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 bonâ 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 to succeed in the solution of problems. But great care must be taken not to introduce sophisms in this reverse 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.

138

ON MATHEMATICAL INSTRUCTION.

Augustis

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 taste 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 to Locke on the Understanding. Honourable exceptions are arising from day to day; and those who guide the

remainder will, if they are wise, look out in time, and see with what favourable eyes the world regards any well-regulated attempt to improve the system. Why are so many proprietary schools erected? The reason is, that parents, who have neither time to choose nor knowledge to guide them in the choice of a place of instruction for their children, find it easier to found a school, and make it good, than run the doubtful chance of placing their sons where they may learn nothing to any purpose. We in this article to make some propose remarks on the manner of teaching mathematics as it is, and as it ought to be.

A very erroneous idea prevails with regard to the object in view, in making mathematical studies a part of education. There are places in abundance where bookkeeping is the great end of arithmetic, land-surveying and navigation of geometry and trigonometry. In some, a higher notion is cultivated; and in mechanics, astronomy, &c., is placed the ultimate use of such studies. These are all of the highest utility; and were they the sole end of mathematical learning, this last would well deserve to stand high among the branches of knowledge which have advanced civilization; but were this all, it must descend from the rank it holds in education. It is no sufficient argument for the introduction of such pursuits that their practical applications are of the highest utility to the public, and profitable to those who adopt them as a profession. The same holds of law, physic, or architecture, which, nevertheless, find no place among the studies of the young. It is considered enough that the lawyer should commence his legal pursuits when his education in other respects is completed; and so would it be with him whose calling requires a knowledge of mathematics, were it not that an important