must have its area equivalent to the rectangle under the radius or the limit of all the successive altitudes and the semicircumference, which limits also the corresponding semiperimeters. Scholium. From this proposition is derived a very simple and elegant method of approximating to the numerical expression for the area of a circle. Let the original polygon be a square, each side of which is denoted by unit; the component sector AOB is therefore a right-angled isosceles triangle, having the perpendicular OC, or the radius of the inscribed circle equal to .5, and the side OA of the circumscribing circle equal to √.5 or.7071067812. But DG, the radius of a circle inscribed in an octagon of .6035533906; and DE the radius of the circle circumscribing that octagon, is = √(OA.DG) = √(.603533906 × .707106812)=.6532814824. Again, the radius of the circle inscribed in a polygon of 16 sides with the same peri meter, is = .603533906+.6532814824 2 = .62844174365; and the radius of the circle circumscribing that polygon, is = √(.6284174365 ×.6532814824)=.6407288619. In like manner, the radii of circles inscribing and circumscribing the polygons of 32, 64, 128, &c. sides, under the same perimeter, are successively found, by an alternate series of arithmetical and geometrical means. It may be observed, that these radii mutually approximate about four times nearer at each step: For (II. 10.) CA2 =OA2-OC =(II. 17.) (OA-OC) (OA+OC); and, for the same reason, GE = DE2-DG2=(DE-DG) (DE+DG). But, CA being double of GE, and CA2=4GE2, it is evident that (OA-OC) (OA+OC)=+(DE-DG) (DE+DG); and since the successive radii must approach on both sides to form the same amount, or OA+OC=DE+DG nearly, it follows that OA-OC=4(DE-DG) nearly. In the subjoined table, where the computation is carried to ten decimal places, this rate of mutual approximation will be found true to the last figure, in the expressions for the radii of the circles attached to all the polygons beyond that of 256 sides. Thus, for the polygon of 512 sides, .6366237671-.6366117828 4 =.0000029960, which is the difference between .6366207710 and .6366177750, the radii of the circles described about and within the polygon of 1024 sides. After five or six terms have been computed, the rest may be found by a simple process, because the mean proportional between two proximate lines is very nearly equal to half their sum, or the arithmetical mean. While each number in the first column, therefore, is always equal to half the sum of the preceding terms in both columns, the corresponding number in the second column may be considered as equal to half the sum of that number and of the term immediately above itself. Thus, .6366207710, the radius of the circle circumscribing the polygon of 1024 sides, is equal to half the sum of .6366177750, the radius of its inscribed circle, and of .6366237671, the radius of the circle circumscribing the polygon of 512 sides. But the final term may be discovered still more expeditiously; for, since the numbers in both columns are formed by taking successive means, those of the second column must each time be diminished by the fourth-part of the common difference, and consequently (V. 21.) the continued diminution will accumulate to one-third of that difference. Wherefore the ultimate radius of the inscribed and circumscribing circles, is the third-part of the sum of a radius of inscription and of double the corresponding radius of circumscription. Thus, stopping at the polygon of 256 sides, .63665878141+2(.6366357516) 3 =.6366197724, the final result. No. of sides of Radius of Inscribed Radius of Circum Hence the radius of a circle, whose circumference is 4, or the diameter of a circle whose circumference is 2, will be denoted by .6366197724; wherefore, reciprocally, the circumference of a circle whose diameter is 1, will be expressed by 3.1415926536, and its area, or that of the ultimate polygon, by .7853981434. In most cases, however, it will be sufficiently accurate to retain only the first four figures. Wherefore 3.1416, multiplied into the diameter of a circle, will denote its circumference, and .7854, multiplied into the square of the diameter, will give the numerical expression for its area. APPENDIX. THE constructions used in Elementary Geometry, were effected, by the combination of straight lines and circles. Many problems, however, can be resolved, by the single application of the straight line or the circle; and such solutions are not only interesting, from the ingenuity and resources which they display, but may, in a variety of instances, be employed with manifest advantage. This Appendix is intended to exhibit a selection of Geometrical Problems, resolved by either of those methods singly. It is accordingly divided into Two Parts, corresponding to the rectilineal and the circular constructions. |