they disclose the fact that the author had no idea in regard to
the real nature of the problem or the meaning of a mathematical
proof. In a few cases the authors were fully aware of the
requirements but were misled by errors in their work. Although
the prize was formally announced more than seven years ago no
paper has as yet been adjudged as fulfilling the conditions.

It may be of interest to note in this connection that a
mathematical proof implies a marshalling of mathematical
results, or accepted assumptions, in such a manner that the
thing to be proved is a NECESSARY consequence. The
non-mathematician is often inclined to think that if he makes
statements which can not be successfully refuted he has carried
his point. In mathematics such statements have no real
significance in an attempted proof. Unknowns must be labeled as
such and must retain these labels until they become knowns in
view of the conditions which they can be proved to satisfy. The
pure mathematician accepts only necessary conclusions with the
exception that basal postulates have to be assumed by common
agreement.

The mathematical subject in which the student usually has to
contend most frequently with unknowns at the beginning of his
studies is the history of mathematics. The ancient Greeks had
already attempted to trace the development of every known
concept, but the work along this line appears still in its
infancy. Even the development of our common numerals is
surrounded with many perplexing questions, as may be seen by
consulting the little volume entitled "The Hindu-Arabic
Numerals," by D. E. Smith and L. C. Karpinski.