elementary mathematical text-books, and the ease with which the
skilful mathematics teachers often cleared away what appeared
to be great difficulties to the students have filled many with
a kind of awe for unusual mathematical ability.

In recent years the unbounded confidence in mathematical
results has been somewhat shaken by a wave of mathematical
skepticism which gained momentum through some of the popular
writings of H. Poincare and Bertrand Russell. As instances of
expressions which might at first tend to diminish such
confidence we may refer to Poincare's contention that
geometrical axioms are conventions guided by experimental facts
and limited by the necessity to avoid all contradictions, and
to Russell's statement that "mathematics may be defined as the
subject in which we never know what we are talking about nor
whether what we are saying is true."

The mathematical skepticism which such statements may awaken is
usually mitigated by reflection, since it soon appears that
philosophical difficulties abound in all domains of knowledge,
and that mathematical results continue to inspire relatively
the highest degrees of confidence. The unknowns in mathematics
to which we aim to direct attention here are not of this
philosophical type but relate to questions of the most simple
nature. It is perhaps unfortunate that in the teaching of
elementary mathematics the unknowns receive so little
attention. In fact, it seems to be customary to direct no
attention whatever to the unsolved mathematical difficulties
until the students begin to specialize in mathematics in the
colleges or universities.