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being homogeneous, and is much denser in its interior than at its surface. CLAIRAUT, therefore, did an unspeakable service to this branch of science, when he showed, that in every case the two fractions just mentioned, though not cqual to one another, must always, when added together, constitute the same sum, that is, išo, orii:. Hence the oblateness appearing from the measurement of degrees to be zia, the increase of gravity from the equator to the poles, or, which is the same, the shortening of the pendulun, must be joWe must have recourse to experiment, then, to discover, whether this be agreeable to the fact, or whether evidences thus brought together from such different regions, conspire to support the same conclusion. LAPLACE, accordingly, from an examination of 37 of the best observations made in different latitudes, from the equator as far as the parallel of 67 degrecs, had obtained a result that agreed very well with the conclusions from the measurement of degrees. But these observations had been most of them made long ago, before the present extreme precision was introduced, and even before the means of comparing the lengths of two rules, or two rods of wood or of metal, was completely understood. It was therefore extremely desirable, that a series of new observations of the same kind should be made in different countries. The National Institute had begun the series at Paris; it had made a part of the Systême Métrique, to determine the relation between the seconds pendulum and the metre; and a number of experiments for that purpose were made by Borda and CASSINI, with every precaution that could ensure exactness.
After quiet was restored to Europe, England had leisure to attend to other objects than those in which the ideas of defence or of conquest were concerned. France and a great part of the Continent had adopted the scheme of uniform measures; in England a plan for the same had been often thought of; it had been more than once undertaken, but never on a right system; and had always fortunately, though perhaps weakly, been aban doned. It was now begun apparently under better auspices; a bill for the purpose was brought into Parliament; and our readers may remember, that it was thrown out in the House of Peers by the opposition of a noble Lord, more remarkable for the ingenuity than the soundness of his opinions. It happened here, however, as appears to us, that his Lordship was entirely in the right; the bill was a crude and imperfect scheme, prepared without due consideration of the various bearings of so nice a question, and consulting partial or present conveniency at the expense of permanent and general utility; having withal no dependence on any of those magnitudes which Nature herself has taken pains to secure against vicissitude and change.
The attention of the men of science about London was now naturally turned to the experiments by which the length of the pendulum may be accurately determined. The nature of the apparatus best fitted for that object is by no means obvious. The French Academicians, just referred to, had indeed employed a very simple one, which seems capable of great exactness. It consisted of a ball of platina suspended by a fine wire, and vibrating about a knife edge, which served as ils axis. The vibrations counted by the person who conducted the experiment, were compared with those of a clock, placed close by, and regulated according to mean solar time. After a sufficient number of such comparisons, the length of the pendulum from the knife edge to the centre of oscillation of the ball, was partly measured and partly calculated ; and thus the quantity required was determined.
Though this method is susceptible of great accuracy, and, in the hands of such men as BORDA and CASSINI, could not fail to lead to a satisfactory conclusion, yet it is right to have so important an element in our researches as the length of the pen. dulum, or the intensity of gravitation, ascertained by experiments made with different instruments; made according to different methods, and particularly not so dependent on the mathematical theory of the centre of oscillation as to be without the possibility of verification by experiment. It must not be supposed, that in laying down this last condition, we mean any thing so absurd, as to question the force of mathematical de monstration. A conclusion purely mathematical, when applied to an object that is also purely mathematical, one that partakes of the same immaterial and impassible nature with itself, is above receiving additional evidence from any source whatever, and despises alike all attempts to increase or diminish its authority. But the same is not exactly the case when the conclusion is applied to a material body; it then partakes of the imperfection of the subject; and thus, in a sphere even of gold or platina, the actual centre of oscillation may not coincide to the ten thousandth part of an inch, with the point which the calculus has determined. In such instances the verification by experiment, if it cannot be called necessary, is at least highly satisfactory.
Among the Mathematicians who endeavoured to resolve the problem on a principle of this kind, the author of the paper which is the subject of this article, came soon to be particularly
distinguished. Captain KATER, to the profession of a soldier, seems early to have united the pursuits of science, and to have acquired uncommon skill and accuracy both in philosophical experiment, and astronomical observation. We understand that in India, when a very young man, he assisted Colonel LAMBTon in the trigonometrical survey of Hindostan, and was extremely useful in a very nice and important part of the work, the selection of the stations where the observations were to be made, and of the summits to be intersected, a matter which requires great judgment; one which, in a mountainous country, and under a vertical sun, must be full of difficulty and danger, and from which we have been sorry to understand that his health had materially suffered.
Captain KATER having returned to England, and resumed the pursuits of science, began to consider how the experiment of the pendulum might best be made in a way to admit of verification by a reverse experiment; and a cylindric rod of brass or of iron readily occurred to him as a body well adapted to that purpose. The impossibility, however, of finding a rod or bar of metal so homogeneous that its centre of oscillation could be determined merely from its dimensions, made him quickly despair of succeeding by such means. It happily occurred to him, in this uncertainty, that there was one property of the centre of oscillation by which its place might be made manifest, whatever were the irregularity in the figure, or the density of the vibrating body.
Huygens, the profound and original author of the Theory of the Pendulum, had demonstrated that the centres of suspension and oscillation are convertible with one another; or that, if in any pendulum the centre of oscillation be made the centre of suspension, the time of vibration will be in both cases the same. Hence, conversely, said Captain KĄTER, if the same pendulum with different points of suspension can be made to vibrate in the same time, the one of these points must be the centre of oscillation when the other is the centre of suspension; and thus their distance, or the true length of the pendulum is found. It is curious to remark, that a proposition, so well known, and affording so direct a solution of the difficulty in which experimenters on this subject had always found themselves involved, was never before, at least in as much as we have been able to discover, applied to a purpose for which, now that the secret is known, it seems so excellently and so plainly adapted. But it is one of the prerogatives of true genius, to find the highest value in things which ordinary men are trampling under their feet.
To reduce the principle just mentioned into a tangible form, some further contrivance was still necessary. We copy thor's description of his convertible pendulum.
· The Pendulum is formed of a bar of plate brass, one inch and a half wide, and one-eighth of an inch thick. Through this bar two triangular holes are made, at the distance of 39.4 inches from each other, to admit the knife edges that are to serve for the axes of suspension in the two opposite positions of the pendulum. Four strong knees of hammered brass, of the same width with the bar, six inches long, and three quarters of an inch thick, are firmly screwed by pairs to each end of the bar; so that when the knife edges are passed through the triangular apertures, their backs may bear steadily against the perfectly plane surface of the brass knees, which are formed as nearly as possible at right angles to the bar. The bar is cut of such a length that its ends fall short of the extremities of the knee-peices about two inches.
Two slips of deal, 17 inches long, are inserted at either end, in the spaces thus left between the knee-pieces unoccupied by the bar, and are firmly secured by screws. These slips of deal are only half the width of the bar ; they are stained black, and a small whalebone point inserted at each end indicates the extent of the arc of vibration.
A cylindrical weight of brass, three inches and a half in diameter, and weighing about two pounds seven ounces, has a rectangular opening in the direction of its diameter, to admit the knee-pieces of one end of the pendulum. This weight, being passed on the pendulum, is so firmly screwed in its place as to render any change inpossible.'
This weight, it must be observed, is not between the knife edges, but is very near to one of them.
A second weight, of about seven ounces and a half, is made to slide on the bar, near the knife edges, at the opposite end; and it may be fixed at any point on the bar by two screws, with which it is furnished. A third weight, or slider, of only four ounces, is moveable along the bar, and is capable of nice adjustment, by means of a screw and a clasp. It is intended to move near the centre of the bar, and has an opening, through which may be seen divisions of · twentieths of an inch engraved on the bar.'
It is by means of this moveable weight that the direction of the vibrations in the two opposite positions of the pendulum are adjusted to one another; after which it is secured immoveably in its place.
The knife edges, or prisms, which make so important a part of this apparatus, and are to serve alternately as the axes of motion, are made of the steel prepared in India, and known by the name of wootz. The two planes which form the edge of each prism are inclined to one another nearly at an angle of 120 degrees. Every precaution was used to render the edges true, or straight, and to give the hardest temper to the steel; and a long series of experiments proves fully that they have been successful. Every precaution was also taken to give stability to the axes of suspension, when the experiments were made: But for the details of these, we find it necessary to refer to the
We come now to the very ingenious method which Captain Kater adopted for determining the number of vibrations made by his pendulum in twenty-four hours. It is no doubt susticiently understood, from what has been already said, that the pendulum was not to be applied to a clock, nor to receive its motion from any thing but its own weight. When experiments of this kind were attempted, it was for a long time supposed that the pendulum might safely be permitted to receive the continuance of its motion from machinery; and that, as it was then in no danger of coming to rest, the results were more to be depend
This conclusion, however, proceeded on a great mistake as to the part which the machinery of the clock performs on such occasions. That machinery is hardly ever, we believe, so nicely adjusted as accurately to restore to the pendulum the motion it loses in each vibration, (from friction about the centre, and from the resistance of the air), without either allowing any defect, or producing any excess. A clock, in general, accelerates the natural motion of the pendulum, and forces it to vibrate faster than it would do if impelled only by its own gravity: In experiments, therefore, where the relation of the length of the pendulum to the time of vibration is to be determined, the clock can only be used to measure out a given portion of time, or to assist in numbering the vibrations,
The manner in which this last can be done, is not so obvious as may be imagined. The mere counting of the vibrations one by one, and marking the number at stated intervals of time, would be a very inconvenient and imperfect way of going to work. As the experiment must be long continued, and frequently resumed, the tedium and irksomeness of counting the vibrations would become great, and, like every labour that is tedious and irksome, must be in danger of being inaccurately performed, more especially by mathematicians, the persons into whose hands the operation is most likely to fall. Even if no error were committed, there would still be an insecurity which nothing could remove. It is, indeed, the business of every experimenter to throw as great a share of the responsibility as he can on his apparatus, or on the physical agents he employs; and as little as possible on himself and his living assistants.