chord.

Exact formulae are:--Area = 1/2a squared([theta] - sin [theta]) = 1/2a squared[theta]
-1/4c squared cot 1/2[theta] = 1/2a squared - 1/2c sqrt(a squared - 1/4c squared). If h be given, we can use
c squared + 4h squared = 8ah, 2h = c tan 1/4[theta] to determine [theta].

Approximate formulae are:--Area = (1/15)(6c + 8c2)h; = (2/3) sqrt(c squared +
(8/5)h squared).h; = (1/15)(7c + 3[alpha])h, [alpha] being the true length of
the arc.

From these results the mensuration of any figure bounded by circular
arcs and straight lines can be determined, e.g. the area of a _lune_
or _meniscus_ is expressible as the difference or sum of two segments,
and the circumference as the sum of two arcs. (C. E.*)

_Squaring of the Circle._

The problem of finding a square equal in area to a given circle, like
all problems, may be increased in difficulty by the imposition of
restrictions; consequently under the designation there may be embraced
quite a variety of geometrical problems. It has to be noted, however,
that, when the "squaring" of the circle is especially spoken of, it is
almost always tacitly assumed that the restrictions are those of the
Euclidean geometry.

Since the area of a circle equals that of the rectilineal triangle whose
base has the same length as the circumference and whose altitude equals
the radius (Archimedes, [Greek: Kyklou metresis], prop. 1), it follows
that, if a straight line could be drawn equal in length to the
circumference, the required square could be found by an ordinary
Euclidean construction; also, it is evident that, conversely, if a
square equal in area to the circle could be obtained it would be
possible to draw a straight line equal to the circumference.
Rectification and quadrature of the circle have thus been, since the
time of Archimedes at least, practically identical problems. Again,
since the circumferences of circles are proportional to their
diameters--a proposition assumed to be true from the dawn almost of
practical geometry--the rectification of the circle is seen to be
transformable into finding the ratio of the circumference to the
diameter. This correlative numerical problem and the two purely
geometrical problems are inseparably connected historically.

Probably the earliest value for the ratio was 3. It was so among the
Jews (1 Kings vii. 23, 26), the Babylonians (Oppert, _Journ. asiatique_,
August 1872, October 1874), the Chinese (Biot, _Journ.