Timeline of Mathematics
Mathematics worldwide from Babylonian tablets of about 1800 BCE through Egyptian, Greek, Chinese, Indian and Islamic advances to modern algebra, logic and computer-assisted proof, ending with the 1995 proof of Fermat's Last Theorem.
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- 34 events c. 1800 BCE – 1995
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c. 1800 BCE: Babylonian scribes calculate with the Pythagorean rule
Old Babylonian scribes wrote numbers in a base-60 place-value system, which survives in modified form in the measurement of time and angles. The clay tablet Plimpton 322, probably from Larsa in southern Iraq, lists 15 rows of numbers related to Pythagorean triples, and a student's tablet now at Yale gives the square root of 2 to about six decimal digits, evidence that the Pythagorean relation was in use over a thousand years before Pythagoras.
Wikipediac. 1550 BCE: Ahmes copies the Rhind Mathematical Papyrus
The Egyptian scribe Ahmes copied the Rhind papyrus from a now-lost original dating to the reign of the pharaoh Amenemhat III. The larger of the two best-known Egyptian mathematical papyri, it begins with a table writing fractions 2/n as sums of unit fractions, then works through problems on sharing loaves, solving linear equations, and finding areas, granary volumes and the slopes of pyramids.
Wikipediac. 300 BCE: Euclid compiles the "Elements"
Drawing on predecessors such as Eudoxus of Cnidus and Theaetetus, Euclid of Alexandria set out plane and solid geometry and elementary number theory in the 13 books of the "Elements", proving its theorems from definitions, postulates and common notions. The oldest surviving large-scale deductive treatment of mathematics, it proves that there are infinitely many primes and has been estimated to be second only to the Bible in number of editions.
Wikipediac. 225 BCE: Archimedes finds the sphere's surface area and volume
In "On the Sphere and Cylinder", Archimedes of Syracuse proved that a sphere has two-thirds of both the volume and the surface area, bases included, of the cylinder that just encloses it. He found such results by a mechanical method of balancing thin slices before proving them by the method of exhaustion; Cicero later found what he took to be his tomb, marked with a sphere and cylinder.
Wikipediac. 250: Diophantus writes the "Arithmetica"
The "Arithmetica" of Diophantus of Alexandria, whose dates are uncertain, solved about 290 problems by algebra, using abbreviations for an unknown and its powers and accepting positive fractions as answers; 6 of its 13 books survive in Greek and 4 more in Arabic translation. Equations to be solved in whole numbers are now called Diophantine, and around 1637 Pierre de Fermat wrote his "Last Theorem" in the margin of a Latin edition.
Wikipedia263: Liu Hui comments on the "Nine Chapters"
Liu Hui wrote a commentary on "The Nine Chapters on the Mathematical Art", an anonymous Chinese collection of problems and methods that reached its final form by about the 1st century CE, explaining why its procedures work and adding a proof of the Pythagorean theorem. The text solved systems of up to five linear equations by the method now called Gaussian elimination and gave rules for adding and subtracting negative numbers.
Wikipedia499: Aryabhata composes the "Aryabhatiya"
Aryabhata's "Aryabhatiya", 121 Sanskrit verses on astronomy and mathematics, covered arithmetic, algebra and plane and spherical trigonometry, including a table of sines and a method, later called kuttaka ("pulverizer"), for whole-number solutions of linear equations. Its letter-based number notation used place value, and its term jya for the sine, garbled in Arabic and Latin translation, became the English word "sine".
Wikipedia628: Brahmagupta sets out rules for arithmetic with zero
In the "Brāhmasphuṭasiddhānta", Brahmagupta gave the earliest known rules for calculating with zero as a number in its own right, with rules for all four operations on negative numbers; Babylonian scribes and the Mesoamerican Long Count calendar had used zero only as a placeholder. His rules match modern ones except for division by zero (he set 0/0 = 0), and the book first stated the quadratic formula explicitly, though for one root only.
Wikipediac. 820: Al-Khwarizmi writes the book that names algebra
Working at the House of Wisdom in Baghdad, Muhammad ibn Musa al-Khwarizmi wrote "al-Jabr", a systematic treatment of linear and quadratic equations that sorted them into six standard forms and justified the solutions geometrically, all in words rather than symbols. The book's title term al-jabr ("restoration") gave the word "algebra", and Algoritmi, the Latin form of his name on a translation of his book on Hindu numerals, gave "algorithm".
Wikipediac. 1070: Omar Khayyam solves cubic equations with conic sections
In a treatise written in Samarkand, the Persian mathematician and poet Omar Khayyam classified cubic equations, identified 14 types that cannot be reduced to lower degree, and constructed positive roots of each as intersections of two conic sections, though he missed cases with three positive roots. He noted that an arithmetic solution eluded him and hoped later mathematicians would find one, which happened in 16th-century Italy.
Wikipedia1202: Fibonacci publishes "Liber Abaci"
Leonardo of Pisa, later known as Fibonacci, learned the "nine Indian figures" as a youth in Bugia (now Béjaïa, Algeria); his "Liber Abaci" taught Hindu–Arabic numerals, zero and place value through commercial arithmetic. Gerbert of Aurillac had used the digits, without zero, on an abacus around 1000, but the book did much to spread them. Its rabbit problem gave the Fibonacci sequence, described in India centuries earlier.
Wikipedia1247: Qin Jiushao generalizes the Chinese remainder theorem
In his "Mathematical Treatise in Nine Sections", the Southern Song mathematician Qin Jiushao gave the dayan method, a general procedure for finding a number with given remainders on division by several numbers; the problem's earliest known statement, a single example, is in the "Sunzi Suanjing" (3rd to 5th century). The book also approximated roots of polynomial equations by a procedure like the one William George Horner published in 1819.
Wikipediac. 1400: Madhava develops infinite series for trigonometric functions
Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics in southern India, is credited with infinite power series for the sine, cosine and arctangent, including a series for π. No surviving work of his states them; later Kerala texts attribute the series to him and give proofs. Isaac Newton, James Gregory and Gottfried Wilhelm Leibniz found the same series independently between 1669 and 1673.
Wikipedia1545: Cardano publishes solutions of cubic and quartic equations
Gerolamo Cardano's "Ars Magna" contained the first published algebraic solutions of cubic and quartic equations, crediting the cubic method to Scipione del Ferro and Niccolò Tartaglia and the quartic to his student Lodovico Ferrari. The book also worked, reluctantly, with square roots of negative numbers; Rafael Bombelli's "L'Algebra" (1572) gave rules for calculating with such complex numbers and used them where the cubic formula requires them.
Wikipedia1614: Napier publishes the first tables of logarithms
John Napier's "Mirifici Logarithmorum Canonis Descriptio" introduced logarithms, with 90 pages of tables that let astronomers and navigators replace multiplication and division by addition and subtraction. Jost Bürgi had devised similar tables independently, printed in 1620; Henry Briggs's base-10 logarithms and the slide rule kept the method central to calculation until electronic calculators took over in the 1970s.
Wikipedia1637: Descartes joins algebra and geometry in "La Géométrie"
René Descartes' "La Géométrie" solved geometric problems by algebra and described curves by equations in two unknown lengths, the germ of coordinate geometry, and introduced x, y and z for unknowns and modern notation for powers. Pierre de Fermat developed the same approach independently in a manuscript circulated from about 1636 but printed only in 1679, and the method fed into the calculus of Isaac Newton and Gottfried Wilhelm Leibniz.
Wikipedia1654: Pascal and Fermat solve the problem of points
Prompted by the writer Antoine Gombaud, Chevalier de Méré, Blaise Pascal and Pierre de Fermat worked out in letters how to divide the stakes of an interrupted game of chance fairly, by counting the ways it could have continued. Their correspondence is often taken as the start of modern probability theory, though Gerolamo Cardano had analyzed dice games about a century earlier; Christiaan Huygens published the first book on the subject in 1657.
Wikipedia1684: Leibniz publishes the first paper on calculus
Gottfried Wilhelm Leibniz's "Nova Methodus pro Maximis et Minimis", in the journal Acta Eruditorum, was the first published work on calculus and used the differential notation (dx) still in use. Isaac Newton had developed his equivalent method of fluxions in 1665–1666 but published it later; a bitter priority dispute followed, and historians now agree that the two developed calculus independently.
Wikipedia1736: Euler solves the Königsberg bridge problem
Leonhard Euler proved that no walk through Königsberg could cross each of its seven bridges exactly once, using only which land masses the bridges joined: such a walk needs zero or two land masses touched by an odd number of bridges, and Königsberg had four. The solution, presented to the St. Petersburg Academy in 1735, appeared in the academy's volume for the following year, printed in 1741, and is regarded as the first theorem of graph theory.
Wikipedia1748: Euler makes the function central to analysis
Leonhard Euler's two-volume "Introductio in analysin infinitorum" made the function the central object of analysis, introduced the exponential and logarithm functions without calculus, took the number e as the base of natural logarithms, and gave Euler's formula linking the exponential function with sine and cosine through complex numbers. The historian Carl Boyer called it "the foremost textbook of modern times".
Wikipedia1801: Gauss publishes "Disquisitiones Arithmeticae"
Written when Carl Friedrich Gauss was 21 and published when he was 24, "Disquisitiones Arithmeticae" built number theory around congruences, the modular arithmetic of remainders. It gave the first proof of the law of quadratic reciprocity, conjectured by Leonhard Euler and Adrien-Marie Legendre, and a condition for constructing regular polygons, such as the 17-sided one, with compass and straightedge.
Wikipedia1824: Abel proves the general quintic unsolvable by radicals
Niels Henrik Abel published, at his own expense and compressed into six pages, a proof that no formula built from the coefficients by arithmetic operations and roots can solve the general polynomial equation of degree five or higher, a question open for more than 250 years. Paolo Ruffini had published an incomplete proof in 1799, and the result is now called the Abel–Ruffini theorem.
Wikipedia1829–1832: Lobachevsky and Bolyai publish non-Euclidean geometry
Nikolai Lobachevsky at Kazan University and, three years later, János Bolyai in an appendix to his father's textbook independently published a geometry in which, through a point not on a given line, more than one line fails to meet it. Carl Friedrich Gauss had similar unpublished ideas; Eugenio Beltrami's models of 1868 showed the new geometry is consistent if Euclid's is, so the parallel postulate cannot be derived from Euclid's other axioms.
Wikipedia1846: Liouville publishes Galois's theory of equations
Joseph Liouville published the memoir of Évariste Galois, who had died after a duel in 1832 at age 20. It showed that a polynomial equation can be solved by radicals exactly when a group of permutations of its roots, now called its Galois group, is "solvable", explaining both the formulas up to degree four and the Abel–Ruffini theorem; its use of groups made Galois one of the founders of group theory.
Wikipedia1847: Boole recasts logic as algebra
George Boole's pamphlet "The Mathematical Analysis of Logic" turned logic into a symbolic algebra, expressing arguments as equations, an approach he set out more fully in "The Laws of Thought" (1854). Successors such as William Stanley Jevons and Charles Sanders Peirce refined it into modern Boolean algebra, which Claude Shannon applied to relay and switching circuits in his 1937 master's thesis.
Wikipedia1859: Riemann links primes to the zeta function's zeros
In his only paper on number theory, Bernhard Riemann extended Leonhard Euler's zeta function to complex numbers and connected the distribution of the primes to the zeros of that function. He remarked that all the nontrivial zeros very probably have real part 1/2; this Riemann hypothesis remains unproved. Jacques Hadamard and Charles de la Vallée Poussin independently used his ideas to prove the prime number theorem in 1896.
Wikipedia1874: Cantor proves the real numbers are uncountable
Georg Cantor proved that the real algebraic numbers can be listed in a sequence, using a proof Richard Dedekind had sent him, but the real numbers cannot, the first proof that an infinite set can be uncountable, and gave a new proof that transcendental numbers exist. In 1878 he used one-to-one correspondences to define when sets have the same size, building a set theory that Leopold Kronecker opposed but that became a foundation of mathematics.
Wikipedia1900: Hilbert poses 23 problems for the new century
Speaking at the International Congress of Mathematicians in Paris, David Hilbert presented 10 unsolved problems; the published version listed 23, among them the continuum hypothesis, the consistency of arithmetic and the Riemann hypothesis. Several shaped 20th-century research: the continuum hypothesis proved independent of the standard axioms of set theory (Kurt Gödel, 1940; Paul Cohen, 1963), while the Riemann hypothesis remains open.
Wikipedia1921: Noether publishes her theory of ideals in rings
Emmy Noether's paper "Idealtheorie in Ringbereichen" ("Theory of Ideals in Ring Domains") built the theory of ideals in commutative rings on the ascending chain condition, proving the Lasker–Noether theorem in full generality; rings with that property are now called Noetherian. Her abstract approach spread through Bartel van der Waerden's textbook "Moderne Algebra", and Nathan Jacobson credited the development of abstract algebra largely to her.
Wikipedia1931: Gödel publishes his incompleteness theorems
Kurt Gödel proved that any consistent formal system whose axioms can be listed by an algorithm and in which elementary arithmetic can be carried out contains true arithmetical statements that it cannot prove; with somewhat more arithmetic, as in Peano arithmetic, it also cannot prove its own consistency. The results are widely taken to show that David Hilbert's program for a complete, consistent axiomatization of mathematics cannot succeed.
Wikipedia1936: Turing shows the decision problem is unsolvable
In "On Computable Numbers, with an Application to the Entscheidungsproblem", Alan Turing modeled computation with abstract machines, now called Turing machines, described a universal machine that can imitate any other, and proved that no algorithm can decide whether any given statement of first-order logic is provable, settling David Hilbert's "decision problem". Alonzo Church independently reached the same answer shortly before.
Wikipedia1976: Appel and Haken prove the four color theorem
Kenneth Appel and Wolfgang Haken proved that four colors suffice to color any map of contiguous regions so that regions sharing a border, not just a corner, differ, a problem Francis Guthrie posed in 1852. It was the first major theorem proved with a computer, which checked 1,936 configurations (1,482 in the published version); after early doubts, a simpler computer proof followed in 1997 and a formal check in the Coq proof assistant in 2005.
Wikipedia1981: Classification of finite simple groups declared complete
Daniel Gorenstein declared that all finite simple groups, the building blocks of finite groups, had been classified: each is cyclic of prime order, alternating, of Lie type, or one of 26 sporadic exceptions, the largest being the Monster. The proof, spread over tens of thousands of pages by about 100 authors, was in print by 1983 except for unfinished work on "quasithin" groups, a gap that Michael Aschbacher and Stephen D. Smith closed in 2004.
Wikipedia1995: Wiles proves Fermat's Last Theorem
Andrew Wiles proved the modularity (Taniyama–Shimura) conjecture for semistable elliptic curves, which by Ken Ribet's 1986 theorem implies Fermat's Last Theorem: for n greater than 2, no nth power of a positive integer is the sum of two such powers. Announced in June 1993, the proof had a gap that Wiles closed in 1994 with his former student Richard Taylor; it was published more than 350 years after Pierre de Fermat's claim.
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