authors as more essential than a like knowledge of the Greek writers. The writing of Latin prose ought to be sedulously cultivated at school, for it is only by practice that any excellence and facility in this exercise can be acquired; and the same may be said of composition in Latin verse, where there appears to be time and talent for this accomplishment. To sacrifice these parts of scholarship to the practice of composition in Greek prose and verse, with a view to University prizes, is a complete perversion of the business of Education, and must interfere with the genuine classical culture of the student.

179 And in the same manner, in the mathematical portion of school teaching, the object ought to be, not so much to teach what will fit the pupil for the University Examinations as for the College Lectures. And as the basis of all real progress in mathematics, the boy ought to acquire a good knowledge of Arithmetic and a habit of performing the common operations of Arithmetic, and of applying the rules in a correct and intelligent manner. This acquirement appears to be often neglected at our most eminent classical schools. Such a neglect is much to be regretted; for the want of this acquirement is a great practical misfortune, and is often severely felt in after life. Many persons who are supposed to have received the best education which the country affords, are, in all matters of numerical calculation, ignorant and helpless, in a manner which places them, in this respect, far below the members of the middle class, educated as they usually are. We are here, however, concerned, not so much with the practical evils arising from the neglect of Arithmetic in our higher education, as with the effect of this neglect in making all sound mathematical education at a later period impossible. And this evil is in no degree remedied by employing the schoolboy on some of the subjects which enter into the University course, as

Geometry and Algebra. These he may speedily learn when he arrives at the University, if he have been properly grounded in mathematical habits: but Arithmetic he cannot then learn to any purpose. Arithmetic

is a matter of habit, and can be learnt only by longcontinued practice. For some years of boyhood there ought to be a daily appropriation of time to this object. Geometry and Algebra do not require so much time. Geometry is a matter of reasoning; and when the proofs are once understood, the student has little more to do. And although Algebra requires, like Arithmetic, the habits of performing operations on symbols, the operations of Algebra are learnt with comparative ease, when those of Arithmetic are already familiar.

180 Indeed we may say that, in general, boyhood is fitted for the formation of practical Habits, and that the aptitude to attend to general Reasonings comes with more advanced youth. In the most natural course of public education, at School we learn to do, at College we learn reasons why we do. At School we learn to construe and to cipher; at College we are invited to follow the speculations of Philologers, and to attend to the proofs of the rules of Arithmetic. And the tastes of boys, for the most part, correspond to this distribution of employments. They can learn to perform and apply the rules of Arithmetic, and they take a pleasure in the correctness of their operations, and in the manner in which the rules verify themselves; but they find it irksome to follow the reasoning of Euclid, where the interest is entirely of a speculative kind. The interest which belongs to demonstration, as demonstration, comes at a later period, when the speculative powers, in their turn, begin to unfold themselves, and to seek their due employment.

181 Perhaps, too, the interest of demonstration is greater when the truth proved is one with which we are already familiar in practice; as when the reasons

are rendered for the common Rules of Arithmetic. In such cases, by having the speculative side of the subject brought before us, we obtain a view altogether new of an object previously quite familiar. Many persons must recollect having experienced this impression who, having learnt as mere Rules, the method of finding the greatest common measure of two numbers, or the third side of a right-angled triangle when two sides are given, have afterwards been introduced to the Demonstrations of these Rules. Demonstrations which are regarded with this interest, are a very effective means of unfolding the reasoning powers. And it is well worth consideration, whether, with a view to the encouragement of such mental processes as these, the mathematical education of boys at school might not be extended to practical methods, much further than is commonly done, at least at Classical Schools. It would appear to me to be a great improvement, if boys were not only made to learn Arithmetic, but also Mensuration at school:-I mean the practical Rules of finding, from the necessary data, the areas of triangles, circles, sectors; the solid contents of prisms, pyramids, cylinders, spheres, and the like. Such knowledge would be, upon innumerable occasions, of great value in the business of life; and would make the proofs which speculative geometry gives, of the truth of such Rules, both much more intelligible, and much more interesting than they generally are. That schoolboys can learn so much of Mensuration as I here speak of, and will usually take a pleasure in learning and applying it, the experience of many of our commercial and other schools abundantly shows.

182 There are other practical matters in mathematics, which might, so far as time allows, be learnt at school; for instance, the use of Logarithmic Tables, and perhaps the solution of triangles by Trigonometrical Tables. There is the more reason for teaching these [PT. I.]

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practical processes, to the schoolboy, inasmuch as if not learnt then, they are rarely performed with facility and correctness by the student at the University: for though the theory of the processes is brought before him, he has not time to familiarize himself with the practice. I am persuaded that if boys at classical schools were well exercised in Arithmetic and Mensuration, with the use of Logarithmic Tables, they would find this a more congenial employment than going over the proofs of geometrical propositions; and would come to the University prepared to pursue their mathematical studies with alacrity and intelligence; instead of finding in them, as they so often do now, a weary and obscure task, which they engage in only as a necessary condition of some other object, and which produces little effect in that education of the reason which is its proper end.

SECT. 5. Of preventing Superficial Reading.

183 We have already said that, inasmuch as, in a good education, we must educate the Reason as well as the Literary Taste, we must require of our students a mathematical combined with a classical culture. To effect this combination is a matter of no small difficulty; among other accounts, on the account just alluded to;-that when we require of our students both classical and mathematical attainments, if one of the two subjects be looked upon with dislike, it will often be attended to in such a manner as to produce little intellectual improvement. The difficulty just spoken of does not at all liberate us from the obligation of pursuing our object. If the difficulty were insurmountable, a Liberal Education would be impossible, and we should have to consider whether it were worth the while keeping up our universities, as means of gratification for the lovers of classical or mathematical pursuits, when they were deprived of ulterior value. But

the fact is, that a Liberal Education is not impossible, as the experience of all ages has shown; and we have not any reason to despair that we may, by a proper adjustment of our educational institutions, educate at the same time the Literary Taste and the Reason. And as a part of these arrangements we must consider in what manner we may provide for a combined classical and mathematical culture, and may avoid the evils which arise from either of the two being forced and superficial, so as to produce no real and permanent impress on the mind.

184 One process by which we may hope to avoid the failures in the business of education which arise from forced and superficial study, is this: we must require evidence of the student's thorough knowledge of the lower parts of each subject, before we allow him to compete for the honours which are assigned to excellence in the higher portions. We must be well satisfied that he can construe common Latin and Greek correctly and well, before we allow him to aim at prizes which are to be won by writing Latin or Greek verses. We must be satisfied that he understands the common algebraical expression of a curve, and a common mechanical problem, before we give him credit because he writes down some wide generalization of modern analysts applied to curves or to mechanical relations. This precaution is far from being so superfluous as might be imagined; especially if the examination be a mere paper examination. For, in mathematics for instance, when the analytical generalization, correctly written, comes before the examiner, it is difficult or impossible for him to know whether the writer understood it, with its reasons and its bearings. And by accepting such a performance, when it is really a mere matter of memory, great injustice may be done to a competitor, who, not aiming at these ambitious generalities, has made himself fully master of the more