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"saving the long progression of the thoughts to remote and first "principles in every case, the mind should provide it several "stages; that is to say, intermediate principles, which it "might have recourse to in the examining those positions that "come in its way. These, though they are not self-evident "principles, yet, if they have been made out from them by a "wary and unquestionable deduction, may be depended on as "certain and infallible truths, and serve as unquestionable "truths to prove other points depending on them by a nearer "and shorter view than remote and general maxims. These <6 may serve as land-marks to show what lies in the direct "way of truth, or is quite beside it. And thus mathema"ticians do, who do not, in every new problem, run it back "to the first axioms, through all the whole train of intermedi"ate propositions. Certain theorems, that they have settled "to themselves upon sure demonstration, serve to resolve to "them multitudes of propositions which depend on them, and "are as firmly made out from thence, as if the mind went "afresh over every link of the whole chain that ties them to "first self-evident principles. Only in other sciences great "care is to be taken that they establish those intermediate "principles with as much caution, exactness, and indifferency, "as mathematicians use in settling any of their great theorems. "When this is not done, but men take up the principles in this
or that science upon credit, inclination, interest, &c. in haste, "without due examination, and most unquestionable proof, "they lay a trap for themselves, and, as much as in them "lies, captivate their understandings to mistake, falsehood, and "error.
I cannot help thinking that Locke's recommendation of the use of intermediate principles must be received with much greater limitations in the case of all the moral sciences than he seems to have been aware of; otherwise he could not have failed to warn his readers, more explicitly and earnestly than he has done, of the extreme difficulty, if not of the impossibility, of establishing, in any of these branches of knowledge, intermediate principles at all analogous to the theorems in mathematics. In mechanical philosophy and chemistry, undoubtedly, there are many intermediate principles which, in the present improved state of these sciences, may be safely assumed as data; but how few, comparatively, are the principles to which we are yet entitled to appeal in any of the branches of moral learning; not excepting even the modern, and sometimes
* Locke's Conduct of the Understanding, § 21.
too oracular science of Political Economy! On all such subjects, Mr. Locke's advice will be found much less favourable to the discovery of truth, than to a display of the disputant's readiness and fluency in the conduct of an oral debate, or in the management of a controversial skirmish in a periodical Journal. I think I have observed a peculiar proneness in mathematicians, on occasions of this sort, to avail themselves of principles sanctioned by some imposing names, and to avoid all discussions which might lead to an examination of ultimate truths, or involve a rigorous analysis of their ideas. The passage quoted from Locke, without any comment, sufficiently accounts for this bias.
As for the metaphysician, he is but too apt in an argument (unless he is much upon his guard against the sin which most easily besets him) to run into the opposite extreme, of disputing vexatiously with his adversary every inch of ground; and, after cavilling at principles which have been sanctioned by the universal consent and experience of ages, to dispute those first principles of human knowledge, which, if they were seriously called in question, would involve all the sciences in complete doubt and uncertainty.
Before dismissing this head, it is proper to take notice of an objection which may occur against the consistency of some of the foregoing remarks; although, in reality, the appearances on which it is founded are necessary consequences of the principles I have endeavoured to establish. I have said, that, of all the branches of human knowledge, mathematics is that in which the faculty of imagination is the least exercised. It is, however, a certain fact, that, in mathematicians who have confined their studies to mathematics alone, there has often been observed a proneness to that species of religious enthusiasm in which imagination is the predominent element, and which, like a contagion, is propagated in a crowd. In one of our most celebrated universities, which has long enjoyed the proud distinction. of being the principal seat of mathematical learning in this island, I have been assured, that if, at any time, a spirit of fanaticism has infected (as will occasionally happen in all numerous societies) a few of the unsounder limbs of that learned body, the contagion has invariably spread much more widely among the mathematicians than among the men of erudition. Even the strong head of Waring, undoubtedly one of the ablest analysts that England has produced, was not proof against the malady, and he seems at last (as I was told by the late Dr. Watson, Bishop of Landaff,) to have sunk into a deep religious melancholy, approaching to insanity.
When Whitfield first visited Scotland, and produced by his powerful though unpolished eloquence such marvellous effects on the minds of his hearers, Dr. Simpson, the celebrated professor of mathematics at Glasgow, had the curiosity to attend one of his sermons in the fields; but could never be persuaded, by all the entreaties of his friends, to hear another. He had probably felt his imagination excited in an unpleasant degree, and with his usual good sense, resolved not to subject himself to the danger of a second experiment. I have observed, too, upon various occasions, the effects of dramatic representations on persons who had spent their lives among calculations and diagrams; and have generally found them much more powerful than upon men devoted to the arts which are addressed to the imagination.
These phenomena tend strongly to confirm a principle which I ventured to state in the concluding Chapter of the first Volume of these Elements; "That by a frequent and habitual "exercise of imagination, we at once cherish its vigour, and "bring it more and more under our command. As we can "withdraw the attention at pleasure from objects of sense, and "transport ourselves into a world of our own, so when we "wish to moderate our enthusiasm, we can dismiss the objects "of imagination, and return to our ordinary perceptions and 66 occupations. But in a mind to which these intellectual vi❝sions are not familiar, and which borrows them completely "from the genius of another, imagination, when once excited, "becomes perfectly ungovernable, and produces something "like a temporary insanity." "Hence" (I have added) "the "wonderful effects of popular eloquence on the lower orders, "effects which are much more remarkable than what it produ❝ces on men of education.
The occasional fits of religious enthusiasm, therefore, to which some mathematicians have been liable, so far from indicating the general predominance of imagination in their intellectual character, are the natural effects of the torpid state in which that faculty is suffered to remain in the course of their habitual studies, and of the uncontrollable ascendant it seldom fails, when strongly excited, to usurp over all the other powers of the understanding, in minds not sufficiently familiarized to its visions and illusions.
Mr. Gray, who appears, from various passages in his works, to have studied the phenomena of the Human Mind much more attentively and successfully than most poets, has, in a passage formerly quoted, struck into a train of thinking, coinciding nearly with the above; and is the only writer in whom I have
met with any observations at all approaching to it. "The pro"vince of eloquence" (he remarks) "is to reign over minds "of slow perception and little imagination; to set things in "lights they never saw them in; to engage their attention by "details of circumstances gradually unfolded; to adorn and "heighten them with images and colours unknown to them; "and to raise and engage their rude passions to the point to "which the speaker wishes to bring them."
It is observed by D'Alembert, in his Elements of Philosophy, (a work abounding with the most profound and original views) among other remarks on what he calls the Esprit Géomètre, That it is not always united with the Esprit Métaphysique. To this observation (which, by the way, corroborates strongly a remark formerly quoted from Descartes,) D'Alembert adds, as a still more curious circumstance, that a genius for mathematics, and a turn for games of skill, however nearly they may at first view seem to be allied to each other, are by no means always to be found in the same individual; and that there is even less affinity or analogy between them than is commonly imagined. The subject may appear to some of very trifling moment; but as D'Alembert has not thought it unworthy of his notice, and as it has led him to an argument which may be extended to some other pursuits of greater importance than those of the gamester, I shall quote it at length. "A mathematical head" (says he) "undoubtedly implies "a propensity to calculate and to combine; but to combine "scrupulously and slowly; examining, one after another, all "the parts and aspects of an object, so as to omit no element "which ought to enter into the computation; and never ven"turing upon a new step, till the last has been well secured. "A turn for play, on the other hand, is founded on a power "of rapid combination, which embraces at a glance, though "vaguely, and sometimes incorrectly, a great number of cir"cumstances and conditions, guided more by a certain natural "quickness improved by habit, than by a scientific application "of general principles. The mathematician, besides, may "command as much time as he pleases, for resolving his pro"blems; repose himself after an effort of study, and begin again with renewed vigour; while the player is obliged to
Gray's Letters, p. 349.
+ L'esprit Géomètre.-I have substituted the word Mathematicians for Geome the last of these expressions being always used in our language in that limited sense in which it was employed in the schools of Ancient Greece. In the best French writers, the title of Geometer is very generally given to mere algebraists, and it is plainly in this extensive acceptation that it is employed by D'Alembert in the present instance.
"resolve his problems on the spur of the occasion, and to "bring all his resources to bear on a single instant. It is not, "therefore, surprising that a great mathematician should, at a "card-table, often sink to the level of mediocrity."
The fact taken notice of in the foregoing passage, is confirmed by my own observations, as far as they have extended. Of the various mathematicians whom I have happened to be acquainted with, (some of them, certainly, of the first eminence,) I cannot recollect one who was at all distinguished as a player at whist. Many of them, at the same time, were fond of the game, and devoted to it regularly a portion of their leisure hours. But all of them, without exception, were mere novices, when compared, not only with professional gamesters, but with such men and women as may be selected to form a cardparty from any large promiscuous assembly.
The only point in D'Alembert's statement, about which I entertain any doubts, relates to the degree of intellectual exertion which he supposes to be implied in the skill of our common card-players. To myself, I must own, the whole seems to resolve into a ready application of established rules, caught from imitation and practice; while, on the other hand, I am disposed to ascribe the failure of the mathematician to his misplaced confidence in the exercise of his own extemporaneous judgment, in cases where he ought to be guided solely by the approved results of more deliberate calculations.
Something of the same sort may be remarked with respect to every other employment of our faculties in which promptitude of decision is indispensably necessary. Wherever this is
the case, a ready application of rules, sanctioned by previous reflection, or by general experience, is far more likely to ensure success, than those hasty and dubious conclusions which are formed under the pressure of present exigencies.
Nor are these the only occasions on which an unseasonable exercise of reasoning and invention is attended with inconvenience. The same effects may be expected wherever the superiority of one man above another depends upon a quickness and facility derived from habitual practice. Whence is it that the mathematician is commonly surpassed in point of rapidity, as an arithmetical calculator, by the illiterate accountant; but because his intellectual activity is adverse to the passive acquisition of a mechanical dexterity? It is owing to a similar cause, that a facility in acquiring languages is seldom combined (at least after years of maturity) with the higher gifts of the mind. The extraordinary promptitude of children in this and other respects, is no doubt owing principally to the suscepti