=652.= Abstruse mathematical researches ... are ... often abused for having no obvious physical application. The fact is that the most useful parts of science have been investigated for the sake of truth, and not for their usefulness. A new branch of mathematics, which has sprung up in the last twenty years, was denounced by the Astronomer Royal before the University of Cambridge as doomed to be forgotten, on account of its uselessness. Now it turns out that the reason why we cannot go further in our investigations of molecular action is that we do not know enough of this branch of mathematics.--CLIFFORD, W. K. _Conditions of Mental Development; Lectures and Essays (London, 1901), Vol. 1, p. 115._ =653.= In geometry, as in most sciences, it is very rare that an isolated proposition is of immediate utility. But the theories most powerful in practice are formed of propositions which curiosity alone brought to light, and which long remained useless without its being able to divine in what way they should one day cease to be so. In this sense it may be said, that in real science, no theory, no research, is in effect useless.--VOLTAIRE. _A Philosophical Dictionary, Article “Geometry”; (Boston, 1881), Vol. 1, p. 374._ =654.= Scientific subjects do not progress necessarily on the lines of direct usefulness. Very many applications of the theories of pure mathematics have come many years, sometimes centuries, after the actual discoveries themselves. The weapons were at hand, but the men were not able to use them. --FORSYTH, A. R. _Perry’s Teaching of Mathematics (London, 1902), p. 35._ =655.= It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications. --WHITEHEAD, A. N. _Introduction to Mathematics (New York), p. 100._