414 THE MATHEMATICAL PRINCIPLES [BOOK III But there are yet other inequalities not observed by former astronomers, by which the motions of the moon are so disturbed, that to this day we have not been able to bring them under any certain rule. For the veloc ities or horary motions of the apogee and nodes of the moon, and their equations, as well as the difference betwixt the greatest eccentricity in the syzygics, and the least eccentricity in the quadratures, and that inequality which we call the variation, are (by Cor. 14, Prop. LXVI, Book I) in the course of the year augmented and diminished in the triplicate proportion of the sun's apparent diameter. And besides (by Cor. 1 and 2, Lem. 10, and Cor. 16, Prop. LXVI, Book I) the variation is augmented and diminished nearly in the duplicate proportion of the time between the quadratures. But in astronomical calculations, this inequality is commonly thrown into and confounded with the equation of the moon's centre. PROPOSITION XXI1L PROBLEM V. To derive the unequal motions of the satellites of Jupiter and Saturn from the motions of our moon. From the motions of our moon we deduce the corresponding motions of the moons or satellites of Jupiter in this manner, by Cor. 16, Prop. LXVI, Book I. The mean motion of the nodes of the outmost satellite of Jupiter is to the mean motion of the nodes of our moon in a proportion compound ed of the duplicate proportion of the periodic times of the earth about the sun to the periodic times of Jupiter about the sun, and the simple propor tion of the periodic time of the satellite about Jupiter to the periodic time of our moon about the earth ; and, therefore, those nodes, in the space of a hundred years, are carried 8° 24' backward, or in antecedentia. The mean motions of the nodes of the inner satellites are to the mean motion of the nodes of the outmost as their periodic times to the periodic time of the former, by the same Corollary, and are thence given. And the motion of the apsis of every satellite in consequential is to the motion of its nodes in antecedentia as the motion of the apogee of our moon to the motion of its nodes (by the same Corollary), and is thence given. But the motions of the apsides thus found must be diminished in the proportion of 5 to 9, or of about 1 to 2, on account of a cause which I cannot here descend to ex plain. The greatest equations of the nodes, and of the apsis of every satel lite, are to the greatest equations of the nodes, and apogee of our moon re spectively, as the motions of the nodes and apsides of the satellites, in the time of one revolution of the former equations, to the motions of the nodes and apogee of our moon, in the time of one revolution of the latter equa tions. The variation of a satellite seen from Jupiter is to the variation of our moon in tne same proportion as the whole motions of their node?