Water in its weight, or air, will be no more deceived by the makers of their instruments than fire is by the alchemists.

1000 pounds. / 1000 pounds. Counterweight. Water. / Here two figures are proposed, that is two water-cisterns with two counterweights that press upon these cisterns with weights equal between themselves, and two tubes of equal emptiness (bore). But one cistern is very wide and the other very narrow. I say that the proportion of the escapes of the water into the air will be such as, oppositely, is the width of one cistern to the other. As if we said: the first cistern is a thousand times wider than the thickness of the tube through which the water escapes, and the weight that presses is 1000 pounds. Therefore the thousandth part of that counterweight presses to the benefit of the water that rises up that tube, which will be one pound of counterweight. Hence it will be able to raise from that level of water, enclosed up along its tube, no more than a single pound of water. But if the said cistern is ten times thicker than the tube through which it drives that water, then a hundred pounds of that counterweight will bear to the benefit of the tube that drives the water. Hence a hundred pounds of water can escape up that tube at one stroke. Now you have seen that over one same thickness of tubes rises water pressed by equal quantities of weight, whose rises have among themselves such proportion in their heights as, oppositely, do the widths of the counterweights and of the waters pressed by them. Among the heights of the waters issuing from tubes of equal thickness, from which the water of the cisterns is driven upward by equal weights, the proportion of height will be such as, oppositely, will be the proportion of the widths of the cisterns pressed. Among the height of the waters that rise out of equal thickness of tubes pressed by equal weights, there will be such proportion as is that of the width of their opposite cisterns, from which such waters are pressed and driven by the weights.

If a droplet of water falls upon the sea when it is calm, it is needful and necessary that the whole surface of the sea gain an imperceptible height; because water cannot be compressed within itself as air can.

a b c d e f n m - 1 g - h f - r K. Here it falls to say how the velocity of the water that runs from a to b will be equal to that which runs enclosed above through b and d, then uncovered through d e, then shut down through e f n, and then uncovered through n m. Here there will be no difference, the whole channel being of equal thickness, and the open channel of equal slope, and the parts of the straight slope being equal, that is a b, d e, n m; for the reach that were longer, the water that moves more becomes faster, by the seventh [proposition] of "On Motion and Weight," which in any case shows that the water nonetheless runs faster at n than at a b. But set the figures apart as below, l g and h f and r K.

a n / m o - f c / I have three tubes, that is a n, m o, f e, which, being of equal mouth at one of their ends, and full of water and loaded on top by a counterweight, one asks which presses the water most upon its bottom.

A column of fraction computations. The figures, from top to bottom, read: 3 3 3 over 1 9 9 (multiplied); 12 11 over 7 1 (multiplied); 6 3 3 over 1 5 30 (multiplied); 1 over 10; 3 3 3 over 1 9 27 (divided); 3 7 7 over 1 9 27 (multiplied); 3 7 over 1 27; 21; 27.