The discovery of the Proton and Neutron occurred in similar reactions, known as the Cloud Chamber Reactions.
The presence of the proton was first detected by Eugen Goldstein using a modified Cathode Ray Tube however, the credit for the discovery usually goes to Ernest Rutherford. Rutherford found that if he bombarded Nitrogen atoms with alpha particles Oxygen was formed as well as a positively charged particle, the proton, was given off.
James Chadwick was a student of Ernest Rutherford and carried on his mentor's studies. He continued to work with the alpha particle. Chadwick found that by bombarding Beryllium atoms with the alpha particle a Carbon atom was formed and a particle approximately the size of a proton but with a zero charge, the neutron.