Shall We Gather at the River

By Bobby Neal Winters

Cowboy looked up from the seat of his skiff at the clear, blue sky and felt the chill wind at it whipped past his face.  It was cold, but no colder than would be expected on a mid-December day.  They’d been no snow yet, but the early morning’s frost, now erased by the sun’s rays, had hinted at things yet to be.

Whenever people asked Cowboy what he was doing out on days like this, he said he was out looking for arrowheads along the creek.  It was a believable story as it was the sort of thing that people did in this part of the country.  And today was a good day for arrowhead hunting in any case. There’d been a long, dry summer which had been relieved recently by some heavy fall rains.  With the grass killed out and the erosion of the rains, there might be some new arrowheads exposed.  

And who knows, he thought, I might even find a few arrowheads.  It was the communing with nature, he liked though.  He loved being in the out of doors with only the sky for a roof.

He pulled his skiff up to the creek bank and carefully stepped out.  It was a beautiful day to be out in the open. The only cloud he saw wasn’t actually a cloud; it was smoke rising from a fire up on the hill a short way ahead.  In other places and times he might’ve thought it was hunters, but here and now Cowboy knew it belonged to Wendell Warthind, who’d been a childhood friend of his father’s.  He had a pretty good idea what Wendell was doing too and he made up his mind to ignore it.  Not ignoring it would make life complicated.  He had every intention of following through with this purposeful ignorance, but events weren’t going to allow it.

Cowboy planted his oar in the mud of the bank and tied his boat to it.  

He’d taken one step, maybe two up the bank when he heard the blast.

The sound could’ve been an explosion.  Cowboy knew what Wendell was doing up there and an explosion was one possible outcome of that.  That wouldn’t happen if Wendel was at his best, but he’d been slipping lately.

But Cowboy’s practiced ear knew differently. While it was technically an explosion, it was one that was more precisely characterized as a shotgun blast.  He pulled is 44 magnum from its holster and began running toward the sound when he heard profanity issuing in Wendell’s voice and several sharper explosions which Cowboy recognized as rifle shots.

He came in sight of an opening in the trees where in he saw Wendall hiding behind a large stump over which he was pointing a thirty-ought-six hunting rifle.  In the middle of the clearing was the campfire whose smoke Cowboy had seen. It was beneath a device that Cowboy recognized as a still.

Wendell hadn’t seen Cowboy yet, so Cowboy slipped behind a tree and looked in the direction that Wendell’s rifle was pointed.  The first thing he saw was a jeep with two flat tires and a couple of bullet holes in the side.  Then he saw a pair of feet beneath the jeep.  He followed those feet up when he saw the top of a blond head.    

When he repositioned himself a little, he saw the whole face and his jaw dropped.  The face belonged to the new preacher over at the Pentecostal Church, the Reverend Mosley.

Mary Beth Mosley.

Cowboy turned his attention back to Wendell  It looked to Cowboy that Wendell had a bead on Mary Beth and that if something weren’t done pretty quick there was going to be a-killing.  It was then that Cowboy pulled out his badge and started talking loud.

“Okay, there, Wendell,” he said with his 44 pointing straight at his father’s old friend.  “This has gone far enough.  Put the gun down.”

Wendell was caught off guard.

“What the hell?” he said as he turned.

“Put that rifle down!”

It came with enough force that Wendell put it down.

Cowboy now turned his attention to the Reverend Mosley.

“Mary Beth,” he said. “Throw that shotgun down.”

Mary Beth stood up from behind the jeep revealing a figure that caused Cowboy to curse it being wasted on a lady preacher.  

She had a double barrel shotgun in her hand.

“Oh, Sheriff,” she said brightly, “I am so glad to see you.”

Cowboy didn’t respond to her brightness.

“Put that shotgun down,” he said.  There wasn’t a hint of a smile.

Her brightness dimmed considerably as she put her shotgun down.

“Okay,” Cowboy said. “You come over here.”

As she got closer to him and farther from the gun, he moved toward Wendell, grabbed his rifle, and tossed it into the woods.  As he did this, he noticed that behind Wendell was a glass three-gallon jug of a clear liquid which he appeared to be protecting like the apple of his eye.

Just at the word was forming in Cowboy’s brain, he heard it coming out of Mary Beth’s mouth.

“Moonshine,” she said. “He’s been making moonshine up here.  I figured that I’d bring my shotgun up here and blow some holes in his still, but he was here.  I was trying to shoot that big jug, but I got him instead.  He had it coming, though, because that’s Satan’s brew.”

It was then that Cowboy noticed that Wendall had been wounded in the leg.

“You all right, Wendell?” he asked.

“I’ve been better,” he said.

Cowboy then went to the jug and smelled it.  

Heaven.

He picked up a tin cup from the ground, tipped a little of the jug’s contents into it, and took a swig.

Rapture.

He looked at the jug, he looked at Wendell’s leg, and he looked at the lady preacher’s jeep.  He finished his cup of whisky with a cough.

“Okay, I need to get you all back to town,” he said.  He reached down and picked up the jug.  With it in one hand and his 44 in the other, he said, “Follow me.”

It only took them a short time to get to the river, but the amount and the viciousness of the fighting made it clear that he wasn’t going to be able to leave them alone together.  Once they arrived, another problem became apparent.

“That is surely Satan’s brew,” the Reverend Mosley said. “We ought to just pour it into the river.”

This caused Cowboy to lick his lips as it was the finest tasting brew he’d had in a long time.  There was no way he could let that happen.  He then looked at the creek and his small boat and a new problem occurred to him.  How was he going to get these two people and the jug of whiskey across the river?  The skiff was only big enough for him and one other thing.

If he took Wendell across first, Mary Beth would dump the whiskey into the creek, and Lord Almighty, that would be a crime.  If he took the whiskey across first and left Mary Beth and Wendell alone together, one might kill the other.  There was only one thing he could do.

“Okay, Mary Beth,” he said, “you come with me.”

He put the jug down, sat in the back of the boat, and invited Mary Beth to sit between his knees which--after looking a little suspicious--she did.

It was a cold day, and it warmed Cowboy up--in more ways than one--to have Mary Beth snuggled between his knees as he paddled across.  Once there, he put her out and made the return trip.  He’d been doing a little thinking on his way across and he’d gotten an idea.  When he got to the other side he had a plan.

“Wendell,” he said, “hand me the jug.”

“Ain’t you afraid she’s going to dump it out?”

“Hand me the jug,” Cowboy repeated.

Wendell gave him the jug, and he went back over.  Once there, he got out, put the jug out of the way on the far bank, and turned to Mary Beth.

“Okay, come back with me,” he said.

“Don’t you trust me with the whiskey?” she asked.  The way she was smiling, she new the answer.

“No,” he said.

She climbed back into the boat and he slipped his knees around her again.  This time there was no protest.  Cowboy rowed back over, and when they arrived at the other side, Cowboy, directed his attention to Wendell again.

“Okay,” he said, “you trade places with her.”

This was done with no more problem than an exchange of hateful stares.

When he rowed Wendell to the other side, it wasn’t nearly as pleasant as it had been with the lady preacher.

“Ain’t you afraid I’m going to run off with the whiskey?” Wendell asked when he got out.  

“Not the way that leg is looking,” Cowboy replied.

He then headed back, fetched Mary Beth, and brought her back, all the while regretting that this would be the last time.  When they arrived at the other side, she didn’t seem to eager to get out either.

Wendell broke the spell, however.

“Where do we go now?” he asked.

“My truck’s up there,” Wendell answered. “I think I’ll take you to the emergency room first and once that’s squared away, I think I’ll take Mary Beth...I mean Reverend Mosley home.”

Cowboy thought he saw Wendell’s eyebrow move ever so slightly.

“Is he going to jail after that?” Mary Beth seemed eager to know.

“You never can tell what’s going to happen next,” he said.  “You never can tell.”

Taking the Measure

By Bobby Neal Winters

Introduction

In mathematics, simple concepts can be quite subtle.  Very early, children take out rulers and begin to measure objects.  A foot ruler will suffice to measure a piece of gum; a yardstick is up to the task of measuring one’s forearm; a tape measure will measure a room. None of this is difficult, though it might become tedious if one worries about accuracy, but that is not my concern.  

Hey, I am a pure mathematician; what can I say?  

Let’s leave the so-called practical concern behind. Consider taking a step from the realm of the concrete to the abstract, from the physical world to the world of pure number.

For example, how would we measure the length of an interval on the number line itself.  This isn’t difficult either.  If we consider the length of the interval of real numbers [1,3], we can recognize immediately that it is two units long.  We figure this out because 3-1=2.  It’s simple.  Indeed, if we want to make a more general formula, the interval of real numbers [a, b] is b-a units long. Only subtraction is involved.

You man have noticed that I am using the notation that [a,b] is the set of all real numbers between a and b, including a and b themselves.  If I write (a,b), this means I am excluding both a and b; (a,b] means I am excluding a; [a, b) means I am excluding b.  And all of these intervals are the same length because they differ only by one or two points.  Recall that, as far back as the time of the Greeks, Euclid was saying “A point is that which has no part,” so one or two (or three or fifty) points removed from an interval have no effect on the total length.  Indeed, one may remove any finite set of points from the interval [1,2] and the result will still be two units long.  We aren’t sweating the small stuff.

The sharp reader--and I know you are out there--may have noticed during the course of that last paragraph that I slipped from the concept of the length of an interval to the length of a set. That’s one small step for a man, but one giant leap for Mankind; or Womankind; on Man-unkind if you are e.e. cummings.

In any case, that one small slight of hand opens a whole different kettle of fish.  In order for the general reader to appreciate the problem, I will now spend a little while talking about a simple set of numbers that many otherwise gentle people hate with a fury: fractions!

 

Being Rational

Fractions, I will be talking about them, are better known as rational numbers.  As I am no longer talking about this on a street corner or in a bar--pssst,wanna take a look at my square root?-- I’d best call things by their scientific names.  Rational numbers are numbers of the form m/n where m and n are both integers but n is not zero.  (Remember, dividing by zero makes green hair grow on your palm. Er...never mind.)  I mentioned earlier that many otherwise gentle people hate rational numbers: mathematicians love them.  Mathematicians are rather like cops, soldiers, morticians, grave robbers, or other people whose jobs force them to see things other people don’t have to, and they have a different standard of nastiness.  Compared to  some of the critters we’ve seen, rational numbers are as tame as puppies.

One thing that most people don’t know is that rational numbers are literally everywhere on the real number line.  Between any two real numbers there is a rational number.  This is easy to see; those of you who aren’t up to a dose of algebra can skip the next paragraph.

Take any two real numbers a less than b.  We know that b-a is positive.  One can choose a positive integer n to be large enough so that n times b-a is greater than one.  This means that there is an integer m that lies between n times a and n times be.  It follows that m/n is between a and b.

Mathematicians have a name for this property of the rational numbers.  We say that the rational numbers are dense in the real numbers.  Mathematicians are also frequently referred to as being dense, usually by their spouses, but that is a matter for another day.

 

The Real Thing

I’ve been referring to the real numbers as well as the rational numbers, and the place that they live together in peace and harmony as the number line.  When I was young and full the brashness rightly common to all newly minted PhDs, I understood all this stuff.  As I’ve had a been of that brashness scuffed off me, I’ve lost much of that understanding.  I try now to approach numbers with a child-like humility.  All of this to say that I am now going to explain some things that bright people like yourselves already know.

All rational numbers are real numbers, but not all real numbers are rational: pi is irrational; the square root of two is irrational.  Those are two examples, but there are lots of others.  Most numbers are irrational. When I say that, I could mean any number of different things.  One thing would refer to the concept of number.  The set of rational numbers is countably infinite, but, by way of contrast, the set of real numbers is uncountably infinite.  I’ve talked a bit about this in other essays.  Another thing that I might mean is that the real numbers are infinitely long and the rational numbers have a length of zero.

In case you are surprised that the rational numbers have a length of zero, you are not alone.  Indeed, one of the early methods of attempting to measure sets called “content” measured the rational numbers as being infinitely long as well.  There were problems with content, however.

 

Piecing It Together

The problem with content can be easily explained.  The content of the rational numbers between zero and one is one; the content of the irrational numbers between zero and one is one.  Now, when I take something that is a foot long and stick it together with something else that is a foot long, I don’t expect the result to still be a foot long; I expect it to be two feet long.  It is at this point that we can start making excuse for poor little content, but that would take me too far afield.  Suffice it to say that content just doesn’t measure up.  (If you missed the pun in the last sentence, I urge you to go back and re-read it.  I am very proud of it.)

We want the sum of the length of disjoint sets to be equal to the length of all of those sets put together, and we wan that to work even when we join together a countably infinite number of sets. A fellow named Lebesgue figured out how to do this and it works.  He cleared up a number of problems that we were having with some very technical mathematics at the same time.  We mathematicians love those among us who clear up technical problems.  We tend to name things after them and teach courses about those things.

I teach a course about what Lebesgue did.  It takes me quite a while to get all of the technical details lined up for the graduate students who take it.  I could dump that all on you right now, but you might whimper; my grad students do.  

Rather than may you whimper, let me show you something cool that was opened up by our old friend Lebesgue.

 

The Un-Measurable

Once upon a time, there was an Italian mathematician by the name of Giuseppe Vitali.  (Let’s just call him Joe, okay?) Joe created a set that not even Lebesgue could measure.  I will show you the set before I let you go.  Seriously, if you haven’t noticed, I’ve trapped you in your seat now.  You are mine! You are mine I say! Bwahahaha!

Er.  Sorry.

Suppose that you could create a set V with the following properties:

1. V is contained in the interval [0,1);
2. There are sets V1, V2, V3, … each geometrically congruent to V such that
1. they are disjoint from each other;
2. there union contains [0,1);
3. there union is contained in [-1,2).

One might innocently look at those conditions and not think much of it.  It all seems so utterly reasonable.  Well, that just goes to show what you know.

Suppose we call the union of all of these sets U.  As union begins with U this sort of works.  On one hand, since U is contained in [-1, 2) and that interval is no longer than three units long, we know that U is no longer than three units long.  On the other hand, since U contains [0,1) and that interval has a length of one unit, we know that U is at least one unit long.

So far, so good.  It is about to get messy.  

Now if things add up the way they are supposed to, the length of U should be the sum of the lengths of V1, V2, V3, … , right?  As each of these sets is congruent to V and they are all disjoint, this should be easy.  We are just adding the length of V to itself an infinite number of times, right?

Again, the alert reader’s eyebrows may just have arched in a Vulcan-like fashion. This is problem that even those pesky Greeks knew about.  If you add up the same number (non-negative) an infinite number of times, the sum is either zero or infinity.  It’s zero if the number you are adding together is zero, and it’s infinity if is positive.  

So if we there is a set V that has the properties of the of one described, either one is less than zero or three is greater than infinity, or, at least, so it seems.

The faint hearted would stop at this point, but not our Joe.

Joe created V just as we describe above, and he did it in stages.  

Joe first broke [0,1) into some disjoint sets.  The first of these was the set of all rational numbers in that interval.  The rest he obtained by taking copies of the rational by sliding them and rotating them through [0,1).

What’s that? A hand went up in the back.  What do I mean by sliding and rotating?

By sliding I mean that you act as if the rationals are marked on a clear ruler the is laying over [0,1) and then sliding that along to cover other numbers.  By rotating, I mean that when we push the numbers past 1, act as if the interval in bent in a circle back on itself and those numbers pop up past 0.  

“It’s all on a circle, Grasshopper.”

Anyway, once you’ve broken the interval [0,1) into pairwise disjoint sets in that way, you choose one number from each of those sets and form V from those points.

The studious drudges who occupy the ranks of the mathematical world can verify that V satisfies the demands that we’ve put on it.  

We are left with a crisis.

 

Resolving the Crisis

I’d said above that we were left with the choice that either one is less than zero or three is greater than infinity.  I left out an alternative.  That is that there are just some sets that can’t be measured. The set V is such a set.

In solving problems, mathematicians often find more than they bargained for.  This is such a case. Above, quite innocently, I told you to form V by just picking a point from each one of the disjoint sets described.  That particular action is now referred to as the Axiom of Choice.  It is a method of creating sets that people hadn’t given too much thought to up until that point.

They started thinking about it.

I may as well, but I will be like Scarlett O’Hara and worry about it another day.

Teaching, Writing, and Mathematics

By Bobby Neal Winters

I have been thinking recently about the writing of mathematics.  This is rendered to being a highly academic exercise as I have not produced any original mathematics to write about for quite some time.  I don’t bring very much credibility with me to this endeavor because during the brief interval I was producing original mathematics I cannot say that I was a very good expositor of it.  

What has happened in to bring my thoughts to the writing of mathematics?  One thing is that I have begun writing myself.  I began writing a weekly column for the local news paper about 10 years ago.  Figure 52 columns a year--I never miss a week--and seven hundred words a column--I usually overrun that barrier--and this comes to about 364,000 words.  I write things besides my column so I will say that I’ve produced about half a million words in the last ten years.  If repetition is the mother of learning--and I believe it is--then I may have learned something about how to write.

In addition to this, it has been my pleasure to teach out of a pair of very well-written books: A Radical Approach to Real Analysis and A Radical Approach to Lebesgue’s Theory of Integration, both written by David Bressoud.  These books are, as I said, well-written, but I think more important to their effect on me is they are written as historical approaches to their particular subjects.  

The historical approach has been very enlightening to me.  Mathematics is a land populated by optimizers and those seeking efficiency through brevity.  A result is discovered and proven with great difficulty.  Then time is spent in organization working out theory wherein the proof of the result seems not only easy but inevitable.  The historical approach allowed me to appreciate that what has been rendered as clear as glass to students of the subject today was once a riddle in a mirror.

As a teacher, it reawakened the excitement of the subject in me.

But in presenting this material to my classes, I’ve reproduced the older proofs of various results that have been included and in doing so I’ve noted a difference in style with much of modern mathematics.  There is a tendency in modern mathematics to drift toward the abstract.  This is a reasonable tendency as the abstract proofs tend to be cleaner and tend to give broader results, i.e. if I prove something about metric spaces, then I prove it about the real numbers, the complex plane, and spaces of functions at the same time.

That is something of an illusion, however.  The abstract structures are out-growths of specific examples.  The examples were originally things that were interesting in themselves and drove the construction of theory to discuss them of abstract structures to more easily prove results about them.

It is somewhat ironic that my experience as a mathematician helped me to be a better writer of things besides mathematics.  As a topologist who studied 3-manifolds, I took trips in my head to places no one had been before and then attempted to explain them to those who’d remained at home.  I learned the art of description as I described these strange places. In the construction of proofs order is important.  In good writing, some things must be explained before others can be understood.

As a writer, it seems to me that in well-written mathematics, well-constructed examples serve the same purpose that metaphors do in good writing.  Much writing seeks to create in you a picture that I have seen in me.  This is true whether I am writing about house cats, building a computer, or mathematics.  If I am to be successful, I must reach you by beginning with a thing you already understand and build from it.  Our shared experiences make communication possible.  Mathematically, the creation of a well-chosen example gives writer and reader a common, shared experience from which the writer may then deviate in order to build.

This technique is especially evident in the proofs in analysis that begin with the proof of a simple, very special case and proceed through stages of increased generality until the most general case is proved.  At its best, one can see the whole of the problem is in that very special case and the succeeding generalizations are simply minor modifications of the original proof.

There have been times when I’ve complained that too much of the time we who teach mathematics treat it as a murder mystery.  We withhold or downplay certain very important details.  We keep things to ourselves as our own little secrets.

I think I know why we do this.  We are trying to teach our students how to think for themselves.  There is a reluctance to “lay out everything plain” because it will deny the students the joy of the gestalt that we ourselves experienced, the joy that led us to become mathematicians.

I must say that there is much virtue to this point of view.  The problem is that it is so easy to do very, very badly.  There is also a tendency--among some in the profession--to use this as a technique to build up their own egos at the expense of their students’. This is far from universal, however.  Much more of a problem is judging difficulty.  What is easy to a professional might be insurmountable to a student.  The teacher’s job is to lay out bread crumbs to tempt the student to the point of gestalt but not so many as to rob the student of the joy of that gestalt. The fat cat will catch no rats.  Not that a cat eats bread crumbs, but sometimes you have to mix your metaphors.

In our teaching, in our writing, we must free ourselves to ape literature.  We we must foreshadow the great mathematical truths to come with smaller, more easily digestible truths.  When the student finally comes to the climax, he must be so prepared that the final step to him is nature, the final joy is true, but he should then be able to reexamine his steps and realize his arrival at this particular point was no accident.

I mentioned at the beginning of this article that when I was producing original mathematics I wasn’t a particularly good expositor of it.  There are those who will read that sentence and recognize me as a master of understatement.  While a careful study of my articles would undoubtedly produce additions to this list, I believe I had two major problems: impatience and an over-reliance on notation.

I believe we can all agree that taking one’s time to do a good job is a virtue.  A lack of patience can cause a lack of proper care.

Creating notation is a way around the difficulties we sometimes encounter in natural language.  Creating a well-chosen metaphor, a well-created example is another.  The example has the advantage that someone might name it after you one of these days.

If I were suddenly given my mathematical life to live over, I hope that I would choose to grasp onto mathematical exposition as the true art that it can be.  To first live the mathematical adventure and then tell the tale, and to understand the telling of the tale is as at least as important as the adventure because its purpose is to convince others to have adventures themselves.

(Bobby Winters is Assistant Dean of the College of Arts and Sciences and Professor of Mathematics at Pittsburg State University. He now holds the title of University Professor.)

Magical Times

By Bobby Neal Winters

A couple of books have come my way lately that are of such a quality I would be remiss if I did not share them with my friends: Isaac Newton by James Gleick and Longitude by Dava Sobel.  I discovered the second while discussing the first with a friend [Maeve for those who are curious] over lunch.

The two books concern roughly the same time and place: England in the late 17th and early 18th centuries.  They also overlap in some of the characters that are discussed: astronomers and scientists.  Together, this pair of books give me a different mental image of the era and the people.

Having been interested in science since grade school, I’ve venerated Isaac Newton as an icon.  For a long time this was done on faith, as I wasn’t familiar with his achievements.  Even after having studied the Calculus in college, I still didn’t know what he’d actually done mathematically.  This was remedied somewhat when I taught a course out of A Radical Approach to Real Analysis by David Bressoud.  Having then seen some of the things that Newton had done (and hadn’t done), I became convinced that he was nuts.

Isaac Newton, while putting a finer point on this hypothesis to be sure, did little to disabuse me of this notion.  For example, in this book we learn that, while studying light, he inserted something like a crochet hook between his eye ball and eye socket, observed rings of light, and took notes about it.  We see the scientific method made perfect, but, to be as eloquent as I can be, ick.

But I don’t mean to diminish Newton’s greatness by highlighting his weirdness. Newton was great and, what’s more, I would put him as perhaps the first and certainly the greatest of the Redneck Mathematicians.

He was born shortly after his father died.  I am thinking less than nine months.  His mother apparently didn’t care too much for him.  She remarried a man who didn’t want Isaac hanging around.  

This is the stuff of country music if I ever heard it.

He inherited his father’s farm, tried his hand at farming, and sucked at it.  His male relatives looked at him, assessed that he was only fit for college, and packed him off to Cambridge.  While he was at Cambridge, it was hit with the Plague, so he came home and invented the Calculus.  At a time when most undergrads are not understanding calculus, he was inventing it.

And he kept it as his little secret.

That is weird.  True, those were different times before openness and communication were valued in science, but Newton took it to an extreme even then.

Newton got a university job and then started doing two things that aren’t really highlighted in his scientific hagiographies: alchemy and Bible Study.  He wanted to turn lead in to gold.  He wanted to use the prophecies of the Old Testament to predict the end of the world.  He didn’t turn lead into gold by the way; but he did predict the world would end in 2060.

While he was busy doing these things, a comet appeared in the sky  Actually, they thought it was two comets, a little one and a big one, but it was really one comet: Halley’s Comet.  Newton was first bothered by John Flamsteed, the Astronomer Royal, who theorized the two comets were one.  Newton more or less blew him off.

Then Edmund Halley comes on the scene and starts a chain of events that climaxes in Newton writing his scientific masterpiece Principia Mathematica.

Gleick does such a good job of drawing the characters here that I will leave the details of this to the interested reader. Suffice it to say, I am thinking of writing a short story about it.

Modern science wouldn’t be the same without these events.  

The importance of technology is made plain in Longitude.  When you are navigating at sea, if you don’t know where you are, their is a pretty good chance that you are going to die one way or the other.  To know your position on the planet earth, you need to know your latitude and your longitude.

Latitude is relatively easy to calculate from direct astronomical observations.  I might write that up sometime.  Longitude is easy to calculate as well, provided you know exactly what time it is at your base  point, i.e. zero longitude.  This was a problem back in the day when clocks were only accurate to about fifteen minutes a day.  That difference doesn’t sound like much to use but for navigational purposes it is quite profound.  Indeed, as little as a few seconds a day drift accumulates over a long sea voyage.

The answer to this problem was ultimately a better clock.  The story of how that clock was created by a carpenter cum clockmaker named John Harrison is told skillfully in Longitude.  Sobel does a great job of making something that could be made dreadfully dull in other hands come to life.  She’s talking about making clocks for heaven’s sake.  How boring is that?

One of her techniques is following the politics of the clock’s gaining acceptance. It is here we run into some of the same characters as we did in Newton’s story, e.g. Flamsteed and Halley.  It’s here that in the juxtaposition of the two stories I gain some insight.  Newton profited by the emergence of institutions such as the Royal Society and the Royal Observatory; Harrison by way of contrast was somewhat blocked by the institutions, though it is much more complicated than that.

Ultimately, I found myself thinking that this was a magical time for England, as there was much, much more going on then than just gravity and time.  Newton knew Jonathon Swift of Gulliver’s Travels fame.  Those pesky colonists over in the New World revolted along about then.  And I am sure that someone who actually knew history could come up with a lot more stuff.

Suffice it to say, these books are interesting, informative, and readable.

Treat yourself to reading them both.

TV Math

While not exactly Redneck Math, this link is interesting.

Drinking Caesar’s Urine

By Bobby Neal Winters

My Burden

Every glass of water you drink contains a few molecules from Caesar’s urine.

There, I have shared my burden with you.  I first heard this myself more than 20 years ago.  A mathematician was talking in the faculty lounge.  He tossed it off, rocked my world, and then the conversation rolled along into other esoteric matters leaving me with this horrific bit of knowledge: I am drinking Caesar’s urine.  Between the period on the end of the last sentence and the “B” at the beginning of this one, I took a sip of coffee; more water, more of Caesar’s urine.

For years I was able to blot this out of my mind, but then it came back at me in the form of a question: how much?  How much of this rare liquid am I getting in every glass of water?  I am a mathematician, so I can do the calculations.  I did them.  The answer is 2100 molecules in every eight-ounce glass of aqua not-so-pura. The point of this essay is to explain how I arrived at that number and explore some of the territory along the way.

Making this calculation requires an assumption, so I will begin by explaining my assumption with the promise that later in essay I will return to justify it.  My assumption is this: Caesar’s urine is uniformly scattered throughout all of the water in the world.  That is a big one, but it is absolutely necessary to the calculation.  The calculation requires a simple formula—you  knew it was coming—that I will give below:

C=fG

Here C is the amount of Caesar’s urine in a glass of water, f is the fraction of the world’s water that is Caesar’s urine, and G is the amount of water in a glass. 

The value for f is one that we will have to calculate.  Here, I am going to limit the scope of my study.  I am only interested in the urine that Caesar excreted upon his assassination.  Calculating the amount of urine Caesar produced over his lifetime would introduce too many assumptions for my comfort-level which is going to be sorely tested in any case.  Consequently, I will focus on that one last bladder-full.  Here I am also having to make the assumption that Caesar had an average sized bladder and that bladder was full.

In any case, the average bladder has the capacity of 350 milliliters.  This is almost exactly the same amount of fluid in a can of Mountain Dew.  Coincidence?  I think not!

I am not too worried about this figure as it is something that I can verify for myself given soft drinks, time, and a calibrated beaker that I am not using for anything else.  Much more of concern is the next number I need, which is the amount of water available on the whole earth. This is a figure I had to look up: 1.36x10^9 km-cubed.  That is to say about one billion cubic kilometers.  This converts to 1.36x 10^21 liters. 

Doing the math—dividing the amount of urine in Caesar’s bladder  by the amount of water in the whole world—this gives us a figure of f=2.58x10^-22.

I have been using scientific notation here, which you might not deal with every day.  The figure for f is incredibly small.  Written out as usual, it would be

0.000000000000000000000258

I meant to have 21 zeros between the point and the 2.  You can count to see if I got it right.  In any case, you will agree that this number is small, so one might wonder at this point whether this whole drinking Caesar’s urine thing is plausible at all.  In order to justify this, I have to make a side trip to the world of the atom.

Avogad Row: The Place Where Chemists on the Skids Wind up

At this point, I get into an area where I don’t feel very comfortable: chemistry.  I never had a chemistry course in college.  I took one as a high school senior of which I only remember two things.  I learned the metric system for the fourth or fifth time and we put some aluminum in some nitric acid. 

Putting aluminum in nitric acid was really, really cool.  I spilled some of the nitric acid on my fingers and it turned them yellow and when we put the aluminum in the acid it got all hot and it looked like it might explode.  Learning the metric system for the fourth or fifth time was somewhat anti-climactic to this.  And come on, we get all of this hype of how the metric system is so much simpler than the so-called British system, but this why does it have to be drilled-in over and over and over.  When I went to college, I couldn’t stand that I might have to learn the metric system yet again, so I opted out of chemistry.  I took physics instead where the first thing they taught me was…the metric system.

I paid for this deficit in my education a few years back when I was made Acting Chair of the Department of Chemistry.  Among the things I learned there was that chemists are much more interested in blowing things up than they are in the metric system, but to blow things up in a predictable orderly way they must have an exhaustive knowledge of the metric system.  That is neither here nor there.

I need chemistry because I want to know how many molecules of water there are in a liter.  Chemists are interested in numbers of molecules because when they mix stuff together the proportions they mix are important.  The reactions go on at an atom-to-atom level. 

Chemists use a special unit to measure substances used in chemical reactions.  That unit is called the mole.  The mole is an odd unit because it is not a unit of volume nor a unit of mass, though in controlled circumstances it can be related to either of those.  Instead, the mole contains a fixed number of molecules.  That number is called Avogadro’s Number. 

Avogadro’s number is the number of molecules in 12 grams of pure carbon 12.  Carbon 12 is an isotope of carbon whose nuclei contain 6 protons and 6 neutrons giving it an atomic weight of 12, hence the name Carbon 12.  Does one get kind of a funny feeling about chemists here?  Do you get the impression they are strange sorts of ducks?  On one hand they have this mind-numbing obsession with the metric system and a total lack of imagination in naming their isotopes, but on the other hand they like to blow things up.

I’ve got all sorts of questions here.  Why carbon?  Why Carbon 12 in particular?  Why 12 grams of Carbon 12?  Were those two 12s chosen at random or is there more going on? Hmmm.  Were the Knights of the Templar involved?

I must put those questions aside to focus on the task at hand.  What is Avogadro’s Number?  Here it is:

                                               NA=6.022x10^23.

Again this is in scientific notation and that 10^23 in there means that if I were to write in out normally there would be 23 places between the 6 and the decimal.  In other worlds this is a big number.

As chemistry arose out of alchemy, I might doubt the veracity of number figured out by people who in their heart of hearts are still looking for the Philosopher’s Stone, but as they love blowing things up so much and this number is so important to that goal, I will accept it.

Aqua Pura

A trip to www.wolframalpha.com will assure us that a liter of water contains 55.5 moles of water, so a quarter liter will contain 13.875 moles and, therefore, 8.33x10^24 molecules.  The fraction of those which are Caesar’s urine is f=2.58x10^-22.  Multiplying these two numbers together rounds off to 2100, which is the number I gave at the beginning of the article.

Mixing

At the beginning of this article, we had to make the assumption that Caesar’s urine was scattered uniformly throughout all of the water in the world.  I promised that this would be justified later, and now is the time when the chickens come home to roost.

I will start with things you already know.  Rain falls from the sky onto the land; it trickles down hill into ditches, creeks, and streams; it gathers into lakes and rivers; it flows into the ocean; and at various points along the way it evaporates back into the air where it turns again into rain.  It is part of a cycle that has been going on for billions of years. 

Of course I’ve left out a few details such as at various places along the way folks—like Julius Caesar for instance—take a drink and then pee, putting it back into the cycle once again.

As a part of this system, many things happen to in and this is where mathematical modeling comes in. In the part of mathematics called Ergodic Theory, they model the behavior of such systems using a function called The Baker’s Transformation.  While this can be described in mathematically excruciating detail—and nobody knows about excruciating detail like a mathematician—you can think of it simply as follows.

A baker takes a ball of dough, stretches it to twice its width, gives it a quarter turn, and then folds it back over onto itself.  If you put a drop of red food coloring on this, it will eventually be spread uniformly throughout the loaf.

The idea is that similar things happen to Caesar’s urine.  It flowed onto the street where he lay dying; it evaporated into the air; wind sheared it and spread it around; and, after a suitable interval of time, it is spread out everywhere.

Mathematicians study strong mixing, weak mixing, and topological mixing depending upon the type of system they are interested in.  The Baker’s Transformation and Caesar’s urine are examples of strong mixing.

Conclusion

It occurs to me that some folks might find the prospect of drinking Caesar’s urine to be disturbing.  In that case, you probably shouldn’t dwell to long upon the fact that you are also drinking the urine of Caesar’s horse, his dog, and of the beggar who was lying on the street in front of Caesar’s house.  We live in an interconnected world.  I’m touched by people who lived thousands of years ago and thousands of miles away.  Those people (or should I say pee-ple) are literally a part of me.

As the author Norman Maclean said, “Eventually all things merge into one and a river runs through it.”

And that river will be carrying some of Caesar’s urine.  I just thought you should know.