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Art. VI. An Account of Experiments for Determining the
Length of the Pendulum Vibrating Seconds in the Latitude of
have been distinguished by a series of Geographical and Astronomical measurements, more accurate and extensive than any yet recorded in the history of science. A proposal made by Cassini in 1783, for comecting the Observatories of Paris and Greenwich by a series of triangles, and for ascertaining the relative position of these two great centres of Astronomical knowledge by actual measurement, gave a beginning to the new operations. The junction of the two Observatories was executed with great skill and accuracy by the geometers of England and France: the new resources displayed, and the improvements introduced, will cause this survey to be remembered as an Era in the practical application of Mathematical science.
A great revolution had just begun to take place in the construction of instruments intended for the measurement of angles, whether in the heavens or on the surface of the earth; and was much accelerated by the experience acquired in this survey. One part of this improvement consisted in the substitution of the entire circle for the quadrant, the semicircle, or other portions of the same curve, as the unity and simplicity of the entire circle, distinguish it above all figures, and give it no less advantage in Mechanicks than in Geometry. Circular instruments admit of being better supported, more accurately balanced, and are less endangered from unequal strain or pressure, than any other. The dilatation and contraction from heat and cold, act uniformly over the whole, and do not change the ratio of the divisions on the circumference.
A geometrical property of the same curve contributes also much to the perfection of those instruments, in which the whole circumference is employed; and though it be quite elementary, and has been long known to geometers, it was first turned to account by artists about the time of which we now speak. The proposition is, that two lines intersecting one another in any point within a circle, cut off opposite arches of the circumference, the sum of which is the same as if they intersected one another in the centre. Hence it follows, that, in a circular instrument, whether the centre about which the index turns be the true centre or not, the mean of the two opposite arcs is the exact measure of the angle to be found. This gives a complete correction for one of the great sources of inaccuracy in the construction of mathematical instruments, since, by opposite readings off, the error in the centering is always corrected. RamsDEN, to whom the art of constructing mathematical instruinents owes so much, was the first ainong modern artists who made an astronomical circle of considerable size. A theodolite, also, which he made for General Roy, who conducted the survey just referred to, was, of its kind, the most perfect instrument yet constructed, and was furnished with the best telescope that had been employed in geodetical observations.
In France, also, the entire circle was introduced, and with a great additional improvement, that of repeating or multiplying the angle to be measured any required number of times. The consequence of this is, that the mean taken by dividing the multiple angle at last obtained by the number of the repetitions, gives the angle with an exactness which would have required a great number of observations, and a great length of time, if other instruments had been used.
The first idea of this excellent contrivance occurred to To BIAS MAYER of Gottingen, whose name is so well known in the history of Astronomy. The instrument was afterwards reconstructed and highly improved by the Chevalier BORDA. In 1787, when the Astronomers of Paris met those of England toward the conclusion of the survey, they were furnished with repeating circles, which was the first time that this instrument had been employed in similar observations.
As an evidence of the increased accuracy now obtained, it may be observed that it was in the survey of the ground between Greenwich and Dover that the excess of the angles of a triangle aboye two right angles arising from the curvature of the surface on which the angles were obseryed, first became an object of actual measurement. On this quantity which has been called the spherical excess, and was measured also by the repeating circle, LE GENDRE, with the ready invention that easily accommodates itself to new circumstances, grounded an admirable rule for reducing the solution of small spherical triangles under the power of plane trigonometry. The accuracy now expected was such, that an error of as many seconds in the measure of an angle as was formerly allowed of minutes, was no longer to be tolerated.
To Great Britain, the operations now entered on were attended with a further advantage, Government having been induced to continue a work so auspiciously begun, by extending a trigo nometrical survey over the whole island, so as to ascertain its topography with more precision than had yet been done with
respect to any tract of equal extent on the surface of the Earth. The survey has accordingly been continued to the present time, and is now carrying on in Scotland under the able direction of Col. Mudge, and by the meritorious exertions of Capt. COLBY, an indefatigable and accurate observer, instructed by much experience, and supported by a zeal and firmness of which there are but few examples.
It was not long after the commencement of this survey, that a system of Trigonometrical and Astronomical operations of still greater extent was undertaken by the French Government.
The want of system in the Weights and Measures of every country; the perplexity which that occasions; the ambiguous language it forces us to speak; the useless labour to which it subjects us, and the endless frauds which it conceals, have been long the disgrace of civilized nations. Add to this, the perishable character thus impressed on all our knowledge concerning the magnitude and weight of bodies, and the impossibility, by a description in words, of giving to posterity any precise information on these subjects, without reference to some natural object that continues always of the same dimensions. The provision which the art of printing has so happily made for conveying the knowledge of one age entire and perfect to another, suffers in the case of magnitude a great and very pernicious exception, for which there is no remedy but such reference as has just been mentioned Philosophers had often complained of these evils, and had pointed out the cure: but there were old habits and inveterate prejudices to be overcome; and the phantom of innovation, even in its most innocent shape, was sufficient to alarm governments conscious that so many of their institutions had nothing but their antiquity to recommend them. At the commencement of the French Revolution the National Assembly was avowedly superior to the last of these terrors, and the Philosophers of France considered it as a favourable opportunity for fixing, with the support of Government, a new system of measures and weights, on the best and most permanent foundation.
Of the quantities which nature preserves always of the same magnitude, there are but few accessible to man, and capable at the same time of being accurately measured. The choice is limited to a portion of the earth's circumference, or to the length of the pendulum that vibrates a given number of times in the course of a solar or syderial day, or any portion of time accurately defined by some of the permanent phenomena of nature. The choice of the French mathematicians fell on the first of these, and was accompanied with this great benefit to science, that it cnforced a very diligent and scrupulous exami.
nation into the magnitude and figure of the earth. The quadrant of the terrestrial meridian was the unit of linear extension which they proposed to assume, and the ten millionth part of it was the standard to which all linear measures were to be referred. The series of difficult and nice observations undertaken with a view to this improvement, carried on in the midst of much intestine disorder with signal firniness and perseverance, and finished, in spite of every obstacle, with all the accuracy that the new instruments and new methods could afford, has raised 10 the men of science * cngaged in it, a monument that can never be effaced. The meridian of Paris continued to Dunkirk, on the one hand, and Solieure on the other, and afterwards extended beyond the latter to the southermost of the Balearic Isles, amounting nearly to an arc of 12 degrees, afforded means more than sufficient for computing the quadrant of the meridian, and thus fixing the standard on sure and invariable principles.
In consequence of this, the figure, as well as the magnitude of the Earih, came to be better known than they had ever been before, because of the new data afforded for entering into combination with the lengths of degrees already measured in different countries. The extent of the arc of the meridian, thus determined, is also about to receive a great increase, by the addition from the British survey, of an arc extending from the parallel of Dunkirk to that of the most northerly of the Shetland Isles; so that the distance between this last parallel and that of Fermentera, nearly a fourth part of the quadrant of the meridian will become known by actual measurement.
But while it is possible to interrogate Nature in two different ways concerning the same thing, curiosity is not to be satisfied without having both hier responses. The pendulum, as is well known, affords the means of determining, not indeed the magnitude, but the figure of the earth; that is, its compression at the Poles, or the ol·lateness of the spheroidal figure into which it is formed. At the Equator, gravitation is weaker than at the Poles; both on account of the centrifugal force which is greatest at the former, and vanishes altogether at the latter, and of the greater distance of the circumference of the Equator from the centre of the mass. If the earth were quite homogeneous, Newton demonstrated, that the same fraction, viz. tso would denote the oblateness of the earth, and the diminution of gravity from the Pole to the Equator. There is, however, good reason to believe, that the earth is very far from
* DỊLAMBRE, MECHAIN, BioT, ARAGO.
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 equal to one another, must always, when added together, constitute the same sum, that is, ašo, or . Hence the oblateness appearing from the measurement of degrees to be yly, the increase of gravity from the equator to the poles, or, which is the same, the shortening of the pendulun, must be: We 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, accordingLv, from an examination of 37 of the best observations made in different latitudes, from the equator as far as the parallel of 67 degrees, 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 eten before the means of comparing the lengths of two rules, or two rods of wood or of metal, was completely understood. It was there fore 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 Systéme 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 abandoned. 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