In Bohr's theory and its developments, there is a lacuna and there is a difficulty. The lacuna has already been mentioned: we do not know in the least why an electron chooses one moment rather than another to jump from a larger to a[Pg 42] smaller orbit. The difficulty is that the jump is usually regarded as sudden and discontinuous: it is suggested that if it were continuous, the experimental facts in the regions concerned would become inexplicable. Possibly this difficulty may be overcome, and it may be found that the transition from one orbit to another can be continuous. But it is as well to consider the other possibility, that the transition is really discontinuous. I have emphasized how little we really know about what goes on in the atom, because I wished to keep open the possibility of something quite different from what is usually supposed. Have we any good reason for thinking that space-time is continuous? Do we know that, between one orbit and the next, other orbits are geometrically possible? Einstein has led us to think that the neighbourhood of matter makes space non-Euclidean; might it not also make it discontinuous? It is certainly rash to assume that the minute structure of the world resembles that which is found to suit large-scale phenomena, which may be only statistical averages. These considerations may serve as an introduction to the most modern theory of quantum mechanics, to which we must now turn our attention.[18] In the new theory inaugurated by Heisenberg, we no longer have the simplicity of the Rutherford-Bohr atom, in which electrons revolve about a nucleus like separate planets.[Pg 43] Heisenberg points out that in this theory there are many quantities which are not even theoretically observable—namely, those representing processes supposed to be occurring while the atom is in a steady state. In the new theory, as Dirac says: "The variable quantities associated with a stationary state on Bohr's theory, the amplitudes and frequencies of orbital motion, have no physical meaning and are of no physical importance" (4, p. 652). Heisenberg, in first introducing his theory, pointed out that the ordinary quantum theory uses unobservable quantities, such as the position and time of revolution of an electron (1, p. 879), and that the electron ought to be represented by measurable quantities such as the frequencies of its radiation (1, p. 880). Now the observable frequencies are always differences between two "terms," each of which is represented by an integer. We thus arrive at a representation of the state of an atom by means of an infinite array of numbers—i.e. by a matrix. If and are two "terms," an observable frequency (in theory) is , where: It is such numbers as (of which there is a doubly infinite series) that characterize the atom, so far as it is observable.