CHAPTER V

APPROXIMATE SOLUTIONS TO SOME UNSOLVED PROBLEMS

63. Rectification of Circular Arcs.-The ratio of the circumference of a circle to its diameter cannot be expressed exactly, in other words the two are incommensurable. The symbol is always used to denote the ratio of the circumference to the diameter and its approximate value is 3.1416 or nearly 31.

The best geometrical constructions hitherto given for finding approximately the length of a circular arc, or for marking off an arc of given length, are those due to Rankine,' and are as follows:-

(a) To draw a straight line approximately equal to a given circular are AB (Fig. 129). Join BA and produce it to D making AD

= AB. With centre D and radius DB describe the arc BC cutting at C the tangent AC to the arc at A. AC is the straight line required.

The error varies as the fourth power of the angle AOB, where O is the centre of the circle of which AB is an arc. When the angle AOB is 30°, AC is less than the arc AB by about 10 of the length of the arc.

14400

=

(b) To mark off on a given circle an arc AB approximately equal to a given length (Fig. 130). Draw a tangent AC to the circle at A, and make AC equal to the given length. Make AD AC. With centre D and radius DC describe the arc CB to cut the circle at B. AB is the arc required.

A Manual of Machinery and Millwork.

The error in (b) as a fraction of the given length is the same as in (a), and follows the same law.

If in Fig. 129 the angle AOB is greater than one right angle and less than two right angles the length of one half of the arc AB should be determined by the construction and the result doubled. If the angle AOB is greater than two right angles the length of one quarter of the arc AB should be determined by the construction and the result quadrupled.

If in Fig. 130, AC is greater than one and a half times the radius OA the arc equal to one half of AC should be determined by the construction and this arc should then be doubled. If AC is greater than three times OA the arc equal to one quarter of AC should be determined by the construction and this arc should then be quadrupled.

The construction in (b) follows easily from that in (a), for if the construction in Fig. 129 be performed and CD be joined (Fig. 131), and the angle CDB be bisected by DE meeting AC at E, a circle with

3AD,

centre E and radius EC will pass through B. Since CD = DB = and since DE bisects the angle CDB, it follows (Euclid VI, 3) that CE 3AE or AE = AC.

=

The following slight modification of Rankine's first construction (a) gives a more approximate result, and is to be preferred, especially when the angle AOB is greater than 60°. Instead of making AD equal to half the chord AB make it equal to the chord AF of half the arc AB (Fig. 132) and proceed as before.

For the case where the angle AOB is 90° the error in Rankine's construction is about 1 in 170 while in the modified construction the error is only about 1 in 2300.

64. To draw a Straight Line whose Length shall be approximately equal to the Circumference of a Given Circle. -Draw a straight line whose length shall be approximately equal to a quarter of the circumference by the modification of Rankine's construction explained in the preceding article; a line four times this in length will be the line required.

The following construction given by the late Mr. T. H. Eagles'

1 Constructive Geometry of Plane Curves, p. 267.

O (Fig. 133) is the centre and

gives a very close approximation. AOB a diameter of the given circle. Draw the tangent AC and make AC equal to three times AB. Draw a radius OD making the

E

B

FIG. 133.

angle BOD 30°. Draw DE at right angles to AB meeting the latter at E. Join EC. The length of EC is very nearly equal to the circumference of the circle. EC is a little longer than the true circumference, the error being about 1 in 21,700.

2

65. To draw a Straight Line whose Length shall represent approximately the Value of .-The circumference of a circle of radius r is 2πr, and therefore a quarter of the circumference is If r = 2, then a quarter of the circumference is equal to . Hence if a quarter of a circle be drawn with radius 2, the length of the arc will be equal to T, and this length may be determined by one of the constructions of Art. 63. Instead of taking a quarter of a circle with a radius 2, a sector whose angle is 60° and radius = 3 may be used, or generally a sector whose angle is n°, and radius 180

=

n

=

may be taken, and the length of its arc will be equal to π.

The circumference of a circle whose radius is 0.5 is equal to and may therefore be found by the construction given in the latter part of Art. 64, by making r equal to 0.5.

66. To find the Side of a Square whose Area shall be approximately equal to that of a Given Circle.-Solving this problem is known as "squaring the circle." O is the centre and AB a diameter of the given circle.

First Method (Fig. 134). Draw AC at right angles to AB and

equal to AO. Draw BC cutting the circle at D. Join BD. BD is the line required. The error in this construction is that BD is too

long by an amount equal to 00164r, where r is the radius of the circle.

Second Method (Fig. 135). Produce AB to E, and make BE equal to three times BO. With centre A and radius AO describe the arc OC. With centre E and radius EA describe the arc AC to cut the former arc at C. Draw CE cutting the circle at D. Join BD. BD is the line required. The error in this construction is that BD is too short by an amount equal to 0.0007r, where r is the radius of the circle.

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67. To draw a Straight Line whose Length shall represent approximately the Value of the Square Root of 7. The area of a circle whose radius is r is r2, and if 8 is the side of a square whose area is equal to that of the circle, then s′′r2, or 8 = r√√ π. If r 1, then 8 = , and this may be found by one of the constructions in the preceding article.

=

68. To find the Side of a Square whose Area shall be approximately equal to that of a Given Ellipse.-If a and b are the semi-axes of an ellipse its area is ab. If r is the radius of a circle equal in area to the ellipse, then r2 = Tab, or r2 = ab, and r is a mean proportional between a and b and may be found as in Art. 12, p. 11. The side of a square whose area is approximately equal to that of the circle may then be found by the construction of Art. 66.

69. To inscribe in a Given Circle a having a Given Number of Sides.-AB (Fig. and O the centre of the given circle. With centre A and radius AB describe the arc BC. With centre B and radius BA describe the arc AC cutting the former arc at C. Divide the diameter AB into as many equal parts as there are sides in the polygon. D is the second point of division from A. Draw CD and produce it to cut the circle at E. The chord AE is one side of the polygon, and the others are obtained by stepping the chord AE round the circle.

Regular Polygon 136) is a diameter

FIG. 136.

B

The above construction is exact for an equilateral triangle, a square, and a hexagon. For a pentagon the central angle AOE is too small, and when the chord AE is stepped round from A five times, the last point will fall short of A by an amount which subtends at O an angle of nearly a quarter of a degree, For a heptagon the central angle AOE is too large, and when the chord AE is stepped round from A seven times, the last point will be beyond A by an amount which subtends at O an angle of a little more than five-eighths of a degree.

Exercises V

1. Draw an arc of a circle of 2 inches radius subtending an angle of 60° at the centre of the circle; then draw by Rankine's construction, a straight line equal in length to the arc.

2. From the circumference of a circle of 2 inches radius cut off, by means of Rankine's construction, an arc equal in length to the radius.

3. Find, by construction, and by calculation, the circular measure of an angle of 45°.

4. Construct an angle whose circular measure is 1.2.

5. Find, by construction, and by calculation, the circumference of a circle whose diameter is 2.75 inches.

6. Find, by construction, and by calculation, the diameter of a circle whose circumference is 6 inches.

7. Find, in the simplest possible way, the diameter of a circle whose circumference is equal to the sum of the circumferences of two circles one of which is 1.75 inches, and the other 1.25 inches in diameter.

8. Draw a square whose area shall be equal to that of a circle whose radius is 1.5 inches.

9. Draw a circle having an area equal to that of a square of 2.25 inches side.

10. Construct a square having an area equal to that of an ellipse whose major and minor axes are 3.5 inches and 2.5 inches respectively.

11. In a circle of 2 inches radius inscribe a regular heptagon.