This determination of the length of the pendulum was by far the most satisfactory that had hitherto been accom plished ; and the same method of operation was followed by Biot in his experiments to determine the intensity of gravity at several stations, both on the continent and in this country. Wc have already mentioned that the unit of measures adopted by the French academicians was the ten millionth part of the quadrant of the circumference of the earth. In order to determine this fundamental quantity, MM. Me chain and Dclambre undertook the measurement of the whole arc, stretching between Dunkirk and Barcelona ; and, often at the peril of their lives, continued their ope rations during all the horrors and anarchy of the Revolu tion. Mechain then conceived the bold idea of extending the arc through Spain to the Balearic Isles. He himself fell a victim to his exertions in the prosecution of this splen did undertaking: but able and not less zealous successors were found in MM. Biot and Arago, who completed it in 1807, by prolonging the measurement to Forntentera, a small island about 25' to the south of lvica. This ample arc afforded sufficient data for determining the magnitude of the earth ; but its figure could not be inferred with equal certainty from the lengths of the consecutive degrees; for where there is an accumulation of denser materials,in the neighbourhood of any station, the plumb-line will be drawn aside from the vertical ; in other words, the direction of gravity will not tend to the centre of the carth,•and thus the length of the degree will not be accurately obtained. The oscillations of the pendulum, on the other hand, are not affected by the direction of gravity; they are only sen sible to its intensity. Hence the reason for measuring a long arc, and for combining the measurement with experi ments on the pendulum. A series of experiments were, therefore, undertaken by Biot to determine the length of the pendulum at the extremities, and some intermediate stations of the arc, viz. Formentera, Figcac, Bordeaux, Clermont, Paris, and Dunkirk. At Formcntera he was assisted by Arago; at the other stations by Mathieu of the Board. of Longitudes.
In England an undertaking of a similar kind, but origi nating in other views, had been begun by General Roy, and carried on under the able direction of Colonel Mudge, till a series of triangles was extended from the south of Eng land to the north of Scotland. On the restoration of peace, an opportunity was taken of connecting the French and English arcs, by which a meridional line has ultimately been obtained, measured with excellent instruments, from the most southerly of the Balearic to the most northerly of the Shetland Islands,—the longest that the finger of ge ometry has ever attempted to trace on the surface of the earth. In order to render the operations as complete as possible, the Board of Longitudes in France were anxious that the length of the pendulum should be determined at some stations on the English arc, in the same manner as it had been done on the Continent. The governments of both countries readily acceded to this wish, and afforded every facility for carrying it into effect. Accordingly, in
the summer of 1817, Biot arrived in England, furnished the same apparatus which he had employed on the Continent,and made a series of observations at Leith Fort, and at Unst, the northern extremity of the English arc.
The apparatus employed by Blot was similar to that of Borda, with some modifications, chiefly with a view to render it more portable. The length of the trial pendulum was reduced from 12 feet to less than 3, or to very nearly that of a clock making 100.000 oscillations in 24 hours ; the number of seconds into which the French academicians, in conformity. with their decimal system of measures, had divided the mean solar day. The planes on which the knife-edges rested were of agate ; and the wire was of cop per, in order to avoid the possibility of its being affected by terrestrial magnetism. The ball was that which had been employed by Borda. The rule was of iron, having its head of tempered steel, so that it might not be penetrated by the knife-edge, which formed the transverse, by means of which it was suspended in the place of the pendulum, and with which it was brought into very close contact. The tongue was protruded by means of a fine screw. The value of its divisions, in a function of the metre, was as certained by a great number of experiments, both before it was carried into Spain, and during the operations at For mentera. But the principal improvement on the apparatus was the introduction of the Comparer; an ingenious instru ment invented by Fortin, a celebrated artist in Paris, for comparing lengths which differ from each other by quan tities extremely small. The shortness of the pendulum rendered it desirable to ascertain the fractions of the divi sions of the rule, with greater accuracy than could be done simply by means of verniers.
The Comparer is composed of a metallic rule, T R, Plate CCCCLVIII. Fig. 7. very straight, and strengthened by cross bars, to prevent it from bending. At one of its extremities is placed a heel 1', with which one of the ends of the rule to be compared is brought into contact. A frame, R R' is moveable along the rule, and can be firmly secured at any point by means of two strong screws R R'. This frame forms the essential part of the Comparer. It is furnished with a fixed pin C, which serves as an axis for the bent lever b C b'. In the instrument employed by Biot, the ratio of the arms of the lever was 1 to 10 ; hence, if the extremity of the short arm was moved forward a space = x, the extremity of the long arm would describe a space = 10 x. A circular scale, D D', divid ed into fifths of a millimetre, was fixed upon the frame, in order to measure the space passed over by the long arm ; a vernier was attached to the arm, which indicated the tenths of these divisions, or the fiftieths of a millimetre ; and the space described by the extremity V being ten times greater than that described by b, each division of the vernier indicated a variation of b equal to the 500th part of a millimetre.