Definition of the sphere.

a - c d - b.

d e - b g - a - c - f. No percussion made by a sphere counts for the whole weight of that sphere unless it is made with the front of the diameter of that sphere, when the diameter lies on the central line of the motion of that sphere. The percussion made by the sphere will be the more powerful the nearer its diameter is to the central line of its motion. The percussion made by the sphere will be of so much less power the more the central line of its motion is remote from the diameter of that sphere. The power of the percussion made outside the central line of the motion is always worth twice the smaller part of the sphere that remains outside the course of the part.

Two figures of spheres with: o.

The particulars conclude the universals.

And the reflections will not be equal.

a - b - c. How impetuses are transmutable powers from one body into another. It is proved. How the impetuses transmitted from one percussor into the other will be of so much greater or lesser power as the percussion is made along lines nearer to the central line of their motion.

The percussion is divided into books, of which the first demonstrates the percussion of two bodies, of which one moves the percussor against the immobile struck body; the other moves the percussors and the struck bodies alternately one against the other; the third is of liquid materials; the fourth of bendable bodies; the fifth...

r - L a h - m p n. On imperfect percussion. The figure a is of invalid percussion upon the site n m at the point p, because of all its weight it strikes only the line r p; for this reason no reflected motion is generated, and where dense bodies do not penetrate the place struck by their impetus, necessity constrains them to consume the acquired impetus by reflected motion. But here, although the place n m struck at p is not penetrated by its percussor, it does not reflect, and for this we have manifest experience that such a percussion is next to nothing, as experience shows us in seeing such a body descend through the air, which at once turns with the arrow of its arc and cord toward the centre of the motion, as is proved in the book of bodies descending through the air. (Three little figures with: a, n, x.) How the immobile jouster leaves the percussor immobile, and the immobile one takes up again the motion left by the one that became immobile.

a - weak - power of 10; b - strong - power of twenty. On the percussion of two moving bodies of various speeds. Moving bodies of various powers are of two kinds: one is of bodies of equal weight and speed; the other of various weights and various speed; another is of equal weight and unequal speed; another of various weights and equal speed; another of various weights and various speeds, that is greater or lesser interchangeably in the one and the other. If you strike the arrival of the ball on the central line of its motion, without doubt it will return back by the same path it came; but if you strike it on the smaller portion, it will not go back, nor forward, but will bend with a transverse motion, losing as much impetus as was the power of the percussion.

n g - S - a - d - b L - r e - t c - m K. First. Every percussion made in the heavy sphere directs the central line of its power toward the centre of the struck sphere. It is proved: let the straightness of the two contrary motions be n m, and the site of the percussion of the two spheres be b. Here it is shown that the power of the percussion is two times more than the smaller portion of the struck sphere, because every balancing of percussion must be made between equal angles of the two powers, as is proved in the book of weights.

That percussor strikes the object with so much less power the more it finds the struck body fleeing with a swifter flight. But that percussor strikes the object with a more intense percussion which finds the object more stable. And it has the greatest percussion in the greatest speeds of dense bodies, being of equal weight and speed. That percussor of less powerful motion will reflect at a smaller distance from the site of the percussion, which strikes with a part nearer to the central line of its motion. And this is proved, because that part of the moving body which is toward the centre of the percussor wants to return back and keep company with the motion of its percussor, while the part from the centre outward, which was not impeded, wants to follow the begun motion; whence of necessity such motion will be revoluble about the centre of its gravity. And if the difference of their powers is unequal, then the more powerful part moves the body somewhat along with it until it consumes the remainder of its impetus. ...able; fifth...

...and with its reflected motion keep company with the incident motion of its victor. But the part from the site of the percussion outward, which was not in the power of the percussor and did not occupy the percussor nor oppose its motion, wants to follow its straight motion through the free air. But being joined to the part that returns back, it is necessary that neither the reflected nor the incident motion be chosen, but that of such a compound a motion of triple composition be made: of which one is reflected motion, another incident, and the third, to annul such contention, will be straight motion, which will carry along a third motion unlike each of the two first, before they struck, and unlike those generated after the percussion, and which will be straight and transverse.

Equal peripheries do not contain equal space unless they are of equal curvature, that is, that their arc has an equal chord and sagitta.

2.

First. The surface of that [cylinder] which will have the diameter of its base equal to its axis is... Quadruple is the surface of the cylinder to its base, when the diameter of that base is equal to the axis of the cylinder.

First. I have proved that such proportion is from circle to circle as from square to square constructed, etc.

Third. By the definition of the circle it is proved that all the straight lines drawn from the periphery of the semicircle to the middle of its straight side are equal to one another.

Fourth - c - n - b a - m. The surface of the cone with the slant-side equal to the diameter of its base is equal to that semicircle which is double the base of that cone. The surface of that cone is equal to that semicircle which is double the... It is proved: all the straight lines produced from the base of the said cone to its apex are equal, by the definition of the cone, and we have already made the slant-side equal to the diameter of that base.