Read Professor Stewart's Hoard of Mathematical Treasures Online
Authors: Ian Stewart
Tags: #Mathematics, #General
c.23,000 BC | Ishango bone records the prime numbers between 10 and 20. Apparently. |
c.1900 BC | Babylonian clay tablet Plimpton 322 lists what may be Pythagorean triples. Other tablets record movements of the planets and how to solve quadratic equations. |
c.420 BC | Discovery of incommensurables (irrational numbers in geometric guise) by Hippasus of Metapontum.* |
c.400 BC | Babylonians invent symbol for zero. |
c.360 BC | Eudoxus develops a rigorous theory of incommensurables. |
c.300 BC | Euclid’s Elements makes proof central to mathematics, and classifies the five regular solids. |
c.250 BC | Archimedes calculates the volume of a sphere, and other neat stuff. |
c.36 BC | Mayans reinvent symbol for zero. |
c.250 | Diophantus writes his Arithmetica - how to solve equations in whole and rational numbers. Uses symbols for unknown quantities. |
c.400 | Symbol for zero re-reinvented in India. Third time lucky. |
594 | Earliest evidence of positional notation in arithmetic. |
c.830 | Muhammad ibn Musa al-Khwarizmi’s al-Jabr w’al-Muqabala manipulates algebraic concepts as abstract entities, not just placeholders for numbers, and gives us the word ‘algebra’. Doesn’t use symbols, however. |
* Hippasus was a member of the Pythagorean cult, and it is said that he announced this theorem while he and some fellow cultists were crossing the Mediterranean in a boat. Since Pythagoreans believed that everything in the universe is reducible to whole numbers, the others were less than overjoyed, and he was expelled. From the boat, according to some versions. | |
876 | First undisputed use of a symbol for zero in positional base-10 notation. |
1202 | Leonardo’s Liber Abbaci introduces the Fibonacci numbers through a problem about the progeny of rabbits. Also promotes Arabic numerals and discusses applications of mathematics to currency trading. |
1500-1550 | Renaissance Italian mathematicians solve cubic and quartic equations. |
1585 | Simon Stevin introduces the decimal point. |
1589 | Galileo Galilei discovers mathematical patterns in falling bodies. |
1605 | Johannes Kepler shows that the orbit of Mars is an ellipse. |
1614 | John Napier invents logarithms. |
1637 | René Descartes invents coordinate geometry. |
c.1680 | Gottfried Wilhelm Leibniz and Isaac Newton invent calculus and argue about who did it first. |
1684 | Newton sends Edmund Halley a derivation of elliptical orbits from the inverse square law of gravity. |
1718 | Abraham De Moivre writes first textbook on probability theory. |
1726-1783 | Leonhard Euler standardises notation such as e, i, π, systematises most known mathematics, and invents a huge amount of new mathematics. |
1788 | Joseph-Louis Lagrange’s Méchanique Analytique places mechanics on an analytic basis, getting rid of pictures. |
1796 | Carl Friedrich Gauss discovers how to construct a regular 17-gon. |
1799-1825 | Pierre Simon de Laplace’s five-volume epic Mécanique Céleste formulates the basic mathematics of the solar system. |
1801 | Gauss’s Disquisitiones Arithmeticae provides a basis for number theory. |
1821-1828 | Augustin-Louis Cauchy introduces complex analysis. |
1824-1832 | Niels Henrik Abel and Évariste Galois prove that the quintic equation is not soluble using radicals; Galois paves the way for modern abstract algebra. |
1829 | Nikolai Ivanovich Lobachevsky introduces non-Euclidean geometry, followed shortly by János Bolyai. |
1837 | William Rowan Hamilton defines complex numbers formally. |
1843 | Hamilton formulates mechanics and optics in terms of the Hamiltonian. |
1844 | Hermann Grassmann develops multidimensional geometry. |
1848 | Arthur Cayley and James Joseph Sylvester invent matrix notation. Cayley predicts that it will never have any practical uses. |
1851 | Posthumous publication of Bernard Bolzano’s Paradoxien des Unendlichen which tackles the mathematics of infinity. |
1854 | Georg Bernhard Riemann introduces manifolds—curved spaces of many dimensions - paving the way for Einstein’s general relativity. |
1858 | Augustus Möbius invents his band. |
1859 | Karl Weierstrass makes analysis rigorous with epsilon-delta definitions. |
1872 | Richard Dedekind proves that √2 × √3 = √6 - the first time this has been done rigorously - by developing the logical foundations of real numbers. |
1872 | Felix Klein’s Erlangen programme represents geometries as the invariants of transformation groups. |
c.1873 | Sophus Lie starts working on Lie groups, and the mathematics of symmetry makes a huge leap forward. |
1874 | Georg Cantor introduces set theory and transfinite numbers. |
1885-1930 | Italian school of algebraic geometry flourishes. |
1886 | Henri Poincaré stumbles across hints of chaos theory and revives the use of pictures. |
1888 | Wilhelm Killing classifies the simple Lie algebras. |
1889 | Giuseppe Peano states his axioms for the natural numbers. |
1895 | Poincaré establishes basic ideas of algebraic topology. |
1900 | David Hilbert presents his 23 problems at the International Congress of Mathematicians. |
1902 | Henri Lebesgue invents measure theory and the Lebesgue integral in his PhD thesis. |
1904 | Helge von Koch invents the snowflake curve, which is continuous but not differentiable, simplifying an earlier example found by Karl Weierstrass and anticipating fractal geometry. |
1910 | Bertrand Russell and Alfred North Whitehead prove that 1 + 1 = 2 on page 379 of volume 1 of Principia Mathematica, and formalise the whole of mathematics using symbolic logic. |
1931 | Kurt Gödel’s theorems demonstrate the limitations of formal mathematics. |
1933 | Andrei Kolmogorov states axioms for probability. |
c.1950 | Modern abstract mathematics starts to take off. After that it gets complicated. |