general formulæ for the
computation of lunar tables. Professor Mayer was employed to calculate
the tables upon these formulæ, and the sum of L.3000 was voted for them
by the British Parliament, to his widow, after his decease. These tables
had been used for ten years, from 1766 to 1776, in computing the
Nautical Almanac, when they were superseded by new and improved tables,
composed by Mr Charles Mason, under the direction of Dr Maskelyne, from
calculations made by order of the Board of Longitude, on the
observations of Dr Bradley. A farther improvement was made by Mason in
1780; but a much more extensive improvement took place in the lunar
calculations by the publication of the tables of the Moon, by M. Bürg,
deduced from Laplace's theory, in 1806. Perfect, however, as Bürg's
tables were considered, at the time of their publication, they were,
within the short period of six years, superseded by a more accurate set
of tables published by Burckhardt in 1812; and these also have since
been followed by the tables of Damoiseau. Professor Schumacher has
calculated by the latter tables his ephemeris of the Planetary Lunar
Distances, and astronomers will hence be enabled to put to the strict
test of observation the merits of the tables of Burckhardt and
Damoiseau.[6]

[Footnote 6: A comparison of the results for 1834, will be found in the
Nautical Almanac for 1835.]

The solar tables have undergone, from time to time, similar changes. The
solar tables of Mayer were used in the computation of the Nautical
Almanac, from its commencement in 1767, to 1804 inclusive. Within the
six years immediately succeeding 1804, not less than three successive
sets of solar tables appeared, each improving on the other; the first by
Baron de Zach, the second by Delambre, under the direction of the French
Board of Longitude, and the third by Carlini. The last, however, differ
only in arrangement from those of Delambre.

Similar observations will be applicable to the tables of the principal
planets. Bouvard published, in 1803, tables of Jupiter and Saturn; but
from the improved state of astronomy, he found it necessary to recompute
these tables in 1821.

Although it is now about thirty years since the discovery of the four
new planets, Ceres, Pallas, Juno, and Vesta, it was not till recently
that tables of their motions were published. They have lately appeared
in Encke's Ephemeris.

We have thus attempted to convey some notion (though necessarily a very
inadequate one) of the immense extent of numerical tables which it has
been found necessary to calculate and print for the purposes of the arts
and sciences. We have before us a catalogue of the tables contained in
the library of one private individual, consisting of not less than one
hundred and forty volumes. Among these there are no duplicate copies:
and we observe that many of the most celebrated voluminous tabular works
are not contained among them. They are confined exclusively to
arithmetical and trigonometrical tables; and, consequently, the myriad
of astronomical and nautical tables are totally excluded from them.
Nevertheless, they contain an extent of printed surface covered with
figures amounting to above sixteen thousand square feet. We have taken
at random forty of these tables, and have found that the number of
errors _acknowledged_ in the respective errata, amounts to above _three
thousand seven hundred_.

To be convinced of the necessity which has existed for accurate
numerical tables, it will only be necessary to consider at what an
immense expenditure of labour and of money even the imperfect ones which
we possess have been produced.

To enable the reader to estimate the difficulties which attend the
attainment even of a limited degree of accuracy, we shall now explain
some of the expedients which have been from time to time resorted to for
the attainment of numerical correctness in calculating and printing
them.

Among the scientific enterprises which the ambition of the French nation
aspired to during the Republic, was the construction of a magnificent
system of numerical tables. Their most distinguished mathematicians were
called upon to contribute to the attainment of this important object;
and the superintendence of the undertaking was confided to the
celebrated Prony, who co-operated with the government in the adoption of
such means as might be expected to ensure the production of a system of
logarithmic and trigonometric tables, constructed with such accuracy
that they should form a monument of calculation the most vast and
imposing that had ever been executed, or even conceived. To accomplish
this gigantic task, the principle of the division of labour, found to be
so powerful in manufactures, was resorted to with singular success. The
persons employed in the work were divided into three sections: the first
consisted of half a dozen of the most eminent analysts. Their duty was
to investigate the most convenient mathematical formulæ, which should
enable the computers to proceed with the greatest expedition and
accuracy by the method of Differences, of which we shall speak more
fully hereafter. These formulæ, when decided upon by this first
section, were handed over to the second section, which consisted of
eight or ten properly qualified mathematicians. It was the duty of this
second section to convert into numbers certain general or algebraical
expressions which occurred in the formulæ, so as to prepare them for,
the hands of the computers. Thus prepared, these formulæ were handed
over to the third section, who formed a body of nearly one hundred
computers. The duty of this numerous section was to compute the numbers
finally intended for the tables. Every possible precaution was of course
taken to ensure the numerical accuracy of the results. Each number was
calculated by two or more distinct and independent computers, and its
truth and accuracy determined by the coincidence of the results thus
obtained.

The body of tables thus calculated occupied in manuscript _seventeen_
folio volumes.[7]

[Footnote 7: These tables were never published. The printing of them was
commenced by Didot, and a small portion was actually stereotyped, but
never published. Soon after the commencement of the undertaking, the
sudden fall of the assignats rendered it impossible for Didot to fulfil
his contract with the government. The work was accordingly abandoned,
and has never since been resumed. We have before us a copy of 100 pages
folio of the portion which was printed at the time the work was stopped,
given to a friend on a late occasion by Didot himself. It was remarked
in this, as in other similar cases, that the computers who committed
fewest errors were those who understood nothing beyond the process of
addition.]

As an example of the precautions which have been considered necessary to
guard against errors in the calculation of numerical tables, we shall
further state those which were adopted by Mr Babbage, previously to the
publication of his tables of logarithms. In order to render the terminal
figure of tables in which one or more decimal places are omitted as
accurate as it can be, it has been the practice to compute one or more
of the succeeding figures; and if the first omitted figure be greater
than 4, then the terminal figure is always increased by 1, since the
value of the tabulated number is by such means brought nearer to the
truth.[8] The tables of Callet, which were among the most accurate
published logarithms, and which extended to seven places of decimals,
were first carefully compared with the tables of Vega, which extended to
ten places, in order to discover whether Callet had made the above
correction of the final figure in every case where it was necessary.
This previous precaution being taken, and the corrections which appeared
to be necessary being made in a copy of Callet's tables, the proofs of
Mr Babbage's tables were submitted to the following test: They were
first compared, number by number, with the corrected copy of Callet's
logarithms; secondly, with Hutton's logarithms; and thirdly, with Vega's
logarithms. The corrections thus suggested being marked in the proofs,
corrected revises were received back. These revises were then again
compared, number by number, first with Vega's logarithms; secondly, with
the logarithms of Callet; and thirdly, as far as the first 20,000
numbers, with the corresponding ones in Briggs's logarithms. They were
now returned to the printer, and were stereotyped; proofs were taken
from the stereotyped plates, which were put through the following
ordeal: They were first compared once more with the logarithms of Vega
as far as 47,500; they were then compared with the whole of the
logarithms of Gardner; and next with the whole of Taylor's logarithms;
and as a last test, they were transferred to the hands of a different
set of readers, and were once more compared with Taylor. That these
precautions were by no means superfluous may be collected from the
following circumstances mentioned by Mr Babbage: In the sheets read
immediately previous to stereotyping, thirty-two errors were detected;
after stereotyping, eight more were found, and corrected in the plates.

[Footnote 8: Thus suppose the number expressed at full length were
3.1415927. If the table extend to no more than four places of decimals,
we should tabulate the number 3.1416 and not 3.1415. The former would be
evidently nearer to the true number 3.1415927.]

By such elaborate and expensive precautions many of the errors of
computation and printing may certainly be removed; but it is too much to
expect that in general such measures can be adopted; and we accordingly
find by far the greater number of tables disfigured by errors, the
extent of which is rather to be conjectured than determined. When the
nature of a numerical table is considered,--page after page densely
covered with figures, and with nothing else,--the chances against the
detection of any single error will be easily comprehended; and it may
therefore be fairly presumed, that for one error which may happen to be
detected, there must be a great number which escape detection.
Notwithstanding this difficulty, it is truly surprising how great a
number of numerical errors have been detected by individuals no
otherwise concerned in the tables than in their use. Mr Baily states
that he has himself detected in the solar and lunar tables, from which
our Nautical Almanac was for a long period computed, more than five
hundred errors. In the multiplication table already mentioned, computed
by Dr Hutton for the Board of Longitude, a single page was examined and
recomputed: it was found to contain about forty errors.

In order to make the calculations upon the numbers found in the
Ephemeral Tables published in the Nautical Almanac, it is necessary that
the mariner should be supplied with certain permanent tables. A volume
of these, to the number of about thirty, was accordingly computed, and
published at national expense, by order of the Board of Longitude,
entitled 'Tables requisite to be used with the Nautical Ephemeris for
finding the latitude and longitude at sea.' In the first edition of
these requisite tables, there were detected, by one individual, above a
thousand errors.

The tables published by the Board of Longitude for the correction of the
observed distances of the moon from certain fixed stars, are followed by
a table of acknowledged errata, extending to seven folio pages, and
containing more than eleven hundred errors. Even this table of errata
itself is not correct: a considerable number of errors have been
detected in it, so that errata upon errata have become necessary.

One of the tests most frequently resorted to for the detection of errors
in numerical tables, has been the comparison of tables of the same kind,
published by different authors. It has been generally considered that
those numbers in which they are found to agree must be correct; inasmuch
as the chances are supposed to be very considerable against two or more
independent computers falling into precisely the same errors. How far
this coincidence may be safely assumed as a test of accuracy we shall
presently see.

A few years ago, it was found desirable to compute some very accurate
logarithmic tables for the use of the great national survey of Ireland,
which was then, and still is in progress; and on that occasion a careful
comparison of various logarithmic tables was made. Six remarkable errors
were detected, which were found to be common to several apparently
independent sets of tables. This singular coincidence led to an
unusually extensive examination of the logarithmic tables published both
in England and in other countries; by which it appeared that thirteen
sets of tables, published in London between the years 1633 and 1822, all
agreed in these six errors. Upon extending the enquiry to foreign
tables, it appeared that two sets of tables published at Paris, one at
Gouda, one at Avignon, one at Berlin, and one at Florence, were infected
by exactly the same six errors. The only tables which were found free
from them were those of Vega, and the more recent impressions of Callet.
It happened that the Royal Society possessed a set of tables of
logarithms printed in the Chinese character, and on Chinese paper,
consisting of two volumes: these volumes contained no indication or
acknowledgment of being copied from any other work. They were examined;
and the result was the detection in them of the same six errors.[9]

[Footnote 9: Memoirs Ast. Soc. vol. III, p. 65.]

It is quite apparent that this remarkable coincidence of error must have
arisen from the various tables being copied successively one from
another. The earliest work in which they appeared was Vlacq's
Logarithms, (folio, Gouda, 1628); and from it, doubtless, those which
immediately succeeded it in point of time were copied; from which the
same errors were subsequently transcribed into all the other, including
the Chinese logarithms.

The most certain and effectual check upon errors which arise in the
process of computation, is to cause the same computations to be made by
separate and independent computers; and this check is rendered still
more decisive if they make their computations by different methods. It
is, nevertheless, a remarkable fact, that several computers, working
separately and independently, do frequently commit precisely the same
error; so that falsehood in this case assumes that character of
consistency, which is regarded as the exclusive attribute of truth.
Instances of this are familiar to most persons who have had the
management of the computation of tables. We have reason to know, that M.
Prony experienced it on many occasions in the management of the great
French tables, when he found three, and even a greater number of
computers, working separately and independently, to return him the same
numerical result, and _that result wrong_. Mr Stratford, the conductor of
the Nautical Almanac, to whose talents and zeal that work owes the
execution of its recent improvements, has more than once observed a
similar occurrence. But one of the most signal examples of this kind, of
which we are aware, is related by Mr Baily. The catalogue of stars
published by the Astronomical Society was computed by two separate and
independent persons, and was afterwards compared and examined with great
care and attention by Mr Stratford. On examining this catalogue, and
recalculating a portion of it, Mr Baily discovered an error in the case
of the star, χ Cephei. Its right ascension was calculated _wrongly_, and
yet _consistently_, by two computers working separately. Their numerical
results agreed precisely in every figure; and Mr Stratford, on examining
the catalogue, failed to detect the error. Mr Baily having reason, from
some discordancy which he observed, to suspect an error, recomputed the
place of the star with a view to discover it; and he himself, in the
first instance, obtained precisely _the same erroneous numerical result_.
It was only on going over the operation a second time that he
_accidentally_ discovered that he had inadvertently committed the same
error.[10]

[Footnote 10: Memoirs Ast. Soc. vol. iv., p. 290.]

It appears, therefore, that the coincidence of different tables, even
when it is certain that they could not have been copied one from
another, but must have been computed independently, is not a decisive
test of their correctness, neither is it possible to ensure accuracy by
the device of separate and independent computation.

Besides the errors incidental to the process of computation, there are
further liabilities in the process of transcribing the final results of
each calculation into the fair copy of the table designed for the
printer. The next source of error lies with the compositor, in
transferring this copy into type. But the liabilities to error do not
stop even here; for it frequently happens, that after the press has been
fully corrected, errors will be produced in the process of printing. A
remarkable instance of this occurs in one of the six errors detected in
so many different tables already mentioned. In one of these cases, the
last five figures of two successive numbers of a logarithmic table were
the following:--

                        35875
                        10436.

Now, both of these are erroneous; the figure 8 in the first line should
be 4, and the figure 4 in the second should be 8. It is evident that the
types, as first composed, were correct; but in the course of printing,
the two types 4 and 8 being loose, adhered to the inking-balls, and were
drawn out: the pressmen in replacing them transposed them, putting the 8
_above_ and the 4 _below_, instead of _vice versa_. It would be a curious
enquiry, were it possible to obtain all the copies of the original
edition of Vlacq's Logarithms, published at Gouda in 1628, from which
this error appears to have been copied in all the subsequent tables, to
ascertain whether it extends through the entire edition. It would
probably, nay almost certainly, be discovered that some of the copies of
that edition are correct in this number, while others are incorrect; the
former having been worked off before the transposition of the types.

It is a circumstance worthy of notice, that this error in Vlacq's tables
has produced a corresponding error in a variety of other tables deduced
from them, _in which nevertheless the erroneous figures in Vlacq are
omitted_. In no less than sixteen sets of tables published at various
times since the publication of Vlacq, in which the logarithms extend
only to seven places of figures, the error just mentioned in the _eighth
place_ in Vlacq causes a corresponding error in the _seventh_ place. When
the last three figures are omitted in the first of the above numbers,
the seventh figure should be 5, inasmuch as the first of the omitted
figures is under 5: the erroneous insertion, however, of the figure 8 in
Vlacq has caused the figure 6 to be substituted for 5 in the various
tables just alluded to. For the same reason, the erroneous occurrence of
4 in the second number has caused the adoption of a 0 instead of a 1 in
the seventh place in the other tables. The only tables in which this
error does not occur are those of Vega, the more recent editions of
Callet, and the still later Logarithms of Mr Babbage.

The _Opus Palatinum_, a work published in 1596, containing an extensive
collection of trigonometrical tables, affords a remarkable instance of a
tabular error; which, as it is not generally known, it may not be
uninteresting to mention here. After that work had been for several
years in circulation in every part of Europe, it was discovered that the
commencement of the table of co-tangents and co-secants was vitiated by
an error of considerable magnitude. In the first co-tangent the last
nine places of figures were incorrect; but from the manner in which the
numbers of the table were computed, the error was gradually, though
slowly, diminished, until at length it became extinguished in the
eighty-sixth page. After the detection of this extensive error, Pitiscus
undertook the recomputation of the eighty-six erroneous pages. His
corrected calculation was printed, and the erroneous part of the
remaining copies of the _Opus Palatinum_ was cancelled. But as the
corrected table of Pitiscus was not published until 1607,--thirteen
years after the original work,--the erroneous part of the volume was
cancelled in comparatively few copies, and consequently correct copies
of the work are now exceedingly rare. Thus, in the collection of tables
published by M. Schulze,[11] the whole of the erroneous part of the _Opus
Palatinum_ has been adopted; he having used the copy of that work which
exists in the library of the Academy of Berlin, and which is one of
those copies in which the incorrect part was not cancelled. The
corrected copies of this work may be very easily distinguished at
present from the erroneous ones: it happened that the former were
printed with a very bad and worn-out type, and upon paper of a quality
inferior to that of the original work. On comparing the first eighty-six
pages of the volume with the succeeding ones, they are, therefore,
immediately distinguishable in the corrected copies. Besides this test,
there is another, which it may not be uninteresting to point out:--At
the bottom of page 7 in the corrected copies, there is an error in the
position of the words _basis_ and _hypothenusa_, their places being
interchanged. In the original uncorrected work this error does not
exist.

[Footnote 11: _Recueil des Tables Logarithmiques et Trigonometriques_.
Par J. C. Schulze. 2 vols. Berlin: 1778.]

At the time when the calculation and publication of Taylor's Logarithms
were undertaken, it so happened that a similar work was in progress in
France; and it was not until the calculation of the French work was
completed, that its author was informed of the publication of the
English work. This circumstance caused the French calculator to
relinquish the publication of his tables. The manuscript subsequently
passed into the library of Delambre, and, after his death, was purchased
at the sale of his books, by Mr Babbage, in whose possession it now is.
Some years ago it was thought advisable to compare these manuscript
tables with Taylor's Logarithms, with a view to ascertain the errors in
each, but especially in Taylor. The two works were peculiarly well
suited for the attainment of this end; as the circumstances under which
they were produced, rendered it quite certain that they were computed
independently of each other. The comparison was conducted under the
direction of the late Dr Young, and the result was the detection of the
following nineteen errors in Taylor's Logarithms. To enable those who
used Taylor's Logarithms to make the necessary corrections in them, the
corrections of the detected errors appeared as follows in the Nautical
Almanac for 1832.


ERRATA, _detected in_ Taylor's _Logarithms_. _London: 4to_, 1792.

                                 °  '  "
 1    _E_        Co-tangent of    1.35.35    _for_ 43671    _read_ 42671
 2    _M_        Co-tangent of    4. 4.49     ---  66976     ----  66979
 3               Sine of          4.23.38     ---  43107     ----  43007
 4               Sine of          4.23.39     ---  43381     ----  43281
 5    _S_        Sine of          6.45.52     ---  10001     ----  11001
 6    _Kk_       Co-sine of      14.18. 3     ---   3398     ----   3298
 7    _Ss_       Tangent of      18. 1.56     ---   5064     ----   6064
 8    _Aaa_      Co-tangent of   21.11.14     ---   6062     ----   5962
 9    _Ggg_      Tangent of      23.48.19     ---   6087     ----   5987
 10              Co-tangent of   23.48.19     ---   3913     ----   4013
 11   _Iii_      Sine of         25. 5. 4     ---   3173     ----   3183
 12              Sine of         25. 5. 5     ---   3218     ----   3228
 13              Sine of         25. 5. 6     ---   3263     ----   3273
 14              Sine of         25. 5. 7     ---   3308     ----   3318
 15              Sine of         25. 5. 8     ---   3353     ----   3363
 16              Sine of         25. 5. 9     ---   3398     ----   3408
 17   _Qqq_      Tangent of      28.19.39     ---   6302     ----   6402
 18   _4H_       Tangent of      35.55.51     ---   1681     ----   1581
 19   _4K_       Co-sine of      37.29. 2     ---   5503     ----   5603


An error being detected in this list of ERRATA, we find, in the Nautical
Almanac for the year 1833, the following ERRATUM of the ERRATA of
Taylor's Logarithms:--

'In the list of ERRATA detected in Taylor's Logarithms, for _cos_. 4°
18' 3", read cos. 14° 18' 2".'

Here, however, confusion is worse confounded; for a new error, not
before existing, and of much greater magnitude, is introduced! It will
be necessary, in the Nautical Almanac for 1836, (that for 1835 is
already published,) to introduce the following:

ERRATUM of the ERRATUM of the ERRATA of TAYLOR's _Logarithms_. For cos. 4°
18' 3", _read_ cos. 14° 18' 3".

If proof were wanted to establish incontrovertibly the utter
impracticability of precluding numerical errors in works of this nature,
we should find it in this succession of error upon error, produced, in
spite of the universally acknowledged accuracy and assiduity of the
persons at present employed in the construction and management of the
Nautical Almanac. It is only by the _mechanical fabrication of tables_
that such errors can be rendered impossible.

On examining this list with attention, we have been particularly struck
with the circumstances in which these errors appear to have originated.
It is a remarkable fact, that of the above nineteen errors, eighteen
have arisen from mistakes in _carrying_. Errors 5, 7, 10, 11, 12, 13, 14,
15, 16, 17, 19, have arisen from a carriage being neglected; and errors
1, 3, 4, 6, 8, 9, and 18, from a carriage being made where none should
take place. In four cases, namely, errors 8, 9, 10, and 16, this has
caused _two_ figures to be wrong. The only error of the nineteen which
appears to have been a press error is the second; which has evidently
arisen from the type 9 being accidentally inverted, and thus becoming a
6. This may have originated with the compositor, but more probably it
took place in the press-work; the type 9 being accidentally drawn out of
the form by the inking-ball, as mentioned in a former case, and on being
restored to its place, inverted by the pressman.

There are two cases among the above errata, in which an error, committed
in the calculation of one number, has evidently been the cause of other
errors. In the third erratum, a wrong carriage was made, in computing
the sine of 4° 23' 38". The next number of the table was vitiated
by this error; for we find the next erratum to be in the sine of 4°
23' 39", in which the figure similarly placed is 1 in excess. A
still more extensive effect of this kind appears in errata 11, 12, 13,
14, 15, 16. A carriage was neglected in computing the sine of 25° 5'
4", and this produced a corresponding error in the five following
numbers of the table, which are those corrected in the five following
errata.

This frequency of errors arising in the process of carrying, would
afford a curious subject of metaphysical speculation respecting the
operation of the faculty of memory. In the arithmetical process, the
memory is employed in a twofold way;--in ascertaining each successive
figure of the calculated result by the recollection of a table committed
to memory at an early period of life; and by another act of memory, in
which the number carried from column to column is retained. It is a
curious fact, that this latter circumstance, occurring only the moment
before, and being in its nature little complex, is so much more liable
to be forgotten or mistaken than the results of rather complicated
tables. It appears, that among the above errata, the errors 5, 7, 10,
11, 17, 19, have been produced by the computer forgetting a carriage;
while the errors 1, 3, 6, 8, 9, 18, have been produced by his making a
carriage improperly. Thus, so far as the above list of errata affords
grounds for judging, it would seem, (contrary to what might be
expected,) that the error by which improper carriages are made is as
frequent as that by which necessary carriages are overlooked.


We trust that we have succeeded in proving, first, the great national
and universal utility of numerical tables, by showing the vast number of
them, which have been calculated and published; secondly, that more
effectual means are necessary to obtain such tables suitable to the
present state of the arts, sciences and commerce, by showing that the
existing supply of tables, vast as it certainly is, is still scanty, and
utterly inadequate to the demands of the community;--that it is
rendered inefficient, not only in quantity, but in quality, by its want
of numerical correctness; and that such numerical correctness is
altogether unattainable until some more perfect method be discovered,
not only of calculating the numerical results, but of tabulating
these,--of reducing such tallies to type, and of printing that type so
as to intercept the possibility of error during the press-work. Such are
the ends which are proposed to be attained by the calculating machinery
invented by Mr Babbage.

The benefits to be derived from this invention cannot be more strongly
expressed than they have been by Mr Colebrooke, President of the
Astronomical Society, on the occasion of presenting the gold medal voted
by that body to Mr Babbage:--'In no department of science, or of the
arts, does this discovery promise to be so eminently useful as in that
of astronomy, and its kindred sciences, with the various arts dependent
on them. In none are computations more operose than those which
astronomy in particular requires;--in none are preparatory facilities
more needful;--in none is error more detrimental. The practical
astronomer is interrupted in his pursuit, and diverted from his task of
observation by the irksome labours of computation, or his diligence in
observing becomes ineffectual for want of yet greater industry of
calculation. Let the aid which tables previously computed afford, be
furnished to the utmost extent which mechanism has made attainable
through Mr Babbage's invention, and the most irksome portion of the
astronomer's task is alleviated, and a fresh impulse is given to
astronomical research.'

The first step in the progress of this singular invention was the
discovery of some common principle which pervaded numerical tables of
every description; so that by the adoption of such a principle as the
basis of the machinery, a corresponding degree of generality would be
conferred upon its calculations. Among the properties of numerical
functions, several of a general nature exist; and it was a matter of no
ordinary difficulty, and requiring no common skill, to select one which
might, in all respects, be preferable to the others. Whether or not that
which was selected by Mr Babbage affords the greatest practical
advantages, would be extremely difficult to decide--perhaps impossible,
unless some other projector could be found possessed of sufficient
genius, and sustained by sufficient energy of mind and character, to
attempt the invention of calculating machinery on other principles. The
principle selected by Mr Babbage as the basis of that part of the
machinery which calculates, is the Method of Differences; and he has in
fact literally thrown this mathematical principle into wheel-work. In
order to form a notion of the nature of the machinery, it will be
necessary, first to convey to the reader some idea of the mathematical
principle just alluded to.

A numerical table, of whatever kind, is a series of numbers which
possess some common character, and which proceed increasing or
decreasing according to some general law. Supposing such a series
continually to increase, let us imagine each number in it to be
subtracted from that which follows it, and the remainders thus
successively obtained to be ranged beside the first, so as to form
another table: these numbers are called the _first differences_. If we
suppose these likewise to increase continually, we may obtain a third
table from them by a like process, subtracting each number from the
succeeding one: this series is called the _second differences_. By
adopting a like method of proceeding, another series may be obtained,
called the _third differences_; and so on. By continuing this process, we
shall at length obtain a series of differences, of some order, more or
less high, according to the nature of the original table, in which we
shall find the same number constantly repeated, to whatever extent the
original table may have been continued; so that if the next series of
differences had been obtained in the same manner as the preceding ones,
every term of it would be 0. In some cases this would continue to
whatever extent the original table might be carried; but in all cases a
series of differences would be obtained, which would continue constant
for a very long succession of terms.

As the successive serieses of differences are derived from the original
table, and from each other, by _subtraction_, the same succession of
series may be reproduced in the other direction by _addition_. But let us
suppose that the first number of the original table, and of each of the
series of differences, including the last, be given: all the numbers of
each of the series may thence be obtained by the mere process of
addition. The second term of the original table will be obtained by
adding to the first the first term of the first difference series; in
like manner, the second term of the first difference series will be
obtained by adding to the first term, the first term of the third
difference series, and so on. The second terms of all the serieses being
thus obtained, the third terms may be obtained by a like process of
addition; and so the series may be continued. These observations will
perhaps be rendered more clearly intelligible when illustrated by a
numerical example. The following is the commencement of a series of the
fourth powers of the natural numbers:--

                   No.     Table.
                    1          1
                    2         16
                    3         81
                    4        256
                    5        625
                    6       1296
                    7       2401
                    8       4096
                    9       6561
                    10    10,000
                    11    14,641
                    12    20,736
                    13    28,561

By subtracting each number from the succeeding one in this series, we
obtain the following series of first differences:

                          15
                          65
                         175
                         369
                         671
                        1105
                        1695
                        2465
                        3439
                        4641
                        6095
                        7825

In like manner, subtracting each term of this series from the succeeding
one, we obtain the following series of second differences:--

                          50
                         110
                         194
                         302
                         434
                         590
                         770
                         974
                        1202
                        1454
                        1730

Proceeding with this series in the same way, we obtain the following
series of third differences:--

                         60
                         84
                        108
                        132
                        156
                        180
                        204
                        228
                        252
                        276

Proceeding in the same way with these, we obtain the following for the
series of fourth differences:--

                         24
                         24
                         24
                         24
                         24
                         24
                         24
                         24
                         24

It appears, therefore, that in this case the series of fourth
differences consists of a constant repetition of the number 24. Now, a
slight consideration of the succession of arithmetical operations by
which we have obtained this result, will show, that by reversing the
process, we could obtain the table of fourth powers by the mere process
of addition. Beginning with the first numbers in each successive series
of differences, and designating the table and the successive differences
by the letters T, D^1 D^2 D^3 D^4, we have then the following to begin
with:--

                  T   D^1   D^2   D^3   D^4
                  1    15    50    60    24

Adding each number to the number on its left, and repeating 24, we get
the following as the second terms of the several series:--

                  T   D^1   D^2   D^3   D^4
                 16    65   110    84    24

And, in the same manner, the third and succeeding terms as follows:--

           No.      T     D^1    D^2    D^3    D^4
            1       1      15     50     60     24
            2      16      65    110     84     24
            3      81     175    194    108     24
            4     256     369    302    132     24
            5     625     671    434    156     24
            6    1296    1105    590    180     24
            7    2401    1695    770    204     24
            8    4096    2465    974    228     24
            9    6561    3439   1202    252     24
           10   10000    4641   1454    276
           11   14641    6095   1730
           12   20736    7825
           13   28561

There are numerous tables in which, as already stated, to whatever order
of differences we may proceed, we should not obtain a series of
rigorously constant differences; but we should always obtain a certain
number of differences which to a given number of decimal places would
remain constant for a long succession of terms. It is plain that such a
table might be calculated by addition in the same manner as those which
have a difference rigorously and continuously constant; and if at every
point where the last difference requires an increase, that increase be
given to it, the same principle of addition may again be applied for a
like succession of terms, and so on.

By this principle it appears, that all tables in which each series of
differences continually increases, may be produced by the operation of
addition alone; provided the first terms of the table, and of each
series of differences, be given in the first instance. But it sometimes
happens, that while the table continually increases, one or more
serieses of differences may continually diminish. In this case, the
series of differences are found by subtracting each term of the series,
not from that which follows, but from that which precedes it; and
consequently, in the re-production of the several serieses, when their
first terms are given, it will be necessary in some cases to obtain them
by _addition_, and in others by _subtraction_. It is possible, however,
still to perform all the operations by addition alone: this is effected
in performing the operation of subtraction, by substituting for the
subtrahend its _arithmetical complement_, and adding that, omitting the
unit of the highest order in the result. This process, and its
principle, will be readily comprehended by an example. Let it be
required to subtract 357 from 768.

The common process would be as follows:--

                        From       768
                        Subtract   357
                                  ----
                        Remainder  411

The _arithmetical complement_ of 357, or the number by which it falls
short of 1000, is 643. Now, if this number be added to 768, and the
first figure on the left be struck out of the sum, the process will be
as follows:--

                        To                 768
                        Add                643
                                          ----
                        Sum               1411
                                          ----
                        Remainder sought   411

The principle on which this process is founded is easily explained. In
the latter process we have first added 643, and then subtracted 1000. On
the whole, therefore, we have subtracted 357, since the number actually
subtracted exceeds the number previously added by that amount.

Since, therefore, subtraction may be effected in this manner by
addition, it follows that the calculation of all serieses, so far as an
order of differences can be found in them which continues constant, may
be conducted by the process of addition alone.

It also appears from what has been stated, that each addition consists
only of two operations. However numerous the figures may be of which the
several pairs of numbers to be thus added may consist, it is obvious
that the operation of adding them can only consist of repetitions of the
process of adding one digit to another; and of carrying one from the
column of inferior units to the column of units next superior when
necessary. If we would therefore reduce such a process to machinery, it
would only be necessary to discover such a combination of moving parts
as are capable of performing these two processes of _adding_ and _carrying_
on two single figures; for, this being once accomplished, the process of
adding two numbers, consisting of any number of digits, will be effected
by repeating the same mechanism as often as there are pairs of digits to
be added. Such was the simple form to which Mr Babbage reduced the
problem of discovering the calculating machinery; and we shall now
proceed to convey some notion of the manner in which he solved it.

For the sake of illustration, we shall suppose that the table to be
calculated shall consist of numbers not exceeding six places of figures;
and we shall also suppose that the difference of the fifth order is the
constant difference. Imagine, then, six rows of wheels, each wheel
carrying upon it a dial-plate like that of a common clock, but
consisting of _ten_ instead of _twelve_ divisions; the several divisions
being marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Let these dials be supposed
to revolve whenever the wheels to which they are attached are put in
motion, and to turn in such a direction that the series of increasing
numbers shall pass under the index which appears over each dial:--thus,
after 0 passes the index, 1 follows, then 2, 3, and so on, as the dial
revolves. In Fig. 1 are represented six horizontal rows of such dials.

Fig. 1.

The method of differences, as already explained, requires, that in
proceeding with the calculation, this apparatus should perform
continually the addition of the number expressed upon each row of dials,
to the number expressed upon the row immediately above it. Now, we shall
first explain how this process of addition may be conceived to be
performed by the motion of the dials; and in doing so, we shall consider
separately the processes of addition and carriage, considering the
addition first, and then the carriage.

Let us first suppose the line D^1 to be added to the line T. To
accomplish this, let us imagine that while the dials on the line D^1 are
quiescent, the dials on the line T are put in motion, in such a manner,
that as many divisions on each dial shall pass under its index, as there
are units in the number at the index immediately below it. It is evident
that this condition supposes, that if be at any index on the line D^1,
the dial immediately above it in the line T shall not move. Now the
motion here supposed, would bring under the indices on the line T such a
number as would be produced by adding the number D^1 to T, neglecting all
the carriages; for a carriage should have taken place in every case in
which the figure 9 of any dial in the line T had passed under the index
during the adding motion. To accomplish this carriage, it would be
necessary that the dial immediately on the left of any dial in which 9
passes under the index, should be advanced one division, independently
of those divisions which it may have been advanced by the addition of
the number immediately below it. This effect may be conceived to take
place in, either of two ways. It may be either produced at the moment
when the division between 9 and 0 of any dial passes under the index; in
which case the process of carrying would go on simultaneously with the
process of adding; or the process of carrying may be postponed in every
instance until the process of addition, without carrying, has been
completed; and then by another distinct and independent motion of the
machinery, a carriage may be made by advancing one division all those
dials on the right of which a dial had, during the previous addition,
passed from 9 to 0 under the index. The latter is the method adopted in
the calculating machinery, in order to enable its inventor to construct
the carrying machinery independent of the adding mechanism.

Having explained the motion of the dials by which the addition,
excluding the carriages of the number on the row D^1, may be made to the
number on the row T, the same explanation may be applied to the number
on the row D^2 to the number on the row D^1; also, of the number D^3 to the
number on the row D^2, and so on. Now it is possible to suppose the
additions of all the rows, except the first, to be made to all the rows
except the last, simultaneously; and after these additions have been
made, to conceive all the requisite carriages to be also made by
advancing the proper dials one division forward. This would suppose all
the dials in the scheme to receive their adding motion together; and,
this being accomplished, the requisite dials to receive their carrying
motions together. The production of so great a number of simultaneous
motions throughout any machinery, would be attended with great
mechanical difficulties, if indeed it be practicable. In the calculating
machinery it is not attempted. The additions are performed in two
successive periods of time, and the carriages in two other periods of
time, in the following manner. We shall suppose one complete revolution
of the axis which moves the machinery, to make one complete set of
additions and carriages; it will then make them in the following
order:--

The first quarter of a turn of the axis will add the second, fourth, and
sixth rows to the first, third, and fifth, omitting the carriages; this
it will do by causing the dials on the first, third, and fifth rows, to
turn through as many divisions as are expressed by the numbers at the
indices below them, as already explained.

The second quarter of a turn will cause the carriages consequent on the
previous addition, to be made by moving forward the proper dials one
division.

(During these two quarters of a turn, the dials of the first, third, and
fifth row alone have been moved; those of the second, fourth, and sixth,
have been quiescent.)

The third quarter of a turn will produce the addition of the third and
fifth rows to the second and fourth, omitting the carriages; which it
will do by causing the dials of the second and fourth rows to turn
through as many divisions as are expressed by the numbers at the indices
immediately below them.

The fourth and last quarter of a turn will cause the carriages
consequent on the previous addition, to be made by moving the proper
dials forward one division.

This evidently completes one calculation, since all the rows except the
first have been respectively added to all the rows except the last.

To illustrate this: let us suppose the table to be computed to be that
of the fifth powers of the natural numbers, and the computation to have
already proceeded so far as the fifth power of 6, which is 7776. This
number appears, accordingly, in the highest row, being the place
appropriated to the number of the table to be calculated. The several
differences as far as the fifth, which is in this case constant, are
exhibited on the successive rows of dials in such a manner, as to be
adapted to the process of addition by alternate rows, in the manner
already explained. The process of addition will commence by the motion
of the dials in the first, third, and fifth rows, in the following
manner: The dial A, fig. 1, must turn through one division, which will
bring the number 7 to the index; the dial B must turn through three
divisions, which will 0 bring to the index; this will render a carriage
necessary, but that carriage will not take place during the present
motion of the dial. The dial C will remain unmoved, since 0 is at the
index below it; the dial D must turn through nine divisions; and as, in
doing so, the division between 9 and 0 must pass under the index, a
carriage must subsequently take place upon the dial to the left; the
remaining dials of the row T, fig. 1, will remain unmoved. In the row D^2
the dial A^2 will remain unmoved, since 0 is at the index below it; the
dial B^2 will be moved through five divisions, and will render a
subsequent carriage on the dial to the left necessary; the dial C^2 will
be moved through five divisions; the dial D^2 will be moved through three
divisions, and the remaining dials of this row will remain unmoved. The
dials of the row D^4 will be moved according to the same rules; and the
whole scheme will undergo a change exhibited in fig. 2; a mark (*) being
introduced on those dials to which a carriage rendered necessary by the
addition which has just taken place.

Fig. 2.

The second quarter of a turn of the moving axis, will move forward
through one division all the dials which in fig. 2 are marked (*), and
the scheme will be converted into the scheme expressed in fig. 3.

Fig. 3.

In third quarter of a turn, the dial A^1, fig. 3, will remain unmoved,
since is at the index below it; the dial B^1 will be moved forward
through three divisions; C^1 through nine divisions, and so on; and in
like manner the dials of the row D^3 will be moved forward through the
number of divisions expressed at the indices in the row D^4. This change
will convert the arrangement into that expressed in fig. 4, the dials to
which a carriage is due, being distinguished as before by (*).

Fig. 4.

The fourth quarter of a turn of the axis will move forward one division
all the dials marked (*); and the arrangement will finally assume the
form exhibited in fig. 5, in which the calculation is completed. The
first row T in this expresses the fifth power of 7; and the second
expresses the number which must be added to the first row, in order to
produce the fifth power of 8; the numbers in each row being prepared for
the change which they must undergo, in order to enable them to continue
the computation according to the method of alternate addition here
adopted.

Fig. 5.

Having thus explained what it is that the mechanism is required to do,
we shall now attempt to convey at least a general notion of some of the
mechanical contrivances by which the desired ends are attained. To
simplify the explanation, let us first take one particular
instance--the dials B and B^1, fig. 1, for example. Behind the dial B^1
is a bolt, which, at the commencement of the process, is shot between
the teeth of a wheel which drives the dial B: during the first quarter
of a turn this bolt is made to revolve, and if it continued to be
engaged in the teeth of the said wheel, it would cause the dial B to
make a complete revolution; but it is necessary that the dial B should
only move through three divisions, and, therefore, when three divisions
of this dial have passed under its index, the aforesaid bolt must be
withdrawn: this is accomplished by a small wedge, which is placed in a
fixed position on the wheel behind the dial B^1, and that position is
such that this wedge will press upon the bolt in such a manner, that at
the moment when three divisions of the dial B have passed under the
index, it shall withdraw the bolt from the teeth of the wheel which it
drives. The bolt will continue to revolve during the remainder of the
first quarter of a turn of the axis, but it will no longer drive the
dial B, which will remain quiescent. Had the figure at the index of the
dial B^1 been any other, the wedge which withdraws the bolt would have
assumed a different position, and would have withdrawn the bolt at a
different time, but at a time always corresponding with the number under
the index of the dial B^1: thus, if 5 had been under the index of the
dial B^1, then the bolt would have been withdrawn from between the teeth
of the wheel which it drives, when five divisions of the dial B had
passed under the index, and so on. Behind each dial in the row D^1 there
is a similar bolt and a similar withdrawing wedge, and the action upon
the dial above is transmitted and suspended in precisely the same
manner. Like observations will be applicable to all the dials in the
scheme here referred to, in reference to their adding actions upon those
above them.

There is, however, a particular case which here merits notice: it is the
case in which 0 is under the index of the dial from which the addition
is to be transmitted upwards. As in that case nothing is to be added, a
mechanical provision should be made to prevent the bolt from engaging in
the teeth of the wheel which acts upon the dial above: the wedge which
causes the bolt to be withdrawn, is thrown into such a position as to
render it impossible that the bolt should be shot, or that it should
enter between the teeth of the wheel, which in other cases it drives.
But inasmuch as the usual means of shooting the bolt would still act, a
strain would necessarily take place in the parts of the mechanism, owing
to the bolt not yielding to the usual impulse. A small shoulder is
therefore provided, which puts aside, in this case, the piece by which
the bolt is usually struck, and allows the striking implement to pass
without encountering the head of the bolt or any other obstruction. This
mechanism is brought into play in the scheme, fig. 1, in the cases of
all those dials in which 0 is under the index.

Such is a general description of the nature of the mechanism by which
the adding process, apart from the carriages, is effected. During the
first quarter of a turn, the bolts which drive the dials in the first,
third, and fifth rows, are caused to revolve, and to act upon these
dials, so long as they are permitted by the position of the several
wedges on the second, fourth, and sixth rows of dials, by which these
bolts are respectively withdrawn; and, during the third quarter of a
turn, the bolts which drive the dials of the second and fourth rows are
made to revolve and act upon these dials so long as the wedges on the
dials of the third and fifth rows, which withdraw them, permit. It will
hence be perceived, that, during the first and third quarters of a turn,
the process of addition is continually passing upwards through the
machinery; alternately from the even to the odd rows, and from the odd
to the even rows, counting downwards.

We shall now attempt to convey some notion of the mechanism by which the
process of carrying is effected during the second and fourth quarters of
a turn of the axis. As before, we shall first explain it in reference to
a particular instance. During the first quarter of a turn the wheel B^2,
fig. 1, is caused by the adding bolt to move through five divisions; and
the fifth of these divisions, which passes under the index, is that
between 9 and 0. On the axis of the wheel C^2, immediately to the left of
B^2, is fixed a wheel, called in mechanics a ratchet wheel, which is
driven by a claw which constantly rests in its teeth. This claw is in
such a position as to permit the wheel C^2 to move in obedience to the
action of the adding bolt, but to resist its motion in the contrary
direction. It is drawn back by a spiral spring, but its recoil is
prevented by a hook which sustains it; which hook, however, is capable
of being withdrawn, and when withdrawn, the aforesaid spiral spring
would draw back the claw, and make it fall through one tooth of the
ratchet wheel. Now, at the moment that the division between 9 and 0 on
the dial B^2 passes under the index, a thumb placed on the axis of this
dial touches a trigger which raises out of the notch the hook which
sustains the claw just mentioned, and allows it to fall back by the
recoil of the spring, and to drop into the next tooth of the ratchet
wheel. This process, however, produces no immediate effect upon the
position of the wheel C^2, and is merely preparatory to an action
intended to take place during the second quarter of a turn of the moving
axis. It is in effect a memorandum taken by the machine of a carriage to
be made in the next quarter of a turn.

During the second quarter of a turn, a finger placed on the axis of the
dial B^2 is made to revolve, and it encounters the heel of the
above-mentioned claw. As it moves forward it drives the claw before it:
and this claw, resting in the teeth of the ratchet wheel fixed upon the
axis of the dial C^2 drives forward that wheel, and with it the dial. But
the length and position of the finger which drives the claw limits its
action, so as to move the claw forward through such a space only as will
cause the dial C^2 to advance through a single division; at which point
it is again caught and retained by the hook. This will be added to the
number under its index, and the requisite carriage from B^2 to C^2 will be
accomplished.

In connexion with every dial is placed a similar ratchet wheel with a
similar claw, drawn by a similar spring, sustained by a similar hook,
and acted upon by a similar thumb and trigger; and therefore the
necessary carriages, throughout the whole machinery, take place in the
same manner and by similar means.

During the second quarter of a turn, such of the carrying claws as have
been allowed to recoil in the first, third, and fifth rows, are drawn up
by the fingers on the axes of the adjacent dials; and, during the fourth
quarter of a turn, such of the carrying claws on the second and fourth
rows as have been allowed to recoil during the third quarter of a turn,
are in like manner drawn up by the carrying fingers on the axes of the
adjacent dials. It appears that the carriages proceed alternately from
right to left along the horizontal rows during the second and fourth
quarters of a turn; in the one, they pass along the first, third, and
fifth rows, and in the other, along the second and fourth.

There are two systems of waves of mechanical action continually flowing
from the bottom to the top; and two streams of similar action constantly
passing from the right to the left. The crests of the first system of
adding waves fall upon the last difference, and upon every alternate one
proceeding upwards; while the crests of the other system touch upon the
intermediate differences. The first stream of carrying action passes
from right to left along the highest row and every alternate tow, while
the second stream passes along the intermediate rows.

Such is a very rapid and general outline of this machinery. Its wonders,
however, are still greater in its details than even in its broader
features. Although we despair of doing it justice by any description
which can be attempted here, yet we should not fulfil the duty we owe to
our readers, if we did not call their attention at least to a few of the
instances of consummate skill which are scattered, with a prodigality
characteristic of the highest order of inventive genius, throughout this
astonishing mechanism.

In the general description which we have given of the mechanism for
_carrying_, it will be observed, that the preparation for every carriage
is stated to be made during the previous addition, by the disengagement
of the carrying claw before mentioned, and by its consequent recoil,
urged by the spiral spring with which it is connected; but it may, and
does, frequently happen, that though the process of addition may not
have rendered a carriage necessary, one carriage may itself produce the
necessity for another. This is a contingency not provided against in the
mechanism as we have described it: the case would occur in the scheme
represented in fig. 1, if the figure under the index of C^2 were 4
instead of 3. The addition of the number 5 at the index of C^3 would, in
this case, in the first quarter of a turn, bring 9 to the index of C^2:
this would obviously render no carriage necessary, and of course no
preparation would be made for one by the mechanism--that is to say, the
carrying claw of the wheel D^2 would not be detached. Meanwhile a
carriage upon C^2 has been rendered necessary by the addition made in the
first quarter of a turn to B^2. This carriage takes place in the ordinary
way, and would cause the dial C^2, in the second quarter of a turn, to
advance from 9 to 0: this would make the necessary preparation for a
carriage from C^2 to D^2. But unless some special arrangement was made for
the purpose, that carriage would not take place during the second
quarter of a turn. This peculiar contingency is provided against by an
arrangement of singular mechanical beauty, and which, at the same time,
answers another purpose--that of equalizing the resistance opposed to
the moving power by the carrying mechanism. The fingers placed on the
axes of the several dials in the row D^2, do not act at the same instant
on the carrying claws adjacent to them; but they are so placed, that
their action may be distributed throughout the second quarter of a turn
in regular succession. Thus the finger on the axis of the dial A^2 first
encounters the claw upon B^2, and drives it through one tooth immediately
forwards; the finger on the axis of B^2 encounters the claw upon C^2 and
drives it through one tooth; the action of the finger on C^2 on the claw
on D^2 next succeeds, and so on. Thus, while the finger on B^2 acts on C^2,
and causes the division from 9 to 0 to pass under the index, the thumb
on C^2 at the same instant acts on the trigger, and detaches the carrying
claw on D^2, which is forthwith encountered by the carrying finger on C^2,
and, driven forward one tooth. The dial D^2 accordingly moves forward one
division, and 5 is brought under the index. This arrangement is
beautifully effected by placing the several fingers, which act upon the
carrying claws, _spirally_ on their axes, so that they come into action in
regular succession.

We have stated that, at the commencement of each revolution of the
moving axis, the bolts which drive the dials of the first, third, and
fifth rows, are shot. The process of shooting these bolts must therefore
have taken place during the last quarter of the preceding revolution;
but it is during that quarter of a turn that the carriages are effected
in the second and fourth rows. Since the bolts which drive the dials of
the first, third, and fifth rows, have no mechanical connexion with the
dials in the second and fourth rows, there is nothing in the process of
shooting those bolts incompatible with that of moving the dials of the
second and fourth rows: hence these two processes may both take place
during the same quarter of a turn. But in order to equalize the
resistance to the moving power, the same expedient is here adopted as
that already described in the process of carrying. The arms which shoot
the bolts of each row of dials are arranged spirally, so as to act
successively throughout the quarter of a turn. There is, however, a
contingency which, under certain circumstances, would here produce a
difficulty which must be provided against. It is possible, and in fact
does sometimes happen, that the process of carrying causes a dial to
move under the index from 0 to 1. In that case, the bolt, preparatory to
the next addition, ought not to be shot until after the carriage takes
place; for if the arm which shoots it passes its point of action before
the carriage takes place, the bolt will be moved out of its sphere of
action, and will not be shot, which, as we have already explained, must
always happen when 0 is at the index: therefore no addition would in
this case take place during the next quarter of a turn of the axis;
whereas, since 1 is brought to the index by the carriage, which
immediately succeeds the passage of the arm which ought to bolt, 1
should be added during the next quarter of a turn. It is plain,
accordingly, that the mechanism should be so arranged, that the action
of the arms, which shoot the bolts successively, should immediately
follow the action of those fingers which raise the carrying claws
successively; and therefore either a separate quarter of a turn should
be appropriated to each of those movements, or if they be executed in
the same quarter of a turn, the mechanism must be so constructed, that
the arms which shoot the bolts successively, shall severally follow
immediately after those which raise the carrying claws successively. The
latter object is attained by a mechanical arrangement of singular
felicity, and partaking of that elegance which characterises all the
details of this mechanism. Both sets of arms are spirally arranged on
their respective axes, so as to be carried through their period in the
same quarter of a turn; but the one spiral is shifted a few degrees, in
angular position, behind the other, so that each pair of corresponding
arms succeed each other in the most regular order,--equalizing the
resistance, economizing time, harmonizing the mechanism, and giving to
the whole mechanical action the utmost practical perfection.

The system of mechanical contrivances by which the results, here
attempted to be described, are attained, form only one order of
expedients adopted in this machinery;--although such is the perfection
of their action, that in any ordinary case they would be regarded as
having attained the ends in view with an almost superfluous degree of
precision. Considering, however, the immense importance of the purposes
which the mechanism was destined to fulfil, its inventor determined that
a higher order of expedients should be superinduced upon those already
described; the purpose of which should be to obliterate all small errors
or inequalities which might, even by remote possibility, arise, either
from defects in the original formation of the mechanism, from inequality
of wear, from casual strain or derangement,--or, in short, from any
other cause whatever. Thus the movements of the first and principal
parts of the mechanism were regarded by him merely as a first, though
extremely nice approximation, upon which a system of small corrections
was to be subsequently made by suitable and independent mechanism. This
supplementary system of mechanism is so contrived, that if one or more
of the moving parts of the mechanism of the first order be slightly out
of their places, they will be forced to their exact position by the
action of the mechanical expedients of the second order to which we now
allude. If a more considerable derangement were produced by any
accidental disturbance, the consequence would be that the supplementary
mechanism would cause the whole system to become locked, so that not a
wheel would be capable of moving; the impelling power would necessarily
lose all its energy, and the machine would stop. The consequence of this
exquisite arrangement is, that the machine will either calculate
rightly, or not at all.

The supernumerary contrivances which we now allude to, being in a great
degree unconnected with each other, and scattered through the machinery
to a certain extent, independent of the mechanical arrangement of the
principal parts, we find it difficult to convey any distinct notion of
their nature or form.

In some instances they consist of a roller resting between certain
curved surfaces, which has but one position of stable equilibrium, and
that position the same, however the roller or the curved surfaces may
wear. A slight error in the motion of the principal parts would make
this roller for the moment rest on one of the curves; but, being
constantly urged by a spring, it would press on the curved surface in
such a manner as to force the moving piece on which that curved surface
is formed, into such a position that the roller may rest between the two
surfaces; that position being the one which the mechanism should have. A
greater derangement would bring the roller to the crest of the curve, on
which it would rest in instable equilibrium; and the machine would
either become locked, or the roller would throw it as before into its
true position.

In other instances a similar object is attained by a solid cone being
pressed into a conical seat; the position of the axis of the cone and
that of its seat being necessarily invariable, however the cone may
wear: and the action of the cone upon the seat being such, that it
cannot rest in any position except that in which the axis of the cone
coincides with the axis of its seat.

Having thus attempted to convey a notion, however inadequate, of the
calculating section of the machinery, we shall proceed to offer some
explanation of the means whereby it is enabled, to print its
calculations in such a manner as to preclude the possibility of error in
any individual printed copy.

On the axle of each of the wheels which express the calculated number of
the table T, there is fixed a solid piece of metal, formed into a curve,
not unlike the wheel in a common clock, which is called the _snail_. This
curved surface acts against the arm of a lever, so as to raise that arm
to a higher or lower point according to the position of the dial with
which the snail is connected. Without entering into a more minute
description, it will be easily understood that the snail may be so
formed that the arm of the lever shall be raised to ten different
elevations, corresponding to the ten figures of the dial which may be
brought under the index. The opposite arm of the lever here described
puts in motion a solid arch, or sector, which carries ten punches: each
punch bearing on its face a raised character of a figure, and the ten
punchy bearing the ten characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. It will
be apparent from what has been just stated, that this _type sector_ (as it
is called) will receive ten different attitudes, corresponding to the
ten figures which may successively be brought under the index of the
dial-plate. At a point over which the type sector is thus moved, and
immediately under a point through which it plays, is placed a frame, in
which is fixed a plate of copper. Immediately over a certain point
through which the type sector moves, is likewise placed a _bent lever_,
which, being straightened, is forcibly pressed upon the punch which has
been brought under it. If the type sector be moved, so as to bring under
the bent lever one of the steel punches above mentioned, and be held in
that position for a certain time, the bent lever, being straightened,
acts upon the steel punch, and drives it against the face of the copper
beneath, and thus causes a sunken impression of the character upon the
punch to be left upon the copper. If the copper be now shifted slightly
in its position, and the type sector be also shifted so as to bring
another punch under the bent lever, another character may be engraved on
the copper by straightening the bent lever, and pressing it on the punch
as before. It will be evident, that if the copper was shifted from right
to left through a space equal to two figures of a number, and, at the
same time, the type sector so shifted as to bring the punches
corresponding to the figures of the number successively under the bent
lever, an engraved impression of the number might thus be obtained upon
the copper by the continued action of the bent lever. If, when one line
of figures is thus obtained, a provision be made to shift the copper in
a direction at right angles to its former motion, through a space equal
to the distance between two lines of figures, and at the same time to
shift it through a space in the other direction equal to the length of
an entire line, it will be evident that another line of figures might be
printed below the first in the same manner.

The motion of the type sector, here described, is accomplished by the
action of the snail upon the lever already mentioned. In the case where
the number calculated is that expressed in fig. 1, the process would be
as follows:--The snail of the wheel F^1, acting upon the lever, would
throw the type sector into such an attitude, that the punch bearing the
character 0 would come under the bent lever. The next turn of the moving
axis would cause the bent lever to press on the tail of the punch, and
the character 0 would be impressed upon the copper. The bent lever being
again drawn up, the punch would recoil from the copper by the action of
a spring; the next turn of the moving axis would shift the copper
through the interval between two figures, so as to bring the point
destined to be impressed with the next figure under the bent lever. At
the same time, the snail of the wheel E would cause the type sector to
be thrown into the same attitude as before, and the punch would be
brought under the bent lever; the next turn would impress the figure
beside the former one, as before described. The snail upon the wheel D
would now come into action, and throw the type sector into that position
in which the punch bearing the character 7 would come under the bent
lever, and at the same time the copper would be shifted through the
interval between two figures; the straightening of the lever would next
follow, and the character 7 would be engraved. In the same manner, the
wheels C, B, and A would successively act by means of their snails; and
the copper being shifted, and the lever allowed to act, the number
007776 would be finally engraved upon the copper: this being
accomplished, the calculating machinery would next be called into
action, and another calculation would be made, producing the next number
of the Table exhibited in fig. 5. During this process the machinery
would be engaged in shifting the copper both in the direction of its
length and its breadth, with a view to commence the printing of another
line; and this change of position would be accomplished at the moment
when the next calculation would be completed: the printing of the next
number would go on like the former, and the operation of the machine
would proceed in the same manner, calculating and printing alternately.
It is not, however, at all necessary--though we have here supposed it,
for the sake of simplifying the explanation--that the calculating part
of the mechanism should have its action suspended while the printing
part is in operation, or _vice versa_; it is not intended, in fact, to be
so suspended in the actual machinery. The same turn of the axis by which
one number is printed, executes a part of the movements necessary for
the succeeding calculation; so that the whole mechanism will be
simultaneously and continuously in action.

Of the mechanism by which the position of the copper is shifted from
figure to figure, from line to line, we shall not attempt any
description. We feel that it would be quite vain. Complicated and
difficult to describe as every other part of this machinery is, the
mechanism for moving the copper is such as it would be quite impossible
to render at all intelligible, without numerous illustrative drawings.

The engraved plate of copper obtained in the manner above described, is
designed to be used as a mould from which a stereotyped plate may be
cast; or, if deemed advisable, it may be used as the immediate means of
printing. In the one case we should produce a table, printed from type,
in the same manner as common letter-press printing; in the other an
engraved table. If it be thought most advisable to print from the
stereotyped plates, then as many stereotyped plates as may be required
may be taken from the copper mould; so that when once a table has been
calculated and engraved by the machinery, the whole world may be
supplied with stereotyped plates to print it, and may continue to be so
supplied for an unlimited period of time. There is no practical limit to
the number of stereotyped plates which may be taken from the engraved
copper; and there is scarcely any limit to the number of printed copies
which may be taken from any single stereotyped plate. Not only,
therefore, is the numerical table by these means engraved and
stereotyped with infallible accuracy, but such stereotyped plates are
producible in unbounded quantity. Each plate, when produced, becomes
itself the means of producing printed copies of the table, in accuracy
perfect, and in number without limit.

Unlike all other machinery, the calculating mechanism produces, not the
object of consumption, but the machinery by which that object may be
made. To say that it computes and prints with infallible accuracy, is to
understate its merits:--it computes and fabricates _the means_ of
printing with absolute correctness and in unlimited abundance.

For the sake of clearness, and to render ourselves more easily
intelligible to the general reader, we have in the preceding explanation
thrown the mechanism into an arrangement somewhat different from that
which is really adopted. The dials expressing the numbers of the tables
of the successive differences are not placed, as we have supposed them,
in horizontal rows, and read from right to left, in the ordinary way;
they are, on the contrary, placed vertically, one below the other, and
read from top to bottom. The number of the table occupies the first
vertical column on the right, the units being expressed on the lowest
dial, and the tens on the next above that, and so on. The first
difference occupies the next vertical column on the left; and the
numbers of the succeeding differences occupy vertical columns,
proceeding regularly to the left; the constant difference being on the
last vertical column. It is intended in the machine now in progress to
introduce six orders of differences, so that there will be seven columns
of dials; it is also intended that the calculations shall extend to
eighteen places of figures: thus each column will have eighteen dials.
We have referred to the dials as if they were inscribed upon the faces
of wheels, whose axes are horizontal and planes vertical. In the actual
machinery the axes are vertical and the planes horizontal, so that the
edges of the _figure wheels_, as they are called, are presented to the
eye. The figures are inscribed, not upon the dial-plate, but around the
surface of a small cylinder or barrel, placed upon the axis of the
figure wheel, which revolves with it; so that as the figure wheel
revolves, the figures on the barrel are successively brought to the
front, and pass under an index engraved upon a plate of metal
immediately above the barrel. This arrangement has the obvious practical
advantage, that, instead of each figure wheel having a separate axis,
all the figure wheels of the same vertical column revolve on the same
axis; and the same observation will apply to all the wheels with which
the figure wheels are in mechanical connexion. This arrangement has the
further mechanical advantage over that which has been assumed for the
purposes of explanation, that the friction of the wheel-work on the axes
is less in amount, and more uniformly distributed, than it could be if
the axes were placed in the horizontal position.

A notion may therefore be formed of the front elevation of the
calculating part of the mechanism, by conceiving seven steel axes
erected, one beside another, on each of which shall be placed eighteen
wheels,[12] five inches in diameter, having cylinders or barrels upon
them an inch and a half in height, and inscribed, as already stated,
with the ten arithmetical characters. The entire elevation of the
machinery would occupy a space measuring ten feet broad, ten feet high,
and five feet deep. The process of calculation would be observed by the
alternate motion of the figure wheels on the several axes. During the
first quarter of a turn, the wheels on the first, third, and fifth axes
would turn, receiving their addition from the second, fourth, and sixth;
during the second quarter of a turn, such of the wheels on the first,
third, and fifth axes, to which carriages are due, would be moved
forward one additional figure; the second, fourth, and sixth columns of
wheels being all this time quiescent. During the third quarter of a
turn, the second, fourth, and sixth columns would be observed to move,
receiving their additions from the third, fifth, and seventh axes; and
during the fourth quarter of a turn, such of these wheels to which
carriages are due, would be observed to move forward one additional
figure; the wheels of the first, third, and fifth columns being
quiescent during this time.

[Footnote 12: The wheels, and every other part of the mechanism except
the axes, springs, and such parts as are necessarily of steel, are
formed of an alloy of copper with a small portion of tin.]

It will be observed that the wheels of the seventh column are always
quiescent in this process; and it may be asked, of what use they are,
and whether some mechanism of a fixed nature would not serve the same
purpose? It must, however, be remembered, that for different tables
there will be different constant differences; and that when the
calculation of a table is about to commence, the wheels on the seventh
axis must be moved by the hand, so as to express the constant
difference, whatever it may be. In tables, also, which have not a
difference rigorously constant, it will be necessary, after a certain
number of calculations, to change the constant difference by the hand;
and in this case the wheels of the seventh axis must be moved when
occasion requires. Such adjustment, however, will only be necessary at
very distant intervals, and after a considerable extent of printing and
calculation has taken place; and when it is necessary, a provision is
made in the machinery by which notice will be given by the sounding of a
bell, so that the machine may not run beyond the extent of its powers of
calculation.

Immediately behind the seven axes on which the figure wheels revolve,
are seven other axes; on which are placed, first, the wheels already
described as driven by the figure wheels, and which bear upon them the
wedge which withdraws the bolt immediately over these latter wheels, and
on the same axis is placed the adding bolt. From the bottom of this bolt
there projects downwards the pin, which acts upon the unbolting wedge by
which the bolt is withdrawn: from the upper surface of the bolt proceeds
a tooth, which, when the bolt is shot, enters between the teeth of the
adding wheel, which turns on the same axis, and is placed immediately
above the bolt: its teeth, on which the bolt acts, are like the teeth of
a crown wheel, and are presented downwards. The bolt is fixed upon this
axis, and turns with it; but the adding wheel above the bolt, and the
unbolting wheel below it, both turn upon the axis, and independently of
it. When the axis is made to revolve by the moving power, the bolt
revolves with it; and so long as the tooth of the bolt remains inserted
between those of the adding wheel, the latter is likewise moved; but
when the lower pin of the bolt encounters the unbolting wedge on the
lower wheel, the tooth of the bolt is withdrawn, and the motion of the
adding wheel is stopped. This adding wheel is furnished with spur teeth,
besides the crown teeth just mentioned; and these spur teeth are engaged
with those of that unbolting wheel which is in connexion with the
adjacent figure wheel to which the addition is to be made. By such an
arrangement it is evident that the revolution of the bolt will
necessarily add to the adjacent figure wheel the requisite number.

It will be perceived, that upon the same axis are placed an unbolting
wheel, a bolt, and an adding wheel, one above the other, for every
figure wheel; and as there are eighteen figure wheels there will be
eighteen tiers; each tier formed of an unbolting wheel, a bolt, and an
adding wheel, placed one above the other; the wheels on this axis all
revolving independent of the axis, but the bolts being all fixed upon
it. The same observations, of course, will apply to each of the seven
axes.

At the commencement of every revolution of the adding axes, it is
evident that the several bolts placed upon them must be shot in order to
perform the various additions. This is accomplished by a third set of
seven axes, placed at some distance behind the range of the wheels,
which turn upon the adding axes: these are called _bolting axes_. On these
bolting axes are fixed, so as to revolve with them, a bolting finger
opposite to each bolt; as the bolting axis is made to revolve by the
moving power, the bolting finger is turned, and as it passes near the
bolt, it encounters the shoulder of a hammer or lever, which strikes the
heel of the bolt, and presses it forward so as to shoot its tooth
between the crown teeth of the adding wheel. The only exception to this
action is the case in which happens to be at the index of the figure
wheel; in that case, the lever or hammer, which the bolting finger would
encounter, is, as before stated, lifted out of the way of the bolting
finger, so that it revolves without encountering it. It is on the
bolting axes that the fingers are spirally arranged so as to equalize
their action, as already explained.

The same axes in the front of the machinery on which the figure wheels
turn, are made to serve the purpose of _carrying_. Each of these bear a
series of fingers which turn with them, and which encounter a carrying
claw, already described, so as to make the carriage: these carrying
fingers are also spirally arranged on their axes, as already described.

Although the absolute accuracy which appears to be ensured by the
mechanical arrangements here described is such as to render further
precautions nearly superfluous, still it may be right to state, that,
supposing it were possible for an error to be produced in calculation,
this error could be easily and speedily detected in the printed tables:
it would only be necessary to calculate a number of the table taken at
intervals, through which the mechanical action of the machine has not
been suspended, and during which it has received no adjustment by the
hand: if the computed number be found to agree with those printed, it
may be taken for granted that all the intermediate numbers are correct;
because, from the nature of the mechanism, and the principle of
computation, an error occurring in any single number of the table would
be unavoidably entailed, in an increasing ratio, upon all the succeeding
numbers.

We have hitherto spoken merely of the practicability of executing by the
machinery, when completed, that which its inventor originally
contemplated--namely, the calculating and printing of all numerical
tables, derived by the method of differences from a constant difference.
It has, however, happened that the actual powers of the machinery
greatly transcend those contemplated in its original design:--they not
only have exceeded the most sanguine anticipations of its inventor, but
they appear to have an extent to which it is utterly impossible, even
for the most acute mathematical thinker, to fix a probable limit.
Certain subsidiary mechanical inventions have, in the progress of the
enterprise, been, by the very nature of the machinery, suggested to the
mind of the inventor, which confer upon it capabilities which he had
never foreseen. It would be impossible even to enumerate, within the
limits of this article, much less to describe in detail, those
extraordinary mechanical arrangements, the effects of which have not
failed to strike with astonishment every one who has been favoured with
an opportunity of witnessing them, and who has been enabled, by
sufficient mathematical attainments, in any degree to estimate their
probable consequences.

As we have described the mechanism, the axes containing the several
differences are successively and regularly added one to another; but
there are certain mechanical adjustments, and these of a very simple
nature, which being thrown into action, will cause a difference of any
order to be added any number of times to a difference of any other
order; and that either proceeding backwards or forwards, from a
difference of an inferior to one of a superior order, and _vice versa_.[13]

[Footnote 13: The machine was constructed with the intention of tabulating
the equation Delta^{7}_{u} = 0, but, by the means
above alluded to, it is capable of tabulating such equations as the
following: Delta^{7}u = a Delta u, Delta^{7}u = aDelta^{3}u,
Delta^{7}u = units figure of Delta u.]

Among other peculiar mechanical provisions in the machinery is one by
which, when the table for any order of difference amounts to a certain
number, a certain arithmetical change would be made in the constant
difference. In this way a series may be tabulated by the machine, in
which the constant difference is subject to periodical change; or the
very nature of the table itself may be subject to periodical change, and
yet to one which has a regular law.

Some of these subsidiary powers are peculiarly applicable to
calculations required in astronomy, and are therefore of eminent and
immediate practical utility: others there are by which tables are
produced, following the most extraordinary, and apparently capricious,
but still regular laws. Thus a table will be computed, which, to any
required extent, shall coincide with a given table, and which shall
deviate from that table for a single term, or for any required number of
terms, and then resume its course, or which shall permanently alter the
law of its construction. Thus the engine has calculated a table which
agreed precisely with a table of square numbers, until it attained the
hundred and first term, which was not the square of 101, nor were any of
the subsequent numbers squares. Again, it has computed a table which
coincided with the series of natural numbers, as far as 100,000,001, but
which subsequently followed another law. This result was obtained, not
by working the engine through the whole of the first table, for that
would have required an enormous length of time; but by showing, from the
arrangement of the mechanism, that it must continue to exhibit the
succession of natural numbers, until it would reach 100,000,000. To save
time, the engine was set by the hand to the number 99999995, and was
then put in regular operation. It produced successively the following
numbers.[14]

                   99,999,996
                   99,999,997
                   99,999,998
                   99,999,999
                  100,000,000
                  100,010,002
                  100,030,003
                  100,060,004
                  100,100,005
                  100,150,006
                    &c. &c.

[Footnote 14: Such results as this suggest a train of reflection on the
nature and operation of general laws, which would lead to very curious
and interesting speculations. The natural philosopher and astronomer
will be hardly less struck with them than the metaphysician and
theologian.]

Equations have been already tabulated by the portion of the machinery
which has been put together, which are so far beyond the reach of the
present power of mathematics, that no distant term of the table can be
predicted, nor any function discovered capable of expressing its general
law. Yet the very fact of the table being produced by mechanism of an
invariable form, and including a distinct principle of mechanical
action, renders it quite manifest that _some_ general law must exist in
every table which it produces. But we must dismiss these speculations:
we feel it impossible to stretch the powers of our own mind, so as to
grasp the probable capabilities of this splendid production of combined
mechanical and mathematical genius; much less can we hope to enable
others to appreciate them, without being furnished with such means of
comprehending them as those with which we have been favoured. Years must
in fact elapse, and many enquirers direct their energies to the
cultivation of the vast field of research thus opened, before we can
fully estimate the extent of this triumph of matter over mind. 'Nor is
it,' says Mr Colebrooke, 'among the least curious results of this
ingenious device, that it affords a new opening for discovery, since it
is applicable, as has been shown by its inventor, to surmount novel
difficulties of analysis. Not confined to constant differences, it is
available in every case of differences that follow a definite law,
reducible therefore to an equation. An engine adjusted to the purpose
being set to work, will produce any distant term, or succession of
terms, required--thus presenting the numerical solution of a problem,
even though the analytical solution be yet undetermined.' That the
future path of some important branches of mathematical enquiry must now
in some measure be directed by the dictates of mechanism, is
sufficiently evident; for who would toil on in any course of analytical
enquiry, in which he must ultimately depend on the expensive and
fallible aid of human arithmetic, with an instrument in his hands, in
which all the dull monotony of numerical computation is turned over to
the untiring action and unerring certainty of mechanical agency?

It is worth notice, that each of the axes in front of the machinery on
which the figure wheels revolve, is connected with a bell, the tongue of
which is governed by a system of levers, moved by the several figure
wheels; an adjustment is provided by which the levers shall be
dismissed, so as to allow the hammer to strike against the bell,
whenever any proposed number shall be exhibited on the axis. This
contrivance enables the machine to give notice to its attendants at any
time that an adjustment may be required.

Among a great variety of curious accidental properties (so to speak)
which the machine is found to possess, is one by which it is capable of
solving numerical equations which have rational roots. Such an equation
being reduced (as it always may be) by suitable transformations to that
state in which the roots shall be whole numbers, the values 0, 1, 2, 3,
&c., are substituted for the unknown quantity, and the corresponding
values of the equation ascertained. From these a sufficient number of
differences being derived, they are set upon the machine. The machine
being then put in motion, the table axis will exhibit the successive
values of the formula, corresponding to the substitutions of the
successive whole numbers for the unknown quantity: at length the number
exhibited on the table axis will be 0, which will evidently correspond
to a root of the equation. By previous adjustment, the bell of the table
axis will in this case ring and give notice of the exhibition of the
value of the root in another part of the machinery.

If the equation have imaginary roots, the formula being necessarily a
maximum or minimum on the occurrence of such roots, the first difference
will become nothing; and the dials of that axis will under such
circumstances present to the respective indices. By previous adjustment,
the bell of this axis would here give notice of a pair of imaginary
roots.

Mr Colebrooke speculates on the probable extension of these powers of
the machine: 'It may not therefore be deemed too sanguine an
anticipation when I express the hope that an compliment which, in its
simpler form, attains to the extraction of roots of numbers, and
approximates to the roots of equations, may, in a more advanced state of
improvement, rise to the approximate solution of algebraic equations of
elevated degrees. I refer to solutions of such equations proposed by La
Grange, and more recently by other annalists, which involve operations
too tedious and intricate for use, and which must remain without
efficacy, unless some mode be devised of abridging the labour, or
facilitating the means of its performance. In any case this engine tends
to lighten the excessive and accumulating burden of arithmetical
application of mathematical formulæ, and to relieve the progress of
science from what is justly termed by the author of this invention, the
overwhelming encumbrance of numerical detail.'

Although there are not more than eighteen figure wheels on each axis,
and therefore it might be supposed that the machinery was capable of
calculating only to the extent of eighteen decimal places; yet there are
contrivances connected with it, by which, in two successive
calculations, it will be possible to calculate even to the extent of
thirty decimal places. Its powers, therefore, in this respect, greatly
exceed any which can be required in practical science. It is also
remarkable, that the machinery is capable of producing the calculated
results _true to the last figure_. We have already explained, that when
the figure which would follow the last is greater than 4, then it would
be necessary to increase the last figure by 1; since the excess of the
calculated number above the true value would in such case be less than
its defect from it would be, had the regularly computed final figure
been adopted: this is a precaution necessary in all numerical tables,
and it is one which would hardly have been expected to be provided for
in the calculating machinery.

As might be expected in a mechanical undertaking of such complexity and
novelty, many practical difficulties have since its commencement been
encountered and surmounted. It might have been foreseen, that many
expedients would be adopted and carried into effect, which farther
experiments would render it necessary to reject; and thus a large source
of additional expense could scarcely fail to be produced. To a certain
extent this has taken place; but owing to the admirable system of
mechanical drawings, which in every instance Mr Babbage has caused to be
made, and owing to his own profound acquaintance with the practical
working of the most complicated mechanism, he has been able to predict
in every case what the result of any contrivance would be, as perfectly
from the drawing, as if it had been reduced to the form of a working
model. The drawings, consequently, form a most extensive and essential
part of the enterprise. They are executed with extraordinary ability and
precision, and may be considered as perhaps the best specimens of
mechanical drawings which have ever been executed. It has been on these,
and on these only, that the work of invention has been bestowed. In
these, all those progressive modifications suggested by consideration
and study have been made; and it was not until the inventor was fully
satisfied with the result of any contrivance, that he had it reduced to
a working form. The whole of the loss which has been incurred by the
necessarily progressive course of invention, has been the expense of
rejected drawings. Nothing can perhaps more forcibly illustrate the
extent of labour and thought which has been incurred in the production
of this machinery, than the contemplation of the working drawings which
have been executed previously to its construction: these drawings cover
above a thousand square feet of surface, and many of them are of the
most elaborate and complicated description.

One of the practical difficulties which presented themselves at a very
early stage in the progress of this undertaking, was the impossibility
of bearing in mind all the variety of motions propagated simultaneously
through so many complicated trains of mechanism. Nothing but the utmost
imaginable harmony and order among such a number of movements, could
prevent obstructions arising from incompatible motions encountering each
other. It was very soon found impossible, by a mere act of memory, to
guard against such an occurrence; and Mr Babbage found, that, without
some effective expedient by which he could at a glance see what every
moving piece in the machinery was doing at each instant of time, such
inconsistencies and obstructions as are here alluded to must continually
have occurred. This difficulty was removed by another invention of even
a more general nature than the calculating machinery itself, and
pregnant with results probably of higher importance. This invention
consisted in the contrivance of a scheme of _mechanical notation_ which is
generally applicable to all machinery whatsoever; and which is exhibited
on a table or plan consisting of two distinct sections. In the first is
traced, by a peculiar system of signs, the origin of every motion which
takes place throughout the machinery; so that the mechanist or inventor
is able, by moving his finger along a certain line, to follow out the
motion of every piece from effect to cause, until he arrives at the
prime mover. The same sign which thus indicates the _source_ of motion
indicates likewise the _species_ of motion, whether it be continuous or
reciprocating, circular or progressive, &c. The same system of signs
further indicates the nature of the mechanical connexion between the
mover and the thing moved, whether it be permanent and invariable (as
between the two arms of a lever), or whether the mover and the moved are
separate and independent pieces, as is the case when a pinion drives a
wheel; also whether the motion of one piece necessarily implies the
motion of another; or when such motion in the one is interrupted, and in
the other continuous, &c.

The second section of the table divides the time of a complete period of
the machinery into any required number of parts; and it exhibits in a
map, as it were, that which every part of the machine is doing at each
moment of time. In this way, incompatibility in the motions of different
parts is rendered perceptible at a glance. By such means the contriver
of machinery is not merely prevented from introducing into one part of
the mechanism any movement inconsistent with the simultaneous action of
the other parts; but when he finds that the introduction of any
particular movement is necessary for his purpose, he can easily and
rapidly examine the whole range of the machinery during one of its
periods, and can find by inspection whether there is any, and what
portion of time, at which no motion exists incompatible with the desired
one, and thus discover a _niche_, as it were, in which to place the
required movement. A further and collateral advantage consists in
placing it in the power of the contriver to exercise the utmost possible
economy of _time_ in the application of his moving power. For example,
without some instrument of mechanical enquiry equally powerful with that
now described, it would be scarcely possible, at least in the first
instance, so to arrange the various movements that they should be all
executed in the least possible number of revolutions of the moving axis.
Additional revolutions would almost inevitably be made for the purpose
of producing movements and changes which it would be possible to
introduce in some of the phases of previous revolutions: and there is no
one acquainted with the history of mechanical invention who must not be
aware, that in the progressive contrivance of almost every machine the
earliest arrangements are invariably defective in this respect; and that
it is only by a succession of improvements, suggested by long
experience, that that arrangement is at length arrived at, which
accomplishes all the necessary motions in the shortest possible time. By
the application of the mechanical notation, however, absolute perfection
may be arrived at in this respect; even before a single part of the
machinery is constructed, and before it has any other existence than
that which it obtains upon paper.

Examples of this class of advantages derivable from the notation will
occur to the mind of every one acquainted with the history of mechanical
invention. In the common suction-pump, for example, the effective agency
of the power is suspended during the descent of the piston. A very
simple contrivance, however, will transfer to the descent the work to be
accomplished in the next ascent; so that the duty of four strokes of the
piston may thus be executed in the time of two. In the earlier
applications of the steam-engine, that machine was applied almost
exclusively to the process of pumping; and the power acted only during
the descent of the piston, being suspended during its ascent. When,
however, the notion of applying the engine to the general purposes of
manufacture occurred to the mind of Watt, he saw that it would be
necessary to cause it to produce a continued rotatory motion; and,
therefore, that the intervals of intermission must be filled up by the
action of the power. He first proposed to accomplish this by a second
cylinder working alternately with the first; but it soon became apparent
that the blank which existed during the upstroke in the action of the
power, might be filled up by introducing the steam at both ends of the
cylinder alternately. Had Watt placed before him a scheme of mechanical
notation such as we allude to, this expedient would have been so
obtruded upon him that he must have adopted it from the first.

One of the circumstances from which the mechanical notation derives a
great portion of its power as an instrument of investigation and
discovery, is that it enables the inventor to dismiss from his thoughts,
and to disencumber his imagination of the arrangement and connexion of
the mechanism; which, when it is very complex (and it is in that case
that the notation is most useful), can only be kept before the mind by
an embarrassing and painful effort. In this respect the powers of the
notation may not inaptly be illustrated by the facilities derived in
complex and difficult arithmetical questions from the use of the
language and notation of algebra. When once the peculiar conditions of
the question are translated into algebraical signs, and 'reduced to an
equation,' the computist dismisses from his thoughts all the
circumstances of the question, and is relieved from the consideration of
the complicated relations of the quantities of various kinds which may
have entered it. He deals with the algebraical symbols, which are the
representatives of those quantities and relations, according to certain
technical rules of a general nature, the truth of which he has
previously established; and, by a process almost mechanical, he arrives
at the required result. What algebra is to arithmetic, the notation we
now allude to is to mechanism. The various parts of the machinery under
consideration being once expressed upon paper by proper symbols, the
enquirer dismisses altogether from his thoughts the mechanism itself,
and attends only to the symbols; the management of which is so extremely
simple and obvious, that the most unpractised person, having once
acquired an acquaintance with the signs, cannot fail to comprehend their
use.

A remarkable instance of the power and utility of this notation occurred
in a certain stage of the invention of the calculating machinery. A
question arose as to the best method of producing and arranging a
certain series of motions necessary to print and calculate a number. The
inventor, assisted by a practical engineer of considerable experience
and skill, had so arranged these motions, that the whole might be
performed by twelve revolutions of the principal moving axis. It seemed,
however, desirable, if possible, to execute these motions by a less
number of revolutions. To accomplish this, the engineer sat down to
study the complicated details of a part of the machinery which had been
put together; the inventor at the same time applied himself to the
consideration of the arrangement and connexion of the symbols in his
scheme of notation. After a short time, by some transposition of
symbols, he caused the received motions to be completed by eight turns
of the axis. This he accomplished by transferring the symbols which
occupied the last four divisions of his scheme, into such blank spaces
as he could discover in the first eight divisions; due care being taken
that no symbols should express actions at once simultaneous and
incompatible. Pushing his enquiry, however, still further, he proceeded
to ascertain whether his scheme of symbols did not admit of a still more
compact arrangement, and whether eight revolutions were not more than
enough to accomplish what was required. Here the powers of the practical
engineer completely broke down. By no effort could he bring before his
mind such a view of the complicated mechanism as would enable him to
decide upon any improved arrangement. The inventor, however, without any
extraordinary mental exertion, and merely by sliding a bit of ruled
pasteboard up and down his plan, in search of a vacancy where the
different motions might be placed, at length contrived to pack all the
motions, which had previously occupied eight turns of the handle, into
five turns. The symbolic instrument with which he conducted the
investigation, now informed him of the impossibility of reducing the
action of the machine to a more condensed form. This appeared by the
fulness of every space along the lines of compatible action. It was,
however, still possible, by going back to the actual machinery, to
ascertain whether movements, which, under existing arrangements, were
incompatible, might not be brought into harmony. This he accordingly
did, and succeeded in diminishing the number of incompatible conditions,
and thereby rendered it possible to make actions simultaneous which were
before necessarily successive. The notation was now again called into
requisition, and a new disposition of the parts was made. At this point
of the investigation, this extraordinary instrument of mechanical
analysis put forth one of its most singular exertions of power. It
presented to the eye of the engineer two currents of mechanical action,
which, from their nature, could not be simultaneous; and each of which
occupied a complete revolution of the axis, except about a twentieth;
the one occupying the last nineteen-twentieths of a complete revolution
of the axis, and the other occupying the first nineteen-twentieths of a
complete revolution. One of these streams of action was, the successive
picking up by the carrying fingers of the successive carrying claws; and
the other was, the successive shooting of nineteen bolts by the nineteen
bolting fingers. The notation rendered it obvious, that as the bolting
action commenced a small space below the commencement of the carrying,
and ended an equal space below the termination of the carrying, the two
streams of action could be made to flow after one another in one and the
same revolution of the axis. He thus succeeded in reducing the period of
completing the action to four turns of the axis; when the notation again
informed him that he had again attained a limit of condensed action,
which could not be exceeded without a further change in the mechanism.
To the mechanism he again recurred, and soon found that it was possible
to introduce a change which would cause the action to be completed in
three revolutions of the axis. An odd number of revolutions, however,
being attended with certain practical inconveniences, it was considered
more advantageous to execute the motions in four turns; and here again
the notation put forth its powers, by informing the inventor, _through
the eye_, almost independent of his mind, what would be the most elegant,
symmetrical, and harmonious disposition of the required motions in four
turns. This application of an almost metaphysical system of abstract
signs, by which the motion of the hand performs the office of the mind,
and of profound practical skill in mechanics alternately, to the
construction of a most complicated engine, forcibly reminds us of a
parallel in another science, where the chemist with difficulty succeeds
in dissolving a refractory mineral, by the alternate action of the most
powerful acids, and the most caustic alkalies, repeated in
long-continued succession.

This important discovery was explained by Mr Babbage, in a short paper
read before the Royal Society, and published in the Philosophical
Transactions in 1826.[15] It is to us more a matter of regret than
surprise, that the subject did not receive from scientific men in this
country that attention to which its importance in every practical point
of view so fully entitled it. To appreciate it would indeed have been
scarcely possible, from the very brief memoir which its inventor
presented, unaccompanied by any observations or arguments of a nature to
force it upon the attention of minds unprepared for it by the nature of
their studies or occupations. In this country, science has been
generally separated from practical mechanics by a wide chasm. It will be
easily admitted, that an assembly of eminent naturalists and physicians,
with a sprinkling of astronomers, and one or two abstract
mathematicians, were not precisely the persons best qualified to
appreciate such an instrument of mechanical investigation as we have
here described. We shall not therefore be understood as intending the
slightest disrespect for these distinguished persons, when we express
our regret, that a discovery of such paramount practical value, in a
country preeminently conspicuous for the results of its machinery,
should fall still-born and inconsequential through their hands, and be
buried unhonoured and undiscriminated in their miscellaneous
transactions. We trust that a more auspicious period is at hand; that
the chasm which has separated practical from scientific men will
speedily close; and that that combination of knowledge will be effected,
which can only be obtained when we see the men of science more
frequently extending their observant eye over the wonders of our
factories, and our great practical manufacturers, with a reciprocal
ambition, presenting themselves as active and useful members of our
scientific associations. When this has taken place, an order of
scientific men will spring up, which will render impossible an oversight
so little creditable to the country as that which has been committed
respecting the mechanical notation.[16] This notation has recently
undergone very considerable extension and improvement. An additional
section has been introduced into it; designed to express the process of
circulation in machines, through which fluids, whether liquid or
gaseous, are moved. Mr Babbage, with the assistance of a friend who
happened to be conversant with the structure and operation of the
steam-engine, has illustrated it with singular felicity and success in
its application to that machine. An eminent French surgeon, on seeing
the scheme of notation thus applied, immediately suggested the
advantages which must attend it as an instrument for expressing the
structure, operation, and circulation of the animal system; and we
entertain no doubt of its adequacy for that purpose. Not only the
mechanical connexion of the solid members of the bodies of men and
animals, but likewise the structure and operation of the softer parts,
including the muscles, integuments, membranes, &c.; the nature, motion,
and circulation of the various fluids, their reciprocal effects, the
changes through which they pass, the deposits which they leave in
various parts of the system; the functions of respiration, digestion,
and assimilation,--all would find appropriate symbols and
representatives in the notation, even as it now stands, without those
additions of which, however, it is easily susceptible. Indeed, when we
reflect for what a very different purpose this scheme of symbols was
contrived, we cannot refrain from expressing our wonder that it should
seem, in all respects, as if it had been designed expressly for the
purposes of anatomy and physiology.

[Footnote 15: Phil. Trans. 1820, Part III. p, 250, on a method of
expressing by signs the action of machinery.]

[Footnote 16: This discovery has been more justly appreciated by
scientific men abroad. It was, almost immediately after its publication,
adopted as the topic of lectures, in an institution on the Continent for
the instruction of Civil Engineers.]

Another of the uses which the slightest attention to the details of this
notation irresistibly forces upon our notice, is to exhibit, in the form
of a connected plan or map, the organization of an extensive factory, or
any great public institution, in which a vast number of individuals are
employed, and their duties regulated (as they generally are or ought to
be) by a consistent and well-digested system. The mechanical notation is
admirably adapted, not only to express such an organized connexion of
human agents, but even to suggest the improvements of which such
organization is susceptible--to betray its weak and defective points,
and to disclose, at a glance, the origin of any fault which may, from
time to time, be observed in the working of the system. Our limits,
however, preclude us from pursuing this interesting topic to the extent
which its importance would justify. We shall be satisfied if the hints
here thrown out should direct to the subject the attention of those who,
being most interested in such an enquiry, are likely to prosecute it
with greatest success.

One of the consequences which has arisen in the prosecution of the
invention of the calculating machinery, has been the discovery of a
multitude of mechanical contrivances, which have been elicited by the
exigencies of the undertaking, and which are as novel in their nature as
the purposes were novel which they were designed to attain. In some
cases several different contrivances were devised for the attainment of
the same end; and that among them which was best suited for the purpose
was finally selected: the rejected expedients--those overflowings or
waste of the invention--were not, however, always found useless. Like
the _waste_ in various manufactures, they were soon converted to purposes
of utility. These rejected contrivances have found their way, in many
cases, into the mills of our manufacturers; and we now find them busily
effecting purposes, far different from any which the inventor dreamed
of, in the spinning-frames of Manchester.[17]

[Footnote 17: An eminent and wealthy retired manufacturer at Manchester
assured us, that on the occasion of a visit to London, when he was
favoured with a view of the calculating machinery, he found in it
mechanical contrivances, which he subsequently introduced with the
greatest advantage into his own spinning-machinery.]

Another department of mechanical art, which has been enriched by this
invention, has been that of _tools_. The great variety of new forms which
it was necessary to produce, created the necessity of contriving and
constructing a vast number of novel and most valuable tools, by which,
with the aid of the lathe, and that alone, the required forms could be
given to the different parts of the machinery with all the requisite
accuracy.

The idea of calculation by mechanism is not new. Arithmetical
instruments, such as the calculating boards of the ancients, on which
they made their computations by the aid of counters--the _Abacus_, an
instrument for computing by the aid of balls sliding upon parallel
rods--the method of calculation invented by Baron Napier, called by him
_Rhabdology_, and since called _Napier's bones_--the Swan Pan of the
Chinese--and other similar contrivances, among which more particularly
may be mentioned the Sliding Rule, of so much use in practical
calculations to modern engineers, will occur to every reader: these may
more properly be called _arithmetical instruments_, partaking more or less
of a mechanical character. But the earliest piece of mechanism to which
the name of a 'calculating machine' can fairly be given, appears to have
been a machine invented by the celebrated Pascal. This philosopher and
mathematician, at a very early age, being engaged with his father, who
held an official situation in Upper Normandy, the duties of which
required frequent numerical calculations, contrived a piece of mechanism
to facilitate the performance of them. This mechanism consisted of a
series of wheels, carrying cylindrical barrels, on which were engraved
the ten arithmetical characters, in a manner not very dissimilar to that
already described. The wheel which expressed each order of units was so
connected with the wheel which expressed the superior order, that when
the former passed from 9 to 0, the latter was necessarily advanced one
figure; and thus the process of carrying was executed by mechanism: when
one number was to be added to another by this machine, the addition of
each figure to the other was performed by the hand; when it was required
to add more than two numbers, the additions were performed in the same
manner successively; the second was added to the first, the third to
their sum, and so on.

Subtraction was reduced to addition by the method of arithmetical
complements; multiplication was performed by a succession of additions;
and division by a succession of subtractions. In all cases, however, the
operations were executed from wheel to wheel by the hand.[18]

[Footnote 18: See a description of this machine by Diderot, in the
_Encyc. Method._; also in the works of Pascal, tom, IV., p. 7; Paris,
1819.]

This mechanism, which was invented about the year 1650, does not appear
ever to have been brought into any practical use; and seems to have
speedily found its appropriate place in a museum of curiosities. It was
capable of performing only particular arithmetical operations, and these
subject to all the chances of error in manipulation; attended also with
little more expedition (if so much), as would be attained by the pen of
an expert computer.

This attempt of Pascal was followed by various others, with very little
improvement, and with no additional success. Polenus, a learned and
ingenious Italian, invented a machine by which multiplication was
performed, but which does not appear to have afforded any material
facilities, nor any more security against error than the common process
of the pen. A similar attempt was made by Sir Samuel Moreland, who is
described as having transferred to wheel-work the figures of _Napier's
bones_, and as having made some additions to the machine of Pascal.[19]

[Footnote 19: Equidem Morelandus in Anglia, tubæ stentoriæ author,
Rhabdologiam ex baculis in cylindrulos transtulit, et additiones
auxiliares peragit in adjuncta machina additionum Pascaliana.]

Grillet, a French mechanician, made a like attempt with as little
success. Another contrivance for mechanical calculation was made by
Saunderson. Mechanical contrivances for performing particular
arithmetical processes were also made about a century ago by Delepréne
and Boitissendeau; but they were merely modifications of Pascal's,
without varying or extending its objects. But one of the most remarkable
attempts of this kind which has been made since that of Pascal, was a
machine invented by Leibnitz, of which we are not aware that any
detailed or intelligible description was ever published. Leibnitz
described its mode of operation, and its results, in the Berlin
Miscellany,[20] but he appears to have declined any description of its
details. In a letter addressed by him to Bernoulli, in answer to a
request of the latter that he would afford a description of the
machinery, he says, 'Descriptionem ejus dare accuratam res non facilis
foret. De effectu ex eo judicaveris quod ad multiplicandum numerum sex
figurarum, _e.g._ rotam quamdam tantum sexies gyrari necesse est, nulla
alia opera mentis, nullis additionibus intervenientibus; quo facto,
integrum absolutumque productum oculis objicietur.'[21] He goes on to
say that the process of division is performed independently of a
succession of subtractions, such as that used by Pascal.

[Footnote 20: Tom. I., p. 317.]

[Footnote 21: _Com. Epist._ tom, I., p. 289.]

It appears that this machine was one of an extremely complicated nature,
which would be attended with considerable expense of construction, and
only fit to be used in cases where numerous and expensive calculations
were necessary.[22] Leibnitz observes to his correspondent, who required
whether it might not be brought into common use, 'Non est facta pro his
qui olera aut pisculos vendunt, sed pro observatoriis aut cameris
computorum, aut aliis, qui sumptus facile ferunt et multo calculo
egent.' Nevertheless, it does not appear that this contrivance, of which
the inventor states that he caused two models to be made, was ever
applied to any useful purpose; nor indeed do the mechanical details of
the invention appear ever to have been published.

[Footnote 22: Sed machinam esse sumptuosam et multarum rotarum instar
horologii: Huygenius aliquoties admonuit ut absolvi curarem; quod non
sine magno sumptu tædioque factum est, dum varie mihi cum opificibus
fuit conflictandum.--_Com. Epist._]

Even had the mechanism of these machines performed all which their
inventors expected from them, they would have been still altogether
inapplicable for the purposes to which it is proposed that the
calculating machinery of Mr Babbage shall be applied. They were all
constructed with a view to perform particular arithmetical operations,
and in all of them the accuracy of the result depended more or less upon
manipulation. The principle of the calculating machinery of Mr Babbage
is perfectly general in its nature, not depending on any _particular
arithmetical operation_, and is equally applicable to numerical tables of
every kind. This distinguishing characteristic was well expressed by Mr
Colebrooke in his address to the Astronomical Society on this invention.
'The principle which essentially distinguishes Mr Babbage's invention
from all these is, that it proposes to calculate a series of numbers
following any law, by the aid of differences, and that by setting a few
figures at the outset; a long series of numbers is readily produced by a
mechanical operation. The method of differences in a very wide sense is
the mathematical principle of the contrivance. A machine to add a number
of arbitrary figures together is no economy of time or trouble, since
each individual figure must be placed in the machine; but it is
otherwise when those figures follow some law. The insertion of a few at
first determines the magnitude of the next, and those of the succeeding.
It is this constant repetition of similar operations which renders the
computation of tables a fit subject for the application of machinery. Mr
Babbage's invention puts an engine in the place of the computer; the
question is set to the instrument, or the instrument is set to the
question, and by simply giving it motion the solution is wrought, and a
string of answers is exhibited.' But perhaps the greatest of its
advantages is, that it prints what it calculates; and this completely
precludes the possibility of error in those numerical results which pass
into the hands of the public. 'The usefulness of the instrument,' says
Mr Colebrooke, 'is thus more than doubled; for it not only saves time
and trouble in transcribing results into a tabular form, and setting
types for the printing of the table, but it likewise accomplishes the
yet more important object of ensuring accuracy, obviating numerous
sources of error through the careless hands of transcribers and
compositors.'


Some solicitude will doubtless be felt respecting the present state of
the calculating machinery, and the probable period of its completion. In
the beginning of the year 1829, Government directed the Royal Society to
institute such enquiries as would enable them to report upon the state
to which it had then arrived; and also whether the progress made in its
construction confirmed them in the opinion which they had formerly
expressed,--that it would ultimately prove adequate to the important
object which it was intended to attain. The Royal Society, in accordance
with these directions, appointed a Committee to make the necessary
enquiry, and report. This Committee consisted of Mr Davies Gilbert, then
President, the Secretaries, Sir John Herschel, Mr Francis Baily, Mr
Brunel, engineer, Mr Donkin, engineer, Mr G. Rennie, engineer, Mr
Barton, comptroller of the Mint, and Mr Warburton, M.P. The voluminous
drawings, the various tools, and the portion of the machinery then
executed, underwent a close and elaborate examination by this Committee,
who reported upon it to the Society.

They stated in their report, that they declined the consideration of the
principle on which the practicability of the machinery depends, and of
the public utility of the object which it proposes to attain; because
they considered the former fully admitted, and the latter obvious to all
who consider the immense advantage of accurate numerical tables in all
matters of calculation, especially in those which relate to astronomy
and navigation, and the great variety and extent of those which it is
professedly the object of the machinery to calculate and print with
perfect accuracy;--that absolute accuracy being one of the prominent
pretensions of the undertaking, they had directed their attention
especially to this point, by careful examination of the drawings and of
the work already executed, and by repeated conferences with Mr Babbage
on the subject;--that the result of their enquiry was, that such
precautions appeared to have been taken in every part of the
contrivance, and so fully aware was the inventor of every circumstance
which might by possibility produce error, that they had no hesitation in
stating their belief that these precautions were effectual, and that
whatever the machine would do, it would do truly.

They further stated, that the progress which Mr Babbage had then made,
considering the very great difficulties to be overcome in an undertaking
of so novel a kind, fully equalled any expectations that could
reasonably have been formed; and that although several years had elapsed
since the commencement of the undertaking, yet when the necessity of
constructing plans, sections, elevations, and working drawings of every
part; of constructing, and in many cases inventing, tools and machinery
of great expense and complexity, necessary to form with the requisite
precision parts of the apparatus differing from any which had previously
been introduced in ordinary mechanical works; of making many trials to
ascertain the value of each proposed contrivance; of altering,
improving, and simplifying the drawings;--that, considering all these
matters, the Committee, instead of feeling surprise at the time which
the work has occupied, felt more disposed to wonder at the possibility
of accomplishing so much.

The Committee expressed their confident opinion of the adequacy of the
machinery to work under all the friction and strain to which it can be
exposed; of its durability, strength, solidity, and equilibrium; of the
prevention of, or compensation for, wear by friction; of the accuracy of
the various adjustments; and of the judgment and discretion displayed by
the inventor, in his determination to admit into the mechanism nothing
but the very best and most finished workmanship; as a contrary course
would have been false economy, and might have led to the loss of the
whole capital expended on it.

Finally, considering all that had come before them, and relying on the
talent and skill displayed by Mr Babbage as a mechanist in the progress
of this arduous undertaking, not less for what remained, than on the
matured and digested plan and admirable execution of what is completed,
the Committee did not hesitate to express their opinion, that in the
then state of the engine, they regarded it as likely to fulfil the
expectations entertained of it by its inventor.

This report was printed in the commencement of the year 1829. From that
time until the beginning of the year 1833, the progress of the work has
been slow and interrupted. Meanwhile many unfounded rumours have
obtained circulation as to the course adopted by Government in this
undertaking; and as to the position in which Mr Babbage stands with
respect to it. We shall here state, upon authority on which the most
perfect reliance may be placed, what have been the actual circumstances
of the arrangement which has been made, and of the steps which have been
already taken.

Being advised that the objects of the projected machinery were of
paramount national importance to a maritime country, and that, from its
nature, it could never be undertaken with advantage by any individual as
a pecuniary speculation. Government determined to engage Mr Babbage to
construct the calculating engine for the nation. It was then thought
that the work could be completed in two or three years; and it was
accordingly undertaken on this understanding about the year 1821, and
since then has been in progress. The execution of the workmanship was
confided to an engineer by whom all the subordinate workmen were
employed, and who supplied for the work the requisite tools and other
machinery; the latter being his own property, and not that of
Government. This engineer furnished, at intervals, his accounts, which
were duly audited by proper persons appointed for that purpose. It was
thought advisable--with a view, perhaps, to invest Mr Babbage with a
more strict authority over the subordinate agents--that the payments of
these accounts of the engineer should pass through his hands. The amount
was accordingly from time to time issued to him by the Treasury, and
paid over to the engineer. This circumstance has given rise to reports,
that he has received considerable sums of money as a remuneration for
his skill and labour in inventing and constructing this machinery. Such
reports are altogether destitute of truth. He has received, neither
directly nor indirectly, any remuneration whatever;--on the contrary,
owing to various official delays in the issues of money from the
Treasury for the payment of the engineer, he has frequently been obliged
to advance these payments himself, that the work might proceed without
interruption. Had he not been enabled to do this from his private
resources, it would have been impossible that the machinery could have
arrived at its present advanced state.

It will be a matter of regret to every friend of science to learn, that,
notwithstanding such assistance, the progress of the work has been
suspended, and the workmen dismissed for more than a year and a half;
nor does there at the present moment appear to be any immediate prospect
of its being resumed. What the causes may be of a suspension so
extraordinary, of a project of such great national and universal
interest,--in which the country has already invested a sum of such
serious amount as L.15,000,--is a question which will at once suggest
itself to every mind; and is one to which, notwithstanding frequent
enquiries, in quarters from which correct information might be expected,
we have not been able to obtain any satisfactory answer. It is not true,
we are assured, that the Government object to make the necessary
payments, or even advances, to carry on the work. It is not true, we
also are assured, that any practical difficulty has arisen in the
construction of the mechanism;--on the contrary, the drawings of all
the parts of it are completed, and may be inspected by any person
appointed on the part of Government to examine them.[23] Mr Babbage is
known as a man of unwearied activity, and aspiring ambition. Why, then,
it may be asked, is it that he, seeing his present reputation and future
fame depending in so great a degree upon the successful issue of this
undertaking, has nevertheless allowed it to stand still for so long a
period, without distinctly pointing out to Government the course which
they should adopt to remove the causes of delay? Had he done this (which
we consider to be equally due to the nation and to himself), he would
have thrown upon Government and its agents the whole responsibility for
the delay and consequent loss; but we believe he has not done so. On the
contrary, it is said that he has of late almost withdrawn from all
interference on the subject, either with the Government or the engineer.
Does not Mr Babbage perceive the inference which the world will draw
from this course of conduct? Does he not see that they will impute it to
a distrust of his own power, or even to a consciousness of his own
inability to complete what he has begun? We feel assured that such is
not the case; and we are anxious, equally for the sake of science, and
for Mr Babbage's own reputation, that the mystery--for such it must be
regarded--should be cleared up; and that all obstructions to the
progress of the undertaking should immediately be removed. Does this
supineness and apparent indifference, so incompatible with the known
character of Mr Babbage, arise from any feeling of dissatisfaction at
the existing arrangements between himself and the Government? If such be
the actual cause of the delay, (and we believe that, in some degree, it
is so,) we cannot refrain from expressing our surprise that he does not
adopt the candid and straightforward course of declaring the grounds of
his discontent, and explaining the arrangement which he desires to be
adopted. We do not hesitate to say, that every reasonable accommodation
and assistance ought to be afforded him. But if he will pertinaciously
abstain from this, to our minds, obvious and proper course, then it is
surely the duty of Government to appoint proper persons to enquire into
and report on the present state of the machinery; to ascertain the
causes of its suspension; and to recommend such measures as may appear
to be most effectual to ensure its speedy completion. If they do not by
such means succeed in putting the project in a state of advancement,
they will at least shift from themselves all responsibility for its
suspension.

[Footnote 23: Government has erected a fire-proof building, in which it
is intended that the calculating machinery shall be placed when
completed. In this building are now deposited the large collection of
drawings, containing the designs, not only of the part of the machinery
which has been already constructed, but what is of much greater
importance, of those parts which have not yet been even modelled. It is
gratifying to know that Government has shown a proper solicitude for the
preservation of those precious but perishable documents, the loss or
destruction of which would, in the event of the death of the inventor,
render the completion of the machinery impracticable.]