Page:The American Cyclopædia (1879) Volume XI.djvu/286

274 MATERA fee, as it contains nearly one half of one per cent. (0-45) of caffeine and 20-88 of caffeo- tannic acid. The amount of the leaves export- ed annually from Paraguay is estimated at over 5,000,000 Ibs. The early Jesuit missionaries, knowing the fondness of the aborigines for the mat6, established plantations of the tree, on which account it is sometimes called Jesuits' tea. The leaves of a related species, ilex cos- tine, furnished the " black drink " or yaupon of the Creek Indians; the leaves of this possess emetic qualities, and the power of resisting them was regarded as a mark of superiority. MATERA, a town of S. Italy, in the prov- ince and 43 m. E. of the city of Potenza, on the Gravina; pop. about 14,000. It is the seat of an archbishop, and has a royal school of belles-lettres, medicine, law, and agriculture, and manufactories of firearms. Near it are the famous caverns of Monte Scaglioso. MATERIA MEDICA. See MEDICINE. MATHEMATICS (Gr. /z(i%a, or fiddqaic, learn- ing), as usually defined, the science of quanti- ties ; or more precisely, the science which de- termines unknown quantities by means of their relations to known quantities. But the ten- dency 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 limit- ed 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 sym- bols. 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 mathe- matics, 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 abstracto, with reference to nothing actual, or in concrete, with reference to existing phenomena. The former, dealing with abstract quantity, . does not imply the idea of matter ; the latter, deal- ing with concrete quantity, embraces the actual material world. The former gives the absolute forms of the universe ; the latter, their illustra- tions by real examples. The elements em- ployed by the former are self-evident princi- ples, suggested or immediately grasped by the reason itself ; the latter applies these principles to natural objects, the properties of which MATHEMATICS must be learned by induction from experience. The former treats of possible, the latter of ac- tual magnitudes. The branches of pure mathe- matics are arithmetic, geometry, algebra, ana- lytical 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 arith- metic lie at the basis of all others. Geome- try 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 tri- angles, quadrilaterals, polygons, &c., according to the number of lines within which they are contained. Solids combine the three dimen- sions of length, breadth, and thickness, and are distinguished as the cube, pyramid, cone, sphere, &c., 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 class- es of geometrical truths. Algebra, analytical geometry, and the differential and integral calculus embrace the entire portion of mathe- matical science in which quantities are repre- sented, not by numbers or diagrams, but by letters of the alphabet. In arithmetic, all prop- ositions 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 arith- metic and geometry are applicable only to spe- cial 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 rep- resentatives of things in general, whether ab- stract 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 knowl - edge, 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 exam- ines, discusses, and develops the properties of geometrical magnitudes, by noticing the changes which take place in their representative alge-