The quadrature of the circle, the square root of two, and the right-angled triangle, by William Alexander Myers.

INTRODUCTION. 55
r, the required radius O'A' by R', and the apothem O'D' by r', we have
O'D 00 - + OD
2 ~ 2
or
R -- 9[1]
In the right triangle OA'O', we have (III. 44),
0'A'2 = 00' X OD',
or
R'= 1R X '; [2]
therefore, -' is an arithmetic mean between R and r, and RB is a geometric mean
between R and r'.
MEASUREMENT OF THE CIRCLE.
The principle which we employed in the comparison of incommensurable
ratios (II. 49) is fundamentally the same as that whicl we are about to apply
to the meastrementof the circle, but we shall now state it in a much more general form, better adapted for subsequent application.
28. Definitions. I. A variable quantity, or simply, a variable, is a quantity which
has different successive values.
II. When the successive values of a variable, under the conditions imposed upon it, approach more and more nearly to the value of some fixed or
constant quantity, so that the difference between the variable and the constant
may become less than any assigned quantity, without becoming zero, the
variable is said to approach indefinitely to the constant; and the constant is
called the limit of the variable.
Or, more briefly, the limit of a variable is a constant quantity to which the
variable, under the conditions imposed upon it, approaches indefinitely.
As an example, illustrating these definitions, let a point be required to move
from A to B under the following conditions: it
shall first move over one-half of AB, that is to C; C C' C"
then over one-half of CB, to C(; then over one- A B
half of C0B, to C; and so on indefinitely; then
the distance of the point from A is a variable, and this variable approaches indefinitely to the constant AB, as its limit, without ever reaching it.
As a second example, let A denote the angle of any regular polygon, and n
the number of sides of the polygon; tlen, a right angle being taken as the
unit, we have (8),
2 4
A - 2 —.
n
The value of A is a variable depending upon n; and since n may be taken so
great that - shall be less than any assigned quantity however small, the value
n