In this passage, Schopenhauer points out one
key distinction between hatred and contempt:
the former is often connected with a desire
to make its object aware of our feeling, but
the latter never is.
If I hate someone, I may wish to let him know
of my hatred.
Part of the pleasure of this emotion, it seems,
is my knowing that he knows how I feel about him.
But contempt rises above all this: if I have
contempt for someone, I care so little for
him that I am truly indifferent to his knowledge
or ignorance of my judgment.
In contempt I have what Schopenhauer calls
“a pure and unsullied conviction of the
worthlessness of another.”
Thinking of Schopenhauer’s influence on
Nietzsche, we might look for connections between
these observations on the psychology of contempt,
and Nietzsche’s description of the Overman’s
attitude toward those whom he knows to be
far beneath him.
The lofty, even noble, contempt of the man
who is supremely confident in his superiority
may have its roots in Nietzsche’s early
study of Schopenhauer.
This passage comes Schopenhauer’s book Studies
in Pessimism, in a section titled “Further
Psychological Observations.” Here’s the quote.
Hatred comes from the heart; 
contempt from the head;
and neither feeling is quite within our control. 
For we cannot alter our heart; its bias is determined
by motives; and our head deals with objective facts and 
applies to them rules which are immutable.
Any given individual is the union of a 
particular heart with a particular head.
Hatred and contempt are diametrically opposed
and mutually exclusive.
There are even not a few cases where
hatred of a person is rooted in nothing but
forced esteem for his qualities.
And besides, if a man sets out to hate
all the miserable creatures he meets, he will
not have much energy left for anything else;
whereas he can despise them,
one and all, with the greatest ease.
True, genuine contempt
is just the reverse of true, genuine pride;
it keeps quite quiet and gives no sign of its existence.
For if a man shows that he despises you, he signifies at least this much regard for you,
that he wants to let you know how little
he appreciates you; and his wish is dictated
by hatred, which cannot exist with real contempt.
On the contrary, if it is genuine, it is simply the conviction 
that the object of it is a man of no value at all.
Contempt is not incompatible with 
indulgent and kindly treatment,
and for the sake of one's own peace and safety, this should not be omitted; it will prevent irritation; and there
is no one who cannot do harm if he is roused to it.
But if this pure, cold, sincere contempt ever shows itself,
it will be met with the most truculent hatred; for the despised person is not in a position
to fight contempt with its own weapons.
