fifteenth part
of a sign" of the zodiac; that is to say, since there are twenty-four,
signs in the zodiac, one-fifteenth of one twenty-fourth, or in modern
terminology, one degree of arc. This is Aristarchus's measurement of the
moon to which we have already referred when speaking of the measurements
of Archimedes.

"If we admit these six hypotheses," Aristarchus continues, "it follows
that the sun is more than eighteen times more distant from the earth
than is the moon, and that it is less than twenty times more distant,
and that the diameter of the sun bears a corresponding relation to the
diameter of the moon; which is proved by the position of the moon when
dichotomized. But the ratio of the diameter of the sun to that of the
earth is greater than nineteen to three and less than forty-three to
six. This is demonstrated by the relation of the distances, by the
position (of the moon) in relation to the earth's shadow, and by the
fact that the arc subtended by the moon is a fifteenth part of a sign."

Aristarchus follows with nineteen propositions intended to elucidate
his hypotheses and to demonstrate his various contentions. These show a
singularly clear grasp of geometrical problems and an altogether correct
conception of the general relations as to size and position of the
earth, the moon, and the sun. His reasoning has to do largely with
the shadow cast by the earth and by the moon, and it presupposes
a considerable knowledge of the phenomena of eclipses. His first
proposition is that "two equal spheres may always be circumscribed in
a cylinder; two unequal spheres in a cone of which the apex is found on
the side of the smaller sphere; and a straight line joining the centres
of these spheres is perpendicular to each of the two circles made by the
contact of the surface of the cylinder or of the cone with the spheres."

It will be observed that Aristarchus has in mind here the moon, the
earth, and the sun as spheres to be circumscribed within a cone,
which cone is made tangible and measurable by the shadows cast by the
non-luminous bodies; since, continuing, he clearly states in proposition
nine, that "when the sun is totally eclipsed, an observer on the earth's
surface is at an apex of a cone comprising the moon and the sun."
Various propositions deal with other relations of the shadows which need
not detain us since they are not fundamentally important, and we
may pass to the final conclusions of Aristarchus,