In Indiana, in 1897, their General Assembly attempted to establish mathematical truth by legislative fiat. It is known to history as The Indiana Pi Bill, but despite the name the main result claimed by the bill was a method to square the circle. However, the bill did contain text that appears to dictate various incorrect values of π, such as 3.2.
The purported attempted legislative fiat of “Pi equals 3.2” was overruled by a mathematicians’ fiat ordering that “Pi equals 3.14159265…” I call this mathematicians’ fiat, The Lilliputian Enactment of Pi. This Lilliputian Enactment for Pi is still wrong, but it is less wrong than the Indiana purported value of 3.2. (Background: “Pi Is Wrong!”, Ersjdamoo’s Blog entry of August 11, 2013.)
John A. Parker of New York City is said to have found his own quadrature value for the “squaring of the circle” problem. Parker’s value was 6561 for diameter and 20612 for circumference. From this Parker ratio of 6561: 20612, a value for Pi of 3.141594269… is deduced. This Parker value differs from that ordered by The Lilliputian Enactment of Pi, 3.141592654…
It may be that even the Parker Pi value is not exact. But the point is that his value is more accurate than the value ordered by The Lilliputian Enactment of Pi. Furthermore, it is asserted that ancient Bronze Age (circa 3600 – 1200 BC) Hebrew and/or Egyptian civilizations had solved or come close to solving the problem of “squaring the circle.” They did this, of course, without the later restricting limitation of only being allowed a compass and a straightedge. (Parker explains in chapter one of his book, The Quadrature of the Circle, how “squaring the circle” cannot be solved if the limitation of compass and straightedge is imposed.)
But “squaring the circle”, in itself, can be solved! Parker argues for this in the second proposition of chapter 2 of his book, Proposition 2-2:
“The circumference of any circle being given, if that circumference be brought into the form of a square, the area of that square is equal to the area of another circle, the circumscribed square of which is equal in area to the area of the circle whose circumference is first given.”
Parker’s “Plate VIII” is hopefully reproduced above. By clicking on the image, an enlargement of same should appear. In Plate VIII, let the circumference of the circle E be given. Let us say its circumference equals 36. Let the circumference (or perimeter) of the square F also equal 36. Since the perimeter of the square F equals 36, each of its sides must equal 9. The area of the square F therefore equals 9 x 9, or 81. (But due to the difference in properties between curved and straight lines, the area of circle E does not equal 81. Background: “Curved Lines Are Different”, Ersjdamoo’s Blog entry of August 14, 2013.) Now let the area of a circle G be equal to 81. Then the area of a square H which circumscribes the circle G equals the area of the circle E.
(Note to above paragraph: I preliminarily calculate the area for circle E above to be 103.1324031. The radius for circle E having circumference 36 would be, r=36/2*Pi. The area therefore would be 103.1324031, from A=Pi * radius squared. Any intelligent comments and/or critiques regarding my calculation are welcomed.)
How does this show that the problem of “squaring the circle” can be solved? By Proposition 2-2 “the existence of a perfect equality between the circle and the square has been clearly shown. For if the circle and the square, or what is the same thing, if the circumference and diameter of a circle be really incommensurable [to the square], as geometers have affirmed, then no circle and square can be exactly equal one to the other. But when it has been demonstrated as has here been done [in Proposition 2-2], that a circle and a square may be exactly equal one to the other, then it is demonstrated, also, that the two are not incommensurable; and with this demonstration the whole theory of mathematicians respecting the non-existence of any expression of numbers by which the circle and the square are made equal is proved to be fallacious.”
However Parker’s Proposition 2-2 does not of itself demonstrate what the true ratio of circumference to diameter (Pi) is.