body present was inhuman enough to offer him a knife. Upon disdaining to be worse than his word, he proceeded immediately to perform his part of the business; that night being the fourth of December, 1805; and in the course of that night he swallowed five knives. Next morning, being the 5th day of the month, the ship's company was anxious to see the performance renewed the second time: by the encouragement of the people, and the assistance of good grog, (and his lot was ordained to be miserable hereafter in consequence of the same,) he swallowed nine that day to his own knowledge; and the spectators informed him afterwards that he swallowed four more, that he knows nothing about: they were all clasp-knives, and some of them very large. Upon the 6th of December he was under the necessity of applying to a doctor, who was surgeon of the ship. The doctor, finding he was in a bad situation, made a strict inquiry of the principal men that were eye-witness to the transaction, which the captain and the rest of the officers found to be a true story. The surgeon, indeed, never neglected to pay the greatest attention, and prescribed what medicines he thought proper towards his relief: but all to no effect. At the expiration of three months, by taking a quantity of oil, he felt them dropping down to his bowels. In a few days after he was able to walk any part of the ship, and in that continuance till the 4th of June following, when he vomited one side of the handle of a knife, marked Cunningham, the same man that it belonged to, in the same ship; and, by asking him if he knew any thing about such a knife, he directly confessed that it was part of the knife Cummings swallowed of his. The surgeon keeps the said piece in his possession. Four months passed without any thing extraordinary having happened. On the 4th of November he passed another piece, the same as the former, with the lining of a knife along with it; two more he passed during that month. In February following vomited another lining of a knife; in the course of that month passed four more pieces, and since nothing extraordinary came away. “ June 12th, 1807, he was discharged the ship, in consequence of his complaint, and likewise being found, at the survey, unserviceable; after which, he was admitted into Guy's Hospital, under the care of Dr. Babington. Great many never believed such a circumstance. After five weeks being in the hospital, was presented out, and was in lodgings for the space of five weeks; but, finding himself getting was obliged to make the second application, and was readmitted under his physician again." FROM THE SAM.. On Machinery for Calculating and Printing Mathematical Tables By Charles BABBAGE, Esq. F.R.S. Lond. and Edin. &c. Among the brilliant inventions which have distinguished the present age, the machinery invented by Mr. Babbage for performing intellectual labour, is entitled to a prominent place. Low as that species of mental exertion undoubtedly is, when the mathematician performs the monotonous round of arithmetical calculations, in which neither the power of combination nor of judgment is called into acVol. I. No.6.-Museum. 3 T tion; yet we were not prepared to see even these humbler functions placed under the surveillance of wheels and pinions: and hence the first intelligence of a calculating machine has been received with as much incredulity by the wise as by the vulgar. The object which Mr. Babbage had in view in constructing this new machinery, was to produce printed copies of any mathematical tables, without the possibility of an error existing in a single copy. This result he proposed to attain solely by machinery, which, at the same time, made the calculations and composed, with a proper type, the tables when computed. At the commencement, of course, of these operations, certain preliminary calculations must be made; and, at intervals, the machinery must be set to these numbers; but, in some cases, when once set, the machine will continue working to the end of the tables. In these contrivances, there are two distinct parts, viz. one, by which the tables are computed, and the other, by which they are composed ; but, as the actual execution of such machinery requires a great ex: pense both of time and money, Mr. Babbage has contented himself with sketches on paper, accompanied by short memoranda, by which his contrivances may be at any time more fully developed ; and, when any new principles have been introduced, he has examined their actions by models executed on purpose. In order, however, to prove the practicability of these views, he has finished a small engine, by which the calculations are accomplished, and which has actually computed the following table, in which the second differences are constant. Table computed by Mr. Babbage's Arithmetical Engine, from the Formula x2 + x + 41. 1373 1447 1523 1601 547 1033 1681 593 1097 1763 1847 1933 2021 a weight, this rate of computation may be maintained for any length of time; and it is probable that few writers are able to copy, with equal speed, for many hours together. Notwithstanding the defective workmanship in this first machine, its computations are wonderfully accurate; and some very skilful mechanics to whom Mr. Babbage has shown it in confidence, are of opinion that it may be carried to any extent. Its parts, indeed, are few, but frequently repeated; and, though it contains many wheels, yet only a few of them move at the same time, so that its simplicity is greater than that of many machines in common and constant use. There is one circumstance in the construction of this machine, which is of considerable importance in making larger ones, viz. that, though its wheels are numerous, yet, by a peculiar contrivance, any error pro, duced by accident, or by a slight inaccuracy in one of them, is corrected as soon as it is transmitted to the next, and in such a manner as effectually to prevent any accumulation of small errors from producing a wrong figure in the calculation. The machine for composing with types, has not yet been actually constructed; but, from several models which Mr. Babbage has made, and from numerous experiments which he has tried, there can be no doubt of its complete success*. When put up, the composing machine will contain about 30,000 types, which are put in their places by children; but the person who attends the engine has a method of ascertaining, in less than half an hour, if any one individual of this number is misplaced. As the engine for computing tables by the method of differences, is the only one which Mr. Babbage has yet completed, he bas, in his letter to Sir Humphry Davy, confined himself to a statement of the powers which that method possesses, and as this statement is highly interesting, we shall give it in his own words. “ I would, however, premise, that if any one shall be of opinion, notwithstanding all the precautions I have taken and means I have employed to guard against the occurrence of error, that it may still be possible for it to arise, the method of differences enables me to determine its existence. Thus, if proper numbers are placed at the outset in the engine, and if it has composed a page of any kind of table, then by comparing the last number it has set up with that number previously calculated, if they are found to agree, the whole page must be correct; should any disagreement occur, it would scarcely be worth the trouble of looking for its origin, as the shortest plan would be to make the engine recalculate the whole page, and nothing would be lost but a few hours' labour of the moving power. Of the variety of tables which such an engine could calculate, I shall mention but a few. The tables of powers and products published at the expense of the Board of Longitude, and calculated by Dr. Hutton, were solely executed by the method of differences; and other tables of the roots of numbers have been calculated by the same gentleman on similar principles. As it is not my intention in the present instance to enter into the theory of differences, a field far too wide for the limits of this letter, and which will probably be yet further extended in consequence of A working-model of this part of the machinery has been recently constructed. the machinery I have contrived, I shall content myself with describing the course pursued in one of the most stupendous monuments of arithmetical calculation which the world has yet produced, and shall point out the mode in which it was conducted, and what share of mental labour would have been saved by the employment of such an engine as I have contrived. The tables to which I allude are those calculated under the direction of M. Prony by order of the French Government;—a work which will ever reflect the highest credit on the nation which patronized, and on the scientific men who executed it. The tables computed were the following. 1. The natural sines of each 10,000 of the quadrant calculated to twenty-five figures with seven or eight orders of differences. 2. The logarithmic sines of each 100,000 of the quadrant calculated to fourteen decimals with five orders of differences. 3. The logarithm of the ratios of the sines to their arcs of the first 5000 of the 100,000ths of the quadrant calculated to fourteen decimals with three orders of differences. 4. The logarithmic tangents corresponding to the logarithmic sines calculated to the same extent. 5. The logarithms of the ratios of the tangents to their arcs calculated in the same manner as the logarithms of the ratios of the sines to their arcs. 6. The logarithms of numbers from 1 to 10,000 calculated to nineteen decimals. 7. The logarithms of all numbers from 10,000 to 200,000 calculated to fourteen figures with five orders of differences. Such are the tables which have been calculated, occupying in their present state seventeen large folio volumes. It will be observed, that the trigonometrical tables are adapted to the decimal system, which has not been generally adopted even by the French, and which has not been at all employed in this country. But, notwithstanding this objection, such was the opinion entertained of their value, that a distinguished member of the English Board of Longitude, was not long since commissioned by our Government to make a proposal to the Board of Longitude of France, to print an abridgment of these tables at the joint expense of the two countries; and £5000 were named as the sum our Government was willing to advance for this purpose. It is gratifying to record this disinterested offer, so far above those little jealousies which frequently interfere between nations long rivals, and manifesting so sincere a desire to render useful to mankind the best materials of science in whatever country they might be produced. Of the reasons why this proposal was declined by our neighbours, I am at present uninformed; but, from a personal acquaintance with many of the distinguished foreigners to whom it referred, I am convinced that it was received with the same good feelings as those which dictated it. I will now endeavour shortly to state the manner in which this enormous mass of computation was executed; one table of which (that of the logarithms of numbers), must contain about eight millions of figures. The calculators were divided into three sections. The first section comprised five or six mathematicians of the highest merit, amongst whom were M. Prony and M. Legendre. These were occupied entire ly with the analytical part of the work; they investigated and determined on the formulæ to be employed. The second section consisted of seven or eight skilful calculators, habituated both to analytical and arithmetical computations. These received the formulæ from the first section, converted them into numbers, and furnished to the third section the proper differences at the stated intervals. They also received from that section the calculated results, and compared the two sets, which were computed independently for the purpose of verification. The third section, on whom the most laborious part of the operations devolved, consisted of from sixty to eighty persons, few of them possessing a knowledge of more than the first rules of arithmetic: these received from the second class certain numbers and differences, with which, by additions and subtractions in a prescribed period, they completed the whole of the tables above mentioned. I will now examine what portion of this labour might be dispensed with, in case it should be deemed adviseable to compute these or any similar tables of equal extent by the aid of the engine I have referred to. In the first place, the labour of the first section would be considerably reduced; because the formulæ used in the great work I have been describing, have already been investigated and published. One person, or at the utmost two, might therefore conduct it. If the persons composing the second section, instead of delivering the numbers they calculate to the computers of the third section, were to deliver them to the engine, the whole of the remaining operations would be executed by machinery, and it would only be necessary to employ people to copy down as fast as they were able the figures presented to them by the engine. If, however, the contrivances for printing, were brought to perfection and employed, even this labour would be unnecessary, and a few superintendants would manage the machine and receive the calculated pages set up in type. Thus, the number of calculators employed, instead of amounting to ninety-six, would be reduced to twelve. This number might, however, be considerably diminished; because when an engine is used, the intervals between the differences calculated by the second section may be greatly enlarged. In the tables of logarithms, M. Prony caused the differences to be calculated at intervals of 200, in order to save the labour of the third section; but as that would now devolve on machinery, which would scarcely move the slower for its additional burden, the intervals might properly be enlarged to three or four times that quantity. This would cause a considerable diminution in the labour of the second section. If, to this diminution of mental labour we add that which arises from the whole work of the compositor being executed by the machine, and the total suppression of that most annoying of all literary labour, the correction of the errors of the press, I think I am justified in presuming, that if engines were made purposely for this object, and were afterwards useless, the tables could be produced at a much cheaper rate: and of their superior accuracy there could be no doubt. Such engines would, however, be far from useless : containing within themselves the power of generating to an almost unlimited extent, tables whose accuracy would be unrivalled, at an expense comparatively moderate, they would become active agents in reducing the abstract a |
