MATHEMATICS. \ 3 



solution, that one of the greatest mathematicians cf the day thought 

 that he had proved its entire impossibility. In the hands of Sylvester, 

 of Kempe, and others, this principle has been developed into a general 

 theory of link-work, on which the last word has not yet been said. 



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 

 Pliicker's complexes and congruencies. I might describe also how 

 Eiemann 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 thus our concep- 

 tion of space, and our power of interpreting otherwise perplexing 

 algebraical expressions, become immensely enlarged. 



Other generalisations 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 authoi-, 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 

 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 ex- 

 pressions. 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 ' quantics.' Studied from this point of view, they all alike give 

 rise to a class of derivative forms, previously unnoticed, but now 



