A history of Greek mathematics - Wilbourhall.org

542 COMMENTATORS AND BYZANTINES
By taking the point
AT on BD such that MN = MA, and
bisecting the angle NMA by the straight line MOP meeting
BD in 0, we find that, if MOP is a plane mirror, the ray BO
is reflected to A.
Similarly, by continually bisecting angles and making more
mirrors, we can get any number of other points of impact. Making
the mirrors so short as to form a continuous curve, we get
the curve containing all points such that the sum of the distances
of each of them from A and B is constant and equal to BQ, BK,
or BN. '
If then ', says Anthemius, ' we stretch a string passed
round the points A, B, and through the first point taken on the
rays which are to be reflected, the said curve will be described,
which is part of the so-called " ellipse ", with reference to
which (i.e. by the revolution of which round BA) the surface
of impact of the saichmirror has to be constructed*'
We have here apparently the first mention of the construction
of an ellipse by means of a string stretched tight round
the foci. Anthemius's construction depends upon two propositions
proved by Apollonius (1) that the sum of the focal
distances of any point on the ellipse is constant, (2) that the
focal distances of any point make equal angles with the
tangent at that point, and also (3) upon a proposition not
found in Apollonius, namely that the straight line joining