Letters, Lies and Calculus

In 1696 Guillaume de l'Hôpital published one of the first calculus textbooks, euphoniously entitled Analysis of the Infinitely Small for the Understanding of Curved Lines--or the Analyse for short.  In the Analyse's pages l'Hôpital laid out a method of figuring out the limits of indeterminate forms that was a huge deal in the burgeoning field of calculus.  It made L'Hôpital a star.

With all the linguistic verve of mathematicians, the rule was dubbed l'Hôpital's Rule, and ever since it has been rammed into the heads of calculus students, where it remains, a bit of discarded fact lodged somewhere between first girlfriend's middle name and capital of Peru.

As you can probably tell from the de appended to l'Hôpital's name (and the frothy wig perched on his head) good old Guillaume was a nobleman.  More than that, he mixed a genuine mathematical curiosity with the ability to straightforwardly explain the stuff he was interested in.  But while l'Hôpital was undoubtably a very good mathematician, and his textbook remained required reading for a hundred years, it turns out that all of the great discoveries that l'Hôpital is known for--including the eponymous rule--weren't actually discovered by l'Hôpital.  He merely owned them.

It all started in the salon of Malebranche, where the aristocratic thirty-something savant l'Hôpital met the 24-year old wannabe math nerd Johann (sometimes John) Bernoulli.  At some point in the night Bernoulli whipped out his 'secret weapon'--an unpublished forumla on how to figure out the radius of the curvature of a curve.   L'Hôpital, impressed, signed Bernoulli up to be his calculus tutor for ten months.  In 1694 l'Hôpital offered Bernoulli a further three hundred francs a year if he would tell him everything he could about this new-fangled calculus--and not tell anyone else.  Bernoulli agreed, and produced a series of brilliant letters explaining everything l'Hôpital could hope to know--and then some.  L'Hôpital would then take the insights Bernoulli told him and pass them off as his own, reaping the fame.

When l'Hôpital died, Johann Bernoulli claimed much of the content of l'Hôpital's work.  The famous textbook?  Actually that amounted to the ten-month course Bernoulli taught l'Hôpital.  The rule?  It should be Bernoulli's Rule.  L'Hôpital's work on conic sections?  That was Bernoulli's work.  But no one believed him.

There was good reason for this.  Johann Bernoulli was an irascible  thin-skinned man who involved himself in quite a few mathematical kerfuffles.  One acrimonious struggle was with his own son Daniel.  To win the argument (against his own son!) over who came up with some principle of hydrodynamics first, Johann resorted to forgery.

So clearly Bernoulli was jealous of his reputation.  Since he didn't claim l'Hôpital's discoveries with any special grievance, people just thought Johann's claim was just Johann being Johann again.

But Johann Bernoulli was right.  And nobody realized until 1922, when Bernoulli's first calculus lectures were discovered in a musty archive somewhere.   They were written before l'Hôpital's textbook.  And they were undoubtedly l'Hôpital's inspiration for the Analyse.  The L'Hôpital's Rule is really Bernoulli's Rule.

But I suspect that renaming l'Hôpital's Rule is just plain greedy.  The Bernoullis claim a menagerie of grey matter so quirky and brilliant that the three generations of genius could easily make up the cast of a Wes Anderson film.  (Bill Murray as Johann Bernoulli; Jason Schwartzman as Daniel Bernoulli.  Right?)  Because of this tons of stuff is already named after them.  There's the Bernoulli Effect.  The Bernoulli Principle.  The Bernoulli Distribution.  The Bernoulli Theorem.  These range over the domains of statistics, fluid dynamics, and calculus--and they are only a small sampling of the discoveries pinned with the Bernoulli name.  Do we really need a Bernoulli Rule?  Really?  The rule itself is confusing enough as it is.  We don't need to go messing around with its name.

My primary source for this story is an article by C. Truesdale.  I learned about l'Hôpital's Rule in Mark Hansen's math for social scientists class.

Plato's Less-Than Ideal Arithmetic

Philosophy is nothing more than footnotes to this guy, so they say.

Plato rightly deserves his central place in the Western canon.  He founded a school--the Akademia--from which we get both the word and the inspiration for the modern academy.  His insistance on the immorality of the soul so suffused the Greek-speaking world--including the authors of the New Testament--that Nietzsche dismissed Christianity as mere 'Platonism for the masses.'  Plato wrote over thirty dialogues that survive as masterpieces of argument and storytelling--a feat made all the more striking by the fact that back when Plato lived there were no paper mills, no printers, no bookstores, and no pens.  Plato pretty much set the aims, the methods, and the questions of philosophy for the next two thousand years.

But despite his heavyweight resume, Plato seems to have flubbed his math a bit.

Here's Plato calculating the exact amount that the philosopher's life is better than the tyrant's, from Book nine of the Republic.

Or if some person measures the interval by which the king is parted from the tyrant in truth of pleasure, he will find him, when the multiplication is complete, living 729 times more pleasantly, and the tyrant more painfully by this same interval.

What a wonderful calculation! And how enormous is the distance which separates the just from the unjust in regard to pleasure and pain!

Yet a true calculation, and a number which nearly concerns human life, if human beings are concerned with days and nights and months and years.

Plato's math, according to the footnotes in the second edition of the Grube translation of the Republic are "hard to follow."  Here's a try.  The tyrant experiences only two-dimensional pleasures, while the philosopher experiences three dimensional pleasures.  Additionally, the philosopher is nine times away from the tyrant in terms of pleasure, so the philosopher's pleasure is represented by a nine-unit cube, while the tyrant's pleasure is represented by a one-unit square.  But Plato flubbed things getting to the number 729, which was sacred to the Pythagoreans.  He miscounted the number of times removed the tyrant was from the philosopher (it should have been five, not six) and multiplied where he should have merely added.  Sadly, it turns out that the philosopher is only 125 times happier than the tyrant!

But we can't blame Plato for having trouble with his sums.  In Plato's time, before zero, before calculators, before arithmetic notation, math was decidedly hard to do.  Here's another example of Plato doing math, from the Republic, Book 8:

Now that which is of divine birth has a period which is contained in a perfect number, but the period of human birth is comprehended in a number in which first increments by involution and evolution, obtaining three intervals and four terms of like and unlike, waxing and waning numbers, make all the terms commensurable and agreeable to one another. The base of these with a third added when combined with five and raised to the third power furnishes two harmonies; the first a square which is a hundred times as great, and the other a figure having one side equal to the former, but oblong, consisting of a hundred numbers squared upon rational diameters of a square (i. e. omitting fractions), the side of which is five, each of them being less by one or less by two perfect squares of irrational diameters ; and a hundred cubes of three. Now this number represents a geometrical figure which has control over the good and evil of births.

What Plato's trying to say--again according the Grube Edition's footnotes--is that the human number is the product of three, four and five raised to the power of four, or (3*4*5)^4, which comes to 12,960,000.  This can be shown geometrically in two ways.  First, by the area of a square with the sides of 3600 or as a rectangle with sides 4800 and 2700.  Simple enough for us moderns.  But we have the ease of working with arabic numerals.  You can see how Plato--even Plato!--can be forgiven for messing up his math.

And you thought math was hard in high school!  Sacrifice a cock to Asclepius in thanks that you were never a math student in ancient Athens.

Nothing Really Matters

One fine night on the Island of the Cyclops, Odysseus the many-schemed plunged a sharpened stick into the eye of the bone-gnawing Polyphemus.  "Who are you to blind me?" Polyphemus bellowed, his story nearly done.  "Who am I?" answered the canny Ithican, "I am no one."  Polyphemus' friends came running to see why Polyphemus had a stick in his eye.  "Who did this to you?" they asked.  Polyphemus, son of Poseidon answered:  "No one!"

Odysseus demonstrated that nothing can really matter.  Today we're going to take a look at one of the greatest inventions in the history of thought:  zero.

Math For (Homeric) Poets

The importance of zero can be seen just by imagining how Odysseus might have done math.  For us this calculation is straightforward:

2345 -
1234

We chop the equation into manageable chunks.  First we subtract one-thousand from two-thousand, then two hundred from three hundred, and so on.

Odysseus--so cunning that he was beloved of the godess of wisdom herself--would have struggled with this simple act of arithmetic.  For him the equation would have been reckoned like this:

Two-thousand-three-hundred-forty-five minus one-thousand-two-hundred-thirty-four.

And done in the head or scratched on clay using primitive numerals--Odysseus being most probably illiterate.

With the development of writing, math became a little easier.  But only a little.  Here's how Caesar would have contended with our arithmetic problem:

MMCCCXLV -
MCCXXXIV

This kind number system is called additive notation because all you do is add up a series of symbols.  One thousand for the Romans is just M.  You want two thousand?  Just write MM.

The number system we use today is a positional notation, and it is infinitely more useful than additive notation.  In positional notation, the value of a given numeral depends where it comes in the figure.  In 1234, the symbol 1 stands for one-thousand.  In 4321, the same symbol 1 simply stands for one.  The same symbol--two different values depending on where in the number the symbol is.

There's a problem with positional numbering systems that you would never think was an actual problem unless you had to deal with it.  How do you represent one hundred and one?  You have one hundred, no tens, and one one?  Does it just look like this?

1 1

Enter The Zero

This is why zero makes all the difference.  Once you have zero, you can just plug it in whenever you have an empty column, and our previously unclear mark above becomes the familiar 101.

It's entirely possible to have a positional numbering system without a zero--though it must be quite awkward.  The Babylonians, Indians and Mayans all stumbled around with positional systems for hundreds of years before coming up with a zero.  The Chinese positional notation system never introduced a zero ever, and it continued to produce stunning mathematicians for longer than America has been putting bacon on hamburgers.

Super Zero

An amazing thing happens when the symbol zero becomes more than just a mark representing a gap in the tens column, and becomes a real number--a number every bit as solid as seven or seventy-seven.  This happened only once in human history, in India around the sixth century.  For comparison, at this time in Europe we were busy lolling around in mud and ruin.

Once zero is recognized as an actual number, a whole new universe of mathematics opens up.  Zero allows us to envision negative numbers.  The positive numbers stretch on infinitely to the right of the zero, the negative numbers stretch on infinitely to the left of the zero, and the zero stands as a fulcrum between plus and minus.  Zero also allows the development of complex algebra, allowing whole extra years of math courses to be added to the school curriculum.

There are a ton of great sources that I consulted for today's post.  Etymology Online and Mediatinker were particularly helpful.  If you're still curious about more of the odds and ends of zero, check out Numbers of Interest on Zero.  If you still haven't had your fill, the always chin-scratchingly good In Our Time has an episode on Zero as well.