Ashes Wrap: Smith century sees Test peter out to draw.
Phrase.org gives a number of rather half-hearted attempts at establishing a possible origin of the expression. It refers to Peter the Apostle and his denial; it refers to saltpetre; and it makes a stab at linking the two together.
The only plausible part of its explanation is the suggestion that it comes from mining :
When my mineral petered why they all Petered me. Now it is dig, dig, dig, drill, drill for nothing. My luck is clean gone - tapered down to nothing.
December 1845 Milwaukee Daily Gazette.
The AmE Ngram rises from 1860ish, a little earlier than the BrE, but both keep pace after that.
Where does this expression actually come from ?