As far as I can tell from Ebenezer’s question, the book is saying that “assay” can be used in those sentences (which I would disagree with, at least in modern English). Is that how you understod it?
Your mentioning of “at least in modern English” shows that in one way or another it could be used as a substitution for ‘an attempt’.
But could it be used, is it correct at all in English? If not in modern English.
Alifathima:
double-cross: deceive, cheat.
cross-question: to question someone again.
The use of ‘assay’ is very restricted as it has a specific meaning - an investigation into the composition of a metal to find out its quality. In no sense therefore could you use it as a synonym for ‘attempt’, especially in the sentences you have quoted.
I believe “assay” originally had a more general meaning of “try, attempt”. It’s interesting that M-W lists this as the first definition, with no qualifications: merriam-webster.com/dictionary/assay (the noun is marked “archaic” in that sense but not the verb).
As far as I know, this general meaning is obsolete (or very unusual). As Alan says, nowadays the word normally used in reference to tests of composition of substances.