3 Correction.

Take a look at Haynes Miller's article about Leray and the invention of sheaves and spectral sequences, especially p.10. He attributes the first use of "spectral sequence" spectral" in this context to Leray in a 1947 conference write-up, and he quotes from a letter of Borel, in which Borel speculates that "spectral" is used by analogy with analysis. Borel points out that in Leray's original formulation, a filtration could be indexed by real numbers, not just by integers, which makes the analogy a bit more appealing.

(I originally wrote that Leray's 1947 paper had the first use of "spectral sequence", but that's not what the article says.)

2 Finally learned how to do hyperlinks right ...

Take a look at Haynes Miller's articlehttp://www-math.mit.edu/~hrm/papers/ss.pdf about Leray and the invention of sheaves and spectral sequences, especially p.10. He attributes the first use of "spectral sequence" to Leray in a 1947 conference write-up, and he quotes from a letter of Borel, in which Borel speculates that "spectral" is used by analogy with analysis. Borel points out that in Leray's original formulation, a filtration could be indexed by real numbers, not just by integers, which makes the analogy a bit more appealing.

1

Take a look at Haynes Miller's article http://www-math.mit.edu/~hrm/papers/ss.pdf about Leray and the invention of sheaves and spectral sequences, especially p.10. He attributes the first use of "spectral sequence" to Leray in a 1947 conference write-up, and he quotes from a letter of Borel, in which Borel speculates that "spectral" is used by analogy with analysis. Borel points out that in Leray's original formulation, a filtration could be indexed by real numbers, not just by integers, which makes the analogy a bit more appealing.