b 2 - a - 4

a b

First - 2 - a - 1 - b n - m 4. Vtu, Vpo, Vio, Mle. Force - Time - Space - Moved body.

First - l b - 2 - a n - m 4. The whole force, half the moved body, in the whole time, twice the space.

Second. The same force, half the moved body, over the whole space, in half the time.

Third - 2 - 4. The same force, the whole moved body, half the space, half the time.

Fourth - 1 - 2. The same force, the whole time, half the body, the whole space.

Fifth - 4 - 4. Double the moved body, double the time, the same space. One po - 1 c - 1 T - 1 s; p, c.

2 4

On the squares of their roots.

1 1. The cannonball travels a mile in 5 tempi, of which tempi one hour is composed of one thousand and eighty; 5 goes 216 times into that 1080, which works out, such motion continuing at 216 miles per hour. The mile is 3000 braccia, and a braccio is understood from the shoulder to the hand of a common man; and the arrow travels 2 miles per tempo, that is 2160 miles per hour. You will find there to be such a proportion from the circumferential line of the sphere of air to the circumferential line of the sphere of earth as is from 5 to 1, putting the whole sphere of earth as one quantity and the sphere of air as 100; for the cube root of a hundred is 5 and the cube root of 1 is 1; and 1 is a side of the square made from that root.

That wave will be of greater size which is generated by a swifter mover. The wave of the air, therefore, will be greatest in its upper part, which is moved by the motion made by the sphere of fire that borders it, which moves more than (two thousand) miles per hour. For I presuppose, as the ancients concluded, that the proportion of the sphere of air is tenfold to the sphere of water, which sphere I reckon to turn as much as is said of the earth, that is 25,000 miles; and because the earth is about as much above the sphere of water as it is below — that is, the highest mountain outside the sphere of water I reckon to be as much above the water as the greatest depth of the sea is below that water — so that we might compute the one lowness of the earth covered by water to be as far beneath the skin of the water as the highest land is above that skin; and, the excess being put back into the deficiency, the skin of the water would remain about equal to the sphere of the earth, were it of perfect sphericity. Now, the sphere of water being tenfold to the earth and incorporated with it, it would be nothing other than the sphere of the earth were it spherical. Therefore, the earth being 25,000 miles, the water lacking which is tenfold the earth, and the air coming to be a hundredfold that earth by such tenfold proportion — and whoever should move the whole circle of the earth in twenty-four hours, it being 24,000 miles, would move a thousand miles per hour. Now what would it be per hour for one who should move over the surface of the air, which is a hundred times more than the surface of the earth, that is a hundred in quantity but not in circuit? Take the cube root of the earth and likewise of the air and make the circles in the one cube and the other, and you will see the true quantities of these bodies of air and earth. And because we say that if the earth were made of perfect sphericity it would be like, or about like, the sphere of water now seen, and then the whole sphere of water would be united and would cover the earth with equal thickness — and because it is supposed ten times more than the earth, and the air ten times more than the water, it follows that the air is a hundred times the sphere of the earth. Now if you imagine separately, in the form of a sphere, the element of earth, which is one quantity, and the element of air, which is 100 of those quantities, and take their cube roots, which is 1 for the earth and 5 for the air, saying '1 times 1 makes 1, and 5 times 5 makes 25, and 5 times twenty-five makes 100,' you find the circuits of their ...

5; 1

one thousand; 80; 03; 1080; 1080; 216; 1080; 555; 2160; 24; 216; 12; 320; 80; 320; 5; 24; 12; 20; 24; (83²⁰; 120; 864; 184; 320; 432; 160; 32; 5184; 1920; 32.