PROF. ELIE CARTAN, who has just been elected a foreign member of the Royal Society, has been in the forefront of French mathematics since 1894, when his doctorate thesis on the structure of continuous groups was first published. In response to continued demand, it was reprinted forty years later. This subject has been Cartan's chief interest throughout his subsequent career as a mathematician. By 1900 Cartan had made fundamental contributions in another field, the problem of Pfaff. Two long memoirs on the subject appeared about this time in the Annales de l'Ecole Normale. It was in the first of these (in 1899) that he developed systematically the theory of 'exterior differential forms' which obey the laws of exterior multiplication of Grassmann. Throughout most of his subsequent work Cartan has found the method of 'exterior derivation' an extremely powerful one.