4 INTRODUCTION path from one of these regions to the other without crossing the line. These two half-planes can be superimposed by turning one of them about the straight line as a hinge. We will first study figures drawn in a plane, the study of which is the subject of plane geometry. 7. The Circle. A circle is the geometric locus of points in a plane situated at a given distance from a given point O (Fig. 3) in this plane.11 Figure 3 A segment joining the center of a circle to a point on the circle is called a radius of the circle. All the radii are therefore equal. This is illustrated by the spokes of a wheel, which has the shape of a circle. According to the preceding definition, in order to express the fact that a point in the plane is on a circle located in that plane, it suﬃces to say that its distance to the center is equal to the radius. Any circle divides the plane of which it is part into two regions, the exterior, unbounded, formed by the points whose distance to the center is greater than the radius and the interior, bounded in all directions, formed by the points whose distance to the center is less than the radius. This interior region is called a disk .12 The locus of points in space situated at a fixed distance from a given point is a surface called a sphere. It is clear that a circle is determined when we are given its plane, its center, and its radius. When no confusion is possible, we denote a circle by the letter denoting its center, or by the two letters denoting one radius, the letter denoting the center being written first. Thus, the circle of figure 4 will be denoted by O, or (if there are several circles with center O) by OA. Two circles with the same radius are congruent: it is clear that they will coin- cide if their centers are made to coincide. Two circles can be superimposed in infinitely many ways: we can move one over the other in such a way that a given point M of the second corresponds with 11 The locus of the points in a plane situated at a fixed distance from a given point outside this plane (if such points exist) is also a circle we will prove this in space geometry. The locus of points in space situated at a given distance from a given point is called a sphere. 12 Hadamard uses the standard French terminology, in which the points on a circle form a set called a circonf´ erence, while the points inside the circle form a set called a cercle. The cognate English terms mean different things. –Transl.

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