Merriam Webster's defines "perchance" as:
"maybe but not definitely"
Similarly, the fifth definition of chance on Merriam Webster has "by chance" defined as:
"in the haphazard course of events"
The famous Shakespeare quote "to sleep, perchance to dream" would sound odd as "to sleep, by chance to dream." Conversely, I wouldn't think twice about asking someone either "do you by chance have a piece of gum," or "perchance do you have a piece of gum?" Would the second example be incorrect? I found this page from "The Grammarist" which would seem to indicate I'm using perchance wrong.
Both words are conditional words dependent on chance. Are they interchangeable, or only interchangeable in certain circumstances?
Best Answer
As the page you link notes, the more current version and near-perfect synonym of perchance is perhaps (though oddly enough, “it may chance” is way more current than “it may hap”). Neither perchance nor perhaps is reliably interchangeable with by chance. “By chance I met him on the street” is not at all the same as “Perchance/Perhaps I met him on the street.” In the former, I definitely did meet him on the street, but the meeting was unplanned and unpredictable. In the latter, I am giving you no assurance that I met him on the street at all.