ANALYSIS (Gr.), the resolution of a whole in to its component parts. In mental philosophy, this term is applied to the logical treatment of an idea so as to resolve it into other ideas which combine to form it. A judgment or proposition may thus also be analyzed. The opposite of A. is synthesis (q.v.); and the opposition of these terms is common in other branches of science as well as in mental philosophy. We speak of an analytic method in science, and of a synthetic method; and both are necessary, the one coming to the assistance of the other to secure against error, and promote the ascertain ment of truth. The analytic method proceeds from the examination of facts to the deter mination of principles ; whilst the synthetic method proceeds to the determination of consequences from principles known or assumed. The test of perfection in a theory is the harmony of the results obtained by the methods of A. and synthesis.
Mathematical A., in the modern sense of the term, is the method of Treating all quan tities as unknown numbers, and representing them for this purpose by symbols, such as letters, the relations subsisting among them being thus stated and subjected to further investigation. It is therefore the same thing with algebra, in the widest sense of that term, although the term algebra is more strictly limited to what relates to equations, and thus denotes only the first part of A. The second part of it, or A. more strictly so called, is divided into the A. of finite quantities, and the A. of infinite quantities. To the for mer, also called the theory of functions, belong the subjects of series, logarithms, curves, etc. The A. of infinites comprehends the differential calculus, the integral calculus, and
the calculus of variations. To the diligent prosecution of mathematical A. by minds of the greatest acuteness, is to be ascribed the great progress both of pure and applied mathematics within the last two centuries.
The A. of the ancient mathematicians was a thing entirely different from this, and consisted simply in the application of the analytic method as opposed to the synthetic, to the solution of geometrical questions. That which was to be proved being in the first place assumed, an inquiry was instituted into those things upon which itidepended, and thus the investigation proceeded, as it were, back, until something was reached which was already ascertained, and from which the new proposition might be seen by necessary consequence to flow. A reversal of the steps of the inquiry now gave the synthetical proof of the proposition. The modern mathematical A. affords a much more easy and rapid means of geometrical questions; but the ancient A. also afforded opportu nity for the exercise of much acuteness, and was the chief instrument of the advance ment of mathematical science until comparatively recent times. The invention of it is ascribed to Plato; but of the works of the ancients on geometrical A. none are extant, except some portions of those of Euclid, Apollonius of Pugs, and Archimedes.