condition of science cannot suddenly approach per- | fection; as all our knowledge and all our discoveries must be made by slow and painful steps; as the human mind requires the strong illuminating power of present knowledge, to penetrate even a short distance into the unknown and unexplored region of discovery, so we may readily suppose that the invention of a machine so truly wonderful as that contrived by Mr. Babbage, must have been the offspring of an old family of machines, professing the same object, but accomplishing far less. That this is so, detracts nothing from the last inventor, for had there been no other calculating machines previous to his, he could have gained nothing from the experience of the past, and consequently his machine, instead of being almost perfect, might have been only a germ for future men of genius to improve upon. But not only machines, but mental processes which give birth to machines, pass through various phases of improvement in the course of ages. There was a time when our forefathers could perform the arithmetical operation of addition, but not that of multiplication. It was discovered, however, that without greatly changing the character of addition, its processes might be facilitated by arranging and committing to memory a sort of table in which the results of addition are presented for the combinations of units from 1 to 12. Thus when we multiply 9 by 6, we add up 9 six times; but in the table we lose sight of the process of addition, and simply take the result as it stands, 9x6 =54. In like manner division was introduced as a speedier method of subtraction; for if we have to divide 30 by 5, we in effect subtract 5 from 30 six times over, by which we separate 30 into six parcels of 5 each. One of the simplest forms of calculating machines, is the Abacus. The school-boys of ancient Greece acquired the elements of knowledge by working on a smooth board with a narrow rim, called the Abar, probably from the combination of A, B, I, the first letters of their alphabet. They were instructed to compute by forming progressive rows of counters, which consisted of small pebbles, of round bits of bone or ivory, or even of silver coins. The same board also served for teaching the rudiments of writing and the principles of geometry, for by strewing the abax with sand, it was easy with a small rod to trace letters, draw lines, construct triangles, or describe circles. It appears also, that the practice of bestowing on pebbles an artificial value, according to the rank or place which they occupied, was practised at a very early period, for Diogenes Laertius states that Solon used to compare the passive ministers of kings or tyrants, to the counters or pebbles of arithmeticians, which are sometimes most important, and at other times quite insignificant. The Romans borrowed their abacus from the Greeks, and to each pebble or counter used on the board, they gave the name of calculus, the diminutive of calr, a stone; and applied the verb calculare to the operations performed with such pebbles; whence the English verb to calculate. (The Greeks also derived their verb, ynpiše, to 1 compute, from Vpos, a pebble.) A small box or coffer, called a loculus, with compartments for holding the counters, was a necessary appendage of the abacus: so that, "instead of carrying a slate and satchel, as in modern times, the Roman boy was accustomed to trudge to school, loaded with his arithmetical board, and his box of counters." The abacus appears to have been divided by means of perpendicular lines or bars: this was improved by dividing the surface of the board by sets of parallel grooves, or by stretched wires, or even by successive rows of holes. It was easy to move small counters in the grooves, to slide perforated beads along the wires, or to stick large nobs in the different holes. To diminish the number of marks required, each column was surmounted by a shorter one, wherein each counter had the same value as five of the ordinary kind, being half the index of the denary scale. Two of these instruments, delineated on the antique monuments, show clearly how they were used. In one the numbers are represented by flattish perforated beads, ranged on parallel wires; and in the other they are expressed by small round counters, moving in parallel grooves. These instruments contain each seven principal bars, expressing in order units, tens, hundreds, thousands, ten-thousands, hundred-thousands, and millions; and above them are shorter bars, following the same progression, but having five times the relative value. With four beads on each of the long wires, and one bead on every corresponding short wire, any number as far as ten millions could be expressed. Immediately below the place of units is a bar with its corresponding branch, both intended to signify ounces, or the twelfth parts of a pound. Fractions of ounces are also expressed by three very short bars behind the rest. During the middle ages, calculations were per formed on the principle of the abacus, by representing numbers by counters placed in parallel rows. It was the usual practice for merchants, auditors of accounts, or judges appointed to decide in matters of revenue, to appear on a covered bank or bench; before them was a table covered with a cloth, resembling the abacus, and distinguished by perpendicular and chequered lines. The Court of Exchequer, which takes cognisance of all questions of revenue, was introduced at the Norman Conquest. A writer of the twelfth century describes this table or scaccarium as being about ten feet long and five broad, with a ledge or border about four inches high, to prevent anything from rolling over, and was surrounded on all sides by seats for the judges, the tellers, and other officers. It was covered every year after Easter Term with fresh black cloth, divided by perpendicular white lines, at intervals of about a foot, and again parted by similar transverse lines. Small coins were used for counters: the lowest bar exhibited pence, the one above it shillings, the next pounds; and the higher bars denoted successively tens, twenties, hundreds, thousands, and ten-thousands of pounds. The first bar, therefore, advanced by dozens, the second and (1) Encyclopædia Britannica, Art. Abacus. third by scores, and the rest by multiples of ten. The teller sat about the middle of the table: on his right hand eleven pennies were heaped on the first bar, and a pile of nineteen shillings on the second; while a quantity of pounds was collected opposite to him on the third bar. For the sake of expedition, he might employ a different mark to represent half the value of any bar; a silver penny for ten shillings, and a gold penny for ten pounds. Offices for changing money were indicated by a chequered board, a sign afterwards adopted for an inn or hostelry. The method of recording numbers by tallies was also introduced into England at the Norman Conquest. Tallies are well seasoned sticks of hazel or willow, Fig. 396, from the French tailler, to cut, because they are cut or squared at each end. The sum of money was marked on the side with notches by the cutter of tallies, and also inscribed on both sides in Roman characters by the writer of tallies. The smallest notch denoted a penny, a larger one a shilling, and one still larger a pound: other notches increasing in breadth denoted a ten, a hundred, and a thousand. The stick was then cleft through the middle by the deputy-chamberlains with a knife and a mallet: the one portion being called the tally, and the other the countertally. The Chinese make use of an ancient instrument called the Schwan-pan, or computing-table, which is similar in shape and construction to the abacus of the Romans. It consists of a small oblong board surrounded by a high ledge, and parted lengthwise near the top by another ledge. It is divided vertically by 10 smooth and slender rods of bamboo, on which are strung 2 small balls of ivory or bone in the upper compartment, and five such balls in the lower and larger compartment: each of the latter on the several bars denoting a unit, and each of the former a 5. As the decimal notation is used in China, this instrument is well adapted for facilitating calculation, and the native traders are said to use it with such rapidity and skill as to astonish Europeans. The Chinese also make use of digital signs for denoting numbers, (as did also the ancient Romans, whence the term digit from digitus a finger.) As, every finger has 3 joints, let the thumb nail of the other hand touch those joints in succession, passing up the one side of the finger, down the middle and again up the other side, and it will give 9 different marks applicable to the denary scale of arrangement. On the little finger those marks signify units, on the next finger tens, on the middle finger hundreds, on the index finger thousands, and on the thumb hundred thousands. With the combined positions of the joints of one hand therefore it is easy to advance by signs as far as a million. The merchants of China are said to conclude bargains with each other by means of these signs, and that often with a fraudulent design they conceal them from the by-standers, by seeming only to seize each others' hands with a hearty grasp. A simple form of arithmetical machine was invented by Baron Napier of Merchiston, the inventor of logarithms. It is called Napier's bones or rods. It consists of a number of rods or square parallelopipeds 3 inches in length and ths of an inch in breadth, Fig. 397. Each of the faces of the parallelopipeds except the index-rod on the left hand is divided into common multiplication table; the only difference being that the right hand or unit figure of the product is placed below the diagonal line, and the 10 belonging to it above it; and when the product is but one figure, it is written below it. With these rods, multiplication and division can be performed by means of addition and subtraction; but as the same figure may occur several times, it is necessary to be provided with 3 or 4 rods of each kind, and to have 4 sides of each rod inscribed with a different set of figures. Multiplication is performed by placing the rods so as to form the multiplicand at the top; then apply a rod with unity at the top to the left hand side, and exactly opposite to each figure will be found the product arising from the multiplication of that figure into the multiplicand; but this must be obtained by addition, in the following manner:-the figure below the diagonal on the right-hand rod, is the unit figure of the product; the figure above the diagonal, added to the figure on the next rod below tne diagonal, gives the next figure of the product; the figure above the diagonal on the second rod, added to the figure below the diagonal on the following rod, gives the third figure of the product; and so on. It will not be necessary to illustrate the use of these rods by examples, for the multiplication table is so well and so generally known, as to render | until two needles pointed to the two figures 5 and 9, the tedious use of Napier's rods unnecessary. The method of performing division therewith, is even more complex than that of multiplication. Many other contrivances for facilitating calculation have been made from time to time, but they take the place of instruments rather than of machines. The first calculating machines properly so called are those by Gersten and Pascal, the former of which is described by Professor Gersten himself, in an early number of the Philosophical Transactions, and the latter is described by Diderot in the Encyclopédie Méthodique. Article Arithmétique. It would not be possible to describe these machines without the aid of elaborate engravings. Their use, if indeed they were ever used practically, was confined to a few simple arithmetical operations, which can be performed much more readily by the pen of a skilful computer. It will, however, be useful to notice the principle upon which these machines were constructed, since it is that which has been subsequently adopted in later and more efficient machines of this class. If we notice the manner in which quantities are combined in the common system of numeration, it will be found that the value of each figure is ten times greater than it would be if it occupied a position one place to the right. Thus, in the number 1829, although 9 is greater than 2, yet the 2 in this position represents a larger sum than the 9, because it occupies a place to the left of the 9. The quantities really expressed by the figures 1000800 1829 are 20 but in practice we omit the cyphers, and place the significant figures side by side preserving their proper position from the right hand. Now if we have a wheel on whose axis is a pinion with leaves or teeth; if these teeth work into another set of teeth or cogs on the periphery of another wheel, and if the teeth on the latter wheel are just ten times as numerous as those on the pinion, this system being made to revolve, the pinioned wheel will revolve just ten times as fast as the other. This produces a kind of analogy between the decimal notation and the working of the wheels; for it takes 10 units to make up 1 figure or unit in the second place in common numeration, and it requires 10 revolutions of the pinioned wheel to impart 1 revolution to the larger wheel. This is the fundamental principle in the calculating machines now under notice. In such machines there are a number of dial-faces, each marked with figures from 1 to 10. These dial-faces are fixed upon wheels, the teeth of which work into the pinions of other wheels, on which are similarly divided faces or discs, so that while one face indicates units, another indicates tens, a third, hundreds, and so on. These wheels and dial-faces may be differently arranged in different machines, but the principle is the same in all. In Gersten's instrument, for example, if 32 were to be added to 59, two dial-faces had to be turned by hand one on each plate: two slides were then adjusted, until two indices pointed to the figures 3 and 2, one on each slide. Both the discs and both the slides were connected with toothed rack-work, which working into each other, turned another dial-plate in such a direction as to show on its face 91, or the sum of 32 and 59. If on the contrary it were required to subtract 59 from 91, indices would be pointed to 9 and 1 on two separate discs, and to 5 and 9 on two separates slides, and the movement of these discs and slides in an opposite direction to the former would turn another wheel, to show 32 on its face, the difference between 59 and 91. The process of multiplication was effected by a kind of reiteration of additions, and that of division by a succession of subtractions. Pascal's machine was constructed for the purpose of performing certain calculations connected with the duties of an office, held in Upper Normandy by Pascal's father. The calculations were reckoned in the currency of France at the time: the denier wheel had 12 teeth, representing the number of deniers in a sol; the sol wheel had 20 teeth, equal to the number of sols in a livre, above which each wheel had 10 teeth, indicating 10, 100, 1000, &c. livres. Connected with the wheels were a number of cylinders. The cylinder expressing deniers had the following figures engraved upon it : 0. 11. 10. 9. 8. 7. 6. 5. 4. 3. 2. 1. 11. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. The cylinder representing the sols: 0. 19. 18. 17. 16. 15. 14. 13. 12. 11. 10. 9. 8. 7. 6. 5. 4. S. 2.1. 19.0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. And when it expressed the units of the livres or Now as matters are so arranged in this machine that a complete turn of the first barrel is necessary to produce a twentieth of the second; a turn of the second to produce a tenth of the third; a turn of the third to produce a tenth of the fourth; and in short, since the barrels follow in their revolutions the proportion which prevails between the different orders of the figures in arithmetic, it is evident that the operations of numbers may be performed by means of the barrels and the figures which are engraved on them. This machine has some ingenious arrangements which we cannot stay to describe: it is, however, tedious in its operations and inferior to the common modes of performing arithmetical calculations. It is introduced here in order to furnish a glimpse of one of the predecessors of Mr. Babbage's machine. It was improved by M. de l'Epine in 1725, and again in 1730 by M. Boitissendeau. Both machines are described in the "Recueil des Machines approuvées." About 1670, Sir Samuel Morland contrived two machines, one for addition and subtraction, and the other for multiplication, division, and the extraction of the square and cube roots. They are described in the Philosophical Transactions, No. 94. Leibnitz purposes of numerical calculation; in the second, these same formula were calculated for values of the variable, selected at certain successive distances; and under the third section, comprising about 80 individuals, who were most of them only acquainted with the first two rules of arithmetic, the values which were intermediate to those of the second section were interpolated by means of simple additions and subtractions. The body of tables thus calculated consisted of 17 MS. folio volumes. They have not been published. The printing was begun, and a small portion stereotyped; but a sudden fall in the value of assignats rendered it impossible for the printer to fulfil his contract with the government. The British government made an offer of 5,000l. towards the completion of the work; but political circumstances appear to have prevented its re-adoption, and it has never been resumed. A similar undertaking was, however, entered upon in England; and Mr. Babbage conceived that the operations performed under the third section might be executed by a machine, which he proposed to call the Difference Engine, on account of the principle upon which its construction is founded. Some notion of this will be gained by considering the series of whole square numbers, is also said to have invented a calculating machine, | formula were so combined as to adapt them to the but no account of it appears to have been published. But machines of this kind, however ingenious, were not directed to the supply of a want which had long been really felt ; namely, the production of arithmetical and other tables which should be rigorously correct. In the navigation of a ship, in the preparation of an almanac, in the higher branches of astronomy, and in other occupations, the use of such tables is very great. As such tables are constructed by human heads and hands, they are all more or less disfigured by errors. Tables of multiplication, of powers and roots, of trigonometrical elements, of logarithms, of the solar, lunar, and planetary motions, &c., have been computed and published in various countries, to the extent of many hundred volumes; and although very great care has been bestowed on their preparation, they all contain errors of greater or less magnitude. In a multiplication table (as far as 100 times 1,000) constructed by Dr. Hutton for the Board of Longitude, 40 errors were discovered in a single page taken at random. In the solar and lunar tables from whence the computations were formerly made for the Nautical Almanac, more than 500 errors were discovered by one person. In the Tables requisite to be used with the Nautical Almanac," more than 1,000 errors were detected. In certain tables published by the Board of Longitude, a table of errata, containing 1,100 errors, was affixed. It was afterwards found necessary to have a list of errata of the errata; and in one instance there was an erratum of the errata of the errata. In cases such as these the sources of error are so numerous that it is difficult to find a remedy for them all, so long as the common modes of calculating, transcribing, and printing are adopted. Some of these errors have been referred to false computation, others to inaccurate transcription; some have been referred to the compositor; others to a displacement of the types by the inking-ball used in the old method of printing, and the incorrect replacement of such types by the pressman. The apparently hopeless task of preparing accurate tables by the common methods led Mr. Babbage, about thirty years ago, to consider whether a machine might not be constructed for computing and printing off mathematical tables. The proposed machine was to be capable not only of performing arithmetical calculations, but all those of mathematical analysis, provided their laws were known; and the principle upon which it was to depend was to be of so general a nature that, if applied to machinery, the latter might be capable of mechanically translating the operations indicated to it by algebraical notation. Mr. Babbage's first attempts originated in the following circumstances. The French government, wishing to promote the extension of the decimal system, had ordered the construction of logarithmical and trigonometrical tables of enormous extent. M. de Prony, who directed the undertaking, divided it into three sections, to each of which was appointed a special class of persons. In the first section, the 1, 4, 9, 16, 25, 36, 49, 64, &c. By subtracting each of these from the succeeding one, a new series is obtained, named the series of first differences, consisting of the numbers, 3, 5, 7, 9, 11, 13, 15, &c. On subtracting from each of these the preceding one, we obtain the second differences, which are all constant, and equal to 2. This succession of operations and their results are represented in the following table :C.. A. Column of B. 1 3 4 26 From the mode in which the columns B and C have been formed, it is obvious that if we wish to pass from the number 5 to the succeeding one 7, for example, we must add to the former the constant difference 2; so, also, if from the square number 9 we would pass to the following one, 16, we must add to the former the difference 7, which difference is, in other words, the preceding difference 5 plus the constant difference 2; or, what is the same thing, to obtain 16, we have only to add together the three numbers, 2, 5, 9, placed obliquely in the direction a b. So also we get the number 25, by summing up | but this portion is capable of calculating to five figures and two orders of differences, and performs its work with absolute precision; but no portion of the printing machinery exists. the three numbers placed in the oblique direction dc. Commencing by the addition 2 + 7, we have the first difference 9 consecutively to 7; adding 16 to 9, we have the square 25. Thus, the three numbers 2, 5, 9, being given, the whole series of successive square numbers, and also that of their first differences, may be obtained by means of simple additions. Now, to reproduce these operations by means of a machine, suppose the latter to have three dials, A, B, C, on each of which are traced, say, 1,000 divisions, over which a needle shall pass; that the two dials C, B, have also a registering hammer, which is to give a number of strokes equal to that of the divisions indicated by the needle; that for each stroke of the registering hammer of the dial c, the needle B shall advance one division, and that similarly the needle A shall advance one division for every stroke of the registering hammer of the dial B. Such is the general disposition of the mechanism. A This being understood, suppose, at the beginning of the operations which are to be executed, we place the needle c on the division 2, the needle в on the division 5, and the needle A on the division 9. Allow the hammer of the dial c to strike: it will strike twice, and at the same time the needle B will pass over two divisions. The latter will then indicate the number 7, which succeeds the number 5 in the column of first differences. If the hammer of the dial в be then allowed to strike, it will strike 7 times, during which the needle a will advance 7 divisions: these added to the 9 already marked by it will give the number 16, which is the square of the number consecutive to 9. If we now recommence these operations, beginning with the needle c, which is always to be left on the division 2, it will be seen that, by repeating them indefinitely, we may successively reproduce the series of whole square numbers by means of very simple mechanism. It would be quite impossible, however, to enter upon the details of the machinery, for the drawings of the various parts, constructed by Mr. Babbage, or under his direction, cover a space of 1,000 square feet. A most valuable feature intended to be introduced into this machine is the power of printing the tables as fast as it calculates them. This was proposed to be accomplished in the following manner:- -When one of the dial wheels is in such a position as to indicate any particular figure of the table, some mechanism at the back raises a curved arm, containing several figure punches. A plate of copper is brought near the bent arm, and by a sudden blow an impression of the required figure is punched in the copper, and the figure so punched corresponds with that indicated on the dial. The plate is shifted from place to place, until it is punched all over with figures arranged in a tabular form. It is then used either as an engraved copper plate, and printed from in that form, or it may be used as a mould in the casting of stereotype plates of the tables, which may thus be multiplied in a permanent form without the slightest chance of an erratum being required. A portion only of this machine has been completed; In 1843 an application was made to Government by the Trustees of King's College, London, to allow the engine as it existed to be removed to the museum of that institution. The request was complied with, and the engine enclosed within a glass case now stands nearly in the centre of the museum. Fig. 398, is a representation thereof, made from a drawing from the engine itself. As this fragment of the difference-engine is public property, it will be interesting to state briefly the circumstances which have interfered with the completion of this great work of genius; and this we are enabled to do with the assistance of a chapter contained in the second Volume of Mr. Weld's "History of the Royal Society," (Svo. London, 1848,) which we understand received the sanction of Mr. Babbage himself, as a true and impartial statement. It appears that on the 3d July, 1822, Mr. Babbage, in a letter to Sir Humphrey Davy, gave some account of a small model of his engine for calculating dif ferences, which "produced figures at the rate of 44 a minute, and performed with rapidity and precision all those calculations for which it was designed. Induced by a conviction," says Mr. Babbage, "of the great utility of such engines, to withdraw for some time my attention from a subject on which it has been engaged during several years, and which possesses charms of a higher order, I have now |