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Use the following slider to explore circle squarings of the type proposed by Goodwin for a variety of different circle radii. The corresponding approximate values for the mathematical constants

2

and π are shown beneath the diagrams:

Out[]=

​
r 2  1.4000, π  3.2000

Computational Explanation

While known as the "Indiana Pi Bill," the main result in Bill #246 of the 1897 sitting of the Indiana General Assembly was a method to square the circle. In particular, section 2 states:

"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…"

To illustrate this claim drafted by mathematical crank Edward J. Goodwin, draw two diagrams: the first showing the proportions claimed by Goodwin (left) and the second showing actual dimensions:

Out[]=

Goodwin asserts that the diameter-to-circumference ratio is 10/(4*8)  (5/4)/4 = 5/16, whereas it is actually 1/π.

Goodwin's assertions are therefore equivalent to the claims that

50

7.071…7

(or

2

1.414…7/51.2

), and

5π/27.853…8

(or

π16/53.2

), as summarized in the following table:

In[]:=

TextGrid{"expression","actual","actual numeric","Goodwin","Goodwin numeric"},"pi",π,N[π],

16

5

,N[16/5],"Pythagorean constant",

2

,N[

2

],

7

5

,N[7/5],Dividers{{Thick,{Thick,True},Thick},{Thick,Thick,True,Thick}},Alignment{Center,{Left,{Center}}}

Out[]=

expression	actual	actual numeric	Goodwin	Goodwin numeric
pi	π	3.14159	16 5	3.2
Pythagorean constant	2	1.41421	7 5	1.4

The motivation for Goodwin's selection of the numbers 5, 7 and 8 for the radius, square edge and quarter-arc length, respectively, appears to be that all are small integers. It is therefore of interest to investigate the edge and quarter-arc lengths obtained by taking the integers closest to their true geometric values for various other integer radii. In particular, compute these values for integer radii from 1 to 1000:

In[]:=

data=Table​​r,​​side=Round[

2

r],​​arc=Round[πr/2],​​sqrt2=sider//N,​​pi=2.arcr,​​Norm

sqrt2

2

,

pi

π

-1​​,{r,1000};

The following table summarizes the results. As can be seen, the total relative error is not even particularly small for a radius of 5. In fact, radii of 7 and 12 give much better approximations:

Out[]=

radius	square edge	quarter-arc	2	π	total relative error
1	1	2	1.	4.	0.400557
2	3	3	1.5	3.	0.0755711
3	4	5	1.33333	3.33333	0.0836411
4	6	6	1.5	3.	0.0755711
5	7	8	1.4	3.2	0.0211344
6	8	9	1.33333	3.	0.0728158
7	10	11	1.42857	3.14286	0.0101605
8	11	13	1.375	3.25	0.0442673
9	13	14	1.44444	3.11111	0.0234754
10	14	16	1.4	3.2	0.0211344
11	16	17	1.45455	3.09091	0.0327659
12	17	19	1.41667	3.16667	0.00816763
13	18	20	1.38462	3.07692	0.0293559
14	20	22	1.42857	3.14286	0.0101605
15	21	24	1.4	3.2	0.0211344
16	23	25	1.4375	3.125	0.0172923
17	24	27	1.41176	3.17647	0.0112362
18	25	28	1.38889	3.11111	0.0203669
19	27	30	1.42105	3.15789	0.00709319
20	28	31	1.4	3.1	0.0166221

To investigate particularly "good" approximations, find the radii giving the incrementally smallest total relative errors for approximating

2

and π:

In[]:=

minima=FoldPairList[Function[{min,y},{If[Last[y]<min,y,Null],Min[Last[y],min]}],Infinity,data]/.NullNothing

Out[]=

{{1,1,2,1.,4.,0.400557},{2,3,3,1.5,3.,0.0755711},{5,7,8,1.4,3.2,0.0211344},{7,10,11,1.42857,3.14286,0.0101605},{12,17,19,1.41667,3.16667,0.00816763},{19,27,30,1.42105,3.15789,0.00709319},{37,52,58,1.40541,3.13514,0.00655872},{39,55,61,1.41026,3.12821,0.00509793},{44,62,69,1.40909,3.13636,0.00398637},{46,65,72,1.41304,3.13043,0.00364676},{51,72,80,1.41176,3.13725,0.0022147},{58,82,91,1.41379,3.13793,0.00120285},{63,89,99,1.4127,3.14286,0.00114448},{70,99,110,1.41429,3.14286,0.00040572},{128,181,201,1.41406,3.14063,0.00032601},{198,280,311,1.41414,3.14141,0.0000763639},{268,379,421,1.41418,3.14179,0.0000676874},{466,659,732,1.41416,3.14163,0.0000377087}}

Summarizing in grid form:

In[]:=

TextGrid[Prepend[minima,header],gridoptions]

Out[]=

radius	square edge	quarter-arc	2	π	total relative error
1	1	2	1.	4.	0.400557
2	3	3	1.5	3.	0.0755711
5	7	8	1.4	3.2	0.0211344
7	10	11	1.42857	3.14286	0.0101605
12	17	19	1.41667	3.16667	0.00816763
19	27	30	1.42105	3.15789	0.00709319
37	52	58	1.40541	3.13514	0.00655872
39	55	61	1.41026	3.12821	0.00509793
44	62	69	1.40909	3.13636	0.00398637
46	65	72	1.41304	3.13043	0.00364676
51	72	80	1.41176	3.13725	0.0022147
58	82	91	1.41379	3.13793	0.00120285
63	89	99	1.4127	3.14286	0.00114448
70	99	110	1.41429	3.14286	0.00040572
128	181	201	1.41406	3.14063	0.00032601
198	280	311	1.41414	3.14141	0.0000763639
268	379	421	1.41418	3.14179	0.0000676874
466	659	732	1.41416	3.14163	0.0000377087

As a result of these straightforward computations, a whole series of bills could be proposed based on much better approximations to circle squaring, for example:

"Furthermore, new investigations have revealed the ratio of the chord and arc of ninety degrees, which is as 659 to 732, and also the ratio of the diagonal and one side of a square which is as 932 to 659, disclosing the fourth important fact, that the ratio of the diameter and circumference is as 466 halves to 732…"

Other Resources

Additional Reading

• Anonymous. "The Mathematical Bill. Fun–Making In the Senate Yester-Day Afternoon—Other Action." Indianapolis News, 13 February, p. 11, 1897.
• Beckmann, P. A History of Pi. New York: Barnes & Noble, pp. 174–177, 1993.
• Berggren, L.; Borwein, J.; and Borwein, P. Pi: A Source Book, 3rd ed. New York: Springer-Verlag, pp. 230–235, 2004.
• Edington, W. E. House Bill No. 246, Indiana State Legislature, 1897. Proceedings of the Indiana Academy of Science, Vol. 45, pp. 206–210, 1935.
• Goodwin, E. J. "Quadrature of the Circle." American Mathematical Monthly, Vol. 1, No. 7, pp. 246–248, 1894.
• Hallenberg, A. E. House Bill No. 246 Revisited. Proceedings of the Indiana Academy of Science, Vol. 84, pp. 376–399, 1974.
• Waldo, C. A. "What Might Have Been." Proceedings of the Indiana Academy of Science, pp. 445–446, 1916.
• Walsh, J. E. The Centennial History of the Indiana General Assembly, 1816–1978. Indianapolis: The Select Committee on the Centennial History of the Indiana General Assembly, pp. 428–429, 1987.

Additional Links

• The Eccentric Crank Who Tried To Legislate The Value Of Pi
• Hoosier State Chronicles: Indianapolis News, Indianapolis, Marion County, 13 February 1897
• Moment of Indiana History: Squaring the Circle
• Wikipedia: Clarence Abiathar Waldo
• Wikipedia: Indiana Pi Bill
• Wikipedia: Squaring the circle

Image Credits

Indiana State Archives and Records Administration, Indiana General Assembly Collection (Cropped)