easiest to the most difficult;-and because they leave the mind to the undisturbed use of its powers. If these are not the subjects and these the means best adapted to bring out and invigorate the reasoning powers, where can we find them? Will you plunge at once into all the intricacies of deep Metaphysics and Theoretical Ethics, where many a sagacious observer has but seen men as trees walking, and many a profound reasoner has but inscribed his reasonings on the sand? Will you grapple, at once, with all the obscurities, the equivocations, the uncertainties of language, considered as an instrument of refined logic? As well might you expect to train the young courser to his utmost speed, by driving him blindfold over ditches, and and rocks, and brambles. As well might you expect the best musical proficiency from the use of a half-finished instrument, with here a broken key and there a discordant string.

Without further illustration, the relation of these studies to the direct improvement of the noblest powers of the human intellect, the reasoning powers, must be apparent. I proceed to offer a few remarks on the second branch of my subject, viz.

II. The relation of these studies to certain intellectual habits which they are fitted to form and strengthen.

The first I will mention is, the habit of patient thought. By this I mean, the habit of fixing the attention upon any subject of thought and of retaining it in mind till all its various bearings have been nicely examined. This constitutes the genius of investigation and reasoning on all subjects. Nor is there, proba bly, any habitude of mind more prolific in discovery and invention; more truly characteristic of a great and powerful intellect than the one now in question.

Now the tendency of Mathematical pursuits to induce this habit of mind, grows, it is conceived, immediately out of the definiteness of the relations to be investigated. Whenever in passing from one step to another of the reasoning, a difficulty arises, it is easy to circumscribe it, and to determine precisely where it begins and where it ends. The inquirer plants himself firmly upon the last position gained; and, if he cannot see clearly where to take the next step, he can at least see clearly where not to take it. If he cannot securely advance, he will at least neither retreat nor wander from the track. If he cannot grasp the true reason, he will at least, not be satisfied with any false one. He is convinced that the true reason is there; that others have perceived it; that it lies within his own reach;

and that he cannot mistake it when it once falls under his notice. He is much in the condition of one who, by turning his eye aside, has lost sight of the distant äeronaut, plunged in the deep blue ether. He knows that the object of his admiration is still there is still visible; and with eager gaze he runs his eye round and round the little circle in which it must be found, till he again catches it. So here, the conviction, I had almost said the consciousness, that important truth is near, is attainable,— indeed so circumscribed that it can scarce fail of being discovered upon close examination, summons up all the energies of the mind and marshals them to the onset, and will not suffer them to decline till the work is done. Frequent exercises of this kind, tend, we think, directly to form that habit of mind to which Newton ascribed his unrivalled success in science. And such exercises are the constant attendants on all Mathematical inquiries.

A second habit of mind naturally induced by these studies, and nearly allied to the former, is that of precision of knowledge. The importance of this habit, and especially of its being early formed, need not be urged. The want of it must inevitably prove fatal to all high attainments. Now the whole tendency of an acquaintance with the exact sciences, manifestly is to make the line of demarkation between the known and the unknown, as distinct as possible. Not to know a thing certainly, in Mathematical reasoning, is not to know it at all. And the least discriminating mind cannot fail to perceive the transition from relations, which are clear and definite to those which are doubtful and uncertain. We think it too obvious to need any proof, that a mind accustomed to carry on its processes in the strong light of a demonstrative science, can never be satisfied to grope its way in the obscure regions of unperceived relations and vague generalities. At least, it will never be satisfied short of the conviction that exact knowledge on the subject is placed beyond the reach of the human faculties. This habit of mind is evidently of the highest importance to intellectual progress. It puts the inquirer in possession of a complete map of his own proficiency, and consequently exempts him from the liability of repeating the work which has already been done. Neither treading round perpetually in the same circle, nor crossing and recrossing a tortuous course, he moves steadily onward to the attainment of what still lies beyond him.

A third habit of mind naturally consequent upon these purVOL. VII. No. 21.

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suits is self-confidence. I do not mean that inflated self-confidence which balloon-like rises higher in proportion to the lightness of the material it carries ;-which springs from a fancied superiority, and leads one to suppose that others must pay as profound a deference to his opinions as he himself does; but that self-confidence which springs from a perception of truth and certainty in the decisions of the understanding. It is the confidence of a far-sighted, discriminating, comprehensive intellect. A man of this description stands erect amid the rush and conflict of opinions. His own he holds with firmness, because they are founded on evidence; and for this very reason he abandons them when that evidence is seen to be insufficient. On the contrary, the want of this confidence necessarily renders one timid, vacillating, servile, and really without opinions, except as they are borrowed to meet the exigencies of the passing occasion. The moral courage of such an one, if he have any, is of that particular kind, which waxes stronger and stronger after the battle is over. Of these two characters, the one may be compared to the rock, which rising from its broad and deep foundation, remains immoveable amid all the agitations of the ocean. The other more nearly resembles the buoy, which floats over the shallow, and rises and falls with the surge.

From the nature of the case, this legitimate self-confidence must enter largely into the aggressive character of the pioneer of Science. Scientific improvement has, for the most part, been obliged to drag public opinion as a heavy weight in its train. And hence, he who would introduce new methods in the place of old ones, is forced to a reliance upon his own resources. He must fearlessly abide by the dictates of his own understanding, even with a host against him.

Now we claim for the Mathematical studies the honor of forming and strengthening, in a peculiar manner, this habit of mind. At the conclusion of the very first proposition in Geometry, the learner adds " quod erat demonstrandum" with as strong an assurance of truth as the philosopher of Syracuse himself. And at every subsequent conclusion, he feels the consciousness of certain knowledge. He rests in no degree upon the mere authority of any one. Nor can the distrust of any one at all impair his confidence. Thus, from step to step, he becomes acquainted with his own strength, and learns to confide in it; and, without this confidence, nothing great and noble can ever be accomplished.

It must be allowed that where the elements of character are not all happily blended (and it is seldom that we find them so blended), this self-confidence is liable to degenerate into obstinacy. But the general benefits conferred by it, are incomparably greater than evils of occasional excess.

I will venture to add, lastly, that these pursuits tend to beget a habit of equanimity; of coolness and moderation in judging on moral questions, which a moment's reflection will show to be of singular importance. Conversant with definite relations, and trained to the business of rigorous demonstration, the mind acquires firmness and stability. Accustomed to conclusions based upon the most perfect evidence, it cannot rest with satfaction in those where the evidence is imperfect. And hence, following the light of demonstrative evidence and the dictates of sound philosophy, we learn to draw our inferences with a cautious reference to the completeness or incompleteness of the proof. And hence, too, in moral reasonings, if our opponent with the same premises arrives at a conclusion different from our own, we see from the nature of the case, that it becomes us to assume the attitude of inquiry, not of dictation; it is our business to examine, not denounce. A consciousness of our own liability to error in these fields of difficult research,-a consciousness quickened and strengthened by a familiarity with strict demonstration,-will tend to inspire us with some little respect and some little charity for the opinions of our rivals and opponents. Accustomed to strict demonstration, we can scarcely fail to be sensible, at every step, of the vast difference between assumption and proof;-between denial and refutation. If such then, is the tendency of these studies, I leave the reader to judge of their importance in this point of view. To me, that can never appear unimportant which has any real tendency to restrain the outbreakings of bigotry and intolerance, and chain the rampant spirit of party and sectarian strife. When I look abroad upon the discordant elements of the social system, and, amid the busiest scenes of active benevolence, can discern so few traces of that christian charity which suffereth long and is kind; which envieth not, and is not puffed up ;-which beareth all things and endureth all things;-I cannot but regard that as preeminently useful which confers coolness of investigation and moderation of judgment. And such, we venture to suggest, is the tendency of the Mathematical studies.

III. But I hasten to offer some remarks upon the third branch of my subject ;—the relation of these studies to a knowledge of Natural Philosophy.

My remarks are not intended to apply to the whole range of that extended science, but only to those parts of it in which the phenomena are dependent upon the action of forces, whose laws have been ascertained and reduced to mathematical expressions. Such, for instance, as the various branches of theoretical and practical Mechanics; the greater portion of Optics ;-and the whole splendid system of Astronomy. These subjects cannot be well understood;-some of them scarcely understood at all, -without at least a pretty intimate acquaintance with the more elementary parts of the Mathematical sciences. It is true that many important facts may be known, without the aid of science of any kind. And, it is also true, that many important facts may be known without a knowledge of Natural Philosophy, which is chiefly conversant with the relations, and principles, and causes of things.

Thus, for example, one may know that a boiler, and a cylinder, and a piston-rod, are parts of a steam engine. These are important facts. But with these only, could he be said to understand the philosophy of that masterpiece of mechanism? Nay, he may watch over the construction of all its parts, and see them fitted together, and yet know nothing of its principles. He cannot be said to understand these, till he knows the origin of motion, and can trace the mechanism by which it is propagated. He must become acquainted with the elastic power and condensation of steam, and must be able to follow the process of motion from this prime mover, through all its successive stages, till it terminates in the varied productions of the manufactory, or the rapid propulsion of the majestic steam-boat. Nor is this all; he must be able to determine the quantity of motion, or in other words, the intensity of the force producing motion, and the manner in which it is modified by different modes of transmission, so as to determine the effective power at the working point. And even further than this, the question of the maximum effect; that is, the particular form and adjustments, and modes of action, by which a given expenditure of power will produce the greatest effect, is one that belongs to the philosophy of the engine, and is to be settled by theoretical Mechanics.