How Did Newton's Second Law Get Its Definition? If I've read Newton's Laws of Motion correctly in the Principia, it seems that Newton attributed the "change in motion" (momentum) to the "impressed force". Mathematically this would be read as $\Delta p \propto F$, right? But then how did it end up as $\frac{\Delta p}{\Delta t} \propto F$?
Also, I've read in other forums that Newton's Second Law is based on Galileo's experiments on falling bodies, where he treated acceleration and mass as distinct parameters. For example, his experiment with comparing free-fall of different masses yielded something along the lines of $\frac{F_1}{F_2} \propto \frac{m_1}{m_2}$ ($\propto a$). This threw me off even further, making me question how (and why) the heck Newton ended up with $\Delta p \propto F$ as his Second Law.
I'm not questioning its validity, I just want to understand how it came to be understood as $\frac{\Delta p}{\Delta t} \propto F$.
 A: Newton didn't say "change in momentum", he said "alteration in momentum", and whichever he said, this means clearly, and with no room for doubt, rate of change of momentum, the limit of small $\Delta t$ of $\Delta P \over \Delta t$. This was understood this way by everyone who read the book, there is no way to misinterpret if you follow the mathematical things.
The experiments of Galileo showed that bodies in gravity have the same acceleration. This means that the Earth is imparting changes in velocity to particles. The notion of "force" is already present to some extent in the theory of statics developed by Archimedes, and gravity produces a steady force in a static situtation, and this force is proportional to the mass. If you know force is proportional to the mass, and the acceleration is the same for all bodies, it is no leap to conclude that a force produces a steady acceleration inversely proportional to the mass.
The second law was not the major innovation in Newton, this was known to Hooke and Halley and Huygens for sure. Newton's innovation is the third law, and the system of the world, the special problems.
A: Isaac Newton, Philosophiae Naturalis Principia Mathematica, Axiomata, sive Leges Motus [1]:
Lex. II.
Mutationem motus proportionalem esse vi motrici impressæ, 
& ﬁeri secundum lineam rectam qua vis illa imprimitur.
[...]

Mutationem: means change, alteration, variation, mutation, ...
Motus: momentum.
So  Mutationem motus means exactly the variation of momentum:
$$
\frac{\Delta p}{\Delta t}
$$
[1] http://la.wikisource.org/wiki/Philosophiae_Naturalis_Principia_Mathematica
A: I will try to explain:
Let us go back to 1600s when force is not defined quantitatively, and try to understand as to how we can arrive to the definition. 
Force: a feel of push or pull is named by humans as "Force".
Lets imagine one situation in which we have a toy gun which is pushing a small marble kept on an ice frame. 
The following things happened:
1. Now the toy gun pushed the marble, using single arrow. 
2. We plotted the instantaneous rate of change of momentum for the object on y axis and on x axis, we plotted Force. 
Now mind here that the force is not defined quantitatively yet; But intuitively, we can say that the toy arrow is applying a push on the marble (thus its applying a force on the marble). 
Therefore on x axis, we plotted force as using one toy shoot. 
Now, we used two toy arrows at the very same time and applied onto the marble. 
Again we plotted rate of change of momentum on y axis, and on x axis, we can intuitively think that if ine toy arrow can provide some value of force, now two toy arrows will be providing twice that amount. 
Therefore, we plotted that on x axis. 
We did similar experiments with 3 toy arrows, 4 toy arrows etc and plotted rate of change of momentum vs this pushing force on y-x axis respectively. 
When we joined all these, for the marble of constant mass, it formed a straight line, passing through zero. 
Thus one thing is clear that the force acting on a body is directly proportional to the rate of change of momentum. 
F directly proportional to d(mv)/dt
Now to eliminate the constant of proportionality, we defined 'Newton' as a unit of force such that when we apply a force of 1 N, its capable of producing the acceleration of 1m/s2 for a body of 1kg mass. 
If on the other hand, lets say we defined the unit of force as Ron, and defined that one ron is that force which will produce an acceleration of 0.5 m/s2 for a mass of 1kg, then we can write F=2ma. 
But for the time being till, unit of force is Newton, lets enjoy with the following equation 
F=ma 
