diff options
| -rw-r--r-- | .gitattributes | 4 | ||||
| -rw-r--r-- | LICENSE.txt | 11 | ||||
| -rw-r--r-- | README.md | 2 | ||||
| -rw-r--r-- | old/71292-0.txt | 3103 | ||||
| -rw-r--r-- | old/71292-0.zip | bin | 62424 -> 0 bytes | |||
| -rw-r--r-- | old/71292-h.zip | bin | 1280612 -> 0 bytes | |||
| -rw-r--r-- | old/71292-h/71292-h.htm | 3974 | ||||
| -rw-r--r-- | old/71292-h/images/figure01.jpg | bin | 247828 -> 0 bytes | |||
| -rw-r--r-- | old/71292-h/images/figure02.jpg | bin | 230629 -> 0 bytes | |||
| -rw-r--r-- | old/71292-h/images/figure03.jpg | bin | 256707 -> 0 bytes | |||
| -rw-r--r-- | old/71292-h/images/figure04.jpg | bin | 245050 -> 0 bytes | |||
| -rw-r--r-- | old/71292-h/images/figure05.jpg | bin | 237382 -> 0 bytes |
12 files changed, 17 insertions, 7077 deletions
diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..d7b82bc --- /dev/null +++ b/.gitattributes @@ -0,0 +1,4 @@ +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/LICENSE.txt b/LICENSE.txt new file mode 100644 index 0000000..6312041 --- /dev/null +++ b/LICENSE.txt @@ -0,0 +1,11 @@ +This eBook, including all associated images, markup, improvements, +metadata, and any other content or labor, has been confirmed to be +in the PUBLIC DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..bdf0e09 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #71292 (https://www.gutenberg.org/ebooks/71292) diff --git a/old/71292-0.txt b/old/71292-0.txt deleted file mode 100644 index ec2ac6f..0000000 --- a/old/71292-0.txt +++ /dev/null @@ -1,3103 +0,0 @@ -The Project Gutenberg eBook of Babbage's calculating engine, by -Charles Babbage - -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you -will have to check the laws of the country where you are located before -using this eBook. - -Title: Babbage's calculating engine - -Author: Charles Babbage - -Release Date: July 28, 2023 [eBook #71292] - -Language: English - -Credits: Laura Natal Rodrigues (Images generously made available by - Hathi Trust Digital Library) - -*** START OF THE PROJECT GUTENBERG EBOOK BABBAGE'S CALCULATING -ENGINE *** - - - 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.] - -*** END OF THE PROJECT GUTENBERG EBOOK BABBAGE'S CALCULATING -ENGINE *** - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the -United States without permission and without paying copyright -royalties. Special rules, set forth in the General Terms of Use part -of this license, apply to copying and distributing Project -Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ -concept and trademark. Project Gutenberg is a registered trademark, -and may not be used if you charge for an eBook, except by following -the terms of the trademark license, including paying royalties for use -of the Project Gutenberg trademark. If you do not charge anything for -copies of this eBook, complying with the trademark license is very -easy. You may use this eBook for nearly any purpose such as creation -of derivative works, reports, performances and research. Project -Gutenberg eBooks may be modified and printed and given away--you may -do practically ANYTHING in the United States with eBooks not protected -by U.S. copyright law. Redistribution is subject to the trademark -license, especially commercial redistribution. - -START: FULL LICENSE - -THE FULL PROJECT GUTENBERG LICENSE -PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK - -To protect the Project Gutenberg™ mission of promoting the free -distribution of electronic works, by using or distributing this work -(or any other work associated in any way with the phrase “Project -Gutenberg”), you agree to comply with all the terms of the Full -Project Gutenberg™ License available with this file or online at -www.gutenberg.org/license. - -Section 1. General Terms of Use and Redistributing Project -Gutenberg™ electronic works - -1.A. By reading or using any part of this Project Gutenberg™ -electronic work, you indicate that you have read, understand, agree to -and accept all the terms of this license and intellectual property -(trademark/copyright) agreement. If you do not agree to abide by all -the terms of this agreement, you must cease using and return or -destroy all copies of Project Gutenberg™ electronic works in your -possession. If you paid a fee for obtaining a copy of or access to a -Project Gutenberg™ electronic work and you do not agree to be bound -by the terms of this agreement, you may obtain a refund from the -person or entity to whom you paid the fee as set forth in paragraph -1.E.8. - -1.B. “Project Gutenberg” is a registered trademark. It may only be -used on or associated in any way with an electronic work by people who -agree to be bound by the terms of this agreement. There are a few -things that you can do with most Project Gutenberg™ electronic works -even without complying with the full terms of this agreement. See -paragraph 1.C below. There are a lot of things you can do with Project -Gutenberg™ electronic works if you follow the terms of this -agreement and help preserve free future access to Project Gutenberg™ -electronic works. See paragraph 1.E below. - -1.C. The Project Gutenberg Literary Archive Foundation (“the -Foundation” or PGLAF), owns a compilation copyright in the collection -of Project Gutenberg™ electronic works. Nearly all the individual -works in the collection are in the public domain in the United -States. If an individual work is unprotected by copyright law in the -United States and you are located in the United States, we do not -claim a right to prevent you from copying, distributing, performing, -displaying or creating derivative works based on the work as long as -all references to Project Gutenberg are removed. Of course, we hope -that you will support the Project Gutenberg™ mission of promoting -free access to electronic works by freely sharing Project Gutenberg™ -works in compliance with the terms of this agreement for keeping the -Project Gutenberg™ name associated with the work. You can easily -comply with the terms of this agreement by keeping this work in the -same format with its attached full Project Gutenberg™ License when -you share it without charge with others. - -1.D. The copyright laws of the place where you are located also govern -what you can do with this work. Copyright laws in most countries are -in a constant state of change. If you are outside the United States, -check the laws of your country in addition to the terms of this -agreement before downloading, copying, displaying, performing, -distributing or creating derivative works based on this work or any -other Project Gutenberg™ work. The Foundation makes no -representations concerning the copyright status of any work in any -country other than the United States. - -1.E. Unless you have removed all references to Project Gutenberg: - -1.E.1. The following sentence, with active links to, or other -immediate access to, the full Project Gutenberg™ License must appear -prominently whenever any copy of a Project Gutenberg™ work (any work -on which the phrase “Project Gutenberg” appears, or with which the -phrase “Project Gutenberg” is associated) is accessed, displayed, -performed, viewed, copied or distributed: - - This eBook is for the use of anyone anywhere in the United States and - most other parts of the world at no cost and with almost no - restrictions whatsoever. You may copy it, give it away or re-use it - under the terms of the Project Gutenberg License included with this - eBook or online at www.gutenberg.org. If you are not located in the - United States, you will have to check the laws of the country where - you are located before using this eBook. - -1.E.2. If an individual Project Gutenberg™ electronic work is -derived from texts not protected by U.S. copyright law (does not -contain a notice indicating that it is posted with permission of the -copyright holder), the work can be copied and distributed to anyone in -the United States without paying any fees or charges. If you are -redistributing or providing access to a work with the phrase “Project -Gutenberg” associated with or appearing on the work, you must comply -either with the requirements of paragraphs 1.E.1 through 1.E.7 or -obtain permission for the use of the work and the Project Gutenberg™ -trademark as set forth in paragraphs 1.E.8 or 1.E.9. - -1.E.3. If an individual Project Gutenberg™ electronic work is posted -with the permission of the copyright holder, your use and distribution -must comply with both paragraphs 1.E.1 through 1.E.7 and any -additional terms imposed by the copyright holder. Additional terms -will be linked to the Project Gutenberg™ License for all works -posted with the permission of the copyright holder found at the -beginning of this work. - -1.E.4. Do not unlink or detach or remove the full Project Gutenberg™ -License terms from this work, or any files containing a part of this -work or any other work associated with Project Gutenberg™. - -1.E.5. Do not copy, display, perform, distribute or redistribute this -electronic work, or any part of this electronic work, without -prominently displaying the sentence set forth in paragraph 1.E.1 with -active links or immediate access to the full terms of the Project -Gutenberg™ License. - -1.E.6. You may convert to and distribute this work in any binary, -compressed, marked up, nonproprietary or proprietary form, including -any word processing or hypertext form. However, if you provide access -to or distribute copies of a Project Gutenberg™ work in a format -other than “Plain Vanilla ASCII” or other format used in the official -version posted on the official Project Gutenberg™ website -(www.gutenberg.org), you must, at no additional cost, fee or expense -to the user, provide a copy, a means of exporting a copy, or a means -of obtaining a copy upon request, of the work in its original “Plain -Vanilla ASCII” or other form. Any alternate format must include the -full Project Gutenberg™ License as specified in paragraph 1.E.1. - -1.E.7. Do not charge a fee for access to, viewing, displaying, -performing, copying or distributing any Project Gutenberg™ works -unless you comply with paragraph 1.E.8 or 1.E.9. - -1.E.8. You may charge a reasonable fee for copies of or providing -access to or distributing Project Gutenberg™ electronic works -provided that: - -• You pay a royalty fee of 20% of the gross profits you derive from - the use of Project Gutenberg™ works calculated using the method - you already use to calculate your applicable taxes. The fee is owed - to the owner of the Project Gutenberg™ trademark, but he has - agreed to donate royalties under this paragraph to the Project - Gutenberg Literary Archive Foundation. Royalty payments must be paid - within 60 days following each date on which you prepare (or are - legally required to prepare) your periodic tax returns. Royalty - payments should be clearly marked as such and sent to the Project - Gutenberg Literary Archive Foundation at the address specified in - Section 4, “Information about donations to the Project Gutenberg - Literary Archive Foundation.” - -• You provide a full refund of any money paid by a user who notifies - you in writing (or by e-mail) within 30 days of receipt that s/he - does not agree to the terms of the full Project Gutenberg™ - License. You must require such a user to return or destroy all - copies of the works possessed in a physical medium and discontinue - all use of and all access to other copies of Project Gutenberg™ - works. - -• You provide, in accordance with paragraph 1.F.3, a full refund of - any money paid for a work or a replacement copy, if a defect in the - electronic work is discovered and reported to you within 90 days of - receipt of the work. - -• You comply with all other terms of this agreement for free - distribution of Project Gutenberg™ works. - -1.E.9. If you wish to charge a fee or distribute a Project -Gutenberg™ electronic work or group of works on different terms than -are set forth in this agreement, you must obtain permission in writing -from the Project Gutenberg Literary Archive Foundation, the manager of -the Project Gutenberg™ trademark. Contact the Foundation as set -forth in Section 3 below. - -1.F. - -1.F.1. Project Gutenberg volunteers and employees expend considerable -effort to identify, do copyright research on, transcribe and proofread -works not protected by U.S. copyright law in creating the Project -Gutenberg™ collection. Despite these efforts, Project Gutenberg™ -electronic works, and the medium on which they may be stored, may -contain “Defects,” such as, but not limited to, incomplete, inaccurate -or corrupt data, transcription errors, a copyright or other -intellectual property infringement, a defective or damaged disk or -other medium, a computer virus, or computer codes that damage or -cannot be read by your equipment. - -1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right -of Replacement or Refund” described in paragraph 1.F.3, the Project -Gutenberg Literary Archive Foundation, the owner of the Project -Gutenberg™ trademark, and any other party distributing a Project -Gutenberg™ electronic work under this agreement, disclaim all -liability to you for damages, costs and expenses, including legal -fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT -LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE -PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE -TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE -LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR -INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH -DAMAGE. - -1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a -defect in this electronic work within 90 days of receiving it, you can -receive a refund of the money (if any) you paid for it by sending a -written explanation to the person you received the work from. If you -received the work on a physical medium, you must return the medium -with your written explanation. The person or entity that provided you -with the defective work may elect to provide a replacement copy in -lieu of a refund. If you received the work electronically, the person -or entity providing it to you may choose to give you a second -opportunity to receive the work electronically in lieu of a refund. If -the second copy is also defective, you may demand a refund in writing -without further opportunities to fix the problem. - -1.F.4. Except for the limited right of replacement or refund set forth -in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO -OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT -LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE. - -1.F.5. Some states do not allow disclaimers of certain implied -warranties or the exclusion or limitation of certain types of -damages. If any disclaimer or limitation set forth in this agreement -violates the law of the state applicable to this agreement, the -agreement shall be interpreted to make the maximum disclaimer or -limitation permitted by the applicable state law. The invalidity or -unenforceability of any provision of this agreement shall not void the -remaining provisions. - -1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the -trademark owner, any agent or employee of the Foundation, anyone -providing copies of Project Gutenberg™ electronic works in -accordance with this agreement, and any volunteers associated with the -production, promotion and distribution of Project Gutenberg™ -electronic works, harmless from all liability, costs and expenses, -including legal fees, that arise directly or indirectly from any of -the following which you do or cause to occur: (a) distribution of this -or any Project Gutenberg™ work, (b) alteration, modification, or -additions or deletions to any Project Gutenberg™ work, and (c) any -Defect you cause. - -Section 2. Information about the Mission of Project Gutenberg™ - -Project Gutenberg™ is synonymous with the free distribution of -electronic works in formats readable by the widest variety of -computers including obsolete, old, middle-aged and new computers. It -exists because of the efforts of hundreds of volunteers and donations -from people in all walks of life. - -Volunteers and financial support to provide volunteers with the -assistance they need are critical to reaching Project Gutenberg™’s -goals and ensuring that the Project Gutenberg™ collection will -remain freely available for generations to come. In 2001, the Project -Gutenberg Literary Archive Foundation was created to provide a secure -and permanent future for Project Gutenberg™ and future -generations. To learn more about the Project Gutenberg Literary -Archive Foundation and how your efforts and donations can help, see -Sections 3 and 4 and the Foundation information page at -www.gutenberg.org. - -Section 3. Information about the Project Gutenberg Literary -Archive Foundation - -The Project Gutenberg Literary Archive Foundation is a non-profit -501(c)(3) educational corporation organized under the laws of the -state of Mississippi and granted tax exempt status by the Internal -Revenue Service. The Foundation’s EIN or federal tax identification -number is 64-6221541. Contributions to the Project Gutenberg Literary -Archive Foundation are tax deductible to the full extent permitted by -U.S. federal laws and your state’s laws. - -The Foundation’s business office is located at 809 North 1500 West, -Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up -to date contact information can be found at the Foundation’s website -and official page at www.gutenberg.org/contact. - -Section 4. Information about Donations to the Project Gutenberg -Literary Archive Foundation - -Project Gutenberg™ depends upon and cannot survive without -widespread public support and donations to carry out its mission of -increasing the number of public domain and licensed works that can be -freely distributed in machine-readable form accessible by the widest -array of equipment including outdated equipment. Many small donations -($1 to $5,000) are particularly important to maintaining tax exempt -status with the IRS. - -The Foundation is committed to complying with the laws regulating -charities and charitable donations in all 50 states of the United -States. Compliance requirements are not uniform and it takes a -considerable effort, much paperwork and many fees to meet and keep up -with these requirements. We do not solicit donations in locations -where we have not received written confirmation of compliance. To SEND -DONATIONS or determine the status of compliance for any particular -state visit www.gutenberg.org/donate. - -While we cannot and do not solicit contributions from states where we -have not met the solicitation requirements, we know of no prohibition -against accepting unsolicited donations from donors in such states who -approach us with offers to donate. - -International donations are gratefully accepted, but we cannot make -any statements concerning tax treatment of donations received from -outside the United States. U.S. laws alone swamp our small staff. - -Please check the Project Gutenberg web pages for current donation -methods and addresses. Donations are accepted in a number of other -ways including checks, online payments and credit card donations. To -donate, please visit: www.gutenberg.org/donate. - -Section 5. General Information About Project Gutenberg™ electronic works - -Professor Michael S. Hart was the originator of the Project -Gutenberg™ concept of a library of electronic works that could be -freely shared with anyone. For forty years, he produced and -distributed Project Gutenberg™ eBooks with only a loose network of -volunteer support. - -Project Gutenberg™ eBooks are often created from several printed -editions, all of which are confirmed as not protected by copyright in -the U.S. unless a copyright notice is included. Thus, we do not -necessarily keep eBooks in compliance with any particular paper -edition. - -Most people start at our website which has the main PG search -facility: www.gutenberg.org. - -This website includes information about Project Gutenberg™, -including how to make donations to the Project Gutenberg Literary -Archive Foundation, how to help produce our new eBooks, and how to -subscribe to our email newsletter to hear about new eBooks. diff --git a/old/71292-0.zip b/old/71292-0.zip Binary files differdeleted file mode 100644 index 9b7e328..0000000 --- a/old/71292-0.zip +++ /dev/null diff --git a/old/71292-h.zip b/old/71292-h.zip Binary files differdeleted file mode 100644 index 4f6f0cc..0000000 --- a/old/71292-h.zip +++ /dev/null diff --git a/old/71292-h/71292-h.htm b/old/71292-h/71292-h.htm deleted file mode 100644 index f91cbf1..0000000 --- a/old/71292-h/71292-h.htm +++ /dev/null @@ -1,3974 +0,0 @@ -<!DOCTYPE html> -<html lang="en"> -<head> -<meta charset="UTF-8"> -<title>Babbage's calculating engine | Project Gutenberg</title> - -<style> - -body { - font-family: "Times New Roman", Times, serif; - margin-left: 10%; - margin-right: 10%; -} - -h1 { - text-align: center; - clear: both; -} -h2 { - text-align: center; - font-weight: bold; - margin-top: 1em; - margin-bottom: 1em; - } - -h3,h4,h5,h6 { - text-align: center; - clear: both; -} - -p { - margin-top: .51em; - text-align: justify; - margin-bottom: .49em; - text-indent:4%; -} - -.nind {text-indent:0%;} - -.hanging2 {padding-left: 2em; - text-indent: -2em; - } - -.right {text-align: right; -} - -hr { - width: 33%; - margin-top: 2em; - margin-bottom: 2em; - margin-left: auto; - margin-right: auto; - clear: both; -} - -hr.r5 {width: 5%; margin-top: 1em; margin-bottom: 1em;} - -.dropcap { - float: left; - clear: left; - font-size: 250%; - margin-top:-.7%; - margin: 0 0.15em 0 0; - padding: 0; - line-height: 0.85em; - text-indent: 0 -} - -sup { - vertical-align: super; - font-size: smaller; -} - -.space-above1 { margin-top: 1em; } -.space-above2 { margin-top: 2em; } -.space-above3 { margin-top: 3em; } -.space-below1 { margin-bottom: 1em; } -.space-below2 { margin-bottom: 2em; } - -table { margin-left: auto; margin-right: auto;} -table.box_border {border-collapse: collapse; - border-spacing: 0px; - border: 1px solid black} - -.tdl {text-align: left;} -.tdc {text-align: center;} -.tdl_ws1 {text-align: left; vertical-align: top; padding-left: 1em;} -.tdr_ws1 {text-align: right; vertical-align: top; padding-right: 1em;} - -.bb {border-bottom: solid thin;} -.bbn {border-bottom: hidden;} -.bt2 {border-top: solid thin;} -.br {border-right: solid thin;} - -.fontsize_100 { font-size: 100%; } -.fontsize_120 { font-size: 120%; } -.no-wrap {white-space: nowrap; } -.center {text-align: center; text-indent: 0em;} - -td.hanging2 { - vertical-align: middle; - text-align: justify; - text-indent: -1em; - padding-left: 1em; - padding-right: .25em; - padding-bottom: .25em; - padding-top: .25em; -} - -div.chapter { - page-break-before: always; - margin-top: 4em - } - -.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} - -.footnote .label {position: absolute; right: 84%; text-align: right;} - -.fnanchor { - vertical-align: super; - font-size: .8em; - text-decoration: - none; -} - -.figcenter { - margin: 3% auto 3% auto; - clear: both; - text-align: center; - text-indent: 0% -} - -.caption {font-weight: normal; - font-size: 90%; - text-align: right; - padding-bottom: 1em;} - -.caption p -{ - text-align: center; - text-indent: 0; - margin: 0.25em 0; -} - - </style> -</head> - -<body> -<p style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of Babbage's calculating engine, by Charles Babbage</p> -<div style='display:block; margin:1em 0'> -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online -at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you -are not located in the United States, you will have to check the laws of the -country where you are located before using this eBook. -</div> - -<p style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: Babbage's calculating engine</p> -<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Author: Charles Babbage</p> -<p style='display:block; text-indent:0; margin:1em 0'>Release Date: July 28, 2023 [eBook #71292]</p> -<p style='display:block; text-indent:0; margin:1em 0'>Language: English</p> - <p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em; text-align:left'>Credits: Laura Natal Rodrigues (Images generously made available by Hathi Trust Digital Library)</p> -<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK BABBAGE'S CALCULATING ENGINE ***</div> - -<div class="chapter"> -<p class="center space-above2 space-below1">THE<br> -EDINBURGH REVIEW,</p> - -<p class="center space-above2 space-below2">JULY, 1834.</p> - -<hr class="r5"> - -<p class="center space-above2 space-below2">No. CXX.</p> - -<hr class="r5"> - -<h1>THE CALCULATING ENGINE</h1> - -<p class="center space-above2 space-below1">BY</p> - -<h2>CHARLES BABBAGE</h2> - -<p class="hanging2"> -Art I.—1. <i>Letter to Sir Humphry Davy, Bart. P.R.S., on the -application of Machinery to Calculate and Print Mathematical Tables</i>. -By CHARLES BABBAGE, Esq. F.R.S. 4to. Printed by order of the House of -Commons. -</p> -<p class="hanging2"> -2. <i>On the Application of Machinery to the Calculation of Astronomical -and Mathematical Tables</i>. By CHARLES BABBAGE, Esq. Memoirs Astron. Soc. -Vol. I. Part 2. London: 1822. -</p> -<p class="hanging2"> -3. <i>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</i>. -Memoirs Astron. Soc. Vol. I. Part 2. London: 1822. -</p> -<p class="hanging2"> -4. <i>On the determination of the General Term of a new Class of Infinite -Series</i>. By CHARLES BABBAGE, Esq. Transactions Camb. Phil. Soc. -Cambridge: 1824. -</p> -<p class="hanging2"> -5. <i>On Errors common to many Tables of Logarithms</i>. By CHARLES -BABBAGE, Esq. Memoirs Astron. Soc. London: 1827. -</p> -<p class="hanging2"> -6. <i>On a Method of Expressing by Signs the Action of Machinery</i>. -By CHARLES BABBAGE, Esq. Phil. Trans. London: 1826. -</p> -<p class="hanging2"> -7. <i>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</i>. London: 1829. -</p> - -<p class="nind space-above3 space-below1"> -<span class="dropcap">T</span>HERE 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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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 -<i>invention</i> of that machine—the <i>improvements</i> 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.<a id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> -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. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a>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.</p></div> - -<p> -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. -</p> -<p> -<i>First</i>, 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. -</p> -<p> -<i>Secondly</i>, 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. -</p> -<p> -<i>Thirdly</i>, 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. -</p> -<p class="space-above2 space-below1"> -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. -</p> -<p> -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 <i>Powers</i>, in which a number is multiplied by -itself successively. In Dodson's <i>Calculator</i> we find a table of -multiplication extending as far as 10 times 1000.<a id="FNanchor_2_1"></a><a href="#Footnote_2_1" class="fnanchor">[2]</a> -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.<a id="FNanchor_3_1"></a><a href="#Footnote_3_1" class="fnanchor">[3]</a> 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. - - -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_2_1"></a><a href="#FNanchor_2_1"><span class="label">[2]</span></a>Dodson's <i>Calculator</i>. 4to. London: 1747.</p></div> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_3_1"></a><a href="#FNanchor_3_1"><span class="label">[3]</span></a>Hutton's <i>Tables of Products and Powers</i>. Folio. -London; 1781.</p></div> - -<p> -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. -</p> -<p> -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. -</p> -<p> -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<a id="FNanchor_4_1"></a><a href="#Footnote_4_1" class="fnanchor">[4]</a> 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. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_4_1"></a><a href="#FNanchor_4_1"><span class="label">[4]</span></a>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.</p></div> - -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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 <i>Nautical -Astronomy</i> 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 <i>Nautical Astronomy</i> 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 <i>Nautical -Geography</i>, 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.<a id="FNanchor_5_1"></a><a href="#Footnote_5_1" class="fnanchor">[5]</a> -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_5_1"></a><a href="#FNanchor_5_1"><span class="label">[5]</span></a>Report of the Committee of the Astronomical Society prefixed -to the Nautical Almanac for 1834.</p></div> - -<p> -Among the tables <i>directly</i> 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. -</p> -<p> -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 <i>time</i> 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. -</p> -<p> -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. -</p> -<p> -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.<a id="FNanchor_6_1"></a><a href="#Footnote_6_1" class="fnanchor">[6]</a> -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_6_1"></a><a href="#FNanchor_6_1"><span class="label">[6]</span></a>A comparison of the results for 1834, will be found in the -Nautical Almanac for 1835.</p></div> - -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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 <i>acknowledged</i> in the respective errata, amounts to above -<i>three thousand seven hundred</i>. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -The body of tables thus calculated occupied in manuscript <i>seventeen</i> -folio volumes.<a id="FNanchor_7_1"></a><a href="#Footnote_7_1" class="fnanchor">[7]</a> -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_7_1"></a><a href="#FNanchor_7_1"><span class="label">[7]</span></a>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.</p></div> - -<p> -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. -<a id="FNanchor_8_1"></a><a href="#Footnote_8_1" class="fnanchor">[8]</a> 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. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_8_1"></a><a href="#FNanchor_8_1"><span class="label">[8]</span></a>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.</p></div> - -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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.<a id="FNanchor_9_1"></a><a href="#Footnote_9_1" class="fnanchor">[9]</a> -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_9_1"></a><a href="#FNanchor_9_1"><span class="label">[9]</span></a>Memoirs Ast. Soc. vol. III, p. 65.</p></div> - -<p> -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. -</p> -<p> -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 <i>that result wrong</i>. 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 <i>wrongly</i>, and yet <i>consistently</i>, 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 <i>the same erroneous numerical result</i>. It was only on -going over the operation a second time that he <i>accidentally</i> -discovered that he had inadvertently committed the same error.<a id="FNanchor_10_1"></a><a href="#Footnote_10_1" class="fnanchor">[10]</a> - - -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_10_1"></a><a href="#FNanchor_10_1"><span class="label">[10]</span></a>Memoirs Ast. Soc. vol. iv., p. 290.</p></div> - -<p> -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. -</p> -<p> -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:— -</p> - -<p class="center space-above1 space-below1"> -35875<br> -10436.</p> - -<p class="nind"> -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 -<i>above</i> and the 4 <i>below</i>, instead of <i>vice versa</i>. 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. -</p> -<p> -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, <i>in which nevertheless the erroneous figures in Vlacq are -omitted</i>. 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 -<i>eighth place</i> in Vlacq causes a corresponding error in the -<i>seventh</i> 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. -</p> -<p> -The <i>Opus Palatinum</i>, 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 <i>Opus Palatinum</i> 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,<a id="FNanchor_11_1"></a><a href="#Footnote_11_1" class="fnanchor">[11]</a> the whole of the -erroneous part of the <i>Opus Palatinum</i> 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 <i>basis</i> and <i>hypothenusa</i>, -their places being interchanged. In the original uncorrected work this -error does not exist. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_11_1"></a><a href="#FNanchor_11_1"><span class="label">[11]</span></a><i>Recueil des Tables Logarithmiques et Trigonometriques</i>. -Par J. C. Schulze. 2 vols. Berlin: 1778.</p></div> - -<p> -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. -</p> - -<p class="center space-above2 space-below2"> -ERRATA, <i>detected in</i> Taylor's <i>Logarithms</i>. <i>London: 4to</i>, -1792. -</p> - -<table class="no-wrap"> -<thead><tr> -<th class="tdc"> </th> -<th class="tdc"> </th> -<th class="tdc"> </th> -<th class="tdc">° ' " </th> -<th class="tdc"> </th> -<th class="tdc"> </th> -</tr> -</thead> -<tbody><tr> -<td class="tdr_ws1"> 1</td> -<td class="tdl_ws1"><i>E</i></td> -<td class="tdl_ws1">Co-tangent of</td> -<td class="tdr_ws1"> 1.35.35</td> -<td class="tdr_ws1"><i>for</i> 43671</td> -<td class="tdr_ws1"><i>read</i> 42671</td> -</tr><tr> -<td class="tdr_ws1"> 2</td> -<td class="tdl_ws1"><i>M</i></td> -<td class="tdl_ws1">Co-tangent of</td> -<td class="tdr_ws1"> 4. 4.49</td> -<td class="tdr_ws1">— 66976</td> -<td class="tdr_ws1">—— 66979</td> -</tr><tr> -<td class="tdr_ws1"> 3</td> -<td class="tdl_ws1"> </td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1"> 4.23.38</td> -<td class="tdr_ws1">— 43107</td> -<td class="tdr_ws1">—— 43007</td> -</tr><tr> -<td class="tdr_ws1"> 4</td> -<td class="tdl_ws1"> </td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1"> 4.23.39</td> -<td class="tdr_ws1">— 43381</td> -<td class="tdr_ws1">—— 43281</td> -</tr><tr> -<td class="tdr_ws1"> 5</td> -<td class="tdl_ws1"><i>S</i></td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1"> 6.45.52</td> -<td class="tdr_ws1">— 10001</td> -<td class="tdr_ws1">—— 11001</td> -</tr><tr> -<td class="tdr_ws1"> 6</td> -<td class="tdl_ws1"><i>Kk</i></td> -<td class="tdl_ws1">Co-sine of</td> -<td class="tdr_ws1">14.18. 3</td> -<td class="tdr_ws1">— 3398</td> -<td class="tdr_ws1">—— 3298</td> -</tr><tr> -<td class="tdr_ws1"> 7</td> -<td class="tdl_ws1"><i>Ss</i></td> -<td class="tdl_ws1">Tangent of</td> -<td class="tdr_ws1">18. 1.56</td> -<td class="tdr_ws1">— 5064</td> -<td class="tdr_ws1">—— 6064</td> -</tr><tr> -<td class="tdr_ws1"> 8</td> -<td class="tdl_ws1"><i>Aaa</i></td> -<td class="tdl_ws1">Co-tangent of </td> -<td class="tdr_ws1">21.11.14</td> -<td class="tdr_ws1">— 6062</td> -<td class="tdr_ws1">—— 5962</td> -</tr><tr> -<td class="tdr_ws1"> 9</td> -<td class="tdl_ws1"><i>Ggg</i></td> -<td class="tdl_ws1">Tangent of</td> -<td class="tdr_ws1">23.48.19</td> -<td class="tdr_ws1">— 6087</td> -<td class="tdr_ws1">—— 5987</td> -</tr><tr> -<td class="tdr_ws1">10</td> -<td class="tdl_ws1"> </td> -<td class="tdl_ws1">Co-tangent of </td> -<td class="tdr_ws1">23.48.19</td> -<td class="tdr_ws1">— 3913</td> -<td class="tdr_ws1">—— 4013</td> -</tr><tr> -<td class="tdr_ws1">11</td> -<td class="tdl_ws1"><i>Iii</i></td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1">25. 5. 4</td> -<td class="tdr_ws1">— 3173</td> -<td class="tdr_ws1">—— 3183</td> -</tr><tr> -<td class="tdr_ws1">12</td> -<td class="tdl_ws1"> </td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1">25. 5. 5</td> -<td class="tdr_ws1">— 3218</td> -<td class="tdr_ws1">—— 3228</td> -</tr><tr> -<td class="tdr_ws1">13</td> -<td class="tdl_ws1"> </td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1">25. 5. 6</td> -<td class="tdr_ws1">— 3263</td> -<td class="tdr_ws1">—— 3273</td> -</tr><tr> -<td class="tdr_ws1">14</td> -<td class="tdl_ws1"> </td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1">25. 5. 7</td> -<td class="tdr_ws1">— 3308</td> -<td class="tdr_ws1">—— 3318</td> -</tr><tr> -<td class="tdr_ws1">15</td> -<td class="tdl_ws1"> </td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1">25. 5. 8</td> -<td class="tdr_ws1">— 3353</td> -<td class="tdr_ws1">—— 3363</td> -</tr><tr> -<td class="tdr_ws1">16</td> -<td class="tdl_ws1"> </td> -<td class="tdl_ws1">Sine of</td> -<td class="tdr_ws1">25. 5. 9</td> -<td class="tdr_ws1">— 3398</td> -<td class="tdr_ws1">—— 3408</td> -</tr><tr> -<td class="tdr_ws1">17</td> -<td class="tdl_ws1"><i>Qqq</i></td> -<td class="tdl_ws1">Tangent of</td> -<td class="tdr_ws1">28.19.39</td> -<td class="tdr_ws1">— 6302</td> -<td class="tdr_ws1">—— 6402</td> -</tr><tr> -<td class="tdr_ws1">18</td> -<td class="tdl_ws1"><i>4H</i></td> -<td class="tdl_ws1">Tangent of</td> -<td class="tdr_ws1">35.55.51</td> -<td class="tdr_ws1">— 1681</td> -<td class="tdr_ws1">—— 1581</td> -</tr><tr> -<td class="tdr_ws1">19</td> -<td class="tdl_ws1"><i>4K</i></td> -<td class="tdl_ws1">Co-sine of</td> -<td class="tdr_ws1">37.29. 2</td> -<td class="tdr_ws1">— 5503</td> -<td class="tdr_ws1">—— 5603</td> -</tr> -</tbody> -</table> - -<p> -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:— -</p> -<p> -'In the list of ERRATA detected in Taylor's Logarithms, for <i>cos</i>. 4° -18' 3", read cos. 14° 18' 2".' -</p> -<p> -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: -</p> -<p> -ERRATUM of the ERRATUM of the ERRATA of TAYLOR's <i>Logarithms</i>. For -cos. 4° 18' 3", <i>read</i> cos. 14° 18' 3". -</p> -<p> -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 <i>mechanical fabrication of tables</i> -that such errors can be rendered impossible. -</p> -<p> -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 <i>carrying</i>. 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 <i>two</i> 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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p class="space-above2 space-below1"> -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. -</p> -<p> -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.' -</p> -<p> -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. -</p> -<p> -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 <i>first differences</i>. 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 <i>second differences</i>. By -adopting a like method of proceeding, another series may be obtained, -called the <i>third differences</i>; 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. -</p> -<p> -As the successive serieses of differences are derived from the original -table, and from each other, by <i>subtraction</i>, the same succession of -series may be reproduced in the other direction by <i>addition</i>. 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:— -</p> - -<table class="no-wrap fontsize_100" > -<thead><tr> -<th style="width:75px">No. </th> -<th style="width:75px"> Table.</th> -</tr> -</thead> -<tbody><tr> -<td class="tdl_ws1"> 1</td> -<td class="tdr_ws1">1</td> -</tr><tr> -<td class="tdl_ws1"> 2</td> -<td class="tdr_ws1">16</td> -</tr><tr> -<td class="tdl_ws1"> 3</td> -<td class="tdr_ws1">81</td> -</tr><tr> -<td class="tdl_ws1"> 4</td> -<td class="tdr_ws1">256</td> -</tr><tr> -<td class="tdl_ws1"> 5</td> -<td class="tdr_ws1">625</td> -</tr><tr> -<td class="tdl_ws1"> 6</td> -<td class="tdr_ws1">1296</td> -</tr><tr> -<td class="tdl_ws1"> 7</td> -<td class="tdr_ws1">2401</td> -</tr><tr> -<td class="tdl_ws1"> 8</td> -<td class="tdr_ws1">4096</td> -</tr><tr> -<td class="tdl_ws1"> 9</td> -<td class="tdr_ws1">6561</td> -</tr><tr> -<td class="tdl_ws1">10</td> -<td class="tdr_ws1">10,000</td> -</tr><tr> -<td class="tdl_ws1">11</td> -<td class="tdr_ws1">14,641</td> -</tr><tr> -<td class="tdl_ws1">12</td> -<td class="tdr_ws1">20,736</td> -</tr><tr> -<td class="tdl_ws1">13</td> -<td class="tdr_ws1">28,561</td> -</tr> -</tbody> -</table> - -<p class="nind"> -By subtracting each number from the succeeding one in this series, we -obtain the following series of first differences: -</p> - -<p class="center text-align:right"> - 15<br> - 65<br> - 175<br> - 369<br> - 671<br> -1105<br> -1695<br> -2465<br> -3439<br> -4641<br> -6095<br> -7825 -</p> - -<p class="nind"> -In like manner, subtracting each term of this series from the succeeding -one, we obtain the following series of second differences:— -</p> - -<p class="center text-align:right"> - 50<br> - 110<br> - 194<br> - 302<br> - 434<br> - 590<br> - 770<br> - 974<br> -1202<br> -1454<br> -1730 -</p> - -<p class="nind"> -Proceeding with this series in the same way, we obtain the following -series of third differences:— -</p> - -<p class="center text-align:right"> - 60<br> - 84<br> - 108<br> - 132<br> - 156<br> - 180<br> - 204<br> - 228<br> - 252<br> - 276 -</p> - -<p class="nind"> -Proceeding in the same way with these, we obtain the following for the -series of fourth differences:— -</p> - -<p class="center text-align:right"> - 24<br> - 24<br> - 24<br> - 24<br> - 24<br> - 24<br> - 24<br> - 24<br> - 24 -</p> - -<p> -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<sup>1</sup> D<sup>2</sup> D<sup>3</sup> D<sup>4</sup>, -we have then the following to begin with:— -</p> - -<table class="no-wrap fontsize_100"> -<thead><tr> -<td class="tdc"> T</td> -<td class="tdc"> D<sup>1</sup></td> -<td class="tdc"> D<sup>2</sup></td> -<td class="tdc"> D<sup>3</sup></td> -<td class="tdc"> D<sup>4</sup></td> -</tr> -</thead> -<tr> -<td class="tdl_ws1"> 1</td> -<td class="tdl_ws1">15</td> -<td class="tdl_ws1">50</td> -<td class="tdl_ws1">60</td> -<td class="tdl_ws1">24</td> -</tr> -</tbody> -</table> - -<p class="nind"> -Adding each number to the number on its left, and repeating 24, we get -the following as the second terms of the several series:— -</p> - -<table class="no-wrap fontsize_100"> -<thead><tr> -<td class="tdc"> T</td> -<td class="tdc"> D<sup>1</sup></td> -<td class="tdc"> D<sup>2</sup></td> -<td class="tdc"> D<sup>3</sup></td> -<td class="tdc"> D<sup>4</sup></td> -</tr> -</thead> -<tr> -<td class="tdl_ws1"> 16</td> -<td class="tdl_ws1">65</td> -<td class="tdl_ws1">110</td> -<td class="tdl_ws1">84</td> -<td class="tdl_ws1">24</td> -</tr> -</tbody> -</table> - -<p class="nind"> -And, in the same manner, the third and succeeding terms as follows:— -</p> - -<table class="no-wrap fontsize_120 box_border"> -<thead><tr> -<th class="tdc bb bt2 br">No.</th> -<th class="tdc bb bt2 br">T</th> -<th class="tdc bb bt2 br">D<sup>1</sup></th> -<th class="tdc bb bt2 br">D<sup>2</sup></th> -<th class="tdc bb bt2 br">D<sup>3</sup></th> -<th class="tdc bb bt2 br">D<sup>4</sup></th> -</tr> -</thead> -<tbody><tr> -<td class="tdr_ws1 bb br bbn"> 1</td> -<td class="tdr_ws1 bb br bbn">1</td> -<td class="tdr_ws1 bb br bbn"> 15</td> -<td class="tdr_ws1 bb br bbn"> 50</td> -<td class="tdr_ws1 bb br bbn"> 60</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 2</td> -<td class="tdr_ws1 bb br bbn"> 16</td> -<td class="tdr_ws1 bb br bbn"> 65</td> -<td class="tdr_ws1 bb br bbn"> 110</td> -<td class="tdr_ws1 bb br bbn"> 84</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 3</td> -<td class="tdr_ws1 bb br bbn"> 81</td> -<td class="tdr_ws1 bb br bbn"> 175</td> -<td class="tdr_ws1 bb br bbn"> 194</td> -<td class="tdr_ws1 bb br bbn"> 108</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 4</td> -<td class="tdr_ws1 bb br bbn"> 256</td> -<td class="tdr_ws1 bb br bbn"> 369</td> -<td class="tdr_ws1 bb br bbn"> 302</td> -<td class="tdr_ws1 bb br bbn"> 132</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 5</td> -<td class="tdr_ws1 bb br bbn"> 625</td> -<td class="tdr_ws1 bb br bbn"> 671</td> -<td class="tdr_ws1 bb br bbn"> 434</td> -<td class="tdr_ws1 bb br bbn"> 156</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 6</td> -<td class="tdr_ws1 bb br bbn"> 1296</td> -<td class="tdr_ws1 bb br bbn"> 1105</td> -<td class="tdr_ws1 bb br bbn"> 590</td> -<td class="tdr_ws1 bb br bbn"> 180</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 7</td> -<td class="tdr_ws1 bb br bbn"> 2401</td> -<td class="tdr_ws1 bb br bbn"> 1695</td> -<td class="tdr_ws1 bb br bbn"> 770</td> -<td class="tdr_ws1 bb br bbn"> 204</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 8</td> -<td class="tdr_ws1 bb br bbn"> 4096</td> -<td class="tdr_ws1 bb br bbn"> 2465</td> -<td class="tdr_ws1 bb br bbn"> 974</td> -<td class="tdr_ws1 bb br bbn"> 228</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 9</td> -<td class="tdr_ws1 bb br bbn"> 6561</td> -<td class="tdr_ws1 bb br bbn"> 3439</td> -<td class="tdr_ws1 bb br bbn"> 1202</td> -<td class="tdr_ws1 bb br bbn"> 252</td> -<td class="tdr_ws1 bb br bbn"> 24</td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 10</td> -<td class="tdr_ws1 bb br bbn"> 10000</td> -<td class="tdr_ws1 bb br bbn"> 4641</td> -<td class="tdr_ws1 bb br bbn"> 1454</td> -<td class="tdr_ws1 bb br bbn"> 276</td> -<td class="tdr_ws1 bb br bbn"> </td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 11</td> -<td class="tdr_ws1 bb br bbn"> 14641</td> -<td class="tdr_ws1 bb br bbn"> 6095</td> -<td class="tdr_ws1 bb br bbn"> 1730</td> -<td class="tdr_ws1 bb br bbn"> </td> -<td class="tdr_ws1 bb br bbn"> </td> -</tr><tr> -<td class="tdr_ws1 bb br bbn"> 12</td> -<td class="tdr_ws1 bb br bbn"> 20736</td> -<td class="tdr_ws1 bb br bbn"> 7825</td> -<td class="tdr_ws1 bb br bbn"> </td> -<td class="tdr_ws1 bb br bbn"> </td> -<td class="tdr_ws1 bb br bbn"> </td> -</tr><tr> -<td class="tdr_ws1 bb br "> 13</td> -<td class="tdr_ws1 bb br "> 28561</td> -<td class="tdr_ws1 bb br "> </td> -<td class="tdr_ws1 bb br "> </td> -<td class="tdr_ws1 bb br "> </td> -<td class="tdr_ws1 bb br "> </td> -</tr> -</tbody> -</table> - -<p> -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. -</p> -<p> -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 <i>addition</i>, and in others by <i>subtraction</i>. 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 <i>arithmetical complement</i>, 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. -</p> -<p> -The common process would be as follows:— -</p> - -<table class="no-wrap fontsize_100"> -<tbody><tr> -<td class="tdl">From</td> -<td class="tdr_ws1">768</td> -</tr><tr> -<td class="tdl">Subtract</td> -<td class="tdr_ws1">357</td> -</tr><tr> -<td class="tdl"> </td> -<td class="tdr_ws1">——</td> -</tr><tr> -<td class="tdl">Remainder</td> -<td class="tdr_ws1">411</td> -</tr> -</tbody> -</table> - -<p class="nind"> -The <i>arithmetical complement</i> 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:— -</p> - -<table class="no-wrap fontsize_100"> -<tbody><tr> -<td class="tdl">To</td> -<td class="tdr_ws1">768</td> -</tr><tr> -<td class="tdl">Add</td> -<td class="tdr_ws1">643</td> -</tr><tr> -<td class="tdl"> </td> -<td class="tdr_ws1">——</td> -</tr><tr> -<td class="tdl">Sum</td> -<td class="tdr_ws1">1411</td> -</tr><tr> -<td class="tdl"> </td> -<td class="tdr_ws1">——</td> -</tr><tr> -<td class="tdl">Remainder sought</td> -<td class="tdr_ws1">411</td> -</tr> -</tbody> -</table> - -<p class="nind"> -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. -</p> -<p> -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. -</p> -<p> -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 <i>adding</i> and -<i>carrying</i> 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. -</p> -<p> -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 <i>ten</i> instead of <i>twelve</i> 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. -</p> - -<div class="figcenter" style="width: 400px;"> -<a id="figure01"></a> -<img src="images/figure01.jpg" width="400" alt="fig01"> -<div class="caption"> -<p>Fig. 1.</p> -</div></div> - -<p> -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. -</p> -<p> -Let us first suppose the line D<sup>1</sup> to be added to the line T. -To accomplish this, let us imagine that while the dials on the line -D<sup>1</sup> 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<sup>1</sup>, 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<sup>1</sup> 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. -</p> -<p> -Having explained the motion of the dials by which the addition, -excluding the carriages of the number on the row D<sup>1</sup>, may be -made to the number on the row T, the same explanation may be applied to -the number on the row D<sup>2</sup> to the number on the row -D<sup>1</sup>; also, of the number <sup>3</sup> to the number on the row -<sup>2</sup>, 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:— -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -(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.) -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -This evidently completes one calculation, since all the rows except the -first have been respectively added to all the rows except the last. -</p> -<p> -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, <a href="#figure01">fig. 1</a>, 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, <a href="#figure01">fig. 1</a>, -will remain unmoved. In the row D<sup>2</sup> the dial -A<sup>2</sup> will remain unmoved, since 0 is at the index below it; the -dial B<sup>2</sup> will be moved through five divisions, and will render -a subsequent carriage on the dial to the left necessary; the dial -C<sup>2</sup> will be moved through five divisions; the dial -D<sup>2</sup> will be moved through three divisions, and the remaining -dials of this row will remain unmoved. The dials of the row -D<sup>4</sup> will be moved according to the same rules; and the whole -scheme will undergo a change exhibited in <a href="#figure02">fig. -2</a>; a mark (*) being introduced on those dials to which a carriage -rendered necessary by the addition which has just taken place. -</p> - -<div class="figcenter" style="width: 400px;"> -<a id="figure02"></a> -<img src="images/figure02.jpg" width="400" alt="fig02"> -<div class="caption"> -<p>Fig. 2.</p> -</div></div> - -<p> -The second quarter of a turn of the moving axis, will move forward -through one division all the dials which in <a href="#figure02">fig. 2</a> -are marked (*), and the scheme will be converted into the scheme expressed -in <a href="#figure03">fig. 3</a>. -</p> - -<div class="figcenter" style="width: 400px;"> -<a id="figure03"></a> -<img src="images/figure03.jpg" width="400" alt="fig03"> -<div class="caption"> -<p>Fig. 3.</p> -</div></div> - -<p> -In third quarter of a turn, the dial A<sup>1</sup>, <a href="#figure03">fig. 3</a>, -will remain unmoved, since is at the index below it; the dial -B<sup>1</sup> will be moved forward through three divisions; -C<sup>1</sup> through nine divisions, and so on; and in like manner the -dials of the row D<sup>3</sup> will be moved forward through the number -of divisions expressed at the indices in the row D<sup>4</sup>. This -change will convert the arrangement into that expressed in -<a href="#figure04">fig. 4</a>, the dials to which a carriage is due, being -distinguished as before by (*). -</p> - -<div class="figcenter" style="width: 400px;"> -<a id="figure04"></a> -<img src="images/figure04.jpg" width="400" alt="fig04"> -<div class="caption"> -<p>Fig. 4.</p> -</div></div> - -<p> -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 <a href="#figure05">fig. 5</a>, 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. -</p> - -<div class="figcenter" style="width: 400px;"> -<a id="figure05"></a> -<img src="images/figure05.jpg" width="400" alt="fig05"> -<div class="caption"> -<p>Fig. 5.</p> -</div></div> - -<p> -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<sup>1</sup>, <a href="#figure01">fig. 1</a>, -for example. Behind the dial B<sup>1</sup> 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<sup>1</sup>, 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<sup>1</sup> -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<sup>1</sup>: thus, if 5 had been under the index of the -dial B<sup>1</sup>, 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<sup>1</sup> 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. -</p> -<p> -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, <a href="#figure01">fig. 1</a>, -in the cases of all those dials in which 0 is under the index. -</p> -<p> -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. -</p> -<p> -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<sup>2</sup>, <a href="#figure01">fig. 1</a>, 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<sup>2</sup>, immediately to the left of B<sup>2</sup>, 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<sup>2</sup> 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<sup>2</sup> 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<sup>2</sup>, 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. -</p> -<p> -During the second quarter of a turn, a finger placed on the axis of the -dial B<sup>2</sup> 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<sup>2</sup> 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<sup>2</sup> 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<sup>2</sup> to C<sup>2</sup> will be -accomplished. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -In the general description which we have given of the mechanism for -<i>carrying</i>, 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 <a href="#figure01">fig. 1</a>, -if the figure under the index of C<sup>2</sup> were 4 instead of 3. The -addition of the number 5 at the index of C<sup>3</sup> would, in this -case, in the first quarter of a turn, bring 9 to the index of -C<sup>2</sup>: 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<sup>2</sup> would not be -detached. Meanwhile a carriage upon C<sup>2</sup> has been rendered -necessary by the addition made in the first quarter of a turn to -B<sup>2</sup>. This carriage takes place in the ordinary way, and would -cause the dial C<sup>2</sup>, in the second quarter of a turn, to -advance from 9 to 0: this would make the necessary preparation for a -carriage from C<sup>2</sup> to D<sup>2</sup>. 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<sup>2</sup>, 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<sup>2</sup> first -encounters the claw upon B<sup>2</sup>, and drives it through one tooth -immediately forwards; the finger on the axis of B<sup>2</sup> encounters -the claw upon C<sup>2</sup> and drives it through one tooth; the action -of the finger on C<sup>2</sup> on the claw on D<sup>2</sup> next -succeeds, and so on. Thus, while the finger on B<sup>2</sup> acts on -C<sup>2</sup>, and causes the division from 9 to 0 to pass under the -index, the thumb on C<sup>2</sup> at the same instant acts on the -trigger, and detaches the carrying claw on D<sup>2</sup>, which is -forthwith encountered by the carrying finger on C<sup>2</sup>, and, -driven forward one tooth. The dial D<sup>2</sup> 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, <i>spirally</i> on their axes, so that they come -into action in regular succession. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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 -<i>snail</i>. 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 <i>type sector</i> -(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 <i>bent -lever</i>, 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. -</p> -<p> -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 <a href="#figure01">fig. 1</a>, -the process would be as follows:—The snail of the wheel -F<sup>1</sup>, 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 -<a href="#figure05">fig. 5</a>. 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 <i>vice versa</i>; 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. - -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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 <i>the means</i> of -printing with absolute correctness and in unlimited abundance. -</p> -<p> -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 <i>figure wheels</i>, 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. -</p> -<p> -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,<a id="FNanchor_12_1"></a><a href="#Footnote_12_1" class="fnanchor">[12]</a> 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. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_12_1"></a><a href="#FNanchor_12_1"><span class="label">[12]</span></a>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.</p></div> - -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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 <i>bolting axes</i>. -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. -</p> -<p> -The same axes in the front of the machinery on which the figure wheels -turn, are made to serve the purpose of <i>carrying</i>. 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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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 <i>vice versa</i>.<a id="FNanchor_13_1"></a><a href="#Footnote_13_1" class="fnanchor">[13]</a> -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_13_1"></a><a href="#FNanchor_13_1"><span class="label">[13]</span></a>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.</p></div> - -<p> -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. -</p> -<p> -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.<a id="FNanchor_14_1"></a><a href="#Footnote_14_1" class="fnanchor">[14]</a> -</p> - -<p class="center text-align:right"> - 99,999,996<br> - 99,999,997<br> - 99,999,998<br> - 99,999,999<br> -100,000,000<br> -100,010,002<br> -100,030,003<br> -100,060,004<br> -100,100,005<br> -100,150,006<br> - &c. &c. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_14_1"></a><a href="#FNanchor_14_1"><span class="label">[14]</span></a>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.</p></div> - -<p> -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 <i>some</i> 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? -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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.' -</p> -<p> -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 <i>true to the last figure</i>. 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. -</p> -<p> -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. -</p> -<p> -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 <i>mechanical notation</i> 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 <i>source</i> of motion -indicates likewise the <i>species</i> 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. -</p> -<p> -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 <i>niche</i>, 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 <i>time</i> 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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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, <i>through -the eye</i>, 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. -</p> -<p> -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.<a id="FNanchor_15_1"></a><a href="#Footnote_15_1" class="fnanchor">[15]</a> 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.<a id="FNanchor_16_1"></a><a href="#Footnote_16_1" class="fnanchor">[16]</a> 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. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_15_1"></a><a href="#FNanchor_15_1"><span class="label">[15]</span></a>Phil. Trans. 1820, Part III. p, 250, on a method of -expressing by signs the action of machinery.</p></div> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_16_1"></a><a href="#FNanchor_16_1"><span class="label">[16]</span></a>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.</p></div> - -<p> -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. -</p> -<p> -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 <i>waste</i> 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.<a id="FNanchor_17_1"></a><a href="#Footnote_17_1" class="fnanchor">[17]</a> -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_17_1"></a><a href="#FNanchor_17_1"><span class="label">[17]</span></a>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.</p></div> - -<p> -Another department of mechanical art, which has been enriched by this -invention, has been that of <i>tools</i>. 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. -</p> -<p> -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 -<i>Abacus</i>, an instrument for computing by the aid of balls sliding -upon parallel rods—the method of calculation invented by Baron -Napier, called by him <i>Rhabdology</i>, and since called <i>Napier's -bones</i>—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 <i>arithmetical -instruments</i>, 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. -</p> -<p> -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.<a id="FNanchor_18_1"></a><a href="#Footnote_18_1" class="fnanchor">[18]</a> -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_18_1"></a><a href="#FNanchor_18_1"><span class="label">[18]</span></a>See a description of this machine by Diderot, in the -<i>Encyc. Method.</i>; also in the works of Pascal, tom, IV., p. 7; Paris, -1819.</p></div> - -<p> -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. -</p> -<p> -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 <i>Napier's -bones</i>, and as having made some additions to the machine of Pascal.<a id="FNanchor_19_1"></a><a href="#Footnote_19_1" class="fnanchor">[19]</a> -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_19_1"></a><a href="#FNanchor_19_1"><span class="label">[19]</span></a>Equidem Morelandus in Anglia, tubæ stentoriæ author, -Rhabdologiam ex baculis in cylindrulos transtulit, et additiones -auxiliares peragit in adjuncta machina additionum Pascaliana.</p></div> - -<p> -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,<a id="FNanchor_20_1"></a><a href="#Footnote_20_1" class="fnanchor">[20]</a> 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, <i>e.g.</i> rotam quamdam tantum sexies gyrari necesse est, -nulla alia opera mentis, nullis additionibus intervenientibus; quo facto, -integrum absolutumque productum oculis objicietur.'<a id="FNanchor_21_1"></a><a href="#Footnote_21_1" class="fnanchor">[21]</a> He goes on to -say that the process of division is performed independently of a -succession of subtractions, such as that used by Pascal. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_20_1"></a><a href="#FNanchor_20_1"><span class="label">[20]</span></a>Tom. I., p. 317.</p></div> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_21_1"></a><a href="#FNanchor_21_1"><span class="label">[21]</span></a><i>Com. Epist.</i> tom, I., p. 289.</p></div> - -<p> -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.<a id="FNanchor_22_1"></a><a href="#Footnote_22_1" class="fnanchor">[22]</a> 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. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_22_1"></a><a href="#FNanchor_22_1"><span class="label">[22]</span></a>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.—<i>Com. Epist.</i></p></div> - -<p> -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 <i>particular -arithmetical operation</i>, 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.' -</p> -<p class="space-above2 space-below1"> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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. -</p> -<p> -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.<a id="FNanchor_23_1"></a><a href="#Footnote_23_1" class="fnanchor">[23]</a> 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. -</p> - -<div class="footnote"> - -<p class="nind"> -<a id="Footnote_23_1"></a><a href="#FNanchor_23_1"><span class="label">[23]</span></a>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.</p></div> - -</div> - -<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK BABBAGE'S CALCULATING ENGINE ***</div> -<div style='text-align:left'> - -<div style='display:block; margin:1em 0'> -Updated editions will replace the previous one—the old editions will -be renamed. -</div> - -<div style='display:block; margin:1em 0'> -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. Special rules, set forth in the General Terms of Use part -of this license, apply to copying and distributing Project -Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ -concept and trademark. Project Gutenberg is a registered trademark, -and may not be used if you charge for an eBook, except by following -the terms of the trademark license, including paying royalties for use -of the Project Gutenberg trademark. If you do not charge anything for -copies of this eBook, complying with the trademark license is very -easy. You may use this eBook for nearly any purpose such as creation -of derivative works, reports, performances and research. Project -Gutenberg eBooks may be modified and printed and given away—you may -do practically ANYTHING in the United States with eBooks not protected -by U.S. copyright law. Redistribution is subject to the trademark -license, especially commercial redistribution. -</div> - -<div style='margin-top:1em; font-size:1.1em; text-align:center'>START: FULL LICENSE</div> -<div style='text-align:center;font-size:0.9em'>THE FULL PROJECT GUTENBERG LICENSE</div> -<div style='text-align:center;font-size:0.9em'>PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK</div> - -<div style='display:block; margin:1em 0'> -To protect the Project Gutenberg™ mission of promoting the free -distribution of electronic works, by using or distributing this work -(or any other work associated in any way with the phrase “Project -Gutenberg”), you agree to comply with all the terms of the Full -Project Gutenberg™ License available with this file or online at -www.gutenberg.org/license. -</div> - -<div style='display:block; font-size:1.1em; margin:1em 0; font-weight:bold'> -Section 1. General Terms of Use and Redistributing Project Gutenberg™ electronic works -</div> - -<div style='display:block; margin:1em 0'> -1.A. By reading or using any part of this Project Gutenberg™ -electronic work, you indicate that you have read, understand, agree to -and accept all the terms of this license and intellectual property -(trademark/copyright) agreement. If you do not agree to abide by all -the terms of this agreement, you must cease using and return or -destroy all copies of Project Gutenberg™ electronic works in your -possession. If you paid a fee for obtaining a copy of or access to a -Project Gutenberg™ electronic work and you do not agree to be bound -by the terms of this agreement, you may obtain a refund from the person -or entity to whom you paid the fee as set forth in paragraph 1.E.8. -</div> - -<div style='display:block; margin:1em 0'> -1.B. “Project Gutenberg” is a registered trademark. It may only be -used on or associated in any way with an electronic work by people who -agree to be bound by the terms of this agreement. There are a few -things that you can do with most Project Gutenberg™ electronic works -even without complying with the full terms of this agreement. See -paragraph 1.C below. There are a lot of things you can do with Project -Gutenberg™ electronic works if you follow the terms of this -agreement and help preserve free future access to Project Gutenberg™ -electronic works. See paragraph 1.E below. -</div> - -<div style='display:block; margin:1em 0'> -1.C. The Project Gutenberg Literary Archive Foundation (“the -Foundation” or PGLAF), owns a compilation copyright in the collection -of Project Gutenberg™ electronic works. Nearly all the individual -works in the collection are in the public domain in the United -States. If an individual work is unprotected by copyright law in the -United States and you are located in the United States, we do not -claim a right to prevent you from copying, distributing, performing, -displaying or creating derivative works based on the work as long as -all references to Project Gutenberg are removed. Of course, we hope -that you will support the Project Gutenberg™ mission of promoting -free access to electronic works by freely sharing Project Gutenberg™ -works in compliance with the terms of this agreement for keeping the -Project Gutenberg™ name associated with the work. You can easily -comply with the terms of this agreement by keeping this work in the -same format with its attached full Project Gutenberg™ License when -you share it without charge with others. -</div> - -<div style='display:block; margin:1em 0'> -1.D. The copyright laws of the place where you are located also govern -what you can do with this work. Copyright laws in most countries are -in a constant state of change. If you are outside the United States, -check the laws of your country in addition to the terms of this -agreement before downloading, copying, displaying, performing, -distributing or creating derivative works based on this work or any -other Project Gutenberg™ work. The Foundation makes no -representations concerning the copyright status of any work in any -country other than the United States. -</div> - -<div style='display:block; margin:1em 0'> -1.E. Unless you have removed all references to Project Gutenberg: -</div> - -<div style='display:block; margin:1em 0'> -1.E.1. The following sentence, with active links to, or other -immediate access to, the full Project Gutenberg™ License must appear -prominently whenever any copy of a Project Gutenberg™ work (any work -on which the phrase “Project Gutenberg” appears, or with which the -phrase “Project Gutenberg” is associated) is accessed, displayed, -performed, viewed, copied or distributed: -</div> - -<blockquote> - <div style='display:block; margin:1em 0'> - This eBook is for the use of anyone anywhere in the United States and most - other parts of the world at no cost and with almost no restrictions - whatsoever. You may copy it, give it away or re-use it under the terms - of the Project Gutenberg License included with this eBook or online - at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you - are not located in the United States, you will have to check the laws - of the country where you are located before using this eBook. - </div> -</blockquote> - -<div style='display:block; margin:1em 0'> -1.E.2. If an individual Project Gutenberg™ electronic work is -derived from texts not protected by U.S. copyright law (does not -contain a notice indicating that it is posted with permission of the -copyright holder), the work can be copied and distributed to anyone in -the United States without paying any fees or charges. If you are -redistributing or providing access to a work with the phrase “Project -Gutenberg” associated with or appearing on the work, you must comply -either with the requirements of paragraphs 1.E.1 through 1.E.7 or -obtain permission for the use of the work and the Project Gutenberg™ -trademark as set forth in paragraphs 1.E.8 or 1.E.9. -</div> - -<div style='display:block; margin:1em 0'> -1.E.3. If an individual Project Gutenberg™ electronic work is posted -with the permission of the copyright holder, your use and distribution -must comply with both paragraphs 1.E.1 through 1.E.7 and any -additional terms imposed by the copyright holder. Additional terms -will be linked to the Project Gutenberg™ License for all works -posted with the permission of the copyright holder found at the -beginning of this work. -</div> - -<div style='display:block; margin:1em 0'> -1.E.4. Do not unlink or detach or remove the full Project Gutenberg™ -License terms from this work, or any files containing a part of this -work or any other work associated with Project Gutenberg™. -</div> - -<div style='display:block; margin:1em 0'> -1.E.5. Do not copy, display, perform, distribute or redistribute this -electronic work, or any part of this electronic work, without -prominently displaying the sentence set forth in paragraph 1.E.1 with -active links or immediate access to the full terms of the Project -Gutenberg™ License. -</div> - -<div style='display:block; margin:1em 0'> -1.E.6. You may convert to and distribute this work in any binary, -compressed, marked up, nonproprietary or proprietary form, including -any word processing or hypertext form. However, if you provide access -to or distribute copies of a Project Gutenberg™ work in a format -other than “Plain Vanilla ASCII” or other format used in the official -version posted on the official Project Gutenberg™ website -(www.gutenberg.org), you must, at no additional cost, fee or expense -to the user, provide a copy, a means of exporting a copy, or a means -of obtaining a copy upon request, of the work in its original “Plain -Vanilla ASCII” or other form. Any alternate format must include the -full Project Gutenberg™ License as specified in paragraph 1.E.1. -</div> - -<div style='display:block; margin:1em 0'> -1.E.7. Do not charge a fee for access to, viewing, displaying, -performing, copying or distributing any Project Gutenberg™ works -unless you comply with paragraph 1.E.8 or 1.E.9. -</div> - -<div style='display:block; margin:1em 0'> -1.E.8. You may charge a reasonable fee for copies of or providing -access to or distributing Project Gutenberg™ electronic works -provided that: -</div> - -<div style='margin-left:0.7em;'> - <div style='text-indent:-0.7em'> - • You pay a royalty fee of 20% of the gross profits you derive from - the use of Project Gutenberg™ works calculated using the method - you already use to calculate your applicable taxes. The fee is owed - to the owner of the Project Gutenberg™ trademark, but he has - agreed to donate royalties under this paragraph to the Project - Gutenberg Literary Archive Foundation. Royalty payments must be paid - within 60 days following each date on which you prepare (or are - legally required to prepare) your periodic tax returns. Royalty - payments should be clearly marked as such and sent to the Project - Gutenberg Literary Archive Foundation at the address specified in - Section 4, “Information about donations to the Project Gutenberg - Literary Archive Foundation.” - </div> - - <div style='text-indent:-0.7em'> - • You provide a full refund of any money paid by a user who notifies - you in writing (or by e-mail) within 30 days of receipt that s/he - does not agree to the terms of the full Project Gutenberg™ - License. You must require such a user to return or destroy all - copies of the works possessed in a physical medium and discontinue - all use of and all access to other copies of Project Gutenberg™ - works. - </div> - - <div style='text-indent:-0.7em'> - • You provide, in accordance with paragraph 1.F.3, a full refund of - any money paid for a work or a replacement copy, if a defect in the - electronic work is discovered and reported to you within 90 days of - receipt of the work. - </div> - - <div style='text-indent:-0.7em'> - • You comply with all other terms of this agreement for free - distribution of Project Gutenberg™ works. - </div> -</div> - -<div style='display:block; margin:1em 0'> -1.E.9. If you wish to charge a fee or distribute a Project -Gutenberg™ electronic work or group of works on different terms than -are set forth in this agreement, you must obtain permission in writing -from the Project Gutenberg Literary Archive Foundation, the manager of -the Project Gutenberg™ trademark. Contact the Foundation as set -forth in Section 3 below. -</div> - -<div style='display:block; margin:1em 0'> -1.F. -</div> - -<div style='display:block; margin:1em 0'> -1.F.1. Project Gutenberg volunteers and employees expend considerable -effort to identify, do copyright research on, transcribe and proofread -works not protected by U.S. copyright law in creating the Project -Gutenberg™ collection. Despite these efforts, Project Gutenberg™ -electronic works, and the medium on which they may be stored, may -contain “Defects,” such as, but not limited to, incomplete, inaccurate -or corrupt data, transcription errors, a copyright or other -intellectual property infringement, a defective or damaged disk or -other medium, a computer virus, or computer codes that damage or -cannot be read by your equipment. -</div> - -<div style='display:block; margin:1em 0'> -1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right -of Replacement or Refund” described in paragraph 1.F.3, the Project -Gutenberg Literary Archive Foundation, the owner of the Project -Gutenberg™ trademark, and any other party distributing a Project -Gutenberg™ electronic work under this agreement, disclaim all -liability to you for damages, costs and expenses, including legal -fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT -LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE -PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE -TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE -LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR -INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH -DAMAGE. -</div> - -<div style='display:block; margin:1em 0'> -1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a -defect in this electronic work within 90 days of receiving it, you can -receive a refund of the money (if any) you paid for it by sending a -written explanation to the person you received the work from. If you -received the work on a physical medium, you must return the medium -with your written explanation. The person or entity that provided you -with the defective work may elect to provide a replacement copy in -lieu of a refund. If you received the work electronically, the person -or entity providing it to you may choose to give you a second -opportunity to receive the work electronically in lieu of a refund. If -the second copy is also defective, you may demand a refund in writing -without further opportunities to fix the problem. -</div> - -<div style='display:block; margin:1em 0'> -1.F.4. Except for the limited right of replacement or refund set forth -in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO -OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT -LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE. -</div> - -<div style='display:block; margin:1em 0'> -1.F.5. Some states do not allow disclaimers of certain implied -warranties or the exclusion or limitation of certain types of -damages. If any disclaimer or limitation set forth in this agreement -violates the law of the state applicable to this agreement, the -agreement shall be interpreted to make the maximum disclaimer or -limitation permitted by the applicable state law. The invalidity or -unenforceability of any provision of this agreement shall not void the -remaining provisions. -</div> - -<div style='display:block; margin:1em 0'> -1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the -trademark owner, any agent or employee of the Foundation, anyone -providing copies of Project Gutenberg™ electronic works in -accordance with this agreement, and any volunteers associated with the -production, promotion and distribution of Project Gutenberg™ -electronic works, harmless from all liability, costs and expenses, -including legal fees, that arise directly or indirectly from any of -the following which you do or cause to occur: (a) distribution of this -or any Project Gutenberg™ work, (b) alteration, modification, or -additions or deletions to any Project Gutenberg™ work, and (c) any -Defect you cause. -</div> - -<div style='display:block; font-size:1.1em; margin:1em 0; font-weight:bold'> -Section 2. Information about the Mission of Project Gutenberg™ -</div> - -<div style='display:block; margin:1em 0'> -Project Gutenberg™ is synonymous with the free distribution of -electronic works in formats readable by the widest variety of -computers including obsolete, old, middle-aged and new computers. It -exists because of the efforts of hundreds of volunteers and donations -from people in all walks of life. -</div> - -<div style='display:block; margin:1em 0'> -Volunteers and financial support to provide volunteers with the -assistance they need are critical to reaching Project Gutenberg™’s -goals and ensuring that the Project Gutenberg™ collection will -remain freely available for generations to come. In 2001, the Project -Gutenberg Literary Archive Foundation was created to provide a secure -and permanent future for Project Gutenberg™ and future -generations. To learn more about the Project Gutenberg Literary -Archive Foundation and how your efforts and donations can help, see -Sections 3 and 4 and the Foundation information page at www.gutenberg.org. -</div> - -<div style='display:block; font-size:1.1em; margin:1em 0; font-weight:bold'> -Section 3. Information about the Project Gutenberg Literary Archive Foundation -</div> - -<div style='display:block; margin:1em 0'> -The Project Gutenberg Literary Archive Foundation is a non-profit -501(c)(3) educational corporation organized under the laws of the -state of Mississippi and granted tax exempt status by the Internal -Revenue Service. The Foundation’s EIN or federal tax identification -number is 64-6221541. Contributions to the Project Gutenberg Literary -Archive Foundation are tax deductible to the full extent permitted by -U.S. federal laws and your state’s laws. -</div> - -<div style='display:block; margin:1em 0'> -The Foundation’s business office is located at 809 North 1500 West, -Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up -to date contact information can be found at the Foundation’s website -and official page at www.gutenberg.org/contact -</div> - -<div style='display:block; font-size:1.1em; margin:1em 0; font-weight:bold'> -Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation -</div> - -<div style='display:block; margin:1em 0'> -Project Gutenberg™ depends upon and cannot survive without widespread -public support and donations to carry out its mission of -increasing the number of public domain and licensed works that can be -freely distributed in machine-readable form accessible by the widest -array of equipment including outdated equipment. Many small donations -($1 to $5,000) are particularly important to maintaining tax exempt -status with the IRS. -</div> - -<div style='display:block; margin:1em 0'> -The Foundation is committed to complying with the laws regulating -charities and charitable donations in all 50 states of the United -States. Compliance requirements are not uniform and it takes a -considerable effort, much paperwork and many fees to meet and keep up -with these requirements. We do not solicit donations in locations -where we have not received written confirmation of compliance. To SEND -DONATIONS or determine the status of compliance for any particular state -visit <a href="https://www.gutenberg.org/donate/">www.gutenberg.org/donate</a>. -</div> - -<div style='display:block; margin:1em 0'> -While we cannot and do not solicit contributions from states where we -have not met the solicitation requirements, we know of no prohibition -against accepting unsolicited donations from donors in such states who -approach us with offers to donate. -</div> - -<div style='display:block; margin:1em 0'> -International donations are gratefully accepted, but we cannot make -any statements concerning tax treatment of donations received from -outside the United States. U.S. laws alone swamp our small staff. -</div> - -<div style='display:block; margin:1em 0'> -Please check the Project Gutenberg web pages for current donation -methods and addresses. Donations are accepted in a number of other -ways including checks, online payments and credit card donations. To -donate, please visit: www.gutenberg.org/donate. -</div> - -<div style='display:block; font-size:1.1em; margin:1em 0; font-weight:bold'> -Section 5. General Information About Project Gutenberg™ electronic works -</div> - -<div style='display:block; margin:1em 0'> -Professor Michael S. Hart was the originator of the Project -Gutenberg™ concept of a library of electronic works that could be -freely shared with anyone. For forty years, he produced and -distributed Project Gutenberg™ eBooks with only a loose network of -volunteer support. -</div> - -<div style='display:block; margin:1em 0'> -Project Gutenberg™ eBooks are often created from several printed -editions, all of which are confirmed as not protected by copyright in -the U.S. unless a copyright notice is included. Thus, we do not -necessarily keep eBooks in compliance with any particular paper -edition. -</div> - -<div style='display:block; margin:1em 0'> -Most people start at our website which has the main PG search -facility: <a href="https://www.gutenberg.org">www.gutenberg.org</a>. -</div> - -<div style='display:block; margin:1em 0'> -This website includes information about Project Gutenberg™, -including how to make donations to the Project Gutenberg Literary -Archive Foundation, how to help produce our new eBooks, and how to -subscribe to our email newsletter to hear about new eBooks. -</div> - -</div> -</body> - -</html> diff --git a/old/71292-h/images/figure01.jpg b/old/71292-h/images/figure01.jpg Binary files differdeleted file mode 100644 index fef36d0..0000000 --- a/old/71292-h/images/figure01.jpg +++ /dev/null diff --git a/old/71292-h/images/figure02.jpg b/old/71292-h/images/figure02.jpg Binary files differdeleted file mode 100644 index c311507..0000000 --- a/old/71292-h/images/figure02.jpg +++ /dev/null diff --git a/old/71292-h/images/figure03.jpg b/old/71292-h/images/figure03.jpg Binary files differdeleted file mode 100644 index a014c2a..0000000 --- a/old/71292-h/images/figure03.jpg +++ /dev/null diff --git a/old/71292-h/images/figure04.jpg b/old/71292-h/images/figure04.jpg Binary files differdeleted file mode 100644 index 25595ed..0000000 --- a/old/71292-h/images/figure04.jpg +++ /dev/null diff --git a/old/71292-h/images/figure05.jpg b/old/71292-h/images/figure05.jpg Binary files differdeleted file mode 100644 index 27bc355..0000000 --- a/old/71292-h/images/figure05.jpg +++ /dev/null |
