THE ELEMENTS OF PLANE AND SOLID GEOMETRY; WITH CHAPTERS ON MENSURATION AND MODERN GEOMETRY. BY ISAAC SHARPLESS, PROFESSOR OF MATHEMATICS IN HAVERFORD COLLEGE, PA. THROUGH its various stages this work has had the benefit of the suggestions and criticisms of Thomas K. Brown, Mathematical Teacher of Westtown School, whose Algebra has been prepared simultaneously with it. The authors have kept in view the object of making these books parts of a connected series of Mathematical text-books. The following volumes are now completed: ELEMENTARY ALGEBRA. By THOMAS K. BROWN. ELEMENTARY PLANE GEOMETRY. BY ISAAC SHARPLESS. ELEMENTS OF PLANE AND SOLID GEOMETRY. By ISAAC SHARPLESS. Copyright, PORTER & COATES. 1879. PREFACE. THIS little work on elementary Geometry is offered to those who desire to make themselves familiar with the general principles of the science in the limited time usually allotted to it. Beginning with simple definitions, and with postulates and axioms obvious to every one, it develops, in an unbroken series of propositions, the essential truths of Geometry. It differs alike from those treatises whose main object is to present the subject in its shortest and simplest form, and from those which are exhaustive and comprehensive in their scope. The student does not meet at the outset, as in most modern Geometries, a collection of theorems involving constructions which he has not been taught to perform, but a simple problem; and, as he needs them, he finds other problems, so that no figure is called for which he has not the means to construct accurately and intelligently. He is impressed with the logical idea that correct conclusions can only be deduced from known premises, and he acquires practical skill in construction by applying the problems to every proposition. While the basis of elementary Geometry is, and must ever be, contained in the works of Euclid, modern geometers, especially in France, have made important additions and corrections. Euclid's methods are sometimes cumbersome, and his omissions, especially in solid Geometry, are serious; yet years of experience attest the beneficial results of his teaching. His students think accurately and scientifically, and their training shows itself in their future work. It has been an aim, in the preparation of this treatise, to incorporate with these advantages the improvements and additions which recent study has suggested. While thus 3 |