« ΠροηγούμενηΣυνέχεια »
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. Την τοιαύτην έφοδον (says Pappus) αναλυσιν καλούμεν, οιον αναπαλιν λυσιν ; a 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.†
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 is probable that a Greek geometer would have used the word Synthesis. Thus, in astronomy, when we attempt from the known phenomena to establish the truth of the Copernican system, we are said to proceed analytically. But the analogy of ancient geometry would apply this word to a process directly the reverse; a process which, assuming the system as true, should reason from it to the known phenomena: After which, if the process could be so reversed as to prove that this system, and this system alone, is consistent with these facts, it would bear some analogy to a geometrieal synthesis.
[This process is termed Analysis; as if we should say, an inverse solution.]
+ 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 supposi tion, 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.
These observations had occurred to me, long before I had remarked, that the celebrated Dr. Hooke (guided also by what he conceived to be the analogy of the Greek geometry) uses the words analysis and synthesis in physics, precisely in the contrary acceptations to those assigned to them in the definitions of Sir Isaac Newton. "The "methods (he observes) of attaining a knowledge of nature may be "two; either the analytic or the synthetic. The first is the proceeding from the causes to the effects. The second, from the "effects to the causes. The former is the more difficult, and sup66 poses the thing to be already done and known, which is the thing "sought and to be found out. This begins from the highest, most "general and universal principles or causes of things, and branches "itself out into the more particular and subordinate. The second "is the more proper for experimental inquiry, which from a true 'information of the effect by a due process, finds out the immediate "cause thereof, and so proceeds gradually to higher, and more re"mote causes and powers effective, founding its steps upon the low"est and more immediate conclusions."*
That Hooke was led into this mode of speaking by the phraseology of the ancient mathematicians, may, I think, be safely inferred from the following very sagacious and fortunate conjecture with respect to the nature of their analytical investigations, which occurs in a different part of the same volume. I do not know, that any thing approaching to it is to be found in the works of any other English author prior to Dr. Halley.
Hooke's Posthumous Works, p. 330.
As this volume is now become extremely rare, I shall transcribe the paragraph which immediately follows the above quotation.
"An inquisition by the former (or analytic) method, is resembled fitly enough by the example of an architect, who hath a full comprehension of what he designs to do, and acts accordingly: But the latter (or synthetic) is more properly resembled to that of a husbandman or gardener, who prepares his ground, and sows his seed, and diligently cherishes the growing vegetable, supplying it continually with fitting moisture, food, and shelter,-observing and cherishing its continual progression, till it comes to its perfect ripe ness and maturity, and yields him the fruit of his labour. Nor is it to be expected, that, a production of such perfection as this is designed, should be brought to its complete ripe Less in an instant; but as all the works of nature, if it be naturally proceeded with, it must have its due time to acquire its due form and full maturity, by gradual growth and a natural progression; not but that the other method is also of excellent and necessary use, and will very often facilitate and hasten the progress. An instance of which kind I de signed, some years since, to have given this honourable society, in some of my lectures upon the motions and influences of the celestial bodies, if it had been then fit; but I understand, the same thing will now be shortly done by Mr. Newton, in a Treatise of his now in the press But that will not be the only instance of that kind which I design to produce, for that I have divers instances of the like nature, wherein, from a bypothesis being supposed, on a premeditated design, all the phenomena of the subject will be a priori foretold, and the effects naturally follow, as proceeding from a cause so and so qualified and limited. And, in truth, the synthetic way, by experiments and observations, will be very slow, if it be not often assisted by the analytic, which proves of excellent use, even though it proceed by a false position; for that the discovery of a negative is one way of restraining and limiting an affirmative."
Change the places of the words analytic and synthetic in this last sentence; and the remark coincides exactly with what Boscovich, Hartley, Le Sage, and many other authors, have advanced in favour of synthetical explanations from hypothetical theories. I shall have occasion afterwards to offer some additional suggestions in support of their opinion, and to point out the limitations which it seems to require.
"What ways the ancients had for finding out these mediums, or "means of performing the thing required, we are much in the dark ; "nor do any of them shew the way, or so much as relate that they "had such a one: Yet 'tis believed. they were not ignorant of some "kind of algebra, by which they had a certain way to help them"selves in their inquiries, though that we now use be much confined "and limited to a few media. But I do rather conceive, that they "had another kind of analytics, which went backwards through al"most all the same steps by which their demonstrations went for"wards, though of this we have no certain account, their writings "being altogether silent on that particular. However, that such a "way is practicable, I may hereafter, upon some other occasion, "shew by some examples; whereby it will plainly appear, how "much more useful it is for the finding out the ways for the solution "of problems, than that which is now generally know and practised “by species.”*
The foregoing remarks, although rather of a critical than of a philosophical nature, may, I hope, be of some use in giving a little more precision to our notions on this important subject. They are introduced here, not with the most distant view to any alteration in our established language (which, in the present instance, appears to me to be not only unexceptionable, but very happily significant of its true logical import,) but merely to illustrate the occasional influence of words over the most powerful understandings; and the vagueness of the reasonings into which they may insensibly be betrayed, by a careless employment of indefinite and ambiguous terms.
If the task were not ungrateful, it would be easy to produce numerous examples of this from writers of the highest and most deserved reputation in the present times. I must not, however, pass over in silence the name of Condillac, who has certainly contributed, more than any other individual, to the prevalence of the logical errours now under consideration. "I know well (says he, on one oc"casion) that it is customary to distinguish different kinds of ana"lysis; the logical analysis, the metaphysical, and the mathematical; "but there is, in fact, only one analysis; and it is the same in all "the sciences." On another occasion, after quoting from the logic of Port Royal a passage in which it is said, "That analysis and syn"thesis differ from each other only, as the road we follow in ascend"ing from the valley to the mountain, differs from the road by which "we descend from the mountain into the valley,"-Condillac proceeds thus: "From this comparison, all I learn is, That the two
*Hooke's Post. Works, p. 68.
Of the illustrations here promised by Hooke of the utility of the analytical method in geometrical investigations, no traces, as far as I have observed, occur in his writings. And it would appear from the following note by the editor, on the passage last quoted, that nothing important on the subject had been discovered among his papers.
"I do not any where find, that this was ever done by Dr. Hooke, and leave the useful. ness therefore to be considered by the learned."
La Logique, Seconde Partie, Chap. vii.
"methods are contrary to one another, and consequently, that if the "one be good, the other must be bad. In truth, we cannot proceed "otherwise than from the known to the unknown. Now, if the "thing unknown be upon the mountain, it will never be found by "descending into the valley; and if it be in the valley, it will not "be found by ascending the mountain. There cannot, therefore, "be two contrary roads by which it is to be reached. Such opinions "(Condillac adds) do not deserve a more serious criticism."
To this very extraordinary argument, it is unnecessary to offer any reply, after the observations already made on the analysis and synthesis of the Greek geometers. In the application of these two opposite methods to their respective functions, the theoretical reasoning of Condillac is contradicted by the universal experience of mathematicians, both ancient and modern; and is, indeed, so palpably absurd, as to carry along with it its own refutation, to the conviction of every person capable of comprehending the terms of the question. Nor would it be found more conclusive or more intelligible, if applied to the analysis and synthesis of natural philosophers; or indeed to these words, in any of the various acceptations in which they have ever hitherto been understood. As it is affirmed, however, by Condillac, that "there neither is, nor can be, more than "6 one analysis," a refutation of his reasoning, drawn from any particular science, is, upon his own principle, not less conclusive, than if founded on a detailed examination of the whole circle of human knowledge. I shall content myself, therefore, on the present occasion, with a reference to the mathematical illustrations contained in the former part of this section.
With regard to the notion annexed to this word by Condillac himself, I am not certain, if, after all that he has written in explanation of it, I have perfectly seized his meaning. "To analyze, (he tells
us, in the beginning of his Logic) is nothing more than to observe "in a successive order the qualities of an object, with a view of giv"ing them in the mind that simultaneous order in which they co"exist." In illustration of this definition, he proceeds to remark, That 66 although with a single glance of the eye, a person may "discover a multitude of objects in an open champaign which he "has previously surveyed with attention, yet that the prospect is 66 never more distinct, than when it is circumscribed within narrow "bounds, and only a small number of objects is taken in at once. "We always discern with accuracy but a part of what we see.”
"The case (he continues) is similar with the intellectual eye. I have, at the same moment, present to it, a great number of the "familiar objects of my knowledge. I see the whole group, but am "unable to mark the discriminating qualities of individuals. To "comprehend with distinctness all that offers itself simultaneously "to my view, it is necessary that I should, in the first place, decompose the mass;-in a manner analogous to that in which a cu
Ibid. Chap. vi.
+ La Logique, Première Partie, Chap. ii.
"rious observer would proceed, in decomposing, by successive steps, "the co-existent parts of a landscape. It is necessary for me, in "other words, to analyze my thoughts."*
The same author afterwards endeavours still farther to unfold his notion of analysis, by comparing it to the natural procedure of the mind in the examination of a machine. "If I wish (says he) to un"derstand a machine, I decompose it, in order to study separately "each of its parts. As soon as I have an exact idea of them all, and 86 am in a condition to replace them as they were formerly, I have a "perfect conception of the machine, having both decomposed and "recomposed it."t
In all this, I must confess, there seems to me to be much both of vagueness and of confusion. In the two first quotations, the word analysis is employed to denote nothing more than that separation into parts, which is necessary to bring a very extensive or a very complicated subject within the grasp of our faculties; a description, certainly, which conveys but a very partial and imperfect conception of that analysis which is represented as the great organ of invention in all the sciences and arts. In the example of the machine, Condillac's language is somewhat more precise and unequivocal; but, when examined with attention, will be found to present an illustration equally foreign to his purpose. This is the more surprising, as the instance here appealed to might have been expected to suggest a juster idea of the method in question, than that which resolves into a literal de-composition and re-composition of the thing to be analyzed. That a man may be able to execute both of these manual operations on a machine, without acquiring any clear comprehension of the manner in which it performs its work, must appear manifest on the slightest reflection; nor is it less indisputable, that another person, without disengaging a single wheel, may gain, by a process purely intellectual, a complete knowledge of the whole contrivance. Indeed, I apprehend, that it is in this way alone that the theory of any complicated machine can be studied; for it is not the parts, separately considered, but the due combination of these parts, which constitutes the mechanism.§ An observer, accordingly,
La Logique, Première Partie, Chap. ii. In this last paragraph, I have introduced one or two additional clauses, which seemed to me necessary for conveying clearly the au thor's idea. Those who take the trouble to compare it with the original, will be satisfied, that in venturing on these slight interpolations, I had no wish to misrepresent his opinion.
† Ibid. Chap. iii.
Ce qu'on nomme methode d'invention, o'est autre chose que l'analyse. C'est elle qui a fait toutes les découvertes; c'est par elle que nous retrouverons tout ce qui a été trouvé. Ibid.
[What we call the Method of Invention, is nothing else but Analysis It is this which has effected all discoveries; it is by this that we shall again find out what has been heretofore known.]
If, on any occasion, a literal decomposition of a machine should be found necessary, it can only be to obtain a view of some of its parts, which, in their combined state, are goncealed from observation.