I presume you will admit, that the circular measure
of an angle is founded on the two well known geometri-
cal propositions —

First, That in circles of the same radius, the angle is
proportional to the arc which subtends it.
Second, That for the same angle, in circles of different
radii, the arc varies as the radius.
Then, If we take the unit of angle to be the angle
which is subtended by an arc of the same length as the
radius ; in this case, the angle being measured by the
ratio of two lines, it can enter a calculation in which we
are dealing with lines.
Let X equal the number of degrees required. Then,

X is subtended by an arc of the same length as the

radius, equal to R. Now, an angle of 186° is subtended

by a semi-circle, that is, by an arc equal to P x R, where

P is equal to 3*14159 &c. you would say, but which I

maintain to be equal to 3-125.
X° R I

Then, on your data, 00= p-^ =^ ; Therefore, X

180°

= = 57°'2QS82 &c. will be the value of the arc,

3-14159 ^^ ^^

and equal to the radius; 57°-29582 x 2 = Ii4°-59i64, will

be the value of the diameter; and Ii4°-59i64 x 3'i4i59 =

359°* 999950376, will be the value of the circumference;

but is not in agreement with the number of degrees into

which the circumference of the circle is divided.
38 THE QUADRATURE OF THE CIRCLE.
