Measuring the Circle:The Story of

Eric Wishnie

Seth Coldsmith

Formal Definition

• The ratio between a circle’s circumference and diameter.

• The ratio between a circle’s area and the square of its radius.

• “ ” first used by William Jones, 1706– Adopted by Leonhard Euler in publications

1730s, ’40s

Discovery as a Constant

• First considered only as 3.

• Egyptians, circa 1650 B.C., recognized a constant when computing the area of a circle

Archimedes circa 240 B.C.

• Used polygons and circles to estimate .

• Involved the use of two similar methods.– Inscribed polygons– Circumscribed polygons

• Provided an upper limit of 3.142857143 and a lower limit of 3.14084507 for .

• Example: Archimedes.gsp

Other advances after Archimedes

• Claudius Ptolemy circa 150 A.D. used his table of chords to estimate to be

• Chinese scholar Zu Chongzi circa 480 A.D. used

for it, but his methods are unknown. He later worked out to be between 3.141596 and 3.141597.

6141.3120

377

14159292.3113

355

Aryabhata

• First to use expression to calculate

• a = length of one side of an inscribed polygon with n sides

• b = length of one side of an inscribed polygon with 2n sides

Brahmagupta circa 650 A.D.

• Also used method of inscribing polygons with doubling numbers of sides to estimate the value of .

• Found the values of

• From these results he concluded that was converging to

65.9 81.9 86.9 87.9

1622.310

Different Sequences Used

• John Wallis in 1650 A.D. proved that /2 = 2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 x …

• Viscount Brouncker proposed a few years earlier the sequence

• Neither sequences were really used.

Timeline of , Part I

• c. 1650 BCE – Egyptians estimate pi as

• 240 BCE – Archimedes uses polygons and circles to estimate upper and lower bounds of , method widely used later in Europe

• c. 530 CE – Aryabhata develops first algorithmic method to calculate

• c. 650 CE – Brahmagupta deduces as converging to

• c. 1650 CE – First numerical methods of calculating developed as sequences

• 1765 CE – Johann Heinrich Lambert proves is irrational

• 1949 – ENIAC (Electronic Numerical Integrator and Computer), computes to 2035 decimal places, 70 hours

1622.310

Timeline of , Part II

• 1987 – University of Tokyo, under Prof. Yasumasa Kanada, calculates to 134,217,000 digits on NEC supercomputer

• 1991 – Gregory and David Chudnovsky calculate to 2,260,321,336 decimal places on home-built supercomputer, 250 hours

• 1999 – Prof. Kanada calculates again, achieves to 206,158,430,000 decimal places

• 2002 – Takahashi Kanada calculates to 1.2411 trillion digits

References

• “The Timeline of Pi.” http://people.bath.ac.uk/lr226/timeline.html

• Berlingholl and Gouvea• Katz, Victor J. A History of Mathematics: Brief

Edition. Pearson/Wesley 2004• Harris, Herman H. Jr., “The History and

Calculation of Pi.” Emporia State Research Studies, Volume 8, Number 1 Graduate Division of the Kansas State Teachers College, Emopria Kansas, September 1959