With respect to some of the other books mentioned by Pappus it is remarked by Dr. Simson's biographer, that "they relate to general problems of frequent recurrence in geometrical investigations and that their use was for the more immediate resolution of any proposed geometrical problem, which could be easily reduced to a particular case of any one of them. By such a reduction the problem was considered as fully resolved; because it was then necessary only to apply the analysis, composition, and determination of that case of the general problem, to this particular problem which it was shown to comprehend." * ""*

From these quotations it manifestly appears, that the greater part of what was formerly said of the utility of analysis in investigating the demonstration of theorems, is applicable, mutatis mutandis, to its employment in the solution of problems. It appears farther, that one great aim of the subsidiary books, comprehended under the title of τόπος αναλυόμενος was to multiply the number of such conclusions as might secure to the geometer a legitimate synthetical demonstration, by returning backwards, step by step, from a known or elementary construction. The obvious effect of this was, at once to abridge the analytical process, and to enlarge its resour'ces; on a principle somewhat analogous to the increased facilities which a fugitive from Great Britain would gain, in consequence of the multiplication of our seaports.

Notwithstanding, however, the immense aids afforded to the geometer by the ancient analysis, it must not be imagined that it altogether supersedes the necessity of ingenuity and invention. It diminishes, indeed, to a wonderful degree, the number of his tentative experiments, and of the paths by which he might go astray;† but (not to mention the prospective address which it supposes, in preparing the way for the subsequent investi

* Ibid. pp. 159, 160.

"Nihil a verâ et genuinâ analysi magis distat, nihil magis abhorret, quam tentandi methodus; hanc enim amovere et certissimâ viâ ad quæsitum perducere, præcipuus est analyseos finis."

Extract from a MS. of Dr. Simson, published by Dr. Traill. See his Account, &c. p. 127.

gation, by a suitable construction of the diagram) it leaves much to be supplied at every step, by sagacity and practical skill; nor does the knowledge of it till disciplined and perfected by long habit, fail under the description of that δύναμις αναλυτική which is justly represented by an old Greek writer, as an acquisition of greater value than the most extensive acquaintance with particular mathematical truths.

According to the opinion of a modern geometer and philosopher of the arst eminence, the genius thus displayed in conducting the approaches to a preconceived mathematical conclusion, is of a far higher order than that which is evinced by the discovery of new theorems. "Longe sublimioris ingenii est," says Galileo, "alieni Problematis enodatio, aut ostensio Theorematis, quam novi cujuspiam inventio: hæc quippe fortunæ in incertum vagantibus obvia plerumque esse solet; tota vero illa, quanta est, studiosissimam attentæ mentis, in unum aliquem scopum collimantis, ratiocinationem exposcit." + Of the justness of this observation, on the whole, I have no doubt; and have only to add to it, by way of comment, that it is chiefly while engaged in the steady pursuit of a particular object, that those discoveries which are commonly considered as entirely accidental, are most likely to present themselves to the geometer. It is the methodical inquirer alone who is entitled to expect such fortunate occurrences as Galileo speaks of; and wherever invention appears as a characteristical quality of the mind, we may be assured, that something more than chance has contributed to its success. On this occasion, the fine and deep reflection of Fontenelle will be found to apply with peculiar force: "Ces hasards ne sont que pour ceux qui jouent bien."

II.

Critical Remarks on the vague Use, among Modern Writers, of the Terms Analysis and Synthesis.

THE foregoing observations on the Analysis and Synthesis of the Greek Geometers may, at first sight, appear somewhat out of place, in a disquisition concerning the principles and rules of the Inductive Logic. As it was, however, from the Mathematical Sciences, that these words were confessedly borrowed by the experimental inquirers of the Newtonian School, an attempt to illustrate their original technical import seemed to form a necessary introduction to the strictures which I am about to offer, on the loose and inconsistent applications of them, so frequent in the logical phraseology of the present times.

Sir Isaac Newton himself has, in one of his Queries fairly brought into comparison the Mathematical and the Physical Analysis, as if the word, in both cases, conveyed the same idea. "As in Mathematics, so in Natural Philosophy, the investigation of difficult things, by the method of Analysis, ought ever to precede the method of Composition. This analysis consists in making experiments and observations, and in drawing conclusions from them by induction, and admitting of no objections against the conclusions, but such as are taken from experiments, or other certain truths. For hypotheses are not to be regarded in experimental philosophy. And although the arguing from experiments and observations by induction be no demonstration of general conclusions; yet it is the best way of arguing which the nature of things admits of, and may be looked upon as so much the stronger, by how much the induction is more general. And if no exception occur from phenomena, the conclusion may be pronounced generally. But if, at any time afterwards, any exception shall occur from experiments; it may then begin to be pronounced, with such exceptions as occur. By this way of analysis we may proceed from compounds to ingredients; and from motions to the forces producing them; and, in general,

from effects to their causes; and from particular causes to more general ones, till the argument end in the most general. This is the method of analysis. And the synthesis consists in assuming the causes discovered, and established as principles, and by them explaining the phenomena proceeding from them, and proving the explanations."

It is to the first sentence of this extract (which has been repeated over and over by subsequent writers) that I would more particularly request the attention of my readers. Mr. Maclaurin, one of the most illustrious of Newton's followers, has not only sanctioned it by transcribing it in the words of the author, but has endeavoured to illustrate and enforce the observation which it contains. "It is evident, that as in Mathematics, so in Natural Philosophy, the investigation of difficult things by the method of analysis ought ever to precede the method of composition, or the synthesis. For, in any other way, we can never be sure that we assume the principles which really obtain in nature; and that our system, after we have composed it with great labor, is not mere dream or illusion." The very reason here stated by Mr. Maclaurin, one should have thought, might have convinced him, that the parallel between the two kinds of analysis was not strictly correct: inasmuch as this reason ought, according to the logical interpretation of his words, to be applicable to the one science as well as to the other, instead of exclusively applying (as is obviously the case) to inquiries in Natural Philosophy.

After the explanation which has been already given of geometrical and also of physical analysis, it is almost superfluous to remark, that there is little, if any thing, in which they resemble each other, excepting this, that both of them are methods of investigation and discovery; and that both happen to be called by the same name. This name is, indeed, from its literal or etymological import, very happily significant of the notions conveyed by it in both instances; but, notwithstanding

* See the concluding paragraphs of Newton's Optics. † Account of Newton's Discoveries.

this accidental coincidence, the wide and essential difference between the subjects to which the two kinds of analysis are applied, must render it extremely evident, that the analogy of the rules which are adapted to the one can be of no use in illustrating those which are suited to the other.

Nor is this all: The meaning conveyed by the word analysis, in Physics, in Chemistry, and in the Philosophy of the Human Mind, is radically different from that which was annexed to it by the Greek Geometers, or which ever has been annexed to it, by any class of modern Mathematicians. In all the former sciences, it naturally suggests the idea of a decomposition of what is complex into its constituent elements. It is defined by Johnson, "a separation of a compound body into the several parts of which it consists."-He afterwards mentions, as another signification of the same word, “a solution of any thing whether corporeal or mental, to its first elements; as of a sentence to the single words; of a compound word to the particles and words which form it; of a tune to single notes; of an argument to single propositions." In the following sentence, quoted by the same author from Glanville, the word analysis seems to be used in a sense precisely coincident with what I have said of its import, when applied to the Baconian method of investigation. "We cannot know any thing of nature but by an analysis of its true initial

In the Greek geometry, on the other hand, the same word evidently had its chief reference to the retrograde direction of this method, when compared with the natural order of didactic demonstration. Τὴν τοιαύτην ἔφοδον (says Pappus) ανάλυσιν καλοῦμεν, οἷον ἀνάπαλιν λύσιν ;

* By the true initial causes of a phenomenon, Glanville means (as might be easily shown by a comparison with other parts of his works) the simple laws from the combination of which it results, and from a previous knowledge of which, it might have been synthetically deduced as a consequence.

That Bacon, when he speaks of those separations of nature, by means of comparisons, exclusions, and rejections, which form essential steps in the inductive process, had a view to the analytical operations of the chemical laboratory, appears sufficiently from the following words, before quoted: "Itaque naturæ facienda est prorsus solutio et separatio; non per ignem certe, sed per mentem, tanquam ignem divinum."