39S THE MATHEMATICAL PRINCIPLES [BOOK III PROPOSITION VIII. THEOREM VIII. Tn two spheres mutually gravitating each towards the other, if tlie matter in places on all sides round about and equi-distant from the centres is similar, the weight of either sphere towards the other will be recipro cally as the square of the distance between their centres. After I had found that the force of gravity towards a whole planet did arise from and was compounded of the forces of gravity towards all its parts, and towards every one part was in the reciprocal proportion of the squares of the distances from the part, I was yet in doubt whether that re ciprocal duplicate proportion did accurately hold, or but nearly so, in the total force compounded of so many partial ones; for it might be that the proportion which accurately enough took place in greater distances should be wide of the truth near the surface of the planet, where the distances of the particles are unequal, and their situation dissimilar. But by the help of Prop. LXXV and LXXVI, Book I, and their Corollaries, I was at last satisfied of the truth of the Proposition, as it now lies before us. COR. 1. Hence we may find and compare together the weights of bodies towards different planets ; for the weights of bodies revolving in circles about planets are (by Cor. 2, Prop. IV, Book I) as the diameters of the circles directly, and the squares of their periodic times reciprocally ; and their weights at the surfaces of the planets, or at any other distances from their centres, are (by this Prop.) greater or less in the reciprocal duplicate proportion of the distances. Thus from the periodic times of Venus, re volving about the sun, in 224 ^oVr? an^ ___i___ respectively. Then because as the distances are increased or di minished, the weights are diminished or increased in a duplicate ratio, the weights of equal bodies towards the sun, Jupiter, Saturn, and the earth, at the distances 10000, 997, 791, and 109 from their centres, that is, at their very superficies, will be as 10000, 943, 529, and 435 respectively. How much the weights of bodies are at the superficies of the moon, will be shewn hereafter. COR. 2. Hence likewise we discover the quantity of matter in the several