Page:The New International Encyclopædia 1st ed. v. 01.djvu/583

cus eharactpristios ; thus, the process by which we recognize that an apple is a tiling whose attributes are sweetness, roundness, rosiness, etc., is said to be a process of analysis. On the contrary, the process by which we recognize that various properties together form the char- acteristics of a single object, is termed synthe- sis; thus the consolidation, in our mind, of the several characteristics of an apple into a single concept, is a synthesis. The two processes are coinplementar}' aspects of the same mental act. It should be borne in mind that analJ^sis does not really destroy the unitj- of a given object; it merely recognizes various distinctions within that unity. Nor does synthesis fuse into indis- tinguisliableness the characteristics it starts with: it correlates them into a unity, but in that unity the identity of each part is full}' preserved.

In mathematics the term analysis is employed, on the one hand, to denote a potent method of discovery and demonstration: on the other hand, and more or less inaptly, to designate collectively several important branches of modern mathe- matics.

The method said to be analytic consists in resolving a given relation into its mathematical elements. Analysis in this sense of the term is sometimes applied to the solution of geometric questions. It consists in assuming a certain relation to be the true answer to the question, and resolving that relation into simple truths. Euclid {Elements, Book XIII.), formulates this idea as follows: "Analysis is the obtaining of the thing sought by assuming it and so reason- ing up to an admitted truth." For example, let the question be, In what ratio does the alti- tude of an isosceles triangle divide its base?

The simple answer that siiggests itself through the inspection of a figure is, that the base is bisected. Assume this to be so. In that case the two triangles into which the altitude divides the given triangle are identically equal, because their sides are respectively equal ; and therefore the two angles made by the altitude and the base are also c(]ual. But the latter conclusion is an evident truth, if we remember that the altitude of a triangle is a line perpendicular to its base. We therefore infer that our assump- tion was correct and that the base is really bisected. Furthermore, by reversing the above piocess we can now demonstrate our assumed truth synthetically; i.e., reconstruct it from the simple, admitted truths, to which the analysis has led.

Now, although the demonstrations of geomet- ric theorems and perhaps most of the theorems them.selves, were originally discovered in the manner just indicated, bj- analysis, most of the ordinary text-book demonstrations are undoubt- edly syntheses, for they gradually lead from the mathematical elements — the axioms — to more or less complex trutlis. Geometry is therefore spoken of as a synthetic science. However, the rcdiictio ad uhsiin'Kin, which is not infrequently employed, is a purely analytical method, differ- ing only in form from the type of analysis con- sidered above. The suggested relation is, name- ly, assumed to be not true, but false, and this is shown to lead to absurd conclusions — the in- ference being that the suggested relation is nec- essarily true.

In designating a part of mathematical science, the term analysis is applied, on the one hand, to the theory of functions (including series, logarithms, curves, etc.), on the other hand, to the mathematics of infinite quantities, com- prising the differential calculus, the integral calculus, and the calculus of variations. Alge- bra, although usually limited to equations, in- cludes in the wider sense of its name the branches just enumerated. Indeed, it is because of their relation to algebra that these branches have lieen united under the general term of math- ematical analysis. Algebra itself, however, is far from being uniformly analytical, and many an instance of pure synthesis may be found in any of the branches of applied algebra, say in analytical geometry. In general, there is no branch of human thought in which the method of analysis, or that of synthesis, is used exclu- sively. The complete abolition of either of these methods would involve not a small diminution in our power of establishing interesting truths.

In discussions concerning the methods of science, the processes of analysis and synthesis are often erroneously identified with those of induction and deduction. The reason of this lies mainly in the fact that there has been consider- able disagi'eement as to the proper definition of the terms in question. The distinction between the tio pairs of antithetic terms becomes perfectly clear, however, if we define analysis as leading from the compound to the elementary, and synthesis as leading from the elementary to the compound; induction as leading from the particular to the general, and deduction as leading from the general to the particular. As thus defined, analysis, as well as synthesis, may be coincident, though not identical, with either induction or deduction. Thus, to turn for an illustration again to mathenuitics, the ordinary demonstration of a geometric theorem is . a deduction ; for what can be more general in character than the axiomatic truths from which the theorem is deduced? But the demonstration is also a synthesis; for what can be more elementary than those axioms which are used in reasoning up to the theorem? On the other hand. Newton's binomial theorem, as often demonstrated in text-books of algebra, presents an instance of synthesis coincident with induction. The general relation expressed by that theorem is induced by the examination of a number of particular instances. But the demonstration is also a true synthesis, for it combines a number of relations into one.

More or less extensive discussions of the analytical processes of philosophy may be found in the following woiks: R. H. Lotze. Logic. English translation (Oxford. 1888): F. H.' Bradley, Principles of Logic (London, 188.3) ; L. T. Hobhouse, Theorij of Knowledge (London, 1896), and Bosanquet, Logic (O.xfoi'd, 18S8). See also articles. Analytic Judgment; Judgment; Knowledge, Theory of, and Logic.