Mathematics (Gr. , or , learning), as usually defined, the science of quantities; or more precisely, the science which determines unknown quantities by means of their relations to known quantities. But the tendency of modern thought is to give the term a much wider meaning, to include under it all exact sciences, and to designate as mathematical every science which can be reduced to a limited number of definite conceptions from which all the propositions which constitute the science can be deduced in accordance with the rules of logic. It is defined by Kant as the science of the laws of space and time, since it treats of the quantities occupying space and time, and representable by diagrams, numbers, or symbols. In this he has been followed by De Morgan and some other mathematicians. But in order to make the science conform to the definition, they have been obliged to regard the idea of number as included or implied in the idea of time. The present tendency of mathematical speculation is to regard mathematics, when considered in its most general form, as a branch of the science of mind, and every mathematical formula as expressing an operation of the understanding.

This doctrine is expressly asserted by Ohm, and seems to be implied, if not expressly stated, in the writings of Grassmann, Peirce, and many other modern mathematicians. - -The science is distinguished as pure or mixed mathematics, according as it treats of laws and relations in alatracto, with reference to nothing actual, or in concreto, with reference to existing phenomena. The former, dealing with abstract quantity, does riot imply the idea of matter; the latter, deal-ing with concrete quantity, embraces the actual material world. The former gives the absolute form, of the universe; the latter, their illustration, by real examples. The elements employed by the former are self-evident principle, suggested or immediately grasped by the reason itself; the latter applies these principles to natural objects, the properties of which must be learned by induction from experience. The former treats of possible, the latter of actual magnitudes. - The branches of pure mathematics are arithmetic, geometry, algebra, analytical geometry, and the differential and in-tegral calculus. Arithmetic is the science and art of numbers. It does not calculate functions or relations, but special values in every case.

Its single elementary idea is one or unity, from which all other numerical values, integral or fractional, are formed. The processes of arithmetic lie at the basis of all others. Geometry measures extension, comparing portions of space with each other. Its elements are not numbers, but lines, surfaces, and volumes or solids. Lines have only the dimension of length, and are either straight or curved. Surfaces embrace both length and breadth, are either plane or curved, and are distinguished as triangles, quadrilaterals, polygons, etc, according to the number of lines within which they are contained. Solids combine the three dimensions of length, breadth, and thickness, and are distinguished as the cube, pyramid, cone, sphere, etc, according as they are bounded by planes, by plane and curved surfaces, or only by curved surfaces. Definitions, or statements of a priori facts, axioms, or statements of self-evident relations, and propositions, deduced from definitions and axioms, as premises, in a series of logical arguments, are the three classes of geometrical truths. Algebra, analytical geometry, and the differential and integral calculus embrace the entire portion of mathematical science in which quantities are represented, not by numbers or diagrams, but by letters of the alphabet.

In arithmetic, all propositions concerning numbers, embracing units of the same kind, are true without regard to the nature of the quantities to which the numbers may be applied. In geometry, every figure represents all the properties inherent in all the figures of its class. But the truths both of arithmetic and geometry are applicable only to special and actual classes of things. Algebra has a broader generalization. Its symbols extend to all objects whatsoever, and do not suggest ideas of particular things. They stand as representatives of things in general, whether abstract or concrete, real or hypothetical, known or unknown, finite or infinite. Having the relation of quantities embodied in an equation of symbols, we may proceed to trace what other truths are involved in the one thus stated, resolving the symbolical assertion step by step into others more fitted for our purpose, thus following long trains of symbolical reasoning, every result of which must express some general truth, though, in the present state of our knowledge, we may not be able to give any actual example of the truth.

Analytical geometry, the application of algebra to geometry, is that branch of mathematical science which- examines, discusses, and develops the properties of geometrical magnitudes, by noticing the changes which take place in their representative algebraic symbols. The geometrical question is solved by resolving the corresponding algebraic equation. Algebra being defined as the ordinary analysis, calculus is the transcendental analysis, and has various applications in the higher departments of the science. The best achievements of modern mathematics are due to it. - To these branches of mathematics the 19th century has added another, the final form of which is as yet undetermined, but the essential characteristics of which are to be found in the "quaternions" of Sir W. Rowan Hamilton, in the Ausdehmmgslehre of H. Grass-mann, and in the " Linear-Associative Algebra " of Prof. Peirce. The great characteristics of this new science are: 1, the introduction of several units differing in quality; and 2, the rigid distinction between the multiplier and the multiplicand, or between the thing which acts and the thing acted upon. In the mathematical sciences, as hitherto treated, xy is always equal to yx; it is a matter of indifference which quantity we regard as multiplier and which as multiplicand.

In the new science the distinction must be always regarded; xy and yx are entirely different things. The second characteristic is really a result of the first. Thus, in geometry, as treated by Grassmann, we have four different units, viz., a point and three mutually perpendicular straight lines. From the combinations of these units all the truths of geometry are deduced. Prof. Peirce, in his work above mentioned, has endeavored to fix a priori the laws which must regulate this introduction of units, and has divided algebra, according to the number of units introduced, into single, double, triple, etc. We can enter into no further explanations of this branch of mathematics, but will remark that as the great event in the intellectual history of ,the 17th century was the invention of the calculus, so perhaps future historians will regard this as the great event in the history of the 19th century. - Algebra and geometry are usually, but not with strict accuracy, regarded as types respectively of analytical and synthetical reasoning. The former has an artificial language. Symbols are operated upon according to certain general rules, while the mind dismisses altogether the conceptions of the things which the symbols represent, whether lines, angles, velocities, forces, or whatever else.

The steps in the processes are merely applications of the rule. The elements are symbols, and the results are only equations. Geometrical reasoning, on the contrary, is concerning things as they are. It retains the conceptions of quantities. It apprehends the nature of the new truths which it introduces at every step. Analysis is therefore the more powerful instrument for the professed mathematician, but geometry is the more effective mode of exercising the reason, and is amore useful part of the gymnastics of education. - Comte, who makes mathematics preeminent in the hierarchy of the positive sciences, introduces a peculiar classification. Abstract mathematics, according to him, embraces ordinary analysis, or the calculus of direct functions, and transcendental analysis, or the calculus of indirect functions. The former includes arithmetic and algebra; the latter, the differential and integral calculus and the calculus of variations. Concrete mathematics embraces synthetic and analytic geometry, the former being either graphic or algebraic, and the latter beting distinguished according as its objects are of two or three dimensions. Comte includes also rational mechanics, or the laws of statics and dynamics, as a department of concrete mathematics.

If the universe were immovable, there would be only geometrical phenomena; but motions are mechanical phenomena. - As commonly explained, the mixed mathematics are the applications of abstract mathematical laws to the objects of nature and art. From the universality and variety of these objects, no strict and comprehensive classification of them has been made. Matter in rest and matter in motion are the primary phenomena in space and time. The laws which rule the one and the forces which impel the other are the first objects of inquiry. Mechanics treats of both, and is divided into statics and dynamics, dealing respectively with the equilibrium and the action of forces. Astronomy, hydraulics, pneumatics, optics, and acoustics may be regarded as subdivisions of dynamics. Surveying, architecture, fortification, and navigation are among the principal applications of mathematics to the arts. - The pure mathematics are merely formal sciences. They occupy and discipline but do not fill the mind. Their entirely formal character will be best appreciated by one or two illustrations. It is a law of falling bodies that the spaces passed through by the falling body are proportional to the squares of the times during which it falls.

It is a law of geometry that the areas of circles are proportional to the squares of their radii. The mathematical formula expressing one of these laws also expresses the other. Let A: a=b2: B2, and we may consider A and a as representing either spaces described by a falling body or areas of circles, and B and o as representing either the times during which it falls or radii of circles. In either case the formula is true. Yet the space described by a falling body and the area of a circle, the time during which a body falls and the radius of a circle, are wholly disparate notions. When we see a person adding a column of numbers, no inspection of the column itself will tell us what the person who wrote it down intended to represent by those numbers. He may have had in his mind sums of money, or yards of cloth, or bushels of wheat. Whatever it was, the process of finding the sum is in all cases the same. Again, an engineer investigating a problem in regard to bridge building, an actuary one in life insurance, a machinist one in mechanics, might all arrive at the same algebraical forinula. The formula expresses merely a relation between the different objects, and the relation in all these cases may be the same, although the objects themselves have nothing 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 concerned. - The attempt has often been made to give to philosophical speculations a mathematical form, in order to give them mathematical 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 schol, combined mathematical with philosophical 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'metl!-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 opinion, 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 metaphysical 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 arithmetic was hardly more than a mechanical calculation by means of the abacus, geometrical methods attained a degree of elegance scarcely-to Ik- surpassed, as appears from the rank still maintained by Euclid. After the decline of Koine, the sciences took refuge among the Arabs, who translated and preserved the literary treasures of Greece. The Arabian philosophers were, however, rather learned than inventive, and added little to the heritage. But they introduced the second great period in tin- progress of mathematics by imparting to Europe the decimal arithmetic and the algebraic 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 Pith century, soon received important improvements. Scipio Ferrea (1505) was the first to solve a cubic equation.

Cardan and rartalea 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 perfection. It was at first regarded merely as a preparatory process in the investigation of a problem, to be afterward exchanged for a geometrical 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 mode of characterizing curves by an equation between 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 principal weapon by which its subsequent splendid triumphs have been achieved. Perceiving the importance of the discovery, Descartes hastened to apply it to questions of the greatest difficulty and generality, and resolved the problems of tangents and of maxima and minima.

The methods of Eoberval and Fermat tended toward 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 quadratures), the inverse of the differential, was improved by Leibnitz and the Bernoullis; Euler extended the theory of analytical trigonometry; Fontaine illustrated that of differential equations; Taylor invented the calculus of finite differences or increments; Cavalieri published his- method of indivisibles; and other improvements were introduced by Kepler, Huygens, and Wallis. The Principia of Newton (1087) 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 mathematics. Euler, D'Alembert, and Daniel Bernoulli were the most distinguished of their successors till near the close of the 18th century.

Euler suggested conceptions in the application 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 calculation to the mechanics of the universe was carried to the highest pitch of generality and symmetry. One of the most remarkable achievements of the science was Leverrier's prediction in 1846 of the place and orbit of the planet Neptune from- the motions of Uranus, announcing before its discovery by the telescope the existence, position, and magnitude of a body beyond the recognized limits of our system, merely as an inference from the perturbations of the outermost planet known to us. Poisson, Airy, Plana, Hansen, Gauss, Adams, De Morgan, and Peirce are among the recent mathematicians who have solved important problems in the physical application of analysis.

Many new mathematical theories have been originated during the last half century. Among them may be mentioned the theory of determinants, the theory of invariants of Messrs. Cayley and Sylvester, that of clinants of Mr. Ellis, and many others. Some of them will probably pass into history only as evidences of the ingenuity of their authors, while others promise to be of great value. As they are of interest to professed mathematicians only, they require no further notice in this work. - Among the greatest works in mathematical literature are the Principia of Newton, the Mechanica of Euler, the Theorie des fonctions and the Mechanique analytique of Lagrange, the Application de Vanalyse a la geometric of Monge, and the Mecanique celeste of Laplace. - See Montucla, Histoire des mathematiques, continued by Lalande (4 vols., Paris, 1799-1802); Bossut, Essai sur l'histoire des mathematiques (2 vols., Paris, 1802); Comte, Philosophic positive, vol. i., and Synthese positive; Libri-Carucci, Histoire des sciences mathematiques en Italie (4 vols., Paris, 1838 - '41); Montferrier, Dictionnaire des sciences mathematiques (2d ed., 3 vols. 4to, Paris, 1844), and Eiicyclopedie mathematique d'apres les prin-cipes de la philosophic des mathematiques de Hoine Wronski (4 vols. 8vo, Paris, 1856-'9); Fries, Die mathematische Naturphilosopjhie (Heidelberg, 1822); Poppe, Geschichte der Mathematik (Tubingen, 1828); Ohm, Versuch eines volkommenen, consequenten Systems der Mathematih (3d ed., Nuremberg, 1853-5); Bartholomaei, Philosophic der Mathematik (Jena, 1860); Davies, "Logic and Utility of Mathematics " (New York, 1851); and Davies and Peck, "Mathematical Dictionary" (New York, 1856). See also the works cited under the title Geometry.