Indiana Pi Bill
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The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the
Indiana General Assembly The Indiana General Assembly is the state legislature, or legislative branch, of the state of Indiana. It is a bicameral legislature that consists of a lower house, the Indiana House of Representatives, and an upper house, the Indiana Senate. ...
, one of the most notorious attempts to establish
mathematical truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, ...
by legislative fiat. Despite its name, the main result claimed by the bill is a method to square the circle, although it does imply various incorrect values of the mathematical constant , the ratio of the
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...
of a circle to its
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
. The bill, written by a physician who was an amateur mathematician, never became law due to the intervention of Prof. C. A. Waldo of
Purdue University Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money ...
, who happened to be present in the legislature on the day it went up for a vote. The mathematical impossibility of squaring the circle using only
compass and straightedge constructions In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
, suspected since ancient times, had been rigorously proven 15 years previously, in 1882 by
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficien ...
. Better approximations of than those implied by the bill have been known since ancient times.


Legislative history

In 1894,
Indiana Indiana () is a U.S. state in the Midwestern United States. It is the 38th-largest by area and the 17th-most populous of the 50 States. Its capital and largest city is Indianapolis. Indiana was admitted to the United States as the 19th s ...
physician Edward J. Goodwin (ca. 1825–1902) believed that he had discovered a correct way of squaring the circle. He proposed a bill to state representative Taylor I. Record, which Record introduced in the House under the long 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". The text of the bill consists of a series of mathematical claims (detailed below), followed by a recitation of Goodwin's previous accomplishments: Goodwin's "solutions" were indeed published in the ''American Mathematical Monthly'', though with a disclaimer of "published by request of the author"."Clearing the Misunderstanding Re My April Fool's `Joke'"
math.rutgers.edu. Upon its introduction in the
Indiana House of Representatives The Indiana House of Representatives is the lower house of the Indiana General Assembly, the state legislature of the U.S. state of Indiana. The House is composed of 100 members representing an equal number of constituent districts. House memb ...
, the bill's language and topic occasioned confusion among the membership; a member from Bloomington proposed that it be referred to the Finance Committee, but the Speaker accepted another member's recommendation to refer the bill to the Committee on Swamplands, where the bill could "find a deserved grave". It was transferred to the Committee on Education, which reported favorably. Following a motion to suspend the rules, the bill passed on February 6, 1897 without a dissenting vote. The news of the bill occasioned an alarmed response from ''Der Tägliche Telegraph'', a
German-language German ( ) is a West Germanic language mainly spoken in Central Europe. It is the most widely spoken and official or co-official language in Germany, Austria, Switzerland, Liechtenstein, and the Italian province of South Tyrol. It is also a c ...
newspaper in Indianapolis, which viewed the event with less favor than its English-speaking competitors. As this debate concluded,
Purdue University Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money ...
professor C. A. Waldo arrived in
Indianapolis Indianapolis (), colloquially known as Indy, is the state capital and most populous city of the U.S. state of Indiana and the seat of Marion County. According to the U.S. Census Bureau, the consolidated population of Indianapolis and Marion ...
to secure the annual appropriation for the
Indiana Academy of Science Indiana () is a U.S. state in the Midwestern United States. It is the 38th-largest by area and the 17th-most populous of the 50 States. Its capital and largest city is Indianapolis. Indiana was admitted to the United States as the 19th st ...
. An assemblyman handed him the bill, offering to introduce him to the genius who wrote it. He declined, saying that he already met as many crazy people as he cared to. When it reached the
Indiana Senate The Indiana Senate is the upper house of the Indiana General Assembly, the state legislature of the U.S. state of Indiana. The Senate is composed of 50 members representing an equal number of constituent districts. Senators serve four-year terms ...
, the bill was not treated as kindly, for Waldo had talked to the senators previously. The Committee on Temperance to which it had been assigned had reported it favorably, but the Senate on February 12, 1897 postponed the bill indefinitely. It had been nearly passed, but opinion changed when one senator observed that the General Assembly lacked the power to define mathematical truth. Influencing some of the senators was a report that major newspapers, such as the ''
Chicago Tribune The ''Chicago Tribune'' is a daily newspaper based in Chicago, Illinois, United States, owned by Tribune Publishing. Founded in 1847, and formerly self-styled as the "World's Greatest Newspaper" (a slogan for which WGN radio and television ar ...
'', had begun to ridicule the situation. According to the ''Indianapolis News'' article of February 13, 1897, page 11, column 3:
... the bill was brought up and made fun of. The Senators made bad puns about it, ridiculed it and laughed over it. The fun lasted half an hour. Senator Hubbell said that it was not meet for the Senate, which was costing the State $250 a day, to waste its time in such frivolity. He said that in reading the leading newspapers of Chicago and the East, he found that the Indiana State Legislature had laid itself open to ridicule by the action already taken on the bill. He thought consideration of such a proposition was not dignified or worthy of the Senate. He moved the indefinite postponement of the bill, and the motion carried.


Mathematics


Approximation of

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: This comes close to an explicit claim that \pi=\frac=3.2, and that \sqrt=\frac\approx 1.429. This quotation is often read as three mutually incompatible assertions. However, they fit together well if the statement about \sqrt 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 In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of 50 (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 Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
' 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 In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
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: This appears needlessly convoluted, as an "
equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
rectangle" is, by definition, a
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
. 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: 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 64=80\times (1-\frac) with 80=64\times (1+\frac ), an approximation that works only for fractions much smaller than \frac. The area found by Goodwin's rule is  \frac times the true area of the circle, which, in many accounts of the Pi Bill, is interpreted as a claim that \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 1-\frac 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 In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
). 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.


Notes


References

* "Indiana's squared circle" by Arthur E. Hallerberg (''Mathematics Magazine'', vol. 50 (1977), pp. 136–140) gives a good account of the bill. * David Singmaster, in "The legal values of pi" (''Mathematical Intelligencer'', vol. 7 (1985), pp. 69–72) finds seven different values of pi implied in Goodwin's work. *
Petr Beckmann Petr Beckmann (November 13, 1924 â€“ August 3, 1993) was a professor of electrical engineering who became a well-known advocate of libertarianism and nuclear power. Later in his life he disputed Albert Einstein's theory of relativity and oth ...
, '' A History of π''. St. Martin's Press; 1971. *


External links

*
The Straight Dope "The Straight Dope" was a question-and-answer newspaper column written under the pseudonym Cecil Adams. Contributions were made by multiple authors, and it was illustrated (also pseudonymously) by Slug Signorino. It was first published in 1973 in ...
â€
Did a state legislature once pass a law saying pi equals 3?
* Snopes.com â€
Alabama’s Slice of Pi: Did the state legislature of Alabama redefine the value of pi according to Biblical precepts?
a related hoax {{Authority control Pi Squaring the circle Indiana General Assembly Political history of Indiana Legal history of Indiana 1897 in American law 1897 in Indiana Pseudomathematics