Approximate formulae are:—Arc = 1⁄3(8c2 - c) (Huygen’s formula); arc = 1⁄45(c - 40c2 + 256c4).
3. Segment.—Data: a, θ, c, c2, as in (2); h = height of segment, i.e. distance of mid-point of arc from chord.
Exact formulae are:—Area = ½a²(θ - sin θ) = ½a²θ - ¼c² cot ½θ = ½a² - ½c √(a² - ¼c²). If h be given, we can use c² + 4h² = 8ah, 2h = c tan ¼θ to determine θ.
Approximate formulae are:—Area = 1⁄15(6c + 8c2)h; = 2⁄3 √(c² + 8/5h²)·h; = 1⁄15(7c + 3α)h, α 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, Κύκλου μέτρησις, 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. asiatique, June 1841), and probably also the Greeks. Among the ancient Egyptians, as would appear from a calculation in the Rhind papyrus, the number (4⁄3)4, i.e. 3.1605, was at one time in use.[1] The first attempts to solve the purely geometrical problem appear to have been made by the Greeks (Anaxagoras, &c.)[2], one of whom, Hippocrates, doubtless raised hopes of a solution by his quadrature of the so-called meniscoi or lune.[3]