THE ELEMENTS. BOOKS II-III 
381 
generally of the same sort as those of Book I. Definition 1, 
stating that equal circles are those which have their diameters 
or their radii equal, might alternatively be regarded as a 
postulate or a theorem; if stated as a theorem, it could only 
be proved by superposition and the congruence-axiom. It is 
curious that the Greeks had no single word for radius, which 
was with them ‘ the (straight line) from the centre 17 tov 
xeurpov. A tangent to a circle is defined (Def. 2) as a straight 
line which meets the circle but, if produced, does not cut it; 
this is provisional pending the proof in III. 16 that such lines 
exist. The definitions (4, 5) of straight lines (in a circle), 
i. e. chords, equally distant or more or less distant from the 
centre (the test being the length of the perpendicular from 
the centre on the chord) might have referred, more generally, 
to the distance of any straight line from any point. The 
definition (7) of the ‘angle of a segment’ (the ‘mixed’ angle 
made by the circumference with the base at either end) is 
a survival from earlier text-books (cf. Props. 16, 31). The 
definitions of the ‘angle in a segment’ (8) and of ‘similar 
segments’ (11) assume (provisionally pending III. 21) that the 
angle in a segment is one and the same at whatever point of 
the circumference it is formed. A sector (ropevy, explained by 
a scholiast as crKVTOTOfUKbs ropevs, a shoemaker’s knife) is 
defined (10), but there is nothing about ‘ similar sectors ’ and 
no statement that similar segments belong to similar sectors. 
Of the propositions of Book III we may distinguish certain 
groups. ‘ Central properties account for four propositions, 
namely 1 (to find the centre of a circle), 3 (any straight line 
through the centre which bisects any chord not passing- 
through the centre cuts it at right angles, and vice versa), 
4 (two chords not passing through the centre cannot bisect 
one another) and 9 (the centre is the only point from which 
more than two equal straight lines can be drawn to the 
circumference). Besides 3, which shows that any diameter 
bisects the whole, series of chords at right angles to it, three 
other propositions throw light on the form of the circum 
ference of a circle, 2 (showing that it is everywhere concave 
towards the centre), 7 and 8 (dealing with the varying lengths 
of straight lines drawn from any point, internal or external, 
to the concave or convex circumference, as the case may be,