Divining the ratio of a circle's circumference to its diameter has been an intellectual challenge for nearly four millennia. There's even an implied specification for p in the Old Testament (1 Kings 7:23 and 2 Chronicles 4:2): Then he made the molten sea; it was round, ten cubits from brim to brim, and five cubits high, and a line of thirty cubits measured in circumference.Gauged empirically at first — perhaps with lengths of rope — then attacked analytically over the years with increasing mathematical sophistication, the noble goal was to find p's exact integer ratio. How close did they get? As the 1600s commenced, p had been revealed to well over 30 decimal places. In the early 1700s, it was known to over 100 decimal places — with no sign of repeating digits. (A repeating digit sequence is the telltale clue of rationality.)
Finally, in the 1760s, Lambert proved p to be an irrational number. A little more than century later, it was proved also to be transcendental, a special breed of irrational. (Transcendental numbers cannot be solutions to algebraic equations with rational coefficients.)
But despite these breakthroughs, mathematical truth soon after experienced an amusing episode of American democracy in action. In 1897, the state of Indiana's House of Representatives voted 67 to 0, proclaiming p equal to 16/5, exactly! The bill was proposed by a physician and amateur mathematician, a Dr. Edwin Goodwin, who claimed this "discovery." And being a loyal Hoosier, he offered Indiana free use of the result — all others would have to pay royalties! Fortunately, last-minute lobbying by an alert mathematics professor kept the bill from making it through the Indiana Senate.