## Now Think Big!

So during the plague, down on the farm in 1666 he used some incorrect numbers and was still working out the mathematics---inventing it as he went along. Even though he never published his results, he worked on the Moon idea off-and-on for years. But eventually, he started to think about the actual force that the Earth would exert on the Moon and the apple.

Now it's model-building time with the crucial idea that the centripetal force that the Moon feels in its orbit is a "regular" acceleration. So with his Second law for the force that the Moon feels is:

```{admonition}   Pens out!
:class: warning

$$F=m_Ma_C$$ 
and using his derived centripetal acceleration from Equation 7.5,
$$a_C(\text{M}) =  \frac{4\pi^2}{k_E}\frac{1}{D_M^2}$$ 

we would find that the force of attraction by the Earth on the Moon is:

$$
\begin{aligned}
F_{\text{Moon due to Earth}} &= m_Ma_c \\
F\text{ tentatively } &=m_M\frac{4\pi^2}{k_E}\frac{1}{D_M^2}
\end{aligned}
$$



But from his Third law, the Moon must exert an identical (in magnitude, but oppositely-directed) force on the Earth,

$$
\begin{align*}
|F_{\text{Moon due to Earth}}|  &= |F_{\text{Earth due to Moon}}| \\
& \propto M_E\frac{4\pi^2}{k_E}\frac{1}{D_M^2}
\end{align*}
$$

and the only way that can happen is if both forces are proportional to *both masses*:

$$
F_{\text{Moon due to Earth}}  = F_{\text{Earth due to Moon}} = M_E m_M\frac{4\pi^2}{k_E}\frac{1}{D_M^2}   \nonumber
$$

Now let's re-think the force on the Moon:

$$
\begin{align}
F_{\text{Moon due to Earth}}  &= m_Ma_C \nonumber \\
F_{\text{Moon due to Earth}}  &= m_M \left[\left(M_E \frac{4\pi^2}{k_E}  \right) \frac{1}{D_M^2}\right]
\end{align}$$

in which everything in the square brackets is a constant…just a number.

```

The normal way of writing this is to take all of the constants in the smooth brackets in Equation \@ref(eq:earthmoonUG) and give them a name: $G$. And therein begins a long history of a constant of nature—still a fundamental constant:
$$
F_\text{M-E}=F_\text{E-M}=G\frac{M_M m_E}{D_M^2}
$$