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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 instruments owes so much, was the first among 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 Tobias 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 above two right angles arising from the curvature of the surface on which the angles were observed, 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, Legendre, with the ready invention that easily accommodates itself to new circumstances, grounded an admirable rule for reducing the solu tion 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 trigonometrical survey over the whole island, so as to ascer
tain 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 Colonel Mudge, and by the meritorious exertions of Captain 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 sidereal 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 enforced a very diligent
and scrupulous examination 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 firmness and perseverance, and finished, in spite of every obstacle, with all the accuracy that the new instruments and new methods could afford, has raised to the men of science engaged 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 southernmost 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 earth, 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 differThe extent of the arc of the meri
* Delambre, Mechain, Biot, Arago.