continuously for a certain time.This will be a finite time, since the total time is finite in which the heavens complete their circular orbit, and consequently the time subtracted from it, during which the one line in its motion cuts the other, is also finite.Therefore there will be a point at which ACE began for the first time to cut BB.
This, however, is impossible.The infinite, then, cannot revolve in a circle; nor could the world, if it were infinite.
(3) That the infinite cannot move may also be shown as follows.
Let A be a finite line moving past the finite line, B.Of necessity A will pass clear of B and B of A at the same moment; for each overlaps the other to precisely the same extent.Now if the two were both moving, and moving in contrary directions, they would pass clear of one another more rapidly; if one were still and the other moving past it, less rapidly; provided that the speed of the latter were the same in both cases.This, however, is clear: that it is impossible to traverse an infinite line in a finite time.Infinite time, then, would be required.(This we demonstrated above in the discussion of movement.) And it makes no difference whether a finite is passing by an infinite or an infinite by a finite.For when A is passing B, then B overlaps A and it makes no difference whether B is moved or unmoved, except that, if both move, they pass clear of one another more quickly.It is, however, quite possible that a moving line should in certain cases pass one which is stationary quicker than it passes one moving in an opposite direction.One has only to imagine the movement to be slow where both move and much faster where one is stationary.To suppose one line stationary, then, makes no difficulty for our argument, since it is quite possible for A to pass B at a slower rate when both are moving than when only one is.
If, therefore, the time which the finite moving line takes to pass the other is infinite, then necessarily the time occupied by the motion of the infinite past the finite is also infinite.For the infinite to move at all is thus absolutely impossible; since the very smallest movement conceivable must take an infinity of time.Moreover the heavens certainly revolve, and they complete their circular orbit in a finite time; so that they pass round the whole extent of any line within their orbit, such as the finite line AB.The revolving body, therefore, cannot be infinite.
(4) Again, as a line which has a limit cannot be infinite, or, if it is infinite, is so only in length, so a surface cannot be infinite in that respect in which it has a limit; or, indeed, if it is completely determinate, in any respect whatever.Whether it be a square or a circle or a sphere, it cannot be infinite, any more than a foot-rule can.There is then no such thing as an infinite sphere or square or circle, and where there is no circle there can be no circular movement, and similarly where there is no infinite at all there can be no infinite movement; and from this it follows that, an infinite circle being itself an impossibility, there can be no circular motion of an infinite body.
(5) Again, take a centre C, an infinite line, AB, another infinite line at right angles to it, E, and a moving radius, CD.CD will never cease contact with E, but the position will always be something like CE, CD cutting E at F.The infinite line, therefore, refuses to complete the circle.
(6) Again, if the heaven is infinite and moves in a circle, we shall have to admit that in a finite time it has traversed the infinite.For suppose the fixed heaven infinite, and that which moves within it equal to it.It results that when the infinite body has completed its revolution, it has traversed an infinite equal to itself in a finite time.But that we know to be impossible.
(7) It can also be shown, conversely, that if the time of revolution is finite, the area traversed must also be finite; but the area traversed was equal to itself; therefore, it is itself finite.
We have now shown that the body which moves in a circle is not endless or infinite, but has its limit.