I am reading a 1892 novel, in which I see the sentence:
For months my life was despaired of, and when at last I came to myself....
Is this an old use of the word "despair"? If the word "of" is dropped, does the meaning of the sentence change?
The "of" would certainly change the meaning of the sentence- in fact it would be meaningless without it.
People will still use "despair of" sometimes. "I despair of you" is the kind of phrase my mother would say when I was young and I hadn't done my chores. I suppose the meaning is close to "I give up" so in the original quote they had given up hope of the protagonist recovering.
The sentence means the narrator's friends or family were fearful that he or she might die. To "despair of" something is to fear for it. I know of no common modern spelling of "despair" that is spelled "dispair." You can drop the word "of" but the verb does not take a direct object. Instead you would use it in a construction like "I despair that you will ever get to be President of the United States." That means the speaker doesn't think you will ever occupy the Oval Office. It may also mean that said speaker deplores said condition.
And, yes, "to despair of" something is a construction more likely to be seen in older writing, but is not archaic and I still see it used frequently today — mostly by people whose writing tends toward the literary turn of phrase.
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