impress on the mind an image of the country; the questions already alluded to, which the learner must prepare himself to answer, will naturally lead him to assist his memory of names and of order by the image of the map, which readily presents itself to his mind; and this again, by being constantly referred to, becomes more distinct in all its parts. From this image he becomes gradually accustomed to read off in his mind as it is related of an actor, who, being obliged in rather a short time to prepare himself for acting a new part, succeeded in committing it to memory, by imagining before him the pages of the book from which he had learned it, and reading off his part in imagination while he was acting. Though this sort of local imagination and association is not very desirable for an actor, whose imagination ought to present him with living scenes instead of the leaves of a book, it is very useful for the geographer, when he wishes to gain that local knowledge of the different objects on the surface of the earth, to which he may refer knowledge of a less general nature regarding the formation of mountains, their mineralogical nature, the still more interesting phenomena of vegetation, &c., and at last those which relate to man himself, and to the physical existence and development of the human species. It may be remarked, that these subjects present themselves in common life to the observation of the uninstructed in a reverse order; but the collecting of facts, and the arranging and digesting of them, must follow an opposite course.

We have one more circumstance to mention before we leave this subject, to which is due no small part of the success with which the method in question has been attended. It will easily be perceived, that though a class be numerous, if every pupil be provided with paper, with the book containing the directions, and with the printed map, they can all go on independently of each other; the teacher can pay sufficient attention to each, though all may be doing something different. This renders practicable the application of a system of education to this branch of instruction which deserves particular attention, and which is generally introduced into the academy at Carlberg. It may be styled the system of free determination*, or voluntary labour,' according to which the pupil is allowed, to a very great extent, to propose to himself the

* 'System der freien Vorsetzung. It is one of the many excellent institutions for which the Kriegsakademie' in Carlberg is indebted to General Lefren and Professor Agren. The latter, after a long stay in Germany, has now returned to his own country, where he is engaged in extensive labours for the improvement of education. The publication of his numerous novel opinions may be looked for with interest.

APRIL-JULY, 1833.

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quantity he wishes to learn; he is not forced on with his class, but is left to do as much as he can by himself. It must be a desirable object in education, to give to the inward impulse of the mind for acquiring knowledge an opportunity of displaying and strengthening itself; but this natural impulse must be guided by reason and a sense of duty, if it is to bring its full advantages. A school in which forethought, duty, and natural impulse, are employed as motives to influence and regulate the conduct and exercises of the pupils, would in this respect be perfect. Examinations may teach the pupil forethought, by showing the advantages of diligence, and the disadvantages of idleness; discipline imposes duties; 'free determination' calls into action natural impulse. Emulation, produced by examinations and fostered by prizes, with strict observance of duty, enforced by discipline, is more common in English schools than free determination.' In this respect the example of the school at Carlberg might be followed.

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The principle of the constructive method' is not quite new in several Pestalozzian schools similar attempts have been made; but as it has never been so thoroughly considered as by Professor Agren, all the attempts have become more or less fruitless. The Pestalozzian principle-' to allow the pupil every possible opportunity of following his own natural impulse for acquiring knowledge, and particularly to treat him more like a rational agent than a passive receiver'—is the origin of the constructive method.

Among the many elementary works on geography which are constantly appearing in Germany, the Lehrbuch der Physischen Geographie,' by Professor Agren, has attracted most attention from those engaged in education. It has been recommended to the Board of Public Instruction in Prussia, by the celebrated geographer, C. Ritter, who assigns the first place to Professor Agren's method among all compendiums on elementary geography that have been published.'

If we are not mistaken, Agren's method will soon be introduced into all schools in Berlin, and from thence will extend itself into the schools of the kingdom of Prussia, and other German states.

ON THE METHOD OF TEACHING THE ELEMENTS OF GEOMETRY. PART I.

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 less 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. Attention 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 an 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

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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 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 billiardtable 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 when he begins an untried exercise of mind; he would study the several parts of a dissected map before he begins to put them together.

Our preliminary method would depend more upon palpable

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

objects than even in the preceding articles: for, whereas in arithmetic, the tangible instruments were only helps to the acquirement of a difficult abstract notion; in geometry, according to our preliminary system, they are the objects whose properties are to be studied.

The first thing to be done is, not to give the notions attached to the words point, line, straight line, surface, and plane surface, for they exist already, but to take care that the ideas are attached to the right words. About the term

point there is no difficulty; we need hardly warn the instructor not to use the definition of Euclid, but to proceed as follows. Instead of digging a pencil, or the end of a pair of compasses into the paper, and calling the visible surface so produced a point, let all the first points shown be made by drawing two intersecting lines slightly upon the paper with a hard and well-pointed pencil, using the hand only, and not the ruler or compasses. Having made several of these, the learner should be required to find the points in which they cut one another, by showing them with a fine needle. When he can do this, he should be allowed to try to draw a line through two or more points, either straight or curved, or composed of both species. A flat ruler should then be given to him, with which he should be shown how to draw a straight line. And here we must observe, that his notion of a straight line will probably be, one which is parallel to the upper and under edges of the paper. Thus he has been told that he cannot write straight without lines, and so on. This misconception must be corrected by drawing lines over the paper in all directions with the same ruler, and applying the term straight to them all. The learner must be made to understand, that the line which comes off the ruler is of the same form in whatever position the ruler may be held, and that a line which is straight in any one position is so in every other; that what he has been accustomed to call a straight line, means a straight line in the same direction as the top of the paper. As to defining a straight line on a plane surface, we think it had better be let alone, unless perhaps the definition ascribed to Plato, or one of his school, be called in as an illustration only, which is, that a straight line is that which can be so held before the eye, that nothing but a point shall be visible; and a plane surface, that which can be made to appear as a straight line in the same manner. But the plane surface may be illustrated by the definition of Euclid. Taking a plane, and also a piece of a cone, cylinder, sphere, or any other which may be at hand, the child may be shown that the edge of the straight