Analysis (Gr., 'taking apart') and its converse Synthesis ('putting together') are now generally used to designate two complementary processes, the correlatives of each other, employed in chemistry, logic, mathematics, and philosophy. Analysis is the resolution of a whole into its component parts, the tracing of things to their source, and so discovering the general principles underlying individual phenomena; Synthesis is the explanation of certain phenomena by means of principles which are for this purpose assumed as established. Analysis, as the resolution of our experience into its original elements, is an artificial separation; while synthesis is an artificial reconstruction. 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 ascertaining of truth. The analytic method proceeds from the examination of facts to the determination of principles, from the individual to the universal; whilst the synthetic method proceeds to the determination of consequences from principles known or assumed, to the individual from the universal. It will thus be seen that they are really two necessary parts of the same method, and that, whereas the value of the synthesis depends on the accuracy of the analysis which has established the principle from which the synthesis sets out, so, on the other hand, an analysis which does not aspire to a synthesis, halts on the way. Synthesis without analysis gives a false science, since it is a pure imagination, based simply on hypothesis; and analysis without synthesis gives an incomplete science. The ideal of science and of philosophy can only be attained by a method which combines the two processes, and the test of the perfection of a theory is the harmony of the results obtained by them.
The part of Herbert Spencer's Psychology which he calls Special Analysis (following the preliminary General Analysis), has 'for its aim, to resolve each species of cognition into its components. Commencing with the most involved ones, it seeks by successive decompositions to reduce cognitions of every order to those of the simplest kind; and so, finally, to make apparent the common nature of all thought, and disclose its ultimate constituents;' while the synthesis describes the nature and genesis of the different modes of intelligence. In Logic, analysis is the division of a concept into the qualities or attributes of which it is constituted (see ABSTRACTION), whilst synthesis is the reverse process of adding together the qualities or attributes which determine a particular concrete. See GENERALISATION, INDUCTION, LOGIC, ALGEBRA, GEOMETRY.
In Grammar, analysis is a term much used since 1852 for the school exercise of distinguishing the different elements composing a sentence, or any part of it. It is allied to logical analysis, being a systematic resolution of the sentence into elements, performing different functions in the expression of thought, with definite relations to the whole sentence and to each other, as subject and predicate, with their respective enlargements. Dr Morell was one of the first of English grammarians to make a systematic use of this method in books for teaching.
Mathematical Analysis, in the modern sense of the term, is the method of treating all quantities 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 analysis. The second part of it, or analysis more strictly so called, is divided into the Analysis of Finite Quantities, and the Analysis of Infinite Quantities. To the former, also called the Theory of Functions, belong the subjects of Series, Logarithms, Curves, &c. The Analysis of Infinites comprehends the Differential Calculus, the Integral Calculus, and the Calculus of Variations (see the several articles). To the diligent prosecution of mathematical analysis 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 analysis 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 it depended, 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 analysis affords a much more easy and rapid means of solving geometrical questions; but the ancient analysis also afforded opportunity for the exercise of much acuteness, and was the chief instrument of the advancement of mathematical science until comparatively recent times. The invention of it is ascribed to Plato; but of the works of the ancients on geometrical analysis none are extant, except some portions of those of Euclid, Apollonius of Perga, and Archimedes.