Volume 3: 1771 Edition of the Encyclopedia Britannica!
Mathematics
Mathematics, originally signified any discipline or learning; but, at
present, denotes that science which teaches, or contemplates, whatever is
capable of being numbered or measured, in so far as computable or
measurable; and accordingly is subdivided into Arithmetic, which has
numbers for its object, and Geometry, which treats of magnitude.
See ARITHMETICK and GEOMETRY.
Mathematics are commonly distinguished into pure and speculative, which
consider quantity abstractedly; and mixed, which treat of magnitude as
subsisting in material bodies, and consequently are interwoven every where
with physical considerations.
Mixed mathematics are very comprehensive; since to them may be referred
Astronomy, Optics, Geography, Hydrostatics, Mechanics, Fortification,
Navigation, etc. See the articles ASTRONOMY, OPTICS, ETC.
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 shewing,
either that our adversary has not stuck to his definitions, or has not laid
down true premisses, or else that he has drawn false conclusions from true
principles; 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 descriptions; and they
will be of the same use as definitions, provided we are consistent 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
appointed professor of mathematics at Cambridge.
The mathematics, he observes, effectually exercise, not vainly delude, nor
vexatiously torment, studious minds with obscure subtilties; but plainly
demonstrate every thing within their reach, draw certain conclusions,
instruct by profitable rules, and unfold pleasant questions. These
disciplines likewise ensure 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 us from a rash presumption, most easily incline us to a due
assent, perfectly subject us to the government of right reason. While the
mind is abstracted and elevated from sensible matter, distinctly views pure
forms, conceives the beauty of ideas, and investigates the harmony of
proportions; the manners themselves are sensibly corrected and improved,
the affections composed and rectified, the fancy calmed and settled, and
the understanding raised and excited to more divine contemplations.