Aristotle on the Continuum

Ed Ockham (I hope he doesn't mind me calling him this) has been posting on Aristotle's argument that the continuum cannot be composed of indivisibles.  He summarises the argument here.  Briefly, it goes as follows:

(1) If a continuum is composed solely of points, these points must be either continuous with one another, or touch, or be in succession.
(2) The points are not continuous
(3) They do not touch
(4) They are not in succession
(5) Therefore the continuum is not composed solely of points.

There is general agreement that the argument fails because (1) is false.  My question is Why did Aristotle put this forward at all?  Here is a comment I made at Ed's:

I agree that the flaw with Aristotle's argument lies with (1) but for me the difficulty is in understanding why he should put this 'definition' forward in the first place!  What can he have in mind that leads him to break down continuity into these three cases?  Can we reconstruct his thought?  

Here is a tentative suggestion.  First of all, given his aversion to the actual infinite, the thought that a continuum could be 'composed' of countably many, let alone uncountably many, discrete points is simply beyond him.  We really do have to wait for Cantor and Dedekind before we can see how to make sense of this.  His idea of a 'part' of a continuum, say a line interval, seems to be a subinterval---something with distinct extremities or endpoints.   Now, if we think of subintervals as possible parts of an interval, then any pair of such subintervals either (1) overlap, (2) touch, or (3) are separated.  So we find three cases here.  But though the second of Aristotle's cases seems to be about touching, and the third seems to admit the possibility of separation in its definition, it's hard to fit overlapping into his first case.  Indeed, all three of his cases seem to come down to contiguity, ie, touching.  Can anyone explain the three-way distinction that Aristotle is drawing here? 

My guess is that he is groping towards an understanding of what we would now call 'connectedness'.  A line interval is connected in the sense that, if it is decomposed into it least two subintervals, then for any subinterval there must be another subinterval contiguous with it---a successor (or predecessor) subinterval, in Aristotle's terminology.    Although this captures, I think, a necessary property of a continuum, it says nothing about its ultimate parts.*  To say that an interval is subintervals 'all the way down', which I think is what Aristotle is doing here, is to beg the question against other possibilities which we can conceive now, but were unimaginable for Aristotle.

But I haven't convinced myself.
 

* This is also true of the modern topological notion of connectedness,  which rests on the fundamental idea of 'open set'.

To ask again: Can anyone explain the three-way distinction that Aristotle is drawing in (1)?