Wednesday, June 27, 2012

Mathematical misnomers

Mathematicians don’t always get their history correct. Maybe that’s why they went into math. Because of this, there are numerous examples of results that have gone by the name of someone other than the original discoverer. Stephen Stigler, author of several books on the often neglected topic of the  history of statistics has gone so far as to suggest (with his tongue firmly planted in his cheek) “Stigler’s Law of Eponymy”: “No scientific discovery is named after its original discoverer.”
Incidentally, Stigler’s Law was first postulated by Robert Merton.
Famous example
The Pythagorean theorem is perhaps the most well known mathematical misnomer. Although is it likely that Pythagoras first proved the theorem, the theorem was known by the Babylonians long before Pythagoras. There is a tablet from Mesopotamia that lists fifteen different Pythagorean triplets. This tablet was inscribed at least a millennia before the birth of Pythagoras.
Pascal’s triangle is yet another case where the proper person has not received full credit. Chu Shih-Chieh tabulated the triangle to show the binomial coefficients in 1303. Pascal got a triangle and a programming language named after him. Contrary to popular belief, Chu Shih-Chieh did not have a dog breed named after him.
This triangle appeared in print in Europe in the Rechnung by Peter Apian in 1527, nearly a hundred years before Pascal’s birth. Peter Apian was a bit luckier than Chu Shih-Chieh. A crater on the moon and a minor planet were named after him. Although Apian was into maps and astronomy and all that, he paid little attention to either his crater or his planet. To the best of my knowledge, he never planned a summer holiday to either.
Our number system (that is, the way we write numbers) has commonly been referred to as the Arabic number system. The Encarta encyclopedia calls our number system the Arabic numbers. Rouse Ball1 refers to our number system as Arabic. This is unfortunately a misnomer.
The number system we use today did make it to Europe from Arabia, but the concepts and symbols are entirely due to the Hindus. Hence it is more appropriate to call our numbers the Hindu system of numerals. It was first introduced to Europe through the translation of a book by Mohammed ibn-Musa al-Kwarizmi, and from him has become known as the Arabic system.
The hapless De Moivre
Ah, poor DeMoivre. So often misunderstood.
A statistician calls the curve the “normal curve”, a teacher calls it the “bell curve”, and a mathematician refers to it as the “Gaussian”. The discovery of this curve rightfully belongs to De Moivre, however. It was De Moivre who first described this as a probability distribution.
De Moivre was also cheated out of recognition for a formula to approximate the factorial. The formula
is known as Stirling’s formula, although De Moivre discovered it earlier. This formula isn’t that well known under either name. I was rather taken by it when I was in high school. I found it in a book somewhere. Perhaps if this formula became part of the normal high school curricula, there would be more people like me?
Ironically, the one relation that bears De Moivre’s name is one that he never explicitly stated. The famous De Moivre’s theorem states that
From reviewing his work, it is certain that De Moivre was aware of this magical result. I, personally, am not aware of any historian that came right out an asked the guy, "Hey Abraham, you ever hear of De Moivre's theorem?" I am guessing that this is yet another historical event that will never be completely resolved.
The Cauchy distribution and its odd properties as a statistical distribution were first written about by Poisson, almost thirty years before Cauchy did any work with this function.
The Cauchy distribution, itself a misnomer, is another name for the mathematical curve known as the witch of Agnesi. A more proper name for this might be the “curve of Grandi”. I will write more about the multiple times this was invented in a future blogpost.
Cauchy did a great deal to make other mathematicians aware the danger of a divergent series. He certainly popularized the ratio test for convergence of a series. Laplace immediately rushed home to recheck his series in Mécanique Céleste after hearing Cauchy give a paper.
Reading Boyer2 however, it is rather difficult to say just who deserves the credit for discovery of what:
“… the familiar ratio test … more frequently is known as Cauchy’s test, despite the fact that it had given by Waring as early as 1776.” [p. 458]
"Gauss as early as 1812, for example, used the ratio test to show [convergence of a hypergeometric series]… This test seems to have been first used long before, in England, by Edward Waring although it generally bears the name d’Alembert or, more occasionally, that of Cauchy.
"[Another convergence test] has come to be known as Cauchy’s criterion, but it was known earlier by Bolzano (and possibly still earlier by Euler). [p. 517]"
Cauchy first used the matrix of partial derivatives that has come to be known as the Jacobian.
Various other misnomers
“Euler’s formula” is the name given to the beautiful relation v + f = e + 2, where v is the number of vertices in a polyhedra, f is the number of faces, and e is the number of edges. As prolific as Euler was, he did not invent the formula. René Descartes first discovered this relationship in 1619. Euler was not born until 1707.
Cartesian coordinates were named after Rene Descartes, but were first used by Nicole Oresme, who lived in the 14th century. Oresme did have a crater on the moon named after him, though. Any guess as to whether he discovered the crater?
Pell's equation, y = ax2 + 1, where a is a non-square integer, was first studied by Brahmagupta  and Bhaskara. Its complete theory was worked out by Lagrange, not Pell.

I am still looking for an un-named discovery to put my name on. Any suggestions?
1)      Rouse Ball, W. W., A Sort Account of the History of Mathematics, 1960, Dover Publications, (unabridged and unaltered republication of the 1908 4th edition)
2)      Boyer, Carl B., A History of Mathematics, 2nd Edition, 1991, John Wiley and Sons

Saturday, June 23, 2012

Our Arabic number system?

Our number system has been inappropriately called the “Arabic number system”. The correct name is the “Hindu number system”, since it actually originated in India.

Positional number systems

In a positional number system like we use today, a single symbol (like “5”) can be used to mean five, or fifty, or five thousand, depending on its position. It is easy for us to take this number system for granted, unless we compare it to the Roman system of numbers.
Our positional system of arithmetic was completely unheard of in thirteenth century Europe. The Roman system, where V meant “five” and L meant “fifty”, was use for notation by European merchants. However, since even adding two numbers together was very cumbersome with Roman numerals, they used the abacus when they wanted to do arithmetic.
Because of this positional notation, there are relatively simple longhand methods for addition, subtraction, multiplication, and division.

Hindu beginnings

The Babylonians had probably the first positional number system around the 19th century BC. Their number system was, however, a sexagesimal system, being based on 60. Remnants of the Babylonian system remain today in the fact that there are 60 seconds in a minute and 60 minutes in an hour.
The Mayan culture independently developed a positional number system that was based on 20 and 18. This system was in use perhaps as early as 400 BC.
A positional number system was also developed in India as early as 594 AD, where it was used to record a date. Numerous other dates were written in positional notation in the 700’s and 800’s. Historians have disputed all these documents. It is possible that they were forgeries actually written at a later date. The first appearance of Indian positional notation that historians have accepted was in 876 AD.


Of the three positional number systems, the Indian system has survived largely because of a good PR man. The Arab al-Khwarizmi wrote a book entitled “Concerning the Hindu Art of Reckoning” sometime in the early 800’s. This book largely was derived upon the earlier work of Brahmagupta.
The account of the Hindu numbers and their use in calculation by al-Khwarizmi was so clear that he was mistakenly credited for developing the system. As a result, we even see historians of mathematics refer to our number system as the Arabic system. [Rouse Ball, p. 166 and p. 167]
Al-Kwarizmi was a Persian from around 800 AD. He belonged to a school in Baghdad that was responsible for the preservation of much of the earlier science. Greece was the center of science up until about 300 AD. India took over, and then the torch was passed to the Persians, thanks to a very forward looking ruler whose name escapes me right now. (This is all from memory, sorry.) We have this king to thank for the knowledge of Euclid's Elements, for the works of Ptolemy and Aristotle, and for the math contributions of the Hindus.
This Persian school was also the birthplace of Algebra. Al-Kwarizmi wrote the first book on algebra. A word or two from the title became our word "algebra".


Fibonacci wrote the book Liber Abaci in 1202, in which he introduced what he called the Arab system of numbers into Italy. Adelard of Bath and John of Seville also introduced the Hindu system to Europe [Boyer, p. 252]
Enter Leonardo di Pisa, also known as Fibonacci. Today, he is known for inventing the Fibonacci series which has to do with counting generations of rabbits. He had a much more significant contribution to European math and commerce, though. He was studying Arabic texts, and came upon a book by Al-Kwarizmi entitled "On the calculation with Hindu numerals".
Getting back to the story, Leonardo was quite taken with the idea of doing calculating with this notation. He understood that, with the Hindu notation, businessmen would no longer need an abacus to multiply two numbers together. They could do long multiplication like we do today. He translated this book by Al-kwarizmi into Latin to make it available to the businessmen of the day.
Leonardo was careful to credit the book to Al-Kwarizmi. He even included the name of the original author in his Latin title: Algoritmi de numero Indorum. Unfortunately for the history of math, the populace did not understand that "Algorithmi" referred to a person. It was assumed that this word was the name for the method of calculation that was introduced to Europe in this book. As an aside, this misconception is the origin of the word algorithm.
Somewhere along the line, Europe also forgot that the original number system came from India, and not Persia. Today, we incorrectly refer to our numbers as Arabic. It is somewhat more correct to refer to our number system as Hindu-Arabic, as they are sometimes called. The accurate description, though, is "Hindu".
The effect of the Hindu number system on European mathematics cannot be over. In the words of Laplace:
The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

Friday, June 15, 2012

Flies walk on the ceiling

A tricky feat for sticky feet?

Flies walk on the ceiling. I can’t. But don’t think it’s for lack of trying!
People naturally assume that flies have this super-human ability – that their feet have some extra-special sticky stuff that enables them to perform this incredible feat. They don’t.
Consider the relative sizes of the fly and the human. I will take a rough estimate of the fly’s length to be 1 cm (musca domesticas[1]), and a human’s length to be 2 m. Humans are roughly 200 times as long as flies.
How about the weight? All else being equal, weight is proportional to volume, and volume is proportional to the cube of the size (length times width times height). Based on that, if a fly weighs one flyweight, then a person weighs 200X200X200 =  eight million flyweights.
Now consider the relationship between the sticky pads a fly uses and those which I might attempt to use. I might requisition pads which are 20 cm in diameter in order to walk on the ceiling. A similarly proportioned fly would have pads about 1 mm in diameter.
Here it starts to get interesting. The sticky force between the sticky pad and the ceiling is proportional to the amount of sticky stuff in contact with the ceiling, which is to say, proportional to the area of the sticky pads. All else being equal, my sticky pads would hold a weight which is 200X200 = 40 thousand times the weight of the fly’s sticky pads.
If the glue on the fly’s feet is capable of tacking ten flyweights to the ceiling (we’ll design in a little safety margin, since flies generally work without a net) then the same glue covering my sticky pads should be capable of holding 400,000 flyweights. But, I weigh in at 8 Mega-flyweights.
When I step out onto the ceiling after carefully collecting the glue from the feet of 40,000 flies, the glue doesn’t hold. I come crashing to the floor, breaking my neck. (Well actually, I only broke my neck in four of the ten trials.) It is all a question of size, not of how good the glue is.
Termination by terminal velocity
This brings me to the second point. If, for some reason, a daredevil fly were to let go of the ceiling and free-fall to the floor, would it get hurt? To determine this, I have carefully performed pteraectomies (wing removal surgery) on flies. (Animal rights activists, remember: This is all in
the name of science. It was performed under anesthesia. I had about eight beers before I got started.) After removing the wings, I dropped the flies from a height of eight feet onto a concrete floor. Those flies which the cat did not eat survived, and actually appeared to enjoy the experiment. (I am thinking of opening a bungee jump for flies. Anyone interested in investing?)
How can this be explained? Am I a wimp compared to these Rambo-esque moscids? Well, probably. But there is another explanation which is more soothing to my ego. Again, it has to do with surface area and volume.
The wind resistance one experiences in free fall is roughly proportional to the surface area which is presented to the air. This is why, if you should happen to inadvertently debark from a plane somewhere over Kansas City without a parachute, you should remember to fall lying parallel to the ground rather than toes pointed earthward. Remembering this at the right time could add milliseconds onto your life!
The same effect of size applies here. The force of the wind resistance on the fly compared to its weight is significantly larger than it is for me. As a result, the terminal velocity[2] of a fly is much less than my (really) terminal velocity.
Even if the fly were to hit the cement at the same speed as myself, much less damage would be done to the fly. This has to do with some really technical stuff like the mass to structural strength ratio, which is too complex to get into here. (That’s just my way of saying that I really don’t
understand it.)
Summarizing, flies can walk on the ceiling and I can’t. If they fall from the ceiling, they don’t get hurt, but I do. Flies are certainly coming out ahead in this comparison, but read on...
Splish, splash
Since its getting into summer, I took a bath this morning. As expected, I climbed out of the tub with no particular difficulty. But as I stepped out, I happened to glance down into the water, and discovered that I had been sharing my morning ablutions with a fly. (Needless to say, the fly was quite disgusted when he realized that he was in the same water as a human being!)
As I toweled myself dry, I happened to notice that he was actually a bit more than disgusted. He was in a state of distress. The fly was hopelessly trapped in the water, a hapless victim of surface tension.
I had a sudden rush of power as I realized that, for me, the surface tension of water meant having to dry myself with a towel. For him, surface tension is a matter of life and death.
Newton, the fly
What if Isaac Newton had been born as a fly? I’m sure that the reader is aware that there has been a movement promulgating the notion that Isaac Newton was indeed a fly, but the movement has yet to gain much of a foothold in conservative states like Wisconsin.
If Newton was a fly, I doubt that gravity would have attracted his attention. We have seen that gravity just does not have the same gravity for a fly as for a human. To take a wild guess, I would say that Flysaac Newton would probably have discovered the laws of surface tension instead.
The point is, physics is an entirely different animal for entirely different animals. When we make big changes in scale, the rules of physics change entirely. The intuition which we have keenly honed for one set of circumstances may no longer apply.
A pressing discussion
I had occasion at work one day to be discussing how wide ink spreads as it passes through the nip point in rollers on a printing press. The first engineer stated that ink is pretty thick, so he would not expect it to spread very much. The second engineer argued that the viscosity of ink goes down as it is worked between the rollers. A third engineer pointed out that ink is not between the rollers for long enough to spread very far at all. The fourth engineer countered that the pressure between the rollers is quite large, so the spread should be large.
I asked the ink spread question of one pressman whose intuition said that ink spreads “gobs and bunches” in the rollers. He related to me an incident where a sparrow thought that it should be on the cover of Newsweek. I mean literally on the cover of Newsweek.
This poor sparrow flew into the rollers of a press, and very quickly experienced the transformation from a three dimensional being to a two dimensional object. In the transformation, there was a significant increase in two of the dimensions, to make up for the sharp decrease in the thickness dimension. Hence, sparrows spread between the rollers of a press, and the implication is that ink does as well.
Contrary to the celebrated sparrow experiment, we have performed direct experiments of the spread of ink in between rollers and concluded that the amount of spread[3] is very small, less than one centimeter. So much for intuition.The moral of this story is that intuition is only valid under the circumstances that the intuition was formed. When venturing into new ground, it is best to test assumptions.

[1] My father taught me to use fancy scientific names when I don’t know what I am
talking about.
[2] Terminal velocity is the top speed that a body falls at. Somebody had a pretty macabre sense of humor to call it terminal velocity!
[3] When the vibrator rollers, which move back and forth, are disabled. The vibrator rollers
accomplish ink spread by physically picking the ink up and moving it laterally.