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discussion of which I do not think it necessary to interrupt any longer the progress of my work.

SECTION III.

Of the Import of the words Analysis and Synthesis, in the Language of Modern Philosophy.

As the words Analysis and Synthesis are now become of constant and necessary use in all the different departments of knowledge; and as there is reason to suspect, that they are often employed without due attention to the various modifications of their import, which must be the consequence of this variety in their applications, -it may be proper, before proceeding farther, to illustrate, by a few examples, their true logical meaning in those branches of science, to which I have the most frequent occasion to refer in the course of these inquiries. I begin with some remarks on their primary signification in that science, from which they have been transferred by the moderns to physics, to chemistry, and to the philosophy of the human mind.

1.-Preliminary Observations on the Analysis and Synthesis of the Greek Geometricians.

It appears from a very interesting relic of an ancient writer," that,

*Preface to the seventh book of the Mathematical Collection of Pappus Alexandrinus. An extract from the Latin version of it by Dr. Halley, is here introduced.

From the preface of Pappus Alexandrinus to the 7th book of his Mathematical Collection. (See Halley's Version and Restitution of Apollonius Pergæus de Sectione Rationis et Spatii, p. xxviii.)

"Resolutio est methodus, quâ à quæsito quasi jam concesso per ea quæ deinde consequuntur, ad conclusionem aliquam, cujus ope Composito fiat, perducamur. In resolutione enim, quod quæritur ut jam factum supponentes, ex quo antecedente hoc consequatur expendimus; iterumque quodnam fuerit hujus antecedens; atque ita deinceps, usque dum in hunc modum regredientes, in aliquid jam cognitum locoque principii habitum incidamus. Atque hic processus Analysis vocatur, quasi dicas, inversa solutio. E contrario autem in Compositione, cognitum illud, in Resolutione ultimo loco acquisitum ut jam factum præmittentes; et quæ ibi consequentia erant, hic ut antecedentia naturali ordine disponentes, atque inter se conferentes, tandem ad Constructionem quæsiti pervenimus. Hoc autem vocamus Synthesin. Duplex autem est Analyseos genus, vel enim est veri indagatrix, diciturque Theoretica; vel propositi investigatrix, ac Problematica vocatur. In Theoretico autem genere, quod quæritur, revera ita se habere supponentes, ac deinde per ea quæ consequuntur, quasi vera sint, ut sunt ex hypothesi, argumentantes; ad evidentem aliquam conclusionem procediJam si conclusio illa vera sit, vera quoque est propositio de qua quæritur : ac demonstratio reciproce respondet analysi. Si vero in falsam conclusionem incidamus, falsum quoque erit de quo quæritur.* In Problematico vero genere,

mus.

From the account given in the text of Theoretical Analysis, it would seem to follow, that its advantages, as a method of investigation, increase in proportion to the variety of demonstrations

among the Greek geometricians, two different sorts of analysis were employed as aids or guides to the inventive powers; the one adapted to the solution of problems; the other to the demonstration of theorems. Of the former of these, many beautiful exemplifications have been long in the hands of mathematical students; and of the latter, (which has drawn much less attention in modern times,) a satisfactory idea may be formed from a series of propositions published at Edinburgh about fifty years ago. I do not, however, know, that any person has yet turned his thoughts to an examination of the deep and subtle logic displayed in these analy tical investigations; although it is a subject well worth the study of those who delight in tracing the steps by which the mind proceeds in pursuit of scientific discoveries. This desideratum it is not my present purpose to make any attempt to supply; but only to convey such general notions as may prevent my readers from falling into the common error of confounding the analysis and synthesis of the Greek geometry, with the analysis and synthesis of the inductive philosophy.

In the arrangement of the following hints, I shall consider, in the first place, the nature and use of analysis in investigating the demonstration of theorems. For such an application of it, various occasions must be constantly presenting themselves to every geometer; when engaged, for example, in the search of more elegant modes of demonstrating propositions previously brought to light; or in ascertaining the truth of dubious theorems, which from analogy, or other accidental circumstances, possess a degree of verisimilitude sufficient to rouse the curiosity.

In order to make myself intelligible to those who are acquainted only with that form of reasoning which is used by Euclid, it is necessary to remind them, that the enunciation of every mathemati cal proposition consists of two parts. In the first place, certain suppositions are made; and secondly, a certain consequence is affirmed to follow from these suppositions. In all the demonstrations which are to be found in Euclid's Elements, with the excep tion of the small number of indirect demonstrations, the particulars involved in the hypothetical part of the enunciation are assumed as

:

quod proponitur ut jam cognitum sistentes, per ea quæ exinde consequuntur tanquam vera, perducimur ad conclusionem aliquam quod si conclusio illa possibilis sit ac oporn, quod mathematici Datum appellant; possibile quoque erit quod proponitur et hic quoque demonstratio reciproce respondebit Analysi. Si vero incidamus in conclusionem impossibilem, erit etiam problema impossibile. Diorismus autem sive determinatio est qua discernitur quibus conditionibus quotque modis problema effici possit. Atque hæc de Resolutione et Compositione dicta

sunto.

* Propositiones Geometrica More Veterum Demonstrata. Auctore Matthæo Stewart, S. T. P. Matheseos in Academia Edinensi Professore, 1763.

of which a theorem admits; and that, in the case of a theorem admitting of one demonstration alone the two methods would be exactly on a level. The justness of this conclusion will, I believe, be found to correspond with the experience of every person conversant with the processes of the Greek geometry.

the principles of our reasoning; and from these principles, a series or chain of consequences is, link by link, deduced, till we at last arrive at the conclusion which the enunciation of the proposition asserted as a truth. A demonstration of this kind is called a Synthetical demonstration.

Suppose now, that I arrange the steps of my reasoning in the reverse order; that I assume hypothetically the truth of the propo sition which I wish to demonstrate, and proceed to deduce from this assumption, as a principle, the different consequences to which it leads. If, in this deduction, I arrive at a consequence which I already know to be true, I conclude with confidence, that the principle from which it was deduced, is likewise true. But if, on the other hand, I arrive at a consequence which I know to be false, I conclude, that the principle or assumption on which my reasoning has proceeded, is false also. Such a demonstration of the truth or falsity of a proposition is called an Analytical demonstration.

According to these definitions of Analysis and Synthesis, those demonstrations in Euclid which prove a proposition to be true, by showing, that the contrary supposition leads to some absurd inference, are, properly speaking, analytical processes of reasoning.-In every case, the conclusiveness of an analytical proof rests on this general maxim, That truth is always consistent with itself; that a supposition which leads, by a concatenation of mathematical deductions, to a consequence which is truc, must itself be true; and that what necessarily involves a consequence which is absurd or impossible, must itself be false.

It is evident, that, when we are demonstrating a proposition with view to convince another of its trath, the synthetic form of reasoning is the more natural and pleasing of the two; as it leads the understanding directly from known truths to such as are unknown. When a proposition, however, is doubtful, and we wish to satisfy our own minds with respect to it; or when we wish to discover a new method of demonstrating a theorem previously ascertained to be true; it will be found, as I already hinted, far more convenient to conduct the investigation analytically. The justness of this remark is universally acknowledged by all who have ever exercised their ingenuity in mathematical inquiries; and must be obvious to every one who has the curiosity to make the experiment. It is not however, so easy to point out the principle on which this remarkable difference between these two opposite intellectual processes depends. The suggestions which I am now to offer appear to myself to touch upon the most essential circumstance; but I am perfectly aware that they by no means amount to a complete solution of the difficulty.

Let it be supposed, then, either that a new demonstration is required of an old theorem; or that a new and doubtful theorem is proposed as a subject of examination. In what manner shall I set to work, in order to discover the necessary media of proof?—

From the hypothetical part of the enunciation, it is probable, that a great variety of different consequences may be immediately deducible; from each of which consequences, a series of other consequences will follow: at the same time, it is possible, that only one or two of these trains of reasoning may lead the way to the truth which I wish to demonstrate. By what rule am I to be guided in selecting the line of deduction which I am here to pursue? The only expedient which seems to present itself, is merely tentative or experimental; to assume successively all the different proximate consequences as the first link of the chain, and to follow out the deduction from each of them, till I, at last, find myself conducted to the truth which I am anxious to reach. According to this supposition, I merely grope my way in the dark, without rule or method: the object I am in quest of, may, after all my labor. elude my search; and even, if I should be so fortunate as to attain it, my success affords me no lights whatever to guide me in future on a similar occasion.

Suppose now that I reverse this order, and prosecute the investigation analytically; assuming, agreeably to the explanation already given, the proposition to be true, and attempting from this supposition, to deduce some acknowledged truth as a necessary consequence. I have here one fixed point from which I am to set out: or, in other words, one specific principle or datum from which all my consequences are to be deduced; while it is perfectly immaterial in what particular conclusion my deduction terminates, provided this conclusion be previously known to be true. Instead, therefore. of being limited, as before, to one conclusion exclusively, and left in a state of uncertainty where to begin the investigation, I have one single supposition marked out to me, from which my departure must necessarily be taken; while, at the same time, the path which I follow, may terminate with equal advantage in a variety of differ ent conclusions. In the former case, the procedure of the understanding bears some analogy to that of a foreign spy, landed in a remote corner of this island, and left to explore, by his own sagacity, the road to London. In the latter case, it may be compared to that of an inhabitant of the metropolis, who wished to effect an escape, by any one of our sca-ports, to the continent. It is scarcely necessary to add, that as this fugitive, should he happen, after reaching the coast, to alter his intentions, would easily retrace the way to his own home; so the geometer, when he has once obtained a conclusion in manifest harmony with the known principles of his science, has only to return upon his own steps (caca regens file vestigia) in order to convert his analysis into a direct synthetica! proof.

A palpable and familiar illustration (at least in some of the most essential points) of the relation in which the two methods now described stand to each other, is presented to us by the operation of unloosing a difficult knot, in order to ascertain the exact process by

which it was formed. The illustration appears to me to be the more apposite, that I have no doubt it was this very analogy, which suggested to the Greek geometers the metaphorical expressions of analysis and of solution, which they have transmitted to the philosophi cal language of modern times.

Suppose a knot, of a very artificial construction, to be put into my hands as an exercise for my ingenuity, and that I was required to investigate a rule, which others as well as myself, might be able to follow in practice, for making knots of the same sort. If I were to proceed in this attempt, according to the spirit of geometrical synthesis, I should have to try, one after another, all the various experiments which my fancy could devise, till I had, at last, hit upon the particular knot I was anxious to tie. Such a process, however, would evidently be so completely tentative, and its final success would, after all, be so extremely doubtful, that common sense could not fail to suggest immediately the idea of tracing the knot through all the various complications of its progress, by cautiously undoing or unknitting cach successive turn of the thread in a retrograde order, from the last to the first. After gaining this first step, were all the former complications restored again by an inverse repetition of the same operations which I had performed in undoing them, an infallible rule would be obtained for solving the problem originally proposed; and at the same time, some address or dexterity, in the practice of the general method, probably gained, which would encourage me to undertake, upon future occasions, still more arduous tasks of a similar description. The parallel between this obvious suggestion of reason, and the refined logic of the Greek analysis, undoubtedly fails in several particulars; but both proceed so much on the same cardinal principle, as to account sufficiently for a transference of the same expression from the one to the other. That this transference has actually taken place in the instance now under consideration, the literal and primitive import of the words are and ivos, affords as strong presumptive evidence as can well be expected in any etymological speculation.

In applying the method of analysis to geometrical problems, the investigation begins by supposing the problem to be solved; after which a chain of consequences is deduced from this supposition, terminating at last in a conclusion, which either resolves into another problem, previously known to be within the reach of our resources; or which involves an operation known to be impracticable. In the former case, all that remains to be done, is to refer to the construction of the problem in which the analysis terminates; and then by reversing our steps, to demonstrate synthetically, that this construction fulfils all the conditions of the problem in question. If it should appear, in the course of the composition, that in certain cases the problem is possible, and in others not, the specification of these different cases, (called by the Greek geometers the

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