The principle is so perfectly general that no particular application of it is possible.

An idea which can be used only once is a trick. If one can use it more than once it becomes a method.

Where should I start? Start from the statement of the problem. ... What can I do? Visualize the problem as a whole as clearly and as vividly as you can. ... What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its purpose on your mind.

If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book ... it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.

Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.

Hilbert once had a student in mathematics who stopped coming to his lectures, and he was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: 'I never thought he had enough imagination to be a mathematician.'

Quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous. The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better.