Chiò's and Dodgson's determinantal identities

In this expository paper, we analyze and compare two determinantal identities constructed in the mid nineteenth century, Chiò's and Dodgson's, from the perspective of their origins in earlier work by J.J. Sylvester and C.G.J. Jacobi. All the known proofs of Chiò's and Dodgson's identities are cited and we present and compare them as modern algorithms for evaluating determinants. Restated as a recurrence, we discuss the role of Dodgson's identity in the development of the alternating sign matrix conjecture in the late twentieth century.