The explicit study of surfaces of constant negative total curvature goes back to the work of Minding in 1838. Thus, in that year, Minding's theorem established the important result that all such surfaces are isometric, that is, they can be placed in one-to-one correspondence in such a way that the metric is preserved. Beltrami subsequently gave the term pseudospherical to these surfaces and made important connections with Lobachevski's non-Euclidean geometry.

It was Bour, in 1862, who seems to have first set down what is now termed the sine-Gordon equation arising out of the compatibility conditions for the Gauss equations for pseudospherical surfaces expressed in asymptotic coordinates.

In 1879, Bianchi in his habilitation thesis presented, in mathematical terms, a geometric construction for pseudospherical surfaces with the same total curvature. This result was extended by Bä in 1883 to incorporate a key parameter which allows the iterative construction of such pseudospherical surfaces. The Bäacklund transformation was subsequently shown by Bianchi, in 1885, to be associated with an elegant invariance of the sine-Gordon equation. This invariance has become known as the Bäacklund transformation for the sine-Gordon equation. It includes an earlier parameter-independent result of Darboux. The Bäacklund transformation has important applications in soliton theory. Indeed, it proves that the property of invariance, under Bäacklund and associated Darboux transformations as originated in, is enjoyed by all soliton equations.