I think swearing is where one technically crosses the Bar on the record of the court. Jurors swear in. Witnesses swear in.
The swearing became compulsory under the Trading with the Enemy Act (for US citizens).
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The tradition of swearing in witnesses began in the Star Chamber with Hywett v. (Sovereign) City of London around 1508. The tradition of swearing compact seems to have began in the Book of Enoch with a third of the angels going into rebellion together, lusting after human women.
Regards,
David Merrill.