The Radon transform apparently was discovered around 1917 if Wikipedia is to be believed. The Cauchy-Crofton theorem is a much older theorem (mid 19th-century). But both ideas are more or less the same.

Did Radon consider his transform as a generalization of the Cauchy-Crofton theorem? Did he not know about the Cauchy-Crofton theorem?

Links to statements of results:

http://en.wikipedia.org/wiki/Crofton_formula

http://en.wikipedia.org/wiki/Radon_transform