In "Number: The Language of Science" (1930), Tobias Dantzig refers to what we call the base case of mathematical induction as "the induction step" (and refers to what we call the induction step as "the recurrence step" or "the proof of the hereditary property"). Was this standard terminology a century ago, or was Dantzig confused?
Note that he wrote this way back when mathematical induction was commonly called complete induction as opposed to Baconian or incomplete induction. Since verification of a single base case could be viewed as a minimalist version of Baconian induction, Dantzig's terminology does not seem totally illogical to me. Perhaps his use of the phrase "induction step" was standard a century ago, and over time its meaning shifted so that it now has the "opposite" meaning (that is, it now refers to the other component of proof by mathematical induction).
I’d be grateful for comments by those who know more history of mathematics than I do, as well as those who can bring a multicultural perspective to this question (what sort of terminology for mathematical induction is used in other languages?).