INTRODUCTION 3

A line which is composed of portions of straight lines is called a broken line.

Other lines, which are neither straight nor broken, are called curves.

5. The part of a straight line contained between two points A and B is called

a segment of the line AB, or the distance AB. We can also consider the portion

of a line which is not limited in one direction, but limited in the other by a point.

This is called a ray.

Figure 1 Figure 2

It follows from the remarks above that two arbitrary rays are congruent.

The distance AB is said to be equal to the distance A B if the ﬁrst segment

can be moved onto the second in such a way that A falls onto A and B onto B .

After such a move, it follows from the two properties which deﬁne the notion

of a straight line that the two segments will coincide entirely. Consequently, the

deﬁnition of equal segments agrees with the deﬁnition of congruent ﬁgures given

earlier.

Two equal segments AB, A B can be made to coincide in two diﬀerent ways;

namely, the point A falling on A and B on B , or A falling on B and B on A . In

other words, one can turn the segment AB around in such a way that each of the

points A, B takes the place of the other.

When two segments AB, BC are on the same line so that one of them is the

extension of the other (Fig. 1), the segment AC is called the sum of the ﬁrst two

segments. The sum of two, and therefore of many, segments is independent of the

order of the summands.9

In order to compare segments, we move them onto the same line, so that they

start from the same point and are oriented in the same direction: for example AB

and AC (Figures 1 and 2). If the points are in the order A, B, C (Fig. 1), the

segment AC is the sum of AB and another segment BC; in this case AC is greater

than AB; if on the contrary, the order is A, C, B, the segment AC is smaller

than AB. In either case, the third segment BC which, when added to one of the

ﬁrst produces the other, is called the diﬀerence of the two segments. Finally, the

points B and C may coincide, in which case we know that the two segments under

consideration are equal.

On each line segment AB, there exists a point M , the midpoint of AB, equally

distant from A and B; every point on the line between A and M is clearly closer

to A than to B, and the opposite is true for points between M and B.

More generally, a line segment can be divided into any number of equal

parts.10

6. The Plane. An inﬁnite surface such that a line joining any two points on

it is contained entirely on the surface is called a plane.

We will assume that there is a plane passing through any three points in space.

A straight line drawn in a plane separates the surface into two regions called half-

planes, situated on diﬀerent sides of the line. One cannot travel along a continuous

9For

two segments this follows immediately from the preceding paragraph.

10We

mean by this that there exist points on AB which divide this segment into equal parts.

The question as to whether these points can actually be determined with the help of available

instruments will be considered later (Book III).