In the last several years, pre-university mathematics education in North
America has been transformed quite dramatically. A new emphasis on preci-
sion of language and concept has become apparent and, unfortunately, this
has often been accompanied by a corresponding reduction of useful exercise
in mathematics. Consequently, exposure to the demands of first-year mathe-
matics courses at a university is often a traumatic experience.
There seem to be two basic problems here. First, the machinery required
to make use of the axiomatic method is being taught without giving signifi-
cant exercise in the axiomatic method itself. Second, the usefulness of mathe-
matics in helping us to understand the world and its inhabitants is not suf-
ficiently emphasized.
The second of these problems is our prime concern in this book. Neverthe-
less, in the context of a variety of applications, we also try to give some feeling
for the limitations, as well as the capabilities, of mathematics as applied to
our environment. Thus, we try to explain and discuss both the mathematics
involved and the situation to which it is applied. This concern for precision
in the use of mathematics has meant that the methods used have had to be
very limited. In a one-term (39-hour) course, for which this book is designed,
it is not possible to develop the methods of the calculus, for example, and also
to apply them. It has therefore been an important factor in collecting mate-
rial that every subject must be amenable to treatment, in the first instance,
by the methods which we can be confident that high school students have met.
This puts a serious limitation on the applications we are able to handle. With
more mathematical techniques at our disposal, a wider variety of physical
problems would be amenable to mathematical analysis.
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