Logical Answer: He recognized he was ineligible as people were taught the meaning of 'natural born citizen' in his era. Arthur covered up the circumstances of his birth (and even went to the extreme of burning all his personal papers before he died).
All the differences of opinion here about what the Founders intended are meaningless banter in light of the Arthur counter-example. These events took place when the original meaning of the phrase was still widely understood and not clouded by post-modern 'thought' and lawyer-ese.
A implies B
~A
Therefore, ~B.
P.S. I do not think ineligibility will bring down the travesty of Obama. But promoting the pretense that he is eligible is simply another Arthur-like lie.