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His , an early application of mathematical inductionto solve 2nd-order equations, has been called "the finest thingachieved in the theory of numbers before Lagrange"(although a similar statement was made about one of Fibonacci's theorems).

Clausen, B. (2000). Christianity aiding the development of science. [On-line] Available URL:

Hippocrates is said tohave invented the proof method.

He was a key figure of functional analysis and integral geometry;he pioneered representation theory, important to modern physics;he also worked in many fields of analysis,soliton theory, distribution theory, index theory, Banach algebra,cohomology, etc.

Eilenberg is considered a founder of category theory,but also worked in algebraic topology, automata theory and other areas.

His approximation 223/71 was the best of his day.

Turing developed a new foundation for mathematicsbased on computation;he invented the abstract Turing machine,designed a "universal" version of such a machine,proved the famous Halting Theorem (related toGödel's Incompleteness Theorem), anddeveloped the concept of machine intelligence(including his famous proposal).

The is an astronomical clockconsidered amazing for its time.

Chern was an important influence in China and a highlyrenowned and successful teacher:one of his students (Yau) won the Fields Medal,another (Yang) the Nobel Prize in physics.

Tusi is most famous for his mathematics.

He is especially noted for his work in algebraic geometry, topologyand fiber bundles, developing his Chern characters(in a paper with "a tremendous number of geometrical jewels"),developing Chern-Weil theory,the Chern-Simons invariants,and especially for his brilliant generalization ofthe Gauss-Bonnet Theorem to multiple dimensions.

But Cardano is most remembered for his achievements in mathematics.

He studied Sanskrit as a child,loved to travel, taught at a Muslim university in India fortwo years (intending to teach French civilization),wrote as a young man under the famous pseudonymNicolas Bourbaki, spent time in prison during World War IIas a Jewish objector, was almost executed as a spy, escaped toAmerica, and eventually joined Princeton'sInstitute for Advanced Studies.

Bombelli is often called the Inventor of Complex Numbers.

Weil proved a special case of the Riemann Hypothesis; he contributed,at least indirectly, to the recent proof of Fermat's Last Theorem;he also worked in group theory, general and algebraic topology,differential geometry, sheaf theory, representation theory, andtheta functions.

Some of his work was borrowed from others,e.g.

He also proved that the Axioms of Choice (AC) and the Generalized ContinuumHypothesis (GCH) were with set theory, but that settheory's own consistency could not be proven.