symmetry, and rational coherence. So signal an improvement could hardly have been effected without considerable inventive powers, and, at all events, the new and increased value which Geometry received at the hands of the systematic compiler, fully entitles him to the credit of originality. Of Euclid's personal history there is but little known. He flourished at Alexandria about 300 years before the commencement of the Christian era, and is said to have enjoyed the friendship of Ptolemy Philadelphus, whom he instructed in geometry. As a proof of the friendly intimacy subsisting between him and that monarch, and the freedom with which they conversed together, it is related that when the latter, grown weary of his slow progress, inquired whether the knowledge of the more important geometrical truths might not be attained by a shorter method, Euclid in reply, assured the king, "that there is no royal road to mathematics." The tedious repetitions and prolix minuteness which exhausted the patience of Ptolemy, are felt still more sensibly by the student of Euclid's Elements at the present day. Geometry, which in the age of Euclid was the central compartment of the mathematical edifice, is now comprised in its vestibule. The vast increase which latter times have witnessed in the scope and application of Mathematics, has, without diminishing in the least. degree the absolute value of elemental geometry, greatly reduced its proportion to the whole science, and renders it desirable to economize as far as possible the time and labour of the student. Besides, the modern mind has, since the discovery of the art of printing, undergone a training which renders it averse from the diffuse style characteristic of lessons orally delivered. The permanence of written language enables the student of books to dispense with those amplifications and reiterations which are acceptable to the hearer of a discourse. Add to this, the influence exercised on intellectual habits by the superior brevity and clearness of the modern arithmetical notation (to say nothing of algebraic symbols), and it will be evident that the present age may, without casting any severe reproach on the Greek Geometrician, naturally express itself dissatisfied with his frequent repetitions and habitual verboseness. The removal of this objection to Euclid's Elements has been a point constantly kept in view in preparing the present edition. Care has been taken to retrench every superfluity, and to get rid of verbiage. By this means a great abridgment has been effected, without omitting a single step in the reasoning, or in the slightest degree impairing its strength and validity. The Six Books of the Elements, here comprised in only 120 moderately-sized pages, are fully as complete as they have ever yet appeared, and, it is hoped, much clearer than in the usual form. They have lost by the curtailment nothing but that tediousness of manner, which, without being really of service to any, is calculated to prove a material hindrance to most understandings. The desire to combine conciseness and clearness in the highest degree in which they are compatible with one another, has led us to adopt a few symbols borrowed from Algebra, but used in forms so simple as to throw no difficulty in the way of learners. This innovation comes to us recommended by the authority of PLAYFAIR. Relations so simple as those of equality, or of greater and less, and such operations as addition and subtraction, cannot be too briefly indicated. The mind is apt to ponder on ordinary language, even when it is most clear and unambiguous, while that which is expressed by appropriate 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 |