How to find the expression of the equation of the fundamental principle of dynamics starting from the Galileo group ?

Question

Good morning all,
I learned quite a while ago that the equation of the fundamental principle of
dynamics in Newtonian mechanics was discovered by studying its 
invariance group which is the Galileo group.

In the beginning, physicists of the $ 17 $ -th and $ 18 $ -th century did not 
yet know the notion of group as it is the case today, but did they nevertheless
 extract the expression of the equation of the fundamental principle of 
dynamics on the basis of the theoretical study of the relative movements of a 
body moving in two Galilean frames of reference in motion of inertia.

Could you explain to me in details, how were they able to extract the 
expression of the equation of the fundamental principle which is 
$ \sum \vec{F} = m \ \vec{a} $, 
from its invariance group which is the Galilean group 
$ (\mathrm{SO} (3) \times \mathbb{R}^3) \times \mathbb{R}^4 $?

Thanks in advance for your answers.