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 ShihChieh 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 ShihChieh 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 ShihChieh. 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 Ball^{1} 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 ibnMusa
alKwarizmi, 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.

Cauchy

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
Boyer^{2} 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 14^{th} 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 = ax^{2} + 1, where a
is a nonsquare integer, was first studied by Brahmagupta and Bhaskara.
Its complete theory was worked out by Lagrange, not Pell.

I
am still looking for an unnamed 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 4^{th}
edition)
2)
Boyer, Carl B., A History of Mathematics,
2^{nd} Edition, 1991, John Wiley and Sons

Wednesday, June 27, 2012
Mathematical misnomers
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.

Arabic
Of the three positional number systems, the Indian
system has survived largely because
of a good PR man. The Arab alKhwarizmi 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 alKhwarizmi 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]
AlKwarizmi 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.
AlKwarizmi wrote the first book on algebra. A
word or two from the title became our word "algebra".

Europe
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 AlKwarizmi
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 Alkwarizmi into Latin
to make it available to the businessmen of the day.
Leonardo was careful to credit the book to AlKwarizmi. 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 HinduArabic, 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?
People naturally assume that flies have this superhuman ability – that their feet have some extraspecial 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 Megaflyweights.
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
velocityWhen 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.
This brings me to the second point. If, for some reason, a daredevil fly were to let go of the ceiling and freefall 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?)
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 Ramboesque 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.)
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.
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.
accomplish ink spread by physically picking the ink up and moving it laterally.
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