Geometric Study · GA 323
The Curve of Cassini
Steiner introduces the lemniscate not as a stretched figure-eight but as one frame of the Curve of Cassini — the locus of a constant product of distances to two foci, just as the ellipse is the constant sum and the hyperbola the constant difference. Sweep the focal distance and watch the oval pinch into the lemniscate, then split into two disjoint loops — the moment continuity “goes out of space.”
Unlike a circle organised around one centre, a Cassini curve is determined through its relationship to two foci. When the product constant K equals the focal distance c, the curve is a lemniscate — the threshold form. When K > c the figure is a closed oval; when K < c it splits into two separate branches. The dashed counter-space ghost stands in for the continuity that is lost at the split.
Counter-space is used here as a contemplative and geometric principle, not as a settled alternative physics. The construction follows questions raised in GA 323 (lectures 9–10, January 1921) and later counter-spatial thought — it is not a completed model given by Steiner.
Source & confidence: the Cassini derivation and the three regimes are stated in GA 323 (L9–L10). The rendered curves are teaching-forms, not an ephemeris. See the live Cosmos for the real-time planetary system, or the full Lemniscatory Geometry article for all four GA 323 studies.