There is a nice recent article about Cauchy's contribution to integral geometry which is probably what you mean by Cauchy's Crofton formula:
Hyksova, M.; Kalousova, A.; Saxl, I. ``Early history of geometric probability and stereology“Early history of geometric probability and stereology.''” Image Anal. Stereol. 31, (2012), no. 1, 1--16. See this link.