archaic language - What is this variant of remember doing?

I remember to have heard that your mother has many suitors

Source

"My friend," said Nestor, "now that you remind me, I remember to have heard that your mother has many suitors, who are ill disposed towards you and are making havoc of your estate. Do you submit to this tamely, or are public feeling and the voice of heaven against you? Who knows but what Ulysses may come back after all, and pay these scoundrels in full, either single-handed or with a force of Achaeans behind him? If Minerva were to take as great a liking to you as she did to Ulysses when we were fighting before Troy (for I never yet saw the gods so openly fond of any one as Minerva then was of your father), if she would take as good care of you as she did of him, these wooers would soon some of them him, forget their wooing."

Answer

This usage, with remember in the sense of "recall, experience a memory" taking an infinitive complement (Direct Object) is now obsolete. Today we would employ a gerund or that complement:

I remember hearing that ...
I remember I have heard that ...

Nowadays infinitive complements with remember are reserved for situations where remember has the sense "recall and perform (an obligation)" ...

Did you remember to wash the dishes?

See this question for more about the present-day distinction between remember to VERB and remember VERBing.