Exact formulae are:—Arc＝𝑎θ, where θ may be given directly, or indirectly by the relation 𝑐＝2𝑎 sin 12θ. Area of sector＝12𝑎2θ＝12 radius × arc.
Approximate formulae are:—Arc＝13(8𝑐2 − 𝑐) (Huygen’s formula); arc＝145(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＝12𝑎2(θ − sin θ)＝12𝑎2θ − 14𝑐2 cot 12θ＝12𝑎2 − 12𝑐 √(𝑎2 − 14𝑐2). If ℎ be given, we can use 𝑐2 + 4ℎ2＝8𝑎ℎ, 2ℎ＝𝑐 tan 14θ to determine θ.
Approximate formulae are:—Area＝115(6𝑐 + 8𝑐2)ℎ;＝23 √(𝑐2 + 8/5ℎ2)·ℎ;＝115(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 (43)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]
[The Greeks were in possession of several relations pertaining to the quadrature of the lune. The following are among the more interesting. In fig. 6, ABC is an isosceles triangle right angled at C, ADB is the semicircle described on AB as diameter, AEB the circular arc described with centre C and radius CA＝CB. It is easily shown that the areas of the lune ADBEA and the triangle ABC are equal. In fig. 7, ABC is any triangle right angled at C, semicircles are described on the three sides, thus forming two lunes AFCDA and CGBEC. The sum of the areas of these lunes equals the area of the triangle ABC.]
As for Euclid, it is sufficient to recall the facts that the original author of prop. 8 of book iv. had strict proof of the ratio being < 4, and the author of prop. 15 of the ratio being > 3, and to direct attention to the importance of book x. on incommensurables and props. 2 and 16 of book xii., viz. that “circles are to one another as the squares on their diameters” and that “in the greater of two concentric circles a regular 2𝑛-gon can be inscribed which shall not meet the circumference of the less,” however nearly equal the circles may be.
With Archimedes (287–212 B.C.) a notable advance was made. Taking the circumference as intermediate between the perimeters of the inscribed and the circumscribed regular 𝑛-gons, he showed that, the radius of the circle being given and the perimeter of some particular circumscribed regular polygon obtainable, the perimeter of the circumscribed regular polygon of double the number of sides could be calculated; that the like was true of the inscribed polygons; and that consequently a means was thus afforded of approximating to the circumference of the circle. As a matter of fact, he started with a semi-side AB of a circumscribed regular hexagon meeting the circle in B (see fig. 8), joined A and B with O the centre, bisected the angle AOB by OD, so that BD became the semi-side of a circumscribed regular 12-gon; then as AB:BO:OA::1: √3:2 he sought an approximation to √3 and found that AB:BO > 153:265. Next he applied his theorem[4] BO + OA:AB::OB:BD to calculate BD; from this in turn he calculated the semi-sides of the circumscribed regular 24-gon, 48-gon and 96-gon, and so finally established for the circumscribed regular 96-gon that perimeter:diameter < 317:1. In a quite analogous manner he proved for the inscribed regular 96-gon that perimeter:diameter > 31071:1. The conclusion from these therefore was that the ratio of circumference to diameter is < 317 and > 31071. This is a most notable piece of work; the immature condition of arithmetic at the time was the only real obstacle preventing the evaluation of the ratio to any degree of accuracy whatever.[5]
No advance of any importance was made upon the achievement of Archimedes until after the revival of learning. His immediate successors may have used his method to attain a greater degree of accuracy, but there is very little evidence pointing in this direction. Ptolemy (fl. 127–151), in the Great Syntaxis, gives 3·141552 as the ratio[6]; and the Hindus (𝑐. A.D. 500), who were very probably indebted to the Greeks, used 62832/20000, that is, the now familiar 3·1416.[7]
It was not until the 15th century that attention in Europe began to be once more directed to the subject, and after the resuscitation a considerable length of time elapsed before any progress was made. The first advance in accuracy was due to a certain Adrian, son of Anthony, a native of Metz (1527), and father of the better-known Adrian Metius of Alkmaar. In refutation of Duchesne(Van der Eycke), he showed that the ratio was < 317120 and > 315106, and thence made the exceedingly lucky step of taking a mean between the two by the quite unjustifiable process of halving the sum of the two numerators for a new numerator and halving the sum of the two denominators for a new denominator, thus arriving at the now well-known approximation 316113 or 335113, which, being equal to 3·1415929. . ., is correct to the sixth fractional place.[8]
- ↑ Eisenlohr, Ein math. Handbuch d. alten Ägypter, übers. u. erklärt (Leipzig, 1877); Rodet, Bull. de la Soc. Math. de France, vi. pp. 139-149.
- ↑ H. Hankel, Zur Gesch. d. Math. im Alterthum, &c., chap, v (Leipzig, 1874); M. Cantor, Vorlesungen über Gesch. d. Math. i. (Leipzig, 1880); Tannery, Mém. de la Soc., &c., à Bordeaux; Allman, in Hermathena.
- ↑ Tannery. Bull. des sc. math. [2], x. pp. 213-226.
- ↑ In modern trigonometrical notation, 1 + sec θ:tan θ::1:tan 12θ.
- ↑ Tannery, “Sur la mesure du cercle d’Archimède,” in Mém. . ..Bordeaux[2], iv. pp. 313-339; Menge, Des Archimedes Kreismessung (Coblenz, 1874).
- ↑ De Morgan, in Penny Cyclop, xix. p. 186.
- ↑ Kern, Aryabhattíyam (Leiden, 1874), trans. by Rodet (Paris,1879).
- ↑ De Morgan, art. “Quadrature of the Circle,” in English Cyclop.; Glaisher, Mess. of Math. ii. pp. 119-128, iii. pp. 27-46; de Haan, Nieuw Archief v. Wisk. i. pp. 70-86, 206-211.
