hod is only an approximation. The study of the tides
is based on accurate observation of their rise and fall on different
places round the earth. To show how these observations are to be made,
and how they are to be discussed and reduced when they have been made,
I may refer to the last edition of the _Admiralty Manual of Scientific
Inquiry_, 1886. For a complete study of the tides at any port a
self-registering tide-gauge should be erected, on which not alone the
heights and times of high and low water should be depicted, but also
the continuous curve which shows at any time the height of the water.
In fact, the whole subject of the practical observation and discussion
and prediction of tides is full of valuable instruction, and may be
cited as one of the most complete examples of the modern scientific
methods.

In the first place, the tide-gauge itself is a delicate instrument;
it is actuated by a float which rises and falls with the water, due
provision being made that the mere influence of waves shall not make
it to oscillate inconveniently. The motion of the float when suitably
reduced by mechanism serves to guide a pencil, which, acting on the
paper round a revolving drum, gives a faithful and unintermitting
record of the height of the water.

Thus what the tide-gauge does is to present to us a long curved line
of which the summits correspond to the heights of high water, while
the depressions are the corresponding points of low water. The long
undulations of this curve are, however, very irregular. At spring
tides, when the sun and the moon conspire, the elevations rise much
higher and the depressions sink much lower than they do at neap tides,
when the high water raised by the moon is reduced by the action of the
sun. There are also many minor irregularities which show the tides to
be not nearly such simple phenomena as might be at first supposed. But
what we might hastily think of as irregularities are, in truth, the
most interesting parts of the whole phenomena. Just as in the
observations of the planets the study of the perturbations has led us
to results of the widest interest and instruction, so it is these
minor phenomena of the tides which seem most pregnant with scientific
interest.

The tide-gauge gives us an elaborate curve. How are we to interpret
that curve? Here indeed a most beautiful mathematical theorem comes to
our aid. Just as ordinary sounds consist of a number of undulations
blended toget