Nine Mathematicians in Order of Birth Year, From Leibniz in 1646 to Knuth in 1938

Leibniz developed the differential and integral calculus while in Paris from 1672 to 1676, and published his differential calculus with his own d notation in 1684. Newton had reached an equivalent method earlier without publishing it, and modern historians hold that the two worked independently. Leibniz's notation proved the more influential for the calculus that followed.

Euler introduced notation still in use today, including f(x) for a function, e for the base of the natural logarithm and i for the imaginary unit. His 1748 Introductio in analysin infinitorum established analysis as the study of functions, and he solved the Basel problem. He kept publishing after losing his eyesight.

At twenty-four Gauss published the Disquisitiones Arithmeticae in 1801, unifying number theory into one systematic framework. The same year he used the method of least squares to predict the position of the dwarf planet Ceres from limited observations. His Theorema Egregium later gave rise to Gaussian curvature.

Babbage began the Difference Engine in 1822 to compute polynomial values automatically by finite differences. From 1837 he designed the Analytical Engine, controlled by punched cards, with a store, a mill, and conditional branching and looping. Neither machine was completed in his lifetime, though a Difference Engine built to his designs was finished between 1989 and 1991.

Hilbert's 1899 Grundlagen der Geometrie set out a complete system of axioms for Euclidean geometry. In 1900 he posed twenty-three unsolved problems at the International Congress of Mathematicians in Paris, including the continuum hypothesis and the Riemann hypothesis. In 1915 he and Felix Klein invited Emmy Noether to Gottingen.

Noether proved the theorem that links the symmetries of a physical system to its conserved quantities, and her 1921 paper Idealtheorie in Ringbereichen refounded ring theory. She worked years at Erlangen without pay before the 1915 invitation to Gottingen, and was dismissed in April 1933 and emigrated to the United States that October.

Largely self-taught and working as a clerk in Madras, Ramanujan wrote to G. H. Hardy in January 1913, and Hardy brought him to Trinity College, Cambridge, in 1914. Their collaboration covered number theory, elliptic functions, continued fractions, infinite series and partition functions. He was elected a Fellow of the Royal Society in 1918.

Von Neumann studied under David Hilbert at Gottingen and in June 1945 wrote First Draft of a Report on the EDVAC, describing a single memory holding both instructions and data. Because he was the sole named author, the design became known as the von Neumann architecture, and a dispute followed over the uncredited contributions of Mauchly and Eckert.

Knuth began The Art of Computer Programming in 1962, with its first volume appearing in 1968, and developed a rigorous mathematical analysis of how an algorithm's running time grows with its input. His 1976 note on Big Omicron, Big Omega and Big Theta fixed the present meaning of the asymptotic notations, and he also wrote TeX and METAFONT.