Proportional Times Explained

Is a cusp a cusp, be it angular or non-angular? From an energetic perspective, the sun’s influence at culmination differs significantly from its morning or afternoon phases. This shift, however, constitutes a property of the object’s position or potency. The mathematical exercise of discerning the six traditional phases of light—the six stages of ascent and descent—remains fundamentally constant, uniform. The illustration appearing below constitutes an indispensable educational tool, for it allows for the visualisation of the geometric or proportional necessity of celestial partition. It demonstrates or makes it possible to confirm that the intermediate cusps have as their sole author the same principle or phenomenon as the angles. Let us analyse each component of the illustration beyond its embedded description. (Click here to see all illustrations within the Lightbox slideshow without leaving this page.)

Why the illustration is helpful: the visual argument

The Diurnal Arcs, or the Foundation of a Celestial Partition Unit (House)

The orange semicircles explicitly represent six diurnal arcs. In this sense, they correspond to six different circles of declination (dates) that displace themselves across the local horizon (green rectangle). This constitutes the fundamental local astronomical reality in accord with the Earth’s rotation, whose axis of inclination relative to the plane of our orbit creates the phenomenon we know or regard as apparent angular motion (diurnal motion).

Temporal or Physical Proportionality

The intermittent or dashed grey arrow lines crossing the arcs are the key. They mark equal time divisions, i.e. proportional fractions of each of the six diurnal arcs. Hence the fraction or time length labels (6/6, 5/6, 4/6, etc.). These clearly show that each cusp corresponds, solely, to one-sixth (1/6) of that diurnal arc (i.e., that ecliptic degree, specifically). The reader may also make use of this presentation by our colleague Santino Mancuso.

The Angle/Cusp Equivalence by Physical Definition

It visually confirms that the Ascendant (ASC) is nothing more than the point where that arc completes six sixths (6/6) of its own nocturnal arc, while the twelfth cusp (first segment of the upper horizon) is the point where it would have completed, that same degree or another, one sixth (1/6) of its own diurnal arc, and so upon. Diurnal motion concedes the understanding of the reason for which this unifies the angles and the intermediate cusps into a cohesive spatiotemporal method of celestial partition.

Addressing the Gansten and Makransky mathematical fallacy

Refutation of Makransky (1995, The American Astrologer, p. 4)

The image directly refutes the idea that “only directions to angles are valid.” By showing that the intermediate cusps are simply angles in motion (measured at proportional time intervals), the distinction presents itself as one entirely arbitrary. If 6/6 (the ASC) is valid, why is 1/6 (12th House Cusp) not?

Refutation of Gansten (2009, Primary Directions, p. 56)

The illustration highlights the fundamental component of the reason why we must adapt the primary directions to, or avoid deviating them from, this phenomenon, that is, the phenomenon that makes primary directions possible in the first place. Nature confirms the diametric opposite of Gansten’s assertion (i.e., he maintains that there is no indication that we should adapt the primary directions to a particular method of celestial partition, or vice versa). If it is true that directions measure the amount of time naturally elapsed (the specific arc of a celestial object on a given horizon), whereas cusps, for their part, define the amount of time elapsed for the cusp degree in question (celestial object on a given horizon), this method, therefore, provides the only calculation where the measurement of the directed arc is geometrically homologous to the natural, and therefore true, definition of a cusp. Linear systems, in turn, would uniform or standarise all ascensional times, i.e. the variable proportions. In even clearer terms: they would falsify the length of each arc, whether through the celestial equator (Regiomontanus) or the prime vertical (Campanus), or a single diurnal arc, the ASC (Alcabitius) or the MC (Koch), two abbreviated forms of the Ptolemaic/Placidian method.

Animation

The Relay Race Motion

As you will lay your eyes upon the first illustration consigned herein, you may imagine the mechanism of action. The cusps will appear displacing from one diurnal arc to the subsequent diurnal arc, that is, not linearly, but in accordance with the geometry of the Earth’s rotation relative to the plane of the local horizon (the reality of the observer). As time elapses or flows (the orange arrows will appear displacing from left to right), each number (cusp) will jump from one diurnal arc to the subsequent diurnal arc. No two numbers (cusps or zodiacal degrees) can or are to occupy the same track (diurnal arc) because once a certain cusp has been ascertained, we seek to discern the next, “successively” (Placidus de Titis, 1814, pp. 2, 5, trans. Cooper), whose diurnal arc necessarily corresponds to a different circle of declination (date). Only within the polar regions can two distinct ecliptic points (zodiacal degrees) have simultaneously fulfilled one-sixth or two-sixths of their own distinct diurnal arcs due to the parallel ascension of the ecliptic relative to the plane of the local polar horizon.

Pedagogical Relevance

The following animation may become the clearest and most accessible visual refutation of two myths insistently reproduced in some schools of traditional topocentric astronomy: that the Ptolemaic/Placidian method (a) trisects a single diurnal arc and (b) is not a natural or organic method of celestial partitioning. While we have scientifically confirmed the reason for which this methodology fulfils Morin de Villefranche’s epistemological requirement (i.e., “there is a unique mode of nature for acting; therefore, there can only be a unique natural system of dividing the Caelum into houses, upon which alone the true principles of this science depends,” 2008, p. 69, trans. Holden), this animation, in turn, will be able to provide the same proof from a visual point of view.

Watch how the cusps shift based upon declination without ceasing to constitute one-sixth (1/6) of their own diurnal arc.

Most methods of celestial partitioning (house division) use ‘flat’ math, but the sky is organic. This simulation shows the natural or ‘Proportional Method’ (Ptolemy/Placidus), whose partitioning exercise is anchored upon the actual behaviour of the ecliptic upon the local horizon.

See the rest of our animations here.