Cor. 2. Similar arcs, as AB and DE, are like parts of the circumferences to which they belong, and similar sectors, as ACB and DOE, are like parts of the circles to which they belong : hence, similar arcs are to each other as their radii, and similar sectors are A B D E to each other as the squares of their radii. above, is to be When it is pro Scholium. The term infinite, employed understood in its limited technical sense. posed to make the number of sides of the polygons infinite, by the method indicated in the Scholium of Proposition VII., it is simply meant to make that number so great that the difference between the areas of the circle and polygon shall be less than any appreciable quantity. We have seen (P. XII.), that when the number of sides 16384, the areas differ by less than the millionth part of a unit. By increasing the number of sides, a still closer approximation may be had. PROPOSITION XIV. THEOREM. The area of a circle is equal to half the product of its circumference and radius. Let be the centre of a circle, OC its radius, and ACDE its circumference: then will the area of the circle be equal to half the product of the circumference and radius. For, inscribe in it a regular polygon ACDE. Then will the area of this polygon be equal to half the pro F duct of its perimeter and apothem, whatever may be the number of its sides (P. VIII.). If the number of sides be made infinite, the polygon will coincide with the circle, the perimeter with the circumference, and the apothem with the radius: hence, the area of the circle is equal to half the product of its circumference and radius; which was to be proved. Cor. 1. The area of a sector is equal to half the product of its arc and radius. Cor. 2. The area of a sector is to the area of the circle, as the arc of the sector to the circumference. PROPOSITION XV. PROBLEM. To find an expression for the area of any circle in terms of its radius. Let be the centre of a circle, and CA its radius. Denote its area by area CA, its radius by R, and the area of a circle whose That is, the area of any circle is 3.1416 times the square of the radius. PROPOSITION XVI. PROBLEM. To find an expression for the circumference of a circle, in terms of its radius, or diameter. Let C be the centre of a circle, and CA its radius. Denote its circumference by circ. CA, its radius by R, and its diameter by D. From the last Proposition, we have, circ. CA = 2xR, or, circ. CA = «D. That is, the circumference of any circle is equal to 3.1416 times its diameter. Scholium 1. The abstract number, equal to 3.1416, denotes the number of times that the diameter of a circle is contained in the circumference, and also the number of times that the square constructed on the radius is contained in the area of the circle (P. XV.). Now, it has been proved by the methods of Higher Mathematics, that the value of · ¶ is incommensurable with 1; hence, it is impossible to express, by means of numbers, the exact length of a circumference in terms of the radius, or the exact area in terms of the square described on the radius. We may also infer that it is impossible to square the circle; that is, to construct a square whose area shall be exactly equal to that of the circle. Scholium 2. Besides the approximate value of π, 3.1416, usually employed, the fractions 2 and 355 are also used, when great accuracy is not required. 113 BOOK VI. PLANES AND POLYEDRAL ANGLES. DEFINITIONS. 1. A straight line is PERPENDICULAR TO A PLANE, when it is perpendicular to every line of the plane which passes through its FOOT; that is, through the point in which it meets the plane. In this case, the plane is also perpendicular to the line. 2. A straight line is PARALLEL TO A PLANE, when it cannot meet the plane, how far soever both may be produced. In this case, the plane is also parallel to the line. 3. TWO PLANES ARE PARALLEL, when they cannot meet, how far soever both may be produced. 4. A DIEDRAL ANGLE is the amount of divergence of two planes. The line in which the planes meet, is called the edge of the angle, and the planes themselves are called faces of the angle. The measure of a diedral angle is the same as that of a plane angle formed by two lines, one drawn in each face, and both perpendicular to the edge at the same point. A diedral angle may be acute, obtuse, or a right angle. In the latter case, the faces are perpendicular to each other. 5. A POLYEDRAL ANGLE is the amount of dir gence of several planes meeting at a common point. between the This point is called the vertex of the angle ; which the planes meet are called edges of the the portions of the planes lying called faces of the angle. Thus, S is the vertex of the polyedral angle, whose edges are SA, SB, SC, SD, and whose faces are ASB, BSC, CSD, DSA. A polyedral angle which has but three faces, is called a triedral A he lines in angle, and edges are angle. POSTULATE. A line may be drawn perpendicular to a p ne from anv point of the plane, or from any point without the plane. PROPOSITION I. THEOREM. If a straight line has two of its points in a plane, it will lie wholly in that plane. For, by definition, a plane is a surface sun, that if any two of its points be joined by a straight line, that line will lie wholly in the surface (B. I., D. 8). Cor. Through any point of a plane, an infinite number of straight lines may be drawn which will lie in the plane. For, if a line be drawn from the given point to any other point of the plane, that line will lie wholly in the plane. Scholium. If any two points of a plane be joined by a straight line, the plane may be turned about that line as an |