Newton's laws of mechanics state:
The first (a.k.a. Galileo's Principle of Inertia) tends to be re-stated, by modern physicists, as saying that there are frames of reference with respect to which it holds true; these are known as inertial frames of reference. The interesting property is then that, if the points which a frame of reference deems to be at rest are deemed to be moving at constant velocity (i.e. in a straight line at constant speed) by some inertial frame, then the original frame is also inertial. The second and third laws implicitly take an inertial frame of reference for granted.
The second law is commonly restated to equate force with rate of change of momentum (which is mass times velocity). Either way, the second is essentially the quantitative definition of force, ready for use by the third. For angular momentum to also be conserved, one must strengthen the third law to assert that the two forces in a pair act along a common line. Interestingly, electromagnetism can appear to violate this principle; but the principle is preserved by the electromagnetic field itself carrying the stray angular momentum.
Newton also developed the mathematical infrastructure, called
differential calculus, for actually solving equations of motion of bodies
governed by the above laws and acted on by forces such as gravitation, whose
effect Newton characterized by the law:
Newton demonstrated that this leads to orbits obeying Kepler's three laws.
Indeed, from Newton's three laws and Kepler's three laws, it is fairly straightforward to infer the inverse-square law, at least if you have Newton's differential calculus at your disposal.
From Kepler's first law, infer that a planet could follow an actually circular orbit, since a circle is simply an ellipse with zero eccentricity (one may do the following with a general elliptical orbit, but it is messier). The two foci of a circle coincide at its centre, so Kepler's first law puts The Sun at the centre; thus the line from such a planet to The Sun would have constant radius. Kepler's second law now tells us that the angular velocity (and, indeed, the speed) of our hypothetical planet is constant. A fairly straightforward application of the differential calculus yields the acceleration of a body in circular orbit at constant angular velocity; it's proportional to the radius divided by the square of the period (actually, one can probably infer that without overtly using differential calculus). Kepler's third law can be restated to say that the square of the period is proportional to the cube of the semi-major axis (in our circular case, this is just the radius), with all planets sharing the same constant of proportionality; our planet's acceleration is proportional to radius divided by square of period, hence to radius divided by cube of radius, i.e. to the inverse of the square of the radius.
Now, Kepler's laws don't involve the mass of the planet at all, so our planet's acceleration doesn't depend on its mass; thus, applying Newton's second law, the gravitational force acting on each planet is simply proportional to its mass. By Newton's third law the same force is likewise acting, in the opposite direction, on the Sun as the gravitational influence of the planet. By an implicit symmetry argument, the force acting on The Sun must be proportional to its mass. We thus infer Newton's law of gravitation from Newton's laws of mechanics and Kepler's laws of planetary motion.
Thus, while Robert Hooke may actually have conjectured every fragment of
Newton's laws – he almost certainly knew of the first, from Galileo, and
there are reasonable grounds to suppose he came up with the inverse square law
(he and Newton fell out over a fierce priority dispute on this exact point),
given which he quite likely did suppose something resembling Newton's second and
third laws – the supreme achievement of Newton was really the development
of a mathematical too, the differential calculus, without which such laws could
serve no practical purpose. Hooke could at best conjecture and speculate; he
lacked the mathematical skill to turn his intuitions into a usable physical
theory. Given Galileo's work on mechanics, and Hooke's extensive (and highly
valuable) experimental work, Newton's laws are reasonable intuitions and
conjectures that we may fairly suppose several investigators may have
independently come to. Given that the
natural philosophers of the day
(including both the gregarious Hooke and even the reclusive Newton) participated
in a lively discourse, it is likely that these ideas emerged collectively from
discussions among many – the exact details of who first articulated each
fragment are relatively unimportant. Furthermore, the study of infinitesimals
had been pursued vigorously (if not particularly rigorously) for much of the
preceding century, so that it is not especially surprising that Newton and
Leibniz came up with the infinitesimal calculus (of which the differential
calculus is one half) at roughly the same time. Thus both halves of what Newton
achieved were ideas whose time had come; yet it is a tribute to his intelligence
that he did indeed bring them forth, credibly as independently as anyone ever
brings forth an idea; and the progress of natural philosophy (which we now call
science) was doubtless brought forward significantly by his combination of
mathematical talent and physical insight. Leibniz was doubtless his peer in the
former, and Hook likely his peer in the latter, but we owe much to the
combination of the two in one mind.
It is worth noting that Newton understood gravitation's
action at a
distance as not being mediated by any intervening thing. Others
subsequently described it in terms of a
gravitational field which
propagated according to a field equation and acted locally on each body. None
the less, until Maxwell's triumph with
electrodynamics gave greater legitimacy to field theories, many scientists
regarded the field theory of gravity purely as a computational technique to
simplify the complexities of computing the dynamics of systems: the vast respect
in which Newton was held long after his death ensured that his belief in action
at a distance (as likewise the corpuscular nature of light) was an orthodoxy
that contrary descriptions had to overcome before gaining acceptance.