hich so clearly suggests the
methods of reasoning of the great astronomer, and so explicitly cites
the results of his measurements, that we cannot well pass it by
without quoting from it at some length. It is certainly one of the most
remarkable scientific documents of antiquity. As already noted, the
heliocentric doctrine is not expressly stated here. It seems to be
tacitly implied throughout, but it is not a necessary consequence of any
of the propositions expressly stated. These propositions have to do with
certain observations and measurements and what Aristarchus believes to
be inevitable deductions from them, and he perhaps did not wish to have
these deductions challenged through associating them with a theory which
his contemporaries did not accept. In a word, the paper of Aristarchus
is a rigidly scientific document unvitiated by association with any
theorizings that are not directly germane to its central theme. The
treatise opens with certain hypotheses as follows:

"First. The moon receives its light from the sun.

"Second. The earth may be considered as a point and as the centre of the
orbit of the moon.

"Third. When the moon appears to us dichotomized it offers to our view a
great circle (or actual meridian) of its circumference which divides the
illuminated part from the dark part.

"Fourth. When the moon appears dichotomized its distance from the sun is
less than a quarter of the circumference (of its orbit) by a thirtieth
part of that quarter."

That is to say, in modern terminology, the moon at this time lacks three
degrees (one thirtieth of ninety degrees) of being at right angles with
the line of the sun as viewed from the earth; or, stated otherwise, the
angular distance of the moon from the sun as viewed from the earth is at
this time eighty-seven degrees--this being, as we have already observed,
the fundamental measurement upon which so much depends. We may fairly
suppose that some previous paper of Aristarchus's has detailed the
measurement which here is taken for granted, yet which of course could
depend solely on observation.

"Fifth. The diameter of the shadow (cast by the earth at the point where
the moon's orbit cuts that shadow when the moon is eclipsed) is double
the diameter of the moon."

Here again a knowledge of previously established measurements is taken
for granted; but, indeed, this is the case throughout the treatise.

"Sixth. The arc subtended in the sky by the moon is a