_Jahresberichte der Deutschen Mathematiker Vereinigung, Bd. 13, p. 367._ =1212.= The results of systematic symbolical reasoning must _always_ express general truths, by their nature; and do not, for their justification, require each of the steps of the process to represent some definite operation upon quantity. The _absolute universality of the interpretation of symbols_ is the fundamental principle of their use.--WHEWELL, WILLIAM. _The Philosophy of the Inductive Sciences, Part I, Bk. 2, chap. 12, sect. 2 (London, 1858)._ =1213.= Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.--COURNOT, A. _Theory of Wealth [N. T. Bacon], (New York, 1897), p. 4._ =1214.= As arithmetic and algebra are sciences of great clearness, certainty, and extent, which are immediately conversant about signs, upon the skilful use whereof they entirely depend, so a little attention to them may possibly help us to judge of the progress of the mind in other sciences, which, though differing in nature, design, and object, may yet agree in the general methods of proof and inquiry.--BERKELEY, GEORGE. _Alciphron, or the Minute Philosopher, Dialogue 7, sect. 12._ =1215.= In general the position as regards all such new calculi is this--That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able--without the unconscious inspiration of genius which no one can command--to solve the respective problems, yea, to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange’s calculus of variations, with my calculus of congruences, and with Möbius’s calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.--GAUSS, C. J. _Werke, Bd. 8, p. 298._