PART I PLANE GEOMETRY INTRODUCTION PRELIMINARY NOTIONS AND DEFINITIONS WHEN We speak of a point or a line, a straight line or a crooked line, everybody knows in a general way what is meant. In the minds of persons not familiar with the strict sense in which these terms are used in Geometry, it may be that the words 'point' and 'line' are associated with marks of some sort, made for example with a pen or pencil; they may think of a point as a dot, and of a line as a long, narrow mark, but in spite of this association the true meaning of the terms 'point,' 'line,'' straight,' 'crooked,' etc., is pretty clearly understood. It is not at all certain that definitions could be given in words which would convey an intelligent idea of the meaning of these terms to a person not already familiar with the thought conveyed by them, still a few words of comment may serve to fix the attention upon some properties of points and lines which are fundamental in the study of Geometry. 1. A point has no magnitude; it has only position. The smallest mark that can be made with the finest pencil or pen will have some magnitude, and consequently is not a true point in the sense in which this term is used in Geometry. We can think of points as having position only, but in making diagrams we are obliged to use pencil or pen marks to represent them. 2. A line has position and length, but no breadth or thickness. Whenever we speak of a line it suggests the idea of extension in one way, namely, length, without regard to breadth or thickness; but if we try to represent a line by a mark, this mark will of necessity have some breadth and so cannot be a true geometrical line. You should endeavor to think of a line without actually drawing a mark to represent it. A line is sometimes spoken of as the path of a moving point. On a line you can choose as many points as you please. Through two points you can draw as many lines, of one kind or another, as you please. 3. In order to tell one point from another we give them names, conveniently the names of the letters of the alphabet, and place the letter by the side of the point bearing its name. •C •D •B Thus we speak of the point A, the point B, etc. If we wish to designate different points by the same letter, P say, we distinguish them as P1, P2, P3, and We denote a line by naming a sufficient number of points on it to distinguish it from every other line. Thus we have in the diagram the line ABC, the line DE, the line FBE. The two points B and E would not distinguish the line DBE from the line FBE, since B and E are points of both lines, or as we say, are common to both R S P lines. The lines PQ and RS in the second diagram are, however, distinguished by naming two points on each. 4. A straight line is such that only one of that kind can pass through the same two points. That is to say, if two straight lines pass through the same two points, they must occupy the same position, and there is no point of one which does not also lie on the other. This is sometimes expressed by saying that, "If two straight lines coincide at two points, they coincide throughout"; or, "Two points determine a straight line," i.e. fix the location of the line and distinguish it from every other straight line. Two different straight lines therefore can have only one point in common; in other words, two straight lines intersect in only one point. Any two straight lines can be made to coincide at every point by placing them so that they coincide at two points. 5. Suppose that on a given straight line we choose two points and mark them by the letters A and B. These will serve to distinguish this straight line from every other, since no other straight line can pass B through both of these points. We may therefore call this line the straight line AB. That portion of the line which lies between the points A and B is called the segment AB of the line. If other points are chosen on the line, as C, D, E, etc., A с B the portion of the line lying between the points A and C is called the segment AC; that between C and D, the segment CD, etc. It should be observed that two points chosen anywhere on a straight line determine it; the line may be as long as you please, i.e. it may be unlimited in extent, yet these two points will distinguish it from every other straight line. When we have in mind only that segment of a straight line lying between |