David E. Bustamante

This work provides the first systematic quantification and geometric evidence of the astronomical precision limitations that linear methods of celestial partition present compared to a non-linear method in oblique latitudes. As opposed to previous critiques, which have focused upon textual or philosophical exegesis, we set forth the necessary mathematical evidence through applied spherical geometry at 57º N. Computational analysis revealed that a linear assumption upon a fundamentally non-linear phenomenon produces a cumulative temporal error of up to 30 minutes (Alcabitius, Regiomontanus) and of up to 72 minutes (Campanus, Koch) between two cusps. These results demonstrate that the underlying assumption—inherent in all uniform partitions—is geometrically untenable in oblique horizons. This error, quantifiable and quantified, clarifies the mathematical justification pursuant to which methods that assume uniformity (Regiomontanus, in particular), were historically superseded, requiring a historical reassessment of their presumed physical accuracy. The analysis confirms that the method of diurnal motion, by recognising all possible circles of declination at any latitude (including polar circles), establishes itself as the historical coordinate system naturally compatible with the true and non-uniform passage of time across the celestial sphere. The Alcabitius and Koch partition methods constitute simplified uniform variations of the complex oblique calculation—originally set forth by Ptolemy in the second century—whereas that of Regiomontanus an also linear variation mathematically equivalent to that used by Campanus in the thirteenth century. Because the time of arrival of the sun or its ecliptic degrees is a daily changing, non-uniform physical reality, this work demonstrates that only a method that is not uniform-dependent can accurately reflect arrival times.