the mere recreation of the school boy. Nor is it in the least unphilosophical to suppose, that what from the difficulty of making the acquisition is now restricted to a small circle of proficients in science, will, in the progress of another century, form the common stock of all well educated men. Nay, I may venture to assert even more than this. Such is the expansive power of intellect, and such the means of its improvement, that it is not chimerical to suppose that those profound reasonings of Newton and Laplace, which have disclosed in all its parts the beautiful mechanism of the heavens, and have conferred the highest honors on the human understanding, may yet become as elementary as those of Euclid now are. This may be expected to result mainly from improvements in the Mathematical Analysis, that is, from the invention of simpler and more direct modes of proof; but, partly also, from improvements in the modes of intellectual culture. So far as this latter branch of the subject is concerned, any anticipated progress must be sought in a careful and exact analysis of the effects produced upon the mind by different intellectual pursuits and different modes of mental occupation. And until some considerable progress is made in settling this question, all our efforts must be merely tentative; and hence often misdirected, and always unaided by the lights of genuine philosophy. It is hardly to be doubted that a successful investigation of this subject would form a distinguished era in the history of the human mind.

The inquiry now suggested is doubtless attended with considerable difficulty. It may require great acuteness of observation, and much patient thought, and much careful attention to the circumstances of the mind's advancement from one degree of strength to another. But we see in it no difficulty so great as to forbid investigation. We cannot readily consent to believe that the subject is so inscrutable as not to be known.

It is with the view of calling the attention of scholars to this general inquiry, and of eliciting a more complete investigation than I am able to make, that I venture to offer the following remarks upon one branch of it, viz. the importance of the Mathematical studies, as a part of a liberal education. I shall consider them

I. First in relation to the immediate discipline of the reasoning powers;—

II. Secondly in relation to certain intellectual habits, which they are fitted to form and strengthen,

III. And thirdly in relation to an acquaintance with Natural Philosophy.

There are other and those vastly important ends secured by the cultivation of these studies, to which I may barely refer hereafter. The considerations now stated appear to me to embrace the principal objects in view, so far as the interests of general Education are concerned.

I. First, then, we are to consider their immediate bearing upon the discipline of the reasoning powers.

"The object of all reasoning," as stated by a very acute Logician, "is merely to expand and unfold the assertions wrapt up, as it were, and implied in those with which we set out," in other words, it is to convince a person, that the thing to be proved is a legitimate consequence of things already admitted. Thus the sole business of reasoning, to whatever subject applied, is to trace out the consequences of general principles. Thus in Geometry, its object is to follow out the results of those first truths called axioms. In the science of Theoretical Astronomy, it traces the consequences of the great law of gravitation; and in Christian Ethics, those of the golden rule of doing to others as we would that others should do unto us; or of some other principles equally general. Such, then, is the object of reasoning, and hence the whole exercise of the reasoning powers must be comprised in the deduction of consequences from admitted truths. And hence, manifestly, he who cannot make these deductions cannot reason; and he who cannot make them well and skilfully, cannot reason well and skilfully.

Now we maintain that the mathematical studies are peculiarly fitted to call into its proper action, and thus to strengthen the power of deducing consequences from admitted general principles. In proof of this assertion, we remark

1. First, the subjects of inquiry are susceptible of being exactly defined. The relations of number and quantity are, from the nature of the case, fixed and definite. These are such as may be perceived with the greatest possible clearness; and may, for the most part, be expressed with the greatest possible accuracy. There are, indeed, many cases, where the relation of quantities cannot be reduced to an expression numerically But this is of no moment in the reasoning, since there is no danger of mistaking an approximate result for an exact

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one. Moreover, the error, wherever it exists, is certainly known to be confined within narrow limits and may be made less than any assignable quantity. Now this clearness of perception and this accuracy of expression are absolutely indispensable to conclusive reasoning. If I clearly perceive that gold is heavier than lead, and that lead is heavier than iron, I infer with absolute certainty that gold is heavier than iron. And this is a process of reasoning. But if, on the contrary, I do not clearly perceive the relative weights of the first and second, nor yet those of the second and third, upon what principle can I infer any thing with regard to the first and third? Again, if the relation were clearly perceived, yet if it could not be accurately expressed, it would be extremely difficult to advance beyond the inference from simple comparison. Now this accurate expression of what is clearly and fully perceived, characterises every branch of Mathematical Science; and to this, it is, if I mistake not, absolutely restricted. The extreme difficulty, not to say impossibility of attaining this precision of thought and expression on all moral questions is proverbial. It is not a little humiliating to reflect, that many a memorable controversy, where talents of the highest order have been enlisted on both sides, has arisen from a misconception of the subject or a misapprehension of the terms employed. And often, the whole result of a splendid intellectual contest, has been to fix somewhat more exactly the sense in which a term shall subsequently be used by the parties; or to correct a misconception which, in the moment of coolness, might readily have been effected without the aid of an exterminating logical warfare. Reasoning with entire accuracy from the same general principles, men might obviously arrive at different conclusions, but how they could arrive at those which are opposing and contradictory, it is not easy to imagine. Supposing, therefore, that disputants are candid, and do not knowingly misrepresent either their subject or their opponents (and this supposition we are in charity bound to make), we need but a single glance at the great field of Ethical polemics, to be convinced that nothing is more difficult than to reason well on moral questions. Ages of conflict have not sufficed for appeasing the spirit of war. Fresh laurels are still won or lost, on the very spot which has a thousand times been covered with the spoils of victory. Whether the cause of this interminable struggle lies in the nature of the subject alone, in the inherent difficulty of apprehending the truth

and embodying it in language, or whether it must be sought in the aberrations of the understanding occasioned by the disturbing influence of self interest, I will not undertake to decide. Certain it is, that the difficulties of ascertaining the truth and of vindicating it, are excessive; and, so far as the world has yet seen, insuperable. And hence the necessity of bringing to those inquiries the results of intellectual discipline derived from the habit of reasoning on subjects where there are no obscurities of thought-no ambiguities of expression. And for such subjects we must look to the exact sciences. Here the relations are all definite, the expressions all clear and exact. On this account alone, we apprehend it will be conceded that the Mathematical studies are peculiarly fitted for the preliminary discipline of the reasoning powers.

But I remark secondly, that they furnish a complete test of the accuracy of every deductive process. From the nature of the case, all true conclusions must be compatible with each other. And, from the nature of Mathematical language, any discrepancy in the results from the same principle are readily detected. In most cases, the inquirer, without leaning upon any authority but that of his own understanding, may be certain of the truth or falsity of his own conclusions. At every step of his progress, may apply to his work an infallible test of its accuracy. I cannot but regard this consideration of the highest importance in the training of the mind to right reasoning. Could such a test be applied to all other subjects, how soon would the wings of many a soaring disputant be clipped? How soon would the arrogant and acrimonious spirit of controversy assume the attitude of docile inquiry after truth?

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I remark, thirdly, that in the Mathematical studies, there is a regular gradation from the simplest and easiest process of reasoning to the most complicated and difficult. The more elementary parts of Geometry and Algebra abound with elegant examples of ratiocination, level to the comprehension of all; while the more abstruse parts of Analysis, demand a grasp and power of intellect possessed by few. Yet from the simplest proposition, which fixes the attention of a school boy, to the sublime conclusions of a Newton, which have surprised and astonished the world, there is a regular succession of steps so related to each other, that he who has taken the first is by that very means prepared to take the second. And he who has taken the second, passes on to the third without obstruction.

And thus, throughout the whole, extended series, the advanced step requires nothing for its attainment but a secure possession of those which have preceded.

The hill of Science is indeed lofty, but not precipitous,rugged, but not abrupt. The ascent is indeed toilsome, but every new position gained affords to the traveller firm footing and a secure resting place. There are no shifting sands, no rolling rocks, no treacherous caverns to endanger his safety. There is no sliding avalanche to bear off, in a moment of convulsion, and to bury in its ruins, the labor of ages.

In this department, especially, what is done, no change can undo. What is acquired becomes an inalienable possession. And acquisition, moreover, is made easy by this felicitous gradation of truths. Every requirement may readily be adapted to the present capabilities of the learner. What to-day is too difficult for his unpractised judgment and reason, will to-morrow yield to the mind's growing strength. As a mental discipline, this consideration, peculiar, so far as I understand the subject, to this study, seems to me of great moment.

I add fourthly, what is no less deserving of notice, these studies open an extended field for intellectual exertion, where there is nothing to disturb the free and unbiassed exercise of the judging and reasoning faculty. On most subjects, the prejudices of education, the intrusions of interest, and even the caprices of fancy become powerful disturbing forces in the intellectual system. They are, moreover, disturbing forces which it is extremely difficult to estimate and counteract; and, in the present constitution of things, impossible to avoid. They necessarily impair that harmony of movement, which would otherwise exist. Here, on the contrary, the mind prosecutes its investigation of truth, unimpeded, undisturbed. Avoiding the sinuosities incident to excited passions and conflicting interests, its course is right onward to the goal. It presents the noble spectacle of an intellect weighing evidence with impartial care, and bending all its energies to the ascertainment of truth. The peculiar advantage of this kind of exercise, as a mental discipline, must be obvious.

To recapitulate, we say that the Mathematical studies are peculiarly fitted to develop the reasoning powers, because the subjects of them are exactly defined;-because they furnish a complete test of the accuracy of every deductive process ;because they present a regular gradation of exercises from the