We had a discussion of this a long time ago, which I am sure nobody remembers.

Near Side and Off Side. Left side and right side. “Near wheel” means that to the coachman's left hand;
and “near horse” (in a pair) means that to the left hand of the driver. In a four-in-hand the two horses on
the left side of the coachman are the near wheeler and the near leader. Those on the right hand side of the
coachman are “off horses.” This, which seems an anomaly, arose when the driver walked beside his
team. The teamster always walks with his right arm nearest the horse, and therefore, in a pair of horses,
the horse on the left side is nearer than the one on his right.

In a discussion for reasons UK vehicles drive on left side of road, it seemed logical to me
to think it had started with drivers of carriages using right hand to lead horses when it
was necessary, and wanting to be close to shoulder of road, rather than in middle where
they would be in danger from oncoming vehicles. Nobody agreed with me.