Last day of class

 Prof Kim listed the following suggestions for a biography:

Women in math

 

1. advantages & disadvantages
2. obstacles to editions
3. parents, teachers & gender stereotypes
4. professional obstacles
5. burial by history
6. picture today
7. promise now
8. future – how can we challenge & take down barriers

 

 Today we learned about three new women mathematicians: Sonia Kovalevskaya, Grace Chisholm Young and Emmy Noether, who is Prof. Kim’s idol.  All three were connected to Göttingen University in Germany. Göttingen University was a concentration of mathematical activity in Europe before the rise of the Third Reich in WW2. With the rise of Third Reich, Jewish and other exiles fled. While Mary Somerville may have popularized math, all of these women were creative & contributed to its progress. 

 

Grace Chisholm Young (1868-1944)
Grace Chisholm was born in England and went to Cambridge University. At age 13 she already displayed a skill in math; by 17 she passed the Cambridge entrance exams. She wanted to study medicine at Cambridge but women were not allowed. Instead, she enrolled in Girton College of Education at Cambridge where her future husband was a tutor. After receiving her degree from Girton, Grace was unable to attend graduate school in England due to her gender. So she moved to Germany to attend Gottigen University, known as a “Mecca for Mathematicians.” Grace was the 1^st woman to get a PhD from there. When she returned to England, she married her old tutor William Young, also a mathematician, and they eventually had six children. She was committed to her family & work and was a tremendous inspiration to her husband’s mathematical career. Before they were married, William didn’t have any groundbreaking work but after hitching himself to Grace he suddenly started having some brilliant work. They collaborated and published some books together, although they may have been mostly Grace’s work.  
Grace worked on computative algebra. She was awarded a prize for her paper on derivates of real functions. She did work on set theory and the history of the Pythagorean theorem. An interesting aside, Prof. Kim knows Young’s granddaughter, Grace Young! That’s only 3 degrees of separation!

 

 

Sonia (Sofia) Kovalevskaya (1850-1891)

According to Prof. Kim, Sonia’s work opened up the field. (She also let the most colorful like of the three.) Born in Moscow, her father, a military officer, was a strict authoritarian. Math ran in the family as her grandfather and great-grandfather were mathematicians. When she was 15, her family was given a textbook, and she became enamored of calculus. The walls of her bedroom were covered by integral & differential calculus. Since Russia did not allow women to travel alone, she pursued a marriage of convenience with Vladimir O. Kovalevsky. After leaving Russia, she went to Heidelberg in Germany with her older sister Anyuta, but found out she could not attend class; she could only attend class unofficially. After 2 years, she left & went to study with the father of mathematical analysis, Carl Weierftraff. By 1874 she produced three original works including one on partial differential equations. Karl explained that any of the three papers should have earned her a doctorate of philosophy in math but the University of Berlin did not allow her to. She edited the journal Acta Mathematica and in 1889, she was awarded tenureship & professorship at Stockholm University. Sonia was the first woman in Europe to earn a doctorate in mathematics. She is quoted as having said:

“Many who have had an opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination.”

MatematikNet.com

 

The Greatest of All Woman Mathematicians:
Emmy Noether (1882-1935)
Amalie, or Emmy as she was known, was not allowed to attend school as a child, although she eventually earned her PhD in 1907.  Her father, Max, was a famous mathematician. She had a thesis Sweden 1^st country offer a job but not allowed to teach. She used to help her father do research, and would sometimes teach for him, but didn’t get paid.  David Hilbert tried to get her a job at Göttingen University.  Her thesis was originally not accepted in 1915 because she was a woman, and when she reapplied in 1919, she was defended by Hilbert who said “ I don’t see why the sex of the candidate is relevant – this is after all an academic institution not a bath house.” She was still not able to officially teach, so while she taught many classes, her name was not advertised on the schedule and she didn’t get paid.

 

Her work in abstract algebra developed the concept of the Noetherian ring. She was known for taking European mathematicians under her wing. Being Jewish, she had to flee persecution, so she came to the US. She taught as Bryn Mar and in 1935, died unexpectedly after surgery. Einstein wrote her obituary in NY times.

  “Asked for a testimony to the effect that Emmy Noether was a great woman mathematician, he said: I can testify that she is a great mathematician, but that she is a woman, I cannot swear.”  E. Landau

 

Prof. Kim sugggested we check out the Klein Four Group. They sing about math. Fun stuff.

In Sympathy

My sympathy goes out to Professor Gottlieb who just lost her father.  We’ve learned in our class how Maria and Hypatia’s fathers played an inspiring role in their following mathematical career paths. As students, we have been impacted by our professor’s father’s leadership in mathematics. His leadership in mathematical advances did not lessen his commitment to students.

For many of the women mathematician we have studied during this course contributed to the advancement of math by writing mathematical concepts in more publicly accessibly ways. As someone who entered this class with trepidation and who had accumulated an intimidated attitude towards math, our professor’s love of math and love of her father helped us to appreciate and learn when mathematical concepts in an encouraging environment. Thank you for your patience and commitment towards teaching math to those of us who questioned our own abilities to learn.

Dec 9th Deflections and Symmetries

Today we found out about the passing of our Professor’s father (see above) from Professor Kim.

We started the class by participating in a mathematical exercise (p.93 in Math Equals) the answer of which was the dates of Mary Somerville’s life 1780 to 1872. Mary also distinguished herself in the physical sciences. The terminology she came up with was used for 50 years. Mary Somerville wrote four popular books about science that popularized and educated people. Mary built on the work of Chladni, a German physicist (p 160 Math Equals). Our teacher explained about these diagrams noting how our text cover was based on these. These nodal patterns were created by sprinkling white sand on glass plates that were vibrated with a violin bow and represent the modes of vibration of the plates. The concept of Chladni’s vibrating plates is rooted in mathematics and spurred the study of waves and acoustics ever since.  Chladi’s law which explains this can be written as

where m is the number of diametric (linear) nodes and n is the number of radial (circular) nodes.

Chladni’s patterns are useful in the manufacturing of fine instruments (violins, chellos and guitars) as a means of testing their acoustic properties as the patterns are predictable. See  Chladni patterns for violin plates for some great videos showing the vibrations in action.  

Isometry

Our teacher explained the concepts of isometry (or symmetry) and deflection. There are 4 types of isometry (reflection, rotation, translation, glide-reflection) as seen in this illustration:
Notice that the figures stay they same shape, they are not distorted. It is said that the distance between the points remains the same. (image from: http://www.spsu.edu/math/tile/defs/symmetrytypes.htm)

Prof. Kim drew the different types of deflection and explained why there were 4 types of deflections with a square (horizontal, vertical, and 2 diagonals) while there were only 3 types of deflection in the case of an equilateral triangle. 

Mary Somerville also used the pendulum in her studies. She used the principles of the force of gravity acting on the pendulum to prove that the earth is a sphere, since if it were perfectly round the gravitational pull with be the same at all points, but in fact the gravitational pull at the poles is different.  Like Mary Agnesi, Mary Somerville also worked on curves, specifically cycloids which is the curved formed by tracing the path of a point on circle as it travled down a path.

Image: wikimedia-commons

Mary also distinguished herself in the physical sciences. One of her books was used as a textbook for 50 years. Her four successful books popularized science and educated people. There’s a college in Oxford named after Somerville. 

Trigonometry, Anyone?

Trigonometry studies triangles. It is useful in astronomy as well as geography and surveying. Early Greek astronomers began the study and it was improved by Indian mathematicians then Arabic mathematicians before heading over to Europe.

We talked about the following functions of angles:

Sin  – sine,   Cos – cosine,  Tan – tangent

which express the ratios of the sides of right triangles.

 SinØ = o/H  opposite/hypoteuse

CosØ=a/H   adjacent/hypoteuse

TanØ= o/a   opposite/adjacent

The teacher shared with us a mnemonic device to remember these 3 formulas:

Some old hag caught a hippie, tripping on acid.

 

Then we moved on to Ada Lovelace. Prof Kim explained how while Mary Somerville’s work mostly aimed to communicate mathematical concepts to wider audience, Lovelace contributed more to the development of mathematical concepts. Her major contribution was the difference algorithm.

We learned about the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8,…..which is continued by added the last number of the sequence to the number immediately preceding it. [0+1=1, 1+1+2, 2+1=3, 3+2=5, 5+3=8 and 8+5=13]. The equation is written as f(n)=f(n-1)+f(n-2) for n>1  It is thought that the Incas used the Fibonacci sequence in an early calculator. We then discussed an example.

Ada worked on recurrence relations, an equation which “defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms.” A specific type of recurrence relation is the difference equation, which was the impetus for George Babbage’s Difference Machine, the precursor to his Analytic Engine spawn the computer revolution, thanks to Ada’s notes.

 

Reproduction of Rabbits

A single pair of rabbits is born at the beginning of the year and rabbits become fertile in their 2nd month. Assuming that after the 2nd month, a pair of rabbits will produce 1 pair of male & female rabbits,  how many pairs will there be at the end of 1 year?  (I asked the teacher how many offspring and the she explained we assume each pair has 2, 1 male & female. I didn’t think to ask how long the gestation period was.) The other assumption is that no rabbits die in that year. Fn=number of rabbits at the end of n
How many rabbits will there be at the end onf one year?

The illustration below shows the first 4 months. At the end of month 4, there are 5 pairs. If we continued the sequence  we can see that at the end of month 5, there’s be 13 pairs, after 6 months, 21……[1,1,2,3,5,8,13,21,34,55,89,144,233]

we can determined the answer to be 233 pairs, or 466 rabbits!

 

 from http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrab.GIF

2 more Women in Math

Mary Fairfax Greig Somerville was born in Scotland in 1780. During that time, girls were not encouraged to learn more than social graces and only enough reading to be able to read the Bible.  Mary did attend a boarding school for a brief time, where she learned a little writing, French and English grammar. When her family took up an apartment in Edinburgh, Mary was able to attend a school for writing and even managed to learn some math (besides teaching herself Latin and Greek). Her first exposure to Algebra was in a fashion magazine.

Mary married at the age of 24 and moved to London. After 3 years her husband died and so returned to Scotland with her 2 children where she intensified her study of math much to the dismay of her relatives. She later married another cousin and gradually formed correspondences with other mathmaticians. She won some prizes (one being on the solutions of a Diophantine equation), published and became quite a celebrity in the 1830s. She also worked for the cause of women’s suffrage and was staunchly anti-slavery, refusing to use sugar.

Her earliest work revolved around magnetism and the sun. She wrote a paper that was published in the Royal Society’s Philosophical Transactions. She was greatly influenced by French math. Her work with physical science helped her to become the first woman elected to the Royal Astronomical Society in 1835.

Enchantress of Numbers

Included  in Mary’s intellectual circle was Lady Byron, called the “Princess of Parallelograms” by her poet husband Lord Byron, who abandoned her after the birth of his daughter, Augusta Ada Byron who herself would become a great mathmatician thanks to her mother’s encouragement and tutelage. Ada married William, Earl of Lovelace and had 3 children. She became a great friend and colleague of Charles Babbage, the father of computers, helping him with his work on the “Analytical Engine.”  Ada could not have her name associated with the project, and had to resort to using her initials A.A.L. on her contribution of explanatory notes which included explanations of looping and recursion and the operations punch cards needed to perform algebraic and trigonomic equations such as for calulating Bernoulli numbers.

Ada died of cancer at the age of 36. In 1980, a programming language was named Ada in honor of her contributions to the development of computer programming. Her life is described in the 1997 film “Conceiving Ada.” You can get it on Netflix!

http://www.towerofhanoi.net

 

We also talked about the Tower of Hanoi puzzle. It involves moving discs from one pole to another, never putting a larger disc on top of a smaller one. The object is to do it in as few moves as possible.  You can try it for yourself at Tower of Hanoi.net. Ths more discs you have, the harder it gets, obviously. The puzzle is an example of a recursive algorithm, which means it falls back on itself, or is defined in terms of itself.  This type of algorithm is frequently used in computer programming.  The game has been popular at least since 1883 when it was published by Mr. Édouard Lucas and you can still buy wooden versions of it on Amazon.com today (anyone need a last minute Christmas gift?)

Original box of the game http://www.cs.wm.edu/~pkstoc/page_1.html

 

 

 

 

 

 

What’s the Difference?

The precursor to Babbage’s Analytical Engine was the Difference Engine also worked on by Ada Lovelace. The Difference Machine could calcultate the difference between numbers, particularly in a series, by performing the operation of addition.  The symbol used to represent difference is Δ. The example used in our text illustrates this concept:

 The third row shows the difference between the numbers in the 2nd row; the fourth row shows the different between numbers in the third row (in this case, they are all the same, 2’s, so it is a constant difference).

n	0        1         2        3        4        5        6
f(n)	0        1         4        9        16      25      36
Δ f(n)	1        3        5        7         9        11
Δ Δ f(n)	2         2        2        2        2

Cryptography et al

Our class November 20th was about Modular Arithmetic, sometimes called “remainder” or “clock” arithmetic, as the “numbers “wrap around” after they reach a certain value, much like the hands of a clock passing twelve.  Apparently we already use modular arithmetic when we look at a clock and try to figure out what time to set it if we want a certain number of hours of sleep. If it is 10pm and we want 8 hours of sleep, what time do we set the clock for? Well, it’s 2 hours to midnight, then we reset and count 6 more (so 6am, right?) It is solving the equation (10+8) mod 12  [Thanks to http://www.math.cornell.edu/~mec/2003-2004/cryptography/diffiehellman/diffiehellman.html for the clear example.)

For example, 7= 1 mod 6. Similarly, 13=1 mod 6. Here is my conceptual understanding:

Recalling the Ceasar Cypher we looked at about awhile back which shifted the alphabet 3 places, this can be written as y = x + 3 (mod 26).   In fact, cryptography is based in modular math.

We discussed how one-way functions are somtimes referred to as “humpty dumpty functions” because they are difficult to reverse or “put back together again.”

I discovered that in computer security systems, a humpty dumpty function is called a “hash.” It is a one-way code that computers use to authenticate messages. It does not encode or decode the message, it just vouches for the message’s intergity, a digital signature.

We analyzed the formula x3 mod P=2 and learned that it is difficult to solve. It also does not have just one solution.

This led to the topic of Cryptography. Cryptography is used in sending and receiving information over the internet – using the mod formulas we were just introduced to. Specifically, the Diffie-Hellman key exchange is the protocol for encypting & decrypting communication between 2 parties using a public key.  The chart (below from Wikipedia) illustrate how this works.

1. Alice and Bob agree to use a prime number p=23 and base g=5.
2. Alice chooses a secret integer a=6, then sends Bob (g^a mod p)
   • 5⁶ mod 23 = 8.
3. Bob chooses a secret integer b=15, then sends Alice (g^b mod p)
   • 5¹⁵ mod 23 = 19.
4. Alice computes (g^b mod p)^a mod p
   • 19⁶ mod 23 = 2.
5. Bob computes (g^a mod p)^b mod p
   • 8¹⁵ mod 23 = 2.

Basically, the private keys that Alice and Bob use must be diffcult to solve, so that an eavesdropper, Eve, could not crack them. For this reason, very high prime numbers are used in the calculations.

Number Theory Redux

This class was a review of the prime and perfect number concepts we discussed last class. Are there an infinite number of prime numbers?  In fact, there are. Euclid first proved this:

Proof that there are an infinite number of primes:

Suppose that there are a finite number of primes,

 p1,p2, … ,pn

Consider the number,

N=p1*p2*…*pn+1

N has to be devisable by a prime, which we will call pk

Since pk is a prime it must be included in the  (p1*p2*…*pn) term.

Since pk is in the (p1*p2*…*pn) term, it must evenly divide it.

This means that pk must also evenly divide 1, which does not make 
 
Therefore our assumption that there are a finite number of primes is incorrect. (from http://www-zeus.desy.de/~brownson/data/infinite.htm)

We also reviewed Mersenne Primes. There are 2 theorums that relate Mersenne Primes and prefect numbers:
Theorem One:  k is an even perfect number if and only if it has the form 2n-1(2n-1) and 2n-1 is prime. You can see a proof that all even perfect numbers are a power of two times a Mersenne prime here.
Theorem Two:  If 2n-1 is prime, then so is n. You can see a proof here.
“So the search for Mersennes is also the search for even perfect numbers!”
(http://primes.utm.edu/mersenne/index.html#theorems) Actually, it is thought there there are no odd perfect numbers.

We reviewed Goldbach’s conjecture that we learned about last class, how every even integer greater than 4 is the sum of 2 odd primes:  6=3+3, 8=3+5, 10=5+5, 12=5+7…

Sophie Germain has also had a set of prime numbers named after her: a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 2 x 11 + 1=23, which is a prime number. Some Germain primes are: 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131. The Germain primes may be infinite, it is not known for sure.

Prime Time: November 13th

Today we learned about number theory, specifically prime numbers (including Mersenne prime) and perfect numbers. 

Prime numbers are numbers that can only be divded by themselves and 1, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,….  
You can listen to a song(!) that explains about prime numbers, their tendency to clump in pairs (twin primes are pairs of primes that differ by two such as 17 & 19 or 41 & 43) and the twin prime conjecture which posits that this behavior continues to infinity.

Perfect numbers are numbers where the sum of its factors are equal to itself: 6 (2×3, 6×1; 2+3+1=6), 28 (4×7, 2×14, 28×1) 4+7+2+14+1=28. It is thought that all perfect numbers are even.

Marin Mersenne was a French theologian, philosopher and mathematician who developed a list of numbers called Mersenne numbers which are positive integers that are one less than a power of 2 or  Mp=2^p -1. If the Mersenne number is prime, then it is a Mersenne prime. Examples: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647…….There may be an infinite number of Mersenne primes. Mersenne primes are related to perfect numbers.

Not ancient history
Here is a great article on Mersenne and the Great Internet Mersenne Prime Search, for huge prime numbers. In fact, the 45th and 46th Mersenne Primes were just found this summer!

Goldbach’s conjecture is the theory that every even integer greater than 4 is the sum of 2 odd primes:  8=3+5 (both 3 and 5 are prime), 10=5+5, 12=5+7, 20= 3+17 and =7+13 
42=5+37 =11+31 =13+29 =19+23

While his thoery has not been proven, you can try out this calculator that will find all the pairs of any even number over 4.

Sophie Redux (Nov. 6th)

Sophie focused on refining her contribution to number theory. She ultimately won a prize for her contribution but was denied entry into the French Academy along with other French mathematicians who were also refused entry despite their brilliant contributions to mathematical thought and progress. In addition to her work with number theory, Sophie also did work with elasticity. Despite her seminal work in this area regarding the elasticity of metals, her name was not one of the 72 inscribed on the Eiffel Tower, the building of which relied on elasticity concepts.  Sophie also did work with soap films and their behavior of finding the shortest distance between 2 points.  She won a prize for her work with Chladni vibrating plates (see below).

Sophie died at age 55 from breast cancer.  Gauss advocated for her to receive an honorary doctoral degree, but her death prevented her from accepting this honor.  Sophieʼs ambition and persistence helped her to gain the respect of male mathematicians of her time.  However, she was unfortunately still perceived as an exception rather than proof that women should be considered equals.

Sophie Germain, 1776-1831 (Nov.4th)

Sophie Germain

Sophie Germain is the second female French mathematician that weʼve studied.  Although she shared the same nationality as Emily, her father was from the merchant class.  Our Professor reminded us that Emilyʼs family would have been embarrassed if they had to work.  While Merchants may have become wealthy from their businesses, their expectations their ideals was that you could gain wealth through your business activities. Although her family was not poor, she did not benefit from tutors like Emily did. Sophieʼs parents gave her less leeway and encouragement regarding her mathematical pursuits. Women in her time were not expected to value intellectual pursuits.  Sophieʼs parentsʼ concern went beyond following the conventions of societal norms and included fears that Sophieʼs intellectual activities would physically and mentally endanger her wellbeing.  Professor Gottlieb explained how their were social myths that included thoughts that womenʼs reproductive systems would be negatively effected by prolonged intellectual activities and that womenʼs minds were less capable of handling the stresses of intense intellectual thought. Sophieʼs parents accordingly discouraged her from pursuing her intellectual activities.  It was only with Sophieʼs persistent insistence that Sophieʼs parents decided to allow her to follow her intellectual trying to write with an inkwell that was frozen, influencing her parents to allow Sophie to a candle to study with.  Before her parents gave up on their battle against her intellectual desires, they used tactics of dissuasion including taking away her clothing, hoping she would be forced to sleep under covers rather than face the cold.  By the time that Sophie was 13 years old in 1789, dangerous times had come to France. Although her family not being royalty were not the main targets of revolutionary discontent, her family non-the-less decided to move to a rural area.  The upsurges against the aristocratic class created a social instability that concerned her family.  Sophie took refuge in her intellectual pursuits, staying inside rather than venturing into the uncertain social climate. She read about Ancient Greece and one particular story has been amplified to play significant narrative influence on her life.  Archimedes was a Greek philosopher who was deeply involved in intellectual pursuits. An account read by Sophie, described Archimedesʼs behavior during a Roman invasion. Allegedly, while Romans were cautioned not to harm the Greek philosopher, the Roman soldier questioned him and without receiving a prompt and adequate response, because Archimedes was deep in intellectual thought, the soldier automatically killed him.  This story is sometimes projected as an inspirational influence on Sophia.  She was fascinated by the possibility that someone could be so lost in thought that worldly things became secondary.

 

 

 

Although her father initially discouraged her from reading, he ironically had an extensive library. Her fatherʼs interests however different from her mathematically inclined interests. So she was unable to use his library. She would stay up late studying.  From age 13 until 18, she accumulated a wealth of knowledge, and also began to feel the limitations that were put on her gender.  She wanted to attend the polytechnic institute but no women were allowed.  Instead, she would get lecture notes from other students.  She used these notes to teach herself mathematics.

Napoleon

Our Professor noted the significance of math during the French Revolution.  She explained how metric system was one of the new measurements of progress that was aligned with revolutionary France. Nowadays metrics has overshadowed the English system in most societies globally.  Outside of military time, English methods of time keeping still persist.   Professor Gottlieb described Napoleonʼs fascination with math.   He aspired to gather talented mathematicians while simultaneously denied womenʼs ability to join the mathematical profession. 

 

From Sophieʼs use of notes from the Polytechnic she was able to understand the theoretical underpinnings of calculus.  Eventually, she wasnʼt satisfied simply studying of other students, but wanted to submit a report to the institute. She had found out about a student who had dropped out. She submitted a paper using his name, Monsieur LeBlanc.  Her report was well-received.  The professor acknowledged that this paper was exceptional in its originality. This provoked the instructorʼs curiosity about the student. Although the lecture course was large, he decided he wanted to meet this talented student.  Sophie met with the professor and revealed her gender.  Luckily this professor accepted her on her merits, although he did not change his position regarding womenʼs entry into the institute.  Sophie continued to use the pen name M. LeBlanc to correspond with other mathematicians.

For five years she corresponded with a German mathematician Gauss on her interest in number theory.  Later the French army occupied Hanover where Gauss lived, and Sophie wrote a letter of concern to the French military asking them to assure Gaussʼs safety. This letter would ultimately reveal her identity to Gauss.  Fortunately, Gauss was accepting of her gender.   He told her that he was still impressed with her.

The Witch of Agnesi and Here’s to you Mrs.Robinson

Maria Gaetana Agnesi is known for her work with the functions of curves - the Curve or Witch (a mistranslation) of Agnesi.

  The wannabe nun known for her curves

Maria Gaetana Agnesi was born in Italy in 1718 and was part of the merchant class. The professor explained that her life was perhaps even more exceptional than Émilie du Châtelet considering that fewer women had access to education in Italy than in France. Women in Italy were typically were not taught to read but rather were trained in embroidery, etiquette and religion and taught to pray.

 Maria Gaetana was the oldest of 21 siblings and her sister Maria Theresa was a talented musician. Maria’s father, Pietro, encouraged her in her studies. Her father was a mathematic and a scholar who was in contact with a large social network due to his mercantilism. He would host large dinner parties which would highlight his children’s skills; they would often include music by his daughter Maria Theresa. Maria Gaetana spoke quite a few languages at an early age. Her father would proudly host Maria Gaetana’s talks on different issues.

 

By the time she was 13 years old, a book had been complied of her various presentations. Among the topics which she focused on were Newton’s law of gravity, laws of motion and celestial mechanics. The law of gravity illustrated that despite variances in weight, gravity dictates that objects fall at the same rate. Galileo is said to have dropped balls off the Tower of Pisa – 1 solid lead ball, 1 hollow which fell at the same speed.  Newton and Galileo’s experiments helped with the evolution of science and Keppler’s use of data also helped expand scientific methods.

 

In 1732 at 14 years old, Maria Gaetana’s mother died while giving birth to her 8th child and Maria Gaetana played a pivotal role in the raising of her 20 siblings. She put great value on the education of her sisters; brothers and she wrote text books at age 17. She compiled a manuscript explaining Newtonian ideas but publishers, although they knew it to be brilliant, shunned it because of her gender.

 

Maria wanted to become a nun but her father discouraged her from this goal. Her book, Analytical Institutions, published in 1748 when Maria was only 30 years old, became the first work in English about calculus. The book brought together many theories of the day such as Newton’s “fluxions,” Liebniz’s differentials and included sections on tangents, inflections, conic sections, infinitesimals, integration, differential equations and the versed sine curve introduced by Fermet. In 1801, a professor from Cambridge translated her work from Italian to English mistranslating the name of the curve to “witch” of Agenesi.

Although denied admission to the French Academy because of her gender, her intellectual skills were recognized by royalty of Austria (who controlled Italy as well as the pope). Maria Gaetana was named honorary professor but it is unclear whether she accepted the position although it is well known that she would lecture in conjunction with and replacing her father. When her father became ill she would take over his lecture. In 1752 her father passed away. She was 34 years old.

 

Seven years after her father passed, she moved to a children’s shelter and cared for children and taught them religious studies. She sold the jewels that were given to her as her wealth declined. For 24 years she worked with the poor. In her later years she was asked about a math problem but it had been over 50 years since she had left the field. She died in 1799 at 89 years old in poverty.

Julia Bowman Robinson was a twentieth century U.S. mathematician.

 

Julia Robinson, who was born 1919, didn’t prove her self to be an exceptional study early on but by the 1940s she had demonstrated her mathematical ability. She attended UC Berkeley during WW2, earning her PhD in 1948, and began working on diophantine equations. She also worked as a researcher on game theory for the RAND Corporation. She worked for years as a visiting lecturer and was only promoted after being elected to the National Academy of Sciences. Although this was hundreds of years after Maria Gaetana, it illustrated the unfortunate bias and ongoing chauvinism that women mathematicians have faced. Our professor relayed a particularly illustrative point when she mentioned how Julia was referred to as Mr. Robinson’s wife rather than being recognized as an independent scholar who had acclaimed notoriety on her own merits.