Changes

Some ancients expressed pi by using fractional approximations. Papyrus of Ahmes, dated c. 1650 B.C., shows that ancient Egyptians had value 3 1/6 = 3.167). The Babylonian value from the same era was 3 1/8 = 3.125.<ref>Boyer, ''A History of Mathematics'', 2nd edition</ref> Both these values are accurate to within 1 percent. Note that the value 22/7 is still used today.

Some ancients expressed pi by using fractional approximations. Papyrus of Ahmes, dated c. 1650 B.C., shows that ancient Egyptians had value 3 1/6 = 3.167). The Babylonian value from the same era was 3 1/8 = 3.125.<ref>Boyer, ''A History of Mathematics'', 2nd edition</ref> Both these values are accurate to within 1 percent. Note that the value 22/7 is still used today.

[[Archimedes]] of Syracuse (287-212 BC) carried out "the first theoretical calculation" of pi.<ref>[http://veling.nl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref> He said it was between 223/71 and 22/7. This is ten times better than the Egyptian and Babylonian values, and it is within 0.04 percent of the correct value. The German-Dutch mathematician Ludolph van Ceulen used Archimedes's formula to calculate pi to 32 digits in 1615. He was so proud of his achievement, he had the digits engraved on his tombstone.

[[Archimedes]] of Syracuse (287-212 BC) carried out "the first theoretical calculation" of pi.<ref>[http://veling.nl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref> He said it was between 223/71 and 22/7. This is ten times better than the Egyptian and Babylonian values, and it is within 0.04 percent of the correct value. The German-Dutch mathematician Ludolph Van Ceulen used Archimedes's formula to calculate pi to 32 digits in 1615. He was so proud of his achievement that he had the digits engraved on his tombstone.

During the plague of 1665-1666, [[Isaac Newton]] was confined to the English village of Woolsthorpe. There he calculated pi to 16 digits. He considered such calculations to be a waste of time. “I am ashamed to tell you to how many figures I carried these computations, having no other business at the time," he wrote.<ref>Beckmann, Petr (2015-01-06). ''A History of Pi'' (Kindle Locations 2249-2250). St. Martin's Press. Kindle Edition.</ref>

During the plague of 1665-1666, [[Isaac Newton]] was confined to the English village of Woolsthorpe. There he calculated pi to 16 digits. Unlike Van Ceulen, Newton did not consider his calculations to be an achievement. “I am ashamed to tell you to how many figures I carried these computations, having no other business at the time," he wrote.<ref>Beckmann, Petr (2015-01-06). ''A History of Pi'' (Kindle Locations 2249-2250). St. Martin's Press. Kindle Edition.</ref>

In 1706, John Machin, secretary of England's Royal Society, developed a quickly converging formula for pi and used it to calculated the first 100 digits. In 1844, the idiot savant Johann Dase of Hamburg used Machin's formula to calculate 200 digits in less than two months.<ref>Beckmann</ref> In contrast, William Shanks spent twenty years calculating pi to 707 places, a task he completed in 1873. In 1945, it was discovered that only the first 527 of Shanks's digits were correct.

In 1706, John Machin, secretary of England's Royal Society, developed a quickly converging formula for pi and used it to calculated the first 100 digits. In 1844, the idiot savant Johann Dase of Hamburg used Machin's formula to calculate 200 digits in less than two months.<ref>Beckmann</ref> In contrast, William Shanks spent twenty years calculating pi to 707 places, a task he completed in 1873. In 1945, it was discovered that only the first 527 of Shanks's digits were correct.