Analysis

ANALYSIS means literally, in the Greek, an unloosening 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 process of reasoning; as, accordingly, in the body of the work {Anal. Prior, i. 32), we find him once using the verb “analyse” of arguments, when they are to be presented in “figure,” 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 wül 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 veιy different results, according to the particular aspect in which it is considered. Hence it becomes impossible, 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 unprejudiced view of the different applications of analysis in science, that one being first and most prominently put forward 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 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; but to arrogate for these exclusively the name of analysis, it cannot be too pointedly declared, is to lose sight of the end in the means.

II. Chemical Analysis and Synthesis. —After mathematics, chemistry is the science in which application has most expressly been made of processes termed analysis and synthesis. In physics, regarded as the science of motion, whether abstractly taken or as manifested actually in natural bodies, the application is universal; the resolution and composition of velocities, motions, and forces being fundamental processes pervading the whole science under all variety of circumstances. There is nothing, however, in such an employment of analysis and synthesis that is not easily intelligible in the light of the processes as practised either in the more general science of mathematics, dealing with relations of quantity in number and form, or in the more special science of chemistry, which deals with those characteristic qualities of actual bodies for which no definite expression in terms of motion can be found.

The concrete substances in nature are found to be such that some by no means in our power can be brought to anything simpler, while others can be broken up into constituents differing in character from the original substances and also among themselves. Hence a division is made of bodies into elements and compounds; elements being all such bodiee, not farther reducible, as are either actually found in nature, or, though not so found, have emerged in the manipulation of actual bodies; compounds, all such as, being actually found, are reducible to two or more different elements, or have by artificial combination been constituted. The process of reduction to elements is called analysis; the process of re-combination or free combination iβ called synthesis. When the analysis is carried out simply with the view of detecting what elements are present in a substance, it is called qualitative; and quantitative, if with the further view of determining the definite proportions (by weight) in which the constituents are present in a definite quantity of the substance. There are corresponding varieties of synthesis.

Now here the subject-matter is so manifestly different from what it is in mathematics, that it is idle to look for exact correspondence in the processes practised under the same names within the two sciences. In fact, however, the correspondence is greater than may at first sight appear. Chemical analysis of a given substance is a process of discovery real and actual, like the analysis of a mathematical problem, and proceeds similarly by taking what is given, and working with it in relation to other substances, to see whether it can be made to yield up aught that is already known, or may be regarded as fixed and certain. Again, just as mathematical synthesis may be a process of invention, either generally, by way of combination of principles, or sometimes specially, in reference to particular questions, so does chemical synthesis give a knowledge of new forms of matter, or haply solve the question as to the constitution of particular substances in hand. Once more, the relation of analysis and synthesis as two complementary phases of one process (instead of their being regarded as two processes) is exhibited as plainly in chemistry as in mathematics. It may seem to be exhibited even more impressively, when the very constituents got out by analysis of a substance are used in the synthesis to give it being again. This circumstance, however, is far from giving to the science of chemistry a character of evidence superior to that of mathematics: its inferiority in this respect is but too well marked, and has a reason that at the same time explains what else is peculiar in its application of analysis and synthesis. The chemist deals with things known only by experience, and connected by way of physical causation: true, they are things with which he can freely experiment—and this gives to chemistry a prerogative character among the natural sciences—but the things are taken as they are found, and experience is constantly disclosing in each new attributes which have simply to be accepted, at least in the present state of our knowledge, by the side of the others. On the contrary, the mathematician deals with things over which he has full power of construction, and whose relations in the fact of constructing he constitutes, whether they are internal or external relations. But positive construction carries with it an insight which is wanting in experiment, be the physical conditions ever so favourable; and thus analysis and synthesis have in mathematics, along with perfect freedom of scope, a determinateness far surpassing anything that is attainable in chemistry.

III. Psychological Analysis and Synthesis.— Passing for the next signal application of analysis from the world of matter to mind, we have here a subject which more perhaps than any other calls for an exercise of the process in order to be scientifically understood. Physical things in their superficial relations lie to a great extent open to direct apprehension, and, whatever deeper connections there may be to be traced out among things the most remote in their nature as apprehended, yet the fact of their separation in space involved in our perception of them is already something done, leaving the scientific function {analytic and synthetic) to be exercised chiefly in „the attempt to comprehend them. Very different is the state of affairs in mind, where everything, as it were, runs or melts into everything else. Even to lay hold of particular mental phenomena, with a view to the explanation of them, implies already an express scientific attitude, which must be called analytic.

Particular mental states being supposed to be got, with such definiteness of apprehension (always more or less imperfect) as the subject-matter admits of, the business of the psychologist becomes substantially one with that of the physical inquirer. Accordingly, it is often urged that complex mental states conform to the two types of mechanical and chemical composition, in the sense that some are to be resolved after the manner of complex phenomena of motion, and others by a process analogous to that employed in chemistry for the qualities of concrete substances. The analogy, however, especially in the second class of states, is decidedly loose. Psychological phenomena of cognition or emotion, held to be developed, under general mental laws, out of simpler states of sense, resemble chemical compounds only in having a character unlike that of any of the elements that go to make them; in particular, they do not admit of that actual resolution into their elements which lends so much evidence to the processes of chemistry. The realm of nature supplies a far apter analogy in the phenomena of organic growth, more especially as mental states do, in fact, stand in direct relation with states of the bodily organism. It is as impossible to make an actual analysis or synthesis of the physiological complex of life as of the psychological complex of mind; and it is only more difficult (the phenomena being undoubtedly more recondite and fluctuating) to practise experiments in psychology than in physiology. But, at all events, there is no new principle involved in the scientific treatment of mind; nor again in the treatment of moral and social questions, for an insight into which psychological knowledge is indispensable.

IV. Logical Analysis and Synthesis.— To logic, taken in its widest sense as the methodology of all science, it belongs to appreciate the general import of all such applications of analysis and synthesis as have now been considered. There remains, however, a special variety which is itself entitled logical analysis and synthesis, and which has the more carefully to be distinguished from the other heads, because it stands in an opposition to them all.

Logical analysis is the same process as that which is otherwise called metaphysical division. (The process called logical division is different. See Logic and Division.) Given, say, a concrete subject like man, this may be divided physically into a number of parts in space, or, as a concept, metaphysically into a number of qualities or attributes,— metaphysically, because none of these has an independent subsistence or physical existence apart. They are distinguished in the way of mental consideration, or, as it is technically called, abstraction; and, this being a thoughtprocess or logical act, the resolution of the given complex into such conceptual elements gets the name also of logical analysis. The corresponding act of synthesis proceeds by the way that is technically called determination; thus the general concept man, to take the traditional example, has the attribute of rational joined to the attributes of animal, or is determined by that addition, and much else has to be added in a similar way before the particular concrete can be determined.

Now it is evident that such analysis and synthesis have an application to any kind of thought that the mind can conceive; and thus logicians, in meaning, as they have commonly done, nothing more by the names, have signalised processes that are in truth of no small account for knowledge in general. There is no kind of scientific inquiry, strictly so called, and whatever be its scope and method, that does not involve at all stages from the first such analysis or abstract mental consideration. Nay, it may be said that science, as opposed to the natural experience of things, or to the artistic interest which centres upon fully bodied-out concretes, is analysis in this present sense, everywhere breaking up to find community of character under the mask of superficial difference, and sifting out the one from the many. But when logicians, not disregarding the various applied methods of the real sciences, or consciously excluding them as lying beyond the province of pure logic, would seek to reduce all scientific procedure to this kind of mental action, the attempt implies a deep misapprehension. It is one thing for the mind to have its subject of inquiιy clearly and sharply defined apart from what else is given therewith, or again to have its existing knowledge always well in hand and sifted out to the uttermost; it is another thing for the mind to be making advances, to be passing out from the known to the unknown, or labouring to bring the unknown into relation with that which is known already. Condillac is the thinker who has most expressly made the attempt to bring all scientific method back to the conception of mere logical analysis, repeating it everywhere throughout his works. The sixteenth chapter of his unfinished treatise, the Langue des Calculs, may especially be noted in this respect; the more because he there endeavours to justify his developed expression for the procedure of all science— that it consists in a continued substitution of identical propositions—by the actual solution of an algebraical problem. Simple, however, though the instance chosen is, he fails to make good his view, appearing to prove it only by leaving out the step of critical moment,

To analysis and synthesis in the specially logical sense is undoubtedly related the distinction that logicians have made of analytic and synthetic method. Without stepping beyond the bounds of logic conceived as a formal doctrine, a fourth department, under the name of Method or Disposing, may be added to the three departments regularly assigned — Conceiving (Simple Apprehension), Judging, Reasoning; and this would consider how reasonings, when employed continuously upon any matter whatever, should be set forth to produce their combined effect upon the mind. The question is formal, being one of mere exposition, and concerns the teacher in relation to the learner. How should results, attained by continuous reasoning, be set before the mind of a learner? Upon a line representing the course by which they were actually wrought out 1 Or always in the fixed order of following from express principles to which preliminary assent is required? If the latter, all teaching becomes synthetic, and follows a progressive route from principles to conclusions, even when discovery (supposing discovery foregone) was made by analysis or regression to principles; of which expository method no better illustration could be given than the practice of Euclid in the demonstrations of his Elements. On the other hand, it may be said that the line of discovery is itself the line upon which the truth about any question can best be expounded or understood, for the same reason that was found successful in discovery, namely, that the mind (now of the learner) has before it something quite definite and specific to start from; upon which view, the method of exposition should be analytic or regressive to principles, at least wherever the discovery took that route. The blending of both methods, where possible, is doubtless most effective; otherwise it depends upon circumstances—chiefly the character of the learner, but also the nature of the subject in respect of complexity—which should be preferred, when one alone is followed.

The question of prime logical, or general, importance remaining is to determine the relation of Analysis and Synthesis as methods of real science, to the ground-processes of all reasoning, known since the days of Aristotle under the names of Induction and Deduction. Much difference of opinion has been expressed on this subject, not only because of the want of agreement as to what should be called analysis and synthesis, but also because of more fundamental disagreement regarding the nature of the inductive and deductive processes.

It was remarked before as somewhat surprising, that Aristotle himself did not more expressly consider the relation, when we have seen that he was familiar with the process of geometrical analysis, under the very name. The distinction, however, upon which he lays so much stress throughout his works, between knowledge from principles, prior or better known by nature, and knowledge of or from facts, prior in experience or relatively to us, has generally been understood to imply a connection of synthesis with deduction, of analysis with induction; so much so indeed, that synthetic and deductive method, analytic and inductive method, have come to be used respectively almost as interchangeable terms. Nor, although Sir William Hamilton seems to wish to reverse the usual association of the terms, when he calls induction a purely synthetic process, and declares it to be erroneously viewed as analytic (Metaphysics, i. p. 102), is he really at variance with the other authorities; his observation having a special reference which the others also might allow. But any sùch association seems to rest upon a misconception, not to be laid to the charge of Aristotle himself. In the sense of analysis and synthesis for which it is important to determine the relation, namely, when they are taken as the means of real discovery in science, the true view rather is that they are the different methods in which reasoning, whether inductive or deductive, must be applied for discovering truth in the form of special or particular questions. Analysis, as well as synthesis, may proceed by way of deduction, as we have seen in the process of mathematics; on the other hand, synthesis as applied in chemistry is as much an inductive act, being strictly experimental, as anything could well be. Induction and deduction are concerned about the relation of the particular and general in thought; analysis and synthesis about the relation of the known and the unknown. The two points of view are of course related to each other: analysis and synthesis, as practised by the human mind, either for purposes of science or in the affairs of life, cannot be worked except under those highest laws of the relation between the particular and general in thought which Aristotle’s genius first was able to extract from the instinctive practice of human reason. But whether the processes are applied singly, or, for greater assurance, conjointly, it depends upon the matter of the inquiry under which laws—those of induction or those of deduction—they shall be worked; and in any case there is implied a peculiar intellectual attitude different from that of mere formal reasoning. It is the difference between the act of finding out and proving. If it should ever become possible to develop a logic of Discovery, it must consist in the formulation of the processes of Analysis and Synthesis, conceived in the general sense attributed to them in the foregoing article. (g. c. r.)