Most early civilizations had counting systems. The Babylonians, Egyptians, and Chinese had calendars. The Egyptians had hieroglyphics in 3400 BC. The Babylonians, Romans, Chinese, Greeks, Mayans, and Hindus all had positional number systems.

The Sumerians used fractions, algebra, multiplication tables, quadratic equations, and could calculate square roots in 3,000 BC.

The Egyptians developed analytical geometry and approximated the number pi in 1650 BC. The Egyptians used the sundial to tell time in 1500 BC.

Pythagoras developed the Phythagorean Theorem of the right triangle and irrational numbers in 500 BC.

Euclid developed geometry, conic section theory, and prime numbers in 300 BC. Euclid and the Greeks created the postulational form of thinking to the bewilderment of most high school scholars. Archimedes followed shortly after Euclid. Archimedes is credited with development of infinite series, mechanics, and hydrostatics.

Hipparchus developed trigonometry in 180 BC. Trigonometry is essential to the study of Astronomy. The first record of surveying land occurred in 75 AD.

The only mathematical accomplishment during the Dark Ages was the development of the Christian calendar.

Fibonacci was the most talented mathematician of the Middle Ages. He is credited with introducing Arabic number notation into Europe in 1220. The Arabic number systems was essential to the advancement of mathematics. The pictorial and graphical number systems used by ancient civilizations were not up to the manipulations required in higher mathematics.

The invention of the printing press helped the development of mathematics. The first arithmetic textbook was printed in 1478. Euclid’s Geometry was first printed in 1482.

Pacioli developed double entry bookkeeping in 1494.

Napier developed logarithms in 1614. He also invented a device used for mechanically multiplying, dividing, and working square roots called Napier Rods. The slide rule was invented in 1622. Pascal invented the first calculating machine in 1642. These devices allowed for more rapid computation and resulted in many new mathematical achievements.

Newton and Liebniz invented functions and calculus in 1682. The first calculus textbook appeared in 1696. Calculus is essential for predicting orbital trajectories and space travel. Probability Theory and combinatorics were developed in 1690.

Number Theory, topology, abstract algebra, vectors, Boolean Algebra, mathematical logic, computer science, and Group Theory were developed in the 1800s. Boolean Algebra is the basis for computer programming.

Graph Theory, Game Theory, fractals, and Chaos Theory were developed in the 1900s. The first book on Graph Theory by Konig was published in 1936. Von Neumann and Morgenstern developed Game Theory in 1944. Lorenz developed Chaos Theory in 1960. Chaos is about finding order in random data. Chaos Theory explains outcomes dependent on initial conditions. Chaos Theory has changed the direction of the science of physics.

References:

H. Eves, An Introduction to the History of Mathematics, Third Edition, 1969, Holt, Rinehart and Winston, Inc.

A. Tucker, Applied Combinatorics, 1980, John Wiley & Sons.