Mathematics

Mathematics (Gr. mathēma, 'learning'), the science which has for its subject-matter the properties of magnitude and number. It is usually divided into pure and mixed or applied; the first including all deductions from the abstract, self-evident relations of magnitude and number, the second the results arrived at by applying the principles so established to certain relations found by observation to exist among the phenomena of nature. The branches of pure mathematics which were first developed were, naturally, Arithmetic, or the science of number, and Geometry, or the science of quantity (in extension). The latter of these was the only branch of mathematics cultivated by the Greeks, their cumbrous notation opposing a barrier to any effective progress in the former science. Algebra or the science of numbers in its most general form is of much later growth, and was at first merely a kind of universal arithmetic, general symbols taking the place of numbers; but its extraordinary development within the last two centuries has established for it a right to be considered as a distinct science, the science of operations. Combinations of these three have given rise to trigonometry and analytical geometry. All those sciences in which a few simple axioms are mathematically shown to be sufficient for the deduction of the most important natural phenomena are regarded as belonging to applied mathematics. This definition includes those sciences which treat of pressure, motion, light, heat, sound, electricity, and magnetism—usually called Physics—and excludes chemistry, geology, political economy, and the other branches of science, which, however, receive more or less aid from mathematics. See GEOMETRY, and the works there cited; as also, besides articles on the subjects named above, and many others, the following: