Involution and Evolution.

309

~and led them to recognize it under the general abstract for mula:

If a = b+c then a2 = b2 + c2+2bc.

You are now in a position to deal with the problem origin

ally proposed: Find the square root of 676.

Evolution.

But if so, the remainder 276 must contain not only the square of that other number, but twice the product of the number 20 and that number. With a view to find that number, try how many times twice 20 are contained in the remainder. The number 6 appears to fulfil this condition. See now, if 276 contains six times 40, together with 6 times 6, or 6 times 46 in all. If so, 6 is the unit figure of the required root. It has now been shown that 676 contains the square of 20, and the square of 6, and twice the product of 20 and 6,

The who.e explanation of this inverse process is evidently deducible from the simple law of involution first described. The reason of the pupil follows every step, and acquiesces in a rule, otherwise prima facie absurd, and therefore hard to remember. All this is of course very familiar and simple to the student of algebra; but I have never been able to understand why it should be postponed to algebra, or why the principles of arithmetic, requiring as they do for their elucidation no use

of symbols, no recondite language, nothing but simple numerical processes, should not be taught on their own merits, and in their own proper place.

Analogous truths in Arithmetic

An appeal to the eye will greatly help the understanding of the rule for the extraction of the square root. A square may be erected on a line divided into two and Geome- unequal parts, and it will be seen to be separable try. into four spaces whose dimensions correspond to the products just given. Afterwards a square on a line divided into three or more parts may be shown, and the dimensions of the several parts may be expressed in numbers. In like manner every proposition in the Second Book of Euclid may be compared with some analogous proposition respecting the powers and products of numbers. But it is important here not to mistake analogy for identity. Some teachers seem to think they have proved the theorems in geometry when they have expressed the corresponding truths in algebraic symbols. The use of the word " Square," both for a four-sided figure and for the second power of a number, is a little misleading; and Euclid's use of the terms Plane and Solid numbers in his Seventh Book would have further mystified students had it been commonly accepted. But since Geometry is founded entirely on the recognition of the properties of space, and Algebra and Arithmetic on those of number, it is necessary to preserve a clear distinction in the reasoning applicable to the two subjects. Except as showing interesting analogies, the two departments of science should be kept wholly separate; and while the truths about the powers and products of numbers should be investigated by the laws of number alone, geometrical demonstrations should be founded rigorously on axioms relating to space, and should not be confused by the use of algebraic symbols.

Our attention to-day has been necessarily confined to the conAlgebra and sideration of a rational way of treating ArithGeometry. metic, the one department of mathematics with which, in a school, the teacher is first confronted. But the same general design should be in the mind of the teacher,

True Purpose of Mathematical Teaching. 311

through Geometry, Algebra, Trigonometry, the Calculus, and all the later stages of mathematical teaching. While constantly testing the success of his pupils by requiring problems to be worked, he will nevertheless feel that the solution of problems is not the main object of this part of his school discipline, but rather the insight into the meaning of processes, and the training in logic. If Algebra and Geometry do not make the student a clearer and more accurate and more consecutive thinker, they are worth nothing. And in the new revolt against the long supremacy of Euclid, as represented in the Syllabus of the " Association for the Improvement of Geometrical Teaching,” the one danger we have to fear is that the demonstrative exercises will be cut up into portions too small to give the needful training in continuity of thought. 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.

office of

We return finally to the fundamental reason for teaching mathematics at all either to boys or men. Is it The true because the doctrines of number and of magni- mathematitude are in themselves so valuable, or stand in any cal teaching. 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 connection 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 premise 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 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 discussing 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 in deed

"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."

Object to be Kept in View.

313

XII. GEOGRAPHY AND THE LEARNING OF FACTS.

In considering the subject of Geography we shall do well to repeat our former question-Why teach it at all? Object to be What purposes do we hope to serve in including kept in view. 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, 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

It is mainly useful as

information.