William Dampier was the first Englishman to explore and map parts of New Holland (Australia). On 4 January 1688, his ship the 'Cygnet' was beached on the northwest coast of Australia, at King Sound near Buccaneer Archipelago on the north-west coast of Australia.
The next English visitor was Lieutenant James Cook (not yet captain), who charted the eastern coast in 1770. Cook's positive report led to the colonisation of New South Wales by England in 1788.
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The first time the British arrived in Australia was when an English pirate named William Dampier landed on the western coast in 1688. Before William Dampier, all of the visitors to Australia had been Dutch explorers. William Dampier was sent to Australia again to give a bit more detail about the land, this time in 1699. He wasn't really interested in the countryside, and did not recommend that Australia (or New Holland, as it was called back then) should be settled. That only changed when Captain Cook came across the eastern coast in 1770, and recommended that Australia be settled by convicts. Actually, it was the botanist with him, Joseph Banks, who said Australia (or New South Wales, as the eastern side was called) should be settled. Banks was apparently reallt keen on the British settling Australia and sending its many convicts over here instead.
If the question means "when did the British settle in Australia", that happened in 1788 when the First Fleet arrived. There were eleven ships in the First Fleet, and they were full of convicts. The British convicts were the ones who colonised Australia.
Australia as a nation became part of the British Empire when the colonies federated on 1 January 1901 to become the Commonwealth of Australia.
However, the continent that is now Australia actually started to become part of the British Empire much earlier, when Lieutenant James Cook (later "Captain Cook") claimed a section of the eastern coast for Great Britain in August 1770.
The process of Australia becoming part of the British Empire was not a single event, but a series of claims.
In 1770, James Cook claimed the east coast of Australia for Britain. In 1788, the first convicts arrived.
As part of the British Empire Australia joined WW1 with the rest of the UK
Australia was once a colony of Great Britain.
north America The British Empire controlled an entire continent in Australia. The British Empire was the largest empire in history.
The indigenous people of Australia, the Aboriginal and Torres Strait Islanders, were the first to inhabit Australia. Whilst it is unknown when they first arrived, estimates vary between 6,000 and 40,000 years.
They heard about it with a StereoGraphy machine from the British.