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

TRIGONOMETRY 187
merely a curiosity, and of no practical or
theoretical interest. The accurate determination of wr is one of the two parts of
the famous problem of squaring the circle.
The other part of the problem is, by the
theoretical methods of pure geometry to
describe a straight line equal in length to the
circumference. Both parts of the problem
are now known to be impossible; and the
insoluble problem has now lost all special
practical or theoretical interest, having become absorbed in wider ideas.
After this digression on the value of 7r, we
now return to the question of the general
definition of the magnitude of an angle, so as
to be able to produce an angle corresponding
to any value u. Suppose a moving point, Q,
to start from A on OX (cf. fig. 27), and to
rotate in the positive direction (anti-clockwise, in the figure considered) round the circumference of the circle for any number of
times, finally resting at any point, e.g. at P
or P' or P" or P"'. Then the total length
of the curvilinear circular path traversed,
divided by the radius of the circle, r, is the
generalized definition of a positive angle of
any size. Let x, y be the coordinates of the
point in which the point Q rests, i.e. in one of
the four alternative positions mentioned in
fig. 27; x and y (as here used) will either x
and y, or x' and y, or x' and y', or x and y".