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A Greek philosopher by the name of Democritus came up with the general idea and the name of atoms (atomos literally means "no cut" -- in other words, indivisible -- in Greek). However, he didn't really have any experimental evidence for this, he just thought it was the way things should work. It took almost 2000 years before scientists demonstrated that this concept was actually pretty accurate. Actually, this concept goes back to the Greeks. Dalton also believed it, but was wasn't the first:

From ChemTeam: Atomic Structure by John Dalton (see link to the left):

"Elements are made up of minute, discrete, indivisible, and indestructible particles called atoms. These atoms maintain their identity through all physical and chemical changes. This, of course, is not a new idea to Dalton. This basic idea goes back to the Greeks. However, please keep in mind that atoms, as such, were not part of the chemical mainstream in the early 1800's. Dalton's idea of an element is what we believe today - an element is a chemical substance that cannot be decomposed further by chemical means (i.e. heat, electricity, reacting with another chemical). This definition traces to Lavoisier. Daltonian atoms are usually taught as being similar to featureless billiard balls. In truth, Dalton never ruled out the possibility of subatomic structure. He just knew that the state of the art in the early 1800's did not allow the physical structure of an atom to be probed."

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