Approximation of π

Although the bill has become known as the "pi bill", its text does not mention the name "pi" at all. Goodwin appears to have thought of the ratio between the circumference and diameter of a circle as distinctly secondary to his main aim of squaring the circle. Towards the end of Section 2, the following passage appears:

Furthermore, it has revealed the ratio of the chord and arc of ninety degrees, which is as seven to eight, and also the ratio of the diagonal and one side of a square which is as ten to seven, disclosing the fourth important fact, that the ratio of the diameter and circumference is as five-fourths to four[.]^[12]

This comes close to an explicit claim that ${\textstyle \pi ={\frac {4}{1.25}}=3.2}$, and that ${\textstyle {\sqrt {2}}={\frac {10}{7}}\approx 1.429}$.

This quotation is often read as three mutually incompatible assertions. However, they fit together well if the statement about ${\displaystyle {\sqrt {2}}}$ is taken to be about the inscribed square (with the circle's diameter as diagonal) rather than the square on the radius (with the chord of 90° as diagonal). Together, they describe the circle shown in the figure, whose diameter is ten and circumference is 32; the chord of 90° is taken to be 7. Both values 7 and 32 are within a few percent of the true lengths for a diameter-10 circle (which does not justify Goodwin's presentation of them as exact). The circumference should be nearer to 31.4159, and the diagonal "7" should be the square root of 50 (${\displaystyle 50=25+25}$) or nearer to 7.071.

Area of the circle

Goodwin's main goal was not to measure lengths in the circle but to find a square with the same area as the circle. He knew that Archimedes' formula for the area of a circle, which calls for multiplying the diameter by one-fourth of the circumference, is not considered a solution to the ancient problem of squaring the circle.

This is because the problem is to construct the area using a compass and straightedge only. Archimedes did not give a method for constructing a straight line with the same length as the circumference. Goodwin was unaware of this central requirement; he believed that the problem with the Archimedean formula was that it gave wrong numerical results; a solution to the ancient problem should replace it with a "correct" formula. So, in the bill that he proposed, without argument, his method:

It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side.^[12]

This appears needlessly convoluted, as an "equilateral rectangle" is, by definition, a square. In simple terms, the assertion is that the area of a circle is the same as that of a square with the same perimeter. This claim results in other mathematical contradictions to which Goodwin attempts to respond. For example, right after the above quotation, the bill goes on to say:

The diameter employed as the linear unit according to the present rule in computing the circle's area is entirely wrong, as it represents the circle's area one and one-fifth times the area of a square whose perimeter is equal to the circumference of the circle.

In the model circle above, the Archimedean area (accepting Goodwin's values for the circumference and diameter) would be 80. In contrast, Goodwin's proposed rule leads to an area of 64. Now, 80 exceeds 64 by one-fifth of 80. Goodwin appears to confuse ${\textstyle 64=80\times {\bigl (}1-{\tfrac {1}{5}}{\bigr )}}$ with ${\textstyle 80=64\times {\bigl (}1+{\tfrac {1}{5}}{\bigr )}}$, an approximation that works only for fractions much smaller than ${\textstyle {\tfrac {1}{5}}}$.

The area found by Goodwin's rule is  ${\textstyle {\tfrac {\pi }{4}}}$ times the true area of the circle, which, in many accounts of the pi bill, is interpreted as a claim that ${\textstyle \pi =4}$. However, there is no internal evidence in the bill that Goodwin intended to make such a claim. On the contrary, he repeatedly denies that the area of the circle has anything to do with its diameter.

The relative area error of ${\textstyle 1-{\tfrac {\pi }{4}}}$ works out to about 21 percent, which is much more grave than the approximate lengths in the model circle of the previous section. It is unknown what made Goodwin believe that his rule could be correct. Generally, figures with identical perimeters do not have an identical area (see isoperimetry). The typical demonstration of this fact is to compare a long, thin shape with a small, enclosed area (the area approaching zero as the width decreases) to one with the same perimeter that is approximately as tall as it is wide (the area approaching the square of the width), obviously of a much greater area.