It is true that most of the work on this problem until, say, Descartes was geometric. Cantor's breakthrough was to think in terms of sets; specifically, countable versus uncountable sets. Lindemann was unaffected by this because he was proving a specific number transcendental (which is harder). If there is a message for P vs. NP it would to prove P is "small" in some sense compared with NP, rather than to find a specific NP problem not in P. Unfortunately, all attempts to transfer the concept of "small" from set theory to computational complexity theory have failed so far.

To bring this answer more into line with the original question, it might be better to compare the classes: algebraic numbers versus all real numbers. The existence of nonalgebraic numbers was conjectured long ago (James Gregory tried to prove $\pi$ transcendental around 1670), proved with some difficulty by Liouville in 1844, then shown to be almost trivial by Cantor 1874.

Another example in the same vein as Edit 3 is polynomial equations solvable by radicals versus all polynomial equations.

Many thanks, David and Andres, for these fascinating updates!

It's even more surprising if you start with the inductive definitions of plus and times. The proof that $ab=ba$ comes as Proposition 72 in the first development of this theory, by Grassmann in 1861.

The proof remains long, but it no longer takes a sizable part of a lifetime to check; see research.microsoft.com/en-us/news/features/…

I realize that MO has changed since the old days, but still I think this question is appropriate. Insightful books on elementary mathematics are quite uncommon, and I'd like to see more of them.

See also hyperbolic-crochet.blogspot.com/2010/07/… , which includes a picture of a paper model made by Beltrami.