CHAPTER XXII.

OPERATIONS FOR DETERMINING THE Figure of tHE EARTH.

An arc of the terrestrial meridian measured in France by Picard and Cassini.—The terrestrial spheroid supposed at first to be prolate.--Proof of the diminution of gravity in the equatorial regions.-Arcs of the meridian measured in Lapland and Peru.-The figure of the earth proved to be oblate.- Trigonometrical operations in England.-Arcs of the meridian measured in various parts of the world.--Great geodetical operations in France and Spain.-Experiments of Captain Sabine on the lengths of pendulums.-Ratio of the earth's equatorial and polar diameters.-Effects of local attractions in geodetical and astronomical observations.

IN the time of Newton the question of the dimensions and figure of the earth particularly engaged the attention of mathematicians, and several efforts were made to determine both, by ascertaining, and comparing together, the lengths of certain portions of the terrestrial meridian. The attempts of the ancient Greeks and Arabians to acquire a knowledge of the earth's magnitude have been already mentioned, and we may observe that similar attempts had, just before the time of which we are speaking, been made by Ricciolus, Fernel, and Snellius, on the continent, and by Norwood in England; but from their labours no satisfactory conclusion could be obtained, on account of the inaccuracy of their itinerary measurements. At length, about the year 1670, the French Academy of Sciences engaged Picard to determine the distance, in the direction of the meridian, from Malvoisine to Amiens, by means of a measured base line, and a series of triangles formed between those places. This base was made a side of one of the triangles, and the angles of all the triangles being taken with instruments, the lengths of the several sides were computed; from which, with the observed azimuths, or bearings of the sides from the meridian, the corresponding arcs of the meridian were found: finally, comparing the length of the whole arc with the difference, in latitude, between its

extremities, it was found that the length of a degree of the meridian, in that district, was 57060 toises. The arc of the meridian measured by Picard was afterward, by Dominicus Cassini, extended from Amiens to Perpignan, towards the south, and by Jaques Cassini, in 1713, from Amiens to Dunkirk, northward; and the lengths of the different degrees of latitude within this extent being separately ascertained, it was found that they diminished in going from north to south, a circumstance which indicated that the polar diameter exceeded, in length, the equatorial diameter of the earth, or that our planet had the form of a prolate spheroid this indication appeared, subsequently, to be confirmed by the results which Cassini de Thury obtained from the measurement of the length of a degree of longitude in the parallel to Brest; for this was found to be shorter by 781 toises than it should have been if the earth were a perfect sphere; and, if such had been the fact, it would also have followed that the earth was elongated in the direction of its axis. Hence arose, in France, a prejudice in favour of this opinion, which was not easily removed, and which operated strongly for a time to retard, on the continent, the acquiescence in the Newtonian theory of gravitation; because, from the latter, as we have seen, in consequence of the earth's rotation on its axis, it should have been found that the equatorial, is longer than the polar diameter, or that the earth is an oblate spheroid. While the opinion prevailed that the polar diameter was the longest, there were not wanting philosophers, like Mairan, in France, who believed that they had discovered a physical reason for it, though it is now known to have had no other foundation than the inaccuracies of the observations and admeasurements; it is, however, but fair, to say that, when the error was detected, it was with the utmost candour acknowledged, and ample justice was rendered to the sagacity of the English philosopher whose theory now, daily, received additional confirmation from the agreement between the consequences drawn from it and the results of operations then constantly carried on with superior diligence and care.

A proposal having been made in the French Academy to ascertain the parallax of Mars, at the time of his opposition to the sun, by his distance from a certain fixed star observed at the

same instant in two places, considerably distant from each other, on the earth; with a view of computing, from thence, the value of the sun's parallax and the distances of the planets from that luminary; M. Richter, one of the members, was appointed to proceed to Cayenne in order to make there the necessary observations, simultaneously with those which were to be made at Paris, and was furnished with a pendulum clock for the purpose of obtaining the right ascensions of stars by their transits over the meridian: on comparing together the times of the transits of particular stars on successive nights, this astronomer was surprised to find that the pendulum, which had been carefully regulated according to mean time at Paris, made, in one day, at Cayenne, a number of vibrations less by 148 than it made at the former place, or that the movement of the clock was too slow, daily, by 2′ 28′′; so that, to cause the pendulum to make the same number of vibrations at both places, it was found necessary, at Cayenne, to diminish its length by above one tenth of an inch. Now, as the effect was too great to be considered as arising from the expansion of the rod by heat, or from a greater resistance of the air in those regions, no other conclusion could be drawn from it than that the force of gravity was less near the equator than in France; and though, at first, some doubt prevailed concerning the justness of this conclusion, because it was supposed possible that the difference in the time of a vibration might be owing to some local cause; yet when, soon afterward, similar results were obtained from observations, made expressly for the purpose, in other parts of the world, it was readily admitted that the fact, in accordance with the Newtonian theory, amounted to a satisfactory demonstration that the equatorial, was longer than the polar diameter of the earth; in consequence of which, terrestrial bodies at the surface of the latter, in the equatorial regions, being further from the centre than in France, were less powerfully attracted by gravity, and more affected by the centrifugal force arising from the earth's daily rotation on its axis.

But the limits of France were, evidently, insufficient to allow any conclusive argument concerning the figure of the earth to be drawn from the variations in the lengths of the degrees of latitude measured in that country alone; and the desire of ob

taining data from more extensive operations induced the French Academy to send some of the distinguished men who were then associated with that learned body to the torrid and frigid zones for the purpose of ascertaining, by their admeasurements, with more precision, the proportion between the equatorial and polar diameters of the earth; the length of a degree measured in one of those regions compared with that of a degree measured in the other, necessarily, presenting a much greater difference than can exist between two degrees measured in any one country. In 1735 M.M. Maupertuis, Clairaut and Lemonnier proceeded to Lapland and, at the same time, M.M. De la Condamine and Bouguer sailed to Peru to fulfil the objects of their respective missions; the former party was joined by M. Celsius, a Swedish philosopher; and the latter, by the Spanish mathematicians Don George Juan and Don Antonio di Ulloa; and each party, labouring for its own glory as well as for the interests of science, prosecuted its operations with singular zeal and assiduity. That in the north encountered, as may be expected, vast difficulties from the nature of those frozen regions; a base, however, was measured upon the ice of the river Torneo, the mercury in Reaumur's thermometer being at 37 degrees below the freezing point, and a series of triangles was extended, from the city of that name, to Pello, towards the north, including an arc of nearly one degree; and the result of the operations was that, in the seventysixth degree of latitude, the length of a degree on the meridian is equal to 57438 toises. The measured arc of the meridian in Peru extended along the valley of the Cordilieres, from Cotchesqui, nearly under the equator, to Tarqui towards the south; and the difference of latitude between the two extremities of the arc was rather greater than 3 degrees: the operations cost the labour of eight years, and the length of a degree at the equator was found to be equal to 56753 toises [=60484.5 English fathoms].

On the return to Paris, of the mathematicians who had been engaged in these distant and laborious surveys, a comparison was made of the results they had obtained; and the earth's compression at the poles was determined from the lengths of the degrees of latitude in the several places: considerable

differences, however, were found in the values of the compression, but the fact itself was fully confirmed. The degree in Peru compared with that in France gave, for the ratio of the equatorial to the polar diameter, 304 to 303; the former degree compared with that in Lapland, gave 211 to 210; while, according to the theory of gravitation, the earth being supposed homogeneous, it should have been as 231 to 230. But some doubts having been raised about the accuracy of the base line, measured in Lapland, the length of a degree in that country was, by M. M. Swanberg and Offerbom, re-measured in 1801, nearly on the site of the former operations but, upon an arc of greater extent, and found to be equal to 57196 toises, [=60956-6 English fathoms,] or about 212 toises less than it had been before made; and this new measure, being compared with that of the degree in Peru, gave the ratio of 334 to 333 for that of the earth's dia

meters.

In 1783, the government of France, at the recommendation of M. Cassini de Thury, proposed to that of England to unite the geodetical operations then in progress, in the former country, with corresponding operations in this, for the purpose of deterinining with accuracy the difference of longitude between Greenwich and Paris; and the proposal, being accepted, was immediately put in execution, under the direction of General Roy who, having measured a base on Hounslow Heath, extended a series of triangles from thence to the coast of Kent, where it was connected with the series formed between Paris and the opposite shores of France: a detailed account of these operations was published in the Philosophical Transactions; but the plan first adopted by government was, afterward, enlarged, and a complete trigonometrical survey of Great Britain was executed chiefly under the direction of General Mudge who, in the course of the proceedings, in the years 1800, 1 and 2, measured, between Dunnose in the Isle of Wight and Clifton in Yorkshire, an arc of the meridian including nearly three degrees of latitude, from which it was found that the length of a degree, in lat. 52° 50′ 30′′, is equal to 60766 English fathoms; and in lat. 51° 2′ 54", to 60884 fathoms". Colonel Colby was subsea Operations for a Trigonometrical Survey of England by Mudge and Dalby.