E3P40D
Demonstration — Part III
Latin
Qui aliquem odio affectum imaginatur, eo ipso etiam odio afficietur (per propositionem 27 hujus) hoc est (per scholium propositionis 13 hujus) tristitia concomitante idea causæ externæ. At ipse (per hypothesin) nullam hujus tristitiæ causam imaginatur præter illum qui ipsum odio habet; ergo ex hoc quod se odio haberi ab aliquo imaginatur, tristitia afficietur concomitante idea ejus qui ipsum odio habet sive (per idem scholium) eundem odio habebit. Q.E.D.
English (Elwes 1883)
He who conceives another as affected with hatred, will thereupon be affected himself with hatred (III. xxvii.), that is, with pain, accompanied by the idea of an external cause. But, by the hypothesis, he conceives no cause for this pain except him who is his enemy; therefore, from conceiving that he is hated by some one, he will be affected with pain, accompanied by the idea of his enemy; in other words, he will hate his enemy in return. Q.E.D.
Modern English
One who imagines another as affected with hatred will, by that very fact, be affected with hatred as well (E3P27), that is, with sadness accompanied by the idea of an external cause (E3P13S). But by hypothesis this person imagines no cause for that sadness except the one who hates him. Therefore, from imagining that he is hated by someone, he will be affected with sadness accompanied by the idea of that person, or, which is the same thing (E3P13S), he will hate that person in return. Q.E.D.