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. 13 It 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. 14 The remark made about straight line segments in footnote 10 holds here as well.

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