P. HISTORY OF ' AATHEMATICAL - School of Mathematics

TEACHINGS OF HISTORY
331
would render the solution of a problem impossible, in some cases
destroy each others effect when conjoined..... The theorems that
are sometimes briefly discovered by the use of this symbol, may be
demonstrated without it by the inverse operation, or some other way;
and tho' such symbols are of some use in the computations in the
method of fluxions, its evidence cannot be said to depend upon any
arts of this kind."
718. Charles Babbage stresses the power of algebraic symbolism
as follows:' "The quantity of meaning compressed into small space
by algebraic signs, is another circumstance that facilitates the reasonings
we are accustomed to carryon by their aid. The assumption
of lines and figures to represent quantity and magnitude, was
the method employed by the ancient geometers to present to the
eye some picture by which the course of their reasonings might be
traced: it was however necessary to fill up this outline by a tedious
description, which in some instances even of no peculiar difficulty
became nearly unintelligible, simply from its extreme length: the invention
of algebra almost entirely removed this inconvenience.....
A still better illustration of this fact is noticed by Lagrange and Delambre,
in their report to the French Institute on the translationof the
works of Archimedes by M. Peyrard." It occurs in the ninth proposition
of the second book on the equilibrium of planes, on which they
observe, 'La demonstration d'Archimede a trois enormas colonnes
in-folio, et n'est rien moin que lumineuse.' Eutochius commence sa
note 'en disant que le theorems est fort peu clair, et il promet de
l'expliquer de son mieux. II emploie quatre colonnes du meme format
et d'un caractere plus serre sans reussir d'avantage; au lieu que
quatre lignes d'algebre suffisent a M. Peyrard pour mettre la verite
du theoreme dans le plus grand jour.' "
But Babbage also points out the danger of lack of uniformity in
notations: "Time which has at length developed the various bearings
of the differential calculus, has also accumulated a mass of materials
of a very heterogeneous nature, comprehending fragments of unfinished
theories, contrivances adapted to peculiar purposes, views
perhaps sufficiently general, enveloped in notation sufficiently obscure,
a multitude of methods leading to one result, and bounded by
the same difficulties, and what is worse than all, a profusion of notations
(when we regard the whole science) which threaten, if not duly
1 C. Babbage, "On the Influence of Signs in Mathematical Reasoning,"
Trana. Cambridge Philosophical Society, Vol. II (1827), p. 330.
2 Otwrages d'Archimede (traduites par M. Peyrard), Tom. II, p. 415.