The name of the North Island is Te Ika o Maui (The Fish of Maui), but is given by Cook as Aeheino Mouwe.

It has been suggested (Rusden) that the name given to him was Tehinga o Maui (The Fishing of Maui), and imperfectly rendered.) which Signifies green Talk or Stone, such as they make their Tools or ornaments, etc., and for the third he pointed to the land on the East side of the Strait; this, he said, was a large land, and that it would take up a great many Moons to sail round it; this he called Aeheino Mouwe, a name many others before had called it by.

That part which borders on the strait he called Teiria Whitte.

After we had done our business upon the Island we returned on board, bringing the old Man along with us, who after dinner went ashore in a Canoe that came to attend upon him.
[February 1770.] Thursday, February 1st.

P.M.having compleated the Ship with wood, and filled all our water, the Boatswain was sent ashore with a party of Men to cut and make brooms, while others were Employ'd about the rigging, fishing, etc.