f 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 app