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](http://www.ams.org/mathscinet-getitem?mr=2952193).” Image Anal. Stereol. 31, (2012), no. 1, 1--16.