all attributes, in fact, which are divisible.There can be nothing divisible in an indivisible thing, but the attributes of bodies are all divisible in one of two ways.They are divisible into kinds, as colour is divided into white and black, and they are divisible per accidens when that which has them is divisible.In this latter sense attributes which are ****** are nevertheless divisible.Attributes of this kind will serve, therefore, to illustrate the impossibility of the view.It is impossible, if two parts of a thing have no weight, that the two together should have weight.But either all perceptible bodies or some, such as earth and water, have weight, as these thinkers would themselves admit.Now if the point has no weight, clearly the lines have not either, and, if they have not, neither have the planes.Therefore no body has weight.It is, further, manifest that their point cannot have weight.For while a heavy thing may always be heavier than something and a light thing lighter than something, a thing which is heavier or lighter than something need not be itself heavy or light, just as a large thing is larger than others, but what is larger is not always large.A thing which, judged absolutely, is small may none the less be larger than other things.
Whatever, then, is heavy and also heavier than something else, must exceed this by something which is heavy.A heavy thing therefore is always divisible.But it is common ground that a point is indivisible.
Again, suppose that what is heavy or weight is a dense body, and what is light rare.Dense differs from rare in containing more matter in the same cubic area.A point, then, if it may be heavy or light, may be dense or rare.But the dense is divisible while a point is indivisible.And if what is heavy must be either hard or soft, an impossible consequence is easy to draw.For a thing is soft if its surface can be pressed in, hard if it cannot; and if it can be pressed in it is divisible.
Moreover, no weight can consist of parts not possessing weight.
For how, except by the merest fiction, can they specify the number and character of the parts which will produce weight? And, further, when one weight is greater than another, the difference is a third weight; from which it will follow that every indivisible part possesses weight.For suppose that a body of four points possesses weight.A body composed of more than four points will superior in weight to it, a thing which has weight.But the difference between weight and weight must be a weight, as the difference between white and whiter is white.Here the difference which makes the superior weight heavier is the single point which remains when the common number, four, is subtracted.A single point, therefore, has weight.
Further, to assume, on the one hand, that the planes can only be put in linear contact would be ridiculous.For just as there are two ways of putting lines together, namely, end to and side by side, so there must be two ways of putting planes together.Lines can be put together so that contact is linear by laying one along the other, though not by putting them end to end.But if, similarly, in putting the lanes together, superficial contact is allowed as an alternative to linear, that method will give them bodies which are not any element nor composed of elements.Again, if it is the number of planes in a body that makes one heavier than another, as the Timaeus explains, clearly the line and the point will have weight.For the three cases are, as we said before, analogous.But if the reason of differences of weight is not this, but rather the heaviness of earth and the lightness of fire, then some of the planes will be light and others heavy (which involves a similar distinction in the lines and the points); the earthplane, I mean, will be heavier than the fire-plane.In general, the result is either that there is no magnitude at all, or that all magnitude could be done away with.For a point is to a line as a line is to a plane and as a plane is to a body.Now the various forms in passing into one another will each be resolved into its ultimate constituents.It might happen therefore that nothing existed except points, and that there was no body at all.
A further consideration is that if time is similarly constituted, there would be, or might be, a time at which it was done away with.
For the indivisible now is like a point in a line.The same consequences follow from composing the heaven of numbers, as some of the Pythagoreans do who make all nature out of numbers.For natural bodies are manifestly endowed with weight and lightness, but an assemblage of units can neither be composed to form a body nor possess weight.