a – K – b – L – c – m – d h g f – i – e – q d. The curve f e is part of the sphere of the water, and no part of water of that sphere will have motion of itself. And no water will move with straight and equal motion unless it is curved with a curvature uniformly non-uniform, [equally] distant from the sphere of the water and from the centre of the world. This non-uniformity will be, in all equal partitions, of excesses always equal, as is 1 2 3 4 5 6 etc., where the excess is always one unit; as you see 1 and 2, that the two exceeds one by one unit, and the 3 exceeds 2 by another unit, and so it continues continuously from unit to unit; and the spaces interposed between these increments are equal, as: for every 100 braccia of the sphere of the water the course of the river approaches one finger toward the centre of the world, and every 3 fingers make a braccio, and 300 braccia make a mile; and this motion of the water, being [slow], makes a mile per hour; and by this slowness the river neither ruins nor bends its straight banks; whence such a motion of the water has a curved surface both in width and in length; which curvatures are always varied, because they always increase in approaching that centre of the world; and this variety is uniform, because the partition and the descent are uniform.

1 . 3 . 6 . 11

2 1 –3; 4 2. It is impossible that weights placed in the air in double proportion of weight, being then joined together, should make motions equal to the antecedent ones; rather they will be as much faster as the weight of the compound is greater than before.

Of the herbs or gravel that impede the motion of the waters.

o – e a – c b – n – m n o. In doubled circles about one and the same centre, the excess of the greater is worth the smaller. It follows that half of the excess is worth half of the smaller circle, and a quarter of the whole greater is worth half of the excess and half of the smaller circle. Therefore from these 3 equalities I shall take an equal part from the 2, and the remainder will be equal; and n will be taken from the quarter-circle a b c, and from the half-excess I shall take m o.