Plane trigonometry, by S.L. Loney.

THE RADIAN.

7

the lines ab and AB must be parallel, and hence
AB OA
ab -OAc ~ (Euc. vi. 4).
ab Oa'
Also the polygon ABCD... being regular, its perimeter,
i.e. the sum of its sides, is equal to n. AB. Similarly for
the inner polygon.
Hence we have
Perimeter of the outer polygon n. AB AB OA
Perimeter of the inner polygon n. ab ab Oa............ ().
This relation exists 'whatever be the number of sides
in the polygons.
Let then the number of sides be indefinitely increased
(i.e. let n become inconceivably great) so that finally the
perimeter of the outer polygon will be the same as the
circumference of the outer circle, and the perimeter of the
inner polygon the same as the circumference of the inner
circle.
The relation (1) will then become
Circumference of outer circle OA
Circumference of inner circle Oa
Radius of outer circle
Radius of inner circle'
Circumference of outer circle
Hence
Radius of outer circle
Circumference of inner circle
Radius of inner circle
Since there was no restriction whatever as to the sizes
of the two circles, it follows that the quantity
Circumference of a circle
Radius of the circle
is the same for all circles.