Wednesday, August 1, 2012

A witch by any other name


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 cross-pollination.
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.

If the circle has radius a, the curve has the formula 

History of the witch
It has been variously reported that Fermat (1601 – 1665) had described this curve well before Agnesi [Maor, Osen, MacTutor on Agnesi, Stigler 1974]. Cajori, however, elaborates that Fermat had worked on was a related curve:

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 Agnésienne [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 pseudo-versiera, and hence establishes another name for the function.
Cauchy distribution
The second appearance of this function is in the field of statistics, where it took on the name “Cauchy distribution”. The Cauchy distribution is a probability density function, similar to the Gaussian, or normal distribution. The formula for it is

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 ill-behaved 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, non-zero 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 Breit-Wigner formula.
A physicist is likely to parameterize the Lorentzian as

Here, the curve is centered at x0 and the “full width at half max” is w. The distance between the half-way 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) Agnesíenne
    6) Pseudo-versiera
    7) Cauchy distribution
    8) Lorentzian
    9) Breit-Wigner 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, Cambridge, 1901
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, Paris, 1890
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. 375-380
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,

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