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Taxi and cab are both clippings from the same word: taxicab.
And taxicab is itself a complex clipping of the word *taximeter-cabriolet,* dating back to the beginning of the 20th century.
Clipping is a process of new word formation by chopping off the beginning of a word (e.g. internet -> net), the end of a word (e.g., doctor -> doc), or even the middle of a word (madam -> ma'am).
Let's break this down: taxi is from the Latin taxare and means to charge, rate; meter is Greek for measure. Cabriolet is from the French capriole, meaning the capering or leaping of a goat, perhaps indicating what kind of ride you should expect. ~WuL