Unit 2: The Complete Quadrangle, Harmonic Separation & Projectivity

Karl Georg Christian von Staudt — Geometrie der Lage (1847), §§ 6–8 (arts. 66–98)

[Folio § 6 | p. 29]

§ 6. Complete Plane Quadrangles and Harmonic Forms (arts. 66–74)

Art. 66. Four points $E, F, G, H$ lying in one plane, no three of which are collinear, determine six connecting lines called sides (Seiten), which group into three pairs of opposite sides:

\[([EF], [GH]), \qquad ([FG], [HE]), \qquad ([EG], [FH]).\]

Such a configuration is called a complete plane quadrangle (vollständiges ebenes Viereck), and the three points in which its three pairs of opposite sides meet are called its diagonal points (Diagonalpunkte). Dually, four coplanar lines, no three concurrent, form a complete plane quadrilateral (vollständiges ebenes Vierseit) having six vertices grouped into three opposite pairs joined by three diagonals.

Art. 67. Let a straight carrier line $\ell$ pass through two diagonal points $A = [EF] \cap [GH]$ and $B = [FG] \cap [HE]$ of a complete plane quadrangle $EFGH$, and let the remaining two opposite sides $[EG]$ and $[FH]$ meet $\ell$ in $C$ and $D$ respectively. Then the four collinear points $A, C, B, D$ are called four harmonic points (harmonisches Gebilde), and $D$ is called the fourth harmonic point (der vierte harmonische Punkt) to $A, B, C$, conjugate to $C$ with respect to the pair $(A, B)$.

Von Staudt §§ 7–8 (Art. 93): Complete Quadrangle Harmonic Fourth & Desarguesian Uniqueness Upper quadrangle EFGH and lower quadrangle E'F'G'H' sharing A, B, C both intersect line AB in the unique fourth harmonic conjugate D = (-6, 0). A B C D E G F H E' G' F' H' Geometrie der Lage §§ 7–8 (Art. 93): Independent quadrangles EFGH and E'F'G'H' determine the identical harmonic fourth D with (A, B; C, D) = −1
**Pure Straightedge Harmonic Fourth (1847 § 6, art. 67 & § 8, art. 93):** Given collinear $A, B, C$, choose $E$ off $\ell$ and $G$ on $[CE]$, intersect $F = [AE] \cap [BG]$ and $H = [BE] \cap [AG]$, and draw $[FH]$ cutting $\ell$ in the unique harmonic fourth $D$.

Art. 70 (Spatial Proof of Uniqueness). Given three distinct collinear points $A, B, C$, the fourth harmonic point $D$ constructed by a complete plane quadrangle $EFGH$ depends solely upon $A, B, C$ and is entirely independent of the choice of the plane and of the quadrangle.

Proof. Suppose first that a second quadrangle $E'F'G'H'$ lies in a different plane $\alpha'$ passing through the same carrier line $\ell = [AB]$, with $[E'F'] \cap [G'H'] = A$, $[F'G'] \cap [H'E'] = B$, and $[E'G']$ passing through $C$. Because $[EG]$ and $[E'G']$ meet at $C$, the lines $[EE']$ and $[GG']$ lie in one plane and meet at a spatial center $S$ outside both $\alpha$ and $\alpha'$. Projecting $\alpha$ onto $\alpha'$ from $S$:

  • Plane $SEE'A$ meets $\alpha'$ in $[AE']$, and plane $SGG'B$ meets $\alpha'$ in $[BG']$; since $F = [AE] \cap [BG]$, ray $[SF]$ passes through $F' = [AE'] \cap [BG']$.

  • Likewise ray $[SH]$ passes through $H' = [BE'] \cap [AG']$.

  • Therefore the plane $SFH$ contains both $[FH]$ and $[F'H']$, so both lines meet the common intersection line $\ell = \alpha \cap \alpha'$ in the exact same point $D = D'$.

If a third quadrangle $E''F''G''H''$ lies in the same plane $\alpha$ as $EFGH$, both yield the same point $D$ as the auxiliary quadrangle $E'F'G'H'$ in the second plane $\alpha'$, hence $D'' = D' = D$. $\blacksquare$

Art. 72. In every harmonic tetrad $(A, B; C, D)$, the roles of the two conjugate pairs $(A, B)$ and $(C, D)$ are completely symmetric, and within each pair the two elements may be interchanged; moreover, $(A, B)$ and $(C, D)$ invariably separate one another.

[Folio §§ 7–8 | p. 38]

§§ 7–8. Perspectivity, Projectivity & the Fundamental Theorem (arts. 75–98)

Art. 76 (§ 7). Every perspective projection between a range of points and a flat pencil of rays, or between two ranges, carries every harmonic tetrad into a harmonic tetrad; for the four projecting planes or rays from an external center carry a complete quadrangle directly into a corresponding harmonic configuration.

Art. 93 (§ 8 — Pure Synthetic Definition of Projectivity). Two uniform fundamental forms are called projectively related (projektivisch) when their elements are placed in one-to-one correspondence so that to every four harmonic elements of the one correspond four harmonic elements of the other.

Art. 96 (Fundamental Theorem of Projective Geometry). A projectivity between two uniform fundamental forms is uniquely and completely determined as soon as three distinct elements $A, B, C$ of the first form are assigned to three arbitrary distinct elements $A', B', C'$ of the second form.

For from $A, B, C$ the fourth harmonic $D_1$ is uniquely determined by straightedge alone, and from any triad among $\{A, B, C, D_1\}$ further harmonic points $D_2, D_3, \dots$ are generated in an everywhere dense harmonic net, each of whose images $D'_k$ is uniquely forced by the corresponding quadrangle construction on $A', B', C'$. Hence if a projectivity fixes three distinct elements of one and the same carrier, it fixes every element of that carrier identically.