EUCLID'S ELEMENTS OF PLANE GEOMETRY. WITH EXPLANATORY APPENDIX, AND SUPPLEMENTARY PROPOSITIONS. ADAPTED FOR THE USE OF SCHOOLS, OR FOR SELF-INSTRUCTION. BY W. D. COOLEY, A.B. LONDON: WHITTAKER AND CO., AVE MARIA LANE. PRINTED BY JANES HOLMES, TOOK'S COURT, CHANCERY LANE. 1840. 503, PREFACE. Euclid's volume on the Elements of Geometry has been regarded for more than two thousand years as the ground-work of Mathematical Science. It has been translated into the language of almost every nation pretending to the least degree of intellectual refinement; and in the more civilized countries of Europe, where mathematical studies have been prosecuted with so much success, and where so many rival treatises have been composed, the work of the Greek Geometrician still retains its pre-eminence, and is generally adopted as the Introduction to the Scientific Course. Such universal and steadfast approbation can be ascribed only to the intrinsic value of the work which is the object of it; and, indeed, such is the excellence of Euclid's Elements, that if all the merit of that work belonged to a single author, he might be deservedly ranked not only as the greatest man of antiquity, but even as one who, in fertility of genius, and in extent of conquest over the domains of demonstration, has far outdone the Newtons, EULERS, LAGRANGES, and LAPLACES of modern times. Euclid was not, however, so much the author as the compiler of the work which bears his name. Geometry had engaged, from an early age, the earnest attention of the most eminent Greek philosophers; but the fruits of their ingenuity lay widely scattered, till Euclid, collecting them together, linked them into a system, and gave to the body of hitherto isolated and disjointed truths, plan, 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 |