Dem.—Produce BC to M. Join HM. Erect BO at right angles to BM. Then, because CH = HF, the angle HCF = HFC, and the angle DCE = ECB (const.). Hence the angle HCD = HBC [I. xxxii.], and the right angles ACD, CBO are equal; therefore the angle ACH is equal to HBO; that is [III. xxxii.], equal to HMB, or to half the angle HCB. Hence ACH is one-third of ACB.
NOTE G.
on the quadrature of the circle.
Modern mathematicians denote the ratio of the circumference of a circle to its diameter by the symbol π. Hence, if r denote the radius, the circumference will be 2πr; and, since the area of a circle [VI. xx. Ex. 15] is equal to half the rectangle contained by the circumference and the radius, the area will be πr2. Hence, if the area be known, the value of π will be known; and, conversely, if the value of π be known, the area is known. On this account the determination of the value of π is called “the problem of the quadrature of the circle,” and is one of the most celebrated in Mathematics. It is now known that the value of π is incommensurable; that is, that it cannot be expressed as the ratio of any two whole numbers, and therefore that it can be found only approximately; but the approximation can be carried as far as we please, just as in extracting the square root we may proceed to as many decimal places as may be required. The simplest approximate value of π was found by Archimedes, namely, 22 : 7. This value is tolerably exact, and is the one used in ordinary calculations, except where great accuracy is required. The next to this in ascending order, viz. 355 : 113, found by Vieta, is correct to six places of decimals. It differs very little from the ratio 3.1416 : 1, given in our elementary books.
Several expeditious methods, depending on the higher mathematics, are known for calculating the value of π. The following is an outline of a very simple elementary method for determining this important constant. It depends on a theorem which is at once inferred from VI., Ex. 87, namely “If a, A denote the reciprocals of the areas of any two polygons of the same number of sides inscribed and circumscribed to a circle; a′, A′ the corresponding quantities for polygons of twice the number; a′ is the geometric mean between a and A, and A′ the arithmetic mean between a′ and A.” Hence, if a and A be known, we can, by the processes of finding arithmetic and geometric means, find a′ and A′. In like manner, from a′, A′ we can find a′′, A′′ related to a′, A′; as a′, A′ are to a, A. Therefore, proceeding in this manner until we arrive at values a(n), A(n) that will agree in as many decimal places as there are in the degree of accuracy we wish to attain; and since the area of a circle is intermediate between the reciprocals of a(n) and A(n), the area of the circle can be found to any required degree of approximation.
If for simplicity we take the radius of the circle to be unity, and commence with the inscribed and circumscribed squares, we have
These numbers are found thus: a′ is the geometric mean between a and A; that is, between .5 and .25, and A′ is the arithmetic mean between a′ and A, or between .3535533 and .25. Again, a′′ is the geometric mean between a′ and A′; and A′′ the arithmetic mean between a′′ and A′. Proceeding in this manner, we find a(13) = .3183099; A(13) = .3183099. Hence the area of a circle radius unity, correct to seven decimal places, is equal to the reciprocal of .3183099; that is, equal to 3.1415926; or the value of π correct to seven places of decimals is 3.1415926. The number π is of such fundamental importance in Geometry, that mathematicians have devoted great attention to its calculation. Mr. Shanks, an English computer, carried the calculation to 707 places of decimals. The following are the first 36 figures of his result:—
