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easy to frame a rule for conversion of ordinary expressions for money into their equivalent decimal expressions.
Thus £17. 165. 7 d.= £17.832, because 16s. =3 florins or £8; 6d. = £'025, and 7 farthings = £.037. .
In like manner £21'367 = £21 + 3 florins or £'3 +1 shilling or £.05 +17 farthings or £017, or in all £21. 75. 414.
Half an hour's practice in conversion and reconversion in this way renders the process familiar. All questions in which the given sum of money does not extend to lower fractions than 6d. can evidently be solved with perfect accuracy by decimals, and without encumbering the mind with the ordinary reduction at all. Nearly all questions in Interest and many in Practice and Proportion can be wrought much more expeditiously by this than by any other method. Precaution is needed in those questions only in which odd pence and farthings
occur and require to be multiplied. Visible These various applications of arithmetic have different relation to degrees of utility; but their value is not to be measured business no test of real by inquiring which of them is most likely to be practiutility. cally useful. The true aim in devising exercises in
practical arithmetic is to cultivate general power, fertility of resource, and quickness in dealing with numbers; the habit of seeing at once all round a new problem, of understanding its bearings, and applying the best rule for its solution. Power of this kind is available, not only in all businesses alike, but in the intellectual and practical life of those boys and girls who are not likely to go to business. And this general quickness and vcrsatility is just as well promoted, we must remember, by working problems which have an abstract look as by solving those in which the phraseology of the counter or the exchange is most ostentatiously used.
One other department of mathematics which has Practical found its way into schools, resembles Arithmetic in being Geometry. an Art and having useful practical applications, and also in furnishing disciplinal and purely intellectual exercise. Demonstrative Geometry has a value for this latter purpose, which, from the days of Plato and Archimedes, has been very generally recognised; but the claims of merely practical geometry as a useful part of both of primary and of secondary instruction appear to me to deserve more consideration than they generally receive. Every scholar should be taught to use the compass and ruler, and the quadrant and scale of equal parts. He should draw simple geometrical figures, as well as talk about them, and recognise their properties. He should know how to measure angles and lines, and to construct ordinary plane figures. In the best schools of Germany, France, and Switzerland, these simple things are taught to every scholar as matter of course. You may hear a teacher dictate to the class directions one by one as to the construction of a figure. “Draw a line 15 centimètres long, then another line upon it at an angle of 35 degrees, then another line of a given length to the right or left, &c., &c.” until the class produces one after another figures which he has pre-determined, and of which the qualities and dimensions are afterwards explained and discussed in the class. The rules for practical geometry are comparatively few and simple; the exercise is interesting, and is a considerable relief from graver employment. It serves to familiarize the scholar with the properties of circles, of triangles, or of parallelograms, and so to make the future scientific study of geometry more intelligible. And for those who may never learn Euclid or even the modern system of demonstrative geometry which seems destined to supersede it,
geometrical drawing will be found to have a value of its own in enabling scholars to judge better of heights and distances, and to know at least the chief properties of plane and solid figures.
Note on the form of Abacus. An ingenious modification of the Abacus, or ball-frame, in use in some of the French schools, possesses some advantages over the square Chinese frame with horizontal bars which is in common use in English schools. It is thus constructed:
A much greater variety of exercises in subtracting and combining numbers can be made by means of this instrument; and the upright lines may be made very useful in explaining the principle of our notation, and the necessity for keeping hundreds, tens and units in columns.
XI. ARITHMETIC AS A SCIENCE.
HAVING sought to lay down some rules by which a teacher may be guided in making the mere arts of computation and measurement effective parts of education, it becomes necessary to consider more fully the claims of Arithmetic as a science, and the reasons for assigning to it as a disciplinal study, even a higher rank than would be due to its practical usefulness.
We should all be agreed that the main purpose of our Science.. intellectual life is the acquirement of truth, and that one of the things we go to school is to learn how to acquire it. The mere accumulation of facts and information does not supply what we want. The difference between a wise man and one who is not wise consists less in the things he knows than in the way in which he knows them. We call arithmetic a science, and science, it may be said, means knowledge. But there is a good deal of knowledge which is not science. Science, properly so called, is organized knowledge, knowledge of things and facts and events in their true relation and co-ordination, their antecedents and consequences,—the recognition of every separate phenomenon in the shifting panorama of life as an illustration of some principle
or law, broader, higher, and more enduring than itself. No number of facts or aphorisms learned by heart makes a man a thinker, or does him much intellectual service. Every particular fact worth knowing is connected with some general truth, and it is in the tracing of the connexion and collocation of particular and separate truths with general and abiding truths that science mainly consists. We may see hereafter that an historical fact is learned to little purpose unless it is seen in its bearing on some political, economic, or moral law. And we have already seen that a grammatical rule has scant meaning or use for us until it is seen as part of the science of language. This distinction runs through all sound and fruitful acquirement, and should always be present in the mind of a teacher. We must learn to see special facts and bits of experience in the light of the larger generalizations by which the world is governed and held together. We have so to teach as to develop the searching and enquiring spirit, the love of truth, and the habit of accurate reasoning. And if Arithmetic can be so taught as to serve this purpose, it has a value which greatly transcends what seem to be its immediate objects, and will be found to affect not the notions about number only, but also those about every other
subject with which the understanding has to deal. Induction Here it seems right to take the opportunity of refertion ring to a distinction much insisted on in books on
education, and on which I have yet said little or nothing ; I mean the distinction between inductive and deductive modes of reasoning. In studying some subjects, the learner begins by acquiring separate facts, and as he goes on learns to group them, to see their resemblances, and to arrive at last at some larger statement of fact which embraces and comprehends them all. This pro