Analysis \A*nal"y*sis\, n.; pl. Analyses. [Gr. ?, fr. ? to unloose, to dissolve, to resolve into its elements; ? up + ? to loose. See Loose.]

1. A resolution of anything, whether an object of the senses or of the intellect, into its constituent or original elements; an examination of the component parts of a subject, each separately, as the words which compose a sentence, the tones of a tune, or the simple propositions which enter into an argument. It is opposed to synthesis.

2. (Chem.) The separation of a compound substance, by chemical processes, into its constituents, with a view to ascertain either (a) what elements it contains, or (b) how much of each element is present. The former is called qualitative, and the latter quantitative analysis.

3. (Logic) The tracing of things to their source, and the resolving of knowledge into its original principles.

4. (Math.) The resolving of problems by reducing the conditions that are in them to equations.

5.
   (a) A syllabus, or table of the principal heads of a discourse, disposed in their natural order.
   (b) A brief, methodical illustration of the principles of a science. In this sense it is nearly synonymous with synopsis.

6. (Nat. Hist.) The process of ascertaining the name of a species, or its place in a system of classification, by means of an analytical table or key.

Ultimate, Proximate, Qualitative, Quantitative, and Volumetric analysis. (Chem.) See under Ultimate, Proximate, Qualitative, etc.

Source: Webster's Revised Unabridged Dictionary (1913)
 

Mathematics \Math`e*mat"ics\, n. [F. math['e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See Mathematic, and -ics.]

That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations.

Note: Mathematics embraces three departments, namely: 1. Arithmetic. 2. Geometry, including Trigonometry and Conic Sections. 3. Analysis, in which letters are used, including Algebra, Analytical Geometry, and Calculus. Each of these divisions is divided into pure or abstract, which considers magnitude or quantity abstractly, without relation to matter; and mixed or applied, which treats of magnitude as subsisting in material bodies, and is consequently interwoven with physical considerations.

Source: Webster's Revised Unabridged Dictionary (1913)