The Indiana Pi Bill is the popular name for Indiana House of Representatives bill #246 of 1897, which is one of the most famous historical attempts to (erroneously) define scientific truth by legislative fiat. The bill never became law thanks to the speedy intervention of a mathematics professor who was incidentally present in the legislature. Official language(s) English Capital Indianapolis Largest city Indianapolis Area  - Total  - Width  - Length  - % water  - Latitude  - Longitude Ranked 38th 94,321 km² 225 km 435 km 1. ...

Legislative history

C. A. Waldo

In 1897, a physician and amateur mathematician from Indiana named Edward J. Goodwin believed that he had discovered a correct way to square circles. He proposed a bill to Indiana Representative T. I. Record which the latter introduced to the House under the title A Bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the Legislature of 1897. 1897 (MDCCCXCVII) was a common year starting on Friday (see link for calendar). ... Official language(s) English Capital Indianapolis Largest city Indianapolis Area  - Total  - Width  - Length  - % water  - Latitude  - Longitude Ranked 38th 94,321 km² 225 km 435 km 1. ... This square and circle have the same area. ...

After a series of mathematical claims (detailed below), the bill also recites Goodwin's previous accomplishments:

... his solutions of the trisection of the angle, doubling the cube and quadrature of the circle having been already accepted as contributions to science by the American Mathematical Monthly ... And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man's ability to comprehend.

These false claims are typical of a mathematical crank. Claims of the trisection of an angle and the doubling of the cube are particularly widespread in crank literature. A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... Doubling the cube is one of the three most famous geometric problems unsolvable by straightedge and compass alone. ... This square and circle have the same area. ... The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ... Crank (or kook, crackpot, or quack) is a pejorative term for a person who writes or speaks in an authoritative fashion about a particular subject, often of a scientific or pseudo-scientific nature, but is perceived as holding false or even ludicrous beliefs. ...

The Indiana House of Representatives referred the bill to the Committee on Swamp Lands (or on Canals, according to some sources), which Petr Beckmann has seen as symbolic. It was transferred to the Committee on Education, which reported favorably, and the bill passed unanimously. One argument used was that Goodwin had copyrighted his discovery, and proposed to let the State use it in the public schools for free. As this debate concluded, Professor C. A. Waldo arrived in Indianapolis to secure the annual appropriation for the Indiana Academy of Sciences. An assemblyman handed him the bill, offering to introduce him to the genius who wrote it. He declined, saying that he already knew as many crazy people as he cared to. The Indiana General Assembly is the state legislature, or legislative branch, of the state government of Indiana. ... Petr Beckmann (1924-1993) was a physicist who defected to the United States from Czechoslovakia in 1963 and became a Professor of electrical engineering at the University of Colorado. ... The Indianapolis skyline Indianapolis is the capital of the U.S. state of Indiana. ...

The Indiana Senate had not yet finally passed the bill (which they had referred to the Committee on Temperance), and Professor Waldo coached enough Senators overnight that they postponed the bill indefinitely.

Approximation of π

Goodwin's model circle as described in section 2 of the bill. It has a diameter of 10 a circumference of 32; the chord of 90° has length 7

Although the bill has become known as the "pi bill", its text does not mention the name pi at all, and 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. Yet towards the end of Section 2 appears the following passage: Lower-case pi The mathematical constant π is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...

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 ...

which comes close to an explicit claim that π = 4/1.25 = 3.2, as well as .

This quote is often read as three mutually incompatible assertions, but they fit together well if the statement about 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 10 and circumference is 32; the chord of 90° is taken to be 7. Both of the values 7 and 32 are within a few percent of the true lengths for a diameter-10 circle (which, of course, does not justify Goodwin trying to pass them off as exact).

Area of the circle

Goodwin's main goal was not to measure lengths in the circle but to square it, which he interpreted literally as finding a square with the same area as the circle. He knew that the traditional formula for the area of a circle calls for multiplying the diameter by one fourth of the circumference. Apparently he believed that the long-standing failure of mathematics to solve the problem consisted in the formula giving the wrong numerical results, and so he made the bold leap to conclude that the traditional formula must be wrong. Instead he proposed, without argument, his own method: This square and circle have the same area. ...

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.

This appears to be needlessly convoluted, as an "equilateral rectangle" can hardly mean anything different from a square. Perhaps Goodwin is trying to make a distinction beween a geometrical figure and a "square" in the sense of a number multiplied by itself. In the rest of the bill, however, it is clear that the assertion is simply that area of a circle is the same as that of a square with the same perimeter. For example, right after the above quote 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 area computed by the traditional formula (accepting Goodwin's values for the circumference and diameter) would be 80, whereas Goodwin's proposed rule leads to an area of 64. Now, 80 exceeds 64 by one fifth of 80, and Goodwin appears to reason from 64 = 80×(1-1/5) to 80 = 64×(1+1/5), an approximation that works only for fractions much smaller than 1/5.

The area found by Goodwin's rule is π/4 times the true area of the circle, which in many accounts of the Pi Bill is interpreted as a claim that π=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 1-π/4 works out to about 21 percent, which is much more grave than the approximations of the lengths in the model circle of the previous section. It remains a mystery what made Goodwin believe that his rule could be correct. In general, figures with identical perimeters do not have identical area, but the typical demonstration of this fact is to compare a long thin shape to one that is approximately as tall as it is wide. Perhaps Goodwin erroneously concluded from this demonstration that as long as both figures are equally wide and tall (as in the case of a circle and a square) equality of perimeters does imply equality of area.

External links

• Full text of the bill

• [1] A narrative about the near-miss passing of the bill.

• The Staight Dope article