Experiments on the Regular Pentagon. 
113 
the promising experiments, and what i,s of more value, trained to go 
over his already acquired knowledge in a systematic way when attacking 
new exercises. The experiments that fail are often as useful to the 
learner as those that succeed, the proof that he has failed as logical as the 
proof that he has succeeded. It is not the result that is of educational 
value, but the method employed in reaching it. Of course, however, a 
student properly trained' and of moderate capacity will soon learn the 
well beaten geometric roads and develop some capacity for actually solv- 
ing originals. But success in actually getting an original is not the only 
basis for estimating a student’s work. One can imagine that a student 
who fails to reach a successful conclusion may deserve a better grade 
than one who does not so fail. 
In conclusion, notice that a strictly deductive geometry must often 
invert the historical order. In many geometries the problem of dividing 
a sect in the way just done is first given and then, later, this division is 
made use of in drawing the regular pentagon. Row, it is very probable 
that the problem of constructing a regular pentagon preceded that of the 
divided sect and gave rise to the need of so dividing, which, when accom- 
plished, rendered the construction of the regular pentagon possible. In 
this way the probable historical order is often obscured and inverted. 
Obviously the historical order should not be followed, at all hazards, even 
if known. It w'ould involve too much waste and the thought sequence 
of any discoverer of a new theorem is seldom known with entire accuracy 
even when thought out in modern times. But such thought sequence is 
precisely the true history of mathematics. Perhaps mathematical teach- 
ing, like so many other things, will profit by following loosely the course 
of those who built up the subject. 
