If time permitted, I might point out how the study of particular geometric figures, such as curves and surfaces, has been in many instances replaced by that of systems of figures infinite in number, and, indeed, of various degrees of infinitude. Such, for instance, are Plücker's complexes and congruences. I might describe also how Riemann taught us that surfaces need not present simple extension without thickness, but that, without losing their essential geometric character, they may consist of manifold sheets; and that our conception of space, and our power of interpreting otherwise perplexing algebraical expressions, become immensely enlarged.
Other generalizations might be mentioned, such as the principle of continuity, the use of imaginary quantities, the extension of the number of the dimensions of space, the recognition of systems in which the axioms of Euclid have no place. But as these were discussed in a recent address, I need not now do more than remind you that the germs of the great calculus of quaternions were first announced by their author, the late Sir W. R. Hamilton, at one of our meetings.
Passing from geometry proper to the other great branch of mathematical machinery, viz., algebra, it is not too much to say that within the period now in review there has grown up a modern algebra, which to our founders would have appeared like a confused dream, and whose very language and terminology would be as an unknown tongue.
Into this subject I do not propose to lead you far. But, as the progress which has been made in this direction is certainly not less than that made in geometry, I will ask your attention to one or two points which stand notably prominent.
In algebra we use ordinary equations involving one unknown quantity; in the application of algebra to geometry we meet with equations, representing curves or surfaces, and involving two or three unknown quantities respectively; in the theory of probabilities, and in other branches of research, we employ still more general expressions. Now, the modern algebra, originating with Cayley and Sylvester, regards all these diverse expressions as belonging to one and the same family, and comprises them all under the same general term "qualities." Studied from this point of view, they all alike give rise to a class of derivative forms, previously unnoticed, but now known as invariants, covariants, canonical forms, etc. By means of these, mathematicians have arrived not only at many properties of the quantics themselves, but also at their application to physical problems. It would be a long and perhaps invidious task to enumerate the many workers in this fertile field of research, especially in the schools of Germany and of Italy; but it is perhaps the less necessary to do so, because Sylvester, aided by a young and vigorous staff at Baltimore, is welding many of these results into a homogeneous mass in the classical memoirs which are appearing from time to time in the "American Journal of Mathematics."