RADIAN MEASVRE. In higher mathematics, especially in anal ytic trigonometry. another unit of angular measure, called the radian, is in general use. This is defined as the angle subtend ed at- the centre of a circle by an arc equal in length to the radius (Fig. 2). The relation of the radian to other angular units is as follows: The radian Anil arc r 1 4 right angles circumference 27rr 27r Therefore, the radian equals X 4 right angles -= X 1 right angle. In degrees oneradian is 2 = 57. +.
cr, more nearly.
57° 17' 44.6666" 7T radians = a quadrant ; 7T radians = 180°, and 27r radians = 360°.
Since 1 radian = 1 ° Io = 7T raThuins and a 7r n° = radians.
180 The word radian is commonly omitted in discus sions of angles: e.g. 7T radians = 1S0' is ex pressed 7T = 180°.
A few of the modern theories concerning the circle are suggested by the following: (1) Co-uxal Circles.—The radical axis XX, (Fig. 3) of two circles of radii r„ is the line perpendicular to their centre line and divid ing this line so that the difference of the squares on the segments equals the difference of the squares on the radii. The common chord of two intersecting cireles is a segment of their radical axis. All circles having a common radical axis pass through two real or two imaginary points, and such a group of circles is ealled a eo•axal system. If two circles are concentric, their radical axis is the line at infinity: therefore, a system of •on•entric circles passes through two imaginary points at infinity. These are called the circular points. The radical axis of two circles is the locus of points from which tangents to the two circles are equal. If a variable polygon inscribed in a circle of a coaxal system moves so that all the side, hut one touch fixed circles of the system, the last side also touches, in every position, a fixed circle of the system (Poncelet's theorem).
(2) Inversion.—Let 0 (Fig. 4) be the centre of a circle of radius r, and P, Q 1.Nvo points on a line through 0, such that OP•OQ = 1' and Q are called inverse points with respect to the circle. Either point is said to be the inverse of the other. The circle and its centre are called the circle •and centre of inversion, and r the constant of inversion. if every point of a plane figure be inverted with respect to a circle, or every point of a figure in respect to a sphere, the resulting figure is called the inverse image of the given one. The inverse of a circle is either a straight line or a circle, according as the centre of inversion is or is not on the given circle. The centre of inversion is then the centre of similitude of the original circle and its in verse; and the circle, its inverse. and the circle
of inversion arc coaxal. The theory of inversion was invented by Stubbs and Ingram in 1842. and has beeri made use of by Lord Kelvin in several important propositions of mathemat in] physics.
(3) Pole and Polar.—The polar of any point P, with respect to a circle, is the perpendicular to the diameter 01' drawn th•nigh the inverse point. Hence the polar of a point exterior to a circle is the chord joining the points of contact of the tangents drawn from the external point. Any point P lying on the polar of a point Q' has its own polar passing through Q'. The polars of any two points, and the line joining the points, form a triangle called the self-recip roral triangle with respect to the circle, the three vertices being the poles of the opposite sides.
(4) Inrolution.—Pair-: of inverse points, P, P': Q. etc., on the same straight line, form a system in involution, the relation between them being OP•01" = 0Q•OQ' = . . = Here the inverse points are visually called con jugate points. .:1ity four points whatever of a system in involution on a straight line have their anharmonic ratio (q.v.) equal to that of their four conjugates.
(5) Xinr-Points Circle.—The intersection of the three altitudes of a triangle is called the or thocentre. The mid-points of the segments from the orthoeentre to the vertices constitute three points, the feet of the altitudes three more, and the mid-points of the sides of the triangle three more—all nine lying on the circumference of a circle, called the nine-points circle.
In Fig. 5. 0 is the orthocentre and K, L, 0, D, AI, E, F arc the nine points.
(6) Seren-Points Circle (Brocard circle). Point S in Fig. 6, so placed that its distances from the sides of the triangle ABC are propor tional to the lengths of the respective sides, is called the symmedian point of the triangle. Lines through this point parallel to the sides cut them in six point-, 1), E', E, F, D', which lie on a circle called the triplieate-ratio or Tucker circle. If lines were drawn through A, C, parallel to the sides of the triangles DEF, !YET', they would intersect one another and PE, DE', in P, L. Al, N. These five points, together with S and the eiremu•entre of the tri angle. lie on a circle called the seven-points or 'Inward circle. 1'. 1'' are called the Droe;ird point,.
'McClelland, Geometry of the Circle I London. 1591 : Casey, to Euclid (Dublin. 155S1: Catalan, Th('on'mrs et meq de pc'ont0 ric (qime»taire (Paris, 1870)•