Addendum, OK, 60NM (one degree) is 111.1 km but still within one night. Some of these records may be indoors/on a track, but on smooth terrain I think possible (heck, cross county ski near the pole and have 24 hours of night (looks like folks have skiied 100km in less than 4 hours)
I watched that video carefully, and he uses the Eratosthenes method of measuring the length of a shadow at mid-day, and not particularly accurately. No (nighttime) stars involved.
ETA also, he had a friend take the other measurement at the same time, so that he didn’t have to wait months for the sun to return to the same declination. So it doesn’t really matter how far he can bicycle in one night.
It’s the same thing - the length of a shadow is directly related to the angle of incident light - you’re measuring the position of a star, just the nearest star, and measuring it less directly than looking at it through a sextant or some such
Well, sure, nights can last for days at the poles, in winter, but I feel like if this story supposedly took place in the Arctic Circle, and the exercise was being carried out by some ancient champion race-walker, those details would have been the story.
The typical use of a sextant was to shoot the sun at local noon, and the height above the horizon was your degrees latitude.
So from there, you could just walk/ride/sail north or south some distance, and repeat the same thing the following day at noon, and figure out what fraction of a degree that you traveled, and work out how far one degree would be.
@CC asked about some not completely specified method where you observe the altitude of some star, then travel some distance (I suppose he did not say it had to be at night or on the same night) and “see when the north star is one degree away from where it was when you started”.
Nobody questions that with a chronometer, sextant, and almanac you can figure out your latitude and longitude. In any case, in modern times the length of a meter was originally determined via careful geodetic surveys (and developing and using instruments more precise than wooden rods).
The thing is, when you’re dealing with stars, it’s not particularly important that you do it all in one night. Indeed, even with the sun or moon, it’s far more important that you know where the sun would be in the sky at that moment if you were observing it from somewhere else than that you make your observation on “the same day” but hours apart.
So not only is it (still, in my view) implausible, even if theoretically possible, to cover 60 miles at speed in the dark on foot (how well lit were ancient roads going on for sixty miles or more?), it’s unnecessary. That’s the real nail in the coffin for the scenario as described. Day-to-day variations in the apparent path across the sky traced by stars are basically negligible given the accuracy (or the lack of it) achievable by ancient astronomers. Variations might be detectable over, say, six months as you observe from the other side of Earth’s orbital path (if it’s one of the closer stars), but not so much in a day of travel around the sun.
So, yes, perhaps an ancient astronomer could hazard the dark and risk breaking a leg for the sake of science. But why?
Of course the angle doesn’t have to be one degree. It is whatever difference in angle the observer can get a useful level of accuracy measuring. One degree might be too small for an ancient observer with simple instruments. Then again with care it might be an easy task. If you could get the angle down small enough whilst maintaining accuracy the distance asked can shrink. You just balance accuracy of the final estimate of the Earth’s diameter against this.
The sun is about 1/2 degree across (as is the moon). A star is a point source. Your measurement margin of error is much smaller with a star using hand-held instruments before precision manufacturing. Plus as others point out, the sun moves in the sky. The stars wheel through the sky on a daily cycle but unlike the moon, sun, and planets, they are always in the same place on the celestial globe. It’s just that in the “Good Old Days” the light to do your work at night was an issue, let along running a marathon or two by starlight.
The more degrees between the two measurement points, the lesser the error - with only one degree, you’d better have some pretty precise measuring tools.