INTRODUCTION. STUDENTS of Geometry should be led to see that while we can not tell what space is, all bodies, the largest and the smallest, exist in space. We can not conceive of any boundaries, or limits, to space. We look into the sky on a clear night, and think of what astronomers tell us about the distances to a few of the stars. But these are not lights placed upon the outer wall of creation. We can not avoid believing that space reaches on and on, and that there is no end to it. Any object that we see, as a house, field, lake, hill, book, fence, tree, is called a body. Every body has size and shape and position. Size is the result of extension. The lot of ground upon which your house stands, extends, or stretches, so many feet along the street, and so many feet back from the street. These distances are found by measuring them. The word dimension is from a Latin word which means to measure. The length and breadth of the lot are its dimensions. By the use of these two dimensions, we are able to find out what is the size of the lot, the amount of ground in it. However, we do this indirectly, and not by laying a square foot or rod down upon the land as many times as possible so as not to overlap, and counting the number of times. Knowing the two dimensions already named, suppose we wish to ascertain the distance from the front lefthand corner to the rear right-hand corner; geometry teaches us how to learn this without going upon the land at all. With a little more knowledge of geometry, we can measure distances over rivers which we can not cross, and find the height of steeples which we can not climb. When we learn the shape and the dimensions of a water-tank, we can calculate how many gallons it will hold, though there is not a drop in it. If it were full, and if we should empty it, counting the gallons, as the farmer counts his bushels of grain from the threshingmachine, there would be no mathematical science about the operation; but when we do this by the use of certain lines, we employ Geometry. This science teaches the manner of such operations, and the reasons underlying them; and we are led to see the propriety of the following definition: Geometry is the science which treats of the properties of figures, their construction, and the indirect measurement of their extension. Geometry deals with lines, surfaces, and solids. We may place before us that object which we learned. in arithmetic to call a cube. It is a body, a figure, a solid. The space which the cube occupies is also called a cube. It is a geometric solid. The cube has, as we see, three dimensions, and they are called length, breadth, and thickness. Now we can conceive of, or imagine, the thickness growing less and less until it can no longer fix the attention. We can think of the magnitude which remains as having but two dimensions. It is a surface, and is the same as one of the original six faces of the cube. We may next imagine what would result if the breadth should do as the thickness did; and with the eye of reason we can see a line. It is the same as one of the edges of the cube. If the line should shorten till the eye can detect no length, the result would be a point. The epithet imaginary is used as definitive of geometric lines, denoting that they are visible only to the imagining, the image-making power of the mind. NOTE TO TEACHERS.-The word is, however, on the ear of the pupil, a symbol for something so nearly synonymous with untrue that it is well for the teacher to give reality to the term by abundant illustration. The shortest distance from A to B is real, "actually being or existing." The plane of this page extended any distance into space is real. In short, all the airdrawn lines and surfaces of geometry,—the axis of the earth, which the pupil has learned to call imaginary, the equator, the tropics, are as real as the Amazon River, or the great plains of Siberia. The figures which we draw upon paper, or blackboard, are but rude, imperfect signs of the perfect figures, which exist only in the mind; though, for all geometric purposes, we may regard the symbol as the thing signified. GEOMETRY. SECTION I.-DEFINITIONS. 1. The term definition denotes a boundary. It is such a description of a thing as will distinguish it from all other things. 2. A line is that magnitude which has length, without breadth; or, it is the path of a point. 3. A surface is that magnitude which has length and breadth, without thickness; or, it is the path of a line. 4. A point is simply position, as the end of a line, or the crossing of two lines. 5. A straight line is such a line that any part will, however placed, coincide with any other part, if its extremities are made to fall on that other part; or, it is the path of a point moving without change of direction. 6. A curve is a line which continually changes its direction. 7. A plane surface is one such that if any two of its points be connected by a straight line, this line will lie wholly within the surface. Thus, the top of a desk, the face of the blackboard, or of a pane of glass, are usually planes. |