are too technical to be detailed here. The extent of his
mathematical knowledge, however, is suggested by the fact that he
computed in great detail the number of grains of sand that would
be required to cover the sphere of the sun's orbit, making
certain hypothetical assumptions as to the size of the earth and
the distance of the sun for the purposes of argument.
Mathematicians find his computation peculiarly interesting
because it evidences a crude conception of the idea of
logarithms. From our present stand-point, the paper in which this
calculation is contained has considerable interest because of its
assumptions as to celestial mechanics. Thus Archimedes starts out
with the preliminary assumption that the circumference of the
earth is less than three million stadia. It must be understood
that this assumption is purely for the sake of argument.
Archimedes expressly states that he takes this number because it
is "ten times as large as the earth has been supposed to be by
certain investigators." Here, perhaps, the reference is to
Eratosthenes, whose measurement of the earth we shall have
occasion to revert to in a moment. Continuing, Archimedes asserts
that the sun is larger than the earth, and the earth larger than
the moon. In this assumption, he says, he is following the
opinion of the majority of astronomers. In the third place,
Archimedes assumes that the diameter of the sun is not more than
thirty times greater than that of the moon. Here he is probably
basing his argument upon another set of measurements of
Aristarchus, to which, also, we shall presently refer more at
length. In reality, his assumption is very far from the truth,
since the actual diameter of the sun, as we now know, is
something like four hundred times that of the moon. Fourth, the
circumference of the sun is greater than one side of the
thousand- faced figure inscribed in its orbit. The measurement,
it is expressly stated, is based on the measurements of
Aristarchus, who makes the diameter of the sun 1/170 of its
orbit. Archimedes adds, however, that he himself has measured the
angle and that it appears to him to be less than 1/164, and
greater than 1/200 part of the orbit. That is to say, reduced to
modern terminology, he places the limit of the sun's apparent
size between thirty-three minutes and twenty-seven minutes of
arc. As the real diameter is thirty-two minutes, this calculation
is surprisingly exact, considering the implements then at
command. But the honor of first making it must be given to
Aristarchus and not to Archimedes.

We need not follow Archimedes to the limits of his
incomprehensible numbers of sand-grains. The calculation is
chiefly remarkable because it was made before the introduction of
the so-called Arabic numerals had simplified mathematical
calculations. It will be recalled that the Greeks used letters
for numerals, and, having no cipher, they soon found themselves
in difficulties when large numbers were involved. The Roman
system of numerals simplified the matter somewhat, but the
beautiful simplicity of the decimal system did not come into
vogue until the Middle Ages, as we shall see. Notwithstanding the
difficulties, however, Archimedes followed out his calculations
to the piling up of bewildering numbers, which the modern
mathematician finds to be the consistent outcome of the problem
he had set himself.

But it remains to notice the most interesting feature of this
document in which the calculation of the sand- grains is
contained. "It was known to me," says Archimedes, "that most
astronomers understand by the expression 'world' (universe) a
ball of which the centre is the middle point of the earth, and of
which the radius is a straight line between the centre of the
earth and the sun." Archimedes himself appears to accept this
opinion of the majority,--it at least serves as well as the
contrary hypothesis for the purpose of his calculation,--but he
goes on to say: "Aristarchus of Samos, in his writing against the
astronomers, seeks to establish the fact that the world is really
very different from this. He holds the opinion that the fixed
stars and the sun are immovable and that the earth revolves in a
circular line about the sun, the sun being at the centre of this
circle." This remarkable bit of testimony establishes beyond
question the position of Aristarchus of Samos as the Copernicus
of antiquity. We must make further inquiry as to the teachings of
the man who had gained such a remarkable insight into the true
system of the heavens.


ARISTARCHUS OF SAMOS, THE COPERNICUS OF ANTIQUITY

It appears that Aristarchus was a contemporary of Archimedes, but
the exact dates of his life are not known. He was actively
engaged in making astronomical observations in Samos somewhat
before the middle of the third century B.C.; in other words, just
at the time when the activities of the Alexandrian school were at
their height. Hipparchus, at a later day, was enabled to compare
his own observations with those made by Aristarchus, and, as we
have just seen, his work was well known to so distant a
contemporary as Archimedes. Yet the facts of his life are almost
a blank for us, and of his writings only a single one has been
preserved. That one, however, is a most important and interesting
paper on the measurements of the sun and the moon. Unfortunately,
this paper gives us no direct clew as to the opinions of
Aristarchus concerning the relative positions of the earth and
sun. But the testimony of Archimedes as to this is unequivocal,
and this testimony is supported by other rumors in themselves
less authoritative.

In contemplating this astronomer of Samos, then, we are in the


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