by the relation Σ*a*²(*p*−*q*) (*p*−*r*) = 4Δ² (see Geometry: *Analytical*),
which is generally written {*ap*, *bq*, *cr*}² = 4Δ², we obtain
{*ap*, *bq*, *cr*}²ρ² = 4Δ² {(*lp*+*mq*+*nr*)/(*l*+*m*+*n*)}², the accents being
dropped, and *p*, *q*, *r* regarded as current co-ordinates. This equation,
which may be more conveniently written {*ap*, *bq*, *cr*}²
= (λ*p* + μ*q* + ν*r*)², obviously represents a circle,
the centre being λ*p*+μ*q*+ν*r*=0,
and radius 2Δ/(λ + μ + ν).
If we make λ = μ = ν = 0,
ρ is infinite, and we obtain {*ap*, *bq*, *cr*}²=0 as the equation to the
circular points.

*Systems of Circles.*

*Centres and Circle of Similitude.*—The “centres of similitude”
of two circles may be defined as the intersections of the common
tangents to the two circles, the direct common tangents giving
rise to the "external centre," the transverse tangents to the
"internal centre." It may be readily shown that the external
and internal centres are the points where the line joining the
centres of the two circles is divided externally and internally in the ratio of their radii.

The circle on the line joining the internal and external centres of similitude as diameter is named the "circle of similitude." It may be shown to be the locus of the vertex of the triangle which has for its base the distance between the centres of the circles and the ratio of the remaining sides equal to the ratio of the radii of the two circles.

With a system of three circles it is readily seen that there are six centres of similitude, viz. two for each pair of circles, and it may be shown that these lie three by three on four lines, named the “axes of similitude.” The collinear centres are the three sets of one external and two internal centres, and the three external centres.

*Coaxal Circles.*—A system of circles is coaxal when the locus
of points from which tangents to the circles are equal is a straight
line. Consider the case of two circles, and in the first place
suppose them to intersect in two real points A and B. Then by
Euclid iii. 36 it is seen that the line joining the points A and B is
the locus of the intersection of equal tangents, for if P be any
point on AB and PC and PD the tangents to the circles, then
PA·PB = PC² = PD², and therefore PC = PD. Furthermore it is
seen that AB is perpendicular to the line joining the centres,
and divides it in the ratio of the squares of the radii. The line
AB is termed the “radical axis.” A system coaxal with the two
given circles is readily constructed by describing circles through the common points on the radical axis and any third point;
the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis—considered as
a circle of infinite radius. In the case of two non-intersecting circles
it may be shown that the radical axis has the same metrical relations
to the line of centres.

There are several methods of constructing the radical axis in this case. One of the simplest is: Let P and P′ (fig. 5) be the points of contact of a common tangent; drop perpendiculars PL, P′L′, from P and P′ to OO′, the line joining the centres, then the radical axis bisects LL′ (at X) and is perpendicular to OO′. To prove this let AB, AB¹ be the tangents from any point on the line AX. Then by Euc. i. 47, AB²=AO²–OB²=AX²+OX²+OP²; and OX²=OD²–DX²=OP²+PD²–DX². Therefore AB²=AX²–DX²+PD². Similarly AB′²=AX²–DX²+DP′². Since PD=PD′, it follows that AB=AB′.

To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle. Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as radii, are members of the coaxal system.

In the case of non-intersecting circles, it is seen that the
minimum circles of the coaxal system are a pair of points I and I′,
where the orthogonal circle to the system intersects the line of
centres; these points are named the "limiting points." In the
case of a coaxal system having real points of intersection the
limiting points are imaginary. Analytically, the Cartesian
equation to a coaxal system can be written in the form
*x*²+*y*²+2*ax*±*k*²=0, where a varies from member to member,
while k is a constant. The radical axis is x = 0, and it may be
shown that the length of the tangent from a point (0, *h*) is *h*² ± *k*², *i.e.* it is independent of a, and therefore of any particular
member of the system. The circles intersect in real or imaginary
points according to the lower or upper sign of *k*², and the limiting
points are real for the upper sign and imaginary for the lower sign. The fundamental properties of coaxal systems may be summarized:—

1. The centres of circles forming a coaxal system are collinear;

2. A coaxal system having real points of intersection has imaginary limiting points;

3. A coaxal system having imaginary points of intersection has real limiting points;

4. Every circle through the limiting points cuts all circles of the system orthogonally;

5. The limiting points are inverse points for every circle of the system.

The theory of centres of similitude and coaxal circles affords
elegant demonstrations of the famous problem: To describe a
circle to touch three given circles. This problem, also termed
the "Apollonian problem," was demonstrated with the aid of conic sections by Apollonius in his book on *Contacts* or *Tangencies*;
geometrical solutions involving the conic sections were also given
by Adrianus Romanus, Vieta, Newton and others. The earliest
analytical solution appears to have been given by the princess
Elizabeth, a pupil of Descartes and daughter of Frederick V.
John Casey, professor of mathematics at the Catholic university
of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are reproduced in his *Sequel to Euclid*; an analytical solution by Gergonne is given in Salmon’s *Conic Sections*. Here we may notice that there are eight circles which solve the problem.

*Mensuration of the Circle.*

All exact relations pertaining to the mensuration of the circle
involve the ratio of the circumference to the diameter. This
ratio, invariably denoted by π, is constant for all circles, but it does not admit of exact arithmetical expression, being of the
nature of an incommensurable number. Very early in the history of geometry it was known that the circumference and area of a circle of radius r could be expressed in the forms 2π*r* and π*r*².
The exact geometrical evaluation of the second quantity, viz. π*r*², which, in reality, is equivalent to determining a square
equal in area to a circle, engaged the attention of mathematicians for many centuries. The history of these attempts, together
with modern contributions to our knowledge of the value and nature of the number π, is given below (*Squaring of the Circle*).

The following table gives the values of this constant and several expressions involving it:—

Number. | Logarithm. | Number. | Logarithm. | ||

π | 3.1415927 | 0.4971499 | π² | 9.8696044 | 0.9942997 |

2π | 6.2831858 | 0.7981799 | |||

4π | 12.5663706 | 1.0992099 | 16π² | 0.0168869 | ̅2.2275490 |

12π | 1.5707963 | 0.1961199 | |||

13π | 1.0471976 | 0.0200286 | √π | 1.7724539 | 0.2485750 |

14π | 0.7853982 | ̅1.8950899 | |||

16π | 0.5235988 | ̅1.7189986 | ^{3}√π |
1.4645919 | 0.1657166 |

18π | 0.3926991 | ̅1.5940599 | |||

112π | 0.2617994 | ̅1.4179686 | 1√π | 0.5641896 | ̅1.7514251 |

43π | 4.1887902 | 0.6220886 | |||

π180 | 0.0174533 | ̅2.2418774 | 2√π | 1.1283792 | 0.0524551 |

1π | 0.3183099 | ̅1.5028501 | 12√π | 0.2820948 | ̅1.4503951 |

4π | 1.2732395 | 0.1049101 | ^{3}√(6π) |
0.2820948 | 1.4503951 |

14π | 0.0795775 | ̅2.9097901 | ^{3}√(34π) |
0.6203505 | ̅1.7926371 |

180π | 57.2957795 | 1.7581226 | log_{e} π |
1.1447299 | 0.0587030 |

Useful fractional approximations are 22/7 and 355/113.

A synopsis of the leading formula connected with the circle will now be given.

1. *Circle.*—Data: radius = a. Circumference = 2π*a*. Area = π*a*².

2. *Arc* and *Sector*.—Data: radius = *a*; θ = circular measure of angle subtended at centre by arc; *c* = chord of arc; *c*_{2} = chord of semi-arc; *c*_{4} = chord of quarter-arc.