Countless millennia ago, hunter-gatherer ancestors evidently experienced the compulsion to add, subtract and divide. Then, when primitive societies gradually formed themselves, the necessity to record quantities in some notational form evolved into an identifiable language. This language was essentially mathematical even if inscribed on wood, bone or clay. Mathematical concepts can thus be said to have long preceded the evolution of writing. 

For educated Greeks in the age of Plato and Aristotle, the credit for evolving a recognisable and sophisticated mathematical language was readily conceded to the Babylonian cultures of Egypt and Mesopotamia. But Greek writers of the sixth century BC, such as Pythagoras and Thales, have left an indelible mark on mathematical history. And, for more than 20 centuries, Euclid of Alexandria's Elements has dominated schoolrooms worldwide. This feature provides a concise overview of some other mathematical milestones through the ages. 

Pythagoras (570 –495 BC) was credited with many mathematical discoveries, including the Pythagorean Theorem, the five regular solids, the sphericality of the Earth and the identity of the morning and evening stars as the planet Venus. 

Archimedes (287-212 BC) is best known for his discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder. He is also celebrated by the story that he determined the proportion of gold and silver in a wreath made for Hieron by weighing it in water. While this could be true, the story that describes him leaping from his bath and running naked through the streets shouting, “I have found it!" is probably a colourful myth. 

Egypt and Rome had no concept of zero. Case in point: The year 2020 in Roman numerals is MM XX. It was not until 400 BC that the Babylonians introduced ᴧ ᴧ to denote an empty place for what became zero. A millennium later, in seventh century AD, zero appears in an equation as 1-1 =0 in a treatise of Brahma by Brahmagupta. 

This era is notable for the discovery of the laws of planetary motion, infinitesimal calculus and gravitational mechanics. 

Johannes Kepler (1571-1630), a mathematics teacher in Austria, was the first to state that orbits of planets were not circular round the sun, but elliptical. His observation: When viewed from above the Sun's North Pole, all planets orbited in a counter-clockwise direction within the ecliptic plane. 

Kepler also explained the movement of comets, and his calculations form the bedrock for his three laws of planetary motion, which are used today by NASA, Arianespace and Roscosmos to predict planetary orbits for the launch of rockets and satellites. 

Pierre de Fermat (1607-1665), was a French mathematician known for his pioneering work on infinitesimal calculus. His paper, "Methodus ad disquirendam maximam et minimam," published in Latin in 1636, includes the concept of adequality, a tool to calculate maxima and minima in calculus. He is most famous for his last theorem, which was finally proven in 1993 by Andrew Wiles of Princeton University. 

As every schoolchild knows, Isaac Newton (1643-1727) was inspired to investigate the laws of motion and gravity whilst sitting under an apple tree. In his seminal three-volume "Mathematical Principles of Natural Philosophy 1687," Newton explained the movement of planets and objects . His calculations analysed the motion of bodies, and his theory of gravitational attraction explained the movement of tides, projectiles and pendulums as well as the orbits of planets and comets. An Italian mathematician, Jacopo Riccati (1676 – 1754), made Newton's work known in Italy, and is best remembered for the differential equation that bears his name. 

Britain's Charles Babbage (1791-1871) is widely regarded as the father of computing based on his invention of the Difference Engine, a mechanical calculator able to process complex calculations in moments. He also developed the concept of the Analytical Engine or “thinking" machine in association with Ada Lovelace, which would respond to problems and solve them automatically. 

Irishman William Rowan Hamilton (1805-1865) is famous for the invention of a mathematical puzzle called the “Icosian Game" as well as his pioneering work on quantum mechanics, which has been celebrated in a commemorative silver Euro coin issued by the Irish Central Bank. Hamilton's work enabled future mathematicians to better understand three-dimensional geometry. Hamilton's contributions are also heralded by a new institute of mathematics at Trinity College Dublin, marking the 200th anniversary of his birth. 

James Maxwell (1831-1879) published "A Dynamical Theory of the Electromagnetic Field" in 1865, in which he stated four equations that govern electromagnetism. He assembled established experimental laws, including Faraday's and Ampere's, and unified them into a set of equations that bore his name. Maxwell also concluded that electromagnetic waves and visible light were the same thing. 

In practice, Maxwell's equations help explain how every form of radiation, including radio, telephone and mobile phone signals work, and are still in use today by scientists and electrical engineers. 

Probability, games of strategy, algorithms and computation and the “butterfly" effect are amongst the notable mathematical advances of the 20^th century. 

French mathematician Félix Boral (1871-1956) is known for his pioneering work on measure theory and probability. Borel explored improbability—when events had a tiny chance of ever occurring—concluding that events with a sufficiently small probability will never occur. In a series of papers in the 1920s he defined games of strategy; one of his books on probability introduced the thought experiment popularly called the infinite monkey theorem. 

Alan Turing (1912-1954), a leading English mathematician, most famously led Britain's war-time code-breaking efforts with a computer that infiltrated German military codes. He also pioneered work developing theoretical computer science, including algorithms and artificial intelligence, for which he is acknowledged as the father of computer science. He will soon appear on the Bank of England's new £50 banknote. 

American mathematician and meteorologist Edward Norton Lorenz (1917-2008) established the theoretical basis of weather and climate forecasting. His talk in the 1960s, “Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?" became known as "The Butterfly Effect," the idea that a tiny change in atmospheric conditions (caused by anything—not just a butterfly) is enough to alter far distant future weather patterns. For his work, he is considered a father of chaos theory. 

The mathematical calculations and theories of the 20^th century underpin modern life, products, manufacturing trading and a range of studies, including population dynamics, chemical reactions and irregular heartbeats. Algorithms and artificial intelligence are the basis of machine learning, with today's practical applications seen in robotics, energy trading and new drug development. The common thread throughout—mathematics. 

Mathematics is a set of tools that has developed and evolved over time, aiding humanity in its development over the countless centuries. Today, it is used in all aspects of our society, whether we are at work, rest and play, including in the desperate fight against COVID-19. One thing is clear, mathematics has a bright and interesting future in the coming centuries in taking humanity to the stars. 

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