# A short history of PI approximations

## © Stu Savory, 2004.

** = 3.14159 26535 89793 23846 26433
83279 50288 41971 ...**

is the ratio of the circumference of any
convex curve of constant diameter to that diameter. Usually a circle
is stated, but this more precise definition includes e.g.
Wankel rotors, and others of that ilk with more than the Wankel's
three lobes :-)

In the bible's old testament, I Kings 7:23 implies
that = 3.

Ahmes, an egyptian scribe, wrote that the area of a circle is
like a square on 8/9 of its diameter,
thus giving = 3.16049...

Archimedes used a 96-sided polygon to limit
to between
3 + 10/71 (3.14085...) and 3 + 10/70 (3.142857...) .
Archimedes also gave the schoolchild's approximation 22/7 (3.142857...)

Ptolomy worked with = 377/120 (3.1416...)

About 480 AD in China, Tsu Ch'ung-Chi gave
= 355/113 (3.1415929...)
which is 7 digits of accuracy.

Not until the 15th century did Al-Kashi reach 16 digits. Ludolph
van Ceulen (1540-1610) reached 20, then 32, then 35 digits. They are
carved on his gravestone in the church in Leyden.

In 1706 John Machin calculated 100 digits.

Finally, Johann Lambertz (1728-1777) proved that
is irrational (i.e. is not expressible
as a fraction). His personal best approximation was 1,019,514,486,099,146 /
324,521,540,032,945. Here is a shorter
proof that is irrational.

In 1853 William Shanks calculated to 707
places, but made a mistake in the 528th position. All these calculations so far were done manually.
The error was not found until 1945 when Ferguson used a desk calculator.

Starting in 1949 computers were used, ENIAC managing 2037 places in 70 hours runtime :-)

In 1897 in the State of Indiana the House of Representatives
(in House Bill 246) proposed to rule that
=3. I guess US politicians were
just as stupid then as they are now, and provably so ;-)

Now go visit my blog please, or look at other interesting maths stuff :-)