Introduction
Mathematics is an abstract science that deals with numbers, shapes, and measurements, practiced by human beings for tens of thousands of years. Through the millennia, new topics and sub-branches have been created, and many of them evolved into dynamic popular branches. In this blog post, we discuss the history and evolution of some of the most popular and important mathematical branches that had got its application in our daily personal or professional lives.
Arithmetic
The oldest branch of mathematics has its earliest proof in Central Africa, from around 20,000 BCE, in a bone named Ishango Bone, having notches to record lunar counts.
In recorded history, the Sumerians were the first to develop a base-60 system, which was used for trading, astronomy, and land measurements. This 60-base system, millennia later, gave rise to the 60-second minute, 60-minute hour, and a 360-degree circle. In Egypt, a decimal system was developed with unique symbols representing 1, 10, 100, 1000, etc.
At the start of the Iron Age, the Chinese invented counting rods to represent numbers with place value, allowing decimal representation and operations like addition, subtraction, and even multiplication. In India, the Sulba Sutras, the manual for constructing Vedic sacrificial altars, described numerical methods, ratios, and even square approximations. In the Mediterranean world, the Greeks had their unique numerical system where 5, 10, 100, etc., were represented with letters.
In the early first millennium CE, the Indians created the numerical representation of zero and introduced the number system we use today. After the Islamic conquests of the Sub-continent, the Arabs adopted the Indian numerals and promoted them across Eurasia, and thus came to be known as the Hindu-Arabic numerals.
Geometry
Around 3000 BCE, the Egyptians used knotted cords to measure fields, boundaries, and canals, representing the earliest evidence of Geometry.
Some of the Babylonian clay tablets from 2000 BCE display right-angle triples like 4-3-5 and 12-5-13, a millennium before the birth of Pythagoras. The Indian Sulba Sutras described geometric diagrams with proper proportions for constructing fire altars. They also showed the transformation between shapes of creating equal areas, methods of creating right angles, and an approximated square root of two.
Around the first millennium BCE, Greece gave birth to several famous mathematicians. Thales developed reasoning in geometry through proportions, while Pythagoras developed theorems on polygons, ratios, and right triangles. Euclid described geometry through axioms and proofs, while Archimedes developed formulae of surface areas and volumes of solid figures like spheres and cylinders.
In classical India, Aryabhata also independently systematized the areas and volumes of various figures and also approximated the value of π.
During the Islamic Golden Age, Ibn al-Haytham established geometric applications to physics, especially in the field of optics.
Algebra
One of the earliest examples of Algebra is found in the Egyptian Rhind Papyrus from around 1500 BCE, showing linear equations for solving daily purposes. The Egyptian algebra later influenced the entire Mediterranean world.
In the 3rd century CE, Diophantus of Alexandria composed Arithmetica, which introduced unknowns and the method to solve determinate and indeterminate solutions. In India, Brahmagupta systematized the rules of zero, negative integers, and algebraic operations.
In the Middle Ages, al-Khwarizmi, in his book al-Jabr, organized the various rules of algebra and described the systematic solutions for the linear and the quadratic equations. Another mathematician, Omar Khayyam, displayed various processes of solving cubic equations.
In the early 13th century, Leonardo of Pisa, better known as Fibonacci, introduced solutions to algebraic equations in Europe. Later, René Descartes formulated coordinate geometry and also introduced variables like x, y, and z to Algebra, which are still in use today.
Trigonometry
The earliest proto-trigonometric tables can be found in the Babylonian astronomical cycles with base-60.
In Hellenistic Greece, Hipparchus invented the first known trigonometric tables using chords of circles. In India, Aryabhata developed sine tables using half-chords. Later, Bhaskara I and II expanded trigonometry with interpolation formulas and sine approximations.
Medieval Islamic scholars like al-Battani and al-Tusi introduced functions like tangent and cotangent, and also developed spherical trigonometry.
During the Renaissance, Regiomontanus wrote De Triangulis, the first major European text dedicated entirely to trigonometry. Later, John Napier, the Scottish mathematician, invented logarithms, thus revolutionizing trigonometry to a different level.
Calculus
The earliest proto-calculus examples are found in the Egyptian papyri and Babylonian tablets, which contained numerical tables for squares, reciprocals, and compound growth.
In ancient Greece, Eudoxus and Euclid used the method of exhaustion to compute areas and volumes by inscribing different polygons. Later, Archimedes advanced the exhaustion method by adding infinite slices to find areas, volumes, and infinite mass.
In medieval India, Madhava discovered infinite expansions of trigonometric functions, including a power series of π, centuries before the Taylor series was invented.
Finally, in the 17th century, Isaac Newton and Gottfried Wilhelm Leibniz formulated calculus in a structured form. Newton introduced fluxions and instantaneous motion, while Leibniz described differentials and integral notations.
Later, Leonard Euler transformed calculus into a more advanced form by systematizing functions, derivatives, and infinite series. He also introduced standardized notations and solved a vast range of differential equations, thus influencing the entire world of science and engineering.
Conclusion
The branch of mathematics has slowly evolved over the last three millennia. From arithmetic to calculus, mathematics now forms an integral part of our lives. Thus, knowing the history of mathematics can help us appreciate the influence of mathematics in developing human civilization.
References
If you like to read my original in-depth article in theindicscholar.com, you can read it here.