Why Is Newton's second law $F=ma$, and how did Newton "Discover it"? I have been thinking lately, why is it that Newton's second law of motion takes the form $F = ma$? It seems to me the idea of a force is a somewhat made up concept, at least the value we assign to it seems made up. Is the a reason that $F= ma$ instead of maybe something  $F = kma$ where $k$ is some constant or $F = ma^2$ or  $F = m^2a$. The fact that it relates mass and acceleration makes sense to me, but it seems as if the type of relation is somewhat arbitrary.
However upon reflection I have come up with a way that I guess you could establish this law by experiment.The basic idea of my experiment would be to get some sort of "pusher" mechanism that would "push", with constant strength, a variety of movable objects that are subject to no external interaction and then measure the acceleration of the objects while they are being pushed. I guess you could make the assumption that a falling object is being "pushed" towards earth somehow and assume that this push is constant. 
Does anyone one know how Newton himself conducted the experiment? I can't seem to find an answer simply by googling it.
 A: 
it seems as if the type of relation is somewhat arbitrary

It's not arbitrary.
It's matched by experiment.
All the alternative formulas do not match experiment as well.  That's why these laws were successful.  This is the rule used in physics to choose which laws are the most successful - they have to be accurate and consistent with experiment to within the margin of error of the experiment.
So not arbitrary, but more survival of the fittest.
A: It is $F=kma$ with $k > 0$, in different units systems. For example, if you measure the mass of something in kg, but measure forces in pounds, you have to use a constant for converting the units.
As for why it's linear like that, it was just an observable fact that if you apply two forces it will accelerate under the vector sum of the forces. This is, of course, approximate. When you get in to special relativity the law gets adjusted to,
$$\vec{F} = \frac{\operatorname{d} \vec{p}}{\operatorname{d} t},$$
where $\vec{p}$ is the relativistic momentum. For more, see Wikipedia's section on relativistic mechanics.
A: According to Newton's second law of motion Force is directly proportional to the rate of change of momentum and when we solve it for a body with fixed mass and travels at uniform acceleration we get $F = kma$ and after doing so many experiments Newton found that the value of k turns out to be 1 so we don't write it every time and the other formulas which you suggested like F=ma² or F=m²a are dimensionally incorrect
