ELEMENTS OF GEOMETRY. INTRODUCTION. 1. EVERY person possesses a conception of space indefi nitely extended in all directions. Material bodies occupy finite, or limited, portions of space. The portion of space which a body occupies can be conceived as abstracted from the matter of which the body is composed, and is called a geometrical solid. The material body filling the space is called a physical solid. A geometrical solid is, therefore, merely the form, or figure, of a physical solid. In this work, since only geometrical solids will be considered, we shall, for brevity, call them simply solids, and we shall define them formally, as follows: Definition. A solid is a limited or bounded portion of space, and has length, breadth, and thickness. 2. The boundaries of a solid are surfaces. Definition. A surface is that which has length and breadth, but no thickness. If a surface is bounded, its boundaries are lines. If two surfaces intersect, their intersection is a line. Definition. A line is that which has length, but neither breadth nor thickness. If a line is terminated, it is terminated by points. Definition. A point has position, but neither length, breadth, nor thickness. 3. If we suppose a point to move in space, its path will be a line, and it is often convenient to regard a line as the path, or locus, of a moving point. If a point starts to move from a given position, it must move in some definite direction; if it continues to move in the same direction, its path is a straight line. If the direc tion in which the point moves is continually changing, the path is a curved line. If a point moves along a line, it is said to describe the line. By the direction of a line at any point we mean the direction in which a point describing the line is moving when it passes through the point in question. Definitions. A straight line is a line which has everywhere the same direction. A curved line is one no portion of which, however short, is straight. A broken line is a line composed of different successive straight lines. 4. Definitions. A plane surface, or simply a plane, is a surface such that, if any two points in it are joined by a straight line, the line will lie wholly in the surface. B Α را B D F E B A curved surface is a surface no portion of which, however small, is plane. 5. Definitions. A geometrical figure is any combination of points, lines, surfaces, or solids, formed under given condi tions. Figures formed by points and lines in a plane are called plane figures. Those formed by straight lines alone are called rectilinear, or right-lined, figures; a straight line being often called a right line. 6. Definitions. Geometry may be defined as the science of extension and position. More specifically, it is the science which treats of the construction of figures under given conditions, of their measurement and of their properties. Plane geometry treats of plane figures. The consideration of all other figures belongs to the geometry of space, also called the geometry of three dimensions. 7. Some terms of frequent use in geometry are here defined. A theorem is a truth requiring demonstration. A lemma is an auxiliary theorem employed in the demonstration of another theorem. A problem is a question proposed for solution. An axiom is a truth assumed as self-evident. A postulate (in geometry) assumes the possibility of the solution of some problem. Theorems, problems, axioms, and postulates are all called propositions. A corollary is an immediate consequence deduced from one or more propositions. A scholium is a remark upon one or more propositions, pointing out their use, their connection, their limitation, or their extension. An hypothesis is a supposition, made either in the enunciation of a proposition or in the course of a demonstration. |