Of the weight of the beam of the balance.

r - b S - n a - t o - s - v

The oblique weight pushes from r to S and from t to v. The pole of these motions is o, and the weights of the arms are n o t v and, on the opposite side, n o r S; the line n o divides these two powers, of which n o r S becomes the victor because it is greater than n o t v. This inequality of weights is the cause of the swinging of the balance, through which swinging the inequalities of the weights of its arms pass from one arm to the other as many times as such swinging lasts.

The weight that the beam of the balance has is divided into two parts, of which one tends toward the centre of the world and the other is accidental, because it moves by transverse motion. But the first, which gravitates toward the centre of the world, consists of equal opposed lateral weights that, with their centres of gravity and their distances, establish the mathematical centre of such a balance. The second mathematical centre may rather be called the mathematical point of the contact that the pole of the balance has with its support; and this lies outside the centre of the natural gravity of the balance by as much space as the upper part of the balance is of very great weight, greater than its lower part, whence the transverse weight of such a balance gives none of its gravity to either of the two aforesaid centres. And this is proved in the sixth proposition of this book, which says: "Every parallel body of uniform heaviness and weight, placed obliquely, has within itself two divided gravities, of which one tends toward the centre of the world and the other is transverse; but the one is natural and simple, and the other accidental and compound." But if such a body, so placed, has free descent through the air, the centres of the two gravities will transform one into the other over some space of motion, and in the end a single centre will remain common to the whole heavy body that descends, and thus with straight motion it will penetrate all the air that lies beneath it.