There's no magic behind it. It was done by non-dimensionalizing the momentum equation in the Navier-Stokes equations.
Starting with:
$$\frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho}\frac{\partial P}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_j x_j}$$
which is the momentum equation for an incompressible flow. Now you non-dimensionalize things by choosing some appropriate scaling values. Let's look at just the X-direction equation and assume it's 1D for simplicity. Introduce $\overline{x} = x/L$, $\overline{u} = u/U_\infty$, $\tau = tU_\infty/L$, $\overline{P} = P/(\rho U_\infty^2)$ and then substitute those into the equation. You get:
$$ \frac{\partial U_\infty \overline{u}}{\partial \tau L/U_\infty} + U_\infty\overline{u}\frac{\partial U_\infty \overline{u}}{\partial L \overline{x}} = - \frac{1}{\rho}\frac{\partial \overline{P}\rho U_\infty^2}{\partial L\overline{x}} + \nu \frac{\partial^2 U_\infty \overline{u}}{\partial L^2 \overline{x}^2} $$
So now, you collect terms and divide both sides by $U_\infty^2/L$ and you get:
$$ \frac{\partial \overline{u}}{\partial \tau} + \overline{u}\frac{\partial \overline{u}}{\partial \overline{x}} = -\frac{\partial \overline{P}}{\partial \overline{x}} + \frac{\nu}{U_\infty L}\frac{\partial^2 \overline{u}}{\partial \overline{x}^2}$$
Where now you should see that the parameter on the viscous term is $\frac{1}{Re}$. Therefore, it falls out naturally from the definitions of the non-dimensional parameters.
The intuition
There's some other ways to come up with it. The Buckingham Pi theorem is a popular way (demonstrated in Floris' answer) where you collect all of the units in your problem in this case $L, T, M$ and find a way to combine them into a number without dimension. There is one way to do that, which ends up being the Reynolds number.
The interpretation of inertial to viscous forces comes from looking at the non-dimensional equation. If you inspect the magnitude of the terms, namely the convective (or inertial term) and the viscous term, the role of the number should be obvious. As $Re \rightarrow 0$, the magnitude of the viscous term $\rightarrow \infty$, meaning the viscous term dominates. As $Re \rightarrow \infty$, the viscous term $\rightarrow 0$ and so the inertial terms dominates. Therefore, one can say that the Reynolds number is a measure of the ratio of inertial forces to viscous forces in a flow.
Best Answer
The first estimate of Avogadro's number was made by a monk named Chrysostomus Magnenus in 1646. He burned a grain of incense in an abandoned church and assumed that there was one 'atom' of incense in his nose at soon as he could faintly smell it; He then compared the volume of the cavity of his nose with the volume of the church. In modern language, the result of his experiment was $N_A \ge 10^{22}$ ... quite amazing given the primitive setup.
Please remember that the year is 1646; the 'atoms' refer to Demokrit's ancient theory of indivisible units, not to atoms in our modern sense. I have this information from a physical chemistry lecture by Martin Quack at the ETH Zurich. Here are further references (see notes to page 4, in German): http://edoc.bbaw.de/volltexte/2007/477/pdf/23uFBK9ncwM.pdf
The first modern estimate was made by Loschmidt in 1865. He compared the mean free path of molecules in the gas phase to their liquid phase. He obtained the mean free path by measuring the viscosity of the gas and assumed that the liquid consists of densely packed spheres. He obtained $N_A \approx 4.7 \times 10^{23}$ compared to the modern value $N_A = 6.022 \times 10^{23}$.