A history of Greek mathematics - Wilbourhall.org

354 HERON OF ALEXANDRIA
reversing the ray so that the eye is at D instead of 0, and the
object at C instead of D, we must have fi > a. But (3 was
less than a, which is impossible. (Similarly it can be proved
that a is not less than ft.) Therefore a. = /?.
In the Pseudo-Euclidean Gatoptrica the proposition is
practically assumed ; for the third assumption or postulate
at the beginning states in effect that, in the above figure, if A
be the point of incidence, CE : EA = DH : HA (where DH is
perpendicular to AB). It follows instantaneously (Prop. 1)
that
ACAE = LDAH.
If the mirror is
the convex side of a circle, the same result
follows a fortiori. Let GA, AD meet
the arc at
equal angles, and CB, BD at
unequal angles. Let AE be the tangent
at A, and complete the figure.
Then, says Heron, (the angles GAC,
BAD being by hypothesis equal), if
we
subtract the equal angles GAE, BAF
from the equal angles GAC, BAD (both
pairs of angles being ' mixed
', be it
observed), we have Z EAC = I FAD. Therefore CA+AD
< CF+FD and a fortiori < CB + BD.
The problems solved (though the text is so corrupt in places
that little can be made of it) were such as the following:
11, To construct a right-handed mirror (i.e. a mirror which
makes the right side right and the left side left instead of
the opposite); 12, to construct the mirror called polyiheoron
('with many images'); 16, to construct a mirror inside the
window of a house, so that you can see in it (while inside
the room) everything that passes in the street;
18, to arrange
mirrors in a given place so that a person who approaches
cannot actually see either himself or any one else but can see
any image desired (a 'ghost-seer').