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Hi

He had canvassed the archives looking for one item in particular. After an hours-long search, but aided by his knowing the week span he was looking for, he found it.

I cannot figure out the bolded part. What does it say?
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"Knowing the week span he was looking for" is a gerund phrase. Gerunds are verbal nouns often identified by the "-ing" ending, which in this case is "knowing". A much more common word would be "knowledge", which would make the sentence say "knowledge of the week span [that] he was looking for". So, the man's knowledge of the approximate dates of his search term helped him search through the archives faster.
Hi

So it means that after many hours of his search during which he actually knew which week to look for, he found it.
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Yes.
Hi

I have one more question. I wrote "which week to look for". But maybe it would be more precise to say: which dates within the given week to look for ...

I suppose it doesn't say he knew which week to look for, but which days within the given week to look for, but maybe it's one and the same Emotion: rolleyes
"which week to look for" would be okay because the week includes all the days within the week. Another way to say it: "he knew which week to look in."
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Thanks a lot!!!