I logged on today with this exact question: Ellipse definition
I found it disconcerting for him to say that it was clear that $a > c$ when $a$ could be equal to $c$ (a straight line) or maybe even less than $c$ (if complex numbers are allowed). So he is assuming that we don't want a straight line, and also that complex numbers aren't allowed. Neither of those assumptions were stated or explained. I don't even know whether complex numbers would work, whether any sum at all could be arrived at. It's also not stated that the formula wouldn't work for a straight line; it's just glossed over by saying it 'clearly' couldn't be a straight line.
I picked up Spivak's book because I had heard it was extremely rigourous, but now I'm wondering a) whether the unstated assumption and lack of addressing conceivable possibilities is common in his book, and b) whether there were any other book recommendations to learn calculus with the requirement of rigour in mind.
I'm a bit hesitant to continue, as I may be unable to tell whether something 'clear' to him is not clear to me due to me not understanding it properly, or due to not being aware of his assumptions. As I'm trying to learn this on my own, that's not a favourable position for me to be in.