Page:EB1911 - Volume 06.djvu/398

 shrubs. From any point in the Pomptine Marshes or on the coast-line of Latium the Circeian promontory dominates the landscape in the most remarkable way.

See T. Ashby, “Monte Circeo,” in Mélanges de l’école française de Rome, xxv. (1905) 157 seq.

CIRCLE (from the Lat. circulus, the diminutive of circus, a ring; the cognate Gr. word is, generally used in the form  ), a plane curve definable as the locus of a point which moves so that its distance from a fixed point is constant.

The form of a circle is familiar to all; and we proceed to define certain lines, points, &c., which constantly occur in studying its geometry. The fixed point in the preceding definition is termed the “centre” (C in fig. 1); the constant distance, e.g. CG, the “radius.” The curve itself is sometimes termed the “circumference.” Any line through the centre and terminated at both extremities by the curve, e.g. AB, is a “diameter”; any other line similarly terminated, e.g. EF, a “chord.” Any line drawn from an external point to cut the circle in two points, e.g. DEF, is termed a “secant”; if it touches the circle, e.g. DG, it is a “tangent.” Any portion of the circumference terminated by two points, e.g. AD (fig. 2), is termed an “arc”; and the plane figure enclosed by a chord and arc, e.g. ABD, is termed a “segment”; if the chord be a diameter, the segment is termed a “semicircle.” The figure included by two radii and an arc is a “sector,” e.g. ECF (fig. 2). “Concentric circles” are, as the name obviously shows, circles having the same centre; the figure enclosed by the circumferences of two concentric circles is an “annulus” (fig. 3), and of two non-concentric circles a “lune,” the shaded portions in fig. 4; the clear figure is sometimes termed a “lens.”

The circle was undoubtedly known to the early civilizations, its simplicity specially recommending it as an object for study. Euclid defines it (Book I. def. 15) as a “plane figure enclosed by one line, all the straight lines drawn to which from one point within the figure are equal to one another.” In the succeeding three definitions the centre, diameter and the semicircle are defined, while the third postulate of the same book demands the possibility of describing a circle for every “centre” and “distance.” Having employed the circle for the construction and demonstration of several propositions in Books I. and II. Euclid devotes his third book entirely to theorems and problems relating to the circle, and certain lines and angles, which he defines in introducing the propositions. The fourth book deals with the circle in its relations to inscribed and circumscribed triangles, quadrilaterals and regular polygons. Reference should be made to the article : Euclidean, for a detailed summary of the Euclidean treatment, and the elementary properties of the circle.

Analytical Geometry of the Circle.

In the article : Analytical, it is shown that the general equation to a circle in rectangular Cartesian co-ordinates is 𝑥2+𝑦2+2𝑔𝑥+2𝑓𝑦+𝑐=0, i.e. in the general equation of the second degree the co-efficients of 𝑥2 and 𝑦2 are equal, and of 𝑥𝑦 zero. The co-ordinates of its centre are –𝑔/𝑐, –𝑓/𝑐;  and  its  radius  is  (𝑔2+𝑓2–𝑐)undefined. The

equations to the chord, tangent and normal are readily derived by the ordinary methods.

Consider the two circles:—

𝑥2+𝑦2+2𝑔𝑥+2𝑓𝑦+𝑐 =0,&emsp;𝑥2+𝑦2+2𝑔′𝑥+2𝑓′𝑦+𝑐′=0.

Obviously these equations show that the curves intersect in four points, two of which lie on the intersection of the line, 2(𝑔 – 𝑔′)𝑥 + 2(𝑓 – 𝑓′)𝑦 + 𝑐 – 𝑐′＝0, the radical axis, with the circles, and the other two where the lines 𝑥2 + 𝑦2＝(𝑥 + 𝑖𝑦) (𝑥 – 𝑖𝑦)＝0 (where 𝑖＝√–1) intersect the circles. The first pair of intersections may be either real or imaginary; we proceed to discuss the second pair.

The equation 𝑥2 + 𝑦2＝0 denotes a pair of perpendicular imaginary lines; it follows, therefore, that circles always intersect in two imaginary points at infinity along these lines, and since the terms 𝑥2 + 𝑦2 occur in the equation of every circle, it is seen that all circles pass through two fixed points at infinity. The introduction of these lines and points constitutes a striking achievement in geometry, and from their association with circles they have been named the “circular lines” and “circular points.” Other names for the circular lines are “circulars” or “isotropic lines.” Since the equation to a circle of zero radius is 𝑥2 + 𝑦2＝0, i.e. identical with the circular lines, it follows that this circle consists of a real point and the two imaginary lines; conversely, the circular lines are both a pair of lines and a circle. A further deduction from the principle of continuity follows by considering the intersections of concentric circles. The equations to such circles may be expressed in the form 𝑥2 + 𝑦2＝2, 𝑥2 + 𝑦2＝2. These equations show that the circles touch where they intersect the lines 𝑥2 + 𝑦2＝0, i.e. concentric circles have double contact at the circular points, the chord of contact being the line at infinity.

In various systems of triangular co-ordinates the equations to circles specially related to the triangle of reference assume comparatively simple forms; consequently they provide elegant algebraical demonstrations of properties concerning a triangle and the circles intimately associated with its geometry. In this article the equations to the more important circles—the circumscribed, inscribed, escribed, self-conjugate—will be given; reference should be made to the article for the consideration of other circles (nine-point, Brocard, Lemoine, &c.); while in the article : Analytical, the principles of the different systems are discussed.

The equation to the circumcircle assumes the simple form 𝑎 + 𝑏 + 𝑐＝0, the centre being cos A, cos B, cos C. The inscribed circle is cos A √ cos B √ + cos C √＝0, with centre ＝＝; while the escribed circle opposite the angle A is cos A √– + sin B √ + sin C √＝0, with centre -＝＝. The self-conjugate circle is 2 sin 2A + 2 sin 2B + 2 sin 2C＝0, or the equivalent form 𝑎cosA 2 + 𝑏cosB2 + 𝑐cosC2＝0, the centre being sec A, sec B, sec C.

The general equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the line infinity in the circular points. Consider the equation

This obviously represents a conic intersecting the circle 𝑎 + 𝑏 + 𝑐＝0 in points on the common chords 𝑙 + 𝑚 + 𝑛＝0, 𝑎 + 𝑏 + 𝑐＝0. The line 𝑙 + 𝑚 + 𝑛 is the radical axis, and since 𝑎 + 𝑏 + 𝑐＝0 is the line infinity, it is obvious that equation (1) represents a conic passing through the circular points, i.e. a circle. If we compare (1) with the general equation of the second degree 𝑢2 + 𝑣2 + 𝑤2 + 2𝑢′ + 2𝑣′ + 2𝑤′＝0, it is readily seen that for this equation to represent a circle we must have

–𝑘𝑎𝑏𝑐＝𝑣𝑐2 + 𝑤𝑏2 – 2𝑢′𝑏 𝑐＝𝑤𝑎2 + 𝑢𝑐2 – 2𝑣′𝑐𝑎＝𝑢𝑏2 + 𝑣𝑎2 – 2𝑤′𝑎𝑏.

The corresponding equations in areal co-ordinates are readily derived by substituting 𝑥/𝑎, 𝑦/𝑏, 𝑧/𝑐 for, , respectively in the trilinear equations. The circumcircle is thus seen to be 𝑎2𝑦𝑧 + 𝑏2𝑧𝑥 + 𝑐2𝑥𝑦＝0, with centre sin 2A, sin 2B, sin 2C; the inscribed circle is √(𝑥 cot A) + √(𝑦 cot B) + √(𝑧 cot C)＝0, with centre sin A, sin B, sin C; the

escribed circle opposite the angle A is √(–𝑥 cot A) + √(𝑦 tan B) + √(𝑧 tan C)=0, with centre – sin A, sin B, sin C; and the self-conjugate circle is 𝑥2 cot A + 𝑦2 cot B + 𝑧2 cot C＝0, with centre tan A, tan B, tan C. Since in areal co-ordinates the line infinity is represented by the equation 𝑥 + 𝑦 + 𝑧＝0 it is seen that every circle is of the form 𝑎2𝑦𝑧 + 𝑏2𝑧𝑥 + 𝑐2𝑥𝑦 + (𝑙𝑥 + 𝑚𝑦 + 𝑛𝑧)(𝑥 + 𝑦 + 𝑧)＝0. Comparing this equation with 𝑢𝑥2 + 𝑣𝑦2 + 𝑤𝑧2 + 2𝑢′𝑦𝑧 + 2𝑣′𝑧𝑥 + 2𝑤′𝑥𝑦＝0, we obtain as the condition for the general equation of the second degree to represent a circle:—

(𝑣 + 𝑤 – 2𝑢′) / 𝑎2＝(𝑤 + 𝑢 – 2𝑣′) / 𝑏2＝(𝑢 + 𝑣 – 2𝑤′) / 𝑐2.

In tangential (𝑝, 𝑞, 𝑟) co-ordinates the inscribed circle has for its equation (𝑠 – 𝑎)𝑞𝑟 + (𝑠 – 𝑏)𝑟𝑝 + (𝑠 – 𝑐)𝑝𝑞＝0, 𝑠 being equal to (𝑎 + 𝑏 + 𝑐); an alternative form is 𝑞𝑟 cot A + 𝑟𝑝 cot B + 𝑝𝑞 cot C＝0; the centre is 𝑎𝑝 + 𝑏𝑞 + 𝑐𝑟＝0, or 𝑝 sin A + 𝑞 sin B + 𝑟 sin C＝0. The escribed circle opposite the angle A is –𝑠𝑞𝑟 + (𝑠 – 𝑐)𝑟𝑝 + (𝑠 – 𝑏)𝑝𝑞＝0 or –𝑞𝑟 cot A + 𝑟𝑝 tan B + 𝑝𝑞 tan C＝0, with centre –𝑎𝑝 + 𝑏𝑞 + 𝑐𝑟＝0. The circumcircle is 𝑎 √(𝑝) + 𝑏 √(𝑞) + 𝑐 √(𝑟)＝0, the centre being 𝑝 sin 2A + 𝑞 sin 2B + 𝑟 sin 2C＝0. The general equation to a circle in this system of co-ordinates is deduced as follows: If be the radius and 𝑙𝑝 + 𝑚𝑞  + 𝑛𝑟＝0 the centre, we have =(𝑙𝑝1–𝑚𝑞1+𝑛𝑟1) / (𝑙+𝑚+𝑛), in which 𝑝1, 𝑞1, 𝑟1 is a line distant  from the point 𝑙𝑝+𝑚𝑞+𝑛𝑟=0. Making this equation homogeneous