Central Events in Mathematics

Murray, C.A. (2003) Human accomplishment the pursuit of excellence in the Arts and Sciences, 800 B.C. to 1950. HarperCollins Publishers.

Year	Country	Event
600 BCE	Greece	Thales founds abstract geometry and deductive mathematics with the “Thales Proposition” (triangles over the diameter of a circle are right-angled), the oldest theorem of occidental mathematics.
520 BCE	Greece	The Pythagorean theorem appears, allegedly proved by Pythagoras.
420 BCE	Greece	Hippias of Elis discovers the quadratix, the first known curve that cannot be constructed with a straightedge and compass.
350 BCE	Greece	Menaechmus makes the first known attempt to investigate the geometry of the cone.
300 BCE	Alexandria	Euclid’s Elements synthesizes and systematizes knowledge of geometry.
260 BCE	Greece	Archimedes calculates the first known value for π.
250 BCE	Greece	Conon of Samos discovers the curve known as the spiral of Archimedes.
232 BCE	Greece	Apollonius of Perga’s Conicorum presents a systematic treatment of the principles of conics, introducing the terms parabola, ellipse, and hyperbola.
50	Greece	Hero of Alexandria discovers the formula for expressing the area of a triangle in terms of its sides.
98	Greece	Menelaus gives the first definition of a spherical triangle and theorems on congruence of spherical triangles, founding spherical trigonometry.
250	Greece	Diophantus discovers solutions to certain equations, known as Diophantine equations, that represent the beginnings of algebra.
490	China	Zu Chongzhi calculates that π lies between 3.1415926 and 3.1415927, by far the most accurate estimate of π to that time.
500	India	Aryabhatiya summarizes Indian mathematical knowledge.
700	India	Over the course of 8C, a full and consistent use of zero develops.
810	Persia	Al-Khwarizmi’s Hisab al-Jabr W’al-Musqabalah gives methods for solving all equations of the first and second degree with positive roots, synthesizes Babylonian with Greek methods, and is the origin of the word algebra.
870	Persia	Thabit ibn Qurra translates Greek mathematical texts into Arabic. His translations will become the major source for European knowledge of Greek mathematics.
1100	Persia	Omar Khayyam is the first to solve some cubic equations.
1120	England	Adelhard of Bath translates an Arabic version of Euclid’s Elements into Latin, introducing Euclid to Europe.
1202	Italy	Leonardo Fibonacci’s Liber Abaci awakens Europe to the advantages of Arabic numerals and computation.
1350	France	Nicole Oresme anticipates coordinate geometry with a plot of time against velocities.
1360	France	Nicole Oresme introduces fractional exponents.
1464	Germany	Regiomontus’s De Triangulis Omnimodus is the first systematic European work on trigonometry as a subject divorced from astronomy.
1491	Italy	Filippo Calandri publishes an account of the modern method of long division.
1494	Italy	Luca Pacioli’s Summa de Arithmetica presents an overview of mathematics handed down from the Middle Ages, becoming one of the most influential mathematics books of its time. It is also the first book to discuss double-entry bookkeeping.
1525	Austria	Christoff Rudolff's Die Coss introduces the square root symbol and introduces decimal fractions.
1535	Italy	Tartaglia discovers a general method for solving cubic equations.
1545	Italy	Girolamo Cardano's Ars Magna is the first book of modern mathematics.
1551	Germany	Rheticus prepares tables of standard trigonometric functions, defining trigonometric functions for the first time as ratios of the sides of a right triangle rather than defining them relative to the arcs of circles.
1557	Wales	Robert Recorde introduces an elongated version of the equal sign into mathematics, and introduces the plus and minus signs into English.
1572	Italy	Rafael Bombelli introduces the first consistent theory of imaginary numbers.
1580	France	Francois Viete introduces a precise analytic definition of π.
1585	Netherlands	Simon Stevin’s De Thiende presents a systematic account of how to use decimal fractions.
1591	France	Francois Viete introduces the systematic use of algebraic symbols.
1613	Italy	Pietro Cataldi develops methods of working with continued fractions.
1614	Scotland	John Napier’s Mirifici Logarithmorum Cationis Descriptio introduces logarithms.
1631	England	William Oughtred's Ciavis Mathematicae summarizes the status of arithmetic and algebra, employing extensive mathematical symbolism.
1635	Italy	Francesco Cavalieri’s Geometria Indivisibilibus Continuorum expounds a method of using “indivisibles” that foreshadows integral calculus.
1637	France	Pierre de Fermat states his Last Theorem.
1637	Netherlands	Rene Descartes’ “La Geometrie,” an appendix to Discours de la Methode, founds analytic geometry.
1637	Netherlands	Rene Descartes’ “La Geometrie” introduces exponents and square root signs.
1638	France	Pierre de Fermat achieves major progress toward differential calculus, determining maxima and minima by procedures used today.
1640	France	Pierre de Fermat founds number theory through his work on the properties of whole numbers.
1648	France	Girard Desargues’s Maniere Universelle de Mr. Desargues pour Pratiquer la Perspective contains Desargues’s theorem, founding projective geometry.
1654	France	Pierre de Fermat and Blaise Pascal found probability theory with methods for judging the likelihood of outcomes in games of dice.
1654	France	Blaise Pascal’s “Traite du Triangle Arithmetique” analyzes the properties of the arithmetical triangle.
1655	England	John Wallis’s Arithmetica Infinitorium introduces concepts of limit and negative and fractional exponents, along with the symbol for infinity.
1657	Netherlands	Christiaan Huygens introduces the concept of mathematical expectation into probability theory
1662	England	John Graunt’s Natural and Political Observations Made upon the Bills of Mortality is the first significant use of vital statistics.
1668	Belgium	Nicolus Mercator calculates the area under a curve, using analytical geometry.
1668	Scotland	James Gregory introduces a precursor of the fundamental theorem of calculus, expressed geometrically.
1669	England, Germany	Isaac Newton’s De Analysi per Aequationes Numero Terminorum Infinitas presents the first systematic account of the calculus, independently developed by Gottfried Leibniz.
1670	England	Isaac Barrow discovers a method of tangents essentially equivalent to those used in differential calculus.
1676	England	Isaac Newton formally states the binomial theorem.
1685	England	John Wallis introduces the first graphical representation of complex numbers.
1687	England	Isaac Newton’s Philosophiae Naturalis Principia Mathematical appears, representing the origin of modern applied mathematics.
1693	England	Edmond Halley prepares the first detailed mortality tables.
1704	England	Isaac Newton’s Enumberatio Linearum Tertii Ordinis describes the properties of cubic curves.
1713	Switzerland	Jakob Bernoulli’s Ars Conjectandi contains Bernoulli’s theorem, that any degree of statistical accuracy can be obtained by sufficiently increasing the observations, thereby also representing the first application of calculus to probability theory.
1715	England	Brook Taylor’s Methodus Incrementorum Directa et Inversa introduces the calculus of finite differences.
1718	England	Abraham de Moivre’s Doctrine of Chances is the first systematic treatise on probability theory.
1720	Scotland	Colin Maclaurin’s Geometrica Organica describes the general properties of planar curves.
1731	France	Alexis Clairaut’s Recherches sur les Conrbes a Double Courbure is a pioneering study of the differential geometry of space curves.
1733	Italy	Girolamo Saccheri’s Euclides ab Omni Naevo Vindicatus inadvertently lays the foundation for non-Euclidean geometry
1770	France	Johann Lambert demonstrates that both π and π² are irrational.
1795	Germany	Carl Gauss proves the law of quadratic reciprocity.
1796	Germany	Carl Gauss discovers a method for constructing a heptadecagon with compass and straightedge and demonstrates that an equilateral heptagon could not be constructed the same way, constituting the only notable advance in classic geometry since ancient Greece.
1797	Norway	Caspar Wessel introduces the first geometric representation of complex numbers employing the x-axis as the axis of reals and the y-axis as the axis of imaginaries.
1799	France	Gaspard Monge introduces advances in projecting three-dimensional objects onto two-dimensional planes, founding descriptive geometry.
1799	Germany	Carl Gauss presents a new and rigorous proof of the fundamental theorem of algebra.
1801	Germany	Carl Gauss’s Disquisitiones Arithmeticae expands number theory to embrace algebra, analysis, and geometry.
1803	France	Lazare Carnot’s Geometrie de Position revives and extends projective geometry.
1807	France	Jean Fourier introduces Fourier’s theorem and the beginnings of Fourier analysis.
1810	France	Joseph Gergonne's Annales de Mathematiques Pures et Appliques is one of the first periodicals devoted to mathematics and becomes highly influential.
1812	France	Pierre Laplace's Theorie Analytiquc des Probabilities introduces the Laplace transform and expands the power of probability theory.
1813	France	Simeon Poisson derives the Poisson distribution.
1817	Czechoslovakia	Bernardus Bolzano develops calculus using a continuous function, dispensing with infinitesimals.
1822	France	Fourier’s Theorie Analytiquc de la Chaleur gives a full presentation of Fourier’s dimensional analysis, using mass, time, and length as fundamental dimensions that must be expressed in consistent units.
1822	France	Jean Poncelet’s Traite des Proprietes Projectives des Figures serves as a foundation of modern geometry.
1823	Hungary	Janos Bolyai develops the first consistent system of non-Euclidean geometry, but publication is delayed until 1832.
1824	Norway	Niels Abel proves the impossibility of a general solution for quin­tic equations.
1825	France	Adrien Legendre’s Traite des Fonctions Elliptiques et des Integrales Euleriennes presents a systematic account of his theory of elliptic integrals.
1825	France	Jean Poncelet and Joseph Gergonne develop the first clear expres­sion of the principle of duality in geometry.
1825	Norway	Niels Abel creates elliptic functions and discovers their double periodicity.
1829	Russia	Nikolai Lobachevsky introduces hyperbolic geometry, replacing Euclid’s parallel postulate and founding one of the most important systems of non-Euclidean geometry
1830	France	Evariste Galois develops group theory, critical later for quantum mechanics.
1843	Ireland	William Hamilton introduces quaternions (algebra with hyper­-complex numbers).
1844	Germany	Hermann Grassmann’s theory of “extended magnitude” general­izes quaternions, creating an algebra of vectors.
1847	England	George Boole’s The Mathematical Analysis of Logic introduces Boolean algebra, systematically applying algebraic opera­tions to logic.
1851	Germany	Bernhard Riemann introduces topological considerations into the study of complex functions and lays the basis for Riemann sur­faces.
1854	Germany	Bernhard Riemann’s Uber die Hypothesen Welche der Geometrie zu Grande Liegen introduces a new non-Euclidean geometry and accelerates the acceptance and potential utility of non-Euclidean geometries.
1857	England	Arthur Cayley introduces the algebra of matrices.
1872	Germany	Felix Klein’s “Erlanger Programm” calls for geometry to be based on groups of transformations.
1872	Germany	Richard Dedekind introduces theory that any rational or irra­tional number can be defined in terms of rationals.
1873	France	August Hermite proves that e is transcendental.
1874	Germany	Georg Cantor’s first formal publication on set theory founds the field.
1881	USA	Josiah Gibbs’s Elements of Vector Analysis introduces a system of vec­tors in three dimensions.
1882	Germany	Carl Lindemann proves that π is transcendental.
1883	Germany	Georg Cantor introduces transfinite set theory.
1884	Sweden	Sonya Kovalevskaya demonstrates that certain kinds of Abelian integrals can be expressed in terms of simpler elliptic integrals.
1895	France	Henri Poincare’s Analysis Situs effectively founds topology (although a few theorems of topology had already been proved).
1899	Germany	David Hilbert’s Grundlagen der Geometrie establishes the basic axiomatic-formalist approach to systematizing mathematics, initi­ated by compactly deriving a formal axiomatic model for Euclid’s geometry.
1902	France	Henri Lebesgue introduces a new theory for integrating discon­tinuous functions.
1906	France	Maurice Frechet introduces a geometry of abstract spaces and the concepts of separability and completeness.
1910	England	Bertrand Russell’s and Alfred Whitehead’s Principia Mathematica represents the best, though flawed, attempt to establish mathemat­ics as branch of logic.
1931	Austria	Kurt Godel demonstrates that any formal system strong enough to include the laws of arithmetic is either incomplete or inconsis­tent.
1934	Russia	Aleksander Gelfond and T. Schneider demonstrate that an irra­tional power of an algebraic number other than zero or one is transcendental.
1936	England	Alan Turing’s “On Computable Numbers” develops the hypothetical Turing machine as a method of determining what kinds of mathematical results can be proved.