the apothem of the second polygon is equal to half the sum of the given apothem and radius. Again, in the right angled triangle GCF (324), FC: FG :: FG: FI. But FC is equal to CB; therefore, FG, the radius of the second polygon, is a mean proportional between the given radius and the apothem of the second. 491. For convenient application of these principles, let us represent the given apothem by a, the radius by r, and the side by s, the apothem of the polygon of double the number of sides by x, and its radius by y. 492. Again, since, in any regular polygon, the apothem, radius, and half the side form a right angled 493. Problem. To find the approximate value of the ratio of the circumference to the diameter of a circle. Suppose a regular hexagon whose perimeter is unity. Then its side is or .166667, and its radius is the same (454). By the formula, a=√4r2—s2, the apothem is Then, by the formula, x=(a+r), the apothem of the regular polygon of twelve sides, the perimeter being unity, is (+3) or .155502. The radius of the Geom.-15 same, by the formula y=√xr, is .160988. Proceeding in the same way, the following table may be constructed: Now, observe that the numbers in the second column express the ratios of the radius of any circle to the perimeters of the circumscribed regular polygons; and that those in the third column express the ratios of the radius to the perimeters of the inscribed polygons. These ratios gradually approach each other, till they agree for six places of decimals. It is evident that by continuing the table, and calculating the ratios to a greater number of decimal places, this approximation could be made as near as we choose. But it has been already shown that the circumference is less than the perimeter of the circumscribed, and greater than that of the inscribed polygon. Hence, we conclude, that when the circumference is 1, the radius is .159155, with a near approximation to exactness. The diameter, being double the radius, is .31831. 494. It was shown by Archimedes, by methods resembling the above, that the value of π is less than 34, and greater than 30. This number, 34, is in very common use for mechanical purposes. It is too great by about one eight-hundredth of the diameter. About the year 1640, Adrian Metius found the nearer approximation, which is true for six places of decimals. It is easily retained in the memory, as it is composed of the first three odd numbers, in pairs, 113|355, taking the first three digits for the denominator, and the other three for the numerator. By the integral calculus, it has been found that is equal to the series 4—3+-+-+, etc. By the calculus also, other and shorter methods have been discovered for finding the approximate value of л. In 1853, Mr. Rutherford presented to the Royal Society of London a calculation of the value of π to five hundred and thirty decimals, made by Mr. W. Shanks, of Houghton-le-Spring. The first thirty-nine decimals are, 3.141 592 653 589 793 238 462 643 383 279 502 884 197. EXERCISES. 495.-1. Two wheels, whose diameters are twelve and eighteen inches, are connected by a belt, so that the rotation of one causes that of the other. The smaller makes twenty-four rotations in a minute; what is the velocity of the larger wheel? 2. Two wheels, whose diameters are twelve and eighteen inches, are fixed on the same axle, so that they turn together. A point on the rim of the smaller moves at the rate of six feet per second; what is the velocity of a point on the rim of the larger wheel? 3. If the radius of a car-wheel is thirteen inches, how many revolutions does it make in traveling one mile? 4. If the equatorial diameter of the earth is 7924 miles, what is the length of one degree of longitude on the equator? QUADRATURE OF CIRCLE. 496. The quadrature or squaring of the circle, that is, the finding an equivalent rectilinear figure, is a problem which excited the attention of mathematicians during many ages, until it was demonstrated that it could only be solved approximately. The solution depends, indeed, on the rectification of the circumference, and upon the following 497. Theorem.-The area of any polygon in which a circle can be inscribed, is measured by half the product of its perimeter by the radius of the inscribed circle. · From the center C of the circle, let straight lines extend to all the vertices of the polygon ABDEF, also to all the points of tangency, G, H, I, K, and L. The lines extending to the points of tangency are radii of the circle, and are therefore perpendicular to the sides of the polygon, which are tangents of the circle (183). The polygon is divided by the lines extending to the vertices into as many triangles as it has sides, ACB, BCD, etc. Regarding the sides of the poly A K E H gon, AB, BD, etc., as the bases of these several triangles, they all have equal altitudes, for the radii are perpendicular to the sides of the polygon. Now, the area of each triangle is measured by half the product of its base by the common altitude. But the area of the polygon is the sum of the areas of the triangles, and the perimeter of the polygon is the sum of their bases. It follows that the area of the polygon is measured by half the product of the perimeter by the common altitude, which is the radius. 498. Corollary.-The area of a regular polygon is measured by half the product of its perimeter by its apothem. 499. Theorem.-The area of a circle is measured by half the product of its circumference by its radius. For the circle is the limit of all the polygons that may be circumscribed about it, and its circumference is the limit of their perimeters. 500. Theorem.—The area of a circle is equal to the square of its radius, multiplied by the ratio of the circumference to the diameter. For, let r represent the radius. Then, the diameter is 2r, and the circumference is TX2r, and the area is 2rr, or r2 (499); that is, the square of the radius multiplied by the ratio of the circumference to the diameter. 501. Corollary. The areas of two circles are to each other as the squares of their radii; or, as the squares of their diameters. 502. Corollary. When the radius is unity, the area is expressed by л. 503. Theorem.-The area of a sector is measured by half the product of its arc by its radius. For, the sector is to the circle as its arc is to the circumference. This may be proved in the same manner as the proportionality of arcs and angles at the center (197 or 202). 504. Since that which is true of every polygon may |