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

But long before the development of the maths 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 maths and its relationship to the world.

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

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 maths enthusiasts to this day.

One apocryphal story of the Pythagoreans illustrates the danger of combining religion and maths. The Pythagoreans idealised 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 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 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.

al-Khwarizmi was a ninth century mathematician who created many of the most basic techniques for how we perform calculations. 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 non-positional systems.

al-Khwarizmi also came up with systems of rules for solving basic equations, like 4x + 8 = 2, or x^{2} - 8 = 4. His work marks the beginning of what we today understand as algebra. Indeed, the word 'algebra' itself 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.

While many of the mathematicians on this list made contributions to a huge number of different fields of maths, 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.

In modern terms, logarithms have a 'base', and the logarithm of a number gives us the power we need to raise the base to to get that number. For example, the base 10 logarithm of 10 is 1, and the base 10 logarithm of 100 is 2, since 10^{1}= 10, and 10^{2}= 100.

One huge reason the logarithm is so useful lies in some of its properties: logarithms turn multiplication into addition, and division into subtraction. To be more specific, the logarithm of the product of two numbers is the sum of the logarithms of the numbers, and similarly, the logarithm of a quotient is the difference of the logarithms.

This, especially in the pre-computer world, makes calculations far easier. Multiplication and division algorithms for large or very precise numbers take much longer than addition or subtraction. So, if someone had to multiply together two large numbers, they could look up the logarithms of the numbers in a table, add those, and then use a table from that sum to get back their result. Devices like slide rules also take advantage of logarithms to allow for quick calculation. This speed up of calculation had very useful applications in science and navigation, in which large numbers of calculations had to be done very quickly.

Many quantities that vary over several orders of magnitude are measured on logarithmic scales, like the Richter scale for earthquakes and the decibel scale for loudness.

Johannes Kepler was a gifted geometer who applied his mathematical abilities to solidify our understanding of the solar system. Kepler worked closely with the great empirical astronomer Tycho Brahe, who kept some of the most meticulous records of the movements of the planets up until that time. By analysing those records, 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.

Kepler's laws are impressive because they are a precise and elegant mathematical description of a physical process. The fact that things in the world, like planets orbiting the sun, follow such laws has been referred to quite elegantly by the 20th century physicist Eugene Wigner as 'the unreasonable effectiveness of mathematics'. Kepler's laws are an early example of that unreasonable effectiveness.

Kepler's laws also set the stage for Newton's development of his laws of motion and especially of his theory of gravity. Kepler's contributions to our understanding of planetary mechanics led to him being the namesake of NASA's main probe searching for planets outside our solar system.

Rene Descartes is most widely known for his contributions to philosophy, in particular his development of the idea of the dualism of mind and body, and for his famous saying 'I think; therefore I am'. However, much of the mathematics we use today owes a great debt to Descartes.

Descartes' primary contribution to mathematics was in the development of analytic geometry. Throughout the history of mathematics until Descartes, there was always a divide between algebra and geometry. On the one hand, we had the symbolic and abstract manipulation of numbers and unknown quantities, and on the other hand, we had the investigation of shapes and solids.

Descartes' analytical geometry unified these two fields. He pioneered the idea of representing algebraic forms and equations using geometric lines and curves on a coordinate plane. His basic ideas are still taught in high school mathematics today, with students learning how to graph an equation like y = 3x + 5 as a line, or an equation like y = x^{2} - 4 as a parabola.

This combination of geometry and algebra was a significant precursor to the later development of calculus, and is such a central idea of modern mathematics that we take it for granted. Descartes' work was so fundamental that we refer to the coordinate system he invented as the 'Cartesian plane'.

The French mathematician Blaise Pascal, like many of the people on this list, contributed to a number of fields of mathematics. Pascal's Triangle provides a remarkably elegant way to calculate binomial coefficients, a set of numbers that are important in algebra and elsewhere. He also developed one of the first mechanical calculators in the world, a distant and primitive relative of modern computers.

Pascal was also one of the originators of probability theory, coming from his analysis of games of chance. Pascal's work on the basics of probability represented the beginning of our ability to understand chance and risk in a mathematical way.

Pascal's work on probability, and his late in life religious revelations, lead to him coming up with Pascal's Wager, an argument for why one should believe in God rooted in the probabilistic idea of expected value.

No list of great mathematicians could be complete without Newton. With his invention of calculus (an achievement shared with our next entry), mathematics was able for the first time to systematically describe how things change across space and time. Newton developed calculus in the context of developing his theories of physics.

The language of calculus is the most natural way to describe motion. A car's speed is the rate at which it is changing position, or the derivative of its position. The acceleration of a ball dropped from a tall building is in turn the rate at which its speed is changing, or the derivative of its speed, and Newton understood that this acceleration was the result of the force of the earth's gravity acting on the mass of the ball.

Newton's physics also represented a milestone in our overall view of the world. Earlier physicists and astronomers, like the previously mentioned Johannes Kepler, understood that the behaviour and movement of objects followed certain patterns. But Newton and the physicists who would follow him understood, with the help of mathematics, the reasons why objects follow those patterns.

Further, Newton's laws were understood to be universal -- the same force of gravity that causes a ball to accelerate as it falls is the force that keeps the moon going around the earth. The idea that the same laws of physics apply everywhere in the universe is a core tenet of science, and it is supported by all existing evidence.

Leibniz independently developed calculus in Germany at the same time Newton was developing it in England, an occasional issue of debate among mathematicians. Leibniz, however, came up with much of the notation for calculus that we continue to use up to the present.

Leibniz also anticipated in many ways a huge number of later mathematical developments. He had a strong belief in rationalism, with a focus on formal symbolism that would later come to fruition in the late nineteenth and early twentieth centuries with the development of modern logic and set theory. Leibniz also had a hand in the improvement of mechanical calculators like the one developed by Pascal.

Thomas Bayes provided one of the most important tools used in probability theory and statistics. It allows us to find more difficult to observe probabilities.

Finding the probability of an event when we have a good understanding of the underlying mechanism tends to be pretty straightforward. Some basic calculations can give you the probability of drawing a full house in a hand of poker, or getting five heads in a row when flipping a coin five times, or of holding a winning lottery ticket.

In most interesting situations, however, we are interested in the reverse problem. Rather than computing probabilities of outcomes based on a known underlying mechanism, we want to gain an understanding of a hidden process based on observed outcomes.

This need to understand a hidden process based on observations underlies situations ranging from medicine (how likely is it a patient has a disease based on a positive test result for that disease?), to the social sciences (what is the best model that explains the relationship between inflation and unemployment, based on historical observations?) to everyday life (what is the likelihood that my date is romantically interested in me, given that she agreed to go to a second bar with me?).

Bayes' Theorem gives us a formal tool that allows us to answer these questions. The theorem lets us calculate the probability that a particular underlying process is happening, given our observed outcome, based on our understanding of the likelihoods of getting our observed outcome in the two cases where our underlying process is true and where it is not true, along with our prior degree of faith in the underlying process.

Bayes' Theorem is an incredibly powerful tool in analysing information to get at the reasons for why that information looks the way it does, and it is also the underlying framework for an entire school of thought in statistics.

Euler took up the reigns of calculus where Newton and Leibniz left off. He introduced what is now the fundamental concept of a function: some kind of rule, or set of rules, used to assign a number to another number. This is a concept used in modern maths to bring together all kinds of disparate things: linear and polynomial equations, trigonometric concepts, and even how we measure geometric distance in the plane can all be represented and understood in terms of functions and their manipulations.

Euler also furthered the theory of power series: a way of representing complicated functions using infinitely long sums of much simpler terms. His work on the power series representations of trigonometric and exponential functions led to, as a special case of a more general and extremely important formula, his famous equation e^{iπ} +1 = 0.

Euler was also one of the most prolific mathematicians of all time, and contributed to a number of fields. His solution to the Konigsberg Bridge Problem is considered one of the earliest results in topology and graph theory.

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