# The Precomputer History of

That the ratio of circumference to diameter is the same (and roughly equal to 3) for all circles has been accepted as "fact" for centuries; at least 4000 years, as far as I can determine. (But knowing why this is true, as well as knowing the exact value of this ratio, is another story.) The "easy" history of concerns the ongoing story of our attempts to improve upon our estimates of . This page offers a brief survey of a few of the more famous early approximations to .

The value of given in the Rhynd Papyrus (c. 2000 BC) is

Various Babylonian and Egyptian writings suggest that each of the values

were used (in different circumstances, of course). The Bible (c. 950 BC, 1 Kings 7:23) and the Talmud both (implicitly) give the value simply as 3.

Archimedes of Syracuse (240 BC), using a 96-sided polygon and his method of exhaustion, showed that

and so his error was no more than

The important feature of Archimedes' accomplishment is not that he was able to give such an accurate estimate, but rather that his methods could be used to obtain any number of digits of . In fact, Archimedes' method of exhaustion would prove to be the basis for nearly all such calculations for over 1800 years.

Over 700 years later, Tsu Chung-Chih (c. 480) improved upon Archimedes' estimate by giving the familiar value

which agrees with the actual value of to 6 places.

Many years later, Ludolph van Ceulen (c. 1610) gave an estimate that was accurate to 34 decimal places using Archimedes' method (based on a -sided polygon). The digits were later used to adorn his tombstone.

The next era in the history of the extended calculation of was ushered in by James Gregory (c. 1671), who provided us with the series

Using Gregory's series in conjunction with the identity

John Machin (c. 1706) calculated 100 decimal digits of . Methods similar to Machin's would remain in vogue for over 200 years.

William Shanks (c. 1807) churned out the first 707 digits of . This feat took Shanks over 15 years -- in other words, he averaged only about one decimal digit per week! Sadly, only 527 of Shanks' digits were correct. In fact, Shanks published his calculations 3 times, each time correcting errors in the previously published digits, and each time new errors crept in. As it happened, his first set of values proved to be the most accurate.

In 1844, Johann Dase (a.k.a., Zacharias Dahse), a calculating prodigy (or "idiot savant") hired by the Hamburg Academy of Sciences on Gauss's recommendation, computed to 200 decimal places in less than two months.

In the era of the desktop calculator (and the early calculators truly required an entire desktop!), D. F. Ferguson (c. 1947) raised the total to 808 (accurate) decimal digits. In fact, it was Ferguson who discovered the errors in Shanks' calculations.

Today, of course, in the era of the supercomputer, hundreds of millions of digits are known. The evolution of the machine-assisted approximations to is summarized on a table on the next page.

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Neal Carothers - carother@bgnet.bgsu.edu