P40: He who imagines he is hated by someone, and believes he has given the other no cause for hate, will hate the other in return.

Dem.: He who imagines someone to be affected with hate will thereby also be affected with hate (by P27), i.e. (by P13 S), with Sadness accompanied by the idea of an external cause.

But (by hypothesis) he imagines no cause of this Sadness except the one who hates him. So from imagining himself to be hated by someone, he will be affected with Sadness, accompanied by the idea of the one who hates him {as a cause of the sadness} or (by the same Scholium) he will hate the other, q.e.d.

Schol. If he imagines he has given just cause for this hatred, he will be affected with Shame (by P30 and P30 S). But this rarely happens (by P25). Moreover, this reciprocity of Hatred can also arise from the fact that Hatred is followed by a striving to do evil to him who is hated (by P39). He, therefore, who imagines that someone hates him will imagine the other to be the cause of an evil, or Sadness. And so, he will be affected with Sadness, or Fear, accompanied by the idea of the one who hates him, as a cause. I.e., he will be affected with hate in return, as above.

Cor. 1: He who imagines one he loves to be affected with hate toward him will be tormented by Love and Hate together. For insofar as he imagines that {the one he loves} hates him, he is determined to hate {that person} in return (by P40). But (by hypothesis) he nevertheless loves him. So he will be tormented by Love and Hate together.

Cor. 2: If someone imagines that someone else, toward whom he has previously had no affect, has, out of hatred, done him some evil, he will immediately strive to return the same evil.

Dem.: He who imagines someone to be affected with Hate toward him, will hate him in return (by P40), and (by P26) will strive to think of everything that can affect {that person} with Sadness, and be eager to bring it to him (by P39). But (by hypothesis) the first thing he imagines of this kind is the evil done him. So he will immediately strive to do the same to {that person}, q.e.d.

Schol.: The striving to do evil to him we hate is called Anger; and the striving to return an evil done us is called Vengeance.