The atom as an ultimate and indivisible particle of matter was a venerable and a viable scientific notion for many years before and after Dalton. For example, Newton's speculations about matter in the Queries at the end of his Opticks included Particles "so very hard, as never to wear or break in pieces; no ordinary Power being able to divide what God himself made one in the first Creation" (Newton, I. Opticks; London, 1704; Query 31). This chapter describes ideas and scientific evidence from the late 19(th) and early 20(th) centuries about the contrary notion, the divisibility of atoms. It is about the notion that the "ultimate" pieces of matter themselves have pieces. It focuses on the electron and the nucleus, with a few words about the proton and neutron as well; it does not treat constituent pieces of nucleons and more exotic particles.