The Evolution of Basic Mathematics: From Counting to Calculus

19th November 2025

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Introduction

Mathematics is the branch of science dealing with numbers, quantities, shapes, and measurements. It’s an abstract science that primarily consists of abstract concepts, which can also be applied in our day-to-day life for understanding and developing other branches of science. It is also the oldest branch of science practiced by human beings for 20,000 to 30,000 years. As human societies developed, mathematics evolved alongside them, branching into hundreds of specialized fields over the centuries. In this blog, we talk about the history and evolution of the nine main branches of mathematics across the millennia. We discuss how they evolved as separate branches in ancient civilizations and gradually developed in classical, medieval, and modern times. These nine branches are those that are taught in nearly every school around the globe. Branches like topology, game theory, discrete mathematics, and computational mathematics are intentionally omitted as they are usually taught at higher institutes like colleges and universities. So, in order to keep this blog accessible and understandable to everyone, only nine of them have been selected.

1. Arithmetic

The oldest proof of arithmetic was found in Central Africa, the Ishango Bone, dating to around 20,000 BCE. In that bone, deliberate notches were marked that may represent doubling sequence, lunar counts, or early tallying. Later, tally marks emerged independently across different prehistoric societies. They were generally used to record items and for other trading purposes.

Around 3000 BCE, the Sumerians developed a base-60 system that became the backbone of Mesopotamian mathematics, which they used for trade, astronomy, and land measurements. This system, millennia later, gave rise to 60-second minute, 60-minute hour, and 360-degree circle. Around the same time, the Egyptians developed a decimal system with separate symbols for 1, 10, 100, and so on. While lacking place value, this system supported large-scale administration, taxation, construction, and record-keeping among the bureaucrats.

At the beginning of the Iron Age, China developed counting rods arranged on a board to represent numbers using place value, centuries before their appearance in the West. Rod numerals allowed both decimal representation and efficient computation with addition, subtraction, and even multiplication techniques. In Vedic India, on the other hand, the Sulba Sutras used numerical methods embedded with geometry for structuring fire altars. Although not a fully developed number system, they developed ratios, square approximations, and applied arithmetic for ritualistic purposes. The Greeks also developed an alternative acrophonic numerical system where 5, 10, 100, and others were based on initial letters of number words.

In the first half of the first millennium CE, the Indians introduced a number system with positional notations and zero as a number, and the Chinese developed counting boards under the Han Dynasty. The counting boards used rods arranged in vertical columns for place value, and were used for multiplication, division, square roots, and even systems of equations.

After the invasion of the subcontinent by the Arabs, they adopted Indian numerals and promoted them across Eurasia, thus known to the world today as Hindu-Arabic numerals.

Later, in 1202, Leonardo of Pisa, better known as Fibonacci, introduced Hindu-Arabic numerals to Europe through his book Liber Abaci. Thus, these numerals slowly replaced Greco-Roman numerals and became the dominant numeral system in the modern world.

2. Geometry

The earliest example of geometry can be found in Ancient Egypt from 3000 BCE. They used knotted cords to survey fields, maintain boundaries, and construct canals around the Nile River. They developed simple mensuration formulas that laid the foundations of geometry later.

Babylonian clay tablets, especially Plimpton 322 (~1800 BCE), recorded sophisticated lists of right-angle triples like 4-3-5 and 12-5-13. Their works show that the Mesopotamian scribes understood the Pythagorean principles, at least a millennium before Pythagoras was born.

During the start of the Iron Age, the Sulba Sutras of Vedic India developed geometric diagrams for constructing fire altars. They also contain an approximated value of √2, transformations between shapes of equal area, and methods of creating right angles.

In Greece, Thales introduced abstract reasoning in Geometry through propositions such as the equality of angles in a semicircle and basic similarity principles. The Pythagoreans, on the other hand, developed theorems related to polygons, ratios, and right triangles. Euclid, in his work Elements, systematized geometry through axioms, definitions, and proofs. Archimedes developed formulae to compute surface areas and volumes of solid figures like spheres and cylinders. He also approximated the value of π

In the classical age, China’s Nine Chapters on the Mathematical Art created independently the areas and volumes of various geometric figures, thereby influencing future East Asian mathematics. In India, Aryabhata also independently introduced the areas and volumes of different shapes and approximated the value of π. He also introduced sine tables, which later became crucial for developing trigonometry.

In the medieval period, Ibn al-Haytham, of modern-day Iraq, integrated geometric applications to physics, especially optics, in his work, also named Optics.

3. Algebra

The Babylonian tablet mentioned in the Geometry section is also very important to Algebra, as the tablet contains quadratic equations solved with geometric reasoning and completing the square methods, thereby acting as one of the earliest examples of algebra known to mankind.

The Rhind papyrus from Egypt, from around 1500 BCE, shows linear equations for practical problem solving regarding sharing, proportions, and rates. The Egyptian algebra later influenced the algebra around the entire eastern Mediterranean coast.

In the 3rd century CE, Diophantus of Alexandra composed Arithmetica, introducing symbolic shorthands for unknowns and methods to solve determinate and indeterminate solutions. In India, Brahmagupta, in his work Brahmasphutasiddhanta, established rules for zero, negative numbers, and algebraic operations, forming a major milestone in global mathematics.

In the Middle Ages, al-Khwarizmi, in his work Al-Jabr, organized algebra as a discipline focused on solving linear and quadratic equations through reduction and balancing. His work gave us the word “algebra” itself. Another Islamic mathematician, Omar Khayyam, classified cubic equations into geometric types and solved them using intersections of conic sections. Although not symbolic, his work represented the most advanced treatment of cubics before the Renaissance.

Fibonacci, in his Liber Abaci, introduced methods of solving algebraic problems in Europe. His works included linear and quadratic equations, commercial arithmetic, and recreational problems like the famous rabbit sequence. By combining Mediterranean techniques with Islamic and Indian knowledge, particularly al-Khwarizmi’s and Brahmagupta’s, Fibonacci catalyzed Europe’s transition toward symbolic algebra.

Later during the Renaissance, Tartaglia discovered methods for solving certain cubic equations, later published by Cardano in Ars Magna, along with Ferrari’s solution to quartics. Much later, Francois Viete introduced systematic symbolic notation using letters for known and unknown quantities, transforming algebra from a rhetorical prose to an abstract language.

Finally, René Descartes unified algebra and geometry through coordinate methods in La Géométrie, introducing modern exponent notation and the convention of x,y for variables.

4. Trigonometry

The earliest example of proto-trigonometric tables can be seen in the Babylonian astronomical cycles working with base-60. Their cuneiform records include lists of reciprocal pairs and angle relations useful for predicting planetary positions.

In Hellenistic Greece, Hipparchus developed the first known trigonometric table by measuring chords in circles, enabling astronomers to convert geometric angles with numerical values. His Table of Chords effectively established trigonometry as a mathematical discipline in the Hellenistic world.

In India, Aryabhata introduced a sine table based on half-chords, shifting trigonometry away from Greek chord geometry toward the modern sine function. His systematic values of sine differences helped Indian astronomers compute planetary motions with remarkable accuracy. Bhaskara I and II further expanded Indian trigonometry with interpolation formulas, sine approximations, and early versions of trigonometric identities.

Islamic scholars like Al-Battani and Al-Tusi formalized trigonometry by introducing functions like tangent and cotangent, developing spherical trigonometry, and creating rigorous tables for astronomical use.

During the Renaissance, Regiomontanus wrote De Triangulis, the first major European text devoted entirely to trigonometry, establishing it as an important tool for navigation and astronomy.

Later, John Napier, the Scottish mathematician, in the 16th century, invented logarithms, thereby revolutionizing trigonometric computation. Logarithmic sine and tangent tables dramatically reduced calculation time and errors, enabling efficient handling of spherical spheres.

5. Probability

The earliest examples of probability come from the Far East in China. Texts like the I Ching explored probability-like patterns through hexagram combinations. In India, epics like the Mahabharata mention weighted dice and gaming odds, and scholars analyzed outcome frequencies, which can be seen as an early form of probability. In the Arabic World, scholars examined risk in trade, inheritance disputes, and legal judgments. Their writings used combinatorial reasoning for dividing wealth or calculating fair shares in uncertain situations. Though not yet a formal theory, these analyses introduced structured thinking about likelihood, expected value, and equitable outcomes in real-world contexts.

In the 16th century, Gerolamo Cardano, an Italian mathematician, became the first one to systematically attempt to quantify dice and card games in his work Liber de Ludo Aleae. He used sample spaces and frequency-based arguments to estimate chances, laying the groundwork for mathematical probability.

Later Pascal-Fermat letters created the foundation of modern probability theory. Their analysis of the problem of points introduced expected value, combinatorial reasoning, and consistent methods for evaluating uncertain outcomes.

In the 17th century, the Swiss mathematician Jacob Bernoulli established the Law of Large Numbers, proving that long-run frequencies converge to true probabilities. His work gave probability a firm theoretical basis applicable to statistics, insurance, and scientific inference. Later, Thomas Bayes, the 18th-century English mathematician, introduced the idea of updating probabilities using new evidence, providing the conceptual foundation for Bayesian inference. His theorem formalized reasoning under uncertainty and linked probability to belief revision.

6. Statistics

The earliest examples of statistical data collection can be found in the censuses conducted in Ancient Egypt, Mesopotamia, and China about 3,000 years ago.

Imperial China later maintained the traditional registers for detailed accounts of its demographics, occupations, and agricultural output. Governments used large-scale datasets for taxation, labor allocation, and social policy.

In India, the Arthashastra described sophisticated data-gathering methods, including crop surveys, population counts, market price tracking, and crime tables. These records supported economic planning and state management.

Millennia later, the seventeenth-century English thinkers like John Graunt and William Petty pioneered “political arithmetic,” using data on births, deaths, and trade to analyze state power. Later, the German mathematician and astronomer Carl Friedrich Gauss introduced the normal distribution while studying astronomical measurement errors. His bell curve model revealed how random variations cluster around a mean, becoming a cornerstone of inferential statistics.In the latter half of the 19th century, Karl Pearson, the British statistician, formalized modern statistics by creating correlation coefficients, chi-square tests, and regression methods. His work transformed statistics into a mathematical discipline rooted in probability theory. Later, another British mathematician, Ronald Fisher, advanced statistical theory through concepts like maximum likelihood estimation, analysis of variance (ANOVA), and experimental design principles. Fisher’s framework became the basis of modern statistical science and still shapes research methodology across disciplines.

7. Logic

Many of the early civilizations developed their own schools of logic. In India, the Nyaya school developed a rigorous system of reasoning centered on perception, inference, comparison, and testimony. It represented one of the earliest and most detailed logical traditions in the ancient world. In Greece, the Aristotelians organized reasoning into structures based on universal and particular propositions. Aristotle’s Organon shaped Western logic for nearly two millennia, defining rules for deduction, classification, and scientific argument. The Stoics later developed propositional logic, focusing on whole statements rather than terms. They introduced logical connectives, inference rules, and truth value analysis far ahead of their time. Later in India, Buddhist thinkers like Dignaga and Dharmakirti created a sophisticated epistemological framework emphasizing perception and inference. They introduced theories of exclusion, fallacies, and valid cognition, shaping logic across India and Tibet. In the Islamic World, Avicenna extended Aristotelian logic by refining modal reasoning, distinguishing between necessity and contingency, and developing new types of syllogisms. His reworked logical framework deeply influenced Islamic and later European scholasticism. Medieval European scholars later integrated Aristotelian and Islamic logic into a unified teaching tradition.

Much later, in the 19th century, George Boole, a British mathematician, revolutionized logic by expressing reasoning through algebraic symbols, turning logical statements into mathematical equations. Boolean algebra introduced operations like AND, OR, and NOT, enabling precise analysis of logical relations.

Gottlob Frege, of Germany, developed predicate logic and argued that mathematics derives from pure logic. His formal system introduced quantifiers and variables, vastly expanding the expressive power of logic. Finally, Bertrand Russell and Alfred North Whitehead attempted to ground mathematics in logic through their Principia Mathematica. Their monumental work formalized set theory, arithmetic, and logic in one system, responding to the foundational crisis. Although incomplete, it shaped twentieth-century logic and influenced computer science and mathematical philosophy.

8. Set Theory

Before the formalization of Set Theory, the early mathematicians of Greece, India, and the Islamic world used set-like concepts when grouping numbers, classifying geometric objects, or distinguishing categories of quantities.

In the 19th century, George Cantor transformed mathematics by showing that different sizes of infinity exist. He introduced one-to-one correspondence to compare infinite sets and demonstrated that real numbers form a larger infinity than natural numbers. This breakthrough challenged long-held assumptions and opened an entirely new domain in mathematical thought. He also defined sets, cardinality, ordered pairs, and transfinite numbers. He built a coherent theory explaining both finite and infinite collections, establishing the basis of modern set theory.

Later, Ernst Zermelo, the German logician, improved set theory by introducing a formal axiomatic system in 1908, which was later strengthened by Abraham Fraenkel and others, creating the Zermelo-Fraenkel axioms with the axiom of choice(ZFC). This became the standard foundation of mathematics, defining how sets behave, how infinite collections operate, and how structure emerges from simple axioms. ZFC remains the dominant formal model today.

9. Calculus

The most ancient form of proto-calculus can be found in Ancient Egypt, where the scribes used practical methods to approximate areas and volumes for fields, granaries, and pyramids. Texts like the Rhind Papyrus show rules for computing slopes and circle areas with near-calculus intuition. The Babylonian tablets also show sophisticated numerical tablets for squares, reciprocals, and compound growth, enabling calculations related to interest, motion, and geometry. Their systematic use of sequences and iterative procedures hints at proto-calculus thinking.

Later, Greek mathematicians like Eudoxus and Euclid used the method of exhaustion to compute areas and volumes by inscribing ever-finer polygons. It laid the foundational groundwork for integration centuries before calculus was formalized. Archimedes advanced the exhaustion method by summing infinitely many slices to find areas, volumes, and centers of mass. His work on spirals, parabolas, and spheres essentially performed an experimental form of integral calculus.

In India, around the late 14th century CE, Madhava, the Kerala mathematician, discovered infinite series expansions of trigonometric functions, including a power series form of π. His results anticipated the Taylor series by several centuries, using iterative corrections and convergence analysis. The Kerala School of Mathematics created a remarkably advanced pre-calculus framework rooted in precise infinite summations.

In the Islamic world, scholars like Ibn al-Haytham and Sharaf al-Din al-Tusi examined the instantaneous rate of change, volumes generated by rotation, and early limit-like arguments.

Finally, in the 17th century, Isaac Newton and Gottfried Wilhelm Leibniz formalized calculus in a structured form. Newton focused on fluxions and instantaneous motion, while Leibniz introduced differentials and integral notation. Together, they unified ideas of change and accumulation, providing rules of derivatives, integrals, and infinite series that became the foundation of modern analysis.

Later, Leonhard Euler transformed calculus into a powerful symbolic discipline by systematizing functions, derivatives, and infinite series. He introduced standardized notation and solved a vast range of differential equations. Euler’s clarity and depth turned calculus from a discovery into a mature mathematical language used across science and engineering.

Conclusion

Mathematics has slowly evolved over the last three millennia. Whether it is arithmetic or trigonometry, statistics or calculus, mathematics now plays an important part in each of our lives – from professional to personal. Thus, knowing the origin of each branch can be very crucial in understanding how the ideas inside the human mind have evolved. Sorry for making such a large blog. For convenience, no formulae or equations have been used. Hope you will not mind the length. So, if you find this blog interesting, please like, share, and subscribe. Also, feel free to comment- whether any criticism, any inquiry, or anything you like. Finally, thank you for reading this blog.