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which, perhaps, often gave more pleasure from the fact that they were 
unexpected and came as surprises. The mathematicians of those days 
resembled the ocean navigators, Columbus, ]\[agellan, Yasco de Gama, 
and others of the same period, who went forth over the world to 
discover new lands. And as these, while bent on some particular 
object, often, unexpectedly, made great discoveries, so the mathe- 
maticians referred to came upon important theorems in mathematics 
in the course of their investigations of geometrical and physical 
problems. And in the same way as the whole extent of the earth is 
known now, so also is the whole range of mathematics mapped out, 
and its various departments have their boundaries assigned. Also, 
like our physical world, the regions are mapped out, though the greater 
part yet remains unoccupied : and that part awaits being opened up 
and brought within the domain of cultivated territory ; and, like 
many districts of the earth which for a long time were believed to be 
barren and uninhabitable, and have now proved to be quite the 
contrary, there are branches of mathematics neglected, and heretofore 
believed to be unproductive, which may turn out to be most fertile 
and yield many results in the future to the investigator. 
Eut in this comparison the similitude only holds up to a certain 
point. The earth is finite, and when we have occupied and exhausted 
it, nothing more remains to be done ; but there is no limit to the 
acquisition of mathematical knowledge. In fact, in mathematics we 
can move from one planet to another, and when we have exhausted 
one solar system, we can take wings and seek another, where we may 
find a fresh field for investigation. 
In the old-fashioned method of proceeding to solve problems, the 
same process was gone through over and over again, the mathematical 
worker being apparently unconscious that he had ever done anything 
like it before. Por instance, in finding the envelope of a system of 
curves or surfaces varying, subject to certain conditions, the same 
process was continually repeated, and the fact that such problems 
depended upon finding the discriminant of an algebraic equation 
seemed to be unknown. Also, the determination of the discriminant 
of every algebraic equation that turned up in the consideration of any 
geometrical problem was treated as a separate question and not as one 
of a class. In this direction the change brought about by the opera- 
tion of the modem mathematical idea has been most striking. How- 
ever, this could hardly have been otherwise. In the days of these 
earlier mathematicians, the starting-point was immediately from 
experience. All the problems in mathematics had their origin in 
