E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
Tether.
He has come to the end of his tether. He has outrun his fortune; he has exhausted all his resources. The reference is to a cable run out to the bitter end (see BITTER END), or to the lines upon lines in whale fishing. If the whale runs out all-the lines it gets away and is lost.
1
Horace calls the end of life ultima linca rerum, the end of the goal, referring to the white chalk mark at the end of a racecourse.