The answer seems obvious enough. A purring cat is a contented cat. This surely must be true. But it is not. Repeated observation has revealed that cats in great pain, injured, in labor and even dying often purr loud and long. These can hardly be called contented cats. It is true, of course, that contented cats do also purr, but contentment is by no means the sole condition for purring. A more precise explanation, which fits all cases, is that purring signals a friendly social mood, and it can be given as a signal to, say, a vet from an injured cat indicating the need for friendship, or as a signal to an owner, saying thank you for friendship given.
Purring first occurs when kittens are only a week old and its primary use is when they are being suckled by their mother. It acts then as a signal to her that all is well and that the milk supply is successfully reaching its destination. She can lie there, listening to the grateful purrs, and know without looking up that nothing has gone amiss. She in turn purrs to her kittens as they feed, telling them that she too is in a relaxed, co-operative mood. The use of purring among adult cats (and between adult cats and humans) is almost certainly secondary and is derived from this primal parent-offspring context.
An important distinction between small cats, like our domestic species, and the big cats, like lions and tigers, is that the latter cannot purr properly. The tiger will greet you with a friendly 'one-way purr'...a sort of juddering splutter...but it cannot produce the two-way purr of the domestic cat, which makes its whirring noise not only with each outward breath (like a tiger) but also with each inward breath. The exhalation/inhalation rhythm of feline purring can be performed with the mouth firmly shut and may be continued for hours on end. In this respect small cats are one up on their giant relatives, but big cats have another feature which compensates for it. They can roar, which is something small cats can never do.
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