Newton's Law of Graviation: Why $G$ and not e.g. $\dfrac{1}{4\pi G_0}$? I've been wondering, in Coulomb's Law, $k_e = \dfrac{1}{4\pi\epsilon_0}$. Therefore, why do we use $G$ in Newton's Law of Gravitation? What if the constant is more like Coulomb's Law, e.g.
$G = \dfrac{1}{4\pi G_0}$ where $G_0$ is some constant.
This would make Newton's Law of Gravitation look like the following:
$$\bf{F}_{12} = -\dfrac{m_1 m_2}{4\pi G_0 |\bf{r}_{12}|^2} \bf{\hat{r}}_{12}$$
$GM = \mu$ is used for calculations but so could $\dfrac{M}{4\pi G_0} = \mu$.
If this is not the case, what is the significance of this definition?
 A: Pure convention. There is no reason alternative conventions couldn't be used, apart from the need to avoid confusion. Newton introduced the constant to make the force law simple, whereas the electrostatic definition with the $4\pi$ is designed to make Poisson's equation (one of the equations for the electric field) look simple. You can write a Poisson equation for the gravitational field as well, and it would look simpler in your convention. (Though note that the Poisson equation for gravity gets modified by general relativity, whereas the one for electromagnetism is exact.) The physics is equivalent in both cases.
Note that in high energy physics one often uses the Planck mass which is related to the Newton constant by (up to a normalisation convention)
$$ m_P^2 = \frac{\hbar c}{G_N} \approx (2.2 \times10^{-8}\ \mathrm{kg})^2. $$
So you could write
$$ F = \frac{\hbar c m_1 m_2}{m_P^2 r^2}, $$
which is closer to what you're doing and, with units where $\hbar=c=1$, is the convention in a lot of high energy physics.
A: The completely symmetric form of Coulomb's constant, such that $E=H$, is $K_c=c/4\pi$.  If you equate the forces of gravity and electricity, you can wrote, eg $M=þQ$, where $þ$ is a constant.  Then $G=c/4\pi þ^2$.  Stoney's mass is then $eþ$ and planck's mass is $eþ/\sqrt{\alpha}$.  
The rationalisation of equations only start when you do, as Heaviside and as Lorentz do, start with maxwell's equations.  People are starting to do this with gravity, see, eg gravitomagnetism on the wikipedia.
SI treats rationalisation of quantities in three different ways according to whether it is gravity (unrationalised, no units), or electricity (rationalised, no extra units), or light (unrationalised with units)
It should be noted that gravitational magnetic-theory is somewhat behind the corresponding electric version, because the anticipated size of the field is so small that only now it is possible to try to detect the field.
One should imagine that $G$ is a 'falle-constant'.  That is, Newton's equation is not usable as a definition of mass in the way that Coulomb's equation might define charge, so the non-variable constants are lumped together in the manner that $K_c$ defines coulomb's constant.  It's only when one has a sufficient theory behind it that one tries to modify the value of $G$.  'Falle-constant' here simply means that the units and value of the constant are 'as they fall'.
