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citizens, therefore, of this renowned Republic neglect the study of Geometry." He concludes with observing, "We well know how much it conduces to the easy acquisition of any kind of knowledge, to have previously learned Geometry."

There are, perhaps, some persons who believe that mathematical studies tend to contract the mind, or to impart to it a distaste, if not an inaptitude for any but dry and abstract pursuits. But to such an objection it may be at once replied, that the studies of early age are calculated rather to develope the powers than to fix the habits of the understanding. It is certainly to be deprecated, that a peculiar and abstract study should engross all the attention of the mind, and exclude from it every other kind of cultivation. But the inconvenience of excessive ardour in such a pursuit is much less to be apprehended in boyhood than during the period of adolescence. In this respect we may compare the study of Mathematics to dancing, fencing, and gymnastic exercises. These exercises are calculated to expand the frame and invigorate the limbs of the growing youth; but if postponed till the period of manhood, they lose half of their efficacy, and are more liable to become a ruling passion and idle amusement.

It must be acknowledged, however, that young people have some difficulty in learning Geometry from the book, not because the reasoning contained in it transcends their mental powers, but because they cannot command the undeviating attention necessary for a study in which success is foiled by a single lapse or oversight. They cannot rein in their minds and proceed step by step with the cool circumspection required in mathematical deductions. They may, nevertheless, be easily trained to the study, and taught to overcome all difficulties by the following method, which will be found to be successful even with the dullest capacities. Let the young learner, instead of reading and committing to memory the proposition and its proof, repeat them, sentence by sentence, after the teacher, who must explain as he proceeds, the reasons of the method followed.

Attention will be thus kept alive, and the learner, taking the expressions from the teacher's mouth, and made to understand perfectly the connexion between the steps of the demonstration, must then repeat sentences in conjunction, until at last he can go through the whole proposition, less by effort of memory than of the reasoning faculty. When a proposition shall have been thus completely learned, let it be gone over again with another diagram, varying as the case may permit, from that previously used. By this means, the young student will perceive the general nature of the abstract reasonings in which he is engaged, and will become familiar with the fact, that the diagram in general represents but one out of innumerable cases. It must not, however, be overlooked, that the diagram, being a sensible image, is a great help to the memory; the capacity of the learner therefore must be consulted, before the diagram be exhibited in such a variety of forms, as to weaken its impression on the mind.**

The adult who desires to instruct himself in Geometry, but dreads the labour of the undertaking, may rest assured that although the ideas of difficulty and dryness are very generally associated with the study of Mathematics, yet there is scarcely any kind of knowledge more easily attainable if properly approached, or which grows more captivating on a little acquaintance. It is true that the following hundred and twenty pages cannot, with any profit, be all perused in a single day. That superficial rapidity, for which unhappily there is so much encouragement in the literary character of the present age, is here quite out of place, where a moment's inattention may sever the chain of thought and interrupt the coherence of ideas on which the intelligibleness of the whole depends. Mathematics are difficult only to those who have no control over their minds, and who will not take pains to habituate them to patience, clearness, and strict method.

* In order that the pupil, in demonstrating the propositions unassisted by the text, may have the benefit of the Diagrams, these have been printed in a separate and convenient form.

In the first place, let the learner, entering on the study of Geometry, pass at once over all metaphysical disquisitions respecting the nature of the simple ideas which are the subjects of his reasoning. He will find that the difficulty of inventing an unexceptionable definition of a point or of a straight line, will not in the least vitiate or obscure his reasoning respecting those abstractions. It will be sufficient if he takes care to discriminate and to understand clearly what are the data, or what is the hypothesis of the proposition before him; in other words, what are the things granted, or what are assumed—what are the indefeasible and defeasible conditions on which he reasons.

His next and his chief care must be not to go too fast. Nothing can be more easy or more evident than mathematical demonstrations taken step by step; but whoever attempts to advance two steps at a time, will be sure of finding how ill so steep and narrow an ascent admits of haste, and will be precipitated to the bottom. The complete conviction which ought to attend each separate deduction is liable to be lost only by hurry and inadvertence, and the moment that conviction fails, it is vain to go further: there is no longer a foundation for any inference. The mental habit thus formed of looking for certainty and of surely finding it by patient seeking, is of inestimable value; and, once acquired, is naturally and imperceptibly transferred from Geometry to other

studies.

The truths of Geometry once learned ought to be perfectly retained; and since there is necessity in each inference, there is no difficulty in retaining mathematical theorems of which the demonstration is once thoroughly understood. In like manner, the whole series of propositions in the Elements may be committed to memory with little effort, being fixed in the mind, not by the labour of learning them by rote, but by their natural and necessary coherence. Thus the study of Geometry is calculated to be the best possible exercise of rational

memory.

The demonstrations being perfectly understood and

retained, the learner should endeavour to dispense with diagrams and letters, and to express the proposition and its demonstration in general terms in the manner exemplified in Appendix (B), page 146. This indeed is at the commencement a task of considerable difficulty; but the exertion which it calls for at first, is amply repaid by its wonderful efficacy in strengthening the hold which the mind has of its mathematical acquisitions, and in giving increased facility in wielding them. Truths once clearly and accurately expressed in general language, are much more likely to be recalled to the mind and revived in meditation, than those the adequate expression of which depends in any degree on diagrams or other physical aids. The practice of demonstrating in general terms, conducts, it may be added, to habitual clearness and precision of language.

The learner, after having made himself master of the Propositions in Euclid's Elements, will do well to exercise his sagacity in the solution of new questions. A little perseverance will suffice to convince him that sagacity and the faculty of discovery, are, like our other faculties, capable of being immeasurably developed by judicious training. Among the Supplementary Propositions appended, he will find several theorems, which though not entering into the consecutive series essential to an elementary system of Geometry, are yet of much importance as giving an insight into the properties of figures, and add greatly to the resources of those who take pleasure in geometrical investigations.

In corroboration of the opinion expressed above, and from which many, perhaps, will be inclined to dissent, respecting the ease with which Geometry may be learned, if only care be taken to avoid hurry, we shall here close our remarks with a passage taken from the late Mr. WALKER'S edition of Euclid. That learned writer, who was an experienced teacher as well as able mathematician, expresses himself on that point in the following

terms:

"If in any part the Student finds himself unable to comprehend what he meets with, he may almost with

certainty conclude, that there is something in the previous part of which his comprehension has not been clear, or which is not perfectly retained. In fact, for one that fails in mathematical studies for want of intellectual abilities, there are hundreds that fail for want of mental patience. And, hazardous as the engagement might appear to some, my experience makes me not afraid to say, that I would undertake to make any one a mathematician, who should only combine patience of mind with the most moderate understanding, and would exert attention to the subject. The young student may also be assured, that in pursuing the study aright, however slow he may seem to proceed for a time, he will find difficulties rapidly lessen, and his progress become as smooth as it is sure."

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