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In its article about how the electron was discovered, Wikipedia says that Thomson and his students performed experiments which suggested that cathode rays were negatively charged "particles". But even after reading about his experiments I am unable to understand how he came to the conclusion that the cathode rays are made up particles. I can see how he came to the conclusion that they are negatively charged, but how did he find out about their particle nature?

The idea that matter was made up of "primordial" particles, and currents in metals consisted of them was well established by then. Stoney suggested the name "electron" in 1891, and Lorentz's theory of electrons dates back to 1892, see Wikipedia's timeline. Thomson himself mentioned Prout as the source for the view, and saw his task at hand more in establishing the "nature" of the particles, not that they are particles. And he is very clear on the hypothetico-deductive approach to it: he does not attempt to "derive" the particle hypothesis from experiments, he only confirms that the experimental results fit well with what is hypothesized. And not just his, but previous ones as well. Here is Thomson's own explanation from Cathode Rays (1897):

"As the cathode rays carry a charge of negative electricity, are deflected by an electrostatic force as if they were negatively electrified and are acted on by a magnetic force in just the way in which this force would act on a negatively electrified body moving along the path of these rays I can see no escape from the conclusion that they are charges of negative electricity carried by particles of matter. The question next arises, What are these particles? are they atoms or molecules, or matter in a still finer state of subdivision? To throw some light on this point, I have made a series of measurements of the ratio of the mass of these particles to the charge carried by it."

His measurements established that $m/e$ for different gases and pressures had the same value, and the paths of particles depended on the density of the medium. He then reasoned further:

"The explanation which seems to me to account in the most simple and straightforward manner for the facts is founded on a view of the constitution of the chemical elements which has been favourably entertained by many chemists: this view is that the atoms of the different chemical elements are different aggregations of atoms of the same kind. In the form in which this hypothesis was enunciated by Prout, the atoms of the
different elements were hydrogen atoms; in this precise form the hypothesis is not tenable, but if we substitute for hydrogen some unknown primordial substance X, there is nothing known which is inconsistent with this hypothesis, which is one that
has been recently supported by Sir Norman Lockyer for reasons derived from the study of the stellar spectra.

If, in the very intense electric field in the neighbourhood of the cathode, the molecules of the gas are dissociated and are split up, not into the ordinary chemical atoms, but into these primordial atoms, which we shall for brevity call corpuscles;
and if these corpuscles are charged with electricity and projected from the cathode by the electric field, they would behave exactly like the cathode rays. They would evidently give a value of $m/e$ which is independent of the nature of the gas and its pressure, for the carriers are the same whatever the gas may be; again, the mean free paths of these corpuscles would depend solely upon the density of the medium through
which they pass."