Democritus might have based his atomism on an argument like this: let all possible cuts be made in a natural body. This leaves either something or nothing. If nothing, then things must be made from nothing, but this is impossible. Therefore there is something unable to be cut: a-tomos.
Notice, however, that this indivisibility is purely factual, not logical. Domocritus cannot argue that there is a logical impossibility of the atom being unable to be cut, only a factual one. This is the difference between the inability of an atom to be divided and the inability of a subsistent point to be divided.
Just Thomism 
pseudonoma said,
April 28, 2014 at 11:46 pm
“…and the inability of a subsistent point to be divided.”
…or, to use a different reference of indivisibility, the indivisibility of prime matter. I wonder which is the more appropriate reference, if either can be said to be. The question is of interest to me because it may (depending on how we interpret the phenomenon of a geometrical entity) have a bearing on the categories which provide your post with its title, namely, the factual and the logical.Does the indivisibility of prime matter not point to another primacy, distinct from both the factual and the logical –say, the metaphysical?
One might order it this way, moving from the more to the less formal according to the order of being:
1. the logical indivisibility of the “subsistent point” ( I confess, I am not sure what you mean here, unless we be Platonists, since a point that subsists seems only to signify an indivisible, i.e. formal, ideal point)
2. the factual indivisibility of the atom
3. the metaphysical indivisibility of prime matter,
Thinking of it this way, I suppose one could make a case that, because Democritus came upon his “uncuttable” by a reduction to material cause, the pure indivisibility which he was seeking wanted to satisify the criterion of an ontological principle. The principle of prime matter is obviously more apt to do this than a logical principle, and it seems the just what Democritus missed in his attachment to the factual was the possibility of establishing real indivisibility on the basis of an a priori principle..