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. (T. As.)

**CIRCLE** (from the Lat. *circulus*, the diminutive of *circus*, a
ring; the cognate Gr. word is κιρκος, generally used in the form
κρίκος), 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 Geometry: *Euclidean*, for a
detailed summary of the Euclidean treatment, and the elementary
properties of the circle.

*Analytical Geometry of the Circle.*

In the article Geometry: *Analytical*, it is shown that the
general equation to a circle in rectangular Cartesian co-ordinates
is *x*^{2}+*y*^{2}+2*gx*+2*fy*+*c*=0, *i.e.* in the general equation
of the second degree the co-efficients of *x*^{2} and *y*^{2} are
Cartesian co-ordinates.
equal, and of *xy* zero. The co-ordinates of its centre
are –*g*/*c*, –*f*/*c*; and its radius is (*g*^{2}+*f*^{2}–*c*)^{½}. The
equations to the chord, tangent and normal are readily derived
by the ordinary methods.

Consider the two circles:—

*x*^{2}+*y*^{2}+2*gx*+2*fy*+*c* =0, *x*^{2}+*y*^{2}+2*g*′*x*+2*f*′*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 *x*² + *y*² = (*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 *x*² + *y*² = 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
*x*² + *y*² 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 *x*² + *y*² = 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
*x*² + *y*² = α², *x*² + *y*² = β². These equations show that the circles touch
where they intersect the lines *x*² + *y*² = 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 Triangle for the consideration
of other circles (nine-point, Brocard, Lemoine, &c.);
while in the article Geometry: *Analytical*, the principles of the
different systems are discussed.

The equation to the circumcircle assumes the simple form
*a*βγ + *b*γα + *c*αβ = 0, the centre being cos A, cos B, cos C. The inscribed
circle is cos ½A √α cos ½B √β + cos ½C √γ = 0, with centre
α = β = γ; while the escribed circle opposite the angle A
Trilinear co-ordinates.
is cos ½A √–α + sin ½B √β + sin ½C √γ = 0, with centre
-α = β = γ. The self-conjugate circle is α² sin 2A + β² sin 2B
+ γ² sin 2C = 0, or the equivalent form a cosA α²
+ *b* cos B β² + c cos C γ² = 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

*a*βγ + *b*γα + Cαβ + (*l*α + *m*β + *n*γ) (*a*α + *b*β + *c*γ) = 0 (1).

This obviously represents a conic intersecting the circle *a*βγ + *b*γα
+ *c*αβ = 0 in points on the common chords *l*α + *m*β + *n*γ = 0, aα + *b*β
+ *c*γ = 0. The line *l*α + *m*β + *n*γ is the radical axis, and since *a*α + *b*β
+ *c*γ = 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
*u*α² + *v*β² + *w*γ² + 2*u*′βγ + 2*v*′γα + 2*w*′αβ = 0, it is readily seen that for
this equation to represent a circle we must have

–*kabc* = *vc*² + *wb*² – 2*u*′*bc* = *wa*² + *uc*² – 2*v*′*ca* = *ub*² + *va*² – 2*w*′*ab*.

The corresponding equations in areal co-ordinates are readily
derived by substituting *x*/*a*, *y*/*b*, *z*/*c* for α, β, γ respectively in
the trilinear equations. The circumcircle is thus seen
to be *a*²*yz* + b²*zx* + *c*²*xy* = 0, with centre sin 2A, sin 2B,
Areal co-ordinates.sin 2C; the inscribed circle is √(*x* cot ½A) + √(*y* cot ½B)
+ √(*z* cot ½C) = 0, with centre sin A, sin B, sin C; the
escribed circle opposite the angle A is √(–*x* cot ½A) + √(*y* tan ½B)
+ √(*z* tan ½C)=0, with centre – sin A, sin B, sin C; and the self-conjugate
circle is *x*² cot A + y² cot B + *z*² 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 *a*²*yz* + *b*²*zx* + *c*²*xy* + (*lx* + *my* + *nz*)(*x* + *y* + *z*) = 0. Comparing
this equation with *ux*² + *vy*² + *wz*² + 2*u*′*yz* + 2*v*′*zx* + 2*w*′*xy* = 0, we
obtain as the condition for the general equation of the second degree
to represent a circle:—

(*v* + *w* – 2*u*′)/*a*² = (*w* + *u* – 2*v*′)/*b*² = (*u* + *v* – 2*w*′)/*c*².

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 ½(*a* + *b* + *c*); an alternative form is *qr* cot ½A + *rp* cot ½B + *pq* cot ½C = 0; the centre is *ap* + *bq* + *cr* = 0, or *p* sin A + *q* sin B + *r* sin C = 0. Tangential co-ordinates.The escribed circle opposite the angle A is –*sqr* + (*s* – *c*)*rp* + (*s* – *b*)*pq* = 0 or –*qr* cot ½A + *rp* tan ½B + *pq* tan ½C = 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 ρ be the radius and *lp* + *mq* + *nr* = 0 the centre, we have ρ=(*lp*_{1}–*mq*_{1}+*nr*_{1})/(*l*+*m*+*n*), in which *p*_{1}, *q*_{1}, *r*_{1} is a line distant ρ from the point *lp*+*mq*+*nr*=0. Making this equation homogeneous