,,the ,,edinburgh ,,review1 ,,july1 #ahcd4 ,no4 ,,cxx4 ,,the ,,calculating ,,engine ,,by ,,charles ,,babbage ,art ,i4--#a4 ,letter to ,sir ,humphry ,davy1 ,bart4 ,p4,r4,s41 on the application of ,machinery to ,calculate and ,print ,mathematical ,tables4 ,by ,,charles ,,babbage1 ,esq4 ,f4,r4,s4 #dto4 ,printed by order of the ,house of ,commons4 #b4 ,on the ,application of ,machinery to the ,calculation of ,astronomical and ,mathematical ,tables4 ,by ,,charles ,,babbage1 ,esq4 ,memoirs ,astron4 ,soc4 ,vol4 ,i4 ,part #b4 ,london3 #ahbb4 #c4 ,address to the ,astronomical ,society1 by ,henry ,thomas ,colebrooke1 ,esq4 ,f4,r4,s4 ,president1 on presenting the first gold medal of the ,society to ,charles ,babbage1 ,esq4 for the invention of the ,calculating ,engine4 ,memoirs ,astron4 ,soc4 ,vol4 ,i4 ,part #b4 ,london3 #ahbb4 #d4 ,on the determination of the ,general ,term of a new ,class of ,infinite ,series4 ,by ,,charles ,,babbage1 ,esq4 ,transactions ,camb4 ,phil4 ,soc4 ,cambridge3 #ahbd4 #e4 ,on ,errors common to many ,tables of ,logarithms4 ,by ,,charles ,,babbage1 ,esq4 ,memoirs ,astron4 ,soc4 ,london3 #ahbg4 #f4 ,on a ,method of ,expressing by ,signs the ,action of ,machinery4 ,by ,,charles ,,babbage1 ,esq4 ,phil4 ,trans4 ,london3 #ahbf4 #g4 ,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 ,treasury1 respecting ,mr ,babbage's ,calculating ,engine1 and to report thereupon4 ,london3 #ahbi4 ,,there is no position in society more enviable than that of the few who unite a moderate independence with high intellectual qualities4 ,liberated from the necessity of seeking their support by a profession1 they are unfettered by its restraints1 and are enabled to direct the powers of their minds1 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 community1 and with the most lasting reputation to themselves4 ,on the other hand1 their middle station and limited income rescue them from those allurements to frivolity and dissipation1 to which rank and wealth ever expose their possessors4 ,placed in such favourable circumstances1 ,mr ,babbage selected science as the field of his ambition2 and his mathematical researches have conferred on him a high reputation1 wherever the exact sciences are studied and appreciated4 ,the suffrages of the mathematical world have been ratified in his own country1 where he has been elected to the ,lucasian ,professorship in his own ,university--a chair1 which1 though of inconsiderable emolument1 is one on which ,newton has conferred everlasting celebrity4 ,but it has been the fortune of this mathematician to surround himself with fame of another and more popular kind1 and which rarely falls to the lot of those who devote their lives to the cultivation of the abstract sciences4 ,this distinction he owes to the announcement1 some years since1 of his celebrated project of a ,calculating ,engine4 ,a proposition to reduce arithmetic to the dominion of mechanism1--to substitute an automaton for a compositor1--to throw the powers of thought into wheel-work could not fail to awaken the attention of the world4 ,to bring the practicability of such a project within the compass of popular belief was not easy3 to do so by bringing it within the compass of popular comprehension was not possible4 ,it transcended the imagination of the public in general to conceive its possibility2 and the sentiments of wonder with which it was received1 were only prevented from merging into those of incredulity1 by the faith reposed in the high attainments of its projector4 ,this extraordinary undertaking was1 however1 viewed in a very different light by the small section of the community1 who1 being sufficiently versed in mathematics1 were acquainted with the principle upon which it was founded4 ,by reference to that principle1 they perceived at a glance the practicability of the project2 and being enabled by the nature of their attainments and pursuits to appreciate the immeasurable importance of its results1 they regarded the invention with a proportionately profound interest4 ,the production of numerical tables1 unlimited in quantity and variety1 restricted to no particular species1 and limited by no particular law2--extending not merely to the boundaries of existing knowledge1 but spreading their powers over the undefined regions of future discovery--were results1 the magnitude and the value of which the community in general could neither comprehend nor appreciate4 ,in such a case1 the judgment of the world could only rest upon the authority of the philosophical part of it2 and the fiat of the scientific community swayed for once political councils4 ,the ,british ,government1 advised by the ,royal ,society1 and a committee formed of the most eminent mechanicians and practical engineers1 determined on constructing the projected mechanism at the expense of the nation1 to be held as national property4 ,notwithstanding the interest with which this invention has been regarded in every part of the world1 it has never yet been embodied in a written1 much less in a published form4 ,we trust1 therefore1 that some credit will be conceded to us for having been the first to make the public acquainted with the object1 principle1 and structure of a piece of machinery1 which1 though at present unknown 7except as to a few of its probable results71 must1 when completed1 produce important effects1 not only on the progress of science1 but on that of civilisation4 ,the calculating machinery thus undertaken for the public gratuitously 7so far as ,mr ,babbage is concerned71 has now attained a very advanced stage towards completion2 and a portion of it has been put together1 and performs various calculations2--affording a practical demonstration that the anticipations of those1 under whose advice ,government has acted1 have been well founded4 ,there are nevertheless many persons who1 admitting the great ingenuity of the contrivance1 have1 notwithstanding1 been accustomed to regard it more in the light of a philosophical curiosity1 than an instrument for purposes practically useful4 ,this mistake 7than which it is not possible to imagine a greater7 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 produce4 ,there are also some persons who1 not considering the time requisite to bring any invention of this magnitude to perfection in all its details1 incline to consider the delays which have taken place in its progress as presumptions against its practicability4 ,these persons should1 however1 before they arrive at such a conclusion1 reflect upon the time which was necessary to bring to perfection engines infinitely inferior in complexity and mechanical difficulty4 ,let them remember that--not to mention the invention of that machine--the improvements alone introduced into the steam-engine by the celebrated ,watt1 occupied a period of not less than twenty years of the life of that distinguished person1 and involved an expenditure of capital amounting to ,l4#ej1jjj4a ,the calculating machinery is a contrivance new even in its details4 ,its inventor did not take it up already imperfectly formed1 after having received the contributions of human ingenuity exercised upon it for a century or more4 ,it has not1 like almost all other great mechanical inventions1 been gradually advanced to its present state through a series of failures1 through difficulties encountered and overcome by a succession of projectors4 ,it is not an object on which the light of various minds has thus been shed4 ,it is1 on the contrary1 the production of solitary and individual thought1--begun1 advanced through each successive stage of improvement1 and brought to perfection by one mind4 ,yet this creation of genius1 from its first rude conception to its present state1 has cost little more than half the time1 and not one-third of the expense1 consumed in bringing the steam-engine 7previously far advanced in the course of improvement7 to that state of comparative perfection in which it was left by ,watt4 ,short as the period of time has been which the inventor has devoted to this enterprise1 it has1 nevertheless1 been demonstrated1 to the satisfaction of many scientific men of the first eminence1 that the design in all its details1 reduced1 as it is1 to a system of mechanical drawings1 is complete2 and requires only to be constructed in conformity with those plans1 to realize all that its inventor has promised4 ,footnote #a3 ,watt commenced his investigations respecting the steam-engine in #agfc1 between which time1 and the year #aghb inclusive1 he took out several patents for improvements in details4 ,bolton and ,watt had expended the above sum on their improvements before they began to receive any return4 ,with a view to remove and correct erroneous impressions1 and at the same time to convert the vague sense of wonder at what seems incomprehensible1 with which this project is contemplated by the public in general1 into a more rational and edifying sentiment1 it is our purpose in the present article4 ,first1 ,to show1 the immense importance of any method by which numerical tables1 absolutely accurate in every individual copy1 may be produced with facility and cheapness4 ,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 extended1 a large portion of which have been produced at the public expense2 by showing also1 that they are nevertheless rendered inefficient1 to a greater or less extent1 by the prevalence of errors in them2 that these errors pervade not merely tables produced by individual labour and enterprise1 but that they vitiate even those on which national resources have been prodigally expended1 and to which the highest mathematical ability1 which the most enlightened nations of the world could command1 has been unsparingly and systematically directed4 ,secondly1 ,to attempt to convey to the reader a general notion of the mathematical principle on which the calculating machinery is founded1 and of the manner in which this principle is brought into practical operation1 both in the process of calculating and printing4 ,it would be incompatible with the nature of this review1 and indeed impossible without the aid of numerous plans1 sections1 and elevations1 to convey clear and precise notions of the details of the means by which the process of reasoning is performed by inanimate matter1 and the arbitrary and capricious evolutions of the fingers of typographical compositors are reduced to a system of wheel-work4 ,we are1 nevertheless1 not without hopes of conveying1 even to readers unskilled in mathematics1 some satisfactory notions of a general nature on this subject4 ,thirdly1 ,to explain the actual state of the machinery at the present time2 what progress has been made towards its completion2 and what are the probable causes of those delays in its progress1 which must be a subject of regret to all friends of science4 ,we shall indicate what appears to us the best and most practicable course to prevent the unnecessary recurrence of such obstructions for the future1 and to bring this noble project to a speedy and successful issue4 ,viewing the infinite extent and variety of the tables which have been calculated and printed1 from the earliest periods of human civilisation to the present time1 we feel embarrassed with the difficulties of the task which we have imposed on ourselves2--that of attempting to convey to readers unaccustomed to such speculations1 any thing approaching to an adequate idea of them4 ,these tables are connected with the various sciences1 with almost every department of the useful arts1 with commerce in all its relations2 but above all1 with ,astronomy and ,navigation4 ,so important have they been considered1 that in many instances large sums have been appropriated by the most enlightened nations in the production of them2 and yet so numerous and insurmountable have been the difficulties attending the attainment of this end1 that after all1 even navigators1 putting aside every other department of art and science1 have1 until very recently1 been scantily and imperfectly supplied with the tables indispensably necessary to determine their position at sea4 ,the first class of tables which naturally present themselves1 are those of ,multiplication4 ,a great variety of extensive multiplication tables have been published from an early period in different countries2 and especially tables of ,powers1 in which a number is multiplied by itself successively4 ,in ,dodson's ,calculator we find a table of multiplication extending as far as #aj times #ajjj4b ,in #agge1 a still more extensive table was published to #aj times #aj1jjj4 ,the ,board of ,longitude subsequently employed the late ,dr ,hutton to calculate and print various numerical tables1 and among others1 a multiplication table extending as far as #ajj times #ajjj2 tables of the squares of numbers1 as far as #be1djj2 tables of cubes1 and of the first ten powers of numbers1 as far as #ajj4c ,in #ahad1 ,professor ,barlow1 of ,woolwich1 published1 in an octavo volume1 the squares1 cubes1 square roots1 cube roots1 and reciprocals of all numbers from #a to #aj1jjj2 a table of the first ten powers of all numbers from #a to #ajj1 and of the fourth and fifth powers of all numbers from #ajj to #ajjj4 ,footnote #b3 ,dodson's ,calculator4 #dto4 ,london3 #agdg4 ,footnote #c3 ,hutton's ,tables of ,products and ,powers4 ,folio4 ,london2 #agha4 ,tables of ,multiplication to a still greater extent have been published in ,france4 ,in #aghe1 was published an octavo volume of tables of the squares1 cubes1 square roots1 and cube roots of all numbers from #a to #aj1jjj2 and similar tables were again published in #ahja4 ,in #ahag1 multiplication tables were published in ,paris by ,voisin2 and similar tables1 in two quarto volumes1 in #ahbd1 by the ,french ,board of ,longitude1 extending as far as a thousand times a thousand4 ,a table of squares was published in #ahaj1 in ,hanover2 in #ahab1 at ,leipzig2 in #ahbe1 at ,berlin2 and in #ahbg1 at ,ghent4 ,a table of cubes was published in #ahbg1 at ,eisenach2 in the same year a similar table at ,ghent2 and one of the squares of all numbers as far as #aj1jjj1 was published in that year1 in quarto1 at ,bonn4 ,the ,prussian ,government has caused a multiplication table to be calculated and printed1 extending as far as #ajjj times #ajjj4 ,such are a few of the tables of this class which have been published in different countries4 ,this class of tables may be considered as purely arithmetical1 since the results which they express involve no other relations than the arithmetical dependence of abstract numbers upon each other4 ,when numbers1 however1 are taken in a concrete sense1 and are applied to express peculiar modes of quantity1--such as angular1 linear1 superficial1 and solid magnitudes1--a new set of numerical relations arise1 and a large number of computations are required4 ,to express angular magnitude1 and the various relations of linear magnitude with which it is connected1 involves the consideration of a vast variety of ,geometrical and ,trigonometrical tables2 such as tables of the natural sines1 co-sines1 tangents1 secants1 co-tangents1 c4 c42 tables of arcs and angles in terms of the radius2 tables for the immediate solution of various cases of triangles1 c4 ,volumes without number of such tables have been from time to time computed and published4 ,it is not sufficient1 however1 for the purposes of computation to tabulate these immediate trigonometrical functions4 ,their squares#d and higher powers1 their square roots1 and other roots1 occur so frequently1 that it has been found expedient to compute tables for them1 as well as for the same functions of abstract numbers4 ,footnote #d3 ,the squares of the sines of angles are extensively used in the calculations connected with the theory of the tides4 ,not aware that tables of these squares existed1 ,bouvard1 who calculated the tides for ,laplace1 underwent the labour of calculating the square of each individual sine in every case in which it occurred4 ,the measurement of linear1 superficial1 and solid magnitudes1 in the various forms and modifications in which they are required in the arts1 demands another extensive catalogue of numerical tables4 ,the surveyor1 the architect1 the builder1 the carpenter1 the miner1 the ganger1 the naval architect1 the engineer1 civil and military1 all require the aid of peculiar numerical tables1 and such have been published in all countries4 ,the increased expedition and accuracy which was introduced into the art of computation by the invention of ,logarithms1 greatly enlarged the number of tables previously necessary4 ,to apply the logarithmic method1 it was not merely necessary to place in the hands of the computist extensive tables of the logarithms of the natural numbers1 but likewise to supply him with tables in which he might find already calculated the logarithms of those arithmetical1 trigonometrical1 and geometrical functions of numbers1 which he has most frequent occasion to use4 ,it would be a circuitous process1 when the logarithm of a sine or co-sine of an angle is required1 to refer1 first to the table of sines1 or co-sines1 and thence to the table of the logarithms of natural numbers4 ,it was therefore found expedient to compute distinct tables of the logarithms of the sines1 co-sines1 tangents1 c41 as well as of various other functions frequently required1 such as sums1 differences1 c4 ,great as is the extent of the tables we have just enumerated1 they bear a very insignificant proportion to those which remain to be mentioned4 ,the above are1 for the most part1 general in their nature1 not belonging particularly to any science or art4 ,there is a much greater variety of tables1 whose importance is no way inferior1 which are1 however1 of a more special nature3 ,such are1 for example1 tables of interest1 discount1 and exchange1 tables of annuities1 and other tables necessary in life insurances2 tables of rates of various kinds necessary in general commerce4 ,but the science in which1 above all others1 the most extensive and accurate tables are indispensable1 is ,astronomy2 with the improvement and perfection of which is inseparably connected that of the kindred art of ,navigation4 ,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 navigator4 ,we feel1 nevertheless1 that the truly national importance which must attach to any perfect and easy means of producing those tables cannot be at all estimated1 unless we state some of the previous calculations necessary in order to enable the mariner to determine1 with the requisite certainty and precision1 the place of his ship4 ,in a word1 then1 all the purely arithmetical1 trigonometrical1 and logarithmic tables already mentioned1 are necessary1 either immediately or remotely1 for this purpose4 ,but in addition to these1 a great number of tables1 exclusively astronomical1 are likewise indispensable4 ,the predictions of the astronomer1 with respect to the positions and motions of the bodies of the firmament1 are the means1 and the only means1 which enable the mariner to prosecute his art4 ,by these he is enabled to discover the distance of his ship from the ,line1 and the extent of his departure from the meridian of ,greenwich1 or from any other meridian to which the astronomical predictions refer4 ,the more numerous1 minute1 and accurate these predictions can be made1 the greater will be the facilities which can be furnished to the mariner4 ,but the computation of those tables1 in which the future position of celestial objects are registered1 depend themselves upon an infinite variety of other tables which never reach the hands of the mariner4 ,it cannot be said that there is any table whatever1 necessary for the astronomer1 which is unnecessary for the navigator4 ,the purposes of the marine of a country whose interests are so inseparably connected as ours are with the improvement of the art of navigation1 would be very inadequately fulfilled1 if our navigators were merely supplied with the means of determining by ,nautical ,astronomy the position of a ship at sea4 ,it has been well observed by the ,committee of the ,astronomical ,society1 to whom the recent improvement of the ,nautical ,almanac was confided1 that it is not by those means merely by which the seaman is enabled to determine the position of his vessel at sea1 that the full intent and purpose of what is usually called ,nautical ,astronomy are answered4 ,this object is merely a part of that comprehensive and important subject2 and might be attained by a very cheap publication1 and without the aid of expensive instruments4 ,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 earth1--such as remarkable headlands1 ports1 and islands2 together with the general trending of the coast between well-known harbours4 ,it is not necessary to point out here how important such knowledge is to the mariner4 ,this knowledge1 which may be called ,nautical ,geography1 cannot be obtained by the methods of observation used on board ship1 but requires much more delicate and accurate instruments1 firmly placed upon the solid ground1 besides all the astronomical aid which can be afforded by the best tables1 arranged in the most convenient form for immediate use4 ,this was ,dr ,maskelyne's view of the subject1 and his opinion has been confirmed by the repeated wants and demands of those distinguished navigators who have been employed in several recent scientific expeditions4#e ,footnote #e3 ,report of the ,committee of the ,astronomical ,society prefixed to the ,nautical ,almanac for #ahcd4 ,among the tables directly necessary for navigation1 are those which predict the position of the centre of the sun from hour to hour4 ,these tables include the sun's right ascension and declination1 daily1 at noon1 with the hourly change in these quantities4 ,they also include the equation of time1 together with its hourly variation4 ,tables of the moon's place for every hour1 are likewise necessary1 together with the change of declination for every ten minutes4 ,the lunar method of determining the longitude depends upon tables containing the predicted distances of the moon from the sun1 the principal planets1 and from certain conspicuous fixed stars2 which distances being observed by the mariner1 he is enabled thence to discover the time at the meridian from which the longitude is measured2 and1 by comparing that time with the time known or discoverable in his actual situation1 he infers his longitude4 ,but not only does the prediction of the position of the moon1 with respect to these celestial objects1 require a vast number of numerical tables1 but likewise the observations necessary to be made by the mariner1 in order to determine the lunar distances1 also require several tables4 ,to predict the exact position of any fixed star1 requires not less than ten numerical tables peculiar to that star2 and if the mariner be furnished 7as is actually the case7 with tables of the predicted distances of the moon from one hundred such stars1 such predictions must require not less than a thousand numerical tables4 ,regarding the range of the moon through the firmament1 however1 it will readily be conceived that a hundred stars form but a scanty supply2 especially when it is considered that an accurate method of determining the longitude1 consists in observing the extinction of a star by the dark edge of the moon4 ,within the limits of the lunar orbit there are not less than one thousand stars1 which are so situated as to be in the moon's path1 and therefore to exhibit1 at some period or other1 those desirable occultations4 ,these stars are also of such magnitudes1 that their occultations may be distinctly observed from the deck1 even when subject to all the unsteadiness produced by an agitated sea4 ,to predict the occultations of such stars1 would require not less than ten thousand tables4 ,the stars from which lunar distances might be taken are still more numerous2 and we may safely pronounce1 that1 great as has been the improvement effected recently in our ,nautical ,almanac1 it does not yet furnish more than a small fraction of that aid to navigation 7in the large sense of that term71 which1 with greater facility1 expedition1 and economy in the calculation and printing of tables1 it might be made to supply4 ,tables necessary to determine the places of the planets are not less necessary than those for the sun1 moon1 and stars4 ,some notion of the number and complexity of these tables may be formed1 when we state that the positions of the two principal planets1 7and these the most necessary for the navigator17 ,jupiter and ,saturn1 require each not less than one hundred and sixteen tables4 ,yet it is not only necessary to predict the position of these bodies1 but it is likewise expedient to tabulate the motions of the four satellites of ,jupiter1 to predict the exact times at which they enter his shadow1 and at which their shadows cross his disc1 as well as the times at which they are interposed between him and the ,earth1 and he between them and the ,earth4 ,among the extensive classes of tables here enumerated1 there are several which are in their nature permanent and unalterable1 and would never require to be recomputed1 if they could once be computed with perfect accuracy on accurate data2 but the data on which such computations are conducted1 can only be regarded as approximations to truth1 within limits the extent of which must necessarily vary with our knowledge of astronomical science4 ,it has accordingly happened1 that one set of tables after another has been superseded with each advance of astronomical science4 ,some striking examples of this may not be uninstructive4 ,in #agfe1 the ,board of ,longitude paid to the celebrated ,euler the sum of ,l4#cjj1 for furnishing general formul for the computation of lunar tables4 ,professor ,mayer was employed to calculate the tables upon these formul1 and the sum of ,l4#cjjj was voted for them by the ,british ,parliament1 to his widow1 after his decease4 ,these tables had been used for ten years1 from #agff to #aggf1 in computing the ,nautical ,almanac1 when they were superseded by new and improved tables1 composed by ,mr ,charles ,mason1 under the direction of ,dr ,maskelyne1 from calculations made by order of the ,board of ,longitude1 on the observations of ,dr ,bradley4 ,a farther improvement was made by ,mason in #aghj2 but a much more extensive improvement took place in the lunar calculations by the publication of the tables of the ,moon1 by ,m4 ,brg1 deduced from ,laplace's theory1 in #ahjf4 ,perfect1 however1 as ,brg's tables were considered1 at the time of their publication1 they were1 within the short period of six years1 superseded by a more accurate set of tables published by ,burckhardt in #ahab2 and these also have since been followed by the tables of ,damoiseau4 ,professor ,schumacher has calculated by the latter tables his ephemeris of the ,planetary ,lunar ,distances1 and astronomers will hence be enabled to put to the strict test of observation the merits of the tables of ,burckhardt and ,damoiseau4#f ,footnote #f3 ,a comparison of the results for #ahcd1 will be found in the ,nautical ,almanac for #ahce4 ,the solar tables have undergone1 from time to time1 similar changes4 ,the solar tables of ,mayer were used in the computation of the ,nautical ,almanac1 from its commencement in #agfg1 to #ahjd inclusive4 ,within the six years immediately succeeding #ahjd1 not less than three successive sets of solar tables appeared1 each improving on the other2 the first by ,baron de ,zach1 the second by ,delambre1 under the direction of the ,french ,board of ,longitude1 and the third by ,carlini4 ,the last1 however1 differ only in arrangement from those of ,delambre4 ,similar observations will be applicable to the tables of the principal planets4 ,bouvard published1 in #ahjc1 tables of ,jupiter and ,saturn2 but from the improved state of astronomy1 he found it necessary to recompute these tables in #ahba4 ,although it is now about thirty years since the discovery of the four new planets1 ,ceres1 ,pallas1 ,juno1 and ,vesta1 it was not till recently that tables of their motions were published4 ,they have lately appeared in ,encke's ,ephemeris4 ,we have thus attempted to convey some notion 7though necessarily a very inadequate one7 of the immense extent of numerical tables which it has been found necessary to calculate and print for the purposes of the arts and sciences4 ,we have before us a catalogue of the tables contained in the library of one private individual1 consisting of not less than one hundred and forty volumes4 ,among these there are no duplicate copies3 and we observe that many of the most celebrated voluminous tabular works are not contained among them4 ,they are confined exclusively to arithmetical and trigonometrical tables2 and1 consequently1 the myriad of astronomical and nautical tables are totally excluded from them4 ,nevertheless1 they contain an extent of printed surface covered with figures amounting to above sixteen thousand square feet4 ,we have taken at random forty of these tables1 and have found that the number of errors acknowledged in the respective errata1 amounts to above three thousand seven hundred4 ,to be convinced of the necessity which has existed for accurate numerical tables1 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 produced4 ,to enable the reader to estimate the difficulties which attend the attainment even of a limited degree of accuracy1 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 them4 ,among the scientific enterprises which the ambition of the ,french nation aspired to during the ,republic1 was the construction of a magnificent system of numerical tables4 ,their most distinguished mathematicians were called upon to contribute to the attainment of this important object2 and the superintendence of the undertaking was confided to the celebrated ,prony1 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 tables1 constructed with such accuracy that they should form a monument of calculation the most vast and imposing that had ever been executed1 or even conceived4 ,to accomplish this gigantic task1 the principle of the division of labour1 found to be so powerful in manufactures1 was resorted to with singular success4 ,the persons employed in the work were divided into three sections3 the first consisted of half a dozen of the most eminent analysts4 ,their duty was to investigate the most convenient mathematical formul1 which should enable the computers to proceed with the greatest expedition and accuracy by the method of ,differences1 of which we shall speak more fully hereafter4 ,these formul1 when decided upon by this first section1 were handed over to the second section1 which consisted of eight or ten properly qualified mathematicians4 ,it was the duty of this second section to convert into numbers certain general or algebraical expressions which occurred in the formul1 so as to prepare them for the hands of the computers4 ,thus prepared1 these formul were handed over to the third section1 who formed a body of nearly one hundred computers4 ,the duty of this numerous section was to compute the numbers finally intended for the tables4 ,every possible precaution was of course taken to ensure the numerical accuracy of the results4 ,each number was calculated by two or more distinct and independent computers1 and its truth and accuracy determined by the coincidence of the results thus obtained4 ,the body of tables thus calculated occupied in manuscript seventeen folio volumes4#g ,footnote #g3 ,these tables were never published4 ,the printing of them was commenced by ,didot1 and a small portion was actually stereotyped1 but never published4 ,soon after the commencement of the undertaking1 the sudden fall of the assignats rendered it impossible for ,didot to fulfil his contract with the government4 ,the work was accordingly abandoned1 and has never since been resumed4 ,we have before us a copy of #ajj pages folio of the portion which was printed at the time the work was stopped1 given to a friend on a late occasion by ,didot himself4 ,it was remarked in this1 as in other similar cases1 that the computers who committed fewest errors were those who understood nothing beyond the process of addition4 ,as an example of the precautions which have been considered necessary to guard against errors in the calculation of numerical tables1 we shall further state those which were adopted by ,mr ,babbage1 previously to the publication of his tables of logarithms4 ,in order to render the terminal figure of tables in which one or more decimal places are omitted as accurate as it can be1 it has been the practice to compute one or more of the succeeding figures2 and if the first omitted figure be greater than #d1 then the terminal figure is always increased by #a1 since the value of the tabulated number is by such means brought nearer to the truth4#h ,the tables of ,callet1 which were among the most accurate published logarithms1 and which extended to seven places of decimals1 were first carefully compared with the tables of ,vega1 which extended to ten places1 in order to discover whether ,callet had made the above correction of the final figure in every case where it was necessary4 ,this previous precaution being taken1 and the corrections which appeared to be necessary being made in a copy of ,callet's tables1 the proofs of ,mr ,babbage's tables were submitted to the following test3 ,they were first compared1 number by number1 with the corrected copy of ,callet's logarithms2 secondly1 with ,hutton's logarithms2 and thirdly1 with ,vega's logarithms4 ,the corrections thus suggested being marked in the proofs1 corrected revises were received back4 ,these revises were then again compared1 number by number1 first with ,vega's logarithms2 secondly1 with the logarithms of ,callet2 and thirdly1 as far as the first #bj1jjj numbers1 with the corresponding ones in ,briggs's logarithms4 ,they were now returned to the printer1 and were stereotyped2 proofs were taken from the stereotyped plates1 which were put through the following ordeal3 ,they were first compared once more with the logarithms of ,vega as far as #dg1ejj2 they were then compared with the whole of the logarithms of ,gardner2 and next with the whole of ,taylor's logarithms2 and as a last test1 they were transferred to the hands of a different set of readers1 and were once more compared with ,taylor4 ,that these precautions were by no means superfluous may be collected from the following circumstances mentioned by ,mr ,babbage3 ,in the sheets read immediately previous to stereotyping1 thirty-two errors were detected2 after stereotyping1 eight more were found1 and corrected in the plates4 ,footnote #h3 ,thus suppose the number expressed at full length were #c4adaeibg4 ,if the table extend to no more than four places of decimals1 we should tabulate the number #c4adaf and not #c4adae4 ,the former would be evidently nearer to the true number #c4adaeibg4 ,by such elaborate and expensive precautions many of the errors of computation and printing may certainly be removed2 but it is too much to expect that in general such measures can be adopted2 and we accordingly find by far the greater number of tables disfigured by errors1 the extent of which is rather to be conjectured than determined4 ,when the nature of a numerical table is considered1--page after page densely covered with figures1 and with nothing else1--the chances against the detection of any single error will be easily comprehended2 and it may therefore be fairly presumed1 that for one error which may happen to be detected1 there must be a great number which escape detection4 ,notwithstanding this difficulty1 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 use4 ,mr ,baily states that he has himself detected in the solar and lunar tables1 from which our ,nautical ,almanac was for a long period computed1 more than five hundred errors4 ,in the multiplication table already mentioned1 computed by ,dr ,hutton for the ,board of ,longitude1 a single page was examined and recomputed3 it was found to contain about forty errors4 ,in order to make the calculations upon the numbers found in the ,ephemeral ,tables published in the ,nautical ,almanac1 it is necessary that the mariner should be supplied with certain permanent tables4 ,a volume of these1 to the number of about thirty1 was accordingly computed1 and published at national expense1 by order of the ,board of ,longitude1 entitled ',tables requisite to be used with the ,nautical ,ephemeris for finding the latitude and longitude at sea4' ,in the first edition of these requisite tables1 there were detected1 by one individual1 above a thousand errors4 ,the tables published by the ,board of ,longitude for the correction of the observed distances of the moon from certain fixed stars1 are followed by a table of acknowledged errata1 extending to seven folio pages1 and containing more than eleven hundred errors4 ,even this table of errata itself is not correct3 a considerable number of errors have been detected in it1 so that errata upon errata have become necessary4 ,one of the tests most frequently resorted to for the detection of errors in numerical tables1 has been the comparison of tables of the same kind1 published by different authors4 ,it has been generally considered that those numbers in which they are found to agree must be correct2 inasmuch as the chances are supposed to be very considerable against two or more independent computers falling into precisely the same errors4 ,how far this coincidence may be safely assumed as a test of accuracy we shall presently see4 ,a few years ago1 it was found desirable to compute some very accurate logarithmic tables for the use of the great national survey of ,ireland1 which was then1 and still is in progress2 and on that occasion a careful comparison of various logarithmic tables was made4 ,six remarkable errors were detected1 which were found to be common to several apparently independent sets of tables4 ,this singular coincidence led to an unusually extensive examination of the logarithmic tables published both in ,england and in other countries2 by which it appeared that thirteen sets of tables1 published in ,london between the years #afcc and #ahbb1 all agreed in these six errors4 ,upon extending the enquiry to foreign tables1 it appeared that two sets of tables published at ,paris1 one at ,gouda1 one at ,avignon1 one at ,berlin1 and one at ,florence1 were infected by exactly the same six errors4 ,the only tables which were found free from them were those of ,vega1 and the more recent impressions of ,callet4 ,it happened that the ,royal ,society possessed a set of tables of logarithms printed in the ,chinese character1 and on ,chinese paper1 consisting of two volumes3 these volumes contained no indication or acknowledgment of being copied from any other work4 ,they were examined2 and the result was the detection in them of the same six errors4#i ,footnote #i3 ,memoirs ,ast4 ,soc4 vol4 ,,iii1 p4 #fe4 ,it is quite apparent that this remarkable coincidence of error must have arisen from the various tables being copied successively one from another4 ,the earliest work in which they appeared was ,vlacq's ,logarithms1 7folio1 ,gouda1 #afbh72 and from it1 doubtless1 those which immediately succeeded it in point of time were copied2 from which the same errors were subsequently transcribed into all the other1 including the ,chinese logarithms4 ,the most certain and effectual check upon errors which arise in the process of computation1 is to cause the same computations to be made by separate and independent computers2 and this check is rendered still more decisive if they make their computations by different methods4 ,it is1 nevertheless1 a remarkable fact1 that several computers1 working separately and independently1 do frequently commit precisely the same error2 so that falsehood in this case assumes that character of consistency1 which is regarded as the exclusive attribute of truth4 ,instances of this are familiar to most persons who have had the management of the computation of tables4 ,we have reason to know1 that ,m4 ,prony experienced it on many occasions in the management of the great ,french tables1 when he found three1 and even a greater number of computers1 working separately and independently1 to return him the same numerical result1 and that result wrong4 ,mr ,stratford1 the conductor of the ,nautical ,almanac1 to whose talents and zeal that work owes the execution of its recent improvements1 has more than once observed a similar occurrence4 ,but one of the most signal examples of this kind1 of which we are aware1 is related by ,mr ,baily4 ,the catalogue of stars published by the ,astronomical ,society was computed by two separate and independent persons1 and was afterwards compared and examined with great care and attention by ,mr ,stratford4 ,on examining this catalogue1 and recalculating a portion of it1 ,mr ,baily discovered an error in the case of the star1 ,cephei4 ,its right ascension was calculated wrongly1 and yet consistently1 by two computers working separately4 ,their numerical results agreed precisely in every figure2 and ,mr ,stratford1 on examining the catalogue1 failed to detect the error4 ,mr ,baily having reason1 from some discordancy which he observed1 to suspect an error1 recomputed the place of the star with a view to discover it2 and he himself1 in the first instance1 obtained precisely the same erroneous numerical result4 ,it was only on going over the operation a second time that he accidentally discovered that he had inadvertently committed the same error4#aj ,footnote #aj3 ,memoirs ,ast4 ,soc4 vol4 iv41 p4 #bij4 ,it appears1 therefore1 that the coincidence of different tables1 even when it is certain that they could not have been copied one from another1 but must have been computed independently1 is not a decisive test of their correctness1 neither is it possible to ensure accuracy by the device of separate and independent computation4 ,besides the errors incidental to the process of computation1 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 printer4 ,the next source of error lies with the compositor1 in transferring this copy into type4 ,but the liabilities to error do not stop even here2 for it frequently happens1 that after the press has been fully corrected1 errors will be produced in the process of printing4 ,a remarkable instance of this occurs in one of the six errors detected in so many different tables already mentioned4 ,in one of these cases1 the last five figures of two successive numbers of a logarithmic table were the following3-- #cehge #ajdcf4 ,now1 both of these are erroneous2 the figure #h in the first line should be #d1 and the figure #d in the second should be #h4 ,it is evident that the types1 as first composed1 were correct2 but in the course of printing1 the two types #d and #h being loose1 adhered to the inking-balls1 and were drawn out3 the pressmen in replacing them transposed them1 putting the #h above and the #d below1 instead of vice versa4 ,it would be a curious enquiry1 were it possible to obtain all the copies of the original edition of ,vlacq's ,logarithms1 published at ,gouda in #afbh1 from which this error appears to have been copied in all the subsequent tables1 to ascertain whether it extends through the entire edition4 ,it would probably1 nay almost certainly1 be discovered that some of the copies of that edition are correct in this number1 while others are incorrect2 the former having been worked off before the transposition of the types4 ,it is a circumstance worthy of notice1 that this error in ,vlacq's tables has produced a corresponding error in a variety of other tables deduced from them1 in which nevertheless the erroneous figures in ,vlacq are omitted4 ,in no less than sixteen sets of tables published at various times since the publication of ,vlacq1 in which the logarithms extend only to seven places of figures1 the error just mentioned in the eighth place in ,vlacq causes a corresponding error in the seventh place4 ,when the last three figures are omitted in the first of the above numbers1 the seventh figure should be #e1 inasmuch as the first of the omitted figures is under #e3 the erroneous insertion1 however1 of the figure #h in ,vlacq has caused the figure #f to be substituted for #e in the various tables just alluded to4 ,for the same reason1 the erroneous occurrence of #d in the second number has caused the adoption of a #j instead of a #a in the seventh place in the other tables4 ,the only tables in which this error does not occur are those of ,vega1 the more recent editions of ,callet1 and the still later ,logarithms of ,mr ,babbage4 ,the ,opus ,palatinum1 a work published in #aeif1 containing an extensive collection of trigonometrical tables1 affords a remarkable instance of a tabular error2 which1 as it is not generally known1 it may not be uninteresting to mention here4 ,after that work had been for several years in circulation in every part of ,europe1 it was discovered that the commencement of the table of co-tangents and co-secants was vitiated by an error of considerable magnitude4 ,in the first co-tangent the last nine places of figures were incorrect2 but from the manner in which the numbers of the table were computed1 the error was gradually1 though slowly1 diminished1 until at length it became extinguished in the eighty-sixth page4 ,after the detection of this extensive error1 ,pitiscus undertook the recomputation of the eighty-six erroneous pages4 ,his corrected calculation was printed1 and the erroneous part of the remaining copies of the ,opus ,palatinum was cancelled4 ,but as the corrected table of ,pitiscus was not published until #afjg1--thirteen years after the original work1--the erroneous part of the volume was cancelled in comparatively few copies1 and consequently correct copies of the work are now exceedingly rare4 ,thus1 in the collection of tables published by ,m4 ,schulze1#aa the whole of the erroneous part of the ,opus ,palatinum has been adopted2 he having used the copy of that work which exists in the library of the ,academy of ,berlin1 and which is one of those copies in which the incorrect part was not cancelled4 ,the corrected copies of this work may be very easily distinguished at present from the erroneous ones3 it happened that the former were printed with a very bad and worn-out type1 and upon paper of a quality inferior to that of the original work4 ,on comparing the first eighty-six pages of the volume with the succeeding ones1 they are1 therefore1 immediately distinguishable in the corrected copies4 ,besides this test1 there is another1 which it may not be uninteresting to point out3--,at the bottom of page #g in the corrected copies1 there is an error in the position of the words basis and hypothenusa1 their places being interchanged4 ,in the original uncorrected work this error does not exist4 ,footnote #aa3 ,recueil des ,tables ,logarithmiques et ,trigonometriques4 ,par ,j4 ,c4 ,schulze4 #b vols4 ,berlin3 #aggh4 ,at the time when the calculation and publication of ,taylor's ,logarithms were undertaken1 it so happened that a similar work was in progress in ,france2 and it was not until the calculation of the ,french work was completed1 that its author was informed of the publication of the ,english work4 ,this circumstance caused the ,french calculator to relinquish the publication of his tables4 ,the manuscript subsequently passed into the library of ,delambre1 and1 after his death1 was purchased at the sale of his books1 by ,mr ,babbage1 in whose possession it now is4 ,some years ago it was thought advisable to compare these manuscript tables with ,taylor's ,logarithms1 with a view to ascertain the errors in each1 but especially in ,taylor4 ,the two works were peculiarly well suited for the attainment of this end2 as the circumstances under which they were produced1 rendered it quite certain that they were computed independently of each other4 ,the comparison was conducted under the direction of the late ,dr ,young1 and the result was the detection of the following nineteen errors in ,taylor's ,logarithms4 ,to enable those who used ,taylor's ,logarithms to make the necessary corrections in them1 the corrections of the detected errors appeared as follows in the ,nautical ,almanac for #ahcb4 ,,errata1 detected in ,taylor's ,logarithms4 ,london3 #dto1 #agib4 ' #a ,e ,co-tangent of #a4ce4ce for #dcfga read #dbfga #b ,m ,co-tangent of #d4 #d4di --- #ffigf ---- #ffigi #c ,sine of #d4bc4ch --- #dcajg ---- #dcjjg #d ,sine of #d4bc4ci --- #dccha ---- #dcbha #e ,s ,sine of #f4de4eb --- #ajjja ---- #aajja #f ,kk ,co-sine of #ad4ah4 #c --- #ccih ---- #cbih #g ,ss ,tangent of #ah4 #a4ef --- #ejfd ---- #fjfd #h ,aaa ,co-tangent of #ba4aa4ad --- #fjfb ---- #eifb #i ,ggg ,tangent of #bc4dh4ai --- #fjhg ---- #eihg #aj ,co-tangent of #bc4dh4ai --- #ciac ---- #djac #aa ,iii ,sine of #be4 #e4 #d --- #cagc ---- #cahc #ab ,sine of #be4 #e4 #e --- #cbah ---- #cbbh #ac ,sine of #be4 #e4 #f --- #cbfc ---- #cbgc #ad ,sine of #be4 #e4 #g --- #ccjh ---- #ccah #ae ,sine of #be4 #e4 #h --- #ccec ---- #ccfc #af ,sine of #be4 #e4 #i --- #ccih ---- #cdjh #ag ,qqq ,tangent of #bh4ai4ci --- #fcjb ---- #fdjb #ah #d;,h ,tangent of #ce4ee4ea --- #afha ---- #aeha #ai #d,k ,co-sine of #cg4bi4 #b --- #eejc ---- #efjc ,an error being detected in this list of ,,errata1 we find1 in the ,nautical ,almanac for the year #ahcc1 the following ,,erratum of the ,,errata of ,taylor's ,logarithms3-- ',in the list of ,,errata detected in ,taylor's ,logarithms1 for cos4 #d #ah' #c1 read cos4 #ad #ah' #b4' ,here1 however1 confusion is worse confounded2 for a new error1 not before existing1 and of much greater magnitude1 is introduced6 ,it will be necessary1 in the ,nautical ,almanac for #ahcf1 7that for #ahce is already published17 to introduce the following3 ,,erratum of the ,,erratum of the ,,errata of ,,taylor's ,logarithms4 ,for cos4 #d #ah' #c1 read cos4 #ad #ah' #c4 ,if proof were wanted to establish incontrovertibly the utter impracticability of precluding numerical errors in works of this nature1 we should find it in this succession of error upon error1 produced1 in spite of the universally acknowledged accuracy and assiduity of the persons at present employed in the construction and management of the ,nautical ,almanac4 ,it is only by the mechanical fabrication of tables that such errors can be rendered impossible4 ,on examining this list with attention1 we have been particularly struck with the circumstances in which these errors appear to have originated4 ,it is a remarkable fact1 that of the above nineteen errors1 eighteen have arisen from mistakes in carrying4 ,errors #e1 #g1 #aj1 #aa1 #ab1 #ac1 #ad1 #ae1 #af1 #ag1 #ai1 have arisen from a carriage being neglected2 and errors #a1 #c1 #d1 #f1 #h1 #i1 and #ah1 from a carriage being made where none should take place4 ,in four cases1 namely1 errors #h1 #i1 #aj1 and #af1 this has caused two figures to be wrong4 ,the only error of the nineteen which appears to have been a press error is the second2 which has evidently arisen from the type #i being accidentally inverted1 and thus becoming a #f4 ,this may have originated with the compositor1 but more probably it took place in the press-work2 the type #i being accidentally drawn out of the form by the inking-ball1 as mentioned in a former case1 and on being restored to its place1 inverted by the pressman4 ,there are two cases among the above errata1 in which an error1 committed in the calculation of one number1 has evidently been the cause of other errors4 ,in the third erratum1 a wrong carriage was made1 in computing the sine of #d #bc' #ch4 ,the next number of the table was vitiated by this error2 for we find the next erratum to be in the sine of #d #bc' #ci1 in which the figure similarly placed is #a in excess4 ,a still more extensive effect of this kind appears in errata #aa1 #ab1 #ac1 #ad1 #ae1 #af4 ,a carriage was neglected in computing the sine of #be #e' #d1 and this produced a corresponding error in the five following numbers of the table1 which are those corrected in the five following errata4 ,this frequency of errors arising in the process of carrying1 would afford a curious subject of metaphysical speculation respecting the operation of the faculty of memory4 ,in the arithmetical process1 the memory is employed in a twofold way2--in ascertaining each successive figure of the calculated result by the recollection of a table committed to memory at an early period of life2 and by another act of memory1 in which the number carried from column to column is retained4 ,it is a curious fact1 that this latter circumstance1 occurring only the moment before1 and being in its nature little complex1 is so much more liable to be forgotten or mistaken than the results of rather complicated tables4 ,it appears1 that among the above errata1 the errors #e1 #g1 #aj1 #aa1 #ag1 #ai1 have been produced by the computer forgetting a carriage2 while the errors #a1 #c1 #f1 #h1 #i1 #ah1 have been produced by his making a carriage improperly4 ,thus1 so far as the above list of errata affords grounds for judging1 it would seem1 7contrary to what might be expected17 that the error by which improper carriages are made is as frequent as that by which necessary carriages are overlooked4 ,we trust that we have succeeded in proving1 first1 the great national and universal utility of numerical tables1 by showing the vast number of them1 which have been calculated and published2 secondly1 that more effectual means are necessary to obtain such tables suitable to the present state of the arts1 sciences and commerce1 by showing that the existing supply of tables1 vast as it certainly is1 is still scanty1 and utterly inadequate to the demands of the community2--that it is rendered inefficient1 not only in quantity1 but in quality1 by its want of numerical correctness2 and that such numerical correctness is altogether unattainable until some more perfect method be discovered1 not only of calculating the numerical results1 but of tabulating these1--of reducing such tallies to type1 and of printing that type so as to intercept the possibility of error during the press-work4 ,such are the ends which are proposed to be attained by the calculating machinery invented by ,mr ,babbage4 ,the benefits to be derived from this invention cannot be more strongly expressed than they have been by ,mr ,colebrooke1 ,president of the ,astronomical ,society1 on the occasion of presenting the gold medal voted by that body to ,mr ,babbage3--',in no department of science1 or of the arts1 does this discovery promise to be so eminently useful as in that of astronomy1 and its kindred sciences1 with the various arts dependent on them4 ,in none are computations more operose than those which astronomy in particular requires2--in none are preparatory facilities more needful2--in none is error more detrimental4 ,the practical astronomer is interrupted in his pursuit1 and diverted from his task of observation by the irksome labours of computation1 or his diligence in observing becomes ineffectual for want of yet greater industry of calculation4 ,let the aid which tables previously computed afford1 be furnished to the utmost extent which mechanism has made attainable through ,mr ,babbage's invention1 and the most irksome portion of the astronomer's task is alleviated1 and a fresh impulse is given to astronomical research4' ,the first step in the progress of this singular invention was the discovery of some common principle which pervaded numerical tables of every description2 so that by the adoption of such a principle as the basis of the machinery1 a corresponding degree of generality would be conferred upon its calculations4 ,among the properties of numerical functions1 several of a general nature exist2 and it was a matter of no ordinary difficulty1 and requiring no common skill1 to select one which might1 in all respects1 be preferable to the others4 ,whether or not that which was selected by ,mr ,babbage affords the greatest practical advantages1 would be extremely difficult to decide--perhaps impossible1 unless some other projector could be found possessed of sufficient genius1 and sustained by sufficient energy of mind and character1 to attempt the invention of calculating machinery on other principles4 ,the principle selected by ,mr ,babbage as the basis of that part of the machinery which calculates1 is the ,method of ,differences2 and he has in fact literally thrown this mathematical principle into wheel-work4 ,in order to form a notion of the nature of the machinery1 it will be necessary1 first to convey to the reader some idea of the mathematical principle just alluded to4 ,a numerical table1 of whatever kind1 is a series of numbers which possess some common character1 and which proceed increasing or decreasing according to some general law4 ,supposing such a series continually to increase1 let us imagine each number in it to be subtracted from that which follows it1 and the remainders thus successively obtained to be ranged beside the first1 so as to form another table3 these numbers are called the first differences4 ,if we suppose these likewise to increase continually1 we may obtain a third table from them by a like process1 subtracting each number from the succeeding one3 this series is called the second differences4 ,by adopting a like method of proceeding1 another series may be obtained1 called the third differences2 and so on4 ,by continuing this process1 we shall at length obtain a series of differences1 of some order1 more or less high1 according to the nature of the original table1 in which we shall find the same number constantly repeated1 to whatever extent the original table may have been continued2 so that if the next series of differences had been obtained in the same manner as the preceding ones1 every term of it would be #j4 ,in some cases this would continue to whatever extent the original table might be carried2 but in all cases a series of differences would be obtained1 which would continue constant for a very long succession of terms4 ,as the successive serieses of differences are derived from the original table1 and from each other1 by subtraction1 the same succession of series may be reproduced in the other direction by addition4 ,but let us suppose that the first number of the original table1 and of each of the series of differences1 including the last1 be given3 all the numbers of each of the series may thence be obtained by the mere process of addition4 ,the second term of the original table will be obtained by adding to the first the first term of the first difference series2 in like manner1 the second term of the first difference series will be obtained by adding to the first term1 the first term of the third difference series1 and so on4 ,the second terms of all the serieses being thus obtained1 the third terms may be obtained by a like process of addition2 and so the series may be continued4 ,these observations will perhaps be rendered more clearly intelligible when illustrated by a numerical example4 ,the following is the commencement of a series of the fourth powers of the natural numbers3-- ,no4 ,table4 #a #a #b #af #c #ha #d #bef #e #fbe #f #abif #g #bdja #h #djif #i #fefa #aj #aj1jjj #aa #ad1fda #ab #bj1gcf #ac #bh1efa ,by subtracting each number from the succeeding one in this series1 we obtain the following series of first differences3 #ae #fe #age #cfi #fga #aaje #afie #bdfe #cdci #dfda #fjie #ghbe ,in like manner1 subtracting each term of this series from the succeeding one1 we obtain the following series of second differences3-- #ej #aaj #aid #cjb #dcd #eij #ggj #igd #abjb #aded #agcj ,proceeding with this series in the same way1 we obtain the following series of third differences3-- #fj #hd #ajh #acb #aef #ahj #bjd #bbh #beb #bgf ,proceeding in the same way with these1 we obtain the following for the series of fourth differences3-- #bd #bd #bd #bd #bd #bd #bd #bd #bd ,it appears1 therefore1 that in this case the series of fourth differences consists of a constant repetition of the number #bd4 ,now1 a slight consideration of the succession of arithmetical operations by which we have obtained this result1 will show1 that by reversing the process1 we could obtain the table of fourth powers by the mere process of addition4 ,beginning with the first numbers in each successive series of differences1 and designating the table and the successive differences by the letters ,t1 ,d#a ,d#b ,d#c ,d#d1 we have then the following to begin with3-- ,t ,d#a ,d#b ,d#c ,d#d #a #ae #ej #fj #bd ,adding each number to the number on its left1 and repeating #bd1 we get the following as the second terms of the several series3-- ,t ,d#a ,d#b ,d#c ,d#d #af #fe #aaj #hd #bd ,and1 in the same manner1 the third and succeeding terms as follows3-- ,no4 ,t ,d#a ,d#b ,d#c ,d#d #a #a #ae #ej #fj #bd #b #af #fe #aaj #hd #bd #c #ha #age #aid #ajh #bd #d #bef #cfi #cjb #acb #bd #e #fbe #fga #dcd #aef #bd #f #abif #aaje #eij #ahj #bd #g #bdja #afie #ggj #bjd #bd #h #djif #bdfe #igd #bbh #bd #i #fefa #cdci #abjb #beb #bd #aj #ajjjj #dfda #aded #bgf #aa #adfda #fjie #agcj #ab #bjgcf #ghbe #ac #bhefa ,there are numerous tables in which1 as already stated1 to whatever order of differences we may proceed1 we should not obtain a series of rigorously constant differences2 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 terms4 ,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 constant2 and if at every point where the last difference requires an increase1 that increase be given to it1 the same principle of addition may again be applied for a like succession of terms1 and so on4 ,by this principle it appears1 that all tables in which each series of differences continually increases1 may be produced by the operation of addition alone2 provided the first terms of the table1 and of each series of differences1 be given in the first instance4 ,but it sometimes happens1 that while the table continually increases1 one or more serieses of differences may continually diminish4 ,in this case1 the series of differences are found by subtracting each term of the series1 not from that which follows1 but from that which precedes it2 and consequently1 in the re-production of the several serieses1 when their first terms are given1 it will be necessary in some cases to obtain them by addition1 and in others by subtraction4 ,it is possible1 however1 still to perform all the operations by addition alone3 this is effected in performing the operation of subtraction1 by substituting for the subtrahend its arithmetical complement1 and adding that1 omitting the unit of the highest order in the result4 ,this process1 and its principle1 will be readily comprehended by an example4 ,let it be required to subtract #ceg from #gfh4 ,the common process would be as follows3-- ,from #gfh ,subtract #ceg ---- ,remainder #daa ,the arithmetical complement of #ceg1 or the number by which it falls short of #ajjj1 is #fdc4 ,now1 if this number be added to #gfh1 and the first figure on the left be struck out of the sum1 the process will be as follows3-- ,to #gfh ,add #fdc ---- ,sum #adaa ---- ,remainder sought #daa ,the principle on which this process is founded is easily explained4 ,in the latter process we have first added #fdc1 and then subtracted #ajjj4 ,on the whole1 therefore1 we have subtracted #ceg1 since the number actually subtracted exceeds the number previously added by that amount4 ,since1 therefore1 subtraction may be effected in this manner by addition1 it follows that the calculation of all serieses1 so far as an order of differences can be found in them which continues constant1 may be conducted by the process of addition alone4 ,it also appears from what has been stated1 that each addition consists only of two operations4 ,however numerous the figures may be of which the several pairs of numbers to be thus added may consist1 it is obvious that the operation of adding them can only consist of repetitions of the process of adding one digit to another2 and of carrying one from the column of inferior units to the column of units next superior when necessary4 ,if we would therefore reduce such a process to machinery1 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 figures2 for1 this being once accomplished1 the process of adding two numbers1 consisting of any number of digits1 will be effected by repeating the same mechanism as often as there are pairs of digits to be added4 ,such was the simple form to which ,mr ,babbage reduced the problem of discovering the calculating machinery2 and we shall now proceed to convey some notion of the manner in which he solved it4 ,for the sake of illustration1 we shall suppose that the table to be calculated shall consist of numbers not exceeding six places of figures2 and we shall also suppose that the difference of the fifth order is the constant difference4 ,imagine1 then1 six rows of wheels1 each wheel carrying upon it a dial-plate like that of a common clock1 but consisting of ten instead of twelve divisions2 the several divisions being marked #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i1 #j4 ,let these dials be supposed to revolve whenever the wheels to which they are attached are put in motion1 and to turn in such a direction that the series of increasing numbers shall pass under the index which appears over each dial3--thus1 after #j passes the index1 #a follows1 then #b1 #c1 and so on1 as the dial revolves4 ,in ,fig4 #a are represented six horizontal rows of such dials4 ,fig4 #a4 ,the method of differences1 as already explained1 requires1 that in proceeding with the calculation1 this apparatus should perform continually the addition of the number expressed upon each row of dials1 to the number expressed upon the row immediately above it4 ,now1 we shall first explain how this process of addition may be conceived to be performed by the motion of the dials2 and in doing so1 we shall consider separately the processes of addition and carriage1 considering the addition first1 and then the carriage4 ,let us first suppose the line ,d#a to be added to the line ,t4 ,to accomplish this1 let us imagine that while the dials on the line ,d#a are quiescent1 the dials on the line ,t are put in motion1 in such a manner1 that as many divisions on each dial shall pass under its index1 as there are units in the number at the index immediately below it4 ,it is evident that this condition supposes1 that if #j be at any index on the line ,d#a1 the dial immediately above it in the line ,t shall not move4 ,now the motion here supposed1 would bring under the indices on the line ,t such a number as would be produced by adding the number ,d#a to ,t1 neglecting all the carriages2 for a carriage should have taken place in every case in which the figure #i of any dial in the line ,t had passed under the index during the adding motion4 ,to accomplish this carriage1 it would be necessary that the dial immediately on the left of any dial in which #i passes under the index1 should be advanced one division1 independently of those divisions which it may have been advanced by the addition of the number immediately below it4 ,this effect may be conceived to take place in either of two ways4 ,it may be either produced at the moment when the division between #i and #j of any dial passes under the index2 in which case the process of carrying would go on simultaneously with the process of adding2 or the process of carrying may be postponed in every instance until the process of addition1 without carrying1 has been completed2 and then by another distinct and independent motion of the machinery1 a carriage may be made by advancing one division all those dials on the right of which a dial had1 during the previous addition1 passed from #i to #j under the index4 ,the latter is the method adopted in the calculating machinery1 in order to enable its inventor to construct the carrying machinery independent of the adding mechanism4 ,having explained the motion of the dials by which the addition1 excluding the carriages of the number on the row ,d#a1 may be made to the number on the row ,t1 the same explanation may be applied to the number on the row ,d#b to the number on the row ,d#a2 also1 of the number ,d#c to the number on the row ,d#b1 and so on4 ,now it is possible to suppose the additions of all the rows1 except the first1 to be made to all the rows except the last1 simultaneously2 and after these additions have been made1 to conceive all the requisite carriages to be also made by advancing the proper dials one division forward4 ,this would suppose all the dials in the scheme to receive their adding motion together2 and1 this being accomplished1 the requisite dials to receive their carrying motions together4 ,the production of so great a number of simultaneous motions throughout any machinery1 would be attended with great mechanical difficulties1 if indeed it be practicable4 ,in the calculating machinery it is not attempted4 ,the additions are performed in two successive periods of time1 and the carriages in two other periods of time1 in the following manner4 ,we shall suppose one complete revolution of the axis which moves the machinery1 to make one complete set of additions and carriages2 it will then make them in the following order3-- ,the first quarter of a turn of the axis will add the second1 fourth1 and sixth rows to the first1 third1 and fifth1 omitting the carriages2 this it will do by causing the dials on the first1 third1 and fifth rows1 to turn through as many divisions as are expressed by the numbers at the indices below them1 as already explained4 ,the second quarter of a turn will cause the carriages consequent on the previous addition1 to be made by moving forward the proper dials one division4 7,during these two quarters of a turn1 the dials of the first1 third1 and fifth row alone have been moved2 those of the second1 fourth1 and sixth1 have been quiescent47 ,the third quarter of a turn will produce the addition of the third and fifth rows to the second and fourth1 omitting the carriages2 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 them4 ,the fourth and last quarter of a turn will cause the carriages consequent on the previous addition1 to be made by moving the proper dials forward one division4 ,this evidently completes one calculation1 since all the rows except the first have been respectively added to all the rows except the last4 ,to illustrate this3 let us suppose the table to be computed to be that of the fifth powers of the natural numbers1 and the computation to have already proceeded so far as the fifth power of #f1 which is #gggf4 ,this number appears1 accordingly1 in the highest row1 being the place appropriated to the number of the table to be calculated4 ,the several differences as far as the fifth1 which is in this case constant1 are exhibited on the successive rows of dials in such a manner1 as to be adapted to the process of addition by alternate rows1 in the manner already explained4 ,the process of addition will commence by the motion of the dials in the first1 third1 and fifth rows1 in the following manner3 ,the dial ,a1 fig4 #a1 must turn through one division1 which will bring the number #g to the index2 the dial ,b must turn through three divisions1 which will #j bring to the index2 this will render a carriage necessary1 but that carriage will not take place during the present motion of the dial4 ,the dial ,c will remain unmoved1 since #j is at the index below it2 the dial ,d must turn through nine divisions2 and as1 in doing so1 the division between #i and #j must pass under the index1 a carriage must subsequently take place upon the dial to the left2 the remaining dials of the row ,t1 fig4 #a1 will remain unmoved4 ,in the row ,d#b the dial ,a#b will remain unmoved1 since #j is at the index below it2 the dial ,b#b will be moved through five divisions1 and will render a subsequent carriage on the dial to the left necessary2 the dial ,c#b will be moved through five divisions2 the dial ,d#b will be moved through three divisions1 and the remaining dials of this row will remain unmoved4 ,the dials of the row ,d#d will be moved according to the same rules2 and the whole scheme will undergo a change exhibited in fig4 #b2 a mark 7997 being introduced on those dials to which a carriage is rendered necessary by the addition which has just taken place4 ,fig4 #b4 ,the second quarter of a turn of the moving axis1 will move forward through one division all the dials which in fig4 #b are marked 79971 and the scheme will be converted into the scheme expressed in fig4 #c4 ,fig4 #c4 ,in the third quarter of a turn1 the dial ,a#a1 fig4 #c1 will remain unmoved1 since #j is at the index below it2 the dial ,b#a will be moved forward through three divisions2 ,c#a through nine divisions1 and so on2 and in like manner the dials of the row ,d#c will be moved forward through the number of divisions expressed at the indices in the row ,d#d4 ,this change will convert the arrangement into that expressed in fig4 #d1 the dials to which a carriage is due1 being distinguished as before by 79974 ,fig4 #d4 ,the fourth quarter of a turn of the axis will move forward one division all the dials marked 79972 and the arrangement will finally assume the form exhibited in fig4 #e1 in which the calculation is completed4 ,the first row ,t in this expresses the fifth power of #g2 and the second expresses the number which must be added to the first row1 in order to produce the fifth power of #h2 the numbers in each row being prepared for the change which they must undergo1 in order to enable them to continue the computation according to the method of alternate addition here adopted4 ,fig4 #e4 ,having thus explained what it is that the mechanism is required to do1 we shall now attempt to convey at least a general notion of some of the mechanical contrivances by which the desired ends are attained4 ,to simplify the explanation1 let us first take one particular instance--the dials ,b and ,b#a1 fig4 #a1 for example4 ,behind the dial ,b#a is a bolt1 which1 at the commencement of the process1 is shot between the teeth of a wheel which drives the dial ,b3 during the first quarter of a turn this bolt is made to revolve1 and if it continued to be engaged in the teeth of the said wheel1 it would cause the dial ,b to make a complete revolution2 but it is necessary that the dial ,b should only move through three divisions1 and1 therefore1 when three divisions of this dial have passed under its index1 the aforesaid bolt must be withdrawn3 this is accomplished by a small wedge1 which is placed in a fixed position on the wheel behind the dial ,b#a1 and that position is such that this wedge will press upon the bolt in such a manner1 that at the moment when three divisions of the dial ,b have passed under the index1 it shall withdraw the bolt from the teeth of the wheel which it drives4 ,the bolt will continue to revolve during the remainder of the first quarter of a turn of the axis1 but it will no longer drive the dial ,b1 which will remain quiescent4 ,had the figure at the index of the dial ,b#a been any other1 the wedge which withdraws the bolt would have assumed a different position1 and would have withdrawn the bolt at a different time1 but at a time always corresponding with the number under the index of the dial ,b#a3 thus1 if #e had been under the index of the dial ,b#a1 then the bolt would have been withdrawn from between the teeth of the wheel which it drives1 when five divisions of the dial ,b had passed under the index1 and so on4 ,behind each dial in the row ,d#a there is a similar bolt and a similar withdrawing wedge1 and the action upon the dial above is transmitted and suspended in precisely the same manner4 ,like observations will be applicable to all the dials in the scheme here referred to1 in reference to their adding actions upon those above them4 ,there is1 however1 a particular case which here merits notice3 it is the case in which #j is under the index of the dial from which the addition is to be transmitted upwards4 ,as in that case nothing is to be added1 a mechanical provision should be made to prevent the bolt from engaging in the teeth of the wheel which acts upon the dial above3 the wedge which causes the bolt to be withdrawn1 is thrown into such a position as to render it impossible that the bolt should be shot1 or that it should enter between the teeth of the wheel1 which in other cases it drives4 ,but inasmuch as the usual means of shooting the bolt would still act1 a strain would necessarily take place in the parts of the mechanism1 owing to the bolt not yielding to the usual impulse4 ,a small shoulder is therefore provided1 which puts aside1 in this case1 the piece by which the bolt is usually struck1 and allows the striking implement to pass without encountering the head of the bolt or any other obstruction4 ,this mechanism is brought into play in the scheme1 fig4 #a1 in the cases of all those dials in which #j is under the index4 ,such is a general description of the nature of the mechanism by which the adding process1 apart from the carriages1 is effected4 ,during the first quarter of a turn1 the bolts which drive the dials in the first1 third1 and fifth rows1 are caused to revolve1 and to act upon these dials1 so long as they are permitted by the position of the several wedges on the second1 fourth1 and sixth rows of dials1 by which these bolts are respectively withdrawn2 and1 during the third quarter of a turn1 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 rows1 which withdraw them1 permit4 ,it will hence be perceived1 that1 during the first and third quarters of a turn1 the process of addition is continually passing upwards through the machinery2 alternately from the even to the odd rows1 and from the odd to the even rows1 counting downwards4 ,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 axis4 ,as before1 we shall first explain it in reference to a particular instance4 ,during the first quarter of a turn the wheel ,b#b1 fig4 #a1 is caused by the adding bolt to move through five divisions2 and the fifth of these divisions1 which passes under the index1 is that between #i and #j4 ,on the axis of the wheel ,c#b1 immediately to the left of ,b#b1 is fixed a wheel1 called in mechanics a ratchet wheel1 which is driven by a claw which constantly rests in its teeth4 ,this claw is in such a position as to permit the wheel ,c#b to move in obedience to the action of the adding bolt1 but to resist its motion in the contrary direction4 ,it is drawn back by a spiral spring1 but its recoil is prevented by a hook which sustains it2 which hook1 however1 is capable of being withdrawn1 and when withdrawn1 the aforesaid spiral spring would draw back the claw1 and make it fall through one tooth of the ratchet wheel4 ,now1 at the moment that the division between #i and #j on the dial ,b#b passes under the index1 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 mentioned1 and allows it to fall back by the recoil of the spring1 and to drop into the next tooth of the ratchet wheel4 ,this process1 however1 produces no immediate effect upon the position of the wheel ,c#b1 and is merely preparatory to an action intended to take place during the second quarter of a turn of the moving axis4 ,it is in effect a memorandum taken by the machine of a carriage to be made in the next quarter of a turn4 ,during the second quarter of a turn1 a finger placed on the axis of the dial ,b#b is made to revolve1 and it encounters the heel of the above-mentioned claw4 ,as it moves forward it drives the claw before it3 and this claw1 resting in the teeth of the ratchet wheel fixed upon the axis of the dial ,c#b drives forward that wheel1 and with it the dial4 ,but the length and position of the finger which drives the claw limits its action1 so as to move the claw forward through such a space only as will cause the dial ,c#b to advance through a single division2 at which point it is again caught and retained by the hook4 ,this will be added to the number under its index1 and the requisite carriage from ,b#b to ,c#b will be accomplished4 ,in connexion with every dial is placed a similar ratchet wheel with a similar claw1 drawn by a similar spring1 sustained by a similar hook1 and acted upon by a similar thumb and trigger2 and therefore the necessary carriages1 throughout the whole machinery1 take place in the same manner and by similar means4 ,during the second quarter of a turn1 such of the carrying claws as have been allowed to recoil in the first1 third1 and fifth rows1 are drawn up by the fingers on the axes of the adjacent dials2 and1 during the fourth quarter of a turn1 such of the carrying claws on the second and fourth rows as have been allowed to recoil during the third quarter of a turn1 are in like manner drawn up by the carrying fingers on the axes of the adjacent dials4 ,it appears that the carriages proceed alternately from right to left along the horizontal rows during the second and fourth quarters of a turn2 in the one1 they pass along the first1 third1 and fifth rows1 and in the other1 along the second and fourth4 ,there are two systems of waves of mechanical action continually flowing from the bottom to the top2 and two streams of similar action constantly passing from the right to the left4 ,the crests of the first system of adding waves fall upon the last difference1 and upon every alternate one proceeding upwards2 while the crests of the other system touch upon the intermediate differences4 ,the first stream of carrying action passes from right to left along the highest row and every alternate row1 while the second stream passes along the intermediate rows4 ,such is a very rapid and general outline of this machinery4 ,its wonders1 however1 are still greater in its details than even in its broader features4 ,although we despair of doing it justice by any description which can be attempted here1 yet we should not fulfil the duty we owe to our readers1 if we did not call their attention at least to a few of the instances of consummate skill which are scattered1 with a prodigality characteristic of the highest order of inventive genius1 throughout this astonishing mechanism4 ,in the general description which we have given of the mechanism for carrying1 it will be observed1 that the preparation for every carriage is stated to be made during the previous addition1 by the disengagement of the carrying claw before mentioned1 and by its consequent recoil1 urged by the spiral spring with which it is connected2 but it may1 and does1 frequently happen1 that though the process of addition may not have rendered a carriage necessary1 one carriage may itself produce the necessity for another4 ,this is a contingency not provided against in the mechanism as we have described it3 the case would occur in the scheme represented in fig4 #a1 if the figure under the index of ,c#b were #d instead of #c4 ,the addition of the number #e at the index of ,c#c would1 in this case1 in the first quarter of a turn1 bring #i to the index of ,c#b3 this would obviously render no carriage necessary1 and of course no preparation would be made for one by the mechanism--that is to say1 the carrying claw of the wheel ,d#b would not be detached4 ,meanwhile a carriage upon ,c#b has been rendered necessary by the addition made in the first quarter of a turn to ,b#b4 ,this carriage takes place in the ordinary way1 and would cause the dial ,c#b1 in the second quarter of a turn1 to advance from #i to #j3 this would make the necessary preparation for a carriage from ,c#b to ,d#b4 ,but unless some special arrangement was made for the purpose1 that carriage would not take place during the second quarter of a turn4 ,this peculiar contingency is provided against by an arrangement of singular mechanical beauty1 and which1 at the same time1 answers another purpose--that of equalizing the resistance opposed to the moving power by the carrying mechanism4 ,the fingers placed on the axes of the several dials in the row ,d#b1 do not act at the same instant on the carrying claws adjacent to them2 but they are so placed1 that their action may be distributed throughout the second quarter of a turn in regular succession4 ,thus the finger on the axis of the dial ,a#b first encounters the claw upon ,b#b1 and drives it through one tooth immediately forwards2 the finger on the axis of ,b#b encounters the claw upon ,c#b and drives it through one tooth2 the action of the finger on ,c#b on the claw on ,d#b next succeeds1 and so on4 ,thus1 while the finger on ,b#b acts on ,c#b1 and causes the division from #i to #j to pass under the index1 the thumb on ,c#b at the same instant acts on the trigger1 and detaches the carrying claw on ,d#b1 which is forthwith encountered by the carrying finger on ,c#b1 and driven forward one tooth4 ,the dial ,d#b accordingly moves forward one division1 and #e is brought under the index4 ,this arrangement is beautifully effected by placing the several fingers1 which act upon the carrying claws1 spirally on their axes1 so that they come into action in regular succession4 ,we have stated that1 at the commencement of each revolution of the moving axis1 the bolts which drive the dials of the first1 third1 and fifth rows1 are shot4 ,the process of shooting these bolts must therefore have taken place during the last quarter of the preceding revolution2 but it is during that quarter of a turn that the carriages are effected in the second and fourth rows4 ,since the bolts which drive the dials of the first1 third1 and fifth rows1 have no mechanical connexion with the dials in the second and fourth rows1 there is nothing in the process of shooting those bolts incompatible with that of moving the dials of the second and fourth rows3 hence these two processes may both take place during the same quarter of a turn4 ,but in order to equalize the resistance to the moving power1 the same expedient is here adopted as that already described in the process of carrying4 ,the arms which shoot the bolts of each row of dials are arranged spirally1 so as to act successively throughout the quarter of a turn4 ,there is1 however1 a contingency which1 under certain circumstances1 would here produce a difficulty which must be provided against4 ,it is possible1 and in fact does sometimes happen1 that the process of carrying causes a dial to move under the index from #j to #a4 ,in that case1 the bolt1 preparatory to the next addition1 ought not to be shot until after the carriage takes place2 for if the arm which shoots it passes its point of action before the carriage takes place1 the bolt will be moved out of its sphere of action1 and will not be shot1 which1 as we have already explained1 must always happen when #j is at the index3 therefore no addition would in this case take place during the next quarter of a turn of the axis2 whereas1 since #a is brought to the index by the carriage1 which immediately succeeds the passage of the arm which ought to bolt1 #a should be added during the next quarter of a turn4 ,it is plain1 accordingly1 that the mechanism should be so arranged1 that the action of the arms1 which shoot the bolts successively1 should immediately follow the action of those fingers which raise the carrying claws successively2 and therefore either a separate quarter of a turn should be appropriated to each of those movements1 or if they be executed in the same quarter of a turn1 the mechanism must be so constructed1 that the arms which shoot the bolts successively1 shall severally follow immediately after those which raise the carrying claws successively4 ,the latter object is attained by a mechanical arrangement of singular felicity1 and partaking of that elegance which characterises all the details of this mechanism4 ,both sets of arms are spirally arranged on their respective axes1 so as to be carried through their period in the same quarter of a turn2 but the one spiral is shifted a few degrees1 in angular position1 behind the other1 so that each pair of corresponding arms succeed each other in the most regular order1--equalizing the resistance1 economizing time1 harmonizing the mechanism1 and giving to the whole mechanical action the utmost practical perfection4 ,the system of mechanical contrivances by which the results1 here attempted to be described1 are attained1 form only one order of expedients adopted in this machinery2--although such is the perfection of their action1 that in any ordinary case they would be regarded as having attained the ends in view with an almost superfluous degree of precision4 ,considering1 however1 the immense importance of the purposes which the mechanism was destined to fulfil1 its inventor determined that a higher order of expedients should be superinduced upon those already described2 the purpose of which should be to obliterate all small errors or inequalities which might1 even by remote possibility1 arise1 either from defects in the original formation of the mechanism1 from inequality of wear1 from casual strain or derangement1--or1 in short1 from any other cause whatever4 ,thus the movements of the first and principal parts of the mechanism were regarded by him merely as a first1 though extremely nice approximation1 upon which a system of small corrections was to be subsequently made by suitable and independent mechanism4 ,this supplementary system of mechanism is so contrived1 that if one or more of the moving parts of the mechanism of the first order be slightly out of their places1 they will be forced to their exact position by the action of the mechanical expedients of the second order to which we now allude4 ,if a more considerable derangement were produced by any accidental disturbance1 the consequence would be that the supplementary mechanism would cause the whole system to become locked1 so that not a wheel would be capable of moving2 the impelling power would necessarily lose all its energy1 and the machine would stop4 ,the consequence of this exquisite arrangement is1 that the machine will either calculate rightly1 or not at all4 ,the supernumerary contrivances which we now allude to1 being in a great degree unconnected with each other1 and scattered through the machinery to a certain extent1 independent of the mechanical arrangement of the principal parts1 we find it difficult to convey any distinct notion of their nature or form4 ,in some instances they consist of a roller resting between certain curved surfaces1 which has but one position of stable equilibrium1 and that position the same1 however the roller or the curved surfaces may wear4 ,a slight error in the motion of the principal parts would make this roller for the moment rest on one of the curves2 but1 being constantly urged by a spring1 it would press on the curved surface in such a manner as to force the moving piece on which that curved surface is formed1 into such a position that the roller may rest between the two surfaces2 that position being the one which the mechanism should have4 ,a greater derangement would bring the roller to the crest of the curve1 on which it would rest in instable equilibrium2 and the machine would either become locked1 or the roller would throw it as before into its true position4 ,in other instances a similar object is attained by a solid cone being pressed into a conical seat2 the position of the axis of the cone and that of its seat being necessarily invariable1 however the cone may wear3 and the action of the cone upon the seat being such1 that it cannot rest in any position except that in which the axis of the cone coincides with the axis of its seat4 ,having thus attempted to convey a notion1 however inadequate1 of the calculating section of the machinery1 we shall proceed to offer some explanation of the means whereby it is enabled1 to print its calculations in such a manner as to preclude the possibility of error in any individual printed copy4 ,on the axle of each of the wheels which express the calculated number of the table ,t1 there is fixed a solid piece of metal1 formed into a curve1 not unlike the wheel in a common clock1 which is called the snail4 ,this curved surface acts against the arm of a lever1 so as to raise that arm to a higher or lower point according to the position of the dial with which the snail is connected4 ,without entering into a more minute description1 it will be easily understood that the snail may be so formed that the arm of the lever shall be raised to ten different elevations1 corresponding to the ten figures of the dial which may be brought under the index4 ,the opposite arm of the lever here described puts in motion a solid arch1 or sector1 which carries ten punches3 each punch bearing on its face a raised character of a figure1 and the ten punches bearing the ten characters1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i1 #j4 ,it will be apparent from what has been just stated1 that this type sector 7as it is called7 will receive ten different attitudes1 corresponding to the ten figures which may successively be brought under the index of the dial-plate4 ,at a point over which the type sector is thus moved1 and immediately under a point through which it plays1 is placed a frame1 in which is fixed a plate of copper4 ,immediately over a certain point through which the type sector moves1 is likewise placed a bent lever1 which1 being straightened1 is forcibly pressed upon the punch which has been brought under it4 ,if the type sector be moved1 so as to bring under the bent lever one of the steel punches above mentioned1 and be held in that position for a certain time1 the bent lever1 being straightened1 acts upon the steel punch1 and drives it against the face of the copper beneath1 and thus causes a sunken impression of the character upon the punch to be left upon the copper4 ,if the copper be now shifted slightly in its position1 and the type sector be also shifted so as to bring another punch under the bent lever1 another character may be engraved on the copper by straightening the bent lever1 and pressing it on the punch as before4 ,it will be evident1 that if the copper was shifted from right to left through a space equal to two figures of a number1 and1 at the same time1 the type sector so shifted as to bring the punches corresponding to the figures of the number successively under the bent lever1 an engraved impression of the number might thus be obtained upon the copper by the continued action of the bent lever4 ,if1 when one line of figures is thus obtained1 a provision be made to shift the copper in a direction at right angles to its former motion1 through a space equal to the distance between two lines of figures1 and at the same time to shift it through a space in the other direction equal to the length of an entire line1 it will be evident that another line of figures might be printed below the first in the same manner4 ,the motion of the type sector1 here described1 is accomplished by the action of the snail upon the lever already mentioned4 ,in the case where the number calculated is that expressed in fig4 #a1 the process would be as follows3--,the snail of the wheel ,f#a1 acting upon the lever1 would throw the type sector into such an attitude1 that the punch bearing the character #j would come under the bent lever4 ,the next turn of the moving axis would cause the bent lever to press on the tail of the punch1 and the character #j would be impressed upon the copper4 ,the bent lever being again drawn up1 the punch would recoil from the copper by the action of a spring2 the next turn of the moving axis would shift the copper through the interval between two figures1 so as to bring the point destined to be impressed with the next figure under the bent lever4 ,at the same time1 the snail of the wheel ,e would cause the type sector to be thrown into the same attitude as before1 and the punch would be brought under the bent lever2 the next turn would impress the figure beside the former one1 as before described4 ,the snail upon the wheel ,d would now come into action1 and throw the type sector into that position in which the punch bearing the character #g would come under the bent lever1 and at the same time the copper would be shifted through the interval between two figures2 the straightening of the lever would next follow1 and the character #g would be engraved4 ,in the same manner1 the wheels ,c1 ,b1 and ,a would successively act by means of their snails2 and the copper being shifted1 and the lever allowed to act1 the number #jjgggf would be finally engraved upon the copper3 this being accomplished1 the calculating machinery would next be called into action1 and another calculation would be made1 producing the next number of the ,table exhibited in fig4 #e4 ,during this process the machinery would be engaged in shifting the copper both in the direction of its length and its breadth1 with a view to commence the printing of another line2 and this change of position would be accomplished at the moment when the next calculation would be completed3 the printing of the next number would go on like the former1 and the operation of the machine would proceed in the same manner1 calculating and printing alternately4 ,it is not1 however1 at all necessary--though we have here supposed it1 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 operation1 or vice versa2 it is not intended1 in fact1 to be so suspended in the actual machinery4 ,the same turn of the axis by which one number is printed1 executes a part of the movements necessary for the succeeding calculation2 so that the whole mechanism will be simultaneously and continuously in action4 ,of the mechanism by which the position of the copper is shifted from figure to figure1 from line to line1 we shall not attempt any description4 ,we feel that it would be quite vain4 ,complicated and difficult to describe as every other part of this machinery is1 the mechanism for moving the copper is such as it would be quite impossible to render at all intelligible1 without numerous illustrative drawings4 ,the engraved plate of copper obtained in the manner above described1 is designed to be used as a mould from which a stereotyped plate may be cast2 or1 if deemed advisable1 it may be used as the immediate means of printing4 ,in the one case we should produce a table1 printed from type1 in the same manner as common letter-press printing2 in the other an engraved table4 ,if it be thought most advisable to print from the stereotyped plates1 then as many stereotyped plates as may be required may be taken from the copper mould2 so that when once a table has been calculated and engraved by the machinery1 the whole world may be supplied with stereotyped plates to print it1 and may continue to be so supplied for an unlimited period of time4 ,there is no practical limit to the number of stereotyped plates which may be taken from the engraved copper2 and there is scarcely any limit to the number of printed copies which may be taken from any single stereotyped plate4 ,not only1 therefore1 is the numerical table by these means engraved and stereotyped with infallible accuracy1 but such stereotyped plates are producible in unbounded quantity4 ,each plate1 when produced1 becomes itself the means of producing printed copies of the table1 in accuracy perfect1 and in number without limit4 ,unlike all other machinery1 the calculating mechanism produces1 not the object of consumption1 but the machinery by which that object may be made4 ,to say that it computes and prints with infallible accuracy1 is to understate its merits3--it computes and fabricates the means of printing with absolute correctness and in unlimited abundance4 ,for the sake of clearness1 and to render ourselves more easily intelligible to the general reader1 we have in the preceding explanation thrown the mechanism into an arrangement somewhat different from that which is really adopted4 ,the dials expressing the numbers of the tables of the successive differences are not placed1 as we have supposed them1 in horizontal rows1 and read from right to left1 in the ordinary way2 they are1 on the contrary1 placed vertically1 one below the other1 and read from top to bottom4 ,the number of the table occupies the first vertical column on the right1 the units being expressed on the lowest dial1 and the tens on the next above that1 and so on4 ,the first difference occupies the next vertical column on the left2 and the numbers of the succeeding differences occupy vertical columns1 proceeding regularly to the left2 the constant difference being on the last vertical column4 ,it is intended in the machine now in progress to introduce six orders of differences1 so that there will be seven columns of dials2 it is also intended that the calculations shall extend to eighteen places of figures3 thus each column will have eighteen dials4 ,we have referred to the dials as if they were inscribed upon the faces of wheels1 whose axes are horizontal and planes vertical4 ,in the actual machinery the axes are vertical and the planes horizontal1 so that the edges of the figure wheels1 as they are called1 are presented to the eye4 ,the figures are inscribed1 not upon the dial-plate1 but around the surface of a small cylinder or barrel1 placed upon the axis of the figure wheel1 which revolves with it2 so that as the figure wheel revolves1 the figures on the barrel are successively brought to the front1 and pass under an index engraved upon a plate of metal immediately above the barrel4 ,this arrangement has the obvious practical advantage1 that1 instead of each figure wheel having a separate axis1 all the figure wheels of the same vertical column revolve on the same axis2 and the same observation will apply to all the wheels with which the figure wheels are in mechanical connexion4 ,this arrangement has the further mechanical advantage over that which has been assumed for the purposes of explanation1 that the friction of the wheel-work on the axes is less in amount1 and more uniformly distributed1 than it could be if the axes were placed in the horizontal position4 ,a notion may therefore be formed of the front elevation of the calculating part of the mechanism1 by conceiving seven steel axes erected1 one beside another1 on each of which shall be placed eighteen wheels1#ab five inches in diameter1 having cylinders or barrels upon them an inch and a half in height1 and inscribed1 as already stated1 with the ten arithmetical characters4 ,the entire elevation of the machinery would occupy a space measuring ten feet broad1 ten feet high1 and five feet deep4 ,the process of calculation would be observed by the alternate motion of the figure wheels on the several axes4 ,during the first quarter of a turn1 the wheels on the first1 third1 and fifth axes would turn1 receiving their addition from the second1 fourth1 and sixth2 during the second quarter of a turn1 such of the wheels on the first1 third1 and fifth axes1 to which carriages are due1 would be moved forward one additional figure2 the second1 fourth1 and sixth columns of wheels being all this time quiescent4 ,during the third quarter of a turn1 the second1 fourth1 and sixth columns would be observed to move1 receiving their additions from the third1 fifth1 and seventh axes2 and during the fourth quarter of a turn1 such of these wheels to which carriages are due1 would be observed to move forward one additional figure2 the wheels of the first1 third1 and fifth columns being quiescent during this time4 ,footnote #ab3 ,the wheels1 and every other part of the mechanism except the axes1 springs1 and such parts as are necessarily of steel1 are formed of an alloy of copper with a small portion of tin4 ,it will be observed that the wheels of the seventh column are always quiescent in this process2 and it may be asked1 of what use they are1 and whether some mechanism of a fixed nature would not serve the same purpose8 ,it must1 however1 be remembered1 that for different tables there will be different constant differences2 and that when the calculation of a table is about to commence1 the wheels on the seventh axis must be moved by the hand1 so as to express the constant difference1 whatever it may be4 ,in tables1 also1 which have not a difference rigorously constant1 it will be necessary1 after a certain number of calculations1 to change the constant difference by the hand2 and in this case the wheels of the seventh axis must be moved when occasion requires4 ,such adjustment1 however1 will only be necessary at very distant intervals1 and after a considerable extent of printing and calculation has taken place2 and when it is necessary1 a provision is made in the machinery by which notice will be given by the sounding of a bell1 so that the machine may not run beyond the extent of its powers of calculation4 ,immediately behind the seven axes on which the figure wheels revolve1 are seven other axes2 on which are placed1 first1 the wheels already described as driven by the figure wheels1 and which bear upon them the wedge which withdraws the bolt immediately over these latter wheels1 and on the same axis is placed the adding bolt4 ,from the bottom of this bolt there projects downwards the pin1 which acts upon the unbolting wedge by which the bolt is withdrawn3 from the upper surface of the bolt proceeds a tooth1 which1 when the bolt is shot1 enters between the teeth of the adding wheel1 which turns on the same axis1 and is placed immediately above the bolt3 its teeth1 on which the bolt acts1 are like the teeth of a crown wheel1 and are presented downwards4 ,the bolt is fixed upon this axis1 and turns with it2 but the adding wheel above the bolt1 and the unbolting wheel below it1 both turn upon the axis1 and independently of it4 ,when the axis is made to revolve by the moving power1 the bolt revolves with it2 and so long as the tooth of the bolt remains inserted between those of the adding wheel1 the latter is likewise moved2 but when the lower pin of the bolt encounters the unbolting wedge on the lower wheel1 the tooth of the bolt is withdrawn1 and the motion of the adding wheel is stopped4 ,this adding wheel is furnished with spur teeth1 besides the crown teeth just mentioned2 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 made4 ,by such an arrangement it is evident that the revolution of the bolt will necessarily add to the adjacent figure wheel the requisite number4 ,it will be perceived1 that upon the same axis are placed an unbolting wheel1 a bolt1 and an adding wheel1 one above the other1 for every figure wheel2 and as there are eighteen figure wheels there will be eighteen tiers2 each tier formed of an unbolting wheel1 a bolt1 and an adding wheel1 placed one above the other2 the wheels on this axis all revolving independent of the axis1 but the bolts being all fixed upon it4 ,the same observations1 of course1 will apply to each of the seven axes4 ,at the commencement of every revolution of the adding axes1 it is evident that the several bolts placed upon them must be shot in order to perform the various additions4 ,this is accomplished by a third set of seven axes1 placed at some distance behind the range of the wheels1 which turn upon the adding axes3 these are called bolting axes4 ,on these bolting axes are fixed1 so as to revolve with them1 a bolting finger opposite to each bolt2 as the bolting axis is made to revolve by the moving power1 the bolting finger is turned1 and as it passes near the bolt1 it encounters the shoulder of a hammer or lever1 which strikes the heel of the bolt1 and presses it forward so as to shoot its tooth between the crown teeth of the adding wheel4 ,the only exception to this action is the case in which #j happens to be at the index of the figure wheel2 in that case1 the lever or hammer1 which the bolting finger would encounter1 is1 as before stated1 lifted out of the way of the bolting finger1 so that it revolves without encountering it4 ,it is on the bolting axes that the fingers are spirally arranged so as to equalize their action1 as already explained4 ,the same axes in the front of the machinery on which the figure wheels turn1 are made to serve the purpose of carrying4 ,each of these bear a series of fingers which turn with them1 and which encounter a carrying claw1 already described1 so as to make the carriage3 these carrying fingers are also spirally arranged on their axes1 as already described4 ,although the absolute accuracy which appears to be ensured by the mechanical arrangements here described is such as to render further precautions nearly superfluous1 still it may be right to state1 that1 supposing it were possible for an error to be produced in calculation1 this error could be easily and speedily detected in the printed tables3 it would only be necessary to calculate a number of the table taken at intervals1 through which the mechanical action of the machine has not been suspended1 and during which it has received no adjustment by the hand3 if the computed number be found to agree with those printed1 it may be taken for granted that all the intermediate numbers are correct2 because1 from the nature of the mechanism1 and the principle of computation1 an error occurring in any single number of the table would be unavoidably entailed1 in an increasing ratio1 upon all the succeeding numbers4 ,we have hitherto spoken merely of the practicability of executing by the machinery1 when completed1 that which its inventor originally contemplated--namely1 the calculating and printing of all numerical tables1 derived by the method of differences from a constant difference4 ,it has1 however1 happened that the actual powers of the machinery greatly transcend those contemplated in its original design3--they not only have exceeded the most sanguine anticipations of its inventor1 but they appear to have an extent to which it is utterly impossible1 even for the most acute mathematical thinker1 to fix a probable limit4 ,certain subsidiary mechanical inventions have1 in the progress of the enterprise1 been1 by the very nature of the machinery1 suggested to the mind of the inventor1 which confer upon it capabilities which he had never foreseen4 ,it would be impossible even to enumerate1 within the limits of this article1 much less to describe in detail1 those extraordinary mechanical arrangements1 the effects of which have not failed to strike with astonishment every one who has been favoured with an opportunity of witnessing them1 and who has been enabled1 by sufficient mathematical attainments1 in any degree to estimate their probable consequences4 ,as we have described the mechanism1 the axes containing the several differences are successively and regularly added one to another2 but there are certain mechanical adjustments1 and these of a very simple nature1 which being thrown into action1 will cause a difference of any order to be added any number of times to a difference of any other order2 and that either proceeding backwards or forwards1 from a difference of an inferior to one of a superior order1 and vice versa4#ac ,footnote #ac3 ,the machine was constructed with the intention of tabulating the equation ,delta#gu #j1 but1 by the means above alluded to1 it is capable of tabulating such equations as the following3 ,delta#gu a ,delta u1 ,delta#gu a,delta#cu1 ,delta#gu units figure of ,delta u4 ,among other peculiar mechanical provisions in the machinery is one by which1 when the table for any order of difference amounts to a certain number1 a certain arithmetical change would be made in the constant difference4 ,in this way a series may be tabulated by the machine1 in which the constant difference is subject to periodical change2 or the very nature of the table itself may be subject to periodical change1 and yet to one which has a regular law4 ,some of these subsidiary powers are peculiarly applicable to calculations required in astronomy1 and are therefore of eminent and immediate practical utility3 others there are by which tables are produced1 following the most extraordinary1 and apparently capricious1 but still regular laws4 ,thus a table will be computed1 which1 to any required extent1 shall coincide with a given table1 and which shall deviate from that table for a single term1 or for any required number of terms1 and then resume its course1 or which shall permanently alter the law of its construction4 ,thus the engine has calculated a table which agreed precisely with a table of square numbers1 until it attained the hundred and first term1 which was not the square of #aja1 nor were any of the subsequent numbers squares4 ,again1 it has computed a table which coincided with the series of natural numbers1 as far as #ajj1jjj1jja1 but which subsequently followed another law4 ,this result was obtained1 not by working the engine through the whole of the first table1 for that would have required an enormous length of time2 but by showing1 from the arrangement of the mechanism1 that it must continue to exhibit the succession of natural numbers1 until it would reach #ajj1jjj1jjj4 ,to save time1 the engine was set by the hand to the number #iiiiiiie1 and was then put in regular operation4 ,it produced successively the following numbers4#ad #ii1iii1iif #ii1iii1iig #ii1iii1iih #ii1iii1iii #ajj1jjj1jjj #ajj1jaj1jjb #ajj1jcj1jjc #ajj1jfj1jjd #ajj1ajj1jje #ajj1aej1jjf c4 c4 ,footnote #ad3 ,such results as this suggest a train of reflection on the nature and operation of general laws1 which would lead to very curious and interesting speculations4 ,the natural philosopher and astronomer will be hardly less struck with them than the metaphysician and theologian4 ,equations have been already tabulated by the portion of the machinery which has been put together1 which are so far beyond the reach of the present power of mathematics1 that no distant term of the table can be predicted1 nor any function discovered capable of expressing its general law4 ,yet the very fact of the table being produced by mechanism of an invariable form1 and including a distinct principle of mechanical action1 renders it quite manifest that some general law must exist in every table which it produces4 ,but we must dismiss these speculations3 we feel it impossible to stretch the powers of our own mind1 so as to grasp the probable capabilities of this splendid production of combined mechanical and mathematical genius2 much less can we hope to enable others to appreciate them1 without being furnished with such means of comprehending them as those with which we have been favoured4 ,years must in fact elapse1 and many enquirers direct their energies to the cultivation of the vast field of research thus opened1 before we can fully estimate the extent of this triumph of matter over mind4 ',nor is it1' says ,mr ,colebrooke1 'among the least curious results of this ingenious device1 that it affords a new opening for discovery1 since it is applicable1 as has been shown by its inventor1 to surmount novel difficulties of analysis4 ,not confined to constant differences1 it is available in every case of differences that follow a definite law1 reducible therefore to an equation4 ,an engine adjusted to the purpose being set to work1 will produce any distant term1 or succession of terms1 required--thus presenting the numerical solution of a problem1 even though the analytical solution be yet undetermined4' ,that the future path of some important branches of mathematical enquiry must now in some measure be directed by the dictates of mechanism1 is sufficiently evident2 for who would toil on in any course of analytical enquiry1 in which he must ultimately depend on the expensive and fallible aid of human arithmetic1 with an instrument in his hands1 in which all the dull monotony of numerical computation is turned over to the untiring action and unerring certainty of mechanical agency8 ,it is worth notice1 that each of the axes in front of the machinery on which the figure wheels revolve1 is connected with a bell1 the tongue of which is governed by a system of levers1 moved by the several figure wheels2 an adjustment is provided by which the levers shall be dismissed1 so as to allow the hammer to strike against the bell1 whenever any proposed number shall be exhibited on the axis4 ,this contrivance enables the machine to give notice to its attendants at any time that an adjustment may be required4 ,among a great variety of curious accidental properties 7so to speak7 which the machine is found to possess1 is one by which it is capable of solving numerical equations which have rational roots4 ,such an equation being reduced 7as it always may be7 by suitable transformations to that state in which the roots shall be whole numbers1 the values #j1 #a1 #b1 #c1 c41 are substituted for the unknown quantity1 and the corresponding values of the equation ascertained4 ,from these a sufficient number of differences being derived1 they are set upon the machine4 ,the machine being then put in motion1 the table axis will exhibit the successive values of the formula1 corresponding to the substitutions of the successive whole numbers for the unknown quantity3 at length the number exhibited on the table axis will be #j1 which will evidently correspond to a root of the equation4 ,by previous adjustment1 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 machinery4 ,if the equation have imaginary roots1 the formula being necessarily a maximum or minimum on the occurrence of such roots1 the first difference will become nothing2 and the dials of that axis will under such circumstances present to the respective indices4 ,by previous adjustment1 the bell of this axis would here give notice of a pair of imaginary roots4 ,mr ,colebrooke speculates on the probable extension of these powers of the machine3 ',it may not therefore be deemed too sanguine an anticipation when ,i express the hope that an instrument which1 in its simpler form1 attains to the extraction of roots of numbers1 and approximates to the roots of equations1 may1 in a more advanced state of improvement1 rise to the approximate solution of algebraic equations of elevated degrees4 ,i refer to solutions of such equations proposed by ,la ,grange1 and more recently by other annalists1 which involve operations too tedious and intricate for use1 and which must remain without efficacy1 unless some mode be devised of abridging the labour1 or facilitating the means of its performance4 ,in any case this engine tends to lighten the excessive and accumulating burden of arithmetical application of mathematical formul1 and to relieve the progress of science from what is justly termed by the author of this invention1 the overwhelming encumbrance of numerical detail4' ,although there are not more than eighteen figure wheels on each axis1 and therefore it might be supposed that the machinery was capable of calculating only to the extent of eighteen decimal places2 yet there are contrivances connected with it1 by which1 in two successive calculations1 it will be possible to calculate even to the extent of thirty decimal places4 ,its powers1 therefore1 in this respect1 greatly exceed any which can be required in practical science4 ,it is also remarkable1 that the machinery is capable of producing the calculated results true to the last figure4 ,we have already explained1 that when the figure which would follow the last is greater than #d1 then it would be necessary to increase the last figure by #a2 since the excess of the calculated number above the true value would in such case be less than its defect from it would be1 had the regularly computed final figure been adopted3 this is a precaution necessary in all numerical tables1 and it is one which would hardly have been expected to be provided for in the calculating machinery4 ,as might be expected in a mechanical undertaking of such complexity and novelty1 many practical difficulties have since its commencement been encountered and surmounted4 ,it might have been foreseen1 that many expedients would be adopted and carried into effect1 which farther experiments would render it necessary to reject2 and thus a large source of additional expense could scarcely fail to be produced4 ,to a certain extent this has taken place2 but owing to the admirable system of mechanical drawings1 which in every instance ,mr ,babbage has caused to be made1 and owing to his own profound acquaintance with the practical working of the most complicated mechanism1 he has been able to predict in every case what the result of any contrivance would be1 as perfectly from the drawing1 as if it had been reduced to the form of a working model4 ,the drawings1 consequently1 form a most extensive and essential part of the enterprise4 ,they are executed with extraordinary ability and precision1 and may be considered as perhaps the best specimens of mechanical drawings which have ever been executed4 ,it has been on these1 and on these only1 that the work of invention has been bestowed4 ,in these1 all those progressive modifications suggested by consideration and study have been made2 and it was not until the inventor was fully satisfied with the result of any contrivance1 that he had it reduced to a working form4 ,the whole of the loss which has been incurred by the necessarily progressive course of invention1 has been the expense of rejected drawings4 ,nothing can perhaps more forcibly illustrate the extent of labour and thought which has been incurred in the production of this machinery1 than the contemplation of the working drawings which have been executed previously to its construction3 these drawings cover above a thousand square feet of surface1 and many of them are of the most elaborate and complicated description4 ,one of the practical difficulties which presented themselves at a very early stage in the progress of this undertaking1 was the impossibility of bearing in mind all the variety of motions propagated simultaneously through so many complicated trains of mechanism4 ,nothing but the utmost imaginable harmony and order among such a number of movements1 could prevent obstructions arising from incompatible motions encountering each other4 ,it was very soon found impossible1 by a mere act of memory1 to guard against such an occurrence2 and ,mr ,babbage found1 that1 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 time1 such inconsistencies and obstructions as are here alluded to must continually have occurred4 ,this difficulty was removed by another invention of even a more general nature than the calculating machinery itself1 and pregnant with results probably of higher importance4 ,this invention consisted in the contrivance of a scheme of mechanical notation which is generally applicable to all machinery whatsoever2 and which is exhibited on a table or plan consisting of two distinct sections4 ,in the first is traced1 by a peculiar system of signs1 the origin of every motion which takes place throughout the machinery2 so that the mechanist or inventor is able1 by moving his finger along a certain line1 to follow out the motion of every piece from effect to cause1 until he arrives at the prime mover4 ,the same sign which thus indicates the source of motion indicates likewise the species of motion1 whether it be continuous or reciprocating1 circular or progressive1 c4 ,the same system of signs further indicates the nature of the mechanical connexion between the mover and the thing moved1 whether it be permanent and invariable 7as between the two arms of a lever71 or whether the mover and the moved are separate and independent pieces1 as is the case when a pinion drives a wheel2 also whether the motion of one piece necessarily implies the motion of another2 or when such motion in the one is interrupted1 and in the other continuous1 c4 ,the second section of the table divides the time of a complete period of the machinery into any required number of parts2 and it exhibits in a map1 as it were1 that which every part of the machine is doing at each moment of time4 ,in this way1 incompatibility in the motions of different parts is rendered perceptible at a glance4 ,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 parts2 but when he finds that the introduction of any particular movement is necessary for his purpose1 he can easily and rapidly examine the whole range of the machinery during one of its periods1 and can find by inspection whether there is any1 and what portion of time1 at which no motion exists incompatible with the desired one1 and thus discover a niche1 as it were1 in which to place the required movement4 ,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 power4 ,for example1 without some instrument of mechanical enquiry equally powerful with that now described1 it would be scarcely possible1 at least in the first instance1 so to arrange the various movements that they should be all executed in the least possible number of revolutions of the moving axis4 ,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 revolutions3 and there is no one acquainted with the history of mechanical invention who must not be aware1 that in the progressive contrivance of almost every machine the earliest arrangements are invariably defective in this respect2 and that it is only by a succession of improvements1 suggested by long experience1 that that arrangement is at length arrived at1 which accomplishes all the necessary motions in the shortest possible time4 ,by the application of the mechanical notation1 however1 absolute perfection may be arrived at in this respect2 even before a single part of the machinery is constructed1 and before it has any other existence than that which it obtains upon paper4 ,examples of this class of advantages derivable from the notation will occur to the mind of every one acquainted with the history of mechanical invention4 ,in the common suction-pump1 for example1 the effective agency of the power is suspended during the descent of the piston4 ,a very simple contrivance1 however1 will transfer to the descent the work to be accomplished in the next ascent2 so that the duty of four strokes of the piston may thus be executed in the time of two4 ,in the earlier applications of the steam-engine1 that machine was applied almost exclusively to the process of pumping2 and the power acted only during the descent of the piston1 being suspended during its ascent4 ,when1 however1 the notion of applying the engine to the general purposes of manufacture occurred to the mind of ,watt1 he saw that it would be necessary to cause it to produce a continued rotatory motion2 and1 therefore1 that the intervals of intermission must be filled up by the action of the power4 ,he first proposed to accomplish this by a second cylinder working alternately with the first2 but it soon became apparent that the blank which existed during the upstroke in the action of the power1 might be filled up by introducing the steam at both ends of the cylinder alternately4 ,had ,watt placed before him a scheme of mechanical notation such as we allude to1 this expedient would have been so obtruded upon him that he must have adopted it from the first4 ,one of the circumstances from which the mechanical notation derives a great portion of its power as an instrument of investigation and discovery1 is that it enables the inventor to dismiss from his thoughts1 and to disencumber his imagination of the arrangement and connexion of the mechanism2 which1 when it is very complex 7and it is in that case that the notation is most useful71 can only be kept before the mind by an embarrassing and painful effort4 ,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 algebra4 ,when once the peculiar conditions of the question are translated into algebraical signs1 and 'reduced to an equation1' the computist dismisses from his thoughts all the circumstances of the question1 and is relieved from the consideration of the complicated relations of the quantities of various kinds which may have entered it4 ,he deals with the algebraical symbols1 which are the representatives of those quantities and relations1 according to certain technical rules of a general nature1 the truth of which he has previously established2 and1 by a process almost mechanical1 he arrives at the required result4 ,what algebra is to arithmetic1 the notation we now allude to is to mechanism4 ,the various parts of the machinery under consideration being once expressed upon paper by proper symbols1 the enquirer dismisses altogether from his thoughts the mechanism itself1 and attends only to the symbols2 the management of which is so extremely simple and obvious1 that the most unpractised person1 having once acquired an acquaintance with the signs1 cannot fail to comprehend their use4 ,a remarkable instance of the power and utility of this notation occurred in a certain stage of the invention of the calculating machinery4 ,a question arose as to the best method of producing and arranging a certain series of motions necessary to print and calculate a number4 ,the inventor1 assisted by a practical engineer of considerable experience and skill1 had so arranged these motions1 that the whole might be performed by twelve revolutions of the principal moving axis4 ,it seemed1 however1 desirable1 if possible1 to execute these motions by a less number of revolutions4 ,to accomplish this1 the engineer sat down to study the complicated details of a part of the machinery which had been put together2 the inventor at the same time applied himself to the consideration of the arrangement and connexion of the symbols in his scheme of notation4 ,after a short time1 by some transposition of symbols1 he caused the received motions to be completed by eight turns of the axis4 ,this he accomplished by transferring the symbols which occupied the last four divisions of his scheme1 into such blank spaces as he could discover in the first eight divisions2 due care being taken that no symbols should express actions at once simultaneous and incompatible4 ,pushing his enquiry1 however1 still further1 he proceeded to ascertain whether his scheme of symbols did not admit of a still more compact arrangement1 and whether eight revolutions were not more than enough to accomplish what was required4 ,here the powers of the practical engineer completely broke down4 ,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 arrangement4 ,the inventor1 however1 without any extraordinary mental exertion1 and merely by sliding a bit of ruled pasteboard up and down his plan1 in search of a vacancy where the different motions might be placed1 at length contrived to pack all the motions1 which had previously occupied eight turns of the handle1 into five turns4 ,the symbolic instrument with which he conducted the investigation1 now informed him of the impossibility of reducing the action of the machine to a more condensed form4 ,this appeared by the fulness of every space along the lines of compatible action4 ,it was1 however1 still possible1 by going back to the actual machinery1 to ascertain whether movements1 which1 under existing arrangements1 were incompatible1 might not be brought into harmony4 ,this he accordingly did1 and succeeded in diminishing the number of incompatible conditions1 and thereby rendered it possible to make actions simultaneous which were before necessarily successive4 ,the notation was now again called into requisition1 and a new disposition of the parts was made4 ,at this point of the investigation1 this extraordinary instrument of mechanical analysis put forth one of its most singular exertions of power4 ,it presented to the eye of the engineer two currents of mechanical action1 which1 from their nature1 could not be simultaneous2 and each of which occupied a complete revolution of the axis1 except about a twentieth2 the one occupying the last nineteen-twentieths of a complete revolution of the axis1 and the other occupying the first nineteen-twentieths of a complete revolution4 ,one of these streams of action was1 the successive picking up by the carrying fingers of the successive carrying claws2 and the other was1 the successive shooting of nineteen bolts by the nineteen bolting fingers4 ,the notation rendered it obvious1 that as the bolting action commenced a small space below the commencement of the carrying1 and ended an equal space below the termination of the carrying1 the two streams of action could be made to flow after one another in one and the same revolution of the axis4 ,he thus succeeded in reducing the period of completing the action to four turns of the axis2 when the notation again informed him that he had again attained a limit of condensed action1 which could not be exceeded without a further change in the mechanism4 ,to the mechanism he again recurred1 and soon found that it was possible to introduce a change which would cause the action to be completed in three revolutions of the axis4 ,an odd number of revolutions1 however1 being attended with certain practical inconveniences1 it was considered more advantageous to execute the motions in four turns2 and here again the notation put forth its powers1 by informing the inventor1 through the eye1 almost independent of his mind1 what would be the most elegant1 symmetrical1 and harmonious disposition of the required motions in four turns4 ,this application of an almost metaphysical system of abstract signs1 by which the motion of the hand performs the office of the mind1 and of profound practical skill in mechanics alternately1 to the construction of a most complicated engine1 forcibly reminds us of a parallel in another science1 where the chemist with difficulty succeeds in dissolving a refractory mineral1 by the alternate action of the most powerful acids1 and the most caustic alkalies1 repeated in long-continued succession4 ,this important discovery was explained by ,mr ,babbage1 in a short paper read before the ,royal ,society1 and published in the ,philosophical ,transactions in #ahbf4ae ,it is to us more a matter of regret than surprise1 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 it4 ,to appreciate it would indeed have been scarcely possible1 from the very brief memoir which its inventor presented1 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 occupations4 ,in this country1 science has been generally separated from practical mechanics by a wide chasm4 ,it will be easily admitted1 that an assembly of eminent naturalists and physicians1 with a sprinkling of astronomers1 and one or two abstract mathematicians1 were not precisely the persons best qualified to appreciate such an instrument of mechanical investigation as we have here described4 ,we shall not therefore be understood as intending the slightest disrespect for these distinguished persons1 when we express our regret1 that a discovery of such paramount practical value1 in a country preeminently conspicuous for the results of its machinery1 should fall still-born and inconsequential through their hands1 and be buried unhonoured and undiscriminated in their miscellaneous transactions4 ,we trust that a more auspicious period is at hand2 that the chasm which has separated practical from scientific men will speedily close2 and that that combination of knowledge will be effected1 which can only be obtained when we see the men of science more frequently extending their observant eye over the wonders of our factories1 and our great practical manufacturers1 with a reciprocal ambition1 presenting themselves as active and useful members of our scientific associations4 ,when this has taken place1 an order of scientific men will spring up1 which will render impossible an oversight so little creditable to the country as that which has been committed respecting the mechanical notation4#af ,this notation has recently undergone very considerable extension and improvement4 ,an additional section has been introduced into it2 designed to express the process of circulation in machines1 through which fluids1 whether liquid or gaseous1 are moved4 ,mr ,babbage1 with the assistance of a friend who happened to be conversant with the structure and operation of the steam-engine1 has illustrated it with singular felicity and success in its application to that machine4 ,an eminent ,french surgeon1 on seeing the scheme of notation thus applied1 immediately suggested the advantages which must attend it as an instrument for expressing the structure1 operation1 and circulation of the animal system2 and we entertain no doubt of its adequacy for that purpose4 ,not only the mechanical connexion of the solid members of the bodies of men and animals1 but likewise the structure and operation of the softer parts1 including the muscles1 integuments1 membranes1 c42 the nature1 motion1 and circulation of the various fluids1 their reciprocal effects1 the changes through which they pass1 the deposits which they leave in various parts of the system2 the functions of respiration1 digestion1 and assimilation1--all would find appropriate symbols and representatives in the notation1 even as it now stands1 without those additions of which1 however1 it is easily susceptible4 ,indeed1 when we reflect for what a very different purpose this scheme of symbols was contrived1 we cannot refrain from expressing our wonder that it should seem1 in all respects1 as if it had been designed expressly for the purposes of anatomy and physiology4 ,footnote #ae3 ,phil4 ,trans4 #ahbj1 ,part ,,iii4 p4 #bej1 on a method of expressing by signs the action of machinery4 ,footnote #af3 ,this discovery has been more justly appreciated by scientific men abroad4 ,it was1 almost immediately after its publication1 adopted as the topic of lectures1 in an institution on the ,continent for the instruction of ,civil ,engineers4 ,another of the uses which the slightest attention to the details of this notation irresistibly forces upon our notice1 is to exhibit1 in the form of a connected plan or map1 the organization of an extensive factory1 or any great public institution1 in which a vast number of individuals are employed1 and their duties regulated 7as they generally are or ought to be7 by a consistent and well-digested system4 ,the mechanical notation is admirably adapted1 not only to express such an organized connexion of human agents1 but even to suggest the improvements of which such organization is susceptible--to betray its weak and defective points1 and to disclose1 at a glance1 the origin of any fault which may1 from time to time1 be observed in the working of the system4 ,our limits1 however1 preclude us from pursuing this interesting topic to the extent which its importance would justify4 ,we shall be satisfied if the hints here thrown out should direct to the subject the attention of those who1 being most interested in such an enquiry1 are likely to prosecute it with greatest success4 ,one of the consequences which has arisen in the prosecution of the invention of the calculating machinery1 has been the discovery of a multitude of mechanical contrivances1 which have been elicited by the exigencies of the undertaking1 and which are as novel in their nature as the purposes were novel which they were designed to attain4 ,in some cases several different contrivances were devised for the attainment of the same end2 and that among them which was best suited for the purpose was finally selected3 the rejected expedients--those overflowings or waste of the invention--were not1 however1 always found useless4 ,like the waste in various manufactures1 they were soon converted to purposes of utility4 ,these rejected contrivances have found their way1 in many cases1 into the mills of our manufacturers2 and we now find them busily effecting purposes1 far different from any which the inventor dreamed of1 in the spinning-frames of ,manchester4#ag ,footnote #ag3 ,an eminent and wealthy retired manufacturer at ,manchester assured us1 that on the occasion of a visit to ,london1 when he was favoured with a view of the calculating machinery1 he found in it mechanical contrivances1 which he subsequently introduced with the greatest advantage into his own spinning-machinery4 ,another department of mechanical art1 which has been enriched by this invention1 has been that of tools4 ,the great variety of new forms which it was necessary to produce1 created the necessity of contriving and constructing a vast number of novel and most valuable tools1 by which1 with the aid of the lathe1 and that alone1 the required forms could be given to the different parts of the machinery with all the requisite accuracy4 ,the idea of calculation by mechanism is not new4 ,arithmetical instruments1 such as the calculating boards of the ancients1 on which they made their computations by the aid of counters--the ,abacus1 an instrument for computing by the aid of balls sliding upon parallel rods--the method of calculation invented by ,baron ,napier1 called by him ,rhabdology1 and since called ,napier's bones--the ,swan ,pan of the ,chinese--and other similar contrivances1 among which more particularly may be mentioned the ,sliding ,rule1 of so much use in practical calculations to modern engineers1 will occur to every reader3 these may more properly be called arithmetical instruments1 partaking more or less of a mechanical character4 ,but the earliest piece of mechanism to which the name of a 'calculating machine' can fairly be given1 appears to have been a machine invented by the celebrated ,pascal4 ,this philosopher and mathematician1 at a very early age1 being engaged with his father1 who held an official situation in ,upper ,normandy1 the duties of which required frequent numerical calculations1 contrived a piece of mechanism to facilitate the performance of them4 ,this mechanism consisted of a series of wheels1 carrying cylindrical barrels1 on which were engraved the ten arithmetical characters1 in a manner not very dissimilar to that already described4 ,the wheel which expressed each order of units was so connected with the wheel which expressed the superior order1 that when the former passed from #i to #j1 the latter was necessarily advanced one figure2 and thus the process of carrying was executed by mechanism3 when one number was to be added to another by this machine1 the addition of each figure to the other was performed by the hand2 when it was required to add more than two numbers1 the additions were performed in the same manner successively2 the second was added to the first1 the third to their sum1 and so on4 ,subtraction was reduced to addition by the method of arithmetical complements2 multiplication was performed by a succession of additions2 and division by a succession of subtractions4 ,in all cases1 however1 the operations were executed from wheel to wheel by the hand4#ah ,footnote #ah3 ,see a description of this machine by ,diderot1 in the ,encyc4 ,method42 also in the works of ,pascal1 tom4 ,,iv41 p4 #g2 ,paris1 #ahai4 ,this mechanism1 which was invented about the year #afej1 does not appear ever to have been brought into any practical use2 and seems to have speedily found its appropriate place in a museum of curiosities4 ,it was capable of performing only particular arithmetical operations1 and these subject to all the chances of error in manipulation2 attended also with little more expedition 7if so much71 as would be attained by the pen of an expert computer4 ,this attempt of ,pascal was followed by various others1 with very little improvement1 and with no additional success4 ,polenus1 a learned and ingenious ,italian1 invented a machine by which multiplication was performed1 but which does not appear to have afforded any material facilities1 nor any more security against error than the common process of the pen4 ,a similar attempt was made by ,sir ,samuel ,moreland1 who is described as having transferred to wheel-work the figures of ,napier's bones1 and as having made some additions to the machine of ,pascal4#ai ,footnote #ai3 ,equidem ,morelandus in ,anglia1 tub stentori author1 ,rhabdologiam ex baculis in cylindrulos transtulit1 et additiones auxiliares peragit in adjuncta machina additionum ,pascaliana4 ,grillet1 a ,french mechanician1 made a like attempt with as little success4 ,another contrivance for mechanical calculation was made by ,saunderson4 ,mechanical contrivances for performing particular arithmetical processes were also made about a century ago by ,deleprne and ,boitissendeau2 but they were merely modifications of ,pascal's1 without varying or extending its objects4 ,but one of the most remarkable attempts of this kind which has been made since that of ,pascal1 was a machine invented by ,leibnitz1 of which we are not aware that any detailed or intelligible description was ever published4 ,leibnitz described its mode of operation1 and its results1 in the ,berlin ,miscellany1#bj but he appears to have declined any description of its details4 ,in a letter addressed by him to ,bernoulli1 in answer to a request of the latter that he would afford a description of the machinery1 he says1 ',descriptionem ejus dare accuratam res non facilis foret4 ,de effectu ex eo judicaveris quod ad multiplicandum numerum sex figurarum1 e4g4 rotam quamdam tantum sexies gyrari necesse est1 nulla alia opera mentis1 nullis additionibus intervenientibus2 quo facto1 integrum absolutumque productum oculis objicietur4'#ba ,he goes on to say that the process of division is performed independently of a succession of subtractions1 such as that used by ,pascal4 ,footnote #bj3 ,tom4 ,i41 p4 #cag4 ,footnote #ba3 ,com4 ,epist4 tom4 ,i41 p4 #bhi4 ,it appears that this machine was one of an extremely complicated nature1 which would be attended with considerable expense of construction1 and only fit to be used in cases where numerous and expensive calculations were necessary4#bb ,leibnitz observes to his correspondent1 who required whether it might not be brought into common use1 ',non est facta pro his qui olera aut pisculos vendunt1 sed pro observatoriis aut cameris computorum1 aut aliis1 qui sumptus facile ferunt et multo calculo egent4' ,nevertheless1 it does not appear that this contrivance1 of which the inventor states that he caused two models to be made1 was ever applied to any useful purpose2 nor indeed do the mechanical details of the invention appear ever to have been published4 ,footnote #bb3 ,sed machinam esse sumptuosam et multarum rotarum instar horologii3 ,huygenius aliquoties admonuit ut absolvi curarem2 quod non sine magno sumptu tdioque factum est1 dum varie mihi cum opificibus fuit conflictandum4--,com4 ,epist4 ,even had the mechanism of these machines performed all which their inventors expected from them1 they would have been still altogether inapplicable for the purposes to which it is proposed that the calculating machinery of ,mr ,babbage shall be applied4 ,they were all constructed with a view to perform particular arithmetical operations1 and in all of them the accuracy of the result depended more or less upon manipulation4 ,the principle of the calculating machinery of ,mr ,babbage is perfectly general in its nature1 not depending on any particular arithmetical operation1 and is equally applicable to numerical tables of every kind4 ,this distinguishing characteristic was well expressed by ,mr ,colebrooke in his address to the ,astronomical ,society on this invention4 ',the principle which essentially distinguishes ,mr ,babbage's invention from all these is1 that it proposes to calculate a series of numbers following any law1 by the aid of differences1 and that by setting a few figures at the outset2 a long series of numbers is readily produced by a mechanical operation4 ,the method of differences in a very wide sense is the mathematical principle of the contrivance4 ,a machine to add a number of arbitrary figures together is no economy of time or trouble1 since each individual figure must be placed in the machine2 but it is otherwise when those figures follow some law4 ,the insertion of a few at first determines the magnitude of the next1 and those of the succeeding4 ,it is this constant repetition of similar operations which renders the computation of tables a fit subject for the application of machinery4 ,mr ,babbage's invention puts an engine in the place of the computer2 the question is set to the instrument1 or the instrument is set to the question1 and by simply giving it motion the solution is wrought1 and a string of answers is exhibited4' ,but perhaps the greatest of its advantages is1 that it prints what it calculates2 and this completely precludes the possibility of error in those numerical results which pass into the hands of the public4 ',the usefulness of the instrument1' says ,mr ,colebrooke1 'is thus more than doubled2 for it not only saves time and trouble in transcribing results into a tabular form1 and setting types for the printing of the table1 but it likewise accomplishes the yet more important object of ensuring accuracy1 obviating numerous sources of error through the careless hands of transcribers and compositors4' ,some solicitude will doubtless be felt respecting the present state of the calculating machinery1 and the probable period of its completion4 ,in the beginning of the year #ahbi1 ,government directed the ,royal ,society to institute such enquiries as would enable them to report upon the state to which it had then arrived2 and also whether the progress made in its construction confirmed them in the opinion which they had formerly expressed1--that it would ultimately prove adequate to the important object which it was intended to attain4 ,the ,royal ,society1 in accordance with these directions1 appointed a ,committee to make the necessary enquiry1 and report4 ,this ,committee consisted of ,mr ,davies ,gilbert1 then ,president1 the ,secretaries1 ,sir ,john ,herschel1 ,mr ,francis ,baily1 ,mr ,brunel1 engineer1 ,mr ,donkin1 engineer1 ,mr ,g4 ,rennie1 engineer1 ,mr ,barton1 comptroller of the ,mint1 and ,mr ,warburton1 ,m4,p4 ,the voluminous drawings1 the various tools1 and the portion of the machinery then executed1 underwent a close and elaborate examination by this ,committee1 who reported upon it to the ,society4 ,they stated in their report1 that they declined the consideration of the principle on which the practicability of the machinery depends1 and of the public utility of the object which it proposes to attain2 because they considered the former fully admitted1 and the latter obvious to all who consider the immense advantage of accurate numerical tables in all matters of calculation1 especially in those which relate to astronomy and navigation1 and the great variety and extent of those which it is professedly the object of the machinery to calculate and print with perfect accuracy2--that absolute accuracy being one of the prominent pretensions of the undertaking1 they had directed their attention especially to this point1 by careful examination of the drawings and of the work already executed1 and by repeated conferences with ,mr ,babbage on the subject2--that the result of their enquiry was1 that such precautions appeared to have been taken in every part of the contrivance1 and so fully aware was the inventor of every circumstance which might by possibility produce error1 that they had no hesitation in stating their belief that these precautions were effectual1 and that whatever the machine would do1 it would do truly4 ,they further stated1 that the progress which ,mr ,babbage had then made1 considering the very great difficulties to be overcome in an undertaking of so novel a kind1 fully equalled any expectations that could reasonably have been formed2 and that although several years had elapsed since the commencement of the undertaking1 yet when the necessity of constructing plans1 sections1 elevations1 and working drawings of every part2 of constructing1 and in many cases inventing1 tools and machinery of great expense and complexity1 necessary to form with the requisite precision parts of the apparatus differing from any which had previously been introduced in ordinary mechanical works2 of making many trials to ascertain the value of each proposed contrivance2 of altering1 improving1 and simplifying the drawings2--that1 considering all these matters1 the ,committee1 instead of feeling surprise at the time which the work has occupied1 felt more disposed to wonder at the possibility of accomplishing so much4 ,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 exposed2 of its durability1 strength1 solidity1 and equilibrium2 of the prevention of1 or compensation for1 wear by friction2 of the accuracy of the various adjustments2 and of the judgment and discretion displayed by the inventor1 in his determination to admit into the mechanism nothing but the very best and most finished workmanship2 as a contrary course would have been false economy1 and might have led to the loss of the whole capital expended on it4 ,finally1 considering all that had come before them1 and relying on the talent and skill displayed by ,mr ,babbage as a mechanist in the progress of this arduous undertaking1 not less for what remained1 than on the matured and digested plan and admirable execution of what is completed1 the ,committee did not hesitate to express their opinion1 that in the then state of the engine1 they regarded it as likely to fulfil the expectations entertained of it by its inventor4 ,this report was printed in the commencement of the year #ahbi4 ,from that time until the beginning of the year #ahcc1 the progress of the work has been slow and interrupted4 ,meanwhile many unfounded rumours have obtained circulation as to the course adopted by ,government in this undertaking2 and as to the position in which ,mr ,babbage stands with respect to it4 ,we shall here state1 upon authority on which the most perfect reliance may be placed1 what have been the actual circumstances of the arrangement which has been made1 and of the steps which have been already taken4 ,being advised that the objects of the projected machinery were of paramount national importance to a maritime country1 and that1 from its nature1 it could never be undertaken with advantage by any individual as a pecuniary speculation1 ,government determined to engage ,mr ,babbage to construct the calculating engine for the nation4 ,it was then thought that the work could be completed in two or three years2 and it was accordingly undertaken on this understanding about the year #ahba1 and since then has been in progress4 ,the execution of the workmanship was confided to an engineer by whom all the subordinate workmen were employed1 and who supplied for the work the requisite tools and other machinery2 the latter being his own property1 and not that of ,government4 ,this engineer furnished1 at intervals1 his accounts1 which were duly audited by proper persons appointed for that purpose4 ,it was thought advisable--with a view1 perhaps1 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 hands4 ,the amount was accordingly from time to time issued to him by the ,treasury1 and paid over to the engineer4 ,this circumstance has given rise to reports1 that he has received considerable sums of money as a remuneration for his skill and labour in inventing and constructing this machinery4 ,such reports are altogether destitute of truth4 ,he has received1 neither directly nor indirectly1 any remuneration whatever2--on the contrary1 owing to various official delays in the issues of money from the ,treasury for the payment of the engineer1 he has frequently been obliged to advance these payments himself1 that the work might proceed without interruption4 ,had he not been enabled to do this from his private resources1 it would have been impossible that the machinery could have arrived at its present advanced state4 ,it will be a matter of regret to every friend of science to learn1 that1 notwithstanding such assistance1 the progress of the work has been suspended1 and the workmen dismissed for more than a year and a half2 nor does there at the present moment appear to be any immediate prospect of its being resumed4 ,what the causes may be of a suspension so extraordinary1 of a project of such great national and universal interest1--in which the country has already invested a sum of such serious amount as ,l4#ae1jjj1--;is a question which will at once suggest itself to every mind2 and is one to which1 notwithstanding frequent enquiries1 in quarters from which correct information might be expected1 we have not been able to obtain any satisfactory answer4 ,it is not true1 we are assured1 that the ,government object to make the necessary payments1 or even advances1 to carry on the work4 ,it is not true1 we also are assured1 that any practical difficulty has arisen in the construction of the mechanism2--on the contrary1 the drawings of all the parts of it are completed1 and may be inspected by any person appointed on the part of ,government to examine them4#bc ,mr ,babbage is known as a man of unwearied activity1 and aspiring ambition4 ,why1 then1 it may be asked1 is it that he1 seeing his present reputation and future fame depending in so great a degree upon the successful issue of this undertaking1 has nevertheless allowed it to stand still for so long a period1 without distinctly pointing out to ,government the course which they should adopt to remove the causes of delay8 ,had he done this 7which we consider to be equally due to the nation and to himself71 he would have thrown upon ,government and its agents the whole responsibility for the delay and consequent loss2 but we believe he has not done so4 ,on the contrary1 it is said that he has of late almost withdrawn from all interference on the subject1 either with the ,government or the engineer4 ,does not ,mr ,babbage perceive the inference which the world will draw from this course of conduct8 ,does he not see that they will impute it to a distrust of his own power1 or even to a consciousness of his own inability to complete what he has begun8 ,we feel assured that such is not the case2 and we are anxious1 equally for the sake of science1 and for ,mr ,babbage's own reputation1 that the mystery--for such it must be regarded--should be cleared up2 and that all obstructions to the progress of the undertaking should immediately be removed4 ,does this supineness and apparent indifference1 so incompatible with the known character of ,mr ,babbage1 arise from any feeling of dissatisfaction at the existing arrangements between himself and the ,government8 ,if such be the actual cause of the delay1 7and we believe that1 in some degree1 it is so17 we cannot refrain from expressing our surprise that he does not adopt the candid and straightforward course of declaring the grounds of his discontent1 and explaining the arrangement which he desires to be adopted4 ,we do not hesitate to say1 that every reasonable accommodation and assistance ought to be afforded him4 ,but if he will pertinaciously abstain from this1 to our minds1 obvious and proper course1 then it is surely the duty of ,government to appoint proper persons to enquire into and report on the present state of the machinery2 to ascertain the causes of its suspension2 and to recommend such measures as may appear to be most effectual to ensure its speedy completion4 ,if they do not by such means succeed in putting the project in a state of advancement1 they will at least shift from themselves all responsibility for its suspension4 ,footnote #bc3 ,government has erected a fire-proof building1 in which it is intended that the calculating machinery shall be placed when completed4 ,in this building are now deposited the large collection of drawings1 containing the designs1 not only of the part of the machinery which has been already constructed1 but what is of much greater importance1 of those parts which have not yet been even modelled4 ,it is gratifying to know that ,government has shown a proper solicitude for the preservation of those precious but perishable documents1 the loss or destruction of which would1 in the event of the death of the inventor1 render the completion of the machinery impracticable4