Characterization of totally unimodular matrices

We exposed the proof or the sufficient condition of this theorem as well as the one of Statement 3 at the Seminary on Combinatorial Problems of the Faculté des Sciences de Paris in January 1962. A. Ghouila-Houri gave independently a characterization [3] which could have been deduced however from our results. R. Gomory gave us recently a new proof of the sufficient condition. His proof, combined with ours, allows us to prove a simpler characterization :