Coddington was born on October 21, 1873, in New York City; her parents were of well-off colonial stock.
In 1898, Columbia University awarded her a master's degree in mathematics, minoring in mechanics and Greek, based on a thesis she wrote at the end of her studies in London concerning the history of determinants.
Her dissertation, A Brief Account of the Historical Development of Pseudospherical Surfaces from 1827 to 1887, concerned the history of non-Euclidean geometry and the pseudosphere,[2][b] also including material on a paper of Albert Victor Bäcklund on hyperbolic geometry, which she translated from Swedish into English.
She participated in several clubs concerning gardening and women's suffrage, and traveled frequently to Europe.
In 1952, Williams set sail for Europe on the RMS Queen Mary, but she broke her hip while on board, and died in a hospital in Paris on August 8, 1952, leaving a large legacy to her godchildren and the societies she belonged to.