It follows that the centrifugal force at the poles, where there is the least motion, would not be equal to the force of gravitation, and at the equator must be exactly equal; but the deflection of a circle from a tangent measures the intensity of the earth’s attraction, and is equal to the versed sine of the arc described during that time, the velocity of the earth’s rotation being known, the value of the arc is deducible. The centrifugal force at the equator is equal to ¹⁄₂₈₉th part of the attraction of gravitation. Again, the uniformity of the earth’s mass becomes an object of consideration. Assuming that the figure of the earth is an ellipsoid of rotation, we will show the relation that form bears to the equal oscillation of a pendulum. Taking the earth as a homogeneous mass, analysis gives us the certainty that if the intensity of gravitation at the equator be taken as unity, the increase of gravity to the poles eliminating the differences of the centrifugal force must be = to 2.5, the ratio of the centrifugal force to that of gravitation at the equator. Now, taking the 2.5 of .346 = 1/115.2, this then must be the total increase of gravitation. Did we know the exact amount of increase at every point, from the equator to the poles, a perfect map of the form of the earth could be produced from calculation; experiment being from physical causes totally impracticable. The following analysis, quoted from an eminent physicist, gives a very lucid idea of the reasoning: “If the earth were a homogeneous sphere without rotation, its attraction on bodies on its surface would be everywhere equal. If it be elliptical and of variable density, the force of gravity ought to increase in intensity from the equator to the pole as _unity plus_ a constant quantity multiplied into the square of the sine of the latitude. But for a spheroid in rotation the centrifugal varies by the law of mechanics, as the square of the sine of the latitude from the equator, where it is greatest, to the poles, where it is least. And as it tends to make bodies fly off the surface, it diminishes the force of gravity by a small quantity. Hence, by gravitation, which is the difference of these two forces, the fall of bodies ought to be accelerated from the equator to the poles proportionably to the square of the sine of the latitude, and the weight of the body ought to increase in that ratio.”