IX.]

METHOD OF ERATOSTHENES.

171

with the axis of the gnomon, so as to cast no shadow, and,
if produced, would strike the earth's centre (-£). In Alex-

Northern Ray

Southern Ray
••A

andria at the same time a ray of the sun (CD\ falling on
the point of the gnomon (D\ would form an angle with the
axis of the gnomon (DG). Now, if the axis of the gnomon at
Alexandria (DG) is produced to the centre of the earth (JE),
the line thus drawn (DGE) would intersect the lines formed by
these two rays (CDH> ABE\ and form alternate angles with
them j one of these angles (BED) being at the centre of the
earth (E\ the other (EDH) at the point where the northern-
most of the two rays meets the point of the gnomon at Alex-
andria (D). But, since alternate angles are equal, and the arcs
of a circle which subtend equal angles bear the same proportion
to the circles of which they are segments, the arc of the celestial
sphere marked by the shadow of the gnomon on the scaphe at
Alexandria (FG) must correspond to the arc of the great circle of
the earth which lies between Syene and Alexandria (BD). Con-
sequently, since it has been already shewn that the shadow of the
gnomon at Alexandria measures one-fiftieth part of the great
circle of the heavens, the arc between Syene and Alexandria
must be one-fiftieth of the great circle of the earth; and as the
distance between Syene and Alexandria is 5,000 stadia, the great