Since the beginning of history P : have managed to catch Q : the Eskimos and Red Indians R : by a very difficulty method S : a few specimens of this aquatic animal The Proper sequence should be:

Q-P-S-R creates: 'Since the beginning of history the Eskimos and Red Indians have managed to catch a few specimens of this aquatic animal by a very difficulty method.' This is grammatically coherent (note: 'difficulty' should be 'difficult' - there's a typo in the original). The subject (Eskimos and Red Indians) is followed by the verb phrase (have managed to catch), then object, then method.