The Witch of (Maria) Agnesi

 Consider a circle that fits perfectly between two parallel lines. One of the easiest pairs of parallel lines to picture is a top and a bottom. In this picture, the point O is the point tangent to the bottom and the point M is the point tangent to the top.

Take any point on the circle and call it A. The line drawn from O to A is called a secant and it will cross the line at the top at a point we call N. We can create a right triangle by making a line parallel to both top and bottom. The point P is the corner of the right triangle that has the 90° angle. For every new A, there is a new N and a new P. We are interested in the curve created by all the points P.
  

This may seem like a very strange and roundabout way to make a curve in the plane. You wouldn't be wrong to think so. In the early days of the xy-plane, mathematicians worked on these kinds of very strange curves. The first work on this was done by Pierre De Fermat, an amateur mathematician working with Rene Decartes at the beginning of the discovery of the Cartesian plane, named for Descartes ("of the cards" in French.) The formula for the curve is given by the formula in x, y and a, where a is the radius of the circle.

The simplest version of the equation is when the radius is ½, so the diameter is 1.

The curves are sometimes called bell-shaped curves, but this is not the formula for the most famous of the bell-shaped curves, the most studied version of that being called the normal curve. This curve is called the Witch of Maria Agnesi.

The name comes about by a mistranslation of Italian to English, a mistranslation that might have been intentional but has most certainly lasted until today.

Fermat's first work on the curve is done is 1630, but in 1703, the Italian Guido Grandi also studies the shape and names it in Latin, the versoria, which is also the name for one of the ropes that holds a sail in place, usually called a sheet in English.

Move forward again to 1748 and a very odd occurrence at the time, an important mathematical paper written by a woman, the Italian mathematician Maria Agnesi. She writes her paper in Italian and calls the curve "versiera", the Italian translation for Grandi's Latin word "versoria". Here comes the confusion, possibly done on purpose.

The Italian for adversary is "aversiera" is sometimes shortened to "versiera". The adversary of God is the Devil, but because this is the feminine form, it can be translated to "witch". The translation by Cambridge professor John Colson, a contemporary of Maria Agnesi, turns the name for a rope on a boat to witch, a play on words that likely was done at the expense of the very rare female mathematician.

So here is to Maria Agnesi, not an originator of an idea but, like so many of us in mathematics, a worker in the field, doing her best to preserve knowledge for future generations. Her name lasts to this day in a somewhat mocking form, but at least we remember her, one of the few women in all of Europe to publish a mathematical work that has lasted from an era when women were not allowed to get a degree from any university on the continent.

Pascal's Triangle long before Pascal.

Blaise Pascal did not call the array of numbers he studied "Pascal's Triangle". In math, it's considered poor form to name something after yourself.

Pascal's Treatise on the Arithmetic Triangle was published posthumously. In it, he gathered together all the facts he knew about the patterns he discovered himself or had read about in other books.  It became the "go to" text for information about the array and other mathematicians started calling it "the triangle of M. Pascal" so much so that it is now the way nearly everyone in the world refers to it.

Nearly everyone. About 100 years before Pascal, the great Italian algebraist Niccolo Fontana, known by his nickname Tartaglia - which means "the stammerer" - did a lot of work with the number pattern and in Italian the array is known as Tartaglia's Triangle.

Several centuries earlier, the Chinese were discovering things about the array, and in Chinese it is known as Yang Hui's Triangle.

But the Chinese from 700 years ago are not the first people to study the numbers and leave a paper trail behind that future generations could find. There are two completely different problems from before the birth of Christ that originate in India whose answers come from the numbers in the array we call Pascal's Triangle.

Let's say we have a spice rack with six flavors: salt, pepper, garlic, nutmeg, curry and basil.  How many different combination of three spices are there? (Note: in this problem, we are not saying how much of any one spice we are using, only if it is used. Two parts salt and one part garlic would taste different from one part salt and two parts garlic, but in this problem we would say that both are salt/garlic combinations.)

If we abbreviate the spices to S, P, G, N, C and B, here are the 20 different groups of three

Salt included

SPG  SPN SPC  SPB
SGN SGC SGB
SNC SNB
SCB

Salt excluded, pepper included
PGN PGC PGB
PNC PNB
PCB

Salt and pepper excluded, ginger included
GNC GNB
GCB

No salt, pepper or ginger
NCB

The other ancient problem from India that uses the binomial coefficients deals with music and rhythm. In the musical notation developed in Europe that is used almost everywhere today, if we say a song is in a rhythm of six beats, all those beats have the same duration. In India, beats can either be short or long. For instance, let's say we had a song that has double hand claps on the second and fourth beat of every measure, so the pattern might the counted out

bump clap-clap bump clap-clap...

In Western music, we would say this is a four beat pattern. In Indian music, they would say it is a six beat pattern.

long short short long short short.

Okay so how many different six beat patterns have three long beats and three short beats? Again the answer is 20.

First beat long
LLLSSS LLSLSS LLSSLS LLSSSL
LSLLSS LSLSLS LSLSSL LSSLLS
LSSLSL LSSSLL
First beat short, second beat long
SLLLSS SLLSLS SLLSSL SLSLLS
SLSLSL SLSSLL
First two beats short, third beat long
SSLLLS SSLLSL
SSLSLL
First three beats short
SSSLLL

At first glance, these problems don't seem to be connected, but in fact they are. If we line up the spices alphabetically in English, we would get.

Basil Curry Garlic Nutmeg Pepper Salt

Take any six beat pattern.

LLSSLS

Think of L as being yes and S as being no

YesYesNoNoYesNo.

Make a spice combination where we only use the spices that correspond to the Yes positions.

Basil Yes, Curry Yes, Garlic No, Nutmeg No, Pepper Yes, Salt No.

In this way, the beat pattern LLSSLS corresponds to the recipe that uses basil, curry and pepper. If two beat patterns are different, they will correspond to different recipes. In math, this kind of matching is called a one to one correspondence, and it is one way to prove that one set of objects has the same number of things as another set.

Tomorrow, we will look at actual geometric triangles, notably triangles with one right angle, known simply enough as right triangles.