Theory of human magic,
By Bellatrix Lestrange
As published in: The Daily Prophet, The Quibbler, Witch Weekly, and The Journal Of Advanced Magical Research
Because most humans (Muggles) lack magic, I postulate that witches and wizards have come by it through crossbreeding with other magical beings, i.e goblins and elves. This began many thousands of years ago, before sentient beings were prone to form prejudice.
If a human breeds with a goblin or elf, the child produced will have magic. If that child breeds with a human and then the child resulting of that union does the same, the human appearance is usually completely restored within the third generation. When those children continue to breed within the pureblood lines, (with other witches and wizards), magic is properly preserved.
The result of Muggle-borns occurs when magic skips several family generations of Muggles. This is a result of wizards breeding frequently with Muggles, who then breed with one another, continuing to weaken the bloodlines.
This is a common occurrence with genetic factors such as hair and eye color or even health conditions. Magic, also being genetic, behaves in a similar fashion. When a Squib is born to wizards, genetics are, of course, also a factor, and it could very well be related to weakening of the bloodlines through wizard/Muggle crossbreeding. If this is true, I propose that goblins and elves are indeed the true purebloods of the magical world.
To allow the bloodlines to remain strong and thus for magic to continue as an active power in the world, it appears most likely that keeping the bloodlines pure between wizards is, indeed, the correct way to accomplish this. Breeding with goblins or elves will also keep the magic flowing strong in wizarding blood. Keep in mind that if the altered appearance of the children is a concern, breeding with wizards for the next few generations should have the third generation or so looking completely human again.
Due to the fact that goblins and elves are the reason wizards quite likely exist, and are most surely the reason they will continue to do so over the centuries, it seems only logical and just that they always be given full and equal rights within our magical society. What is given, after all, can eventually fade if not helped to thrive. Wizards did not begin alone, and nor shall they thrive alone.