de this by three hundred and
eighty-six, and thus we see that the tidal efficiency of the sun is
less than half that of the moon.

When the solar tide and the lunar tide are acting in unison, they
conspire to produce very high tides and very low tides, or, as we
call them, spring tides. On the other hand, when the sun is so placed
as to give us a low tide while the moon is producing a high tide, the
net result that we actually experience is merely the excess of the
lunar tide over the solar tide; these are what we call neap tides. In
fact, by very careful and long-continued observations of the rise and
fall of the tides at a particular port, it becomes possible to
determine with accuracy the relative ranges of spring tides and neap
tides; and as the spring tides are produced by moon plus sun, while
the neap tides are produced by moon minus sun, we obtain a means of
actually weighing the relative masses of the sun and moon. This is one
of the remarkable facts which can be deduced from a prolonged study of
the tides.

The demonstration of the law of the tide-producing force is of a
mathematical character, and I do not intend in these lectures to enter
into mathematical calculations. There is, however, a simple line of
reasoning which, though it falls far short of actual demonstration,
may yet suffice to give a plausible reason for the law.

The tides really owe their origin to the fact that the tide-producing
agent operates more powerfully on those parts of the tide-exhibiting
body which are near to it, than on the more distant portions of the
same. The nearer the two bodies are together, the larger
proportionally will be the differences in the distances of its various
parts from the tide-producing body; and on this account the leverage,
so to speak, of the action by which the tides are produced is
increased. For instance, if the two bodies were brought within half
their original distance of each other, the relative size of each body,
as viewed from the other, will be doubled; and what we have called the
leverage of the tide-producing ability will be increased twofold. The
gravitation also between the two bodies is increased fourfold when the
distance is halved, and consequently, the tide-producing ability is
doubled for one reason, and increased fourfold again by another;
hence, the tides will be increased eightfold when the distance is
reduced to one half. Now, as eight is the cube of two, this
illustration may be t