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 related to each other as mutually inverse processes, expository convenience alone might be pleaded in favour of the parallel treatment; but the two are in practice often em-OJoyed as strictly complementary processes, in support of each other on the same occasion · or, in other words, the composition 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 (remarks 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 Elements of Euclid, containing so many examples of geometrical 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 therefrom, 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 independent 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 process bound up with the use of analysis. These are his words: “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.” The two processes are involved together in what he calls the τόπο? άναλνόμχ,νος, 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 “the first in order of the books written by the ancient geometers to facilitate and promote the method of resolution or analysis.” 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, remarking of both that the last stage in the analysis (αναλύσει) is the first in the production or construction (γεv∈σeι). 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 enunciation, namely, theorems and problems. The distinction was not marked by Euclid himself, nor is it in any sense radical, for either kind of proposition may easily be transformed into the expression of the other; but, as commonly accepted, it amounts to this—that a theorem, is given cut as an assertion to be accepted, and has to be shown true; a problem is given out as an act to be done, and has to be shown possible. In the case of a theorem, Euclid accordingly, after enunciating the proposition, proceeds generally to show, with more or less of construction on a particular diagram, and working always with fixed definitions, that the assertion follows deductively from certain truths, either assumed as evident (axioms), or formerly proved therefrom, and seen to be applicable to the present case by inspection of the figure as constructed. The grounding propositions are allowed by the reader as they are brought forward, though he may for the moment have not the least idea whither the author is tending, and at the end the conclusion is accepted, because the successive premises, being allowed, have been combined logically. In the case of a problem, after an express construction for which no reason is given, the object is to show that what has been brought to pass really supplies what was sought; but the procedure is not different from what it was in the case of a theorem, because the object is attained by showing again that certain truths allowed, in their particular application to the figure constructed, involve as a conclusion some relation which the figure is seen to exhibit. Now if this is Euclid’s procedure in general—there is an exception, afterwards to be noted, where he proves his point indirectly—it is undeniably synthetic, in any meaning that can be ascribed to that term, the result being obtained by a massing or combining of elements or conditions. But on Euclid’s part the process is one of demonstration, not of discovery. Still less is the reader’s mind in the attitude of discovery: he is led on to a result which is indeed indicated, but by a way which he does not know, and, as it were, blindfold. There must, however, have been discovery before there could be such demonstration ; or how should the proposition admit of definite enunciation at the beginning? Thus there is, in the background, an earlier question of procedure or method, and it is this that the ancient geometricians had chiefly in view when speaking of analysis and synthesis.

Now, some propositions are so simple that they must have been seen into almost as soon as conceived, and conceived as soon as the human mind began to be directed to the consideration of forms or figures; in which case no method of discovery, to speak of, can have been necessary. There is, again, another class of propositions, more complex though still simple, which probably were established by a process of straightforward synthesis. An inquirer must have in his head some knowledge in the shape of principles more or less fixed, or he would not be an inquirer; and either the accidental combination of such principles may lead in his mind to particular results, or the first time a particular question suggests itself to him, it may be seen at once to involve, or to follow from, certain of the principles. Many propositions in the Elements, giving the most apparent properties of triangles, circles, &c., it can hardly be doubted, were arrived at by this way of discovery, even when a more elaborate process of synthesis was employed for their formal demonstration; as, for example, in the case of the famoue fifth proposition of Book I. But the same process of direct composition (understood always as joined with inspection) is no longer applicable, or is not effective, when the question is of less obvious properties, or of construction to be made under special conditions. To discover the fact or the feasibility in such cases is so much the real difficulty, that the question of demonstration becomes of merely secondary importance. And there is even a still prior question of discovery; for it has to be determined that some points rather than others should be made the subject of express inquiry. This, however, may be left aside. To any one engaged in geometrical inquiry, in the constant inspection of figures for the understanding of their properties and mutual relations, questions must incessantly be occurring—so incessantly and inevitably that it is needless, if it were not vain, to seek out a reason for the particular suggestions. As in all discovery to the last, so more especially at the first stages, there is an element of instinctive tact in the mind’s action which eludes expression; and there is also an element of what might be called chance, were it not that those only get the benefit of it who are consciously on the look-out, either generally or in some special direction. A particular question being started by whatsoever suggestion, hew shall the mind arrive at certain knowledge regarding it? Such, practically, is the form which is assumed by geometrical inquiry.

Besides the thing sought there is nothing else given, or at least there is nothing else immediately given or suggested. But the mind is supposed to have some knowledge pertaining to the matter—though not extending to the particular aspect of it—in question, also some knowledge of such matters generally. In such circumstances the aim of the inquirer must be to bring what is sought into some definite relation with what is known. Direct composition or synthesis of the known, with more or less of construction, if it led to that which is sought as a result, would determine the relation for the inquirer, and determine it in like manner for all who allow the principles whence the conclusion is logically deduced, being thus at one stroke both discovery and demonstration. But synthesis, arbitrarily made, as it must be where the question is at all difficult, may fail, however often it is attempted. Without a proper start it avails nothing; and what is to determine the start? There is always one course open. Let the objective itself be made the starting-point, and let it be seen whether thence it may not be possible by some continuous route to get upon known ground. In other words, a thing sought, when itself assumed, may admit of being brought into relation, upon some side or other, with the body of ascertained knowledge. If it can be so brought, through whatever number of steps, there is then attained as a result what before it was impossible to light upon as a beginning; and now nothing hinders from making the start originally desired, and from reaching as a proper conclusion the assumed beginning, if the path struck out before is measured over again in the opposite direction. The course thus becomes once more synthetic, but only because of what was first accomplished. Till the point in question was made to yield up its own secret by a process fitly called analysis or resolution, nothing certain could be determined. At the analytic stage, however, the line taken may be twofold. The proposition, assumed at starting as something definite to work from, cither may be held as following deductively from some other, which again is dependent on still another or others, till one is worked up to that is known to be true; or it may be taken as itself a premiss leading deductively to some other proposition, which in turn, by one or more steps, leads to a true proposition as conclusion. In either case the implication is that a proposition must itself be true, if by any line of formally correct logic it leads to a proposition known to be true. And though the expression must be modified for questions in the form of problems, requiring something to be done—to which form of question, indeed, the analytic process is peculiarly applicable—the point of logical principle remains there exactly the same.

But is the process, thus stated as it was understood by the ancient geometricians, logically valid? In the first of the two alternative forms, it is valid: the proposition assumed at starting will undoubtedly be true, if a proposition on which it is shown to be ultimately dependent is true. At the same time, there 13 in this case no guarantee that the most effective line for establishing it has been taken, in view of the well-known logical principle that the same conclusion may follow from different premisses. In the other form of the process, where the proposition assumed is itself used as a premiss, the case as to validity is otherwise. As Aristotle first clearly apprehended and showed, it is quite possible to reach a (materially) true conclusion by strict logical deduction from premisses either one or both false; and thus the mere fact that the proposition assumed is found, in combination with others, to lead to a conclusion known to be true, does nothing to establish its own character. Yet although the process of analysis thus carried out by way of deduction, as formulated by Euclid and (in one of his expressions) by Pappus, is theoretically faulty, through neglect or ignorance of Aristotle’s observation, the practice of Euclid is not therefore invalidated. It was his habit, as Pappus also enjoins, to follow up the analysis by a synthesis consisting in a reversal of it, and this would effectively get rid of error; since the result of the analysis, if it did not follow from the assumed premiss by true implication, but only accidentally, could not itself, when in turn used as a premiss for the synthesis, be made to yield the original proposition as a legitimate conclusion. In order, however, to validate this form of analysis it is not necessary to resort to the laborious expedient of retracing the whole path synthetically. As Duhamel, in his treatise Des Méthodes dans les Sciences de Raisonnement (pt. i. c. 5), has pointed out, it is enough if, at the different stages of the deduction, the inquirer assures himself, as he easily may do where it is the fact, that there is perfect “reciprocity” among the propositions successively obtained from the one first assumed; meaning that, in the circumstances of the deduction, each may as well follow from the one coming after as it is fitted to yield that. And the same simple expedient suffices equally to obviate the less grave defect above noted in analysis carried out by regression from consequents to conditions, or conclusions to premises; reciprocity, if it can be made out here at the different stages, will guarantee the exclusive validity of the line of reasoning taken. So may analysis become perfectly independent as a method of discovery, and give as much insight as synthesis, where this is directly applicable, does; while it is—what synthesis is not directly—applicable to every kind of question, however complex.

It is unnecessary, for the purposes of the present article, to enter further into details respecting the methods anciently practised in geometry. Let it suffice to mention only the method of indirect proof known as rechιctio ad absurdum, employed sometimes by Euclid in the Elements. This conforms to the type of analysis in that it starts from the question to be determined, though it is peculiar in following out, not the assumption itself, but what is thereby suggested as excluded, with the final result that the point in question is established upon the ruin of every other supposition. It is a method of discovery as well as a method of demonstration; while the previous argument has shown that analysis, directly practised, may be made a method of demonstration by itself, besides being the most potent and unfailing instrument of discovery. Also it was seen before that synthesis may be a method of discovery, though it is more frequently employed as a method of demonstration in sequence upon discovery by analysis. To insist thus upon the double character alike of analysis and synthesis, as practised in geometry, is of vital importance, because of the change in application which the terms have undergone among mathematicians. In modern times analysis has come to mean the employment of the algebraical and higher calculus, and synthesis any direct treatment of the properties of geometrical figures, in the manner of the ancients, without the use of algebraical notation and transformations. The excuse for the change lies in the fact that, while the Greeks had only extremely undeveloped means of analysis, they gave the highest possible finish and exactness to their synthetic demonstrations of geometrical propositions, seldom being content to let their discoveries rest upon the ground of that analysis by which they were made. But though it has this excuse or motive, the change involves a misunderstanding, as all mathematicians allow who have turned their minds seriously to consider the rationale of their practice. It is, in the first place, clear that only by the process described above, rightly called analysis, can anything be determined about the more complex properties and relations of geometrical figures; haphazard synthesis is of no avail. The ancients therefore, in their geometry, had an analysis. It is next to be remarked that the algebraical solution of problems is not so exclusively analytic in character that it may not in simple cases assume the form of direct (algebraical) synthesis; and in all cases, for verification, it admits of being followed up by an exposition that is truly synthetic. The moderns, therefore, in their calculus, are not without their synthesis. Furthermore, the ancients, however little progress they made, comparatively speaking, in the general science of calculation, and however their special methods for the resolution of geometrical questions, even as involving direct figured construction, still more as applying calculation, fell short of the variety and pliability of modern devices, yet had their own analytical weapons, though they cannot be specified here. For our present purpose it is equally unnecessary to enter into detaile as regards the modern devices, whether belonging to the lower or higher analysis, or as regards the principle for applying them developed by Descartes and his successors;...