DEFINITIONS. PROPOSITIONS 1-37 WITH RIDERS .154 .162 APPENDIX III. RADICAL AXIS. DEDUCTIONS...., 214 .218 CIRCLES AND THEIR SCRIBED REGULAR POLYGONS, REGULAR POLYGONS AND THEIR SCRIBED EUCLID'S ELEMENTS. BOOK I. DEFINITIONS. 1. A point has position, but it has no magnitude. A point is indicated by a dot with a letter attached, as the point P. .P The dots employed to represent points are not strictly geometrical points, for they have some size, else they could not be seen. But in geometry the only thing connected with a point, or its representative a dot, which we consider, is its position. 2. A line has position, and it has length, but neither breadth nor thickness. Hence the ends of a line are points, and the intersection of two lines is a point. A line is indicated by a stroke with a letter attached, as the line C. C Oftener, however, a letter is placed at each end of the line, as the line AB. A -B The strokes, whether of pen or pencil, employed to represent lines, are not strictly geometrical lines, for they have some breadth and some thickness. But in geometry the only things connected with a line which we consider, are its position and its length. 3. If two lines are such that they cannot coincide in any two points without coinciding altogether, each of them is called a straight line. Hence two straight lines cannot inclose a space, nor can they have any part in common. Thus the two lines ABC and ABD, which have the part AB in common, cannot both be straight lines. A Euclid's definition of a straight line is that which lies evenly to the points within itself.' B 4. A curved line, or a curve, is a line of which no part B is straight. Thus ABC is a curve. c 5. A surface (or superficies) has position, and it has length and breadth, but not thickness. Hence the bound. aries of a surface, E and the intersection of two surfaces, are lines. Thus AB, A B ACB, and DE are lines. 6. A plane surface (or a plane) is such that if any two points whatever be taken on it, the straight line joining them lies wholly in that surface. This definition (which is not Euclid's, but is due to Heron of Alexandria) affords the practical test by which we ascertain whether a given surface is a plane or not. We take a piece of wood or iron with one of its edges straight, and apply this edge in various positions to the surface. If the straight edge fits closely to the surface in every position, we conclude that the surface is plane. 7. When two straight lines are drawn from the same point, they are said to contain a plane angle. The straight lines are called the arms of the angle, and the point is called the vertex. B Thus the straight lines AB, AC drawn from A are said to contain the angle BAC; AB and AC are the arms of the angle, and A is the vertex. An angle is sometimes denoted by A С three letters, but these letters must be placed so that the one at the vertex shall always be between the other two. Thus the given angle is called BAC or CAB, never ABC, ACB, CBA, BCA. When only one angle is formed at a vertex it is often denoted by a single letter, that letter, namely, at the vertex. Thus the given angle may be called the angle A. But when there are several angles at the B same vertex, it is necessary, in order to avoid ambiguity, to use three letters to express the angle intended. Thus, in the A -C annexed figure, there are three angles at the vertex A, namely, BAC, CAD, BAD. Sometimes the arms of an angle have D several letters attached to them ; in which case the angle may be denoted in various ways. Fig. 1. Fig. 2. Fig. 3. E F D Thus the angle F (fig. 1) may be called AFC or BFC indifferently; the angle G (fig. 2) may be called AGB or CGB; the angle A (fig. 3) may be called BAC, FAG, DAE, FAC, GAB, and so on. It is important to observe that all these ways of denoting any particular angle do not alter the angle ; for example, the angle BAC (fig. 3) is not made any larger by calling it the angle FAG, or the angle DAE. In other words, the size of an angle does not depend on the length of its arms; and hence, if the two arms of one angle are respectively equal to the two arms of another angle, the angles themselves are not necessarily equal. As a further illustration, the angles A, B, C with unequal arms |