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Expansion of geometry, square and triangular numbers, Pythagoras’ theorem

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Euclid brought the works of previous mathematicians together into one book. The book is one of the earliest examples of a textbook built on deductive mathematics.

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275 - 195 BC Eratosthenes of Cyrene - Greece

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275 - 195 BC Eratosthenes of Cyrene - Greece

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He calculated the circumference of the Earth, measured the tilt of the Earth’s axis of rotation, estimated the distance to the sun and created some of the first maps of the world. He also invented the “Sieve of Eratosthenes “, an efficient way to calculate prime numbers

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(to date) Known as the first female mathematician to have made significant contribution to maths. Not much of her work survived but she may have assisted her father in producing new versions of Euclid's Elements. She became head of the Platonist school around 400 AD, which was a remarkable achievement for a woman then.

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He established Indian numerals to the Islamic world, which found its way to the Christian world. He also contributed to the evolution of cryptography. His book titled Manuscript on Deciphering Cryptographic Messages advanced the rise of cryptanalysis. He popularized several methods of decoding cyphers, such as frequency analysis.

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He is considered the first mathematician to use irrational numbers as solutions and coefficients to equations methodically. Fibonacci later embraced this method.
He effortlessly worked on algebraic equations having powers higher than x2, solved sets of non-linear simultaneous equations with three unknown variables.

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He measured the solar year was 365 days, 5 hours, 46 minutes, and 24 seconds. He missed out by 2 minutes and 22 seconds! He produced several Trigonometric relationships including solving the equation sin x = a cos x and formula of a right-angled triangle.

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First discovery of the "Pascal's Triangle"

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He was an astronomer and mathematician. He invented torquetum that took and converted measurements into the horizon, equatorial, and ecliptic. In simple words, it was an analogue computer.

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1170-1250 AD Leonardo of Pisa (Fibonacci)

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1170-1250 AD Leonardo of Pisa (Fibonacci)

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Fibonacci Sequence of numbers, advocacy of the use of the Hindu-Arabic numeral system in Europe, Fibonacci’s identity (product of two sums of two squares is itself a sum of two squares)

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Chinese version of Pascal's Triangle

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Logarithms which was an important aid to calculation. Also popularised the use of the decimal point.

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Tartaglia Triangle (earlier version of Pascal's Triangle)

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Creation of the cartesian coordinate system as a means of connecting algebra and geometry.

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The development of calculus and its notation

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Pascal's Triangle
Addition:
Pascal started studying maths only at the age of 12 with geometry. He may have invented the first digital calculator, mainly to help his father with his work collecting tax. Pascal made significant contributions in projective geometry. The triangle now known as Pascal's Triangle was attributed to him because of its use in several of his work, even though he never claimed the triangle to be his.

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Binomial expansion of fractional indices, Maclaurin series

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The argument between Leibniz and Newton over who discovered calculus. Great to bring up when discussing the two types of notation for calculus. Leads to academic integrity, and many ToK questions.

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De Moivre's Formula - linking trigonometry with complex numbers as well as important contributions to the fields of probability and statistics

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founded the studies of graph theory, topology, and many discoveries in analytic number theory, complex analysis, infinitesimal calculus

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Simpson’s rule for numerical integration

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As a child she excelled at languages. When her parents denied her request to become a nun, she focused her studies on theology and maths. She wrote her first book on differential calculus as a teaching text and continued to publish improved versions of her work which she would became very well known for. One of her noted work is a discussion on the cubic curve now known as the "witch of agnesi".

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In her teens (some time after the French Rev), she decided she wanted to become a mathematician after reading about Archimedes' life in a book on the history of mathematics. Her family initially opposed this but seeing her passion gave way. She made notable contributions in number theory and curvature of surfaces, and attempted to prove Fermat's last theorem.

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- proved the binomial theorem.
- group theory
- Abelian functions

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1777 - 1855 Carl Friedrich Gauss - Germany

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1777 - 1855 Carl Friedrich Gauss - Germany

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The sum of 1 to n and many other contributions to the fields of number theory, algebra, geometry and astronomy

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At the insistence of her mother, Ada studied maths and music from a young age, even though she preferred other subjects. In her late teens she met Charles Babbage and became interested in the Difference Engine (polynomial solving calculator) on display at his place. She started studying maths in greater depth and contributed to refining the programme to Babbage's Difference Engine.

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Pioneered the visual presentation of statistical data.

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Her interest in maths started as a child and by the age of 11 she was able to understand some calculus methods. Years later she was allowed to study maths openly and under the  tutilage of Aleksandr Strannoliubskii she studied analytic geometry, differential and integral calculus. Her notable works were in differential equations.

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Georgy Fedoseevich Voronoy (1868 – 1908)

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Georgy Fedoseevich Voronoy (1868 – 1908)

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Best known for Voronoi diagram

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Correlation, regression and standard deviation and pioneered the use of the questionnaire

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Amazing contributions to symmetry, rings, group theory, and gauge theory.

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A genius, mathematician who did not receive any formal Maths training. Gave almost 4000 equations and identities during his short life. Some are not proven until recently and are found very useful in different fields in modern science.

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American mathematician whose calculations of orbital mechanics as a NASA employee were critical to the success of the first and subsequent U.S. crewed spaceflights (depicted in movie “Hidden Figures”)

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Proved Fermat's Last Theorem after a few reviews and rewrites.

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First woman to win the Field's Medal (2014). Her significant contribution in the geometry of Riemann Surfaces.