system
Given the following post, respond as if you're commenting on Reddit:
human
How was Pi "discovered" and how was it plugged into formulae correctly? How was it able to be used before the invention of calculators?
gpt
Take a length of rope. Tie one end to a pole and to the other end attach a stick. Pull the rope taut and use it to draw a circle in the sand. Measure how many rope lengths the perimeter of the circle is. Get the answer "roughly six" (so about three times the *diameter*). Be happy for a while. Quoting the Old Testament:

I Kings 7: 23, 
> And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about
 
(KJV).

Time passes, more measurements are made because people are curious and besides more precise values for the perimeter-diameter ratio are useful in engineering, architecture, astronomy and whatnot. Come up with approximations like 25/8 (Babylonia ~1800 BCE), 256/81 (Egypt ~1600 BCE), and 339/108​ (India ~500 BCE).

In the 3rd century BCE, Archimedes proves the inequality 223⁄71 < π < ​22⁄7 by inscribing and circumscribing a circle with regular 96-gons. In the 2nd century CE, Ptolemy gives the approximation 377/120, the first one accurate to three decimals. By the sixth century, Indian mathematicians know the value of π to five decimals, and the Chinese to seven decimals. 

After that, no further progress is made until the 13th century (probably because nobody needed that much precision anyway) when Indian mathematician Madhava discovers the infinite series for π and uses it to calculate an approximation accurate to 11 decimal places. Circa 1600 van Ceulen computes 35 decimals, in 1789 Jurij Vega 126 correct decimals, and in 1841 William Rutherford 152 correct decimals. I'll quote Wikipedia for the ultimate result before electronic computers:

> The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places. This was accomplished in 1873, with the first 527 places correct. He would calculate new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of π until the advent of the electronic digital computer three-quarters of a century later.