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SHARPLESS AND BROWN'S

Mathematical Text-Books.

ELEMENTARY ALGEBRA. BY THOMAS K. BROWN. ELEMEMTARY PLANE GEOMETRY. BY ISAAC SHARPLESS. ELEMENTS OF PLANE AND SOLID GEOMETRY. By ISAAC SHARPLESS.

GEOMETRY AND TRIGONOMETRY. BY ISAAC SHARPLESS. GUMMERE'S SURVEYING. Revised by ISAAC SHARPLESS,

Copyright,

PORTER & COATES.

1879 and 1882.

PREFACE.

THIS 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

increasing the scope of the work, its size has been kept down by such an arrangement of the problems and theorems as secures the simplest demonstrations.

PART I. treats of Plane Geometry. The order of propositions here adopted, seems to accomplish, as fully as possible, the two ends of keeping practice always in advance of theory, and removing difficulties, as much as strict logic will permit, from the path of the beginner. These ends gained, the introduction to the science becomes interesting and suggestive. Analyses of the proof, showing at a glance the relations of the different parts of the demonstration, are given at the close of some of the propositions; thus the student has the choice. of two statements, which will assist each other in giving a correct understanding of the methods. At the end of each book is a collection of exercises for original investigation and practical application. They are so explained that the average student should be able to solve them unaided--not so difficult as to cause him to give up in despair, nor so easy as to be of no interest or value. A collection of rules for the mensuration of plane surfaces, and a number of examples in them, complete Part I. This part is bound separately for the convenience of those who desire such a limited course.

PART II. contains three books, treating respectively of the geometry of planes, of solids, and of spherical geometry, with accompanying exercises. The rules for the measurement of geometrical solids are collected; a few additional rules in mensuration, not previously referred to, are proved, and numerous examples given.

PART III. contains an introduction to Modern Geometry, the name usually given to the discoveries in pure geometrical science, made since the advance in this direction was stayed, for a time, by the brilliant prospects opened by the Analytical Geometry of DESCARTES. For much of this part the author is indebted to the TRAITÉ DE GÉOMÉTRIE of ROUCHÉ ET DE COMBEROUSSE.

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