Intrinsic Spherical Geometry vs. Euclid’s Planar Elements

Emancipating the Sphere from Stereometry

Prior to Menelaus, Greek treatises on the sphere—such as Autolycus’s On the Moving Sphere and Theodosius’s Sphaerica—studied great and parallel circles exclusively as three-dimensional Euclidean sections of a solid ball. Neither Autolycus nor Theodosius ever defined a spherical triangle.

In Book I, Definition I of his Sphaerica (Halley 1758, p. 1), Menelaus performs a quiet axiomatic revolution: he defines the triangulum sphaericum intrinsically on the two-dimensional curved surface \(S^2\) as the region bounded by three minor arcs of great circles (arcus circulorum magnorum… semper minores semicirculo).

What Survives and What Breaks Without the Parallel Postulate

Because two distinct great circles on \(S^2\) always intersect in two antipodal points, Euclid’s Fifth Postulate fails globally on the sphere. Menelaus systematically rebuilds Euclid’s Elements Book I on \(S^2\), revealing with razor-sharp precision which theorems belong to neutral/absolute metric geometry and which depend on zero curvature:

Euclidean Plane (\(K = 0\)) vs. Menelaus’s Sphere (\(K = +1\))

Geometric Principle	Euclid’s Elements (Book I & VI)	Menelaus’s Sphaerica (Book I & III)
Geodesic sides	Straight line segments (I Def. 4)	Minor arcs of great circles (I Def. 1–2)
Congruence (SAS, SSS, ASA)	I.4, I.8, I.26	I.4, I.8, I.14–16 (holds intrinsically!)
Triangle inequality	I.20: \(a + b > c\)	I.5: \(\widehat{AB} + \widehat{BC} > \widehat{AC}\) (and \(a+b+c < 2\pi\))
Exterior & interior angle sum	I.16 & I.32: \(\text{ext} > \text{int}\), \(A + B + C = \pi\)	I.11: \(\text{ext} < \text{int}_1 + \text{int}_2\), \(A + B + C > \pi\) (excess)
Equiangular triangles (AAA)	VI.4: Similar, arbitrary scale!	I.17: Strictly congruent (scale fixed by sphere radius!)
Transversal compound ratio	\(\frac{AF}{FB}\frac{BD}{DC}\frac{CE}{EA} = -1\) (Lemma I)	III.1: \(\frac{\sin AF}{\sin FB}\frac{\sin BD}{\sin DC}\frac{\sin CE}{\sin EA} = -1\)