says Mr. Bertrand Russell in a typical passage. A strain of romantic eloquence seems, indeed, to be inseparable from the writings of mathematicians on their subject. But the analogy can be pressed more closely. There are elegant and inelegant mathematical demonstrations, those which merely “command assent,” as Lord Rayleigh said, and those which provide a very high degree of æsthetic satisfaction. In these latter demonstrations the mind seems to be moving with more swiftness and freedom; the whole demonstration seems to flower in a natural and spontaneous way; we have the impression of _inevitability_. Mathematical elegance, as Poincaré has put it, “n’est autre chose que la satisfaction due à je ne sais quelle adaptation entre la solution que l’on vient de découvrir et les besoins de notre esprit.” It is as if there were a mode of living natural to the human spirit, an _unadapted_ life, a life free from the necessity of accommodating itself to the elements, so largely alien, of the actual world. Mathematics is the expression of this life so far as the intellect is concerned. Is it too much to say that music is a fuller embodiment of this free life? If we are to say this we must acknowledge that more than the intellect is capable of this free life, that there is a logic of the emotions as well as of the mind. This assumption is not difficult to make; indeed, if we reflect on our experience of some compositions, such as, to take the same example, the first movement of the C minor Symphony, it is difficult to avoid making it. And, in considering the matter from this point of view, we may gain some results useful for musical criticism in general. The theme of the movement in question is characteristic of many of Beethoven’s themes in that it does not serve merely as a kind of structural skeleton on which a composition is to be built. In this respect it differs from, for instance, many of Bach’s themes. The theme immediately, in its ominous and arresting quality, throws the mind into a certain state of expectancy, a state where a large number of happenings, but only the happenings belonging to a certain class, can logically follow. As an analogous vague yet restricted initial preparation we may instance the entry of the witches at the beginning of _Macbeth_. As the music proceeds this rich, but more or less definite, state in the hearer becomes more and more precise, more and more subtle. It is, as it were, explored and shown in all its height and depth. What was pregnant in the theme is exhibited to us in all its extent, its definiteness, and its force. The theme was the entrance to a world. And we have the consciousness of logic, of _inevitability_, because at no point are we constrained. We exult because we are free; this is how we, too, would move but for our fetters, our alien, arbitrary fetters from which, for this time, we have been freed. And in none of this, unless we have incurably literary minds, are we ever reminded of experience.