First book of the waters.

The point has no middle; but it is itself the middle of everything.

The point has no middle; but it is itself the middle, and nothing can be smaller.

The point is that least.

A point is that which the mind does not divide. A point is that which has no part.

The point is that than which nothing can be called smaller, and it is the common boundary of nothing with the line; nor is it nothing nor line, nor does it occupy place between nothing and the line. Therefore the end of nothing and the beginning of the line are in contact with each other, but not joined; and in such contact is the point, divider of the continuation of nothing with the line. It follows that the point is less than nothing, and if all the parts of nothing are equal to one, it will be concluded all the more that all the points too are equal to a single point, and one point is equal to all.

And from this it follows that many points imagined in continuous contact do not compose the line, and consequently many lines in continuous contact with their sides do not make a surface, nor many surfaces in continuous contact make a body, because among us bodies are not composed of incorporeal things.

The point is that which has no centre, because it is all centre, and nothing can be smaller.

The point is that which has no middle, and nothing can be smaller.

The contact of the liquid with the dense is a surface common to the liquid and to the dense, and the like the lighter liquids have with the heavier.

All the points are equal to one, and one to all.

New tower founded partly upon the old.

Nothing has a surface with the thing, and the thing has a surface with nothing, and the surface of the thing is not part of that thing. It follows that the surface of nothing is not part of that nothing; whence it is necessary that a single surface be the common boundary of two things that are in contact, as the surface of the water is not part of the water, and consequently is not part of the air, nor do other bodies interpose between them. What is it, then, that divides the air from the water? It is necessary that it be a common boundary that is neither of air nor of water, but is incorporeal, because a body interposed between two bodies prevents their contact, which does not happen with water and air, wherefore they are in contact without interposition of any medium. Therefore they are joined, and the air will not be raised nor moved without the water, nor can the flat thing be lifted from the other without drawing it after itself between the air. Therefore a surface is the common boundary of two bodies that are not continuous, and it partakes neither of the one nor of the other, because if the surface were part, it would have divisible thickness, which not being divisible, nothing divides such bodies one from the other.

The point has no middle in its quantity, nor the line in width, nor the surface in depth; but the body has a middle in thickness, width and depth.

Of the elements.

The bodies of the elements are joined, and in them there is neither gravity nor levity; gravity and levity are generated in the mixture of the elements.

The point is that which has no middle.

The line is a length (extension) made by the motion of the point, and its ends are the points.

The surface is an extension made by the transverse motion of the line, and its ends are the lines.

The body is a quantity made by the motion that has the face of the surface, and its ends are the surfaces.

A point is that which has no middle, and from this it follows that it has no width, length or depth.

A point is that which has no middle, whence it is indivisible in every aspect, and nothing is smaller than it.

A line is a length made by the motion of the point, whence it has no width or depth.

A body is a length, and has width with depth, made by the motion from the width of the surface.

The line is made by the motion of the point.

The point has no middle.

A line is length without width or depth.

A surface has width, length, without depth.

A body has width, length and depth.

A body is a length that has width with depth.

The body is made by the motion that has the length and width of the surface, and its ends are the surfaces.

A line is a length made by the motion of the point.

A body is a length with width and depth made by the motion of a face of the surface, and its ends are the surfaces.