This function appeared as an example in a previous post on regression. Aside from that interesting bit of trivia, this function has an
interesting history under a variety of aliases and misnomers. The function
has surfaced in several different fields, apparently with little
crosspollination.

Witch of
Agnesi

The
function first occurred as a solution to a geometric puzzle.

The curve called
the Witch of Agnesi is defined as a locus of points based on a circle. The
figure below shows how this locus of point is derived. A lines is drawn from
a point A on the circle so that it intersects first the circle (at B), and
then the tangent line (at C). This line is tangent to the circle directly
opposite point A. A line perpendicular to the tangent line is drawn through C
(line CD), and a line parallel to the tangent line is drawn through B (line
BD). The intersection of these two lines is one point on the witch of Agnesi.
The witch is made up of all such points D as created by all lines that go
through A.


History of the witch

Note
that this function is shaped quite differently from the witch because of the
minus sign in the denominator. Fermat’s function has two poles. Thus, there
is some question about whether Fermat had actually worked on the witch.

Stigler
[1974, and 1999] also reports that Newton had worked on this curve some time
before 1718, but that this work was not published until 1779 (posthumously).
Stigler does not identify the work, but it could have been “Geometrica
Analytica”.

Stigler
also added Leibniz and Huygens to the list of early investigators.

A
mathematics professor at the University of Pisa by the name of Luigi Guido
Grandi offered a construction of the curve in 1703 and 1710 [Cajori, MacTutor
on Grandi and Agnesi]. Grandi referred to the curve as the versiera, from the
Latin verb for “turn”.

Maria
Agnesi is indirectly responsible for the popularization of the name “witch of
Agnesi”. She wrote a very popular calculus textbook in 1748. The two volume
set was a unified treatment of algebra and the fledgling subject of calculus.
In this book, she referred to the curve as the versiera, as had Grandi.

The
curve became known as a “witch” due to a mistranslation of Agnesi’s textbook.
The British mathematician John Colson translated Agnesi’s work into English
sometime before 1760, but this was not published until 1801 [MacTutor on Agnesi]. He learned Italian specifically
for this task, so it is understandable that he made some translation errors. He mistook versiera for avversiera, which means “devil woman”,
or “witch”. Somehow this mistranslation stuck, and the curve became known as
the witch of Agnesi.

Thus,
we see that the name “witch of Agnesi” is both a mistranslation and a misnomer.
Not only is it not a witch, but it was not invented by Agnesi. It might be
more appropriate to refer to it as the “curve of Grandi”.

There
are two additional names given to this curve: Cubique d’Agnesis and
AgnÃ©sienne [Smith, and Wolfram MathWorld].

In
a text by Longchamps and subsequently by Basset, there is a description of a
different, but similar, derivation of the witch of Agnesi. In this case, the
definition is such that the curve lay all below the top of the circle. The
subsequent equation
is of the same shape as the versiera as defined by Agnesi. Both Longchamps and Bassett both refer to this as the witch of Agnesi. Loria disagrees with the name and states that this curve is not a versiera, but coins the term pseudoversiera, and hence establishes another name for the function. 
Cauchy
distribution

The
Cauchy distribution most commonly makes its appearance as an example of a
pathological distribution. Despite its gross similarity to the normal curve (it has
wider tails); it is as illbehaved as Paris Hilton.

Technically
speaking, it does not have a mean, since the integral used to compute the
mean from a distribution is undefined. This, perhaps, is a technicality,
since the distribution is symmetric about x = 0, so the mean could be defined
as being 0.

More
troublesome is the fact that the standard deviation of the distribution is
infinite. Since the standard deviation of a distribution is a measure of its
width, the Cauchy distribution paradoxically has an infinite width.

This
pathological behavior of the Cauchy distribution makes it a wonderful example
of when the central limit theorem does not apply. The central limit theorem
states that the distribution of the sum of random numbers tends to look more
and more like a Gaussian as more and more random numbers are added together.
This applies for random numbers drawn from any distribution, provided that
distribution has a finite, nonzero standard deviation.

On
the other hand, if two samples from a Cauchy distribution are added, the
distribution of the sum is another Cauchy distribution. It follows that the
sum of an arbitrary number of Cauchy distributed variables also follows a
Cauchy distribution.

Interestingly
enough, calling this curve the Cauchy distribution is yet another misnomer, as I have reported earlier.
According to Stigler, Poisson had published a paper in 1824 where he
described how this was an example of a distribution where the central limit
theorem did not work. Cauchy did not work with the distribution until 1853.
It would then be more accurate to refer to this as the Poisson distribution,
but of course that name has already been taken.

Lorentzian

The
third place where the witch raised her pointed little hat is in the field of
physics. Maor makes the following comment about the witch:

It is
somewhat of a mystery why this particular curve, which rarely shows up in
applications, has interested mathematicians for so long.

He
does comment in a footnote that the witch is identical to the Cauchy
distribution.

Maor’s
claim about the witch seems to also hold true for the Cauchy distribution. A
book by Trivedi is a practical book on statistics. A quick look at the index
under the heading “distribution”, reveals 24 different distributions, but
does not include the Cauchy distribution. It would seem that as a
distribution, its only claim to fame is as an example of bad behavior.

But,
I disagree with Maor’s comment that the witch rarely shows up in
applications. People who deal with spectroscopy are familiar with this curve
as the Lorentzian.

The
IR spectrum of a molecule is used by chemists as a fingerprint to identify
and quantify a compound. Each of the bonds in a molecule has a specific
resonance, generally in the infrared. Under ideal conditions, these
resonances show up as narrow spikes. As the molecules of a rarefied gas come
closer together (higher pressure), collisions between the molecules will
compress the molecular bonds by varying amounts. In this way, the spectral
spikes are broadened into what is called the Lorentzian.

This
spectral shape is named after the Nobel prize winning Dutch physicist Hendrik
Lorentz. The formula also shows up in scattering theory, where it has become
known as the BreitWigner formula.

A
physicist is likely to parameterize the Lorentzian as


Here,
the curve is centered at x_{0}
and the “full width at half max” is w.
The distance between the halfway points on either side of a peak is a
convenient measure of the width. It is all the more convenient, since the
width of a Lorentzian cannot be measured by the standard deviation. Also,
this measure can be readily estimated from a plot.

Summary

Here
is a list of the names given the curve:

1) Versoria (Latin)

2) Versiera (Italian)

3) Witch of Agnesi

4) Cubique d’Agnesi

5) AgnesÃenne

6) Pseudoversiera

7) Cauchy distribution

8) Lorentzian

9) BreitWigner formula

It
is interesting that I have not found a single reference that mentions all
three of the main names (Witch of Agnesi, Cauchy distribution, and
Lorentzian). I have only found references that include any two of the three.

Bibliography

Basset,
Alfred Barnard, An Elementary Treatise
on Cubic and Quartic Curves,
Boyer,
Carl, A History of Mathematics,
second edition, John Wiley, 1991
de
Longchamps, M. G., Essai sur la
geometrie de la regle et de l'equerre,
Loria,
Gino, Spezielle algebraische und
transscendente ebene kurven, B. G. Teubner, 1902
Maor,
Eli, Trigonometric Delights,
Princeton University Press, 1998, pps 108 – 111
Miller,
Jeff, Earliest Known Uses of Some of
the Words of Mathematics, http://members.aol.com/jeff570/w.html
Osen,
Lynn M., Women in Mathematics, MIT
Press, 1974
Singh,
Simon, Fermat’s Enigma, Anchor
Books, 1998
Smith,
History of Mathematics, Vol II,
Ginn and Company, 1953, p. 331
Stigler,
Stephen M. Studies in the History of
Probability and Statistics. XXXIII Cauchy and the Witch of Agnesi: An
Historical Note on the Cauchy Distribution, Biometrika, Vol. 61, No. 2
(Aug., 1974), pp. 375380
Stigler,
Stephen M., Statistics on the Table,
Harvard Press, 1999
Trivedi,
Kishor Shridharbhai, Probability &
Statistics with Reliability, Queueing and Computer Science Applications,
1982, Prentice Hall
Wolfram
MathWorld, Witch of Agnesi,

Wednesday, August 1, 2012
A witch by any other name
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Very good summaryhits all the high points. Another appearance of the Lorentzian shape: Consider a horizontal line of light strung (in the y direction) a height h above a horizontal matte reflecting plane (x,y). The intensity of reflected light as a function of x is proportional to 1/(x^2 + h^2).
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