circle are reversed in the order in which they stand on the common clock faces, because the motion of the hours becomes reversed by making the dial revolve instead of the hands. The register is composed of three distinct parts: a light brass circular magazine, half an inch wide, and of the same depth, divided into 12 compartments; a lid or cover with the same number of elliptical perforations; and an hour circle. Fig. 4 shows the hour circle, to the outer rim of which are attached three crutches, b, b, b, to receive and support the magazine, which is guided into its place, and kept steady there, by six small shoulders or projections, c, c, c, on its inner edge, which embrace the sides of the crutches. Fig. 3 represents the magazine or outer circle, with its front plate removed, exhibiting the division into compartments. The partitions are made by two or three wires rivetted or screwed into the two sides of the circle. Fig. 2 is a front view of the register with all its parts connected: c, c, c, are three studs or buttons which keep the lid affixed to the magazine. In this view are shown the elliptical openings into the magazine. These openings are 10 minutes in their shortest, and 20 in their longest, diameter. The rim moves so easily in and out of the crutches, as to admit of its being detached, or put on, without affecting the going of the clock. Immediately above the meridian of the clock, fig. 1, a hole, a, is bored through the wooden door, of a diameter equal to 10 minutes measured on the register circle. The centre of this hole must exactly coincide with the centre of the openings in the rim of the register. It only remains to furnish the watchman with a sufficient number of light spherical wooden balls of a diameter equal to the holes in the clock front, and to instruct him to drop one into the hole, a, at each hourly visit. The elliptical shape of the holes in the face of the rim will allow the ball to pass into the register either five minutes before or five minutes after the exact hour stroke. At the expiration of the watch, the door is unlocked, the rim removed from the crutches, and the face or lid slipped from the studs. The absence or presence of a ball in each compartment indicates the regularity or the neglect with which the duty has been performed. Reference to the Plate. Fig. 1, the head of the clock with the register adapted ready for use. Fig. 2, the hour and register circles in working order. Fig. 3, the register without its plate or cover. Fig. 4, the hour circle with the crutches ready to receive the register, fig. 3. Fig. 5, a section in perspective of fig. 2, fixed to the hour arbor, d, of the clock, as in fig. 1. Fig. 6, one of the crutches attached to the hour circle, fig. 4, and which carry the register, fig. 3. 7 ARTICLE VII. Further Observations on Fluxions. By Alex. Christison, Esq. Professor of Humanity, Edinburgh. MY DEAR SIR, (To Dr. Thomson.) Edinburgh, Oct. 13, 1815. To some remarks on Euclid's definition of proportion, in his fifth book, to be inserted, if you choose, in your Annals of Philosophy, I subjoin the deduction of fluxions from the definition which you published in May. 1st × A 3d x A 21 × B 4th x B If the equality of the products of the means and of the extremes be assumed as the criterion of proportionality, it is evident that the formula demonstrates Euclid's property, in his fifth definition, book fifth, with regard both to commensurables and to incommensurables; and that the formula is applicable even to abstract numerical and algebraical quantities: but it will be found that the criterion above-mentioned is not so convenient as Euclid's for demonstration. We cannot, however, apply Euclid's criterion or property so universally as the other; for we cannot, in demonstration, apply it to abstract numerical and algebraical quantities : we can apply it to those quantities only in which, as in geometrical magnitudes, the slowest learner sees that the first and the third have a necessary dependance on each other, as also the second and the fourth. A learner understands immediately Euclid's definition if he be directed to prop. 33, book 6; for supposing the angle at the centre, the first term is an arch of the one circle, and the third term is the corresponding angle of that arch. Now it is impossible for the slowest learner to conceive that he can double, &c. the arch or first term without doubling, &c. the corresponding angle or third term. The same may be said with regard to the second and the fourth terms, which belong to the other circle. If the one circle be laid on the other, and if the multiple of the one arch be equal to the multiple of the other, the multiple of the one angle must also be evidently equal to the multiple of the other; and if greater, greater; and if less, less: consequently the quantities are, by the definition, proportional. The equimultiples of the first and third, and of the second and fourth, can be exhibited to the learner without taking any particular numbers as multipliers: but it is impossible, I think, to do so with regard to any abstract numerical and algebraical quantities which are to be proved proportional; and if we take parts, we abandon Euclid's definition. Euclid's definition, then, is not applicable to all proportional quantities; but it is perfect if it be limited by its proper range: it admits, but it does not need, demonstration; it includes incommensurables; and it de-. monstrates simply and rapidly where some eminent mathematicians demonstrate complexly and tediously. As fluxions are a part of the theory of rates, I subjoin an algebraical investigation and demonstration of the fluxional problem. Definition. Fluxions is a method for finding the rate of change in a quantity and its function. Problem. To find the fluxion of x"; n being any positive, integral, number; and or 1 being assumed, as it may be, for the fluxion of x, while x varies uniformly. x" may be represented thus, x x x x x x x ....with x as often employed as there are units in n; and if it is only the first x that varies, the fluxion must be 1x1; but if every x varies in succession, the fluxion must be n xx. Q. E. I. and D. - 1 If i = 1 * = 1 x 1 be represented by a square whose side is 1, na may be represented by an oblong with 1 for one of its sides, and n x for the other; consequently, As the square the oblong :: 1 x : n x2 - 1 x. If a learner find any difficulty in conceiving that if x vary uniformly, or 1 is the fluxion of x, he will easily learn that, in comparing the rate of variation or change, in an uniform motion of 16 feet in a second, with the law of descent of heavy bodies, the rates at the end of each successive second are, as 1 : 2, as 1 : 4, as 16, &c., the space gone over by the uniform motion being represented by x. If those ancients, such as Archimedes, who understood varying quantities, had subjoined the rate of variation, they would have taught us fluxions; but perhaps they had not a proper notation. In your 33d number the reader may insert &c. after great, p. 179, 1. 2; reduced for referred, p. 179, 1. 33; being = 1, p. 180, 1. 30, after coefficients; rigour for vigour, p. 181, 1.4; letter for latter, p. 182, 1. 7; and he may efface the words between the first rate and of, p. 182, 1. 41. I am, my dear Sir, yours faithfully ARTICLE VIII. Letter from Wm. Henry, M. D. F.R. S. correcting some defective Statements in different Histories of the Introduction of Bleaching by Oxymuriatic Acid. DEAR SIR, (To Dr. Thomson.) Manchester, Oct. 1815. The fourth volume of Mr. Parkes's useful work, lately published, contains an account of the introduction of the mode of bleaching by oxymuriatic acid into this country, which, though correct in the main, is not altogether so. It resembles, indeed, so closely, in several respects, a statement published some years ago in Dr. Rees's Cyclopædia, that it is probable the historical information of both was derived from the same source. You will, therefore, oblige me, if you consider the subject of sufficient importance, by admitting into your Journal the substance of a representation, which I addressed several years ago, to the Rev. Dr. Rees, in behalf of the claims of a person in whose reputation I may naturally be supposed to feel some interest. But, independently of this interest, it does appear to me, that the public ought to be set right respecting the real history of this invention. The credit which a man of science derives from contributing to the improvement of the useful arts, is often (as in this case) the only reward he receives; and it is the duty of the historian of those arts, first to make himself thoroughly master of the facts, and then to detail them with fairness and impartiality. I have chosen this time for bringing the matter before the public, because all the parties concerned are still living, some of them at a very advanced age, and may readily be called upon for farther evidence, if it should be thought necessary. I am, dear Sir, yours very truly, WM. HENRY. (To the Rev. A. Rees, D.D. F.R.S. &c.) REV. SIR, Dec. 1809. Observing that the early volumes of your Cyclopædia are about to be reprinted, I am induced to fulfil an intention, which I have long entertained, of addressing a few lines to you respecting the article BLEACHING, published in vol. iv. part 2, 1st edition. The writer of that article, in assigning to different persons their shares of merit, in the introduction of the mode of bleaching by oxymuriatic acid and its compounds, has made a distribution, which is very far from being an equitable one. Of the part which was taken by Mr. Watt of Birmingham, in the application of Berthollet's important discovery, far too little is said; and of Mr. Henry's share in the improvement not the smallest notice is taken, though it could not fail to be known to the writer of the article, who, at that period, was himself engaged in this town in pursuit of the same object, and was in habits of occasionally communicating with Mr. Henry on the subject. The truth is, that next to Mr. Watt, Mr. Henry was at least equally early with any other person in applying the discovery to practice. In proof of this I might appeal to the general notoriety of the fact in this town and neighbourhood: but I depend chiefly for its establishment, on a number of letters from Mr. Watt to Mr. Henry, written in the year 1788, which are now before me. They form The person alluded to is Mr. Thomas Henry, F. R. S., President of the Literary and Philosophical Society of Manchester. |
