, let us bear in mind that we owe them
to the most solid and severe processes of human thought.

Yet the principle can in most cases be made so clear that the reader
will not be asked to take much on trust. It is, for instance, a matter
of common knowledge that gold is soft enough to be beaten into gold
leaf. It is a matter of common sense, one hopes, that if you beat a
measured cube of gold into a leaf six inches square, the mathematician
can tell the thickness of that leaf without measuring it. As a matter of
fact, a single grain of gold has been beaten into a leaf seventy-five
inches square. Now the mathematician can easily find that when a single
grain of gold is beaten out to that size, the leaf must be 1/367,000 of
an inch thick, or about a thousand times thinner than the paper on
which these words are printed; yet the leaf must be several molecules
thick.

The finest gold leaf is, in fact, too thick for our purpose, and we turn
with a new interest to that toy of our boyhood the soap-bubble. If you
carefully examine one of these delicate films of soapy water, you notice
certain dark spots or patches on them. These are their thinnest parts,
and by two quite independent methods--one using electricity and the
other light--we have found that at these spots the bubble is less than
the three-millionth of an inch thick! But the molecules in the film
cling together so firmly that they must be at least twenty or thirty
deep in the thinnest part. A molecule, therefore, must be far less than
the three-millionth of an inch thick.

We found next that a film of oil on the surface of water may be even
thinner than a soap-bubble. Professor Perrin, the great French authority
on atoms, got films of oil down to the fifty-millionth of an inch in
thickness! He poured a measured drop of oil upon water. Then he found
the exact limits of the area of the oil-sheet by blowing upon the water
a fine powder which spread to the edge of the film and clearly outlined
it. The rest is safe and simple calculation, as in the case of the
beaten grain of gold. Now this film of oil must have been at least two
molecules deep, so a single molecule of oil is considerably less than a
hundred-millionth of an inch in diameter.

Innumerable methods have been tried, and the result is always the same.
A single grain of indigo, for instance, will colour a ton of water. This
obviously means that the grain contains billions of molecules which
spread through th