Amalie Emma Noether: (1882-1935) Germany
Amalie Noether was a pioneering mathematical researcher. She was born in Erlangen, Germany on March 23, 1882. She was named Amalie, but always called "Emmy". She was the eldest of four children, and Mathematics seemed in her bloodline. Her father was Max Noether, a noted mathematician of his time, and her brother Fritz also made a career of mathematics.
At the age of 18, Noether chose to take classes in mathematics at the University of Erlangen. Her brother, Fritz, was a student there, and her father was a professor of mathematics. Due to the fact she was a woman, the university refused to let Noether take classes. They did grant her permission to audit classes, however. She sat in on classes for two years, and then took and passed the exam that would permit her to be a doctoral student in Mathematics. This put her in good standing with the school. After five more years of study, she was granted the second ever degree to a woman in the field of Mathematics. The first had graduated a year earlier.
Emmy Noether made several major advances in abstract algebra. She originated novel reasoning methods, especially one based on "chain conditions.” Also, her perseverance about generalization led to a unified theory of modules and Noetherian rings. Her methods tended to unify disparate areas (algebra, geometry, topology, logic) and led eventually to modern category theory. Also, her origination of homology groups revolutionized topology.
Emmy Noether's work has found various applications in physics, and she made direct advances in mathematical Physics as well. Noether's Theorem established that certain symmetries imply conservation laws. This theorem has been noted as the most important theorem in physics since the Pythagorean Theorem.
Emmy Noether was not just great in Mathematics, she was a great person. She was generous with students and colleagues, even allowing them to claim her work as their own. Noether was close friends with other greatest mathematicians of her generation. Hilbert, von Neumann, and Weyl were among her friends. Weyl once said he was embarrassed to accept the famous Professorship at Göttingen because Noether was his "superior as a mathematician." Few would dispute that Emmy Noether was the greatest female mathematician ever.
Archimedes of Syracuse (287-212 BC) Greek domain
Archimedes is commonly recognized to be the greatest of ancient mathematicians. He studied at Euclid's school (likely after Euclid's death), but his work far bested the works of Euclid. His achievements are particularly impressive given the lack of proper mathematical notation during his life. His proofs are noted not only for brilliance but for unsurpassed simplicity.
Archimedes made advances in number theory, Algebra, and analysis, but is most renowned for his many theorems of plane and solid Geometry. He was first to prove Heron's formula for the area of a triangle. One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series. Many of Archimedes' discoveries are known only in the second-hand: Pappus reports that he discovered the Archimedean solids. Thabit ibn Qurra used Archimedes’ method to construct a regular heptagon. Also, Alberuni credits the Broken-Chord Theorem to him.
Archimedes anticipated integral calculus, most notably by determining the centers of mass of hemisphere and cylindrical wedge, and the volume of two cylinders' intersection. Although Archimedes made little use of differential calculus, He was similar to Newton in that he used his (non-rigorous) Calculus to discover results and then devise rigorous geometric proofs for publication. His original achievements in Physics include the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, and war machines. Also, Archimedes discovered formula for the volume and surface area of a sphere, and may have been first to notice and prove the simple relationship between a circle's circumference and area. For these reasons, π is often called Archimedes' constant. His approximation 223/71 < π < 22/7 was the best of his day. (Apollonius soon surpassed it, but only by using Archimedes' method.)
In the 20th century, modern technology led to the discovery of new writings by Archimedes, hitherto hidden on a palimpsest. It included a note which may imply an understanding of the distinction between countable and uncountable infinities (a distinction which wasn't resolved until Georg Cantor, who lived 2300 years after the time of Archimedes). Although Newton may have been the most important mathematician, and Gauss the greatest theorem prover, it is widely accepted that Archimedes was the greatest genius who ever lived. Unfortunately, Archimedes was too far ahead of his time.
References
http://fabpedigree.com/james/greatmm.htm
