The Beginnings of the Domesticated Cat
Regardless of how they were domesticated, it was certainly to the purpose of hunting vermin, and to some small degree for companionship. Essentially, as the human population grew and expanded, cats were naturally attracted to both the extra discarded food, and the increasing population of rats and mice. Slowly, as the cats proved that they were useful, humans began feeding them, hoping they'd stay. They did.
Why Domestication Took Longer With Cats Than Other Animals
Cats are not considered fully domesticated. In fact, cats are said to be the most domestic of the wild animals, and the most wild of the domestic. There are a couple of factors that have led to this development. First of all, with the exception of the lion and the cheetah, cats are solitary creatures. This is unlike any other domesticated animal. The survival of the cat does not depend upon a group; they answer only to themselves, as a general rule. Horses and dogs, though, follow a leader, an alpha. This natural group instinct makes it very easy for humans to become the alpha, the pack or herd leader, so to speak. A cat, not possessing this instinct, does not follow so easily.
Secondly, cats, until the late 19th century, were not selectively bred. Humans exerted virtually no control over their choice of mates. All other domesticated animals had been subjected to selective breeding for thousands of years before this. Since the more docile animals were chosen to breed, a more domestic temperament was naturally the result.
It could be argued that domestication, then, did not begin until humans began selectively breeding cats. It was not until the first pedigreed (meaning selectively bred) cats, came about that the feline became truly domesticated. The older the pedigree, the more domesticated the cat, in a manner of speaking.
The difference in temperament between pedigreed and non-pedigreed cats has not been officially studied, but experienced breeders who have both have been known to make some telling observations. The pedigreed cats are definitely more 'domesticated' than the non-pedigreed. This has long been the observation among cat fanciers and breeders alike.
Compared to dogs, goats, sheep, and horses, cats have been domesticated for only a short while. However, it can already be seen how selective breeding has changed and altered the cat and its personality. Based on this, most breeders would say that every effort should be made to breed as much for temperament as for do color and type.
Though the truth about the origins of the domestic cat will likely always be shrouded in mystery, it can be stated with a fair degree of certainty that the domestication of the modern feline began in or around ancient Egypt. This domestication spread, until the appearance of what is now seen as the modern cat.