If we have not yet discovered a method of separating quarks from each other in protons and neutrons, how did scientists discover that protons and neutrons were composed of quarks?
Quantum mechanics theorizes about quarks to explain the phenomena observed when things like protons and neutrons collide, decay, and in general interact.
Quarks have never been observed directly, but every particle that a neutron or a proton can decay into can be explained in terms of "quark channels." These "channels" are simply different combinations of quarks from within the proton/neutron combining such that their possibilities constructively create different particles (or destructively prevent certain particles from existing).
Quarks explaining flawlessly the generation of these post-decay particles. Thus scientists are confident that the quark model is a good one.
So the short answer is that a theory based on a quark model popped up in quantum mechanics that assumed that protons and neutrons were made of smaller quantum entities. Thus, using quantum mechanics, these quantum entities' interactions were analyzed and the theory further asserted that if this was the case, other particles would be created under certain situations (collisions, etc.). Being able to detect these other particles would help support the quark theory. Gradually, every particle that was ever produced could be explained through quark model quantum interactions and a strong level of confidence was gained in the idea of a quark.
The existential trouble you may have wrapping your head around a quark may be a result of not being introduced to all the problems concerning different "reality models" of quantum mechanics and how those reality models deal with the highly stochastic quality of quantum processes. To help with the question you ask here as well as further questions about quantum reality, Nick Herbert's _Quantum Reality: Beyond the New Physics_ should be helpful. Richard P. Feynman's _QED: The Strange Theory of Light and Matter_ should provide some interesting details about quantum mechanics (the last section of the book diverts its attention from QED and starts talking about the quark).
Ted Pavlic, Electrical Engineering Undergrad Student, Ohio St.
Quark theory or QCD (Quantum Chromo Dynamics)provides an explanation of how matter can be constructed from smaller bits; namely quarks. As you correctly state a naked quark or single quark has never be observed; and can never be observed according to the theory. Quarks do not like to be revealed.
Before QCD, Physics had a very large number of particles; often described as the particle zoo! This new theory of quarks was not only able to explain why all the existing particles existed, but also predict new, as yet unknown particles. Thus through it's predictive power it gained recognition as a good theory.
So in fact we have not discovered that n or p are made of quarks, but n and p are both part of the particle zoo, and their masses can be consistently explained by the quark theory.
It is also possible using the quark model of n & p to gain further insight into the structure of these particles.
In summary, Scientists never discovered that neutrons and protons were made of quarks, but discovered that the vast variety of other particles which can be created in particle accelerators could be explained in this way, and neutrons and protons also fitted the bill.
Andrew Couch, M.S., Physics Grad, Manchester, UK
Indeed if current theory is correct, the issue of colour confinement will ensure that we will never separate quarks from one another (and certainly to the highest limit of modern particle accelerators this has proven to be the case). However it is not nessesary to observe quarks individually to infer their existance, indeed quarks existance was discovered in much the same way as rutherford discovered that the atom was made of constituent parts.
Basically, as I am sure you will be aware from rutherfords experiment, if you fire particles at a target (like an atom or a proton in the case of quarks) and it is made of smaller componants, then when one of the incoming particles hits off one of these componants it is scattered significantly. If it hits nothing it passes through barely scattered. However if a particle were truely elementary you would expect scattering to be small.
In the SLAC in 1968 scientists found evidence for large scattering when they fired electrons at protons. However the existance of quarks had been strongly anticipated in the theoretical community for some time before that, particularly by Murray Gell-Mann who gave them their name, and the evidence from SLAC only confirmed their theory. Their reason for suspecting quarks existance was due to the ordering exotic, or "strange" particles took when graphed appropriately (Mathematically speaking a representation of an SU(3) symmetry - though weakly broken due to masses), so perhaps you could say that it the existance of quarks was discovered through theory.
Alan Cooney, B.A., Physics Grad Student, Ill.
Deep inelastic scattering experiments at Stanford Linear Accelerator Center (SLAC) were used to analyse the structure of the proton.
They studied the prosess
electron + proton -> electron + anything
They discovered that at very high energies the process would lose it's depenence on some low energy parameters. This phenomenon is known as scaling.
The simpliest explanation of this scaling comes from Feynman's parton model, where the proton was assumed to consist of point-like constituents. This model predicted many of the experimental results at SLAC.
This picture was improved upon by using a non-abelian gauge theory, Quantum Chromodynamics (QCD). QCD is very much like quantum electrodynamics, but instead of having one charge there are 3 colour charges and the gauge bosons (photons for QED, gluons for QCD) carry charge. (the gauge group is SU(3))
In this theory the partons are quarks. This theory gives rise to what is known as asymptoyic freedom. That is at high energies the quarks can be treated as free, but at low energies the force between quarks gets stronger and stronger. This explains why quarks are always confined in hadrons at low energies. This force is often called the strong force.
Dispite it's succsess, QCD at low energies is difficult to work with. One standard method is to use large supercomputers to do the work.
The standard model, which QCD is part of, describes all known particles to date. So far no experimental eveidence has been discoved that puts dout on the standard model.
Andrew James Bruce, Physics graduate, UK
'To myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.'