Proof.—He, who affects pleasurably or painfully the object of our love, affects us also pleasurably or painfully—that is, if we conceive the loved object as affected with the said pleasure or pain (III. xxi.). But this pleasure or pain is postulated to come to us accompanied by the idea of an external cause; therefore (III. xiii. note), if we conceive that anyone affects an object of our love pleasurably or painfully, we shall be affected with love or hatred towards him. Q.E.D.
Note.—Prop. xxi. explains to us the nature of Pity, which we may define as pain arising from another's hurt. What term we can use for pleasure arising from another's gain, I know not.
We will call the love towards him who confers a benefit on another, Approval; and the hatred towards him who injures another, we will call Indignation. We must further remark, that we not only feel pity for a thing which we have loved (as shown in III. xxi.), but also for a thing which we have hitherto regarded without emotion, provided that we deem that it resembles ourselves (as I will show presently). Thus, we bestow approval on one who has benefited anything resembling ourselves, and, contrariwise, are indignant with him who has done it an injury.
PROP. XXIII. He who conceives, that an object of his hatred is painfully affected, will feel pleasure. Contrariwise, if he thinks that the said object is pleasurably affected, he will feel pain. Each of these emotions will be greater or less, according as its contrary is greater or less in the object of hatred.
Proof.—In so far as an object of hatred is painfully affected, it is destroyed, to an extent proportioned to the strength of the pain (III. xi. note). Therefore, he (III. xx.) who conceives, that some object of his hatred is painfully affected, will feel pleasure, to an extent proportioned to the amount of pain he conceives in the object of his hatred. This was our first point. Again, pleasure postulates the existence of the pleasurably affected thing (III. xi. note), in proportion as the pleasure is greater or less. If anyone imagines that an object of his hatred is pleasurably affected, this conception (III. xiii.) will hinder his own endeavour to persist; in other words (III. xi. note), he who hates will be painfully affected. Q.E.D.
Note.—This pleasure can scarcely be felt unalloyed, and without any mental conflict. For (as I am about to show in Prop. xxvii.), in so far as a man conceives that something similar to himself is affected by pain, he will himself be affected in like manner; and he will have the contrary emotion in contrary circumstances. But here we are regarding hatred only.
PROP. XXIV. If we conceive that anyone pleasurably affects an object of our hate, we shall feel hatred towards him also. If we conceive that he painfully affects that said object, we shall feel love towards him.
Proof.—This proposition is proved in the same way as III. xxii., which see.
Note.—These and similar emotions of hatred are attributable to envy, which, accordingly, is nothing else but hatred, in so far as it is regarded as disposing a man to rejoice in another's hurt, and to grieve at another's advantage.
PROP. XXV. We endeavour to affirm, concerning ourselves, and concerning what we love, everything that we can conceive to affect pleasurably ourselves, or the loved object. Contrariwise, we endeavour to negative everything, which we conceive to affect painfully ourselves or the loved object.