TITLE: Who discovered this definition of Stiefel-Whitney classes?
QUESTION [9 upvotes]: I would define Stiefel-Whitney classes as the pullbacks of generators of $H^*(BO, \mathbb{Z}/2)$ under a classifying map, and I gather this is a pretty common definition. 
However, the book "Characteristic classes" by Milnor-Stasheff contains a different definition, as ``eigenvalues'' for the Steenrod squares acting on the fundamental class of a manifold in its normal bundle. I want to attribute this construction to somebody for a paper, but I'm not sure what the correct attribution is. Who is the discoverer of this definition?

REPLY [5 votes]: The AMS has spent a lot of money and effort over many years to develop a fine tool that easily allows for the investigation of such questions: it is called Math Reviews (MathSciNet).  The oldest mention of SW classes in the reviews seems to be a review by Whitney of a 1947 paper by Wu.   There is a fantastic review (in German) by Hirzebruch of Milnor and Stasheff's book, giving some history - e.g. the SW classes go back to the mid 1930's.  I believe Milnor and Stasheff themselves discuss how to think about SW classes as obstructions, and I am pretty sure that this finding this interpretation was quite close to the discovery of the formula for the SW classes via Steenrod squares: this type of question is how the `reduced squares' were discovered in the first place.
Added later: ... see Thom, René
Espaces fibrés en sphères et carrés de Steenrod. (French)
Ann. Sci. Ecole Norm. Sup. (3) 69, (1952). 109–182. 
In particular, section 3 of chapter 2 carefully develops the formula for the SW classes via Steenrod squares, and compares it to earlier constructions.  Chapter 1 discusses what folks might now call the Thom isomorphism (though the Gysin sequence is already known).
Remark: the basic theory of fiber bundles and classifying spaces is just being figured out at this same time.