Taking the reciprocals of the last result, we have

KH+HM 1/KII+HM

1

If then we know the reciprocals of the areas of inscribed and circumscribed polygons of n sides, the reciprocals of those of 2n sides may be determined by the above established relations.

100. A Numerical Computation. radius is unity we inscribe a square, and then circumscribe a square about it, we would have for the area of the inscribed square, a = 2; and for the area of the circumscribed square, A=4; for the area of the inscribed octagon, a' = √8.

- If in a circle whose

2

FIG. 167.

13

a3 and 4 are respectively the areas of regular polygons of 32768 sides, inscribed within and circumscribed about a circle; the square on the radius being the unit.

The area of the circle lies between the areas of a, and A, and is nearer to either of them than they are to each other.

The same may be said of a, and A1, ɑ, and A2, ɑg and A, etc., as far as the computations may be continued.

139

In our investigation we have carried the approximation to a3 and 4, and find that they are the same for seven decimal places. The area of the circle will be the same for seven decimal places.

Letting K represent the area of a circle and R its radius, we will have:

or

K = (3.1415926...) (R),

K

R2

=

3.1415926...

The area of a circle is an exact thing and the square on the radius is an exact thing. The ratio of the two, by common consent, is represented by the Greek letter (#) and is an incommensurable number to which we may approach as near as time and patience will allow, but which we can never express in integers or their fractional parts.

NOTE. The quadrature of a circle (i.e., the determination of its area in terms of any given square) has demanded the attention of students of mathematics for 4000 years or more, and has had .expended upon it a vast amount of time.

The earliest statements that we have give the ratio as 3, later (1700 в.c.) as about 3.16, still later (220 в.c.) as 31. The most remarkable ratio of whole numbers that approximate to π (1600 A.D.) is, which when expressed decimally agrees with π, expressed decimally, to the sixth decimal place.

113

355 may be easily remembered by noting the fact that if we write the first three odd numbers twice each, and then divide the last three by the first three, thus, 113)355, we have the ratio. If 3.1415926 be expressed as a continual fraction, it gives :

The terms of the series,,, V., etc., are (beginning with the fourth term) alternately the arithmetical and the geometrical means of the two preceding terms; and is known as Schwab's series.

Remark. - Because of the fact that we are unable to express the area of a circle in terms of the square on the radius exactly, it does not follow that other areas bounded by curved lines, or by a combination of curved and straight lines, cannot be expressed exactly in terms of a square on some given line, or a rectangle, or a parallelogram, which may be converted into a square.