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

11The

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.

12Hadamard

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.