330 Mr. BaspaceE on the Influence of Signs 
pher. . “ La troisiéme raison,” observes M. Degerando, ‘est dans 
la propriété qu’ a l’algébre de ne saisir, dans les idées des quantités, 
que certains rapports généraux, de ne présentir ainsi 4 notre esprit 
que les considérations qui lui sont vraiment utiles dans les re- 
cherches auxquelles il se livre. De la il arrive que notre atten- 
tion se trouve débarassée d’un grand nombre d’idées accessoires, 
qui etrangéres au but de ses méditations, n’auroient servi qu’ a 
la distraire *.” 
The quantity of meaning compressed into small space by 
algebraic signs, is another circumstance that facilitates the 
reasonings we are accustomed to carry on by their aid. The 
assumption of lines and figures to represent quantity and magni- 
tude, was the method employed by the ancient geometers to pre- 
sent to the eye some picture by which the course of their rea- 
sonings 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, and presented to the eye a picture per- 
fect in all] its parts, disclosing at a glance, not merely the conclusion 
* Des Signes et l'art de Penser, p. 214. tom. II. 
+ The difficulty which many students experience in understanding the propositions 
relating to ratios as delivered in the fifth book of Euclid, arises entirely from this cause, 
and the facility of comprehending their algebraic demonstrations forms a striking contrast 
with the prolixity of the geometrical proofs. 
A still better illustration of this fact is noticed by Lagrange and Delambre, in their 
report to the French Institute on the translation of the works of Archimedes by M. Peyrard. 
It occurs in the ninth proposition of the 2nd book on the equilibrium of planes, on which 
they observe, “ La demonstration d’Archimede a trois énormes colonnes in-folio, et n’est 
rien moin que lumineuse.” Eutochius commence sa note “en disant que le theoréme est fort 
peu clair, et il promet de l'expliquer de son mieux. Il emploie quatre colonnes du méme 
format et d’un charactére plus serré sans reussir d’avantage; au lieu que quatre lignes 
d’algebre suffisent a M. Peyrard pour mettre la verité du theoréme dans le plus grand jour,” 
Ouvrages d’Archimede traduites par M. Peyrard, p. 415. tom. II. 
