the absolute constant of curvature
The three classical geometries — hyperbolic, Euclidean, elliptic/spherical — are not three unrelated worlds but a single family tuned by ONE number. Call it the curvature K (or work with a length k related to it). When K is negative you are in hyperbolic geometry; when K is exactly zero you are in flat Euclidean geometry; when K is positive you are in spherical/elliptic geometry. The same triangle-area law writes all three at once: area is proportional to the angle defect (or, for positive K, the excess), and the constant of proportionality is set by K. Euclid is simply the borderline case K = 0, where the defect collapses to zero and the angle sum locks at 180 degrees.
The deepest consequence is what Gauss and others called the existence of an ABSOLUTE unit of length. In Euclidean geometry there is no natural yardstick — you may rescale the whole plane and nothing intrinsic changes, which is why blueprints and scale models work. But in a curved geometry the constant K builds a definite length into the fabric: a specific distance can be singled out purely by the angles it generates (recall the angle of parallelism Pi(p), which ties a length to an angle with no arbitrary ruler). Lambert foresaw this; he noted a non-Euclidean plane behaves like a sphere of radius 1/sqrt(K), real for the sphere and 'imaginary' for the hyperbolic plane.
Why call it 'absolute'? Because the geometry fixes the unit, not the other way round — the constant is woven into the space rather than chosen by a surveyor. As triangles shrink relative to 1/sqrt(K), every geometry looks Euclidean (small regions of a sphere or hyperbolic plane are nearly flat), which is why we discovered flat geometry first and why the universe looks Euclidean on human scales. One honest boundary: WHY a surface carries a given K — the differential-geometric account of curvature — belongs to differential geometry; here K is the single dial that organises the three geometries into one continuous family.
Write the unified triangle-area law as area = (1/|K|) * |defect or excess| (angles in radians). For a sphere of radius R, K = 1/R^2, so 1/|K| = R^2 and area = R^2 * excess — exactly Girard. For the hyperbolic plane with K = -1/k^2, area = k^2 * defect. As K -> 0 the constant 1/|K| blows up while the defect shrinks to 0, and the product approaches the finite Euclidean area, with the angle sum frozen at 180 degrees.
One constant K dials the family: K < 0 hyperbolic, K = 0 Euclidean, K > 0 spherical — and it builds an absolute length into curved space.
Only flat (K = 0) Euclidean geometry has no absolute length and lets you freely rescale; both curved geometries fix a natural unit. The question of WHY a space has its curvature belongs to differential geometry, not here.