How did Newton discover his second law? I've always assumed/been told that Newton's 2nd law is an empirical law — it must be discovered by experiment. If this is the case, what experiments did Newton do to discover this? Is it related to his studies of the motion of the moon and earth? Was he able to analyze this data to see that the masses were inversely related to the acceleration, if we assume that the force the moon on the earth is equal to the force the earth exerts on the moon? 
According to Wikipedia, the Principia reads:

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

Translated as: 

Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.

My question is how did Newton come to this conclusion? I get that he knew from Galileo Galilei the idea of inertia, but this doesn't instantly tell us that the change in momentum must be proportional to the net force. Did Newton just assume this, or was there some experiment he performed to tell him this?
 A: Newton's 1st and 2nd laws weren't particularly revolutionary or surprising to anyone in the know back then. Hooke had already deduced inverse-square gravitation from Kepler's third law, so he understood the second law. He just could not prove that the bound motion in response to an inverse square attraction is an ellipse.
The source of Newton's second law was Galileo's experiments and thought experiments, especially the principle of Galilean relativity. If you believe that the world is invariant under uniform motion, as Galileo states clearly, then the velocity cannot be a physical response because it isn't invariant, only the acceleration is. Galileo established that gravity produces acceleration, and its no leap from that to the second law.
Newton's third law on the other hand was revolutionary, because it implied conservation of momentum and conservation of angular momentum, and these general principles allow Newton to solve problems. The real juicy parts of the Principia are the specific problems he solves, including the bulge of the Earth due to its rotation, which takes some thinking even now, three centuries later.
EDIT: Real History vs. Physicist's History
The real history of scientific developments is complex, with many people making different contributions of various magnitude. The tendency in pedagogy is to relentlessly simplify, and to credit the results to one or two people, who are sort of a handle on the era. For the early modern era, the go-to folks are Galileo and Newton. But Hooke, Kepler, Huygens, Leibniz and a host of lesser known others made crucial contributions along the way.
This is especially pernicious when you have a figure of such singular genius as Newton. Newton's actual discoveries and contributions are usually too advanced to present to beginning undergraduates, but his stature is immense, so that he is given credit for earlier more trivial results that were folklore at the time.
To repeat the answer here: Newton did not discover the second law of motion. It was well known at the time, it was used by all his contemporaries without comment and without question. The proper credit for the second law belongs almost certainly to the Italians, to Galileo and his contemporaries.
But Newton applied the second law with genius to solve the problem of inverse square motion, to find the tidal friction and precession of the equinoxes, to give the wobbly orbit of the moon (in an approximation), to find the oblateness of the Earth, and the altitude variation of the acceleration of gravity g, to give a nearly quantitative model of the propagation of sound waves, to find the isochronous property of the cycloid, and a host of other contributions which are so brilliant ad so complete in their scope, that he is justly credited as founding the modern science of physics.
But in physics classes, you aren't studying history, and the applications listed above are too advanced for a first course, and Newton did indeed state the second law, so why not just give him credit for inventing it?
Similarly, in mathematics, Newton and Leibniz are given credit for the fundamental theorem of calculus. The proper credit for the fundamental theorem of calculus is to Isaac Barrow, Newton's advisor. Leibniz does not deserve credit at all. The real meat of the calculus however is not the fundamental theorem, but the organizing principles of Taylor expansions and infinitesimal orders, with successive approximations, and differential identities applied in varied settings, like arclength problems. In this, Newton founded the field.
Leibniz gave a second set of organizing principles, based on the infinitesimal calculus of Cavalieri. Cavalieri was Galileo's contemporary in Itali, and he either revived or rediscovered the ideas originally due to Archimedes in "The Method of Mechanical Theorems" (although he might not have had access to this work, which was only definitively rediscovered in the early 20th century. One of the theorems in Archimedes reappear in Kepler's work, suggesting that perhaps the Method was available to these people in an obscure copy in some library, and only became lost at a later date. This is pure speculation on my part. Kepler might have formulated and solved the problem independently of Archimedes. It is hard to tell. The problem is the volume of a cylinder cut off by a prism, related to the problem of two cylinders intersecting at right angles). Cavalieri and Kepler hardly surpassed Archimedes, while Newton went far beyond. Leibniz gave the theory its modern form, and all the formalism of integrals, differentials, product rule, chain rule, and so on are all due to Leibniz and his infinitesimals. Leibniz was also one of the discoverers of the conservation of mechanical energy, although Huygens has his paws on it too, and I don't know the dates.
The mathematicians' early modern history is no better. Again, Newton and Leibniz are given credit for theorems they did not produce, and which were common knowledge.
This type of falsified history sometimes happens today, although the internet makes honest accounting easier. Generally, Witten gets credit for everything, whether he deserves it or not. The social phenomenon was codified by Mermin, who called it "The Matthew principle", from the biblical quote "To those that have, much will be given, and to those that have not, even the little they have will be taken away." The urge to simplify relentlessly reassigns credit to well known figures, taking credit away from lesser known figures.
The way to fight this is to simply cite correctly. This is important, because the mechanism of progress is not apparent from seeing the soup, you have to see how the soup was cooked. Future generations deserve to get the recipe, so that we won't be the only ones who can make soup.
A: First of all, it would be preposterous to think that there was a simple recipe that Newton followed and that anyone else can use to deduce the laws of a similar caliber. Newton was a genius, and arguably the greatest genius in the history of science.
Second of all, Newton was inspired by the falling apple - or, more generally, by the gravity observed on the Earth. Kepler understood the elliptical orbits of the planets. One of Kepler's laws, deduced by a careful testing of simple hypotheses against the accurate data accumulated by Tycho Brahe, said that the area drawn in a unit time remains constant.
Newton realized that this is equivalent to the fact that the first derivative of the velocity i.e. the second derivative of the position - something that he already understood intuitively - has to be directed radially. In modern terms, the constant-area law is known as the conservation of the angular momentum. That's how he knew the direction of the acceleration. He also calculated the dependence on the distance - by seeing that the acceleration of the apple is 3,600 times bigger than that of the Moon.
So he systematically thought about the second derivatives of the position - the acceleration - in various contexts he has encountered - both celestial and terrestrial bodies. And he was able to determine that the second derivative could have been computed from the coordinates of the objects. He surely conjectured very quickly that all Kepler's laws can be derived from the laws for the second derivatives - and because it was true, it was straightforward to prove him this conjecture.
Obviously, he had to discover the whole theory - both $F=ma$ (or, historically more accurately, $F=dp/dt$) as well as a detailed prescription for the force - e.g. $F=Gm_1m_2/r^2$ - at the same moment because a subset of these laws is useless without the rest.
The appearance of the numerical constant in $F=ma$ or $p=mv$ is a trivial issue. The nontrivial part was of course to invent the mathematical notion of a derivative - especially because the most important one was the second derivative - and to see from the observations that the second derivative has the direction it has (from Kepler's law) and the dependence on the distance it has (from comparing the acceleration of the Moon and the apple falling from the tree).
It wasn't a straightforward task that could have been solved by anyone but it was manifestly simple enough to be solved by Newton. So he had to invent the differential calculus, $F=ma$, as well as the formula for the gravitational force at the same moment to really appreciate what any component is good for in physics.
A: Ok, a bit more searching and I came across the Stanford Encyclopedia of Philosophy:

In other words, the measure of the change in motion is the distance between the place where the body would have been after a given time had it not been acted on by the force and the place it is after that time. This is in keeping with the measure universally used at the time for the strength of the acceleration of surface gravity, namely the distance a body starting from rest falls vertically in the first second. The only special provision that Newton has to make is for non-uniform continuously acting forces, for which, in accord with Lemma 10, he takes the distance AB to vary “at the very beginning of the motion in the squared ratio of the times.”[21]
If this way of interpreting the second law seems perverse, keep in mind that the geometric mathematics Newton used in the Principia — and others were using before him — had no way of representing acceleration as a quantity in its own right. Newton, of course, could have conceptualized acceleration as the second derivative of distance with respect to time within the framework of the symbolic calculus. This indeed is the form in which Jacob Hermann presented the second law in his Phoronomia of 1716 (and Euler in the 1740s). But the geometric mathematics used in the Principia offered no way of representing second derivatives. (Newton employed curvature — that is, the circle “touching a curve” — in place of the second derivative with respect to distance throughout the Principia). Hence, it was natural for Newton to stay with the established tradition of using a length as the measure of the change of motion produced by a force, even independently of the advantage this measure had of allowing the law to cover both discrete and continuously acting forces (with the given time taken in the limit in the continuous case).
Under this interpretation, Newton's second law would not have seemed novel at the time. The consequences of impact were also being interpreted in terms of the distance between where the body would have been after a given time, had it not suffered the impact, and where it was after this time, following the impact, with the magnitude of this distance depending on the relative bulks of the impacting bodies. Moreover, Huygens's account of the centrifugal force (that is, the tension in the string) in uniform circular motion in his Horologium Oscillatorium used as the measure for the force the distance between where the body would have been had it continued in a straight line and its location on the circle in a limiting small increment of time; and he then added that the tension in the string would also be proportional to the weight of the body. So, construed in the indicated way, Newton's second law was novel only in its replacing bulk and weight with mass.[22]

I find this a bit hard to follow, but sounds like Newton is relying a lemma (assumption) that the distance an object falls varies to the squared ratio of the times, and making arguments about circular motion. So he essentially came to this conclusion on the basis of astronomical observations alone, is that correct? And how would you explain this to a high school student?
A: Newton had much precedent. He didn't devise the 1st and 2nd Postulates in a vacuum.
Regarding the 1st Postulate:

*

*John Philoponus (ca. 490-570) first devised the notion of inertia.

…rest is found in all things. For the perpetually moving heavens partake in rest, because the very persistence of perpetual motion is rest.[In De anima, 75, 11].
  …the celestial bodies are, if I may say so, motionless in their motion.[In Meteorologica, 11, 31]
  
  Regarding the 2nd Postulate:


*

*Jean Buridan (ca. 1295-1358) devised the notion of momentum and how it charges, which is what Newton called force.

It must be imagined that a heavy body acquires from its primary mover, namely from its gravity, not merely motion, but also, with that motion, a certain impetus such as is able to move that body along with the natural constant gravity. And because the impetus is acquired commensurately with motion, it follows that the faster the motion, the greater and stronger is the impetus. Thus the heavy body is moved initially only by its natural gravity, and hence slowly; but it is then moved by that same gravity as well as by the impetus already acquired, same gravity as well as by the impetus already acquired, and thus it is… continuously accelerated to the end.[Qu. De caelo et mundo (1942), 180.]
  

A: I don't if this is what Newton did, but I could prove it with an elaborate thought experiment. Think of an infinitely long smooth (frictionless) plane on which you could roll a ball. Imagine adding a ramp to this plane and roll a ball down that peak and onto the plane.
Clearly, as Galileo had said it would keep on rolling towards infinity, but if you think about it what makes the ball start rolling in the first place? There has to be something pushing it or pulling it to make it do that. So, if you see the start then it has to be something to do with falling down. As, Galileo's experiments at Pisa had shown when 2 objects of different masses fall they hit the ground at the same moment and appear to fall at the same rate. So, the thing pulling them must adjust itself by how heavy that object is...
So, this means that I don't have to care about how heavy my ball is. So, to just check if there is some relation over here. I could make a smooth ramp out of wood and just time a ball rolling down it from the same height again and again and again. I could also note where the ball is at different moments by putting some sort of scale next to that plane.
This means that I can correlate how fast it starts to go with what is pulling it. Now, this proves that there is something pushing it and that this ball seems to accelerate at the same rate, but what happens when they hit something? If we have balls of the same size and different mass and something that can give away down there then can we see how far the object gets moved over a rough surface? (how much work it does)
Does that mean somehow that these heavier balls have gained more inertia during the same time? So, that would mean acceleration and mass are correlated!

So, essentially, I would go on like this looking at things, observing them and just testing my thoughts out until I could find something that could explain (a bunch of laws, perhaps?) why that damn ball behaves the way it behaves. :D
[Note: I am sorry for any spelling or grammatical errors. I am slightly dyslexic.]
A: I'm tempted to assume that this doesn't have its origin in orbits. However, it can of course be used to describe why they occur in conjunction with a theory of gravity.
Newton would have been familiar with many textbook mechanics scenarios:

... the force pulling an object to the ground. He also calculated the centripetal force needed to hold a stone in a sling, and the relation between the length of a pendulum and the time of its swing. http://www.newton.ac.uk/newtlife.html

and hence may have been able to compare these forces - the weight of a large object, and the centripetal force of a stone - by their effects on known objects: snapping wires, uncoiling (or deforming) springs, etc.
From there you can start to quantify forces and compare them to changes in velocity and begin to postulate proportionality.
That's my take on it anyway - hope this helps!
A: Newton discovered $F=ma$ because it is an "economizer of thought," as Ernst Mach would say. To "economize thought" means to succinctly summarize the results of physical experiments or observations. Since there are many ways of "saving the phenomena" of experiments or observations, there are also many theories and thus many corresponding physics formulae.
For example, consider the following three theories of gravity applied to planetary motion:


*

*epicyclic theory

*Newton's $F\propto1/r^2$ theory

*Einstein's theory of General Relativity (GR)


All three of these theories can explain, within certain limits, a given set of observations of the motions of the planets, but they all use completely different mathematical formulae:


*

*The epicyclic theory basically uses a complex Fourier series (cf. this).

*Newton's theory uses a simple algebraic equation.

*GR uses tensors.


Newton thought that his universal theory of gravitation, $F=Gm_1m_2r^{-2}$, was uniquely, exactly, and logically deduced from Kepler's observations, but this clearly is false because Kepler's observations showed perturbations from a perfect $1/r^2$ law due to the solar system being comprised of many masses. It is also false because, e.g., Einstein's GR theory superseded Newton's theory of gravitation.
Thus, one physics theory (e.g., $p=mv$) is not more logically correct than another (e.g., $p=m+v$), although one might certainly be better at summarizing the results of experiments and observations than another.
Physics formulae do not derive from mathematics like a geometric proof derives from Euclid's axioms. Physics formulae derive from observations and experiment; mathematics does not force a physics formula to be a certain way.
For an excellent book on this whole topic, see The Aim & Structure of Physical Theory by the French physicist, historian, and philosopher of physics, Pierre Duhem.
A: Jerry Schirmer and Tobais Kienzler offer the what seems to me to be a pretty good answer. 
Jerry says:

It's kinetmatics to determine the acceleration of the moon. Geometry says that the acceleration of a circular orbit is v2r. You can measure the distance to the moon via parallax, and if you know the distance, you can infer the velocity from the length of the month. Newton's 2nd law is more a definition than a statement. Once you have the law of inertia, then you just presuppose when something deviates from constant motion, there must be some force, and the more deviation you get, the more force. It's circular unless you just define force this way.

Tobias Says:

This may sound weird, but I never understood what's so special about it: There is momentum, and if it isn't constant there is a cause defined as force, and measurable by observing the change of momentum. The great thing however is the idea of generalization to obtain e.g. the law of gravity as something valid for all kinds of matter and not just the one observed in one experiment

So is N2 really a way of defining force in terms of change in momentum? I had always heard it was a relationship that had to be proven by experiment, and that's certainly the only way I've seen it taught in school—via experiment. 
A: It could just well be that Newton was given the idea of the inverse square law from a contemporary genius, Robert Hooke. See here for more details:
http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation
PS: Even if Hooke originated the idea (how?) it was Newton's genius that propelled the inverse square law to the far reaches of time and space!
A: Keep reading the pricipia, say the first 20 or 30 pages... What theorems do Newton prove immediately after the ennuntiation of the laws? It is a good guess that the process of proving such theorems led him to depth thinking about the laws and axioms need for it.
An improvement respect to wikipedia/wikibooks is, as of today, the Newton Project, http://www.newtonproject.sussex.ac.uk/ , where you can check the "diplomatic" versions, the pre-release versions of the texts, with corrections and variations by Newton himself.
