Of the Properties of the Circle . What is meant by Squaring a Circle Recapitulation of the Truths contained in the Fourth Section 173 GEOMETRY. INTRODUCTION. IF, without regarding the qualities of bodies, viz their smoothness, roughness, colour, compactness, tenacity, &c., we merely consider the space which they filltheir extension in space-they become the special subject of mathematical investigation, and the science which treats of them, is called Geometry. The extensions of bodies are called dimensions. Every body has three dimensions, viz: length, breadth, and depth.* In speaking of a wall or a house, for instance, you can form no idea of it without conceiving it to extend in length, breadth, and *The term height, is sometimes used for breadth. But height and breadth express originally the same thing. When a surface is placed horizontally, we speak of its length and breadth; and when perpendicularly, of its length and height. depth; and the same is the case with every other body you can think of. The limits or confines of bodies are called surfaces (superficies), and may be considered independently of the bodies themselves. So you may look at the front of a house, and inquire how long and how high is that house, without regarding its depth; or the length and breadth of a field, without asking how deep it goes into the ground, &c. In all such cases, you merely consider two dimensions. A surface is, therefore, defined to be an extension in length und breadth without depth. The limits or edges of surfaces are called lines, and may again be considered independently of the surfaces themselves. You may ask, for instance, how long is the front of such a house, without regarding its height; or how far is it from Boston to Roxbury, without inquiring how broad is the road? Here, you consider evidently only one dimension; and a line, therefore, is defined to be an extension in length without breadth or depth. The beginning and end of lines are called points. They merely mark the positions of lines, and can, therefore, of themselves, have no magnitude. To give an example : when you set out from Boston to Roxbury, you may indicate the place you start from, which you may call the point of starting. If this chances to be Marlborough Hotel, you do not ask how long, or broad, or deep that place is; it suffices for you to know the spot where you begin your journey. A point is therefore defined to be mere position, without either length or breadth. Remark. A point is represented on paper or on a board, by a small dot. A line is drawn on paper with a pointed lead pencil or pen; and on the board, with a thin. mark made with chalk. The extensions of surfaces are indicated by lines; and bodies are represented on paper or on the board, according to the rules of perspective. In order to begin the study of Geometry, it is necessary, first, to acquaint ourselves with the meaning of some terms, which are frequently made use of in books treating on that science. Definitions. A line is called straight, when every part of it lies in the same direction, thus, Any line in which no part is straight, is called a curve line. A geometrical plane is a surface, in which two points being taken at pleasure, the straight line joining them will lie entirely in that surface.* A surface in which no part is plane, is called a curved surface. Any plane surface, terminated by lines, is called a geometrical figure. The simplest geometrical figure, terminated by three straight lines, is called a triangle. A geometrical figure, terminated by four straight lines, is called a quadrilateral, by 5, a pentagon-by 6, an hexagon-by 7, an * The teacher can give an illustration of this definition, by taking, any where on a piece of paste board, two points and joining them by a piece of stiff wire. Then, by bending the board, the wire which represents the line will be off the board, and you have a curved surface; and by stretching the board, so as to make the wire fall upon it, you have a plane. |