A while ago, my wife bought me a book about the history of pi.
It was a cute book that just talked about its history, importance,
etc...

It talked about how people had long been (since at least Roman times)
wrestling with a formula for "squaring a circle". That is, given a
circle of known area (or maybe just a known radius), determining the
dimensions of a square with the same area. Clearly, you'll run smack
into pi if this is your pursuit, but I never understood why, thousands
of years ago, this was such a burning question. Note, I might have it
backwards here and the question was how to "circle a square" (given a
square of known area, what would the radius be of a circle with that
same area).

Either way, my question is:
"Why was this circle-square area equivalency equation a useful and
important question to resolve thousands of years ago."

Greetings, and thank you for your question.

The earliest believed usage of some kind of pi dates back to the
ancient Egyptians. There were signs of the usage of pi by the
Egyptians in 950 BC while they were building the Great Kingdom of
Solomon.

Pi was originally used for practical problems, such as building, and
it was only with the scholars of Ancient Greece that people began to
explore 'pure' mathematics. I found this information on a page titled
'The Discovery Of Pi':

[ http://www.ga.k12.pa.us/academics/MS/8th/haddad/stwk97/mathjess/body.htm
]

Now to your question of the circle-square relation problem of the
past. An internet company called Orbik have a logo that shows a circle
becoming a square. A page on their website tells about their logo, and
this is an extract:

"Egyptians, Ancient Romans, and later Italians all were intrigued by
the puzzling relationship between a circle and its area as a square."

If you would like to read this page on the Orbik website yourself, it
can be found here:

[ http://orbik.com/company/history/logo/ ]

I found a page where a mathematician, Dr. David Harbater, tells of the
fascination in ancient times of wanting to 'square the circle'. He
says:

"The ancient Greeks had posed various problems, namely to do various
constructions with a straight edge and compass, and there were all
sorts of constructions they could do, but one of the things they could
never figure out how to do was to square the circle."

[ http://www.msri.org/activities/jir/bwachtel/CantSquaretheCircle.html
]

Squaring the circle appears to be one of the most famous problems of
all time.

[ http://members.aol.com/iterate/Pi.htm ]

That page gives a LOT of detail about possible reasons the Ancient
Greeks had for wanting to 'square the circle'. While it is widely
accepted today that squaring the circle is impossible, the author of
that page feels that modern mathematicians misunderstand what the
Ancient Greeks actually wanted to achieve.

"[the Ancient Greeks] were an extraordinary people, naming whole
branches of knowledge we venerate. Hardly the kind to waste their time
on a fool's errand."

I hope this at least helped you in your quest to find the answer to
this problem. I don't think we will ever truly know why it was so
important to find the answer, but hopefully the things I mentioned
here will shed some more light on the subject. You may also like to
see:

Squaring a circle
[ http://www.cut-the-knot.com/impossible/sq_circle.shtml ]

History and oddities of pi
[ http://www.inwit.com/inwit/writings/historyandodditiesofpi.html ]

Uselessness of pi
[ http://www.go2net.com/useless/useless/pi.html ]

I searched for a number of things on Google, including:
pi usage in history
"usage of pi"
"circle to square" history
why square a circle

Best wishes,
jon-ga

---

Request for Answer Clarification by wanderingjew-ga on 06 Jun 2002 15:04 PDT

jon-ga, thank you for your response. I hope you can help me get an
answer to my question.

To reiterate, "Why was this circle-square area equivalency equation a
useful and important question to resolve thousands of years ago?”

Thank you for the catalog of relevant links. However, I have already
read a book on the history of pi, I understand why the ancient Greek
mathematicians were consumed with this (they were mathematicians!),
and I am possessed of the mathematical wherewithal to understand the
intractable issues involved in attempting to square a circle.

The closest I came to an answer was in your second paragraph where you
state “Pi was originally used for practical problems, such as
building…”

This is exactly what I’m looking for, the specific practical problems
that squaring a circle (or at least approximating it) solves. What
were the Egyptians building and in what manner were they building it
such that squaring a circle would be useful or important?

Or, as I learned from the book, ancient Romans also had a practical,
real world reason for squaring a circle. The book indicates that
approximation methods for accomplishing this were found in Roman
manuals. What were these manuals for? What were they trying to do that
made squaring a circle a worthwhile pursuit?

---

Clarification of Answer by jon-ga on 06 Jun 2002 16:14 PDT

Sorry my original answer was not what you were looking for. Really,
the equation was not for any apparently useful reason.

"It is widely known that one of mankind's greatest
intellectual/spiritual quests was to try to square the circle - that
is, to construct a square of exactly the same circumference of a given
circle. It was a "spiritual" endeavour because the circle symbolised
Heaven and the square symbolised Earth - thus it would represent
Heaven coming down to Earth. The intellectual aspect of the quest, at
least, lasted for over 2,000 years (until 1882, when Lindemann proved
it to be impossible)."

[ http://www.geocities.com/ResearchTriangle/Forum/5375/index.htm ]

So one reason was spiritual, and the other was intellectual. I can
find no mention anywhere that people needed the answer of the problem
for any practical reason. I would love to point to an example of a
person needing the equation to build an ancient temple or other
similar structure, but such an example does not exist. I can find no
evidence that this problem existed for any reason other than the
exploration of geometry. Squaring the circle was one of three
classical problems, along with doubling the cube and trisecting the
angle.

I hope this will answer your question.

---

Clarification of Answer by jon-ga on 06 Jun 2002 16:29 PDT

Just to add something, you may find this page interesting: 
 
[ http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html
]
 
It gives a lot of information about squaring the circle, and also
backs me up on the idea that the quest to do so was purely
mathematical.

---

Clarification of Answer by jon-ga on 06 Jun 2002 16:48 PDT

Yes, another clarification! This one, I believe, proves that the
problem did not occur for any kind of construction reason.

From: A History of Greek Mathematics, Vol. 1, From Thales to Euclid,
by Sir Thomas Heath, 1921.
Reprinted by Dover Publications, 1981, ISBN 0-486-24073.

"There is presumably no problem which has exercised such a fascination
throughout the ages as that of rectifying or squaring the circle; and
it is a curious fact that its attraction has been no less (perhaps
even greater) for the non-mathematician than for the mathematician. It
was naturally the kind of problem which the Greeks, of all people,
would take up with zest the moment that its difficulty was realized.
The first name connected with the problem is Anaxagoras, who is said
to have occupied himself with it when in prison."

Pure pursuit of knowledge is fine, but I'd argue more for the
practical side.

As craftsmen have known for a long time, you can draw big circles
anywhere with a string, a stick, and a helper. Note this diagram of
Stonehenge:

http://www.the-timeless-dimension.com/shmap.jpg

But trial and error could use up a lot of string (and helpers, too).
The written mathematics of Pi gives us an easier way to:

- Consistently repeat a useful technique.
- Do accurate planning (even on parchment).
- Reduce costs in time, labor, and materials.
- Transfer that knowledge to new generations and across cultures.

It's another tool in the belt. 

Now, what if "Pi" could only be purchased through Home Depot? 

huntsman

"Frustra laborant quotquot se calculationibus fatigant pro inventione
quadraturae circuli."

(Futile is the labor of those who fatigue themselves with calculations
to square the circle.)
- Michael Stifel (1544) 
http://www.go2net.com/useless/useless/pi.html

Couldnīt resist )

THX1138

The oldest known mathematical document, called the Rhind papyrus,
gives an approximation for squaring a circle.  It seems it's a problem
that's been wrestled with for all recorded history.  I've never seen
anything mentioning a practical application for squaring a circle,
however.

Now, this is just my opinion, but I think the reason squaring the
circle has taken on such significance is because it is so difficult to
solve.  Many people (the Greeks especially) have produced endless
useless geometric relationships, but they were all easily determined
so you never hear about them.  Just like today Fermat's last theorem
(still not solved) receives considerably more attention than ,say,
standard deviation (way more useful).

Forgive me for not providing definitions.

I would also not ignore that Pi possibly had divine significance to
some cultures, i.e. the Ancient Egyptians.  Another number of similar
significance was the "golden ratio" or phi.  These two numbers were
used to correlate all sorts of relationships.  This page has examples
of a good number of them and how they have been used:

http://library.thinkquest.org/C005449/home.html

Unfortunately I can find no reference to what significance these
numbers actually played in peoples lives.  Some speculate there was
none, and some claim the numbers achieved god status.  It all seems
highly speculative.

Just to throw this in, here is a rather unbiased study of the
appearance of pi and phi in the pyramids:

http://www.emis.de/journals/NNJ/Reynolds.html

Just my 2 drachmas.

Just to add something, you may find this page interesting:

[ http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html
]

It gives a lot of information about squaring the circle, and also
backs me up on the idea that the quest to do so was purely
mathematical.

Well, wanderingjew, we can be pretty sure they weren't trying to
figure out whether two extra-large round pizzas are a better deal than
the rectangular party pizza.  Beyond that, anything else is pure
conjecture.

However, it is worth noting that the Romans had a fundamentally
"blue-collar" ethos.  They were not (for the most part) great artists;
they WERE (for the most part) great engineers.  Practicality was a
large issue in their overall scheme of things.

When building their large viaducts and aqueducts, the Romans planned
ahead to the extent of custom-cutting and numbering huge stone blocks
at the quarry, to be fitted into specific portions of a given arch or
other structure.  Now, given this context, imagine yourself a Roman
builder planning - say - a round temple.  You will want to know how
many square feet of stone will be necessary for this structure, in
order to estimate costs and place your order at the quarry.  Pi not
being available, you pull out your tables of approximations, and base
your calculations on those.

Now, I know a number of modern-day contractors.  Aside from getting
and keeping good workers, one of the biggest keys to success in that
business is being able to accurately estimate costs.  If you can do
that, you can project your profits accurately, and chances are you
stay in business longer!

As I said at the top, this is conjecture on my part, but I think it's
a plausible example.

Thank you for an interesting (if not wholly answerable) question!

This is just a comment that I thought might be helpful.

I think something like Pi was no only needed for designing and
structure but lots of other things. I am positive it was used in
astronomy and mapping the sky just like it is today
[http://homeworktips.about.com/gi/dynamic/offsite.htm?site=http%3A%2F%2Fmathforum.org%2Fdr.math%2Ftocs%2Fpi.middle.html]
 
But as I was earching I found other interesting things you might
consider looking at like as to who discovered PI:
 
Excellent site: http://www.wsu.edu/DrUniverse/pi.html

@ the bottom of the page:
"Okay, back to who discovered pi. Professor Kallaher says that it was
probably discovered sometime after people started using the wheel.
        Probably what happened was someone was playing around with a
wheel. Maybe he or she noticed that if he or she doubled the diameter
of a wheel, the circumference would also double. In other words, it
would travel twice as far in one rotation. In OTHER words, the
circumference divided by the diameter yielded a number, a CONSTANT,
that had nothing to do with the wheel's size. "Hmmm," he or she
thought, "VERY interesting. This undoubtedly has implications." Or
something along those lines.
        Anyway, the people of Mesopotamia (now Iran and Iraq)
certainly knew about this ratio. The Egyptians knew it, too, says
Professor Kallaher. They gave it a value of 3.16. Later, the
Babylonians figured it to 3.125.
        But it was the Greek mathematician Archimedes who really got
serious about the ratio. He was the one who figured that the ratio was
less than 22/7, but greater than 221/77.
        But pi wasn't called "pi" until William Jones, an English
mathematician, started referring to the ratio with the Greek letter
pi, or "p", in 1706. Even so, pi really didn't catch on until the more
famous Swiss mathematician Leonhard Euler picked up on it in 1737."

Sorry I didn't really find anything on what you asked, but that was a
very thought-provoking question.

Thanks for asking and have a great day!
~Angel1

I would like to correct a previous comment by kaitou-ga: Fermat's last
theorem was solved by English mathematician Andrew Wiles in May 1995.
To quote the following web site:

http://www.cs.unb.ca/~alopez-o/math-faq/mathtext/node9.html

"In summary:

Both manuscripts have been published. Thousands of people have a read
them. About a hundred understand it very well. Faltings has simplified
the argument already. Diamond has generalized it. People can read it.
The immensely complicated geometry has mostly been replaced by simpler
algebra. The proof is now generally accepted. There was a gap in this
second proof as well, but it has been filled since October 1994."

We may never know whether the builders actually used Pi or not, but
whatever their technique, the Hagia Sophia is a beautiful example of a
circle *on* a square:

Monolithic Dome Articles 
"Hagia Sophia in Istanbul, Turkey" 
http://www.monolithicdome.com/articles/hagiasophia/ 

Stock Photography - Hagia Sophia Mosque 
http://www.anthroarcheart.org/tblz357.htm 

huntsman