inomial theorem can give (without hesitation) the
square of 21, or of 21.5, or any similar quantity. With practice and
reflection, results which seem astonishing may be attained.

(_f_) KEEP THE MIND ACTIVE AND ALERT.--Do not simply sit and gaze upon
a book, expecting to have ideas come to you, but exert the mind. Study
is active and intelligent, not dreamy. By this is not meant that haste
is to be practised. On the contrary, what might perhaps be called a
sort of dreamy thinking often gives time and opportunity for ideas to
clarify and take shape and proportion in the mind. We often learn most
in hours of comparative idleness, meditating without strenuous mental
activity upon what we have read. Such meditation is of the greatest
value, but it is very different from the mental indolence of which the
poet speaks when he says:

"'Tis thus the imagination takes repose
In indolent vacuity of thought,
And rests and is refreshed."

{40} This is beneficial to the proper extent; but it is rest, not study.

(_g_) WHEN YOU MEET WITH DIFFERENCES OF OPINION UPON A SUBJECT, REFLECT
UPON THE REASONS WHICH MAY CAUSE INTELLIGENT MEN TO ARRIVE AT DIFFERENT
CONCLUSIONS.--These reasons are:

1. One or both may fail to grasp all the pertinent facts, or even the
problem itself, or may assume, as true, facts or principles which are
really erroneous. This should easily be ascertainable.

2. One or both may reason incorrectly even from accurate premises.
This also should be discoverable.

3. One or both may see facts out of proportion--may lack a true mental
balance or perspective.

4. One or both may illustrate the inherent stubbornness or
imperviousness of the human mind.

Whether the student can discover the last two sources of error will
depend upon his own mental characteristics. He must not forget,
however, that on many matters no definite demonstrable conclusion is
possible, and that the result must remain more or less a matter of
opinion.

(_h_) REMEMBER THAT A STATEMENT IS NOT A PROOF. MANY STUDENTS THINK
THEY PROVE A STATEMENT BY MERELY REPEATING IT IN DIFFERENT WORDS. YOU
DO NOT UNDERSTAND A CONCLUSION UNLESS YOU CAN SEE THE STEPS IN ITS
LOGICAL DEMONSTRATION.

{41}

It is quite surprising how many students commit this error. For
instance, if I am asked why can I see through glass and I reply,
because it is transparent, I am giving no reason at all, for
transparent means what can be seen thr