MATHEMATICS, originally signified any discipline or learning ; but, at pre sent, denotes that science which teaches, or contemplates, whatever is capable of numbered or measured, in so far as it is computable or measurable ; and, accordingly, is subdivided into, arithmetic, which has numbers for its object, and geometry, which treats of magnitude: See ARITHMETIC and GEOMETRY.
Mathematics are commonly distinguish ed into pure and speculative, which con sider quantity abstractedly ; and mixed, which treats of magnitude as subsisting in material bodies, and consequently are in terwoven every where with physical con siderations. Mixed mathematics are very comprehensive ; since to them may be re ferred astronomy, optics, geography, by drography, hydrostatics, mechanics, forti fication, navigation, &c. See ASTRONOMY, OPTICS, &C.
Pure mathematics have one peculiar advantage, that they occasion no disputes among wrangling disputants, as in other branches of knowledge ; and the reason is, because the definitions of the terms are premised, and every body that reads a proposition has the same idea of every part of it. Hence it is easy to put an end to all mathematical controversies, by sheaving either that our adversary has not stuck to his definitions, or has not laid down true premises, or else that he has drawn false conclusions from true princi ples ; and in case we are able to do neither of these, we must acknowledge the truth of what he has proved.
It is true, that in mixed mathematics, where we reason mathematically upon physical subjects, we cannot give such just definitions as the geometricians : we must, therefore, rest content with de scriptions ; and they will be of the same use as definitions, provided we are con sistent with ourselves, and always mean the same thing by those terms we have once explained. Dr. Barrow gives a most
elegant description of the excellence and usefulness of mathematical knowledge, in his inaugural oration, upon being ap pointed Professor of Mathematics at Cambridge.
The mathematics, he observes, effec tually exercise, not vainly delude, nor vexatiously torment, studious minds with obscure subtilties ; but plainly demon strate every thing within their reach, draw certain conclusions, instruct by pro fitable rules, and unfold pleasant ques tions. These disciplines likewise enure and corroborate the mind to a constant diligence in study ; they wholly deliver us from a credulous simplicity, most strongly fortify us against the vanity of scepticism, effectually restrain as from a rash pre sumption, most easily incline us to a due assent, perfectly subject us to the govern ment of right reason. While the mind is abstracted and elevated from sensi ble matter, distinctly views pure forms, conceives the beauty of ideas, and in vestigates the harmony of proportions ; the manners themselves are sensibly corrected and improved, the affections composed and rectified, the fancy calm ed and settled, and the understanding raised and excited to more divine con templations.