The terms ‘continuity,’ ‘contiguity’ (or touching), and ‘next-in- succession’ defined and distinguished. These definitions suffice to show that a continuum (such as length, time, movement) cannot be constituted by indivisibles (points, ‘nows,’ stations) or be resolved into them. Nor can two points (nows, stations) be continuous or contiguous one with another (231 a 21–29).
Further demonstration that points cannot by contiguity form a continuum; for indivisibles must either be in the same proper place (i.e. be positionally identical) or else be entirely isolated (they cannot occupy different places without having intervals between them), whereas the successive parts of a continuum occupy different places but have nothing between them (a 29–b 6).
A line cannot be constituted by a succession of points which are ‘next without contact’; for between ‘nexts’ there is nothing of their own category, and what lies between any two points is linear extension which is divisible at intermediate points (see above). So that points lie between any two points, and no point is next to any other (b 6–18).
A formal proof that the argument holds equally for spatial magnitude, time, and motion, and that all three hang together. It rests on two axioms:
The first: That when motion is taking place, something is moving from here to there and vice versa.
The second: That the mobile or subject which experiences the motion cannot simultaneously be in the act of moving towards a given position and in the state of being already at it.
Chapter IArgument (continued)
The steps of the argument are: (1) If L, a component of motion, is itself a motion, then (by axiom ii) after L has started and before it has finished, P (the mobile)is (by axiom i) past the start and short of the finish of A (the distance). Therefore A is divisible in correspondence with L; and so likewise are B and C with M and N. (2) If it were still maintained that A, etc., need not be (divisible) distances but might be (indivisible) ‘terms’ in the distance, it would involve one or other of the following impossibilities: (a) If L, etc., were motions, P would be in motion (while L was in progress) without moving from A; and so with M and N, and B and C. (b) If L, etc., were not motions, P would never be in motion but would accomplish the motion without moving. Therefore both distance and motion must be divisible (b 18–232 a 18).
Time is divisible if distance and motion are, and vice versa, for if the whole of the length A is traversed in time T, a part of it would be traversed (at equal speed) in less than T. Or if the whole time T were occupied in traversing the distance A, then in part of the time less than A would be traversed (a 18–22).
Note.—The absence of method in the system of lettering in the Greek text makes the discussion in this and the following chapter unnecessarily difficult to follow. Therefore an entirely independent system of lettering has been adopted in the translation. But for purposes of comparison duplicate diagrams are given showing the two systems of lettering side by side. Where there is no diagram the Greek letters are given in brackets after the English.