TRANSCENDENTAL GEOMETRY:
A Second Ray Approach to the Antahkarana


"INCLUSIVE REASON produces that inherent divine capacity which enables the detail of the sensed Whole to be grasped in meticulous entirety." Esoteric Psychology II, p.393.

The antahkarana (from "antar"=between; "karana"=to do, to act) is the bridging, and thus healing process of integration, fusion and identification with the Source of Being --the One from Whom all things proceed (manifestation), and to Whom all things return (abstraction). This "channel of light and life" (Rays and Initiations, p.486) may also be defined as that state of consciousness "which transmutes knowledge into wisdom within a flash of time; which changes sensitivity into love within an area of space; which alters sacrifice into bliss where neither time nor space exists" (Discipleship New Age, p.248). The antahkarana is essentially "the science of light manifestation" (Education New Age 143-144).

In the outgoing process of manifestation, a point (0) of pure subjective abstraction (the Source or Root) expands to become a Sphere of the Absolute --the as yet unmanifest Father-Mother (1) of the Universe. At the very periphery of the Sphere of the Absolute a point of objective concretion occurs and becomes the Germ of a Sphere of Manifestation-- the Mother (2), as an equal but differentiated co-creative Principle with the Father.

The growth and development of the Germ of manifestation engenders the material world molded after the Father's archetypal world. In this ever-expanding and spiral-like process of manifestation the Word or Logos (3) of the Father serves as the Divine Intermediary Son or Christ Principle. The Christ Principle (Cosmic Hermes), procreated by the Divine Parents, is the Sound after which all manifestation will take material form.

In a mysterious way, the Father, Mother and Son are One but also Three. The Mother is one with the Father, as well as the Son. Contemplation of the geometric configuration known as the Vesica Piscis may provide an insightful clue to "the doctrine of the trinity... the esoteric basis of occultism" (Esoteric Astrology, p.98). The Vesica Piscis, very frequently seen in Gothic architecture, is the fish-like surface created when two equal circles overlap so that the center of each lies on the circumference of the other. That is exactly the proposed relationship existing between Father and Mother: the center of Each lying on the circumference of the Other, or as expressed from a Second Ray perspective, "a point within a circle and yet myself" (Esoteric Psychology II, p.355) The interface between Them is the Son, the Christ Principle, and thus the Three Principles coexist as One within the Vesica Piscis. Furthermore, the Vesica Piscis model allows us to visualize the material world in manifestation as being only a part of the Divine Mother. This concrete manifested aspect of the Mother Principle has been traditionally associated with the moon, the lunar or generative principle. But there is an unmanifest aspect of the Divine Mother, co-equal to the Father, that is also symbolically portrayed to us in the Vesica Piscis model. The Agni Yoga books make frequent reference to this often ignored aspect of the Divine Mother, the Mother of the World.

Likewise, at the human level, the concrete, manifested aspect of the Mother is a human Individuality (Jivatma or Ego) --an Individuality manifesting from incarnation to incarnation through numerous personalities. Initially, these personalities are but faint and distorted reflections of the true Source --the Father-Mother Monad. These personalities represent the moon or lunar aspect at the human level. However, the Mother Principle is not identical with the unregenerate personality. It is only as the integrated, Soul-infused personality is RE-GENERATED that the true Mother Principle is gradually expressed in the human experience. Integration of the personality occurs as the Germ of the Mother grows and develops out toward its periphery in a clockwise, ever-expanding, spiral-like pattern. This is the Mutable Cross experience. Soul infusion of the personality occurs as, at the point of maximum extension from the Motherly Germ, the human Individuality reverses its course and, in a counter-clockwise, ever-contracting, spiral-like pattern traces its path back to its Mother's Germ that is also one with the Father. This is the Fixed Cross experience. Once the Prodigal Son or Soul has thus redeemed the manifested (lunar) aspect of the Mother Principle, then the Mother (regenerated Personality) is ready to meet the Father --the Assumption of the Mother. This is the Cardinal Cross experience wherein Father and Mother meet and an immortal Son is immaculately conceived "from above," called by the Gnostics the "Body of Resurrection." The immortal, solar Body of Resurrection is the alchemical Magnum Opus of the Path of Regeneration represented by Horus in the Egyptian Osiris-Isis mythology.

The Law of Analogy
We are told that "the task of the Christ is to establish relationships" and that "the major linking agent in the universe is the energy of love-wisdom (Rays and Initiations, p.89). In fact, from the angle of the esoteric sciences LOVE and REASON [RATIO] are synonymous terms. The REASON [RATIO] referred to is that "pure intuitional infallible comprehension which grasps cause and effect simultaneously, and sees why and whence and to what end all things are moving" (Esoteric Psychology II, p.395).

Our spatial sense of order operates by means of perceptual contrasts. The most fundamental perceptual operation is the RATIO, symbolically represented by the relationship A:B (read as "A is to B"). We perceive an object ("A") in relation to what it is not ("B"). The perception of spatial order through the study and contemplation of relationships within and among forms is called GEO-METRY (Geo=Gaia=Space). Therefore, the study and practice of spatial order, as in the contemplation of geometrical ratios, may provide an important means to bridge the gap between the archetypal source of forms and the forms themselves.

If we are not familiar with neither A nor B but have some familiarity with C and what C is not (object D), such that C represents some sort of shadow or reflection of A, we can then infer or deduct A and B by the proportional relationship A:B::C:D (read as "A is to B as C is to D"). This comparative relationship of two RATIOS is an ANA-LOGY.

The Law of Analogy is expressed in the Hermetic axiom: "As Above, So Below." The esoteric goal of transcendental geometry is to lead the mind back toward the sense of Oneness through a succession of analogous relationships. In the words of Thomas Taylor, "Geometry enables its votary, like a BRIDGE, to pass over the obscurity of material nature, as over some dark sea, to the LUMINOUS regions of perfect reality."

The Generative Process
Geometrically, the generative process in space may be illustrated by the forms generated within the Vesica Piscis. The Vesica Piscis is the geometric Mother of the five regular polyhedra or Platonic solids: tetrahedron, hexahedron (cube), octahedron (double pyramid), dodecahedron and icosahedron. A regular polyhedron is a perfectly symmetrical solid in which all faces are identical and regular (all sides equal) polygons. There are five and only five regular polyhedra; these five polyhedra form the basis for all volumetric forms in space.

The five regular polyhedra may be construed as the fundamental archetypes of the Mother Principle, macrocosmically and microcosmically considered. For instance, the number of petals within the five chackras up the spine (representing the Mother Principle) may be related to the five regular polyhedra according to their corresponding number of edges, faces and vertices.

It can be geometrically proven that the five regular polyhedra are derived from the three "sacred generative roots" of "irrational" (that is, transcendental) numbers implicitly defined in the Vesica Piscis. These three irrational numbers are: the square root of 2, the square root of 3, and the square root of 5. Interestingly, the numbers 2,3 and 5 are the only numbers required for the division of the octave into musical scales.

The Regenerative Process
So far, we have considered the outward process of generation into form-matter. That is, from the Fatherly ROOT to the Motherly GERM, generating forms within the Womb of Space-Gaia (the Vesica Piscis), mediated through the Archetypal World of the Logos --the Word. However, eventually the generative process reverses into the regenerative Path of Return wherein Osiris and Isis conceive the regenerated Horus "from above." This reversal occurs after the lunar aspect of the Mother Principle has been exhausted and does not provide any further incentive for experience. Then the redemptive process of regeneration begins with the yearning of the Prodigal Son to be lead from the shadows into the self-luminous realm of the Light, from the ectypal to the Archetypal, and from death to the immortal life of the Spirit.

The Path of Return or Regeneration is revealed as a result of an authentic and vivid twofold re-cognition: first, the epistemological identity of subject and object in the act of cognition, and second, the realization of an all-inclusive UNITY within the diversity of manifestation.

"Ana-logus," as strictly regarded by enlightened Greek philosophers, was a three-term proportion, A:B::B:D. The transition from a four-term "discontinuous" analogy (A:B::C:D), the usual way an unregenerate but integrated personality perceives the world, to the three-term "continuous" analogy (A:B::B:D), the way a soul-infused personality knows, is accomplished by mastering the technique of knowledge-through-identity (B=C). It is the Knower himself (B) who builds the bridge in the analogy between the Above (A) and the Below (D). Thus, the seemingly irreconcilable extremes (A and D) are bound or related by a single mean term (B), the Soul, as the Logos or Christ Principle linking the extremes. The knower no longer stands objectively outside the act of cognition as in the four-term, disjunctive mode of perception. In the three-term analogy, the knower is subjectively fused with the soul of the object to be known. Or in the words of Paul the Apostle (Ephesians III, 17-18), "that you, being rooted and grounded in LOVE, may be able to comprehend with all saints what is the breadth, and length, and depth, and height," in a clear reference to the perfect three-dimensional cube as the symbol of the truly regenerated Soul-infused Personality.

But a three-term analogy is not the limit of knowledge. There is a step further: the realization that the smaller must be to the larger as the larger is to the whole (smaller:larger::larger:whole). That is, D=A+B, so that A:B::B:(A+B). This unique geometric analogy of two terms (A and B) is referred to as "the Golden Proportion." Furthermore, when it is realized that the WHOLE is identical to the ONE (that is, A+B=1), and that "... there is no identity apart from universality and no appreciation of the universal apart from the individual realisation" (Rays and Initiations, p.106), the mathematical metaphor for the Holy Trinity stands revealed and a key to the archetypal world is obtained. Then, "all that remains is a point of light. This point is conscious, immutable and aware of the two extremes of divine expression: the sense of individual identity and the sense of universality. These are fused and blended in the ONE. Of this ONE the divine Hermaphrodite [HERMES-APHRODITE] is the concrete symbol" (Rays and Initiations, p.106; bracket is the author's).

The Golden Analogy is a unique proportion of two terms, and is designated by the Greek letter phi, representing the irrational number 1.618...

[1 + Squareroot(5)]/2
Phi is the only geometric partitioning of Unity and the only continuous proportion that yields a geometric progression of numbers wherein the terms representing the manifested Mother are an exact, continuous proportional reflection of the unmanifest Father --the apotheosis of creation.

Thus, the formula is discovered to conceive the immortal body of resurrection as the perfect manifestation of the Father's archetype (Logos) in the world of forms. The soul-infused Personality (the Mother), as the Weaver of the antahkarana, is ready for its crucial final bridging work.

Building the Antahkarana
We have arrived to the point where the architectural design is finished, but still the actual engineering work of bridging needs to be completed. Although, technically speaking, the bridging work is done entirely on mental matter, pure intellectual work to develop the abstract mind is not enough to bridge the gap between the higher ("noetic") and the lower ("phrenic") mind. Intellectual effort to develop abstract thought is undoubtedly necessary but definitely not sufficient to bridge and heal the gap between the formless archetypal world of the Spiritual Triad and the world of perfect forms of which the immortal solar body of resurrection is a symbol. The solar body is made up of perfectly redeemed substance from the atomic subplanes.

As a result of many lives of disciplined spiritual aspiration and purity of purpose, the causal body becomes a suitable "substance of the flame which lights the way" (Esoteric Astrology, p. 274) so that the perfecting or "telestic" work may be accomplished. This relatively perfect stage is arrived at the third initiation wherein "the personality is irradiated by the full light of the soul and the three personality vehicles are completely transcended" (Rays and Initiations, p.278). The causal body has to faithfully reflect the light of the Jewel in the Lotus as a "radiant circle of pure white light" (Light of the Soul, p.296), a microcosmic counterpart of a spotless full moon. The moon itself, as the "shadow of the monad" (Rays and Initiation, p.5), is transmuted, transformed and transfigured. The generative permanent atoms within the causal vehicle are set afire and radioactively sterilized within the now transformed "solar lotus" (Rays and Initiations, p.6).

However, the successful completion of the Magnum Opus is not possible without the evocative inflow of the group-conscious Triadal INTUITION. The synthetic, intuitive sense of direct cognition is evoked according to a sixfold technique that varies slightly depending on the builder's psychological ray structure. A crucial part for the correct projection of the invoking soul-infused Personality into the realm of the Spiritual Triad (wherein the intuitive buddhic sense dwells) is the enunciation of a "Word of Power." We are told, "each disciple, having successfully organised the bridge substance, having brought into activity the will aspect, and being consciously aware of process and performance, proceeds now to move the organised substance forward, so that from the centre of force which he has succeeded in accumulating there appears a line of light-substance or projection. This is sent forward upon a Word of Power, as in the Logoic creative process" (Rays and Initiations, pp.490-491).

The Word of Power used in the act of upward projection by a second ray disciple is: "I SEE THE GREATEST LIGHT." Sight is a sense that "gives an idea of proportion (Esoteric Psychology II, p.562) But, what is "the greatest light?" As we are told that "beyond the mental plane there is no light but Life itself" (Rays and Initiations, p.74), the "greatest light" must be found within the abstract mental plane. It is proposed that an aid that may be used by second ray disciples in building the antahkarana is to visualize proportional relationships, as in the study and practice of transcendental geometry.

The study and practice of transcendental geometry promotes the audial, intuitive aspect of the mind, what the Egyptians called the "intelligence of the heart." The second ray disciple needs to truly think in his heart to build the antahkarana. To see the greatest light reveals the real ("root") world of causal relationships, enabling disciples "to see in the light the relation of the whole" (Rays and Initiations, p.516). The Pythagoreans understood this and aimed at the esoteric understanding of mathematics and geometry to attain what the Gnostics called "epopteia": the complete sight of direct cognition within the causal world of harmonious relationships. The mind becomes the sixth sense, the sense of the intuition. This intuitive aspect of the mind conveys an adjusted sense of right proportion (humility), and reveals the profound simplicity of true causal relationships.

Finally, we are told that "the three spiritualising processes," that is, transmutation ("the process whereby the energy of the three lower centres is carried up into the three higher centres"), transformation and transfiguration (of the mortal body into the immortal body of resurrection), constitute "a paralleling activity to the task of building the antahkarana" (Rays and Initiations, p. 278-279). Prospective pyrotechnicians should be warned about the dangers of attempting to prematurely raise the spereima, the kundalini fire of the as yet dormant, coiled Mother Principle. We are thus cautioned, "How this fire at the base of the spine can be aroused, the form its progression should take (dependent upon the Ray), the blending of the fire with pranic fire and their subsequent united progression, are things of the past with the many, and fortunately for the race, the work was achieved without conscious effort. The second blending with the fire of manas has to be effected. Scarcely as yet men succeeded in directing the fire up more than one channel of the threefold column; hence two-thirds of its effect in the majority is yet confined to the stimulation of the organs of race propagation. Only when the fire has circled unimpeded up another channel is the complete merging with the fire of manas effected, and only when it progresses GEOMETRICALLY up all three is the true kundalini fire fully aroused, and therefore able to perform its work of cleansing through the burning of the confining web and of the separating particles. When this is accomplished the threefold channel becomes one channel. Hence the danger... Only when buddhi has assumed an ever-increasing control of the personality, via the mental plane (hence the need of building the antaskarana), will the personality respond to that which is above, and the lower fires mount and blend with the two higher" (Treatise Cosmic Fire, pp.139-140).

Reference:
Lawlor R. Sacred Geometry: Philosophy and Practice. Thames and Hudson Ltd, London. 1982.
Pryse JM. The Apocalypse Unsealed. Kessinger Publishing Co, Kila, Montana (USA). 1910.

THOTh, last revised 01SEP96