  Galileo Galilei believed that stars were distant suns whose sizes, measured
via his telescope, were a direct indication of distance -- fainter stars
(appearing smaller in the telescope) were farther away than brighter ones.
Galileo argued in his Dialogue that telescopic observation of a chance
alignment of a faint (distant) and bright (closer) star would reveal annual
parallax, if such double stars could be found. This would provide support both
for Galileo's ideas concerning the nature of stars and for the motion of the
Earth. However, Galileo actually made observations of such double stars, well
before publication of the Dialogue. We show that the results of these
observations, and the likely results of observations of any double star that
was a viable subject for Galileo's telescope, would undermine Galileo's ideas,
not support them. We argue that such observations would lead either to the more
correct conclusion that stars were sun-like bodies of varying sizes which could
be physically grouped, or to the less correct conclusion that stars are not
sun-like bodies, and even to the idea that the Earth did not move. Lastly, we
contrast these conclusions to those reached through applying Galileo's ideas to
observations of visible stars as a whole.
