| c. 30000 BC | | Palaeolithic peoples record tallies on bone in central Europe and France; one wolf bone has 55 cuts arranged in groups of five – the earliest counting system. |
| c. 3400 BC | | The first symbols for numbers, simple straight lines, are used in Egypt. |
| c. 3000 BC | Babylon | The Sumerians of Babylon develop a sexagesimal (based on 60) numbering system. Used for recording financial transactions, the order of the numbers determines their relative, or unit value (place-value), although no zero value is used. It continues to be used for mathematics and astronomy until the 17th century AD. |
| c. 300 BC | Egypt | Alexandrian mathematician Euclid sets out the laws of geometry in his Stoicheion/Elements; it remains a standard text for 2,000 years. He also sets out the laws of reflection in Catoptrics. |
| c. 250 BC | | Greek mathematician and inventor Archimedes, in his On the Sphere and the Cylinder, provides the formulae for finding the volume of a sphere and a cylinder; in Measurement of the Circle he arrives at an approximation of the value of p; in The Sand Reckoner he creates a place-value system of notation for Greek mathematics; and in Floating Bodies, the first known work on hydrostatics, he discovers the principle that bears his name – that submerged bodies are acted upon by an upward or buoyant force equal to the weight of the fluid displaced. |
| c. 230 BC | Greece | Greek scholar Eratosthenes of Cyrene develops a method of finding all prime numbers. Known as the sieve of Eratosthenes it involves striking out the number 1 and every nth number following the number n. Only prime numbers then remain. |
| c. 230 BC | Egypt, Ptolemaic Kingdom | Alexandrian mathematician Apollonius of Perga writes Conics, a systematic treatise on the principles of conics in which he introduces the terms ‘parabola’, ‘ellipse’, and ‘hyperbola’. |
| c. 190 BC | China, Former Han Empire | Chinese mathematicians use powers of 10 to express magnitudes. |
| 200 | | The Greek astronomer and mathematician Claudius Ptolemy produces many important geometrical results with applications in astronomy. |
| 598–665 | | Indian mathematician and astronomer Brahmagupta uses negative numbers in mathematics, and introduces a method of approximation for calculating the sines of small angles. |
| c. 680 | India | Indian mathematicians develop what is now known as the Hindu-Arabic decimal place–value system. Using a multiplicative system to base ten, and employing the number zero, this becomes the basis of the modern Western number system. |
| 816 | | The Council of Chelsea, a church council convened at King Offa's palace, introduces the anno Domini system of dating into England. |
| 1225 | | The Italian mathematician Fibonacci writes Liber quadratorum/The Book of the Square, the first major Western advance in arithmetic since the work of Diophantus a thousand years earlier. |
| 1335 | England | The English abbot of St Albans Richard of Wallingford writes Quadripartitum de sinibus demonstratis, the first original Latin treatise on trigonometry. |
| 1591 | | The French mathematician François Viète writes In artem analyticam isagoge/Introduction to the analytical arts, in which he uses letters of the alphabet (x and y are now standard) to represent unknown quantities. Before this, equations had been written out in long descriptive sentences. |
| 1614 | Scotland | The Scottish mathematician John Napier invents logarithms, a method for doing difficult calculations quickly. His results are published in Mirifici logarithmorum canonis descriptio/Description of the Marvellous Rule of Logarithms. |
| 1622 | England | English mathematician William Oughtred invents an early form of circular slide rule, adapting the principle behind Scottish mathematician John Napier's ‘bones’. |
| 1642 | | French mathematician Blaise Pascal, aged only 19, builds the first calculating machine to help his father, the Intendant of Rouen, with tax calculations. It performs only additions. |
| 1653 | France | The French mathematician Blaise Pascal publishes his ‘triangle’ of numbers. This has many applications in arithmetic, algebra, and combinatorics (the study of counting combinations). |
| 1654 | France | The French scientist and mathematician Blaise Pascal publishes a treatise on hydrostatics, in which he recognizes that force is transmitted equally in all directions through a fluid, the principle that is named after him. |
| 1666 | | In order to calculate the Moon's orbit accurately, English mathematician and physicist Isaac Newton completes the development of a new type of mathematics, calculus or ‘fluxions’, to add infinitesimally small elements of the orbit together. |
| 1669 | England | The English mathematician John Wallis publishes his Mechanica/Mechanics, a detailed mathematical study of mechanics. |
| 1673 | | German mathematician Gottfried von Leibniz presents a calculating machine to the Royal Society. It is the most advanced yet, capable of multiplication, division, and extracting roots. |
| 1679 | | German mathematician Gottfried von Leibniz introduces binary arithmetic (a number system to the base two), in which only two symbols are used to represent all numbers. It will eventually be adopted for use in digital computers. |
| 1684 | | The German philosopher and physicist Gottfried Wilhelm Leibniz invents the differential calculus, a fundamental tool in studying rates of change, independently of Newton. |
| 1687 | | The English mathematician Isaac Newton publishes Philosophiae naturalis principia mathematica/The Mathematical Principles of Natural Philosophy, his most important work. It presents his theories of motion, gravity, and mechanics, which form the basis of much of modern physics. |
| 1707 | | The French mathematician Abraham de Moivre uses trigonometric functions to represent complex numbers for the first time, in the form cos z + i(sin nx) where cos x is the real part of the complex number, i(sin x) is the imaginary part, and i is the square root of -1. |
| 1715 | | The English mathematician Brook Taylor publishes Methodus incrementorum directa et inversa/Direct and Indirect Methods of Incrementation, an important contribution to Scottish mathematician Colin Maclaurin's ‘fluxions’ and to Isaac Newton's calculus. |
| 1733 | | French mathematician Abraham de Moivre first describes the normal (‘bell-shaped’) distribution curve. Later, in 1820, the discovery is credited also to the German mathematician and physicist Carl Friedrich Gauss. |
| 1743 | | French mathematician Jean d'Alembert expands Newton's work on dynamics in his Traité de dynamique/Treatise on Dynamics. He states a principle that the external and inertial forces acting on a solid object in motion are in equilibrium. |
| 1744 | | French mathematician Jean d'Alembert publishes Traite de l'equilibre et du mouvement des fluides/Treatise on Equilibrium and on Movement of Fluids, applying his principle to the motion of fluids. |
| 1746 | | French mathematician Jean d'Alembert further develops the theory of complex numbers. |
| 1748 | Italy | Maria Gaetana Agnesi of Italy writes Instituzioni analitiche/Analytical Institutions. It contains an analysis of a curve that becomes known as ‘the witch of Agnesi’. |
| c. 1750 | France | The French mathematician Jean d'Alembert works with other mathematicians, including Leonhard Euler, Joseph, comte de Lagrange, and Pierre, marquis de Laplace, on the ‘three-body problem’, applying calculus to problems of celestial mechanics. |
| 1763 | England | Thomas Bayes, the English mathematician and theologian, publishes ‘An Essay Towards Solving a Problem in the Doctrine of Chances’. This includes Bayes' theorem, which is important in statistics. |
| 1785 | | French Enlightenment philosopher Jean-Marie-Antoine-Nicolas Caritat, marquis de Condorcet, publishes ‘Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix/Essay on the Application of the Analysis to the Probability of Decisions Made to the Plurality of Voters’, a major advance in the study of probability in the social sciences. |
| 1799 | Germany | The German mathematician Carl Friedrich Gauss proves the fundamental theorem of algebra: that every algebraic equation has as many solutions as the exponent of the highest term. |
| 1799–1825 | | The French mathematician and physicist Pierre-Simon Laplace publishes the five-volume Traité de mécanique céleste/Celestial Mechanics, which applies calculus to the motions of celestial bodies and Isaac Newton's theories of the Solar System to show how its stability is implicit in the law of gravitation. |
| 1801 | Germany | German mathematician Carl Friedrich Gauss publishes Disquisitiones Arithmeticae/Discourses on Arithmetic, which deals with relationships and properties of integers and leads to the modern theory of algebraic equations. |
| 1815 | | The English physician and philologist Peter Roget invents the ‘log-log’ slide rule, although he will always be best remembered as the author of Roget's Thesaurus. |
| c. 1820 | | The German mathematician Carl Friedrich Gauss reintroduces the normal distribution curve (‘Gausian distribution’) – a basic statistical tool. |
| 1822 | | French mathematician Augustin-Louis Cauchy formulates the basic mathematical theory of elasticity; he defines stress as the load per unit area of the cross-section of a material, agreeing with Poisson's conclusions of 1811. |
| 1823 | England | English mathematician Charles Babbage begins construction of the ‘difference engine’, a machine for calculating logarithms and trigonometric functions. |
| 1824 | | French scientist Sadi Carnot publishes Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance/Thoughts on the Motive Power of Fire, and on Machines Suitable for Developing that Power, a pioneering study of thermodynamics in which he explains that a steam engine's power results from the decrease in temperature from the boiler to the condenser. He also describes the ‘Carnot cycle’ whereby heat is converted into mechanical motion and mechanical motion converted into heat – the basis of the second law of thermodynamics. |
| 1825 | | French physicist André Ampère publishes Electrodynamics, in which he formulates the mathematical laws governing electric currents and magnetic fields. It lays the foundation for electromagnetic theory. |
| 1827 | Germany | German mathematician Carl Friedrich Gauss introduces the subject of differential geometry that describes features of surfaces by analyzing curves that lie on it – the intrinsic-surface theory. |
| 1829 | | Russian mathematician Nikolay Ivanovich Lobachevsky develops hyperbolic geometry, in which a plane is regarded as part of a hyperbolic surface shaped like a saddle. Austrian mathematician János Bolyai publishes a treatise on non-Euclidean geometry in 1832. It is the beginning of non-Euclidean geometry. |
| 1837 | | The French mathematician Siméon-Denis Poisson publishes Recherches sur la probabilité des jugements/Researches on the Probabilities of Opinions, in which he establishes the rules of probability and describes the Poisson distribution for a discrete random variable. |
| 1845 | | English mathematician Arthur Cayley publishes Theory of Linear Transformations. He studies compositions of linear transformations. |
| 1847 | | The English mathematician George Boole publishes The Mathematical Analysis of Logic, in which he shows that the rules of logic can be treated mathematically. Boole's work lays the foundation of computer logic. |
| 1913 | | English philosopher Bertrand Russell publishes the final volume of Principia Mathematica/Principles of Mathematics in collaboration with another English mathematician and philosopher, Alfred North Whitehead. They attempt to derive the whole of mathematics from a logical foundation. |
| 1923 | | German mathematician Hermann Oberth publishes Die Rakete zu den Planetenräumen/The Rocket into Interplanetary Space, a treatise on space-flight in which he is the first to provide the mathematics of how to achieve escape velocity. |
| 1931 | | Austrian mathematician Kurt Gödel publishes ‘Gödel's proof’ (On Formally Undecidable Propositions of Principia Mathematica and Related Systems). His proof questions the possibility of establishing dependable axioms in mathematics, showing that any formula strong enough to include the laws of arithmetic is either incomplete or inconsistent. |
| 1936 | England | The English mathematician Alan Turing supplies the theoretical basis for digital computers by describing a machine, now known as the Turing machine, capable of universal rather than special-purpose problem solving. |
| 1937 | | The US mathematician Georges Stibitz builds the first binary circuit that can add two binary numbers based on Boolean algebra. Consisting of batteries, lights, and wires, it is instrumental in the development of subsequent electromechanical computers. |
| 1961 | USA | US meteorologist Edward Lorenz discovers a mathematical system with chaotic behaviour, leading to a new branch of mathematics known as chaos theory. |
| 1980 | | Mathematicians worldwide complete the classification of all finite and simple groups, a task that has taken over 100 mathematicians more than 35 years to complete. The results take up more than 14,000 pages in mathematical journals. |
| 1994 | | The English mathematician Andrew Wiles proves Fermat's last theorem, a problem that had remained unsolved since 1637. |
| 26 June 1997 | | The English mathematician Andrew Wiles is awarded the Wolfskehl Prize for solving Fermat's last theorem. The most notorious problem in mathematics, the Last Theorem was created in the 17th century by the French judge Pierre de Fermat, who studied mathematics in his spare time. In 1908 the German industrialist Paul Wolfskehl bequeathed DM100,000 (£1 million by today's value) to be given to the first person to prove it. |
