Another good example of a Judiciary non-sequitur. They had no real proof of where he was born, and they just assumed that a citizen by the 14th amendment was the same thing as a "natural" citizen.
Shoddy and incompetent work, but the courts never have to answer for their blunders.
BTW, there have been claims President Arthur was born in Canada.
Well aware of it. This was actually alleged at the time, but President Arthur denied it. He also took great pains to muddle any efforts to find out more details about him. Among other things, he deliberately misled people as to his date of birth. He burned all his documentation before his death. He acted like a man with something to hide, and as Leo Donofrio pointed out, it appears that he had good reason to do so.
The Constitution provides no mechanism for a candidate to prove his eligibility.
To me, the obvious reason for this is that the Founders thought the Electoral College would function as intended, that the Electors would actually elect the President they thought best suited to lead the nation. Thus the electors would decide whether a candidate met the specified criteria.
The EC, of course, only functioned as intended twice, electing Washington. It was simply one of the Founders more poorly designed systems. Just couldn’t work.