The Geometry of Arrival: Quantitative Evidence for Non-Linearity in Celestial Partitioning

This is a summary of the paper Astronomical Fidelity in Historical Coordinate Systems of Celestial Partitioning: Quantitative Comparison of Linear vs. non-Linear Measurements (2025-11-11), a recent methodological discovery whose supplemental material has been made available herein for full independent reproducibility of the results, thereby complying with the principles of transparency of scientific research. Such paper analyses the limitations of linear methods of celestial partitioning compared to non-linear methods in topocentric astronomy (or astrology), and emphasises the necessity for a historical reassessment of these methodologies based upon mathematical evidence.

Spoiler: the seventh point of this summary is key, although its comprehension is dependent upon reading the rest.

1. Fundamental definitions

Time of Arrival or Arrival Time

Refers to the exact amount of time required for a point of the ecliptic—or a celestial object lodged therein—to move or displace from the eastern horizon to a certain point above the plane of the local horizon, or from the western horizon to a certain point below it. Although it tends to be measured as a certain amount of time after rise or after set, it can be measured also as a certain amount of time after culmination (from the midheaven or MC) or after anti-culmination (from the Imum Coeli or IC). Therefore, both zodiacal degrees (points of the ecliptic) and planetary bodies are treated indistinguishably as celestial objects whose time of arrival at a certain region of the sky can be not only observed but also measured unequivocally anywhere on Earth, including the polar regions. To understand polar arrival times, please see relevant section in the original document (paper).

Linear Methods of Celestial Partition

Are those that assume uniformity of ascensional times for all ecliptic (zodiacal) degrees contained between an Ascendant (ASC) and a Midheaven (MC) and/or between the Imum Coeli (IC) and the Ascendant (ASC). While this uniformity is achieved through the same mechanism of action (i.e. the undifferentiated trisection of varying diurnal arcs by use of great circles), the frame of reference through which it is achieved (e.g. celestial equator, prime vertical, the time of culmination of a sole diurnal arc) is dependent upon the methodology employed (e.g. Regiomontanus, Campanus, Alcabitius, Koch).

Non-Linear Methods of Celestial Partition

Are two alone, the original Ptolemaic or Placidian method, and the one conceived by Wendel Polich. The latter, however, constitutes a linear, albeit unnecessary, expression of the former by use of cones, as opposed to great circles and/or true, natural curves. The original method (curves) then remains as the sole true non-linear, and therefore natural, method of celestial partition.

2. Historical Context of Celestial Partitioning

This section discusses the historical relationship between astronomy and astrology and the evolution of celestial coordinate systems in their attempt to ascertain the true position of a point of the ecliptic upon the local horizon (commonly, a house cusp).

• Astronomy and astrology were indistinguishable until the 17th century.
• Coordinate systems of celestial partition (commonly, house systems) consist of spatiotemporal measurements of the angular movement of the ecliptic relative to the plane of a local horizon.
• Eade (1984) emphasised the fascination with the complexities of systems of celestial coordinates/house division.
• The necessity for simpler linear approximations arose due to the complexity of calculations before logarithms were finally set forth by Napier in 1614, which would have helped Placidus de Titis by the late 17th century.
• Historical reliance upon simplifications is well-documented, with the Regiomontanus’ method being a notable example.
• Primary directions expert, Anthony Louis LaBruzza, explains the reason for which Regiomontanus could not make use of the Placidus method of celestial partition under its organic, Ptolemaic form.

3. Discrepancies in Methodologies

This section highlights the flaws in historical methodologies of celestial partitioning and their implications.

• The paper addresses defects in the documentation of celestial partitioning methods.
• Previous analyses focused upon textual analysis (retreating to celestial inference or interpretation) rather than mathematical proof and/or adherence to the physical basis of the traditional symbol (periods of light intensity or stages of solar influence).
• The shift from linear to non-linear methods in the 18th and 19th centuries is examined.
• The paper asserts that the difference in methodologies is a matter of mathematical integrity (concerning cuspal degrees), not of philosophical preference (methodological relativism or “everything goes”).

4. What is the Burden of Proof?

This section outlines the criteria for evaluating celestial partitioning methods and the challenges of geometric certainty with regard to cuspal discernment.

• The Michelsen-Houlding Burden of Proof is introduced as a criterion for evaluating methods, drawing upon authors Neil F. Michelsen and Deborah Houlding.
• Houlding (1998) noted the lack of geometric justification for claims of superiority amongst methods.
• Michelsen (2009) emphasised the necessity for evidence that a method explains more observable
• The paper meets both standards by demonstrating a systematic cumulative error in all linear methodologies, validating the non-linear one.

5. Components of Celestial Partitioning

This section displays the fundamental elements involved in celestial partitioning and their relevance.

• The celestial sphere is filled with no less than 360 circles of declination, recognised as diurnal arcs. Therefore, that is, due to the Earth’s tilt, all bear a different ascensional time.
• The tropical zodiac or ecliptic and its constituent degrees serves as fixed markers with constant celestial properties (their declination) relative to the plane of a local horizon.
• Unequal hours are defined by the varying lengths of daylight throughout the year.
• The Ascendant and Midheaven are natural cusps that reflect the time of arrival of ecliptic points to a certain place in the local horizon.
• The rest of cusps are, too, natural, provided the celestial engineer has respected the time of arrival of these points to said place in the local horizon.

6. Geometric Computations

This section presents the calculations of spherical geometry involved in determining cuspal times of arrival using different historical methods.

• The diurnal motion (natural) method is used as a control for testing other methods.
• A specific example is provided for Kodiak, Alaska, calculating the amount of time required for 25º Gemini to become the twelfth house cusp (i.e. twelfth cusp time of arrival).
• The equatorial method (conceived by Regiomontanus) results in a 30-minute discrepancy compared to the diurnal motion method.
• The prime vertical method (conceived by Campanus) yields a 72-minute discrepancy.
• Alcabitius and Koch, while deriving from the diurnal motion method, rely upon two distinct forms of abbreviating the measurement of all necessary times of arrival, also leading to unacceptable temporal discrepancies, inasmuch as they, too, violate the principles of non-uniform (seasonal) time.

7. Special Implications (Cause of Temporal Discrepancies)

The text explains the mathematical principles underlying the temporal discrepancies in the  calculations.

• Regiomontanus and Campanus methods project uniformed segments of space from or upon the celestial equator or the prime vertical, respectively, onto the ecliptic, leading to inaccuracies that become greater with latitude/obliquity.
• Discrepancies become significant at latitudes above 35º, affecting major population
• The actual time for ecliptic points to move from the horizon to other regions of it—in order to become house cusps—is non-linear or not linearly proportional to segments of either right ascension (equator) or altitude (prime vertical).
• Alcabitius and Koch’s sole semiarcs do not share—because only in polar regions would it be possible due to parallel ascension—the same time of ascension (seasonal time) of the other two intermediate diurnal arcs whose ascertainment is necessary.
• Primary motion indicates that the actual time of arrival differs significantly from calculated times, with discrepancies of up to 72 minutes.
• The focus upon primary directions without verifying original cuspal calculations have an inherent tendency to lead to false positives due to the non-exact nature of celestial inference (interpretation), as opposed to the exact nature of cuspal ascertainment (mathematics). The practise allows, always, a certain amount of time for the event to come about, irrespective of the calculation.

8. General Implications (Contradictive Practise)

This section summarises the findings and their implications for modern astronomical practises, including software.

• The calculations fail to accurately model the physical phenomenon of celestial motion, leading to significant errors in the times of arrival relative to the house system-calculated cusps upon or against which primary directions are conducted.
• Only non-linear methods can accurately reflect the non-uniform passage of time or non-linear displacement of the ecliptic, and, therefore, the actual time of arrival of a point.
• Historical methodologies that rely upon linear assumptions are deemed geometrically untenable.
• The findings necessitate a reassessment of the purported accuracy of certain celestial partitioning methods.
• The proposed formal definition of the celestial partition unit (i.e., each one-sixth of the relevant diurnal/nocturnal arc) aims to clarify methodologies in modern practise, especially relative to primary directions (forecasting technique).

9. Compared Quantitative Analysis and Data Tables

The data analysis occupies itself with the methodological conflict, equator vs. obliquity, or Regiomontanus vs. Ptolemy/Placidus.

• Tables provide detailed comparisons of diurnal arcs, time intervals, and the constant verifiable discrepancies in arrival times.
• The analysis is based upon a control latitude of 57º N, revealing the cumulative systematic errors (inaccurate times of arrival) in linear methods.
• Table Zero lists reference dates and universal times for mean diurnal arcs across zodiac signs (ecliptic segments).
• Tables 1-5 detail the necessary measurements of all relevant diurnal arcs and temporal discrepancies between Placidus and Regiomontanus on the basis of verifiable times of arrival.
• The average values are derived from calculations at the 15º median degree of each sign (ecliptic segment) due to each point of the ecliptic/degree being a curve, not a straight line.
• Use of the starting and ending points/degrees of an ecliptic segment/sign leads to highly distorted mean time of arrival, for the total daylight amount of time of the first degree differs greatly from that of the last degree.

10. Logical Fallacies in Historical Arguments

The text identifies logical fallacies in the historical arguments traditionally set forth in support of linear methodologies.

• Circular reasoning is evident in the reliance upon the celestial equator as the primary frame of reference through which ascertain the position of celestial objects that lie upon the ecliptic, not the equator.
• Contradictory methodologies arise when natural phenomena, observable and mathematically verifiable, is ignored in favour of anecdotal evidence and/or prone-to-false positives cases.
• Misapplication of Occam’s Razor favours simpler, inaccurate models over less simple, accurate ones.

See post Logical inconsistencies and methodological fallacies in coordinate systems (4 min read) in order to understand this point better.

11. Conclusion on Methodological Validity

The paper concludes with a call for revaluation of topocentric coordinate systems (methods of celestial partition) on the basis of geometric accuracy.

• Linear methods fail to reflect the actual non-uniform motion of celestial bodies.
• A formal definition of the celestial partition unit is proposed, emphasising the necessity of accurate timing/temporal measurement.
• The integrity of celestial partitioning must align with observable physical mechanics to ensure reliability in topocentric/astrological practise.