=1219.= Symbolism is useful because it makes things difficult. Now in the beginning everything is self-evident, and it is hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we must invent a new and difficult symbolism in which nothing is obvious.... Thus the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions.--RUSSELL, BERTRAND. _International Monthly, 1901, p. 85._ =1220.= The employment of mathematical symbols is perfectly natural when the relations between magnitudes are under discussion; and even if they are not rigorously necessary, it would hardly be reasonable to reject them, because they are not equally familiar to all readers and because they have sometimes been wrongly used, if they are able to facilitate the exposition of problems, to render it more concise, to open the way to more extended developments, and to avoid the digressions of vague argumentation.--COURNOT, A. _Theory of Wealth [N. T. Bacon], (New York, 1897), pp. 3-4._ =1221.= An all-inclusive geometrical symbolism, such as Hamilton and Grassmann conceived of, is impossible.--BURKHARDT, H. _Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 5, p. 52._ =1222.= The language of analysis, most perfect of all, being in itself a powerful instrument of discoveries, its notations, especially when they are necessary and happily conceived, are so many germs of new calculi.--LAPLACE. _Oeuvres, t. 7 (Paris, 1896), p. xl._ CHAPTER XIII MATHEMATICS AND LOGIC =1301.= Mathematics belongs to every inquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form the syllogism cannot be directly compared with all experience, or it would not have required an Aristotle to discover it. It must be transmuted into all the possible shapes in which reasoning loves to clothe itself. The transmutation is the mathematical process in the establishment of the law.--PEIRCE, BENJAMIN. _Linear Associative Algebra; American Journal of Mathematics, Vol. 4 (1881), p. 97._