"'Let the jury consider their verdict,' the King said, for about the twentieth time that day.
The process of impeachment in the House of Representatives and the subsequent trial in the Senate is often described in terms of a judicial proceeding. The House assumes the role of grand jury, returning an indictment, and then acts as the prosecutor when the matter goes to the Senate, which is described as a jury.
Rolling in the chief justice of the United States as presiding officer (so the vice president is not placed in the awkward position of either protecting his boss or using the opportunity to steal his job) bolsters the view that the Senate Chamber becomes a court of law.