MATHEMATICS (aciaricris, or aciOssa), a 'name given in the first instance to a branch of knowledge, not as descriptive of its subject matter, but of the methods and consequences of learning it. The word actensis, and the Latin diseiplina by which it has been rendered, have been the origin of the vernacular terms mathematics and discipline, the meanings of which have long since separated. The properties of space and number, the subject-matters of the actOscrir, have usurped the name ; so that anything which relates to them, however learnt, is called mathematics : the Latin word, on the contrary, still retains the signi fication of a corrective process; and, in speaking of any branch of knowledge, is applied when power of mind is derived from the methods of learning it, as well as knowledge from the results.
The original use of the word mathematics cannot be gathered, so far as we can find, from any express contemporary authority ; a few pass ages in which the term is used without explanation, as one of notoriety, being all that can be cited, and mostly from Plato. Later writers, as for instance Anatolius (cited by Heilbronncr), A.D. 270, give the deri vation above alluded to. But before the time of Anatoliue the meaning of the word had been extended : thus the book of Sextus Empiricus "against the mathematicians" is, as Vossius remarks, directed as much against grammarians and musicians as against arith meticians and geometers. And John Tzetzes, in the 12th century, includes under the paanyara nearly what the universities afterwards called by the name of arts ; calling grammar, rhetoric, and philosophy, the disciplines (uaOhyara), and arithmetic, music, geometry, aud astro nomy, arts (wixecii) included under philosophy.
The distinction between the old and new meaning of mathematics is most requisite to be kept in mind, because arguments are frequently urged for and against mathematics, in which the discipline is con founded with the communication of facts and processes about space and number; and because it is our intention in the present article, con fining ourselves to the most important view of the science, as well as to the etymological meaning of its name, to offer a few remarks on the discipline called mathematics.
In the time of Plato, which was probably that of the application of words which imply " the discipline" to that one exercise of mind which consists in making deductions by pure reasoning from the self-evident properties of space and number, it is probable that such restriction of the word was easily justifiable. At present we have, besides mathe
matics, also physics, the study of antiquity, grammar, &c., which have all been made disciplines, but not one of which was then entitled to that appellation. Nevertheless it has happened that writers, misled partly by the name of mathematics and partly by the pre-eminence of mathematical reasoning in strictness and connection, have spoken as if it were the only cultivator of the pure reasoning power.
Much discussion has arisen upon the question whether those primary propositions which, from our clear apprehension and willing admission of them, are called self-evident, are notions inherent in the mind, or deductions of early experience. Except to mention this controversy, we have here nothing to do with it. The certainty of these proposi tions is all that we want, and this is conceded by both sides. The con sideration however of the fundamental supports of mathematical reasoning is useful and interesting, and, as a safeguard, even necessary. It is not long since a school of metaphysicians existed who imagined that because all mathematical definitions are precise, therefore the exact sciences are founded upon definition. It was not to them a necessary result of the constitution of our faculties that the three angles of every triangle make up the same amount, but a consequence of definition, which might have been something else, upon different suppositions. We can hardly undertake to explain what we do not understand : if the opinion we have quoted have not the meaning we have given to it, there is in it some confusion of terms. We recom mend every beginner in the subject to seek no knowledge about the character of fundamental propositions until he shall have become well acquainted with their consequences. He must take care to admit nothing which is not, or cannot be made, most evidently true; and he will find that all axioms, as they are called, have the highest sort of certainty, namely, that they cannot be imagined otherwise.