DYNAMICS  ,

ANALYTICAL; GYROSCOPE; HARMONIC ANALYSIS; WAVE; HYDROMECHANICS ; ELASTICITY; MOTION, LAWS OF; ENERGY; ENERGETICS; ASTRONOMY (Celestial Mechanics) ; TIDE . Mechanics (including dynamical astronomy) is that subject among those traditionally classed as " applied " which has been most completely transfused by mathematics—that is to say, which is studied with the deductive spirit of the pure mathematician, and not with the covert inductive intention overlaid with the superficial forms of deduction, characteristic of the applied mathematician . Every branch of physics gives rise to an application of mathematics . A prophecy may be hazarded that in the future these applications will unify themselves into a mathematical theory of a hypothetical substructure of the universe, uniform under all the diverse phenomena . This reflection is suggested by the following articles: AETHER; MOLECULE; CAPILLARY ACTION; DIFFUSION; RADIATION, THEORY OF; and others . The applications of mathematics to statistics (see STATISTICS and PROBABILITY) should not be lost sight of ; the leading fields for these applications are insurance, sociology, variation in zoology and economics . The History of Mathematics.—The history of mathematics is in the main the history of its various branches . A short account of the history of each branch will be found in connexion with the article which deals with it . Viewing the subject as a whole, and apart from remote developments which have not in fact seriously influenced the great structure of the mathematics of the European races, it may be said to have had its origin with the Greeks, working on pre-existing fragmentary lines of thought derived from the Egyptians and Phoenicians . The Greeks created the sciences of geometry and of number as applied to the measurement of continuous quantities . The great abstract ideas (considered directly and not merely in tacit use) which have dominated the science were due to them—namely, ratio, irrationality, continuity, the point, the straight line, the plane . This period lasted' from the time of Thales, c .

600 B.C., to the

capture of Alexandria by the Mahommedans, A.D . 641 . The medieval Arabians invented our system of numeration and developed algebra . The next period of advance stretches from the Renaissance to Newton and Leibnitz at the end of the 17th century . During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation . The 18th century witnessed a rapid development of analysis, and the period culminated with the genius of Lagrange and Laplace . This period may be conceived as continuing throughout the first quarter of the loth century . It was remarkable both for the brilliance of its achievements and for the large number of French mathematicians of the first rank who flourished during it . The next period was inaugurated in analysis by K . F . Gauss, N . H .

Abel and A . L . Cauchy . Between them the general theory of the complex variable, and of the various " infinite" processes of mathematical analysis, was established, while other mathematicians, such as Poncelet, Steiner, Lobatschewsky and von Staudt, were founding modern geometry, and Gauss inaugurated the differential geometry of surfaces . The applied mathematical sciences of light, electricity and electromagnetism, ' Cf A Short History of Mathematics, by W . W . R . Ball . and of heat, were now largely developed . This school of mathematical thought lasted beyond the middle of the century, after which a change and further development can be traced . In the next and last period the progress of pure mathematics has been dominated by the critical spirit introduced by the German mathematicians under the guidance of Weierstrass, though fore-shadowed by earlier analysts, such as Abel . Also such ideas as those of invariants, groups and of form, have modified the entire science .

But the progress in all directions has been too rapid to admit of any one adequate characterization . During the same period a brilliant

group of mathematical physicists, notably Lord Kelvin (W . Thomson), H . V . Helmholtz, J . C . Maxwell, H . Hertz, have transformed applied mathematics by systematically basing their deductions upon the Law of the conservation of energy, and the hypothesis of an ether pervading space . translations into French and Italian) . (A . N .