MATHEMATICS

From Agepedia

MATHEMATICS. If we call every thing, which we can represent to our mind as composed of homogeneous parts, a magnitude, mathematics, according to the common definition, is the science of determining magnitudes, i. e. of measuring or calculating. Every magnitude appears as a collection of homogeneous parts, and may be considered in this sole respect; but it also appears under a particular form or extension in space, which originates from the composition of the homogeneous parts, and to which belong the notions of situation, proportion of parts, &c. Not only all objects of the bodily world, but also time, powers, motion, light, tones, &c, may be represented and treated as mathematical magnitudes. The science of mathematics has to do only with these two properties of magnitudes, the quantity of the homogeneous parts, which gives the numerical magnitude, and the form, which gives the magnitude of extension. This is one way, and the most common, of representing the subject: there are others more philosophical, but less adapted to the limited space which can be allowed to so vast a subject, in a work like the present. In investigating these two properties of magnitudes, the peculiar strictness of the proofs of mathematics gives to its conclusions and all its processes a certainty, clearness and general application, which satisfies the mind, and elevates and enlarges the sphere of its activity.* (See Method, Mathematical.) According as a magnitude is considered merely in the respects abovementioned, or in connexion with other circumstances, mathematics are divided into pure and applied. Pure mathematics are again divided into arithmetic (q. v.), which considers the numerical quality of magnitudes, and geometry (q. v.), which treats of magnitudes in their relations to space. In the solution of their problems,* As a branch of intellectual culture, mathematics has great excellences and great defects. Its certainty,the precision of its signs never conveying more nor less than the meaning intended, its completeness in itself, and independence of all other branches, distinguish it from every other science, and nothing accustoms the young mind more to precision and exactness of thought and expression than the study of mathematics. But, dn the other hand, these very excellences render it liable to give a partial direction to the mind, to withdraw it from, and unfit it for pursuits of a different character. Hence so many great mathematicians have appeared to be wholly unfitted for other studies. On the whole, however, its advantages are so great that it can never be dispensed with in a liberal education. Nothing expands and elevates the mind more than the acquisition of a mathematical truth, a law which is obeyed throughout the universe. The study of the conic sections, as has been already observed (see Cone), affords a fine illustration of this influence. And there are few instances in which there will be much danger of the pupil being indtily absorbed in the study. the common mode of numerical calculation, and also algebra (q. v.), and analysis (q. v.), are employed. To the applied mathematics belong the application of arithmetic to political, commercial and similar calculations ; of geometry to surveying (q. v.), levelling, &c. ; of pure mathematics to the powers and effects, the gravity, the sound, &c, of the dry, liquid and aeriform bodies in a state of rest, in equilibrium or in motion, in one word, its application to the mechanic sciences, (see Mechanics, Hydraulics, Hydrostatics, &c.); to the rays of light in the optical sciences (see Optics, Dioptrics, Perspective, &c.) ; to the position, magnitude, motion, path, &c, of heavenly bodies in the astronomical sciences (see Astronomy), with which the measurement and calculation of time (see Chronology) and the art of making &undials (see Vial) are closely connected. The name of applied mathematics has sometimes been so extended as to embrace the application of the science to architecture, navigation, the military art, geography, natural philosophy, &c.; but in these connexions it may more conveniently be considered as forming a part of the respective sciences and arts. It is to be regretted that there is as yet no perfectly satisfactory work, treating of the history of this science, so noble in itself, and so vast in its application: even Kastner and Montucla leave much to be desired. The establishment of mathematics on a scientific basis probably took place among the Indians and Egyptians. The first developement of the science we find among the Greeks, those great teachers of Eurone in almost all branches. Thales, and moi'1 particularly Pythagoras, Plato, Eudoxus, investigated mathematics with a scientific spirit, and extended its domain. It appears that geometry, in those ages, was more thoroughly cultivated than arithmetic. The ancients, indeed, under stood by the latter something different from that which we understand by it. In fact, we have not a clear idea of the ancient arithmetic. Their numerical calculation was limited and awkward, sufficient ground for which might be found m their imperfect way of writing numbers, if there was no other reason. Euclid's famous Elements, a work of unrivalled excellence, considering the time of its origin, the ingenious discoveries of Archimedes, the deep investigations of Apollonius of Perga, carried the geometry of tho ancients to a height which has been the admiration of all subsequent times. Sine* then it has been made to bear more on astronomy and has become more connected with arithmetic. Among the Greek mathematicians are still mentioned Eratosthenes, Conon, Nicomedes, Hipparchus, Nicomachus, Ptolemy, Diophantus, Theon, ProcJus, Eutocius, Papus and others. It is remarkable that the Romans showed little disposition for mathematics ; but. the Arabians, who learned mathematics, like almost all their science, from the Greeks, occupied themselves much with it. Algebra (q. v.) and trigonometry owe them important improvements. Through the Arabians, mathematics found entrance into Spain, where, under Alphonso of Castile, a lively zeal was displayed for the cultivation of this science. After this, it found a fertile soil in Italy; and in the convents a monk would sometimes follow out its paths, without, however, adding to its territory. This was reserved for later ages. Mathematics owes much to Gmiinden, Peuerbach, llegiomontanus, Pacciolo, Tartaglia, Cardanus, Macrolycus, Vieta, Ludolphus de Ceulen, Peter Nuiiez, Justus Byrge, and others. To this period, however, all mathematical operations of any extent required a weary length of detail; when, in the seventeenth century, Napier, by the introduction of logarithms, immensely facilitated the process of calculation; and Newton and Leibnitz, by their infinitesimal calculus, opened the way into regions, into which, before them, no mathematician attempted to penetrate. From this time, the science obtained a wonderful extension and influence, by the labors of such minds as Galilei, Torricelli, Pascal, Descartes, L'Hopital, Cassini, Huyghens, Harriot, Wallis, Barrow, Halley, James and John ernouilli, and others. Thus it becan <e possible for Manfredi, Nicoli, Nic. and Dan. Bernouilli, Elder, Maclaurin, Taylor, Bradley, Clairaut, D'Alembert, Lambert, Tobias Mayer, Kastner, Hindenburg (the inventor of the combinatory analysis), Lagrange, Laplace, Legendre, Gauss, Bessel, and the later mathematicians in the eighteenth, and in our century, to make great advances, and to give us satisfactory conclusions, not only respecting our earth, but also the heavenly bodies, the phenomena and powers of nature, and their useful application to the wants of life, to establish firmly so many notions, previously vague, and to correct so many errors. (See the articles on these mathematicians, and the works mentioned in the articles on the various branches of mathematics.) The o lumber of mathematical manuals in;Meases daily, without, however, much sur passing the best of the earlier ones in perspicuity, novelty and method, or rendering them unnecessary to the thorough student.