@tchrist
Italian language uses the apostrophe to indicate the elision of one o more letters at the and of a word. In the case of *de'* the apostrophe is used as "apocope postvocalica".
I have noticed that "apocope" is an English word too ("the loss of a sound or sounds at the end of a word, e.g., in the derivation of *curio* in *curiosity*" - OED), and "postvocalica" literally means "after vowel".
In Italian language is not common truncate *dei* to write *de'*, but in Florentine dialcet is.