MATHEMATICS (Gr. mathema, learning), the science which has for its subject-matter the properties of magnitude and number. It is usually divided into pure and mixed; the grst 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, natually, 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 (q.v.), or the science of numbers in its mos. 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 grigonometry (q.v.) and Analytical Geometry. The
Differential and Integral Calculus (q.v.) makes use of the operations or processes of geonie try, algebra, and analysis indifferently; the calculus of finite differences is in part included under algebra, and may be considered as an extension of that science; and the calculus of variations is based upon the differential calculus. The term " mixed mathematics" is calculated to lead to error; "applied mathematics" is a more appropriate name. This.
portion of mathematics includes all those sciences in which a few simple axioms are mathematically shown to be sufficient for the deduction of the most important natural phenomena. This definition includes those sciences which treat of pressure. motion, light, heat, sound, electricity and magnetism—usually called excludes chemistry, geology, political economy, and the other branches of science, which, how ever, receive more or less aid from mathematics. For a notice of the separate sciences, see ASTRONOMY, OPTICS, MECHANICS, HYDROSTATICS, HYDRODYNAMICS, HEAT, ACOUS TICS, ELECTRICITY, MAG:NETISM, CIC.