Anthony P. Solli
Activity I
The scribe Ahmes assumed that the area of a circular field with a diameter of nine units is the same as the area of a square with a side of eight units.
Have students calculate the area of both figures first using the Egyptian ¹. 3 1/6. Than ¹ = 3.14, as known today, and compare the results with the Egyptian’s approximation.
Activity II
A Babylonian scribe found the area of a circle by taking three times the square of the radius.
Have students calculate the area of the circle first using the Babylonian method A = 3r2, and than by today’s method of A = ¹r2.
Activity III
Heron chose a specific case in which the sum of the circumference, diameter, and area of a circle was 212. Heron’s instructions “Multiply 212 by 154, add 841, take the square root and subtract 29 and divide by 11” will be a circle with diameter 14.
Have students read and following Heron’s instructions to calculate the diameter. Is it 14?
Activity IV
The circumference of a wheel with a diameter of 4 feet is given by Vitruvius as 12.5 feet, implying a value of 3.125 for ¹.
Have students using the formula C = ¹d, calculate the value
Now have the students calculate the circumference of the wheel using ¹= 3.14 given a diameter of 4 feet.
Compare: Was Vitruvius off by a great amount?
Activity V
In the
Nine Chapters on the Mathematical Art
, the area of the circle was found by taking three fourths the square on the diameter or onetwelfth the square of the circumference.
Have students figure the value of ¹ using the following information:
A = (3/4)(d)2 = (1/12)(C)2
¹ should be about 3.
Activity VI
One statement in the
Aryabhatiya
to which Hindu scholars have pointed with pride is as follows: Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000.
Have students follow the above instructions and calculate ¹.
Activity VII
V10= 3.162277... is sometimes known as the Hindu value.
Have students calculate areas of circles, and circumferences of circles using V10 for ¹.
Compare results with calculations done using 7 ~ 3.14.
Activity VIII
Using John Wallis’ infinite product:
2/¹ = (1/2)(3/2)(3/4)(5/4)(5/6)(7/6) ....
have students calculate ¹ using hand calculators.
How far does one have to carry out the product to approach 3.14?
Activity IX
Using William Brouncker’s expression:
(figure available in print form)
have students calculate ¹ using hand calculators,
How far does one have to carry out the expression to approach 3.14?
Activity X
Using Leibniz’s infinite series:
¹/4 = 1/1 1/3 + 1/5 1/7 + 1/9 ....
have students calculate ¹ using hand calculators.
How far does one have to carry out the infinite series to approach 3.14?
Activity XI
Using Leonard Euler’s infinite series:
¹ = 1 + 1/2 + 1/3 + 1/4 1/5 + 1/6 + 1/7 + 1/8 + 1/9 1/10 + ....
where the sign of a term, after the first two, is determined as follows: If the denominator is a prime of form 4m 1, a minus sign is used, if the denominator is a prime of form 4m + 1, a plus sign is used, and if the denominator is a composite number, the sign indicated by the product of the signs of its components is used.
Have students calculate ¹ using hand calculators.
How far does one have to carry out the infinite product to approach 3.14?
Activity XII
Have students using hand calculators choose one of these Fourier series (named after Joseph Fourier), to compute ¹ to any desired number of decimal places:
/2 = (2/1)(2/3)(4/3)(4/5)(6/5)(6/7) ...
¹2/8 = 1 + 1/9 + 1/25 + 1/49 +...