Pythagoras: "As the story goes, a follower of Pythagoras was investigating the ratio of the length of the diagonal of a square to the length of the sides of that square. He then discovered that there was no way to express this as the ratio of two whole numbers. In modern terminology, this follower had figured out that the square root of 2 is an irrational number."
Euclid: "Euclid introduced the idea of rigorous proof: Starting with a handful of assumed axioms about the basic nature of points, lines, circles, and angles, Euclid builds up ever more complicated ideas in geometry by using pure deductive logic to combine insights from previous results to understand new ideas."
Archimedes: "He's best known for his contributions to our early understanding of physics by figuring out how levers work and in the famous legend of his discovery of how water is displaced by a submerged object. [...] He was able to estimate the value of pi to a remarkably precise value and to calculate the area underneath a parabolic curve."
Muhammad ibn Musa al-Khwarizmi: "His greatest contributions were in the realm of developing formal, systematic ways of doing arithmetic and solving equations. Al-Khwarizmi's writings introduced the Hindu-Arabic decimal number system we use today to Europe, and this system makes it far easier to add, subtract, multiply, and divide quantities of any size than using Roman numerals or other nonpositional systems. [...] His work marks the beginning of what we today understand as algebra. Indeed, the word "algebra" comes from part of the title of his book on solving equations, and the word "algorithm," meaning a systematic set of rules used to solve a problem, descends from al-Khwarizmi's name."
John Napier: "John Napier created one incredibly important concept: the logarithm. The logarithm of a number, roughly speaking, gives us an idea of the order of magnitude of that number."
Johannes Kepler: "Kepler was able to confirm and refine the Copernican view of the solar system: The planets move around the sun, and the time it takes a planet to move around the sun is described by precisely-defined mathematical laws based on the shape of the planet's elliptical orbit."
