Adrien-Marie Legendre, a mathematician, asserted that Germain's theory was not universal. He showed that her equation could not represent all plane curves of any given degree.
Germain proved her theorem by contradiction. She assumed that her equation represented all curves of degree n and tried to prove this assumption. If it were not true, then there would be a curve which could not be expressed using her formula. But she failed to find such a curve. This proves that her formula does represent all curves of degree n as claimed.
Legendre argued that the proof given by Germain was not satisfactory because it made use of an argument called "abduction". Abduction is a logical fallacy in which something is inferred from what is considered likely to be true. In other words, abduction is the process of assuming what should be shown.
Here is how Legendre explained abduction: "If we wanted to demonstrate that all men are mortal, we could do so either by arguing that every man is mortal or else by arguing that some man is mortal. Since we have no reason to believe that any particular man is immortal, we can conclude that none of them is immortal, which shows that all men are mortal."
Polytechnique (Polytechnic School) In the late 18th and early 19th century, Marie-Sophie Germain was a self-taught French mathematician who worked on numerous groundbreaking mathematical ideas. She published several papers while she was a student at the Polytechnique school in Paris, where she learned mathematics from some of the best teachers of her time.
Germain was interested in both analysis and geometry, two of the most important branches of mathematics. She is known for developing the theory of functions of a complex variable, which is a crucial tool for studying analytic curves like the lemniscate and other interesting shapes. She also proved that any closed curve can be divided into 2n pieces without overlapping or crossing each other for some natural number n. This property is called decomposability by parts and it's still useful in mathematics and engineering today.
Besides being one of the first women to work in academia, Germain was also one of the first female scientists in France. Her work on functions of a complex variable was widely cited by her male colleagues and influenced many other mathematicians including Pierre de Fermat and Carl Friedrich Gauss.
Germain died at the age of 36 after giving birth to her only child. But even though she died before reaching maturity, she left an influential mark on modern mathematics through her work.
Sophie Germain was a political revolutionary. She overcame societal stereotypes as well as a lack of formal education to become a recognized mathematician. Her work in number theory is well recognized, but her work in elasticity theory is also highly important in mathematics. In addition, she made important contributions to the understanding of hyperbolic functions and played an important role in the founding of the French Institute of Science.
Germain became interested in mathematics at a very young age. This interest came from reading scientific books that her father would bring home from his job as a school teacher. At the age of 11, she wrote her first paper on calculus. This paper was published in a mathematical journal and it showed significant progress in her thinking about infinitesimals. A year later, at the age of 12, she proved her first theorem on power series. This proof used advanced algebraic techniques that were beyond what most mathematicians at the time could understand or prove.
In 1770, when she was only 15 years old, she wrote a letter to Jean-Antoine Caritat, marquis de La Condamine asking him for advice on how to proceed with her studies. Three years later, she sent another letter repeating some of the questions she had in mind. He invited her to come to Paris and study with him and his students. This is exactly what she did when she went to live with her uncle who was a senator.
Germain was never married and never had children. Her parents provided financial assistance for her mathematics study. She also composed intellectual works.
Germain was born on 4 May 1776 in Neuchâtel, Switzerland. Her father was a wealthy notary public while her mother came from a middle-class family. She had two older brothers who died when they were young. Her mother took care of her while studying to be a teacher. When Germain was 12 years old, her mother went to Paris to study and left her daughter in the care of their neighbor. Germain wrote that she felt lonely but also proud because this showed that her parents expected great things from her.
At age 16, Germain started taking lessons from a mathematician named Pierre de Fermat. He taught her calculus and trigonometry and inspired her to learn more about mathematics. After one year, she decided that it was not for her and stopped going to his classes. However, she remained a fan of his work and described many mathematical problems before they were known or even thought of.
In 1795, Germain returned to Paris where she became friends with another mathematician named Mary Somerville. They shared ideas about mathematics and science and talked about them over dinner parties.
Germain was unable to pursue a profession in mathematics due to gender bias. She did, however, work independently throughout her life. Germain was born in Paris, France, on April 1, 1776. Her father was a wealthy merchant who had several offices around Paris.
Germain showed an interest in mathematics from an early age. When she was only 12 years old, she wrote a letter to the French Academy of Sciences asking about the possibility of applying for membership. The letter was considered important enough to be published in one of their journals. This indicates that at a very young age, she knew about the academy and its rules regarding membership.
In these days before computers, mathematicians of the time worked out problems by writing down all possible answers and checking them against each other and against reality. If a solution wasn't found, it was posted on paper with other unsolved problems in order to help others working on them. Germain wrote two papers while still in high school. The first investigated how many different ways there are to arrange n objects into a line where n is large compared to the number of objects. The second examined how much more difficult it is to find the greatest common divisor of two numbers than it is to find either of those numbers individually.
She died in Paris at the age of 62.
Sophie never heard back from him regarding her final email since he had left number theory to become a professor of astronomy at the University of Gottingen. However, around 12 years later, she wrote to the mathematician Legendre about her most important contribution in number theory. She told him that she was not sure if what she had discovered was new or not, but it seemed interesting enough to write down. Legendre replied saying that it was new and that he would be happy to publish any paper she might send him. No such paper has been found yet, but it's possible that she sent nothing more than an abstract since there is no record of further contact between them.
In any case, Sophie Germain went on to other things and didn't work anymore on her project on perfect numbers. But several mathematicians after her have continued to study these numbers...