SCHOOL A BEING BOOKS I. & II. OF EUCLID. EUCLID'S ELEMENTS. WITH NOTES, EXERCISES AND EXPLANATIONS BY CHARLES MANSFORD, B.A., MATHEMATICAL TUTOR IN THE WESTMINSTER TRAINING COLLEGE. SCHOOL BOOK DEPÔT, TRAINING COLLEGE, CENTRAL AGENCY:- HAMILTON, ADAMS, & Co., 32, PATERNOSTER ROW, E.C. E.C. JOHN MENZIES & CO. EDINBURGH & GLASGOW:- W. STEWART & Co., 32, NEW BRIDGE THE EDUCATIONAL TRADING COMPANIES AND DEPÔTS. AND ALL BOOKSELLERS. PRINTED BY WILKINS AND ELLIS, DERBY. [1375] PREFACE. WHATEVER may be the value of the objections brought against Euclid's system of Geometry, the Elements ' still maintain their position in Schools and Universities as the standard text-book both for teaching and reference. Undoubtedly there are great advantages in thus retaining Euclid's book, and many of the objections urged against it would lose much of their force if it were more carefully and thoroughly taught. When a book on such a subject is put into the hands of a pupil, and he is required to master a whole body of new truths in their proper connections while no attempt is made to exhibit the relations of its several parts, or to supply applications and illustrations as the learner proceeds; no wonder that the subject should prove wearisome and unfruitful. An attempt is here made to give a wider view of the subject, and a clearer explanation of its several parts than is usually presented; and it is hoped that the course adopted will be found helpful to pupils and suggestive to teachers. The riders to each proposition are such as can be, in nearly every instance, deduced from the proposition to which they are attached, and are illustrative of it. The notes are explanatory of the more difficult and important points in the demonstrations, or are designed to bring out new relations between the several parts of the subject. The exercises at the end are carefully graduated and arranged under definite heads; and it is hoped that with the explanations given, a pupil who has mastered the propositions will be able to work any exercise in the book. Where, however, any special difficulty seemed to demand it, a short explanation has been added for the pupil's guidance. The text is Simson's, arranged in paragraphs according to approved modern methods. INTRODUCTION. THE collection of truths which constitutes the work on Plane Geometry, known as 'Euclid's Elements,' is arranged in six books, each of which commences with certain definitions. The first book contains also postulates and axioms, which, with the accompanying definitions, underlie the whole subject, and form the groundwork of all the truths established in the following books. Hence it is that the definitions are placed at the very commencement, and it is important that their meaning and their proper relation to the propositions which follow should be clearly understood. DEFINITIONS. Let it be observed then, that the definitions have reference to things which exist in the mind, and not to things which we see around us. A sheet of paper for instance, which is usually taken to represent a plane surface, always has some roughness of face, and some warping of its surface, and is never found perfectly flat and even. The same is true of all other surfaces. But although we cannot find a surface perfectly level in every direction and perfectly smooth in every part, we can easily imagine a "Plane superficies in which any two points being taken, the straight line between them shall lie wholly in that superficies." Def. 7. This notion or conception is what Euclid |