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The student of ripe years
the deductions of his reason. is always disposed to appeal to experience; and having proved two lines to be equal, he measures them, in order to confirm the demonstration. The celebrated PASCAL, when a boy, being debarred the use of books of a scientific character, in order that he might concentrate his attention on literature, is said to have arrived, by force of meditation alone, at the remarkable property of the triangle enunciated in the 32nd proposition of the first book of Euclid. Many other examples might be adduced to prove with what success Mathematics may be cultivated at an early age.
The study of Geometry, so far from adding to the difficulties of education, would rather afford relief to the learner, by bringing a new faculty into play, and thus introducing an important variety into the routine of business. It might be made to lighten the drudgery encountered in learning Arithmetic, at present taught so unphilosophically, and to exercise at once the student's sagacity and his skill in numeration.
In the foregoing remarks, the practical value of mathematical science has not been made an object of consideration; and the early study of it has been recommended solely on account of its efficacy in developing the intellectual powers. On this point there cannot be a more weighty authority than DUGALD STEWART, who seems to have had no predilection for Mathematics, and who yet, in his Philosophy of the Human Mind,' speaks of that study in the following terms:-"The intellectual habits of the metaphysician afford little or no exercise to that species of attention which enables us to follow long processes of reasoning, and to keep in view all the various steps of an investigation till we arrive at the conclusion. In Mathematics such processes are much longer than in any other science; and hence the study of it is peculiarly calculated to strengthen the power of steady and concatenated thinking a power, which in all pursuits of life, whether speculative or active, is one of the most valuable endowments we can possess."
But besides the invigoration of the reasoning faculty resulting from this exercise, there is also a peculiar kind of experience derived from mathematical demonstrations, which is not to be had from any other source, and which is calculated to make a great impression on the intellectual character; we mean, the sense of complete certainty -the thorough conviction which attends mathematical deductions. On all other subjects men deal with opinions more or less true-with principles and conclusions alike liable to be controverted; and whoever would wish in this chaos of doubt and uncertainty, to quicken his perception of the different shades of evidence, let him begin by enuring his mind to the complete, dispassionate, rational conviction to be found in Mathematics.
What a faint conception must the man ignorant of Mathematics have of the laws of the Universe! Or, if he supposes them to bear any resemblance to human laws, what an inadequate idea must he entertain of their certainty, immutability, and harmony with one another! No one who is not conversant with abstract and demonstrable truth, can even imagine the solidity and sublime ingenuity with which the whole fabric of nature is put together. Yet how few men are used to recognize how much fewer are able to give an account of the laws of the natural phenomena of which they are constant spectators! If, at the age when the attention is directed with the greatest curiosity to the external world, we were enabled to comprehend the laws by which it is regulated, with what improved discrimination should we turn from this class of truths, to the numerous verbal tenets which occasion such vehement and interminable disputation in society!
The moral influence of mathematical studies did not escape the penetration of PLATO, who, chiefly on that account, decides that they shall be cultivated in his imaginary Republic. Geometry," he says, "is the knowledge of that which is eternal: it disposes the mind to the contemplation of truth, and gives it a philosophical habit, which withdraws our thoughts from what is low, to dwell on what is elevated. Let not the
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
The demonstrations being perfectly understood and