Why doesn't Newton define F=m(a^2)

[prosteve037]

Galileo attributed changes in motion to forces in a general way. But, at least in my understanding, it was not Galileo but Newton who attributed gravitational accelerations to gravitational forces. Galileo determined that in the absence of friction or resistance, all objects fall at the same rate. He determined that the relationship between time, t, of fall and height, h, of fall of an object was $h = at^2/2$. But Galileo did not conclude that they fall at the same rate due to gravitational force being proportional to mass.

AM

Ahh so it was Newton who made the connection. But I still don't see how he'd figured a way to have a relative scale of forces. Did Newton use the weights of the bodies or something like spring scales to do this?

Also, based on DH's post Newton seemed to have related the change in momentum to the force, without reference to it's rate change with respect to time. If this is the case, how did we come to interpret it as $\textit{F}\propto{\frac{Δp}{Δt}}$?

 

[prosteve037]

Also, was weight seen as the force due to gravity by Galileo's time? I've read that there was confusion among physicists at the time about the nature of weight; I wasn't sure if it was seen as synonymous with the gravitational force though.

 

[Rap]

Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?

Newton needed to name the quantity ma because it entered into the physical description of things. Since ma corresponds to our intuitive notion of force, he named it force.