marked by the centres of two wooden tubes, and have since been more permanently ascertained by the centres of two iron cannon sunk in the ground.

Experiments and observations, of the kind which we are now considering, seldom fail to benefit science, not only directly, but indirectly, by the collateral objects to which they lead. A pyrometer, constructed by Ramsden, for the purpose of ascertaining the expansion of solid bodies by heat, is probably the best instrument of the kind which has yet been made, and is one of the monuments of skill and genius that will long preserve the memory of that incomparable artist.

It was not till the summer of 1787 that the measurement was resumed, by actually extending a series of triangles from Greenwich to Dunkirk. For this purpose, signals were erected, in such conspicuous situations, and at such distances, as were judged convenient: the straight lines joining these points formed a set of triangles, the angles of which were measured by a theodolite which Ramsden had constructed, and which was carried, successively, to all the stations. In these triangles, therefore, which were so formed that one side was always common to two of them, all the angles became known; and a side of one of them being also given, viz. the base on Hounslow-heath, the sides of all the rest could be found by trigonometrical computation. So also, the bearing of any one of

the sides, in respect of the meridian, being known by observation, the bearings of all the rest with respect to the same meridian were determined.

The theodolite by which the angles of these triangles were measured, was superior to any thing that had ever been used in geodetical observations, and might be compared with the best instruments of astronomy. Ramsden had exerted himself to the utmost both in the design and execution of it: he had united in it the powers of a theodolite, a quadrant, and a transit instrument; and had made it capable of measuring horizontal angles to fractions of a second. It was furnished with a telescope of a much higher magnifying power than had ever been before applied to observations purely terrestrial; and by this superiority in its construction, even if it were the only one, we are persuaded that the surveys made with it are more accurate than any other. The French academicians, for example, who joined General Roy at Dover, as we shall see presently, employed in their measurement a very excellent instrument, a circle of repetition, of the kind invented by Borda; and by taking the same angle several times over on different parts of the limb, they could diminish the error arising from the division of the instrument to an indefinite degree. But there was another error which they could not diminish, viz. that which arose from the

small power of their glasses, and the consequent largeness of the real diameters of the objects which appeared to them as points or lines. Though their observations were therefore extremely good, and far exceeding any that had been made in France previous to the introduction of Borda's circle, they do not seem to equal those in the English survey. As England was later in undertaking works of this sort than France, and some other nations on the Continent, it seemed but just that she should aim at superior excellence; and possessing, as Cassini says, the first artist in the world, it was not difficult for her to attain it.

The high power of the telescope just mentioned, obviated many of the difficulties concerning the signals employed to distinguish the precise point at each station that was to be intersected, from the rest. When the object to be intersected was not the spire of a steeple, a flag staff was commonly used; but when the distance was great, or the weather not very favourable, lights were employed, and the observations were made in the night. These lights were either reverberatory lamps, or white lights, (so called from their extreme brilliancy,) fired at a particular time previously agreed on. The signals made in this manner were visible at a great distance even in bad weather. Cassini says, that he hardly expects to be believed, when he

tells, that he observed one from about Calais, which was fired on the opposite shore, about forty miles off, and in bad weather.

The precaution taken, of placing the great theodolite at all times with its centre exactly perpendicular to the point that was to be intersected from the other stations, deserves also to be mentioned. Though the allowance to be made for the distance of the instrument from the angular point is easily computed, yet it is difficult to avoid some error in doing so; and the frequent recurrence of such errors is a source of inaccuracy which it is much better to have entirely cut off,

From all these circumstances, added to others which we cannot here enumerate, the angles were generally observed with such accuracy, as to manifest the effect of the earth's sphericity, by giving the sum of the three angles of a triangle somewhat greater than 180°, and that even where the sides did not exceed 15 or 20 miles. This excess above 180° is produced by the plane of the instrument at the three angular points of the triangle not being parallel to itself, but perpendicular to three lines which meet (at least nearly) in one point, the centre of the earth. It is called the spherical excess; and it was in this survey that there came, for the first time, to be any question concerning the quantity of it, in each triangle. The instruments used in former surveys had never

been accurate enough to bring a quantity so small as hardly ever to amount to 4′′ in one triangle, to be an object of investigation. In the observations made for verifying the meridian of Paris about fifty years ago, the error in the three angles of a triangle often amounts to 20 or 30 seconds; and then, of course, no question could occur about a correction which cannot exceed the tenth part of that quantity. But in General Roy's observations, the error in the three angles of a triangle never reaches 3"; and, therefore, the spherical excess is of importance to be ascertained. In justice to the French academicians who co-operated with the General, it must be observed, that the angles taken by them with Borda's instrument, were accurate within 1" or 2′′ on each angle, so that they found equal reason for employing the spherical excess.

This remark is applicable to all the measurements made in France since the period we are here treating of. They have all been made with the repeating circle, and seem to have reached a very high degree of accuracy.

The introduction of a new element, into trigono metrical computation is of great importance, and will probably be found to mark a precise era in all measurements relating to the figure of the earth. The spherical excess in any triangle has a given relation to the area of that triangle; for it is to 180° as that area is to the area of a great circle