Equal circles are those the diameters of which are equal, or the radii of which are equal. 2. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle. 3. Circles are said to touch one another which, meeting one another, do not cut one another. 4. In a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal. 5. And that straight line is said to be at a greater distance on which the greater perpendicular falls. 6. A segment of a circle is the figure contained by a straight line and a circumference of a circle. 7. An angle of a segment is that contained by a straight line and a circumference of a circle. 8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined. 9. And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference. H. E. II. I 10..A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the circumference cut off by them. II. Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another. DEFINITION I. Ἴσοι κύκλοι εἰσίν, ὧν αἱ διάμετροι ἴσαι εἰσίν, ἢ ὧν αἱ ἐκ τῶν κέντρων ἴσαι εἰσίν. Many editors have held that this should not have been included among definitions. Some, e.g. Tartaglia, would call it a postulate; others, e.g. Borelli and Playfair, would call it an axiom; others again, as Billingsley and Clavius, while admitting it as a definition, add explanations based on the mode of constructing a circle; Simson and Pfleiderer hold that it is a theorem. I think however that Euclid would have maintained that it is a definition in the proper sense of the term; and certainly it satisfies Aristotle's requirement that a "definitional statement" (opioTikòs λóyos) should not only state the fact (rò oT) but should indicate the cause as well (De anima 11. 2, 413 a 13). The equality of circles with equal radii can of course be proved by superposition, but, as we have seen, Euclid avoided this method wherever he could, and there is nothing technically wrong in saying "By equal circles I mean circles with equal radii." No flaw is thereby introduced into the system of the Elements; for the definition could only be objected to if it could be proved that the equality predicated of the two circles in the definition was not the same thing as the equality predicated of other equal figures in the Elements on the basis of the Congruence-Axiom, and, needless to say, this cannot be proved because it is not true. The existence of equal circles (in the sense of the definition) follows from the existence of equal straight lines and 1. Post. 3. The Greeks had no distinct word for radius, which is with them, as here, the (straight line drawn) from the centre ǹ EK TOû KÉνTρOV (evbeîa); and so definitely was the expression appropriated to the radius that ex Tou KéνTPOV was used without the article as a predicate, just as if it were one word. Thus, e.g., in III. I ÈK KÉνTрov yap means "for they are radii": cf. Archimedes, On the Sphere and Cylinder 1. 2, ή ΒΕ ἐκ τοῦ κέντρου ἐστὶ τοῦ....κύκλου, ΒΕ is a radius of the circle. DEFINITION 2. Εὐθεῖα κύκλου ἐφάπτεσθαι λέγεται, ἥτις ἁπτομένη τοῦ κύκλου καὶ ἐκβαλλομένη οὐ τέμνει τὸν κύκλον. Euclid's phraseology here shows the regular distinction between aтeσbαι and its compound épánтeobai, the former meaning "to meet" and the latter "to touch." The distinction was generally observed by Greek geometers from Euclid onwards. There are however exceptions so far as anтeobα is concerned; thus it means "to touch" in Eucl. iv. Def. 5 and sometimes in Archimedes. On the other hand, éþá¬τeσðaɩ is used by Aristotle in certain |