154 Proceedings of the Royal Irish Academy. 
field of discovery in an important branch of mathematical know- 
ledge. 
When we have got a clear view of the modern mathematical idea; 
we are enabled thereby to dispense to a great extent with long calcu- 
lations, which after all only serve to obscure the view, and prevent us 
from obtaining an insight into the relations connecting quantities. 
The modern mathematical idea gives us the power to hold together in 
the mind a large extent of knowledge, and perceive the connexion and 
mutual interdependence of the different departments of that know- 
ledge. We are thus enabled to predict, or at least form, an opinion of 
some value as to the result of certain transformations, and hence we 
can avoid a large amount of useless labour, and direct our attention to 
those points which are most likely to repay a careful study. This 
power is evidently of much more value than the ability to make long 
investigations, or fill pages with work, the faculty of correct working 
being easily attained by the force of habit, and not appearing to differ 
essentially from that of performing complicated sums in commercial 
matters. On looking back through the vast tomes of work written on 
mathematical subjects, especially on the physical side, during the pre- 
ceding century, we see a great number of investigations commenced, 
which lead to practically no result. We see, thus, the importance of 
being able to direct, both in pure and mixed mathematics, our steps 
into those paths which are most likely to lead to valuable results. Of 
course there is naturally an indisposition on the part of many mathe- 
matical workers to destroy as useless investigations which have led to 
no result ; and this may account for the quantity of work published 
which lapses into oblivion, only the residue of grain being winnowed 
from the chaff by time, and becoming part of the mental store of the 
mathematician. Such investigations, however, in some cases may not 
be altogether lost, as they may contain the germs of ideas which may 
fructify in the minds of other workers in the same field of labour. 
It seems now that the theory of invariants and the other products 
of the modern mathematical idea are as necessary a part of mathe- 
matical knowledge as the differential and integral calculi. We have 
had many other new ideas introduced into mathematics, as, for instance, 
Hamilton's Quaternions and Grassman's method ; but we have no con- 
viction of their forming a necessary part of our knowledge. There has 
been, in fact, of late years too great a tendency to originate such 
calculi. They seem liable to degenerate into mere playthings, and I 
believe, also, that they have not availed to obtain any new results, but 
rather prove prejudicial to the endeavour to do so. 
