are purely synthetical. A very eminent mathematician of the present times has even gone so far as to mention it "as a model of geometrical synthesis." He would, I apprehend, have expressed his idea more correctly, if, instead of the epithet geometrical, he had employed, on this occasion, logical or metaphysical; in both of which sciences, as was formerly observed, the analytical and synthetical methods bear a much closer analogy to the experimental inductions of chemistry and of physics, than to the abstract and hypothetical investigations of the geometer.

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The abuses of language which have now been under our review. will appear the less wonderful, when it is considered that mathematicians themselves do not always speak of analysis and synthesis with their characteristical precision of expression; the former word being frequently employed to denote the modern calculus, and the latter, the pure geometry of the ancients. This phraseology, although it has been more than once censured by foreign writers, whose opinion might have been expected to have some weight, still continues to prevail very generally upon the Continent. The learned and judicious author of the History of Mathematics conplained of it more than fifty years ago; remarking on the impropriety of calling by the name of the synthetic method, that which employs no algebraical calculus, and which addresses itself to the mind and to the eyes, by means of diagrams, and of reasonings expressed at full length in ordinary language. It would be more exact," he observes farther, "to call it the method of the ancients, which (as is now universally known) virtually supposes, in all its synthetical demonstrations, the previous use of analysis. As to the algebraical calculus, it is only an abridged manner of expressing a process of mathematical reasoning; which process may, according to circumstances, be either analytical or synthetical. Of the latter, an elementary example occurs in the alge braical demonstrations given by some editors of Euclid, of the propositions in his second book."

This misapplication of the words analysis and synthesis, is not, indeed, attended with any serious inconveniences, similar to the errors occasioned by the loose phraseology of Condillac. It were surely better, however, that mathematicians should cease to give it the sanction of their authority, as it has an obvious tendency,besides the injustice which it involves to the inestimable remains of Greek geometry,-to suggest a totally erroneous theory, with respect to the real grounds of the unrivaled and transcendent powers possessed by the modern calculus, when applied to the more complicated researches of physics.‡

* M. Lacroix. See the Introduction to his Elements of Geometry. + Histoire des Mathématiques, par Montucla, tome premier, pp. 175, 176. In the ingenious and profound work of M. De Gerando, entitled, " Des Signes et de l'Art de Penser, considérés dans leur rapports mutuels," there is a very valuable chapter on the Analysis and Synthesis of metaphysicians and of geome

SECTION IV.

THE CONSIDERATION OF THE INDUCTIVE LOGIC RESUMED.

1.-Additional Remarks on the distinction between Experience and Analogy. Of the grounds afforded by the latter for Scientific Inference and Conjecture.

IN the same manner in which our external senses are struck with that resemblance between different individuals which give rise to a common appellation, our superior faculties of observation and reasoning, enable us to trace those more distant and refined similitudes which lead us to comprehend different species under one common genus. Here, too, the principles of our nature, already pointed out, dispose us to extend our conclusions from what is familiar to what is comparatively unknown; and to reason from species to species, as from individual to individual. In both instances, the logical process of thought is nearly, if not exactly the same; but the common use of language has established a verbal distinction between them; our most correct writers being accustomed (as far as I have been able to observe) to refer the evidence of our conclusions, in the one case, to experience, and in the other to analogy, The truth is, that the difference between these two denominations of evidence, when they are accurately analyzed, appears manifestly to be a difference, not in kind, but merely in degree; the discriminating peculiarities of individuals invalidating the inference, as far as it rests on experience solely, as much as the characteristical circumstances which draw the line between different species and different genera."

This difference in point of degree, it must at the same time be ters. (See vol. iv. p. 172.) The view of the subject which I have taken in the foregoing section, has but little in common with that given by this excellent philosopher; but in one or two instances, where we have both touched upon the same points (particularly in the strictures upon the logic of Condillac,) there is a general coincidence between our criticisms, which adds much to my confidence in my own conclusions.

In these observations on the import of the word analogy, as employed in philosophical discussions, it gives me great pleasure to find, that I have struck nearly into the same train of thinking with M. Prévost. I allude more particu larly to the following passage in his Essais de Philosophie.

"Le mot analogie, dans l'origine, n'exprime que la resemblance. Mais l'usage l'applique à une ressemblance éloignée d'ou vient que les conclusions analogiques sont souveut hasardées, et ont besoin d'êtres déduites avec art. Toutes les fois donc que, dans nos raissonnemens, nous portons des jugemens semblables sur des objets qui n'ont qu'une ressemblance éloignée, nous raissonnons analogiquement. La resemblance prochaine est celle qui fonde la première généralisation, celle qu'on nomme l'espèce. On nomme éloignée la resemblance qui fonde les généralisations superieures, c'est-à-dire, le genre et ses divers degrès. Mais cette definition n'est pars rigoureusement suivie.

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Quoiqu'il en soit, on conçoit des cas, entre lesquels la ressemblance est si parfaite, qu'il ne s'y trouve aucune difference sensible, si ce n'est celle du tems et du lieu. Et il est des cas dans lesquels on apperçoit beaucoup de resemblance, mais

remembered, leads where it is great, to important consequences. In proportion as the resemblance between two cases diminishes in the palpable marks which they exhibit to our senses, our inferences from the one to the other are made with less and less confidence ; and therefore it is perfectly right that we should reason with more caution from species to species, than from individual to individual of the same kind. In what follows, accordingly, I shall avail myself of the received distinction between the words experience and analogy; a distinction which I have hitherto endeavored to keep out of view, till I should have an opportunity of explaining the precise notion which I annex to it. It would, in truth, be a distinction of important use in our reasonings, if the common arrangements, instead of originating, as they have often done, in ignorance or caprice, had been really the result of an accurate observation and comparison of particulars. With all the imperfections of these arrangements, however, a judicious inquirer will pay so much regard to prevailing habits of thinking, as to distinguish very scrupulously what common language refers to experience from what it refers to analogy, till he has satisfied himself, by a diligent examination. that the distinction has, in the instance before him, no foundation in truth. On the other hand, as mankind are much more disposed to confound things which ought to be distinguished, than to distinguish things which are exactly or nearly similar, he will be doubly cautious in concluding, that all the knowledge which common language ascribes to experience is equally solid; or that all the conjectures which it places to the account of analogy are equally suspicious.

A different idea of the nature of analogy has been given by some writers of note; and it can not be denied, that in certain instances it seems to apply still better than that proposed above. The two accounts, however, if accurately analyzed, would be found to approach much more nearly than they appear to do at first sight; or rather, I am inclined to think, that the one might be resolved into the other, without much straining or over refinement. But this is a question chiefly of speculative curiosity, as the general remarks which I have now to offer, will be found to hold with respect to analogy, considered as a ground of philosophical reasoning, in whatever manner the word is defined; provided only it be understood to express some sort of correspondence or affinity between two suboù l'on decouvre aussi quelques differences indépendantes de la diversité tu temps et du lieu. Lorsque nous ferons un jugement général, fondé sur la première espèce de ressemblance, nous dirons que nous usons de la méthode d'induction. Lorsque la seconde espèce de ressemblance autorisera nos raisonnemens, nous dirons que c'est de la méthode d'analogie que nous faisons usage. On dit ordinairement que la méthode d'induction conclut du particulier au général, et que la méthode d'analogie conclut du semblable au semblable. Si l'on analyse ces definitions, on verra que nous n'avons fait autre chose que leur donner de la précision." -Essais de Philosophie, tome ii. p. 202.

See also the remarks on induction and analogy in the four following articles of M. Prévost's work.

jects, which serves, as a principle of association or of arrangement, to unite them together in the mind.

According to Dr. Johnson, to whose definition I allude more particularly at present, analogy properly means "a resemblance between things with regard to some circumstances or effects; as when learning is said to enlighten the mind; that is, to be to the mind what light is to the eye, by enabling it to discover that which was hidden before." The statement is expressed with a precision and justness not always to be found in the definitions of this author; and it agrees very nearly with the notion of analogy adopted by Dr. Ferguson, that "things which have no resemblance to each other may nevertheless be analogous; analogy consisting in a resemblance or correspondence of relation." (Principles of Moral and Political Science, vol. i. p. 107.) As an illustration of this, Dr. Ferguson mentions the analogy between the fin of a fish and the wing of a bird; the fin bearing the same relation to the water which the wing does to the air. This definition is more particularly luminous when applied to the analogies which are the foundation of the rhetorical figures of metaphor and allusion; and it applies also very happily to those which the fancy delights to trace between the material and the intellectual worlds; and which, as I have repeatedly observed, are so apt to warp the judgment in speculating concerning the phenomena of the human mind.

The pleasure which the fancy receives from the contemplation of such correspondences, real or supposed, obviously presupposes a certain disparity or contrast in the natures of the two subjects compared; and, therefore, analogy forms an associating principle, specifically different from resemblance, into which Mr. Hume's theory would lead us to resolve it. An additional proof of this is furnished by the following consideration. That a resemblance of objects or events is perceived by sense, and accordingly has some effect even on the lower animals; a correspondence, or, as it is frequently called, a resemblance of relations, is not the object of sense, but of intellect, and consequently, the perception of it implies the exercise of reason.

Notwithstanding, however, the radical distinction between the notions expressed by the words resemblance and analogy, they may often approach very nearly to each other in their meaning; and cases may even be conceived in which they exactly agree. In proof of this, it is sufficient to remark, that in objects, the parts of which respectively exhibit that correspondence which is usually distinguished by the epithet analogous, this correspondence always deviates, less or more, from an exact conformity or identity; insomuch, that it sometimes requires a good deal of consideration to trace in detail the parallel circumstances, under the disguises which they borrow from their diversified combinations. An obvious instance of this occurs when we attempt to compare the bones and joints in the leg and foot a man with those in the leg and foot of

a horse. Were the correspondence in all the relations perfectly exact, the resemblance between the two objects would be manifest even to sense; in the very same manner that, in geometry, the similitude of two triangles is a necessary consequence of a precise correspondence in the relations of their homologous sides.*

The following passage from Buffon, although strongly marked with the author's characteristical spirit of system, is yet I presume, sufficiently correct in the outline to justify me for giving it a place in this note, as an illustration of what I have said in the text on the insensible gradations which fix the limits betwees resemblance and analogy.

"Take the skeleton of a man; incline the bone of the pelvis; shorten those of the thighs, legs, and arms; join the phalanges of the fingers and toes; lengthen the jaws by shortening the frontal bones: and lastly, extend the spine of the back. This skeleton would no longer represent that of a man; it would be the skeleton of a horse. For, by lengthening the back-bone and the jaws, the number of the vertebræ, ribs, and teeth would be increased; and it is only by the numbers of these bones, and by the prolongation, contraction, and junction of others, that the skeleton of a horse differs from that of a man. The ribs, which are essential to the figure of animals, are found equally in man, in quadrupeds, in birds, in fishes, and even in the turtle. The foot of the horse, so apparently different from the hand of a man, is composed of similar bones, and, at the extremity of each finger we have the same small bone resembling the shoe of a horse which bounds the foot of that animal. Raise the skeletons of quadrupeds, from the ape kind to the mouse, upon their hind-legs, and compare them with the skeleton of a man; the mind will be instantly struck with the uniformity of structure observed in the formation of the whole group. This uniformity is so constant, and the gradations from one species to another are so imperceptible, that to discover the marks of their discrimination requires the most minute attention. Even the bones of the tail will make but a slight impression on the observer. The tail is only a prolongation of the os coccygis or rump-bone, which is short in man. The orang outang and true apes have no tail, and in the baboon and several other quadrupeds its length is very inconsiderable. Thus, in the creation of animals, the Supreme Being seems to have employed only one great idea, and, at the same time, to have diversified it in every possible manner, that men might have an opportunity of admiring equally the magnificence of the execution and the simplicity of the design."-Smellie's Translation.

As a proof that the general conclusion in which the foregoing extract terminates, requires some important qualifications and restrictions, it is sufficient to subjoin a few remarks from a later writer, who, with the comprehensive views of Buffon, has combined a far greater degree of caution and correctness in his scientific details. "It has been supposed by certain naturalists, that all beings may be placed in a series or scale, beginning with the most perfect, and terminating in the most simple, or in the one which possesses qualities the least numerous and most common, so that the mind, in passing along the scale from one being to another, shall be nowhere conscious of any chasm or interval, but proceed by gradations almost insensible. In reality, while we confine our attention within certain limits, and especially while we consider the organs separately, and trace them through animals of the same class only, we find them proceed, in their degradation, in the most uniform and regular manner, and often perceive a part or vestige of a part in animals where it is of no use, and where it seems to have been left by Nature, only that she might not transgress her general law of continuity. "But, on the one hand, all the organs do not follow the same order in their degradation. This organ is at its highest state of perfection in one species of animals; that organ is most perfect in a different species, so that, if the species are to be arranged after each particular organ, there must be as many scales or series formed, as there are regulating organs assumed; and in order to construct a general scale of perfection, applicable to all beings, there must be calculation made of the effect resulting from each particular combination of organs,—a calculation which it is needless to add, is hardly practicable.

"On the other hand, these slight shades of difference, these insensible gradations continue to be observed, only while we confine ourselves to the same com