engage vs interest

engage

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

  • 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 guarantee or promise (to do something). 

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

  • 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. 

interest

verb
  • To engage the attention of; to awaken interest in; to excite emotion or passion in, in behalf of a person or thing. 

noun
  • A great attention and concern from someone or something; intellectual curiosity. 

  • An involvement, claim, right, share, stake in or link with a financial, business, or other undertaking or endeavor. 

  • The price paid for obtaining, or price received for providing, money or goods in a credit transaction, calculated as a fraction of the amount or value of what was borrowed. 

  • Condition or quality of exciting concern or being of importance. 

  • Any excess over and above an exact equivalent 

  • The persons interested in any particular business or measure, taken collectively. 

  • Something or someone one is interested in. 

  • Attention that is given to or received from someone or something. 

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