Who claimed Australia for Great Britain?

In 1770, Lieutenant James Cook (not yet a Captain) claimed "New South Wales", which was actually Australia's eastern half, for Britain. He claimed the east coast under instruction from King George III of England.

At that time, no-one had claimed the western half or the south. In 1829, Captain Charles Fremantle was sent to take formal possession of the remainder of New Holland which had not already been claimed for Britain under the territory of New South Wales. On 2 May 1829, Captain Fremantle raised the Union Jack on the south head of the Swan River, thus claiming the territory for Britain.

The southern regions were claimed by virtue of establishing more colonies. Had the French beaten the British to these regions, Australia could well be one-third French by now ...