An introduction to mathematics, by A. N. Whitehead.

192 INTRODUCTION TO MATHEMATICS
In this formula co, ci, c2, c3, etc., and also
el, e2, e3, etc., are constants, chosen so as to
suit the particular function. Again we have
to ask, How many terms have to be chosen?
And here a new difficulty arises: for we can
prove that, though in some particular cases
a definite number will do, yet in general all
we can do is to approximate as closely as
we like to the value of the function by taking more and more terms. This process of
gradual approximation brings us to the consideration of the theory of infinite series, an
essential part of mathematical theory which
we will consider in the next chapter.
The above method of expressing a periodic
function as a sum of sines is called the "harmonic analysis" of the function. For example, at any point on the sea coast the tides
rise and fall periodically. Thus at a point
near the Straits of Dover there will be two
daily tides due to the rotation of the earth.
The daily rise and fall of the tides are complicated by the fact that there are two tidal
waves, one coming up the English Channel,
and the other which has swept round the
North of Scotland, and has then come southward down the North Sea. Again some high
tides are higher than others: this is due to
the fact that the Sun has also a tide-generating influence as well as the Moon. In this
way monthly and other periods are intro