CIRCLE (from Lat. cireulus, dim. of circus, Gk. dpKos, rk og. Kpkos, kriko.c, circle). The locus (q.v.) of all points in it plane at an equal finite distance from a fixed point in that plane. The fixed point is called the centre, and the spate inelosed. Or, more properly, its ure, the area of the circle. The segment of any straight line intercepted by the circle (AR in Fig. 11 is called a chord. Any chord passing the centre. 0, is called a diameter. as _VW. The centre bisects any diameter, and the halves are called radii. Any line drawn from an external point cutting the circle. as PQ, is called a secant : and any line which has contact with the circle, but does not intersect it when produced. as 11'1'. is called a tattrent. Any por tion of the area limited by two ra dii. as OA and Olt, is called a sector: and any port ion of the circle, 11A'A, is called an are. A chord is said . to divide the area into segments: the segments are equal if the chord is a diameter. A plane passing through the centre of a sphere cuts the surface in a circle called a great cir cle of the sphere. Circles of longitude are great circles. Other circles of a sphere are called small circles. .Ancient writers usually called the circle, as above defined, a circumference, the word `circle' being applied to the space inclosed. In modern geometry, at least alcove the elements, the word 'circumference' is not used, and the word 'circle' applies to the curve.
coirdinate geonit•try (see AN.kLYTIC EONI , the circle ranks as a curve of the second circler (see Cunv•), and belongs to the conic sec• lions; the section of a right circular cone, per• pendi•ular to the axis of the cone, being a circle. The Cartesian equation of the circle, taking its centre as the origin, is F = r'. The con structions of Euclidean geometry being ;incited to the use of two instruments, the straight-edge and the compasses, the circle and the straight line are the two basal elements of plane geom etry. .‘ few of the leading properties of the circle are: (I I The ratio of the eiremnfe•ence to the diameter is a constant; this is designated by the symbol 7. This ratio is approximately 3.141592: 3.1410 :111(1 even 35 are sufliciently ac curate for ordinary purposes: thus the area of a circle of radius 5 inches is 3.1116 X 5' square inches, or 78.54 square inches. The ratio 7r has an interesting history. The papyrus of Alines (q.v.) (before 'Lc. 1700) contains the value or 3.1605: Archimedes (n.c. 2S7-212) de scribed it as lying between If- and 3H-; the Almagest (q.v.) gives it as :30 —- — = 3.14166;
60 60•60 the Romans often used Aryabhatta (q.v.) gave 3.1416 ; Ilhaska ra ( q.v.) . 3.14166 ; and the Chinese of the sixth century A.D., Lit dolph van Ceulen (1586) computed 7 to 35 decimal plaees, and in recent times it has been carried to over 700 places. In 1794 Legendry proved that 7r is an irrational number. Fur• it is not only incommensurable—that is, not expressible as the quotient of two inte oers—but it has been proved by Lindemann (1882) to be transcendental. This means that r cannot be a root of an algebraic equation with integral coefficients. Certain irrational (incommensurable) numbers may be represented by elementary geometric lines: e is repre sented by the diagonal of a square of side 1; but 7r , being transcendental, cannot be repre sented by any construction depending solely upon the straight-edge and compasses. It requires a transcendental curve, such as the integraph of Abdank-Allakanowiez.
Thus. through labors like those of nauss, Her mite, and Lindemann, the true nature of r has been determined, and efforts at circle-squaring by the instruments of elementary geometry have been proved futile. Modern analysis has shown r to be expressible by certain infinite series; e.g.
r = 4 1 — — 1 1 — — — — •• (Leibnitz); 3 or in the form of a continued fraction, as in 1 (11rouneker) + 2 4- 25 2 49 2 +...; or of a continued product, as r • .) • 4 • 4 • (i • (i • 8 • • • 2 3 3.5• 7 • 7 • • • The method of approximating this ratio com monly used before the introduction of calculus (q.v.) consisted in computing the perimeters of the circumscribed and inscribed polygons of a circle of diameter 1. For. since the length of the circumference in this case is the desired ratio, the value of 7r lies between the values of the perimeters of the given polygons. A history of the development of this important problem of geometry will be found in Itudio, Archimedes, Huygens. Lambert, Legendre: vier Abhandlungen iibcr die Kreismessung (Leipzig. 1892).
(2) The centre of the circle is a centre of symmetry, and any diameter is an axis of sym metry (q.v.).
(3) The perimeter of a circle of radius r is 2rr, and its The area is greater than that of any plane figure of the same perimeter.
(4) Concentric circles—that is, those having the same centre—never intersect.
(5) Circles are similar figures (see SIMILAR nY), and their areas are proportional to the squares of their radii or diameters.
(6) Arcs of a circle are proportional to the angles subtended at the centre. and conversely. This property forms the basis of angular meas ure.