most perfect examples of strict inference; compelled habitually to fix his attention on those conditions on which the cogency of the demonstration depends; and, in the mistaken or imperfect attempts at demonstration made by himself or others, he is presented with examples of the most natural fallacies, which he sees exposed and corrected." My Edinburgh reviewer* expressed a wish, that these latter "novel assertions had been explained and exemplified;" and obviously, was really at a loss to understand them, although they refer to the daily occurrences of the lecture-room. This is a curious proof how entirely practical teaching is lost sight of, amid the speculations of his school. I may observe, too, as I have done elsewhere†, that reasoning, as a practical habit, is taught with peculiar advantage by mathematics, because we are, in that study, concerned with long chains of reasoning, in which each link hangs from all the preceding. "The language contains a constant succession of short and rapid references to what has been proved already; and it is justly assumed, that each of these brief movements helps the reasoner forwards in a course of infallible certainty and security. Each of these hasty glances must possess the clearness of intuitive evidence, and the certainty of mature reflection: and yet must leave the reasoner's mind entirely free to turn instantly to the next point of his progress. The faculty of performing • Review of Thoughts on the Study of Mathematics, p. 127. + Mechanical Euclid, with Remarks on Mathematical Reasoning, p. 144. such processes well and readily, is of great value;" and this faculty can hardly be acquired and cultivated in any other way, than by the study of mathematics. I shall not pursue the consideration of the beneficial intellectual influence of mathematical studies. It would be easy to point out circumstances, which show that this influence has really operated;-for instance, the extraordinary number of persons, who, after giving more than the common attention to mathematical studies at the University, have afterwards become eminent as English lawyers. It would be easy, also, to gather together a "cloud of witnesses," who have spoken with admiration and enthusiasm of mathematics as a discipline of the mind. But this would be a very idle mode of treating the subject; for it might be possible also, to adduce a large bulk of testimony on the other side. And what could be inferred from this array of cloud against cloud? Except we can get some clear insight into the subject ourselves, we can never know whether the authors we adduce, are not speaking from views as vague and confused as our own. When any one will point out any other study, as a mode of practically teaching reasoning, which he maintains to be preferable to mathematics, we may be tempted to make the comparison; but this has not been done, so far as I know. It may be said, that mathematical reasoning is but one kind of reasoning, and that the study and practice of this alone, ought not to be spoken of as the cultivation of the reasoning power in general. To this, I reply, that the faculty of reasoning, so far as it can be disciplined by practical teaching, receives such a discipline from mathematical study. If, for instance, any one says, "Why do you not cultivate the habit of inductive as well as of deductive reasoning?" I answer, that the only cultivation of which inductive reasoning admits, is that which is supplied by deductive reasoning. For when we collect a new truth by induction from facts, what is the process of our minds? We acquire a new and distinct view, or hit upon a right supposition; and we perceive that, in the consequences of our new notions, the observed facts are included. The former part of this process, the new and true idea suited to the emergency, the happy guess, no teaching can give the student. All that we can do is, to fix the idea when he has it, and to teach him to test his hypothesis by tracing its consequences. And this, the cultivation of deductive habits does. teach men to invent new truths; we cannot even give them the power of guessing a riddle. But those who have been inventors, have always had, not only that native fertility of mind which no education can bestow, but also a talent of clearly and rapidly applying their newly-sprung thoughts, in which half their power consisted, and which is precisely that faculty which mathematical habits may improve. And the distinctness of the fundamental ideas, a state of thought essential alike to sound reasoning from old truths, and to the discovery of new, is not unprovided for by the study of mathematics; for though deductive habits do not give distinct fundamental ideas We cannot they demand them; and, by the constant appeal to such ideas, they fix and develope them. A perception of the truth of mathematical axioms cannot be conveyed into the mind by reasoning; but still, the mathematical reasoner usually sees more clearly than other men, the necessary truth of his axioms*. Other persons may have the idea of space, as well as the geometer;-the idea of force and matter, as well as the mechanician; but these ideas shine with a clearer and steadier light in the minds of those who constantly work by such lamps, and therefore, carefully tend and trim them. Since the study of mathematics is thus useful, not only in teaching habits of deduction, which are exemplified in its proofs, but also in leading men to the distinct ideas which are expressed in its definitions and axioms, we learn a lesson respecting the kind of mathematics which we may most advantageously introduce in our education. For since those clear ideas upon which the several mathematical sciences depend are a valuable mental possession, both on their own account, and as examples of such a class of elements of truth, we ought not to be content with one or two such ideas and their consequences, but should introduce the student to a wider range of mathematical proof. We shall thus succeed best in repressing the evil consequences which might arise from confining ourselves to one kind of reasoning. We ought, therefore, to include in our course, not * On this subject, see the Remarks at the end of the Mechanical Euclid. only pure mathematical sciences, geometry, arithmetic, and algebra, the consequences of the fundamental ideas of space, number, and quantity; but we ought also to admit the consequences of other ideas, which lead to rigorous mathematical sciences, such as the ideas of pressure and matter, of rigidity and fluidity, of velocity and force; of which ideas, the developements are found in the sciences of Mechanics and Hydrostatics. This maxim I have already urged, in a former publication on this subject*. And I rejoice to say, that a recent alteration in the examinations of the University of Cambridge, by which certain portions of Mechanics and Hydrostatics are introduced into the lower Examinations for Degrees, has made our system, what appears to me, on the grounds just stated, a better intellectual education than it was before. I shall not here dwell upon the intellectual effect of the practical teaching of Greek and Latin, but proceed to consider the effect of the two systems of instruction in another point of view. SECT. 3.-OF THE EFFECT OF PRACTICAL AND SPECULATIVE TEACHING ON THE PROGRESS OF CIVILISATION. If I were to begin by asserting that the progress of civilisation is essentially connected with the prevalent education, the assertion would probably be assented to; but at the same time, it would probably also be understood in so general and indistinct a manner, that no real use could be made of it in our argument. • Thoughts on the Study of Mathematics; reprinted at the end |