The 1925 paper "On quantum mechanics" by M. Born and P. Jordan, and the sequel "On quantum mechanics II" by M. Born, W. Heisenberg, and P. Jordan, developed Heisenberg's pioneering theory into the first complete formulation of quantum mechanics. The Born and Jordan paper is the subject of the present article. This paper introduced matrices to physicists. We discuss the original postulates of quantum mechanics, present the two-part discovery of the law of commutation, and clarify the origin of Heisenberg's equation. We show how the 1925 proof of energy conservation and Bohr's frequency condition served as the gold standard with which to measure the validity of the new quantum mechanics.