CIRCLE, in geometry, a plane figure con tained by one line, which is called the ((circum ference" and is such that all straight lines drawn from a certain point (the ((centre") within the figure to the circumference are equal to one another. According to this definition of Euclid, which is remarkable for its perspicuity and precision, the circle is the space enclosed, while the circumference is the line that bounds it. The circumference is, however, frequently called the circle. Still no confusion ever arises from this usage.
The properties of the circle are investigated in books on geometry and trigonometry. Prop erly the curve belongs to the class of conic sec tions and is a curve of the second order.
The Cartesian equation of a circle with centre at the origin and radius R is Its polar equation is P=R. The general Cartesian equation of a circle is +c=-o; the general polar equation is P sin (d+a)--a+bPi. From the standpoint of pro jective geometry, a circle is any conic passing through a certain pair of imaginary points at infinity, known as the circular points.
The celebrated problem of gsquaring the circle" has given rise to extraordinary geo metrical labors, and even now there are to be found, as in the case of the problem of per petual motion, those who profess to have solved it.
The question is to construct by rules and compass a square whose area shall be equal to the area of a circle. It is not possible to do so. This was only shown within the last 36 years by Lindemann. All that can be done is to ex pre-, approximately the ratio of the length of the circumference of the circle to and to deduce the area of the figure from this approximation. This ratio, which is known as rr has, however, been determined to a degree of exactness more than sufficient for all prac tical purposes. If the diameter be called unity, the length of the circumference of the circle is 3.1415926535...; and the area of the circle is found by multiplying this number by the square of the radius. Thus the area of a circle of 2 feet radius is 3.14159 X 4, or 12.56636 square
feet, approximately.
r can be approximated by series or con tinued fractions or products, as follows: For trigonometrical calculations the whole circumference of the circle is divided into 360 equal parts or arcs, called degrees ,• each degree is divided into 60 minutes, and each minute into 60 seconds. The angles subtended at the centre by these arcs are called respectively degrees, minutes and seconds of angle. A centesimal System of circular division is sometimes used. For purposes of pure mathematics the unit is the radian, i.e., the angle subtending an arc equal to the radius, or 360°. Since every circle 2ir has an equation of the form P=x'- ax +by (-41, it follows that if two circles have the equation PO and ()=0 (of the same form), if they intersect, the line P—(,)==0 will pass through all their finite points of intersection. Even if these points are imaginary, P —Q-0 will be real. will be in this case the locus of the points from which the tangents to the two circles are of equal length. In either case, it is known as the radical axis of the two circles.
Let us consider the equation of a circle in polar co-ordinates; it is psin + a)=a +bp".
1 If we substitute for p, we get — (6-Fa) fa .
p' + -7- or ' sin +a) =-- b This is also P the equation of a circle. It follows that if we take any circle, such as the unit circle about the origin, and exchange every point for one which is on the same radius vector from the origin, but at a distance from the origin which, when multiplied by the distance from the origin of the first point, gives the square of the radius of the circle, we shall retain all circles un changed in form. This transformation is called an inversion. The study of all properties of figures which remain invariant under all in versions is the important branch of mathe matics known as inversion geometry. The polar of a point is the line connecting the real or imaginary points of tangency of the tangents to the circle of reference from the point. A point is said to be the pole of its polar.