Book I of the Waters.

The point is that of which nothing can be said smaller; it is the boundary of nothing and the beginning of the thing, without occupation of place.

The line is a length born together with the motion of the point and terminated at the end of the motion of that point; than which line nothing can be said narrower.

The surface is composed of width with length; it is generated by the transverse motion of the line, and its extremities are lines; than which nothing can be said thinner.

The body is a length that has in itself width with depth, created by the motion of the width of the surface, and its bounds are surfaces.

Every body always has quantity, but not always weight; because the line and the surface are incorporeal quantities and consequently have no weight; and likewise the earth has no weight, nor water, nor the other elements in themselves.

The line is an extension generated by the motion of the point, than which nothing can be narrower; and the point is that which the mind does not divide, and nothing smaller has place in that mind.

The point is the common boundary between nothing and the front of the line; it is neither 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 between themselves, but not conjoined; and in such contact is the point. Of which point nothing is the brother, because all the points that can be imagined are equal to a single point, as all the zeros of arithmetic are in value equal to a single zero. And for this reason it does not follow that many points imagined in contact one with another in a straight line compose the line; and consequently many lines joined together along their sides, one after another, will never make a surface, nor will many equal surfaces set wholly one upon another ever make a body; because among us no body is composed of incorporeal things.

The times of dividing things successively by half are equal, both for him who begins with the mind to divide the universe, and for him who begins at that same time to divide with the mind the natural point. Therefore from the natural point to the mathematical point there is infinite proportion as regards dividing. As it would be to divide the infinite by half; two finite parts will be made of it, and so it will be done in the same time with the natural point; because in the one and the other division made in equal number successively, the mathematical point will never be reached. Therefore things of almost infinite proportion have an equal number of divisions made in equal time with the mind, since in act they cannot be made; and there is no other difference save that the infinite has greater parts than the natural point.

Every continuous quantity is divisible to infinity; therefore the divided of that quantity will never be brought to the point, which is given as the extremity of the line. It follows that the width and thickness of the natural line is divisible to infinity.

It is asked whether all infinities are equal, or one greater than another. It is answered that every infinite is eternal, and eternal things are of equal permanence, but not of equal length of age; because that which in act was first begun to be divided has passed more ages, but the times to come are equal.

Body is called that which is enclosed within the surface, in which our seeing terminates. Surface we call that space which is surrounded by lines. Line is that which is surrounded by space.

No element has in itself gravity or levity unless it moves. The earth is in contact with the water and with the air and has in itself neither gravity nor levity; it feels nothing from the water nor from the air that surrounds it except by accident, which is born from their motion. And this is taught us by the leaves of the herbs grown upon the earth that is in contact with water and air, which do not bend except by the motion of the air or of the water.

By the foregoing we shall say that gravity is an accident created by the motion of the lower elements into the higher.

Levity is an accident created by the rarer element drawn beneath the less rare, which then moves because it cannot resist.

Levity is not generated except together with gravity, nor gravity without levity. It is proved. Let air be pushed beneath the water by blowing into a tube; then such air will acquire levity while standing beneath the water, and the water will acquire gravity through having beneath it the air, a body rarer and lighter than itself.

Therefore levity is born from the heavy, and the heavy from the light. And in one and the same time they give birth to one another, repaying the benefit of their being; and in one and the same instant they destroy one another as avengers of their deaths.

Levity and gravity in the same time are mother and daughters of one another.

Gravity and levity in the same instant give birth to one another, repaying the benefit of their being, and in one and the same instant they destroy one another with a common vengeance for their death.

Levity and gravity are caused by motion immediately.

Motion is created by heat and by cold.

Motion is an accident born of inequality of weight or of force.

The air has in itself no natural place and always rests upon a body denser than itself, never upon one lighter that is in contact with it except by violence.

The motion of the elements is born from the sun.

The heat of the universe is generated by the sun.

The light and the heat of the universe come from the sun, and the cold and the darkness from the privation of the sun.

Every motion of the elements is born from heat and from cold.

Gravity and levity are created in the elements.

...which then acquires weight, which is generated immediately when such an element lacks resistance; which resistance being overcome by the weight, it does not change without change of substance; it changes, and acquires the name of levity.