ca. 2000 BC
- Babylonians solve quadratics in radicals.
ca. 300 BC
- Euclid demonstrates a geometrical construction for solving a quadratic.
- Arab mathematicians reduce:
ux + vx = w
to a quadratic.
- Omar Khayyam (1050-1123) solves cubics geometrically by
intersecting parabolas and circles.
- Al-Kashi solves special cubic equations by iteration.
- Nicholas Chuqet (1445?-1500?) invents a method for solving
- Scipione del Ferro (1465-1526) solves the cubic:
x + mx = n
but does not publish his solution.
- Girolamo Cardan (1501-1576) gives
the complete solution of cubics
in his book, The Great Art, or the Rules of Algebra. Complex
numbers had been rejected for quadratics as absurd, but now they are
needed in Cardan's formula to express real solutions. The Great
Art also includes the solution of the quartic equation by
Ludovico Ferrari (1522-1565), but it is played down because it was
believed to be absurd to take a quantity to the fourth power, given
that there are only three dimensions.
- Michael Stifel (1487?-1567) condenses the previous eight formulas for
the roots of a quadratic into one.
- Francois Viete (1540-1603) solves the casus irreducibilis
of the cubic using trigonometric functions.
- Viete solves a particular
45th degree polynomial
equation by decomposing it into cubics and a quintic. Later he gives a
solution of the general cubic that needs the extraction of only a single
- Albert Girard (1595-1632) conjectures that the
degree equation has
n roots counting multiplicity.
- Rene Descartes (1596-1650) gives his rule of signs to determine the
number of positive roots of a given polynomial.
- Isaac Newton (1642-1727) finds a recursive way of expressing the sum of
the roots to a given power in terms of the coefficients.
- Newton introduces his iterative method for the numerical approximation
- Ehrenfried Walther von Tschirnhaus
(1646-1716) generalizes the linear
substitution that eliminates the
x^(n-1) term in the
nth degree polynomial to eliminate the
x^(n-3) terms as well. Gottfried
Wilhelm Leibniz (1646-1716) had pointed out that trying to get rid
x^(n-4) term usually leads to a harder equation
than the original one.
- Michael Rolle (1652-1719) proves that
f'(x) has an odd
number of roots in the interval between two successive roots of
- Edmund Halley (1656-1742) discusses interative solutions of quartics
with symbolic coefficients.
- Daniel Bernoulli (1700-1782) expresses the largest root of a polynomial
as the limit of the ratio of the successive power sums of the roots.
- Leonard Euler (1707-1783) tries to
find solutions of polynomial
equations of degree
n as sums of
roots, but fails.
- Halley solves the quadratic in trigonometric functions.
- Colin Maclaurin (1698-1746) generalizes Newton's relations for powers
greater than the degree of the polynomial.
- Etienne Bezout (1730-1783) tries to find solutions of polynomial
equations of degree
n as linear combinations of powers
nth root of unity, but fails.
- Euler tries to find solutions of polynomial equations of degree
n as linear combinations of powers of an
nth root, but fails.
- Joseph Louis Lagrange (1736-1813)
expresses the real roots of a
polynomial equation in terms of a continued fraction.
- Lagrange expands a function as a series in powers of another function
and uses this to solve trinomial equations.
- Lagrange shows that polynomials of degree five or more cannot be
solved by the methods used for quadratics, cubics, and quartics.
He introduces the Lagrange resolvent, an equation of degree
- Euler gives series solutions of:
m+n m n
x + ax + bx = 0
- John Rowning (1699-1771) develops the first mechanical device
for solving polynomial equations. Although the machine works
for any degree in theory, it was only practical for quadratics.
- Lagrange finds a stationary solution of the three body problem that
requires the solution of a quintic.
- Erland Samuel Bring (1736-1798) proves that every quintic can be
z + az + b = 0
- Jean Baptiste Joseph Fourier (1768-1830) determines the maximum
number of roots in an interval.
- Paolo Ruffini (1765-1822) publishes
the book, General Theory
of Equations, in which the Algebraic Solution of General Equations
of a Degree Higher than the Fourth is Shown to Be Impossible.
- Carl Friedrich Gauss (1777-1855)
proves the fundamental theorem
of algebra: Every nonconstant polynomial equation has at least
- Gauss solves the cyclotomic equation:
z = 1
in square roots.
- William George Horner (1768-1847) presents his rule for the efficient
numerical evaluation of a polynomial. Ruffini had proposed a similar
- Niels Henrik Abel (1802-1829)
publishes Proof of the Impossibility
of Generally Solving Algebraic Equations of a Degree Higher than the
- Augistin-Louis Cauchy (1789-1857) determines how many roots of a
polynomial lie inside a given contour in the complex plane.
- Evariste Galois (1811-1832) writes
down the main ideas of his theory
in a letter to Auguste Chevalier the day before he dies in a duel.
- Friedrich Julius Richelot (1808-1875) solves the cycolotomic equation:
z = 1
in square roots.
- George Birch Jerrard (1804-1863) shows that every quintic can be
z + az + b = 0
- Karl Heinrich Graeffe (1799-1873) invents a widely used method to
determine numerical roots by hand. Similar ideas had already been
suggested independently by Edward Waring (1734-1798), Germinal
Pierre Dandelin (1794-1847), Moritz Abraham Stern (1807-1894),
and Nickolai Lobachevski (1792-1856). Johann Franz Encke
(1791-1865) later perfects the method.
- Pafnuti Chebyshev (1821-1894)
generalizes Newton's method to make
the convergence arbitrarily fast and uses this to approximate the
roots of polynomials.
- L. Lalanne builds a practical machine to solve polynomials up to
- Gotthold Eisenstein (1823-1852)
gives the first few terms of a
series for one root of a canonical quintic.
- Josef Ludwig Raabe (1801-1859) transforms the problem of finding
roots to solving a partial differential equation, obtaining
explicit roots for a quadratic.
- James Cockle (1819-1895) and
Robert Harley (1828-1910) link a
polynomial's roots to differential equations.
- Carl Johan Hill (1793-1863) remarks that Jerrard's 1834 work is
contained in Bring's 1786 work.
- William Hamilton (1805-1865) closes some gaps in Abel's impossibility
- Johannes Karl Thomae (1840-1921) discovers a key ingredient for the
representation of roots using Siegel functions.
- Camille Jordan (1838-1922) shows that algebraic equations of any degree
can be solved in terms of modular functions.
- Ludwig Sylow (1832-1918) puts the finishing touches on Galois's proofs
- Hermann Amandus Schwarz (1843-1921) investigates the relationship
between hypergeometric differential equations and the group structure
of the Platonic solids, an important part of Klein's solution to
- Felix Klein (1849-1925) solves
the icosahedral equation in terms of
hypergeometric functions. This allows him to give a closed-form
solution of a principal quintic.
- Tables, nomograms, and various mechanical devices are constructed
for solving trinomial equations.
- Ferdinand von Lindemann (1852-1939) expresses the roots of an
arbitrary polynomial in terms of theta functions.
- Carl Woldemar Heymann (1885-1910) solves trinomial equations using
- Vincenzo Mollame (1848-1912) and Ludwig Otto Hoelder (1859-1937)
prove the impossibility of avoiding intermediate complex numbers
in expressing the three roots of a cubic when they are all real.
- Karl Weierstrass (1815-1897) presents an interation scheme that
simultaneously determines all the roots of a polynomial.
- David Hilbert (1862-1943) proves that for every
there exists an
nth polynomial with
rational coefficients whose Galois group is the symmetric
Sn. (The same is true for the alternating
- Johann Gustav Hermes (1846-1912) completes his 12-year effort to
65537th root of unity using
- Emory McClintock (1840-1916) gives series solutions for all the
roots of a polynomial.
- Leonardo Torres Quevedo (1852-1936) builds a machine for the
mechanical calculation of the real and complex roots of an
arbitrary trinomial equation.
- Klein, Leonid Lachtin (1858-1927), Paul Gordan (1837-1912),
Heinrich Maschke (1853-1908), Arthur Byron Coble (1878-1966),
Frank Nelson Cole (1861-1926), and Anders Wiman (1865-1959)
develop the fundamentals of how to solve a sextic via Klein's
- Robert Hjalmal Mellin (1854-1933) solves an arbitrary polynomial
equation with Mellin integrals.
- R. Birkeland shows that the roots of an algebraic equation can be
expressed using hypergeometric functions in several variables.
Alfred Capelli (1855-1910), Guiseppe Belardinelli (1894-?), and
Salvatore Pincherle (1853-1936) express related ideas.
- Andre Bloch (1893-1948) and George Polya (1887-1985) investigate the zeros of
polynomials of arbitrary degree with random coefficients.
- Richard Brauer (1901-1977) analyzes Klein's solution of the quintic
using the theory of fields.
- Scientists at Bell Labs build the Isograph, a precision instrument
that calculates roots of polynomials up to degree 15.
- Emil Artin (1898-1962) uses field theory to develop the modern
theory of algebraic equations.
- Vladimir Arnol'd, using results of Andrei Kolmogorov (1903-1987),
shows that it is possible to express the roots of the reduced
7th degree polynomial in continuous functions of two variables,
answering Hilbert's 13th problem in the negative.
- Hiroshi Umemura expresses the roots of an arbitrary polynomial
through elliptic Siegel functions.
- Peter Doyle and Curt McMullen construct a generally convergent,
purely iterative algorithm for the numerical solution of a
reduced quintic, relying on the icosahedral equation.
- David Dummit and (independently) Sigeru Kobayashi and Hiroshi Nakagawa
give methods for finding the roots of a general solvable quintic in