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symbols, reaches the understanding at once, unincumbered as it were by matter and the weight of words.

The intention of these retrenchments and abbreviations being not so much to save space as to obviate tediousness and confusion, not a syllable has been omitted which seemed in any degree conducive to the evidence of the demonstrations; the greatest care has been taken to preserve inviolate the rigorous reasoning of the original, and, at the same time, to give it such a uniform method as may enable the learner to seize it more readily, and to contract from it more certainly a consentaneous habit of thinking. The references to Definitions, Axioms, and antecedent Propositions, which, wanting altogether in the Greek original, have been industriously, and often needlessly multiplied by the recent editors of Euclid's Elements, will be here found sufficiently numerous to satisfy those who are not wholly deficient in memory and attention.

The appended chapters will, it is hoped, be found to be of signal service to the student, by teaching him to examine critically the reasoning of geometricians; by pointing out many of the by-paths branching off from the high road of Geometry which he has just explored; and by discussing those questions, which, from the obscure or imperfect manner in which they have been treated in the Elements, may have left him doubtful or dissatisfied. He is particularly recommended to study with care Appendix (C) to the Second Book, and also that part of Appendix (E) which has for its object to elucidate the definition of Proportion, as framed to embrace incommensurable quantities.

INTRODUCTION.

REMARKS ON THE STUDY OF MATHEMATICS.

As practical utility has been the object constantly aimed at in the preparation of this volume, perhaps it may not be thought amiss if we here offer a few remarks on the nature of mathematical studies, and the best method of prosecuting them with success.

It is to be lamented that the study of Mathematics has not yet attained its due place in the course of education, and that it is made to yield precedence to studies far less capable of strengthening and training the powers of the understanding. Ancient usage maintains as absolute an authority in the routine of education as in matters of less moment. A course of instruction planned according to the dictates of reason, and the example of enlightened antiquity, would hardly make head now-adays against the practice derived from those rude ages when the necessity of having recourse, for the sake of mental culture, to the writings of the ancients, rendered every other object of education subordinate to that of acquiring the Greek and Latin languages. What is now called a liberal education, has for its purpose not so much to develope, invigorate, and discipline the intellectual faculties, as merely to acquire a stock of a certain kind of knowledge, in former times indispensable, but now much sunk in relative value, and, what is a very important consideration, which might be acquired with greater advantage if sought for with less jealous and engrossing avidity.

Those who are willing to sacrifice, or at least to postpone the study of Mathematics to that of Classical Literature, are in the habit of enlarging on the efficacy of the latter in refining the taste and awakening a variety of sentiments. But they do not and cannot pretend to say, that those ends are only to be attained through means of the dead languages. The study of Greek and Latin at an early age, considered as a mental exercise, induces the dangerous habit of learning by rote; and with respect to its immediate and appropriate result, it only teaches several names for the same idea. But what is gained for moral perception or literary taste, by thus fatiguing the intellect? Let any one who has read the Æneid at fifteen, peruse the same poem again when he is five-and-twenty, and he will be surprised to find how all the colours have changed in the interval; he will learn from the experiment, to what an extent the just appreciation of a poetical masterpiece depends on the full developement and maturity of the reader's feelings; and that the merits of a composition which owes not a little of its beauty to the vividness, truth, and delicacy with which it pourtrays human passions, can be thoroughly felt and sincerely acknowledged by adults alone. To suppose that the pregnant thoughts and deep feelings which constitute the charm of the higher order of literary compositions are not in a great measure lost on the minds of youth, is to imagine that the human breast has not its seasons, that boyhood can comprehend manhood, and that our feelings are only awakened by experience, independent of age and physical developement.

Mathematical and Literary pursuits differ widely from each other in this respect, that the former require no auxiliary or preliminary knowledge; they refer neither to natural facts nor historical events; have no relation to the interests or passions which agitate mankind; and afford no room for the play of the imagination, for rhetorical flourishing, and the illusive colouring of language. Consequently, the boy of ten years and the man of forty are perfectly on a level in commencing the study of Mathematics; for if the adult has the advantage in steady attention and the vigour of the reasoning faculty, which it is the special object of the study to strengthen, the boy, on the other hand, gets more readily into the way of dealing with abstractions, and acquiesces at once in

the deductions of his reason.

The student of ripe years 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.

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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

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