Apparently this seeming contradiction hinges on an ambiguity in one of Euclid's postulates, the assumption that a straight line can be indefinitely extended. This is true of a great circle, in the sense that it just keeps winding around the sphere as far as you want to go. But it is false if interpreted as saying that you keep encountering new points, or that the total length of the line is infinite. That latter interpretation was used, at least tacitly, in reaching a reductio ad absurdum from the obtuse-angle hypothesis.
Étienne Bézout (1730-1783) was a professor of mathematics in naval and artillery academies. He took his teaching and research responsibilities there quite seriously, so his writings have a more elementary and practical flavor than those of, say, Fermat (a lawyer for whom mathematics was always a hobby) or Euler (a full-time pure mathematician at the elite academic level). His name is forever attached to the fundamental theorem about the solutions of simultaneous polynomial equations -- although the main idea goes back at least to Newton and a completely rigorous formulation and proof did not appear until decades later (19th century). Restricted to two unknowns, and expressed in the geometrical language of the time, the simplest form of Bézout's theorem is:
A curve of degree m intersects a curve of degree n in at most mn points. Here it is assumed that the curves do not have components that coincide (so that their number of points in common is infinite).
For example, a conic section is a curve of degree 2, represented by an equation of second order in two variables. It is clear that a noncircular ellipse intersects 4 times with the 90 degree rotation of itself; this is the expected generic behavior. A circle rotated coincides with itself; this case is regarded as "cheating" because the two equations are not independent. An ellipse and a displaced rotated ellipse may intersect in only two points (or 3, or 1, or not at all).
Just as for polynomial equations in one variable, it is possible to improve the theorem to:
A curve of degree m intersects a curve of degree n in EXACTLY mn points, if one knows how to count points correctly.
To understand what this means, consider the cases we already know:
Euler worked at the very beginning of modern ideas of mathematical rigor. He helped to create those ideas, but at the same time his own tastes and talents seem to have been as close to those of a theoretical physicist like Paul Dirac or Richard Feynman as to those of a typical modern pure mathematician. By this I mean that he trusted symbols and their formal manipulation far beyond the realm in which they could be logically justified (at the time). If an infinite series diverged, his reaction was not to discard it as undefined, but to assume that it did have a meaning and to attack the mystery of what its meaning was. This attitude often yielded true and useful results that were not rigorously justified until later centuries.