prepare them by a good fundamental instruction for a progress in the higher subjects, when the university course brings them to that stage. We propose in this paper to give the substance of Dr. Whewell's observations under this head as applied to the study of mathematics. 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 liabit of performing the common operations of arithmetic, and of applying the rules in a correct and intelligent manner. Now arithmetic is a matter of habit, and can be learnt only by long-continued 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. We may say, generally, that boyhood is fitted for the forniation of practical habits, and that the aptitude to attend to general reasonings comes with more advanced youth. Hence it follows, that in the most natural course of 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 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. Demonstrations regarded with this interest, are a very effective means of unfolding the reasoning powers. With a view to the encouragement of such mental processes as these, Dr. Whewell considers that the mathematical education of boys at school might be extended to practical methods, much further than is commonly done, at least at classical schools; and, that it would be a great improvement, if boys were not only made to learn arithmetic, but also mensurationthat is, 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, and will usually take pleasure in learning and applying it, the experience of many of our commercial and other schools abundantly shows. Other practical matters in mathematics 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 practical processes to the schoolboy, because, 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. And 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 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. Such is the mathematical course which Dr. Whewell recommends to be taught at school, as a preparation for the mathematics which are to be studied at the university. He considers that mathematics cannot be studied to any purpose at the university, except an effectual beginning is made at school. This, he says, is true, even of speculative portions of mathematics, such as geometry; but 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. And he adds, “ 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,” -he proceeds to observe, (and we must remember, that his observations are the result of no ordinary experience,)“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 fundholders, 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 bearing, 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.” And in another place he remarks, that not only do young men highly intelligent and well instructed in the classics, come to the university, most of them knowing little of the mathematics, many nothing ; but such persons have often acquired a repugnance to mathematics, from the aspect under which they have seen the subject at school; and would in many cases, study it with more advantage if they were to begin it afresh, when they arrive at the university, provided time enough could be given to it. Whilst pointing out the unsuccessful manner in which mathematical studies have hitherto been conducted at our great schools, Dr. Whewell bears honourable testimony to them in other respects, which must not be omitted, because it affords the best grounds for believing that this particular defect will be remedied, as soon as a suitable remedy can be devised. “I have,” he says, " the highest opinion of many of the excellent men who are at the head of our several great schools; and so far as I am acquainted with the present state of those institutions, I admire greatly both the improvements of system and the elevation of their general spirit and character, which have taken place in many of them of late years. But still, they may be regarded as exclusively classical schools. No other subject than classical studies, attains comparatively, any hold upon the minds of the scholars. In particular, ihe mathematics are not, at most of these schools, taught generally and effectually as part of the business of the school. Whatever means have hitherto been employed in teaching arithmetic in the great schools, it is plain that they are insufficient with regard to the majority of the scholars; and that new and better means should be devised and introduced.” Dr. Whewell does not profess to be sufficiently well acquainted with the system of great schools to be able to suggest how this may best be done; but he conceives that it must be done by making arithmetic a part of the system of the school; not a mere appendage to the ordinary work of the school, enjoined it may be and provided for, but not looked upon with any regard and respect by those who govern the school and direct the minds of the boys. Arithmetic, and, when that has been mastered, geometry, mensuration, algebra, and trigonometry, in succession, should, he says, form a part of the daily business of every school which is intended to prepare students for the university; and he especially urges that arithmetic and a portion of Euclid's geometry should be taught to all the scholars in our great classical schools -taught so as to be familiarly used and permanently possessed, and, with that view, made a part of the daily system of those schools. He observes, that a great part of the vice of the mode in which such branches of learning are now taught at these schools is, that they are taught, not as valuable for their own sakes, but as means of passing examinations in the universities. Hence boys are not taught things the most fit for boys, and in the manner most fit (as the practical teaching of arithmetic is); but they are taught, as much as possible, in the manner most resembling the teaching of the university. And undoubtedly the teachers, looking only to the boys' university career in what they teach, think this a great improvement on the system of teaching matbematical subjects in these schools ; but this notion Dr. Whewell condemns, as altogether a mistake. Boys, he says, should be taught arithmetic and geometry, and, it may be, algebra and trigonometry, in the great classical schools, in the same way in which they are in the best commercial schools : at any rate, in some way in which the knowledge, and not the passing of examinations, is regarded as the valuable result. In concluding this abstract of Dr. Whewell's remarks upon school instruction in mathematics as preparatory to a university course, we would briefly remind our readers of the great importance which he assigns to early instruction in arithmetic, and to a knowledge of arithmetic, as the foundation of all real progress in mathematics. His observation that arithmetic is of a practical nature, and therefore peculiarly suited to the tastes of boys, is well worthy of notice, as also that it is a matter of habit, and can be learned only by long-continued practice. Hence the necessity of a daily appropriation of time during some years of boyhood to this object. Though we quite agree with Dr. W. in thinking that there should be as nearly as possible a daily appropriation of time to this employment, yet we believe that the period so set apart, if well employed, need not be long. How such period may be most effectively employed so as to afford daily improvement to all, appears to be the main problem, upon solving which depends the success of any remedy for that neglect of arithmetic in our great schools of which Dr. W. complains. For its solution we should be disposed to refer the heads of those institutions to the practical skill and experience of those who have of late been so successful in teaching the elements of arithmetic and mathematics in our best national and training schools. The first part of arithmetic is very much a matter of mere drill, and readiness in such necessary processes as multiplication and division, and familiarity with the tables and with the elements of fractions, are most easily and perfectly acquired, in the first instance, by repetition and by being questioned in classes, and by working out sums aloud with the help of the black-board, as in our national schools. Thus boys are made to understand the rules of arithmetic, and how to apply them to particular questions, with least sacrifice of time by the master, and with most ease to themselves. Then, when sums are set them to be done separately by themselves, they have some chance of making them out; and, if they cannot succeed at first, the master quickly discovers it, and when he next uses the black board, he takes occasion to address himself to the correction of such mistakes, or to the removal of such difficulties as may have hitherto prevented boys from working out the sums by themselves. Now, the junior boys at our public schools might thus be taught this elementary knowledge by masters who have been thoroughly habituated to the most approved modes of teaching arithmetic at the best training schools. Such masters might act as non-commissioned officers in our army, and simply go through the processes of teaching, in the presence and under the eye of the higher mathematical tutor, who would be responsible for the discipline and general control of the boys. This would prevent the evil which often arose from the old practice of handing over boys to the writing master, who generally gave instruction also in figures, but who was unable, from his inferior position, to maintain that order, which is absolutely essential in any exercise where strict and undivided attention is indispensable, as in arithmetic, if any real progress is to be made. It is plain that the mere presence of the regular mathematical tutor, when subordinate teachers are employed in going through the elementary and more mechanical processes, would effectually remedy any evil of this kind; whilst it would at once enable our great schools to partake of the benefits to be derived from the recent undoubted improvements upon the old method of communicating an elementary knowledge of arithmetic. When the boys had thoroughly mastered these first steps, and acquired some facility in performing the common operations of arithmetic, and in applying the rules in a correct and intelligent man would be not only easy but interesting, and we should hear no more of the complaint in which Dr. Whewell is tempted to indulge, namely, that “now boys in general are more slow in understanding any portion of mathematics than they were thirty years ner, further progress ago.” Correspondence. BIBLE LESSON. a Rev. SIR,In a former Number of your you inserted notes of a Bible lesson, and promised that a series of them would follow; but, as they have not come, I send you the following lesson, hoping that it may påve the way Your obedient Servant, THOMAS RANKINE. Sr. Mark i. 40. Christ cleanseth the Leper. 1. The meaning of the principal words. 2. The chief points to be illustrated. 3. The application, or the practical lessons. 1. Leper-beseeching-cleansed--testimony-publish. 2. The leper— leprosy—its character, small beginning—sometimes inflicted as a punishment (Examples, Miriam and Gehazi)-the posture of the leper-his prayer—Jesus toucheth him—His compassion for him-his cure-to show himself to the priest—why (Lev. xiv.)leprosy the type of sin. 3. (a.) The one cure for leprosy—but one for the disease of our soul. (b.) Humility, in approaching, due to GOD_kneeling in His presence. (c.) Encouragement from the compassion of our blessed Lord. (d.) Faith from the poor leper. |