Tuesday, October 25th, 2016

Pi, irrationally


Today is Pi (Π) day, although I think the real Pi day should wait until 3/14/15 9:26 AM. But that’s just me.

There’s a story that circulates every year about this time about a state that tried to make Π = 3. Most of the stories have the wrong state and the wrong value.  As near as I can find, here is the actual story about Indiana’s attempt to make Π = 3.2:

It happened in Indiana. Although the attempt to legislate pi was ultimately unsuccessful, it did come pretty close. In 1897 Representative T.I. Record of Posen county introduced House Bill #246 in the Indiana House of Representatives. The bill, based on the work of a physician and amateur mathematician named Edward J. Goodwin (Edwin in some accounts), suggests not one but three numbers for pi, among them 3.2, as we shall see. The punishment for unbelievers I have not been able to learn, but I place no credence in the rumor that you had to spend the rest of your natural life in Indiana.

Just as people today have a hard time accepting the idea that the speed of light is the speed limit of the universe, Goodwin and Record apparently couldn’t handle the fact that pi was not a rational number. “Since the rule in present use [presumably pi equals 3.14159…] fails to work …, it should be discarded as wholly wanting and misleading in the practical applications,” the bill declared. Instead, mathematically inclined Hoosiers could take their pick among the following formulae:

(1) The ratio of the diameter of a circle to its circumference is 5/4 to 4. In other words, pi equals 16/5 or 3.2

(2) The area of a circle equals the area of a square whose side is 1/4 the circumference of the circle. Working this out algebraically, we see that pi must be equal to 4.

(3) The ratio of the length of a 90 degree arc to the length of a segment connecting the arc’s two endpoints is 8 to 7. This gives us pi equal to the square root of 2 x 16/7, or about 3.23.

There may have been other values for pi as well; the bill was so confusingly written that it’s impossible to tell exactly what Goodwin was getting at. Mathematician David Singmaster says he found six different values in the bill, plus three more in Goodwin’s other writings and comments, for a total of nine.

Lord knows how all this was supposedly to clarify pi or anything else, but as we shall see, they do things a little differently in Indiana. Bill #246 was initially sent to the Committee on Swamp Lands. The committee deliberated gravely on the question, decided it was not the appropriate body to consider such a measure and turned it over to the Committee on Education. The latter committee gave the bill a “pass” recommendation and sent it on to the full House, which approved it unanimously, 67 to 0.

In the state Senate, the bill was referred to the Committee on Temperance. (One begins to suspect it was silly season in the Indiana legislature at the time.) It passed first reading, but that’s as far as it got. According to The Penguin Dictionary of Curious and Interesting Numbers, the bill “was held up before a second reading due to the intervention of C.A. Waldo, a professor of mathematics [at Purdue] who happened to be passing through.” Waldo, describing the experience later, wrote, “A member [of the legislature] then showed the writer [i.e., Waldo] a copy of the bill just passed and asked him if he would like an introduction to the learned doctor, its author. He declined the courtesy with thanks, remarking that he was acquainted with as many crazy people as he cared to know.”

The bill was postponed indefinitely and died a quiet death. According to a local newspaper, however, “Although the bill was not acted on favorably no one who spoke against it intimated that there was anything wrong with the theories it advances. All of the Senators who spoke on the bill admitted that they were ignorant of the merits of the proposition. It was simply regarded as not being a subject for legislation.”

Here is another telling of the story, complete with the actual bill and pictures. We’ll skip ahead to the appearance of Professor Clarence Abiathar Waldo, who was doing more than just “passing through” as the other version says (he was lobbying for the state university budget):

Professor Waldo had been an instructor of mathematics (and Latin) at several seminaries, institutes and colleges in the Midwest for more than 20 years. He had also been in administration, as a Registrar and Vice President at other institutions, which may explain why he had been given the task of keeping track of the University’s appropriation. He was the author of a book titled Manual of Descriptive Geometry.2

He was astonished to find the General Assembly debating mathematical legislation. Naturally, he listened in. Naturally, he was horrified. He heard a Representative speak for the bill:

The case is perfectly simple. If we pass this bill which establishes a new and correct value of pi, the author offers our state without cost the use of his discovery and its free publication in our school textbooks, while everyone else must pay him a royalty.3

After the debate, a Representative offered to introduce him to Dr. Goodwin. Professor Waldo replied that he was already acquainted with as many crazy people as he cared to know.

That evening, Professor Waldo “coached” (as he put it) the Senators about the bill. Still, on February 11 the bill was introduced in the Senate and referred to the Committee on Temperance. With a speed we can only admire, the committee reported the bill favorably the next day, and sent it to the Senate floor for debate.4

This time its reception was different. According to the Indianapolis News report of February 13, quoted by Edington (p. 209),

…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 propostion was not dignified or worthy of the Senate. He moved the indefinite postponement of the bill, and the motion carried.

The Indianapolis Journal had Senator Hubbell saying that “the Senate might as well try to legislate water to run up hill as to establish mathematical truth by law.”

If the “water run uphill” line sounds familiar, you probably read Robert Heinlein, whose character Lazarus Long in Time Enough for Love said, “If you pray hard enough, you can make water run uphill. How hard? Why, hard enough to make water run uphill, of course!” (Long also said, “Be wary of strong drink. It can make you shoot at tax collectors – and miss.”) The reason I mention this is because Heinlein referenced the law in Stranger in a Strange Land, only said the law was to make Π = 3 and it took place in Tennessee. The problem with Heinlein fanatics (as opposed to fans like me) is that they tend to believe everything they read – ironically you could call them Heinlein fundamentalists. This has probably added to the confusion.

Speaking of fundamentalists, some claim the Bible incorrectly says Π = 3. For those of you that think so, atheists and fundamentalists alike, you might find this article by Elizabeth Stapel interesting.

“And he [Hiram] made a molten sea, ten cubits from the one rim to the other it was round all about, and…a line of thirty cubits did compass it round about….And it was an hand breadth thick….” — First Kings, chapter 7, verses 23 and 26

The bowl is said to have had a circumference of thirty cubits and a diameter of ten cubits. The diameter is said to be “from one rim to the other”, so this would be the outer diameter; that is, the diameter of the outer mold used to make the bowl.

The circumference is not specified as being the inner or outer circumference, but since using the outer circumference would give us the “ideal” bowl (with no width or thickness), let’s instead use the inner circumference, which also, reasonably, would have been the circumference of the mold used to form the inside of the bowl. That is, we will use the two measurements which were necessary for the casting of the piece.

Using eighteen inches for one cubit, we have the following:

outer diameter: 10 cubits, or 180 inches
outer radius: 5 cubits, or 90 inches
inner circumference: 30 cubits, or 540 inches

To find the “Jewish” or “Bible” value for pi, we need to have the inner radius. Once we have that value, we can plug it into the formula for the circumference and compare with the given circumference value of 540 inches.

Since the thickness of the bowl is given as one handsbreadth, then the inner radius must be:

90 – 4 = 86 inches

Let’s do the calculations:

inner radius: 86 inches
inner circumference: 540 inches

The circumference formula is C = 2(pi)r, which gives us:

540 = 2(pi)(86)
540 = 172(pi)

Solving, we get pi = 540/172 = 135/43 = 3.1395348837…, or about 3.14.

(You’ll want to read the entire article for the diagrams and the full story.)

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