446 THE MATHEMATICAL PRINCIPLES [BOOK III upon the situation of the moon's apogee in respect of the sun, which is in its greatest quantity when the moon's apogee is in the octants of the sun, and vanishes when the apogee arrives at the quadratures or syzygies ; and it is added to the mean motion while the moon's apogee is passing from the quadrature of the sun to the syzygy, and subducted while the apogee is passing from the syzygy to the quadrature. This equation, which I shall call the semi-annual, when greatest in the octants of the apogee, arises to about 3' 45", so far as I could collect from the phenomena : and this is its quantity in the mean distance of the sun from the earth. But it is increased and diminished in the reciprocal triplicate proportion of the sun's distance, and therefore is nearly 3' 34" when that distance is greatest^ and 3' 56" when least. But when the moon's apogee is without the octants, it becomes less, and is to its greatest quantity as the sine of double the distance of the moon's apogee from the nearest syzygy or quad rature to the radius. By the same theory of gravity, the action of the sun upon the moon is something greater when the line of the moon's nodes passes through the sun than when it is at right angles with the line which joins the sun and the earth ; and hence arises another equation of the moon's mean motion, which I shall call the second semi-annual ; and this is greatest when the nodes are in the octants of the sun, and vanishes when they are in the syzygies or quadratures ; and in other positions of the nodes is propor tional to the sine of double the distance of either node from the nearest syzygy or quadrature. And it is added to the mean motion of the moon, if the sun is in antecedentia, to the node which is nearest to him, and subducted if in consequential and in the octants, where it is of the greatest magnitude, it arises to 47" in the mean distance of the sun from the earth, as I find from the theory of gravity. In other distances of the sun, this equation, greatest in the octants of the nodes, is reciprocally as the cube of the sun's distance from the earth ; and therefore in the sun's perigee it comes to about 49", and in its apogee to about 45". By the same theory of gravity, the moon's apogee goes forward at the greatest rate when it is either in conjunction with or in opposition to the sun, but in its quadratures with the sun it goes backward ; and the ec centricity comes, in the former case, to its greatest quantity ; in the latter to its least, by Cor. 7, 8, and 9, Prop. LXVI, Book 1. And those ine qualities, by the Corollaries we have named, are very great, and generate the principal which I call the semi-annual equation of the apogee ; and this semi-annual equation in its greatest quantity comes to about 12° 18', as nearly as I could collect from the phenomena. Our countryman, HorroXj was the first who advanced the theory of the moon's moving in an ellipsis about the earth placed in its lower focus. Dr. Halley improved the notion, by putting the centre of the ellipsis in an epicycle whose cen