For is not life one prolonged succession of problems that demand to be solved? To be sure, most of these problems are not of the mathematical order and do not admit of solution by the methods of mathematics. But it is a thoroughly good thing for a man not to be a coward or a sluggard when he is brought face to face with any hard problem.
The truly liberalizing power of mathematics, however, is felt only when two things are attained. The first of these is a certain amount of free and joyful movement in the handling of mathematical symbols and formulæ. The other is a certain grasp upon the beautiful ideas and the wonderful laws which are represented by these symbols and formulæ. A friend of mine, who stands in the very front rank of the world's great mathematicians (a rank so thin that two men could probably count its numbers on the fingers of their two hands), has recently declared that for him the higher mathematics is chiefly an æsthetical affair; and that no man ought to study it who does not rejoice in the beauty of the ideas with which it deals. Now, of course, it cannot be maintained that such very high mathematics shall be made a necessary part of all liberal culture. But, in my opinion, it is desirable for one in pursuit of this culture to go far enough in mathematics to get some glimpse of the ideality, and the beautiful ideality, of the world in which mathematical conceptions reign supreme.
