CIRCLE, a plane figure bounded by a curved line, which returns into itself, called its circumference, and which is everywhere equally distant from a point within it called the center of the circle. The circumference is sometimes itself called the C., but this is improper; C. is truly the name given to the space contained within the circumference. Any line drawn through the center, and terminated by the circumference, is a diame ter. It is obvious that every diameter is bisected in the center. (See Alto, Citonn.) In co-ordinate geometry, the C. ranks as a curve of the second order, and belongs to the class of the conic sections. It is got from the right cone by cutting the cone by a plane perpendicular to its axis. The C. may be described mechanically with a pair of com passes. fixing one foot in the center, and turning the other round to trace out the cir cumference. The C. and straight line are the two elements of plane geometry, and those constructions only are regarded as being properly geometrical which can be effected by their means. As an element in plane geometry, its properties are well known and investigated in all the text-books. Only a few of the leading properties will here be stated.
1. Of all plane figures, the C. has the greatest area within the same perimeter.
2. The circumference of a C. bears a certain constant ratio to its diameter. This constant ratio, which mathematicians usually denote by the Greek letter 7r, has been determined to be 3.14159, nearly, so that, if the diameter of a C. is 1 foot, its circum ference is 3.14159 feet; if the diameter is 5 ft., the circumference is 5 X 3.14159; and, in general, if the 4iameter is expressed by 2r (twice the radius), then c (circumference) = 2r X 7r. Archimedes, in his book De Di,nensione Circuli, first gave a near value to the ratio between the circumference and the diameter, being that of 7 to 22. Various closer approximations in large numbers were afterwards made, as, for instance, the ratio of 1815 to 5702. Vieta, in 1579, showed that if the diameter of a C. lie 1000, etc., then the circumference will be greater than 3141.5926535, and less than 3141:5926537. This approximation he made through ascertaining the perimeters of the inscribed and circum scribed polygons of 393,216 sides. By increasing the number of the sides of the poly gons, their perimeters are brought more and more nearly into coincidence with the cir cumference of the circle. The approximation to the value of 7r has since been carried
(by M. de Lagny) to 128 places of figures. It is now settled that 7r belongs to the class of quantities called incommensurable (q.v.), i.e., it cannot be expressed by the ratio of any two whole numbers, however great. In general, it may be considered that 3.14159 is a sufficiently accurate value of r.
Though the value of 7r was at first approached by actually calculating the perimeter of a polygon of a great number of sides, this operose method was long ago superseded by modes of calculation of a more relined character, which, however, cannot here be explained. Suffice it to say, that various series were formed expressing its value; by taking more and more of the terms of which into account, a closer and closer approach to the value might be obtained. We subjoin one or two of the more curious.
1 1 1 17r = 4( 1 —• 3 5 7 — — 9 11 + etc.).
Q 1 1 1 i_ 1 1 i_1 it = 41.3 -1- 3.5 — 3.5.7 5.7.9 — 7.9.11 ' 9.11.13 3. The area of a C. is equal to 7r multiplied by the square of the radius (=.7rr'); or 7r i to the square of the diameter multiplied by —; i.e., by .7854. Euclid has proved this 4 , „ ,..
by showing that the area is equal to that of a triangle whose base is the circumference, and perpendicular height the radius of the circle.
4. It follows that different circles are to one another as the squares of their radii or diameters, and that their circumferences are as the radii or diameters.
The C. is almost always employed to measure angles, from its obvious convenience for the purpose, which depends on the fact demonstrated in Euclid (book iv. prop. 33), that angles at the center of a C. are proportional to the arcs on which they stand. It follows, from this, that if circles of the same radii be described from the vertices of angles as centers, the arcs intercepted between the lines, including the angles, are always proportional to the angles. The C. thus presents us with the means of compar• ing angles. It is first necessary, however, to graduate the C. itself; for this purpose its circumference is divided into four equal parts, called quadrants, each of which obviously subtends a right angle at the center, and then each quadrant is divided into degrees, and each degree into minutes, and so on. The systems of graduation adopted are various, and will now be explained.