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der, 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 expressions 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 and Avos, 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 digi 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 shewn) 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 appears from Pappus, wrote thirty-three different treatises (known among mathematicians by the name of τόπος αναλυόμενος.) 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 by 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 shewn 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 enun"ciation 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 determina"tion of that case of the general problem, to this particular problem "which it was shewn 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 aimof the subsidiary books, comprehended under the title οι τοπος αναλυομενος, 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 multi plication 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 experi ments, and of the paths by which he might go astray; but (not to
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, 160
"Nihil a verâ et genuinâ analysi magis distat, nihil magis abhorret, quam tentandi methodus; hanc enim amovere et certissimâ viâ ad quaesitum perducere, praecipuus est analyseos finis."}
Extract from a MS. of Dr. Simson, published by Dr. Traill. See his account, &c. p. 127.
[Nothing is more unlike the true and genuine Analysis, than the method of tentation; for to set this aside, and to point by a sure and direct path to what is sought, is the chief object of Analysis.]
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 durauis avaλution, 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 diplayed 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. (1) Longe sublimioris ingenii est (says Galileo) alieni Proatis enodatio, aut ostensio Theorematis, quam novi cujuspiam "inventio haec quippe fortunæ in incertum vagantibus obviae "plerumque esse solent; tota vero illa, quanta est, studiosissimam "attentæ mentis, in unum aliquem scopum collimantis, rationem ex"poscit." 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 (2) Ces hasards ne sont que pour ceux qui jouent bien.”
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,
*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 duvaus Eugerinn.
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.
(1) [The solution of a Problem or the demonstration of a Theorem which is offered by some one else, is a much greater proof of superior genius, than any new invention: for these instances of good fortune often present themselves to a mind ranging about without any certain plan; whereas the former, whatever it may be, exhibits the utmost energy of a vigorous understanding sharpening itself to one steady aim.]
(2) [These chances occur to those only who are good players.]
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 argu"ing from experiments and observations by induction be no demon"stration 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 excep"tion shall occur from experiments; it may then begin to be pro"nounced, with such exceptions as occur. By this way of analysis "we may proceed from compounds to ingredients; and from mo"tions 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 disco"vered, and established as principles, and by them explaining the "phenomena proceeding from them, and proving the explana"tions."*
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 evi"dent, that as in Mathematics, so in Natural Philosophy, the inves"tigation 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 princi"ples which really obtain in nature; and that our system, after we "have composed it with great labour, is not mere dream or illu
* See the concluding paragraphs of Newton's Optics.
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; in as much 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 com"pounded 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 ele"ments; 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 ana"lysis of its true initial causes."t
Account of Newton's Discoveries.
+ By the true initial causes of a phenomenon, Glanville means, (as might be easily shewa by a comparison with other parts of his works) the simple laws from the combination of which it results, and from a previous kuowledge 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 fol lowing words, before quoted. “Itaque naturae facienda est prorsus solutio et separatio ; non per ignem certe, sed per mentem, tanquam ignem divinum."