This occurred to me this morning, when I saw the following three-fold characterization of the conic sections, which may amuse. (I'm basically copying this from Leila Schneps' lecture notes.)
Conic sections can be defined geometrically, analytically, or algebraically.
- Geometrically: a conic section is the curve obtained when a plane intersects with a cone. The conic section is an ellipse, parabola, or hyperbola according to whether the plane is less steeply slanted, exactly as steeply slanted, or more steeply slanted than the cone's generating line.
- Analytically: given a point F the focus and a line D the directrix, a conic section is the locus of points P such that d(P,F)/d(P,D) is a constant e, the eccentricity. The conic section is an ellipse, parabola, or hyperbola according to whether the eccentricity is less than, equal to, or greater than 1.
- Algebraically: a conic section is the solution set of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. The conic section is an ellipse, parabola, or hyperbola according to whether B2 - 4AC, the discriminant, is negative, zero, or positive.
The three definitions are equivalent.
It's fitting that the quote is in Latin, because I don't know Latin, and I also don't believe that mathematics is divided into only these three parts. In particular, you all know that I like probability and combinatorics, which don't naturally fit into this tripartite scheme. But it's a nice division of what one might call "classical" mathematics (it's roughly the content of the semi-standard first-year graduate mathematics curriculum, for example).
Calling the second definition "analytic" is a bit of a stretch, though.