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 degree seven.

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.

Charles Hermite (1822-1901), Leopold Kronecker (1823-1891), and Francesco Brioschi (1824-1897) independently solve a quintic in Bring-Jerrard form explicitly in terms of elliptic modular functions.

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 proof.

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.