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27(5 MATHEMATICS mula. The formula expresses merely a rela- tion between the different objects, and the relation in all these cases may be the same, although the objects themselves have noth- ing in common. The engineer, the actuary, and the machinist would each interpret the formula in accordance with the nature of the objects about which he was specially con- cerned. The attempt has often been made to give to philosophical speculations a mathe- matical form, in order to give them mathemat- ical certainty. Thus Pythagoras sought in the ideas of order and harmony mysteriously attached to numbers the reasons for great cos- mical phenomena. Plato, who forbade any one unacquainted with geometry to enter his school, combined mathematical with philo- sophical doctrines especially in his " Timams," the most obscure of his dialogues. The Neo- Platonists revived the Pythagorean mystical views of numbers. In modern times Spinoza, Wolf, and Herbart have been chiefly distin- guished for introducing the mathematical meth- od into ethical and metaphysical systems. The latter wrote a work on psychology abounding in algebraic formulas. These attempts have led to no important results. The definitions, axioms, and processes of mathematics deal with objects of sense, which are known with perfect exactitude, which are apprehended as precisely the same by all, concerning which as phenomena there can be no such thing as opin- ion, but only absolute certainty, and the reality of the relations between which can be doubted only by disputing the validity of all human ideas. In none of the most scientific meta- physical and moral systems have the definitive and axiomatic elements been thus precisely and authoritatively determined. The history of mathematics may be divided into three great periods, each characterized by the introduction of important new methods. In the first, the era of Greek and Roman supremacy, geometry was almost exclusively cultivated. While arith- metic was hardly more than a mechanical cal- culation by means of the abacus, geometrical methods attained a degree of elegance scarcely to be surpassed, as appears from the rank still maintained by Euclid. After the decline of Rome, the sciences took refuge among the Arabs, who translated and preserved the liter- ary treasures of Greece. The Arabian philoso- phers were, however, rather learned than in- ventive, and added little to the heritage. But they introduced the second great period in the progress of mathematics by imparting to Europe the decimal arithmetic and the alge- braic calculus, both of which were perhaps of Indian origin. The latter, diffused in Italy by Leonardo, a merchant and traveller of Pisa, early in the 13th century, soon received im- portant improvements. Scipio Ferrea (1505) was the first to solve a cubic equation. Car- dan and Tartalea disputed the honor with him and with each other, while Ferrari solved the biquadratic equation, and Vieta (1600), Girard, and Harriot entered upon the general theory of equations. The algebraic analysis was thus brought nearly to its present state of perfec- tion. It was at first regarded merely as a pre- paratory process in the investigation of a prob- lem, to be afterward exchanged for a geometri- cal construction and synthetic proof. But it gradually supplanted diagrams as a medium of demonstration, being found to surpass them in force and compass. With Descartes begins a great revolution of mathematical science. His tween two variable magnitudes revolutionized the mode of conceiving geometrical questions. Symbolical language, found adequate for every purpose, soon became the general medium of mathematical inquiry, and has been the prin- cipal weapon by which its subsequent splendid triumphs have been achieved. Perceiving the importance of the discovery, Descartes hasten- ed to apply it to questions of the greatest diffi- culty and generality, and resolved the problems of tangents and of maxima and minima. The methods of Roberval and Fermat tended to- ward the discovery of the differential calculus, which was made independently by Newton (under the form of fluxions) and by Leibnitz. Already Napier had invented logarithms, and Newton the binomial theorem ; Mercator had accomplished the quadrature of the hyperbola, and Wallis the quadrature of many other curves while seeking that of the circle. The integral calculus (the Newtonian method of quadra- tures), the inverse of the differential, was im- proved by Leibnitz and the Bernoullis ; Euler extended the theory of analytical trigonom- etry ; Fontaine illustrated that of differential equations; Taylor invented the calculus of finite differences or increments ; Cavalieri pub- lished his method of indivisibles; and other improvements were introduced by Kepler, Huygens, and Wallis. The Principia of New- ton (1687) has gained for him the title of " the profoundest of geometers as well as the first of natural philosophers;" and his influence combined with that of Leibnitz in preparing for the achievements of the mixed mathe- matics. Euler, D'Alembert, and Daniel Ber- noulli were the most distinguished of their successors till near the close of the 18th cen- tury. Euler suggested conceptions in the ap- plication of analysis which others elaborated in almost every part of mathematical science ; D'Alembert established a principle by which every dynamical question was resolved into a statical one; Daniel Bernoulli received ten prizes from the French academy of sciences ; and other contemporaries, as Clairaut and Mac- laurin, were extending the application of mathematics to mechanics and physics. In the period embracing the latter part of the 18th and the early part of the 19th century, the names of Lagrange and Laplace had no rivals. By them the application of all modes of calcu- lation to the mechanics of the universe was carried to the highest pitch of generality and