Neutral (Absolute) Geometry

Neutral geometry (also called absolute geometry, a term of Bolyai) is what remains of Euclidean geometry when the parallel postulate is withheld: the body of theorems provable from Hilbert's incidence, order, congruence, and continuity axioms alone (Groups I–III + V), with Group IV — the parallel axiom — deliberately omitted. It is the common trunk from which Euclidean and hyperbolic geometry branch: every theorem here holds both in the plane you learned in school and in the hyperbolic plane. Understanding exactly how far one can go without the fifth postulate is the whole point — it locates the precise logical work that the postulate does.

References: Euclid I.1–28; Greenberg, Euclidean and Non-Euclidean Geometries, chs. 3–4; Hartshorne, Geometry: Euclid and Beyond, ch. 3.

1. What Counts as Neutral

Euclid himself proves Propositions I.1 through I.28 without the parallel postulate — its first use is in I.29 (alternate angles and parallels). Those first 28 propositions are therefore theorems of neutral geometry. The rough boundary:

The pattern to notice: neutral geometry gives existence and inequalities; the parallel postulate upgrades them to uniqueness and equalities.

2. Congruence and the Base Theorems

Because Hilbert takes SAS as an axiom (Group III), the standard congruence criteria are neutral theorems.

Congruence criteria. Two triangles are congruent under any of SAS, ASA, SSS, or SAA (side–angle–angle). AAA is not a congruence criterion.

That last remark foreshadows the entire non-Euclidean story: in neutral geometry one cannot prove that equal angles force equal sizes, and in hyperbolic geometry AAA turns out to be a congruence criterion — there are no similar, non-congruent triangles there.

Other neutral staples:

• Isosceles triangle theorem (Euclid I.5, pons asinorum): base angles of an isosceles triangle are equal, and conversely.
• Triangle inequality (I.20): any side is shorter than the sum of the other two.
• Existence and uniqueness of perpendiculars: through any point there is a unique line perpendicular to a given line.

3. The Exterior Angle Theorem

The workhorse of neutral geometry is the weak (inequality) form of the exterior angle theorem — the strong form (exterior angle equals the sum of the two remote interior angles) requires the parallel postulate.

Exterior Angle Theorem (neutral). An exterior angle of a triangle is greater than either remote interior angle.

Proof sketch (Euclid I.16). Let the exterior angle at $C$ be formed by extending $BC$ to $D$. Let $M$ be the midpoint of $AC$; extend $BM$ to $E$ with $ME=MB$. Then $△AMB\cong △CME$ by SAS (vertical angles at $M$), so $\angle BAC=\angle MCE$. But $\angle MCE$ is part of the exterior angle $\angle ACD$, and by common notion 5 the whole exceeds the part, so $\angle ACD>\angle BAC$. The other remote angle is handled symmetrically. $■$

This proof uses betweenness ("$\angle MCE$ is part of $\angle ACD$"), a place where Euclid quietly relies on Pasch's axiom — one of the gaps Hilbert closed.

Corollary (existence of parallels). Given a line $\ell$ and a point $P$ not on it, drop a perpendicular from $P$ to $\ell$ at $Q$, then erect the line $m$ through $P$ perpendicular to $PQ$. By the exterior angle theorem $m$ cannot meet $\ell$ (a meeting would create a triangle with two right angles, contradicting the theorem). So at least one parallel to $\ell$ through $P$ exists. Neutral geometry proves parallels exist; it is silent on whether the parallel is unique.

4. Saccheri Quadrilaterals and the Angle-Sum Bound

To probe the angle sum without assuming the fifth postulate, Saccheri (1733) studied a quadrilateral $ABDC$ with two equal sides $AC=BD$ perpendicular to the base $AB$. The two angles at the top, $C$ and $D$, are called the summit angles.

Saccheri's theorem (neutral). In a Saccheri quadrilateral the summit angles are equal to each other. There are exactly three logical possibilities:

• Hypothesis of the Right Angle (HRA): the summit angles are right;
• Hypothesis of the Obtuse Angle (HOA): they are obtuse;
• Hypothesis of the Acute Angle (HAA): they are acute.

Saccheri hoped to derive a contradiction from HOA and HAA and thereby prove the parallel postulate (HRA). He succeeded with HOA — but only by tacitly assuming lines are infinite (see below) — and, despite heroic effort, could not refute HAA. He reluctantly declared it "repugnant to the nature of the straight line" and published anyway. In fact HAA is hyperbolic geometry and is perfectly consistent; Saccheri had proved deep hyperbolic theorems while trying to destroy them. See discovery-history.md.

The three hypotheses correspond exactly to the three geometries by curvature:

$$\text{HOA}\leftrightarrow K>0(\text{elliptic}),\text{HRA}\leftrightarrow K=0(\text{Euclidean}),\text{HAA}\leftrightarrow K<0(\text{hyperbolic}),$$

a correspondence made quantitative in constant-curvature.md.

5. The Saccheri–Legendre Theorem

The single most important quantitative result of neutral geometry bounds the angle sum from above.

Saccheri–Legendre Theorem. In neutral geometry the angle sum of any triangle is at most $\pi$ (two right angles).

Proof idea. Suppose some triangle had angle sum $\pi +\epsilon$ with $\epsilon >0$. A midpoint-and-extend construction produces a new triangle with the same angle sum but one angle at most half the original smallest angle. Iterating, one obtains a triangle whose angle sum is still $\pi +\epsilon$ yet which has two angles summing to more than $\pi$ — contradicting the exterior angle theorem. The Archimedean axiom (continuity, Group V) is what licenses the "iterate until small enough" step. $■$

Note this rules out HOA in the presence of the Archimedean/line-infinitude axioms: if lines are genuinely infinite, the obtuse case is impossible. That is why elliptic geometry must modify Euclid's Postulate 2 (unbounded lines) rather than merely deny Postulate 5 — elliptic geometry is not a neutral geometry. Hyperbolic geometry, by contrast, keeps all of neutral geometry intact.

The angle defect. Define the defect of a triangle as

$$\delta (△)=\pi -(\alpha +\beta +\gamma )\ge 0.$$

Saccheri–Legendre says $\delta \ge 0$ always. The defect is additive: if a triangle is split by a cevian into two sub-triangles, the defects add, $\delta ={\delta }_1+{\delta }_2$. This additivity is the neutral-geometry seed of the fact that defect is proportional to area in hyperbolic geometry (see hyperbolic-geometry.md) — area is the only additive, motion-invariant quantity it can be.

6. The Fork in the Road

Neutral geometry proves:

• a parallel to a given line through an external point exists (§3);
• every triangle has angle sum $\le \pi$, i.e. defect $\ge 0$ (§5).

It leaves exactly one question open — and it is a genuine bifurcation, not an oversight:

Is the parallel unique (equivalently, is the defect always zero)?

Answering "yes" is the parallel postulate and yields Euclidean geometry. Answering "no" — many parallels, positive defect — yields hyperbolic geometry, every bit as consistent, as the models will prove. Both branches sit atop the identical neutral trunk; that shared trunk is why so much school geometry survives the passage to the hyperbolic plane unchanged.