Are Atoms Illogical?

The word "atom" comes from the Greek for uncuttable.  The idea in the doctrines of Democritus and Lucretius was that there was a basic unit of material existence that had no parts, was complete unity in itself.  Descartes, in his book The Principles of Philosophy (his book about physics), argues that the such an atom is a logical impossibility.  His argument is based on an insult to God's omnipotence along the lines of the inability to create a rock so heavy He can't lift it.  The idea of an atom would require God's ability to create an entity He could not divide.  Since an all-perfect God was a logical necessity according to Descartes, atoms had to go.

We can create a non-theological version of the argument from another element of Descartes' views on physics.  According to Descartes, the primary property of a material entity is extension, having size.  If we posit for the argument's sake that classical atoms exist, then they have to have size.  This size could be mapped onto the real numbers, that is anything that has a size can be measured and we can determine how big this size is.  But for every real number, there is a real number that is half as large, that is we can split any real number in half.  Couldn't we then split the space occupied by the atom in half and talk about, say, the left half of the atom and the right half of the atom?  Would this undermine classical atomism? 

Could Democritus respond to this secular Cartesian argument?  What about something Leibnizian along the lines of the division would not create things out of the "parts" of the atom because they have no properties since the atom itself and not its parts are the sole possessors of properties?  The parts, in essence, are not parts.  But wouldn't they have properties of being to the left of or to the right of the other part?

We know the ancient view is factually wrong, but is it logically possible?