AD and BD in E and F, and the straight line joining these points must (VI. 2.) be equal to half the base AB. Wherefore the triangle EDF, repeated about the vertex D, would form a regular polygon with twice as many sides as before, but under the same extent of perimeter, since each of

those sides has only half

the former length.

Cor. 1. Hence DG, the radius of the circle inscribing the derived polygon, is half of CD, that is, half of the sum of OC and OA, the radii of the circles inscribing and circumscribing the given polygon. Again, since AOD is evidently isosceles, AD2=20A.CD (II. 23. cor.), and consequently DE the radius of the circumscribing derived polygon, being the half of AD, is a mean proportional between OA and DG, the radius of the circle circumscribing the given polygon, and the radius of the circle inscribing the derived polygon.

Cor. 2. Hence the area of a circle is equivalent to the rectangle under its radius, and a straight line equal to half its circumference. For the surface of any regular circumscribing polygon, being composed of triangles such as EDF, which have all the same altitude DG, is equivalent (II. 5.) to the rectangle under DG, and half the sum of their bases, or the semiperimeter of the polygon. Therefore the circle itself, since it forms the ultimate limit of the polygon,

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 OA+OC .5+.707106812

2

=

2

the same perimeter, is = .6035533906; and DE the radius of the circle circumscribing that octagon, is =√(OA.DG)= √(.603533906 X .707106812)=.6532814824. Again, the radius of the circle inscribed in a polygon of 16 sides with the same peri.603533906+.6532814824

meter, is =

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.) CAOA2-OC2 =(II. 17.) (OA~OC) (OA+OC); and, for the same reason, GE2= DE”—DG2=(DE—DG) (DE+DG). But, CA being double of GE, and CA-4GE2, it is evident that (OA—OC) (OA+OC)=4(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 QA-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 cir

cumscribing 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.