members of their body undertook the measurement of two meridian arcs-one in the neighbourhood of the equator, the other in a high latitude; and so arose the celebrated expeditions of the French Academicians. In May 1735, MM. Godin, Bouguer, and De la Condamine, under the auspices of Louis XV., proceeded to Peru, where, assisted by two Spanish officers, after ten years of laborious exertion they measured an arc of 3° 7′ intersected by the equator. The second party consisted of Maupertuis, Clairaut, Camus, Lemonnier, and Outhier, who reached the Gulf of Bothnia in July 1736; they were in some respects more fortunate than the first party, inasmuch as they completed the measurement of an arc near the polar circle of 57' amplitude and returned to Europe within sixteen months from the date of their departure.

The measurement of Bouguer and De la Condamine was executed with great care, and on account of the locality, as well as the manner in which all the details were conducted, it has always been regarded as a most valuable determination. The southern limit was at a place called Tarqui, the northern at Cotchesqui. A base of 6272 toises was measured in the vicinity of Quito, near the northern extremity of the arc, and a second base of'5260 toises near the southern extremity. The mountainous nature of the country made the work very laborious, in some instances the difference of heights of two neighbouring stations exceeding a mile. The difficulties with which the observers had to contend were increased by the opposition of the more ignorant of the inhabitants, and they were at times in danger of losing their lives. They had also much trouble with their instruments, those with which they were to determine the latitudes proving untrustworthy. But their energy and ingenuity were equal to the occasion, and they succeeded by simultaneous observations of the same star at the two extremities of the arc in obtaining very fair results. The whole length of the arc amounted to 176,945 toises, while the difference of latitudes was 3° 7' 3". In consequence of a misunderstanding that arose between De la Condamine and Bouguer, their operations were conducted separately, and each wrote a full and interesting account of the operation. Bouguer's book was published in 1749; that of De la Condamine in 1751. The toise used in this measure was ever after regarded as the standard toise, and is always referred to as the Toise of Peru.

The party of Maupertuis, though their work was quickly despatched, had also to contend with great difficulties. They were disappointed in not being able to make use of the small islands in the Gulf of Bothnia for the trigonometrical stations, and were forced to penetrate into the forests of Lapland. They commenced operations at Tornea, a city situated on the mainland near the extremity of the gulf. From this, the southern extremity of their arc, they carried a chain of triangles northward to the mountain Kittis, which they selected as the northern terminus. In the prosecution of this work they suffered greatly from cold and the bites of flies and gnats. The latitudes were determined by observations with a sector (made by Graham) of the zenith distance of a .and 8 Draconis. The base line was measured on the frozen surface of the river Tornea about the middle of the arc; two parties measured it separately, and they differed by about 4 inches. The result of the whole was that the difference of latitudes of the terminal stations was 57' 29'6, and the length of the arc 55,023 toises. In this expedition, as well as in that to Peru, observations were made with a pendulum to determine the force of gravity; and these observations coincided with the geodetical results in proving that the earth was an oblate and not prolate spheroid.

In 1740 was published in the Paris Mémoires an account, by Cassini de Thury, of a remeasurement by himself and

Lacaille of the meridian of Paris. With a view to determine more accurately the variation of the degree along the meridian, they divided the distance from Dunkirk to Collioure into four partial arcs of about two degrees each, by observing the latitude at five stations. The anomalous results previously obtained by J. and D. Cassini were not confirmed, but on the contrary the length of the degree derived from these partial arcs showed on the whole an increase with increasing latitude. In continuation of their labours, Cassini and Lacaille further measured an arc of parallel across the mouth of the Rhone. The difference of time of the extremities was determined by the observers at either end noting the instant of a signal given by flashing gunpowder at a point near the middle of the arc.

While at the Cape of Good Hope in 1752, engaged in various astronomical observations, Lacaille measured an arc of meridian of 1° 13′ 17′′, which gave him for the length of the degree 57,037 toises-an unexpected result, which has led to the modern remeasurement of the arc by Sir Thomas Maclear.

Passing over the measurements made between Rome and Rimini and on the plains of Piedmont by the Jesuits Boscovich and Beccaria, and also the arc measured with deal rods in North America by Messrs Mason and Dixon, we come to the commencement of the English triangulation. In 1783, in consequence of a representation from Cassini de Thury on the advantages that would accrue to science from the geodetic connection of Paris and Greenwich, General Roy was with the king's approval appointed by the Royal Society to conduct the operations on the part of this country,-Count Cassini, Mechain, and Delambre being appointed on the French side. And now a precision previously unknown was brought into geodesy by the use of Ramsden's splendid theodolite, which was the first to make the spherical excess of triangles measurable. The wooden rods with which the first base was measured were speedily replaced by glass rods, which again were rejected for the steel chain of Ramsden. The details of this operation are fully given in the Account of the Trigonometrical Survey of England and Wales. Shortly after this, the National Convention of France, having agreed to remodel their system of weights and measures, chose, as applicable to all countries, for their unit of length the tenmillionth part of the meridian quadrant. In order to obtain this length precisely, the remeasurement of the French meridian was resolved on, and deputed to Delambre and Mechain. The details of this great operation will be found in the Base du Système Métrique Décimale. The arc was subsequently extended by MM. Biot and Arago to the island of Iviza.

The appearance in 1838 of Bessel's classical work entitled Gradmessung in Ostpreussen marks an era in the science of geodesy. Here we find the method of least squares, a branch of the theory of probabilities, applied to the calculation of a network of triangles and the reduction of the observations generally. This work has been looked on as a model ever since, and probably it will not soon be superseded as such. The systematic manner in which all the observations were taken with the view of securing final results of extreme accuracy is admirable. The triangulation, which is a small one, extends about a degree and a half along the shores of the Baltic in a N.N.E. direction. The compound bars with which he measured his base line may be understood by the following brief description. On the surface of an iron bar two toises in length is laid a zinc bar, both being very perfectly planed and in free contactthe zinc bar being slightly shorter than the iron bar. They are united at one end only, and as the temperature varies the difference of length of the bars as seen at the other end varies; this difference of length is a thermometrical indica-,

the actual instrument P and Q are finely engraved dots at the distance of 10 feet apart. In the measurement the bars when aligned do not come into contact; an interval of six inches is left between each bar and its neighbour. This small space is measured by an ingenious micrometrical arrangement constructed on exactly the same principle as the bars themselves. The triangulation was computed by least squares. The total number of equations of condition for the triangulation is 920; if therefore the whole had been reduced in one mass, as it should have been, the solution of an equation of 920 unknown quantities would have occurred as a part of the work. To avoid this an approximation was resorted to; the triangulation was divided into twenty-one parts or figures; four of these, not adjacent, were first adjusted by the method explained, and the corrections thus determined in these figures carried into the equations of condition of the adjacent figures. The average number of equations in a figure is 44; the largest equation is one of 77 unknown quantities.1

Airy's Zenith Sector is too well known to need description. The vertical limb is read by four microscopes; altogether, in the complete observation of a star there are 10 micrometer readings and 12 level readings. In some recent observations in Scotland for latitude the Zenith Telescope has been used with very great success; it is very portable; and a complete determination of latitude, affected with the mean of the declination errors of two stars, is effected by two micrometer readings and four level readings. The observation consists in measuring with the telescope micrometer the difference of zenith distances of two stars which cross the meridian, one to the north and the other to the south of the observer at zenith distances which differ by not much more than 10' or 15', the interval of the times of transit being not less than one nor more than twenty minutes. The advantages are that, with simplicity in the construction of the instrument and facility in the manipulation, refraction is eliminated (or nearly so, as the stars are generally selected within 25° of the zenith), and there is no large divided circle. The telescope, which is counterpoised on one side of the vertical axis, has a small circle for finding, and there is also a small horizontal circle. This instrument is universally used in American geodesy.

The United States Coast Survey has a principal triangulation extending for about 9° 30' along the coast, but the final results are not yet published.

In 1860 was published F. G. Struve's Arc du Méridien de 25° 20' entre le Danube et la Mer Glaciale mesuré depuis 1816 jusqu'en 1855. This work is the record of a vast amount of scientific labour and is the greatest contribution yet made to the question of the figure of the earth. The latitudes of the thirteen astronomical stations of this arc were determined partly with vertical circles and partly by means of the transit instrument in the prime vertical. The triangulation, a great part of which, however, is a simple chain of triangles, is reduced by the method of least squares, and the probable errors of the resulting distances of parallels is given; the probable error of the whole arc in length is 6.2 toises. Ten base lines were measured. The sum of the lengths of the ten measured bases is 29,863 toises, so that the average length of a base line is 19,100 feet. The azimuths were observed at fourteen stations. In high latitudes the determination of the meridian is a matter of great difficulty; nevertheless the azimuths at all the northern stations were successfully determined, the probable error of the result at Fuglences being 0"53.

1 See the volume of the Ordnance Survey, entitled Account of the Principal Triangulation of Great Britain and Ireland, by Captain A. R. Clarke, R.E., F.R.S., 1858.

Mechanical Theory.

Newton appears to have been the first to apply his own newly-discovered doctrine of gravitation, combined with the so-called centrifugal force, to the question of the figure of the earth. Assuming that an oblate ellipsoid of rotation is a form of equilibrium for a homogeneous fluid rotating with uniform angular velocity, he obtained the ratio of the axes 229: 230, and the law of variation of gravity on the surface. A few years later Huyghens published an investigation of the figure of the earth, supposing the attraction of every particle to be towards the centre of the earth, obtaining as a result that the proportion of the axes should be 578:579. In 1740 Maclaurin wrote his celebrated essay on the tides, one of the most elegant geometrical investigations ever made. He demonstrated that the oblate ellipsoid of revolution is a figure which satisfies the conditions of equilibrium in the case of a revolving homogeneous fluid mass whose particles attract one another according to the law of the inverse square of the distance; he gave the equation connecting the ellipticity with the proportion of the centrifugal force at the equator to gravity, and he determined the attraction on a particle situated anywhere on the surface of such a body. Some few years afterwards Clairaut published (1743) his Théorie de la Figure de la Terre, which contains, among other results, demonstrated with singular elegance, a very remarkable theorem which establishes a relation between the ellipticity of the earth and the variations of gravity at different points of its surface. Assuming that the earth is composed of concentric ellipsoidal strata having a common axis of rotation. each stratum homogeneous in itself, but the ellipticities and densities of the successive strata varying according to any law, and that the superficial stratum bas the same form as if it were fluid, he proves the very important theorem contained in the equation

Where g, g' are the amounts of gravity at the equator and at the pole respectively, e the ellipticity of the meridian, and m the ratio of the centrifugal force at the equator to g. Clairaut also proved that the increase of gravity in proceeding from the equator to the poles is as the square of the sine of the latitude. This, taken with the former theorem, gives the means of determining the earth's ellipticity from observation of the comparative force of gravity at any two places. Clairaut would seem almost to have exhausted the subject, for although much has been written since by mathematicians of the greatest eminence, yet, practically, very little of importance has been added. Laplace, himself a prince of mathematicians, who had devoted much of his own time to the same subject, remarks on Clairaut's work that "the importance of all his results and the elegance with which they are presented place this work amongst the most beautiful of mathematical productions" (Todhunter's History of the Mathematical Theories of Attraction and the Figure of the Earth, vol. i. p. 229).

The problem of the figure of the earth treated as a question of mechanics or hydrostatics is one of great difficulty, and it would be quite impracticable but for the circumstance that the surface differs but little from a sphere. In order to express the forces at any point of the body arising from the attraction of its particles, the form of the surface is required, but this form is the very one which it is the object of the investigation to discover; hence the complexity of the subject, and even with all the present resources of mathematicians only a partial and imperfect solution can be obtained, and that not without some labour. We may, however, here briefly indicate the line of reasoning by which some of the most important of the results we

We take into account the rotation of the earth by subtracting the centrifugal force fu2= F from X. Now, the surface of constant density upon which the point f, 0, h is situated gives (1 − 2e) ƒdƒ+hdh = 0; and the condition of equilibrium is that (X-F)df+Zdh = 0. Therefore, (X-F)h = Zƒ(1 − 2e),

which, neglecting small quantities of the order e2 and putting w2 = 42, gives

the land by canals—be a spheroid with small ellipticity, then the law of gravity will be that found above.1

An important theorem by Jacobi must not be overlooked He proved that for a homogeneous fluid in rotation a spheroid is not the only form of equilibrium; an ellipsoid rotating round its least axis may with certain proportions of the axes and a certain time of revolution be a form of equilibrium.2

Local Attraction.

In speaking of the figure of the earth, we mean the surface of the sea imagined to percolate the continents by canals. That this surface should turn out, after precise measurements, to be exactly an ellipsoid of revolution is a priori improbable. Although it may be highly probable that originally the earth was a fluid mass, yet in the cooling whereby the present crust has resulted, the actual solid surface has been left in form the most irregular. It is clear that these irregularities of the visible surface must be accompanied by irregularities in the mathematical figure of the earth, and when we consider the general surface of our globe, its irregular distribution of mountain masses, continents, with oceans and islands, we are prepared to admit that the earth may not be precisely any surface of revolution. Nevertheless, there must exist some spheroid which agrees very closely with the mathematical figure of the earth, and has the same axis of rotation. We must conceive this figure as exhibiting slight departures from the spheroid, the two surfaces cutting one another in various lines; thus a point of the surface is defined by its latitude, longitude, and its height above the spheroid Call this height for a moment n; then

of reference. of the actual magnitude of this quantity we can generally have no information, it only obtrudes itself on our notice by its variations. In the vicinity of mountains it may change sign in the space of a few miles; n being

6

0

if its differential coefficient with respect to the former be zero at a certain point, the normals to the two surfaces then will lie in the prime vertical; if the differential coefficient of a with respect to the longitude be zero, the

two normals will lie in the meridian; if both coefficients are zero, the normals will coincide. The comparisons of terrestrial measurements with the corresponding astronomical observations have been accompanied with discrepancies. Suppose A and B to be two trigonometrical stations, and that at A there is a disturbing force drawing the vertical through an angle 8, then it is evident that the

ever

the equator. Since sec = (1+e+e

which expression contains the theorems we have referrea to as discovered by Clairaut.

The theory of the figure of the earth as a rotating ellipsoid has proved an attractive subject to many of the greatest mathematicans, Laplace especially, who has devoted a large portion of his Mécanique Céleste to it. In English the principal existing works on the subject are Sir George Airy's Mathematical Tracts, where the subject is treated in the lucid style so characteristic of its author, but without the use of Laplace's coefficients, Archdeacon Pratt's Attrac tions and Figure of the Earth, and O'Brien's Mathematical Tracts; in the last two Laplace's coefficients are used. In the Cambridge Transactions, vol. viii., is a valuable essay by Professor Stokes, in which he proves, without making any assumption whatever as to the ellipticity of internal strata, or as to the past or the present fluidity of the earth

A', whose distance from A is rd, when r is the earth's radius; and similarly if there be a disturbance at B of the amount S', the apparent zenith of B will be really that of some other place B', whose distance from B is rd. Hence we have the discrepancy that, while the geodetical measurements deal with the points A and B, the astronomical observations belong to the points A', B. Should 8, & be equal and parallel, the displacements AA', BB' will be equal and parallel, and no discrepancy will appear. The nonrecognition of this circumstance often led to much perplexity in the early history of geodesy. Suppose that, through the unknown variations of n, the probable error of an observed latitude (that is, the angle between the normal to the mathematical surface of the earth at the given point and that of the corresponding point on the spheroid of reference) be e, then if we compare two arcs of a degree

See also a paper by Professor Stokes, in the Cambridge and Dublin Mathematical Journal, vol. iv. 1849.

See a paper in the Proceedings of the Royal Society, No. 123 1870, by I. Todhunter, M.A., F.R.S.