Hmm... I wonder why, when observing the stars, they didn't realize that the reason the stars do not change their positions is because they are light years away! The are too far away for us to notice their movements. If the stars and constellations were in or even slightly near our galaxy, then yes they should, and would, change.
I think the book might be referencing the Precession of the Equinoxes problem.
The stars should be moving over time. The precession is slow. 25,920 yrs = 360° rotation. Divide that up, and the axis of the earth moves at one degree per 72 years in respect to the stars, or 1/4th of a degree every 18 years. Yet the North Star is in the same place it was 72 and 18 years ago. The North Star has not been documented to move, despite the theory that the star configurations were different eons ago. Ancient monuments that were built to point directly at the North Star are, in fact, still lined up with the North Star.
The stars should be moving over time. The precession is slow. 25,920 yrs = 360° rotation. Divide that up, and the axis of the earth moves at one degree per 72 years in respect to the stars, or 1/4th of a degree every 18 years. Yet the North Star is in the same place it was 72 and 18 years ago. The North Star has not been documented to move, despite the theory that the star configurations were different eons ago. Ancient monuments that were built to point directly at the North Star are, in fact, still lined up with the North Star.
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Your question is actually more powerful than that: why doesn't Polaris move if we rotate around the Sun?
The answer lies with the fact that Polaris wobbles in a well understood, measurable fashion:
https://www.lpi.usra.edu/education/skytellers/polaris/Polaris is only the North Star because we’re currently using it as such. 3000 years ago the North Star was Thuban in the constellation of Draco. In 13,000 years we’ll be pointing at Vega as the North Star. And then about 10,000 years after that it will be Thuban again, followed by Polaris in another 3000 years.
Indeed, s=r*\theta, so one second of arc does not apply equally to all bodies at different "r" away, insofar as "s" is concerned. One can only assume your argument to be correct by also automatically assuming that all stars exist on a surface equidistant from Earth.