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tenanced by the looseness of expression of many elementary works. They are mostly founded upon the analogy existing between the algebraical expression a x a, or a', called a square, and the geometrical square described upon the line which contains a units. Against this confusion of terms the teacher must be on his guard, and should carefully avoid that symbolical notation recommended in some books, by which A Bo is made to stand for the square described on A B. If a short symbol for this be used it might be A Bl, and the necessity of proving that the number of 1 is contained in a is a x a, would not be concealed under a petitio principii.
With regard to the fifth book of the Elements, we recommend the teacher to substitute for it the common arithmetical notions of proportion. Admitting that this is not so exact as the method of Euclid, still, a less rigorous but intelligible process is better than a perfect method which cannot be understood by the great majority of learners. The sixth book would thus become perfectly intelligible.
It would much benefit the pupil if solid geometry were introduced at a more early period. There is nothing in the elementary propositions which requires more than the first book of Euclid ; and by a judicious use of the real figures, instead of perspective drawings, the subject might be amazingly simplified.
We come now to the subject of algebra, regretting that the limits of this article will not permit us to discuss the subject upon the scale which it deserves. The great drawback to the proper attainment of this science is the miserable previous instruction in arithmetic. When
this defect is remedied, and not till then, can we expect any better results. It is the practice not to let the pupil proceed to the principles of equations until he can work questions in all the previous rules of a nature which very rarely occur in practice. And of these rules themselves it must be observed that, in order to preserve analogies with arithmetic, their meaning is usually distorted. To the unintelligible way in which the negative sign is used we have already alluded. We shall now explain our views as to the manner of proceeding.
The new symbols of algebra should not be all explained to the student at once. He should be led from the full to the abridged notation in the same manner as those were who first adopted the latter. For example, at this period he should use aa, aaa, &c., and not a", a', and should continue to do this until there is no fear of that confusion of 2 a and a”, 3 a and a', &c., which perpetually occurs. Whenever any new symbol is introduced, not a step should be made until it has been rendered familiar by finding its arithmetical value in particular cases. This indeed is the first exercise; algebraical expressions increasing in complexity are given, and also certain values for the letters, and the student is left to find the corresponding arithmetical value of the expression. Whenever a negative result occurs it should be thrown aside as an impossibility, the pupil being told at the same time that use will be afterwards made of such expressions when he can understand what they mean in the solution of a problem. The leading principles of the solution of equations of the first degree might then be easily established, and applied to some numerical equations. The four rules should follow,
the principles being previously explained, and all negative results avoided. The student is then in possession of the means of solving an equation of the first degree in which some of the given quantities are literal, and may be supplied with examples a little more likely to aid his future studies than the conundrums about posts and saddles which we have instanced.
At this stage of his progress the pupil should be set to work a problem in which a negative result occurs. It should then be pointed out to him that there is a misconception of the problem itself, and the manner of rectifying that error will show, in the course of several examples, what is the meaning of the negative answer. At the same time it will be easy to explain by examples the nature of the wrong suppositions which lead to results of the form 5. He should then examine for himself what change is produced in a process which sets out with some assumption as to b-- a, when this has been incorrectly written for a — b. By comparing the true and false processes he will deduce the rules according to which negative quantities must be treated, in order that their introduction may not affect the soundness of the conclusion. He is thus placed in the same condition as to results with the pupil who has pursued the common method; with this difference however, that he can explain conclusions which the other cannot, and has never believed that, a - b meaning a diminished by b, there can be such a thing as — a, or a quantity less than nothing.
The view which is generally taken of expressions of the first and second degree is too confined for the future purposes of the mathematical student. It is this : what
values of x will make the expressions a x- b, arb x + c, &c., equal to nothing : whereas, it is necessary to inquire what values of x make these expressions positive, negative, or nothing. All that is learnt appears to have no higher view than enabling the student lo solve the pretty problems which we have mentioned, and not to simplify the higher parts of the science. This is too much the fault of the education of our schools in general. It is not recollected that they cannot expect to make learned inen ; but they may make good learners, and at the same time produce such a desire for knowledge as shall lead the individual to devote himself to study, where it is not matter of compulsion, as in the Universities, and still more amid the occupations of life. The great mistake lies in a notion that they are to teach the greatest possible number of bare facts before the pupil arrives at the age of sixteen ; whether he will leave school with the desire of adding one more bit of knowledge to his stock, or with the power to do so if he has the will, does not seem to be considered of any importance. Again we call upon all who still adhere to the old system to reflect a little on their own interest. The number of new methods of teaching proposed every day shows the existence of a general feeling that some change is requisite. The Universities, which have made great advances within the last twenty years, may be proposed to the schools as an example for their imitation. And let them recollect, that, the demand existing, the question is not whether they will supply what is asked for or something else, but whether the public must come to them or go elsewhere.
ON THE STUDY OF GEOGRAPHY.
By G, LONG.
(From the Quarterly Journal of Education, No. XIII.)
A COMMERCIAL country, with numerous and extensive foreign possessions,—a country whose soldiers and ships are found on almost every coast, and whose travellers visit every country, would seem peculiarly adapted to be the centre of geographical knowledge. That Great Britain has made, and is daily making, very large additions to our knowledge of the earth's surface, is a fact which will be generally admitted ; and that hitherto all these accumulated facts have been turned to very little account in systematizing our knowledge, is another fact which is equally indisputable. The nation that has now for several centuries made discovery, colonization, and foreign conquest, whenever opportunity offered, part of its political system, had not, three years ago, even a geographical society, and at present there is not, we believe, a single public teacher of geography in the universities and colleges of Great Britain, with the exception of the professor lately appointed in the London University *. The London Geographical Society now forms a point of union for those who are interested in the knowledge of the earth's surface, and by its Journal it invites and offers facilities to the publication of many valuable contributions, which otherwise would never appear. The formation of a library and a collection of maps, which also are part of the Society's plan, together with the
* Geography has been publicly taught at the London Mechanics’ Institute for some time.