A posting from the usenet group alt.fan.cecil-adams

From: msbrader@interlog.com (Mark Brader)
Subject: Legislating pi 
Date: Wed, 13 Mar 1996 02:08:35 GMT
Lines: 353

> > > Wasn't it the Indiana legislature that, for a brief period, also
> > > declared pi equal to 3.14?
> > Tennessee.  

Indiana.  3.2, to the extent that you can get a value for pi out of the bill
at all.  Would have declared, but the bill was killed in the state senate.

> Cite?

The main Usenet FAQ list.  The alt.folklore.urban archives.  Or the
references given in the bibliography in the following, which is a
partial rewrite of some articles I have previously posted to Usenet.
The actual bill appears in the last part of the article.


The Case of Indiana vs. Pi
by Mark Brader, msbrader@interlog.com


Edington, Will E.: "House Bill No.  246, Indiana State Legislature, 1897",
    Proceedings of the Indiana Academy of Science (PIAS), 1935.

Singmaster, David: "The Legal Values of Pi",
    Mathematical Intelligencer, vol. 7 (1985) #2, p.69-72.

Dudley, Underwood: "Legislating Pi", p.192-7 of his book "Mathematical
    Cranks", Mathematical Association of America, 1992, ISBN 0-88385-507-0.

Edington's article is based on "the bill itself, ..., the Journals of
the House and Senate for 1897, and the files of the three Indianapolis
papers for January and February, 1897."  He also draws on a 1916 art-
icle by C. A. Waldo (see below) which appeared in the same publication.

Incidentally, both Edington and Dudley are/were from De Pauw University
of Greencastle, Indiana.


The bill's author was one Edwin J. Goodwin, M.D., who evidently was a
crank mathematician.  He lived in the village of Solitude, which is in
Posey County, Indiana, about 20 miles west of Evansville.  And the
epoch-making suggestion that he put to Taylor I. Record, the state
Representative for the county, was this:  If the State would pass an
Act recognizing Goodwin's discovery, then he would allow all Indiana
textbooks to use it without paying him a royalty.

Record went along, introducing the bill into the state House on the
18th of January, 1897; the legislators then referred it to the House
Committee on Canals, also called the Committee on Swamp Lands.  On the
19th, that committee's chairman, Representative M. B. Butler of Steuben
County, reported back with a recommendation to refer the bill to the
Committee on Education.

On the 19th, two Indianapolis papers had carried brief notices about a
bill telling how to square the circle.  (That's the name used for the
problem, which in ancient times was to be solved by a compass-and-
straightedge construction, of finding a square with area equal to a
given circle.)  Then on the 20th, the Indianapolis Sentinel ran the
following article:


	  Claims Made That This Old Problem Has Been Solved.

   The bill telling how to square a circle, introduced in the House by
   Mr. Record, is not intended to be a hoax.  Mr. Record knows nothing
   of the bill with the exception that he introduced it by request of
   Dr. Edwin Goodwin of Posey County, who is the author of the demon-
   stration.  The latter and State Superintendent of Public Instruction
   Geeting believe that it is the long-sought solution of the problem,
   and they are seeking to have it adopted by the legislature.  Dr.
   Goodwin, the author, is a mathematician of note.  He has it copy-
   righted and his proposition is that if the legislature will indorse
   the solution, he will allow the state to use the demonstration in
   its textbooks free of charge.  The author is lobbying for the bill.

Another Indianapolis daily ran a more educated article, even mentioning
Ferdinand Lindemann's proof, just 13 years before, that pi is actually
transcendental (which implies that squaring the circle with a compass
and straightedge must be impossible).  Unfortunately, this was a German-
language paper, and so their article didn't get a lot of notice among
English speakers.

And the Committee on Education's chairman, Representative S. E. Nichol-
son of Howard County, reported back on the 2nd of February "with the
recommendation that said bill do pass."

The bill duly came up for second reading on the 5th, and passed.  Nich-
olson then moved the suspension of the constitutional rule "requiring
bills to be read on three [separate] days", allowing an immediate third
(final) reading.  The suspension passed by 72-0, and the bill itself,
by 67-0.  The Indianapolis Journal called it "the strangest bill that
has ever passed an Indiana Assembly."

However, by sheer chance, it happened that a real mathematician, Prof.
C. A. Waldo of the Indiana Academy of Science, had been present in the
House that day.  To quote Edington quoting Waldo:

   ...imagine [the author's] surprise when he discovered that he was
   in the midst of a debate upon a piece of mathematical legislation.
   An ex-teacher from the eastern part of the state was saying:  "The
   case is perfectly simple.  If we pass this bill which establishes
   a new and correct value for pi, the author offers ... its free
   publication in our school text books, while everyone else must pay
   him a royalty."

Waldo was then shown a copy of the bill and asked if he wanted to meet
its author.  He replied that he was already "acquainted with as many
crazy people as he cared to know."

Fortunately, Indiana has a bicameral legislature, and by the time the
bill got to the Senate, Waldo had been able to make sure that the
Senators were "properly coached".  The bill got its first reading in
the Senate on the 11th of February.  Apparently in fun, they referred
it to the Committee on Temperance.  And the next day its chairman,
Senator Harry S. New of Marion County, reported back "with the recom-
mendation that said bill do pass."

According to the Senate Journal for the 12th of February, the bill
was read a second time, an attempt to strike out the enacting clause
failed, and finally the bill was postponed indefinitely.  But this
leaves out all the good stuff, as reported in the Indianapolis News
on the 13th:

   The Senators made bad puns about it, ridiculed it, and laughed over
   it.  The fun lasted half an hour.  [Then] 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 moved the
   indefinite postponement of the bill, and the motion carried.  ...

To which Dudley comments that here it is almost 100 years later and the
bill is *still* indefinitely postponed.  As to the Senators' attitude
to the bill, the Indianapolis Journal reported:

   All of the senators who spoke on the bill admitted that they were
   ignorant of the merits of the proposition.  [In the end,] it was
   simply regarded as not being a subject for legislation.

And so ends the tale of, as Waldo put it, "the epoch making discovery
of the Wise Man from the Pocket."  Goodwin continued promoting his
discovery for a long time afterwards, but never came so close to

As Allan Adler said in a Usenet posting, "and before we laugh too hard
at the legislature of Indiana or at the state of education in 1897, I
think we should have a moment of silence as we contemplate what fate
the bill might have if it were brought up for a referendum today."


First, let it be clearly understood that everyone agrees that the bill
was mathematical nonsense.  But that doesn't mean we can't have fun
thinking about it.

In Singmaster's article, he has fun by taking each individual mathe-
matical statement in it at face value, and comparing it with a true
statement involving pi to derive a supposed value for pi.  Dudley also
goes along with this approach, citing Singmaster.

My opinion, on the other hand, is that it's more fun to try to recon-
struct Goodwin's thinking, and Singmaster and Dudley don't provide a
fair representation of what that was.  I think my annotations, given
below, do.

It's clear from the text of the bill that Goodwin's version of geometry
had lots more deviations from reality than simply the shape of a circle.
So what I have tried to do below is to identify *each* of those differ-
ences, and to see just how self-consistently the bill can be interpreted.

Now, the term "pi" can be defined in many ways, all of which are neces-
sarily equivalent in real mathematics.  For example, it is the ratio of
the area of a circle to the square of its radius, and also the ratio of
the circumference and diameter of a circle.  Then there are possible
definitions based on infinite series or on probabilities, having nothing
to do with geometry.  In Goodwin-land, these definitions are not at all
equivalent, and this is the source of Singmaster's many values.

But the definition usually given for pi is the ratio of the circumfer-
ence and diameter of a circle, and that's the definition that I'm using
from here on.  Accepting that as the definition and accepting standard
ratio operations on Goodwin's "ratio of the diameter and circumference",
the only conclusion can be that the bill assigns the value 3.2 to pi.

Most people writing about the bill have considered it as an *attempt
to legislate* the value of pi.  To Dudley, this is a "falsehood" which
"all of us who revere reason have a sacred duty" to put down.  As
Dudley says:

    The bill that the Indiana House of Representatives passed was not
    one setting the value of pi by law.  It was one that gave the state
    the privilege of using the proper value of pi for free.

And so, he says, when the Senate didn't vote on the bill,

    There was only a refusal by [the lawmakers] to take [Goodwin]'s
    value, or values, [of pi] as a gift.

Now, this is an accurate description of the bill's *intent*.  But all
the same, it *was* a bill "setting the value of pi", in the sense that
sections 1 and 2 of the bill really would have *enacted* Goodwin's
formulas.  Therefore it is also quite accurate to say that the bill
would have set the value of pi to 3.2.


In the annotations, A, r, d, c, and s are respectively the circle's
area, radius, diameter, circumference, and the side of the inscribed

#            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.

You normally have to pay royalties on mathematical truths?  Hmm.
The estate of Euclid must be doing well by now...

#      SECTION 1.
#            Be it enacted by the General Assembly of the State
#      of Indiana: It has been found that a circular area is to
#      the square on a line equal to the quadrant of the circum-
#      ference, as the area of an equilateral rectangle is to
#      the square on one side.

The last 14 words are a remarkable way of saying "as 1 is to 1".

In other words, this says that A = (c/4)^2, which is the same as
A = (pi*r/2)^2 = (pi^2/4)*r^2 instead of the actual A = pi*r^2.

#                          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.

The standard formula A = pi*r^2 is interpreted as A = d*(c/4), which
is correct.  As the author has just explained, he thinks that the d
factor should really be c/4, so the ratio of the area by his formula
to the area by the standard formula is c/(4*d), that is, pi/4.

Now, since he believes pi = 3.2, this ratio is 3.2/4, which is 4/5.
Therefore the area by his rule is 1/5 smaller than the standard area,
or 1-1/5 of the standard area.  Now he apparently thinks that the
reciprocal of this is 1+1/5, and thus that the standard area is 1/5
larger than his area.

#                          This is because one-fifth of the
#      diameter fails to be represented four times in the
#      circle's circumference.

In other words, c = d * (1-1/5) * 4; consistent with pi = c/d = 3.2.

#                          For example: if we multiply the
#      perimeter of a square by one-fourth of any line one-fifth
#      greater than one side, we can in like manner make the
#      square's area to appear one fifth greater than the fact,
#      as is done by taking the diameter for the linear unit
#      instead of the quadrant of the circle's circumference.

He says that if we consider the area of a square of side x to be
(4*x)*(x/4) and we replace the second x by (1+1/5)*x, we get an
area 1/5 too large, and this is analogous to using d in place of
c/4 with the circle.

#      SECTION 2.
#            It is impossible to compute the area of a circle
#      on the diameter as the linear unit without tresspassing
#      upon the area outside the circle to the extent of including
#      one-fifth more area than is contained within the circle's
#      circumference, because the square on the diameter produces
#      the side of a square which equals nine when the arc of
#      ninety degrees equals eight.

I can only assume that "nine" is a mistake for "ten".  See also the
annotations below.

#                          By taking the quadrant of the
#      circle's circumference for the linear unit, we fulfill
#      the requirements of both quadrature and rectification of
#      the circle's circumference.

Getting repetitive here...

#                          Furthermore, it has revealed the ratio
#      of the chord and arc of ninety degrees, which is as seven
#      to eight,

That is, he says s/(c/4) = 7/8.  Hmm.

#                 and also the ratio of the diagonal and one side
#      of a square which is as ten to seven,

That is, he says d/s = 10/7.  Stop turning over, Pythagoras!

#                                            disclosing the fourth
#      important fact, that the ratio of the diameter and circum-
#      ference is as five-fourths to four;

This is the closest thing in the bill to an actual value for pi.
The author combines the last two results to get d/c = (10/7)*(7/8)/4,
which he reduces only as far as (5/4)/4.  Of course this is 5/16,
And if d/c = 5/16, then presumably pi = c/d = 16/5 = 3.2.

#                          and because of these facts and the
#      further fact that the rule in present use fails to work
#      both ways mathematically, it should be discarded as wholly
#      wanting and misleading in its practical applications.


#      SECTION 3.
#            In further proof of the value of the author's pro-
#      posed contribution to education, and offered as a gift
#      to the State of Indiana, is the fact of his solutions of
#      the trisection of the angle, duplication of the cube and
#      quadrature of the circle having been already accepted as
#      contributions to science by the American Mathematical
#      Monthly, the leading exponent of mathematical thought in
#      this country.

Surprised?  I was.  It turns out that in its early days the A.M.M. had
a department called "Queries and Information" which must have been a
prototype for sci.math. :-)  By accepting Goodwin's three constructions
for *that* department, in 1894-95, they were basically treating them
as unpaid advertising; certainly not as refereed papers or other
"contributions to science".

#                          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.

"Given up as unsolvable" is not the same as "proved unsolvable"!

Mark Brader           "Sir, your composure baffles me.  A single counter-
msbrader@interlog.com  example refutes a conjecture as effectively as ten.
                       ... Hands up!  You have to surrender."   
                       -- Imre Lakatos

My text in this article is in the public domain.

Back to the Gödel, Escher, Bach page.