taining the right angle, the cone is said to be a right-angled A. B cone, because then a vertical section of the figure always exhibits a right angle (CAD) at the apex, each of the angles BAC and BAD containing 45°. Similarly, if the fixed side be less than the other side, the cone is said to be obtuse-angled; and, if greater, acute-angled; a vertical section of the cones exhibiting respec tively an obtuse and an acute angle. It is obvious that just as the circle is only a kind of polygon (p. 29), so the cone is only a species of pyramid, and may be treated in exactly the same way. The volume of a cone is equal to one-third of the product of the area of its base and its altitude. A In other words the volume of a cone is exactly equal to onethird of the volume of the cylinder which circumscribes it. The name cone is generally understood to imply only a right or perpendicular cone, as already defined, i.e., one whose apex is exactly over the centre of the base, and whose axis is therefore perpendicular to the base. Sometimes, however, it is used to denote an oblique cone (fig. II.), the volume of which is equal to one-third of the product of the area of the base, and the perpendicular height AB. B It is evident that the convex surface of a right cone is in the form of a sector of a circle, the radius being the slant height, and the length of the arc the circumference of the base. Hence, to find the area of the convex surface of a cone, multiply the circumference of the base by half the slant height. The cone has been already referred to in the lesson on the ellipse (p. 35) in the mensuration of superficies, where a brief explanation is given of the conic sections, viz., the ellipse (A), parabola (B), and hyperbola (C), which are formed by the intersection in different ways of a plane and a cone. You will learn more about these figures when you get into more advanced parts of mathematics, which will enable you to investigate all their properties, but here it will be sufficient to say that the interest and importance of these figures is increased when you know that all the planets move round the sun in elliptical orbits, and that the path of a projectile, whether it be a bullet shot from a rifle, a ball from a cannon, or a stone thrown from the hand, is a parabola. EXERCISES (O). 1. A cone is 30 ft. high, and the diameter of its base is 1 yard; find its volume. 2. Find the volume of a cone whose height is 8 yds. 1 ft., and circumference of base 6 yds. 2 ft. 3. The diameter of the base of a cone is 9 ft., and its height is equal to the circumference of the base; find its value at 1s. per cubic foot. 4. A cylindrical bottle 4 in. in diameter is filled with wine to the height of 7 in.; how many times will this fill a conical wine glass, 2 in. deep, and 2 in. in diameter at the top. 5. A cone is 2 ft. high, and its volume is 8 cubic ft.; find the circumference of its base. 6. A cone measures 11 yds. round at the base, and its slant height is 8 ft.; find its solidity. 7. The slant height of a cone is 13 ft. 5 in., and the diameter of its base is 7 ft. 4 in.; find the cost of plastering its surface at 1s. per square foot. 8. What is the cost of cleaning the curved surface of a cone whose slant height is 18 ft. 8 in., and diameter 13 ft. 4 in., at 1s. per square foot. 9. The slant height of a spire is 45 ft., and the circumference of the base 30 ft.; the cost of painting it is £18, 5s. 7d.; at what rate per square foot is this charged? LESSON XIX. THE SPHERE. A sphere is a body which is perfectly round in all directions, as a ball or globe. Of course you are all able to recognise a sphere by its shape the moment you look at it; but, in order to prevent confusion and mistake, it is necessary, as in the figures we have previously examined, to associate the name of the figure with an accurate definition, such as either of the two following: 1. A sphere is a solid body bounded by one surface, all points of which are equally distant from a point within the body called its centre. 2. A sphere is a body generated by the revolution of a semicircle about its diameter, which remains fixed. We have already spoken of the figures which are formed when a plane intersects a cone. When a plane intersects a sphere it always makes a circle, which is greater or smaller according as the intersection passes through the centre of the sphere, or is more remote from it. Thus the greatest circle that can be made in a sphere is one formed by a plane passing exactly through its centre; this C is called a great circle of the sphere. The segment of a sphere is any part of it, A or B, cut off by a plane. If the plane pass through the centre (C), it will divide the sphere into two equal parts called hemispheres. Now, in speaking of previous figures in this book, we have been careful, as far as possible, to explain to you the reason of the rules which are laid down, and in treating of solid bodies we have spoken first of their volume and afterwards of the area of their surface. In the case of the sphere, however, there is so much that you will not be able at present to understand, that we shall be obliged to give you rules without complete proof, and it will be better also to speak first of the surface of a sphere, and afterwards of its volume. The area of the surface of a sphere may be conveniently found in five different ways, which however, of course, all amount to the same thing, and each of them, as we shall show below, is only a statement in words of the more convenient mathematical expression— surface 4 πr2. 1. The surface of every sphere is equal to the area of four great circles of the sphere, i.e., of four circles, such as already described, formed by a plane passing exactly through the centre of the sphere. But the radius of such a circle as this is the same as the radius of the sphere itself, and its area is therefore πr2; and the area of four such circles is 4πr2. 2. The surface of every sphere is equal to the area of the circle whose diameter is double the diameter of the sphere. But the radius of such a circle is equal to the diameter of the sphere, i.e. = 2r. and areaπ x radius 2 =π × (2r)2 =πx 4r2 =(as before) 4πr2 3. The surface of every sphere is equal to the convex surface of the circumscribed cylinder. Now this convex surface is equal to the circumference multiplied by the height; but the circumference is the same as that of the sphere (viz., 2πr), and the height is equal to the diameter (viz. 2r), so that area=2πrx 2r =(as before) 4πr2 4. The surface of every sphere is equal to the circumference of the sphere multiplied by its diameter. This is exactly the same as the preceding rule: 5. The surface of every cylinder is equal to the square of the diameter multiplied by 3·14159. D But the square of the diameter is (2)2, which multiplied by 3.14,1594r2 x or (as before) = We can now proceed to investigate the method for finding the volume of a sphere. The rule for this is usually stated in one of four convenient forms, which, however, as might be expected, are only different ways of stating the same thing, and as will be shown, they are all equiva lent to the convenient mathematical expression volume = 3 1. Rule I.-The volume of a sphere is equal to the surface of the sphere multiplied by one-third of the radius. Now we can suppose that a sphere, instead of being bounded by one convex surface is bounded by an infinite number of small faces or plane surfaces; the whole sphere would then consist of an infinite number of pyramids, having these small faces for their bases, their common vertex the centre of the sphere, and their altitude the radius of the sphere. But the volume of a pyramid is equal to the area of its base multiplied by one-third of its height; and therefore the volume of the whole sphere is equal to the sum of the areas of all these small faces (i.e., the whole surface of the sphere), multiplied by one-third of the radius, which 2. It will be seen, however, that in this expression the only variable quantity is r3, and that Rule II.-The volume of a sphere is equal to the cube of the radius multiplied by 4-18878. |