To return to my introduction of e using the most roundabout method possible I’d like to imagine the problem of telling someone just where it is you’ve been stranded in a broken car on the New York Thruway. Actually, I’d rather imagine the problem of being stranded in a broken car on the New Jersey Turnpike, as it’s much closer to my home, but the Turnpike has a complexity I don’t want distracting this chat, so I place the action one state north. Either road will do.
There’s too much toll road to just tell someone to find you there, and the majority of their lengths are away from any distinctive scenery, like an airport or a rest area, which would pin a location down. A gradual turn with trees on both sides is hardly distinctive. What’s needed is some fixed reference point. Fortunately, the Thruway Authority has been generous and provided more than sixty of them. These are the toll plazas: if we report that we are somewhere between exits 23 and 24, we have narrowed down our location to a six-mile stretch, which over a 496-mile road is not doing badly. We can imagine having our contact search that.
But the toll both standard has many inconveniences. The biggest is that exits are not uniformly spaced. At the New York City end of the Thruway, before tolls start, exits can be under a mile apart; upstate, where major centers of population become sparse, they can spread out to nearly twenty miles apart. As we wait for rescue those twenty miles seem to get longer.
I admit I can’t imagine us actually giving our location as “between exit 23 and 24”, and don’t imagine you do either. We might say we were halfway between the two exits. We could suggest to our contact that we were at the location of a hypothetical “exit 23 1/2”, the fraction saying how far along the way toward the higher exit we should go past the lower one. There is still the non-uniform distance problem: “exit 23 1/2” would be farther from its surrounding exits than “exit 24 1/2” would be, but we are interested in picking out a spot on the road rather than picking out a spot easily.
It occurs to me that since there is a correlation between how far one travels and the toll one pays, we might alternatively have given directions based on how much we would pay to get from the nearest plaza to the location. But this has many complications in use, the most interesting to me being that travel between exit 25A and exits 23, 24, and 26 are free. The mention of exit 25A is also the complication I didn’t wish to deal with on the Turnpike: there are several suffixed exits on both roads, but the Turnpike has exits 14 A, B, and C, 15 X, and eastern and western spurs. The Thruway adds just a few ‘A’ exits and then offshoots to neighboring toll roads.
These non-uniform lengths are inconvenient, though. We could pin down our location as above to something like “three miles north of exit 23”. Since this is about where the Thruway shifts from running mostly northwards (paralleling the Mohawk River and Hudson River) to running mostly westward (paralleling the Erie canal) this provides plenty of chances for a pedantic contact to claim the location just can’t be found as it would be somewhere hovering between the Northway and Route 9. I don’t want to be rescued by someone arguing points like that either. But we want some measuring scheme with a finer resolution, and after thinking of the plaza-rich environs of Westchester County, settle on miles past, or to, the nearest toll plaza.
A mile is still not a very precise measurement for the location of a car, at least to arrange a meet-up with someone come to dig us out of the snow. (It is the latter half of October already; I’ve see flakes falling by this part of the season.) But one advantage miles have over toll booths is that we can easily imagine a half of a mile, and have a good idea of where it should be. The car might be two and two-thirds miles north of exit 23. Or we can imagine saying it was actually 2.671 miles north. We could be even more exact if that were necessary and we knew the location to that precision.
Obviously, what’s been set up here is a coordinate system, one in which the zero point is wherever along the road the toll plaza is considered to be. There’s a certain ambiguity here, since the toll plaza is with few exceptions not on the road, and even if we pretend its location to be where the onramp merges with the road we still have the ambiguity about where it merges into the road: is it the start or end of the acceleration lane, or the point where the average driver merges in?
There is the obvious way to avoid thinking about all these complications, and that is to just rely on the mile markers which were there all along. These identify approximately the distance from the reference zero point the Thruway Authority has chosen to use (the boundary between the Bronx and Westchester counties, it turns out). Name any distance between zero and 496 miles and we have a distinct spot along the main line of the road. And, each spot where the car is has some unique — at least up to the length of our car, and the measurement errors of the highway’s length — number describing how many miles it is to the zero marker. With that data we can wait for rescue with confidence.