It is always gratifying to discover that it is within one's ability to appreciate
a mathematics book and to read it with pleasure. I have often dreamt what a
joy it would be to get to know some of the elementary gems that are surely
present in every branch of mathematics, only to be dismayed by the literature I
have been able to find. Undoubtedly the gems are there, but they often lie buried
in textbooks or comprehensive reference works. One is frequently left with the
unhappy choice of undertaking a prolonged study of the field or giving up the
idea altogether. While it takes a knowledgeable scholar to write something out of
the ordinary, the dedicated specialist can get carried away with discussions that
one comes to appreciate only after long and serious study. Unfortunately, this
makes it very difficult for general readers to disentangle the elementary gems of
their heart's desire. On top of this, what passes for a proof is often so concise or
sketchy that it is readily understandable only to someone who already knows the
subject.
1 would dearly love to be able to promise that you will find no such frustrations
in the present work. What I can promise is a collection of essays that does not
attempt to cover a large amount of material, and that each topic has been extricated
from the mass of material in which it is usually found and given as elementary
and full a treatment as is reasonably possible. There is no sense pretending there
is any way around the need to lay foundations for one's proofs, but by selecting
from the most accessible topics I hope that many readers will be able to delight
in these gems with a minimum of preliminaries.
With one exception, everything that is beyond a high school background is
developed as the need arises; the basic properties of the nine-point circle are
covered in so many places that their proofs have not been included here. You
should be advised, however, that derivations given in one essay are not repeated
in a later one. Thus, while sampling from the first few essays might not incur a
great loss of continuity, the later essays do make use of concepts and results that
have been considered in earlier discussions, and you might have to look in an
earlier essay for explanations of an item that is encountered later on.