First book. The point has no middle, and its boundaries are nothing. The adversary says that nothing is the void, and that the void does not exist in nature; and if the point were the void it would not have being, and, not existing, the point would not be in being. It is answered that if the void existed it would be place with occupation of place, and the point, which is in a place without occupying place, is not the void — the point that exists is in a place without occupying place, and for this reason the point is not the void. The point has no middle, and its boundaries are nothing. The adversary says nothing would be the void and the void does not exist in nature, and, not existing, the point too would not exist. It is answered that if the void existed it would give place with occupation of place, whence it follows that the point, which is in a place without occupying place, is not the void; for if it were the void it would have a middle and be divisible. Whence it follows that nothing, being in a place without occupying place, cannot be the void. Nothing, the boundary of the point, being situated in a place without occupation of place, cannot be the void. The point has no middle, and its boundaries are nothing. The adversary says that nothing and the void are alike, and that, the void not existing in nature, nothing also does not exist. Nothing has no middle, and its boundaries are nothing. The adversary says nothing and the void are one and the same thing with two names, which are spoken and do not exist in nature. It is answered that if the void existed it would give the place that surrounds it, and nothing exists without occupation of place. It follows that nothing and the void are not alike, because the one is divisible to infinity and nothing is not divided, for no thing can be less; and if a part of it were given, it would be equal to the whole and the whole to the part.

The collapses of mountains over cavernous places shut in the air of their caverns, which, in order to escape, breaks the earth and generates earthquakes. The adversary says this cannot be, because either the whole mountain covering the cavern falls, or only the inner part falls; and if it all falls, then the compressed air escapes through the opening of the uncovered cave; and if only the inner part falls, then the compressed air flees back into the void that the falling earth leaves of itself.

Water that runs through a canal of uniform course and width will be higher in the middle and at the sides, forming in the middle a hill between two valleys, and at the ends two half-hills at the borders of its banks.

3000 / 16 / 18000 / 3000 / 48000 / 20 / 00000 / 960000 (a multiplication reaching 9600⁰⁰.

There are 2 settlings made by arches: the first is that walls settle sooner at the surface than in the middle, and for this reason the surfaces divide their joints.

a - c - b.

n c - p f - m g. If the circular ring is 1/4 of the whole larger circle, the sector c f g p is likewise 1/4 of the said circle. It follows that the circular ring and the sector are equal to each other. Therefore, taking away the portion c f g from the circular ring and the same portion from the sector, there remains the remainder of the sector, which is rectilinear, equal to the remainder of the curvilinear ring. But remember that the curvature of the portion of the circle must always be of the same curvature as the ring.

a – d b - c. Let the third of a circle a b c d be taken, from which the rectilinear triangle a c d is cut.

b a. Let a square be given equal to the 2 lunes a b.

x d c. Let there also be given a square equal to the two helices c d.

The continuous mover always gives continuous motion to that which is joined with it. The converse follows, namely that the discontinuous mover produces discontinuous motions. Therefore every effect partakes of its cause. Among bodies of various colours, deprived of one same light, such proportion will be from shadow to shadow as there is from colour to colour of those bodies.

The centre of every suspended heavy body is entirely throughout the whole central line of its support, and it is entirely in every part of it.

f n m - 2 - l c l b l a - 2 d. a is worth b, d is worth c, c b is worth f (that is 2), and m is worth three. Therefore f n against n m are 3 spaces against two spaces n m, because a d are 3 pounds against 2 pounds in b c.

1 1 1 - 2 1 1 - 3 4; 1 1 1 - 2 2; 1 1 1 - 1.