= Solution
For point masses $m_1,m_2$ separated by distance $r$, <Newton's law of universal gravitation> gives an attractive central <force> of magnitude
$$
\boxed{F=G\frac{m_1m_2}{r^2}.}
$$
The vector force on body 2 is $-Gm_1m_2(\mathbf r_2-\mathbf r_1)/|\mathbf r_2-\mathbf r_1|^3$. For extended bodies one adds their elemental interactions. The <shell theorem> explains why a spherically symmetric body acts externally like its total <mass> placed at its centre.
The law has experimental and astronomical support, rather than a deductive proof from definitions. <Kepler's laws> provide one test. For a circular orbit of radius $r$ and period $T$, the required <centripetal acceleration> is $4\pi^2r/T^2$. With $T^2\propto r^3$, it varies as $r^{-2}$. The full elliptical-orbit analysis, with the <law of areas> identifying a central interaction, gives the same inverse-square dependence and predicts the orbit's form.
The Earth–Moon comparison connects celestial motion to terrestrial falling. Using illustrative mean values $R_\oplus\simeq6.37\times10^6\,\mathrm m$, $r_M\simeq3.84\times10^8\,\mathrm m$ and $g_0\simeq9.81\,\mathrm{m\,s^{-2}}$, the inverse-square prediction is
$$
g_0\left(\frac{R_\oplus}{r_M}\right)^2\simeq2.70\times10^{-3}\,\mathrm{m\,s^{-2}}.
$$
The circular estimate $4\pi^2r_M/T_M^2$, with $T_M\simeq27.3$ days, is about $2.72\times10^{-3}\,\mathrm{m\,s^{-2}}$. Their agreement is the relevant check; the numbers are approximate mean-orbit values, not exact observations of a circular Moon.
The <Cavendish experiment> adds a laboratory test. Small suspended masses twist a fine fibre when larger source masses are moved nearby. Comparing the <torque> with the suspension's calibrated restoring response measures the weak attraction between laboratory bodies. Reversing source positions helps separate the gravitational signal from a persistent instrumental offset. Planetary perturbations and additional laboratory geometries provide further tests of the common law. Its excellent accuracy in ordinary weak-field conditions does not make it an exact theory in every regime.
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