be something singular in the causes that have promoted the study of a particular branch of science in distant countries, when no concert or peculiar influence can be supposed to have acted exclusively on them.

the

Delambre insists at some length on the operations in practical geometry, or what the French call Geodesie, that have been lately carried on for purpose of ascertaining different points relative to the figure of the earth. The first of the operations to which he refers, is that which was undertaken both by France and England in 1787, for the purpose of ascertaining the distance between the meridian of Paris and that of Greenwich. He observes, with respect to these," that, considering the advanced state of the arts and sciences, it was to be expected that the English would endeavour to surpass every thing of the same kind that had yet been done: they succeeded in doing so the theodolite of Ramsden, the Indian lights used for signals, the new apparatus employed in the measurement of the basis, produced a degree of accuracy hitherto unknown. The French, on their side, had only angles to measure; and the repeating circle which Borda had invented, though not of so imposing a form as the theodolite of Ramsden, contained in its construction a principle which assured to it a precision at least equal to that

of the latter instrument, and more independent of the skill of the artist."

We believe, that this encomium on the repeating circle of Borda is very fairly due to it. That circle puts it in our power, not merely to take a mean of a great number of observations, but, as those observations are made without being read off till we come to the last, the error of reading off is no greater for all the observations put together, than it would have been for one observation only; so that, when divided into as many parts as there have been observations made, the result almost vanishes. The repeating circle, therefore, gives a mean of the errors of observation, and of the division of the instrument and the error of reading off, it goes near to annihilate entirely. This seems to be the true light in which these instruments should be viewed; and as they are now made by Troughton, with all the accuracy which that excellent artist gives to whatever passes through his hands, we should think it highly expedient that a comparison was instituted between them and the theodolite of Ramsden, for which the trigonometrical survey affords so good an opportunity.

The success of the measurement of the distance between the meridians of Greenwich and Paris, led to the operation on which the new system of measures was founded. The unit fixed on was a qua

drant of the meridian; and, under the impossibility of measuring the whole, the largest arch accesible, that between Dunkirk and Barcelona, was chosen. The operations for this purpose began under the direction of Mechain and Delambre, in 1792, and were not concluded till 1799. Of these, Delambre gave an account, in a work that was mentioned in a former Number of this Review. The coincidence of two different bases of 12,000 metres each, and distant 700,000 from one another, demonstrated the extreme accuracy with which the whole had been conducted. Two degrees have been since added, by the continuation of the same meridian to the Balearic Isles.

The same spirit has spread into other parts of Europe; and has produced important improvements in the science of geography. The astronomer Swanberg measured over again, in 1802, the degree that had been measured in Lapland by Maupertuis, and a party of the French and Swedish academicians. Their measure made the degree of the meridian which cuts the polar circle, to be 57,405 toises,-considerably greater than it was found possible to reconcile, by any theory, with the magnitude of degrees measured in lower latitudes. Melanderjelm, a Swedish astronomer, known by several valuable works, proposed to repeat the measurement; and the operation was committed to Swanberg and three others, who, using every pre.

caution, and employing the circle of Borda, found the degree, in the latitude of 66° 20′, to be only 57,209 toises; less by 196 toises than the old measurement; agreeing perfectly with other observations; and giving, for the compression at the poles, about one 330th of the earth's semidiame

ter.

The measurement of Maupertuis and his colleagues occasioned much confusion and debate for near seventy years; and proves, in a remarkable manner, how much worse an inaccurate experiment may often prove than no experiment at all. The great trigonometrical operations carrying on in England are also mentioned by Delambre; though he seems not perfectly informed of their extent. He mentions some in Germany and Switzerland, with which we are not acquainted.

Among the improvements that respect this state of practical geometry, where its operations, by aiming at great accuracy, connect it with more profound and difficult researches, Delambre, with great propriety, reckons the theorem of Legendre, by which the calculation of spherical triangles is reduced, in all the cases of such measurements as we now refer to, to plane trigonometry. The same excellent geometer has extended his theorem to triangles on the surface of a spheroid. (Vid. Mémoire sur les Transcendantes Elliptiques, 1 vol. 4to.)

The enumeration which Delambre gives of the works containing improvements and discoveries in algebra, is very extensive, and includes several treatises which have not yet found their way into this island. Of those on which we can add, to the short notice which our author gives, some particu→ lars from our own knowledge, we shall select one or two examples. Lagrange, having accepted the office of professor in the Polytechnic school, composed, for the instruction of his pupils, the work which he calls Calcul des Fonctions, intended as a commentary and supplement to the Théorie des Fonctions Analytiques, which he had before published. These works are both of great value, on account of the new and accurate view which they give of the principles of differential calculus, or of what we call the method of fluxions. For many years, the French mathematicians, and indeed the mathematicians of the Continent in general, gave themselves little trouble about the principles of the new geometry; and, though they extended its methods, rules, and applications, much beyond what was done in England, they were not so successful in explaining and demonstrating the fundamental truths of the science, as Newton and his followers. This, we believe, will be generally allowed; and, till a very late period, scarcely admits of any exception. Euler himself, though such a master of the Calculus as to have hardly any equal, yet, in the