Dr. Wallis afterwards very justly remarks, "that inductions of this sort are of frequent use in mathematical demonstrations; in which, after enumerating all the possible cases, it is proved, that the proposition in question is true of each of these considered separately, and the general conclusion is thence drawn, that the theorem holds universally. Thus, if it were shown, that, in all right-angled triangles, the three angles are equal to two right angles, and that the same thing is true in all acute-angled, and also in all obtuse-angled triangles; it would necessarily follow, that in every triangle the three angles are equal to two right angles; these three cases manifestly exhausting all the possible varieties of which the hypothesis is susceptible."

My chief motive for introducing this last passage was to correct an idea, which it is not impossible, may have contributed to mislead some of Wallis's readers. As the professed design of the treatise in question was to expound the logic of Aristotle, agreeably to the views of its original author; and as all its examples and illustrations assume as truths the Peripatetic tenets, it was not unnatural to refer to the same venerated source, the few incidental reflections with which Wallis has enriched his work. Of this number is the foregoing remark, which differs so very widely from Aristotle's account of mathematical induction, that I was anxious to bring the two opinions into immediate contrast. The following is a faithful translation from Aristotle's own words:

Nor

"If any person were to show, by particular demonstrations, that every triangle, separately considered, the equilateral, the scalene, and the isosceles, has its three angles equal to two right angles, he would not, therefore, know that the three angles of a triangle are equal to two right angles, except after a sophistical manner. would he know this as an universal property of a triangle, although, beside these, no other triangle can be conceived to exist; for he does not know that it belongs to it quá triangle: nor that it belongs to every triangle, excepting in regard to number; his knowledge not extending to it as a property of the genus, although it is impossible that there should be an individual which that genus does not include."

For what reason Aristotle should have thought of applying to such an induction as this the epithet sophistical, it is difficult to conjecture. That it is more tedious, and therefore less elegant

Δια τούτο ονδ' αν τίς δείξη καθ' έκαστον το τρίγωνον αποδείξει η μια η έτερα, ότι δύο ορθας έχει έκαστον, το ισόπλευρον χωρίς, και το σκαληνόν, και το ισοσκελές ουπω οίδε το τρίγωνον ὅτι δυο ορθαίς ίσον, ει μη τον σοφιστικόν τρόπον· ουδε καθόλου τρίγωνον, ουδ' ει μηδεν εστι παρά ταύτα τρίγωνον έτερον ου γαρ, ή τρίγωνον οιδεν. ουδε παν τρίγωνον, αλλ' η κατ' αριθμόν· κατ' είδος δε ου παν, και ει μηδεν εστιν ο ουκ ode.-Analyt. Poster. lib. i. cap. v.

I have rendered the last clause, according to the best of my judgment; but in case of any misapprehension on my part, I have transcribed the author's words. It may be proper to mention, that this illustration is not produced by Aristotle as an instance of induction; but it obviously falls under his own definition of it, and is accordingly considered in that light by Dr. Wallis.

than a general demonstration of the same theorem is undoubtedly true; but it is not on that account the less logical, nor, in point of form, the less rigorously geometrical. It is, indeed, precisely on the same footing with the proof of every mathematical proposition which has not yet been pushed to the utmost possible limit of generalization.

It is somewhat curious that this hypothetical example of Aristotle is recorded as a historical fact by Proclus in his commentary on Euclid. "One person, we are told, (I quote the words of Mr. Maclaurin,) discovered that the three angles of an equilateral triangle are equal to two right angles; another went farther, and showed the same thing of those that have two sides equal, and are called isosceles triangles; and it was a third that found that the theorem was general, and extended to triangles of all sorts. In like manner, when the science was farther advanced, and they came to treat of the conic sections, the plane of the section was always supposed perpendicular to the side of the cone; the parabola was the only section that was considered in the right-angled cone, the ellipse in the acute-angled cone, and the hyperbola in the obtuse-angled. From these three sorts of cones the figures of the sections had their names for a considerable time, till, at length, Apollonius showed that they might all be cut out of any one cone, and, by this discovery, merited in those days the appellation of the Great Geometrician." (Account of Sir I. Newton's Phil. Discoveries, book i. chap. 5.)

It would appear, therefore, that, in mathematics, an inductive inference may not only be demonstratively certain, but that it is a natural, and sometimes, perhaps, a necessary step in the generalization of our knowledge. And yet it is of one of the most unexceptionable inductive conclusions in this science, (the only science in which it is easy to conceive an enumeration which excludes the possibility of any addition,) that Aristotle has spoken,—as a conclusion resting on sophistical evidence.

So much with respect to Aristotle's induction, on the supposition that the enumeration is complete.

In cases where the enumeration is imperfect, Dr. Wallis afterwards observes, "That our conclusion can only amount to a probability or to a conjecture; and is always liable to be overturned by an instance to the contrary." He observes, also, "That this sort of reasoning is the principal instrument of investigation in what is now called experimental philosophy; in which, by observing and examining particulars, we arrive at the knowledge of universal truths." (Institutio Logica. See the chapter De Inductione et Exemplo.) All this is clearly and correctly expressed; but it must not be forgotten, that it is the language of a writer trained in the schools of Bacon and of Newton.

Even, however, the induction here described by Dr. Wallis, falls

greatly short of the method of philosophising pointed out in the Novum Organon. It coincides exactly with those empirical inferences from mere experience, of which Bacon entertained such slender hopes for the advancement of science. "Restat experientia mera; quæ si occurrat, casus; si quæsita sit, experimentum nominatur. Hoc autem experientiæ genus nihil aliud est, quam mera palpatio, quali homines noctu utuntur, omnia pertentando, si forte in rectam viam incidere detur; quibus multo satius et consultius foret, diem præstolari aut lumen accendere, deinceps viam inire. At contra, verus experientia ordo primo lumen accendit, deinde per lumen iter demonstrat, incipiendo ab experientia ordinata et digesta, et minime præpostera aut erratica, atque ex ea educendo axiomata, atque ex axiomatibus constitutis rursus experimenta nova, quum nec verbum divinum in rerum massam absque ordine operatum sit." (Nov. Org. Aph. lxxxii.)

It is a common mistake, in the logical phraseology of the present times, to confound the word experience and induction as convertible terms. There is, indeed, between them a very close affinity; inasmuch as it is on experience alone that every legitimate induction must be raised. The process of induction, therefore, presupposes that of experience; but, according to Bacon's views, the process of experience does by no means imply any idea of induction. Of this method, Bacon has repeatedly said, that it proceeds "by means of rejections and exclusions" (that is, to adopt the phraseology of the Newtonians, in the way of analysis) to separate or decompose nature; so as to arrive at those axioms or general laws, from which we may infer, in the way of synthesis, other particulars, formerly unknown to us, and perhaps placed beyond the reach of our direct examination. (Nov. Org. Aph. cv. ciii.)

But enough, and more than enough, has been already said to enable my readers to judge, how far the assertion is correct, that the induction of Bacon was well known to Aristotle. Whether it be yet well know to all his commentators, is a different question; with the

"Let it always be remembered, that the author who first taught this doctrine (that the true art of reasoning is nothing but a language accurately defined and skilfully arranged,) had previously endeavored to prove, that all our notions, as well as the signs by which they are expressed, originate in perceptions of sense; and that the principles on which languages are first constructed, as well as every step in their progress to perfection, all ultimately depend on inductions from observation; in one word, on experience merely."-Aristotle's Ethics and Politics, by Gillies, vol. i. pp. 94, 95.

In the latter of these pages, I observe the following sentence, which is of itself sufficient to show what notion the Aristotelians still annex to the word under consideration. "Every kind of reasoning is carried on either by syllogism or by induction; the former by proving to us, that a particular proposition is true, be. cause it is deducible from a general one, already known to us; and the latter demonstrating a general truth, because it holds in all particular cases.

It is obvious, that this species of induction never can be of the slightest use in the study of nature, where the phenomena which it is our aim to classify under their general laws, are, in respect of number, if not infinite, at least incalculable and incomprehensible by our faculties.

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.

I.-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 Appollonius 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 procedimus. Jam 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,

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 analytical 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 mathematical 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 exception 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 ogor, 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

Propositiones Geometrica More Veterum Demonstratæ. 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.