Ideality is preëminently the foundation of Mathematics. - Benjamin Peirce

I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry. Whether I shall have the satisfaction of taking part in their exposition, or whether that will remain for some more profound expositor, will be seen in the future.

Ascend with me above the dust, above the cloud, to the realms of the higher geometry, where the heavens are never clouded; where there is no impure vapour, and no delusive or imperfect observation, where the new truths are already arisen, while they are yet dimly dawning on the world below; where the earth is a little planet; where the sun has dwindled to a star; where all the stars are lost in the Milky Way to which they belong; where the Milky Way is seen floating through space like any other nebula; where the whole great girdle of nebulae has diminished to an atom and has become as readily and completely submissive to the pen of the geometer, and the slave of his formula, as the single drop, which falls from the clouds, instinct with all the forces of the material world.

The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics. In every form of material manifestation, there is a corresponding form of human thought, so that the human mind is as wide in its range of thought as the physical universe in which it thinks.