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

limited principles which he pretends to know, and is able to apply them in their proper significance.

185 We should, by these considerations, be led to recommend that, in the general Examinations of a University, students should be so examined that it shall appear that they are fully possessed of their lower mathematical subjects, before they are allowed to compete for the prizes which are assigned to high mathematical proficiency. Retaining the distinction which we have already explained, of Permanent Studies and Progressive Studies in mathematics, we may say, that all students should be ascertained to have attained a sound knowledge of the Permanent Mathematical Studies, before they are admitted as competitors to those examinations for honours, in which an acquaintance with Progressive Mathematical Studies is required. If this condition were established and enforced, we cannot doubt but that it would prevent students from hurrying on, in their reading, to the widest generalizations and newest methods of analytical authors, leaving the elementary principles and their simpler applications very imperfectly possessed and understood; a line of reading which there is reason to believe is now not

uncommon.

186 Something of the same kind may be said of classical studies: but in these, the precipitate advance from the lowest to the highest usually takes place at an earlier period; often, as we have said, at school, before the student arrives at the University. The Examination by which this subversion of the due succession of the student's classical labours is to be prevented, ought therefore, in the state of things which here prevails, to take place at an early period of the student's University progress. Indeed, considering that the evil which it is sought to remedy, is one which prevails rather in the conduct of classical studies before the University career than during its progress, it would

seem that the purpose would be best answered by placing the Examination at the time when the student enters upon the University. An Examination taking place at that time, in which the power of construing Greek should be required, and a correct and familiar acquaintance with Latin, would secure to the Universities those conditions without which they cannot effect-ually discharge their office, namely, a higher Classical teaching than is given in Schools; and would in a short time produce a material improvement in those schools in which there exists such perversions of sound educational methods as I have spoken of.

187 Such an Examination, requiring a familiar acquaintance with Latin, and a competent knowledge of Greek, in every student who was admitted to reside in the University, might advantageously be combined with a thorough examination in Arithmetic; of which, for reasons above mentioned, a full and ready knowledge should be considered as requisite for every one entering upon his university career. If the students of our Universities began their College and University life with the attainments required in such an Examination as I have just described, (which are by no means beyond the reach of ordinary schoolboys,) and if, from the point thus secured, they were conducted onwards by a progressive scheme of College Lectures and Examinations to the final Examinations of the University, framed upon such principles as have been explained, I do not think it can be doubted that the Education of a great part of our students would be much more complete and satisfactory than it now is, without any impediment of any disadvantageous kind being thrown in the way of a love of knowledge and a love of honour such as now prevail.

I have spoken of a progressive scheme of College Lectures and College Examinations, by which men should be led to the final Examinations of the