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It most likely came from the custom in Edo (Tokyo) of people walking on the left side of the street. It is thought that this was because most people are right handed and samurai (who had to carry their swords with them at all times) had theirs on their left; if they passed people on the right it was harder to draw the swords, and people could bump into the protruding swords too on the often very narrow streets of Edo. So they took the British model of left-side roads. They took the British model for railroads too for the most part.

The interesting thing is, pedestrians in the current day Tokyo/Kanto region and most of the country pass each other on the left side (e.g. on escalators), but if you go to the Kansai (Kyoto-Osaka) region, where samurai were not as prevalent as they were in Edo, they still pass each other on the right. So it's likely that whoever made the transportation decisions in the early Meiji period were former samurai.

Another theory is that the new government leaders simply went by what the British did because the British government actively supported their cause when they were trying to bring down the Tokugawa shogunate, but I'm rather skeptical about that since the once the new government took over, they took ideas from several European sources. The parliamentary system was based on the Prussian model for example.

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