Text: Let a b a' represented the path of the earth in its orbit about the sun s, between the point in its entire orbit most remote from the sun at a and that nearest the sun at a', or between the higher and lower vertices of the ellipse. In pursuing the path from a toward a', the earth constantly approaches the sun until it arrives at a', where there is exhibited the greatest predominance, if any, of the force of gravitation over that of repulsion, or of centripetal over centrifugal force, obeyed by a velocity of the earth in direction of the sun, greater than at any other point through which it has passed. If it is true that the approximation of the earth to the sun in occasioned by the predominance of the power of gravitation over that of repulsion, then, the earth increasing in velocity from a to a', it is evident, that the predominance has been increasing from a to a', and that the earth from the point a' must be impelled toward the sun by a force greater than has affected its movement in any other point of its orbit from a to a'. Now, the path of the earth from a to a' is that of a spiral closing upon s, and if the nature of the forces acting upon the earth, resulting in that path, are supposed to be the same at a' as at a, the continued path of the earth should be upon the prolongation of the spiral in the direction o, in obedience to the constantly-increasing predominance of the power of gravitation, or centripetal force. But at a' the earth begins to depart from the sun upon thepath a' b' a; which, having accomplished, its entire orbit has become an ellipse composed of two spirals, a a' and a' a, one closing and one expanding relatively to s, symmetrical with each other; for the completion of which the power of repulsion, or the centrifugal force, from a' to a, must exceed that of gravitation, by the same mesure in which the power of gravitation exceeded that of repulsion to perfect the path a b a'. It follows that the forces of gravitation and repulsion expended in forcing the earth to the completion of its entire orbit have been equal. Since the pwer of gravitation is at its maximum at a' where the earth has acquired its greatest velocity in the direction of the sun, the fact, that at this point the earth begins to recede from the sun, shows that the power of repulsion at this point is equal to that of gravitation. At a', from whence might have been expected the most rapid approach to the sun, the opposing forces have become equal, and a' has, in terms of mechanics, become a dead point as to motion , in direction of a right line passing through a' and the sun's center, and in direction of the sun generally as relates to that part of the orbit already accomplished. By a like process of reasoning a becomes another dead point in the reverse. It necessarily follows, that the opposed forces affecting the point a, or the earth, at a and a' being equal, they must be equal at any and every intermediate point between a and a'; and as the peculiarity of the ellipse is, that as to its path it approaches to and departs from the sun, or the focus to which one of the opposed forces is always directed, the orbit, being the result of opposed forces always equal, can only result by reason of a constant change of position of those forces relatively to each other.

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Source: 159