41 
edge of number combinations and of a practical facility In their rapid and 
accurate manipulation. Very properly in the high school the disciplinary 
idea predominates, but even here it is a question whether the time is not 
near at hand when some of the older mathematical subjects taught should 
be in a measure set aside and other newer ones substituted which are 
of equal disciplinary value, but whose knowledge content is greater. 
In the earlier portion of the college course the disciplinary idea still 
strongly predominates, but if mathematical study is continued through the 
last two years and into graduate work it becomes a cult or a profession 
or a necessary adjunct to a profession. 
In its development we may roughly divide mathematics into three 
general subjects: 
Arithmetic. 
Geometry. 
Algebra. 
Yet again these, especially in their higher ranges, continually overlap 
each other. The theory of numbers seems to belong to arithmetic, yet 
some of the problems like that of prime numbers, which were among 
the earliest propounded, demand now for their approximate solution, after 
twenty-five centuries of development, the highest powers of analysis. 
In analytics. geometry and algebra melt into each other, while in the 
modern group theory the three which in their earlier manifestations seem 
so diverse in spirit and purpose form one grand generalization. 
As a discipline these studies need no apology. Their influence in the 
‘development of the reasoning powers is unquestioned. They exercise the 
muscles and sharpen the teeth of the logical faculty. They furnish the 
growing mind with exercise in useful knowledge with reference to which 
it can have absolutely no prejudice and, while leading to certain truth, 
generate confidence in intellectual powers. They develop the inventive 
faculty by sharpening the powers of comparison, by diversifying and en- 
riching the powers of attack, by developing the power of long continued 
attention and concentration. 
As a cult, pursued for its own sake, it furnishes one of the highest 
occupations of the intellect. The mind revels in the realm of pure thought, 
and each triumph must bring the thinker closer to that all-pervading 
intelligence whose very existence and activity entirely removed from 
chance and imperfect knowledge must be conditioned by mathematical 
necessity. 
