The true office of mathematical teaching.

Euclid, with all his faults, obliges the learner to keep his mind fixed not only on the separate truths, but also on the links by which a long succession of such truths are held together. It is well to simplify the science of geometry, and to arrange-as the authors of the Syllabus have done-its various theorems in a truer order. But since it is not geometry, but the mental exercise required in understanding geometry, which the student most wants to acquire, a system of teaching which challenged less of fixed attention and substituted shorter processes for long would possibly prove rather a loss than a gain.

We return finally to the fundamental reason for teaching mathematics at all either to boys or men. Is it because the doctrines of number and of magnitude are in themselves so valuable, or stand in any visible relation to the subjects with which we have to deal most in after life? Assuredly not. But it is because a certain kind of mental exercise, of unquestioned service in connexion. with all conceivable subjects of thought, is best to be had in the domain of mathematics. Because in that high and serene region there is no party spirit, no personal controversy, no compromise, no balancing of probabilities, no painful misgiving lest what seems true to-day may prove to be false to-morrow. Here, at least, the student moves from step to step, from premiss to inference, from the known to the hitherto unknown, from antecedent to consequent, with a firm and assured tread; knowing well that he is in the presence of the highest certitude of which the human intelligence is capable, and that these are the methods by which approximate certitude is attainable in other departments of knowledge. No doubt your mere mathematician, if there be such a person, he who expects to find all the truth in the world formulated and demonstrable in the same way as

True purpose of Mathematical teaching. 343

the truths of mathematics, is a poor creature, or to say the least a very incomplete scholar. But he who has received no mathematical training, who has never had that side of his mind trained which deals with necessary truth, and with the rigorous, pitiless logic by which conclusions about circles and angles and numbers are arrived at is more incomplete still; he is like one who lacks a sense: for him "wisdom at one entrance" is "quite shut out;" he is destitute of one of the chief instruments by which knowledge is attained.

Nor is it enough to regard mathematical science only in its far-reaching applications to such other subjects as astronomy and physics, or even in its indirect efficacy in strengthening the faculty of ratiocination in him who studies it. There is something surely in the beauty of the truths themselves. We are the richer-even though we look at them for their own sakes merely-for discerning the subtle harmonies and affinities of number and of magnitude, and the wonderful way in which out of a few simple postulates and germinating truths the mind of man can gradually unfold a whole system of new and beautiful theorems, expanding into infinite and unexpected uses and applications. And as we look on them we are fain to say as the brother in Comus said of a kind of philosophy which was novel to him, and which perhaps he had hitherto despised, that it is indeed

"Not harsh or crabbed as dull fools suppose,

But musical as is Apollo's lute,

And a perpetual feast of nectar'd sweets

Where no crude surfeit reigns."

XII. GEOGRAPHY AND THE LEARNING
OF FACTS.

Object to be kept in

view.

It is

mainly

useful as information.

IN considering the subject of Geography we shall do well to repeat our former question-Why teach it at all? What purposes do we hope to serve in including it in our course? We have seen in reference to the teaching of languages and of mathematics, that although there were two distinct purposes to be kept in view,—the practical and useful application of those studies on the one hand, and the indirect mental discipline afforded by them on the other, in both cases the second object was more important than the first. Here, however, it is not so. Our main object in teaching Geography is to have certain facts known, because those facts, however learned, have a value of their own. We live in a beautiful and interesting world; one marvellously fitted to supply our wants and to provide us with enjoyment; and it seems fitting, if we would be worthy denizens of such a home, that we should know something about it, what it looks like, how big it is, what resources it contains, and what sort of lives are lived in it. To know these things is the first thing contemplated in teaching Geography. If there be mental exercise, and good training in the art of thinking and observing to be got out of these studies,

Purpose of Teaching Geography.

345

they are the secondary not the primary objects which we want to attain. Yet even here in the one department of teaching in which mere information, as distinguished from scientific method or intellectual training, is relatively of the most importance, there are as in other subjects right ways and wrong, intelligent and unintelligent methods. The incidental and indirect effect of teaching on the formation of mental habits is not to be disregarded; and though much of the result we hope to Yet partly gain belongs to the region of the memory only, we shall as discipline. be all the better for enquiring whether there is not also room here for appeal to the judgment and to the imagination; whether in short, Geography may not be a really educational instrument, as well as a mass of facts which have to be mastered and committed to memory.

considered

teach.

It is the more important to think thus about Geo- Geography graphy; because I have observed that this is the favour- generally ite subject often with the worst and most mechanical easy to teachers. It is in fact the one subject in which the maximum of visible result may be attained with the minimum of intellectual effort. To give a few names of places and point them out on the map, is the easiest of all lessons, and, what is more to the purpose, it makes a great show when it is learned. And when I ask a teacher what is the favourite subject of pursuit in his school and he answers Geography, and afterwards I find that what is called Geography merely means the knowledge of a number of names, and the power to identify their position on the map, I always draw a very unfavourable inference respecting the character of that school as a place of intellectual training; for I know that such information may have been imparted without the least exertion of educating power on the master's part; and that a good deal of such knowledge may easily

How to arrive at right methods.

co-exist, in the learner's mind, with complete mental inaction and barrenness.

Nevertheless, it would of course be wrong to undervalue the subject, (1) because, if rightly taught, it may be very stimulating and helpful to mental development, and (2) because it is better to have it taught wrongly than not taught at all. For even information as to the position of places on the globe is useful to everybody; useful especially to Englishmen, who are fortunate enough to be 'citizens of no mean city' and to belong to a race which dominates a larger portion of the earth's surface, and has more varied and interesting relations with distant parts of this planet, than any other people in ancient or modern times.

Now in considering how we should teach Geography, we may usefully fall back on a principle we have had before-that we should begin with what is known and what is near, and let our knowledge radiate from that as a centre until it comprehends that which is larger and more remote. This principle is specially applicable to the present subject. You want of course to give right general notions of the surface and configuration of the earth, and of the meaning and use of a map. The best way to begin is to draw a little ground-plan of the schoolroom; and put into it one after another, as the children watch you and make their suggestions, the desks, the tables, and other articles. Train them to observe you as you draw, and to correct you if you put a door into the wrong place, or make the line which represents a desk of disproportionate length. Then try a map of the surroundings of the school-room, its playground, the street in which it stands, the principal roads near, and put in one after another the church, the railway station, a river, a bridge, and other familiar objects, at the same time inviting