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 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 at 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