Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/837

Rh A N A A N A 793 tion; for, on the assumption that the system of nature and the system of religion must both spring from one causal source, his argument acquires rather an inductive cha racter. Accordingly, it is interesting to see how, in con nection with his sense of analogy, he practically raises, in tiis Introduction, the question which the general theory of fnductive logic, as now understood, has first to consider, the question, namely, &quot;whence it proceeds that likeness thould beget that presumptive opinion and full conviction which the human mind is formed to receive from it ; &quot; though he would not take it upon him to say &quot; how far the extent, compass, and force of analogical reasoning can be reduced to general heads and rules, and the whole bo formed into a system.&quot; (o. c. R.) ANALOGY, in Comparative Anatomy, is equivalent to &quot; similarity of function.&quot; See ANATOMY. ANALYSTS means Literally, in the Greek, an unloosen ing or breaking-up, understood of anything complex in which simpler constituents or elements may thus be brought to view. It is this general sense that must be supposed to have been present to the mind of Aristotle when he gave the name of Analytica to the great logical work in which he sought to break up into its elements the complex pro cess of reasoning ; as, accordingly, in the body of the work {Anal. Prior, i. 32), we find him once using the verb &quot; analyse &quot; of arguments, when they are to be presented in &quot; figure,&quot; or brought to the ultimate formal expression in which they can best be tested or understood. Obviously any more special sense that may be ascribed to the process of analysis must vary with the kind of complex to be resolved. Mental states, material substances, motions of bodies, relations of figures, are but a few examples of the complex things or subjects that fall to be analysed, if there is to be any scientific comprehension of them. Nor is it only that the analysis will be into constituents differing from each other as much as the complex subjects differ ; for the same subject may be analysed in different ways, and with very different results, according to the particular aspect in which it is considered. Hence it becomes im possible, or at least very difficult, to describe the process in any terms fitting equally all the variety of its applications. It is from taking stand by some particular application, and either overlooking all others, or trying to force them within the frame of the one, that different writers have given such discrepant accounts of the process discrepant often to the extent of being mutually exclusive. The express object of the present article will, on the contrary, be to give an un prejudiced view of the different applications of analysis in science, that one being first and most prominently put for ward which was earliest recognised and practised, namely, mathematical analysis. The other applications, selected for their representative character, will, as they follow, naturally suggest the consideration how far the difference of matter in the various sciences tends to modify the nature of the process which is called analysis in all. By the side of Analysis, at the different stages, we shall at the same time treat of the related process called, after the Greek, Synthesis, which means a putting together or compounding. If analysis and synthesis were merely re lated to each other as mutually inverse processes, exposi tory convenience alone might be pleaded in favour of the parallel treatment ; but the two are in practice often em ployed as strictly complementary processes, in support of each other on the same occasion; or, in other words, the com position in synthesis may be a direct re-composition of the (principles or elements then and there got out by analysis. .As a matter of course, therefore, the foregoing general iremarks apply also to synthesis, especially the remark as to the modifying effect of difference in the subject-matter worked with. I. Mathematical Analysis and Synthesis. In the Ele ments of Euclid, containing so many examples of geometri cal propositions variously established, there is a scholion near the beginning of Book XIII. which distinguishes two general methods for the treatment of particular questions, under the names of Analysis and Synthesis. In analysis, it is said, the thing sought is taken for granted, and consequences are deduced from it which lead to some truth recognised; synthesis, on the other hand, starts from that which is recognised, and deduces consequences there from, till the thing sought is arrived at. With more detail, but some wavering in his use of terms, Pappus of Alex andria (about 380 A.D.) describes the two processes at the beginning of Book VII. of his Mathematical Collections. He appears, however, to regard synthesis not at all as an in dependent process to be applied alternatively with analysis for the solution of particular questions (which is the view suggested by Euclid), -but rather as a complementary pro cess bound up with the use of analysis. These are his words : &quot; In synthesis, putting forward as done the thing arrived at as ultimate result in the way of analysis, and disposing now in a natural order as antecedents what were consequents in the analysis, we put them together, and finally come at the construction of the thing sought.&quot; The two processes are involved together in what he calls the TOTTOS dvaXvo/xevos, or, as we may call it, one general Method of Analysis, the use of which for the solution of problems, he says, has to be learned after the Elements, having been developed by Euclid himself, Apollonius of Perga, and Aristaeus the elder. In a similar sense, Robert Simson, its modern editor, speaking of the Euclidean book of Data, calls it &quot; the first in order of the books written by the ancient geometers to facilitate and promote the method of resolution or analysis.&quot; Beyond Euclid, however, the invention of the method was carried back by the tradition of antiquity to Plato. The philosopher, whom we know to have been an ardent student of geometry, and otherwise a discoverer in the science, is said by Diogenes Laertius (III. i. 19), to have devised the method for one Leodamas, and is further said by Proclus (Comm. in Eucl., ed. Basil, p. 58) to have made much use of it himself. Though the report is a loose one, it may well be that this method of analysis was first expressly formulated by the theoretic genius of Plato, especially in view of a passage (Eth. Nicom. iii. 5) in Aristotle, which has not been sufficiently noticed, showing that in his time, before Euclid was born, it was currently employed by geometricians. Aristotle there compares the gradually regressive process of thought, whereby the means of effecting a practical end is discovered, to the mathematical way of inquiry upon a diagram, re marking of both that the last stage in the analysis (draAvcrei) is the first in the production or construction (ycvrei). However surprising it may be thought that Aristotle in his logical works makes so little of a process which thus must have been familiar to him, the fact that it was familiar carries it back at least to the time of Plato. In truth it must have been practised earlier still, from the very beginnings of scientific geometry, though it may have had to wait some time to be formulated. Taking analysis and synthesis, thus defined, either as distinct processes or as conjoined in one method, called analytical, we have next to see how they were brought to bear by the ancients in treating geometrical questions. Propositions such as those contained in the Elements fall into two classes with respect to the form of their enuncia tion, namely, theorems and problems. The distinction was not marked by Euclid himself, nor is it in any sense radi cal, for either kind of proposition may easily be trans formed into the expression of the other; but, as commonly accepted, it amounts to this that a theorem, is given out I. 100