the earth on its axis follows as a necessary consequence in
explanation of the seeming motion of the stars. Here, then, was
the heliocentric doctrine reduced to a virtual demonstration by
Aristarchus of Samos, somewhere about the middle of the third
century B.C.

It must be understood that in following out the, steps of
reasoning by which we suppose Aristarchus to have reached so
remarkable a conclusion, we have to some extent guessed at the
processes of thought- development; for no line of explication
written by the astronomer himself on this particular point has
come down to us. There does exist, however, as we have already
stated, a very remarkable treatise by Aristarchus on the Size and
Distance of the Sun and the Moon, which 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 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


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