Theorem: sign cusps are shared by two adjacent signs.
Given: (a) 0 degrees of any sign belongs to that sign, and the point 30 degrees therefrom belongs to the next sign;
(b) any point belonging to a sign which begins at
90°-(30*k)°, k integer, has its antiscion in a point belonging to a sign which begins at 60°+(30*k)°; e.g., each point of Aries has its antiscion in a point of Virgo;
(c) points at 90°-x and 90°+x are in antiscion.
Proof: from (c), the points at 90°-(30*k)°+30° and 90°+(30*k)°-30° are in antiscion. The latter is equal to 60°+(30*k)° and so is the first point of the sign in antiscion with that sign whose first point lies at 90°-(30*k)°. Now from (a), 90°-(30*k)°+30° is the first point of the sign next after this; however, by (b) it is also the last point of the sign which begins at 90°-(30*k)°. QED.
Comment: if signs were closed only on their first point, i.e. Aries [0, 30), Taurus [30, 60), Gemini [60, 90), etc., then either the first point of a sign would have no antiscion or else we would say that the first point of Virgo is in antiscion with Taurus, the first point of Leo with Gemini, etc., which we do not. Therefore sign intervals are closed at both ends and cusps are shared.
