that is needed. In no case are questions asked whose answers are implied by the form of the questions. The arrangement of the text is adapted to three grades of courses: (a) A minimum course, consisting of Chapters I to VI, without the problems and applications at the end of each chapter. This would provide about as much material, theorems, constructions, and originals as is found in the briefest books now in use. (b) A medium course, consisting of Chapters I to VI, including a reasonable number, say one half or two thirds, of the applications at the end of each chapter. This would fully cover the college entrance requirements. (c) An extended course, including Chapter VII, which contains a complete review, together with many additional theorems and a large number of further applications. This would provide ample work for the strongest high schools, and for normal schools in which more mature students are found or more time can be given to the subject. Chapter VII gives a complete treatment of the incommensurable cases, though not based on the formal theory of limits. It is believed that for high school pupils the notion of a limit is best studied as a process of approximation, and that the best preparation for the later understanding of the theory is by a preliminary study of what is meant by "approaches," such as is given in Chapters III and IV. Acknowledgment is due to Miss Mabel Sykes, of Chicago, for the use of a large number of drawings and designs from her extensive collection; also to numerous commercial and manufacturing houses, both in this country and in Europe, through whose courtesy many of the patterns were obtained. CHICAGO AND BOSTON, January, 1910. H. E. SLAUGHT. N. J. LENNES. PLANE GEOMETRY. CHAPTER I. RECTILINEAR FIGURES. INTRODUCTION. 1. Elementary geometry is a science which deals with the space in which we live. It begins with the consideration of certain elements of this space which are called points, lines, planes, solids, angles, triangles, etc. Some of these terms, such as point, line, plane, are here used without being defined in a strictly logical sense. Their meaning is made clear by description and by concrete illustrations like the following. 2. Certain portions of space are occupied by objects which we call physical solids, as, for instance, an ordinary brick. That which separates a solid from the surrounding space is called its surface. This may be rough or smooth. If a surface is smooth and flat, we call it a plane surface. A pressed brick has six plane surfaces called faces. Two adjoining faces meet in an edge. Three edges meet in a corner. The brick is bounded by its six faces. Each face is bounded by four edges, and each edge is bounded by two corners. 1 3. If instead of the brick we think merely of its form and magnitude, we get a notion of a. geometrical solid, which has the three dimensions, length, breadth, and thickness. The faces of this ideal solid are called planes. These are flat and have length and breadth, but no thickness. The edges of this solid are called lines. They are straight and have length, but neither breadth nor thickThe corners of this solid are called points. They have position, but neither length, breadth, nor thickness; that is, they have no magnitude. ness. 4. It is possible to think of these concepts quite independently of any physical solid. Thus we speak of the line of sight from one point to another; and we say that light travels in a straight line. The term straight line is doubtless connected with the idea of a stretched string. Of all the lines which may be conceived as passing through two fixed points that one is said to be straight between these points which corresponds most nearly to a stretched string. Likewise a plane may be thought of as straight or stretched in every direction, so that a straight line passing through any two of its points lies wholly in the plane. 5. If one of two intersecting straight lines turns about their common point as a pivot, the lines will continue to have only one point in common until all at once they will coincide throughout their whole length. Hence, Two straight lines cannot have more than one point in common unless they coincide and are the same line; that is, two points determine a straight line. This would not be so if the lines had width, as may be seen by examining the figures. |