INTRODUCTION 5

Figure 3b

a point M of the ﬁrst (Fig. 3b). To do this it suﬃces to move radius OM onto

radius O M (Fig. 3b), which is possible since the two segments are equal.

8. A portion of a circle is called an arc (ApB, Fig. 4).

From the fact that two circles can be superimposed in inﬁnitely many ways,

it follows that it is possible to compare circular arcs from the same circle or in

two equal circles, in the same way in which we compare line segments. For this

purpose one transports the two arcs to a position where they have the same center, a

common endpoint, and they lie on the same side of this common endpoint. Suppose

that the two arcs in this positions are AB, AC; we will say that the ﬁrst arc is

greater than the second if, starting from the point A and moving along

arc13

AB

we encounter point C before point B (Fig. 5); the ﬁrst arc is said to be smaller, if

on the other hand the order is ABC (Fig. 4).

Figure 4

Figure 5

8b. We can also deﬁne the sum of two arcs AB, BC (Fig. 4) on the same

circle (or on equal circles) by moving them end-to-end.

A circular arc AB can be divided into two or more equal

parts,14

just as a line

segment can.

13It

is essential here to specify the direction of motion (which was not necessary in the case

of a straight line), because the points A, B divide the circle into two arcs, and the order in which

we encounter the points B, C depends on the direction of motion.

14The

remark made about straight line segments in footnote 10 holds here as well.