engage vs engross

engage

verb
  • To engross or hold the attention of; to keep busy or occupied. 

  • To enter into conflict with (an enemy). 

  • To come into gear with. 

  • To enter into (an activity), to participate (construed with in). 

  • To draw into conversation. 

  • To mesh or interlock (of machinery, especially a clutch). 

  • To attract, to please; (archaic) to fascinate or win over (someone). 

  • To guarantee or promise (to do something). 

  • To arrange to employ or use (a worker, a space, etc.). 

  • To bind through legal or moral obligation (to do something, especially to marry) (usually in passive). 

  • To enter into battle. 

engross

verb
  • To completely engage the attention of. 

  • To write (a document) in large, aesthetic, and legible lettering; to make a finalized copy of. 

  • To monopolize; to concentrate (something) in the single possession of someone, especially unfairly. 

How often have the words engage and engross occurred in a corpus of books? (source: Google Ngram Viewer )