P. HISTORY OF ' AATHEMATICAL - School of Mathematics

332 A HISTORY m' :\L\THEMATICAL NOTATIONS
corrected, to multiply our difficulties instead of promoting our
progress," p. 326.
719. E. Machi discusses the process of clarification of the significance
of symbols resulting from intellectual experimentation, and
says: "Symbols which initially appear to have no meaning whatever,
acquire gradually, after subjection to what might be called intellectual
experimenting, a lucid and precise significance. Think only of the
negative, fractional and variable exponents of algebra, or of the cases
in which important and vital extensions of ideas have taken place
which otherwise would have been totally lost or have made their appearance
at a much later date. Think only of the so-called imaginary
quantities with which mathematicians long operated, and from which
they even obtained important results ere they were in a position to
assign to them a perfectly determinate and withal visualizable meaning.
. . . . Mathematicians worked many years with expressions like
cos x+ v=1 sin x and with exponentials having imaginary exponents
before in the struggle for adapting concept and symbol to each other
the idea that had been germinating for a century finally found expression
[in 1797in Wessel and] in 1806 in Argand, viz., that a relationship
could be conceived between magnitude and direction by which -v=1
was represented as a mean direction-proportional between +1 and
-I."
720. B. Branford expresses himself on the "fluidity of mathematical
symbols" as followaf "Mathematical symbols are to be
temporarily regarded as rigid and fixed in meaning, but in reality
are continually changing and actually fluid. But this change is so
infinitely gradual and so wholly subconscious in general that we are
not sensibly inconvenienced in our operations with symbols by this
paradoxical fact. Indeed, it is actually owing to this strange truth
that progress in mathematical science is possible at all. An excellent
instance is the gradual evolution of algebra from arithmetic-a clear
hint this for teachers."
721. A. N. WhiteheadS writes on the power of symbols: "By
relieving the brain of all unnecessary work, a good notation sets
it free to concentrate on more advanced problems, and in effect
increases the mental power of the race. Before the introduction of the
1 E. Mach, Space and Geometry (trans. T. J. McCormack, 1906), p. 103, 104.
• Benchara Branford, A Study oj Mathematical Education (Oxford, 1908),
p.370.
8 A. N. Whitehead, An Introdudion to Mathematics (New York and London,
1911), p. 59.