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is very often done, at the bottom. Thus 6 should be written 68, and not 6.8, or 6,8, the second being reserved to signify six times eight. This will save some confusion when the student comes to the subject of Algebra.

As soon as the meaning of the decimal point has been established, the first exercise should be upon the effect of removing it any number of places to the right or left. Thus the student should write such series as :00681, 0681, '681, 6·81, &c., both in the decimal and common notation, and should be required readily to assign the multiplication or division, which will reduce any one of this series to any other. This is not much attended to in ordinary works on the subject. In demonstrating the rules, the student should reduce the decimal notation to the common fractional one, which removes all difficulty, if he understand the methods for ordinary fractions. The only rule which will cause any embarrassment is that for division. This must be preceded by the rule for turning any common fraction into a decimal, approximately. We would recommend that all rules relative to circulating decimals, as they are called, should be entirely omitted. They are of no use in practice, and the theory cannot be understood previously to that of geometric series in general. The reduction of common into decimal fractions may be thus simply taught. It is wished to ascertain how many thousandths of the unit is contained in the seventh. The division of by Too, with which the student is already familiar, gives or 1424, or one-seventh contains 142 thousandths and six-sevenths of a thousandth part, or 143 thousandths nearly, that is 143. The rule should not be applied until the student has worked a large num

ber of examples in this way. The division of one decimal by another amounts to the reduction of a given fraction to a decimal, as may be shown by the common method. Thus 014 divided by 6.81 amounts to the reduction of to a decimal fraction. This method should supersede the common one for some time, until the student can nearly guess in what part of the quotient the decimal point will be found.

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As our object is rather to point out the way in which the first principles may be taught, than to give a detailed course of instruction, we shall not enter further upon the subject of arithmetic. We must observe that the want of a familiar acquaintance with common and decimal fractions is the source of nine out of ten of the difficulties which are commonly found in the study of algebra. Still more is this defect the reason why so few are perfectly at their ease in the processes of commercial arithmetic. It seems to be a tacit agreement between elementary writers on this last subject and their readers, that no knowledge of decimal fractions shall be assumed to be possessed by the latter. Thus it is that many cumbrous processes are introduced, involving pounds, shillings, pence, and farthings, which, if the reader had only the most moderate knowledge of decimal fractions, might be greatly simplified. This will never be remedied, until rules are not only learnt, but understood.

We shall proceed, in our next Article, to the elementary study of Geometry.

See the "Penny Magazine, vol. ii. No. 52." tions of Arithmetic, No. 1."

"Simplifica

ON THE METHOD OF TEACHING THE
ELEMENTS OF GEOMETRY.

BY A. DE MORGAN.

(From the Quarterly Journal of Education, No, XI.)

THE Science of geometry holds, in some respects, the middle rank between arithmetic, in its widest sense, and natural philosophy, or physics. It consists in the discovery and establishment of the properties of space, or of matter, considered only as that which occupies space. As a part of education, it has always been selected as the medium in which the young might be trained to strict and formal reasoning; and though this is the ground on which it is most defensible as a study, the actual knowledge gained by it is not therefore of small importance.

In our preceding articles, on the Teaching of Arithmetic, we could reasonably suppose that the subject was capable of being so treated, that any parent might enable himself to instruct his own children. Here this is not the case; it would require a treatise to develop our method, so that a grown person, ignorant of geometry, might undertake the task of teaching by it. We suppose a knowledge of at least six books of Euclid, and shall, therefore, content ourselves with merely indicating many things, as perfectly well known to the reader.

We shall consider our present subject under two heads, the first relating to the manner of teaching the terms and the facts of geometry, the second to the method of deducing them from one another by reasoning.

It has not been usual to make this division. Atten

tion to what is called the rigorous geometrical method has generated an aversion to communicating the truths of geometry in any other form than that in which they have been delivered by Euclid, so that those who have neither time nor capacity to study the strictest books, have always been left without any accurate knowledge of some of the most essential properties of matter, viz., those involved in its form or shape. This has not been the case with the mechanical properties; here we have popular works in abundance, which do not refuse to exhibit the phenomena of a screw, because the reader cannot connect it geometrically with the inclined plane, or to talk of the various laws of mechanics, because that universal recipient, the principle of virtual velocities, is above the capacity of a beginner. We would not, however, be understood to depreciate the reasoning or to deny its utility in the smallest degree; we only say, that one who is never likely to reason upon them is better off with a knowledge of the facts than with nothing at all, and that with children a preparatory course of experimental geometry is the best introduction to the severer study.

An editor of Euclid, in the last century*, who deserved credit for a careful edition of the whole of the Elements, in criticising Clairaut for his avowed departure from the strictness of the Grecian model, makes the following remarks, which though not without their force, when directed against experimental geometry as an ultimate course of study, lose their ironical character and become serious earnest, when applied to the same as a preparatory method.

*Elements of Euclid, with Dissertations, &c., by James Williamson, M.A., Fellow of Hertford College, Oxford. Clarendon Press, 1781.

"Elements of geometry carefully weeded of every proposition tending to demonstrate another; all lying so handy that you may pick and choose without ceremony. This is useful in fortification; you cannot play at billiards without this. You only look through a telescope like a Hottentot until this proposition is read, with many such powerful strokes of rhetoric to the same purpose. And upon such terms, and with such inducements, who would not be a mathematician? Who would go to work with all that apparatus which I have described as necessary for understanding Euclid, when he has only to take a pleasant walk with Clairaut upon the flowery banks of some delightful river, and there see, with his own eyes, that he must learn to draw a perpendicular before he can tell how broad it is?" &c.

Let the faults of this style be upon their author; he expresses to the letter what we should wish to do with children, not instead of, but previous to, anything else. If the facts were well selected, leaving out those which are only useful in demonstrating others, and not conspicuous for themselves alone; if their truth were made manifest by measurement, and their utility by application, whether to a billiard-table or a bastion, a telescope or the measurement of an inaccessible distance; we may ask in the terms of our quotation, "with such inducements, who would not be a mathematician?" that is, what child of moderate powers would not be interested in the announcement, that his separate truths are parts of one chain, and that it may be shown that one follows from another; and who would not desire to follow this chain and acquire a new faculty? The consequence of such a previous discipline would be, that the student would not have to learn a new language at the moment

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