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Galileo's gravity

I came across a quote in a text (Ralph, et al, _World Civilizations_,
Vol. 1) used in our World Cultures Course that is news to me:

"Galileo seems to have a broader conception of a universal force of
gravitation than Leonardo da Vinci, for he perceived that the power
which holds the moon in the vicinity of the earth and causes the
satellites of Jupiter to circulate around that planet is essentially
the same as the force which enables the earth to draw bodies to its

I was under the impression that it was Newton who first conceived of
the earth's gravity extending to the Moon and beyond. Galileo still
had some Aristotelian cobwebs in his cosmology in that he believed
the "natural state" of heavenly objects was uniform circular motion.
Does anyone know if Galileo really preempted Newton? Thanks.

Jeff Braun
University of Evansville