Maria Gaetana Agnesi was one of the most important figures
in the 18^{th} century.
Around this time in Italy, where the Renaissance had its origin, women
were making their mark in the academic world. Women of intellect were admired by men and were never mistreated
for being educated or intellectual.
This enabled women to do things that they would've never thought
possible before, such as working with mathematics, medicine, literature, and participating
in the arts. This attitude also
opened the door for Maria. Agnesi
and changed the way we look at mathematics.

Maria Agnesi was born in Milan, Italy, on
May 16, 1718. She was the eldest
of 21 children delivered by three successive wives of Pietro Agnesi, son of a
wealthy silk merchant. Pietro was
a professor of mathematics and provided his daughter with an excellent private education. "She was recognized as a child of
prodigy very early; spoke French by the age of five; and had mastered Latin,
Greek, Hebrew, and several modern languages by the age of nine." [Osen,
40] This earned her the appellation
"Oracle of Seven Languages".

In 1738, at the age of twenty, she wrote, Proposition
Philosophicae, a collection of essays on philosophy and natural science based
on the discussions of the intellectuals who gathered at her father's home and
the inspiration of Isaac Newton.
In many of these essays, she talked about how strongly she believed that
women should be educated. Shortly
following that she began working on her most important work, Analytical Institutions. When this book was published, a great
deal of excitement struck the academic world because it was the first and most
complete works on finite and infinitesimal analysis. The first section of the book was dedicated to the analysis
of finite quantities, and elementary problems of maxima, minima, tangents, and inflection
points. The second section deals
with the analysis of infinitely small quantities and the third about integral
calculus. Finally, the last section
talks about the inverse method of tangents and differential equations. This was originally designed as a
textbook for her younger brothers.

After the success of her book, on the
nomination of Pope Benedict XIV, Maria was elected into the Bologna Academy of
Science. Her name was added to the
faculty roll and the university sent her diploma dated October 5, 1750. However, there is a debate on whether or
not Maria excepted this position. It
is known that when Agnesi's father became ill in 1748, she lectured in his place,
but there is controversy if she accepted the

position.

Her behavior implies that she was not
dedicated to mathematics, which more than likely explains why she gave it up
after her father's death. It seems
as if her father was her inspiration for her interests in mathematics. Many historians debate on whether or not
she was truly dedicated. What
we've implied is that the book that she wrote was meant to educate her brothers,
not for the public's use. The fact
that she gave it up when her father passed on, leaves us to believe that she
was not willing to spend the rest of her life doing mathematics. Instead, she devoted the rest of her
life to religious studies and charitable works. She died before her 81st birthday in May 1799, but her
contribution to math made it possible to clearly integrate the knowledge of
calculus.

The solution that she is remembered for is
called "the witch of Agnesi". This name was not given to this curve because she was a witch. The word witch originated from the word
"versiera", which meant to turn in Latin. It so happened that a similar sounding Italian word
"avversiera" appears in the English translation by Reverend John
Colson. This translation error was
made when Analytical Institutions was being converted from Italian to English. Historians believe that Colson simply
looked into an Italian-English dictionary and found "versiera"
translated as "witch". Even
though this slip-up seems like a bad thing, it is what she is most popular for
because the title of the curve stands out.

Now
that we know what the Witch of Agnesi is, let’s try to explain how we graph

it, and what it means.
Throughout the sources on the versiera, there are several

ways to generate the graph and the equations.

The
algebraic expression that generates the curve is

xy^2=a^2(a-x),

where
a is the diameter of the circle, or the height of the curve.

We then need to take the inverse of the expression
because Maria Agnesi used the x-axis as the vertical axis and the y-axis as the
horizontal.

yx^2=a^2(a-y)

For the example shown, a=1,2,3