respect to the pole (suppose 1°), we require to travel over a longer measured space at one point than at another, we assign to that point a less curvature. Hence we conclude that the curvature of a meridional section of the earth is sensibly greater at the equator than towards the poles; or, in other words, that the earth is not spherical, but flattened at the poles, or, which comes to the same, protuberant at the equator.

(175.) Let NA B D E F represent a meridional section of the earth, C its centre, and NA, BD, GE,

na

arcs of a meridian, each corresponding to one degree of difference of latitude, or to one degree of variation in the meridian altitude of a star, as referred to the horizon of a spectator travelling along the meridian. Let n N, a A, b B, dD, g G, e E, be the respective directions of the plumb-line at the stations N, A, B, D, G, E, of which we will suppose N to be at the pole and E at the equator; then will the tangents to the surface at these points respectively be perpendicular to these directions; and, consequently, if each pair, viz. n N and a A,

b B and d D, g G and e E, be prolonged till they intersect each other (at the points x, y, z), the angles NxA, By D, GE, will each be one degree, and, therefore, all equal; so that the small curvilinear arcs NA, BD, GE, may be regarded as arcs of circles of one degree each, described about x, y, z, as centres. These are what in geometry are called centres of curvature, and the radii x N or x A, y B or y D, z G or ≈ E, represent radii of curvature, by which the curvatures at those points are determined and measured. Now, as the arcs of different circles, which subtend equal angles at their respective centres, are in the direct proportion of their radii, and as the arc NA is greater than BD, and that again than GE, it follows that the radius N must be greater than By, and By than Ez. Thus it appears that the mutual intersections of the plumb-lines will not, as in the sphere, all coincide in one point C, the centre, but will be arranged along a certain curve, xyz (which will be rendered more evident by considering a number of intermediate stations). To this curve geometers have given the name of the evolute of the curve NABDGE, from whose centres of curvature it is constructed.

(176.) In the flattening of a round figure at two opposite points, and its protuberance at points rectan_ gularly situated to the former, we recognize the distinguishing feature of the elliptic form. Accordingly, the next and simplest supposition that we can make respecting the nature of the meridian, since it is proved not to be a circle, is, that it is an ellipse, or nearly so, having NS, the axis of the earth, for its shorter, and EF, the equatorial diameter, for its longer axis; and that the form of the earth's surface is that which would arise from making such a curve revolve about its shorter axis NS. This agrees well with the general course of the increase of the degree in going from the equator to the pole. In the ellipse, the radius of curvature at E, the extremity of the longer axis is the least, and at that of the shorter axis, the greatest it admits, and the

form of its evolute agrees with that here represented.* Assuming, then, that it is an ellipse, the geometrical properties of that curve enable us to assign the proportion between the lengths of its axes which shall correspond to any proposed rate of variation in its curv ature, as well as to fix upon their absolute lengths, corresponding to any assigned length of the degree in a given latitude. Without troubling the reader with the investigation, (which may be found in any work on the conic sections,) it will be sufficient to state that the lengths which agree on the whole best with the entire series of meridional arcs which have been satisfactorily measured, are as follow: :

The proportion of the diameters is very nearly that of 298 299, and their difference

very little greater than góō·

of the greater, or a

(177.) Thus we see that the rough diameter of 8000 miles we have hitherto used, is rather too great, the excess being about 100 miles, or th part. We consider it extremely improbable that an error to the extent of five miles can subsist in the diameters, or an uncertainity to that of a tenth of its whole quantity in the compression just stated. As convenient numbers to remember, the reader may bear in mind, that in our latitude there are just as many thousands of feet in a degree of the meridian as there are days in the year (365): that, speaking loosely, a degree is about 70 British statute miles, and a second about 100 feet; and that the equatorial circumference of the earth is a little less than 25,000 miles (24,899).

(178.) The supposition of an elliptic form of the earth's section through the axis is recommended by its

The dotted lines are the portions of the evolute belonging to the other quadrants.

t See Profess. Airy's Essay before cited.

simplicity, and confirmed by comparing the numerical results we have just set down with those of actual measurement. When this comparison is executed, discordances, it is true, are observed, which, although still too great to be referred to error of measurement, are yet so small, compared to the errors which would result from the spherical hypothesis, as completely to justify our regarding the earth as an ellipsoid, and referring the observed deviations to either local or, if general, to comparatively small causes.

(179.) Now, it is highly satisfactory to find that the general elliptical figure thus practically proved to exist, is precisely what ought theoretically to result from the rotation of the earth on its axis. For, let us suppose the earth a sphere, at rest, of uniform materials throughout, and externally covered with an ocean of equal depth in every part. Under such circumstances it would obviously be in a state of equilibrium; and the water on its surface would have no tendency to run one way or the other. Suppose, now, a quantity of its materials were taken from the polar regions, and piled up all around the equator, so as to produce that difference of the polar and equatorial diameters of 26 miles which we know to exist. It is not less evident that a mountain ridge or equatorial continent, only, would be thus formed, from which the water would run down to the excavated part at the poles. However solid matter might rest where it was placed, the liquid part, at least, would not remain there, any more than if it were thrown on the side of a hill. The consequence, therefore, would be the formation of two great polar seas, hemmed in all round by equatorial land. Now, this is by no means the case in nature. The ocean occupies, indifferently, all latitudes, with no more partiality to the polar than to the equatorial. Since, then, as we see, the water occupies an elevation above the centre no less than 13 miles greater at the equator than at the poles, and yet manifests no tendency to leave the former and run towards the latter, it is evident that it must be

retained in that situation by some adequate power. No such power, however, would exist in the case we have supposed, which is therefore not conformable to nature. In other words, the spherical form is not the figure of equilibrium; and therefore the earth is either not at rest, or is so internally constituted as to attract the water to its equatorial regions, and retain it there. For the latter supposition there is no primâ facie probability, nor any analogy to lead us to such an idea. The former is in accordance with all the phenomena of the apparent diurnal motion of the heavens ; and, therefore, if it will furnish us with the power in question, we can have no hesitation in adopting it as the true one.

(180.) Now, every body knows that when a weight is whirled round, it acquires thereby a tendency to recede from the centre of its motion; which is called the centrifugal force. A stone whirled round in a sling is a common illustration; but a better, for our present pur

pose, will be a pail of water, suspended by a cord, and made to spin round, while the cord hangs perpendicularly. The surface of the water, instead of remaining horizontal, will become concave, as in the figure. The centrifugal force generates a tendency in all the water to leave the axis, and press towards the circumference; it is, therefore, urged against the pail, and forced up its sides, till the excess of height, and consequent increase of pressure downwards, just counterbalances its centrifugal force, and a state of equilibrium is attained. The experiment is a very easy and instructive one, and is admirably calculated to show how the form of equilibrium accommodates itself to varying circumstances. If, for example, we