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Pi to 7 decimal digits is:

3.1415926

Many people are familiar with Pi day. Celebrated on March 14 as per American date format, the holiday brings attention to the fact that the date resembles the decimal value of the natural constant Pi. This year was notable in particular, as Pi was approximated to four decimal places when written in two-digit year date notation:

3-14-15

Reviewing the history of Pi and the history of the Gregorian calendar, it seems that Pi may have been known to sufficient digits to mark 3-14-1592 just as the relevant calendar was becoming available. Might there have been anybody involved with both projects, who may have noticed that the date matched Pi to six decimal places at the time? Since the seventh Pi digit is 6, it would round up the previous digit. Therefore, for purposes of this question 3-14-1593 could also be a valid date.

Of course, hte premise of this question hinges on the use of the MM-DD-YYYY date format being in use. I cannot find any reference to this, so references supporting or denying usage of this date format would be appreciate.

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  • $\begingroup$ Dunno, but we can be pretty sure nobody back then was enlightened enough to demand celebration of $\tau$ day :-) $\endgroup$ – Carl Witthoft Feb 2 '18 at 13:00
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There could not have been a $\pi$ day in 1592 regardless of calendar conventions for the simple reason that there was no such thing as $\pi$ back then. The symbol was introduced by William Jones in 1706 and did not come into common usage until after 1737, when Euler popularized it in his texts. This was similar to zero, which got a placeholder symbol long before it was recognized as a number.

Still, they could have celebrated the number without the name, right? Still no, because $\pi$ and $e$ were first acknowledged as "some kind of numbers" in print by James Gregory in The True Squaring of the Circle and of the Hyperbola published in 1667, and even Jones wrote in 1706 that "the exact proportion between the diameter and the circumference can never be expressed in numbers". Even irrationals like $1+\sqrt{5}$ or $\sqrt[3]{2}$ were treated with gloves at the time and called "deafmute", and for them one could at least write formulas.

So what was $\pi$ in 1592, what was Viète approximating? As Jones said, and as Euclid defined in the Elements, it was a proportion or a ratio of circumference to diameter, and "approximating" is a modernization of what Viète, and Archimedes before him, were doing. According to Euclid, a ratio is a "relation" of magnitudes or numbers, and ratios can be compared to other ratios using an ingenious procedure invented by Eudoxus and described in the Elements. The inequalities with bounds like $3/1 < \pi < 22/7$ were understood as such comparisons rather than "approximations" of a non-existent "number".

Finally, as can be seen from Viète's approach, he was relying on geometry and expressed the bounds in terms of radicals (Archimedes did in terms of integer ratios), so even if something so abstract as a relation between two geometric magnitudes got to be celebrated, decimal digits in the bounds would not have stood out for pointing to the occasion.

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    $\begingroup$ Thank you. This is a remarkably informative answer, one of the best that I've seen on HSM! It really goes into detail about the thoughts and processes that have occurred in the history of mathematics, much more than I've ever seen addressed so concisely. Thank you! $\endgroup$ – dotancohen Aug 25 '15 at 8:25
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François Viète computed $\pi$ to nine decimal places in 1573. He was also intimately aware of the Gregorian calendar, though as a denouncer rather than a supporter of the calculations used to obtain it.

(That said, you could just as well celebrate Ultimate Pi Day in the Julian calendar. What's more important is that you were using Anno Domini years, which were widespread in Europe from the time of the Carolingian Renaissance — six centuries earlier.)

I don't know for sure whether the MM-DD-YYYY convention was widespread or even existent in 1592, but it would surprise me very much if it was.

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    $\begingroup$ Thank you! I'll try to do some more research into the MM-DD-YYYY date format as that would be needed to accept an answer. $\endgroup$ – dotancohen Aug 23 '15 at 18:58
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    $\begingroup$ There seems to be no written evidence of the MM-DD-YYYY date format being used earlier than the 18th century. Lots of speculation, but no written documents. $\endgroup$ – dotancohen Aug 25 '15 at 8:23
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Ludolph van Ceulen (Köln / Cologne) had calculated (using the method of Archimedes) 20 digits of $\pi$ in 1596. Toward the end of his life he knew even 35 digits. Therefore in Germany $\pi$ has also often been called "Ludolphs Zahl". Obviously this was always considered a number or constant although not being named $\pi$. To answer the question: It is highly probable that he knew enough digits in 1592.

But much earlier the Chinese Tsu Ch’ung-Chih (430-501) had found the very good approximation 355/113 which amounts to 3.14159292... (and not so on). In possession of decimal digits and the inconsistent MM-DD-YYYY date format he could have been the first to satisfy the request.

This accomplishment however has been forgotten soon, and his compatriot Liu Hwuy used $157/50 = 3.14$ in the 7th century. The Indian Brahmagupta had $\sqrt{10} = 3.16...$ in the 7th century. Then followed a really dark age. The Byzantine Michael Psellus, 11th century put $\sqrt{8} = 2.828...$, and Franco de Liège, also 11th century, $(9/5)^2 = 3.24$, both worse than the Old Egypts. Only the Dutch Adriaan Metius in 1585 rediscovered $355/113 = 3.1415929...$. So he would have been able in 1593 to celebrate the pi-day.

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