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is never obliterated by subsequent agencies. But the views which have been presented in the preceding pages show us this influence operating still more power. fully in another way. If the scholars who come from the great schools to the university are not in a great degree afterwards moulded by the university system; if they are not engaged upon new subjects and modes of study; if they obtain university honors, and college emoluments, merely by continuing the pursuit of their school-boy labors; if, having done this, they become so numer. ous in the governing body of the university as to be able to control and direct its measures; if they exercise this power so as to perfect the next generation of school boys from being constrained to any studies except those of the schools; then the university is no longer a place of higher education, supplying the deficiencies of the schools, balancing their partial system, liberalizing their necessarily narrow plan, converting the education of the grammar school into a university education; the university then is merely an appendage to the great schools, rewarding their best scholars, but teaching them nothing; giving prizes, but giving these to proficiency acquired at school, exercising little influence to modify or correct, but much to confirm the impressions made by the mere classical education of boyhood.
After what I have already said, my readers will not be surprised at my again saying that the mathematics ought to be taught at school, so far as to be a preparation for the mathematics which are to be studied at the university; nor at my adding that the present mathematical teaching at several of our great schools fails of satisfying this condition with regard to a great number of their scholars, many of them very well instructed in the classics. Nor shall I here attempt further to illustrate these propositions. That mathematics is a necessary portion of a liberal education, I have endeavored to show in this first part. But mathematics cannot be studied to any purpose at the university, except an effectual beginning is made at school. This is true, even of speculative portions of mathematics, such as geometry, in which the main point is to be able to understand and to state the proofs of the propositions which belong to the science. It is still more true of practical sciences, such as arithmetic, algebra, and practical trigonometry, in which the learner has to apply rules and to perform operations which it requires considerable time and application to learn to apply and to perform correctly, and still more, to perform both correctly and rapidly. If this is not learnt during the period of boyhood, at least with regard to arithmetic, it is never learnt; and when this is the case, all real progress in mathematics is impossible. Yet how imperfectly arithmetic is generally learnt at our great schools, is remarkable to the extent of being curious, besides being, as I conceive it is, a great misfortune to the boys. The sons of great merchants, bankers, and fund-holders, when they leave school, are very generally incapable of calculating the discount upon a bill, and often not able to add up the sums of an account. And few indeed of the sons of our great landowners can calculate the area of a field of irregular, or even of regular form, and given dimensions. This appears to be a lamentable state of things on every account; in its first and lowest bearings, because such ignorance is a great impediment in the practical business of life; in the next place, because arithmetic is in itself a good discipline of attention and application of mind, and when pursued into its applications, an admirable exercise of clearness of head and ingenuity; in the next place, because, as the boys of the middle classes at commercial schools are commonly taught arithmetic (and generally mensuration also) effectively and well,
the boys from the great schools have, in this respect, an education inferior to that which prevails in a lower stage of society; and in the next place, again, because the want of arithmetic makes it impossible that such young men should receive a good education at the university. On all these accounts, it appears to me in the highest degree desirable, that arithmetic, at least, should hold a fixed and prominent place in the system of our great schools.
Arithmetic, and when that has been mastered, geometry, mensuration, algebra, and trigonometry in succession, should form a part of the daily business of every school which is intended to prepare students for the university. I am aware that it has been said that any substantial attention to such subjects interferes with the classical teaching; because the classes of boys framed according to their knowledge of Greek and Latin will differ from the classes according to their knowledge of mathematics. Of course this is a difficulty; but one which should be overcome. It has hitherto in a great measure been overcome in the university and in our colleges. It is a difficulty which, if we yield to it, and allow it to deter us from the attempt to improve our education, will make it impossible for us to have a liberal education; because it will exclude all but one element. At this rate, we shall teach our boys Greek and Latin, and not teach them anything else, for fear it should interfere with Greek and Latin; and this, during the first eighteen or nineteen years of life, when they might learn the elements of all human knowledge and acquire habits which would lead them into any part of literature or science, according to their intellectual tendencies.
Arithmetic has usually been a portion of education on somewhat different grounds, namely, not so much on account of its being an example of reasoning, as on account of its practical use in the business of life. To know and to be able familiarly to apply the rules of arithmetic is requisite on innumerable occasions of private and public business; and since this ability can never be so easily and completely acquired as in early youth, it ought to be a part of the business of a boy at school. For the like reasons mensuration ought to be learnt at an early period; that is, the rules for determining the magnitude in numbers of lines, spaces, and solids, under given conditions; a branch of knowledge which differs from geometry as the practical from the speculative, and which, like other practical habits, may be most easily learnt in boyhood, leaving the theoretical aspect of the subject for the business of the higher education which comes at a later period. There is another reason for making arithmetic a part of the school learning of all who are to have a liberal education, namely, that without a very complete familiarity with actual arithmetical processes, none of the branches of algebra can be at all understood. Algebra was, at first, a generalization and abstraction of arithmetic; and whatever other shape it may take by successive steps in the minds of mathematicians, it will never be really understood by those students who do not go through this step. And, as we have already said, there is, in a general education, little or nothing gained by going beyond this. The successive generalizations of one or another new calculus may form subjects of progressive study for those whose education is completed, but cannot enter into a general education without destroying the proportion of its parts.
It is not quite so necessary that geometry should be well studied at school as it is that arithmetic should be well taught there; because in geometry the learner has only to understand and to remember, whereas in arithmetic he has to work in virtue of acquired habit. A student at the university, if he had very good mental talents, might perhaps go forward and acquire a good knowledge of
mathematics, even if he had his geometry to begin after his arrival. Still it is not very likely that he would do so. The habits of mental attention and coherence of thought should be cultivated before the age of eighteen, or they will hardly be cultivated to much purpose. It appears to be, in the present state of things, quite necessary that youths who are to come to the university should become masters of some considerable portion of Euclid before they come. Indeed this appears to be the more necessary now, because, so far as I can judge, boys in general are more slow in understanding any portion of mathematics than they were thirty years ago. It may be that I am mistaken, but so it appears to me; and I do not conceive it to be at all improbable that a long coninuance of mere classical learning, of the kind which I have already attempted to characterize, should have led to that which not I alone think likely to result from such an education; namely, an incapacity for all continuous thought and all intellectual labor. I do not think it at all incredible that a long course of indulgence in the pleasures of taste and imagination, without any corresponding exercise of the reason, may have emasculated the intellects of the rising generation, so that they prove feeble in comparison with their fathers, when they are called to any task requiring continuous and systematic thought.
In the treatise (Part I. and II.), from which the foregoing extracts are taken, Dr. Whewell maintains the supremacy of mathematical study in the cultivation of the reasoning faculty over the classics or natural science, and as a useful gymnastic of the mind, far superior to logic itself. In this field he encountered an antagonist at least worthy of his steel.
SIR WILLIAM HAMILTON.
In an elaborate essay in the Edinburgh Review for January, 1836, Sir William Hamilton examines the claims set forth by Dr. Whewell, and summons a cloud of witnesses to the soundness of his own views in contradiction of those claims.
How opposite are the habitudes of mind which the study of the Mathematical and the study of the Philosophical sciences* require and cultivate, has attracted the attention of observers from the most ancient times. The principle of this contrast lies in their different objects, in their different ends, and in the different modes of considering their objects;-differences in the sciences themselves, which calling forth, in their cultivators, different faculties, or the same faculty in different ways and degrees, determine developments of thought so dissimilar, that in the same individual a capacity for the one class of sciences has, not without reason, been considered as detracting from his qualification for the other.
It may be proper here to remark upon the vague universality which is given to the terms philosophy and philosophical in common English; an indefinitude limited specially to this country. Mathematics and Physics may here be called philosophical sciences; whereas, on the Continent, they are excluded from philosophy, philosophical being there applied emphatically to those sciences which are immediately or mediately mental. Hegel, in one of his works, mentions that in looking over what in England are published under the title of "Philosophical Transactions," he had been unable to find any philosophy at all. This abusive employment of the words is favored, I believe, principally, at Cambridge; for if Mathematics and Physics are not philosophical, then that university must confess that it now encourages no philosophy whatever. The history of this insular peculiarity might easily be traced.
As to their objects.-In the first place:-The Mathematical sciences are limited to the relations of quantity alone, or, to speak more correctly, to the one relation of quantities-equality and inequality; the Philosophical sciences, on the contrary, are astricted to none of the categories, are coëxtensive with existence and its modes, and circumscribed only by the capacity of the human intellect itself. In the second place :-Mathematics take no account of things, but are conversant solely about certain images; and their whole science is contained in the separation, conjunction, and comparison of these. Philosophy, on the other hand, is mainly occupied with realities; it is the science of a real existence, not merely of an imagined existence.
As to their ends, and their procedure to these ends.-Truth or knowledge is, indeed, the scope of both; but the kind of knowledge proposed by the one is very different from those proposed by the other.-In Mathematics, the whole principles are given; in Philosophy, the greater number are to be sought out and established.-In Mathematics, the given principles are both material and formal, that is, they afford at once the conditions of the construction of the science, and of our knowledge of that construction (principia essendi et cognoscendi). In Philosophy, the given principles are only formal-only the logical conditions of the abstract possibility of knowledge. In Mathematics, the whole science is virtually contained in its data; it is only the evolution of a potential knowledge into an actual, and its procedure is thus merely explicative. In Philosophy, the science is not contained in data; its principles are merely the rules for cur conduct in the quest, in the proof, in the arrangement of knowledge; it is a transition from absolute ignorance to science, aud its procedure is therefore ampliative. In Mathematics we always depart from the definition; in Philosophy, with the definition we usually end.-Mathematics know nothing of causes; the research of causes is Philosophy; the former display only the that (rò ori); the latter mainly investigates the why (7ò dióri).—The truth of Mathematics is the harmony of thought and thought; the truth of Philosophy is the harmony of thought and existence. Hence the absurdity of all applications of the mathematical method to philosophy.
It is, however, proximately in the different modes of considering their objects that Mathematics and Philosophy so differently cultivate the mind.
In the first place:-Without entering on the metaphysical nature of Space and Time, as the basis of concrete and discrete quantities, of geometry and arithmetic, it is sufficient to say that Space and Time, as the necessary conditions of thought, are, severally, to us absolutely one; and each of their modifications, though apprehended as singular in the act of consciousness, is, at the same time, recognized as virtually, and in effect, universal. Mathematical science, therefore, whose notions (as number, figure, motion) are exclusively modifications of these fundamental forms, separately or in combination, does not establish their universality on any a posteriori process of abstraction and generalization; but at once contemplates the general in the individual. The universal notions of philosophy, on the contrary, are, with a few great exceptions, gencr. alizations from experience; and as the universal constitutes the rule under which the philosopher thinks the individual, philosophy consequently, the reverse of mathematics, views the individual in the general.
In the second place :-In Mathematics, quantity, when not divorced from form, is itself really presented to the intellect in a lucid image of phantasy, or in a sensible diagram; and the quantities which can not thus be distinctly construed imagination and sense, are, as only syntheses of unity, repetitions of identity,
adequately, though conventionally, denoted in the vicarious combination of a few simple symbols. Thus both in geometry, by an ostensive construction, and in arithmetic and algebra, by a symbolical, the intellect is relieved of all effort in the support and presentation of its objects; and is therefore left to operate upon these in all the ease and security with which it considers the concrete realities of nature. Philosophy, on the contrary, is principally occupied with those general notions which are thought by the intellect but are not to be pictured in the imagination; and yet, though thus destitute of the light and definitude of mathematical representations, philosophy is allowed no adequate language of its own; and the common language, in its vagueness and insufficiency, does not afford to its unimaginable abstractions that guarantee and support, which, though less wanted, is fully obtained by its rival science, in the absolute equivalence of mathematical thought and mathematical expression.
In the third place :-Mathematics, departing from certain original hypotheses and these hypotheses exclusively determining every movement of their procedure, and the images or the vicarious symbols about which they are conversant being clear and simple, the deductions of the sciences are apodictic or demonstrative; that is, the possibility of the contrary is, at every step, seen to be excluded in the very comprehension of the terms. On the other hand, in Philosophy (with the exception of the Theory of Logic), and in our reasonings in general, such demonstrative certainty is rarely to be attained; probable certainty, that is, where we are never conscious of the impossibility of the contrary, is all that can be compassed; and this also, not being internally evolved from any fundamental data, must be sought for, collected, and applied from without.
From this general contest it will easily be seen, how an excessive study of mathematical sciences not only does not prepare, but absolutely incapacitates the mind, for those intellectual energies which philosophy and life require. We are thus disqualified for observation, either internal or external-for abstraction and generalization-and for common reasoning; nay disposed to the alternative of blind credulity or of irrational skepticism.
But the study of mathematical demonstration is mainly recommended as a practice of reasoning in general, and it is precisely, as such a practice, that its inutility is perhaps the greatest. General reasoning is almost exclusively occupied on contingent matter; if mathematical demonstration therefore supplies, as is contended, the best exercise of practical logic, it must do this by best enabling us to counteract the besetting tendencies to error, and to overcome the principal obstacles in the way of our probable reasonings. Now, the dangers and difficulties of such reasoning lie wholly-1, in its form-2, in its vehicle-3, in its object-matter. Of these severally.
1. As to the form:-The study of mathematics educates to no sagacity in detecting and avoiding the fallacies which originate in the thought itself of the reasoner.-Demonstration is only demonstration, if the necessity of the one contrary and the impossibility of the other be, from the nature of the objectmatter itself, absolutely clear to consciousness at every step of its deduction. Mathematical reasoning, therefore, as demonstrative, allows no room for any sophistry of thought; the necessity of its matter necessitates the correctness of its form, and, consequently, it cannot forewarn and arm the student against this formidable principle of error.
2. In regard to the vehicle:-Mathematical language, precise and adequate, nay, absolutely convertible with mathematical thought, can afford us no example of those fallacies which so easily arise from the ambiguities of ordinary language;