496. Historical Note. The development of geometry was aided very decidedly by the appearance of problems whose solution seemed to defy the powers of the best mathematicians. Among these difficult problems none has exerted a greater influence, or has attracted greater attention throughout the ages, than the famous problem of "the squaring of the circle." In its original form the problem meant the construction of a square which should have an area equal to that of a given circle. As will now be readily understood, this question involves the determination of π, or the construction of a line equal to the circumference of the given circle. The history of this problem extends over a period of four thousand years, and may be divided into three periods :

1. From the earliest times to the 17th century A.D. During this time the value of was computed by means of polygons, as in Proposition VIII. 2. From the beginning of modern mathematics (calculus, etc.) to 1766. During this period the value of was computed more accurately by algebraic methods.

3. The modern period. In 1766 Lambert proved that is irrational. This prepared the way for the modern discovery that cannot be constructed with ruler and compasses. The proof was given by Lindemann in 1882. Hence it is impossible to construct geometrically a line equal to the circumference of a given circle, or a square having an area equal to that of a given circle.

The earliest reference to the number is found in the Rhind Papyrus (British Museum), written by Ahmes, an Egyptian scribe, about 1700 в.с. His rule for finding the area of a circle consists in squaring eight ninths of the diameter, which makes π = 3.1604.

The Bible mentions the number in two places: 1 Kings vii, 23; and 2 Chronicles iv, 2, giving ᅲ = 3 (probably a Babylonian method). Archimedes (born 287 в.с.), the greatest mathematician of antiquity, by a method similar to that of Proposition VIII, proved that the value of lies between 34 and 311, or, in decimals, between 3.1428 and 3.1408. Ptolemy, a great astronomer (150 A.D.), found that ᅲ = 3 = 3.14166. The Hindus used π = √ 10 = 3.1623; also, ᅲ = 7 = 3.1416. Metius of Holland (1625 A.D.) found ᅲ = 11 = 3.1415929, and this decimal is correct through the sixth place.

355

3927

Ludolph van Ceulen (Leyden, 1610) carried the value of to thirtyfive decimal places.

By improved methods of calculation the value of has been carried in recent years by Shanks to 707 decimal places. The symbol ᅲ, in the present sense, was used for the first time by William Jones, in 1706. It can be shown that is neither a rational fraction nor a surd. It cannot be expréssed exactly by an algebraic number. It is therefore called a transcendental number.

The correct value of to ten places of decimals is 3.1415926535.

EXERCISES

1. In a certain city whose diameter is about 10 km., it is desired to construct a circular boulevard around the outskirts. Estimate the errors in the length of the inner curb which would arise from using the successive approximations to the value of in § 463.

2. Plot a graph showing successive approximations of ᅲ (§463). Suggestion. On the horizontal axis lay off distances representing the number of sides in the polygons used, and on the vertical axis lay off the corresponding values of π. Use a large scale on the vertical axis.

B

3. Many attempts have been made to construct a line equal to the length of a circle. The following approximate construction is one of the simplest. It is due to Kochansky (1685). At the extremity A of the diameter AB of a given circle of radius R draw a tangent CD, making ∠COA = 30° and CD=3R. Prove that BD = R134 - 2√3 = 3.1415 R; that is, BD = the circumference, very nearly.

CA

0

D

4. Archimedes (250 в.с.) stated the proposition that the area of a circle is equal to that of a triangle whose base is the length of the circle, and whose altitude is the radius. Explain. Construct this triangle approximately by using Ex. 3.

5. Since it is possible to transform any triangle into a square, construct a square approximately equal in area to that of a circle by using the method of Ex. 3.

6. A very old Egyptian manuscript, written about 1700 в.с., contains this rule for finding the area of a circle; that is, for “squaring a circle." From the diameter of a circle subtract one ninth of the diameter, and square the remainder. To what value of does this construction correspond?

7. Hippocrates (430 в.с.), a great Greek geometrician, tried to "square the circle." The following theorem illustrates his method of approaching the problem: If on the sides of an inscribed square as diameters semicircles are described, the area of the four crescents lying without the circle equals the area of the inscribed square. Give proof.

REVIEW EXERCISES

1. What are the important topics of Book V?

2. What use is made of regular polygons in Book V?

3. Give a brief description of the process by which the length

and the area of a circle are determined geometrically.

4. Define π. Prove that it is a constant.

5. Give two formulas for the length of a circle.

6. Give three formulas for the area of a circle.

7. How is the area of a sector found?

8. How is the area of a segment found?

9. If the radius of a circle is multiplied by 2 (3, 4, n), what is

the effect on the length of the circle? on the area?

10. The areas of two regular octagons are as 1:16, and the sum

of their perimeters is 25. How long is a side of each?

11. A side of a regular hexagon is 12. Find the radius of the inscribed circle.

12. A circle and a square have equal areas. Which has the greater perimeter?

13. A circle and a square have equal perimeters. Which has the greater area?

14. The area of a circle is to be divided into 3 (4, 5, n) equal parts by means of concentric circles. If the radius of the given circle is 100, what are the radii of the concentric circles?

15. The arch of a bridge has the form of a circular arc. Its span (chord) is 280 ft., and the greatest height of the circular arc above its chord is 80 ft. Find the radius of the circle of which the arc is a part.

16. The radius of a circle is 10. Find the area lying between two parallel sides of the inscribed regular hexagon.

17. The apothem of a regular hexagon is 4. Find the area of the hexagon.

18. Find the number of degrees in the central angle of a sector if its perimeter is equal to the circumference of the circle of which it forms a part.

19. Show by the figure that the value of C:D lies between 3 and 4. (Use the perimeters of the polygons.)

MISCELLANEOUS EXERCISES

COMPOSITE FIGURES

TREFOILS

The equilateral triangle may be used as the foundation of numerous ornamental designs, which are called trefoils.

These figures are often seen in decorative patterns, and are frequently introduced in the construction of church windows.

1. Construct an equilateral triangle. With each vertex as a center, and with one half of a side as a radius, describe arcs as indicated in Figs. 1 and 2. Let 2 a represent the length of a side of the equilateral triangle. Find the perimeter and the area of the figure bounded by the arcs.

FIG. 1

FIG. 2

FIG. 3

FIG. 4

2. Modify the preceding exercise by using the mid-points of the sides as centers, as indicated in Figs. 3 and 4.

3. Inscribe an equilateral triangle in a circle of radius 2 a. Using the mid-point of each radius of the triangle as a center, and a as a radius, describe circles. A symmetric pattern will result. Find the perimeter and the area of the trefoil and of the shaded part of the figure.

ia

4. The perimeter of a certain church window is made up of three equal semicircles the centers of which form the vertices of an equilateral triangle which has sides 34 ft. long. Find the area of the window and the length of its perimeter. (Harvard College Entrance Examination paper.)

5. If the area of the trefoil in Fig. 3 above is 50 sq. ft., how long is one side of the equilateral triangle? (Take π = 22.)

6. Assuming that the area of the trefoil in each of Figs. 1, 2, and 4 above is 50 sq. ft., find in each case one side of the equilateral triangle.

QUATREFOILS

A square may be used as the foundation of ornamental figures, which are usually called quatrefoils.

7. Construct a square. The length of a side is 2a. In Figs. 1 and 2 use the vertices as centers and one half of a side as a radius.

FIG. 1

FIG. 2

FIG. 3

FIG. 4

In Figs. 3 and 4 use the mid-point of each side as a center. In each case find the perimeter and the area of the figure bounded by the arcs.

exercise. Find the area and the perimeter of the figure bounded by

the arcs. (The construction lines indicate an

equilateral triangle, from which the relation

of the arcs can be inferred.)

A

9. In the adjoining figure find the perimeter and the area of ABCD...K, if the side of the square is 2a and the radius of the circle isa.

K

10. The adjoining cross-shaped figures are obtained by using the

vertices of the square as centers

and one half of a diagonal as

a radius. Find the perimeter and

the area of each figure if the diagonal equals 2 a.

(The second figure was given on

page 230 for the purpose of determining the area of a regular octagon.)