of the sphere; and it was from this theorem that General Roy deduced the rule which he has given for computing the spherical excess, independently of the angles themselves. For this purpose, the area of the triangle is to be estimated as if it were rectilineal; and it is sufficient to do this even by a very rude approximation, because it requires an area of about 75 square miles to produce a single second of spherical excess.

As it was necessary that the French geometers should unite with the English in carrying into full effect the plan which they themselves were the first to propose, three distinguished members of the Academy of Sciences, Cassini, Mechain, and Legendre, met General Roy and Dr Blagden at Dover, where measures were concerted for the corresponding observations to be made on the coasts of France and England. The French academicians were furnished by General Roy with white lights, to be fired on their side, while corresponding observations were made at Dover and FairlightDown, on the coast of England. The operations on both sides succeeded perfectly, notwithstanding that the weather was by no means favourable. The three academicians above named, having crossed the Channel again, after their observations were finished, repaired to London, and appear to have been highly gratified by the objects they saw, and the reception they met with in that metropolis. It

is painful to reflect, that this is the last amicable interview which has taken place even among the men of letters of the two countries; and that the hostile armies of both nations are now encamped on the very ground which was the theatre of these scientific operations.

Besides measuring all the angles in the triangles that have been mentioned, it was necessary to fix the bearing of some one of the sides of those triangles in respect of the meridian. This was done by observing the azimuth of the pole-star, relatively to the given line, at its greatest distance from the meridian, both on the east and west sides. This method of ascertaining the angle which any line on the earth's surface makes with the meridian, we apprehend to be greatly preferable, for expedition as well as accuracy, to any other that is known. It cannot, however, be practised to advantage, but with such an instrument as the great theodolite, which answers for a transit, and carries a telescope of power sufficient to render the pole-star visible during the day. This, therefore, is one of the circumstances on account of which we think the British survey entitled to a preference above every other.

That a check might not be wanting on any errors that had crept into a work of such variety and extent, General Roy caused a second base to be measured on the flat ground of Rumney-Marsh,

which was not far distant from the southern extremity of the series of triangles. When the length of this base, as actually measured, was compared with that deduced by connecting it with the base on Hounslow-heath, the two results were found to differ only by twenty-eight inches, which must appear very inconsiderable, when we reflect that the two lines are more than sixty miles asunder. There was reason, nevertheless, to suspect that this base of verification was not so correctly measured as that on Hounslow-heath.

The conclusions deduced from all these observations, as far as respects the relative position of the observatories of Greenwich and Paris, are, first, that the distance between their parallels of latitude is 963954 feet, = 182.567 miles, which corresponds on the earth's surface to an arch of 2° 38′ 26′′ in the heavens, (the difference of latitude,) and therefore the length of a degree of the meridian in the latitude 50° 10′ comes out = 60843 fathoms = 69.14 miles.

Again, the perpendicular from the tower of Dunkirk on the meridian of Greenwich is found to be 547058 feet; from which, subtracting 9080 feet, the distance of Dunkirk east of the meridian of Paris, we have the perpendicular let fall from the point in the meridian of Paris, which is in the parallel of Greenwich, on the meridian of this last

=

= 537978 feet 101.89 miles. * The General also having determined, from the length and azimuth of one of the lines in the survey, (between Botley-Hill and Goudhurst in Kent,) to how many fathoms a degree of longitude in that parallel corresponds, has from thence deduced the difference of longitude of Greenwich and Paris 2° 19′ 51′′, or, in time, 9m 19′′.4; which agrees with the conclusion which Dr Maskelyne had before drawn from data purely astronomical. It must, however, be observed, that Legendre deduces from the same measurement a result considerably different, and makes the difference of longitude of the two observatories 9 21′′, (Mém. de l'Acad. 1788,) which is 1 second in time greater than the preceding.

But though nothing certainly can exceed the

* Another result, not uninteresting, is the breadth of the English Channel where it is narrowest. The line from the Keep of Dover Castle to the station at Blancnez is 116660 feet 22.095 miles. The South Foreland appears to be about two miles nearer to Blancnez, if we measure on the map which accompanies the survey. The least breadth of the Channel, therefore, does not exceed twenty miles ;-a narrow but a strong barrier,-one of those indelible lines which nature has kindly traced out on the surface of the earth, to resist the ambition, and preserve the independence of nations.

accuracy of General Roy's observations, we cannot bestow praise equally unqualified on the methods by which the results are deduced from them. The General has made use, as before mentioned, of the spherical excess, for the purpose of estimating the accuracy of his observations; yet he has not derived, from the introduction of that new element, all the advantage which it is capable of affording. He has great merit in being the first to make use of it, though he did not perceive the whole of its importance. This was indeed first made known by a theorem of Legendre, in the Memoirs of the Academy of Sciences for 1787, from which it appears, that if each of the angles of a small spherical triangle be diminished by one-third of the spherical excess, the sines of the angles thus diminished will be very nearly proportional to the lengths of the sides themselves; so that the computations with respect to such spherical triangles may be made by the rules of plane trigonometry. General Roy was probably unacquainted with this theorem, which is not of very easy investigation; and though he has virtually employed it in part, because he always reduced the angles of his triangles to 180° before he used them in calculation, yet he derives no benefit from it in many of the cases where it is of the greatest importance. These are when two angles only of a triangle have been observed, and it is required to find the third angle; or, again, in calcu