Monday, August 22, 2011

The Shoulders of Giants

The Shoulders of Giants

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.--1 Kings 7:23

The verse above frequently comes up in arguments between atheists and believers.  It implies that the value of the mathematical constant Pi is 3 as opposed to 3.14159... .
Some atheists use it as an example of a mistake the Bible which proves that the entire Bible is a lie and that, therefore, Christianity is a sham.  Believers, when they deal with this,  have been known to erect huge houses of cards in order to preserve the truth of the Bible on the atomic level.  
Let us put the polemics aside for the moment and just take a look as this issue.
Isaac Newton, the physicist and mathematician, is credited with saying that if he saw further it was because he’d stood on the shoulders of giants.  Newton had discovered the Law of Gravitation that bears his name along with the part of mathematics known as the Calculus and, in doing that, had benefited from the work of ancient Greek minds such as Archimedes along with the coordinate geometry of Rene Descartes, who had himself benefited from algebra that had come to the West from the Arabs.  
While the typical high school or college student might wish that these people, be they giants or not, had never existed, without them we would be without many of the modern amenities that we take for granted, particularly those amenities that require having satellites in orbit.  The mathematics is complicated but very useful.  It is also very beautiful.
Newton referred back to the geometry of the Greeks.  It is somewhat ironic that, although Newton did invent the Calculus and the Calculus is the way we prove Newton’s results today, he demonstrated them himself with synthetic geometry in the manner of the Greeks.  The Greeks mathematics for its own sake; they were interested in it for its beauty.  They pursued geometric truths because they found them to be beautiful.
Again, this is ironic because, when you are in the business of building a civilization, geometry is very useful.  If you are going to have a civilization, you will need at least one city to prove you are civilized.  That city will need palaces and temple and streets.  Geometry is necessary for the construction of all those things.  If your civilization isn’t going to starve, you will need a calendar  in order to keep track of time.  For a good calendar, you are going to need astronomy, and in order to do astronomy well, you are going to need geometry.
When the Greeks were doing geometry, they ran into the number Pi.  The number Pi, as you may recall, is defined to be the circumference of a circle divided by its diameter.  The word circumference comes from the Latin and obscures the Greeks referred to the circumference as the perimeter.  The word perimeter begins with the letter pi in Greek from there is it easy to see why we call it Pi.  
To me, it is an amazing fact that this ratio of circumference to diameter is the same regardless of whether the circle is ten inches across or ten miles across.  While my students have doubted my word on many things--in particular on the value of doing daily homework assignments--they’ve never doubted me on this thoroughly amazing fact.  The ratio of the circumference to the diameter of the circle is constant.
It is even more amazing that they believe this because it’s not true some other models of geometry.  Pi isn’t constant in either hyperbolic or spherical geometry.  You may have heard the old joke that if you tell people there are a billion-billion-billion stars in the universe they will believe you, but if you tell them the paint is wet they’ll have to touch it.
Pi is interesting mathematically because it’s an irrational number.  Irrational numbers are typically explained in school to be numbers whose decimal representation never repeats.  And that is true but it makes use of a way of expression numbers that was foreign to the Ancients.  The Ancients didn’t have decimals.  They didn’t really have fractions the way we have them. They worked with ratios.
A rational number is a number that can be expressed as a ratio of two whole numbers.  Pi cannot be expressed that way so it is irrational.  The Pythagoreans--those clever Greeks again--proved a result that in modern language and concepts boils down to the fact that the square root of two is irrational.  This is a comparatively easy exercise when one has modern notation. This irrationality of Pi does not have such an easy proof.
I’ve worked through a particular proof that Pi is irrational and satisfied myself that I understood every step and had a clear enough ideal of the “big picture” to present it to a class of seniors.  When I did this, they had the same expression on their faces as I presented it as my cat has when he watches me program the DVR.  It requires quite a bit of mathematical machinery and is not something the Greeks would’ve come up with.  The Greeks didn’t come up with anything else either.  
The Ancients, Archimedes among them, did come up with ratios that approximated Pi.  None of these was exact, of course, but only an approximation.  As I have said, there is no exact such ratio.
Let us now talk about the Biblical context in which this arises.  Solomon had hired builders from Tyre to built a temple to YHWH and while they were at it to build a house for him.  This is an interesting use of the word house as palace would be a better description.  This makes me think the PR people were already at work in those days.  This section of the Book of First Kings is describing the palace in what I as a reader consider to be excruciating detail.  In this, the passage has much in common with modern writing of the same type.
In any case, this molten sea is a round pool.  It is described as having a circumference of 30 cubits and a diameter of 10 cubits.  This would give us Pi=30/10=3, which we mentioned earlier.
From the modern point of view, this is redundant.  We know that when you give the diameter, you also give the circumference and vice versa.  The folks writing this description either didn’t know this or were wedded to redundancy.  Given the style of many of the drier patches of scripture, there is a case to be made for the latter, but I believe they just didn’t know.  Solomon had to hire this construction done.  Likely, they simply didn’t have the construction expertise available in Israel.  Without a well-developed practice of construction, they simply didn’t have the geometrical chops to know.
Now I would like you to do a thought experiment.  Act as if you are going to measure this yourself.  I would go to the hardware store and get a tape measure to measure the circumference and then measure the diameter.  I would have to use the tape measure because the pool is round.  Tape measures hadn’t been invented in those days and neither had hardware stores.  Likely they would have used a piece of cord and then measured the cord.  They would’ve measured it in cubits.
A cubit is an ancient form of measure that is based on the length of a man’s forearm.  I belief I am safe in saying that forearm length was as variable in antiquity as it is now, so in order for this method to be useful, they would’ve probably had to set a standard cubit at least for a given job.  That having been said, we can estimate the cubit to be about 18 inches.  This puts the pool at being about 180 inches in diameter.  Using the exact value for Pi, this would make the pool about 31 and a half cubits in circumference.  
In other words, the description given in scripture is about one and a half cubits off. In my mind, this was an amount that they would’ve noticed in the measuring.  They could’ve put down the correct number quite easily, I have to assume they didn’t because they just didn’t care.
They didn’t care because giving the correct value of Pi was not the purpose of this text.  I challenge you to read the seventh chapter of First Kings.  I double-dog dare you.
This text is about the opulence of Solomon’s Palace.  LIke the Russia in the cable TV commercial, Solomon could’ve said, “Opulence, I has it.”  Those of you who are Bible scholars know that Solomon taxed Israel to the point of ruin and his son finished the job.  In this text, you see part of where the money went.
Pi, schmi, Solomon had a swimming pool that was 15 feet across 47 feet around and seven and a half feet deep.  In a desert, if you got this you can say, “It’s good to be King!”
In conclusion, I’ve got to say that for me this is a non-issue, but I do understand there are those for whom it is. In my opinion, the Bible is not about Pi.  It is about the fact that we all stand on the shoulders of giants and that we should never forget that.


k thompson said...

makes me glad i majored in history!

Bobby Winters said...

What's your favorite part of history?