830 MATHEMATICS. 



bisects both the chord, and the arc which it subtends. Every tan- 

 gent is perpendicular to the radius drawn to its point of contact, 

 that is, the point where it touches the circle. Arcs of circles are 

 used to measure angles ; the vertex of the angle being at the centre 

 of the circle. An angle is said to be inscribed in a circle, when 

 it is formed by two chords meeting on the circumference. A polygon 

 is said to be thus inscribed, when all its vertices lie in the circum- 

 ference : and it is said to be circumscribed about the circle, when all 

 its sides are tangent to the circumference. The ratio of the diameter 

 of a circle to its circumference, is nearly as 7 to 22 ; more nearly as 

 113 to 355; and still more nearly as 1 to 3-141596. The measure 

 of a circle, is the product of the circumference by half the radius ; or 

 of half the circumference by half the diameter. 



4. The study of Solid Jingles, and Polyedrons, requires some 

 preliminary explanations. When a given straight line meets a plane, 

 and is perpendicular to two other straight lines lying in the plane, 

 and meeting the former at its foot, then the given line is perpendicu- 

 lar to all other lines in the plane, and to the plane itself. A line is 

 parallel to a plane, and two planes are parallel to each other, when 

 they do not meet, however far extended or produced. Two planes 

 are perpendicular to each other, when either one of them contains 

 any line which is perpendicular to the other. The angle formed by 

 two planes, is measured by that of two lines, one in each plane, and 

 both perpendicular to the line of intersection of the planes, and meet- 

 ing it at the same point. A solid angle, is one formed by three or 

 more planes, all meeting at the same point; which is the vertex both 

 of the solid angle, and of the plane angles that enclose it ; as the 

 angle at D, or those at A, B, and C. (PI. VII., Fig. 5.) 



A Polyedron, is a solid, bounded on all sides with planes, called 

 its faces, the terminating lines of which constitute its edges. A 

 cube has six equal squares for its six faces. A pyramid, Fig. 5, is 

 a solid enclosed by several triangular planes, proceeding from a com- 

 mon vertex, to the sides of a polygon which forms the base. A 

 prism, Fig. 6, is a polyedron, the ends of which are similar poly- 

 gons, and the sides are parallelograms. The measure of a prism is 

 the product of its base by its altitude ; which is the perpendicular 

 distance between its two bases. The measure of a pyramid is the 

 product of its base by one-*third of its altitude, or perpendicular 

 height. The unit of solidity, in these measurements, is a cube, 

 each of whose edges is equal to the unit of length, and each of its 

 faces equal to the unit of surface. 



5. The Three Round Bodies, technically so called, are the cone, 

 cylinder, and sphere. A cone, PI. VII., Fig. 7, is a solid gene- 

 rated by a right angled triangle, (ACD), revolving around one of its 

 legs, or shorter sides, (DC), which remains stationary as an axis. A 

 cylinder, Fig. 8, is a solid generated by the like revolution of a 

 rectangle, about either one of its sides : and a sphere, is a solid 

 generated by the revolution of a semicircle, around one of its diame- 

 ters, which remains stationary as an axis. All possible sections of a 

 sphere, by a plane, are circles ; and planes cutting a cone, or cylinder, 

 perpendicular to its axis, produce also circular sections. A zone, of 



