For the first time in all of nVidia's years I'm looking to ATI for an upgrade on the suggestion of the excellent comparisons here (and user feedback). Lucky Thirteen it ain't quite.
On to the article: I believe there's an error on the first page. The NV1 does use polygons, which seems fairly obvious (so does the Saturn). Reportedly it uses four-vertex polygons, and you achieve a triangle by putting two points at the same coordinate, nasty. Still a polygon, however.
The "quadratic texture mapping" mentioned is, to take a wild guess, simply a name for the NV1's texture mapping. Compare it with the nasty affine texturing on the original PlayStation where textures seem to fall in on themselves as depth information isn't taken into account, and you see the draw of a system that trades correct triangles for better texture rendering (and lower polygon usage and likely increased speed, since the rectangle, bane of mid-90s gamers, takes up two polygons when you're using triangles, instead of just one with the NV1). I don't know how the use of quads was thought produce better-looking texturing, but that seems to have been the assertion.
The Firing Squad covers this interesting period of nVidia's history in greater depth.