what analogous to the increased facilities which a fugitive from Great Britain would gain, in consequence of the multiplication of our sea-ports.

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 investigation, 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, fall under the description of that δυναμις 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 first 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 Pro

"Nihil a verå 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.

+ See the preface of Marinus to Euclid's Data.. In the preface to the 7th book of Pappus, the same idea is expressed by the phrase duvais superinn.

blematis enodatio, aut ostensio Theorematis, quam novi cujuspiam inventio: hæc quippe fortunæ in incertum vagantibus obviæ plerumque esse solent; toto tota vero lila, quanta est, studiosissimam attentæ mentis, in unum aliquem scopumcol limantts, rationem 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."

Not having the works of Galileo at hand, I quote this passage on the authority of Guido Grandi, who has introduced it in the preface to his demonstration of Huyghens's Theorems concerning the Logarithmic Line.-Vid. Hugenii Opera Reliqua, Tom. I. p. 43.

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

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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

See the concluding paragraphs of Newton's Optics,

have composed it with great labour, is not a 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 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

* Account of Newton's Discoveries.