GREEK philosophers twenty-five centuries ago held that the gods work by geometry and by arithmetic: mathematics is the corner-stone of the temple of science. During the past two years a remarkable development has taken place in theoretical physics through the application of mathematical methods to the problems of the quantum. The introduction of the new quantum mechanics associated with the names of Heisenberg and of Schrbdinger, and the revival of interest in the spinning electron, bid fair to initiate a new era in quantum theory. It is a striking testimony to the powers of generalisation inherent in mathematics that the method of matrices, due originally to Cayley, has proved a useful weapon of attack in the hands of Born and Jordan. After a prolonged discussion of the relative merits of language and mathematics as elementary disciplines, Willard Gibbs is reported to have said: “Mathematics is a language.” Unfortunately it is too frequently “a tongue not understanded of the people,” and for that reason it is not now our usual custom to publish in NATURE articles involving advanced and unfamiliar mathematical methods. The present stage of development is, however, exceptional, and as many of our readers are deeply interested in the new ideas, we have to some extent departed from precedent and are publishing certain articles by eminent authorities in which mathematical treatment is a necessity.