The 12 mathematicians who unlocked the modern world

Mathematics is an increasingly central part of our world and an immensely fascinating realm of thought.

But long before the development of the math that gave us computers, quantum mechanics, and GPS satellites, generations of brilliant minds — spanning from the ancient Greeks through the eighteenth century — built up the basic mathematical ideas and tools that sit at the foundation of our understanding of math and its relationship to the world.

Here are 12 of the most brilliant of those minds and some of their contributions to the great chain of mathematics.

The Pythagoreans (5th Century BC)

Some of the earliest mathematicians were Pythagoras and his followers. Mixing religious mysticism with philosophy, the Pythagoreans' contemplative nature led them to explorations of geometry and numbers.

The most famous result attributed to Pythagoras is the Pythagorean theorem: for a right triangle, the sum of the squares of the two shorter legs that join to form the right angle is equal to the square of the long side opposite that angle. This is one of the fundamental results in plane geometry, and it continues to fascinate mathematicians and math enthusiasts to this day.

One apocryphal story of the Pythagoreans illustrates the danger of combining religion and math. The Pythagoreans idealized the whole numbers, and viewed them as a cornerstone of the universe. Their studies of geometry and music centered on relating quantities as ratios of whole numbers.

As the story goes, a follower of Pythagoras was investigating the ratio of the length of the long side of an isosceles right triangle to the length of one of the two shorter sides, which have the same length as each other. 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.

According to the legend, when the follower who discovered this fact revealed it to his peers, the idea that there could be irrational numbers — numbers that can't be expressed as a ratio of two whole numbers — was so shocking to the Pythagoreans that he was taken out on a boat and murdered by drowning.

Euclid (c. 300 BC)

Euclid was one of the first great Greek mathematicians. In his classic "Elements," Euclid laid the framework for our formal understanding of geometry. While earlier Greek philosophers like the Pythagoreans investigated a number of mathematical problems, 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. This process of using rigorous proof to build new results out of existing results introduced in the "Elements" has remained perhaps the most central guiding principle of mathematics for over two millennia.

Archimedes (c. 287-212 BC)

Archimedes was possibly the greatest mathematician of all time. 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: While taking a bath, Archimedes watched the water sloshing up to the top of his tub, and in the excitement of his discovery, he ran through the streets naked and shouting "Eureka!"

As a mathematician, however, Archimedes was able to outdo even his own accomplishments in physics. He was able to estimate the value of pi to a remarkably precise value and to calculate the area underneath a parabolic curve.

What is truly amazing about these accomplishments is that he made these calculations using techniques surprisingly close to those used by Newton, Leibniz, and their heirs in the development of calculus about 1,800 years later. He found these values by approximating them with measurements of polygons, adding more and more refined shapes, so that he would get closer and closer to the desired value. This is strongly reminiscent of the modern idea of an infinite limit. As far as his mathematical sophistication was concerned, Archimedes was nearly two millennia ahead of his time.