382 
EUCLID 
and proving that they are of maximum or minimum length 
when they pass through the centre, and that they diminish or 
increase as they diverge more and more from the maximum 
or minimum straight lines on either side, while the lengths of 
any two which are equally inclined to them, one on each side, 
are equal). 
Two circles which cut or touch one another are dealt with 
in 5, 6 (the two circles cannot have the same centre), 10, 13 
(they cannot cut in more points than two, or touch at more 
points than one), 11 and the interpolated 12 (when they touch, 
the line of centres passes through the point of contact). 
14, 15 deal with chords (which are equal if equally distant 
from the centre and vice versa, while chords more distant 
from the centre are less, and chords less distant greater, and 
vice versa). 
16-19 are concerned with tangent properties including the 
drawing of a tangent (17); it is in 16 that we have the 
survival of the ‘angle of a semicircle’, which is proved greater 
than any acute rectilineal angle, while the ‘ remaining ’ angle 
(the ‘angle’, afterwards called KeparoeiSrjs, or ‘hornlike’, 
between,the curve and the tangent at the point of contact) 
is less than any rectilineal angle. These ‘ mixed ’ angles, 
occurring in 16 and 31, appear no more in serious Greek 
geometry, though controversy about their nature went on 
in the works of commentators down to Clavius, Peletarius 
(Pettier), Vieta, Galilei and Wallis. 
We now come to propositions about segments. 20 proves 
that the angle at the centre is double of the angle at the 
circumference, and 21 that the angles in the same segment are 
all equal, which leads to the property of the quadrilateral 
in a circle (22). After propositions (23, 24) on ‘ similar 
segments ’, it is proved that in equal circles equal arcs subtend 
and are subtended by equal angles at the centre or circum 
ference, and equal arcs subtend and are subtended by equal 
chords (26-9). 30 is the problem of bisecting a given arc, 
and 31 proves that the angle in a segment is right, acute or 
obtuse according as the segment is a semicircle, greater than 
a semicircle or less than a semicircle. 32 proves that the 
angle made by a tangent with a chord through the point 
of contact is equal to the angle in the alternate segment;