A few highlights in the timeline

1545 Cardan publishes "Ars Magna," containing solutions to
     general 3rd and 4th degree polynomial equations

1572 Bombelli publishes "Algebra", clarifying use of
     complex numbers

1799, 1808, 1813 Ruffini publishes incorrect (but almost
     correct) proofs that there is no general solution of
     5th degree polynomial equations by radicals

1824 Abel, at age 22, publishes the first correct proof of 
     the general unsolvability of 5th degree polynomials by
     radicals.  (Abel died at age 27 of tuberculosis.)

1832 Galois, at age 21, dies in a duel over a woman.
     His papers published in 1846 contained a complete theory
     of which polynomial equations can be solved by
     radicals, and how.

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1746 d'Alembert publishes the first fairly good proof of
     the Fundamental Theorem of Algebra.  The computations
     are right, but some crucial concepts are missing.

1814 Argand publishes a simpler and clearer version 
     of d'Alembert's proof.  Concepts still missing.

1872 Dedekind publishes the first clear explanation of what
     the real numbers (and hence the complex numbers) are.
     This leads to a filling-in of the holes in many proofs,
     including proofs of the Fundamental Theorem of Algebra.



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1666, 1669 -- Newton writing materials about calculus.  Much
     of it, however, was just circulated among a few colleagues,
     and was not published more widely until decades later,
     thus justifying Leibniz's claim to independent discovery

1673 Leibniz makes what is apparently the first use of the
     word "function" (or more precisely, a Latin equivalent).

1684 Leibniz publishes his ideas about calculus, before Newton.
     For years, most Germans and other continental Europeans
     gave Leibniz credit, while the English gave Newton credit.
     Newton and Leibniz themselves did not take part in this
     debate.  (In retrospect, we can see that Newton came up
     with the ideas first, but Leibniz made those ideas
     available to the rest of the world first.  Also, Newton
     developed more applications -- e.g., the movement of the
     planets -- but Leibniz may have come up with better
     notations and a cleaner theory.)

1734 Berkeley publishes "The Analyst", which includes an
     attack on the idea of infinitesimals used by Newton and
     Leibniz.

1748 Euler clarified the notion of function, and also gave
     the first use of "continuous"

1807 Fourier's memoir on propagation of heat in solid bodies.
     Fourier's paper used expansions of functions as 
     trigonometric series (what we now call Fourier series).
     Fourier's computations were right, but his concepts of
     function, convergence, integral, etc., were unclear.
     He claimed that "every" function can be represented
     by a trigonometric series; this claim made it more evident
     to mathematicians of that time that they did not have
     a good understanding of what is a "function."

1821 Cauchy's book on analysis, developing basic theorems
     of calculus more rigorously.  It was around this time
     that epsilon-delta proofs came into use, thus making
     infinitesimals unnecessary.  After this, infinitesimals
     gradually disappeared from math books.

1828 Dirichlet proves that many functions can, indeed, be
     represented by trigonometric series, but some can't

1829 Dirichlet discovers that the characteristic function of
     the rationals is discontinuous everywhere.  (See also 1875)

1859-1861 Weierstrass lecturing and writing about the
     foundations of analysis.  Weierstrass discovers a function
     that is continuous but nowhere differentiable.

1872 Dedekind publishes the first clear explanation of what
     the real numbers (and hence the complex numbers) are.
     This leads to a filling-in of the holes in many proofs.

1875 Thomae's example of a function that is continuous at
     every irrational number, and discontinuous at every rational.

1961, 1963 Abraham Robinson invents nonstandard analysis,
     in which infinitesimals make sense and Leibniz's theories
     are carried out rigorously.  It's an alternative approach
     to many things, including calculus -- just as rigorous
     as the epsilon-delta approach, in some ways simpler, in
     some ways more complicated.

1976 Keisler publishes a calculus book which uses infinitesimals
     instead of epsilons and deltas.  It doesn't catch on.