Siogiouos or determination) becomes an indispensable requisite towards a complete solution.

The utility of the ancient analysis in facilitating the solution of problems, is still more manifest than in facilitating the demonstration of theorems; and, in all probability, was perceived by mathematicians at an earlier period. The steps by which it proceeds in quest of the thing sought, are faithfully copied, as might be easily shown, from that natural logic which a sagacious mind would employ in similar circumstances; and are, in fact, but a scientific application of certain rules of method, collected from the successful investigations of men who were guided merely by the light of common sense. The same observation may be applied to the analytical processes of the algebraical art.

In order to increase, as far as the state of mathematical science then permitted, the powers of their analysis, the ancients, as ap pears from Pappus, wrote thirty-three different treatises, (known among mathematicians by the name of τοπος αναλυομενσο;) of which number there are twenty-four books, whereof Pappus has particularly described the subjects and the contents. In what manner some of these were instrumental in accomplishing their purpose, has been fully explained by different modern writers; particularly

the late very learned Dr. Simson of Glasgow. Of Euclid's Data, for example, the first in order of those enumerated by Pappus, he observes, that "it is of the most general and necessary use in the solution of problems of every kind; and that whoever tries to investigate the solutions of problems geometrically, will soon find this to be true; for the analysis of a problem requires, that consequences be drawn from the things that are given, until the thing that is sought be shown to be given also. Now, supposing that the data were not extant, these consequences must, in every particular instance, be found out and demonstrated from the things given in the enunciation of the problem; whereas the possession of this elementary book supersedes the necessity of any thing more than a reference to the propositions which it contains."

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

* Letter from Dr. Simson to George Lewis Scott, Esq. published by Dr. Traill. See his Account of Dr. Simson's Life and Writings, p. 118.

↑ Ibid. pp. 159, 169.

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, comprchended under the title of 10.10; arahvoueros, 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 resources; on a principle somewhat 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 knowl edge of it, till disciplined and perfected by long habit, fall under the description of that δυναμις αναλυτικη, which is justly represented by an old Greck 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 Problematis enodatio, aut ostensio Theorematis, quam novi cujuspiam inventio: hæc quippe fortunæ incertum vagantibus obviæ plerumque esse solent; tota vero illa, quanta est, studiosissimam attentæ mentis, in unum aliquem scopum collimantis, 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

*“Nihil a verà 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 duvapis čuperikη.

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 Huygens's Theorems concerning the Logarithmic Line.-Vid. Hugenii Opera Reliqua, tom. i. p. 43.

geometer. It is the methodical inquirer alone, who is entitled to expect such fortunate occurrences as Galileo speaks of: and wher ever 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: "Des hasards ne sont que pour ceux qui jouent bien."

2.-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 inconsistent application of them, so frequent in the logical phraseology of the present time.

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 rather 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.' See the concluding paragraphs of Newton's Optics.

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 endeavored 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."— (Account of Newton's Discoveries.) 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 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 clements 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 can not know any thing of nature but by an analysis of its true initial causes."*

In the Greek geometry, on the other hand, the same word evi dently had its chief reference to the retrograde direction of this method, when compared with the natural order of didactic demonstration. Την τοιαύτην έφοδον, says Pappus, ανάλυσιν καλούμεν, διον avazakiv hvoir; & passage which Halley thus translates: hic processus analysis vocatur, quasi dicas, inversa solutio. That this is the primitive and genuine import of the preposition ava, is very generally admitted by grammarians; and it accords, in the present instance, so happily with the sense of the context, as to throw a new and strong light on the justness of their opinion.t

In farther proof of what I have here stated with respect to the double meaning of the words analysis and synthesis, as employed in physics and in mathematics it may not be superfluous to add the following considerations:-In mathematical analysis, we always set out from a hypothetical assumption, and our object is to arrive at some known truth, or some datum, by reasoning synthetically from which we may afterwards return, on our own footsteps, to the point where our investigation began. In all such cases, the synthesis is infallibly obtained by reversing the analytical process; and as both of them have in view the demonstration of the same theorem, or the solution of the same problem, they form, in reality, but different parts of one and the same investigation. But in natural philosophy, a synthesis which merely reversed the analysis would be absurd. On the contrary, our analysis necessarily sets out from known facts; and after it has conducted us to a general principle, the synthetical reasoning which follows, consists always of an application of this principle to phenomena, different from those comprehended in the original induction.

*

In some cases, the natural philosopher uses the word analysis, where it it is probable that a Greek geometer would have used the 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."

The force of this preposition, in its primitive sense, may perhaps, without any false refinement, be traced more or less palpably, in every instance to which the word analysis is with any propriety applied. In what Johnson calls for example, "the separation of a compound body into the several parts of which it consists," -we proceed on the supposition, that these parts have previously been combined, or put together, so as to make up the aggregate whole, submitted to the examination of the chemist; and consequently, that the analytic process follows an inverted or retrograde direction, in respect of that in which the compound is conceived to have been originally formed. A similar remark will be found to apply, mutatis mutandis, to other cases, however apparently different.