SEGMENT

(Lat.) In Geometry, a part cut off from a figure by a lino or plane. The segment of a circle is a part of the area comprised between an arc and its chord; faid segments of difl'erent circles are said to be similar when their arcs have the same ratio to the circumferences of their respective circles, or when they contain the same number of degrees. The area of a circular segment may be found as follows: — Let P K be the chord of a circle whose centre is (J, and let A B be tiie diameter perpendicular to P R, and meeting it in Q; and put x = KQ,y = P Q, z = A P, »• = A C the radius, and!r = 3141.^9, the ratio of the circumference to the diameter. Now the area of the segment PA R is evidently equal to the difference between the sector A P C R and tlie triangle P C R; but the area of the sector = rz (the radius into half the arc), and the triangle = y (r —.i); therefore the segment = r % — r y + yx,. W hen the radius of the circle and the chord of the segment (.that is, r and y) are known, this expre ion is easily calculate^ from a table of sines, for the angle A C P is found from its sine — y -7- r; and when the angle is given the arc z is obtained from this proportion, x: sr r:: A C P: 180°. With respect to X, we have from the property of the circle x — The superficial and solid contents of a segment of a sphere are found thus: — Let S be the spherical surface of the segment of the sphere whose centre is C by the plane P Q R. The symbols having the same values as above, the circumference of the circle P Q R is 2 s-.5/ / and the differential of the surface is d S = 2 t y d z. But from the nature of the circle, y dz = r d x; hence d S=2t r d X, and S = 2r r Xj so that the surface IICO SELENOGRAPHY. of the segment is found in terms of the radius of the sphere and the altitude of the segment. Let U denote the solid contents of the segment; then, since the area of the circle P Q R is ^y'^, the differential of the segment is d U = fry'' d x = 2 sr r x d x — irx'^ d x; whence IJ =■ ax' {r — ^x). [s. 1113]