CHAMBERS'S EDUCATIONAL COURSE,-EDITED BY W. AND R. CHAMBERS. ELEMENTS OF PLANE GEOMETRY, ACCORDING TO EUCLID, AS IMPROVED BY SIMSON AND PLAYFAIR; WITH SEVERAL NEW IMPROVEMENTS AND ADDITIONS. By A. BELL, FORMERLY MATHEMATICAL MASTER IN DOLLAR INSTITUTION. SECOND EDITION. EDINBURGH: PUBLISHED BY WILLIAM AND ROBERT CHAMBERS; AND SOLD BY ALL BOOKSELLERS. 1837. HARVARD COLLEGE LIBRARY Graduate School of Education EDINBURGH: PRINTED BY W. AND R. CHAMBERS, 1 PREFACE. THIS treatise of Plane Geometry consists of the usual Six Elementary Books, a book on the Quadrature and Rectification of the Circle, a book on Geometrical Maxima and Minima, an exposition of the method of Geometrical Analysis, and an additional Second and Fifth Book. There is likewise added a Treatise on Plane Trigonometry, and an explanation of the Algebraical Principles employed in the fifth book. The basis of the first six books is the Elements of Geometry of Euclid, as given in the very correct edition by Simson, with the improved fifth book by Playfair, and the other improvements of the latter geometer contained in his original edition of Euclid's Elements. None of the propositions of that edition are here suppressed, but proposition H of the sixth book is inserted at the end of the additional second book with a new demonstration, and instead of it a more important proposition is substituted at the end of the sixth book. There are several improvements in this edition of the six books. Many additional and useful definitions have been added, which tend to improve the language of geometry in respect to conciseness and precision. Several propositions and corollaries have been inserted, as being valuable on account of their practical utility, or as rendering the treatise more complete. Two additional demonstrations are given of the fifth proposition of the first book, the common demonstration of which, from its tediousness, more than from any real difficulty, is rather an obstacle to the progress of young students. One of these demonstrations, against which I know of no objection, by the aid of an additional and admitted axiom, namely, the twelfth, is an easy and direct deduction from the fourth proposition. Numerous scholia have been added explanatory of the utility or connection of some of the propositions. The book on the quadrature of the circle, which is also given in Playfair's edition, is here considerably altered and improved. A new and rigorous demonstration is given of the tenth proposition of this book, which is of great utility in theory as well as in practice. The book on Geometrical Maxima and Minima is taken chiefly from Legendre's Geometry, with some alterations, particularly of the demonstration of the tenth proposition, which, in his treatise, is adapted to the particular view taken by him of the theory of proportion. An entirely new treatise of Plane Trigonometry has been composed for this edition. For an account of the additional second and fifth books, reference is made to the introductory remarks to them. A considerable number of exercises are added after the several books, some of which are new, the most important of these being the 12th, 15th, and 23d of those at the end of the sixth book. The solution of exercises affords the best criterion of the industry and talents of the pupil; for in this exercise memory is of less service than usual, the inventive and reasoning faculties being chiefly exerted. In this matter, a few rules may be useful to the pupil, although he must be left chiefly to his own resources. In order that nothing may be wanting to render the present treatise in every respect complete, an Introductory Discourse has been composed, affording an account of the chief objects of mathematical science, and which, with the History of Geometry which follows, will, it is hoped, tend to animate youth with a prospect of the advantages and pleasures to be derived from the prosecution of this branch of useful study. Solid and Spherical Geometry, also Spherical Trigonometry, Conic Sections, &c. which compose the higher branch of Geometry, will form the matter of a subsequent volume. EDINBURGH, September 1, 1836. GEOMETRY. INTRODUCTORY DISCOURSE. THE study of mathematical science, of which Geometry is a branch, possesses several important advantages. Whether it be considered in reference to its practical utility, in its application to several important arts or as a powerful, and the only adequate, instrument of investigation in the study of several classes of physical phenomena or as an efficient instrument of intellectual culture—or merely in reference to the numerous and striking abstract truths which it makes known, it will, without hesitation, be admitted to be worthy of a prominent place in every course of useful and liberal education. The brief discussion of these points, with a sketch of the history of Geometry, will form the subject of this introduction. OBJECTS OF MATHEMATICAL SCIENCE. The science of mathematics investigates the various relations of measurable quantity. Space, time, force, velocity, and motion, are the principal objects about which it is conversant. Our knowledge of the objects of mathematics is obtained from experience, and its axiomatic principles are necessarily involved in our conceptions of these objects. Although the definitions of any of these objects are not necessarily confined to any single property, still every definition must express some characteristic property, and it cannot therefore be arbitrary. Theoretical Geometry treats of the properties of magnitudes, and Practical Geometry of their construction. There are three kinds of magnitudes; they are of one, two, and three dimensions respectively, or lines, surfaces, and solids. Our conceptions of magnitude, and of space generally, are undoubtedly arrived at by first acquiring a knowledge of |