Page:Dialogues Concerning Two New Sciences (1914).djvu/62

34 to avoid the conclusion that these lines are built up of an infinite number of indivisible quantities because a division and a subdivision which can be carried on indefinitely presupposes that the parts are infinite in number, otherwise the subdivision would reach an end; and if the parts are infinite in number, we must conclude that they are not finite in size, because an infinite number of finite quantities would give an infinite magnitude. And thus we have a continuous quantity built up of an infinite number of indivisibles.

But if we can carry on indefinitely the division into finite parts what necessity is there then for the introduction of non-finite parts?

The very fact that one is able to continue, without end, the division into finite parts [in parti quante] makes it necessary to regard the quantity as composed of an infinite number of immeasurably small elements [di infiniti non quanti}. Now in order to settle this matter I shall ask you to tell me whether, in your opinion, a continuum is made up of a finite or of an infinite number of finite parts [parti quante].

My answer is that their number is both infinite and finite; potentially infinite but actually finite [infinite, in potenza; e finite, in atto]; that is to say, potentially infinite before division and actually finite after division; because parts cannot be said to exist in a body which is not yet divided or at least marked out; if this is not done we say that they exist potentially.

So that a line which is, for instance, twenty spans long is not said to contain actually twenty lines each one span in length except after division into twenty equal parts; before division it is said to contain them only potentially. Suppose the facts are as you say; tell me then whether, when the division is once made, the size of the original quantity is thereby increased, diminished, or unaffected.

It neither increases nor diminishes.

That is my opinion also. Therefore the finite parts [parti quante] in a continuum, whether actually or potentially present, do not make the quantity either larger or smaller; but it is perfectly clear that, if the number of finite parts actually contained