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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

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    $\begingroup$ I am looking at the translation of the original paper by Radon. It has only three references: To Minkowski, Funk, and von Weyl. Crofton is not mentioned in the text, and the only mention of Cauchy is irrelevant to your question. This makes it rather unlikely that he knew about it, I think. $\endgroup$
    – Harald Hanche-Olsen
    Commented Nov 27, 2009 at 20:04
  • $\begingroup$ Ah, thanks I didn't know there was a translation available. I'm reading it now, too. $\endgroup$
    – Ryan Budney
    Commented Nov 27, 2009 at 21:53

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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.” Image Anal. Stereol. 31, (2012), no. 1, 1--16.

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I am looking at the translation of the original paper by Radon. It has only three references: To Minkowski, Funk, and von Weyl. Crofton is not mentioned in the text, and the only mention of Cauchy is irrelevant to your question. This makes it rather unlikely that he knew about it, I think. – Harald Hanche-Olsen Nov 27 2009 at 20:04

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