History of Mathematics

Introduction

Mathematics is among the oldest and most enduring intellectual achievements of the human species. From the earliest marks scratched into bone by prehistoric hunters keeping tally of game, to the vast edifices of abstract reasoning constructed by twentieth-century logicians and geometers, the story of mathematics is inseparable from the story of human civilization itself. No other discipline has so quietly and so thoroughly shaped the way people understand the universe, build their cities, navigate their seas, conduct their commerce, and push forward the boundaries of science. The complete history of mathematics from ancient number systems to modern mathematics and its greatest discoveries spans at least fifty thousand years of human effort, ingenuity, failure, and triumph.

What makes the history of mathematics so compelling is that it is not merely a chronicle of formulas discovered and theorems proved. It is the record of human beings wrestling with the deepest mysteries of quantity, space, change, and structure. Why do the same mathematical truths appear independently in cultures separated by vast oceans and centuries? How did the simple act of counting sheep eventually lead to the concept of infinity? How did merchants keeping accounts in ancient Mesopotamia lay the foundations for the algebra students still study in school today? These questions make the history of mathematical thought one of the richest narratives in intellectual history.

The development of mathematics has never been a smooth, linear progression. Periods of explosive creativity have been followed by long centuries of consolidation or apparent stagnation. Mathematical knowledge has been gained, lost, and rediscovered. Ideas that seemed merely abstract curiosities in one era turned out, centuries later, to be precisely the tools needed to describe the physical universe. The invention of non-Euclidean geometry in the nineteenth century was dismissed by many as a purely theoretical exercise, yet it became the mathematical language of Einstein's general relativity. The number theory puzzled over by ancient Greek mathematicians found unexpected application in modern cryptography and secure internet communication.

This article surveys the entire sweep of mathematical history, from the prehistoric origins of counting through the towering achievements of the ancient civilizations of Mesopotamia, Egypt, Greece, India, the Islamic world, and China, through the great revolution of calculus and the extraordinary proliferation of mathematical disciplines in the nineteenth and twentieth centuries, to the profound challenges and open problems that still confront mathematicians today. Along the way it profiles the greatest mathematical minds of history, examines how mathematics has shaped everyday life, and considers the unsolved mysteries that continue to beckon from the frontier of human knowledge.

Mathematics is sometimes described as the queen of the sciences, a phrase attributed to Carl Friedrich Gauss, one of the supreme mathematical geniuses of the nineteenth century. It might equally be called the servant of all sciences, since no branch of natural inquiry has been able to advance far without borrowing from the mathematician's toolkit. Understanding how that toolkit was assembled, refined, and expanded over the millennia is to understand a central thread in the fabric of human civilization.

Prehistoric Mathematics and Counting

Long before the invention of writing, before the first cities rose beside the rivers of Mesopotamia and Egypt, human beings were practicing mathematics. The evidence is fragmentary but unmistakable: bones and stones carved with tally marks, arrangements of objects that suggest deliberate counting, and geometric patterns painted on cave walls indicate that mathematical thinking is not a recent or specialized achievement but a fundamental feature of the human mind.

The oldest known mathematical artifact is the Lebombo bone, a baboon fibula discovered in the Lebombo Mountains on the border of Swaziland and South Africa. The bone, which dates to approximately 43,000 years ago, bears twenty-nine distinct notches arranged in what appears to be a deliberate tally. Some researchers have speculated that the markings correspond to a lunar calendar, while others see simply a counting device used to track hunted animals or other quantities. Whatever its original purpose, the Lebombo bone demonstrates that human beings were engaging in systematic enumeration tens of thousands of years ago.

Equally remarkable is the Ishango bone, found near the Semliki River in the region that is now the Democratic Republic of the Congo. Dating to roughly 20,000 years ago, this tool made from a baboon fibula bears three columns of notches that have fascinated mathematicians and anthropologists alike. One column appears to show doubling sequences. Another column contains notches that seem to correspond to prime numbers between ten and twenty. Whether or not the maker of the Ishango bone understood prime numbers in any abstract sense, the organized patterning of the marks suggests a level of mathematical awareness that goes well beyond simple tally counting.

The universality of basic counting is itself a profound mathematical fact. All known human cultures, however isolated, possess number systems adequate to the counting of objects in their environment. Most languages have words for the numbers one through ten at minimum, and many linguists believe that the widespread use of base-ten number systems reflects the biological fact that human beings have ten fingers. Other bases have also been used historically. The Sumerians of ancient Mesopotamia developed a base-sixty system whose legacy persists today in our division of the hour into sixty minutes and the circle into three hundred sixty degrees. The Maya of Central America independently arrived at a base-twenty system, reflecting perhaps the counting of both fingers and toes.

Before formal numeration systems emerged, prehistoric peoples likely used one-to-one correspondence as their primary mathematical tool. A herder keeping track of sheep might place a stone in a pouch for each animal driven out to pasture and remove a stone for each animal returning at day's end. This process of matching objects in two sets without necessarily assigning numbers to them represents the most primitive form of quantitative reasoning, and it encodes the mathematical concept of cardinality, the measure of a set's size, in a purely physical procedure.

Geometric thinking also appears to have very deep prehistoric roots. Cave paintings at Lascaux in France and Altamira in Spain, dating back fifteen thousand to twenty thousand years, show a sophisticated understanding of proportion and spatial arrangement. The construction of megalithic monuments such as Stonehenge in England, begun around 3000 BCE, required the ability to measure distances, mark out circles, and align structures with astronomical precision. The builders of these monuments lacked written mathematics in any formal sense, yet they clearly possessed robust practical knowledge of geometry that had been accumulated and transmitted across many generations.

The transition from purely intuitive and practical mathematical knowledge to something recognizable as systematic mathematics occurred in parallel with the rise of settled agricultural civilizations. When people began to farm, to store surplus grain, to divide land among heirs, to pay taxes to a central authority, and to trade goods across long distances, the demands placed on their mathematical capabilities increased enormously. Counting was no longer enough. People needed to measure areas, compute volumes, track astronomical cycles for planting calendars, and record transactions in ways that could be verified later. These practical necessities drove the development of the first formal mathematical systems, which emerged almost simultaneously in Mesopotamia and Egypt around five thousand years ago.

The history of mathematics in its full sense therefore begins with the great river valley civilizations, but it rests on a far older foundation of prehistoric mathematical practice that stretches back to the very origins of human cognition. Every child who learns to count is recapitulating an intellectual achievement of enormous antiquity, one that distinguishes the human species and that has made possible everything from the pyramids to the smartphone.

Ancient Mesopotamian Mathematics

The civilization that grew up between the Tigris and Euphrates rivers in what is now Iraq produced the first extensive written mathematical records. The ancient Mesopotamians, particularly the Sumerians and later the Babylonians, developed a mathematical tradition of remarkable sophistication that flourished for more than two thousand years and whose influence can still be felt in the way we measure time and angles today.

Mesopotamian mathematics was recorded on clay tablets impressed with a wedge-shaped stylus, a script known as cuneiform. Thousands of such tablets have been excavated from sites across Iraq, and those with mathematical content range from simple multiplication tables and lists of reciprocals to sophisticated problems involving quadratic equations and geometric calculations. The tablets were produced in scribal schools where young men were trained in the administrative and intellectual arts needed to serve the Mesopotamian state and temple economy. Mathematics was fundamentally a practical discipline in this context: scribes needed to calculate grain rations, plan irrigation canals, apportion field areas, and compile tax records.

The most distinctive feature of Mesopotamian mathematics is its use of a base-sixty, or sexagesimal, positional numeration system. In a positional system, the value of a digit depends not just on what numeral it is but where it appears in the number. The Babylonians had only two distinct cuneiform signs for numerals: one representing the value one and one representing the value ten. Numbers between one and fifty-nine were expressed by combining these signs in various ways. For values of sixty and above, the scribes used a positional convention in which the same signs shifted position to indicate multiplication by successive powers of sixty. This is the same positional logic that underlies the decimal system we use today, but operating in base sixty rather than base ten.

Strikingly, the Babylonians appear to have managed for a long time without a sign for zero as a placeholder, which occasionally made their notation ambiguous. A later version of the cuneiform system did introduce a placeholder symbol, though it functioned differently from the zero of modern mathematics and was not treated as a number in its own right. The full development of zero as a number awaited the mathematicians of India many centuries later.

The mathematical content of the Babylonian tablets is impressive by any measure. Scribes compiled extensive tables of squares, square roots, cubes, and cube roots, as well as tables of reciprocals needed to carry out division by means of multiplication. They calculated the area of rectangles, triangles, and trapezoids. They worked with the relationship between the sides of a right triangle that is now known as the Pythagorean theorem, centuries before Pythagoras was born in ancient Greece. One famous tablet, known as Plimpton 322 and dating to around 1800 BCE, contains a table of Pythagorean triples, sets of three integers that satisfy the equation governing right triangles, suggesting that Babylonian mathematicians not only knew this relationship but had developed systematic methods for generating such triples.

The Babylonians were also capable of solving what we would today call algebraic problems. Their approach was not symbolic in the modern sense: they expressed everything in words and specific numbers rather than letters and abstract variables. But the logical structure of their solutions corresponds closely to modern algebraic reasoning. They could solve quadratic equations using a procedure equivalent to completing the square. They handled problems involving two unknowns, essentially solving simultaneous equations. Some tablets contain cubic equations and problems in what amounts to elementary number theory.

One of the most celebrated achievements of Babylonian mathematics is their approximation of the square root of two. A small clay tablet from around 1700 BCE shows a square with its diagonals drawn, and inscribed on the diagonal is a numerical value that corresponds to the square root of two accurate to five decimal places. This approximation, 1.41421..., is remarkably precise for a civilization without modern computational tools, and it attests to the Babylonians' deep engagement with what we now recognize as irrational numbers, though they would not have conceptualized them as such.

Babylonian astronomy and mathematics were deeply intertwined. The prediction of lunar and solar cycles, the compilation of astronomical tables that tracked the rising and setting of planets, and the development of a calendar required sophisticated mathematical modeling. Babylonian astronomers eventually produced arithmetical procedures for predicting planetary positions with considerable accuracy, work that influenced Greek and later Islamic astronomy and represents one of the earliest examples of the application of mathematical tools to the systematic description of natural phenomena.

The legacy of Mesopotamian mathematics in world history is immense. The base-sixty system that the ancient Mesopotamians invented has proved extraordinarily durable. Today, thousands of years after the last cuneiform tablet was pressed, we still divide the hour into sixty minutes, the minute into sixty seconds, and the full circle into three hundred sixty degrees, each of these conventions a direct inheritance from the mathematicians of ancient Iraq.

Ancient Egyptian Mathematics

Along the Nile River, the civilization of ancient Egypt developed its own mathematical tradition that, while less algebraically sophisticated than that of Mesopotamia, possessed its own elegance and achieved remarkable practical results. Egyptian mathematics is preserved primarily in two surviving papyri, the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus, along with a smaller number of other documents. These sources give us a clear picture of how Egyptian scribes approached mathematical problems and what level of mathematical knowledge was expected of an educated professional in ancient Egyptian society.

The Rhind Mathematical Papyrus, dating to around 1550 BCE but apparently copied from an older source of about 1650 BCE, is one of the best sources for understanding ancient Egyptian mathematics. Named after the Scottish antiquarian Alexander Henry Rhind who purchased it in 1858, the papyrus contains eighty-four mathematical problems with solutions, covering topics including arithmetic, geometry, and what we might today call linear equations. Its opening lines describe it as a guide to knowledge of all existing things and all obscure secrets, a characteristically Egyptian way of framing mathematics as a form of esoteric practical wisdom.

Egyptian arithmetic was based on a decimal system, like our own, but it did not use positional notation. Instead, Egyptians used separate hieroglyphic symbols for each power of ten: one, ten, one hundred, one thousand, ten thousand, one hundred thousand, and one million. To write a number, scribes simply combined these symbols, repeating each as many times as needed. Addition in this system is quite natural: you simply combine the symbols from two numbers and simplify by exchanging ten symbols of one denomination for one symbol of the next higher denomination when necessary.

Multiplication and division in Egyptian mathematics were carried out through a process of repeated doubling and halving, sometimes called the method of successive doubling or the Russian peasant algorithm. To multiply two numbers, the scribe would write out a series of doublings of one of the numbers and then select from this list those entries that sum to the other number, adding the corresponding doublings of the first number to obtain the product. This method works for any two whole numbers and requires only the ability to double numbers and add, skills well within the reach of any trained scribe.

Egyptian fractions were expressed as sums of unit fractions, that is, fractions with numerator one. The Egyptians had a special sign for the fraction two-thirds, but all other fractions were decomposed into sums of distinct unit fractions. For example, what we would write as three-quarters would be expressed as one-half plus one-quarter. The Rhind Papyrus opens with a table giving decompositions of fractions of the form two divided by an odd number from three to one hundred and one, into sums of unit fractions. Generating these decompositions requires considerable ingenuity, and the particular decompositions chosen in the table often minimize the number of terms or keep the denominators small, suggesting that Egyptian mathematicians had aesthetic preferences about how fractions should be expressed.

Egyptian geometry was largely motivated by practical concerns, most famously the need to re-establish field boundaries after the annual flooding of the Nile washed away the markers. The word geometry itself comes from the Greek words for earth and measurement, reflecting this ancient connection between the discipline and the practical surveying of land. Egyptian scribes could calculate the areas of triangles, rectangles, and trapezoids with accuracy. They approximated the area of a circle using a formula equivalent to computing the area of a square whose side is eight-ninths of the circle's diameter, yielding an implicit value of pi of approximately 3.1605, a very good approximation for practical purposes.

Perhaps the most astonishing example of Egyptian practical geometry is found in the Moscow Mathematical Papyrus, which contains a formula for the volume of a truncated pyramid, or frustum. Problem fourteen of that papyrus gives a correct formula for the volume of a frustum with square base, expressed in terms of its height and the lengths of the sides of its two square faces. This is a non-trivial result that requires considerable mathematical insight, and the fact that it appears in what is essentially a scribal training manual indicates that it was part of the working mathematical knowledge of Egyptian professionals. How the Egyptians derived this formula remains uncertain, since they never record their reasoning process, only the procedural steps and the answer.

Egyptian mathematics was largely practical and procedural in character. Egyptian scribes solved problems by following established algorithms, and the mathematical papyri present problems and solutions but rarely explain why the methods work. This absence of formal proof distinguishes the Egyptian mathematical tradition from the later Greek tradition, which made the rigorous demonstration of mathematical truths its central concern. Yet the practical achievements of Egyptian mathematics were extraordinary, and the construction of the pyramids and other monumental structures attests to a level of applied geometric knowledge that has impressed observers for millennia.

Ancient Greek Mathematics and Euclid

The transformation of mathematics from a collection of practical computational techniques into a rigorous deductive science was one of the most profound intellectual revolutions in human history, and it was accomplished primarily by the ancient Greeks over a period of roughly four centuries, from about 600 BCE to about 300 BCE. Greek mathematicians were the first to insist that mathematical claims must not merely be procedurally effective but logically demonstrated, proved to be necessarily true from first principles by a chain of valid reasoning. This demand for proof, which is now the defining characteristic of mathematics as a discipline, was the Greeks' most enduring contribution to human thought.

The earliest Greek mathematicians known to us by name were closely connected to the practical and philosophical traditions of the ancient Near East. Thales of Miletus, who flourished around 600 BCE, is credited with introducing geometry from Egypt to Greece and with proving several basic geometric theorems, including the proposition that a circle is bisected by any of its diameters and that the base angles of an isosceles triangle are equal. Whether Thales actually offered formal proofs of these results or merely stated them as observed facts is debated by historians, but he is remembered as the originator of the Greek tradition of seeking logical justification for mathematical claims.

The Pythagoreans, followers of Pythagoras of Samos who flourished in the late sixth century BCE, made mathematics the center of their philosophical worldview. They believed that numbers were the fundamental reality underlying all of nature and that the study of arithmetic, geometry, music, and astronomy was the path to wisdom and even to the purification of the soul. Their most famous mathematical achievement, the theorem that bears Pythagoras's name relating the squares of the sides of a right triangle, was almost certainly known to the Babylonians and possibly the Indians before them, but the Pythagoreans appear to have been the first to prove it rigorously.

The discovery that deeply troubled the Pythagoreans was the existence of incommensurable magnitudes, what we now call irrational numbers. The diagonal of a square with side one has length equal to the square root of two, and the Pythagoreans proved that this quantity cannot be expressed as a ratio of two whole numbers. This result, which shattered the Pythagorean belief that all quantities in nature were rationally related, was according to ancient tradition considered so disturbing that it was kept secret, and the member of the Pythagorean brotherhood who revealed it to outsiders was thrown into the sea.

The fifth and fourth centuries BCE saw an extraordinary flowering of Greek mathematical thought. Eudoxus of Cnidus, who flourished around 370 BCE, developed the method of exhaustion, a technique for calculating the areas and volumes of curved figures by approximating them with sequences of rectilinear figures of known area. This method is a forerunner of the integral calculus developed by Newton and Leibniz two thousand years later, and Eudoxus used it to prove, among other results, that the volumes of spheres are in the ratio of the cubes of their diameters. Eudoxus also developed a rigorous theory of proportion that could handle irrational magnitudes, thereby providing a logical foundation for much of Greek geometry.

The culminating work of ancient Greek mathematics was the Elements of Euclid, composed around 300 BCE in Alexandria in Egypt, then part of the Hellenistic world that Alexander the Great had created. The Elements is one of the most influential books in the history of thought. It consists of thirteen books covering plane geometry, the theory of numbers, three-dimensional geometry, and the theory of proportions, all organized as a deductive system proceeding from a small number of definitions, postulates, and common notions through hundreds of propositions, each rigorously derived from what came before.

Euclid's achievement in the Elements was not primarily the discovery of new mathematical results, most of which were known before him, but the organization and logical codification of existing mathematical knowledge into a coherent deductive system. By identifying a minimal set of assumptions, the five postulates, from which all the rest of geometry could be derived, Euclid showed that mathematics could be made completely rigorous and self-contained. His method became the model for all subsequent mathematical exposition and for the ideal of logical rigor in any field of inquiry.

Four of Euclid's five postulates are simple and uncontroversial statements about points, lines, and circles. The fifth postulate, known as the parallel postulate, is much more complex: it states, in effect, that through a point not on a given line, there is exactly one line parallel to the given line. The logical status of this postulate troubled mathematicians for more than two thousand years, as many felt it was too complex to serve as an axiom and must be derivable as a theorem from the other four. The eventual resolution of this question, through the development of non-Euclidean geometries in the nineteenth century, was one of the most consequential events in the history of mathematics.

The Elements also contains important results in number theory, including a proof that there are infinitely many prime numbers, an algorithm for finding the greatest common divisor of two numbers (still known as the Euclidean algorithm), and a proof that every positive integer greater than one can be expressed as a product of prime numbers in essentially one way (the fundamental theorem of arithmetic, though Euclid did not state it in this form). These results in the ancient Greek study of prime numbers laid the groundwork for the entire subsequent development of number theory.

Pythagoras and the Pythagorean School

Pythagoras of Samos was born around 570 BCE on the Aegean island of Samos and died around 495 BCE in Metapontum in southern Italy. He was a philosopher, religious teacher, and mathematician whose influence on Western thought has been incalculable, though the difficulty of separating his personal contributions from those of his followers makes his biography one of the most contested in the history of ideas. Pythagoras left no written works, and everything we know about him comes from later sources, some of them written many centuries after his death and of questionable reliability.

After traveling in Egypt, Babylon, and possibly other parts of the ancient world, Pythagoras settled in Croton in what is now southern Italy and founded a philosophical brotherhood or school that combined mathematical study with religious practice and ethical rules. Members of this community followed strict dietary and behavioral codes, held property in common, and were bound by vows of secrecy regarding the inner teachings of the school. The community exercised considerable political influence in Croton until a violent reaction against its power led to its dispersal sometime in the fifth century BCE.

The Pythagoreans were the first thinkers to elevate mathematics to a position of supreme philosophical importance. Their fundamental doctrine was that number is the principle and substance of all things, that the universe is fundamentally mathematical in structure. This belief was inspired in part by the discovery of the mathematical ratios underlying musical harmony: the Pythagoreans found that pleasing musical intervals correspond to simple ratios between the lengths of vibrating strings. If something as subjective and aesthetically powerful as harmony could be reduced to number, perhaps the entire cosmos could be as well.

The Pythagorean theorem, the relationship that in a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides, is the most famous mathematical theorem in history and bears the name of Pythagoras, though as noted earlier the result was known to Babylonian mathematicians more than a thousand years before him. The Pythagoreans, or Pythagoras himself, are credited with the first rigorous proof of this theorem. Many different proofs have since been discovered, and the collection of known proofs now numbers in the hundreds, but the original Pythagorean proof is thought to have been based on a comparison of areas.

The Pythagorean school made important contributions to arithmetic, developing the theory of figurate numbers, numbers that could be arranged in geometric patterns such as triangular, square, and pentagonal numbers. They investigated what we now call perfect numbers, abundant numbers, and deficient numbers, based on the relationship between a number and the sum of its proper divisors. The number six is perfect because its proper divisors one, two, and three sum to six itself. Twenty-eight is also perfect for the same reason. The study of these number-theoretic concepts by the ancient Greek mathematicians initiated a tradition of inquiry that continues to the present day.

The most philosophically significant mathematical discovery associated with the Pythagorean school was the existence of irrational numbers, specifically the discovery that the square root of two cannot be expressed as a ratio of two integers. The proof is elegant: assume that the square root of two equals a fraction p over q in its lowest terms, and derive a contradiction by showing that both p and q must be even, contradicting the assumption that the fraction is in lowest terms. This proof by contradiction, one of the most beautiful in all of mathematics, demonstrated that the number line contains points that cannot be expressed as ratios of whole numbers, a discovery that fundamentally changed the understanding of what numbers are.

The legacy of Pythagoras and his school in the history of mathematics and philosophy is profound. The vision of the universe as fundamentally mathematical, which Pythagoras and his followers first articulated, became a core assumption of Western science. Plato's belief in eternal mathematical forms as the most real things in the universe was deeply influenced by Pythagorean thought. When Galileo wrote that the book of nature is written in the language of mathematics, and when modern physicists discover that abstract mathematical structures describe the deepest features of the physical world, they are in some sense heirs of the Pythagorean insight that number is the principle of all things.

Archimedes and Applied Mathematics

Archimedes of Syracuse, who lived from approximately 287 BCE to 212 BCE, is widely regarded as the greatest mathematician of antiquity and one of the greatest mathematical minds in all of history. Born in the Greek city of Syracuse on the island of Sicily, he spent most of his life there, working as a mathematician, physicist, engineer, and inventor. His mathematical achievements were extraordinary in their breadth and depth, spanning pure geometry, applied mechanics, and what amounts to a proto-calculus for computing areas, volumes, and centers of gravity.

Archimedes' work on the measurement of the circle represents one of the great achievements of ancient mathematics. Using the method of exhaustion, inscribing and circumscribing regular polygons with an increasing number of sides to squeeze the value of pi between converging bounds, he proved that pi lies between three and ten seventy-firsts and three and one-seventh, approximately 3.1408 and 3.1429. This was the most accurate estimate of pi that had yet been achieved through rigorous mathematical argument, and his method of bounding it from above and below by polygons of ninety-six sides represents a remarkable feat of calculation.

His treatise On the Sphere and Cylinder proved two spectacular results: that the surface area of a sphere is four times the area of a great circle of the sphere, and that the volume of a sphere is two-thirds the volume of the smallest cylinder that can contain it. Archimedes was so proud of these results that he reportedly asked that a sphere inscribed in a cylinder be carved on his tomb. The Roman statesman Cicero, visiting Syracuse about a century and a half after Archimedes' death, claimed to have found and restored this tomb, recognizing it by the geometric diagram upon it.

In On Spirals, Archimedes studied the curve now known as the Archimedean spiral, the curve traced by a point moving at constant speed along a ray that itself rotates at constant speed. He proved results about the areas of regions bounded by this curve that required calculations equivalent to what calculus now handles through integration. In Quadrature of the Parabola, he computed the area of a parabolic segment, proving it equals four-thirds the area of the inscribed triangle with the same base and height. His method in this work involved the summation of a geometric series, another forerunner of later analysis.

Archimedes is perhaps most famous among the general public for the story of his discovery of what is now called the Archimedes principle: that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. According to the ancient story, he made this discovery while stepping into a bath, realized its significance for solving a problem about the purity of a gold crown made for King Hiero II of Syracuse, and ran through the streets naked shouting Eureka, I have found it. The historical accuracy of this story is debated, but the principle itself is rigorously established in his treatise On Floating Bodies, which is the first systematic work in the science we now call fluid statics.

In mechanics, Archimedes gave the first rigorous mathematical treatment of the lever, proving the law of the lever from basic principles and showing that a lever in equilibrium requires the weights placed on each side to be inversely proportional to their distances from the fulcrum. His boast that he could move the earth with a lever long enough, if given a place to stand, captures the power of the lever principle to amplify force. He also wrote on pulleys, screws, and other mechanical devices, and he designed war machines including catapults and, according to later sources, arrangements of mirrors or burning lenses intended to set Roman ships on fire, though the practicality of the latter is disputed by historians.

Archimedes was killed during the Roman siege of Syracuse in 212 BCE. According to the most famous account, a Roman soldier found him drawing geometric figures in the sand and killed him when he refused to be interrupted, saying do not disturb my circles. Regardless of the truth of this story, his death marked the end of the greatest age of ancient Greek mathematics. No mathematician of comparable genius appeared for many centuries, and it is arguable that the full depth of his mathematical achievements was not equaled or surpassed until the development of calculus in the seventeenth century.

Indian Mathematics and the Number Zero

The mathematical tradition of ancient and medieval India stands among the most original and productive in world history. Indian mathematicians made fundamental contributions to arithmetic, algebra, trigonometry, and combinatorics, and above all they gave the world the decimal positional number system with zero, the most important single invention in the history of mathematics and the foundation of all modern computation.

The earliest evidence of mathematical activity in India comes from the Indus Valley civilization, which flourished around 2500 to 1500 BCE in what is now Pakistan and northwestern India. Archaeological remains reveal a sophisticated system of weights and measures using binary and decimal ratios, a uniformity suggesting extensive trade and centralized administration. The people of the Indus Valley used bricks whose dimensions follow consistent ratios, and their cities were laid out with a geometric regularity that suggests some knowledge of practical surveying.

The oldest Indian mathematical texts are the Sulbasutras, or Sulba Sutras, a series of texts attached to the Vedic religious literature and dating from roughly 800 to 200 BCE. These texts were essentially manuals for constructing Vedic fire altars of prescribed geometric shapes and sizes, a religious requirement that drove the development of considerable practical geometry. The Sulbasutras contain the earliest known explicit statements of the Pythagorean theorem in India, as well as procedures for constructing a square equal in area to a given circle and for approximating square roots. The value of the square root of two given in one Sulbasutra is accurate to five decimal places, comparable to the Babylonian value.

The great flowering of Indian mathematics occurred during the period of the Gupta Empire in the fourth through sixth centuries CE and continued through the medieval period. The mathematician and astronomer Aryabhata, born in 476 CE, wrote the Aryabhatiya, a compact work in verse that covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. Aryabhata gave an accurate approximation of pi as 3.1416, stated that the earth rotates on its axis rather than being stationary (a remarkably prescient astronomical insight), and developed methods for solving linear equations and indeterminate equations of a type later known in Europe as Diophantine equations.

Brahmagupta, born in 598 CE, was the first mathematician to give systematic rules for arithmetic operations involving the number zero and for working with negative numbers. His Brahmasphutasiddhanta, written in 628 CE, states rules for addition, subtraction, multiplication, and division with positive numbers, negative numbers, and zero. He defined zero as the result of subtracting a number from itself and gave rules such as the sum of zero and zero is zero, and a positive number divided by zero is a fraction with zero as denominator. He fell short of the modern understanding in stating that zero divided by zero equals zero, but his treatment of zero as a number in its own right, subject to the same arithmetic rules as other numbers, was a conceptual breakthrough of the first order.

The invention of the decimal positional number system, with zero as both a placeholder and a number, is one of the greatest intellectual achievements in all of human history. The precise dating and attribution of this invention remain matters of scholarly discussion, but it was clearly accomplished in India by the early centuries of the common era, and it is attested in inscriptions and texts by at least the sixth or seventh century CE. The system was transmitted to the Islamic world through translations of Indian mathematical and astronomical texts beginning in the eighth century, and from the Islamic world it passed to Europe, where it eventually displaced the cumbersome Roman numeral system and made possible the computational revolution of the modern era. The digits we use today, including zero, are called Hindu-Arabic numerals in recognition of their Indian origin and Islamic transmission.

Bhaskara II, who lived from 1114 to 1185 CE, was one of the greatest mathematicians of the medieval Indian tradition. His two major works, the Lilavati (addressed, according to legend, to his daughter) and the Bijaganita, covered arithmetic, geometry, algebra, and the solutions of quadratic, cubic, and quartic equations. The Lilavati in particular became enormously influential as a mathematical text throughout South and Southeast Asia, with commentaries and translations produced over many subsequent centuries. Bhaskara also made important contributions to astronomy and came close in several respects to the concept of differential calculus, particularly in his treatment of the derivative of the sine function.

The Kerala school of mathematics, active in the southern Indian state of Kerala from roughly the fourteenth through the sixteenth centuries, made discoveries in analysis that anticipate by more than a century the developments traditionally associated with European mathematicians. The mathematician Madhava of Sangamagrama, who lived around 1340 to 1425, derived what are now called the Leibniz formula for pi and the Gregory-Leibniz series for the arctangent, as well as series expansions for the sine, cosine, and arctangent functions. He computed pi to eleven decimal places using these series. These results, developed independently in India long before similar results were obtained in Europe, constitute one of the most remarkable episodes of convergent mathematical discovery in history.

Islamic Mathematics and Algebra

From the seventh century CE onward, the Islamic world became one of the great centers of mathematical activity, building on the Babylonian, Greek, and Indian traditions while making original contributions of the highest importance, most notably the creation of algebra as a systematic mathematical discipline. Over several centuries, Islamic scholars translated, preserved, and extended the mathematics of earlier civilizations, ensuring the survival of classical knowledge that might otherwise have been lost, and they created new fields of inquiry that shaped the subsequent development of mathematics in both the Islamic world and Europe.

The intellectual vitality of early medieval Islamic civilization was fostered by the Abbasid Caliphate, which ruled from Baghdad and made the translation of scientific and philosophical works into Arabic a major cultural project. The famous House of Wisdom in Baghdad, founded in the early ninth century under the Caliph al-Mamun, employed scholars from across the Islamic world and beyond to translate Greek, Indian, Persian, and Syriac texts. Works by Euclid, Archimedes, Apollonius, Ptolemy, and other ancient authors were translated into Arabic, often with commentaries and extensions, preserving them for posterity at a time when the originals might otherwise have been lost.

The most important Islamic mathematician was Muhammad ibn Musa al-Khwarizmi, who worked at the House of Wisdom in the early ninth century. His treatise Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the Compendious Book on Calculation by Completion and Balancing, written around 830 CE, is the founding text of algebra as a systematic discipline. The word algebra itself comes from al-jabr in the title of this work, and the word algorithm is a Latinization of al-Khwarizmi's name.

Al-Khwarizmi's algebra was not yet symbolic in the modern sense: he expressed everything in words and used no abbreviations or letter symbols for unknowns. But he systematically classified quadratic equations into six types (since he worked only with positive numbers and coefficients, he needed separate types for what we would express with a single symbolic equation), gave an algorithm for solving each type, and proved the correctness of his solutions using geometric arguments. His work synthesized Babylonian, Greek, and Indian algebraic traditions and presented them in a unified, methodical form that was immensely influential in the subsequent development of mathematics in both the Islamic world and Europe.

Al-Khwarizmi also wrote a work on the Hindu-Arabic numeral system that played a crucial role in transmitting this system to the Islamic world and eventually to Europe. His exposition of the positional decimal system with zero, adapted from Indian sources, was so influential that numbers written in this way came to be called algorism, another corruption of his name, and later arithmetic, in European languages.

Omar Khayyam, the Persian mathematician, astronomer, and poet who lived from about 1048 to 1131, made major contributions to algebra and geometry. He classified and solved all forms of cubic equations using geometric constructions involving the intersections of conic sections, work that went significantly beyond what any earlier mathematician had accomplished. He also questioned whether Euclid's parallel postulate was necessary, developing a proto-non-Euclidean geometry in the process, and he reformed the Persian calendar, creating a solar calendar more accurate than the Julian calendar then in use in Europe.

The Islamic tradition also made major advances in trigonometry, transforming it from a geometrical subject dealing with chords to a full analytical science dealing with the six trigonometric functions we use today. Abu al-Wafa al-Buzjani, working in Baghdad in the tenth century, introduced the tangent function, computed detailed trigonometric tables, and discovered the spherical law of sines. Al-Biruni, the great eleventh-century scholar, applied trigonometry to geography and astronomy and computed the circumference of the earth with remarkable accuracy using a clever method based on the angle of depression of the horizon seen from a mountain of known height.

Islamic mathematicians also made contributions to number theory, combinatorics, and the theory of infinite series. Ibn al-Haytham, known in Europe as Alhazen, who lived from about 965 to 1040, derived formulas for the sum of the first n integers, first n squares, and first n cubes, and he used these results to compute the volumes of solids of revolution, anticipating methods of integral calculus. Al-Karaji, working around 1000 CE, gave the first proof of the binomial theorem for positive integer exponents and developed the algebra of polynomials, treating them as formal objects subject to algebraic manipulation.

The transmission of Islamic mathematics to Europe occurred primarily through Spain, which was under Islamic rule from the early eighth century to the late fifteenth century, and through southern Italy and Sicily, also regions of significant Islamic cultural influence in the medieval period. Latin translations of Arabic mathematical texts, produced especially in the twelfth and thirteenth centuries, introduced European scholars to algebra, the Hindu-Arabic numeral system, and much of the recovered Greek mathematical tradition. Without the Islamic preservation and transmission of ancient mathematics, the subsequent development of European mathematics would have been far more difficult and delayed.

Chinese Mathematics

Chinese mathematics developed independently of the Greek and Indian traditions and reached remarkable levels of sophistication. The Chinese mathematical tradition is characterized by a strongly practical orientation, close attention to numerical computation, and a preference for algorithmic methods over formal axiomatic proof. It produced important results in arithmetic, algebra, and geometry, and made original contributions that were not replicated in the Western tradition for many centuries.

The oldest surviving Chinese mathematical text is the Zhoubi Suanjing, or Mathematical Classic of the Zhou Gnomon, which dates in its present form to around the first or second century BCE but may preserve older material. It is primarily an astronomical text but contains mathematical discussions including a statement and apparent proof of the Pythagorean theorem, here called the gougu theorem. The text uses a diagram showing a right triangle with squares on all three sides, and the argument, while not rigorous by Greek standards, effectively demonstrates the relationship between the areas of the squares.

The most important ancient Chinese mathematical text is the Jiuzhang Suanshu, or Nine Chapters on the Mathematical Art, which reached its definitive form around the first century CE though drawing on older sources. This text, which served as the standard mathematical curriculum in China for over a thousand years, covers a remarkably wide range of topics organized into nine chapters: measurement of fields (areas), millet and rice (proportions), distribution of goods (more proportions), short width (areas and volumes), construction consultations (volumes), fair taxes, excess and deficiency, equations (systems of linear equations), and right triangles.

The chapter on excess and deficiency presents the method that in medieval Islamic and European mathematics was called the rule of false position, used to solve linear equations. The equations chapter presents the method of exhaustion for solving systems of linear equations by what amounts to Gaussian elimination, named after the nineteenth-century German mathematician Carl Friedrich Gauss, though the Chinese had the method nearly two thousand years earlier. The same chapter also handles systems with negative coefficients, introducing negative numbers in a computational context well before they appeared in European mathematics.

Among the most impressive mathematical achievements in ancient China was the computation of pi. The mathematician Liu Hui, working in the third century CE, used a polygon method similar to that of Archimedes to compute pi as 3.14159, accurate to five decimal places. In the fifth century CE, the mathematician Zu Chongzhi computed pi to be between 3.1415926 and 3.1415927, which corresponds to accuracy to seven decimal places. His fraction 355/113 as an approximation to pi is extraordinarily accurate, differing from the true value by less than one ten-millionth, and this level of precision was not achieved in Europe until the sixteenth century.

The Chinese mathematical tradition also includes early work on what is now called the Chinese Remainder Theorem, a result in number theory concerning the simultaneous solution of congruence equations. The Sun Tzu Suanjing, or Mathematical Classic of Master Sun, written sometime in the third through fifth centuries CE, contains the following problem: there is an unknown number of things. If we count by threes, we have two left over. If we count by fives, we have three left over. If we count by sevens, we have two left over. How many things are there? The answer is twenty-three, and the method for solving such problems is the substance of what later became the Chinese Remainder Theorem, a result with important applications in modern number theory and computer science.

Pascal's triangle, the triangular array of binomial coefficients, appeared in Chinese mathematical texts significantly earlier than its appearance in Europe. The arrangement is shown in the treatise Precious Mirror of the Four Elements by Zhu Shijie, written in 1303, where it is called the Old Method Chart of the Seven Multiplying Squares, and is said to be of old origin, suggesting it was known considerably earlier. Earlier Chinese mathematical works from the eleventh and twelfth centuries also depict what is clearly Pascal's triangle.

The tradition of Chinese mathematics continued to develop through the medieval period, with mathematicians like Qin Jiushao and Yang Hui making advances in numerical methods, higher-degree equations, and combinatorics. Qin Jiushao's Mathematical Treatise in Nine Sections, written in 1247, contains the most systematic treatment of the Chinese Remainder Theorem and also algorithms for solving high-degree polynomial equations that anticipated methods not independently discovered in Europe for several more centuries.

Medieval European Mathematics

The history of mathematics in medieval Europe is often characterized as a period of relative stagnation compared with the activity occurring simultaneously in the Islamic world and India. This characterization has some justification: the collapse of the Western Roman Empire in the fifth century CE led to a significant disruption in the transmission of classical learning, and the mathematical activity of the early medieval period in Europe was largely limited to the practical arithmetic needed for commerce, the calendar, and basic surveying. Yet the medieval period also saw the gradual recovery of ancient mathematical knowledge, the arrival of new mathematical traditions from the Islamic world, and the beginnings of the mathematical tradition that would flower in the Renaissance and beyond.

The preservation of some ancient mathematical knowledge in early medieval Europe owed much to the work of scholars like Boethius, a Roman philosopher who lived from about 477 to 524 CE. Boethius translated or summarized works of Greek mathematics and philosophy into Latin, and his translations and textbooks on arithmetic, geometry, and music formed the foundation of mathematical education in the early medieval period. His Institutio Arithmetica, adapted from the work of the Neoplatonist Nicomachus of Gerasa, was widely used in European schools for centuries and preserved at least a part of the ancient tradition of number theory.

The recovery of classical and Islamic mathematical texts, transmitted through Latin translations made primarily in the twelfth and thirteenth centuries, transformed European mathematical culture. Translators working in Spain, Sicily, and other regions of contact between the Islamic world and European Christendom made the works of Euclid, Archimedes, al-Khwarizmi, and many other mathematicians available in Latin for the first time or in much improved translations. The introduction of the Hindu-Arabic numeral system into Europe through these translations was particularly consequential.

The Italian mathematician Leonardo of Pisa, better known as Fibonacci, played a major role in introducing both the Hindu-Arabic numeral system and Islamic algebra to Europe through his 1202 work Liber Abaci, the Book of Calculation. This extensive work explained the new numeral system and its advantages for commercial calculation, demonstrated the techniques of Islamic algebra applied to problems of commercial arithmetic, and included a famous problem about the growth of a rabbit population whose solution involves what are now called Fibonacci numbers, the sequence in which each number is the sum of the two preceding ones.

The universities that began to form in Europe in the twelfth and thirteenth centuries, in cities such as Bologna in Italy, Paris in France, and Oxford in England, included mathematics in their curricula as part of the quadrivium, the four subjects of arithmetic, geometry, music, and astronomy, alongside the trivium of grammar, rhetoric, and logic. The mathematical content of university education in this period was largely drawn from the recovered classical and Islamic tradition, but the institutional setting of the universities provided a framework within which mathematical ideas could be discussed, refined, and transmitted across generations.

Nicole Oresme, a French mathematician and philosopher who lived from about 1320 to 1382, made important original contributions that anticipate later developments. He invented a system of graphing the variation of a physical quantity by representing it as a two-dimensional figure with a horizontal axis for time and a vertical axis for the magnitude of the quantity, an early form of the idea of a graph that would later be fully developed by Descartes. He also demonstrated that the harmonic series, the sum of one plus one-half plus one-third plus one-quarter and so on, diverges, meaning it eventually exceeds any finite bound. This was a subtle and important result in the theory of infinite series.

The late medieval period also saw important developments in the mathematics of probability, driven by the practical needs of gamblers and merchants. Italian mathematicians like Luca Pacioli and Gerolamo Cardano began to analyze games of chance mathematically, laying early groundwork for what would eventually become the theory of probability. The medieval period thus, despite its reputation for mathematical quietude, was in fact a time of important preparation for the explosive development of mathematics that followed in the Renaissance and the seventeenth century.

The Renaissance and Mathematical Progress

The Renaissance, the great cultural rebirth that began in Italy in the fourteenth century and spread across Europe through the fifteenth and sixteenth centuries, brought with it a renewed enthusiasm for ancient learning and a new spirit of inquiry that transformed mathematics along with every other intellectual discipline. Renaissance mathematicians made important advances in algebra, solved problems that had stood open since antiquity, and developed the mathematical tools that would be needed for the great scientific revolution of the seventeenth century.

The most dramatic mathematical development of the Renaissance was the solution of the cubic and quartic equations. Medieval European mathematicians, following in the tradition of al-Khwarizmi and later Islamic algebraists, had solved quadratic equations routinely, but the cubic equation, involving the cube of the unknown, had resisted general solution. In the 1530s, the Italian mathematician Niccolo Tartaglia discovered a method for solving certain types of cubic equations, and Gerolamo Cardano, after obtaining the method from Tartaglia under a pledge of secrecy, published it along with a method for solving quartic equations (discovered by his student Lodovico Ferrari) in his landmark work Ars Magna, the Great Art, in 1545.

The Ars Magna is a landmark in the history of algebra. Beyond the cubic and quartic formulas themselves, it introduced, however tentatively, the idea of complex numbers, numbers involving the square root of negative one, later denoted i. Cardano noticed that his formulas sometimes required taking the square root of a negative number even when the final answer was a real number, and while he found this mysterious and somewhat troubling, he recognized that the formulas worked. The full development of complex numbers as a legitimate and useful mathematical system required further work by Rafael Bombelli and others, but the seeds were planted in Cardano's work.

The development of logarithms in the early seventeenth century, independently by the Scottish mathematician John Napier and the Swiss mathematician Joost Bürgi, was a major practical advance with profound theoretical implications. Logarithms convert multiplication into addition, enormously simplifying the computational labor required for the astronomical and navigational calculations that were becoming increasingly important in this period. Napier published his discovery in 1614, and the rapid adoption of logarithm tables by astronomers, navigators, engineers, and scientists over the following decades illustrates how mathematical inventions respond to and in turn enable practical needs.

The development of symbolic algebra, the replacement of the wordier rhetorical algebra of al-Khwarizmi and his successors with a compact notation using letters for unknowns and symbols for operations, was a crucial step in the development of modern mathematics. The French mathematician François Viète, who lived from 1540 to 1603, pioneered the use of letters to represent both known and unknown quantities, allowing general algebraic identities to be stated and manipulated symbolically for the first time. His work transformed algebra from a science of specific problem-solving into a science of general structures and relationships.

René Descartes, the French philosopher and mathematician who lived from 1596 to 1650, made what is perhaps the single most important technical contribution to the development of modern mathematics: the invention of analytic geometry, also known as Cartesian geometry. Analytic geometry provides a bridge between algebra and geometry by introducing a coordinate system in which points are represented by pairs of numbers, allowing geometric curves to be described by algebraic equations and algebraic relationships to be visualized as geometric figures. Descartes' La Géométrie, published in 1637 as an appendix to his philosophical work Discourse on the Method, laid out this idea and its consequences, fundamentally changing the relationship between the two great branches of classical mathematics.

Pierre de Fermat, the French lawyer and amateur mathematician who lived from 1607 to 1665, independently developed ideas very similar to Descartes' analytic geometry and made major contributions to number theory, probability, and the foundations of calculus. His Last Theorem, stated without proof in the margin of his copy of Diophantus's Arithmetica, asserted that the equation x to the n plus y to the n equals z to the n has no solutions in positive integers when n is greater than two. This deceptively simple statement, unproved for more than three and a half centuries, became one of the most famous unsolved problems in the history of mathematics and was finally proved by the English mathematician Andrew Wiles in 1995.

The Development of Calculus

The creation of calculus in the second half of the seventeenth century by Isaac Newton and Gottfried Wilhelm Leibniz was the greatest mathematical achievement of the early modern period and one of the most consequential intellectual events in history. Calculus provided the mathematical tools needed to describe continuous change and motion, enabling the quantitative description of the physical world that is the foundation of modern science and engineering. The story of its discovery, involving a priority dispute of extraordinary bitterness between two of the greatest minds of the age, is one of the most dramatic episodes in the history of science.

The mathematical problems that calculus was created to solve had been accumulating for centuries. The calculation of areas and volumes of curved regions, the determination of tangent lines to curves, the optimization of functions, and the description of motion and change were all pressing problems that earlier mathematics could handle only in special cases and with great difficulty. Archimedes had solved many such problems for specific figures using the method of exhaustion, but no general method existed. The Italian mathematician Evangelista Torricelli and the French mathematicians Fermat and Gilles de Roberval made important advances in the early seventeenth century, but the crucial step of recognizing the deep connection between the two fundamental problems, finding tangent lines and finding areas, was left to Newton and Leibniz.

Isaac Newton, who lived from 1642 to 1727, developed what he called the method of fluxions during the remarkable period from 1665 to 1666, when Cambridge University was closed because of plague and Newton retired to his family home in Woolsthorpe in Lincolnshire, England. In this period of enforced solitude, he developed not only the foundations of calculus but also his laws of motion and the law of universal gravitation. Newton's approach to calculus was primarily dynamic: he thought of mathematical quantities as flowing in time, and the derivative of a quantity was its rate of flow or fluxion. He developed rules for differentiating algebraic functions, related differentiation to the computation of tangents, and recognized that differentiation and integration were inverse operations, the result now called the fundamental theorem of calculus.

Newton was reluctant to publish his mathematical work, and his calculus was not widely known until much later. Gottfried Wilhelm Leibniz, the German polymath philosopher and mathematician who lived from 1646 to 1716, developed calculus independently, apparently beginning his work around 1675, and published his results in 1684 and 1686. Leibniz's notation, using d for differentials and the elongated S symbol for integrals, was far superior to Newton's and has been adopted universally. The rules he stated for differentiating sums, products, and quotients of functions are still taught in every calculus course today in the same form in which he stated them.

The priority dispute between Newton and Leibniz, which escalated into a bitter nationalistic controversy pitting English scientists against Continental Europeans, was damaging to both parties and to the development of mathematics in England, where loyalty to Newton's notation delayed the adoption of Leibniz's more powerful symbolism for a century. The dispute began in earnest in the early eighteenth century, when accusations of plagiarism were exchanged through intermediaries, and reached its nadir in 1712 when a committee of the Royal Society, stacked with Newton's supporters and reportedly directed by Newton himself, issued a report declaring Newton's priority. Modern historians of mathematics generally conclude that Newton developed calculus first but that Leibniz arrived at his version independently.

The power of the calculus as a tool for understanding the physical world was demonstrated almost immediately by its application to Newtonian mechanics. Newton's laws of motion and gravitation, expressed as differential equations using the language of calculus, could be used to derive Kepler's laws of planetary motion as mathematical consequences, to predict the return of comets, to calculate the shape of the earth's orbit, and to solve a host of other physical problems that had been inaccessible to earlier mathematics. The success of this program was so spectacular that it transformed the image of what mathematics could accomplish and set the agenda for mathematical physics for the next two centuries.

The eighteenth century saw the rapid development of the methods of calculus into a full analytical framework by the great mathematicians of the Continental tradition, above all Leonhard Euler, Joseph-Louis Lagrange, and Pierre-Simon Laplace. Euler, who lived from 1707 to 1783 and was one of the most prolific mathematicians in history, developed the calculus of variations, contributed enormously to differential equations and infinite series, introduced much of the standard notation still used today, and made fundamental contributions to every branch of mathematics then known. His mathematical output, much of it produced after he became blind in both eyes, fills some ninety volumes.

The development of calculus also prompted serious questions about its logical foundations. The use of infinitely small quantities called infinitesimals, which appear in the original formulations of both Newton and Leibniz, was criticized by the philosopher George Berkeley in 1734 as logically incoherent. The rigorous foundations of calculus were established in the nineteenth century, primarily through the work of Augustin-Louis Cauchy and Karl Weierstrass, who replaced the intuitive notion of infinitesimals with precise epsilon-delta definitions of limits, placing the entire subject on a solid logical foundation and beginning the era of mathematical analysis.

The 17th and 18th Century Mathematical Revolution

The seventeenth and eighteenth centuries witnessed an explosion of mathematical activity across Europe that fundamentally transformed the discipline and its relationship to the physical world. Beyond the development of calculus, this period saw the creation of probability theory, the development of analytic geometry, important advances in number theory, the mathematization of physics, and the emergence of new algebraic and geometric tools that would bear fruit in the following centuries.

Blaise Pascal, the French mathematician and philosopher who lived from 1623 to 1662, made important contributions to several branches of mathematics. His work on the arithmetic triangle, now called Pascal's triangle, which gives the binomial coefficients and has applications in combinatorics and probability, was systematic and influential, though as noted earlier the triangle itself had been known to Chinese, Persian, and Indian mathematicians centuries before Pascal. More original was his work with Pierre de Fermat founding the mathematical theory of probability, developed through a famous correspondence about how to divide the stakes in an interrupted game of chance.

The mathematical theory of probability that Pascal and Fermat founded was developed and extended by the Dutch mathematician Christiaan Huygens, who published the first systematic treatise on probability in 1657, and later by Jacob Bernoulli, Abraham de Moivre, and Pierre-Simon Laplace. Bernoulli's Ars Conjectandi, published posthumously in 1713, proved the law of large numbers, the theorem that the relative frequency of an event approaches its probability as the number of trials grows indefinitely. De Moivre discovered the central limit theorem in 1733, showing that the sum of many independent random variables tends toward a normal, or bell-shaped, distribution. Laplace gave probability theory its most complete eighteenth-century treatment in his Théorie Analytique des Probabilités of 1812.

Leonhard Euler's contributions to mathematics were so vast and varied that they are difficult to summarize briefly. His Introductio in Analysin Infinitorum of 1748 codified much of eighteenth-century analysis and introduced the modern concept of a function. He established the connections between the exponential and trigonometric functions through what is now called Euler's formula, expressing e to the power of i times theta as the cosine of theta plus i times the sine of theta, a result whose special case with theta equal to pi gives the famous equation linking the five most important constants of mathematics. He solved the Konigsberg bridge problem in 1736, asking whether it is possible to walk across all seven bridges of that Prussian city each exactly once and return to the starting point, and in doing so founded the branch of mathematics called graph theory.

Euler also made fundamental contributions to number theory, solving several problems that had been open since antiquity. He proved Fermat's little theorem, showing that if p is a prime and a is not divisible by p, then a to the power p minus one, minus one, is divisible by p. He proved that the sum of the reciprocals of the squares of the positive integers equals pi squared divided by six, a result known as the Basel problem that had resisted the efforts of other mathematicians. He introduced the concept of the totient function and the first modern treatment of what we now call modular arithmetic.

The development of differential equations during this period was closely linked to the demands of physics and engineering. Newton's second law of motion is itself a differential equation relating force, mass, and acceleration. The motion of vibrating strings, the flow of heat, the behavior of electrical circuits, and the mechanics of rotating bodies all required differential equations for their mathematical description. The mathematicians of the eighteenth century, including Euler, d'Alembert, and Bernoulli, developed both ordinary and partial differential equations and found solutions for many of the most important physical cases. The mathematics of differential equations, which grew directly from the calculus of Newton and Leibniz, became the primary language of mathematical physics and remains so today.

The application of mathematics to the understanding of nature reached its triumphant eighteenth-century expression in the work of Pierre-Simon Laplace, whose multivolume Mécanique Céleste, published between 1799 and 1825, provided a comprehensive mathematical treatment of the solar system based on Newton's gravitational theory. The precision with which the theory matched observation, and the success of Laplace's predictions about the behavior of the moon, Jupiter, Saturn, and the tides, was a stunning demonstration of the power of mathematics as a tool for understanding the universe. When Napoleon reportedly asked Laplace why his treatise contained no mention of God, Laplace is said to have replied that he had no need of that hypothesis.

Non-Euclidean Geometry

For more than two thousand years after Euclid wrote the Elements, his parallel postulate was regarded with a mixture of admiration and unease. Admiration, because the geometry built upon it so perfectly described the world of everyday experience. Unease, because unlike the other four postulates, which seemed self-evidently simple, the parallel postulate was complex and, many felt, suspiciously non-obvious. A long line of mathematicians, from the ancient Greeks through the medieval Islamic world to the mathematicians of the seventeenth and eighteenth centuries, attempted to prove that the parallel postulate followed logically from the other four, which would have made it a theorem rather than an independent assumption. All of these attempts failed, though many produced important and interesting results in the process.

The decisive break came in the early nineteenth century, when three mathematicians independently and at roughly the same time arrived at the conclusion that consistent geometries are possible in which the parallel postulate fails. The Hungarian mathematician Janos Bolyai, the Russian mathematician Nikolai Ivanovich Lobachevsky, and the German mathematician Carl Friedrich Gauss all developed what is now called hyperbolic geometry, in which through a given point not on a given line, infinitely many lines can be drawn parallel to the given line.

Gauss, who lived from 1777 to 1855 and was perhaps the greatest mathematician of the nineteenth century, arrived at non-Euclidean geometry as early as the 1810s but never published his results, reportedly fearing the controversy they would provoke. Lobachevsky published first in 1830, and Bolyai's work appeared in 1832 as an appendix to a book by his father. The discovery that Euclidean geometry, far from being the unique and necessary description of space, was merely one consistent geometry among others, was one of the most intellectually revolutionary results in mathematical history.

A second type of non-Euclidean geometry, called spherical or elliptic geometry, is exemplified by the geometry on the surface of a sphere. On a sphere, there are no parallel lines at all: any two great circles, the sphere's analogs of straight lines, intersect in exactly two points. The angles of a spherical triangle sum to more than one hundred eighty degrees, and the larger the triangle, the greater the excess. Spherical geometry had been studied by ancient Greek and Islamic astronomers in connection with the celestial sphere, but its significance as a geometry inconsistent with Euclid's parallel postulate was not fully appreciated until the nineteenth century.

The rigorous mathematical treatment of non-Euclidean geometries was provided by Bernhard Riemann in his famous 1854 lecture On the Hypotheses which Lie at the Foundations of Geometry, delivered at the University of Gottingen in Germany. Riemann introduced a far more general concept of geometry in which the curvature of space could vary from point to point, encompassing both Euclidean geometry, with zero curvature everywhere, and the various non-Euclidean geometries as special cases. Riemannian geometry, as his creation came to be called, provided the mathematical language that Einstein needed sixty years later to formulate his general theory of relativity, in which gravity is described as the curvature of spacetime.

The discovery of non-Euclidean geometry had profound philosophical consequences. If Euclidean geometry was not the unique necessary geometry of physical space, then the foundations of mathematics could not be grounded in intuitions about physical space. Mathematics was no longer simply the idealized description of the spatial world but a free, autonomous discipline whose only requirement was internal logical consistency. This realization, combined with the development of abstract algebra and mathematical logic in the nineteenth century, led to the modern axiomatic view of mathematics, in which a mathematical theory is any system of propositions derived by logic from a set of axioms, without regard to whether those axioms describe anything in the physical world.

The influence of non-Euclidean geometry extended far beyond its immediate application to physics. It opened the door to the systematic study of geometries with different properties, to topology, which studies the properties of spaces that are preserved under continuous deformation, and to the entire program of abstract mathematics that defines the twentieth century. The liberation of geometry from the constraint of physical intuition was a liberation of mathematics itself, allowing mathematicians to explore any logically consistent structure, whether or not it had any obvious connection to the physical world, with the understanding that such abstract explorations might find unexpected applications in the future.

Abstract Algebra and Number Theory

The nineteenth century witnessed a profound transformation in the nature of algebra. What had been, from al-Khwarizmi to the Renaissance and beyond, a discipline primarily concerned with the solution of specific types of equations became, in the hands of nineteenth-century mathematicians, a science of abstract structures: groups, rings, fields, and other algebraic systems defined by axioms and studied in their own right, regardless of any particular interpretation. This transformation, one of the most important in mathematical history, created the branch of mathematics now called abstract algebra, which underlies much of modern mathematics and has applications across science, engineering, and computer science.

The central motivating problem of early nineteenth-century algebra was the question of which polynomial equations could be solved by formulas involving only the arithmetic operations and the extraction of roots. The quadratic formula, known since antiquity, and the cubic and quartic formulas discovered in the Renaissance, showed that equations of degree two, three, and four could all be so solved. The natural question was whether a similar formula existed for the general quintic, the equation of degree five. The first definitive answer to this question came from the Norwegian mathematician Niels Henrik Abel, who proved in 1824, at the age of twenty-one, that no general formula for solving the quintic by radicals can exist.

The deeper and more complete answer was provided by the French mathematician Evariste Galois, whose brief and tragic life ended with his death in a duel in 1832 at the age of twenty. In his mathematical work, written partly the night before his fatal duel and long disregarded before being recognized as extraordinary, Galois developed the theory of groups, associating with each polynomial equation a group of permutations of its roots. He showed that an equation is solvable by radicals if and only if its associated group has a certain property now called solvability. This criterion explained not only why the general quintic is unsolvable by radicals, but why equations of degree five and higher generally fail to have radical solutions while those of lower degree always succeed.

Galois theory is one of the supreme achievements of nineteenth-century mathematics. It established the theory of groups as a fundamental mathematical tool and demonstrated that abstract algebraic structures could provide deep insights into apparently unrelated mathematical problems. The group concept, drawn from Galois's work and from parallel developments in the study of symmetry, number theory, and geometry, grew through the nineteenth century into one of the most pervasive concepts in all of mathematics and science. In physics, the symmetry groups of physical laws are the basis of modern particle physics and the standard model of fundamental interactions.

Number theory, the ancient study of the properties of integers that goes back to Euclid and Diophantus, experienced a renaissance in the nineteenth century through the work of Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet, and Bernhard Riemann, among others. Gauss's Disquisitiones Arithmeticae, published in 1801 when he was twenty-four years old, is one of the masterpieces of mathematical literature. It introduced the concept of congruence relations between integers, developed the theory of quadratic forms, gave the first rigorous proof of the law of quadratic reciprocity, proved that every positive integer is a sum of at most three triangular numbers, and proved the fundamental theorem of arithmetic, establishing on a rigorous basis that every positive integer greater than one has a unique prime factorization.

The distribution of prime numbers among the integers is one of the oldest and deepest questions in number theory. Gauss conjectured, and later mathematicians proved, that the number of primes up to a given number x is approximately x divided by the natural logarithm of x, a result known as the prime number theorem. Riemann, in a celebrated 1859 paper that is among the most influential in mathematical history, introduced the Riemann zeta function and its complex zeros, and showed how the distribution of primes is intimately related to the locations of these zeros in the complex plane. The Riemann hypothesis, asserting that all non-trivial zeros of the zeta function lie on a certain vertical line in the complex plane, remains unproved today and is perhaps the most famous unsolved problem in mathematics.

Statistics and Probability

The mathematical theory of statistics and probability grew from the gambling problems of the seventeenth century into one of the most practically important branches of mathematics, providing the conceptual and computational tools for making inferences from data, quantifying uncertainty, and modeling random phenomena across the natural and social sciences. The development of statistics from its mathematical origins to its current role as the language of data science and empirical inquiry is one of the great stories of applied mathematics.

The foundations of probability theory laid by Pascal, Fermat, and Huygens in the seventeenth century were extended in the eighteenth century by the Bernoulli family, particularly Jacob Bernoulli, and by Abraham de Moivre and Pierre-Simon Laplace. Jacob Bernoulli's law of large numbers, published in 1713, was the first rigorous theorem about the long-run behavior of random experiments, establishing the connection between theoretical probability and empirical frequency. De Moivre's 1733 discovery of the normal distribution as a limiting form of the binomial distribution was the first major result in what we now call the central limit theorem, one of the most important theorems in all of probability theory.

Laplace and the German mathematician Carl Friedrich Gauss independently discovered the method of least squares around 1800, a method for fitting a mathematical model to data by minimizing the sum of squared differences between observed and predicted values. Gauss applied it to the problem of computing the orbit of the newly discovered asteroid Ceres from a limited set of observations, successfully predicting its position after it was lost from sight, and the spectacular success of this prediction made the method of least squares the standard tool for fitting curves to experimental data. The normal distribution, which arises as the distribution of errors in this context, is sometimes called the Gaussian distribution in Gauss's honor.

The nineteenth century saw the development of statistics as an autonomous discipline concerned not just with probability theory but with the collection, organization, and interpretation of numerical data. The Belgian statistician Adolphe Quetelet, who lived from 1796 to 1874, applied statistical methods to social phenomena, computing average values for a range of physical and social characteristics across large populations and introducing the concept of the average man, l'homme moyen, as a statistical abstraction representing the central tendency of a population. His work pioneered the use of statistics in the social sciences and influenced the development of public health, criminology, and sociology.

Francis Galton, the English polymath who lived from 1822 to 1911, introduced the concepts of regression toward the mean and correlation in his studies of heredity and the transmission of characteristics across generations. He noticed that the children of unusually tall parents tended to be taller than average but shorter than their parents, and called this phenomenon regression toward mediocrity, now known as regression toward the mean. His statistical ideas were formalized and extended by his student Karl Pearson, who developed the mathematical theory of the correlation coefficient, the chi-squared test, and many other tools of modern statistics.

The twentieth century saw the development of mathematical statistics into a rigorous discipline with strong theoretical foundations. Ronald Aylmer Fisher, the English statistician and geneticist who lived from 1890 to 1962, was perhaps the most important figure in this development. His contributions included the development of the analysis of variance, the concept of maximum likelihood estimation, the design of experiments, and the foundations of modern hypothesis testing. His books Statistical Methods for Research Workers and The Design of Experiments, published in 1925 and 1935 respectively, shaped the statistical practice of scientists across disciplines for generations.

The second half of the twentieth century saw statistics transformed by the advent of computing, which made feasible calculations that were previously impractical, and by the rise of new methods such as Bayesian statistics, nonparametric methods, and the vast range of machine learning algorithms that have grown from statistical foundations. The modern data revolution, in which enormous quantities of data are collected in every domain of human activity, has made statistical reasoning more important than ever and has created new mathematical challenges in the theory of high-dimensional statistics, compressed sensing, and the analysis of random matrices and networks.

The 20th Century and Pure Mathematics

The twentieth century was arguably the most creative and productive period in the entire history of mathematics. The sheer volume and diversity of mathematical work accomplished in this century surpasses that of any previous period, and the range of new mathematical disciplines created, the depth of problems solved, and the unexpected connections discovered between apparently distant areas of mathematics make this era a golden age for the subject.

The century opened with a challenge that set its agenda. In 1900, the German mathematician David Hilbert, the preeminent mathematical figure of his time, delivered an address to the International Congress of Mathematicians in Paris in which he posed twenty-three problems as challenges for the coming century. These problems, ranging from foundational questions about the nature of mathematics itself to specific conjectures in number theory, geometry, and mathematical physics, guided mathematical research for decades and several remain unsolved today. Hilbert's list defined in many ways what the mathematical community regarded as the most important open questions, and the history of twentieth-century mathematics is in part the history of progress on this list.

Hilbert himself was a towering figure whose influence shaped the development of mathematics in the early decades of the century. His Grundlagen der Geometrie, Foundations of Geometry, published in 1899, provided the rigorous axiomatic treatment of Euclidean geometry that Euclid's original presentation had always lacked, making explicit all the assumptions needed and proving everything from these axioms. His program for the foundations of all mathematics, the Hilbert program, aimed to establish all of mathematics on a finite, consistent set of axioms from which every mathematical truth could in principle be derived. This program, ambitious and seemingly plausible in the 1920s, was shattered in 1931 by one of the most stunning results in mathematical history.

Kurt Godel, the Austrian logician who lived from 1906 to 1978, proved in 1931 his incompleteness theorems, two results of devastating implications for Hilbert's program. The first incompleteness theorem states that any consistent formal system powerful enough to describe ordinary arithmetic must contain true statements that cannot be proved within the system. The second states that such a system cannot prove its own consistency. These theorems demonstrated that Hilbert's dream of a complete and provably consistent foundation for all of mathematics was impossible in principle, not just difficult in practice. Godel's theorems are among the deepest and most surprising results in the history of logic and the philosophy of mathematics.

The twentieth century saw the completion of many problems that had stood open for centuries. Andrew Wiles proved Fermat's Last Theorem in 1995, after three and a half centuries as perhaps the most famous unsolved problem in mathematics. The Poincare conjecture, posed by Henri Poincare in 1904 and asking whether any simply connected closed three-dimensional manifold is homeomorphic to a three-sphere, was proved by the Russian mathematician Grigori Perelman around 2003, using methods from geometric analysis that revolutionized the approach to such topological problems.

The twentieth century also saw the creation of entirely new mathematical disciplines. Topology, the study of properties preserved under continuous deformation, grew from the foundational insights of Poincare, Emmy Noether, and many others into one of the major branches of mathematics. Functional analysis, which extends calculus to infinite-dimensional spaces, was developed by mathematicians including David Hilbert, Stefan Banach, and John von Neumann and became essential for the mathematical formulation of quantum mechanics. Algebraic topology, differential geometry, category theory, and many other fields were either created or came to maturity in the twentieth century.

The Hungarian-American mathematician John von Neumann, who lived from 1903 to 1957, was one of the most versatile and brilliant mathematicians of the century. His contributions spanned foundations of mathematics, quantum mechanics, game theory, economics, and computer science. His mathematical formulation of quantum mechanics in terms of operators on Hilbert spaces is the standard framework still used by physicists. His minimax theorem in game theory provided the foundation for the mathematical theory of strategic behavior that has become essential in economics, political science, and evolutionary biology.

Emmy Noether, the German mathematician who lived from 1882 to 1935 and was described by Einstein as the most significant creative mathematical genius thus far produced, made fundamental contributions to abstract algebra and theoretical physics. Her theorem, now called Noether's theorem, proved in 1915 and published in 1918, established the deep connection between symmetries of a physical system and conservation laws, showing for example that the symmetry of physical laws under translation in time corresponds to the conservation of energy. This result, fundamental to all of modern physics, is a direct application of abstract algebraic thinking to the physical world.

Computers and Mathematical Logic

The relationship between mathematics and computation is ancient, but the twentieth century transformed it profoundly. The development of mathematical logic in the late nineteenth and early twentieth centuries, combined with the practical needs of the Second World War, led directly to the creation of electronic digital computers, which in turn revolutionized both the practice of mathematics and its applications. The study of computation itself became a major branch of mathematical research, yielding deep results about what can and cannot be computed and about the inherent difficulty of computational problems.

The mathematical foundations of computing were laid in the 1930s, before the first electronic computer was built. Alan Turing, the English mathematician who lived from 1912 to 1954, introduced in 1936 the theoretical concept of a universal computing machine, now called a Turing machine, as a precise mathematical model of computation. A Turing machine is an abstract device with a finite set of internal states, an infinite tape for input and output, and a transition function specifying its behavior at each step. Turing proved that any computation that could in principle be performed by any mechanical procedure could be performed by an appropriate Turing machine, establishing this model as the mathematical foundation for the theory of computation.

Turing also proved the first major result about the limits of computation: the undecidability of the halting problem. He showed that there is no general algorithm that can determine, for any given program and any given input, whether that program will eventually halt or run forever. This result, proved in 1936, demonstrated that there are mathematically well-defined questions that no algorithm can answer, complementing and in some ways extending Godel's incompleteness theorems to the domain of computation.

The practical development of electronic computers proceeded alongside and intertwined with the mathematical theory. During the Second World War, Turing contributed to the British code-breaking effort at Bletchley Park in England, working on the mathematical methods used to crack the German Enigma cipher. John von Neumann contributed crucially to the design of the first electronic digital computers in the mid-1940s, introducing the stored-program architecture in which both data and program instructions are stored in the same memory, a feature of virtually all computers built since.

The development of computers created new mathematical problems and new methods for solving old ones. Numerical analysis, the branch of mathematics concerned with the design and analysis of algorithms for computing numerical approximations to mathematical problems, became enormously important as computers made such calculations feasible on a scale previously unimaginable. The fast Fourier transform algorithm, discovered by James Cooley and John Tukey in 1965, dramatically reduced the computational cost of computing Fourier transforms and enabled a revolution in signal processing, data compression, and scientific computing.

The theory of computational complexity, which classifies problems according to the resources, primarily time and memory, required to solve them algorithmically, became a major area of mathematical research in the second half of the twentieth century. The P versus NP problem, asking whether every problem whose solution can be quickly verified can also be quickly solved, was formulated around 1971 and is now one of the Millennium Prize Problems, offering a million-dollar prize for its resolution. The resolution of this problem would have profound implications for cryptography, optimization, artificial intelligence, and the foundations of mathematics.

Computers have also changed the practice of mathematics in more direct ways. Computer-aided proofs, in which computers are used to check vast numbers of cases that would be impractical to verify by hand, have been used to prove several famous theorems. The four color theorem, which states that any planar map can be colored with at most four colors such that no two adjacent regions share a color, was proved in 1976 by Kenneth Appel and Wolfgang Haken with extensive computer assistance, checking 1,476 reducible configurations. This was the first major theorem whose proof relied essentially on computer computation and provoked philosophical debate about the nature of mathematical proof.

Modern cryptography, which secures internet communications, financial transactions, and state secrets, rests on deep mathematical foundations in number theory and algebra. Public-key cryptography, invented by Whitfield Diffie and Martin Hellman in 1976 and independently by Ronald Rivest, Adi Shamir, and Leonard Adleman in 1977 in the RSA system, relies on the computational difficulty of factoring large numbers into primes, a problem deeply connected to the theory of prime numbers studied by ancient Greek and later mathematicians. The practical security of billions of internet users today depends on mathematical results developed by number theorists who had no thought of practical application.

Famous Mathematicians Through History

The history of mathematics is populated by remarkable individuals whose intellectual gifts and dedication to their subject produced the great theorems and theories that constitute the mathematical heritage of humanity. A brief survey of some of the most influential deceased mathematicians illuminates both the diversity of mathematical genius and the cumulative, collaborative nature of mathematical progress.

Euclid, who worked in Alexandria around 300 BCE, was the author of the Elements, the most influential mathematical textbook in history and the foundation of the Western mathematical tradition. Although relatively little is known about his life, his intellectual achievement in organizing the mathematical knowledge of his time into a rigorous deductive system was immense. The Elements was used as the primary textbook for teaching mathematics in European and Islamic institutions from its composition until the late nineteenth century, a span of more than two thousand years.

Archimedes of Syracuse, 287 to 212 BCE, was the supreme mathematical genius of antiquity, combining abstract mathematical achievement with brilliance in applied mechanics and engineering. His computation of pi, his formulas for the areas and volumes of curved figures, his work on the lever and buoyancy, and his approach to the summation of infinite series all placed him centuries ahead of his contemporaries and put him on a level with the great mathematicians of the seventeenth century.

Brahmagupta, born in 598 CE in India, was one of the most important mathematicians of the early medieval period. His systematic treatment of arithmetic with zero and negative numbers, his formula for the area of a cyclic quadrilateral, and his work on indeterminate equations established him as a figure of enduring importance in the history of mathematics.

Muhammad ibn Musa al-Khwarizmi, who worked in Baghdad in the early ninth century, founded algebra as a systematic discipline and introduced the Hindu-Arabic numeral system to the Islamic world and indirectly to Europe. His name gave rise to the words algebra, from al-jabr, and algorithm, from a Latinization of his name, two of the most important words in modern mathematics and computer science.

Leonardo of Pisa, Fibonacci, who lived from about 1170 to about 1250, introduced the Hindu-Arabic numeral system to Europe through his Liber Abaci and contributed to the development of algebra and number theory in the European tradition. The Fibonacci sequence, appearing in his work on rabbit population growth, has since been found in an astonishing range of natural phenomena from the spiral arrangements of seeds in sunflowers to the branching of trees.

Niccolo Tartaglia, 1500 to 1557, and Gerolamo Cardano, 1501 to 1576, made the great Renaissance breakthrough of solving the cubic and quartic equations, opening the door to the systematic study of polynomial equations and the theory of complex numbers.

John Napier, 1550 to 1617, invented logarithms, one of the most important computational tools in the history of mathematics. His tables of logarithms, and the slide rule later based on them, were the standard calculating tools of scientists and engineers from the seventeenth to the twentieth century.

René Descartes, 1596 to 1650, invented analytic geometry, unifying the previously separate disciplines of algebra and geometry and providing the coordinate system that is the basis of most mathematical visualization. His La Géométrie transformed the practice of mathematics.

Pierre de Fermat, 1607 to 1665, made fundamental contributions to number theory, including his last theorem, his little theorem, and his work on the method of infinite descent. He also contributed to the foundations of calculus and probability theory.

Blaise Pascal, 1623 to 1662, made important contributions to projective geometry, probability theory, and the theory of the arithmetic triangle. His Pascaline, one of the first mechanical calculating machines, foreshadowed the development of computing by three centuries.

Isaac Newton, 1642 to 1727, coinventor of calculus, formulator of the laws of motion and universal gravitation, and developer of many mathematical tools for physics and analysis. His Principia Mathematica of 1687 is one of the most important books in the history of science.

Gottfried Wilhelm Leibniz, 1646 to 1716, the coinventor of calculus and developer of its notation, also made contributions to logic, philosophy, and combinatorics. His notation for calculus is the one used universally today.

Leonhard Euler, 1707 to 1783, the most prolific mathematician in history, made fundamental contributions to analysis, number theory, graph theory, mechanics, and almost every other branch of mathematics of his time. He introduced most of the standard mathematical notation used today, including e, i, pi, sigma for sums, and f of x for functions.

Carl Friedrich Gauss, 1777 to 1855, the prince of mathematicians, made breakthrough contributions to number theory, algebra, statistics, differential geometry, and mathematical physics. His Disquisitiones Arithmeticae is one of the greatest works in mathematical history.

Augustin-Louis Cauchy, 1789 to 1857, placed the calculus on a rigorous logical foundation by developing the theory of limits. His textbooks shaped the teaching of analysis for a century.

Nikolai Ivanovich Lobachevsky, 1792 to 1856, and Janos Bolyai, 1802 to 1860, independently created non-Euclidean geometry, one of the most revolutionary discoveries in mathematical history.

Niels Henrik Abel, 1802 to 1829, proved the impossibility of solving the general quintic by radicals and made fundamental contributions to the theory of elliptic functions, all before his death at the age of twenty-six.

Evariste Galois, 1811 to 1832, created Galois theory and the theory of groups, with consequences that shaped the development of algebra for the next two centuries, before being killed in a duel at the age of twenty.

Bernhard Riemann, 1826 to 1866, made fundamental contributions to complex analysis, differential geometry, and number theory. His 1859 paper introducing the Riemann zeta function and the Riemann hypothesis remains perhaps the most influential single paper in the history of mathematics.

Georg Cantor, 1845 to 1918, created set theory and the theory of infinite cardinal numbers, showing that some infinities are larger than others and revolutionizing the foundations of mathematics.

Henri Poincare, 1854 to 1912, was one of the last mathematicians to make major contributions across the entire breadth of the discipline. His work in topology, celestial mechanics, mathematical physics, and the foundations of mathematics was enormously influential.

Emmy Noether, 1882 to 1935, revolutionized abstract algebra and made fundamental contributions to theoretical physics through her theorem linking symmetries and conservation laws.

Alan Turing, 1912 to 1954, created the theoretical foundations of computer science, proved the undecidability of the halting problem, and contributed to the practical development of computing. He was also crucial to the British code-breaking effort in the Second World War.

John von Neumann, 1903 to 1957, made major contributions to quantum mechanics, game theory, economics, and computer science, and contributed to the design of the first modern computers.

Mathematics in Everyday Life

Mathematics pervades everyday life to a degree that most people rarely notice. From the moment a clock radio switches on in the morning to the late-night calculation of a restaurant bill, ordinary human existence is structured by mathematical relationships, measurements, and computations. The history of applied mathematics is in many ways the history of civilization itself: every major development in technology, commerce, navigation, architecture, medicine, and communication has depended on mathematical tools developed by the mathematicians whose work is described in this article.

The most basic and universal application of mathematics to everyday life is money: counting, accounting, and arithmetic. The need to track quantities of goods, record transactions, and compute profits and losses was one of the primary drivers of the development of writing and numerical notation in ancient Mesopotamia and Egypt. The Hindu-Arabic numeral system, with its decimal positional notation and zero, displaced the Roman numeral system in Europe precisely because it made commercial arithmetic so much more efficient. Double-entry bookkeeping, introduced in Italian city-states in the medieval period and described by Luca Pacioli in 1494, applied mathematical structure to commercial record-keeping in a way that contributed directly to the development of modern capitalism.

Architecture and construction have always required substantial mathematical knowledge. The pyramids of Egypt required the ability to measure distances and angles with precision, to calculate volumes of stone, and to plan the logistics of enormous construction projects. The great cathedrals of medieval Europe required an understanding of the geometric principles of arches and vaults sufficient to ensure structural stability. Modern architecture uses calculus-based structural analysis, finite element methods, and computer-aided design software built on a mathematical foundation that encompasses linear algebra, differential equations, and numerical analysis.

Navigation, before the advent of GPS technology, required sophisticated mathematical tools for determining position and plotting courses. The development of trigonometry, logarithms, and the calculus were all driven in part by the needs of navigators. The sextant, chronometer, and navigational tables used by maritime nations from the seventeenth century onward embody centuries of mathematical development. Today, the Global Positioning System, GPS, works by solving a set of simultaneous equations relating the measured distances from a receiver to multiple satellites, a direct application of three-dimensional coordinate geometry.

Medicine and public health rely heavily on statistical and probabilistic reasoning. Clinical trials use statistical methods to determine whether treatments are effective, comparing outcomes in treatment and control groups and using probability theory to assess the likelihood that observed differences could be due to chance. Epidemiology, the study of the distribution and determinants of disease in populations, is essentially applied statistics. The modeling of infectious disease spread, including the famous concept of the reproduction number R, the expected number of secondary cases per primary case, is directly derived from the mathematical theory of differential equations.

Music, which the ancient Pythagoreans regarded as the most mathematical of the arts, rests on mathematical relationships at every level. The harmonic series, the sequence of frequencies at integer multiples of a fundamental frequency, determines the structure of musical tones. The equally tempered scale used in Western music since the seventeenth century divides the octave into twelve equal semitones, with a frequency ratio between adjacent notes of the twelfth root of two. Digital audio processing, which underlies modern recording and playback technology, uses the discrete Fourier transform and other signal processing algorithms grounded in advanced mathematics.

Computers, which have become the central technology of modern life, are built on a foundation of mathematical logic. Boolean algebra, developed by the English mathematician George Boole in the mid-nineteenth century as a formal algebra of logical propositions, turns out to describe the behavior of electronic switching circuits exactly. Every operation performed by a computer, every calculation, every database query, every web page served, every video game frame rendered, is ultimately a sequence of logical and arithmetic operations that can be described in the language of mathematics.

The internet, which has transformed communication, commerce, and culture, depends on mathematics at every level: in the algorithms that route data packets, in the cryptographic protocols that secure transactions, in the search engine algorithms that index and retrieve information, and in the statistical and machine learning methods that power recommendation systems, fraud detection, and natural language processing. The mathematics of networks, graphs, information theory, and optimization, much of it developed in the twentieth century, is the invisible infrastructure of the digital world.

Unsolved Problems in Mathematics

Despite more than five thousand years of mathematical achievement, the frontier of the subject stretches further than ever, and many of the most important and beautiful problems in mathematics remain unsolved. The existence of deep, easily stated questions that resist the efforts of the best mathematical minds for decades or centuries is one of the most fascinating features of mathematics as a discipline. Unsolved problems serve as beacons guiding mathematical research, and the methods developed in attempting to solve them often produce mathematical progress of great importance even when the original problem remains open.

The Riemann hypothesis is widely regarded as the most important unsolved problem in mathematics. Stated by Bernhard Riemann in 1859, it asserts that all non-trivial zeros of the Riemann zeta function, a complex-valued function closely related to the distribution of prime numbers, lie on the critical line in the complex plane, the line where the real part of the complex variable equals one-half. The hypothesis has been verified computationally for trillions of zeros, and virtually every result in analytic number theory concerning the distribution of primes either assumes the Riemann hypothesis or can be strengthened if it is true. Despite this, no proof has been found in more than a century and a half of effort.

The P versus NP problem asks whether every computational problem whose solution can be verified quickly, in polynomial time, can also be solved quickly. More formally, it asks whether the complexity class P, the set of problems solvable in polynomial time, equals the complexity class NP, the set of problems whose solutions can be verified in polynomial time. Almost all computer scientists believe that P is not equal to NP, meaning that there exist problems that are easy to verify but fundamentally difficult to solve, but a proof of this belief remains elusive. The resolution of P versus NP would have profound implications for cryptography, optimization, artificial intelligence, and the foundations of computer science.

The Birch and Swinnerton-Dyer conjecture concerns elliptic curves, a type of cubic algebraic curve with a rich mathematical structure that is connected to many areas of number theory. The conjecture relates the rank of the group of rational points on an elliptic curve, a number measuring how many rational solutions the defining equation has, to the behavior of an associated complex-valued function, the L-function of the curve, near a specific point. This conjecture, formulated in the 1960s, is one of the seven Millennium Prize Problems for which the Clay Mathematics Institute has offered a prize of one million dollars, and despite important partial results it remains unproved.

The Navier-Stokes existence and smoothness problem asks whether the equations governing the flow of a viscous, incompressible fluid, the Navier-Stokes equations, always have smooth solutions that exist for all time when given smooth initial conditions in three spatial dimensions. These equations describe the flow of water, air, and other fluids and are fundamental to engineering, meteorology, and oceanography. In two dimensions, smooth global solutions are known to exist, but the three-dimensional case remains open. The question is whether the equations might develop singularities, points where the velocity becomes infinite, in finite time.

The Hodge conjecture is a problem in algebraic geometry concerning the relationship between the topology of algebraic varieties, geometric objects defined by polynomial equations, and their algebraic structure. It predicts that certain topological classes, cohomology classes, on a smooth projective algebraic variety over the complex numbers can be represented by algebraic cycles. This conjecture, which connects two major branches of geometry in a deep and non-obvious way, has been proved for algebraic varieties of dimension two, the Lefschetz theorem on (1,1) classes, but the general case remains open.

Goldbach's conjecture, formulated by the Prussian mathematician Christian Goldbach in a 1742 letter to Euler, states that every even integer greater than two is the sum of two prime numbers. This statement is one of the most simply stated open problems in number theory and has been verified by computer calculation for all even numbers up to very large bounds. Despite this, a general proof remains elusive. The best result toward Goldbach's conjecture is Chen's theorem, proved by the Chinese mathematician Chen Jingrun in 1966, which shows that every sufficiently large even number is the sum of a prime and a number that is either prime or the product of two primes.

The twin prime conjecture asserts that there are infinitely many pairs of prime numbers that differ by two, such as three and five, eleven and thirteen, and seventeen and nineteen. This conjecture, like Goldbach's, is easy to state and has been verified for all twin primes found to date, which extend to extraordinarily large numbers, yet remains unproved in general. Important progress was made in 2013 when the American mathematician Yitang Zhang proved that there are infinitely many pairs of primes differing by at most seventy million, a finite bound rather than exactly two but a qualitative breakthrough over what was previously known.

The abc conjecture, formulated independently by Joseph Oesterle and David Masser in 1985, is a deep conjecture about the relationship between the prime factorizations of three positive integers a, b, and c that satisfy the equation a plus b equals c. The conjecture has major implications for number theory, implying as consequences several other important results including a strong form of Fermat's Last Theorem. A claimed proof by the Japanese mathematician Shinichi Mochizuki, published in 2012, has not been generally accepted by the mathematical community as of the present date due to the extraordinary difficulty of verifying the highly original and complex methods he employs.

The existence of these and many other unsolved problems demonstrates that mathematics is very much a living discipline, one whose frontier continues to advance but whose depths remain inexhaustible. Each generation of mathematicians inherits a vast legacy of established knowledge and an even vaster terrain of unexplored territory. The history of mathematics is not a closed book but an ongoing story, and its greatest chapters may well lie ahead.

Sources

https://mathshistory.st-andrews.ac.uk/ https://www.ams.org/publicoutreach/math-history/ https://www.claymath.org/millennium-problems/ https://www.maa.org/press/maa-reviews/history-of-mathematics https://aleph0.clarku.edu/~djoyce/java/elements/elements.html https://www-history.mcs.st-and.ac.uk/Chronology/index.html https://www.pbs.org/wgbh/nova/physics/math-and-the-rise-of-civilization.html https://www.math.ubc.ca/~cass/Euclid/elements.html https://plato.stanford.edu/entries/mathematics-ancient/ https://iascud.net/history-of-mathematics-overview/

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Accuracy Audit

The following audit verifies fifteen key factual claims in this article against authoritative non-Wikipedia sources. Web searches were conducted in May 2026. Two errors were found and corrected before this section was appended.

Claim 1: Lebombo bone dates to approximately 43,000 years ago. Status: CONFIRMED. Multiple radiocarbon datings place the bone at 43,000 to 44,200 years old. Sources: African Heritage (afrolegends.com), History of Information (historyofinformation.com).

Claim 2: Ishango bone dates to roughly 20,000 years ago. Status: CONFIRMED. The Ishango bone is consistently dated to approximately 20,000 years before present. Source: University at Buffalo Mathematics of the African Diaspora (math.buffalo.edu).

Claim 3: Plimpton 322 dates to around 1800 BCE and contains Pythagorean triples. Status: CONFIRMED. The tablet is dated to between 1822 and 1762 BCE and contains a table of Pythagorean triples. Sources: MacTutor History of Mathematics (mathshistory.st-andrews.ac.uk), ISAW NYU (isaw.nyu.edu).

Claim 4: Aryabhata was born in 476 CE and gave pi as 3.1416. Status: CONFIRMED. Aryabhata was born in 476 CE and gave the approximation 62832/20000 = 3.1416. Sources: Britannica (britannica.com), MacTutor (mathshistory.st-andrews.ac.uk).

Claim 5: Brahmagupta was born in 598 CE and wrote the Brahmasphutasiddhanta in 628 CE. Status: CONFIRMED. Both dates are attested in multiple scholarly sources. Source: MacTutor History of Mathematics (mathshistory.st-andrews.ac.uk).

Claim 6: Madhava of Sangamagrama (c.1340-1425) computed pi to eleven decimal places using infinite series. Status: CONFIRMED. Madhava is credited with computing pi to eleven decimal places and with the series expansions for sine, cosine, and arctangent. Source: Story of Mathematics (storyofmathematics.com), University of Montana (scholarworks.umt.edu).

Claim 7: Zu Chongzhi computed pi accurate to seven decimal places and gave the fraction 355/113. Status: CONFIRMED. Zu Chongzhi's bounds 3.1415926 and 3.1415927 establish seven decimal place accuracy. The fraction 355/113 is correct to six decimal places. Source: MacTutor (mathshistory.st-andrews.ac.uk), Mind Your Decisions (mindyourdecisions.com).

Claim 8: Archimedes proved pi lies between three and ten seventy-firsts and three and one-seventh using 96-sided polygons. Status: CONFIRMED AFTER CORRECTION. The original article incorrectly stated "three and ten-sevenths" as the lower bound. The correct lower bound is 223/71, which equals three and ten seventy-firsts (approximately 3.1408). The upper bound, 22/7 = three and one-seventh (approximately 3.1429), was correct. This error has been corrected in all English-language files.

Claim 9: Al-Khwarizmi wrote his algebra treatise around 830 CE in Baghdad. Status: CONFIRMED. The Kitab al-Jabr wal-Muqabala was written around 830 CE at the House of Wisdom in Baghdad under Caliph al-Mamun. Sources: Library of Congress (loc.gov), History Rise (historyrise.com).

Claim 10: Evariste Galois died in a duel in 1832 at the age of twenty. Status: CONFIRMED. Galois was born in 1811 and died on 31 May 1832, aged twenty. Source: MacTutor (mathshistory.st-andrews.ac.uk), Britannica (britannica.com).

Claim 11: Kurt Godel published his incompleteness theorems in 1931. Status: CONFIRMED. Godel's 1931 paper shattered Hilbert's program by proving that any consistent formal system sufficient for arithmetic contains unprovable truths. Sources: Stanford Encyclopedia of Philosophy (plato.stanford.edu), American Mathematical Society (ams.org).

Claim 12: Andrew Wiles proved Fermat's Last Theorem, published in 1995. Status: CONFIRMED. Wiles's proof appeared in the May 1995 issue of the Annals of Mathematics. Sources: PBS NOVA (pbs.org), University of Oxford (ox.ac.uk).

Claim 13: Yitang Zhang proved in 2013 that there are infinitely many prime pairs differing by at most seventy million. Status: CONFIRMED. Zhang's paper, received by the Annals of Mathematics in April 2013, proved this result. Source: Quanta Magazine (quantamagazine.org), Annals of Mathematics (annals.math.princeton.edu).

Claim 14: The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken with computer assistance, checking 1,476 reducible configurations. Status: CONFIRMED AFTER CORRECTION. The original article stated "approximately one thousand four hundred," which understates the number. The correct count is 1,476 configurations. This error has been corrected in all English-language files. Source: Georgia Tech (thomas.math.gatech.edu).

Claim 15: Grigori Perelman proved the Poincare conjecture around 2003. Status: CONFIRMED. Perelman posted his final paper to arXiv on July 17, 2003, and the proof was confirmed by the mathematical community. Source: Clay Mathematics Institute (claymath.org), Science.org.

Claim 16: Leonhard Euler's collected works fill approximately ninety volumes. Status: CONFIRMED. Euler's complete works fill about 90 volumes. Much of this output was produced during the final decades of his life when he was totally blind. Source: MacTutor (mathshistory.st-andrews.ac.uk), leonhardeuler.com.

Claim 17: The Rhind Mathematical Papyrus dates to around 1550 BCE and contains 84 problems. Status: CONFIRMED. The papyrus dates to approximately 1550 BCE and contains 84 mathematical problems. Source: Britannica (britannica.com), British Museum (britishmuseum.org).

Corrections Applied to All English-Language Files:

1. Archimedes lower bound for pi changed from "three and ten-sevenths" to "three and ten seventy-firsts" (the correct expression for the fraction 223/71). 2. Number of reducible configurations checked in the four color theorem proof changed from "approximately one thousand four hundred" to "1,476."

Accuracy Audit

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