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time; but as it is, we can only recommend instructors either to reject this book, and substitute the numerical definition of proportion, or, if they retain the book, to make the numerical definition accompany it. Many will not agree with us in the first recommendation, and we feel, as much as they do, the hiatus which would be made in the system by attending to it. But surely it is no compensation to the pupil for an employment of time which brings in no knowledge, (which is his case if he read the fifth book without understanding it,) that his instructor can appreciate the superior completeness and rigour with which all the demonstrations of the sixth book might be given, if the student could only comprehend that which he does not comprehend, namely, the fifth book. We would say to all, teach the fifth book, if you can; but we would have all remember that there is an if.
We now come to solid geometry, which we may observe is seldom or never taught before plane trigonometry: that is, a purely conventional arrangement has placed a very easy part of the subject after one of much greater difficulty; so that, in fact, access to the easier part is practically forbidden to all who do not first master the harder. The propositions contained in the first and second books are sufficient for the establishment of as much of the eleventh as is necessary for the purposes of spherical trigonometry, that is, of the first elements of astronomy; and the same pains which are taken with the fourth book, which is of very little use, would, if applied to the eleventh, most materially increase the power of the student to comprehend popular works on physics.
The main difficulty is one which is not in the subject, but in the manner of treating it, namely, the substitution of drawings upon paper instead of the solid objects which are considered. Yet a few pieces of card, or even of the very paper on which the student looks with despair at right angles which are acute, and lengths the relative magnitudes of which have changed places, would be sufficient for the formation of bona fide prototypes of these perspective anomalies. We should like to know to how many mathematical teachers per cent, it has occurred, instead of drawing one plane inclined to another on a paper, to fold the paper itself, and place the two folds at the required angle? Would it give too much trouble? Does the pupil say his proposition as well without it?
The eleventh book of Euclid may, in our opinion, be abandoned with advantage in favour of more modern works on solid geometry, particularly that of Legendre, which the English reader will find in Sir David Brewster's Translation. If Euclid be adhered to, the first twenty-three propositions of the eleventh book are sufficient for common purposes. We need hardly repeat, that the ocular demonstrations should be made to precede all others, which cannot, of course, be done without lines which are really in different planes.
So long as geometry is made a mere exercise of memory, it is idle to expect that the pupil should make any step for himself in the solution of a problem which is not in the book; and as there are no rules by which such a thing can be done, we find accordingly that this is an exercise almost unknown in the geometrical classes of schools. But supposing the pupil to be taught on a rational system, there is nothing to prevent his being tried with easy deductions from time to time, except the difficulty of procuring the problems in cases where the teacher cannot invent them. The works of Messrs. Bland and Creswell, published at Cambridge, bearing the titles of “ Geometrical Problems,” and “Deductions from Euclid,” contain problems perhaps of too difficult a cast; but from them a judicious teacher might select some which would suit the capacity of his pupils. Previous to this, the pupils should have been accustomed to retrace the steps of the several propositions of Euclid from the end to the beginning, whenever this inversion will not affect the reasoning. This will accustom them to the analytical method, by which alone they can hope 10 succeed in the solution of problems. But great care must be taken not to introduce sophisms in this rererse process. For example, in the forty-seventh proposition of the first book, it is shown that the square on each side of the right-angled triangles is equal to one of the rectangles into which the square on the hypotenuse is divided. The concluding argument is therefore of this form: A is equal to B, and C is equal to D; therefore, the sum of A and C is equal to the sum of B and D. Assuming the result, that A and C together are equal to B and D together, it cannot, therefore, be assumed that A is equal to B, and C to D, but only that if A be equal to B, C is equal to D. Of all the exercises which we have proposed, this is the one which requires most care on the part of the instructor.
Our readers will see that we have throughout advocated the union of the forms of logic with the reasoning of geometry. We are convinced that it would be advantageous to make the former science systematically a part of education. If we except Oxford, there is no place in this country where it is still retained ; and unfortunately
for the study, it is there more an act of memory about things called moods and figures, than an exercise of reasoning. What we have proposed would tend to improve the indefinite straggling form in which the reasoning of Euclid is presented to the young, and would provide a safeguard against the many misconceptions to which it gives birth. We have said nothing of the other advantages of logic, as they have no relation to the subject of this article.
ON MATHEMATICAL INSTRUCTION.
By A. DE MORGAN.
(From the Quarterly Journal of Education, No. II.)
It is matter of general remark that mathematical studies do not yield that pleasure to the young which the more intelligent and well inclined among them derive from every other part of their education. If the opinions of a number of youths could be collected, at the period when their education is just completed, it would be found that, while nearly all profess to have derived pleasure from their classical pursuits, the very name of mathematics is an emblem of drudgery and annoyance. In saying this we are not speaking of the Universities, in which the choice of studies is so far left to the laste of each individual, that no one can have those feelings against any particular study which arise from the remembrance of its having been forced upon them. Our remarks apply to the hundreds of schools with which the country is studded, where, in fact, the great majority of the educated portion of the community receive the knowledge which entitles them to be thus styled in most of which something is taught under the name of mathematics, bearing much the same likeness to an exercise of reason that a table of logarithms does 10 Locke on the Understanding. Honourable exceptions are arising from day to day; and those who guide the