THE MATHEMATICAL BASIS OF ARABIC PARTS
AND A PARADIGM SHIFT IN ASTROLOGY
Copyright © 2005 All Rights Reserved
by David Cochrane
An Arabic Part forms an isosceles trapezoid with the 3 points used in the Arabic Part formula. This fact has enormous implications regarding why Arabic Parts are important, how they work, and even the fundamental assumptions of astrology! However, the underlying geometry of Arabic Parts has been almost completely overlooked throughout the entire history of astrology.
An isosceles trapezoid is a four-sided figure which has two opposite sides of the same length, and the shape is vertically symmetrical. Note that an isosceles trapezoid is similar to a parllelogram, but has very different properties from a parallelogram. A parallelogram has 4 parallel sides and does not have symmetry, except in the case of a rectangle. A rectangle is simultaneously a parallelogram and an isosceles trapezoid.
A remarkable property of an isosceles trapezoid is that it has two pairs of equal angles. For example, if you measure the angles between the PF (Part of Fortune), Sun, Moon, and Asc, you will find two pairs of parallel angles. In the author's birth chart (May 1, 1949, 4:26 AM, East Meadow, NY, USA), for example, rounded to the nearest degree the Asc is 3 Aries, the Sun is 11 Taurus, the Moon is 15 Gemini, and the PF is 7 Taurus. Therefore, the Asc-PF angle is 34 degrees and Sun-Moon angle is 34 degees. Also, the Asc-Sun angle is 38 degrees and the Moon-PF angle is 38 degrees. There are always two pairs of angles that are identical in an isosceles trapezoid, and because the Asc, Sun, Moon, and PF form an isosceles trapezoid this is also true among the 3 points and the Arabic Part used.
In the previous paragraph I used the PF formula of Asc+Moon-Sun. The reader may know that the PS (Part of Spirit), whose formula is Asc+Sun-Moon, is regarded by many authorities as being the PF if the Sun is in the first six houses. Very interestingly, the PS also forms an isosceles trapezoid with the Sun and Moon. In the author's chart, the PS is 29 Aquarius. The PS-Asc angle is 34 degrees and the Sun-Moon angle is 34 degrees. Also the PS-Sun angle is 72 degrees and the Asc-Moon angle is 72 degrees.
Given three points, there are three other points that form an isosceles trapezoid with these three points. In the case of what is widely regarded as the three most important points in the chart (the Sun, Moon, and Asc), two of the points are the PF and PS. Both ancient and modern astrologers very rarely use the third point. The formula for the third Arabic Part that can be constructed with the Sun, Moon, and Asc is Sun+Moon-Asc. When I first began experimenting with this Arabic Part, I called it the Part of Integration but then I noticed that it was already listed in my software and has the name Part of Retribution, and this Arabic Part is attributed to al Biruni. I have not yet researched al Biruni's work to see if he used this Arabic Part because it is the third isosceles trapezoid that can be formed with the Sun, Moon, and Asc, or perhaps someone simply wanted to experiment with the third formula that is possible in adding two of the three points and subtracting the third. I will use al Biruni's term, the Part of Retribution (PR). There is an elegant relationship between PF, PS, and PR; the formula involves adding two of the three points and subtracting the third. This elegant and simple formula calculates the three points which form isosceles trapezoids with any three points. In the author's chart, PR is 23 Cancer. The two pairs of identical angles are: Asc-Sun and Moon-PR are 38 degrees, and Asc-Moon and Sun-PR are 72 degrees.
Because most ancient astrologers appear to have not noticed, or were not concerned, with the fact that PF and PS form isosceles trapezoids, it is not surprising that the PR was largely overlooked. Most Arabic Parts formulae begin with the Ascendant, so subtracting the Asc instead of adding it may not have been an intuitively obvious thing to do. From the standpoint of the geometry involved, however, the PR would appear to be a natural corollary of the PF and PS.
As the reader no doubt noticed, particular angles are repeated in a given chart; in the author's charts, these angles are 34 degrees, 38 degrees, and the sum of these two angles which is 72 degrees. This repetition of particular angles can be viewed as a consequence of the three isosceles trapezoids having overlapping sides. Even more important is this vitally important fact: whenver two angles are the same in chart, there are two pairs of identical angles. As soon as two angles are the same, then an isosceles trapezoid is formed and there are two pairs of identical angles. This is similar to a crystal-forming function in nature, and is entropy-defying and life-enhancing.
The reason that I propose viewing the formation of two pairs of angles as entropy-defying whenever one pair of angles forms is this: one can view the double pairs of angles in an isosceles trapezoid as a huge resonance; from the viewpoint of wave theory, then, there is as a double-resonance, which in turn creates a huge astrological resonance. One can argue that this double resonance, from the standpoint of wave theory (or harmonic astrology, which is founded on the concept of waves), is far more powerful than the usual application of wave theory to support the concept of minor aspects. In fact, from the point of view of wave theory, the PF, PS, and PR are arguably the most important derived points in the chart! Thus, we have a curious consonance formed between John Addey's harmonic theory and the ancient system of Arabic Parts. The consonance is extraordinary because resonating angles are extraordinarily powerful according to wave theory, as the strking of a tuning fork at one end of a room and the other tuning fork echoing back the same note testifies. That we can use harmonic theory retrospectively to identify Arabic Parts as the most important derived points in the chart produces perfect alignment of ancient astrological methods and harmonic astrology.
This perfect alignment, however, contrasts with the growing philosophical chasm that has developed between some adherents to ancient methods and those that espouse modern "scientific" methods. For example, in the book The Arabic Parts: Lost Keys to Prediction Robert Zoller refers to Kepler's astrology as "distorted" and John Frawley has similar scathing remarks about Kepler's introduction of minor aspects to astrology. Kepler's explicit description of minor aspects is the first known clear articulation of the concept, and minor aspects are a fundamental concept in the harmonic astrology further developed by John Addey in the 20th century.
In the article The Newtonian Merry-go-round Bernadette Brady has identified major shifts in astrological through in the 20th century. Brady bases her thesis on a study of articles in major British articles. My less formal, personal anecdotal observations since the early 1970's in the United States confirms Brady's historical trends. According to Brady we are now in a cycle where disillusionment with simple Newtonian proofs of astrology is inspiring us to develop greater interest in qualitative, rather than quantitative models, and many astrologers now have forsaken the view that astrology just needs more research to be validated in simple experiments. I agree with Brady's observations. This movement away from astrology as a simple science is huge, and there is a growing wave of interest in pursuing more subtle views of astrology, incorporating concepts such as divination, chaos theory, etc. Works by Patrick Curry, Geoffrey Cornelius, Garry Phillipson, and many others are creating a new framework for understanding astrology in the 21st century.
BASIS OF ARABIC PARTS IN EUCLIDEAN GEOMETRY