2 o8 THE POPULAR SCIENCE MONTHLY. 



If time permitted, I might point out how the study of particular 

 geometric figures, such as curves and surfaces, has been in many in- 

 stances replaced by that of systems of figures infinite in number, and, 

 indeed, of various degrees of infinitude. Such, for instance, are Pliick- 

 er'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 num- 

 ber 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 mathe- 

 matical 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 quan- 

 tity ; in the application of algebra to geometry we meet with equa- 

 tions, reju-esenting curves or surfaces, and involving two or three un- 

 known 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, re- 

 gards all these diverse expressions as belonging to one and the same 

 family, and comprises them all under the same general term " quali- 

 ties." Studied from this point of view, they all alike give rise to a 

 class of derivative forms, previously unnoticed, but now known as in- 

 variants, covariants, canonical forms, etc. By means of these, mathe- 

 maticians 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 

 < renimny 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 clas- 

 sical memoirs which are appearing from time to time in the " American 

 Journal of Mathematics." 



