Final answer:
British anthropologists recognized the Islanders' equal mental capacity through studies by W.H.R. Rivers and others, which showed that sensory perception was influenced similarly across different populations, regardless of race or culture.
Step-by-step explanation:
British anthropologists were persuaded of the Islanders' intellectual equality through pioneering studies that showed both environmental and biological factors affected sensory perception similarly in different populations. Specifically, the work of anthropologists like W.H.R. Rivers, who engaged with Melanesian populations, provided evidence that effectively debunked concepts of racial inferiority. These scientific observations were used to challenge and renounce the prejudiced view of the 'noble savage', a notion that undermined the mental capabilities of non-European societies by assuming a biological and intellectual hierarchy favoring Europeans.
Rivers' research on the inheritance of sensory capabilities among Melanesians during the Torres Strait Island Expedition of 1898 demonstrated that environmental and biological influences shaped the senses of Melanesians in much the same way as they did for the British. This equalizing revelation in the field of psychological anthropology led to the understanding that mental capacity was not disparate across cultures or races. Thus, the idea of 'inferior mental capacity' of the islanders, as compared to Europeans, was scientifically dispelled.