f light passing from the sun to the moon. At this moment, then,
the imaginary lines joining the sun, the moon, and the earth, make a
right angle triangle. But the properties of the right-angle triangle had
long been studied and were well under stood. One acute angle of such a
triangle determines the figure of the triangle itself. We have already
seen that Thales, the very earliest of the Greek philosophers, measured
the distance of a ship at sea by the application of this principle. Now
Aristarchus sights the sun in place of Thales' ship, and, sighting the
moon at the same time, measures the angle and establishes the shape of
his right-angle triangle. This does not tell him the distance of the
sun, to be sure, for he does not know the length of his base-line--that
is to say, of the line between the moon and the earth. But it does
establish the relation of that base-line to the other lines of the
triangle; in other words, it tells him the distance of the sun in terms
of the moon's distance. As Aristarchus strikes the angle, it shows that
the sun is eighteen times as distant as the moon. Now, by comparing the
apparent size of the sun with the apparent size of the moon--which, as
we have seen, Aristarchus has already measured--he is able to tell us
that, the sun is "more than 5832 times, and less than 8000" times larger
than the moon; though his measurements, taken by themselves, give
no clew to the actual bulk of either body. These conclusions, be it
understood, are absolutely valid inferences--nay, demonstrations--from
the measurements involved, provided only that these measurements have
been correct. Unfortunately, the angle of the triangle we have just seen
measured is exceedingly difficult to determine with accuracy, while at
the same time, as a moment's reflection will show, it is so large an
angle that a very slight deviation from the truth will greatly affect
the distance at which its line joins the other side of the triangle.
Then again, it is virtually impossible to tell the precise moment when
the moon is at half, as the line it gives is not so sharp that we can
fix it with absolute accuracy. There is, moreover, another element of
error due to the refraction of light by the earth's atmosphere. The
experiment was probably made when the sun was near the horizon, at which
time, as we now know, but as Aristarchus probably did not suspect, the
apparent displacement of the sun's position is considerable; and this
displa