According to Wikipedia, the Bohr-Mollerup Theorem (discussed previously on MO here) was first published in a textbook. It says the authors did that instead of writing a paper because they didn't think the theorem was new.
What other examples are there of significant theorems that first saw the light of day in a textbook? (I'm assuming Wikipedia is right about Bohr-Mollerup.)
I recognize that the word "significant" is imprecise; I have in mind theorems that mathematicians have picked up on and used in their own work, but I'm open to other interpretations.
Best Answer
I recall that, and Wikipedia independently confirms that L'Hôpital's rule first appeared in a textbook, apparently the first textbook on differential calculus: Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes published by Guillaume de l'Hôpital and made up of content mostly provided by Johann Bernoulli, who was on retainer to l'Hôpital, more or less, for this purpose.