or even imagined beforehand. Who, for instance, would have supposed that the calculus of forms or the theory of substitutions
would have thrown much light upon ordinary equations; or that Abelian functions and hyperelliptic transcendents
would have told us anything about the properties of curves;
or that the calculus of operations would have helped us in
any way towards the figure of the earth?" A second reason
in favour of the pursuit of advanced mathematics, even when there is no promise of practical application, is this, that mathematics, like poetry and music, deserves cultivation for its own sake.
The great characteristic of modern mathematics is its generalising tendency. Nowadays little weight is given to isolated theorems, "except as affording hints of an unsuspected new sphere of thought, like meteorites detached from some
undiscovered planetary orb of speculation." In mathematics, as in all true sciences, no subject is considered in itself alone, but always as related to, or an outgrowth of, other things. The development of the notion of continuity plays a leading
part in modern research. In geometry the principle of continuity,
the idea of correspondence, and the theory of projection
constitute the fundamental modern notions. Continuity asserts itself in a most striking way in relation to the circular points at infinity in a plane. In algebra the modern idea finds
expression in the theory of linear transformations and invariants, and in the recognition of the value of homogeneity and
symmetry.